UNITED STATES DEPARTMENT OF COMMERCE

Maurice H. Stans, Secretary

NATIONAL BUREAU OF STANDARDS • Lewis M. Branscomb, Director

NBS TECHNICAL NOTE 530

ISSUED NOVEMBER 1970

Nat. Bur. Stand. (U.S.), Tech. Note 530, 162, (Nov. 1970) CODEN: NBTNA

Systems Analysis of Inland Consolidation

Centers for Marine Cargo

Richard H. Jordan, Project Leader

Technical Analysis Division

Institute for Applied Technology

National Bureau of Standards

Washington, D.C. 20234

Technical Analysis Division

Institute for Applied Technology

M. C. Stark

C. O. Bunn

J. L. Donaldson

W. J. Obright

H. R. Millie

Applied Mathematics Division

Institute for Basic Standards

J. Gilsinn

A. J. Goldman

W. A. Horn

"*E«U Of NBS Technical Notes are designed to supplement the Bureau's regular publications program. They provide a means for making available scientific data that are of transient or limited interest. Technical Notes may be

listed or referred to in the open literature.

For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C, 20402.

(Order by SD Catalog No. CI 3.46:530). Price $1.25
PREFACE

In July of 1967, the United States Maritime Administration asked the Technical Analysis

Division, Institute for Applied Technology of the National Bureau of Standards, to develop

analytical techniques to optimize the location and characteristics of inland centers to con- solidate less -than- carload lots of cargo into "full" container loads for export and to unload

and distribute containerized import cargo. It is presumed that such centers, if feasible,

would encourage the use of containers and this, in turn, would promote the United States

Merchant Marine. The study was undertaken by the National Bureau of Standards under Purchase

Order No. P1-MA68-112, dated 1 August 1967.

The results of this study were communicated to the Maritime Administration in a report

dated August 1968. In view of the importance and frequent occurrence of facility-location

problems in governmental decision-making, the Bureau's Technical Analysis and Applied Mathematics Divisions found it appropriate to carry the work somewhat further than had been

possible during the study for the Maritime Administration. These extensions included computer

implementation and exercise of additional features of the mathematical model developed,

design of improved summary-level output formats, and execution of more and better-based

illustrative computations. The present document, which replaces the August 1968 report, covers these additions and also incorporates a few changes in the exposition of the material

reported previously.

Appreciation is expressed to Mr. Paul Mentz of the Office of Research and Development

of the Maritime Administration, Project Engineer, for his guidance and helpful suggestions

in the conduct of the research, and to Mr. Maitland Pennington, Mr. Howard Marsden,

Mr. Thomas Fay, and Mr. John Norris of the Office of Maritime Promotions, whose participation

made possible a valuable understanding of the practical aspects of the problem.

Special acknowledgement is given to Mr. John Frazier, Mr. C. Nelson Bean, and

Mr. Roupen Berberian of the Delaware River Port Authority for their wholehearted cooperation

in making available unique data on shippers which had not been collected elsewhere, without

which the success of the study would have been severely handicapped.

Appreciation is also expressed to Dr. George Suzuki for his technical editing of the

report

.

To Mrs. Verna Durkay goes the thanks of all participants for her part in arranging and

preparing the many drafts and final version.

W. Edward Cushen

Chief, Technical Analysis Division

li
SYNOPSIS

This Technical Note documents a study, carried out for the U.S. Maritime Administration,

to develop analytical techniques for use in optimizing the locations and characteristics of

inland centers to facilitate the flow of containerizable marine cargo,* Such centers would

perform the consolidation of small lots of break-bulk general cargo into container loads for

export; for the reverse flow, they would carry out the handling and unloading of import con- tainers for cargo distribution.** Performance of these functions inland, rather than exclusively at or near the ports involved, should result in savings to the shipping community from

transporting cargo over land in full containers rather than as more costly less-than-carload

(LCL) lots.

A mathematical model and associated solution technique have been developed, implemented

in a digital computer program to a point compatible with the kinds of information available,

and exercised using the body of data and background material accumulated during the factfinding phases of the study. These sample calculations appear to establish the feasibility

and value of our general approach and also aid in pinpointing the types of additional data

needed for a more definitive analysis. The numerical results, although only tentative and

illustrative, are highly encouraging as regards the economic value of the "inland center"

concept

.

Conclusions

The basic conclusions of this study can be summarized as follows:

1. The present line-haul container rates and consolidation costs at inland consolidation

centers, when compared with cost of shipment in less-than-container load lots of break-bulk,

appear to give a distinct monetary advantage to the shipping community from the use of inland

consolidation centers. This conclusion remains intact under a variety of perturbations to our basic data set.

2. A computerized mathematical model to guide the selection of consolidation center

locations is both feasible and useful; its use, level of detail, and the interpretations given

its outputs must be duly sensitive to the quality of the data available. Relatively simple

modifications could provide additional outputs bearing on the levels of service to particular

shipper groups or ports

.

3. For a definitive analysis to aid in realizing the full potential of a system of con- solidation centers, as envisioned in this study, it is essential to have available pertinent

data on which to base accurate, definitive, and sound judgments. For a port authority, a freight forwarder, an exporter or transportation company interested in containerization, or a government agency promoting such a program, it is essential to know points of origin, routes

of transportation, times in transit, pertinent rates, volumes, seasonal variations, and points

of destination of present and future flows of export and import cargo. These data are not

presently available. The data are at least as important as the means by which they are manipulated.

4. The cost of acquiring land and operating a center, and thus the cost to a shipper of

using that center, can vary appreciably depending on precise center location, but given

reasonably acute acquisition choices and operating practices, this variation should not be

so appreciable as to affect a shipper's choice of center. With this understood, it is not

necessary for the mathematical model to pinpoint the exact geographical location of each

center in order to indicate how to achieve near-minimun total costs.

The balance of this document, which reports fully the fact-finding and model-related

work described just above, is relatively lengthy. It contains considerable technical detail,

describes a number of data interpretations and modeling possibilities alternative to those

actually adopted, and includes a good deal of material which (although relevant) proved

peripheral to the main course of the study. This initial synopsis has therefore been

*In view of the general importance and prevalence of facility-location decisions, some extensions to the work were carried out as an in -house project.

**See Chapter II for a fuller discussion of centers' functions.

iii
inserted to provide the reader with a synoptic overview of the study's products and findings,

as well as a guide to the body of the text. The object here is compactness, even at the cost

of oversimplifying the description of the following chapters' contents.

Solution Concept

Each application of the model requires a specified value of

n = number of inland centers to be treated.

Thus applications would typically be arranged in sequence, with n varied systematically to cover some numerical range of interest.

The major outputs of the mathematical model are:

(a) For each shipper or aggregated class of shippers,* an assignment either to one of

the n inland centers or to "his" port of use.

(b) For each of the n inland centers, a location.

(c) For each of the n centers, a measure of capacity ("sizing").

These outputs are required to satisfy (to within a specified tolerance) the following

natural constraints

:

(a') Each shipper should be assigned to that center most economical for him (or to his

port of use if that choice is the most economical)

.

(b') Each center should be optimally located relative to the set of shippers it serves.

(c 1 ) Each center should be optimally sized relative to the volume of its patronage.

The terms "most economical" in (a 1 ), and "optimally" in (b*) and (c'), refer to a "generalized cost function" which will be described below. There are three other constraints

which can be imposed if desired:

(d') Each center is required to achieve at least a certain minimum level of total

patronage. (This serves as one control over the range of reasonable values for n.)

(e*) A center is placed "off limits" to a particular category ("co-containerizable

class") of cargo if its patronage within that category is insufficient to permit (on the

average) reasonably rapid accumulation of reasonably full container-loads.

(f ') A center may be constrained to be in a specified geographical subregion described

as a convex polygon.

The satisfaction of these constraints provides the rationale for an iterative solution

procedure which begins with a trial set of "guessed" locations for the centers. The procedure

involves the alternation of two types of steps; the assignment (and sizing) step, and the

location step.

In the assignment step, the locations and use-costs for the n centers are taken as "given"; that is, they are obtained from the previous location step.** In the simplest

case, the assignment step simply consists of assigning the shippers to centers or ports so

as to satisfy (a') above. If however the "patronage by category" constraint (e 1 ) is also to be enforced, then the sequence of shipper-by-shipper assignments is followed by a check for

violations of this extra condition. For each cargo category, the center (if any) exhibiting

the most severe violation is declared off-limits to that category, and the corresponding

shippers are reassigned in accordance with (a 1 ) to the centers still open to them. This

*We use "shipper" as a generic term for an exporter or importer of containerized cargo in

LCL quantities.

**This is not strictly accurate; for details see p. 85 on "sub- subs teps."

IV
process continues until a pattern of assignments which satisfies (e') is achieved. (Data

considerations did not permit bringing the logic to satisfy constraint (c')--i.e., the centersizing technique described in Chapter VI I --to a definite form incorporable in the computer

program. How this logic would enter the assignment step is sketched on pp. 69-70. )

In the location step, the set of patrons for each of the n centers is taken as "given,"

i.e., it is obtainable from the previous assignment step. Using these assignments, the location for each center is determined so as to satisfy (b'); "geographical" constraints (f) can

also be imposed if desired. (The logic for imposing (f) is worked out, but is not included

in the present computer program.)

The alternation of assignment and sizing steps with location steps continues until all of

(a'), (b*), and (c 1 ) are simultaneously satisfied within their tolerance limits. The nature

of the process is such that (e') and (P), if imposed, are also satisfied at this point. If

the "total patronage" constraint (d') is to be enforced but in fact is violated, then new locations for the offending centers are chosen at random and the iterative process resumes. Should

a prescribed number of such "restarts" occur without a successful termination, this is regarded

as probable evidence that the stipulated number (n) of centers is incompatible with the minimum

patronage level per center imposed in (d'), and the search for a solution is abandoned.

As noted earlier, this procedure has been implemented as a computer program (in FORTRAN),

and applied to a set of illustrative data. (Because these data did not break down centerrelated costs into fixed and variable elements, the "optimal sizing" logic could not be exercised.) The illustrative calculations involved 3 ports (Baltimore, Philadelphia, and New

York), 1400 aggregated shippers* in the East and Midwest, and up to 12 inland centers.**

Running times on the Bureau's UNIVAC 1108 computer^ were of the order of 1-3 minutes per

passage through the alternating-step process to "convergence."

More than one such passage is generally advisable, because all the constraints together

(except the geographical restrictions (f'))are typically not enough to single out a unique

configuration of centers. The final configuration reached by the iterative process will in

general depend on the initial "guesses" as to center locations; exploration of this point

(pp. 94-98j indicates that these final configurations can be quite distinct in their geographical distributions, and can differ fairly significantly in desirability as measured by the

"index of performance" described below. Thus several passages through the iterative process,

using distinct starting guesses, are recommended. (Optional routines for generating such

guesses are included in the program.) Even with this multiplicity, computer running times were found to be quite tolerable.

This capability for rapid solution, which permits examining a variety of parametrically

defined cases with a reasonable outlay of resources, seems a vital feature in assuring the

feasibility and usefulness of the entire approach. The price for this capability, of course,

is that the model operates at a rather gross level of detail. There is no explicit representation of freight transport networks, of commodity-dependent, freight rate structures in all

their changeable complexity, of the schedules and itineraries of container-carrying ships.

No account is taken in the formal model of an inland center's potential value as a possible

reducer of pilferage, as a marshalling point (for full -container shipments) or as a convenient

I focus for information on shipment and ship movements. The pattern of demand for consolidation

and break-bulk services is treated as "given" as regards both geographical distribution and

volume; the very difficult area of demand forecasting, including the influence of the centers'

existence and locations on the demand pattern, was beyond the scope of this study. '

'

*The aggregation procedure is described on pp. 87-88.

**The present computer program can handle up to 3 ports, 3000 shippers, and 50 inland centers;

tradeoffs among these limits can be easily calculated so that changes would require relatively little recoding.

"^Conversion to other machines should not in principle be difficult; some of the necessary

caveats are given on p. 82. "^Adaptation of the model to situations in which the demand for centers' services vary over

the years, and new centers are gradually introduced, is sketched on pp. 72-73 but has not

been programmed. Ihis does not however account for the interactions between the demand and

the supply of containerization services. v
Most to the point, the center locations output by the computer must be regarded as speci

fying only in general terms where each center should be placed. Precise identification of the

best site in each computer- specified locale would require a fine-grained analysis of that

locale with respect to land-cost variations, access points to particularly attractive transportation facilities, and the like. Such "fine-tuning" appears unnecessary (see p. 20) and

indeed inappropriate at the system-wide planning level, lying more in the province of the

interests that undertake to initiate and operate a particular center in a particular locale.

It appears possible (see p. 26) to place most centers on or near trunk-line railyards, con- tiguous to major highways; this should not be surprising in view of the natural tendency of

centers of import-export activity to be located near major transportation facilities. Such

locations should make quite feasible the use of unit trains to carry the containerized cargo

between center and port.

Sample Computer Output

Now that the basic ideas of the solution process have been described, much of what follow can be conveniently based on references to a sample output sheet from the computer program.

Such a sheet appears here as Figure A (and later as Figure 11 of the main text)

The first line shows that n=4 in the case under analysis. The fifth line shows that 5 sets of initial locations for the centers are to be tried; only the outcome from the first of

these sets appears in the Figure. The third line declares that the "patronage by category"

constraint is in force; the greatest allowable average delay to cargo while enough accumulates

to yield an adequately full 40'x8'x8' container is an adjustable parameter whose nominal value"

(not shown in the Figure) was taken as 1.5 days, a figure compatible with the information

gathered in the study's fact-finding phase.* If a center does not receive enough export cargo

in a certain category (including U.S. port involved, and area of destination) to keep the

average delay down to 1.5 days or less, the center is forbidden to serve that category.

The fourth line of the Figure declares that the "total patronage" constraint is in force;

each center was required to handle on the average the equivalent of 10 90% -full containers per

day. This lower limit of 10 was based on the impression gained from our fact-finding efforts

concerning a reasonable cutoff point for economical operation.** Although up to 3 "restarts"

would have been permitted in trying to satisfy this constraint, only one restart was in fact

required. The printout shows that after the first termination of the alternating step process

inland center No. 4 had only about 1/4 as much volume of business as the constraint demanded.

Giving this one center a new (random) initial location and re-entering the iterative process

led to a satisfactory configuration.

The printout's second line shows that initial locations for the centers were chosen at random in the region containing the shippers; these "first guesses" are specified in the first

display on the output sheet."1" The computer program permits as an alternative the initial

placement of the n inland centers near n randomly selected shippers. Another option is to input initial trial locations for all centers. Some centers can if desired be held at fixed

locations; they are then exempt both from random initialization and from the location-changing

parts of the iterative process. This option permits savings in computer time when one is

confident where certain centers should go

§

The second display in Figure A shows the final locations of the 4 inland centers, togethe

with information on their respective hinterlands (the smallest rectangle with sides parallel

to the coordinate axes containing all patrons) , the average "payload-fullness" (load factors)

of the containers they stuff, their average daily levels of activity and the division of that

*Results of varying this value are reported in Chapter IX, p. 109.

**Chapter IX reports the results of varying this limit down to 8 containers a day and up to

12 containers a day.

tThe (X,Y) -coordinate system used is specified on p. 87.

"^Computational experiments on the relative merits of the two alternatives proved inconclusive; they are reported on pp. 94-98.

Computational experiments based on this idea are reported on pp. 98-100.

vi
Ithis Program computes optimal locations for numbers of centers from 4 to 4 at intervals of i. ;the program uses initialization 2. ;the program prohibits centers from accepting cargo for a co-container i zable class if the accumulation delay is too great, centers are reinitialized 3 times or until each center receives at least 432000.00 pounds of cargo.

5 initializations are performed for each number of centers,

jthe transportation rate for containerized freight is .0000155.

the inventory-type carrying charge is .0004

the projected export load factor is .900. the maximum holding time is 4. days. the average import load factor is .822.

INITIALIZATION FOR 4 CENTERS

CENTER

12

3

it

555.02

339.09 262.94

135.04

184.25 454.70

121.03 179.92

CENTER 4 RECEIVING 119217.00 POUNDS AND LOCATED AT 169.750 208.930 REINITIALIZED TO 741.921 420.389

CENTER

555.21 246.79

456.68 303.65

591.09 189.97 770.00 333.37

MIN X OF PATRONS

253.75 - 301.88 - 469.88 - 607.25 -

MAX X OF

PATRONS 874.12

903.00

886.37

.952.87

MIN Y OF PATRONS

141.57 - 150.70 - 100.47 139.28 -

MAX Y OF

PATRONS 484.07 508.04

485.21

509.18

POUNDS

INPUT 1831374.20

2234762.20

941051.82

1603106.30

CONTAINERS LOAD PER CENT PER CENT INDEX OF

OUTPUT FACTOR EXPORT IMPORT PERFORMANCE 43.37

52.59 23.26

36.95

.8798

.8853 .8430

.9039

76.00

82.82 92.06

82.76

24.00

17.18 7.94

17.24

.6471 .5062 .4918

.4037

PLACE

I PHILADELPHIA

ALLENTOWN PA HARRIS BURG

SW. NEW YORK

BUFFALO

. PITTSBURGH

' NE. OHIO

CLEVELAND

DETROIT

SW. OHIO

NW. OHIO

S. MICHIGAN

INDIANAPOLIS

t CHICAGO

: MILWAUKEE

CENT. ILL.

X Y DELTA NUMBER OF CENTERS WITHIN DELTA OF (X,Y)

108.50 203.22 50. NONE

135.62 247.74 50. NONE

169.75 208.93 50. NONE

249.49 355.66 30. NONE

298.37 393.80 30. NONE

367.50 236.33 50. 1 410.00 300.00 20. NONE

456.75 303.68 30. 1 536.37 359.63 50. NONE

592.37 188.38 50. 1 610.16 332.32 50. NONE

650.00 350.00 50. NONE

690.37 189.52 50. NONE

770.00 333.37 50. 1 784.87 412.14 50. NONE

837.37 195.23 50. NONE

PORT

12

3

112.80 200.80 52.50 255.00

188.00 156.20

MIN X OF

PATRONS

87.50 -

157.50

MAX X OF

PATRONS

310.38

.00 784.87

MIN Y OF

PATRONS

93.62 -

102.75

MAX Y OF

PATRONS

371.04

.00 484.07

POUNDS

INPUT 906451.54

.00 99656.02

CONTAINERS LOAD PER CENT PER CENT INDEX OF

OUTPUT FACTOR EXPORT IMPORT PERFORMANCE

21.40

.00 2.42

.8826

.0000 .8586

86.93

.00 64.11

13.07 100.00 35.89.8064

.0000

.8381

I VARIABLE COST

I TOTAL COST

COST WITH CENTERS

121277.23 157432.18

COST WITHOUT CENTERS

252665.13

288820.07

INDEX OF PERFORMANCE .545087

Figure A. Sample Output

VII
activity between import and export trade. The last display gives similar information for the

(fixed) centers regarded as associated with each of the three ports.*

The intermediate display, with first column headed "PLACE", was inserted to permit quick

visualization of the final configuration of centers. The user can specify the names of several cities or subregions, a nominal location for each, and a tolerance distance (in miles) fo

each. The display will then describe how many of the inland centers lie within the tolerance j distance of each of the specified locations, thus yielding a rough mental map of where the

centers fall. (Actual map output, though a desirable adjunct, is not available from the

present program.)

Index of Performance

U'e focus next on the last three lines of Figure A. They depend on the "generalized cost

function," whose description we continue for the moment to defer.

"Total cost" refers to the sum over all shippers of the generalized costs associated witl

their shipments. 'Variable cost" refers to those ingredients of "total cost" which depend on

tne locations of the centers (and on their capacities, should data suitable for exercising the model's "sizing logic" become available). In particular, total cost may contain summands

,

depending on the number (n) of inland centers, which are not included under variable cost;

many but not all of these summands involve "fixed costs" associated with centers, which can- not be separated out using the presently available data.

Imagine now that "total cost" is evaluated according to each of two conceivable scenarios

In the first case, there are no inland consolidation centers at all; LCL cargo is containerized (if export) or broken down (if incoming) at a center located at the U.S. port of entry or egress. In the second case, n inland centers are available and are utilized by shippers

according to the criteria described above; these n centers are located and sized (and the

port-located centers are sized) in the way sketched previously.

The index of performance for a given value of n is taken to be the ratio of the numerical'

value of "total cost" under the second scenario, to its value under the first scenario. Thus

a high value (close to 1) of this index indicates relatively little savings to the shipping

community from consolidation centers, while a low value (close to 0) indicates a great savings

TO the extent that the model reflects all relevant costs and benefits, the "best" value for

the number n of inland centers would be that which minimizes this index of (mis) performance.

As noted earlier, however, present data on center-associated costs are given on a "per con- tainer" basis and so do not separate out fixed costs; hence the present computer program

lacks the essential ingredient needed to show how the index of performance begins to vary in

the unfavorable direction as inappropriately large values of n are postulated.

Figure A indicates an index of performance of roughly 0.545 for 4 well-located inland

centers.** In general, our illustrative computations yielded indices of performance between

0.51 and 0.63. These values are strikingly low , representing very substantial economic

benefits accruing to the shipping community from the presence of the inland centers. That

this savings results from the presence of inland centers rather than centers located in port

areas is demonstrated in Chapter IX. The index of performance with only port centers available is about .95, indicating relatively little savings attributable to these centers.

Even taking into account the uncertainties in and incompleteness of the data employed in our illustrative work, plus the fact that the numerator and denominator of the present index

of performance refer only to the U.S. -incurred portion of the generalized cost of origindestination movement, these index values are so_ low as to create a strong presumption that

the establishment of inland centers will provide worthwhile savings to the shipping community,

Our assorted sensitivity analyses, noted in preceding and subsequent asides, maintain this

picture. The question of how these savings might be divided among the elements of the

community (shippers, center operators, transport operators, etc.) has been excluded from

*A full list of presently and potentially available output information is given on pp. 76-78.

**The importance of having n well-located centers, and not just n centers which "exist" but

are randomly located, is investigated on pp. 93-94.

viii
this study as far as possible*; our concern is with the size of the pie, not how it gets

sliced up.

Number of Centers ; Distribution of Demand

Because the "fixed-cost" element of center-associated cost was not available, the only

factor limiting the number (n) of inland centers in our illustrative analyses was the total

patronage constraint. As indicated in the discussion on pp. 101-102, this factor leads to the conclusion that as many as 7 or 8 inland centers might be justified for the particular

region and demand pattern under scrutiny, a finding quite compatible with the "computerized

map analysis" reported on pp. 92-93, and summarized in Table A (Table 18 of the main text).

This is in moderate contrast with the indication in Chapter VI, based on a hand analysis

performed before our shipper- related data were assimilated into computer processable form,

that 10 or more centers might well be justifiable. As might be expected, the contrast proves

to be attributable to differences in underlying assumptions. The "map analysis" in Chapter VI

treats all flow from the shippers in question as grist for the "consolidation center" mill.

All later calculations, in comparison, worked only with estimates of the "LCL component" of

that flow,** reducing the total volume by over 50%.

The line of approach leading through Chapter VI to Table A, and the accompanying Table B (Table 19 of the main text) , provides useful initial impressions of the likely general locations for centers. The total average daily volume (of estimated LCL cargo) associated with

j all the shippers in our data base is about 225 90 1 -full containers. (The nominal weight of a

'I full 40'x8'x8' container is taken as 48,000 pounds.) Of this total, about 67%, or 150 con- tainers per day, are associated with the 8 areas marked in Table A.

There is already a center at the port in Philadelphia. Three of the remaining areas,

: namely Chicago, Cleveland, and Pittsburgh, would individually support small to medium-sized

centers based on the threshold level of 10 containers /day. The other 4 areas are at best

j marginal per se, but centers located in them could expect to attract some patrons from

neighboring suEregions

.

Of course this material, based solely on flow volumes at shipper locations, does not

J take into account the effects on center location of such factors as port usage patterns and

freight rates for containerized vs. LCL cargo. Thus such initial impressions on likely

center locations required (and received) corroboration and refinement from use of the full

model. We emphasize again that our work involved "as is" use of our shipper-related data;

no account is taken of volume growth since the year 1964, to which those data refer, + of

future growth in LCL cargo stimulated by the existence of inland centers, or of potential

use of the centers by full-container shippers in order to take advantage of possible special

freight schedules or reduced rates as for unit trains. In particular, the numerical results

must be regarded as illustrative of what the methodology can provide, not as definitive.

Generalized Cost

We turn at last to a description of the costing procedure used in assigning shippers to "best" centers, in finding "best" locations and sizes for centers, and in calculating the

index of performance. The description will be primarily in terms of export cargo, with

unobvious differences for the import flow noted as they arise.

The accompanying Figure B (Figure 10 of the main text) shows those portions of the overall movement of cargo which are "costed out." The second line in the Figure refers to cargo

which is consolidated at its U.S. port of exit, the first line to cargo which is containerized at an inland center (or possibly some other port)

.

*See however pp. 7-8 and p. 63 on questions of ownership.

**The estimation step is described on pp. 87-88.

"(•Calculations with an assumed higher total demand level are reported on pp. 104-105.

Additional calculations, with the levels of demand varied nonuniformly by commodity class,

are reported on p. 108.

IX
Table A. Geographical Distribution of Shipments*

.00

.00

.00

.41

.15

.60

.00

.01

.01

.34

.49

1.28

.14

.00

.80

.09

.00

.36

.00

5.59

8 4.54

.00

.00

.65 28^32

.09

.48

.39

.06

.00

.00

.25

1.06

1.01

.60

1.12

.62

.00

.00

.76

3.78

1.57

.77

.02

i Philadelphia-Allentown

2 Western New York

3 Pittsburgh -Johns town

4 Southwest Ohio

.00

.00

.32

6 5.38

2.49 ^00

1.41

h 5.11

3.58 4.77

3.93

1.77

1.59

.00

.00

4.65

.00

.00

2.17

8.45

7.82

1.97

.00

.00

5 'Cleveland

6 Detroit

7 Chicago

8 Milwaukee

.00

.00

.29

3.16

7.74

.08

.00

.00

.00

.00

2 59

2

1 .47

1 61 2 72

.37

4.73

.95

2.29

.00

.00

N

.00

.02

1.07

.13

.02

.70

.94

.30

*Each entry gives the average daily number of 90% full containers associated with the

corresponding rectangle of the 10x10 grid overlaid on the Delaware Port Authority's

hinterland.

Table B. Average Daily Number of 90% -Full Containers

Imported or Exported for Each of the Areas Listed

Philadelphia 37.02

Allentown, Pennsylvania 7.21

Harrisburg 3 . 46

Southwestern New York 2.52

Buffalo 2.59

Johnstown, Pennsylvania 6.82

Pittsburgh 7.57

Northeastern Ohio .76

Cleveland 25.79

Detroit 11.04

Southwestern Ohio 12.33

Northwestern Ohio 3.31

Southern Michigan .92

Indianapolis 1.83

Chicago 27.98

Milwaukee 7.03

Central Illinois .80

Other 62.41
Consolidation Centers Port

Shipper

break-bul

transport

*, Stuffed

into

Containers

Marshalling

at Center

Containerized

transport

Shipper break -bulk transport

Stuffed

into

Containers

Port

Marshalling

Subsystem included in the model

Figure B, Subsystem included in the model

The costs associated with a shipment naturally depend, in part, on what commodity is

.involved. The mathematical model per se is indifferent to what classification of commodities

is employed, but a definite selection was required for use in the illustrative calculations.

This selection, largely dictated by the need for a simple nominal set of freight rates for

LCL cargo, led to a crude classification into 8 categories. Rail rates for container loads,

and truck rates for noncontainer loads in each category were obtained for movement from one selected hinterland point (Cleveland) to 4 ports (New York, Philadelphia, Baltimore, Norfolk),

These were converted into 4 "per mile" rates which were averaged to provide single per mile

rates for each of the 8 categories, both containerized and LCL. The details of this process*

and of the various assumptions involved are given in Chapter V; what is most noteworthy is

that the rate for containerized cargo** (0.135<f/cwt/mi.) is so much less than those for LCL

'cargo (an average of 0.644<f:/cwt/mi.) , providing the basis for the low recorded values of the

index of performance .

The movement of export cargo can be broken into three stages, at each of which cost is

incurred

shipper-to-center movement (uncontainerized)

,

passage through the consolidation center,

center- to-port movement (containerized).

(If consolidation occurs at the center associated with the port of exit, the last stage is

regarded as vacuous.) The costing of the second stage is somewhat complicated, and will be

discussed last.

*Some of our calculations assumed LCL (uncontainerized) movements to be by rail, others by

truck. Since truck rates are typically the lower of the two for short trips (say less than

300 miles; see Ref. 15, p. 135) the former choice tends to overcharge LCL shipments to nearby centers but not the typically longer LCL movements to ports, thus raising the index

of (mis) performance of the system of centers.

**Center locations proved insensitive to variations of ± 20% in the rate, see p. 103.

+Results of varying the LCL truck transportation rates for each commodity class by random

values less than ± 10% are reported on p. 108.

XI
The freight rates described above determine the monetary costs associated with the firsi

and third stages of export cargo movement. A circuity factor (we used the I.C.C. value of

1.13) enters in passing from Euclidean to line-haul distance. In exercising the model the same factor was used for both rail and truck line-haul distance. The costing of the incontainer movement requires a little care, however; an item in a 90%-full container must for

example be charged 10/9 the charge it would incur as part of a perfectly full container.

Thus this aspect of the costing includes multiplication by an appropriate inflation ratio

(essentially but not exactly a reciprocal average load factor) which depends on the patronage

pattern and dispatch policy of the center -- i.e., on the typical "fullness" of the containe

it sends out with material from the particular co-containerizable class in question.

In addition to the direct monetary costs associated with shipper-to-center and center-t< port movement, there are "time costs" to be considered. (Their inclusion motivates the

adjective in "generalized costs.") Representing these in a way commensurable with ordinary

costs requires both a means of converting from time-units to dollars, and a calculable

expression for time in transit. Our approach to both subjects is taken from Meyer et al.*

Conversion from time to dollars is accomplished in two steps. First, the shipment quan

tity involved is multiplied by an "average value per pound" factor to obtain a dollar value;

for our illustrative calculations, values for these multipliers for each of our 8 commodity

classes were found by averaging the $/lb. ratios for the shippers in each class.** Second,

this dollar value is multiplied by a factor proportional to the amount of time in question;

the factor of proportionality signifies a representative charge (per unit time) for interest

risk, obsolescence, and the like. The value of 10% per year (0.004/day, based on a 150-day

effective year) suggested by Meyer et al was used in our numerical work, though values

specific to commodity classes might be developed.

Meyer et al also provide a formula for converting from line -haul distance to time in

transit. It takes into account not only average speed but also delay in terminals and (for

appropriate modes) time lost at sidings, interchanges, and switching points. The parameter

values given by this source lead for the rail mode to the formula,

': t = 0.0059 d + 2,

where t denotes travel time (in days) and d denotes line-haul distance (in statute miles)

Further assumptions by the study team yielded the analogous equation

t = 0.0014 d + 1

for the truck mode.++

This completes the account of the costing method applied to the shipper-to-center and

center- to-port movement of cargo. Costing of the cargo's movement through a center is discussed next.

Costing of Center Use

The mathematical model does not contain a detailed simulation of the sequence of operations in a consolidation center. Instead, formulas are provided for three cost elements

associated with passage of material through the center. These are

accumulation delay (waiting for enough co-containerizable material to accumulate to

yield a sufficiently full container)

,

b. service delay (a general queuing-type concept), and

*Ref. 15, p. 135.

**The values are given in Table 17 (p. 90). One of our sensitivity analyses is equivalent t varying these values.

+The results of varying this factor by ± 10% are recorded on pp. 109-110.

' r +Chapter IX reports the results of varying the two sets of coefficients by ± 10%.

xn

!

::: A
s: c. monetary cost.

These will be taken up in turn. The conversion of the two delay elements to dollar terms is

parried out by the same method applied above to time in transit from shipper to center or center to port.

Average accumulation delay for a shipment depends both on the hold-dispatch policy of the

center, and on the time distribution of arrivals at the center of material co-containerizable

tfith the given shipment. We have assumed a simple but plausible type of hold-dispatch policy

describable by two parameters,

H = maximum holding time,

L = "target" or "projected" load factor.

The policy is that a shipment is held until either there is enough co-containerizable cargo

to fill a fraction L or more of a container (which is then dispatched) or a time H has elapsed,

whichever comes first. Our nominal values for these parameters, based on information gathered

during the study, are those indicated in the eighth line of Figure A's sample output:

H = 4.0 days, L = 0.90.

fin our illustrative calculations, sensitivity analyses (reported on pp. 102-103) indicate that

reasonable variations in these parameters lead to changes of order 0.01 in the index of performance (which itself lies between 0.51 and 0.63). Sensitivity is greater to changes in L than to variations in H. Striving for higher load factors (thus decreasing center-to-port

transport costs, at the expense of higher accumulation delays) appeared advantageous, but the

(differences involved remained small.

A full treatment of arrivals of LCL cargo at the consolidation center might require information on seasonal fluctuations, size variations among lots from the same shipper, etc. In fact, however, the only relevant data available for each shipper were total yearly tonnage and mean frequency of shipment. The present model employs perhaps the simplest nontrivial matheStical representation determined by such data: the shippers to the center, of material in

ch co-containerizable class, are regarded as an aggregate Poisson source of equal shipments,

e Poisson assumption is especially important in simplifying the delay-related and inflationratio calculations which must be redone each time the assignment step changes a center's set

of users.

With the formulations of hold-dispatch policy and shipment -arrival distribution just

described, explicit formulas are obtainable for average accumulation delay (for a co- icontainerizable class at a center) and also for average load factor and average inflation

ratio. This applies to exports; accumulation delay (in the U.S.) is of course not relevant

jlfor imports, and in our illustrative work a load-factor of 0.822 (the average load factor in

rthe second half of 1967 for inbound containers in the North Atlantic trade) was applied to

I all imports.* One more special point: accumulation delay is taken as zero for outbound cargo

• containerized at its port of exit.

To discuss average service delays at a facility, one must consider both its capacity and

its level of activity. Let Au

denote the mean arrival rate of LCL cargo at the center,

-measured in "containerfuls" (units of 48,000 lbs.); it is calculable from the frequencies

and sizes of shipments by the center's users. Let Ac denote the mean outflow rate of con- tainers, taking inflation ratios into account. The "sizing" of the center is represented by

j'two design parameters Ku

and Kc, representing the center's respective throughput capacities

for (a) the processing stages before accumulation delay becomes meaningful, and (b) for those

following the point at which the dispatching rules first permit formation and dispatch of a container; these parameters are assumed chosen in the same ratio as Ay/A-. The relation

'between mean waiting time and excess capacity is taken for simplicity as that for a simple

queue poisson arrivals , exponential service times). The result is that the average delay for

cargo in its uncontainerized form is the reciprocal of Ku

- Au, the average post-containerization delay is the reciprocal of Kc

- Ac, and the average overall service delay is taken as the sum of the two. *The model's sensitivity to this factor was tested, and the results are recorded on p. 109.

xiii
The monetary costs to users of a center are calculated by prorating the total cost of

center operations (perhaps multiplied by a "profit factor") among its patrons in proportion

to the volumes of center output attributable to them. The total cost is hypothesized to be

expressible as a sum of three terms: a constant, a "fixed-cost" term depending only on capacity (say, proportional to it), and a "variable-cost" term depending only on operating

level (say, proportional to it). Capacity might be represented by K^ and throughput perhaps

by Ac alone. The "sizing optimization" would then consist of selecting Kq to minimize the sum of the just -mentioned fixed-cost element and the dollar equivalent of the service delays

for all patrons.

As noted earlier, this treatment was not in fact employed in our illustrative calculations, except for the calculation of accumulation delays, load factors, and inflation ratios. 1 • The reason is that the careful analysis of center-related costs carried out prior to model

development and reported in Chapter III, did not separate out fixed-cost and variable-cost

elements in the required way; after model development, the time and resources remaining did

not permit performance of this extra data-analysis step. Thus the "sizing" portion of the

location-sizing step could not be programmed or applied. Instead, a nominal service delay

of 0.125 days inferred from the material of Chapter III was applied across the board. As for

the monetary cost associated with using a center, Chapter III develops the representative

values of Table C (Table 3 of the main text), yielding a nominal cost of $94.28 per container :: this too was applied uniformly.* (Chapter III gives no reason to anticipate significant

variation of cost with center location -- assuming prudent site selection and design -- but

as a precaution a means for treating such variations has been included in the mathematical

model and computer program.

)

Table C. Summary of Representative Costs Attributed to Major Fixed and Variable

Expenses Contributing to the "User Fee" for Stuffing One 40'x8'x8' Container

Building $7.00

Land 2.00

Site Preparation 3.00

Maintenance and Utilities 3.00

Equipment - Forklifts 1.38

Yard Mule 1.20

Administration 25.00

Direct Labor 51.70

Total $94.28 ' :

t It is to be emphasized that these are representative costs and subject to variations

in locations of Centers, ownership, policies of operations, etc.

++ In comparison with this estimate of $94.00, a well-known steamship company

advertises that it will stuff a 24-foot container for $60, which when con- verted to a 40-foot container, is $100.

^Chapter IX reports the effects of varying these center processing times and costs

model is relatively insensitive to quite large changes in these parameters.

The

changes these

xiv

i

t:

n

:

11]

tt
Sequence of Center Operations

The preceding description treated a consolidation center as an object for mathematical

lodeling; its activities were necessarily represented in only an abstract and symbolic way.

[Ms synoptic chapter should not leave the reader without a more "physical" picture of such

l center and its operations. The picture described below and extracted from Chapter I is

:he general sequence and means of handling cargo; other configurations see some use today and

"urther modifications in the future cannot be ruled out:

The consolidation center shed is a structure some 120 feet wide, 16 feet high, and a Length consistent with the desired freight handling capacity. (See sections on the shed,

^iapter III.)

Gates for the unloading and loading of freight are nearly continuous along the length

m either side. It is a clear span structure to permit free movement around the floor of

orklifts without interference from vertical supporting beams.

At one end there is office space for clerical work and customs personnel. On at least

:wo sides of the building there is parking space for trucks and containers awaiting loading

and unloading or over-the-road haulage. There may be a railroad spur along one side of the

building or extending part way into the shed from one end.

Cargo is generally brought to the center by trucks . These trucks are either parked

priefly in the yard awaiting a gate or are backed up immediately to a gate for unloading.

The unloading is generally done by the trucker. The cargo is then sorted by destination

md piled along the center line of the shed, one pile for each destination. This sorting .d piling is done using forklifts for the movement.

When a container load for a specific destination is collected, a container is backed up

to a gate (the containers are kept on wheels) using an over-the-road tractor or a yard tracer and the container is stuffed. It is then moved out on line-haul over the road to the

3ort or to a railroad yard for loading on a flatcar for the line haul . The latter would

srobably be done by center tractors as described in the Chapter on costs or by a trucking

(Firm under contract to the center.

If a rail spur is run to the center, cargo is unloaded from boxcars through a gate or

lirectly onto the floor if the spur runs into the shed. It is handled from here as described

onder the truck operation. For inland consolidation centers it is not anticipated that

receipt of less -than- carload lots will commonly be by rail; the great majority will be by

truck. Boxcar loads do arrive today at port consolidation centers and the cargo is stuffed

Into containers as described under the truck operation. But with the growth of inland con- solidation centers it is anticipated that this mode of operation will become much less common.

If a spur is constructed to the consolidation center (which can be done through negotiation with a railroad, particularly if the center has enough business in prospect), then the

containers can be stuffed while on flatcars on the spur alongside the consolidation shed.

; [his method of loading has been tried but is not a common practice and, according to converjsations with operational personnel, has not been particularly successful.

It is also conceivable that containers on wheels can be end-stuffed at gates as described above for over-the-road line haul and then loaded on flatcars using ramps and yard

miles (as done generally by the railroads at the railyards) . However, railroads prefer that

stuffed containers be hauled by tractor to the railyard. It seems most practical to build

the center at the railyard and thus eliminate spur construction or haulage to the railyard.

(See Chapter III for a more detailed discussion.)

For a diagram of the flow see Figure C.

xv
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XVI
Data Considerations

The previous material has alluded to or implied several areas in which better data would

be valuable. Here we focus on additional major instances of data deficiencies, and note the

makeshift devices used to bridge these gaps in our illustrative calculations. The principal

related section of the main text is Chapter IV.

For comprehensive planning connected with international maritime commerce, it seems evident that regularly collected nationwide data are needed on goods movements with respect

to ultimate source (domestic or foreign), mode(s) and route of transportation, destination

(foreign or domestic), dollar value, commodity class, seasonal variations in flow, and the

like.* No systematic studies have been conducted which incorporate all or even a significant

part of these data. This situation seems to us a matter for serious concern.

It is perhaps symptomatic that the closest approximation to our needs was not a study

conducted by a Federal agency, but rather the excellent survey of 1964 cargo flows conducted

by the Delaware River Port Authority. This survey proved indispensable for our purposes;

however, it lacked information on the foreign destinations of the cargo, and on seasonal or other variations over time in size and frequency of shipments. These omissions are of course

not mentioned as criticisms of the survey, which was designed for a specific purpose other

than the present study.

For studies specifically relating to containerization, still more information is re- quired. Which commodity classes (or subclasses) are containerizable? Which combinations

of individually containerizable cargo types can compatibly be shipped in a common container?

What are the flow data when specialized to the LCL shipments which require the services of a consolidation center?

The Delaware River Port Authority survey data gave the commodity class (3-digit code)

for each entry. Relying principally on the judgment of that Authority, we eliminated some commodity classes as representing mainly bulk commodities rather than general cargo.

Next came the elimination of those commodities considered unsuitable for containerizing.

Although this elimination step was made on the basis of consultation with the Maritime

Administration and use of Port of New York Authority studies, the degree of arbitrariness

involved was distinctly disturbing.

The thinning-out of reported shipment flows, to reflect only the LCL component requiring

consolidation services, was performed in the following crude but reasonable way. The average

weight per shipment, for each shipper, could be calculated from the available data. If this

average exceeded the nominal weight level (48,000 lbs.) corresponding to a full 40' container,

the shipper was classified "non-LCL" and removed from the list of shippers of concern. All

cargo from all of the remaining shippers is assumed to require consolidation at a center or

at the port. This criterion errs in one direction by discarding the occasional LCL shipments

of typically heavy shippers, but errs in the opposite direction by including in the centers'

potential market all of the occasional full -container movements by "LCL on the average"

shippers . **

The final "makeshift device" involves the question of which of a set of essentially co- located shippers could be regarded as producing "co -containerizable" LCL material, and hence

could be aggregated. Although such information would surely be available in principle from

an adequate data base for a consolidation-related study, it was not in fact deducible from

the data at our disposal.

One would ideally also like a conceptual "explanation" of these data in terms of more

fundamental variables, since mechanical extrapolation of observed trends is dangerous for

planning purposes when underlying factors may change.

**It also ignores the possibility of some LCL cargo being noncontainerizable simply because

its route lacks container-handling facilities.

xvn
If cargo from several shippers is to be co-containerizable, it must

(a) move in the same direction (into or out of the U.S.) through the same U.S. Port,

(b) be of physical natures permitting joint presence in a common container without

damage, and

(c) involve the same overseas distribution/ concentration point (or perhaps port-ofcall area)

The available data permitted the straightforward application of criterion (a)

Regarding (b), there was the question of which combinations of commodity classes, each

individually containerizable, would admit joint containerization. The variety within each

commodity class precludes a really clearcut answer. Without a detailed study on this point,

it seemed necessary to choose as working hypothesis one of the two logical extremes, either

that any containerizable commodity class's material can be combined in a container with any

other class's, or that no two commodity classes can share a container. The first of these

alternatives was believed nearer the truth and was adopted. There is an obvious need for

better data and further study to permit more satisfactory treatment of this point.

Since no information bearing on (c) was at hand, fictitious "data" of the appropriate

kind were generated by the following process. It was arbitrarily assumed that each of the

three U.S. ports in question served overseas points (capable of handling containerized cargo)

which constituted up to at most 20 co-containerizable categories. The exporters using each

port were assigned at random, one by one, to some one of 20 categories, and the same was done

for the importers. (An alternate random assignment to 20 categories and a random assignment

to 15 categories were tried and are recorded on pp. 111-112.)

