NBSIR 75-666

Post Optimality and Parametric

Analysis with tlie National Bureau of

Standards' Linear Programming Subroutine

RYSMPX

Tyrone B. Ayers

Applied Mathematics Division

February 1975

Final

Technical Report to Computer Services Division

Center for Computer Sciences and Technology

I

NBSIR 75-666

POST OPTIMALITY AND PARAMETRIC

ANALYSIS WITH THE NATIONAL BUREAU OF

STANDARDS' LINEAR PROGRAMMING SUBROUTINE

RVSMPX

Tyrone B. Ayers

Applied Mathematics Division

February 1975

Final

Technical Report to Computer Services Division

Center for Computer Sciences and Technology

14.

"•tAU 0»

U.S. DEPARTMENT OF COMMERCE, Rogers C.B. Morton, Secretary

James A. Baker, III, Under Secretary

Dr. Betsy Ancker-Johnson. Assistant Secretary for Science and Technology

NATIONAL BUREAU OF STANDARDS. Ernest Ambler, Acting Director
'if"
CONTENTS

1. Introduction _ 1

2. Sensitivity Analysis on Cost and RHS Vectors (RANGES) 2

3. Parametric programming on Cost Vector (PAROBJ) or on RHS Vector (PARRHS) 3

h. Deck Set-up and parameter Settings k

5. Output 8

5.1 Output from RANGES 8

5.2 Output from PAROBJ and PARRHS 8

6. References l8

Appendix A: Listing of RANGES 19

Appendix B: Listing of PAROBJ 25

Appendix C: Listing of PARRHS 33

Appendix D: Tiaiing Considerations 39

LIST OF FIGURES

Figure 1: PAROBJ Output - Regular format 10

Figure 2: PARRHS Output - Regular Format ,12

Figure 3: PAROBJ Output - Compressed Format 14

Figure k: PARRHS Output - Compressed Format 16

ABSTRACT

This report is a sequel to NBS Report IO695 (February 1972), The

National Bureau of Standards' Linear and Quadratic Prograrruning

Subroutines, which documented one phase of an effort to provide

users of the facility operated by the National Bureau of Standards

Computer Services Division, with reliable clearly -described

solution algorithms for selected frequently -arising classes of

special mathematical problems. The present report presents

subroutines which perform post-optimality analysis and parametric

programming studies on linear programming problems solved by

the National Bureau of Standards' RVSMPX subroutine. (The present

versions of these codes use internal storage only.

)

Keywords: Algorithms; post-optimality analysis; parametric

programming; linear programming.

ACKNOWLEDGEMMT

The author would like to acknowledge the aid provided by

W.G. Hall^ R.H.F. Jackson^ and P.B. Saunders of the Applied

Mathematics Division in the formulation and testing of the

subroutines discussed in this report.

1. INTRODUCTION

This report documents another phase of a continuing effort to provide

users of the computer facility operated by the National Bureau of

Standards' Computer Services Division with reliable, clearly -described

solution algorithms for selected, frequently -arising classes of mathematical

problems. This phase of the effort has increased NBS ' s capabilities

in the area of linear programming by producing three subroutines:

RANGES, a subroutine which provides post-optimality ranges for the

objective function coefficients (costs) and for the constants which

constitute the right -hand -sides (RHS) of the linear program's constraints;

PAROBJ, a subroutine which performs a parametric study of the effects

of changes in the costs; and PARRHS, a subroutine which performs a similar parametric study for changes in the RHS.

The subroutines were written in FORTRAN V, solve the problems "in core,"

and were designed to be used with RVSMEX, a subroutine for performing

the revised simplex method of linear programming. RVSMEX is described

in the previous report [2], and familiarity with that document is

assumed here.

Sections 2 and 3 of this report discuss, respectively, the methods

used in RANGES, and those employed for PAROBJ, and PARRHS. Section h

presents the deck set-up, and parameter settings required to use the

routines, and Section 5 provides sample outputs with explanations.

The appendices contain program listings, and also the results of sample

problem runs made for purposes of code verification and timing.
2.

2. SMSITIVIOTY ANALYSIS ON COST AND RHS VECTORS (RANGES)

The input to RANGES^ consisting of a linear programming problem together

with its solution, is the output from RVSMPX (see [2]). The output

from RANGES is a set of upper and lower bounds. An upper -lower bound

pair appears for each objective - function coefficient and also for each

constraint right hand side, and represents the maximum range over which

that quanti.ty can vary (with all other problem data fixed) without ^ forcing a chan^- from the optimal basis for the original LP problem.