In this synopsis we have attempted to give the reader an overview of the study and its

principal findings, as well as a guide to the main text. The presentation has been largely

structured around the mathematical model, its computer-pro gram embodiment, and the illustrative calculations performed with it; the dependence of these developments on the more empirically oriented material in Chapters I - VI has been traced out. No attempt has been

made above to provide guidance to the technical appendices (A-G) , since these are adequately

introduced at the appropriate points in the main text.

xvm
TABLE OF CONTENTS

Page

Chapter I. INTRODUCTION 1 The Problem and the Objective l General Approach 2 Bounds of the Problem 3 Specific Considerations 3 Cost - A Neasure of Effectiveness 4 Symmetry 4 Customs 4 Ship Scheduling 4 Effect on Community 4 Marshalling Function 5 The Flow of Cargo Through the Center 5

Chapter II. FUNCTION AND OPERATION OF CONSOLIDATION CENTERS 7 Holding of Cargo as a Function of Consolidation Centers 8 The Function of a Consolidation Center as an Intelligence Center 10

Chapter III. COST OF CONSOLIDATION CENTERS 11

Measure of Unit Production 11 Center Throughput Capacity 11 Consolidation Service Charge 12 Components of Cost 12

The Building 16

Land 18

Site Construction and Preparation 20

Maintenance, Repair, and Utilities Service 22 Equipment 22 Administration ..... 26

Direct Labor 27

Chapter IV. DATA: SOURCE, ADEQUACY, AND PREPARATION 30

Chapter V. RATES 38

Chapter VI. MAP ANALYSIS OF DEMAND DATA 44

Summary - Map Analysis 48

Chapter VII. HIE MATHEMATICAL MODEL 50 Scope of Model 50

Decision Criteria 51 Assignment Principle 51 Location Principle 51 Sketch of Solution Process 54 Possible Difficulties 55 Treatment of Transportation Costs 55 Dollar Costs 56

Time Costs 57 Generalized Costs 58 Center- to -Port Costs 58 Resume 59

The Location Step 60

Solution Method 60

Manhattan Metric 61

Restrictions on Locations 62

Location- Dependent User Costs 62

Treatment of Costs Incurred at Center 63

Time Costs 64

Dispatching Policies 64

Evaluation of L~c and t a 65

Minimum Patronage Restrictions 67

Evaluation of Service Delay 68

xix
Table of Contents (Cont.)

Page

Chapter VII (Cont.)

Monetary Processing Cost 69

Sizing of Centers 70 Enlargement of Assignment Step 71 The Multi-Period Case

' 72 Management of Containers 73

Chapter VIII. THE COMPUTER PROGRAM 75 General Program Description 75 Input Data for Program 76

Output Information 77 Other Potential Outputs 78

Program Options 78

General Remarks 79

Input Formats 79

Conversion to Other Computers 82

Subroutines and Their Functions 83

Chapter IX. ILLUSTRATIVE APPLICATION OF THE MODEL 86

Shipper Related Data 87

Other Input Parameters 89

Computer "Map Analysis" 92

Effects of the Location Step and Minimum Patronage Constraint 93

Effects of Random Initial Locations 94

Systematic Selection of Possible Initial Configurations 98

The Number of Centers 101

Sensitivity to Dispatching Policy 102

Sensitivity to Rate for Containers 103

Sensitivity to Level of Total Demand 104

Location-Dependent Processing Costs 104

Sample Program Output 105

Further Sensitivity Analyses 107

Sensitivity to LCL Transportation Rates 108

Sensitivity to Level of Demand by Commodity Class 108

Sensitivity to Import Load Factors 109

Sensitivity to Maximum Allowable Average Accumulation Delay 109

Sensitivity to Inventory Carrying Charge 109

Sensitivity to Parameters Converting Travel Distance to Travel Time Ill

Sensitivity to Co-containerization Classification Ill

Sensitivity to Processing Times and Costs 112

Sensitivity to Total Patronage Constraint 113

Only Port Centers 115

Concluding Remarks 115

APPENDICES

A. Optimum Location o± Centers in a Network 117

B. Minimization of Total Cost 121

C. The Constrained Location Step 123

D. Evaluation of Mean Accumulation Delay 125

E. Alternative Dispatching Policies 129

F. Rigorous Treatment of Inflation Ratio 131

G. Applicability of Trip-end Distribution Models 133

References . . 135

Relevant Background References ' 137

xx
LIST OF FIGURES

Figure Page

1 Export Cargo Flow Through a Consolidation Center (Import flow is the reverse of this flow) 6

2 Construction Cost Factors 15

3 Shed 19

4 Cost of Land 21

5 Site Preparation 23

6 Maintenance Cost 24

7 Labor Costs 28

8 Area Covered by Study 31

9 Exports in Pounds Average Per Day and 40'x8'x8' Containers Loaded to 51,000

Pounds Cargo bet. 46 § 47

10 Subsystem Included in the Model 53

11 Sample Output 106

LIST OF TABLES

Table Page

1 Notations 13

2 Suggested Values 14

3 Summary of Representative Costs Attributed to Major Fixed and Variable Expenses

Contributing to the "User Fee" for Stuffing One 40'x8'x8' Container 29

4 Philadelphia-Camden, Schedule S--Exports 703, 1962 33-34

5 Philadelphia-Camden, Schedule T--Imports 303, 1964 35-36

6 Counties Used for Location of Shippers . 37

7 Plans for Piggyback Trailers/Containers 40

8 Truck LCL and Rail Container Rates 41

9 Truck LCL Rates from Cleveland to 4 Ports in Cents per Hundredweight 42

10 Truck LCL Rates from Cleveland to 4 Ports in Cents per Hundredweight per

Computed Mile 43

11 Rail Container Rates (Plan 11%), Cleveland to Four Ports 43

12 Illinois, Summary, Towns Shipping One or More Containers per day average, One

8-hour Shift 45

13 Indiana, Summary, Towns Shipping One or More Containers per day average, One

8-hour Shift 45

xxi
List of Tables (Cont.)

Table Page

14 Ohio (plus two towns in West Virginia), Summary, Towns Shipping one or More

Containers per day average, One 8-hour Shift 46

15 Pennsylvania, Summary, Towns Shipping One or More Containers per day average,

One 8-hour Shift . . . 47

16 Transportation Rates (cents/hundredweight/mile) . 90

17 Average Unit Values of Cargo ($/lb.) 90

18 Geographical Distribution of Shipments 92

19 Average Daily Number of 90% -Full Containers Imported or Exported for Each of

the Areas Listed 93

20 Effect on Index of Performance of Optimizing Center Locations 94

21 Effect of the Minimum Total Patronage Constraint on the Index of Performance . 94

22 Effects of Randomizations of Initial Locations on Index of Performance .... 95

23 Variations in Rough Locations of Centers (4 Centers) 96

24 Variations in Rough Locations of Centers (5 Centers) 96

25 Variations in Rough Locations of Centers (6 Centers) 97

26 Variations in Rough Locations of Centers (7 Centers) 97

27 Initializing by Splitting a Good Center into Two Located Near Each Other ... 99

28 Effects on Index and Location of Initializing Centers in Cities Chosen from the

Map Analysis 100

29 "New" Center Initialized on Line West of Pittsburgh 100

30 "New" Center Initialized on Line East of Cleveland 101

31 Computer Evaluation of the Locations of Centers Suggested by the Manual Analysis

of Chapter VI 102

32 Variation of Index of Performance with Dispatching Policy 103

33 Variation of the Index of Performance with Transportation Rate for Container

Movement 103

34 Sensitivity to Total Demand (6 Centers) 105

35 Sensitivity to Total Demand (7 Centers) 105

36 Variation of Index of Performance with Random Perturbations of Un containerized

Transportation Dollar Rates 108

37 Variation of Index of Performance with Random Perturbations of Demand by

Commodity Class 109

38 Sensitivity to Import Load Factor 110

39 Sensitivity to Maximum Allowable Average Accumulation Delay (Days) 110

xxii
List of Tables (Cont.)

Table Page

40 Sensitivity to Inventory Carrying Charge Factor 110

41 Sensitivity to Factors for Converting Travel Distance to Travel Time Ill

42 Sensitivity to Co-containerization Classification 112

43 Sensitivity to Processing Times (Days) and Costs ($) 112

44 Variations in Rough Locations of Centers (6 Centers) with Minimum Total

Patronage Constraint 113

45 Variations in Rough Locations of Centers (7 Centers) with Minimum Total

Patronage Constraint 114

46 Sensitivity to Minimum Total Patronage Constraint 114

xxm

SYSTEMS ANALYSIS OF INLAND CONSOLIDATION CENTERS FOR MARINE CARGO*

R. H. Jordan, M. C. Stark, C. 0. Bunn, J. L. Donaldson, W. J. Obright, H. R. Millie

J. Gilsinn, A. J. Goldman, W. A. Horn

This Technical Note documents a study, carried out for the U.S. Maritime

Administration and completed by interested Bureau staff, to develop analytical

techniques to optimize the locations and characteristics of inland consolidation

centers for marine cargo. Such centers would consolidate less-than-container

lots of cargo into "full" container loads for export, and would unload and

distribute containerized import cargo.

After discussing the nature and scope of the study problem and outlining the

functions and operations of the centers, the paper reports the fact-finding phase

of the analysis. Successive chapters present and analyze data on: initial and

operating costs for centers , current demand for their services as derived from a recent survey by the Delaware River Port Authority of its hinterland, and relevant

(ground) transportation rates for containerized and uncontainerized material.

Reported next is the development of a mathematical model for estimating good

locations and sizes for the consolidation centers. The selective implementation

of this model as a computer program is described in sufficient detail to guide

prospective users. A final chapter describes the illustrative application of this

program to the data at hand, with results quite encouraging for the "inland

center" concept.

Keywords: Transportation; maritime, cargo; containerization; systems analysis;

mathematical models; optimal locations.

Chapter I. INTRODUCTION

In order to exploit the full potential of future U.S. foreign trade and a compatible

merchant marine industry, the Maritime Administration has undertaken a comprehensive research

program. A part of this research pertains to the consolidation and containerization of small

shipments of break-bulk general cargo into full loads, each load destined for a designated

overseas distribution point, and the reverse, the receipt of full loads from overseas at domestic centers for unloading and distribution. On August 1, 1967, the National Bureau of

Standards was requested by the Maritime Administration to develop analytical techniques to optimize the characteristics and location of such consolidation centers, first on a regional

and then on a nationwide basis.

The study was conducted in two phases. The first consisted of a detailed definition of

the problem and the development of an approach and analytical techniques to solve the problem.

Phase II consisted of further development of Phase I techniques and their application in

determining the most productive locations and characteristics of consolidation centers in a selected market region serving a specific port facility. A Phase III similar to Phase II was planned to cover the problem nationwide but because of lack of nationwide data the third

i phase was postponed.

The specifications of the study are outlined in the Maritime Administration Purchase

Order, "Scope of Work, Systems Analysis of Inland Consolidation Centers," dated August 1, 1967.

The Problem and the Objective

It has been customary to transport general break-bulk cargo to ports in boxes and crates

of different sizes and shapes depending on the nature of the unit objects being shipped.

This procedure requires handling numerous small units and results in excessive time and

labor costs to the shipper. These costs are further escalated because of pilferage and

damage.

*Sponsored by the United States Maritime Administration, Washington, D. C. , 20235.
Recently there has been a strong and rapidly accelerating trend toward the containerization of break-bulk general cargo. That is, the individual units of cargo are loaded or "stuffed" into large containers, generally from 20 to 40 feet in length and 8 feet by 8 feet

in breadth and height, each destined for a specific foreign distribution point, and the con- tainer is handled as one unit in transport over the land, loading onto ships, transport at sea, and unloading. This procedure can yield large savings in time and labor costs and

presumably decrease losses from pilferage and damage.

The containers are commonly stuffed at a consolidation point in a seaport facility.

It is believed by some segments of the transportation industry that consolidation and con- tainerization at common centers , optimally sized and located in relation to the shippers and

to the pertinent ports, would be less costly to those shipping in less than carlcad lots and

would thus encourage foreign trade and foster growth of the United States Merchant Marine.

If this consolidation and containerization is performed at an inland point, handling may be

minimized and the economy of movement in full container loads can be exploited. The cargo

of shippers with less -than- carload lots can be merged with others to make up full containers.

The shipper who has sufficient cargo for a given foreign distribution point to fill a con- tainer might well bypass the inland consolidation centers but, on the other hand, he may

benefit by taking advantage of lower cost rates to the port as in the case of the speed and

lower cost of unit trains. The study includes the flow of both export and import cargo.

The goals of the study are to analyze the functions of consolidation centers and to develop methods for determining their size and location. The research includes means of

estimating the cost of the use of centers in relation to the location of potential shippers.

The results of this endeavor provide tools for comparing various proposed configurations of

the fixed consolidation center concept and comparing these configurations with the present

break-bulk system of inland distribution.

Also the techniques can be used with forecast data to establish guidelines for the

determination of new environmental features, such as direct lines from inland centers to ports, improved feeder highways, and probable site locations of new centers.

General Approach

The approach has been the development of a general procedure for locating centers

,

utilizing costing techniques that involve transportation costs, center costs, and the location of exporters (and importers) and their volume of business. A symbolic representation

of the system was developed and an algorithm for computer application was formulated and

exercised. This development of the model was evolutionary. A preliminary model was con- structed, its strengths and weaknesses considered, and improved versions developed. In this

manner, a series of progressively improved models was evolved. During this process, the

function of consolidation centers and the effects of various modes of operation and ownership

have become increasingly clear. To develop the model, the problem was divided into three

parts: the necessary handling and transportation of cargo from the shippers to the consolidation centers; the handling at the centers for the necessary consolidation; and transportation

to the ports. The problem is one of providing a system which maximizes the effective movemen

of shipments , determining locations of consolidation centers that are efficient with respect

to time and overall costs, and reducing transportation costs through moving full loads

insofar as possible over optimum routes

In addition to the development of the general location model for solution by automatic

data processing techniques, data on exporters and importers and their cargos were obtained

for the area from Indiana throughout the Midwest to Philadelphia -Camden (see Chapter IV,

Data: Source, Adequacy, and Preparation). These data were used in exercising the computer

model for consolidation center location in relation to shippers and ports . As well as giving

guidance for computer runs and for checking computer answers , these data were used manually

for a graphic portrayal of export origins, indicating cargo tonnages, and generally, by [jjj inspection, probable location of consolidation centers. Although the "map analysis" lacks some of the rigor and refinements of the computer model, it has provided an intimate knowledge

of the data and a direct feel for the problem not always attainable from sophisticated

automated approaches
The scope of the problem is defined as follows:

1. Only small shipments, less than container lots, of break -bulk containerizable cargo

are considered. It is assumed that if a shipment to a specific overseas destination

is large enough to fill a container, the container will be filled at the point of

origin (the factory) and shipped by the most favorable route (perhaps through a con- solidation center) to the port facility.

2. The ownership of consolidation centers was not originally considered a problem

affecting the study nor a part of the research. However, the importance of the

effects of different ownerships has considerably increased as the study has progressed. These effects in relation to the function of centers are discussed later,

although no attempt is made to recommend any one type of ownership.

3. Furthermore, in exercising the model it is assumed that consolidation centers do not

own or control the transportation systems draying cargo to them, the line haul systems to the domestic ports, or the containers. Ownership of these systems can make

a marked difference in the location, size, and function of centers; but because the numerous possibilities are a study in themselves, they are not researched in depth

but are discussed generally to emphasize their importance. No decisions are reached

as to the ownership plans most benefiting the shipping community.

4. Political, sociological, and economic effects on a community are not treated.

5. Legal aspects such as responsibility for losses and damage to cargo, changes required

in documentation, etc., are not considered.

6. Origin and destination data for break-bulk general cargo by commodity class, volume,

and frequency required as inputs to this study were to be provided and, to the extent

possible, were provided by Maritime Administration or sources designated by the

Maritime Administration. Lack of adequate origin and destination data, particularly

foreign destination data, has proved to be a major problem and necessitated substantial modification of the original plans for conducting the study.

7. All cargo considered in this study is assumed to be containerized by the time it is

loaded on the ship.

8. Bulk cargo or cargo not subject to containerization is eliminated from consideration.

9. The advantages of reduction of packing costs and pilferage are very important arguments for containerization but they are not important factors in function, location, or size of consolidation centers and therefore are not discussed. The reduction in

damage is a debatable point under present methods of stuffing (insurance rates have

generally not been reduced) . It is therefore omitted from consideration in this

limited study.

Specific Considerations

In preparation for the study, a broad survey of background information was conducted.

In this survey it became clearly apparent that there were many considerations bearing either

directly or indirectly on the problems that influence the scope of the study, and to a cer- tain extent, the mode of operation of the consolidation concept. Also, because of the

exploratory nature of the study, it was realized that these considerations would come into

| sharper focus as the study unfolded; new ones would appear and some would not retain their

original significance, assuming either greater or lesser importance. This has proved true. The most important of these considerations are: ownership of centers; length of time of holding cargo awaiting consolidation; the center as a shipping intelligence information point;

cost of centers including cost of acquisition and operation as a basis for determining charges

levied on the user; and rates and tariffs. These problems are discussed in the following

chapters on "Function and Operation of Centers," "Cost of Consolidation Centers," and "Rates."
Other considerations are measures of effectiveness, symmetry of export and import flows,

customs, ship scheduling, effects of centers on communities, and the marshalling function.

These considerations are discussed in the following sections.

Cost - A Measure of Effectiveness

At the inception of the study it was believed that the best way to determine the number, :l size, and optimum location of centers, was through a comparison of respective total costs to the shipping community. These costs include the direct cost of transportation of cargo to the

centers and from the centers to the port, and the cost of the centers including their operation which determines the charge to the user for the center services . The time required for

shipment, while not always expressed in dollars, can in principle be converted to dollars.

It is still clear that these costs are significant determinants of the location of center;

However, it soon became apparent that these direct costs are not by any means the only

deciding factors. Others, such as traffic congestion in terminal areas, selection of shipping

routes and selection of domestic ports sometimes are more important than direct costs incurrec

within the continental United States. Furthermore, as will be detailed in the chapter on "Rates," rates are difficult to obtain, change constantly, may be altered by the very construe

tion and operation of a center, and, in regard to the use of containers, are in the midst of

a very marked policy evolution. Therefore, while rates have been used in the exercise of the

model and must be considered a very important factor, they have not been used in the detail

expected at the outset of the study. Given the time and the money to determine more exact

rates, they could be used more nearly as anticipated.

Symmetry

A consolidation center may also receive import cargo in containers from ships and reverse

the functions previously described to forward cargo in the manner best suited to the shipper

and the consignee. Except for the delay in the center awaiting a container load for export

(there is no analogous wait in the domestic center concerning imports) , this flow appears to

be a "mirror image" of the flow to the ships insofar as modeling is concerned. However, an examination of the study data shows that optimum location of centers for imports is not

necessarily best for exports because importers are not always exporters and vice versa. The

model can be run to locate export centers or import centers alone, or, on the assumption that

all centers will handle both exports and imports, it can be run to co-locate export-import

centers simultaneously.

Customs

Space considerations for customs personnel and cargo handling are part of the costing in

Chapter III. It is assumed here that customs inspection will be a part of all centers but

will not affect location of centers.

Ship Scheduling

A knowledge of the departure and arrival of ships at pertinent ports is essential to the'

effective operation of a consolidation center. This knowledge is particularly important in l relation to departures ; it allows a scheduling of the arrival of export cargo at the center

to avoid storage at the center and permits an optimum scheduling of the departure of contain-"

ers from the center for most efficient marshalling and loading on the ship. More will be saic

of the information on sailings in Chapter II, "Function and Operation of Consolidation Center:

Effect on Community

Containerization centers may well affect the communities in which they are built, particularly small communities. The study did not consider these effects. For example, it is

well recognized that traffic congestion plays a definite part in the location of consolidatioi

centers. However important these effects are, they are specific to an area and each case must be considered separately. Therefore, they are recognized but omitted from the model.

di
Marshalling Function

Full containers, wherever loaded, must be held until time for them to be dispatched to the ship. This is a storage function, however temporary. Just prior to bringing containers

to the crane that hoists them onto a ship, the containers will be retained at a place where

they can be dispatched to the crane for a precise arrival time. This is a buffer function.

In this report the storage function is considered to be performed at the consolidation center

and land for this function is included in the land requirements of the center in the cost

estimates of Chapter III. The buffer function must be performed at or near the docks for

precision in timing. This function does not affect the location of the consolidation centers

and is not included in the center cost estimates of Chapter III.

The Flow of Cargo Through the Center

By reading the various sections of this report, the flow of cargo through a consolidation

center can be deduced. However, for convenience early in the report, a concise description in one place is advantageous. The picture described here is the general sequence and means of

handling cargo although it would be erroneous to assume that other configurations are not used

today or will not be devised in the future.

The consolidation center shed is a structure some 120 feet wide, 16 feet high, and a length consistent with the desired freight handling capacity. (See sections on the shed,

Chapter III.) Gates for the unloading and loading of freight are nearly continuous along the

length on either side. It is a clear span structure to permit free movement around the floor

of forklifts without interference from vertical supporting beams. At one end there is office

space for clerical work and customs personnel. On at least two sides of the building there

is parking space for trucks and containers awaiting loading and unloading or over-the-road

haulage. There may be a railroad spur along one side of the building or extending part way

into the shed from one end.

Cargo is generally brought to the center by trucks. These trucks are either parked

briefly in the yard awaiting a gate or are backed up immediately to a gate for unloading.

The unloading is generally done by the trucker. The cargo is then sorted by destination and

piled along the center line of the shed, one pile for each destination. This sorting and

piling is done using forklifts for the movement.

When a container load for a specific destination is collected, a container is backed up

to a gate (the containers are kept on wheels) using an over-the-road tractor or a yard tractor

and the container is stuffed. It is then moved out on line-haul over the road to the port or

to a railroad yard for loading on a flatcar for the line haul. The latter would probably be

done by center tractors as described in the Chapter on costs or by a trucking firm under con- tract to the center.

If a rail spur is run to the center, cargo is unloaded from boxcars through a gate or directly onto the floor if the spur runs into the shed. It is handled from here as described

under the truck operation. For inland consolidation centers it is not anticipated that re- ceipt of less-than-carload lots will commonly be by rail; the great majority will be by truck.

Boxcar loads do arrive today at port consolidation centers and the cargo is stuffed into con- tainers as described under the truck operation. But with the growth of inland consolidation

centers it is anticipated that this mode of operation will become much less common.

If a spur is constructed to the consolidation center, which can be done through negotiation with a railroad, particularly if the center has enough business in prospect, then the

containers will be stuffed while on flatcars on the spur alongside the consolidation shed.

This method of loading has been tried but is not a common practice and, according to conversations with operational personnel, has not been particularly successful.

It is also conceivable that containers on wheels can be end-stuffed at gates as described

above for over-the-road line haul and then loaded on flatcars using ramps and yard mules (as

done generally by the railroads at the railyards) . However, railroads prefer that stuffed

containers be hauled by tractor to the railyard. It seems most practical to build the center

at the railyard and thus eliminate spur construction or haulage to the railyard. (See Chapter

III for a more detailed discussion.) For a diagram of the flow see Figure 1.

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Chapter II. FUNCTION AND OPERATION OF CONSOLIDATION CENTERS

One of the objectives of this study is the determination of the functions of consolidation centers. The basic definition is given by the Maritime Administration in the "Scope of

Work" as, "An Inland Consolidation Center is a terminal for consolidation of containerizable

cargo, located away from the containership berth, which performs these functions:

1. "Receives export cargo as presented by the shipper or forwarder.

2. "Processes, holds, and consolidates such cargo (to the extent practicable) into

containers suitable for foreign trade, each of which contains cargo for only a single destination.

3. "Schedules and routes such containers to the ship berth in a manner which will

minimize traffic congestion in the port.

4. "Receives import cargo in containers from the ship, and reverses the above

functions to forward the cargo in the manner best suited to the shipper or his

forwarder."

This definition is an accurate general statement, but a detailed study of consolidation

indicates that certain functions and concepts of operations should be added. These functions

are related to:

1

.

Ownership

2. Holding of cargo awaiting consolidation (storage)

3. Interchange of information (intelligence center).

There are at least three major plausible ownership arrangements for consolidation centers, and probably a number of combinations or variations of these three. One is ownership

by a not-for-profit organization, perhaps a city, state, or regional port authority*; the

second is by a private concern such as a freight forwarder; and finally by a cooperative

composed of a number of shippers working together for their common benefit. Each of these

types of ownership can have effects upon policy of operation and thus upon the. functions

conducted by the center.

It is not to be inferred here that the type of ownership be dictated by the government, nor is it within the scope of this study to decide which type of ownership would be of the

greatest benefit to the general economy. Rather, the aim is to point out some possible

influences of ownership upon functions so that comparisons for formulation of policy can be

made.

The ownership concept in one of its simplest forms is by a not-for-profit organization,

such as a port authority. The principal motivation would be to attract business to the area of the center. It is assumed that enough revenue would be received to cover operational expenses, retirement of mortgages and other expenses, with perhaps enough excess cash

flow to cover expansion and emergencies. Also, although it need not be limited in this

manner, it is assumed for this study that the organization would not own a fleet of trucks

to collect less -than- container loads; instead cargo would be brought to the center by trucks

owned by exporters or by local trucking companies. Finally, it is assumed that the not-forprofit center would neither own nor control, that is, keep track of and manage, the movement

of the containers used in transportation nor would it own or manage the line haul equipment

to the domestic ports. The center would charge a "user fee" for its services.

*The term "port authority" here is not restricted to a marine or oceanic port but is used in

a more general sense as, for example, an authority managing the airports of an area.
The shipper would gain through:

1. The opportunity to consolidate in full loads which he might not otherwise be able

to accomplish (a number of long haul transportation companies are reluctant or

refuse to handle less-than-carload lots).

2. Paying a lower rate on. the line haul because of consolidated full loads.

5. Less pilferage.

4. Less breakage (hopefully).

This is the type of ownership assumed in this study for exercising the model, although with

the proper reinterpretations the model can be used under any form of ownership.

A second means of ownership is by a private concern. This appears to be the direction

in which the consolidation concept is moving. There are many variations of this type of

ownership. Let us say, for example, that the owner and operator is a freight forwarder.

He owns a fleet of trucks to collect cargo or he contracts with a transportation company to bring cargo to his center. (The latter can be done at lower than prevailing rates by a freight forwarder but not by certain other owners such as a trucking company, and is an incentive to attract customers.) He ships line haul to ports in full containers hopefully,

owned by one or by various transportation companies. The forwarder is charged the container

rate for the line haul, but he in turn charges the shipper the break-bulk less-than-container

load rate. The difference is the basis for his profit. This is common practice today. In addition he may make a direct charge for stuffing containers.

The freight forwarder benefits by:

The difference between the full container line haul rates and the less-than-carload

rates, and perhaps by charging for stuffing the containers.

The shipper benefits by:

1. Less pilferage,

2. Less breakage,

3. An opportunity to consolidate less-than-carload lots,

4. And perhaps by lower rates on the local short-haul to the center.

A third concept very similar in benefits to the first is a cooperative center owned

and operated by the shippers themselves. The advantages to the shipper would be similar to those of the not-for-profit plan, although the shippers would probably have a greater voice

in policies and management. Otherwise rates and user fees might well be almost the same.

Holding of Cargo as a Function of Consolidation Centers

The holding of cargo awaiting consolidation for specific overseas distribution points

in full containers is a function of a consolidation center. How long it should be held is

the question. The policy adopted has a direct and important bearing on the center capacity,

number of shippers using the center, number of centers, and their operation.

On consideration of the problem certain facts are apparent:

1. Some cargo for out-of-the-way ports might have to be held for months awaiting

enough to make up a full container load.

2. The larger the center's volume of business, generally speaking, the shorter the

wait to fill a container.

3. The longer the cargo is held, the greater the need for storage space.
4. If cargo is stored in a building other than the consolidation center, there will be

at least one extra handling and an extra transportation link. (If stored in the

consolidation center building these two factors may or may not be true depending on such factors as labor arrangements and amount of cargo.)

5. Shippers cannot afford to have some commodities held because of competition,

disadvantages of large inventories, perishability, etc.

During the study, discussions were held with consolidation companies concerning the

holding of cargo to accumulate full loads, and their written opinions were examined, as in

the hearings before the ICC for expansion of the New England Freight Forwarders, Inc. 1 There is an almost unanimous opinion among these companies that cargo cannot be held any

appreciable time awaiting full loads. It appears that one week is close to the maximum and

that two or three days or less is generally considered most desirable. (Although aircraft

cargo is not part of this study, it was found that the aircraft consolidation centers visited

hold cargo for no more than a matter of a few hours .

)

One consolidation company stated emphatically that "if you hold cargo you are dead!"

Therefore, the present opinion among operators appears to be that consolidation centers

are not storage centers, except for short periods for customs examination, actual stuffing

operations, or cases where it is known that containers can be filled in a reasonable time.

Storage is not considered a function of a center by consolidation operators.

On this basis the following guidelines were adopted; cargo is to be held no longer than

one week and most cargo will be held no longer than three days; all centers will operate

essentially in this manner; no storage capacity in the center shed will be included for purposes other than holding freight on the shed floor in piles for specified distribution points

for a minimum time, although costs resulting from delay will be considered in the exercising

of the model. A means of introducing them is detailed in the mathematical model in Chapter

VII. (This guideline may appear from a theoretical sense to be an oversimplification, but

it is a convincing one to a practical operator working against intense competition. If in

the future it is found that this guideline is, indeed, an oversimplification, delay time

factors can be implemented, given the necessary data and time for programming.)

The goal of minimum holding can be achieved in at least two ways, both of which affect

the functions of the centers and are chiefly policy decisions:

1. A consolidation center can accept cargo only to those foreign distribution points

to which the center can maintain a flow of sufficiently full containers. Consolidation centers generally follow this practice today by advertising shipments to certain ports only. Under special arrangements, less-than-carload lots could be

accepted to other distribution points but no attempt would be made to consolidate

by storing cargo beyond certain stated periods. The cargo would go in less-thanfull containers perhaps under higher rates, or by ordinary LCL transportation.

On the other hand, centers strategically located might accept cargo for consolidation for those distribution points not served by others. A center, say in Toledo,

might serve certain European points but not South American points served by, say

New Orleans, and vice versa. If a consolidating company should own a number of

centers covering a large area of the country, it would probably send cargo to the

center most likely to fill a container for the desired destination. One such com- pany has stated this to be its future operating procedure.

2. A second means of minimizing the holding of cargo for consolidation is a part of a third possible function of a center and is treated in the following section.

1 Interstate Commerce Commission, New England Forwarding Company, Inc., Extension No. FF-96

(Sub-No. 2).
The Function of a Consolidation Center as an Intelligence Center

It appears that a consolidation center, regardless of ownership or other operational

policy, should act as a shipping intelligence center if it is to operate efficiently. The

center would hold on file all sailings from pertinent ports, departure dates, and overseas

ports of call. These dates would be made available by the center to all exporters in the area served by the center and would be revised and updated as times for the actual sailings

become more certain. The first forecasts might be weeks ahead of time and quite general as to the exact date, while the last would be several days prior to the actual sailing and

quite precise. This procedure is used at present by steamship companies.

Similarly, shippers would notify the center of proposed shipments and would consult

with the center concerning the probability of filling containers for specified overseas distribution points on chosen dates. The cargo would not be forwarded to the center (nor

perhaps even manufactured) until requested, thereby reducing holding time and storage space

at the center.

The shipper would decide whether to hold cargo at the factory until it could be con- solidated; whether to ship direct in less than carload lots; whether to ship to another

center or possibly ship in containers but in less-than-full containers. A decision of this

kind is strictly a business decision predicated upon competition, type of commodity,

prevailing local rate structures, cost of storage, space at the factory, inventory situations

and many other factors specific to the individual shipper. It would be difficult if not

impossible in a general model such as described in this paper to take all of these factors

fully into account.

10
Chapter III. COST OF CONSOLIDATION CENTERS

It is assumed that consolidation centers must collect sufficient revenue to meet their

capital and operating costs plus an appropriate level of profit. This operation assumes a "user fee" collected for the service rendered either directly or indirectly for stuffing the

containers and associated services.

It is the purpose of this chapter to assess the cost of land acquisition, construction,

operation, and maintenance of consolidation centers to determine the effects of these costs

upon the location and use of centers.

This assessment of costs of building and operation is discussed under 7 major headings:

1. The building (the shed or shell)

2. The land

3. Site improvement, grading, etc.

4

.

Maintenance

5

.

Equipment

6. Administration

7. Labor

The total cost associated with each of these major factors is divided by the expected

number of containers to be stuffed to determine the unit cost of stuffing and thus arrive at

the user charge. In addition, the comparative effect of each is studied to determine the

sensitivity of each contributing factor to the total cost per unit.

Prior to a discussion of these major cost factors, certain measures of performance and

costing concepts pertinent to the problem are discussed.

Measure of Unit Production

Because the goal of the chapter is the determination of a user charge (average cost)

for stuffing a container, it is logical and convenient to use the container as the unit of measure of production. However, because of the various sizes of containers, a decision must

be made in selecting the unit. For the report, the 40' x 8' x 8' Group I container is

selected as the unit of measure of production. (United States of America Standards Institute,

MH5. 1-1965 Specification Table 3.1.3.1.) A container stuffed to an average density of

24 short tons (48,000 lbs.) per container is the average density, although the referenced

table cites a maximum gross weight of 33.6 tons for the container and contents.

No inference is intended nor is any recommendation to be assumed that this container is

better than another type or that any particular size or standard is endorsed. The cost model

is capable of adjustment and use with any other unit capacity if such becomes desirable.

In this study, the 40' x 8' x 8' container loaded to 48,000 pounds as a unit of measure

is designated the Design Equivalent Container and is hereafter referred to as a D.E.C.

Center Throughput Capacity

The measure of throughput capacity for a center in this report will be the number of

D.E.C. 's per week , each week considered as 5 days, one 8-hour shift per day. Annual or Haily

capacities could be used with equal facility, but for this report, all center capacities are per week. Hence a unit of capacity will be one container (D.E.C.) per week. The symbol K will be used to represent the specific capacity of a center.

Since the cost of providing consolidation center services depends significantly upon

the utilization of facilities, it is assumed throughout this chapter that each center will

operate at or near its designed capacity. That is to say, a center will be designed with a capacity near its expected operating level. The requirements for each proposed center will

be estimated and the centers will be designed to operate at this level on the average.

11
Consolidation Service Charge

It is assumed that the consolidation center must obtain revenue to cover its costs and

that this revenue will be derived by charging its customers a service fee. This service fee

will be expressed in terms of dollars per D.E.C., and prorated among customers.

The general approach for estimating the service charge is to consider the fixed and

variable costs of operating the center, including an appropriate return on investment. The

pricing policy is simply to charge each container a fee derived from total cost for a year

divided by the expected annual throughput of loaded containers, where total cost includes

return on investment. Somewhat more specifically, if the annual throughput is X containers,

the annual fixed cost is F dollars and the average direct cost of consolidating services per

container is v dollars, then the service fee per container is

r = |+ v [3.1]

F consists of return on capital investments, insurance, interest payments, maintenance of

grounds, and administrative overhead, v is determined by direct labor and other direct

operating costs per container. Equipment cost may or may not be variable depending on the manner in which the equipment is acquired and maintained, i.e., purchase or rental. It is

considered a fixed cost in this study.

Since this study does not presume consolidation centers of fixed or predetermined sizes or location, it is desirable to develop the various cost elements on the basis of a unit of

capacity. In this way a service charge can be estimated for consolidation centers of any particular size. To account for regional variations in costs, a regional cost factor is utilized

In the work to follow, there is no suggested rate of return on investment. The proper

choices for these rates are considered to be beyond the scope of this study. However, effects

of several illustrative rates of return on that portion of service charge attributable to

specific features of the consolidation center are given. It will be noted also that the cost

components considered here are not exhaustive. The principal components are included, however.

Components of Cost

In order to analyze the service costs that would be incurred and to obtain better understanding of the basis for deciding upon the location and size of consolidation centers, it is

useful to partition the per-container charge, r, into its principal components. For this we

partition r into

r = rB

+ r

L

+ r

p

+ rM+ r

E

+ rA+ v [3.2]

where r^ = service charge per container attributable to costs associated with the building,

r^ = service charge per container attributable to costs associated with land,

rp

= service charge per container attributable to costs associated with site preparation,

r*, = service charge per container attributable to costs associated with maintenance,

rg = service charge per container attributable to costs associated with equipment,

r^ = service charge per container attributable to costs associated with administration,

v = service charge per container attributable to costs associated with labor and other

direct operating costs as described above.

To illustrate the general approach, consider the land component. Suppose that in order to develop a consolidation center with a capacity of D containers per week, a land area of A

acres is required (for building, parking, roadway, etc.). Then a = A/D denotes the land

requirement per unit of capacity. If land costs C dollars per acre, then Ca is the investment in land per unit of capacity. Then r

T = I Ca/52 is the service charge per container for

a rate of return on investment of I per year. The factor 52 is involved since I is an annual rate and capacity is related to one -week period.

12
The symbols used in this chapter are listed in Table 1. Table 2 gives the specific

values of the parameters used in this report. These values represent approximations of

current practices and costs derived from observations at several operating enterprises,

construction data from numerous sources, and on occasion, judgment of the project engineer.

These values are specified as initial approximations to be used in testing the general

methodology.

Table 1. Notations

a. Total land area required for a consolidation center per unit of capacity

aL

= a

B

+ V + a

L"

a

R Land area required for building.

a

T

' Land area required for receiving and discharging noncontainerized freight including

driveways and parking for trucks and railroad cars per unit of capacity.

a^" Land area for temporary parking, inbound and outbound, per unit of capacity.

C

T Cost per acre of land.

y Total cost of building shell only, exclusive of the double-high floor and mechanicalelectrical construction costs.

Cp Cost per unit area of double-high floor plus mechanical -electrical construction costs.

CL' Cost per acre of site preparation, including field surveys.

Cp" Cost per acre for paving and storm drainage construction.

Cw' Cost per acre of annual maintenance and repair of parking and roadway areas.

C,* Cost per acre of annual maintenance and repair of freight shed facility.

C," Cost per acre of annual maintenance and repair of electrical and other utility services.

Cwm Cost per acre of electrical power and service (not including power requirement within

shed.)

C Total costs allocated to each average manhour of direct labor.

e Cost index factor which converts annual cost variants between any two given years and

any two geographical areas.

a Constant factor applied to type of shed construction.

I Return on investment (land and site preparation)

.

Ig Return on investment plus depreciation on building. m Number of manhours required to stuff a container and handle a DEC unit of freight.

13
Table 2. Suggested Values

a. 0.455 acres

a

R 0.0098 acres = 427 square feet

a

L

' 0.0082 acres

a," 0.0275 acres

b 0.0883

C, $20,000 per acre

C

F $3.50 per square foot for the Washington, D. C, geographical area at a building cost index factor of unity

C

p

' $11,000 per acre

C

p

" $29,000 per acre for the Washington, D. C, area at a construction cost index

factor of unity.

C.' $125 per acre per year

C* $0,198 per square foot per year. ($8,625 per acre per year)

C," $141 per acre per year

CM

m $1750 per acre per year, based upon a use factor of 1/3 ^rea illuminated

during nonworking night periods) and $0.03 per kwh utility rate.

C $5.50 per manhour

e 1.00 for March 1968, Washington, D. C, area

m 9 hours (m = 9) which is a most likely allowance of 3 hours for a 3 -man team

per DEC. Reported productivity varies from a low of 3 manhours per container

(unloading tinplate from trucks) to a high of 25 manhours (unloading beer

cases from railroad boxcars) . Other classes of freight may be expected to fall within this range.

14
lo in o o o

o O tH i—i to

LO o o

o

1 CJ I* o 15 n

1—1 l<

1 • 1 o

ipE rH x

i£ 1 1- /a- 18

'O O

i o

(A

5hOM

O

O

u

oo

+-> cou

CD

5-t

•H

15
The Building

The costs of owning and operating a consolidation center vary from one place to another.

Construction costs have been estimated to vary as much as 45% between some of the southern

states and New York City. Figure 2 shows location factors assigned to various states which

are adapted from Department of the Air Force criteria. The location factors shown are multiples of 1.00 which has been assigned to Washington, U. C. In a countrywide study these

factors should be taken into account. However, in the area under study in this report (see

Chapter IV) they are much the same and are considered constant.

It is possible to estimate construction costs with more assurance than is possible with

other elements of the cost model. The industrywide trend at the present time is toward a simple structure with many common characteristics:

(1) The building shell is a metal prefabricated structure about 16 feet high and with

about 120-foot -wide clear span framing. The length varies dependent upon the freight

handling capacity required of the facility.

(2) The building shell rests upon a "double high" foundation and floor about 4 ft. 2 in. above the finished roadway height for truck-trailer unloading. The floor is usually

bituminous concrete suitable for forklift truck operation.

(3) There is no automated freight handling or sorting equipment. Some operators con- sider the possible use of an under-floor dragline or dolly towline, but the largest operation

observed (1096 feet in length) was experiencing no difficulties with a forklift operation

only and no future plans to convert to a dragline operation were under consideration, although

there appear to be some possible advantages to limited automation in a freight shed which is

very large and/or which requires a complex sorting of freight analogous to an airline freight

operation. The approximations do not include any automatic freight handling or sorting

equipment. Inclusion of a towline would not significantly affect the unit cost of the cost

model for a large facility, however.

(4) There is a minimum of mechanical -electrical construction requirements. Freight

sheds are either unheated or spot-heated in a minimal manner. Electrical construction is

austere and minimum plumbing is provided. No air conditioning is included except in office

areas

(5) Overhead door openings are nearly continuous along the exterior walls. Access to rail cars are either from outside the building shell or from a freight car spur which

extends into the building shell, being either a single or double track.

(6) There does not appear to be any reason why the building shell, mechanical facilities

inside the building, plus the double-high floor, should cost more than $8 per square foot.

(7) A large facility should probably include automatic data processing for office

procedures as well as external information. The possible requirement is not included in the

cost model because:

(a) Industry decisions (or trends toward an industry or government decision)

are not clear with respect to ownership and management of containers

(an automated data processing system seems mandatory if the centers

control their containers and if the operation is of significant size, say

handling 20,000 containers);

2 "Military Construction Pricing Guide, AFP 88-16, March 1968.

3 With respect to "General Purpose Warehouses" for Air Force construction, a Congressional

cost limitation of $8 per square foot is currently applicable to facilities constructed

in the United States less Alaska (including Hawaii) regardless of size of geographical

cost index. AFP 88-16, March 1967, "Military Construction Pricing Guide," Department

of the Air Force.

16
(b) Research on sizes, standards, and cost-benefit relationship regarding con- tainer ownership and fleet control are not a part of the study-mission;

(c) Some freight consolidation centers of medium and small freight -handling

capabilities do function profitably without automatic data processing and

information systems and there is no clear and readily available evidence

for introduction of automatic data processing systems.

There are technical considerations which permit small economies of scale in the con- struction of a facility. 4 It has been recognized that many different classes of construction

costs and equipment fabrication costs may be correlated by equations of the form:

y = «A

b

[3.3]

where y is the cost, a is a constant for the type of construction involved, A is the square

footage of the unit, and b is also a constant, the factor of economics of scale. Engineering authors refer to the exponent b as the "slope," since in logarithmic form it is the

slope of a linear function. Much research has been done to determine the value of b for

various classes of construction. 5

It has been pointed out by Chilton that although the median value of b for 36 curves

was 0.66, several curves had slopes between 0.88 and 1.0. He explained that these curves represented plants which were composed predominantly of multiple units. 6 Based upon manufacturers' and contractors' data, costed out for several sizes of consolidation center

freight sheds, a value of b = 0.883 has been adopted here. This relates to the cost of

! building shell only, exclusive of the double high floor and mechanical -electrical construction. It can be noted that if the cost (y) is known for a facility with a square footage A

and it is required to estimate the cost (y') of a similar facility with area A', then

The costs of the double high floor and mechanical -electrical construction can be

influenced by local and regional cost variations, but do not appear to be significantly

affected by economies of scalei A linear (direct) relation to area is assumed. The regional

i cost factors shown in Figure 2 may apply reasonably to both shed, flooring, and mechanicalelectrical construction costs in order to adjust from one region to another. Local variations of these costs are not subject to precise analysis and a generalization is assumed.

The service charge attributable to the construction cost of the freight shed including

shed and floor is:

rB

= ^B + Wil [3 - 5]

or since

A

B

= agK [3.6]

(aa£ K^ 1 + C

Fa

B

r^ =

) $& [3.7]

Cecil H. Chilton has presented a good description of the characteristics of the planningestimating process, the problems involved, the meaning and use of the "six- tenths rule"

and some of the caveats to be considered. The numerous representative curves for equipment

items for chemical plants are shown. Some statements are made about William's six-tenths

rule and its usefulness for entire plants. ( Chemical Engineering, June 1949, pp 97-106).

^This relationship is discussed by Roger Williams, Jr., Chemical Engineering , June 1947, p 102.

6 Chilton, Chemical Engineering , April 1950, pp 112-4.

7 Numerous trade sources were used including major manufacturers who expressed reluctance to be specifically quoted due to competitive considerations.

17
!:

t

where: r

R

= Consolidation charge per container attributable to the building construction

costs a and b as previously defined.

AR

= Total area covered by shed for shed with capacity K.

cL, = Area of shed required per unit of capacity (427 sq. ft. or .0098 acres).

K = Capacity in containers per week.

C

p

= Cost per unit area for flooring and mechanical -electrical construction

($3.50 per square foot at Washington, D. C).

e = Regional construction cost factor ( e = 1 for Washington, D. C.)

I R

= Rate of return on investment (including depreciation).

a = Constant for shed type (a = $8.47)

The building area value, a

R

= 427 square feet, represents an average of three field observations of operating freight consolidation sheds analogous to the considerations of this

present study. There is no representation whatsoever that this value is an optimum for all

specific local conditions. It is believed however that this number is a reasonable approximation of real -world operations today.

Figure 3 graphically portrays r

R through Equation 3.7) for a variety of Center capacities H

and for I R

= 101 and 20%. For Figure 3 , the values used were:

a = $8.47

ag <r 427 sq. ft.

b = .883

C

p

= $3.50

e = 1.00

In summary, it should be noted that economics of scale in building construction are relatively small and that the cost of stuffing a container is not particularly sensitive

to the size of the building as long as it operates close to design capacity. A reasonable

cost of stuffing a container attributable to building construction is about $7.00 (L= .15).

Land

Land costs vary widely according to location. Since consolidation centers require

relatively large amounts of land, these cost differences can affect significantly the choice

of location centers. Land costs also can have an important effect upon the feasibility of a particular design and upon consolidation user charges.

The cost component associated with land (r

T ) depends on the land area required, the

cost per acre of land, and return on investment. Hence

C

L

a

L

Z rL

= ^

[3.8]

where a, = Total land required per unit of capacity, in acres

C, = Cost per acre of land

I = Return on investment

For this study, C, = $20,000 and a

T = .0455 acres are assumed. aT can be further partitioned - into

18
19
a„ = Land for building (.0098 acres)

a, '= Land for receiving and discharging empty containers (.0082 acres)

a

T "= Land for temporary storage of inbound and outbound containers and freight

(.0275 acres)

The cost relation [3.8] assumes that there is no economy of scale in the neighborhood of the

capacity that might be built at any center.

Figure 4 shows the effect upon consolidation charge of varying land costs and two

different assumed values of I. The land area (a^) which was used is .0455 acres as shown

in Table 2.

The selection of the value .0455 acres per container as used in Figure 4 is the result

of substantial field observations of operating facilities and consideration of the engineering design characteristics of a functional facility. No single freight consolidation

operation was accepted as a typical prototype for the cost model. The values suggested for

a

T and its components (also used in what follows) represent what are believed to be adequate

but not excessive space provisions for a consolidation center operating at the capacity (K)

,

which refers to outbound freight but which includes space for unloading inbound freight.

A different treatment of variable costs is discussed later in this chapter.

It should be clearly understood that the land requirements for a consolidation center

are necessarily determined from consideration and evaluation of numerous engineering and

economic factors which influence and govern the functional characteristics of a facility.

Each site is unique and will require an individual evaluation. However, it is also reason- able to assume that land at nominal cost can be found in most areas in which containerization

centers, as envisioned in this study, are concerned and that excessive costs can be avoided.

It should be noted that this part of the cost of stuffing a container at the reasonable

figure of $20,000 per acre will mean a charge of about $2.00 (I=.1D) per container. This

figure should not vary appreciably unless extremes in land costs are considered.

Site Construction and Preparation

There is no generally applicable procedure which will result in a thoroughly defensible

estimate of site construction and preparation costs. Each site may be expected to present

a unique situation. It is reasonable to presume that a consolidation center site will be at or closely adjacent to both rail and highway main transportation arteries. Under such

conditions, demolition and removal cost might be substantial, or on vacant flat ground,

would be negligible. The costs discussed here include all demolition and removal costs

lumped as a generalized expected cost. In addition, field surveys, engineering layouts,

clearing, fencing, rough- and- fine grading, paving, and electric lighting are all consolidated into one item (shown on Table 2 as $11,000 per acre suggested value). Experience from

numerous sources entered into the selection of a specific figure which, as stated, may vary

widely from one place to another because of different and unpredictable site conditions, but f the figure is thought to be a fair estimate of expected costs.

rP

- ^ cp' aL

+ V ( V+ V^ Ei [3 - 9]

where r = The portion of consolidation charge per container attributable to site

p construction and preparation.

C '= Cost per acre of demolition, removal, field surveys, fencing, and other

p elements ($11,000 per acre).

C "= Paving cost per acre ($29,000 per acre).

I = Return on investment for site construction and preparation.

20
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construction is not included because this is generally a negotiated affair with the railroad -}- and the cost can vary widely; if operations are large it may cost the center nothing (see

section on Equipment, this Chapter).

Figure 5 shows the effect upon service fees due to variations in the costs of site prep- aration and paving based upon, the suggested values of Table 2. The cost of stuffing a con- tainer does not appear to be particularly sensitive to the costs involved in site preparatioi

A reasonable estimate for the cost of stuffing a container attributed to the cost of site

preparation appears to be about $3.00 (I = .10)

Maintenance, Repair, and Utilities Service

Operational costs of this nature are treated as fixed costs for purposes of our model.

They may be expected to approximate the same annual total irrespective of the flow-through

volume of freight handled. Paint will peel, concrete will spall, electric service lines wiL

break as the result of seasons and weather and not so much as the result of the usage of a facility. This might well not be true of a highly automated operation but for the type of

operation described here it is a close approximation. -

Some generalized statistical information is available for guidance with respect to such

expected costs. Four major categories are recognized: paved areas and roadways, freight she<-

maintenance and repair, electrical and utility services maintenance and repair, and electri- ; cal power costs and service. Utility costs for service for plumbing and heating other than ! electrical are not significant to the cost model. The service charge per container attribut;^

able to maintenance, repair, and utility costs is:

rM

" KV + Si"

+ V)L*l' + a

L")

+ V ^51 [3 - 10]

E

where C,' = Cost per acre for maintenance and repair services related to paved areas and roadways ($125 per acre per year)

Q." = Cost per acre for maintenance and repair services related to electrical

and utility services ($141 per acre per year).

C," = Cost per acre for electrical power ($1750 per acre per year)

C~,* = Cost per acre for maintenance and repair services related to freight shed

($8625 per acre per year or $0,198 per square foot per year).

One element of this cost equation is known to vary widely across the U. S. There is about

300 percent increase in the cost of electrical service when comparing Eugene, Oregon, with

Belmont, Massachusetts. 9

Figure 6 is plotted from Formula 3.10 to illustrate the effect of these costs

upon consolidation center user fees. Parameter values used to develop Figure 6 are listed

on Table 1 and identified in Table 2. A reasonable estimate of the cost ascribed to the

stuffing of a container using the above suggested values is about $3.00.

Equipment

There is a requirement for equipment to haul cargo in and adjacent to consolidation

centers. This equipment is part of the overall cost. There are two categories, the forklifts used inside the centers, and yard equipment for moving containers within the confines

of the parking lots for temporary storage awaiting line haulage.

Forklifts are almost universally used in the movement of cargo on pallets inside the

centers. They are manufactured to carry various maximum loads, generally 2,000, 4,000,

6,000, or 8,000 pounds. One is used with each team of from one to three men loading a con- tainer. (General ly 2 laborers and one driver compose a team. ) M Post Engineering Repairs and Utilities- -Annual Summary of Operations, Office of the Chief ' of Engineers , Department of the Army, Fiscal Year 1966.

9 Federal Power Commission, 1964. National Power Survey , Part 1, Table 10, p. 34. 22 JL

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Forklifts may be rented or purchased outright. They are considered to last about 5 years.

If purchased they cost approximately:

2,000 lb. capacity $5,300

4,000 " " 7,200

6,000 8,800

8,000 9,400

However, the trend appears to be toward leasing, and this study assumes leasing as a basis for cost estimates. The cost varies according to the environment in which they are

jused. If the environment is clean and floors are in good condition, they rent for about

"'$100 a month for 170 hours running time, the equivalent of approximately one eight-hour

-i shift per day for a month.

Thus in a clean operation in which 40 containers are stuffed per day (one shift) , and

ithe maximum use of forklifts are attained, ten gates are used and ten forklifts are required.

:This operation assumes an average of two hours for stuffing each container, or four containers

Iper shift at each of the ten gates. One extra lift would be held in reserve for emergency.

Thus monthly rental for the eleven lifts would be $1100.00 for stuffing 800 containers or

I $1.38 per container.

In contrast, under less favorable conditions at $147 per month, the cost would be $1617

i for stuffing 800 containers or $2.02 per container. A suggested figure is $1.75 per container.

(This figure does not include the operator whose pay is included under direct labor t)

In the containerization yard, containers are usually moved about on wheels using either

over-the-road tractors or yard mules. In this study it is assumed that containers will be

, moved by a tractor or "yard mule", costing about $8,000, and that the containers will be kept

| on wheels.

It is also assumed here that two tractors can handle an operation of at least 40 con- tainers per shift. This size operation is within bounds of a large operation, considered

j in this study to be about 40 containers per shift.

Tractors or yard mules cost about $3.00 an hour exclusive of the driver.10 Again assuming close to full design operation, the cost per container in a 40-container-per-shift

operation, would amount to $1.20 per container, plus $2.20 for the drivers.

Up to this point in this equipment discussion, it has been assumed that containers will

be moved on the line-haul over roads by tractor-trailer. Many containers however will move

by rail, particularly because of the very attractive rail rates.

There are five different container rates (plans) offered by the railroads. The selection of the rate to be used can affect the functions and the locations of the centers and

the cost of stuffing containers as discussed below.

Railroads prefer not to handle containers beyond the rail yards.11 They prefer that the

shippers or freight forwarders assume the responsibility of hauling containers to and from the

point of stuffing. However, they will do so, as in their plan II in which they offer a com

plete service between loading docks of shippers and receivers in separate terminal areas

;

but it is expensive and used generally for specialized operations.

The cheapest plans are Plan IIJ^ and Plan III in which the costs are about the same. Both require the shipper to pick up the empty containers next to the unloading ramp at the

railroad yard. The railroad loads and unloads from the flatcars without charge (today most

railroads accept only containers on wheels because they do not have cranes for loading con- tainers not on chasses. They move them up a ramp onto the flatcars using a yard mule.)

These two plans are the most suitable to the consolidation concept developed in this study,

and Plan llh is selected for exercising the computer model. (See Chapter V on Rates.)

10 "Inland § Maritime Transportation of Unitized Cargo." NAS, NRC Publication 1135, p. 80.

11 Conversations with railroad personnel.

25
However, these two plans, as stated, require that the shipper pick up the container

from the railroad yard and transport it to the place of stuffing and then return it to the

yard. Because a shipper bringing a less -than- carload lot to the center should not be responsible for hauling a container to the center nor for returning the container, stuffed with

several shippers' goods, this function is a shuttle link that becomes the responsibility of

the center.

There are alternative ways to handle this situation. First, the center could own, rent, or contract for tractors for the local hauling. At $3.00 an hour rental (the same as for the

yard tractor costing about $8,000), an hour round trip to the railroad yard and a 40-containe

per day operation would require 40 tractor hours or $120 per day or $3.00 per container, plus

$5.50 for the driver.

Second, it is possible that a negotiated arrangement could be made with the railroad to run a spur to the center, particularly if the operation were of sufficient size, say, 40

twenty-foot containers (20 flatcars) per day. Under certain situations railroads would be

willing to do this, but the possibility and cost depend on the particular situation. The

few present similar operations end- stuff the containers on the cars through the regular shed

doors, the track running parallel to the length of the shed. A few side loading containers

are used and this may come to be a common technique as it appears highly practical for

medium to large operations. Because of the greatly varying situations necessitating negotiated agreements and costs, no costing is given here, although it is probable in a large

operation that it would be nominal.

A third solution, and probably the most practical and desirable, is the location of the

center adjacent to the railroad yards. Railroads often have considerable land holdings

adjacent to their yards and will at times sell or lease at a nominal price to a corporation,

provided it appears that the business will be appreciable. There are many precedents for

this type operation, for example, grain elevators. An operation of this kind would be

highly appropriate for the unit train operation.

To summarize, the cost of stuffing a container attributed to forklifts varies according

to the environment in which it is used but a representative figure is about $1.75 per container. A representative cost attributed to yard equipment for shuttling containers is about

$1.20 per container plus $2.20 for the driver. If the center is located adjacent to a railroad yard, there should be no cost in moving containers to the flatcars, and if ordinary

over- the- road tractor equipment is used on a one hour round trip to the yard from the center,

a representative cost per container would be about $3.00 plus $5.50 for the drivers.

No costs are included for the case of the railroad spur to the center because of the

negotiated costs which would vary widely according to the specific situation.

From a practical point of view, the location of consolidation centers at railroad yards,

but also easily accessible to major highways, appears to be the most desirable arrangement

and is recommended in this study.

Administration

Administrative overhead is considered a fixed cost dependent on the design capacity per

week of the consolidation center. This cost does not vary, although the center may not be

operating all times at its capacity. The cost of overhead, then, is dependent on the

expected level of operations of the center. A sufficient number of clerks and administrators

are employed to handle that amount of cargo.

It is most difficult to estimate this cost; it is a study in itself. It is complex

because of the possible differences in the mode of operation of the center, for example, if

the center owns its containers and must keep track and route them, the costs are greater.

Furthermore, if this process is computerized, cost will be different. It is further complicated by determining the costs that must be allocated to central headquarters operation, if

such centralization exists.

26
However, for planning, some estimates must be made. These were obtained by talking with

administrative personnel in the business. They are admittedly rough because the administrators have difficulty themselves in breaking them down. They estimate that the cost is about

$1300 per week for a small operation of say 10 containers per day (consisting of one administrator, one supervisor, two clerks, one secretary, and three checkers) or about $5000 for

a large operation of 40 per day. This assumes that the costs are essentially linear with

increasing capacity within a reasonable span of operation, say from 50 to 200 containers per

week.

CA

Then r = —

[3.11]

If Ca, cost of administration is $5000 and K, capacity per week, is 200, the rA

, a representative cost attributed per container, is $25.00.

Direct Labor

Reference is made to the second major design assumption of the cost model (page 12)

where it was assumed that v (direct costs per container) is related to the number of containers stuffed during the operating year and included direct labor including benefits, payroll

taxes, and insurance.

No attempt has been made to determine statistical averages which relate labor productivity to specific classes of freight. However, it takes much longer to load or unload

certain commodities than others because of weight, packaging, and configuration. There is a wide range of variable costs dependent upon this factor. In one medium-sized operation

observed during the study, 15,000 lbs. were loaded per manhour and in another only 2,000 per

manhour

.

Information of this nature can be obtained and used in the operation of the mathematical

model at such time as the cost of statistical research into labor productivity is justified

and commensurate with benefits to be derived from the more precise data.

However, in the representative operations observed, there were either two or three men

working per container and it took from one to about six hours to load a container. In one

case observed, for example, cartons of beer were being loaded by two men using a forklift

in about three hours. Some companies work only an eight -hour day except during rush periods

where they will work two eight-hour shifts; other companies work around the clock. The more

hours worked, it is claimed, the less the requirement for container parking areas representing a saving which more than makes up for the overtime.

Because of the diverse nature, even within the same company, of the work and pay

arrangements, it is difficult to pinpoint labor costs. Some companies work through a con- tracting company that not only hires but also furnishes forklifts and similar equipment and

handles labor disputes. Other companies furnish their own equipment and do their own hiring.

However, generally speaking, the costs are much as follows. I LA and other East Coast union

labor cost about the same. The basic pay for loading is about $3.60 per hour with fringe

benefits that increase it to approximately $5.50 per hour. The forklift operator receives

about $.10 more in basic pay. Usually a team is composed of either one or two men plus a forklift operator. (A checker is also part of the team but is considered here under administration and supervision.)

Workers are generally paid whether they are loading or not, so in the calculations made

here, as well as throughout the study, it is assumed that no more labor is hired than can be

used and that operations are close to design capacity for an eight-hour shift. Furthermore,

I LA or other union labor are assumed, although nonunion labor cost can be less, and can be

employed in certain areas for certain kinds of work. (It should be noted that, according to companies visited during this study, where cheap labor is available, as in depressed areas, productivity is often low, to the detriment of the company and can more than offset the

saving from the low cost labor.)

Under these conditions and assuming three hours to stuff a 40-foot container, the cost

would be about $49.50 straight labor (see Figure 7).

27
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28
The service charge in direct labor attributable to variable costs is expressed as:

v = mC^ [3.12]

where m = Number of manhours of direct labor required to stuff a container and handle the

D.E.C. unit of freight (m = 9 hours).

C = Total costs allocated to each average manhour of labor (C = $5.50) .

It is obvious from Table 3 and from Figure 7 that the cost of labor and administration

required to handle a D.E.C. unit of freight is the major contributing element of the revenue required for the use of the facility.

Table 3 summarizes representative costs for each of the seven major factors influencing

the cost of stuffing containers and thus the determination of consolidation service fees.

Finally, it must be stressed again that the figures presented are approximations and

can vary considerably from those actually found in any specific situation. They are, however, illustrative and are presented with this understanding.

Table 3. Summary of Representative Costs Attributed to Major Fixed and

Variable Expenses Contributing to the "User Fee" for stuffing

One 40'x8*x8' Container*

Building $ 7.00 (I = .15)

Land 2.00 (I = .10)

Site Preparation 3.00 (I = .10)

Maintenance and Utilities 3.00

Equipment - Forklifts 1.38

Yard Mule 1.20

Administration 25.00

Direct Labor 51.70

Total $94.28**

*It is to be emphasized that these are representative costs and subject as noted in this chapter to variations in locations of centers, ownership,

policies of operations, etc.

**In comparison with this estimate of $94.00, a well-known steamship company

advertises that it will stuff a 24-foot container for $60, which when con- verted to a 40 -foot container, is $100.

29
Chapter IV. DATA: SOURCE, ADEQUACY, AND PREPARATION

Although it has been recognized by those engaged in .export -import transportation work,

the study reemphasized the lack of origin- route-destination data on cargo. Quantitative

solutions to important questions affecting our foreign trade cannot be determined without a concentrated effort to compile .more exact and comprehensive knowledge of the subject.

Tins lack is felt by those who are planning and investing, not only in consolidation

centers, but also in port authorities, new export businesses, transportation equipment, and

by those government agencies engaged in the encouragement of these enterprises. Much of

this planning is based on rules ^L thumb and inadequate data with important gaps. The

available data inevitably lack the breadth to allow tracing goods from the point of manufacture to the ultimate destination. Bills of lading at a port show where goods were picked

up, but the point of pickup may be a warehouse near the port, whereas the goods may have

been manufactured in a city a thousand miles away.

The exporter, himself, while knowing the source, may not know the route or mode of

transfer of his exports if, as is often the case, this part of the business is handled by a freight forwarder. If the routing is handled by a freight forwarder, he in turn may not

know the ultimate source, nor the ultimate destination because the cargo may be handled

overseas by a second freight forwarder. There is no central source or ready means of

determining point of origin, routing, and destination.

Finally, in relation to lack of data on routing, the actual domestic port used must be

a part of the data collected. The reason for a shipper or forwarder selecting a certain port

in this country is often a matter of personal choice which is not subject to logical analysis. The shipper or freight forwarder may not choose the closest port or the least expensive route. The choice may be traditional or based on a personal relationship with the

transporter, frequency of sailings, availability of special loading equipment, or anticipation of less congestion.

To attain systematic planning for efficient handling of cargo, as in this consolidation

study, these factors should be known. The ultimate source, mode, and route of transportation, foreign destination, and seasonal variation in flow are all important factors in the

location, size, numbers, function, and operation of consolidation centers.

No comprehensive, systematic nationwide studies have been conducted incorporating all or even a significant part of these data although a few limited surveys such as the Warrior12 study have been made. The only study approaching the desired adequacy is the excellent sur- vey conducted by the Delaware River Port Authority utilizing 1964 data, described below and

used as the data source for the illustrative location of centers described in this paper.

The Delaware River Port Authority study consisted of 5100 interviews with exporters

and/or importers tributary to the Philadelphia- Camden port. The interviewers were guided

by a questionnaire completed during the interviews. About 3500 of the interviews gave

pertinent data which were placed on magnetic tape.

The hinterland area covered by the study was all of Pennsylvania; western New York

State; all of Delaware; eastern shore, Maryland; the West Virginia panhandle; all of Ohio;

the northern two-thirds of Indiana and Illinois; southern Michigan; and southeast Wisconsin.

See Figure 8

For the southern New Jersey, Delaware, and Pennsylvania territory nominally within 100

miles of the port, all known shippers were surveyed. State lists of shippers were available

for this area. Outside the immediate area peripheral to the port, sampling procedures were

used which provided smaller samples as the distance from the port became greater.

12The S.S. Warrior, An Analysis of an Export Transportation System from Shipper to Consignee

(NAS-NRC Pub. 339, out of print).

30
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31
In western Pennsylvania, the balance of Delaware, and Maryland, a complete census was

taken of all firms that employed over 100 employees. In West Virginia, western New York,

Ohio, Michigan, Indiana, Illinois, and Wisconsin, all firms with over 500 employees were

interviewed.

With the smaller plants, i.e., with fewer employees, random samples were used. In western Pennsylvania, Delaware, and Maryland, 50% of those firms having 50 to 100 employees were sampled and 251 of those firms having up to 49 employees. In the other seven states,

25% of those firms having between 250-499 employees were sampled and 101 of those firms

having 100 to 249 employees. In using the data, compensation was made for the sampling,

i.e., from the sample design described above an item was counted 1, 2, 4, or 10 times.

An intermediate computer program was written to "clean up" the data from the Delaware

River Port Authority study. All information irrelevant to the study was eliminated and

miscellaneous editing was performed. Pertinent information was put onto a new tape in

concise and orderly form to facilitate its use in later processing. Basically each "record" on the tape reported the commodity (by 3-digit code), location of the domestic shippers,

the commodity classes shipped, the value of the commodity, the yearly tonnage shipped, the

average frequency of shipment, domestic port through which it was exported, how it was

packaged, the transportation mode used for shipping in this country and the sample factor,

and the annual value of the shipments. Data for both exports and imports were processed.

The desired information not provided was the seasonal or other variations over time, and

the foreign destination. Although the lack of this information was a distinct handicap,

the study survey was by far the best that has been made in this field and cannot be criticized for these omissions; it was designed and used for specific purposes other than a con- solidation study.

Certain information collected during the survey pertinent to this study was omitted

during the editing, for example, that for shipments of bulk commodities. This elimination was based principally on the judgment of the Delaware River Port Authority. Also eliminated were those commodities considered unsuitable for containerizing. This elimination was made

on the basis of consultation with the Maritime Administration, Port of New York Authority

studies, and on clues in the source data such as the manner in which the goods were packaged.

However, it must be understood in both of these elimination processes some arbitrary decisions were found necessary so that the work might proceed. A study of what is and what is

not containerizable is a lengthy project in itself and beyond the bounds of this research.

Furthermore, even after lengthy research, much is still left in doubt.

Also to simplify the process, the judgments were limited to "all" or "none" except in cases of considerable doubt when it was assumed that 50% could be containerized. See Tables

4 and 5 for a list of the commodities and how they were classed.

The location of each shipper was taken from the Port Authority data in latitude and

longitude and then converted by a subroutine to corresponding X-Y coordinates. A straight

line distance was then computed between the shipper location and the ports in question, and

entered in the data bank. Actually, as a simplification, the distance-to-ports used were

not those from the exact location of the shipper, but rather those from the center of the

corresponding county. There were 409 counties involved. (In the Philadelphia area, because of the concentration of shippers, Philadelphia County was divided into 11 districts,

each of which was considered as a county.) See Table 6 for a summary.

Data on costs of consoliuation centers are treated in Chapter III, titled "Cost of

Consolidation Centers" and data on rates and their application in Chapter V. titled "Rates."

32
Table 4 Philadelphia-Camden

Schedule S -- Exports 703

1962

005 Animals, edible

010 Meat, fresh frozen

013 Meat, & prods canned

017 Meat prod othrwis pre

*020 Animal oils fats, ed

033 Cond & evap milk

035 Dried milk

037 Cheese

039 Dairy products, nee

040 Fish, fresh frozen

043 Fish, canned

045 Fish prod othrwis pre

049 Shellfish & prods

050 Eggs & egg prods

055 Ed animal prods nee

060 Hides & skins, raw

065 Leather & mfgrs.

075 Furs & manufactures

080 Tallow, inedible

090 Animals, inedible

094 Shells, unmfgrd.

098 Animal prods ined nee *100 Corn

*101 Rice

*102 Barley & rye

*103 Wheat

*104 Oats

107 Wheat flour & semol

108 Grain sorghums

"'"109 Flour & grains nee *110 Animal feeds, nee

120 Veg, fresh, frozen

123 Vegetables, canned

127 Veg. preps, nee

130 Fruits, fresh froz.

132 Bananas

133 Fruits, dried evap.

135 Fruits, canned

136 Fruit juices

137 Fruits & preps

140 Nuts & preps

150 Veg oils fats, ed

160 Coffee

161 Cocoa beans shells

165 Tea

167 Table bev, mats nee

170 Spices

180 Sugar

185 Molasses ed. honey

190 Spirits, liquors wines

195 Bev , syrups, nee

200 Rubber, crude gum

201 Synthetic rubber

203 Rub scrap & rec. 205 Rubber tires & tubes

206 Aircraft tires tubes

207 Rubber mfgrs, nee

210 Naval store gum resin

220 Drugs, herbs, leaves, rts. -231 Soybeans

232 Flaxseed

233 Copra

235 Oilseeds, nee

240 Veg. oils, ined. 50%

250 Veg. dyeing tan mats.

260 Seeds

280 Tobacco, unmfgrd.

285 Tobacco, mfgrd.

"290 Molasses, ined.

297 Veg. prod. ined. nee

300 Cotton, unmfgrd.

310 Cotton, 'semimfgrs.

320 Cotton mfgrs. cotton rags

324 Hemp, manila, abaca, unmfgrd.

326 Sisal, henequen, jute, umrifg,

328 Veg fibers, unmfgrd. nee

331 Burlap & jute baggings

335 Veg. fiber, semi & mfgr. nee

340 Wool, unmfgrd.

350 Wool, semi & mfgrd.

381 Man-made fibers & mfgrs.

390 Textile prod, nee

°400 Logs

°405 Posts, poles & piling

°408 Wood unmfgrd. nee

413 Lumber & shingles

416 Wood, cont. plywd. veneers

°417 Railroad ties

421 Wood mfgrs. nee

430 Cork & mfgrs.

°440 Pulpwood

441 Wood pulp

445 Paper base stocks, nee

450 Stand, newsprint paper

460 Paper board

475 Paper & prods, nee *501 Anthracite coal

'"502 Bituminous coal & lignite

*503 Coal & coke briquets

*504 Coke

»506 Motor fuel & gasoline

*508 Aviation motor fuels

*510 Gas oil distil fuel oil

"511 Petroleum, crude

*512 Jet fuels all types

*513 Kerosene

-Sulk

°Not considered containerizable

.

(% if given is % of commodity considered containerizable)

33
Table 4 Continued

*514 Residual fuel oil bunker

516 Petroleum Asphalt

517 Lub oils & greases

518 Alphatic naptha

*520 Petroleum prods nee

521 Aviation lub oils

*522 Natural gasoline

523 Building cement 50 %

*526 Stone & mf:grs. nee

530 Glass & products

*540 Clays & earths

543 Brick & tile

547 Clay products, nee «548 Gypsum or plaster rock

*550 Sulphur

"551 Limestone, crushed

553 Salt 50%

*554 Sand, gravel, crushed rock

"555 Nonmetal, min, mfgrs, nee *600 Iron ore & concentrates

°601 Pig iron & spong iron

°602 Iron & steel scrap

603 Iron steel semifin. prods. 50%

°605 Iron steel castgs. forgs.

606 Tools & basic hardware

607 Hsld, kitchen, hosp, utens

608 Iron steel pipe tub tubing. 50%

609 Rid fin. steel mill prods 50%

°610 Bridges, prtble knekdwn nee

611 Metal mfgrs parts nee "613 Manganese

"614 Chrome

615 Ores & metals, nee 50%

"617 Aluminum ores & scrap

618 Aluminum ore & semifab.

"620 Copper ore cone, scrap

622 Refined copper, crude

624 Copper, semifabricated

632 Copper base alloys sem. "640 Lead ores & scrap

*642 Lead & allys, semifab.

*652 Nickel ore scrap semi.

"662 Tin ore, cone, scrap

*670 Zinc ore, cone, scrap

672 Zinc crd. semif. fms.

"682 Nonfer ore metis mfgrs.

690 Precious metis mfgrs.

701 Gen. elec. mach appar

705 Spec elec mach. appar

708 Radio comm trans receiv

710 Engines, turbines, nee 50%

°722 Constr. & mining mach.

731 Machine tools 50%

739 Ammo & rifle machines

740 Textile shoe machinery

742 Ind & off machines nee

'"770 Agri machines & trac.

781 Auto, trk bus & trlrs.

°784 Military water craft

°785 Merchant vesls & pts

°786 Railway locos & pts

787 Auto, trk bus & trlrs pts

°788 Military auto trks bus

789 Mil auto trk bus trlr pts

°793 Aircraft cough fit civ.

794 Aircraft & pts, nee 50%

796 Vehicles & parts, nee 507<> 801 Coal-tar products

802 Benzol or benzene

806 Other coal-tar prods

807 Toluene or toluol

810 Med & phar preps

825 Sulphuric acid

826 Alcohol

827 Sodium hydroxide

828 Other indus chemicals

833 Military gases

837 Synthetic resins

844 Chemical specs, nee 845 Carbon black

847 Pigments, paints, varn

849 Ammonium sulfate

*851 Nitrogenous fert.

*852 Phosphate rock

854 Superphosphate

"855 Potash fert. matls.

859 Fertilizer & mats. 50%

862 Dynamite

863 Explosives

865 Soap & toilet preps

901 Gen. misc. commods. nee

903 Small arms 909 Spec. misc. commods. nee

34
Table 5 Philadelphia- Camden

Schedule T -- Imports 303

1964

005 Animals, edible 236

010 Meat prods., fresh 240

018 Meat prods., nee 250

020 Animal oils, fats, ed. 260

033 Cond. & evap. milk 280

035 Dried milk 285

037 Cheese *290

039 Dairy prods., nee 297

040 Fish, fresh, frozen 300

047 Fish, prods., nee 310

049 Shellfish & prods. 320

050 Eggs & egg prods. 324

055 Animal prods, ed., nee 326

060 Hides & skins, raw 328

065 Leather & mfgrs. 331

075 Furs & mfgrs. 335

090 Animals, ined. 340

094 Shells, unmfgrd. 350

095 Animal prods., ined. nee 381

*1CG Corn 390

*101 Rice °400

*102 Barley & rye °405

*103 Wheat °408

»104 Oats 413

107 Wheat flour & semolina 416

*109 Flour grain preps, nee °417

*110 Animal feeds, nee 421

120 Veg. fresh, frozen 430

125 Veg. & preps., nee °440

130 Fruits & preps, fresh 441

132 Bananas, fresh 445

136 Fruit juices 450

138 Fruits & preps., nee 457

140 Nuts & preps. *501

150 Veg. oils, fats, ed. *502

160 Coffee, raw green *503

161 Cocoa beans & shells *504

165 Tea *507

167 Table bev. preps, nee

170 Spices *510

*180 Sugar *511

185 Syrups & prods., ed. *512

190 Spirits, liquors, wines *513

195 Bev., syrups, nee *514

200 Rubber, crude & gums 516

201 Synthetic rubbers *519

203 Rubber scrap -•520 205 Rubber tires, tubes 523

207 Rubber mfgrs., nee °526

210 Naval stores, gums, resins 530

220 Drugs, herbs, leaves, roots, crude *540

*231 Soybeans 543

232 Flaxseed 547

233 Copra *548

234 Castor beans *550

Oil seeds, nee

Veg. oils, fats, waxes, inedible

Veg. dyeing, tanning matls.

Seeds except oilseeds

Tobacco, unmfgrd.

Tobacco, mfgrd.

Molasses, ined.

Veg. prods., ined., nee

Cotton, unmfgrd.

Cotton, semimfgrd.

Cotton mfgrs. incl. rags

Hemp, manila, abaca, unmfgrd.

Sisal, henequen, jute, unmfgrd.

Veg. fibers, unmf . , nee Burlap & jute bagging

Veg. fibers, nee Wool, unmfgrd.

Wool, nee Man-made fibers & mfgrs.

Textile prods., nee Logs

Posts, poles & piling

Wood, unmfgrd. , nee Lumber & shingles

Wood contnrs., plywd. , veneers Railroad ties

Wood mfgrs. ,nec Cork & mfgrs.

Pulpwood

Wood pulp

Paper base stock, nee

Std. newsprint paper

Paper prods. & mfgrs. nee Anthracite coal

Bituminous coal, lignite

Coal & coke briquets

Coke incl. petroleum coke

Gasoline & other motor fuels except

jet fu e

1

Gas, oil, distil, fuel oil

Petroleum, crude

Jet fuels, all types

Kerosene

Residual fuel oil

Petri, asphalt & prods.

Lub. oils & greases

Petroleum prods., nee Building cement

Bldg, other stone, nee Glass & glass prods.

Clays & earths

Brick & Tile

Clay prods. , nee Gypsum, plaster rock

Sulphur

*Bu Ik °Not containerizable (or as noted 50% is estimated to be containerizable)

35
Table 5 Continued

*551 Limestone, crushed

°553 Salt 50%

*554 Sand, gravel, crushed rock

*555 Non-metallic minerals, mfgrs., nee *600 Iron ore & concentrates

°601 Pig iron & sponge iron

°602 Iron & steel scrap

603 Iron & steel semifin. prods.

°605 Iron & steel castgs., forgs.

606 Tools & basic hardware

607 Hshld., kitchn., hosp. uten.

608 Iron & steel pipe tubing

609 Rid. fin. steel mill prods.

612 Metal mfgrs. & pts., nee *613 Maganese

*614 Chrome

°615 Ferroalloys, nee 50%

*617 Bauxite crude aim. scrap

618 Alum, metals, semifab.

*620 Copper ore, cone, scrap

622 Ref. copper crude forms

624 Copper, semifab. forms

632 Copper base alloy semifab.

*640 Lead ores cone. 642 Lead alloys crude smfb.

*652 Nickel ore cone, scrap smfb.

*660 Tin ore cone, scrap

665 Tin metal crude smfb. fms.

*670 Zinc ore cone, scrap

672 Zinc crude semifab.

*682 Other nonferrous ores 690 Precious metals & mfgrs.

700 Elec. machy. & appar.

710 Engines, turbs., pts., nee 50%

730 Machine tools & parts

740 Textile, sewing, shoe machy., parts

745 Machinery & parts, nee

°770 Agricultural machy., pts.

°780 Autos, trucks, busses

782 Auto, truck, bus pts., acces.

°783 Merchant vessels & parts

°786 Railway locos, cars, parts

°790 Aircraft & parts 50%

°796 Vehicles &. parts, nee 50%

802 Benzol or benzine

805 Coal-tar prods.

810 Med. & phar. preps.

825 Sulphuric acid

827 Sodium hydroxide

829 Industrial chemicals

848 Pigments, paints, varn.

849 Ammonium sulfate

*851 Nitrogenous fertilizers

*852 Phosphate rock

854 Superphosphate

*855 Potash fertilizer matls.

°859 Fertilizer matls., nee 50%

860 Misc. chemical prods.

900 Commods. misc., nee

920 Articles, U. S., returned

36
Table 6. Counties Used for Location of Shippers

Survey

Area

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

Number of Counties or Districts

Included and Assigned

11 (7 districts in Phila. County)

4

8

9 (all 67 counties

in Pennsylvania)

10

9

10

12

8 (of 21 in New Jersey

3

9 (of 24 in Maryland)

17

45

26

(all 88 counties

in Ohio)

24 (of 62 in New York)

4 (of- 55 in West Virginia)

20

21 (of 83 in Michigan)

9

55 (of 92 in Indiana)

65

(of 102 in Illinois)

Total

22 (of 71 in Wisconsin)

409

Location

Philadelphia

Pennsylvania

Pennsylvania

Pennsylvania

Pennsylvania

Pennsylvania

Pennsylvania

Pennsylvania

New Jersey

Delaware

Maryland

Ohio, NE

Ohio, S

Ohio, NW

New York, W

West Virginia, Panhandle

Michigan, SE

Michigan, SW

Indiana, SW

Indiana, Central

Illinois, Chicago

Illinois, Central

Wisconsin, SE

37
Chapter V. RATES J $

it

iti As previously outlined, a basic incentive for containerization is that commodities can

be transported more cheaply when stuffed into "full" containers than when handled in small

lots. According to where a shipper is located in relation to the port and an inland consoli

dation center, it may be economically advantageous for the shipper to use the consolidation

center.

In the original concept, the model was to utilize actual transportation rates to be

applied to the commodity shipment data. This would involve rail and truck rates, container

and noncontainer, by each class or commodity.

As the study progressed and the rate structure was examined in more detail, it became

clear that the situation was enormously complicated, requiring the amassing of container

rates, less -than- container rates and local drayage rates between all possible pairs of

places. The problem becomes even more complicated when it is realized that initially it is

not known where the consolidation centers will be. Thus there could be literally an infinite

number of points and an infinite number of each kind of rate for each commodity.

Even with the most complete present rate data, the following problems would present

themselves:

1. An existing commodity rate may be artificial in that it is geared to a special

condition or is induced by a competitive situation.

2. An absence of established rates may exist between hinterland points.

3. An absence of container rates may exist where no container service is yet in

operation.

4. The effect of future competition on rates cannot be accurately anticipated.

5. Probable, unforeseeable, major changes may take place in the overall rate

structure as the technology and the use of containerization spreads.

6. The actual building of a center will undoubtedly influence, in itself, the rate

structure to and from the center.

Despite these problems it was deemed highly desirable to use some kind of real rate

data to exercise the model. This was in lieu of a completely arbitrary Rl and R2 (one rate

for all noncontainer traffic and another rate for all container traffic) , which had been

used in the operation of the model in its preliminary runs. That is, to debug the computer

program, artificial shipment data were used (simulated with a random number generator). It was actually the ratio of Rl to R2 that was instrumental in determining center locations as the result of the operation of the model.

Because neither the time nor the resources were available to develop the complete rate

details deemed desirable, compromises were necessary. Rail rates were obtained from one selected hinterland point (Cleveland) for container and noncontainer loads to four selected

ports (New York, Philadelphia, Baltimore, and Norfolk) for an arbitrary, small number of

commodity groups. All commodities were forced into these limited groups. The rates were

then converted to a basis of equivalent rate per pound per computed mile. The computed mile

is in effect the air line distance as computed by the computer program from the latitude and

longitude data furnished.* Thus the program is able to take a prospective consolidation cen- ter at any point (x,y), compute the distance to the port and to a shipper in question, and

apply an equivalent rate per mile to arrive at a rate to be evaluated by the selection

I

*An alternative computation leading to the same total transportation cost could be made by

applying a circuity factor which would yield a somewhat greater number of route miles and

result in a correspondingly lower rate per route mile. (See Chapter VII on the Mathematical

Model .

)

38
criteria in the optimization program. Comparison of rates will dictate use of the particular consolidation center or will deny it, as determined for each shipper. The noncontainer

rates were furnished with various weight breaks and with a minimum charge per shipment. The

information about shipments from shippers provided insufficient information for a determination of a distribution of shipment sizes to significant destinations. Available data

represent, for each shipper, annual export tonnage to domestic ports.

Also, container rates for one group of commodities were different according to whether

the shipment was insured fully by the carrier or only to a certain amount. The shipment

data gave no clue on this distinction. Furthermore, an agreement for a lower rate if the

particular commodity in question is less than fully insured is an arbitrary arrangement that

could not be predicted for the 5 to 10 year future and beyond. Again, a simple average of

the two furnished rates was used.

The container rates used are for the trailer-on-flatcar, the so-called piggy-back mode

of operation. In brief, five basic plans for piggy-backing trailers or containers are currently in general use and are included in interstate rail and truck rate structures.

(See Table 7.) These plans may be modified to fit particular situations.

A strong assumption exists that consolidation centers will be located near rail lines.

The ideal site would apparently be adjacent to a siding at the railroad yard. This location

would obviously offer the maximum advantages of time and economy for all parties concerned.

Conversations with representatives of the railroads indicate that the railroads greatly

prefer the shippers to do their own handling of containers away from the terminal yard.

However, because of the pressures of competition and expediency, they will collect and

deliver containers locally, but with an additional charge. The various rail rates generally

reflect the amount and nature of the services performed. The cheapest rail rates are without

the local drayage service.

Plan 11%, or some modification thereof, seems to be the most appropriate of the various

plans for pricing the line haul transportation of containers from prospective consolidation

centers to the vicinity of the port or marshalling area. Plan 11%, often referred to as "ramp to ramp" service, is the simplest of the rail rate plans in concept and provides the

most ready yardstick for measuring direct line-haul charges unencumbered by the inclusion or exclusion of other cost factors.

Under Plan 11%, the carrier furnishes the flatcars. The containers are purchased and

maintained by a transportation company but are under control of the railroad during transit.

It is assumed in this exercise that they are not owned by the consolidation center. This

assumption eliminates the need for analyzing container inventory and capital costs. In the

overall system analysis, the costs of the containers themselves would have a very slight

effect of favoring shorter line-haul distance, that is, pulling the consolidation center

toward the port. However, when this factor is part of the rate for transportation service

provided by the carrier, such considerations are automatically taken care of.

This treatment is entirely appropriate because physical containers are the same kind of

item in the total transportation picture as are railroad flatcars, locomotives, roadbeds,

and bridges. It was not contemplated that the present study should be involved in consideration of these rudiments of the transportation service.

The list of pertinent Schedule S export commodities used in this study included 201

commodities. Having in mind Cleveland exports, because the representative rates were based

on Cleveland rates, the commodities were assigned to seven arbitrary groups which were

deemed to include the great majority of Cleveland exports. All other commodities were

arbitrarily assigned to the designated group 8 for which the rate was taken as an average of

the rates of the other seven groups. Table 8 defines the rate groups and shows, for lessthan- truckload shipments, the derived equivalent rate by motor carrier per ton -mile* for

each group. As for container rates by rail (freight -all -kinds) , a flat figure per ton-mile

was applied to all groups.

*Tons are 2000 lb short tons. Miles are 5280 ft statute miles.

39
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to P 3 Po to to U C

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•Hp

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to ~^^ CO §1 u p 3 x 4-> rH P 4-> P 4-J Ph

Ph

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r—^ CO H > . P E o

to bo3P r-{ . >v P -H

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bfl

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O E P M ^ i—i bO • -H P P CO p cO O P

cO P -H

rHp 4-J bZ Oh CO CO p P -H p +-> o to •H

4-J 13

40
Table 8. Truck LCL and Rail Container Rates

LCL Rate

Group in <£/cwt/mile

1 Automotive parts .758

2 Chemicals, n.o.i. .693

3 Iron or steel articles .418

4 Foodstuffs .540

5 Petroleum products .581

6 Alcoholic beverages .892

7 Rubber articles .628

8 All other commodities assigned to average .644 (average)

Sample reading: 0.644<f/cwt/mi = $0. 00644/cwt/mi = 12.88<j:/Short Ton-mile

Container rate f.a.k. = . 135<J:/cwt/mi

f.a.k. = freight all kinds

A supporting tabulation, Table 9, shows the source data for the uncontainerized lessthan-truckload rates. Rates for the seven groups are shown by weight breaks from Cleveland

to each of the four ports (New York, Philadelphia, Baltimore, and Norfolk). The rates were

converted to a per-cwt-mile basis with the assumption that the majority of shipments would

be in the 5000 lb. or over category and thus eligible to receive the most favorable rate for

comparative purposes. Then the per- ton-mile rates to the four ports were averaged to provide

a single per-ton-mile rate for each commodity group to the average port, this figure to be

used by the computer program for application to each shipper. Table 10 shows this development.

Similarly, full -container rates were obtained for transport on flatcars from Cleveland

to the four ports named above. These source rates were in two categories, single -container

and two-container rate. The two-container rate was used in view of the anticipated volume

of consolidation center output. Table 11 shows these figures. Again, these rates were con- verted to a per-ton-mile basis and averaged over the four ports to provide a single per-tonmile rate to be applied by the computer program in the optimization routine.