That is to say, the set of variables in the optimal basis will remain

the same so long as no cost or RHS is changed beyond its bounds. It

should be noted that in the case of a change in an RHS, although the basis

remains the same, the values of the variables in the basis would change.

Correspondingly, in the case of a change in a cost, the values of the variables

in the basis as well as the membership of the basis will remain the same,

while the value gf the objective function changes.

Note, that due to round-off error the computed bound values may occasionally

vary from the actual values by as much as 3 times the epsilon (EPS)

computed in RVSMPX. However, the computed values, when substituted for

their corresponding cost or RHS, do indeed yield the same optimal basis.

A number of computer runs were made to check the "correctness" of the

code. The times for these runs ('which, by the way, employed the very

same problems that were used to verify RVSMPX in [2],) are reported in

Appendix D. A potentially useful statistic from these runs is that

the time consumed by RANGES was on the average roughly 20^ of that required

by RVSMPX to solve the original problem. . _

This technique is described in detail in [l] and [3]
3.

3. PARAMETRIC PROGRAMMING ON

COST VECTOR (PAROBJ) OR RHS VECTOR (PARRHS)

Consider the linear programming problem given in matrix notation as

max cx^ subject to Ajc:b and x > 0,

where ":" represents <, >, or =. If B is the basis matrix, and and

Xg are the corresponding subvectors of c and-jX respectively, then the

solution to this problem is given by x^ = B with optimal objective

value of ex.

B is Parametric programming on^the cost vector is concerned with replacing c in the above problem by c = c + 6" f , where f is an arbitrary vector

supplied by the user and is a nonnegative scalar parameter. PAROBJ

solves this problem by first solving the original problem using RVSMPX,

then computing the largest value of ^ for which the current optimal

basis remains optimal. Call this value . A new basic solution is

rfl£LX then calculated which is optimal in some new interval of P values beginning

with P' . The upper bound of that new interval is the new value of P

max max This procedure continues until either a.) the problem becomes unbounded

(linear programming theory assures that it will remain so for all ]^rger

values of f^) , or b. ) the last of the critical values of Pis reached (the same solution remains optimal for all larger values of p) , or c.) a userspecified limit on 6" is exceeded.

If the solution to the original problem is unbounded, it is still possible

that for some positive value of 6, the solution to the modified problem

"becomes" bounded. If such is the case for the original linear programming

problem, then PAROBJ will start the parametric study with the smallest

positive value of P for which a bounded, feasible solution is obtained.

As with RANGES, a number of sample problems were run on PAROBJ for the

purpose of code verification and timing. The results of these runs are reported in appendix D.

* -x- -xPARRHS is a subroutine written to perform parametric programming on the

RHS vector. This is the situation in which vector b of the original

problem stated in the beginning of this section is replaced by

b = "b + P where r is an arbitrary vector supplied by the user, and

F is as before. All the comments stated above for PAROBJ with regard to

boundedness, apply for PARRHS except that the concern here is with feasibility.

*By way of clarification, RVSMPX accepts the A matrix in non-canonical form,

so that the constraints need not be all equalities or inequalities.
k.

h. DECK SET-UP Am PARAMETER SETTINGS

The previous sections of this report discussed the subroutines in general

teiTOS^ stressing the functions they perform. This section will provide

the reader with the information needed actually to use the subroutines.

The deck set-ups will be discussed first, followed by the form of the

call to the subroutines, followed further by an explanation of each of

the parameters employed. , .

RANGES is used in what can be termed a "post-RVSMPX" mode. In the user's

main program, after the call to RVSMPX In the form described in [2], the

user simply inserts a FORTRAN CALL statement of the following form:

GALL RANGES (A,MA,B,MB,MT,NT,L,X,RHSUP,RHSL0W,C0EUP,C0EL0W,NSBT),

and the range information is printed out.

The observant reader will note that many of the parameters in the above

CALL statement appear identical to those in the GALL statement for RVSMPX.

In most cases they are exactly the same. There are, however, some variations, which are listed below and should be NOTED by the user.

Both PAROBJ and PARRHS operate in a "super -RVSMPX" mode. The user-supplied

main program referred to earlier and discussed in [2] should pass control

to one of these routines, which then takes charge oi' the parametric study

and uses RVSMPX as a subroutine. This, of course, means that any parameters

required by RVSMPX which would normally be passed directly to it, must now

be passed through the parametric programming routines. The FORTRAN GALL

statements, as they snould appear in the main program, are:

CALL PAROBJ (A,MA,B,MB,MT,WT,L,X,TOLP, INV,F,BCUKD,r:SEl)

,

CALL PARRHS (A,MA, B,MB,Mr,NT,L,X,TOLP, INV, BOUND,R,NSET )

.

Again it should be noted that the similarity between these CALL statements and

the one for RVSMPX is not accidental. A full understanding of the parameters,

especially as they relate to the various subroutines, is absolutely imperative.

Listed below are the parameters that appear in RANGES, PAROBJ, and PARRHS.

After each variable name, there appears an (r), an (O), or an (h) if the

variable is a parameter of either RANGES, PAROBJ, or PARRHS, respectively. If

the variable pertains to all three subroutines without variation, no letters

will appear. But if variations are present, an explanation will be given.

The ra and n used with reference to dimensions have the same meaning as in the

RVSMPX documentation: m is the number of constraints, and n is the number

of real (original) variables.

•

• ^ -xA (r) This is the augmented constraint matrix. It snould be dimensioned

at least (m+l) by (n+l) and is unchanged by the subroutine.
5.

(O) Because of the nature of PAROBJ, A must be dimensioned at least (m+2)

by (n+l) to provide storage for the original coefficients of the

objective function. A is unchanged by PAROBJ.

(e) Because of the methods used in storing the original coefficients of

the objective function and in dealing with an originally infeasible problem^

A must be dimensioned at least (m+2) by (n+2). A is unaltered by PARRHS.

MA This variable equals the A/alue of the first dimension of A. It is

unchanged upon exiting from the subroutines.

B (r) This matrix provides storage for the inverse of the basis matrix

and should be dimensioned (m+2) by (m+2). B is unchanged by the subroutine.

(0,H) Since the subroutines perform parametric studies which cover several

bases^ B will contain the inverse of the last optimal basis matrix upon

exiting from the subroutines. B should be dimensioned at least (m+2) by (m+2).

MB This is the value of the first dimension of B. It is not altered by

the subroutines.

ivrr (R) This is the number of rows of information in the A matrix, i.e.,

Xm+l). MT is not changed by RANGES.

(0,H) Recall that the A matrix for PAROBJ and PARRHS has (ra+2) rows. Hence

MT must equal m+2. Upon exiting from the subroutine, MT is unaltered.

NT (R^O) This variable equals the number of columns of information in

the A matrix, i.e., (n+l). It is unchanged by the subroutines.

(H) Recall that the A matrix for PARRHS has (n+2) columns. Therefore, KT

must equal (n+2). Upon exiting from the subroutine, NT is not changed.

L (R) This is a multipurpose vector providing work space needed by RVSMPX.

L is not changed in any way by RANGES.

(0,H) In order to maintain a certain consistency in output formulation,

the user-supplied values of L only function for the initial reporting of the

problem, after which certain elements of L are automatically set by the

subroutines. Upon exiting, L(5)^ L(8)^ L(13)^ and L(i4) will equal 1.

L{h) and L(12) will equal 0. The value of all other L(I), I < 1^, will be

the same as they are upon returning from RVSMPX.

X (R) Just as in RVSMPX, X provides storage for the solution of the

problem. X is not changed by RANGES.

(0,H) These subroutines perform a parametric study over a series of optimal

bases. Therefore, the solution in X upon exiting from the subroutines

will not be the original solution, but rather the solution corresponding

to the final optimal basis of the parametric study.

TOLP (0,H) This is the variable used by RVSMPX to compute the value of

EPS and CEPS. Its value on input should be exactly the same as it is for

RVSMPX. TOLP is not changed by the subroutine.
6.

INV (OjH) This is the switch allowing the user to supply RVSMPX with

an initial feasible solution. As in RVSMPX, INV may still be used to

indicate a user-supplied initial basis. The value of INV is altered

in the subroutines.