41
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42
Table 10. Truck LCL Rates from Cleveland to Four Ports

in Cents per Hundredweight per Computed Mile

1

1 Automotive

Parts

Chemicals N.O

w/o Released

Rel. to 50<f/lb

.1.

Avg.

Iron

Steel

Art. Foodstuffs

Petroleum

-Prod.

Alcoholic

Bev.

Rubber

Art. Overall

Avg.

New York .720 .720 .589 .654 .388 .514 .550 .842 .589 .608

Philadelphia .750 .750 .620 .685 .397 .533 .575 .883 .620 .635

Baltimore .835 .835 .696 .766 .456 .592 .641 .981 .696 .710

Norfolk .728 .728 .603 .666 .432 .519 .559 .864 .603 .624

Average .758 .693 .418 .540 .581 .892 .628 .644

Sample Reading: 0.644*/cwt/mi = $0. 00644/cwt/mi = 12.88cf:/Short Ton-mile

Table 11. Rail Container Rates (Plan llh) , Cleveland to Four Ports

1 -Trailer Rate

(Min. 38500 lb)

2 -Trailer Rate

(Min. 73500 lb)

Computed

Miles

Equiv. Rate

in cents/cwt

per Computed Mile

New York $232.78 $380.07 407 .127*

Philadelphia 219.39 358.44 360 .135*

Baltimore 206.00 336.81 309 .148*

Norfolk 249.26 405.82 426 .130*

Average

Sample conversion:

$/load #/cwt */$

380.07 x 100 x 100

73500 x #/load

407

miles

127 */cwt/mi

135*

It is assumed that all containers will be filled to the minimum weight.

(In some cases container might be only 2/3 full.) Excess weight above

minimum is extension of same rate on per pound basis.

43
Chapter VI. MAP ANALYSIS OF DEMAND DATA

While the demand data described in Chapter IV were being converted from their original

magnetic tape form to edited form, and the mathematical location model and its computer

implementation were in progress, a careful manual map analysis of these data (export only) was also undertaken. Requiring about 5 man-weeks, this analysis proved quite worthwhile,

providing insights and visual comprehensibility not so easily gained by the machine approach.

Such an exercise yields convenient representations of the amount of containerizable

export cargo by day and by geographical distribution of its origins. It displays clustering

around industrial areas, and location in relation to trunk transportation routes (which can

be superimposed on the maps). Such information gives a rather reliable basis for inferring

first impressions concerning number, size, and location of consolidation centers. These

rough calculations have value in checking initial outputs of the computerized models for

general reasonableness, and in estimating the number and kinds of machine runs to be made.

They are particularly important in providing preliminary insight as to the number of consolidation centers appropriate for the export flow; earlier speculation on this point had ranged

from some hundreds to ten or fifteen. Finally, such a manual -map operation provides a more

intimate familiarity with the data than could be attained by the "impersonal" handling

associated with the (computing) machine work.

To keep the different phases of the research in proper perspective, however, it must be

remembered that inferences about the proper configuration of the consolidation center system,

drawn from inspection of the results of the map analysis, involve the transport rate structure

and user fees only in an intuitive, implicit way. Explicit quantitative consideration of

these factors involves a mass of calculations calling for the electronic computer, and it is

just such calculations which comprise the computerized model discussed in Chapter VII.

The actual conduct of the map exercise involved the following steps:

1. The total annual number of pounds of export commodities for each exporter in each

city or town was totaled. (Only annual figures and only weight, not cubic, are available.)

2. The total annual weight in pounds was divided by 250 working days per year (one

8 -hour shift per day) to determine the average pounds exported per export location,

generally a town, per day.

3. For a measure of capacity more easily visualized, the average pounds were then

divided by 51,000 lbs,* a reasonable figure for a "fully" loaded 40' x 8' x 8' container to determine the average number of container loads stuffed per day in one 8-hour shift. (The figure 51,000 lbs is now known to be higher than the

average outbound load to Europe today. For six months (July-December, 1967) the

average was 16,755 lbs. per 20-foot container or 33,510 lbs. for a 40' x 8' x 8' container.1

)

4. The number of container loads per day per shift were then compared with representative numbers stuffed by small, medium, and large consolidation centers visited

during the study. A center stuffing 10 per day (one shift) for this study was

arbitrarily considered a small operation, 20 per day a medium, and 40 per day a large operation. (Much larger operations apparently are feasible and three shifts

rather than one are common.

)

5. The average number of containers stuffed during one shift per day for the cities and

towns involved were then plotted on a map.

* It should be noted that 51,000 lbs. are used here, not the 48,000 lbs. in Chapter III. This discrepancy should cause no problems in that both are used only as measures of amount

of cargo not the actual number of containers stuffed.

3 "North Atlantic Container Statistics Report for Six Months Period Ending Dec. 31, 1967,"

O.M.P. MARAD.

44
The limitations of this procedure are readily apparent;

1. Each container was considered stuffed "full" regardless of the ultimate destination

of the cargo. In other words, some containers were in effect loaded with cargo for more than one ultimate distribution point. (This same limitation is true of the

machine solution as ultimate origin-destination figures are not available, nor are they known. See Chapter IV titled "Data: Source, Adequacy, and Preparation")

2. The figures are averages computed from annual exports. They do not reflect daily

or seasonal fluctuations.

3. The number of exporters that would actually use a consolidation center, particularly

from towns averaging less than one container load per day, is not known because

transportation rates to the centers and from the centers to the ports are not

introduced in the manual computations. The time limitations imposed on the study

precluded the very time consuming manual handling of rates.

The analysis was done by states, principally to test progressively the value of the woik

in relation to allotted time and the resume is by state (and then summed for a total picture).

This procedure is not without other merit in that state borders can affect transportation

rates and thus specific locations, for example, the Gary, Indiana- Chicago, Illinois complex.

After elimination of bulk commodities and those not adapted to containerization, and

after application of the sample blow-up factor (see Chapter IV), the data yielded the results

shown in tabular form (see tables 12, 13, 14, and 15) and are represented pictorially on a map (Figure 9)

.

Table 12

Summary, Towns Shipping One or Illinois

More Containers per Day Average, One 8-hour Shift

City

Average

lbs. per day

Containers per Day

(40x8x8 @ 51000 lbs)

Chicago 1,837,170 36.0

Granite City 80,000 1.6

Aurora 73,200 1.4

Rockford 72,400 1.4

Galesburg 64,400 1.3

Evans ton 54,700

2,181,870

1.1

42.8

All other towns

56 towns less than

921,180

one container each per day.

18.1

Table 13

Indiana

Summary, Towns Shipping one or more Containers per Day Average, one 8 -hour Shift.

Average

City lbs. per day

Indianapolis 231,000

East Chicago 128,000

Lafayette 129,000

Michigan City 93,200

581,200

Ail other towns 343,960

(43 towns, less than one container each per day)

45

Containers per

(40x8x8 @ 51000

Day

lbs)

4.5

2.5

2.5

1.8

11.3

6.7
Ohio (plus

Summary, Towns Shipping One or

Table 14

two towns in West Virginia)

More Containers per Day Average, One 8-hour Shift

City

Average

lbs. per day

Containers per Day

(40x8x8 @ 51000 lbs)

Cleveland 1,562,336 31

Toledo 1,133,663 22

E. Liverpool 640,000 13

Columbiana 576,000 11 Cincinnati 548,007 11 Woodville. 400,000 8 Dayton 365,443 7 Middletown 220,000 4 Wickliffe 216,666 4 Painesville 191,428 4 Fairport 171,428 3 Youngs town 164,477 3 Columbus 151,770 3 Canton 115,827 2 Mansfield 110,881 2 Defiance 96,000 2 Salem 88,644 2 Lancaster 80,240 2 Newark 70,088 1 Williston 66,560 1 Akron 61,312 1 Troy 52,822 1 22 towns 7,083,592 138 containers

All other towns 618,618 10 containers

(71 towns, less than one container each per day)

46
46a

Table 15 Pennsylvania

Summary, Towns Shipping One or More Containers per Day Average, One 8 -hour Shift

City

Average

lbs. per day

Containers per Day

(40x8x8 @ 51000 lbs)

Pittsburgh 3,230,000 63.4

Philadelphia

(not environs)

2,500,000 49.0

Johnstown 600,000 11.8

Bethelehem 371,000 7.3

Franklin 215,000 4.2

Lebanon 186,800 3.7

Steelton 180,000 3.5

Templeton 180,000 3.5

New Kensington 174,700 3.5

Cressona 174,500 3.4

Oil City 158,500 3.1

Norristown 136,400 2.7

Lockhaven 133,000 2.6

Connellsville 132,000 2.6

Ambridge 120,000 2.4

Fairless Hills 120,000 2.4

Lancaster 116,000 2.3

Erie 115,400 2.3

E. Pittsburgh 100,000 2.0

Ford City 95,200 1.9

Bridgeport 94,000 1.8

Hempfield Township 92,700 1.8

Borough Township 92,500 1.8

Clearfield 92,200 1.8

Beaver 92,000 1.8

Beaver Falls 80,600 1.6

Clairton 80,000 1.6

Pottstown 72,000 1.4

Glassport 66,800 1.3

York 60,600 1.2

Leetsdale 60,000 1.2

Morrisville 56,000 1.1

32 towns 9,891,900 196.0

All other towns 1,771,048 32,7

(296 towns, less than one container each per day)

47
Summary - Map Alia lysis

m

According to the Delaware River Port Authority data, a daily average of 23,393,368 pounds

of cargo is exported from the four states of Illinois, Indiana, Ohio, and Pennsylvania

amounting to 458 40-foot containers with a preponderance, 11,662,948 pounds, from Pennsylvania

As expected, the greatest amount is generated from the industrial areas and most of that

originates near the coastal areas- -in this case, Philadelphia- Camden. At the time of data

collection, 1964, very little of this cargo was containerized.

The major exporting areas by states were:

Illinois

Chicago (very little originates in the rest of the state)

Indiana

Gary-Michigan City

Indianapolis

Lafayette

(Small shipments originate throughout the state)

Ohio

Toledo

Cleveland

Youngs town- Salem- E.Liverpool

Cine innati-Middletown-Dayton

(Small amounts originate over the whole state)

Pennsylvania

Pittsburgh-Johnstown

Harrisburg- Lebanon- Pottsville

Philadelphia and environs, bounded by Lancaster, Trenton, Bethlehem

(Very little originates in Central Pennsylvania)

According to operational personnel interviewed, it appears that about ten 40'x8'x8' con- tainers per eight-hour day must be stuffed to support a center, and under the constraints

detailed above, it would appear from "inspection on the basis of the distribution and volume

of these data that a center would be located in each of the following areas.

1. Chicago -Gary

2. Indianapolis- Lafayette

3. Toledo-Gibsonburg

4. Cleveland- Elyria-Euclid-Fairport

5. Cincinnati-Middletown-Dayton

6. Columbus-Lancaster-Newark

7. Mansfiela-Wooster-Canton

8. East Liverpool -Columbiana- Youngs town

9. Pittsburgh and environs

10. Johnstown

11

Harrisburg- Lebanon- Pottsville-Lancaster

12. Philadelphia-Camden-Trenton and environs

Trunk lines of major railroads interlace these states, and in many cases the cities of

each area are located along a major trunk line. This is to be expected. It is assumed, and

recommended, that the center or centers in each area be located at a railroad yard (also with

major road access) to take advantage of the low rail container rates (See Chapter V on Rates,

48

ie.
Chapter III, Cost of Consolidation Centers). A seaport area such as Philadelphia-Camden, in

which a center would be located, would possibly be an exception. A fine-tuned study of each

area would decide upon the exact location of each center.

A striking result of this graphical plotting exercise is the adaptability of distribution

of exporters to unit train operation. Examples are the Perm Central (Northern route) from

Chicago through Toledo, Cleveland, Buffalo, etc., to New York City and Boston; the Perm

Central (Southern route) from St. Louis through Indianapolis, Dayton, Columbus, Pittsburgh,

Harrisburg, and on to the Philadelphia area; the Baltimore § Ohio -Chesapeake § Ohio from

St. Louis, Cincinnati, to Washington, Baltimore, and connecting to Philadelphia and New York;

and Norfolk § Western from Cincinnati through to Norfolk. The distribution of export centers

could hardly be more admirably suited to unit train operation. A study of train routes in

relation to volume and domestic ports of embarkation should certainly be made to expedite

this type of long haul transportation.

49
Chapter VII. THE MATHEMATICAL MODEL

i

i In this chapter and the next two, the following work will be described.

(a) The mathematical model and technique developed for determining good locations and

capacities for cargo consolidation centers,

(b) The computer program written to implement a version of (a) appropriate to the

data at hand, and

(c) The results of applying this program in illustrative exercises based on the data

delineated in Chapters III, IV, and V.

Chapters VII and VIII present some technical mathematical and computer- oriented material

which goes more naturally with the exposition here than it would elsewhere in the report,

but which can be omitted without loss of management -level understanding. Additional technical material is presented in the Appendices.

A comment is in order at the outset, concerning the level of precision to be attributed

to the model's outputs (results). This level must reflect both the accuracy of input data,

and the level of aggregation of the model as a whole. Hence, the center locations obtained

by the computer must be regarded as specifying only in general terms where each center is to be placed. Precise identification of the best site in each computer-specified locale would

require a fine-grained study of that locale with regard to land cost variations, access points to particularly attractive transportation facilities, and the like. Such "fine

tuning" appears unnecessary and indeed inappropriate at the system -wide planning level,

lying more in the province of the interests that undertake the initiation and operation of a particular center in a particular locale.

Scope of Model

Any system is imbedded in one or more larger ones, so that isolating it as the focus of

a study is almost certain to involve some distasteful excisions. To place the model to be

described in better perspective, it may be useful to list some of the requirements for an analysis that would do full justice to the ramifications of, and influences on, consolidation

center location and sizing:

(a) Models to predict U.S. exports and imports, year by year over the planning period,

by season, (containerizable) commodity class, shipper location, U.S. port of departure, and overseas port of delivery.

(b) Predictions of which U.S. and overseas ports will develop capability for efficient

handling of containerized cargo, and how this capability will develop over time.

(c) Models for the growth, deployment, and itineraries of the container -carrying vessel

fleet.

(d) Computer representations of relevant U.S. transportation networks, and their evolution over the design period.

(e) Computer representation of (land) transportation rates for the various commodity

classes, and for containers.

(f) A model describing redistribution of empty containers may prove necessary.

Clearly these items are not independent; one would need a "supermodel" to combine them

and their interactions with models of shipper* behavior to determine volumes and frequencies

of LCL shipments to centers, all as affected by the locations and sizing of those centers.

*"Shipper and/or importer" is actually meant, but the terminology here will generally be

export -oriented.

50
Some of the items concern policy questions not within the scope of this study. Some

involve (and are currently involving) massive research projects of their own. Some involve

masses of data too enormous to permit the rapid computer manipulations needed to accomplish

the location of centers under a variety of scenarios. One might possibly formulate such a battery of models (though this may not even be reasonable as a goal), but securing the

necessary information to validate them would be yet another monumental task.

Appreciation of this situation led to two basic modeling decisions. First, no attempt

would be made to venture beyond the project's assigned scope into the area of demand forecasting. Therefore, the pattern over time of demand for containerizat ion -consolidation

services--by shipper location, commodity class, U.S. port of departure, and overseas port of

delivery- -would be regarded as an input to the model. The computer can be used to work out

the consequences of many alternative assumptions about these critical inputs.

Second, it was judged essential to avoid developing and manipulating computer representations of the nation's complex and extensive freight transportation networks, or of its

transportation rate structures (which have become notorious for their lack of descriptioneasing regularities). As regards the first factor, the matter of centers' proximity to appropriate transportation facilities is left to the type of "fine-tuning" mentioned earlier.

(A discussion of locating centers in a network is given in Appendix A.) As for transportation rates, they are treated on a "distance" basis in the manner described below. Such a basis of course represents a considerable simplification of reality, but one which appears

reasonable in view of the apparent absence of alternatives permitting rapid calculations.

Decision Criteria

In determining "good" or "optimal" locations and capacities for centers on a generalized

cost basis, it is necessary to try to be explicit as to how "goodness" is to be measured.

The factors should be those which depend sensitively on location and sizing (the topics for

decision) , a test which appears to rule out such items as pilferage reduction. The obvious

remaining indicators of system performance are the resources of time and money expended in

the movement of the cargo from shipper to center to port of departure, so defined as to include the costs of initiating and operating the centers themselves. Our objective, therefore, is to minimize a total generalized cost which has both dollar and time components.

Even this, however, is not quite an accurate description of the situation. The difficulty is that "cost minimization" suggests the existence of a single decision-maker with the

authority to determine which center each shipper should use so as to minimize the system-wide

cost, even if some shippers are thereby not assigned to the centers they would most prefer.

Such models are briefly discussed in Appendix B.) It did not seem appropriate to base the

model on such a concept. On the contrary, it seemed best to adhere as closely as possible

to the following:

Assignment Principle : In an "optimal" solution, the assignment of shippers to centers

should be compatible with the centers' locations in that, given these locations, no shipper

should prefer a center other than the one to which the solution assigns him.

This desirable attribute of a "solution" has associated with it the following logical

difficulty: whether or not a shipper prefers a particular center depends on the generalized

cost of his using that center, which may in turn depend on the extent to which other shippers

patronize it. (With insufficient patronage, processing costs may be excessively high, or

there may be unacceptable delay awaiting arrival of sufficient additional cargo for the same destination to permit stuffing a reasonably full container, or perhaps excessive cost to the

center- -ultimately passed on to users --in sending out insufficiently full containers to avoid such delays.) (Such factors are discussed in Chapter II, "Function of Consolidation

Centers.")

This point will be discussed again later. The next observation here is that the "solution" concept should reflect not only the assignment principle, but also the complementary

Location Principle : In an "optimal" solution, the location of each center C should be

compatible with the locations and relative importance (as customers) of the shippers assigned

to C by the solution, in the sense of minimizing the total cost associated with this patronage

51
In applying this principle, the set of shippers assigned to C is regarded as temporarily

"known." Thus the patronage pattern at C is known, and hence the total processing cost at C

(assuming a design properly matched to the patronage pattern) is known.* Thus the principle

calls for locating C so as to minimize the total transportation cost involved in the use of

C by its (known) clientele. Note that this is to hold for each center.

Figure of Merit

A full evaluation of the costs and benefits stemming from any particular configuration

of consolidation centers would of course be a complex matter, especially if it is to deal

carefully with how these costs and benefits are distributed among the various interests

involved (shippers, transport operators, center operators, etc.). For the present study it was considered important, despite the risk of over-simplification, to be able to attach a single numerical "rating" or "score" to each situation studied.

The total generalized cost to the shipping community,

^W53 + WS)] [7 - 0]

in the later notation of [7.33], is a natural candidate for such a figure of merit (more

precisely, "demerit"). Its appropriateness is evidenced by the fact that, very roughly

speaking, the main thrust of the model's calculations is to reduce this quantity, taking

into account the interactions among the independent decision-makers involved.

However, [7.0] as it stands has some deficiencies as a figure of merit. Its absolute

magnitude will not be too reliable, in view of data uncertainties and the unlikeliness that

all relevant cost elements can be incorporated in the numerical work. It does not directly

make vivid the relative merits of different configurations of centers, or the effects of

changes in the values of the model's parameters, or the general benefits from a system of

inland centers versus not having such centers.

Accordingly, the figure of merit chosen is not the total generalized cost itself but

rather a modification of it, "normalized" to facilitate comparative evaluations of the types

mentioned above. This normalized version is obtained by dividing the total generalized cost,

for the configuration of centers to be evaluated, by the corresponding cost for the case in

which no inland consolidation centers are available. Thus the figure of merit, to be interpreted on a "high values bad, low values good" basis, will normally assume values between

and 1; the former extreme represents an unrealistic ideal state, while the latter extreme

represents the (presumably unrealistic) possibility of no economic advantage from inland

centers.

To convey a more accurate impression of the meaning of this scoring function, it may be

useful to discuss the scope of the word "total" in the phrase "total generalized cost."

Here one must distinguish between the conceptual mathematical model (the subject of the

present chapter) , and its present computer implementation as restricted by data considerations and other practical factors. What follows refers to the more comprehensive of the

two, the mathematical model.

First, "total" shares with "generalized" the intent of reflecting "time costs" as well as "money costs." Second, it suggests a comprehensive aggregation over all relevant cargo,

as indicated by the summation over all shippers S (exporters and importers) in [7.0]. This

objective is itself less than "total," since direct and indirect effects on sectors of

society outside the shipping community are not considered. Moreover even the limited objective is not fully attained in the present model concept, which focuses on LCL shipments to the exclusion of full -container shipments which do not require a center's consolidation

services but might benefit from its marshalling activities and associated unit train rates

to ports.

The term "total" also suggests a treatment of the total movement of cargo, from origin

to destination. How far the present analysis falls short of this ideal is schematized in

Figure 10 (drawn for simplicity for exports only) . In evaluating the numerator of the

* Possible effects of location (independent of patronage) upon processing costs, and a way

to handle them within the approach described here, are discussed later.

52
f pf

•H P) -H •H CL, 13 O rC! T3 C -H

nj CO B) OX O o co J -i

bO

— <

Pi H 4-1 4-> i—1

4-1 bo

f-t !hhU c o O o3 o •H

ft ft

X(/)

o3 2

ft +J T—1 O 03 ft Xtou

«k

n

+-)

S-H ~<7T oQ Td

(/ (DOC s

4-1 +-> -h

4-1

C 03 3 -H 4->

+-i 4J

C

co o

T3

u

(1) •H

"" > ». ^C

•H

o3

+-> rtou

M

Pi •H

rH *-c i—1 03 +-> +-> i-H 81

-CrtC s 10 CD o

•H & u

4-> I/) 03 h

'O

•H 4-> S

10

4->

r-l PJ !h P! O •r1 0) T3

t/>

u

O

C c <+-l 4-> -H +-1 T33 o "+j C 03 ^H u 3 -H 4-> o

CO o

ft 3

r—1 u s U

Jk Jh •H

+-> ^ 1 >—i

4-1

4-1 O 3I

10

to 03 S

co

S-i PQ +J r^

1—1 3

XIME3

<D

Sh a

^1 !h ftft

ft

ft > ' H •H X .n

co CO

-3O

£=

P! •Hs3

i-Hu

•Hs

4-J

I/) 3

CO

Sh H

ft

P3

53
figure of merit, i.e., the generalized cost for a system of inland centers, alternatives A

and B are both available to shippers, who choose between them (and for A, select a particular

center) on a generalized-cost basis. In evaluating the denominator, only alternative B is

regarded as available.

Two technical points regarding Figure 10 may be noted here. For alternative A, the cost

associated with the marshalling function will not be separated out from the general center

processing cost described in Chapter III; thus economies of scale from full-container shippers

using the marshalling facilities are not explicitly considered, nor do such shippers (in the

present model) influence the locations or the number of centers. And for alternative B, the

delay to LCL cargo -- in waiting for the accumulation at the port of enough cargo to be mergec

with it — is treated as negligible. To the extent that this assumption is optimistic, the

present calculation of the figure of merit is biased against the use of inland centers.

Sketch of Solution Process

The two principles (Assignment and Location) correspond, respectively, to the two type:

of computational step whose alternation constitutes the solution technique. The process is

to be applied for each of a number of values of

n = number of centers

in order to determine which number of centers is "best" in the sense of yielding lowest cost

after optimization of center locations. More precisely, n is the number of centers to be

located; there may also be centers at some or all ports whose presence there is taken as "given" by the model.

The solution method for a particular value of n begins with an initial or "trial" set of

locations for the n centers. This may represent a "best guess" by the user of the model, or

a set of random choices made by the computer, or perhaps a systematic choice such as placing

the n centers initially at the n "heaviest" shippers.

With these locations regarded as fixed, an assignment step (whose nature is discussed

later) is performed to produce an assignment of shippers to centers, which satisfies the

assignment principle. Next, this just-produced assignment is regarded as fixed, i.e., the

patrons of each center are regarded as known, and so a location step (whose nature is also

discussed below) is performed to select, for each center, a new location which satisfies the

location principle. These center locations are then taken as fixed, another assignment step

is performed, and so on, with assignment steps and location steps alternating. The process

is terminated when the center locations- -and thus the assignments as well- -have "settled down"

(to within a prescribed tolerance level) , indicating a situation in which (as desired) both

principles are satisfied to a good approximation; no shift of any center's location will significantly reduce the cost to that center's users, and no shipper can achieve a non-negligible

saving by transferring his patronage to some center other than the one to which he is assigned

The final assignment determines each center's patronage, and therefore its appropriate sizing

as well.

The preceding description was oversimplified, for ease of initial exposition. First,

assigning of shippers to centers was spoken of as if such an assignment were irrevocable.

Such an approach might be plausible if we are optimizing for some single designated "target

year," or if a shipper must sign some sort of long-term commitment to avail himself of a center's services. But it seems more likely that a shipper will be relatively free to shift

his patronage from one center to another during the planning period, and may well have an incentive to do so (perhaps because the second center has just become operational, or because

exports from its vicinity to certain destinations have grown great enough to permit rapid

filling of containers). It turns out, however, that the difficulties raised by this timevarying ("dynamic") aspect of the problem are predominantly technical- - i. e. , neither practically nor conceptually of great significance. Therefore discussion of their treatment is

deferred to the end of this chapter. Until then the language used will be as if the "single

target year" objective were in force.

Second, the natural concept of a "shipper," as an institutional entity with a specific

location, requires refinement. Such a "shipper" may have exports in several commodity

classes, involving different freight rates, on their way to a center in uncontainerized form.

54
He may have exports to different foreign distribution points, or for the same foreign distribution point but via different U.S. ports of departure; such cargo categories are not co- containerizable,* can suffer different delays at a center in waiting for enough of a containerful to build up, and may in fact be sent to different centers. This makes it convenient

to fractionate ordinary "shippers" into subentities, each characterized by

(a) a single location (essentially),

(b) a single commodity class, and

(c) a single combination of overseas distribution point and U.S. port -of- departure.

It is these subentities which will be referred to as "shippers" from now on. Shippers which

agree in items (b) and (c) , and have near y locations, can of course be aggregated if one is

willing to accept the resulting loss in accuracy for the sake of the resulting reduction in

the number of "shippers," and thus in the computational labor required. Note that while the

above-mentioned fractionation is conceptually natural and analytically convenient, it requires more care in interpreting data and/or in designing questionnaires for the purpose of collecting data on the demand for consolidation services.

Possible Difficulties

The practical reader, observing the requirement that this alternating sequence of assignment steps and location steps "settle down," may well question whether the results of this

process will in fact settle down, and (more important) whether they will do so in few enough

steps to make the solution method realistically feasible. A mathematical analysis of such

questions is often possible, but generally difficult. It is preferred here to adopt a "try

it and see" attitude, referring the reader to our actual computational experience (Chapter IX)

for affirmative evidence of computational feasibility. Such evidence does not provide the same certitude as would a theoretical proof, that rapid settling-down will occur for inputdata combinations other than those specifically tested, but it does produce a rather powerful

intuitive conviction that this will be the case for "reasonable" inputs to the model.

A potentially more dangerous difficulty arises from quite another source. It is a wellknown property, of "successive improvement" solution methods for the optimal location of more

than one facility, that the final pattern of locations can depend on the initial locations

used. In particular, an unfortunate choice of initial guesses on suitable locations for

centers might yield a final configuration which--though obeying both the assignment and location principles --corresponded to a total generalized cost distinctly greater than the true

minimum possible. For proper assurance of achieving the true minimum (generalized cost) or close to it, it was considered essential to explore the possible seriousness of this problem.

These explorations are reported later (Chapter IX) , but a general recommendation is a common- sense precaution that the method be applied using several initial sets of trial locations

(for each n)

.

Treatment of Transportation Costs

Our main aim in the balance of this chapter is to describe, in turn, the location step

and the assignment step whose alternation constitutes the model's solution method. Recall

that the location step is used in the following setting: For each center C, the set of

shippers using C can for the moment be regarded as "given," and the problem is to locate C so as to minimize the total (generalized) transportation cost associated with this usage.

A description of the location step must therefore begin with a discussion of how transportation costs (in dollars and time) are represented in the model. This is the subject of

the present section. Considerable use of mathematical notation becomes necessary at this

point.

Consider any particular shipper S among the users of center C. Our previous "fractionation" of shippers ensures that S's usage of the center involves a definite commodity class,

k~, and a definite U.S. port of departure, P~. S's generalized transportation costs can be

broken up (at least in principle) into two parts, relating respectively to shipper- to- center

movement, and to center-to-port movement. Symbolically,

*They may in fact be co-containerizable, depending on break-bulk policies, if their ports of

arrival are on a common sequence of ports -of-call from their (common) U.S. port of departure.

What follows can readily be modified to reflect this possibility.

55
T

C

(S) = T(S,C) + T(C,P

S

), [7<1]

where

T

r

(S) = transportation cost to shipper S, using center C,

T(S,C) = (generalized) cost of S-to-C movement,

T(C,P

S

) = (generalized) cost of C-to-P

s movement.

Note that T(S,C) and T(C,Pe) have both dollar and time components, and that each depends on

the location of C, which is to be chosen in the location step to minimize

IS

T

C

CS),

the sum being taken, over all users S of C.

Dollar Costs

We must now develop formulas for T(S,C) and T(C,Pg), beginning with the former. The

dollar component of this generalized cost is taken as a product

where

M (S) d (S,C), [7.2]

d (S,C) = distance (in miles) from S to C by mode m

I M (S) = money cost (per mile) for hauling S's material by mode m.

The underlying assumption that the cost of a movement should be proportional to distance

moved is in line with general regulatory policy.

The second factor in [7.2] will now be decomposed further intc

d (S,C) = c d(S,C) [7.3]

where

c = "circuity factor" for mode m, m

J '

d(S,C) = short- line distance from S to C.

For example, ICC line-haul unit costs for rail have been based14 on the value cm

= 1.13. In principle, c could be varied from territory to territory to reflect regional differences

in the circuity of the transportation network. At any rate, combining [7. 2], and [7.3] gives

the formula

M

m

(S)cmd(S,C) [7.4]

for the dollar component of the generalized shipper- to- center transportation cost T(S,C).

There is still the matter of how a numerical value for M (S) is to be found. One

171 possibility is

M

m

(S) = R

m

(k

s )A(S) [7.5]

where

R (k) = rate of mode m, per commodity unit per mile, for moving commodity k,

A(S) = amount shipped by S. [7.6]

^Interstate Commerce Commission, Rail Carload Unit Costs by Territories for the Year 1963,

Statement No. 5-65, March 1965.

56
However, the implied assumption of a constant rate R (k) , independent of shipment size, may

not be tenable.* If the available data provide the distribution of S's shipments, among the

size categories corresponding to different rate levels, then M (S) can be found by expanding

the right-hand side of [7.5] to a sum of products, one per size category. But if only the

frequency of S's shipments is known, then about the only thing to do is to calculate S's

average shipment size, and use the rate corresponding to that size in [7.5].

Time Costs

Now turn to the time component of the generalized cost T(S,C). It is represented

mathematically in the form

V

k(S) rk(S)

A (S)tm

(S,C) [7.7]

where for typographical simplicity we have written k(S) for k-, and where

V, = average (per unit) value of commodity k,

I, = inventory-type carrying cost factor for commodity k,

t (S,C) = average time in transit from S to C by mode m, m

while A(S) is as in [7.6]. Here I, is intended as a representative charge (in percent per

unit time) for interest, risk, obsolescence, and the like.

Next the factor t (S,C) in [7.7] will be split

rial in Meyer et aT. 1 ^ With d (S,C) as in [7.2],

further, in a way based on similar

material in Meyer et al. With d (S,C) as in [7.21, the formula reads

t ( S ,C) = dmCS.C) + <US,C) + drcCS.C) + dm(S,C) +

, g] nr ' J a ma mi m s m' L J m m m m

where

a = mean speed (velocity) of mode m, m

a = fraction of "road" time spent on sidings,

g = mean time per interchange on mode m,

i = mean distance between interchanges,

Y = mean time per switching operation for mode m (primarily for rail)

,

s = mean distance between switching points,

6 = mean delay in origin and destination terminals of mode m.

Note the assumption that delay in transit depends primarily on movement length (as an indicator of number of "opportunities" for delay) rather than shipment size. Note also that

[7.8] can be rewritten as

t (S,C) = a d(S,C) + 6 , [7.9] nr ' ' m

v ' ' m* L J

where a = (1 + a )/a + 3 /& + y /s . m *• nr' m nr • m 'nr m

*In such cases the fractionation of shippers may have to be modified. Customers might wish

to take advantage of reduced "bulk rates" to ship cargo, for several destinations, to the

center together. On the other hand, there would then be an extra operation, of sorting out

the shipment by destination, to be performed at the center.

15J. R. Meyer et ad, The Economics of Competition in the Transportation Industries, Harvard

U. Press, 1964. See Chapter VII. Pp. 188-196 describe an application of this type of

approach, including empirically based estimates of the parameters in [7.8] for rail and

truck. On page 192, a value of 10% per year for 1^ is suggested as reasonable.

57
Generalized Costs

We now combine [7.3], [7.4], [7.7J, and [7.9] into a formula for the generalized transportation cost T(S,C). The result is a linear function of distance d(S,C),

T(S,C) = MS) d(S,C) + K2 (S), [7.10]

where the coefficients Kj and K2 do not depend on distance and are given by

MS) =cm

{Mm(S) + V

k(s) A(S)I k(s)

am

} [7.11]

K^S ) =V

R(S)

I k(s) A(S)6m

. [7.12]

If [7.5] can be used, then [7.11] can be converted to

Kl( S) = A(S)cm{Rm[k(S)] +V

k(s)

I k(s)am

}. [7.13]

The preceding paragraph does not quite tell the whole story. The coefficients K: and

K2 depend on the mode m, and the choice of mode may well depend on the distance d(S,C) to be

covered. We should really have a mode -dependent notation Kj (S) and K

2 (S) , and should

complicate [7.10] to indicate use of that mode which yields the lowest generalized cost:

T(S,C) = minm

{Kim

(S) d(S,C) + K2m

(S)}. [7.14]

The method used to carry out the location step, however, depends critically on the

linearity of [7.10]. This method is itself an iterative one, initialized with a starting

guess as to a good location for C. It is suggested that this starting guess be used to determine a best mode m(S) for each user S of C, and that the location step be carried out

using K, ,<,-> as coefficients in [7.10]. This yields a location for C which is optimal (costminimizing J

J if the shippers' "best-mode" choices for the starting -guess location remain valid

for this "optimized" location. If such is not the case, take the new location as a new

starting guess, altering the mode choices appropriately, and repeat the process. One might

instead check for changes in "best modes" during the location step, interrupting the step to interject such changes as they occur.

(There is reason to expect that this problem will not prove too serious. For, changes

in best mode will typically occur for S if two modes are nearly "tied" in attractiveness,

but then it doesn't matter too much which of them is chosen. The computational experiments

needed to back up this intuitive argument have not, however, been performed.)

We have now discussed the portion T(S,C) of shipper S's generalized transportation

cost, which refers to shipper-to -center movement of goods. The total cost of such movements

to C is

£T(S,C) = £{Ki(S) d(S,C) + K2 (S)}

S S

= XKi(S) d(S,C) + £K2 (S), [7.15]

S S

where the sums are over all users S of C. The second sum at the end of [7.15] is a constant

(ignoring the mode-choice complication expressed in [7.14]), and so can be ignored in

applying the location step to minimize total generalized transportation cost.

Center-to -Port Costs

It remains to consider the second transportation- cost portion, the one which refers to

center- to -port movement. But this can be conceptually transformed to the type of situation

already discussed, so that the previous material can be applied to it. That is, the move- ment from center to ports can be regarded as a ports -to -center movement in reverse, and the

latter can be treated by the formulas developed above for shipper- to-center movement, but

with "port" replacing "shipper." Of course, the transportation rates appropriate to

58

si
containerized cargo must now be used.* Further, [7.7] must be changed to

[I k VkVC ' P)] VC ' P) ' [7.16]

where the sum is over all commodity classes k, and

A, (C,P) = amount of commodity class k shipped through U.S. port P by users of C.

With these changes, one again ends up with a linear function of distance,

T(C,P) = Ki(P) d(C,P) + K2 (P). [7.17]

The resulting analog of [7.15] is

IpT(C.P) = IpK^P) d(C,P) + I pK2 (P), [7.18]

and again the constant second sum in [7.18] can be ignored in the location step. The discussion previously centered on [7.14] applies here as well, but now concerning the choice

of mode for shipment from center to ports.

Resume

In summary, then, the location step applied. to center C is aimed at finding a location

for C to minimize a weighted sum of distances from C,

T(C) =

l s KitS) d(S,C) +l p Ki(P) d(C,P), [7.19]

where the first sum is over the (temporarily fixed) set of shippers assumed to patronize C,

and the second sum is over the set of U.S. ports serving these shippers.

From [7.19] we see that the shippers and the ports enter the problem in exactly the same

way. It is convenient to have a notation which treats them uniformly. Suppose then that the

shippers and ports involved with center C are located respectively at the (known) points

X1? X2 , , X ,p-. (Note that these points, and the value of q(C), can vary during the overall solution TM J process as assignment steps change the patronage pattern of C. Note also

that coincidence of two shippers, or of a shipper with a port, might lead to situations like

[Xj = X2 .) Furthermore, we can define positive constants Wi , w2 , ... w rr

-, by

w. = Kx (S) if X- is the location of shipper S,

w. = Kx (P) if X. is the location of port P.

Then the location step's purpose is to find a location for center C which minimizes the

function** F(C) defined by

F(C) = WidCXi, C) + w2 d(X2 , C) + ... + w

q(c) d(Xq(;c)

, C). [7.20]

*The reduction in rate may not be entirely passed along to the shipper, but the location

step maximizes the total "transportation savings pie," without regard to how it might be

sliced up among the various interested parties. Note that this criterion differs from that

of finding the profit-maximizing location for a center, an objective which would normally

involve attempts to gain additional users, contrary to the "known users" setting of the

location step.

**In the following notation, as in the preceding, we will be somewhat loose in failing to distinguish between a center and its location.
The Location Step

The problem faced by the location step, and described in somewhat abstract form by [7.2C

above, has a long history and (together with various special cases) has led to a number of

publications i\ the mathematics and economics literature. These will not be be summarized.

Instead, the interested reader is referred to the surveys by Witzgall16 and Hargrave.17 Rathei

the concern is with finding an effective method for solving the problem.

The existing theory is adequate to show that there is exactly one optimal (= costminimizing) location for C. But no explicit formula for this location in terms of the proble

data consisting of the w.'s and X-'s, has ever been found except for very specia] cases, and

it is generally believed1 that no such formula exists. Thus the method sought must necessaril;

be some sort of iterative "successive approximations" technique.

Assume some (x,y) -coordinate system chosen to identify locations (of shippers, ports,

and centers) , and let

\ = (- ,7i)

be the coordinates of X. (shipper or port, as the case may be). Also let

C

(t

) =(kCt\ y

(t)

) [7.21]

represent the estimated location for C at the t-th stage of the iterative process, which

terminates when "settling down" is indicated by d(c(t), c( t+1 )) dropping below some prescribe

threshold value. An iterative solution process will then give the coordinates x

1- J and y^

of a "new estimate" C^ * of the optimal location as computationally convenient expressions

in the coordinates x^ * and y^' ' of the "current" estimate C^ . Moreover, it will have the

property that if C^ ' = C^ , i.e., if the locations obtained in two successive stages

coincide, then this location is the optimal one.

Solution Method

The solution method is based on considering what mathematical conditions must hold if

(and only if) some point C = (x,y), not at one of the points X^, is to minimize F(C) as given by [7.20]. From the well known formula

dCX^C) = /(x

i

- x)* + (y7 - yY , [7.22]

it can be shown that these conditions (obtained by equating the partial derivatives of F(C) to ) are

I

iw.(xi

- x)/d(X.,C) = 0, [7.23]

X i

w

i

Cy - y^/dCX^C) = 0. [7.24]

These equations can be solved for the coordinates of C = (x,y), with the result

x =

[I iw

ix

i/d(X

i ,C)]/[Iiwi /d(Xi ,C)] [7.25]

7 = [Iiwiyi/dCX

i ,C)]/[Xiw:/dOC

J

,C)] [7.26]

Of course these equations contain x and y in their right-hand sides as well as on the

left, as coordinates of the "C" appearing there. However, they suggest basing an iterative

solution method on the equations

1d

C. Witzgall, Optimal Location of a Central Facility: Mathematical Models and Concepts,

National Bureau of Standards Report 8388 (6/30/65).

17W. W. Hardgrave, Location -Allocation Problems: A Survey, Operations Research 16_ (1968),

Supplement 1, p. B- 84 (Abstract)

60
*

(t+1)

= [liW^/dCXpC^/tl^/dCX^C^], [7.27]

7

(t+1)

= il^^/d^C^h/ll^/d^C^. [7.28]

This solution method, due to Kuhn and Kuenne^is the one employed.

Some precautions are required, though. For, suppose that C , lies very close to some

X. , say Xi . Then the quantity d(Xi,(> -*), which appears as a denominator in [7.27] and

[7.28], is very small, and so the resulting quotients can be numbers too enormous to be

handled properly by the computer. Now Xi may in fact be the optimal position for C (i.e.,

the cost -minimi zing location of a center may well be "at" one of the shippers or ports it

serves), a possibility simply not provided for in the analysis starting with [7.25]. Fortunately, the theory provides a separate test for this possibility;* this test is a bit

time-consuming, and so is applied only when "triggered" by the proximity of cWto Xi (or some other Xi) . If the test is not satisfied, i.e., if the sequence of tentative locations

for C is only "passing by" Xi , then the numerical difficulty mentioned above is avoided by

replacing [7.27] with the algebraically equivalent

x

(t+1)

= [wlXl + dCX!,C Ct)

D I i> fiVdCxi ,c

Ct)

o]

* [Wl + d(X1( C

(t)

) l ±>1 w^dfX^C^)], [7.29]

and similarly for [7.28].

Manhattan Metric

Satisfactory starting guesses, C = (x^ , y^ J

)

, can be obtained by solving a related

problem, namely, replacing the straight-line distance formula [7.22] by

d*(X.,C) = \x±

-x\ +| yi-y| [7.30]

and choosing C^ ^as a location for C which minimizes

f(C) = w1 d*(X1 ,C) + w2 d*(X2 ,C) + ... + w

q(c) d*(X

q(c)

,C)

with the d's given by [7.30]. This formula [7.30] is known as the "Manhattan metric,"

since it corresponds to travel distance in a network of streets (like much of Manhattan's)

running in two perpendicular directions.

This procedure is useful because the cost minimization problem for the Manhattan metric

does not require an iterative solution method, but instead can be solved in the following

simple way.** Form the sum

W = wj + w2 + + w rn . [7.31]

Renumber the points X. = (x.,y.) so that xj<_ x2 ± • • .<. x rrv Now begin adding up wl9 w 2 ill ..., q (_LJ stopping as soon as the cumulative sum equals, or exceeds W/2. If the sum exceeded W/2 when w. was added (but was aW/2 after the preceding addition of w._J, then the first coordinate x^

of C is given by x = x-. If the sum equaled W/2 when w. was added, then x^ ' can be taken

as any number between x. and x.

1

, say their average. The second coordinate y

' of C is

found analogously. 1

18 H. W. Kuhn and R. E. Kuenne, An Efficient Algorithm for the Numerical Solution of the

Generalized Weber Problem in Spatial Economics, J. Regional Science 4_ (1962), pp. 21-33.

*The test is described in the next chapter; its application requires reassembling the

"fractionated" shippers associated with X lt **Witzgall, op cit , pp. 11-13. Again coincident shippers should be aggregated before applying

this methoaT

61
Instead of using the Manhattan metric approach only to provide a starting guess for each J center's location at the beginning of the location step, there is computational advantage in

exploiting it more fully as follows: Preface the solution method as described so far with a preliminary phase, in which location steps and assignment steps still alternate, but where

the location steps find center locations in terms of the Manhattan metric only. Switch over

to the "real" straight-line distances only after this first phase settles down.

Restrictions on Locations

There is a further element of sophistication which might well prove desirable, but which

has been carried only in part to completion. Namely, general policy decisions may place, on

the locations of centers, additional conditions relating to governmental and/or geographical

subdivisions. Such a condition might for example take a form like:

There shall- be at least one center in the northern part of (particular) state z, and at least one in its southern part.

Clearly this type of restriction must be accomodated in the location step rather than

the assignment step. However, in the location step a center is "identified" by the set of

shippers currently assumed to patronize it, rather than in any intrinsic way. So to implement a constraint reading "region R should contain at least one center," it is convenient to select some relatively prominent shipper S located fairly centrally* in R, and to rephrase

the constraint as "the center currently assumed to serve S must be located in R." To avoid

undue mathematical difficulty it must also be assumed that R has a simple geometrical shape

--that of a convex polygon.** An approach to this "constrained" version of the location step

is given in Appendix C.

Location-Dependent User Costs

In the preceding discussion of the location step, the assumed objective was to minimize

the total generalized cost to the (known) users of the center being located. This ignores

the possibility that the within-center cost, also, might depend directly on the center's

location. Such dependence might enter the dollar component through variations (between locations) in land acquisition costs when constructing the center, or wage scales for the labor

it employs. (See Chapter III, Cost of Consolidation Centers.) The time component might be

affected by differences, among locations, in accessibility to transportation facilities.

As emphasized earlier, no attempt will be made to cope with locational effects at a fine-grained level. It is assumed that the choice of a specific site for a center, within

a demarcated area, would be a judicious one. With this understanding, the influence of location on user fees can be treated in a rough way as follows

The region of interest in locating a particular center C is assumed divided into subregions R\ , R2, etc., such that (a) each of these subregions R- is a convex polygon, and

(b) labor costs and the costs of (careful) land acquisition can be regarded as sensibly con- stant for center locations in R. , though these constants can, for instance, differ between

Rj and R2

. Thus the dollar component of processing cost (and hence, user fee) will be

essentially constant within each R.

Now solve, for each R. in turn the type of problem described in the last subsection:

Find a location for C. constrained to be within R., which minimizes the total generalized

transportation cost to the center's users. Let T. be the minimum- cost level corresponding to

C-, and let U- be that level of total user costs associated with having C in R. . Then choose

the location of C to be a C for which 1

J T. + u\ = min. (T

i

+ U. )

[7.32]

*If there is no such shipper, or analogous set of shippers assigned to a common center, then

the justification for the constraint would be quite questionable.

**It appears that states and/or natural subdivisions of them can typically be rather well

approximated by such shapes.

62
Treatment of Costs Incurred at Center

Now that the discussion of the location step has been completed, the assignment step

must be described. Recall that, roughly speaking,* this step associates each shipper with the

center he would most prefer. The locations of all centers are regarded as known.

It is assumed that the most preferred center, for shipper S, is the one whose use by S leads to the lowest sum of(a) the generalized cost of shipper- to- center and center-to-port

transportation, and (b) the generalized cost incurred by S at the center. If, as before,

the symbol C stands for an arbitrary center, and if

P

r

(S) = generalized "processing" cost incurred by S in using C,

then the center C(S) most preferred by S is characterized by the condition

T

ccs)

(s) + p

c(S)

(s) = min

c [Vs) + p

c

(s)] - [ y - 33 ]

The previous text included a mathematical formulation for the generalized transportation

cost Tp(S), defined in [7.1]. Before the assignment step can be described, a corresponding

'formulation of the generalized processing cost P

r

(S) must be developed. Like T

r

(S), it has

'both a monetary component and a time component. The conceptual analysis is much more com- plicated here, however, because the two components are strongly interrelated and because both

of them are bound up with considerations of centers' sizing, pricing, and operating policies.

Consider just the matter of pricing, for example. A number of quite different scenarios

are possible, at least in principle. One might imagine unified control (or regulation) of

the entire system of centers as leading to the offering of a common schedule of prices by

all centers. (This could certainly simplify the assignment step, since for each shipper S

the monetary part of Pp(S) could be dropped from the comparison of centers in [7.33].) Or,

such unified control might lead to different prices at different centers, so set as to maximize the centers' total profit.

At the opposite extreme, one might conceive of individually operated centers, each

striving to maximize its own profits. Here the appropriate mathematical model for pricing

might well be that of an n-person non-cooperative game.

19 Unfortunately such "games" do not

i always have solutions (in a sense meaningful for the present discussion), and even when

1 solutions exist, they may not be unique.

+

Intermediate between these two extremes (unified control vs. independent centers) is of

i! course the case of a number of subsystems of centers, each with its own unified management.

i: This far -from -exhaustive treatment of pricing (and its dependence on assumptions concerning the ownership of centers) should really be expanded to include alternative possibilities

as to price regulation. And much of the discussion would have to be repeated in describing

alternative approaches to the analytical treatment of selecting sizes for centers, or to the

mathematical representation of operating policies.

Investigating so great a variety of speculative possibilities, though desirable in the name of "comprehensiveness," is not really practical. Instead, a number of guidelines to be

explained as they arise below were introduced in order to arrive at a definite model for

analys is

.

Complications have been alluded to earlier, and will be treated in detail shortly.

19 See R. D. Luce and H. Raiffa, Games and Decisions, Wiley and Sons (1957} Chapters 4, 5, 7. §For games with just 2 players, the chances that a solution will exist are quite good; see

K. Goldberg, A. J. Goldman, and M. Newman, The Probability of an Equilibrium Point, Journal

of Research NBS, 72B(1968), pp. 93-101. But the probability of existence decreases exponentially with the number of players (paper by M. Dresher of the RAND Corporation, presented at a 6/68 Conference on Combinatorial Mathematics, at Yale TIniversitvl

.

tSuggestions to avoid this difficulty have been offered, but are not entirely convincing.

See Luce and Raiffa, op cit , p. 173.

63
Time Costs

The time delays involved in the passage of goods through a center can be roughly classified into two categories: accumulation delays (waiting for enough co-containerizable material

to arrive to yield a sufficiently full container), and service (or "queueing') delays. These

categories will be treated in turn. Note that the time component of the generalized cost

to shipper S incurred at center C is

VsdW® [t a

(s ' c) + VS > C "> r 7 - 34 i

where the product of the first three factors (all defined earlier) yields a cost per shipment

per unit of delay, and

t (S,C) = average accumulation delay per unit,

3.

t (S,C) = .average service delay per unit.

Thus we must develop formulas for t and t . 3. S Dispatching Policies

Accumulation delays clearly depend on the center's operating policies concerning holding times and container load factors. If the latter are too high (too close to 1), there may

be excessive delays to a user S while the center awaits enough cargo co-containerizable with

his.* If the load factor is set too low (too close to 0), there will be excessive costs for

the transportation of an inordinate number of containers, typically skimpy on payload.**

A plausible type of dispatching policy can be described by a pair of quantities,

L = "target" load factor,

H

= maximum holding time.

The policy is that a shipment is held until either there is enough co-containerizable cargo

to fill the fraction L of a container (which is then dispatched) , or a time H has elapsed,

whichever comes first. This is not fully realistic; for example if L = 0.75 and 3 preannounced containerfuls of co-containerizable material arrived during the same morning,

the center would presumably send out 3 full containers rather than 4 containers each 3/4

full of payload. Still, such (L,H)-type policies seem to be reasonable approximations of

what one might expect.

Some policies might admit considerable sophistication. For example, one can conceive

of the commodity classes being grouped into priority categories, each consisting of commodity

classes with similar sensitivities to delay, i.e., V^1 1, - values . The p-th priority category

would have a policy described by some (I. ,H ), with both parameters presumably lower for

high-priority categories. One interfering complication, however, is that co-containerizable

cargo could contain material from several different priority categories, and it is not

immediately clear what priority should be assigned to such a composite; giving it the

priority of the most delay- sensitive category present might mean that in the long run too

much low-priority material gets carried along on a high-priority basis. Alternatively, one might consider having a separate policy (L ,H ) for each c-th class of co-containerizable

cargo. This too seems somewhat awkward, since such a class can contain cargo elements with

quite different sensitivities to delay.

§

*And also excessive cost to the center (ultimately passed on to users) for storage of as yet.

insufficiently full containers or their contents.

**And there will be a similar inflation of cost of the overseas break-bulk operation, let

alone the costs associated with acquisition and/or use of the containers.

§ Partial exception to this statement can be taken, in that shippers with co-containeriz;)Me

cargoes may be implicitly "synchronized" by trying to meet the same sailings.

64
For the present study, however, it seems best to avoid complications and to consider a single pair (L,H) for each center.* (It might vary from center to center.) Recall that L is

a target load factor; because of the effect of H, the quantity

L = actual average load factor for c-th co-containerizable class

will in general be less than L.

The evaluation of Lc , along with that of the average accumulation delay t a (S,C), will be

described below. But first we note the important quantity

I c

= 1/L. [7.35]

which gives** the inflation ratio in converting from incoming quantities (of cargo in the

c-th co-containerizable class, measured in containerfuls) to outgoing quantities (measured in

containers). If c(S) is the unique co-containerizable class to which shipper S belongs (by

virtue of the fractionation of shippers), and subscript "C" designates the center involved,

then the term T(C,Pg) in [7.1] should be multiplied by Ic rg^before use in the location step

and probably also when forming Tq(S) for use in the assignment step via [7.33]. The situation

is not quite so clear in the latter instance, because the increase in Tq(S) is due to elements

of the center's policy, and so might not be passed on (at least not entirely) to shippers. At

present, though, it appears most consistent to perform this modification of T

r

(S) for the

assignment step, as well as for the location step.

Evaluation of Lr and ta

The evaluation of the (actual) average load factor L

c

, and average accumulation delay

ta(S,C) will now be described. Since the delay depends on the co-containerizable class c(S)

to which (fractionated) shipper S belongs, the notation will be changed from t (S,C) to

t (c,C) with the understanding that

a

ta(S,C) = ta (c(S),C). [7.36]

Both quantities to be evaluated depend not only on the parameters (L,H) describing

center C's dispatching policy, but also on the distribution (over times and shipment sizes)

of new arrivals at C of class c cargo. This distribution is a composite of the analogous

distributions for the individual shippers belonging to the c-th class. For general distributions, the evaluation of ta

(c,C) and Lc would require a "Monte Carlo" simulation. But since

the actual distributions are not presently available to us, it is reasonable to proceed under

simplifying assumptions which have been observed to hold approximately in analogous real-life

situations.

The mathematically simplest assumption, of course, is to treat the inflow of class c

jmaterial to C as taking place continuously over time at a uniform rate. This however is too

simple an assumption; under it the target load factor would either be consistently unattainable

ji(so that dispatch times would be governed solely by the maximum holdover H) , or consistently

feasible (in which case the role of H would never come into play). Nor does such a formulation

jseem reasonable, to represent arrivals from a number of users whose shipping schedules are presumably independent .

§

The type of formulation proposed for use here has become rather traditional, in the light

both of practical experience and of theoretical development. It represents arrivals arising

in such a way that the probability of at least one arrival, in any time interval of duration

x, is

1 - exp(-xx) [7.37]

*Another direction of possible sophistication, based on a suggestion by P. B. Mentz of the

U. S. Maritime Administration, is described in Appendix E. **Actually [7.35] is an approximation; the rigorous version is given in Appendix F. §Support for this simplification comes from the observation that priority differences seem

likely to involve distinctions between typical holding times of 1-2 days vs. 3-4 days.

Cargo so delay-sensitive as to find such a distinction significant seems likely to follow

the air route rather than marine routes.

65
Here X is a positive parameter, with the interpretation that 1/A is the average interval

between successive arrivals.

There are (at least) two ways of applying this Poisson distribution of arrivals to our

situation. One is to treat the arrivals at C, of shipments from each shipper, as Poisson

distributed.* For each shipper S in class c, data like those of Chapter IV provide the tota! volume and frequency of S's shipments; treating these : hipments as uniform in time and size

leads to a mean spacing between S's shipments, which can be equated to l/A^ and a mean

shipment size a-. Suppose there are m shippers belonging to co-containerization class c. If all of them had the same mean shipment size a, then their "resultant" would be a Poissoi

process with shipment size a and parameter A = ^cA^, the sum being over all shippers S in

the c-th class.

The evaluation of L is relatively direct in this case. Start counting time from the

arrival of the first class c shipment (of size a) following the dispatch of a class c container. Let m be the largest integer such that ma~<L. Then the maximum holding time (H) comes into effect if, and only if, fewer than m additional shipments arrive during the next

time interval of duration H. By known properties of the Poisson distribution, the probabilit

of this event is

m-1 _M Pm_ 1

(H) = exp (-H) £N=0 T/Nl (H = A qH) , [7.38]

l

i a function of m and H which has been extensively tabulated. In this situation the container',

payload will be (N + l)^ for some N between and m-1 inclusive, while in the complementary

case it will be (m + l)a. Therefore,

L = a {exp(-H) £ (N + 1)H

N

/N! + [1 - p (H) ] (m+1) }

[7.39]

c N=0 m~ 1

This can be rewritten

r

c

= * fHPm- 2

(IT) + Vl® + [1 _ Pm.!(H)](m+l)}. [7.40]

The evaluation of t (c,C) is more complicated and is gi\en in Appendix D; here only the

result is given,

ta (c,C)= (l/^Cm-l) U-pm

(H)) + Hpml (H)]. [7.41]

r

The preceding formulas for L , I , and t were based on the assumption that (m+1) a <_ 1. In case (m+l)a > 1, L is in effect replaced by ma, and this can be accomplished by replacing m with m-1 in the formulas.

The corresponding formulas for L and t , when_the users of C in the c-th co-containeri

zation class have different average snipment sizes a^ can be written out explicitly and

evaluated using a computer, but are very much more complicated. Until the quality of empiri

cal data appears to warrant this extra substantial effort, it is suggested that a fictitious mean shipment size at C for the c-th class be formed as a weighted average

i\

li

a

c

= £s

X S Ws = C1AC

)ZS

X S

a

s

, [7.42]

where the sum_is over all users of C belonging to the c-th class, and that [7.40] and [7.41]

be used with a as a. Still more complex models, which account explicitly for size fluctuations in shipments from individual users, can readily be devised (e.g., using "truncated

stuttering Poisson distributions") but much less readily analyzed. This degree of sophistication will not be considered any further here. At the other extreme in model refinement,

one might consider dropping the dependence of A and a upon the center C, i.e., treating

them as equal for all centers, with values calcSlated c as averages over all shippers in the

c-th class, rather than over all users of C in the c-th class.

*Here the distribution of arrivals at C is identified with that of arrivals at the consolida' > tion phase of processing.

66

; «!

;j k
A final note concerning accumulation delays and container load factors: In dealing with

iports, accumulation delays at the U.S. centers should of course not be included in the

meralized costing. Impacts of overseas accumulation delays on import patronage at U.S.

enters may, however, warrant inclusion in a more refined model. Average container load

ictors remain relevant, but would presumably have to be estimated directly from empirical

; iita rather than derived from a submodel of the sort described above for exports. This

Lstinction is based on an impression that data on arrivals (frequencies and volumes) of LCL

lipments at overseas consolidation-containerization centers will not be readily available.

Minimum- Patronage Restrictions

This is a convenient point at which to interrupt the discussion of (generalized) costs

lcurred at a center, and to take up one of the difficulties described earlier when the

esignment principle was introduced. Consider a particular center C, a co-containerizable

Lass c, and a shipper S belonging to c. In considering a particular C for possible selection

in the assignment step), one of the factors a shipper S must consider is the typical accumuition delays his shipments would encounter at C. But this depends on the degree of patronage

j: C by other members of c!

Of course excessive accumulation delays can be ruled out if the center sets low enough

ilues for one or both of the parameters characterizing its dispatching policy --namely the

target^ load factor L and the maximum holding time H. But then the average realized load

ictor Lc would be very low, resulting in high costs associated with an excessive number of

! ontainers. This is clearly just the same problem (insufficient patronage) from an alternate viewpoint.

The recommended treatment of this problem involves setting a minimum acceptable level

lij one of the ways described below) on the patronage of C by shippers in the c-th class.

: that level is not attained, it is assumed that C will not offer service to class c shipnts.

In more detail: The assignment step for all centers is first performed without imposing

le minimum patronage levels. Then, for each co-containerizable class c, the following

rocess is carried out. A check is made on whether every center, to which shippers in the

;-th class have been assigned, has a patronage level which meets the "minimum acceptable"

riterion. If this is not the case, the center at which the criterion is most severely

tolated is declared "off limits" to class c shippers, and the former class c patrons of

iat center are reassigned. This continues (with previous "off limits" restrictions remainng in force) until all class c shippers are assigned to centers with adequate class c itronage

.

This approach seems both more manageable and more realistic than more esoteric alternates, for example, the use of integer linear programming methods such as described in

>pendix B. Note that the assignment step has now (like the location step) become an itera- ve process.

It remains to describe how the "adequate patronage" criterion, for a specific center C

id a co-containerizable class c, is to be established. The following three proposals are

.1 essentially equivalent, so that the choice among them might be made on a basis of

laturalness":

(a) Put an upper limit on Pjjj.i (A CH), the probability of dispatching a container whose

load factor is less than the target level.

(b) Put an upper limit on the average accumulation delay t (c,C).

(c) Put a lower limit on the average realized load factor, L . A related alternative is based on the presumed saving in transportation cost, from

lipping containerized rather than uncontainerized material over part of the trip to the

$-. port of departure. The idea is to choose the lower limit, in (c) , as that level of Lc

: which the monetary effects of shipping containers largely consisting of "filler" in effect

sduce this saving to a specified fraction of its original value. Since the saving depends

67
on a shipper's location, this criterion would most naturally apply to individual shippers

rather than to a whole co-containerizable class. But perhaps this concept is somewhat too

complex to be a likely basis for actual practice.

Evaluation of Service Delay

The next topic is the evaluation of t s = t s (S,C), the average service delay per unit in

curred by shipper S in the movement of his cargo through center C. In the absence of relevai

empirical data, the treatment of this point must necessarily be speculative, based on con- siderations of mathematical simplicity and of analogy to apparently similar situations which

have been examined in some detail.

It seems natural to express t as a sum t = t + t , and thus total time in the

center as

s s su sc'

ta (S,C) * t

s (S,C) = ta

+ t

s = t su

+ t a

t sc

[7.43]

where t su refers to the time between (1) physical arrival at the center and (2) completion o processing to the point where accumulation delay (ta) becomes meaningful. Similarly, t sc refers to the period between (1) the time at which the dispatching rules first permit

containerization, and (2) the actual departure of the container from the center.

For incoming cargo, there is a question of whether "excess capacity" should refer to al '. arriving material, or should be treated on a shipper-by-shipper basis or at least separately,

for each group of shippers which might utilize separate receiving and/or storage facilities-,

e.g., shippers using the same transportation mode to transfer their goods to the center. Th

same question arises for outgoing material, where a category might consist of the users of a common U.S. port of departure, or (more grossly) of a common means of transit from center to port. .: Pending the gathering of information on which to base a more refined treatment, we shal

interpret "excess capacity" on a "total volume" basis for both inflow and outflow material.

Thus t and t , and hence their sum t = t (S,C), will be treated as independent of the

shipper- identity S. For the present, the relation between "excess capacity" (Key) and

average wait (w) will be taken as that for a "simple" queuing process (Poisson distribution

of arrivals, exponential distribution of "service" times): 20 w=l/Kex

. [7.44]

For the application of [7.41], set

K (C) = mean handling capacity for uncontainerized material at center C,

K (C) = analogous handling capacity, for outgoing (containerized) material,

A(c,C) = A(c,C)a(c,C)

= mean arrival rate at C of material in c-th co-containerizable class.

Then the over-all mean arrival rate at C is

yyo = La(c,o, [7.45]

i

-

while the mean outflow of containers, taking into account the inflation factors I = I(c,C),

is

* The point here is a well-known one: if the handling capacity of a facility is merely

matched without "slack" to nominal input rate, then normal fluctuations in arrival patter

can easily lead to lengthy queues and serious delays. "Capacity" here refers mean throug j. put capability per unit time.

20 See for example, M. Sasieni, A. Yaspan, and L. Friedman, Operations Research - Methods an Problems, Wiley (1959), p. 133. In [7.44], w includes the duration of service time.

68

:f

pe

si

,4
A

C

(C) = I c I(c,C)A(c,C). [7.46]

\pplication of [7.44]* gives

t su(C) = [Ku(C) - A

u (C)]-\ [7.47]

tsc(C) = [Kc(C) - A

c (C)]"

1

. [7.48]

[ |tn inflation ratio for center C can be defined by

1(C) = A

c (C)/Au

(C). [7.49]

.t is reasonable that input and output capacities, K (C) and K (C) , would be chosen in the

;ame ratio 1(C) as the input and output volumes. Then combination of the preceding formulas

eads to

t

s (C) = [1 + 1(C)] [Kc

(C) - A

c (C)T ' [7.50]

Four brief comments are in order before leaving this topic. First, the preceding

liscussion referred to exports, involving the center in the receipt of LCL cargo and the

ispatch of containers. It seems likely that the queueing delays suffered by imports would

jie largely independent of those for exports, i.e., that (especially with proper peak-traffic

lanagement) the two flows would not interfere much with each other. In this case, a separate

'"orrnula like [7.50] would be required for imports.

Second, the capacity level K (C) in [7.50] is a design parameter rather than an empirial datum; finding an appropriate value for it is one of the problems ("center sizing") to be

reated in this study. Selecting such a value is a question that will be returned to later.

Third, both 1(C) and A (C) in [7.49] depend on which shippers patronize center C. But

his information is the output of the assignment step, raising the question of how [7.50] can

e used in the course of that step. This question, too, will be taken up shortly.

Fourth, it is interesting to observe a basic difference between the factors governing

ccumulation delay t and service delay t . Roughly speaking, to minimize t a shipper would

end to choose a heavily patronized center. But then there would be a tendency for t to be

arge —or for there to be extra cost associated with additional capacity needed to rule out

xcessive service delays. Of course this last effect might be more than counteracted by the

eduction in unit cost due to spreading the fixed cost over numerous users.

Note that with eq. [7.41] for t and eq. [7.50] for t at hand, the time component [7.34]

f the genoralized cost to shipper S incurred at center C can be calculated. The monetary

omponent will he treated next.

Monetary Processing Cost

The dollar component of the generalized processing cost Pp(S) appearing above [7.33] will

h denoted Mr(S) . It represents the user's fee that would be paid by shipper S for employing

lie services of center C. As part of the transportation system, one would expect consolida-

'ion centers to have price structures reflecting "cost of service" considerations more emphatially than a "value of service" concept. Thus it seems clear that user fees should, to a ^nsiderable extent, reflect the costs incurred by the center itself.

i The area of pricing policy has already been mentioned, explicitly, as one in which simlifying assumptions (subject to later modification if required) must be introduced in order

d get on with the analysis. The two assumptions made at the outset are (a) that a center's

,cpenses are regarded as allocated among its patrons in proportion to their volumes of

usiness, and (b) that the fee charged a user is directly proportional to "his" share of the

'[.located cost; the constant -n of proportionality (a "profit parameter" among the model

lputs) being the same for all .centers.

ji\ fuller analysis would reflect the fact: that T7.44] is not strictly applicable to yield

7.48], since cne associated queue nas non-Poisson inputs.

69
It should be noted that assumption (a) is not without plausible alternatives. For

example, it might well be the case that some monetary cost elements are roughly proportional

to the time periods required for the associated operations. This would lead to terms

TTtys.ot^cs.o + k

c

t sc (s,o + k

a

t a

cs,c)]

in Mp(S), where the k's are appropriate constants. Such terms might be treated most con- veniently in conjunction with the cost-of -delay portion of Pr

(S), and will net be discussed

further here.

Another point concerning (a) is the question of whether the "user quantities" used for

cost allocation should refer to incoming or outgoing material. In general different allocations would result from the two choices, because of the differences in inflation ratio among

co- containerizable cargo classes. A suitably detailed costing model would permit the allocation to be based upon both quantities, involving separate allocations for the costs of the

operations up through stuffing, and for the remaining operations. But since the cost

information in Chapter III is developed on a "per container stuffed" basis, the allocation

will be treated in terms of outgoing quantities. That is, if shipper S uses center C and is in the co-containerizable class c(S), he will be assigned a fraction

I(c(S),C) A(S)/A

c

(C) [7.51]

of the center's cost, where A (C) is defined by [7.46].

It remains to develop an expression for the costs incurred by the center. Since the

letters "C" and "K" are already in use, the letter "E" (for "expense") will be used to designate this total cost. The approach chosen is a common one, in which E is expressed as the sum of a fixed cost F which depends on the throughput capacity of the center, and a variable for operating) cost V, which depends on average realized throughput. Thus,*

E

c

= F(K

c

(C)) + V(A

C

(C)). [7.52]

With further study, the description of center operations given in Chapter II could pre- sumably be used to divide each of the cost elements developed in Chapter III into "fixed"

and "variable" portions, associated with capacity and with operating level, respectively.

However, available information suggests limiting the treatment to a linear cost equation,

E

c

= b£C) + b

F (C)Kc

(C) + bv(C)A

c

(C), [7.53]

where b ,b„, and b„ are constants which might in principle vary with center location. This

would be valid for all values of A^C) no less than some minimum level ATM^ (say, 50 contain- ers /week) constituting a minimum overall patronage threshold for economical operation of a center.

i In summary, then, the monetary component of the generalized cost to S of using C is givi

by

M

C

(S) = 7rEc I(c(S),C) A(S)/A

c

(C), [7.54]

where E

f

is given by [7.52].

Sizing of Centers

The line of analysis developed above reduces the "sizing problem" to the selection of a appropriate level for the output capacity K (C) of a center C, whose patronage pattern and mean output rate A (C) are known. c

A plausible objective, in selecting this level, is to minimize the total generalized

cost associated with the center's use. Since accumulation delays do not depend on K (C)

,

only service delays and monetary costs need be considered. The expression to be minimized

will be taken to include processing (dollar costs) rather than user fees; the transition to the latter, if preferred, would only require inserting a multiplier it at appropriate points.

*A more detailed equation might involve additional fixed cost and variable cost terms,

depending on K (C) and A (C) , respectively.
To simplify the notation, references to the identity of the center (C) will be dropped

in the remainder of this section. Moreover, let

V* =

^S

l MS) V

k(S) k(S)

v k(S)A ^>

where the sum is over all shippers using the center. Then K is to be chosen, subject to K

> A , to minimize a function

c c' G(KJ = E + V*t

s

, [7.55]

which by [7.50] and [7.52] can be written

G(K

c

) = F(KJ + V(A

c

) + V*(l + I)/k

Q

- A

c

). [7.56]

In general, the minimization can be carried out by equating to zero the partial derivative of G with respect to K . For example, in the linear case [7.53] the solution is

K

c

= A

c

+ [V*(l + I)/bF

]^. [7.57]

This minimizing K

c

is then substituted into [7.52] for use in obtaining the user fee by

[7.54], and is substituted into [7.50] to obtain the average service delay at C.

Enlargement of Assignment Step.

There is only one major point left to discuss in describing the main logic of the mathematical model and technique. It concerns a difficulty already alluded to above: the purpose

of the assignment step is to assign each shipper S to that center whose use by S would involve

the smallest generalized cost. Yet the generalized cost to S of using a particular center C involves quantities- -mean accumulation delay ta (S,C), mean service delay ts (S,C)=ts (C) and

monetary service cost Mp(S)- -which themselves depend both directly and indirectly (via the

optimal sizing KC

(C)) on the patronage pattern of C. Specifically, they depend on the mean

output rate Ac(C) and overall inflation ratio 1(C) of C, as well as (see [7.54]) the explicit

inflation ratio I(c(S),C), and these in turn depend on the patronage pattern of C. Thus the

information required by the assignment step appears to be that which is to be produced by the

step!

The solution proposed for this difficulty will come as no surprise to the reader who has

followed the unfolding of the model thus far. It involves increasing still further, and

perhaps quite heavily, the iterative calculations to be performed within the assignment step.

Specifically, each pass through the assignment step (after completing a pass through the

location step) begins with a set of patronage patterns "left over" from the last previous

assignment step. For each shipper S, the total generalized cost of using each center C is

calculated on the basis of this "left-over" pattern* of patronage for C. Once every shipper

has been assigned, there is now an "observed" set of patronage patterns for the centers. These

are used to determine new generalized costs which provide the decision basis for a new assignment of shippers to centers, and so on. The process terminates when the variable portion of

the total cost for all shippers changes less than 0.5 percent from one run through the shippers

to the next. The assignment corresponding to the cost is then the main output of that particular pass through the assignment step.** After the combined process (of location and

assignment steps) has settled down to a solution, one would also want to record

(a) the optimized capacity K (C) for each center C,

(b) the associated cost Ep for each center, and also

(c) the sum of the generalized costs to all users.

* Alternatively, the first cycle of assignments within an assignment step (especially the

first one) might be based solely on transportation costs. Including nominal values (the same for all centers) of other ingredients of generalized cost would of course be useless,

since they would not affect choices among centers.

**Although there is nothing in the model to insure that the assignments settle down as the

cost does, our experience with the model is that in general the assignments of less than

one percent of the shippers (involving less than one percent of the total volume) change

in two successive passes with a change of less than .5 percent in the variable cost.

71
This last quantity, in particular, should provide guidance as to the proper number of center

Limitations of time and data permitted only very limited implementation of these concept

Specifically, our inability to separate a center's total monetary processing costs into fixedl

cost and variable-cost components, as in [7.52], precluded -any numerical treatment of center I sizing and capacity. This in turn ruled out treatment of the interactions, with center capaci

and patronage, of user's service delay (cf. [7.50]) and user's processing cost (see [7.54]).

Thus the present assignment step is "enlarged" only as regards the accumulation-delay element

of generalized processing cost. Further gathering and/or interpretation of cost data, to permit explicit programming of and experience with the "center sizing procedure," would seem

to have very high priority if development or application of the model is to be carried forwar

It should be noted that two other iterative processes, within the assignment step, must

be appropriately meshed with the process introduced above. One is the procedure, already

described, for placing some centers "off limits" to some co-containerizable cargo classes in

order to insure enough patronage to prevent excessive average accumulation delays. The other

insures that each center has at least the minimum total patronage A^n mentioned below [7.53]

At present this condition is tested only when all other criteria for termination are satis fie

Those centers (if any) which violate it have their locations re-initialized, and the computation starts afresh with a configuration consisting of the new locations for the "lean" center

plus the "good" old locations for the remaining (nonviolating) centers. This procedure is

repeated until there are no violations of the minimum total patronage restriction, or until

a specified maximum number of repetitions is reached.

There is still another way in which the generalized cost to a shipper of using a center

can depend on the patronage of C: the rates for center- to -port movement of containerized

cargo may, in fact presently do, involve volume discounts. For example, a published schedule

of rates* for transcontinental rail movement of nonrailroad -owned containers is:

Shipment Size Rate per Carload**

1

- 10 cars $1320

11 - 20 cars 1220

21 - 30 cars 1120

31 + cars 1020

This dependence of transport cost on LCL center patronage can in principle be accommodated

in the present conceptual model, especially if one ignores tactical possibilities such as sending some containers to the port by an indirect route so that they can be combined with

containers bound for another port, thus meriting a reduced rate for part of their trip. A

more serious difficulty, however, concerns those shippers with full container- loads who

utilize the marshalling capability of the center; their cargo is not only attracted to the

center by the rate discounts for large shipments to the port, but also helps to determine

what rates the (containerized) cargo from the LCL shippers will be subject to. The present

model does not deal with the centers' marshalling activities, and considerably more analysis

(and preferably, data) would be needed before attempting a treatment of those activities.

The Multi -Period Case.

Earlier in the present chapter, it was asserted that no basic difficulty not already

present would arise in extending the solution process to a situation involving several time

periods , and taking into account the possibility that a shipper might shift his patronage

from one center to another as circumstances change. The reasoning on which this assertion

was based will now be given.

For the situations considered up to now, a single "run" of the computer program would

require as inputs the number (n) of centers to be considered, as well as demand, cost, and

freight rate data for the single time period in question. In the multi-period case, data (on

demand, cost, and rates) must of course be provided for each time period in the planning peric

* Provided by the Project Engineer, P. B. Mentz of the Office of Research and Development,

U.S. Maritime Administration.

**One carload (85 feet flatcar, 150 K-lb. capacity) = Two 40- foot containers.

72
If these data remain constant over time, or can be generated by the computer from initial

values using simple projection formulas , the demands on computer memory and the associated

programming effort could be drastically reduced.

Also required, as an input, would be the time period in which each center is to begin

operation. (For simplicity of discussion, it is assumed that no center is phased out during

the planning period.) A more sophisticated version might involve constraints on how fast each

center could expand to full capacity, but this refinement will not be considered here.

As before, the solution process involves an alternating sequence of assignment steps and

location steps. An assignment step treats the locations of all centers as given, and produces

an assignment of shippers to centers for each time period, in conformity with the assignment

principle. A location step takes as given the patronage pattern over time for each center,

and determines a location for each which minimizes the total (over time) location- dependent

cost for its users. Once again, constraints on the locations of individual centers may be

imposed.

There is nothing really new to be said about the location step. Each of the "weights"

w. in [7.20] would be a sum of terms, determined by the patronage patterns of center C in the

various time periods. One might wish to apply discount factors to these time series.

The assignment step would still be an iterative process. It would involve a sequence

of assignments of the type previously described, one for each time period in turn. Of course

no shipper is to be assigned to a center during a time period before the latter goes into

operation!

The only point appearing to offer difficulty is how to treat the evolution over time of

the capacity of a center C. Suppose a value K (C;t) has been determined for this capacity

in time period t. If the optimized value for time period t+1 is less than K (C;t), then it seems best to set

K

c (C;t+l) = K

c (C;t).

Because only data for a single time period were available, this concept (in particular,

the initialization of iterative cycles) has not been thought through in full detail, but it

appears feasible with suitable reprogramming . Some aggregation of time periods to reduce

computer running time and memory storage requirements might prove advisable.

Management of Containers.

This final topic pertains to the likelihood of patronage patterns such that some centers

will typically have more containers arriving than leaving, while the reverse will be true

at other sites. Moreover, depending on such factors as balances of trade, the system of

U.S. centers as a whole may display a net surplus or deficit of containers.

The treatment of this imbalance has not been emphasized because it seems unlikely to exert a serious influence on the location and sizing of consolidation centers. Suppose,

however, that the center locations have been determined, and the patronage patterns of each

predicted. Then the following submodel, though not suitable for the day-to-day management

of the container supply, can provide planning- level guidance on the topic.

Since patronage patterns have been estimated, it is known which centers will over a time period be net sources of empty containers, and which will be net sinks; empty containers

will be shipped from the former to the latter to redress the balance. Let

e. = excess of containers at i-th source,

d. = deficit of containers at j-th sink,

c. . = cost of shipping an empty container from i-th source to j-th sink.

The quantities Cjj should be calculable from the center-to-center distances, and the applicable freight rates (or rates for whatever special arrangement might prove appropriate)

.

73
Suppose first that the container supply for the U.S. system of centers as a whole is in

balance:

y.e. = y.d.. [7.58]

Consider an optimization problem involving the variables

x. . = number of empty containers to be sent from i-th source to j-th sink,

"-J Jr. which must obey the conditions

x. - > 0,

V -x. . = e.

,

L

i n i = d..

The objective is to minimize the total cost of the balance restoring movements,

C = J- .c -x. .. *-\j lj lj

i This problem falls under the heading of linear programming transportation problems; it

has an extensive theory and powerful solution methods, embodied in readily available computer

programs which can handle situations with rather large numbers of sources and/or sinks.

If [7.58] fails because there is an overall excess of containers, one introduces a fictitious extra sink whose "deficit" is just the right amount to restore the total balance.

The "cost of shipping" from a source to this fictitious destination might represent the cost

of storing an idle container, or might be a negative number standing for a fee paid the

system by some external entity for a container or its use. Similarly, an overall deficit of • containers can be accommodated in the submodel by introducing a fictitious source, "shipments ( from which represent acquisitions of additional containers. The variations on this basic

theme, which can be handled by essentially the same techniques, are quite numerous. 21

Although the above material referred only to the U.S. system of centers, the mathematical

formulation also applies to the world-wide system, with excesses and deficits at centers

relative to those containers of concern to the United States. Such a model would provide

guidance for transoceanic shipment of empty containers between specific centers and/ or ports,

to restore total balance.

- The minimized value of C represents an additional cost (possibly negative, i.e., a benefit) associated with the system of centers. Questions of ownership and policy, concerned witi

how this cost's burden might be distributed, are beyond the scope of this discussion.

Combining these themes with that of the preceding section leads to "multi-period transportation problems," whose solution is not yet routine but seems quite feasible. 22

2! L. R. Ford, Jr., and D. R. Fulkerson, Flows in Networks, Princeton U. Press (1962).

22 D. Eklof, The Multi-Period Transportation Problem, Johns Hopkins U. doctoral thesis (1967)

See also W. A. Horn, Determining Optimal Container Inventory and Routing, National Bureau

of Standards Report 9936, 10/68.

74
Chapter VIII. THE COMPUTER PROGRAM

This chapter contains a description of the FORTRAN computer program written to implement

i version, appropriate to the data at hand, of the mathematical model described in Chapter VII. llustrative applications of this program are presented in the following Chapter IX while an

irmotated listing of the computer code itself is retained at the National Bureau of Standards

md the Maritime Administration. Occasional references to Chapter VII are made below, but

amiliarity with that chapter, though helpful, is not essential in reading the present one.

The sequence of presentation for the following material is based on a desire to serve

Jiree classes of readers who require progressively finer levels of detail.

First are those readers seeking a management- level overview of the program. For this

jurpose, we have provided an informal specification of the kinds of input data required, the

ypes of output information produced, and also the additional information of likely interest

/hich is implicitly available from the program and could be explicitly output with a minimum

of recoding.

Second are those who will be charged with the actual operation of the program "as is",

e. , setting up specific "runs", interpreting and reporting the results, and perhaps carrying

put the (routine) conversion of the code for use on computers other than the one (UNIVAC 1108)

cor which it was written. The chapter goes on to provide information needed by such readers:

:he output formats, the allowable ranges of values for the various data inputs, and the specific

Eorms of the necessary control cards and data decks. Additional information for associated

bff-line calculations is given early in the next chapter.

The text of the report includes a number of model refinements and features which have not

Seen included in the present code. Moreover, this code has been written to fit within "interlal memory" without any special use of memory-saving techniques. Those who might be called

apon to incorporate such modifications --refined or additional model features and/or relaxation

p£ limits on problem size--constitute the third class of readers we have in mind. For them,

the balance of this chapter was to have contained a rather detailed account of the logic of the

current code, its breakdown into subroutines, numerical problems encountered and how they are circumvented, etc. This goal of full detail has proven unattainable within the time available;

^instead, the program was provided with extensive "comment cards" which, together with the

partial description written below, should provide sufficient detail for the experienced

programmer.

IGeneral Program Description

The basic flow of the program involves iteration of the following two steps:

1. Assignment of shippers to centers, so as to minimize total generalized costs between

each shipper and "his" port via the center.

2. For a given assignment, location of each center so as to minimize the costs to its users.

Step 1 is performed by subroutine ASSIGN, and step 2 by subroutine LOCATE. The process is

started with an initial location of centers given by subroutine INITAL.

Subroutine ASSIGN tests each center in turn, as the center which shipper i should use

in order to minimize the total generalized cost to his U.S. port of departure. The best of

these is compared with the cost of direct shipment to that port. (Other ports are also

tested as possible consolidation points.) The shipper is then assigned to that center or

port which yields minimum generalized cost. This basic logic has been refined to insure

jsatisfaction of two conditions relating to patronage at any center for any class of (cocontainerizable) cargo: First* that the cost elements governing the assignment of snippers

to the centers, insofar as they depend on the class's total patronage at the center, have

75
numerical values consistent with the patronage pattern of the assignment itself. Second, tH a center serve a cargo class only if there is sufficient patronage to provide (on the avera, acceptably prompt filling of containers.

The generalized cost is made up of several factors: the dollar costs of transportation

from shipper to center and center to port, the dollar cost of the time spent in shipper-to- ; ; center and center- to-port* transportation, and the cost of the accumulation delay** at the

center. Center-to-port costs take into account the cargo inflation factor which results

from a load factor < 1 on outgoing containers. Processing -cost elements which depend on center location can be taken into account. it

[Not represented in the present code are those generalized cost elements related to center- size -dependent service delays or processing costs at the center. The inclusion

of these factors is discussed at the conceptual level in Chapter VII, and their future

incorporation (if desired) seems quite feasible given the requisite data.]

The process used in subroutine LOCATE is the iterative one developed by Kuhn and Kuennc and independently by other authors, 2k to calculate the point C which minimizes a function oj

the form

f(C) =

^ i

w

id(C,X

i

),

where d(C,X.) is the Euclidean distance between points C and X., and w. is a "weighting

factor" associated with point X.. In our problem, C represents a center's location, the

points X- are the locations of the shippers using that center (and the U.S. ports of entry

or departure for those shippers' cargo), while w. is the product of (a) the generalized

shipping rate 5 between C and X. and (b) the quantity shipped.

Iteration of the two steps --assignment of shippers to centers in subroutine ASSIGN, anc relocation of centers in subroutine LOCATE- -continues until the total generalized cost to a]

shippers changes less than 0.51 from one iteration to the next. The program then prints the

desired information, and goes on to repeat the whole process for the next number of centers

to be investigated.

Input Data for Program

The input information required by the program is listed below. Details, on the prepare

tion of the data cards themselves, are given later in the chapter. Model modifications to accommodate more refined empirical data would of course involve corresponding refinements

in this list.

1. The locations of the U.S. ports involved.

2. For each shipper: location, U.S. port used, class of commodity shipped, quantity

(lbs.) shipped or received per year, reciprocal of average time (days) between successive

shipments, co-containerization class. tt

*The present code reflects differences, among delay sensitivities of various commodity

classes, in treating shipper-center movements' but not center-port movements.

**Defined in the previous chapter.

23H. W. Kuhn and R. E. Kuenne, An Efficient Algorithm for the Numerical Solution of the

Generalized Weber Problem in Spatial Economics, J. Reg. Sci. 4 (1962), pp. 21-33.

2I+L. Cooper, Location-Allocation Problems, Oper. Res. 11_ (1963), pp. 331-343; W. Miehle,

Link-Length Minimization in Networks, Oper. Res. 6_ (1958), pp. 232-243; F. P. Palermo, A

Network Minimization Problem, IBM J. Res. Dev. _5 (1961), pp. 335-337.

§"Generalized" in that it includes a term to represent the cost of time spent in transit.

tThese quantities are measured in weight units throughout; some of the necessary conversion

are given early in the next chapter.

++Shipments are "co-containerizable" if their natures and overseas distribution points permi

their being consolidated together.

76
3. Number of U.S. ports involved, of exporters, of commodity classes, of co-containerizle cargo classes per port. Set of values of n (= number of inland centers) for which

ogram is to be run, specified by starting level, final level, and increment.

4. The transportation rate per pound-mile for a full container; the LCL rate per pound

le for each commodity class

.

5. The dollar value per pound for each commodity class, and the average value per pound

tr all cargo. Miltiplicative factors (for rail and truck, respectively) for converting

stance traveled into time consumed, and an inventory carrying factor which (together with

e dollar values) is used to convert from time consumed to dollar costs.

6. The fraction of total nominal cargo to be regarded as LCL material requiring con-

•lidation. (In the present code this same fraction is applied to every shipper; varying it

•ovides a means of testing sensitivity to the level of demand for consolidation services.)

7. The value to be attributed to "work-days per year" at the centers.

8. The nominal weight (lbs.) of a full container.

9. The "target" load factor and maximum permitted holding time (in days) for outgoing

mtainers, and the average load factor for incoming containers.

10. If center- related costs are assumed, to- vary with location among the "boxes" of a ctangular grid: the number of rows and columns of the grid, and the incremental unit cost

iplicable for locations in each box.

In addition to these basic data, there are several other input categories:

11. Maximum allowable average accumulation delay (if desired).

12. Minimum acceptable total patronage at a center in terms of containers filled to the

irget load factor (if the minimum patronage restriction is to be applied)

.

i 13. A list of cities, their locations, and a tolerance distance in miles; for use in

rinting out which cities have centers nearby, when this output is desired.

14. Average cost and average time to stuff a container at a center.

15. Specification of the desired version of the program and of the outputs to be

rinted. (The available alternatives are described below.)

16. If random initial locations for the centers are to be used: the "seed" or "kick-off"

5tting for the pseudo-random number generator (so that the "random" factors can be duplicated

i other computer runs if desired, or duplication can be avoided if preferred).

itput Information

The user of the program can specify which of the following types of information are to

5 printed out. Any subset of these categories can be chosen.

1. A repetition of the input data for each shipper. (Selection of this option of course

Bads to relatively voluminous output.)

2. For each number of centers considered, the initial trial locations for all centers.

3. For each center: its final (optimized) location, its zone of patronage (the smallest

j^ctangle, with sides parallel to the coordinate axes, which contains all the center's users),

Jhe mean daily volume of business (lbs. of payload) , daily output (of containers), average

atgoing load factor, distribution of center input tonnage between import and export (parentages) , total generalized cost to users expressed as a fraction of the cost to them if

hey had to ship LCL direct to the U.S. ports they use.