RHSUP (r) This vector provides storage for the upper bounds on the

ranges of the right-hand sides of the given problem being solved. Since

each right-hand side will have an upper bound RHSUP should be dimensioned

at least m in the main program, where m is the number of rows in the

unaugraented A matrix,

RHSLOW (r) This vector provides storage for the lower bounds on the

ranges of the right-hand sides. Like RHSUP, RHSLOW should be dimensioned

at least m in the main program.

COEUP (R) This vector provides storage for the upper bounds on the

ranges of the coefficients of the objective function of the given

problem. COEUP should be dimensioned at least n in the main program,

where n is the number of columns in the unaugmented A matrix.

COELOW (r) This vector provides storage for the lower bounds on the

ranges of the coefficients. Like COEUP, COELOW should be dimensioned

at least n.

R (h) This Is the vector supplied by the user to indicate the rates

of change of each of the right-hand sides with respect to the scalar P. Therefore, R must be dimensioned at least m. R should be set up so that

the 1th element of R is the rate of change of the ith right-hand sid^.

For example, if the ori^jinal right-hand side vector Is b=( 1, 2, 3^7^ 5

)

ana if R is (0,2, -3,. 6) , then the new right hand side vector b varies

as follows:

b^ = b^ + 0 e = 1

. .

. b^ = b^ + 2 e = 2 + 2 e

; = b^ - 3 P = 3.7 - 3 <P

'

: b^ = b^ + 0.6 P = 5 + 0.6(=^

where <P > 0. The subroutine does not change R.

F (o) This is the vector supplied by the user to indicate the rates of

change of each of the objective function coefficients with respect to

the scalar (p. Therefore, F must be dimensioned by at least n. F should

be set up so that the jth element of F is the rate of change of the jth

cost coefficient. For example, if the original cost vector is c = (2,3,1.6,-4)

and if F is (2,0, -.5, 5), then the new cost vector c^ varies as follows:
7.

c^+2p=2+2P

+ 0 = 3

- 0.5 = 1.6 - 0.5 ^

c + 5 P = -4 + 5

<^

c 3

c k

where > 0. F is not altered by the subroutine.

BOIMD (0,H) This is the user-specified limit on the non-negative scalar

parameter The scalar is incremented up to the value of BOUND. Once

the value of <? exceeds BOUND, the subroutines will terminate by reporting

that the limit has been exceeded and that the current basis remains optimal

at that limit. However, the subroutines may terminate before reaches

1) the largest critical value of 6 is less than BOUND,

2) the problem becomes infeasible (PARRHS),

3) the problem becomes unbounded (PAROBJ).

The value of BOUTNfD is unchanged by tne subroutines.

NSET (r) This parameter is a user-specified switch. If the user sets

NSET = 0, ranges are calculated for both right-hand sides and objective

function coefficients. If NSET = 1, ranges are determined only for the

right-hand sides. If NSET = 2, ranges are obtained only for the objective

function coefficients.

BOUT© if:

(0,H) For these subroutines, this user-specified switch determines the

output format. If NSET = 0^ a detailed format is used. If NSET = 1, a compressed tabular surrimary format is used.
5. OUTPUT

This section of the report will discuss the output from the three subroutines^

and illustrate them through actual output samples.

- 5.1 THE OUTPUT FROM RANGES

The printing of the output is controlled by RANGES. Information on the original right-hand side vector is printed first, followed by information

on the objective function coefficient vector. The user may obtain

information on both the right-hand side and the objective function

coefficient vedtors or either of them separately, depending on his specified

'yalue for the parameter NSET (see section h) . The output is printed in

an El^.8 forinat as follows:

-.14845019+02 LE RHS 1 LE POS . INFINITY

.54586695+01 LE RHS 2 LE POS . INFINITY

.30275573+01 LE RHS 3 LE .15395222+02

.15161557+02 LE RHS 4 LE POS,. INFINITY

NEG. INFINITY LE COEFFICIENT 1 LE -.14055693+00

-.20215248+01 LE COEFFICIENT 2 LE -.64413020+00

NEG. INFINITY LE COEFFICIMT 3 LE -.54610899+01

NEG. INFINITY LE COEFFICIENT 4 LE -.40288613+01

-.12985771-01 LE COEFFICIENT 5 LE .57053266+00

Line 1, for example, reads -14.845019 < RHS 1 < POSITI^/E INFINITY.