77
4. For each shipper: the center he uses, and the generalized cost incurred as a fraction of that which would be incurred in the absence of inland centers.

5. For each commodity class: the total generalized cost incurred in the processing and

movement of that class's shipments, as a fraction of the cost: in the absence of inland center:

6. For each co-containerization class: the average load factors of containers, and

total generalized cost as a fraction of that in the absence of inland centers.

7. The final random number generated, for use in continuing the pseudo- random sequence « in later "runs." (The initial "seed" must be recorded separately by the user if it is to be

employed again.)

8. Information as in 3 for each port center.

9. The number of centers within a specified distance of each of a list of cities.

Other Potential Outputs

A number of quantities of potential interest, in addition to those listed above, are explicitly or implicitly calculated during the program's operation and could be provided as outputs with relatively little reprogramming. While the present output repertoire was felt

to be most suitable for the current exercising of the model and to illustrate adequately

the types of information that could be obtained, a brief sketch of readily available further | outputs may be of value here.

For each shipper, the model calculates and could print out the accumulation delay, time | in transit to the center and from the center to port, dollar cost of transportation between

shipper and center and between center and port, and total dollar cost. These outputs might

be used to analyze which shippers derive the most benefit from the system of centers and how

this benefit is divided between dollar and time components.

The distributions of benefits among commodity classes, among co-containerizable cargo

classes, and among ports seem likely to be of interest. One might want to know which centers serve which ports (to various degrees), and which commodity classes would constitute the bulk

of particular centers' business. As an aid to such analyses, printouts could be obtained-

for each commodity class, or co-containerizable class, or port--of percent of cargo quantity

associated with each center, as well as overall average time delays and dollar costs.

Program Options

Besides the choice of outputs described above, a number of additional options are available to the user of the program:

1. The model may be used to evaluate a specific set of locations for centers. In this

case, of course, no attempt is made to optimize the center sites.

2. Four different procedures for choosing initial center locations have been incorporated*:

(a) Initial locations can be specified on punched- card input.

(b) They can be randomly chosen, within the rectangle bounded by the maximum and

minimum x and y coordinates of the shippers' locations.

(c) They can be chosen near shippers, selected at random from the shipper

population.

(d) They can be placed near the shippers with the greatest annual cargo flows (in

weight)

* Still other procedures are plausible, but these four seemed representative.

78
3. The program can be instructed to impose a restriction which prohibits a center from

accepting cargo from a co-containerizable class if the center's patronage for that class is

so low that mean accumulation delay (for an adequately full container-load) would exceed the

maximum allowed.

4. To speed up the calculations, ordinary (Euclidean) distances may be replaced

throughout by the "Manhattan metric" distances described in Chapter VII.

5. To take into account those elements of center-associated costs (e.g., pertaining to

land acquisition and for labor) which might prove sensitive to rough geographical location,

the region can be divided into rectangles each of which has a different associated additional

cost per container. These cost increments are entered into the assignment -step and locationstep calculations.

6. Since different initializations may lead to different locally optimal solutions, the

program may be directed to try more than one set of initial locations for the same number of

centers

.

7. The program may start out using the Manhattan metric and only use the Euclidean

3tric after the center locations have settled down under the Manhattan metric.

8. The program may enforce a minimum total patronage restriction on each of the centers.

This is accomplished by reinitializing centers which receive too little throughput, and then

starting the assignment- location process again.

9. The program may be required to keep the locations of some centers fixed. The fixedlocation centers affect the assignment and enter into the total cost figures, but are never

allowed to be relocated.

General Remarks

In concluding this management-level overview of the program, it must be repeated that

model outputs are only as good as the input data. Since actual locations of centers may

\ depend on many factors not explicitly represented in the computer program, such as access to

Irailroads or interstate highways, availability of land in the exact area chosen, and so forth,

the program outputs can only be used as a guide to location. Also, since no work was done on

demand forecasting, the model's results will reflect only the existing distribution of demand.

Runs testing the sensitivity to demand can be made, but these do not include changing patterns of demand (resulting from the existence of centers , for instance) . The next chapter

includes a description of the data manipulations , nominal values of parameters , and the

sources of the data. Further efforts to obtain more precise values for some of these data

appear highly desirable.

Input Formats

Passing to a more detailed phase of the exposition, this section describes the input

formats required in setting up a problem to run on the computer. Shipper-related information

i constitutes a tape input, while all other data are entered on punched cards.

The card formats will be described first. All numbers should be right justified in

1 their fields. The number of decimal places for each floating-point input is given below.*

Card input is from logical unit 5. Users should consult subroutine INPUT in the program

listing for further information.

E.g., the phrase "3 dec. places" applied to a variable with a 6-digit field implies a number of decimal form (ab.cde). (The decimal point is counted as one field.)

79
Card No. Columns Information

1 1-10 Smallest number of centers considered.

11-20 Largest number of centers considered (<50).

21-30 Increment in number of centers considered.

31-40 Number of 'commodity classes (<100)

41-50 Number of co-containerizable classes per U.S. port (5.20).

51-60 Number of exporters.

2 1-5 Successive entries of array IDO, specifying program options.

6-10 See program listing's comments for instruction code.

71-75

3 1-5

6-10 Continuation of the entries in the array IDO. 11-15

4 1-10 Fraction of nominal cargo to be treated as LCL. (4 dec. places)

11-20 Number of working days per year.

5 1-20 Transportation rate ($/lb-mi) f r a full container (7 dec. places)

6 1-10 LCL transportation rate for each commodity class (7 dec. places).

11-20

71-80

1-10 Inventory type carrying charge (yr. -i )(4 dec. places).

11-20 Factor for converting distance to rail time ( days /s tat. mi.) (4 dec. places)

21-30 Factor for converting distance to truck time (days /s tat. mi.) (4 dec. places)

1-10 Average value/lb. ($/lb) , all cargo (3 dec. places).

1-10 Average value ($/lb) for each commodity class (3 dec. places).

11-20

71-80

10 1-10 Circuity factor used to reflect the difference between straight line

distance and the actual distance upon which rates are based (3 dec.

places)

80
Card No. Columns Information

11 1-10 Target load factor for outgoing containers (3 dec. places).

11-20 Maximum holding time (days) at all centers (3 dec. places).

21-30 Nominal weight (lbs) in a full container.

12 1-10 Average load factor for incoming containers (3 dec. places).

13 1-10 Average cost to stuff a container at an inland center (3 dec. places)

.

11-20 Average amount of time (in days) to stuff a container (3 dec. places).

21-30 Number of days spent in rail terminals (3 dec. places).

31-40 Number of days spent in truck terminals (3 dec. places).

14 1-10 x-coordinate* of first port (2 dec. places).

11-20 y-coordinate* of each port (2 dec. places)

.

NOTE: This is followed by a similar card for each of the other ports.

(At present, number of ports <_ 3.) These must be followed by a card

with 7/8 in column 1, E0F in column 3.

15 1-24 Random number "seed". (Needed only if location initialization 2 or 3 is

used or if the minimum total patronage restriction is imposed.)

16 1-10 Maximum allowable average delay at any center (3 dec. places).

NOTE: This card is needed only if ID0(3) > 0.

17 1-10 Minimum allowable total patronage at an inland center, in containers per

day (5 dec. places).

NOTE: This card is needed only if ID0(17) > 0.

18 1-12 City name

21-30 x-coordinate of city location (2 dec. places).

31-40 y-coordinate of city location (2 dec. places).

41-50 Tolerance distance (2 dec. places).

NOTE: This is followed by similar cards for each other city for which

it is desired to print out the number of centers within the specified

distance of the city. The final card must have 7/8 in column 1, E0F

starting in column 3. These cards are needed only if IDO(18) > 0.

19 1-10 Number of rows in grid.

11-20 Number of columns in grid.

NOTE: This card is needed only if IDO(12) > 0, i.e., location-dependent

costs are considered. The number of boxes in the grid must be <_ 100.

20 1-10 Cost increment (F/container) for successive boxes in first row of grid

T-i or. (2 dec. places).

'. NOTE: Followed by similar cards for the remaining rows. Needed only if

71 : IDOC12) > 0.

* The coordinate system is described early in the next chapter.

81
The shipper- related input information is on a tape to be mounted on logical unit 7. Exporter data appear in a block at the beginning of the tape, and importer data in a block

at the end. At present the number of shippers must be <_3000. There is one tape record per

shipper, with the following format:

Columns Information

1-11 x-coordinate of location (2 dec. places).

12-22 y-coordinate of location (2 dec. places).

24 U.S. port used.

26-28 Commodity class.

30-40 Quantity (lbs./yr.)

41-47 Number of shipments/yr. (0 dec. places).

48-50 Co- containerization class.

Conversion to Other Computers

The present program is written for the National Bureau of Standards' UNIVAC 1108

computer, under the EXEC II operating system. Its "language" is FORTRAN V , which contains

the instructions of FORTRAN IV as a subset and has some additional features. If the program

is to be run on some other computer $vith at least FORTRAN IV capability), the following

machine -dependent features may need alteration.

1. The UNIVAC 1108 at NBS has 65,000 36-bit words of storage, of which about 50,000

words are available under EXEC II. Reduction of array dimensions might be necessary or expansions might become possible for the core storage of some other machine.

2. On our machine, card input is from logical unit 5, and printer output is on unit 6. Unit 7 has been used in the program for tape input. These selections of logical input/output

units might vary for another computer.

3. The present code contains comments written on program instruction cards following

a 7/8 punch.

4. The present code includes FORTRAN "READ" statements of the foim

READ (unit, format no., END=k)

Such a statement yields a transfer of control to statement number k when an end-of-file

condition is encountered in the input.

5. The code employs a random number routine (RANDNO) written in 1108 assembly language.

A similar routine would be needed on another machine.

6. The 1108 allows and correctly calculates expressions involving mixed-mode arithmetic.

In view of possible conversion difficulties, an attempt has been made to replace such statements throughout the present code, but some may have survived.

7. The 1108 FORTRAN V allows the use of Hollerith characters appearing between quote

marks instead of the nH. . . form. This avoids the necessity of counting characters to be

printed.

82
Subroutines and their Functions

The final level of detail to be presented consists of brief descriptions of the

program's various subroutines and their roles.

1. MARAD: This is the main routine, which controls the iterations through assignment

and location steps alternately. It also calls the input routine and the routine which calculates initial center locations.

2. CDIST: This subroutine calculates an array DIST used for a rapidly calculable

approximation to Euclidean distances. (Thus it is not performed if the Manhattan metric is to be used.)

3. D(Xi, X 2 , Y 1} Y 2): The function D calculates the distance between points (Xi,Yi)

and (X 2 , Y 2). If the Manhattan metric is involved, the exact formula

D =| Xj - X 2 | +|Yi - Y 2 |

is employed. If Euclidean distance is involved, the quantities

M = max {|Xj - X 2 |, \Y l - Y 2 |} , m = min {|Xi - X 2 |, |Y X - Y 2 |} ,

are first calculated. Then the exact result

D = M {1 + (m/M) 2

} 1

is approximated by replacing the proper fraction m/M by the nearest fraction of the form

1/1000, I an integer. The approximate square root is obtained from the above-mentioned array

DIST, which contains the 1001 values

{1 + (I/1000) 2 }^ , I = 0, 1,... 1,000 .

4. INPUT: Card and tape program inputs are read in by this routine. Transportation

rates (per quantity unit per mile) are incremented to include transit time factors. The

annual nominal cargo levels are multiplied by the input parameter WLCL (to obtain the fraction attributed to LCL cargo requiring consolidation at a center or port) , and then divided

by the "working days per year" factor to obtain an average daily level. In addition, the

reprinting of desired input data (to "label" the output fully) is carried out by this subroutine.

5. INITAL. This subroutine calculates initial locations for each number of centers.

There are four initialization procedures; ID0(1) controls which one of these is used for a particular run. The first reads user-specified initial positions from punched cards.""

(WARNING: a center should never be initially located exactly at any shipper. The location

subroutine is such that such a center will never move.) A second initialization option

randomly locates centers in the area bounded by the minimum and maximum x and y coordinates

of shippers. The third initialization procedure locates centers near randomly chosen

shippers. The final method locates centers initially near the "heaviest" shippers. For the

purposes of this last initialization, the tonnage shipped by all shippers at one point is

aggregated to decide which locations ship the greatest amount of cargo.

6. LOCATE: This subroutine performs the location step; for a given assignment of

shippers to centers it calculates, for each center in turn, the location of that center which

is best for its users. Accumulation delay need not be taken into account, since it is con- stant (independent of center location) for a given assignment. However, inflation factors

(describing container load factors less than 1) must be taken into account in costing cargo

movement between center and port. LOCATE calls the location subroutines appropriate to the

metric used.

83
7. KK: This subroutine involves the use, for each center C, of the Kuhn-Kuenne procedur 1 (cited earlier) to choose the location of C so as to minimize a function of the form

F(C) = JjWjdCCX^

-(t) „(t). This process is an iterative one; if (xi,y-j_) are the coordinates of X-j_, and (x^ , y ) are

the coordinates of the t-th stage estimate C^J of the optimal location, then the coordinates

of the next estimate c( t+ lJ are given by

x U+1) =[[

iwixi /d(X

i ,C

(t)

)]/[[ iVd ^X

i' c(t) ^>

y

(t+1)

=[I iw

iy i/d(X1

,c

(t)

)]/[I i

v/i /d(x

ic

(t)

)].

This process obviously will run into difficulty if at some (n-th) iteration, C

J is

very close to some point Xj among the X^'s, so that dfcM, Xj)<<1. This will certainly

occur, for example, if C is "homing in" on the optimal solution, which happens to be at the

point Xj. It is therefore important to have a test (necessary and sufficient condition) for

the optimal C to be at X,.. Kuhn and Kuenne ( op . cit .) show that the following condition con- stitutes such a test:

(w

l)

2 L Hin Vx

i

" XjVdQC^Xj)]2 -* [I i7, Iw

i (y i

- yI )/d(X

i ,X

I )]

2

.

In our implementation of this algorithm, whenever any d(C , Xj) becomes less than 0.1,

the preceding test is applied to the point Xj to see whether the center should be located at

Xt. If the test fails, C is allowed to draw closer to Xj in its passage to the solution.

(In this case, as an extra precaution, a program halt accompanied by an error message occurs

if convergence is not achieved after another 35 iterations. This precaution has proven

unnecessary so far.)

Termination of the iterations through subroutine KK occurs in one of two ways:

1. if the location C^ differs very little from the point C obtained in the

previous iteration,

2. if the partial derivatives of F(C) evaluated at C , namely

^ i (w

i /d(C

(t)

,X

i ))(x

i

- x

(t)

), [ i (w

i /d(C (t)

,X

i ))(yi

- y

(t)

),

are both sufficiently close to zero.

The second section of KK is also employed for each center in turn if IDO(12)>0, i.e.,

when possible dependence of unit processing costs on center location is to be considered.

Here the total shipping area is divided into a grid of rectangles. For each of these

rectangles, the optimal location within that rectangle is determined by the method described

in Appendix C, and the resulting minimized generalized cost for transportation is added to the location- dependent cost associated with the rectangle. The optimal location, in the

rectangle for which the sum is least, is then selected.

8. SORT: This subroutine and the next are used when distances are to be measured by

the Manhattan metric, rather than the Euclidean one. SORT is used first to sort the

x- coordinates of the shippers and ports using a given center, and then to sort the

corresponding y-coordinates.

9. MEDIAN: Finding the optimal center location under the Manhattan metric reduces to

finding the x and y medians of the shippers and ports weighted by their annual quantities of

LCL cargo. This subroutine calculates these x and y medians.

84
10 . ASSIGN: in this subroutine "current" locations for all centers^are given^ For^

bof^nt! So

h

tne%t

ayiSd

1inrtne

e

sUmallesrgenorali Zeo cost is chosen.

Per reasons explained in ^^l^lk^r^AW^^sls^^ll Till

^S^-eacnlnlurrnsirUng'o^a'nSe/ofluo-substlps. These two levels of detail wtll

be explained in sequence.

Each substep begins with a specification for ^.^^g^^^^t^rS'a

which centers are "off limits" to that class because of^fg^rH^ed throughout

prompt or economical accumulation of materi al) . These pronib itions

tronage pattern

?he substep, which ends with ^^^5^1 centSs nave insufficient patronage from

from this assignment may be such that additional centers na ^ ^^

serving

some of the co-contamenzable classes they serve, *°* e*^ = . is n^w als0 declared

it which is the worst violator of ^e patronage adequacy criterion is now ^

unavailable to the class. These additional Prohibitions silent step terminates when no

make up the specifications for the next substep The assignment^ P

s whose

further prohibitions are ^9u^^' 1 - e -

tf^a? termination after some finite number of subcargo they can hand e expedxt ously

;J^e *f^rmi ^ co . COntainerizable class has too fiSL^oU^^^^ * a single center.

The particular condition, which in the present code.triggers ^^f-^l^ter^o

a co-containerizable class, is that the mean delay in waiting for the targe

di££erent

of a containerful to accumulate exceeds a sP^^^^

el

^ ifcalculated using the Poissoncenters could easily be inserted. The mean accumulation delay is caicui

cSnciudes the

arrivals model and the (L,H) dispatching policy des ^^^^^'Scept to remark that this

description of the breakdown of an assignment step ^^eps^excep ^.^ q£

^^:i^^^l^^^^^(^

Sa, Patronage imposed.

Each sub-substep begins with.assumed rt^£%£^*<^^^ used*"

tion ratio, for each co-contaxnerizable cargo class at each center..^ ^^ ^.^ ^

in calculating the generalized costs to a shipper oi usi g centers. When

him. On the basis of these costs, the shippers a^a?^^ °^y is used to calculate

Sds is complete, the patronage pattern from the "f^^f^S^ The substep terminates

the accumulation delays and inflation ratios for the next sub jubstp^ ^ ^

Both the initialization J*£ «TM£.S^oft^STSW-?^S^

further. As to the former: in the first sub suDsreprespectively, so that

step, the accumulation delays and inflation ratios are set at ^an p ^ ^ ^

only generalized transportation costs ^er thislirs^ seto ^ inflation ratios for a recursion: it would be simplest to take th^

acTM^^Lf^gern for the assignments found

new sub-substep, to be those corresponding to the P^ronage patter

ut| m over . in the last sub-substep. This ''total updating, however was ±°^ ^

in£lation

adjustment which failed to Y^^^^U^^^c^^thDse assumed throughout

ratios for a new step are calculated ^o^g^ ^Ing

8

t0 the patronage pattern resulting

equations [7.40] and [7.41]).

12. OUTPUT: This subroutine prints the desired outputs.

* Experimentation with alternative termination criteria might be desirable.
Chapter IX. ILLUSTRATIVE APPLICATION OF THE MODEL

This chapter describes the application of the computer program described in Chapter VIII

to illustrative situations based on the data of Chapters III to V. The discussion of

"results" is aimed at giving the reader a clearer picture of what kinds of qualitative and

quantitative inferences can be drawn from the outputs of the model. It must be emphasized,

however, that the particular results described below are illustrative (of what the methods

developed in this study are capable of), but not definitive; they are based on input data

with serious deficiencies (e.g., concerning the overseas destinations of exports), on a computer program which does not reflect the final phase of model development,* and on numerical explorations severely restricted (for practical reasons) to only a fraction of the cases

that could and would be covered in a full-scale application effort.

Despite these limitations, it is felt that the usefulness and feasibility of the study's

approach--and the desirability of more intensive data collection coupled with further model

refinement and implementation- -have in fact been established. The reader will of course wish

to judge for himself whether this feeling is justified.

The first sections below deal with input data. They state what values were selected as "most reasonable" nominal ones for transportation rates, average dollar values for commodity

classes, load factors for containers arriving from overseas, and other input parameters.

Also included is a description of the manipulations performed to bridge the gaps in the

empirical information available, and thus to arrive at a semblance of a suitable data base.

Taken up next are the results from applying the program, under the above-mentioned

"nominal" values of all input parameters. These computer runs were aimed primarily at investigating the effects of altering the number (n) of inland centers. Another objective

was to evaluate the practical seriousness of the theoretical danger that plausible initial

guesses concerning center locations could lead to far-from-optimal final results. In addition, these runs established that the solution method- -involving many iterative calculations whose good behavior could not be guaranteed in advance--really would converge to an

answer with a tolerable expenditure of computer time. Specifically, the computer runs averaged about 1.5 minutes per case. This was roughly doubled when the minimum total patronage restriction was in force.

The results described next in the chapter pertain to "sensitivity analyses" carried out

to study the effects of plausible perturbations of selected input parameters from their

nominal values. Within the time available, it was possible to perform such analyses only on

a quite limited scale.

Other computer runs illustrate the capability for and consequences of treating the

possibility of location-dependent summands in centers' unit processing costs.

Throughout this chapter the index of performance by which runs of the model are compared

is the ratio of the total cost when inland centers are available, to the total cost if there

are no inland centers. The total cost, for shippers offered no advantage by inland centers,

is the same whether there are such centers or not. For shippers who would avail themselves

of the centers, the total cost of doing so consists of a generalized cost of transportation

to the center (containing both a dollar and a time component) , a cost of accumulation delay

at the center, a generalized cost for processing at the center (containing both a time and a dollar component), and a generalized cost for transportation in a container to the port. The

corresponding cost when there are no centers consists of a generalized transportation cost

LCL to the port, and a generalized processing cost for stuffing the container at the port.**

The index of performance is calculated as the sum over all shippers of their total cost when

inland centers are available, divided by the corresponding sum when there are no inland

centers. Note that good performance of the system of centers is indicated by low values of

this index, poor performance by values near 1.

* The computer program does not attempt to optimize center sizes, nor does it evaluate the

effects of center size on processing cost and time.

** Similarly for imports.

86
Typically, for the computer runs made using the Delaware Port Authority hinterland data

described in Chapter IV, the value of the index of performance was between 51 and 63. This

low figure would represent a substantial savings to the shipping community from such centers

Even with all the disclaimers and cautions that must accompany numerical results from an illustrative example only partially based on empirical data, the impressive magnitude of these_

reductions^ to?al generalized cost provides a yerxJ*icourag3gg indication of the economic

worth ofthl inland consolidation center concept. (It should be recalled that this study has

refrained from stipulations as to how these savings might be distributed among the sectors of

the shipping community, which would include the inland centers themselves.)

Shipper Related Data

The shipper-related data employed have been described in detail in Chapter IV. The

nresent studfwas confined to those shippers sending or receiving containerizable commodities

SrouS Se ports of Baltimore, Philadelphia, and/or New York. For each of those shippers

in the area surveyed by the Port of Delaware River Authority and for their usage of the

Si^gh ports ^mentioned,- the following pieces of information were extracted from data

tapes supplied by the Authority:

Total yearly tonnage,

Total yearly dollar value,

Mean frequency of shipment.

The shippers in each survey area (these areas are specified in Chapter IV) were ''moved"

slightly so as to lie at a single centroid point chosen in this area The centroids (hence,

?he aoproximSe shipper locations) are initially identified by latitude and longitude, but

these

PSoX2es a?e then linearly converted to (x,y) coordinates. The conversions are

given by:

1° latitude = 68.5 mi., 1° longitude = 52-S.mi.,

and the origin of the (x,y) -system is at

Latitude: 37° N. , Longitude: 73° W.

At this ooint the list of shippers numbered over 3,000 exporters and over 900 importers.

Further salification was required! if computer running times and memory storage requirements

were S be kept Sthin acceptable bounds. The simplification process will be described next.

The first step was based on the observation that many of the shippers ' cargo movements

been nerfomed Zs Se results reported below are not directly comparable with those from

tne m?p SSysis, which is based on all reported (export) cargo movement rather than an

estimated LCL component of it.

This thinning-out was performed in the following crude but reasonable way The average

_ t °hp nort This criterion errs in one direction by discarding the occasional LLL snipments

of t^i?ally hSvJ snippers, but errs in the opposite direction by including m the centers

* Recall from Chapter IV that the decisions as to what should be regarded as containerizable, were far from cut and dried.

-A potential bias against inland centers is introduced here, in that their value relative

to usage of ports other than these three is not scored.

87
potential market all of the occasional full-container movements by "LCL on the average"

shippers.* Further work aimed at a more refined treatment of this step seems desirable.**

The second step in the simplification process involved aggregation rather than elimination of shippers. All shippers lying in the same survey area (hence, essentially co-located)

and belonging to the same co-containerizable class were aggregated to form a single new shipper. The difficulty here is that information on common membership of shippers in a co- containerizable class, although surely available in principle from an adequate data base, was

not in fact deducible from the data at our disposal. Thus some "creative" operations on the

data were required. These are detailed next; at any rate, the joint effect of the elimination

and aggregation steps was to reduce the numbers of exporters and importers to tolerable levels

(roughly 1025 and 375, respectively).

If cargo of several shippers is to be co-containerizable, it must

(a) move in the same direction (into or out of the U.S.) through the same U.S. port,

(b) be of physical natures permitting joint presence in a common container without

damage , and

(c) involve the same overseas distribution/concentration point (or perhaps port-of-call

area)

The available data permitted the straightforward application of criterion (a)

Regarding (b) , there was the question of which combinations of commodity classes, each

individually containerizable, would admit joint containerization. The variety within each

commodity class precludes a really clearcut answer. Without a detailed study on this point,

it seemed necessary to choose as working hypothesis one of the two logical extremes; either

that any containerizable commodity class's material can be combined in a container with any

other class's, or that no two commodity classes can share a container. The first of these

alternatives was believed nearer the truth and was adopted. There is an obvious need for

better data and further study to permit more satisfactory treatment of this point.

Since no information bearing on (c) was at hand, fictitious "data" of the appropriate

kind were generated by the following process ; it was arbitrarily assumed that each of the

three U.S. ports in question served overseas points (capable of handling containerized cargo)

which constituted up to at most 20 co-containerizable categories. The exporters using each

port were assigned at random, one by one, to some one of 20 categories, and the same was done

for the importers. The results of varying the "20" as well as the particular assignment to categories are described later in this chapter.

In summary, the following steps were performed on the data from the Delaware River Port

Authority:

1

The following information for each shipper was taken from the data tape

a. shipper location code

b. total yearly tonnage

c. total yearly dollar value

d. mean frequency of shipment

e. commodity sent

f. U.S. port of debarkation or entry

g. sample factor

2. Shipments of non-containerizable commodities were eliminated.

*It also ignores the possibility of some LCL cargo being noncontainerizable simply because

its route lacks container-handling facilities.

**The program provides an option whereby the demand of each shipper may be multiplied by a constant factor. A run using this option to increase demand by 25 percent is described

later in this chapter.

88
3. Shipments not departing or arriving at New York, Philadelphia, or Baltimore were

eliminated.

4. Each tonnage and frequency figure was multiplied by the appropriate sample factor.

5. The average shipment size (total yearly tonnage divided by mean frequency shipment) was calculated; those shippers sending more than 48,000 pounds on the average were eliminated.

6. The commodity class (see Table 8) for each shipper was found.

7. The average dollar value per pound for each commodity class was determined.

8. Each of the remaining shippers was located at the centroid of the study subarea

containing it.

9. All shippers in the same subarea who ship the same commodity class through the same

U.S. port were aggregated into one shipper whose total tonnage iandf frequency were

the sums of the constituent shippers' tonnages and frequencies.

10. Each of the aggregate shippers was randomly assigned to one of 20 co-containerization

categories

.

The input tape to the model then contained "for each (aggregate) shipper:

x- coordinate of location

y-coordinate of location

port of U.S. entry or debarkation

commodity class

total pounds shipped per year

number of shipments per year

co-containerization category.

Other Input Parameters

The computer runs were based on conventions of 250 work-days per year, and 48,000 lbs.

as nominal weight for a full 40' container, consistent with Chapter III.

J. Norris of the Maritime Administration's Office of Maritime Promotions provided the

study with a suitable set of rough transportation rates based on an 8-fold commodity classification. The rate for cargo in a full container is taken as

0.135 cents/hundredweight/mile,

while the rates for uncontainerized cargo are given in Table 16. This subject was discussed

in Chapter V.

Equation [7.4] gives the dollar component of travel cost, as a transportation rate times

the amount of cargo shipped times the Euclidean distance multiplied by a circuity factor.

The value 1.13 discussed in Chapter VII is used for the circuity factor.

Equation [7.8] shows how time in transit can be estimated as a linear function of distance traveled. These times are required in calculating the generalized costs of the shippercenter and center-port movements. The values suggested by Meyers et_ al_ (op.cit. , p. 192)

lead to a conversion factor of .0059 days /mi. for the rail mode. By omitting those delays

not involved in truck transport, the same equation yields .0014 days /mi. for truck. The

value suggested by Meyers et al for the constant term 6 in equation [7.8], for the train mode,

is 2 days; a value 6=1 day was felt to be representative for the truck mode. Within the

program, it was in effect assumed that all uncontainerized shipments use the truck mode,

while containers move by rail. Thus material containerized or broken out at a center uses

the truck mode for the uncontainerized portion of its trip and the rail mode for the con- tainerized portion.

89
Meyers et al suggests that for trips less than 300 miles, the truck mode is usually

cheaper than rail, and the reverse for trips of over 300 miles. Most of the shipper-tocenter trips (which are LCL) are less than 300 miles (many within one city) , so the truck

mode is most appropriate for these trips. The rates quoted in Table 10 are based on an average of rates to the 4 ports of New York, Philadelphia, -Baltimore, and Norfolk, all trips

of more than 300 miles. Therefore the rates used in the model will overestimate the cost of

short trips. The direction of this bias is further enhanced by the fact that all these rates

are for interstate trips, which are more expensive than intrastate hauls. Thus our use of

the truck LCL rates given in Table 10 biases to some extent against the use of inland centers

and raises their index of performance.

Recall that inventory- type charges I]< are employed to convert from time (delays) to dollars. Meyers et_ al (p. 192) suggest a figure of 10%/year, which gives a factor of

0.0004/day.* Appearing as a multiplier is the unit value/unit quantity (V^) ; numerical

values for these were found by averaging the $/lb. ratios for shippers in each commodity

class, and these values appear in Table 17.

Table 16. Transportation Rates (cents/hundredweight/mile)

Commodity Classt Rate

1 0.758

2 0.693

3 0.418

4 0.540

Commodity Class^ Rate

5 0.581

6 0.892

7 0.628

8 0.644

Table 17. Average Unit Values of Cargo ($/lb.)

Commodit)' Class* Value Commodity CI asst Value

1 2.334 5 0.091

2 1.788 6 20.029

3 1.111 7 0.887

4 0.834 8 2.933

+These numbered classes are identified in the corresponding table of

Chapter V.

*The use here of a 250 day effective year can of course be questioned; sensitivity to alternate figures could of course be tested, but off-hand is not expected to be substantial

Of course factors dependent on commodity class would give a more realistic picture of the

situation.

90
The nominal load factor for inbound containers of all co-containerizable classes was taken as the single figure given in the document

North Atlantic Container Statistics

Report for 6 -month Period Ending 12/31/67

(Office of Maritime Promotions, MARAD)

Of course the extension to all trade areas of results for the North Atlantic trade is

questionable, and no account is taken of the effects of "filler" in the containers observed.

(In the model, "load factors" are really "payload factors.")

The average figure given in the document, with 20' -containers as nominal, was 8.81 long

tons. Doubling for conversion for 40 -containers, and recalling the 48,000 lb. nominal weight

assumed for such a container, leads to a load factor of

2 x 8.81 x 2,240/48,000 = 0.822.

The nominal values used for the parameters describing the centers' dispatching policy

represented what appeared to be the central tendency of the information and impressions gained

during the study. These values are

L = "target" load factor = 0.90,

H = maximum holding time = 4 days.

For all the runs which are discussed in this chapter , centers were required to process enough

cargo in each co-containerization class so that the average accumulation delay experienced

by each class is less than 1.5 days. Whenever a center violated this condition for some

class, the center at which cargo of that class experienced the largest accumulation delay was prohibited from accepting cargo of that class.

Chapter III provides nominal values for the cost and time to process one container:

$94.28 and .125 days, respectively. Since all cargo is assumed to be containerized before

going on board ship, and since center processing costs and times are being treated as independent of the center at which they occur, the values of center processing cost and time are

constant and do not enter into the optimizing process.* They do, however, contribute to the

index of performance and are added in at the end when it is computed.

It is recognized that many of the decisions described in this section and the preceding

one are somewhat arbitrary, and no doubt can be criticized on a variety of grounds. They were

selected by the project team as the "most reasonable" choices within the time and information

available, and do appear fairly adequate for the illustrative aims of the computer exercises

reported here. Further sensitivity runs to probe the criticality of the various assumptions

made would of course be desirable. At any rate, the necessity to work through the detailed

specification of a complete set of input parameters, made particular needs for better

empirical data much more vividly apparent.

* There is a provision in the program to consider center processing costs which vary for

different geographical areas as described in Chapter VII. A run of the computer model

was made to illustrate the use of this provision, and the results are described later in

this chapter.

91
Computer "Map Analysis"

As noted above, the manipulation of the shipper input data was performed only after the

(manual) map analysis of Chapter VI had been carried out. In order to display the distributio

of the shipments (both exports and imports) of the revised demand data, two computer runs were

made. The first divided the -rectangle bounding the shippers into a 10x10 grid and calculated

the output, measured in 90% -full 48,000 pound containers of export or import cargo, associated

with each grid element. The results of this run are included in Table 18. The second run calculated the amount of cargo originating or arriving near each of several cities. Its results are given in Table 19.

The information from these two runs can be used in two ways. First, it provided guesses

as to good starting locations for centers. As can be seen in Table 18, there are approximate1> 8 areas from which the bulk of cargo is exported or imported - Philadelphia, western New York,

Pittsburgh, Cleveland, southwestern Ohio, Detroit, Chicago, and Milwaukee. The total amount

of cargo associated" with all shippers is about 225 901-full containers per day. Of this total,

about 67%, or 150 containers per day, originates in the 8 areas above when Harrisburg and

Allentown are included with Philadelphia, the southwestern New York area and Buffalo are lumped together as "western New York", and Pittsburgh and Johnstown are coupled as the

Pittsburgh area.

Second, this information also gave an indication what numbers of centers were reasonable

candidates for analysis in computer runs of the mathematical model. There is already a center

located at the port in Philadelphia. With the cargo originating in the Philadelphia area subtracted from the total, the other 7 of the 8 areas listed produce a daily average of 103

out of the remaining 177 containers. Three of these areas, namely, Chicago, Cleveland, and

Pittsburgh, would individually support small to medium sized centers. The other four areas appear at best marginal. From an examination of the two tables it seems unlikely that the

study area would support more than 7 or 8 inland centers if each is typically to process at least 10 90%-full containers of LCL cargo per day.

Table 18 . Geographical Distribution of Shipments*

.00 .49

.00 1.28

.00 5.59

8 .41 4.54

.15 .65

.60 .14

.00 .00

.01 .80

.01 ,09

.34 .00

.36

.00

.00

.00

7 28.32

.09

.48

.39

.06

.00

.00

.25

1.06

1.01

.60

1.12

.62

.00

.76

3.78

1.57

.77

1.41

4 5.11

3.58 4.77

.00

.00

.32

6 5.38,1

2.49 /Too

00 02

3.93

1.77

1.59

.00

.00

4.65

.00

.00

2.17

8.45

7.82

1.97

.00

.00

00

00

.29

3.16

7.74

.08

.00

.00

.00

.00

2 59

2

4 47

1 61 2 74

.37

4.73

.95

2.29

.00

.00

.00

.02

1.07

.13

.02

.70

21 09

] 23 04

-i

,94

.30

Philadelphia-Allentown

Western New York

Pittsburgh -Johns town

Southwest Ohio

5 Cleveland

6 Detroit

7 Chicago

8 Milwaukee

N

*Each entry gives the average daily number of 901 full containers associated with the corres- ponding rectangle of the 10x10 grid overlaid on the Delaware Port Authority's hinterland.

92
Table 19. Average Daily Number of 90%-Full Containers

Imported or Exported for Each of the Areas Listed

Philadelphia 37.02

Allentown, Pennsylvania 7.21

Harrisburg 3 . 46

Southwestern New York 2.52

Buffalo 2.59

Johnstown, Pennsylvania 6.82

Pittsburgh 7.57

Northeastern Ohio .76

Cleveland 25.79

Detroit 11.04

Southwestern Ohio 12.33

Northwestern Ohio 3.31

Southern Michigan .92

Indianapolis 1.83

Chicago 27.98

Milwaukee 7.03

Central Illinois

Other

.80

62.41

When the results of the two computer map analysis runs are compared with the hand calculations in Chapter VI, the most striking observation is the great reduction in total demand

(by a factor exceeding 2) introduced in the process of passing to the LCL component of the

cargo. This occurs despite the use in the computer runs of a nominal figure of 43,200 pounds

per container (90%-full 48,000 pound containers), as compared with the 51,000 pounds per

container used in the hand analysis. As a result there are only two centers in Ohio, one medium and one small, after non-LCL shipments are deleted, whereas there are six listed in

Chapter VI. Since the map analysis in Chapter VI was done only for Pennsylvania, Ohio,

Indiana, and Illinois, it had no opportunity to exhibit centers in Milwaukee or western New

York.

In summary, although the hand analysis in Chapter VI indicated that the study area could

easily support 10 or more centers, passing to the LCL component of demand as is done here

leads to a more conservative estimate: that present levels of demand probably justify at most

7 or 8 centers. This of course makes no allowance for subsequent growth of shipping volumes,

especially for LCL cargo stimulated by the existence of the inland centers.

Effects of the Location Step and Minimum Patronage Constraint

A natural question is whether the optimization process embodied in the model's iterations

through its "location step" is not more or less of a frill. That is, would not the mere

availability of the inland centers, even if haphazardly located, secure essentially the same savings? Results bearing on this question are given in Table 20, which compares some values

of the index of performance when centers were located at random with those obtained when the

model was used to optimize starting with these locations.

The relatively low value of the index for the randomly located centers indicates that

the very existence of centers provides a great savings, as much as one third. An additional

reduction of about 1/10 in the index is provided by the optimizing procedure. The optimization

procedure also provides a better chance that the centers will process at least the minimum

amount of cargo- -10 containers per day- -indicated during the study as representing a profitable

level of operation.

Table 21 shows the effects on the index of performance of requiring that centers meet the

minimum total patronage constraint. This constraint reduces the index (for results from the

optimization) by approximately an additional 5 percent. Although at first glance such a constraint might be expected to raise the index, it actually has the effect of lowering it

for the following reason: In general, the index of performance is lower when more cargo is

attracted to centers, and the minimum total patronage constraint has the effect of shifting

"small" centers to points where they may attract more patronage.

93
Table 20. Effect on Index of Performance of Optimizing Center Locations*

With Locations Optim].zed from

No. of Centers Random in Area Random Locations

4 .695 .579

5 .683 .577

6 .668 .571

7 .646 .562

The random center configurations referred to in the second column were

used as initializations for the optimizations to which corresponding

figures in the third column refer. The figures are each averaged over

4 cases.

Table 21. Effect of the Minimum Total Patronage Constraint on the Index

of Performance

No. of Centers

4

5

6

7

No Constraintt Patronage Constrainttt

.579 .553

.577 .550

.571 .530

.562 .523

t The figures for the case of no minimum total patronage constraint are averages over 4 cases.

tt The figures for the runs with the minimum total patronage constraint

in effect are averages over 5 cases (each involving up to three

initializations to secure satisfaction of the constraint)

Effects of Random Initial Locations

Recall that the solution obtained by the computer program can depend on the initial

"guesses" as to center locations. Thus it is dangerous to examine only the results from a single initial guess since that guess might have been an unfortunate one leading to far from

optimal results.

As noted earlier, several methods for generating initial guesses (if they are not

supplied by the user) have been included in the program. Two are random in nature; one

("random at shippers") places the n centers near n randomly selected shippers, while the

other ("random in area") places them at random in the smallest rectangle, with sides parallel

to the axes of the (x,y) -coordinate system, which contains all the shippers.

Initial runs were performed, in part, to determine the effects of the random variations

in initialization within each of the two random methods. While the resultant configurations

of center locations were of course of great interest, it was felt even more important to get

insight into the resultant fluctuations in the benefits provided by the system.

Table 22 presents the indices of performance found in these limited runs. The first

thing to strike the eye is the low values- -all between .5 and .63- -recorded for the index of

performance. These indicate very substantial economic benefits accruing from the presence

94
of the inland centers. Such an impression is reinforced by the fact that the index of performance, as presently calculated, carries an anti-system bias in that accumulation delays

for containerization at the ports are taken as zero.

Some immediate caveats are in order. There are of course all the previously-mentioned

uncertainties and arbitrariness in the data and input parameters employed. Another point is

that both numerator and denominator of the index of performance refer only to the U.S.-

incurred portion of the generalized costs of origin-to-destination movement; one should con- sider how the results would be modified if the benefits of containerization for the over- water trip segment and overseas- land segment were taken into account.

Allowing for all these factors, the reported values still appear small enough to warrant

the inference in the Summary that the establishment of inland centers promises wholesale

advantages to the shipping community.

From an examination of the indices of performance in Table 22, there seems no reason to anticipate order- of-magnitude variations in the solutions obtained from different random

initializations. The serious fluctuations are at about the 10% level, which seems significant

enough to merit continuing the practice of multiple initializations for each case studied.

Table 22. Effects of Randomizations of Initial

Locations on Index of Performance

Initialization No. of

Method Centers

Random 4 in 5 Area 6

7

Random 4 at 5 Shipper 6

7

Values of Index Avg. Min.

.587,* .568, .600, .562 .579 .562

.626, .562, .544, .575 .577 .544

.564, .551, .577, .589 .571 .551

.571, .570, .565, .541 .562 .541

.559, .616, .655 .610 .559

.545, .572, .578 .565 .545

.550, .551, .566 .556 .550

.566, .559, .617 .579 .559

* Each value of the index refers to a separate initialization.

The previous material referred to the effects of random initialization on the index of

system performance. It is also necessary to consider the effects on the locations of centers.

For this purpose, a rough geographical classification of locations was developed, and is

employed in the illustrative Tables 23-26. Examination of these tables reveals unanimity

on some points, e.g., the need for one or more centers in the general areas of Chicago,

Cleveland, and Pittsburgh. Other areas, in which a center was located starting from a substantial number of different initial configurations, are: southwestern Ohio, western New

York, and Philadelphia. Milwaukee, although marginal, would probably also support a small

center.

Perhaps the most striking difference between the location patterns obtained from the

two random initialization methods is in their respective treatments of the eastern

Pennsylvania area. Heavy concentration of centers here is especially odd in view of the

proximity of the port's facility. For an explanation, recall from Chapter IV the sampling

scheme used by the Delaware Port Authority. There are two factors in it which might bias the

distribution of shippers in favor of Philadelphia. The first is the fact that Philadelphia

is divided into 11 study subareas, whereas all other subareas are full counties. Since a "shipper", for the purpose of the model, is an aggregate of all shippers in a subarea who

ship the same commodity through the same U.S. port, the definition of 11 subareas in

Philadelphia would tend to produce as many as 11 times more shippers in the Philadelphia

area as in an area of comparable size elsewhere. Secondly, the sampling scheme only

samples larger (as measured by number of employees) firms in areas outside the immediate

95
Table 23. Variations in Rough Locations of Centers (4 Centers)

Area

E. Pennsylvania

(including Philadelphia)

Western New York

Greater Pittsburgh

N.E. Ohio

Cleveland

Detroit

S.W. Ohio

N.W. Ohio

South Central Michigan

Central Indiana

Chicago

Milwaukee

Other

Random in Area**

Random at Shipper**

1 1

1 1

1*

Table 24. Variations in Rough Locations of Centers (5 Centers)

Area

E. Pennsylvania

f including Philadelphia)

Western New York

Greater Pittsburgh

N.E. Ohio

Cleveland

Detroit

S.W. Ohio

N.W. Ohio

South Central Michigan

Central Indiana

Chicago

Milwaukee

Other

Random in Area

1

1 1,1* 2

Random at Shipper

1 1 2

1

1*

* Center received no cargo.

1 Each column corresponds to a separate initialization.

96
Table 25. Variations in Rough Locations of Centers (6 Centers)

Random at Random in Area _Shipjoer_

Area _ — 1 2 3 E. Pennsylvania

(including Philadelphia)

Western New York

Greater Pittsburgh

N.E. Ohio

Cleveland

Detroit

S.W. Ohio

N.W. Ohio

South Central Michigan

Central Indiana

Chicago

Milwaukee

Other

1

1 1

1 1

1111

11* 1 3,1* 2

Table 26. Variations in Rough Locations of Centers (7 Centers)

Random at Random in Area _Shipper

Area ' 2 3 3

E. Pennsylvania

(including Philadelphia)

^ l Western New York ? 1 1 Greater Pittsburgh

N.E. Ohio -, 1 12 11 l x Cleveland ! Detroit ^ S.W. Ohio

N.W. Ohio

South Central Michigan

Central Indiana 1

1 1 1 1

1 Chicago

Milwaukee

' „ -. 3 1,1* 3 2 2 1 Other

* Center received no cargo.

97
tion methods

methods

values of

some Philadelphia area. Such a sampling scheme may bias the distribution of commodities since

aggregates

commodities may be produced or imported mainly by smaller firms. SinTM shippers are

tion of

handling the same commodity, the sampling procedure may again Mas thHistSbu- shippers in favor of Philadelphia where every

1 known exporting or importing fim was interrogated.

the relative

Therefore for both of these reasons, the model's inoSt data may e1aggTMatf number of shippers in the Philadelphia area. Such a bias wouldILd the random-at-shippers

the

method of getting an initial configuration of centers to start out on up with

average with unduly many center sites in the Philadelphia area, and thus to tend to end configurations suffering the same bias.

Sthn/

eSU^

S d

°i n0t TV pemit a Clear selection between the two random initializadf tr rS

S Va

J UeS °

f the inde* of Performance are approximately the same for the" rwo JfA-F Z Par

K 1C^f vaxues CaS6S reP° rted here the random- in- area method gives average

purely

of the index which decrease with the number of centers. This would be expected on theoretical grounds since (a) the computer program does not incorporate Sy dependence fn S?SSin

£,

COStS TM Center size > TM* ^ the TMniTMm total patronage^estricSon was no?

ILll » above.

e ra"dom-^- area ^thod also avoids the bias towards Philadelphia mentioned However, the random- in-area method has a higher incidence of centers receTvina no S?£, ^ ^

ayb/ rectangle with

exP laTMed ^ the fact that this method locates randomly In thlsmaRest

If the

sides parallel to the axes which contains the region bounding the shippers

of the

region

rectangle

bounding the shippers were roughly oval in shape, then points near tte Vomers would be apt to be bad poinds for locating ceAters?

mc .,F; possibleom the results here, it is suggested that both methods be used to get an idea of

choice

combinations of initial locations to be tried. The next section wmtescribe the and results of such combinations of locations.

section will describe the

Systematic Selection of Possible Initial Configurations

In testing the random initialization procedures, many different "anrvi" «n,i „ c- tions were found from a variety of starting nositinns 7thTM !i g -, al ^f^raof performance for the random start i Self

^m f

in several cases the index

possible- the U*UlhlnA„?u itself was quite good (low), improvement was still

clearlv verv JS? S

happening by pure chance on an unimprovable configuration is

the nth center ? Three *^&^%S£^&^'£?2&1"°- *°*

1. It was located near one of the n-1 positions already found " ^^£TMt^^l^-£ d

^TM"* analysis was

3. It was placed systematically in areas which did not yet have centers.

SfTStert rfU'bSfSS^S otn St'

6 " l0C

hatl °nS f°J

n " X C6nterS be C« He close

a good starting approximation ^dnas in prac? ce'ied'to c^f^T

^^

Ukely t0 Pr°Vide

involving only small adjustmeAts in the initial n-1 loca?ions ° ^^ ^S^01*

ginal^lf'^rSts'of^cfal^ "JTM ?

nter WaS t0 l0Cate * — — «* the orithis method ended'p'with'^centerf n"ea 'TEHnT *? ^^S? 27 " In *eneral > extra center ends up in Mi Iwaukee Cleveland InTpir^h V^w10

?

1S O11 ^ ' where the

cargo to support two small centers- hofJfif ^

Pittsburgh are both associated with enough

does not seem too prom! ng since'i doel "nollelT^

5 ^ ^ ^^ '" n0t " TMS meth°d

most cases.

eS n0t lead t0 significantly new configurations in

98
Table 27. Initializing by Splitting a Good

Center into Two Located Near Each Other*

5 Centers

Western New York

Greater Pittsburgh

Cleveland

i 7 1 Southwest Ohio 1

x

2 111 12 11

Chicago

Milwaukee

Index

111 1

1

536 .537 .539 .536

* Based on a 4 -center configuration with locations in

Pittsburgh, Cleveland, southwest Ohio, and Chicago.

Se columns correspond respectively to initializations

of the fifth center near each of the first 4, in turn.

A, exaBinat.cn of the resuUs %££<£?%£* Sd^dliftsbSrS should

random initial locations strongly indieates that Chicago ^ n ^ requlre ^ SS»bSifcSS,.

B^cS'S tne assignment process but are never

some locations to have centers

moved in the location step

Table 28 gives the results of runs made to ^^%TMiyVZ map^alysif £?

are associated with at least between S a^lO «J^g Y ^

indicated centers are Table 19). Line I of Table 28 gives the -gdex °± pert

are £reed

^

n centers

fivpH- Line II gives the corresponding index wnen uie ui nrH res in the two lines is

fallowed to move. The difference between the ^"f^.^^^is only about half

quite small, and since the computer TM^^v!^£B t00i for investigating starting con- that of regular runs ^Procedure prwides a valuable to ^ ^

con£i tl0n

figurations indicated by the distributi^ of shippers

southwestern Ohio and Chicago; for

fcenterslfZXZ&TM^%^'3^£Sl2Sra&

M

DeSeihfigo

OI

ano SeTlS oftn°e£ ^"process at least the desired mintcargo per day (10 .containers)

.

r „„-;i,i a initial renter configuration were carried out using

Two other investigations of possible initial center coniig

centers and investitive procedure of fixing some of the centers. T^ese involved fixing n center ^

gating

results

several

of initializing

points on

centers

a line asat starting

eveniy bPfT?

pc intsoothe

T

nth^enter.^ ^

aeneral the resulting location of

of Pittsburgh from central Ohio to centr£ J££°«.

^

*£

r^ same rough area. The

the last center is somewhat east of its mitialization but in tne s g ^ initialiindices are slightly higher than those in Table 28 ^^Jj*TM^ border east of

present set of data, using the bes ^own?^ center lo^tions as ^ start ^ ^ ^ ^

£^10^^^^

western New York, and a corresponding lowering of the index irom about
Table 28. Effects on Index and Location of Initializing

Centers in Cities Chosen from the Map Analysis

E. Pennsylvania

(including Philadelphia)

Western New York

Greater Pittsburgh

N.E. Ohio

Cleveland

Detroit

S.W. Ohio

N.W. Ohio

South Central Michigan

Central Indiana

Chicago

Milwaukee

Other

5 Centers

F* F r r

6 Centers 7 Centers

1 F1 F

1

1 F1 F

1

1 F1 F

1

1 1

1 F1 F1

I. Index-some centers fixed .540 .537 .523

II. Index-no centers fixed .533 .535 .522

522 .522

520 .522 517

Table 29. "New" Center Initialized on Line West of Pittsburgh

Western New York

Greater Pittsburgh

N.W. Pennsylvania

Cleveland

Central Ohio

S.W. Ohio

N.W. Ohio

North Central Indiana

Chicago

Milwaukee

Central Illinois

5 Centers

f** f f f f r r r r r

i £

i £i £

i £i £

i £

i £

i £i £i £

i £i £

i £

i £i £

6 Centers

f f f f f f

l rl r 1 l r

l rl

r f f f f f f

l rl rl rl r

l r

l r

1 £1 £

1 £1 £1 £

1 £

1 £

1 £1 £1 £1 £ /

1 £1 £1 £1 £1 £1 £

Index 536 .539 .538 .538 .539 522 .523 .523 .523 .524

* The superscript F indicates those centers which were fixed for the runs of line I. ** The superscript f indicates those centers whose locations were fixed.

100
Table 30. "New" Center Initialized on

Line East of Cleveland

York

; sburgh

5 Centers

Western New

Greater Pitl

Cleveland

11F

*

1F

1

1F

1F

1

1F

1 F

Detroit

S.W. Ohio 1F

1F

1F

Chicago 1F

1F

1F

Mi lwaukee

Other

Index .523 -523 .526

* The subscript F indicates those centers whose

location is fixed.

The Number of Centers

.n obviously important question is that of^most gP^^-^SSS^^

for the number (n) of inland centers to be established As n incre ' ^ x val the

portion of total generalized cost should decrease But for n past some cr ^^

generalized processing-cost portion is e?pectodtouo«e since tne g ^ nage

Sill tend to contain some ^^^f^^^^^eSe Sese two conflicting trends

by some of the co-contamenzable cargo class ^^y serve

d roces sing costs, as

total generalized cost is minimized.

The mathematical TM*£^^TMJ^J^SlftS^^,

SnSTe^^

dependent processing costs and times It does

, ^ver conta p^

^^ & tions based on adequate patronage. The first P

ro^?"s ,^Il "- c

. t great . This prohibition

co-containerization class if the expected fJ^^f^Zr It should be noted here that was imposed in all the computer runs reported in ^<W»- aken to be a linear function of

^he cost of accumulation delay CJike other tmec^^^^,2»s finite at the

the delay time for values less than the threshold, ^s

C Sa5 the threshold , this linear

threshold level. For values of delay dose o ^tless than the thresn

o

^^

approximation is probably too low ^^/^ Sfect of Sis underestimation 'of the

increase as the delay approached the threshold. The ettect or x advantageous than

cost of accumulation delay is to make a larger number of centers £° Jter< it really is.' Tne second prohibition method^J^^^ ^J^garfad as unecononiSince any center processing fewer than 10 containers per da

y was * cessed too few

cal, whenever the final configuration of centers contained centers w p^

^^

containers, all such centers were randomly ^initialized. iTM

configuration. This

in all of our work because of the additional computer time required.

E

, Me tn have 7 or 8** carefully placed centers which meet the minimum

Although it 1 %Poss

^

1^

e

r ^ph^inaf1

° catgnfo£ Table 28 do), such a configuration is

total patronage restriction (the 7 final ^cations or configurations of 7 centers

unlikely to be reached accidentally. Only one out/^t Jje

conr^u

configuration

had all centers meeting the restriction after three "ini^alizations^ ^ ^ ^^

resulted in two centers m ^

p£*^t*f^^ to confirm the intuitive conclusion

froTtne computer ma/analySs! £t°S given study area and demand pattern would support

about 7 or 8 centers at most.

I^rfTguration of « centers would include one in the Philadelphia area.

101
locations as a starting configuration and onI?,^a tZ

hef

locatlons; the other used the

not significantly lower than ?K,« to" centers On t^lF^,TM?" is lo"er > ""' m one run and 6 in the other nmr-P«Pri *u2

centers. On the other hand, only 5 centers

this it is clear that ^erSSoS^rfom^5 ^ nTTM °l

TM contai^rs P^r d om

the LCL component) have sig^fica^tVredS the estiva\7l?ZT would ******* ^

(t ° iso1

^

support- -on the basis of the cargo fiow „ th I S how many centers this area

should be noted that the present modePgive no "Ire^V/ZV^ ?° -^J?" A^in " the centers m order to take advantage of possible sneci a

: rJn^

C°ntainer shiPPe^ utilizing

such as those for unit trains. The inclusion of u h £n f > ?!1 S l"

transP°rtation rates -crease the number of centers which the^a ^T^S^^^TM tend t0

^ r Tf

ble

n

1, CoiT1Puter Evaluation of the Locations of Centers Suggested by the Manual Analysis of Chapter VI

No Optimized

Optimization Locations

E. Pennsylvania ^ (including Philadelphia)

Central Pennsylvania i Western New York i

Pittsburgh

Cleveland

Detroit

Southwest Ohio

Northwest Ohio

Indianapolis

Chicago

1

1

1 1 1 1

1

1 1 1 1

1 1

1 1

Index en r .516 .524

Sensitivity to Dispa tching Policy

-t-f =-ss^^^-^^:^:r^^zr^- -

parameters

1 *** " '^ TMthTM*^ -del, a dispatching policy is described by the two

L = "target" load factor (a dimensionless proper fraction)

H - maximum allowable holding time (days).

The nominal values selected for these parameters were

L = 0.90, H = 4 days.

and MSS^r&^^VS "Sg^ °f

^

*— ^ L = *«

unlikely, since rather full containers are desirable IS T P0ssible but ^eem somewhat

rate for transporting containerized material Th^ ?TM Pa^-e^Ploitation of the reduced

3 days and 5 days respectively. Beyond the 5 dS iSt rW V?

U6S ° f H W6re taken as might be accompanied by the cost of extra storage JapabilSies'

5 gGS °f eXtra delay

the target load factor than to the max^I^g &.^^£^S^°
- TMc+c at the exuense of higher accumulation delays) seems to (decreasing center- to-port costs, ^he^xp^e

time does not seem to make up, in extra

be a good policy. Increasing the maximum hoiaiag

transportation cost, for the extra

Table 32. Variation of Index of Performance with Dispatching Policy

L

.95

.95

.90

.90

.90

.85

.85

H

4

5

34**

5

3

4

4 Centers

.573*

.575

.577

.579

.580

.583

.587

6 Centers

.567

.568

.571

.571

.571

.577

.577

7 Centers

.558

.558

.561

.562

.563

.568

.569

Tj^e indices listed here are averages oyer 4 cases The « set

of initial locations was used for each of the 7 policies tested.

**Nominal case.

Sensitivity to Rate for Containers

" The second sensitivity analysis was concerned with the (monetary) component of *J*JES ff«£SS£T£ tt£J^&££^'*^ economic

motivation for consolidation of LCL cargo in containers.

Table 33 records the indices oi^Pf*"^ «rate

value of the containerized freight rate. A change o i.c&uses &bout &

SSIrorlhrfrSeTt SJan^eastfdfnoSfSrfalter the amount of carg

o

attracted to centers.

Table 33 Variation of the Index of Performance with Transportation

iaDX

Rate for Container Movement

Rates

.0000108

.0000122

.0000135t

.0000148

.0000162

5 Centers

.538

.562

.577

.597

.615

7 Centers

.530

.545

.562

.579

.599

t Nominal case. The value of the rate is in $/lb.

103
Sensitivity to Level of Total Demand

The third sensitivity analysis was concerned with the level of demand. Forecasting cargo

demand for a future year was not within the scope of this project. However, it is clear that

the proper number of centers and their index of performance depend on demand level. Also, the

admittedly approximate operations which were performed on the Delaware Port Authority data have

significantly affected the level of assumed LCL demand. Therefore, some analysis of the effect*

of alternative levels of demand is necessary to give insight into the consequences of both

present data uncertainties and future growth.

Tables 34 and 35 record the results of increasing demand by 25% over the "nominal" level

used in all other work. This increase is accomplished by increasing both the total yearly vol- ume and the total yearly shipment frequency for each shipper by 25%. Thus the average shipment

size remains the same; each shipper just ships more often. The same initial locations were

used for both the nominal demand pattern and the "raised by 25% version." Both runs produced

the same final configurations. The realized load factors remained substantially the same since

the target load factor and average shipment sizes did not change. The number of containers

processed by all centers increased by about 25% although the increase is not distributed evenly

over the centers. (For instance, in Table 34, the center in Western New York gained only about

11%, while the one at Pittsburgh gained over 33%.) The center at Milwaukee, which was marginal

at the nominal demand level, became much more robust once demand had been raised. Further investigations would be needed to evaluate whether the study area v/ould support more than the seven inland centers or whether the seven centers would just each process more. The increase

in demand level produced a small decrease in the index of performance, but this is not

especially significant in comparing situations with different demand patterns.

Location-Dependent Processing Costs

Four computations are made to illustrate the treatment of monetary processing costs which

depend on center location. The region of interest was divided into 4 rectangles by a 2x2 grid.

It was first assumed that the processing cost per container would vary by the following additive quantities:

N.W. Rectangle = $2.00

S.W. Rectangle = -6.50

N.E. Rectangle = -3.00

S.E. Rectangle = 5.00

These quantities were arbitrary, but (based on Chapter III and infonaation gained during the

study) are thought to be of reasonable order of magnitude.

The computation was made for 7 centers initialized as in Table 28. The index of perfor- mance rose, but only by about .02 percent which is probably less than the resolution capability

of the model. No significant change in the locations of the seven centers was noted. The

reduced cost in the southwest rectangle did not draw any other centers to join the one in southwest Ohio. The increased cost in the southeast rectangle did not drive out its single center

at Pittsburgh, nor did the increased cost of the northwest rectangle drive out the centers of

Detroit, Chicago, and Milwaukee. For this particular example the variation in locationdependent costs from -6.5 to +5.0 dollars per container did not produce a significant change

in either locations of centers or the index of performance. This range of variation is about

that to be expected from the material in Chapter III. The remaining calculations described

below were made in order to test the logic of the model; the values of the incremental cost in

the southwestern rectangle are an order of magnitude greater than ones which one might reasonably expect.

A second computation was made with the cost in the southwestern rectangle very large

($9,999,999.99 per container) and the costs of the other rectangles as above. This drove the

center originally in southwest Ohio out of business. A third computation with the incremental

cost in this southwest rectangle at $480 per container did not change its location but reduced

the southwestern Ohio center's patronage to less than 4 containers per day. A fourth computation with the incremental cost in the southwest rectangle at $800 per container drove all

patronage except one shipper away from the southwestern Ohio center and moved its location to that shipper.

104
Table 34. Sensitivity to Total Demand

(6 centers)

Nominal Demand Demand Raised by 25%

Containers/day Load Factor Containers/day Load Factor

Western New York 21.51 .8842 23.99 .8924

Pittsburgh 36.09 .8930 48.34 .8922

Cleveland 55.25 .8764 68.37 .8746

Southwest Ohio 17.40 .8851 23.58 .8552

Chicago 25.39 .9020 31.29 .9086

Milwaukee 12.35 .8669 14.67 .8838

Index .520 .517

Table 35. Sensitivity to Total Demand

(7 centers)

Nominal Demand Demand Raised by 25%

Containers/day Load Factor Containers/day Load Factor

Western New York 20.41 .8928 24.00 .8922

Pittsburgh 39.77 .8896 51.74 .8873

Cleveland 42.36 .8492 47.56 .8519

Detroit 14.10 .8816 22.27 .8561

Southwest Ohio 16.29 .8746 21.82 .8574

Chicago 24.09 .9087 30.10 .9092

Milwaukee 11.22 .8875 14.38 .8893

Index 517 .514

Sample Program Output

Figure 11 contains part of a sample program output. The program contains options to print more information than that recorded here; this particular output contains only those

types of information which were desired for the runs described in this chapter.

The upper section contains the run description, which is printed once at the beginning

of a run. This particular run is going to perform 5 separate initializations of type 2 ("random in area") for 4 centers. Both types of patronage restrictions are in force. The

dispatching policy and containerized freight rates are printed for use in the sensitivity

analyses

.

The next section of output contains the initial center locations. The coordinate system

used in the output is the one -described earlier in this chapter.

The next line is printed whenever a center violates the minimum total patronage restriction (if imposed) , and is therefore reinitialized to a new location.

105
THIS PROGRAM COMPUTES OPT WAAL LOCATIONS FOR NUMBERS OF CENTERS FROM 4 TO 4 AT INTERVALS OF 1. THE PROGRAM USES INITIALIZATION 2. THE PROGRAM PROHIBITS CENTERS FROM ACCEPTING CARGO FOR A CO-CONTAINER I ZABLE CLASS IF THE ACCUMULATION DELAY IS TOO GREAT. CENTERS ARE REINITIALIZED 3 TIMES OR UNTIL EACH CENTER RECEIVES AT LEAST 432000.00 POUNDS OF CARGO.

5 INITIALIZATIONS ARE PERFORMED FOR EACH NUMBER OF CENTERS.

THE TRANSPORTATION RATE FOR CONTAINERIZED FREIGHT IS .00001j5.

THE INVENTORY-TYPE CARRYING CHARGE IS .000"+ THE PROJECTED EXPORT LOAD FACTOR IS .900. THE MAXIMUM HOLDING TIME IS 4. DAYS. THE AVERAGE IMPORT LOAD FACTOR IS .822.

INITIALIZATION FOR 4 CENTERS

CENTER X Y

1 555.02 184.25

2 339.09 454.70

3 262.94 121.03

4 135.04 179.92

CENTER 4 RECEIVING 119217.00 POUNDS AND LOCATED AT 169.750 208.930 REINITIALIZED TO 741.921 420.389

CENTER

32

14

355.21 246.79

456.68 303.65

591.09 189.97 770.00 333.37

MIN X OF MAX X OF MIN Y OF MAX Y OF PATRONS PATRONS PATRONS PATRONS

253.75 - 874.12 141.57 - 484.07

301.88 - 903.00 150.70 - 508.04 469.88 - 886.37 100.47 485.21 607.25 - .952.87 139.28 - 509.18

POUNDS CONTAINERS LOAD PER CENT PER CENT INDEX OF

INPUT OUTPUT FACTOR EXPORT IMPORT PERFORMANCE 1831374 20 43 37 .8798 76 00 24 00 .6471 2234762 20 52 59 .8853 82 82 17 18 .5062 941051 82 23 26 .8430 92 06 7 94 .4918 1603106 30 36 95 .9039 82 76 17 24 .4037

PLACE X DELTA NUMBER OF CENTERS WITHIN DELTA OF (X,Y) PHILADELPHIA 108.50 203.22 50. NONE ALLENTOWN PA 135.62 247.74 50. NONE

HARRIS BURG 169.75 208.93 50. NONE

SW. NEW YORK 249.49 355.66 30. NONE BUFFALO 298.37 393.80 30. NONE

PITTSBURGH 367.50 236.33 50. 1 NE. OHIO 410.00 300.00 20. NONE CLEVELAND 456.75 303.68 30. 1 DETROIT 536.37 359.63 50. NONE

SW. OHIO 592.37 188.38 50. 1 NW. OHIO 610.16 332.32 50. NONE

S. MICHIGAN 650.00 350.00 50. NONE

INDIANAPOLIS 690.37 189.52 50. NONE

CHICAGO 770.00 333.37 50. 1 MILWAUKEE 784.87 412.14 50. NONE

CENT. ILL. 837.37 195.23 50. NONE

PORT X Y MIN X OF 1AX X OF MIN Y OF MAX Y OF POUNDS CONTAINERS LOAD PER CENT PER CENT INDEX PATRONS PATRONS PATRONS PATRONS INPUT OUTPUT FACTOR EXPORT IMPORT PERFORM

1 112.80

2 52.50

200.80 255.00 87.50 - 310. 38

00

93. 62 - 371.04 906451.54

.00 21.40

.00 .8826.0000 86.93.00 13.07 100.00

.8064

.0000

3 188.00 156.20 157.50 - 784. 87 102. 75 - 484.07 99656.02 2.42 .8586 64.11 35.89 .8381

COST WITH CENTERS COST WITHOUT CENTERS VARIABLE COST 121277.23 252665.13 TOTAL COST 157432.18 288820.07

INDEX OF PERFORMANCE .545087

Figure 11. Sample Output

106
The next section contains the output information for each center: final location, zone

i of attraction, the number of pounds of cargo processed per day, the number of containers

stuffed (or unstuffed) per day, the average load factor of these containers, the percentages

of the cargo which are export and import, respectively, and the index of performance for the

patrons of each center.

The next section gives a rough geographical description of the locations of the centers.

Several cities (16 in this example) are chosen together with a tolerance distance, labeled

DELTA. The number of centers which are "near" (within the tolerance distance of) each of the

cities is tabulated.

The next section contains the same information for port centers as is printed for the

inland centers. Here it should be noted that port center 2 received no patronage. The

asterisks and 100.00 percent import result from this fact and should be ignored.

The final section contains information on the index of performance of the system and the

breakdown of the costs which make up this index. The total variable cost is that cost which

enters into the optimization phase- -transportation costs (except the constant term of equation [7.8]), and center accumulation delay costs at inland centers.

Further Sensitivity Analyses

After the sensitivity analyses described above were performed and a preliminary report

completed, occasion arose to make further improvements in the model as well as further

sensitivity analyses.

The computer program was expanded to include two versions of the equation [7.8] which

gives travel time as a function of line haul distance; one of these is applied to containerized freight (in effect assumed to go by rail), the other to uncontainerized (moved by truck).

Therefore since two values of the Sm of equation [7.8] were needed, one for rail and one for

truck, this factor had to be included in the assignment step, whereas previously it only had

to be added in at the end when evaluating the index of performance. The circuity factor

(see [7.3]) is now used in calculating the travel time as well as travel cost. These changes

affect the values of the index of performance, typically raising them, but they still remain

generally between .55 and .63. The changes also decrease the differences among the indexvalues for different sets of initial locations. They do not, however, have any appreciable

effect on the center locations determined.

The following sections describe the nine additional sensitivity analyses performed.

Since the changes described above are included in the computer runs for these analyses , the

indices of performance which follow are not properly comparable to those of the previous

sections. For this set of runs the nominal case had the minimum total patronage constraint

in operation. All other cases used the center locations determined for the nominal case as starting configuration, and the minimum total patronage constraint was not imposed except in

the analysis summarized in Table 46.

The general findings are essentially the same as those noted earlier for the previous

set of sensitivity analyses. Final center locations are most sensitive to the set of initial

locations; relatively large variations in other parameters produce only small variations

either in locations or in the index of performance. Throughout our tests, sensitivity of the

index to parameter changes never significantly exceeded its sensitivity to the choice of

initial locations and most often was significantly less. Because the tables which follow do

not include the evidence for this assertion (specifically, the index-values arising from

different location-initializations) , we illustrate the point here with a fairly typical

example: Three runs of the model with different sets of initial locations for five centers

produced indices of performance of .5969, .5775, and .5901. A decrease in the import load

factor by 10% produced indices of .5998, .5813, and .5906 for the same three sets of initial

locations. The maximum raise in index is .0038, occurring for the second set of initial

locations. The variation in index due to initial location set, however, is much larger:

.0194 for the base set of runs and .0185 for the runs with lower import load factor.

107
Sensitivity to LCL Transportation Rates

The LCL transportation rates listed in Table 16 are averages over the commodities in

each of the eight commodity classes. As described in Chapter V, transportation rates depend

in reality on such factors as origin and destination, nature of the commodity, and the

packaging of goods; Table 16' s figures are representative averages over all these factors.

The present analysis was performed to investigate sensitivity to deviations from these nominal

rates. Each of the eight rates was given a random perturbation uniformly distributed between

±101 of the nominal value in Table 16; the eight perturbations were independent. Two sets of

perturbed rates (8 rates each) were formed in this way.

The results are reported in Table 36. (The new transportation rates for the results are given in footnotes.) Both sets of variations in transportation rates happened to reduce the

total cost without centers, of the set in line 1 by only 0.41 and that in line 3 by 4%.

However, the commodity mix is such that the numerator of the index of performance for line 3 was reduced relatively less than was the denominator, causing higher values of the index.

On the whole, the index appears less sensitive to changes or moderate errors in the LCL

transportation rates than to different starting positions.

Table 36. Variation of Index of Performance with Random Perturbations

of Uncontainerized Transportation Dollar Rates*

4 Centers 5 Centers 6 Centers 7 Centers

1 ** .600 .591 .596 ,580

2.+ .607 .588 .594 .577

3.++ .611 .602 .605 .590

* All entries are averages over 3 cases

**The new transportation rates for this case are .696, .730, .450,

.508, .566, .959, .671, and .615 cents per hundredweight mile for

each of the eight commodity classes, respectively.

+ Nominal case. ++The new transportation rates for this case are .801, .640, .380,

.553, .548, .955, .604, and .586 cents per hundredweight mile for

each of the eight commodity classes, respectively.

Sensitivity to Level of Demand by Commodity Class

An earlier sensitivity analysis varied the level of total demand while keeping its distribution constant. However, it is reasonable to suppose that the demands for different

commodities have different growth characteristics. This sensitivity analysis, therefore, was concerned with variation of demand by commodity class. No detailed forecasting of demand

level by commodity class was attempted, such an analysis requiring resources and data far

greater than those available. Rather, a growth factor between -101 and +10% was chosen

randomly for each of the eight commodity classes.

Table 37 records the results of two such runs. (The growth factors chosen are given in

footnotes.) Both lines 1 and 3 have lower total cost without centers than the base run,

line 1 by 51 and line 3 by only 0.4%. However, the commodity mix is such that line 3 did not

have a commensurate reduction in the cost with centers, while line 1 did. (The measure of

disbenefit is reduced only if the savings with centers available is a larger fraction of the

cost with centers available, than the savings without centers available is of the total cost

without centers.) The most significant inference which can be drawn from the results in

Table 37 is that the index of performance is not very sensitive to random variations in the

level of demand by commodity class.

108
Table 37. Variation of Index of Performance with Random Perturbations

of Demand by Commodity Class*

4 Centers 5 Centers 6 Centers 7 Centers

1.** .595 .584 .589 .573

2.t .607 .588 .594 .577

3. ++ .604 .594 .599 .584

* All entries are averages over 3 cases.

** The growth factors used in this case are +.05731, -.07523, -.08977,

+.02383, -.05636, +.07106, -.03835, and -.08930 for the eight commodity

classes, respectively.

t Nominal case.

+t The growth factors used in this case are -.08188, +.05303, +.07668,

-.05893, -.02567, +.07539, +.06786, and -.04501 for the eight com- modity classes, respectively.

Sensitivity to Import Load Factors

As described earlier, our nominal value for the import load factor was taken from the

North Atlantic Container Statistics (Report for 6 -month period ending 12/31/67) . This figure

only represents container trade levels in the North Atlantic; although one might expect the

bulk of container import cargo through the ports of New York, Philadelphia, and Baltimore to result from the North Atlantic trade, this load factor figure may be quite different for

other trade areas. Also, in 1967 container trade was still somewhat of a fledgling industry,

and we have no reason to expect that load factors will remain at the 1967 level as the industry grows. For these reasons, the model's sensitivity to the value of the import load factor

was tested. The results of this analysis are summarized in Table 38. An increase of 10% in

the nominal import load factor brought about approximately a 1% drop in the index of performance, while a decrease of 10% brought an even smaller increase in the index. For this

analysis it seems clear that for a rather wide range of average import load factors (from 751

full to 90% full), the model remains relatively insensitive.

Sensitivity to Maximum Allowable Average Accumulation Delay

The dispatching policy used in the model involves holding cargo until either a full

enough container accumulates at the center or cargo has been held for too long a time. In addition to this maximum holding time, if the average time cargo of one co-containerization

class is held at a center would be greater than some threshold value, that center is

prohibited from accepting cargo of that class. In this way, the number of centers available

to a given cargo class may be reduced, so that the remaining centers have a better chance of

accumulating full enough containers within the maximum allowable holding time.

The nominal value of the threshold for the average accumulation delay is 1.5 days

.

Table 39 gives the results of varying this value. As the threshold was decreased to 1 day,

the value of the index was increased about 1%. A corresponding 1% decrease in the index was observed when the maximum allowable average accumulation delay was raised to 2 days. Thus,

allowing some extra time for cargo to accumulate seems better than forcing cargo to be sent

to a less convenient center where a higher load factor can be achieved. However, the variations in the index of performance with rather wide changes in the average accumulation delay

threshold are less than those observed for different starting configurations.

Sensitivity to Inventory Carrying Charge

In converting various time components (e.g., travel times, processing times, and accumulation delay times) to dollar equivalents for inclusion in the generalized cost function, an inventory- type charge is used. This charge represents the fraction of the value of the cargo

109
Table 38. Sensitivity to Import Load Factor, *

Import

Load Factor 4 Centers 5 Centers 6 Centers 7 Centers

.740 • .602 .591 .595 .581

.822** .607 .588 .594 .577

.904 .593 .582 .587 .572

* All entries listed here are averages over 3 cases.

** Nominal case.

Table 39. Sensitivity to Maximum Allowable Average Accumulation Delay (days)*

Delay 4 Centers 5 Centers 6 Centers 7 Centers

1.0 .606 .593 .601 .587

1.5** .607 .588 .594 .577

2.0 .592 .581 .587 .571

* All entries are averages over 3 cases.

** Nominal case.

being handled which is to be charged for each unit of time delay. Variation of the inventory

charge is equivalent to variation of the dollar values of all commodity classes by a constant

factor, since these two quantities always appear within the model as a product.

Our nominal value of the inventory charge is 10% per year, i.e., 0.00041 per day for a 250-day year. Table 40 gives the results of varying this charge by ±10%. An increase of 101

in the inventory carrying charge brings a 1 to 2% increase in the index of performance. A

decrease of 10% produces a decrease of more than 2% in the index. Again it should be noted

that the model is not greatly sensitive to this parameter.

Using a center is time-consuming since it involves both a roundabout route to the port

and the experiencing of accumulation delay. Thus an increase in the inventory carrying charge

might be expected to cause not only an increase (as observed) in the (mis-) performance index,

but also a diversion of traffic from inland centers to containerization at the ports.

Sensitivity in this regard also proved quite slight, the patronage at centers remaining

almost the same for corresponding runs of all three cases.

Table 40. Sensitivity to Inventory Carrying Charge Factor*

Value Fraction/day 4 Centers 5 Centers 6 Centers 7 Centers

.00036 .588 .574 .580 .564

.00040** .607 .588 .594 .577

.00044 .608 .598 .600 .587

* All entries are averages over 3 cases

** Nominal case.

110
Sensitivity to Parameters Converting Travel Distance to Travel Time

Equation [7.8] estimates time in transit as a linear function of distance. Meyer et al

(p. 135, Ref. 15) provide values of the parameters for the rail mode. By omitting those^elay

factors inappropriate to the truck mode, and estimating appropriate terminal delays, corre- sponding parameters were constructed for the truck mode. Equation [7.8] can be rewritten

(cf. [7.9])

t = a d +6 [9.1] m m m m

L J where tm is the time to go a distance dm by mode m. The nominal values of am and 6m for the

rail mode are given in Meyer et al as .0059 days /mi. and 2 days, respectively. The nominal

values chosen for truck were 70~0T4 days /mi. and 1 day, respectively. Shippers are charged

truck rates for uncontainerized shipments and rail rates for those in containers. The tradeoff is one of cost versus time, with truck faster and rail cheaper. The nominal values represent averages of values which may vary for different railroads and different times of the year.

Table 41 shows the variation of the index of performance with changes of ±10% in the

parameters a„, and 6m

. The values of the index are almost identical to those produced by ±10%

changes in the interest carrying charges, although these latter affect the cost of accumulation delays and center processing times, while the variations in am and 6m do not. Again it

should be noted that the variation in the index is not very large, i.e., the model is not very

sensitive to the coefficients in the travel time equation.

Table 41. Sensitivity to Factors for Converting Travel Distance to Travel Time*

4 Centers 5 Centers 6 Centers 7 Centers

Nominal Values - 10% .585 .575 .576 .564

Nominal Values .607 .588 .594 .577

Nominal Values +10% .608 .598 .602 .587

* All entries are averages over 3 cases.

Sensitivity to Co-containerization Classification

Within the model, the accumulation delay is calculated separately for each (co-containerization class, U.S. port) pair. Although there are some exceptions it was assumed in our illustrative calculations (see page 88) that most types of commodities can be loaded into the same container. On the other hand, shipments for different destinations (for our purposes,

foreign decontainerization points) cannot be put into the same container if efficiency of

unloading is desired. Therefore, for the most part, our co-containerization classification

scheme is a surrogate for the foreign destination of each shipment. This information is

unavailable from the Delaware Port Authority data; a hypothetical version was concocted by

randomly assigning a class to each shipment.

The nominal number of such classes was 20. Table 42 gives the variation in the index of

performance for (a) a different randomly assigned classification with 20 classes, and (b) also

for one with 15 classes. The new set of 20 classes produces a rise of less than 1% in the

index. The 15 classes reduce the index by 1 to 2%. (This is the appropriate direction of

change; classifying the same amount of cargo into fewer co-containerization classes should

reduce the accumulation delay since there should be more cargo available in any one class.)

The model therefore appears not very sensitive either to the classification scheme or to the

number of classes.

Ill
Table 42. Sensitivity to Co-Containerization Classification*

Classification 4 Centers 5 Centers 6 Centers 7 Centers

15 classes .593 .580 .583 .567

Nominal (20 classes) .607 .588 .594 .577

20 classes .604 .594 .598 .583

| Sensit

c

fixed

tente

* All entries are averages over 3 cases.

Sensitivity to Processing Times and Costs

Chapter III provides nominal values of .25 days and $94.28 to process a 40'x8'x8' container. Since all cargo entering the model is eventually containerized and these processing

times and costs are taken independent of the center at which the cargo is containerized,

these cost elements need not enter into the optimization process. However, they do affect

the index of performance and are added into both numerator and denominator of that index at the end of the model run. The formula for the index of performance thus becomes

(Ti + T

p

)/(T2 + T

p

)

where T 2 is the total variable cost with centers available, T2 is the corresponding cost

when centers are not available and T is the processing (generalized) cost for stuffing con- tainers . "

Table 43 records the results of varying the cost to process a container by ±10% and

increasing the time to process a container by a factor of 2 (from 3 hours to 6) . The variation in the cost of processing a container affected the index more than the change in time

did. On the other hand the total range of index values changes only about 1% , which indicates

that the model is quite insensitive to these parameters over the range under consideration.

Table 43. Sensitivity to Processing Times (Days) and Costs ($)*

Time Cost 4 Centers 5 Centers 6 Centers 7 Centers

.125 84.852 .604 .585 .591 .573

.250 84.852 .606 .587 .592 .575

.125** 94.280 .607 .588 .594 .577

.250 94.280 .609 .590 .595 .578

.125 103.708 .610 .591 .597 .580

.250 103.708 .612 .593 .599 .581

* All entries are averages over 3 cases

** Nominal case.

112
Sensitivity to Total Patronage Constraint

Since the center costing analysis described in Chapter III does not separate out the

fixed cost element from the variable cost element of centers , we were unable to exercise the

center sizing portion of the model as described on pages 70 and 71. As a result, for each

number n of centers there is a set of center locations which under our present computer program will reduce or at least not increase the index of performance beyond the best set of

n-1 locations, so that we are presently unable to base the choice of the best number of centers on minimum cost alone. As a reasonable alternative, we can require each center to process at least a given minimum amount of cargo. The number of centers is thus limited by

the number which can process this minimum cargo level.

Recall that the logic is as follows: centers are located by the alternation of assignment and location steps as described previously. When this process has settled down, the

final set of centers is checked to see if each obeys the minimum patronage restriction. The

locations of any centers which fail this test are randomly reinitialized, and the assignmentlocation process is started again. This is continued up to a specified number of tries or

until all centers obey the restriction. Such a procedure may be described as testing several

different initial configurations, tentatively retaining at each point those center locations

from the last stage that worked out well.

Tables 44 and 45 record the center locations resulting from runs changing the nominal

constraint of 10 containers/day to 8 and 12, respectively. It should be noted that only one of the six-center configurations and none of the seven-center ones processed enough cargo at all centers, and this single six-center configuration worked for a constraint of only

8 containers/day. However, further inspection of the two tables suggests the best configurations for 6 and 7 centers respectively as: Cleveland, Chicago, Southwestern Ohio, Greater

Pittsburgh, Western New York, and Detroit for 6 centers, with Milwaukee added when 7 centers

are being located.

Table 44. Variations in Rough Locations of Centers (6 Centers)

with Minimum Total Patronage Constraint

Area Minimum Patronage = 8 Minimum Patronage = 10* Minimum Patronage = 12

1 1 1

1 111

E. Pennsylvania

(incl. Philadelphia)

Western New York 1 1

Greater Pittsburgh 1 1

N.E. Ohio

Cleveland 1 1 1

Detroit 1

S.W. Ohio 1 1 1

N.W. Ohio

1**

South Central Michigan

Central Indiana

Chicago 1

Milwaukee

Other 1**

* Nominal case

**Not enough total patronage

1

1**

1 1

2** 2** 1

i ** 111

1** ]_i

113
Area

Table 45. Variations in Rough Locations of Centers (7 Centers)

with Minimum Total Patronage Constraint

Minimum Patronage = 8 Minimum Patronage = 10* Minimum Patronage = 12

E. Pennsylvania

(incl. Philadelphia)

Western New York 1 1

Greater Pittsburgh 1 1 1

N.E. Ohio

Cleveland 1 1 1

Detroit 1 1

S.W. Ohio 1 1

N.W. Ohio

South Central Michigan

Central Indiana 1 1

Chicago 1 1

Milwaukee 1 1

Other 1 1 1 1,1

* Nominal case,

Table 46 gives the indices of performance for this set of runs. No real conclusion as to dependence of index on level of patronage constraint can be drawn from this table.

Although the original initial locations of centers are the same for each of the three rows,

the process of reinitializing "bad" centers in effect tries several initial configurations.

As noted earlier, the model is more sensitive to initial configuration than to any other

single factor. The variations listed in Table 46 are as much a reflection of different

"initial" sets of center locations as they are of the patronage constraint. Most locations

which fail to attract enough patronage process only about 2 or 3 containers a day. In only

a very few cases does one location satisfy a constraint of 8 containers but not of 12. This

is a consequence of the particular shipment distribution with which we work.

Table 46. Sensitivity to Minimum Total Patronage Constraint*

Constraint 6 Centers 7 Centers

8 containers/day .586 .584

10 containers/day** .594 .577

12 containers/day .582 .583

* All entries are averages over 3 cases.

** Nominal case.

114
Only Port Centers

In addition to the sensitivity runs just discussed, one run was made allowing centers

only in the port areas. In this run, shippers have a choice between shipping directly to their own port, or consolidating at one of the other port centers and only then shipping on

to their final U.S. port of debarkation. This run was made to investigate the value of

inland versus port centers. The index of performance was .951, which when compared with

the values of less than .63 for inland centers, indicates that the greater savings come from

the inland centers, rather than just from freedom to consolidate at ports other than that of

egress from or entry to the United States.

Concluding Remarks

The exercises with the model which were described above, serve to illustrate the ways in

which it can be used. They provide only a sample of the types of analyses which should be

done in using the model described here. Clearly more initial configurations should be tested,

since the model seems most sensitive to this factor. All work should be done with the minimum

total patronage restriction in force. For the Delaware River Port Authority data, the process

described in Appendix G should be used to provide origin-destination input (U.S. shipper

linked with specific foreign port) . Sensitivity to changes in combinations of factors could

also be checked in a more complete analysis.

Present program output could be expanded to include any time or cost components for any

shipper or for groups of shippers, aggregated by port, center, or commodity class shipped.

These items are all computed within the model and it could be reprogrammed with a minimum of

effort to print them out. In addition, the inclusion in the model of a capability to output

a map showing center locations would facilitate quick analysis and encourage more general

usage.

The exercises performed so far have focused attention on some of the items which should

be included in future data collection efforts and which are lacking in the Delaware River

Port Authority data. The need for knowledge of the origin and destination of each shipment

has been stressed above. The Delaware data contain only a general yearly description of

shipments. More specific information on the distribution over the year of shipments (including

such things as seasonal variations), and more information on the variation of shipment size, are also needed to permit realistic calculation of actual accumulation delays. Further work

is necessary on the definition of which commodities are co-containerizable.

For a more realistic evaluation of the benefits of inland consolidation centers , and a better estimate as to their appropriate size and number, it would be necessary to predict

future foreign cargo demands. In order to do this, data on past export and import demand

trends are needed. Moreover, the present model only considers LCL shipments. It is

necessary to separate out the costs and benefits of the marshalling function before full

container- load shipments can be included in the analysis. Clearly such shipments will

affect the size and possibly the location of centers. It is also necessary to separate out

the variable and fixed cost portions of center processing costs, as given in equation [7.52], so that the processing cost (including cost of processing time) is not just a constant rate

per pound of cargo processed but depends on the total amount being processed. This would

make possible exercising the logic described in Chapter VII for optimal sizing of centers.

With these developmental features and studies added to the basic methodology developed

in this report, the model should provide a powerful tool for guiding and executing analytical

investigations and data-collection efforts aimed at determining favorable configurations for

a system of inland consolidation centers.

115

APPENDIX A. OPTIMUM LOCATION OF CENTERS IN A NETWORK

The problem to be considered here is that of locating centers in a network so as to minimize the sum of the associated transportation costs to users. It is assumed that the

center location-dependent portion of each user's cost (combining dollar and time components),

which is the only portion that figures in this problem, can be expressed- -as in the main text

--as a multiple of user-center distance plus a multiple of center -port distance.

We begin with the case of a single center (n=l) , the situation of concern in the location step of the mathematical model described in Chapter VII. Here the problem is that of

choosing the center location C so as to minimize a function of the form

f(C) = I.wid(X

i ,Q, [A.l]

where the X-^ are the locations in the network of shippers (or importers) and ports, and thew^

are appropriate positive numerical "weights" reflecting shipment volumes and freight rates. The main novelty is that "d" now represents distance via a shortest path in the network.

If some X^ is an interior point of a network link with endpoints e' and e", we can con- ceptually split the link into two links, one with e' and X-; as endpoints and the other -with

Xi and e" as endpoints. Since this process can be repeated so long as there are any X^'s

which are interior points (rather than endpoints) of links, it can for convenience be assumed

in advance that every X- is a vertex (node) of the network.

The main known theoretical result on this optimal location problem is due to Hakimi, 25 who showed that the search for optimal locations for C can be confined to the vertices of the

network. (The same conclusion persists for cost functions more general than [A-l]

,

26 namely,

those such that cost per unit distance along each link is a non- increasing function of distance traveled.

)

This result justifies the following 3-step method for finding the optimal location C. The method is unpleasantly brute-force, but appears computationally feasible for problems of

reasonable size.

STEP 1. For each vertex v^ of the network, set up a "location" in computer memory and

initialize the value uj of that location's contents at zero.

STEP 2. For each X^ in turn: Determine d(X-,Vj) for all vertices v,- . (This is not done

vertex-by-vertex, but efficiently at once by a "labeling" shortest path algorithm. For a network with about 1000 nodes and twice as many links, this would take 1-2 seconds per X. on

a fast machine.)J Add w.d(X.,v.) to the current value of u-.

l

v i- y j STEP 3. When Step 2 is complete, find the smallest u^ . Choose C to be the corresponding vy

The proof of Hakimi' s result is brief enough to warrant presentation here. The generic

symbol for a vertex of the network will be "v", and [A.l] is rewritten as

f(C) = Iwy

d(v,C) [A. 2] where w = if vertex v is not an X.

.

v 1 Consider any point x of the network. It lies on some link, with certain vertices v and v as endpoints, so that r

d(v

r

,x) + d(x,v

s

) = d(v

r

,v

s

). [A.3]

25 S. L. Hakimi, Optimum Location of Switching Centers and the Absolute Centers and Medians of

a Graph, Operations Research, 12 (1964), pp. 450-459.

26A. J. Goldman and P. R. Meyers, A Domination Theorem for Optimal Locations, Operations

Research 13 (1965), p. B-147 (Abstract).

27 C. Witzgall, On Labelling Algorithms for Determining Shortest Paths in Networks, NBS Report

9840 (5768).

117
The vertices v of the network can be divided into 2 classes; the set V(r) of vertices v such

that a shortest path between x and v passes through vr , and those (not in V(r)) such that a shortest path between x and v passes through v . It follows that

d(v,x) = d(v,v

r

) + d(v

r

,x) (v in V(r)),

d(v,x) = d(v,v

g

) + d(v

s ,x) (v in V(s))

Hence

f(x) =

l y w

vd(v,x)

= ^V(r) w

v

[d(v ' v

r

} + d ( v

r

> x)] + £v(s)

W

v

[d(v 'V + d Os

' x)] '

If [A. 3] is used to substitute for d(v ,x), the result is

f(x) = Iv(f) w

v^ dCv, V + d(Vx^

+ ^V(s)

W

v

[d(v,V

s

) + d(vs'V " d Or

> x)]-

It follows from the definition of "d" as distance via a shortest path, that

d(v,v

s

) + d(v

s ,vr

) >_d(v,v

r

).

Thus the last formula yields

£ W 1 IV (r) Vd(v 'V + d ( v

r' x)] + ^V(s) Vd(v 'V " d(^ v

r'x^

= y, r

, . w d(v,v j + J",,, -, w dfv.v ) + [ ], rr sW - J. rr .w ] dfv ,x) LV(r) v

* ' r

J LV(s) vv

' r

J L ^V(r) v ^V(s) v

J * r' J

= f(v ) + [ }",,, >. w - ), T , .w ] dfv ,x). i. t

j l i-y^-f) v i-v(s) v

J v r' '

It can be assumed without loss of generality that

**V(r) v — ^V(s) v

This and the preceding inequality together imply f(x) >_ f (v ). That is, there is a vertex

(Vr) which is at least as good a location for the center as x. But x was an arbitrary

point of the network. So the search for optimal locations can be confined to vertices, and

the proof is complete.

Because shipment volumes (and/or freight rates) are not perfectly known or predictable,

the generalized problem in which the w 's are random variables is of interest. Frank28 treated the case in which they are independent random variables. This, however, is not the case for our situation; the w corresponding to a particular U.S. port of departure is

determined by the w 's corresponding to those shippers using that port. In a later paper29

Frank considered the case in which the w 's are correlated multinormally-distributed random

variables, showing in particular how to reduce, to the numerical evaluation of a multiple

integral of multivariate normal type, the determination of a point C in the network which

for a fixed number R maximizes

Prob {f(C) <_ R}.

(In general such a C cannot be taken to be a vertex.) It is not clear how much error is

introduced because the normality assumption permits negative values for the w 's.

Next consider the problem of locating n centers, where n> 1. The natural generalization

of [A. 2] in this direction is

f(d, ..., Cn

) = luv min.d(v,C.), [A.4]

28H. Frank, Optimum Locations on a Graph with Probabilistic Demands, Operations Research 1£

(1966), pp. 409-421.

29H. Frank, Optimum Locations on Graphs with Correlated Normal Demands, Operations Research

14 (1966), pp. 552-557.

118
corresponding to the idea that each vertex v is assigned to (or chooses) a center nearest

him. Hakimi^°has shown that his theorem for n = 1 generalizes to this case, i.e., in minimizing £ it can be assumed that each C- is at a vertex. Computational methods have been

explored by Singer31 (only the abstract-'of this paper has been published).

The proof that attention can be confined to vertex locations runs as follows. Consider

any points x1

, . . . , x of the network. Let V be the set of all vertices, and V(i) the set of all vertices v for which

d(v^c : ) = min. d(v, x,).

It follows, from [A. 4], that

f(xlf ..., xn) =

Iv(l)w

v

d(v,Xl ) + [V-V(i)W

v

min

j> ld(V,X

j

]

f '(xi) +

IV-V(i) w

v m^j>1 dCv,x

j

),

where f ' is the objective function for the 1- center problem with weights

w' = w

y

if v is in V(i),

w' = otherwise.

By the previous result for the 1-center case, there is a vertex vi such that f'.(vi) <_ f ' (x{) , so that

f(vi, x2 , ... , x

n

) = lyr^ \ min [d(v,vi), min.

1 d(v, x.)]

+ ^V - V(i) w

v min ^ d Cv > vi)» min

j>!

d (v> xj)]

< f'(vi) + J,, , jr . w min. d(v,x.)

1 f(xi, ..., x

n

).

That is, xx can be replaced in (xi, . .

. , x ) by some vertex vi . In the same way it can be

shown that x2 in (v1} X2, ..., x ) can be replaced by some vertex v2 and so on, completing

the proof.

Unfortunately, the applications arising in the present study involve a more complicated

function than [A. 4] . This is because a vertex might represent a U.S. port of departure, or

the locus of several shippers perhaps using different U.S. ports, or possibly even both.

To represent this, replace the simple "weights" w by weights

W CV>P)

= "weight" associated with shipment from v due to leave CONUS* via port p, w*(v,p) = "weight" associated with same shipment in its center- to -port movement.

Then [A. 4] is replaced by

f(d, ..., Cn

) =

IVjp mui-j [w(v,p) dfv.Cj) +w*(v,p) dCC^p)]. [A.5]

It will now be shown that Hakimi's result for [A. 4] remains true for [A.5], i.e., that

in minimizing f as given by [A.5], attention can be confined to vertex locations for centers.

As above, it suffices to show that for any points xi , . .

. , x of the network, there is a vertex vi such that

30S. L. Hakimi, Optimal Distribution of Switching Centers in a Communication Network and Some

Related Graph Theoretic Problems, Operations Research 13_ (1965), pp. 462-475.

31 S. Singer, Multi-Centers and Multi-Medians of a Graph, with an Application to Optimal Warehouse Location, Operations Research 16^ (1968), p. B- 87 (Abstract).

* Continental U. S.

119
f Ol , X2 ,

. . . , X

n

) < f (Xj , . .

. , xn

)

Let V now denote the set of all ordered pairs (v,p), where v ranges over all vertices

and p ranges over all vertices representing ports. Let V(i) now denote the subset of V con- sisting of those pairs (v,p) for which

w(v,p) d(v,xj) + w*(v,p) d(x!,p) = min. [w(v,p)d(v,x.) + w*(v,p)d(x. ,p) ]

J > J *

[t follows that id

f(x 1} ..., x) = Lfn [w(v,p)d(v,Xi) + w*(v,p)d(x!,p)] nu tei + ly-V(i) min

j >

! C w Cv , p) d Cv ,x^ ) + w*(v,p)d(Xj,p)]. [A.6]

As in the previous proof, it suffices to show that the first sum in [A.6] can be written

f ' (Xi), where

I f(x) =

£ v

w' vd(v,x)

is the objective function of a 1-center problem with suitable weights w' . This however is

readily done by setting* w

' v

=

£ {w(v ' p): (v,p) inV(i>}+ I(w*(u,v).: (u,v) in V(i)}.

It would be of evident interest to find an effective computational method for determining vertex locations to minimize the function f of [A. 5]. The priorities of the present

study, however, ruled out undertaking a research effort in this direction.

*A sum over the empty set is taken to be zero.

120
APPENDIX B. MINIMIZATION of TOTAL COST

This Appendix contains a brief discussion of the situation in which some single decisionmaker, based on cost minimization, could impose an assignment of shippers to centers.

There is a fairly extensive literature dealing with a simpler situation involving not

three types of points --origins (shippers), intermediate points (centers), and U.S. ports

(destinations) --but only two, namely, supply points (origins) and markets (destinations).

Each destination has a prescribed demand, and the problem is that of locating the supply

points, thought of as production facilities ("plants'), so as to meet the demands at minimum

cost. Both production and transportation costs are involved, but since supply points (centers) are assumed to be known in advance, this situation (Hitchcock- Koopman's model) does

not apply to the problem on hand.

The paper most akin to the approach in the body of the present report is that of Cooper, 32 who also employs an alternating sequence of assignment and location steps, with the KuhnKuenne iterative method used for the latter. (Hartley 33 has proposed a similar approach to center location.) Balinski 31

* gives a mixed- integer linear programming formulation:

Minimize 5". .ex.. + T-f-ysubject to ^Xjj = 1 ( a11 J)»

ix

ij lYi I 1 (all i»j)»

y- . = or 1.

Here f- is the fixed cost associated with using plant location i, variable x^- is the fraction of market j's demand (D-) which is to be supplied from the i-th location, while the cjj

represent the appropriate transportation and (variable) production costs. While Balinski

proposes a "partitioning" mixed- integer algorithm for this problem, Efroymson and Ray35 develop what appears to be a more promising approach from a computational viewpoint, using a "branch and bound" technique together with several ingenious simplifications. The paper of

Sharp et al 36 illustrates recent work featuring nonlinear production costs rather than the

sharp dTscontinuity of a fixed cost. Further references can be found in the bibliographies

of the cited papers.

One might hope to adapt the "suppliers and markets" model to the problem of locating con- solidation centers. There are three difficulties, however. First, the model above presupposes knowledge of a finite and manageably small set of "allowed" locations for plants. This

is not the way locations for centers are treated in the present study, though a change on this

point would seem reasonable if a decisively better solution method would result. Second, the

number of shippers and importers in the present study puts the problem well beyond the size

range of the cases treated in the papers cited. Third, there is the matter of twisting the

models around to make them represent the presence of two classes of customers (shippers/

importers, and ports), with each member of the first class associated to a definite member

of the second. This however turns out to give no real trouble, as will be seen below.

32 L. Cooper, Location -Allocation Problems, Operations Research 11 (1963) , pp. 331-343. Also

Cooper, Heuristic Methods for Location-Allocation Problems, SIAM Review 6_ (1964), pp. 37-53;

Solutions of Generalized Locational Equilibrium Models, J.Reg. Sci. 1_ (1967), pp. 1-18.

33H. 0. Hartley, Optimisation of the Location of Serving Centers, 7/68, manuscript communicated by the author.

3kM. Balinski, Integer Programming: Methods, Uses, Computations, Management Sci. 12 (1965),

pp. 253-313.

35M. A. Efroymson and T. L. Ray, A Branch-Bound Algorithm for Plant Location, Operations

Research 14 (1966) , 361-368.

36J. F. Sharp, J. C. Snyder, and J. H. Green, An Algorithm for Solving the Multi-facility

Production-Transportation Problem with Nonlinear Production Costs, Operations Research 16_ (1968), p. B-87 (Abstract).

121
giver

We conclude by presenting a mixed- integer linear programming formulation, patterned

after that of Balinski, but including all three types of points (shippers/importers, ports,

and centers) and explicitly including the problem of sizing the centers. It assumes given a manageably small set of possible center locations, and the possible sizes (levels of capacity)

for each.

The model's data are as follows:

b., = fixed charge for operating capacity increment k of center at location j,

v., = unit variable cost for such operation,

t. . = unit transport cost for use by shipper i of center at location j (including

1

-' center-to-port cost),

a., = size of k-th capacity increment at j,

s. = "supply" total to be sent by the i-th shipper.

The model's discrete variables are the

d., = 1 if k-th increment at j is operated, otherwise,

while its continuous non-negative variables are the

x. ., = amount sent by i-th shipper to center at j -th location, attributed to k-th incre1JK ment.

The objective function to be minimized is then

^jk jk jk ^ijk^ jk iy ijk'

The supply balance equations read

Ljkx

ijk =si (alii),

while the capacity constraints can be expressed as

V .x. ., < a., d., (all j ,k)

L i ljk — jkjk ^ •" ' The requirement that d. , , = unless the k-th increment at j is used to capacity, can be

written as a linear constraint

ft

th

I :a

'i

ti

ajkdj,k+ l^ix

ijk ( a11 J> k> What still would require investigation is the computational feasibility of this model, in

view of the large number of shippers involved.

122
APPENDIX C: THE CONSTRAINED LOCATION STEP

The main text gave an iterative method, specified by formulas [7.27] and [7.28], for

finding the location C of a center so as to minimize a total (variable) transportation cost

given by

f(C) = Iw,d(X.,C). [C.l]

" represents Euclid

ecting shipment volumes

Here the X- are the locations of shippers (or importers) and ports, "d" represents Euclidean

distance, and the w. are appropriate positive numerical "weights" refle

and freight rates.

The text also noted the desirability of being able to solve a constrained version of

this problem, in which C is restricted to be within a prescribed convex polygon R. This is

the problem to be treated here. The solution method will be informally described for the case of a general polygon R, to the point where its computational feasibility should be

apparent. More detail will be given for the special case--R a rectangle with sides parallel

to the axes--for which computer implementation has actually been carried out.

Let V, , ... , V, be the vertices of R, in (say) counter-clockwise order around its

perimeter. The polygon R is assumed to be "given" by listing (in order) the coordinates of

these vertices V-. Let S. denote the side joining V. and V. , (where V, , signifies V, )

.

The equation of the line carrying S. can be found, say in the form

a.x + b-y = c

J Y 3 by standard analytic-geometry techniques. Substitute the coordinates of any vertex other

than V- and V. -, into a.x + b.y: if the result is > c, replace (a., b., c) by their negatives. With this accomplished, R itself can be characterized as the set of points (x,y) for

which

a.x + bjY £ Cj (j= 1, 2, ..., k). [C.2]

The process for solving the constrained location problem begins by using the method,

given in the main text, for solving the corresponding unconstrained problem. Let C*=(x*,y*)

be the resultant location. If (x*,y*) satisfies the k conditions [C.2], then C* lies in R

and the constrained problem is also solved.

Suppose this is not the case. Then those j's, for which [C.2] are violated, correspond

to those sides S. of R which are "visible" (in an obvious sense) from C*. And it is known37 that the optimal-' location within R must occur on one of these sides. Thus it suffices to find the f-minimizing location C- along each of the sides S. visible from C*, and to choose

as C that C for which f(C) is smallest.* 3

3 3 The solution of each of the one- dimensional optimization problems, i.e., the minimization

of f(C) along a side S. of R, can be carried out (for example) as follows. Perform a transformation of coordinates to make S. lie along the x-axis; this will yield new coordinates

for the X.'s. Now use only the first equation [7.27] of the iterative process, with yC t )=0, to minimize f(C) along the (new) x-axis. If the minimizing point is in the segment S.,

choose it as C.; if it lies outside S-, choose the endpoint of S. closer to it as C. Then

3

3

v

j j reverse the coordinate transformation to find the "true" (i.e., original) coordinates of C.

Assume now, in particular, that R is the set of points (x,y) satisfying

x £x£x , y £ y .S y ,

i.e., a rectangle with sides parallel to the axes. Then the characterization [C.2] of R can be written out explicitly as a set of four conditions:

37A. J. Goldman, A Theorem on Convex Programming, paper delivered to the Mathematical Association of America, 1963 Annapolis Meeting.

* More elegant methods may be possible, but this one will surely do.

123
+ X <_ X ,

y iy

+

>

-x <_ -x ,

-y <-y".

If the solution C* = (x*,y*) of the unconstrained problem lies outside R, then it can violate at most two of the above conditions (e.g., either the first or the third must hold).

Thus at most two sides of R, involving at most three vertices, can be visible from C*. No

transformation of coordinates is required; the minimization of f (C) along a horizontal side

of R can be carried out using [7. 27] --with yiX) set at whichever of y~ or y

+ applies --while

the minimization along a vertical side can be done using [7-28] • If desired the appropriate

partial derivative could first be evaluated at the two endpoints of a side, so as to identify

an endpoint optimum at once.

pri1

124
APPENDIX D: EVALUATION OF MEAN ACCUMULATION DELAY

This appendix contains an evaluation of the mean accumulation delay, t a , under a dispatching policy of the type (L,H) formulated in Chapter VII. As in the main text's discussion

prior to [7.38], the actual situation is approximated by Poisson arrivals (with parameter

*• = -O °f shipments of uniform size a. Moreover, m denotes the largest integer such that

ma~<L.

Let v+l (a random variable) denote the number of shipments in a dispatched container,

and w the average waiting time of those shipments. Then the quantity to be evaluated is m

t = E(w) =

I E(w|v= N) Prob (v=N). [D.l] a N=0

Suppose first that N = m (i.e., the container has a "normal" payload (m+l)a). Let z. be the period between the arrival of the i-th shipment entering the container, and the next

arrival after that, for l<i<m. Since the (m+l)-st shipment suffers no accumulation delay,

the total accumulation delay for all shipments in the container is I

, iz., so that the mean delay per shipment is m W= (I iz.)/(m+l). [D.2]

1

1

Thus the summand in [D.l] corresponding to N=m is given by the multiple integral

-, m m

(m+1)"

1

/ $ izp exp(-A£ z ±

) Xm ^ ... dzm

[D.3]

where the integration is over the region in (z, , ... , z ) - space defined by m

all z- > , I

z. < H. [D.4]

The change of variable y. = Xz- converts this to

,

i m m

X'^m+l)" 1

/ (I i7i) exp (-1 y±

) dYl ... dym

, [D.5]

with the integration over m

all y. > , I y. <_ H = HA. [D.6]

1

1

1

Before proceeding with the main derivation, it will be useful to evaluate certain

special integrals over the m-dimensional region described by [D.6]. In the first of these,

the integrand is 1, so that the integral is just the (hyper-) volume of the region.

Multi-dimensional analytical geometry gives

Vm

= / 1 dy

x

... dym

= H m

/m! [D.7]

as the value of this volume.

The second special integral is m

Em

= |exp [-1 y±

) dyj ... dym

. [D.8]

To evaluate it, note first that

125
H

E

1 =

J o exp(-y) dy = 1 - exp(-H). [D.9]

Next, for m>l perform the y -integration in [D.8] first, obtaining

m-1 m-1

Em

= /{l - exp (-H +

I y.)} exp (- £ y/)dy

1

... dy^,

where the integration is over the (m-1) -dimensional analog of [D.6] . This yields

E = E , - exp(-H)V ,

m m-1 FV J m-1

From this recursion (in m) , the initial condition [D.9], and the evaluation [D.7] of the V

integrals, we obtain

E = 1 - p -(H)" [D.10]

where p , (H) is as in [7.38] of the main text.

Mil

The third special integral is

L = fy dy, ... dy . [D.ll] m •''m 7 1

J m

L J

To evaluate it, perform the y -integration first. The result is

m-1

Lm

= d/2) /(H-X

i

y.)^ djn ... dym _ r

where the region of integration is the (m-1) -dimensional analog of [D.6]. Now perform the

y -.-integration; its result is

m-2

Lm

= (1/6)/ (H- ^

y.)3 dy,... dym _ 2

where the integration is over the (m-2) -dimensional analog of [D.6]. From these two steps,

the pattern of results from the remaining successive integrations is evident; the final step

will be

H _

Lm

= (l/m!)J (H-yi ) m

dyi

°

[D.12]

=H m+1

/(m+ l)!.

We return now to the evaluation of [D.5], the term of [D.l] corresponding to N-m. From

[D.5] , this term is

A'-Vl)"1

I

i jYi exp (- I y.)dy

1 ... dym

. [D.13]

The integral in [D.13] is independent of i, so [D.13] becomes

-1 m

\\ m Jym

exp (-£ y±

) d

7;L ... dym

, [U.14]

126
and performance of the y -integration yields (if m>l)

, m-1 _ m-1 m-1

fcA~"Sn /[l - (exp(-H +

I yi ))(l + H - I y.)] exp(- I y.) dy

x ... dym_ 1

,

where the integration is over the (m-1) -dimensional analog of [D.6]. Simplifying, we obtain

(for m>l)

-, m-1

^A"

1m[Em _ 1

- exp(-H)-(l+H)Vm_ 1 + expf-H)/^ yi)dy

1 ... dy^] . [D.15]

The integral in [D.15] is in turn reducible to (m-l)L -, , and so the preceding evaluations

for the E, V and L integrals yield

^""^[l-p.tH) - exp(-H) [(l+FTjH^Voii-l)! -

(m-l)if/m!]] m-2

-1 A'Vl-p^OT) - exp(-H)Ff/m!]

-^'^[I-PjjjOO]. [D.16]

This formula can also be verified (starting at [D.14]) to hold when m=l.

Next consider the N-th summand in [D.l] for some N<m, corresponding to a dispatched

container containing N+l shipments. For lfijvN, let z. be the period between the arrival of

the i-th and (i+l)-st of these shipments, and let zN+, be the period between the arrival of

the last of these shipments and the first shipment- arrival after the dispatch. The

total accumulation delay for all shipments in the dispatch is

N N

I iz + (N+l) (E-l z ),

1

1

1 1

so that the mean delay per shipment is

-. N N

w= (N+l)"

1 I iz + H - I z..

1

x

1 x

Thus the N-th summand in [D.l] is

i N N N+1 N+1

/[(N+l)" 1 I iz. + H - ^z.] A N 1 expd-xl zj dz

x ... dzN+1 ,

where the integration is over the region in (z-. , ... , zN+1 )

- space defined by

N N+1

N l all z. ^ 0, £ z. <_H < £ z..

l xl

x

l

The change of variable y. = Ax. converts this to

1

, N N N+1

A"

1 /[(N+1)' 1

I iy. + H - I y.] exp (-£ y.) dy

1 ... dyN+r

N N

As before, the terms £ iy. and £ y. can be replaced, respectively, by N(N+l)yM

/2 and NyN , yielding 1 1

1 N+1

A"

1 /(H-NyN

/2) exp (- ^ y.) dy

x ... dyN+r

First perform the yN

, -integration, which has range

>w H - ly±-

127
The result is

•1 i X /(H - Ny

N

/2) exp (-H + ^ y.) exp (- tp±

) dy

x ... dyN

,

where the integral is over the N-dimensional analog of [D.6]. This immediately simplifies to

-1 T-'expC-H) /(IT - Ny

N

/2) dy

x ... dyN

= A~

x exp(-H) [HVN

- NL

N

/2]

= A _1

exp(-H) [H

N+1

/N! - (N/2)H

N+1 /(N+1)!]

^A"1ejq>(-H)[rf

W'/N! + H

N+1

/(N+1) !]

From [D.16] and [D.17]

,

t =U _im[l-pm

(H)] +

15 A"

1 exp(-H)[H I FT/N! + I FT V(N+1)!]

-v,-l

kX

x [m(l-pm

(H)) + Hp^^H) + pm(H) - 1]

[D.17]

as a

lisp

1

flit

tai

it

or finally

t a =U

-1

[Cm-l)(l-pm

(H)) + Hp^^H); [D.18]

128
APPENDIX E: ALTERNATIVE DISPATCHING POLICIES

The main text proposed a particular type of dispatching policy, called an (L,H) policy,

as a reasonable and tractable approximation to what might be expected. Here a container is

dispatched whenever its load factor has reached the target level L, or some of its cargo has

waited for the maximum holdover time H, whichever comes first.

Suppose however that very soon after the arrival of the first increment for some container, the target level L has already been reached. There would clearly be good economic reason to retain the container in hopes of soon filling it further, rather than dispatching

it immediately. How might the (L,H) type of dispatching policy be altered to include such

cons iderat ions ?

Perhaps the simplest approach is to adjoin a minimum holding time h, with h<H, to the

parameters describing the policy: The container would not be sent out until its firstarrived increment had suffered a wait of at least h, unless of course an as-full-as-possible

container -load were achieved prior to that. If M is the largest integer for which Ma~<l (so

that M>m, with equality only if (m+l)a>i), then a container "as full as possible" of payload

is of course one loaded with M of the (idealized) shipments of uniform size a". A value

h=(l day) does not seem inconsistent with the information gathered during the Project, though

a larger value than this would appear questionable.

The derivations, for an (L,H) policy, to obtain the average realized load factor Lc (in

Chapter VII, see [7.3 SJ). the inflation ratio T c (in Appendix F) , and the average accumulation

delay ta(in Appendix D) , can be adapted to yield the same quantities for an (L,H,h) policy.

One critical change is that the number v+1 of size a~ shipments in a container -load, which

previously could be at most m+l(m, if (m+l)a>l), can now be as much as M. For m<N<M-l (where

(m+l)a~<l) , we have the additional non-zero probabilities

ProMv= N} = expt-ITlh^/N! (K = A h)

,

as well as

Prob {v=M-l} = l-pj^Qi).

This yields (for (m+l)a<l)

L~c

= a£

N

(N+l) Prob {v=N}

m-1 _.. m-1 _N

= afexp(-H) I (N+1)FT/N! + (m+1) exp (-h) [IT/m! + I (F7N!)[l-p ^(TT-h)]]

o o M-2

+ exp(-E)£ (N+1)F7N! + M[l-p -(H)]};

m+1 m' L

I = (l/i)L (N+l)"

1 Prob {v=N}

(1/a) {exp(-H) I (N+l)"

1 Jf/N! m" 1 -If*.

o

+ (m+l)" i exp(-h)[h%! +

I (h

N/N!)[l-pm_ 1

_ N

(H-h)]]

o M-2 , ,, , + exp(-E) I (N+D'V/N! +M" i [l-p (h)]}.

m+1

These expressions can readily be simplified as for the (L,H) policy; for example

L

c

= i-{(l+H)pm_ 2 (H) + M[l-pm. 2 (h)]

+ H [^.3 (H) - pm . 2 (h)] + pm _ 2 (h) + m P^Ch)} .

129
A generalization is a policy represented by a decreasing function of the waiting time w such that a container is dispatched as soon as it is filled to a fraction f(w) after a wait of w.

A more sophisticated approach might be based upon the following concept. Let IV denote

the cost of delay (in dollars per unit quantity per unit time) for the material already

accumulated for a container, and let A be the amount (a fraction of a containerful) of this

material. The cost of waiting an additional time t would be IVAt. If N additional shipments for the container arrive in that time--an event with probability exp(-A t) (A t)^/N!--

and A + Na <_ 1, then the center-to-port transportation cost for the original material is

reduced from R (the rate for a container*) to RA/(A + Na~) . Thus the expected saving for the

original cargo is

[N

{R - RA/(A + Na)} exp(-A

c

t) (A ct)

N

/N! - IVAt

= Ra exp(-A

c

t) IN [N/(A + NT)] (A c

t) N/N! - IVAt.

The general idea is to choose t so as to maximize this quantity.

This last notion has been recorded for completeness of documentation, but has not been

examined to the point where the advisability of ultimately including some version of it in

the computer program can be regarded as established.

* A more careful treatment would take the LCL container rate schedule into account.

130
APPENDIX F: RIGOROUS TREATMENT OF INFLATION RATIO

In [7. 35] of the main text, the equation

Ic= 1/L

C

[F.l]

was proposed for the inflation ratio to be used in converting from incoming quantities (of

cargo in the c-th co- containerizable class, measured in containerfuls) to outgoing quantities

(measured 1 in containers).J Here I" is the mean load factor for class s. c This is not quite correct; I c should be_the mean value of the reciprocal of the random

variable L (the load factor) whose_ mean is L . The discrepancy arises from the fact that

the reciprocal of a mean (here, 1/LC) is only Dy accident equal to the mean of the corresponding reciprocal (here, of 1/LC )

, otherwise underestimating the latter.*

As in the material around [7.38] and [7.39] in the main text, the notations

a = shipment size

m = largest integer with ma<L

will be employed. The number v+1, of shipments which make up the payload of an outgoing

container, is then a random variable with values between 1 and m+1 inclusive. If a is

measured in containerfuls, then

L

c

= (v+l)I ,

and so

1/L

c

= (1/IXv+l)"1

.

Thus the rigorous expression for the inflation ratio is m

I = (1/a) I (N+l)

i Prob(v=N}. [F.2] c N=0

As_jioted in the main text, the probability of a "normal" container- load (v=m) is

1-p , (H) , while for l<N<m we have

Prob{v=N} = exp(-H) #/N!.

It follows that

-i m~l -1 _

M

Ic

= (l/DKm+D'^l-p ^CH)) + exp(-H) I (N+l)"

1 TT/W.}. [F.3]

o

Since

(N+l)"

1 #/N! = H"1 H^VCN+l)!,

the summation in [F.3] can be rewritten

-, m-1 _ M, , H" 1

I H

N VCN+1)! =H _1

{pm

(H)- exp(-H)} exp(H).

o

Thus [F.3] yields

Ic

= (1/a) {(m^l)" 1

(1 - p^ttO) + H

"X [pm

(H) - exp (-U)]} [F.4]

That the direction of error is under-estimation follows from the Cauchy-Schwartz Inequality.

131

APPENDIX G: APPLICABILITY OF TRIP-END DISTRIBUTION MODELS

As noted in Chapter III, the available demand data included the U.S. member of each

origin- destination pair (origin for exports, destinations for imports), but not the overseas

member. Thus some arbitrary assumptions, detailed in Chapter IX, were required in order to exercise the model.

Quite late in the study, it was found that for each U.S. port, data on the total amount

shipped to each overseas port might well have been available. From the study data, the amount

shipped by each shipper to each of the three U.S. ports is known. The available 'data" would

then have included "trip end totals" at each of a number of U.S. shippers (referred to here

as origins although for import cargo they are actually destinations) and foreign ports

(referred to here as destinations) . The remaining problem would be that of somehow using

these totals to attribute a numerical value to the flow volume between each individual origindestination pair.

The very same problem arises in what has become a somewhat stylized approach to the planning of urban transport systems. 38 The mathematical models developed for treating it are called trip-end distribution models, and include among their input data either observed or estimated values of the quantities

r. = volume of flow from i-th origin,

c = volume of flow to i-th destination;

satisfying the obvious balance condition

The outputs of these models are proposed values for the quantities

x. . = flow volume between i-th origin and j-th destination,

which of course must satisfy the conditions

x

±

. > 0, [G.l]

Y.x.. = r., [G.2]

y.x.. = c. [G.3]

The overwhelming majority of trip-end distribution models have been of the type known as gravity models. 39 Here the model inputs include quantities

K. . = "conductance" between i-th origin and i-th destination.

The basic idea is to mimic Newton's Law of Gravitation by setting

x. . = r.c.K. .

.

However, this equation is in general inconsistent with [G.l] through [G.3], and so is

replaced by

x. . = r* c* K. ., [G.4]

38 B. M. Levin and R. E. Schofer, The Urban Transportation Planning Process, Socio-Econ. Plan.

Sci. Vol. 1 (1967), pp. 185-197.

39New work along this line, plus a useful bibliography, is given by P. S. Loubal, A Mathematical Model for Traffic Forecasting, 5/68, Berkeley doctoral thesis.

133
where r* and c* are quantities to be determined so as to make [G.4] consistent with [G.l]

through [G.3].

The system of equations [G.l] through [G.4] is nonlinear. It is solved by an iterative

process in which one starts with r* = r. and c* = c, computes the x. . by [G.4], finds r^'s

so that (with the "current" c^-values) the results of [G.4] will satisfy [G.2], finds c*'s

so that (with the "current" revalues) the results of [G.4] will satisfy [G.3], and so on, alternately imposing [G.2] and [G.3]. The convergence of this process, and the existence

and uniqueness of x.

. 's satisfying [G.l] through [G.4], have only recently been put on a firm basis. Gravity-1 models differ among themselves predominantly in how values for the con- ductances K. . are constructed from the distance, travel time and travel cost associated with

the (i, j) -tin? origin-destination pair.

The availability of data on overseas trip-end totals became known too late to warrant a literature search as to whether these concepts have previously been applied in the context

of export-import trade. Though the techniques are presumably "better than nothing," their

transferability from an urban transportation setting to overseas movements of cargo obviously

cannot be asserted with any great confidence, pending actual empirical testing.

It is natural to ask how one might proceed to attribute values to the flows x. ., when

not even the data needed to estimate the K- -'s are at hand. 40 With such a dearth * of information, the most that can be hoped for is a-'method which is systematic and reproducible.

Clearly the array X of x.-'s must satisfy conditions [G.l] through [G.3]. Let d(X,Y) denote

some measure of the "distance" between two such arrays, X = (x..) and Y = (y-0; here d(X,Y)

might be interpretable as indicating the seriousness of assuming the flow pattern to be X

when in fact it is Y. Then the worst possible error that could result, from assuming the

flow pattern to be X, is

F(X) = max

y

d(X,Y),

and a conservative policy would be to choose X so as to minimize this worst possible error,

i.e. to reduce it to

min„ maxY

d(X,Y)

The tractability of the mathematical problem of determining the minimizing X, of course,

depends on the formula chosen for d(X,Y) . For the not unreasonable choice

d(X,Y) = max. . I x. . - y. . I ,

this problem has been proved equivalent to the solution (which can be carried out by standard

methods) of the following linear program in the x. .'s and one extra variable z:

Minimize z

subject to [G.l] through [G.3] as well as the constraints

z >M- . - x. .

,

- ij ij

z >x- . - m. . , - ij ij

where the constants M. . and m. . are given by

M^ = min (r^c.),

nUj = max (0, r

±

+ c. - S),

and S is the common value of the two sides of [G.0].

4tThe following material is adapted from A. J. Goldman and P. R. Meyers, Minimax Error Selection of a Bivariate Distribution with Given Marginals, manuscript in progress.

134
REFERENCES

1. Interstate Commerce Commission, New England Forwarding Company, Inc., Extension No. FF-96

(Sub-No. 2).

2. Military Construction Pricing Guide, Department of the Air Force, AFP 88-66, March 1967.

3. Military Construction Pricing Guide, Department of the Air Force, AFP 88-66, March 1968.

4. Cecil H. Chilton, Chemical Engineering, June 1949, pp. 97-106.

5. Roger Williams, Jr., Chemical Engineering, June 1947, p. 102.

6. Chilton, Chemical Engineering, April 1950, pp. 112-4.

7. Numerous trade sources

8. Post Engineering Repairs and Utilities, Annual Summary of Operations, Office of the Chief

of Engineers, Department of the Army, Fiscal Year 1966.

9. Federal Power Commission, 1964, National Power Survey, Part 1, Table 10, p. 34.

10. Inland and Maritime Transportation of Unitized Cargo. NAS, NRC Publication 1135, p. 80.

11. Conversations with Railroad Personnel.

12. The S. S. Warrior, An Analysis of an Export Transportation System from Shipper to Consignee (NAS-NRC Publication 339) out of print.

13. North Atlantic Container Statistics Report for Six Months Period Ending Dec. 31-, 1967,

O.M.P. MARAD.

14. Interstate Commerce Commission, Rail Carload Unit Costs by Territories for the Year 1963,

Statement No. 5-65, March 1965.

15. J. R. Meyer e_t al, The Economics of Competition in the Transportation Industries, Harvard

U. Press, 1964,"Chapter VII, pp. 188-196.

16. C. Witzgall, Optimal Location of a Central Facility: Mathematical Models and Concepts,

National Bureau of Standards Report 8388 (6/30/65).

17. W. W. Hardgrave, Location-Allocation Problems: A Survey, Operations Research 16_ (1968),

Supplement 1, p. B-84 (Abstract).

18. H. W. Kuhn and R. E. Kuenne, An Efficient Algorithm for the Numerical Solution of the

Generalized Weber Problem in Spatial Economics, J. Regional Science 4_ (1962), pp. 21-33.

19. R. D. Luce and H. Raiffa Games and Decisions, Wiley and Sons (1957) Chapters 4, 5, 7.

20. M. Sasieni, A. Yaspan, and L. Friedman, Operations Research - Methods and Problems, Wiley

(1959), p. 133.

21. L. R. Ford, Jr., and D. R. Fulkerson, Flows in Networks, Princeton U. Press (1962).

22. D. Eklof, The Multi-Period Transportation Problem, Johns Hopkins U. doctoral thesis (1967)

23. H. W. Kuhn and R. E. Kuenne, An Efficient Algorithm for the Numerical Solution of the

Generalized Weber Problem in Spatial Economics, J. Reg. Sci. £ (1962), pp. 21-33.

24. L. Cooper, Location-Allocation Problems, Oper. Res. 11_ (1963), pp. 331-343, W. Miehle,

Link-Length Minimization in Networks, Oper. Res. 6 (1958), pp. 232-243; F. P. Palermo, A

Network Minimization Problem, IBM J. Res. Dev. 5_ "(1961) , pp. 335-337.

135
References (cont .

)

25. S. L. Hakimi, Optimum Location of Switching Centers and the Absolute Centers and Medians

of a Graph, Operations Research, 12 (1964), pp. 450-459.

26. A. J. Goldman and P. R. Meyers, A Domination Theorem for Optimal Locations, Operations

Research 13 (1965), p. B-147 (Abstract).

27. C. Witzgall, On Labelling Algorithms for Determining Shortest Paths in Networks, NBS

Report 9840 (5/68).

28. H. Frank, Optimum Locations on a Graph with Probabilistic Demands, Operations Research

14 (1966), pp. 409-421.

29. H. Frank, Optimum Locations on Graphs with Correlated Normal Demands, Operations Research

14 (1966), pp. 552-557.

30. S. L. Hakimi, Optimal Distribution of Switching Centers in a Communication Network and

Some Related Graph Theoretic Problems, Operations Research 13_ (1965), pp. 462-475.

31. S. Singer, Multi-Centers and Multi-Medians of a Graph, with an Application to Optimal

Warehouse Location, Operations Research 16 (1968), p. B-87 (Abstract).

32. L. Cooper, Location-Allocation Problems, Operations Research 11_ (1963), pp. 331-343.

Also Heuristic Methods for Location-Allocation Problems, SIAM~~Review 6_ (1964, pp. 37-53);

Solutions of Generalized Locational Equilibrium Models, J. Reg. Sci. 7_ (1967), pp. 1-18.

33. H. 0. Hartley, Optimisation of the Location of Serving Centers, 7/68, manuscript

communicated by author.

34. M. Balinski, Integer Programming: Methods, Uses, Computations, Management Sci. 12 (1965),

pp. 253-313.

35. M. A. Efroymson and T. L. Ray, A Branch-Bound Algorithm for Plant Location, Operations

Research 14 (1966), 361-368.

36. J. F. Sharp, J. C. Snyder, and J. H. Green, An Algorithm for Solving the Multi-facility

Production-Transportation Problem with Nonlinear Production Costs, Operations Research

16 (1968), p. B-87 (Abstract).

37. A. J. Goldman, A Theorem on Convex Programming, paper delivered to the Mathematical

Association of America, 1963 Annapolis meeting.

38. B. M. Levin and R. E. Schofer, The Urban Transportation Planning Process, Socio-Econ.

Plan. Sci. Vol. 1 (1967), pp. 185-197.

39. P. S. Loubal, A Mathematical Model for Traffic Forecasting, 5/68, Berkeley doctoral thesis,

40. A. J. Goldman and P. R. Meyers, Minimax Error Selection of a Bivariate Distribution with

Given Marginals, manuscript in progress.

136
RELEVANT BACKGROUND REFERENCES

Flower, Walter C. II - Work of the Freight Forwarder, March 10, 1964

Locklin, D. Philip - Economics of Transportation , Richard D. Irwin, Inc., 1966.

Selogie, Louis A. - An Engineering Analysis of Cargo Handling - VII: Information - Communication Network, University of California, Los Angeles,

Report 61-65, December 1961

Craig, D.C. , Ross, Martin, and Rolston, B.F. - Inland Marine Freight

Centers Progress Report No. L, American University, October 28, 1967

American Association of Port Authorities - 1967 Annual Report, Committee II - Standarization and Special Research

Business Week - August 26, 1967, Page 124, Speeding the Schedule of

Seagoing Cargo

Meyer, John R. , Peck, Merton J., Stenason, John, and Zwick, Charles - The Economics of Competition in the Transportation Industries, Harvard Economic

Studies, 1964

Maritime Administration, Department of Commerce, Office of Research and

Development, November 1964 - Cargo Movement in International Trade, A Summary

of Available Data

Dillon, E. Scott, Ebel, Francis G. , and Goobeck, Andrew R. - Ship Design

for Improved Cargo Handling, October 11, 1961

Maritime Cargo Symposium Proceedings, Long Beach, California, September

17-18, 1964

Maritime Administration, Department of Commerce, Office of Maritime

Promotion, March 3, 1967. Program Memorandum, Ports and Systems Program

Bureau of Census, Department of Commerce. Guide to Foreign Trade

Statistics: 1967

Highway Research Board, Record No. 82, 1965 - Freight Transportation,

9 Reports

National Academy of Sciences - National Research Council, Publication 1135,

1963 - Inland and Maritime Transportation of Unitized Cargo

Ramm, Dorothy V., Library of Transportation Center, Northwestern University,

April 14, 1967 - A Bibliography on Economics of Containerization

Northwestern University, Library of Transportation Center, April 20, 1967 - Ph. D. Dissertations on Intercity and Local Freight Transportation 1961-66.

137
Interstate Commerce Commission, Bureau of Economics, Statement No. 67-1,

February 1967 - Air-Truck Coordination and Competition

Interstate Commerce Commission, Bureau of Economics, Statement No. 66-1,

December 1966 - Piggyback Traffic Characteristics

Department of Commerce - 1963 Census of Transportation, Parts 1 and 2,

Commodity Groups, Parts 3 and 4, Shippers Groups and Production Areas

Maryland Port Authority - Decade for Port Progress, 1967-1977

Grossman, William L. , New York University, 1959 - Fundamentals of

Transportation

Corps of Engineers, Department of the Army - Waterborne Commerce of

the United States, Calendar Year 1960, Part 1, Waterways and Harbors, Atlantic

Coast

Port of Seattle Reporter, June 1, 1967 - Pan American World Airways - Clipper Cargo Horizons, October 1967

Port of Seattle, March 1967 - Seattle Container Gateway Port of the

North Pacific Range

Truck Trailer Manufacturers Association, October 1967 - Watch Your

Weight

National Academy of Sciences - National Research Council, Publication 745,

1959 - Maritime Transportation of Unitized Cargo

United States of America Standards Institute - Specifications for Cargo

Containers, USASI MH 5.1 - 1965

United States Senate, Ninetieth Congress, Committee on Commerce, Subcommittee on Merchant Marine and Fisheries, Hearings, July 13, 14 and 17, 1967 - Standarization of Containers

138

* U. S. GOVERNMENT PRINTING OFFICE : 1970 431-991/109
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the world's literature and critically evaluated. Product Standards. Provide requirements for sizes, types, quality and methods for testing various industrial products. These standards are developed cooperatively with interested Government and industry groups

and provide the basis for common understanding of product characteristics for both buyers and sellers. Their use is voluntary.

Technical Notes. This series consists of communications and reports (covering both other agency and

NBS-sponsored work) of limited or transitory interest. Federal Information Processing Standards Publications. This series is the official publication within

the Federal Government for information on standards

adopted and promulgated under the Public Law

89-306, and Bureau of the Budget Circular A-86

entitled, Standardization of Data Elements and Codes

in Data Systems.

Order NBS publications from: Superintendent of Documents

Government Printing Office

Washington, D.C. 20402
U.S. DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20230

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PENALTY FOR PRIVATE USE, $300