This means that the reported final basis will remain optimal as long as RHS 1 (the first element of the right-hand side vector of the problem

being solved) is greater than -14.85019. Similarly, bounds are reported

for each remaining element of the right-hand side vector, i.e., RHS 2,

RHS 3, and RHS 4.

Line 5 reads ' • NEGATIVE INFINITY < COEFFICIENT 1 < -.l4055693.

This means that coefficient 1, the first element of the cost vector of

the problem being solved, has no lower bound and has an upper bound of

-.14055693. Following line 5 are the bounds on the remaining elements of

the cost vector.

5.2 THE OUTPUT FROM PAROBJ AND PARRHS

All output is completely controlled by the subroutines, and the output

formats for the two routines are nearly identical. The first item to be

printed is a statement that the first critical value of (? is zero. With

tnis value of 6 , the solution to the parametric programming problem is the

same as the solution to the simple linear prograrimiing problem, and that
solution Is printed next. A formula for the calculation of the objective

function value for all values of from zero through the next critical

value is then printed.

A cycle of output then begins. The "current" critical ^alue of P is prin"*-

along with the basis change required when passing through that value for

The "current" basis is then given, followed by the objective function

calculation formula. This output grouping is repeated until one of the

termination conditions in section 3 is met. This output forirat is

illustrated in Figures 1 and 2.

A more compressed tabular format is also available. The value of NSET

(see Section k) controls the output format selection. Tiie compressed

format is illustrated in Figures 3 and h.
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Figure 1: PAROBJ Output

Regular Format (c-iiti ::^.ed)
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u.

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.ocIaoo • «

tfl J> Z to ^ -0 < (M

as - u —

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in

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Figure 2: PARRHS Outpat

Re>5ulax- FoiMiat
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Figure 2: PARRHS Output

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bJ *

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6. REFERENCES

1. Hadley, G.^ Linear Programming , Addison-Wesley , Reading, Mass.,

1962.

2. Hall, W.G., Jackson, R.H.F., Saunders, P.B., The National Bureau

of Standards' Linear and Quadratic Programraing Subroutines , NBS

Report No. 1069^, February 1972.

3. Simonnard, M., Linear Frogramming . Prentice Hall, Englewood Cliffs,

N.J. , 1966.
APPENDIX A: Listing of RANGES
20.

_l VS W _1 _l IT L. OU.—I 3-r~— OM 0(SI — — — — (SifsiiMi-ir^nr^-) i z >-

r»- ^CNj— — —.J-, rgrgy — ncoOfN)

3- o*^rg3-i/)—.(Nf^iNjoaoooo

O ODOOO — — OODOOOOO

O ODOOOOOCODOOOOO O DCODDOOOOOOOOOO

— O — — —— —— — — ODOOOp o oooooooooooooob o

o

UU_l_li5_) JU.U-Q. _ _j3-i^oo — oi/i— o-> — _l

Csl 3--0&'O^O— riozr^— Q_Q-*< — — — — PMrgonni/)- zx

O -o^^lO— o— r^3"i/ii/iLnN0r><N)*o

csj -of^J^or^^^o^^^Mn^-^^JOO — — — CMof^a- — in - — — ooooo o ooooo— ooooooooo o ooooooooooooooo o ooooooooooooooo

O LflOOODOOOOOOOOOOOO

O O — fMO^-lPNOt^OJO^O — *N)'^^Ln>0

O OOOOOOOOOO— — — — — — —

— — — — —— — OOOOCOOO OOOOOOOOOOOOOOO DCOOOOOOOOOOCOO OOOOOOOOOOOOOOO

_l _J _l _l U. _1 15 _)oooo — oin •OUNlTIO-— rlC MO — — cgrsifsioma- —

o «

»cf\ri c \r ^ <• ^ ^ o ^ ifi

l/> — LP'NJ— «^0<^'*1— M — Of^- DO^or^c- — (vjCDOCC-OO

oc. oo c — — coDocc: COOOOOCOODOOOOO ODOODOCDOCOOOCO

— — -r — — q: C — — — c CO C O (TO

c c o c o c c cc o o c- D c o OOOOOOOOOOOOOOO ooooooooooooooo — o

_ _ IT _J —

O — _J — O O C- ^ ITi OLfia: OincCi— r^r^Ij —

— — — rsjfvcs'^. ^ _j r: cc

— c • • _l cs m LJ bJ r o • — o —

3-3- — r f^LT — fV'Cf^ —

r-. — 3- fs' — — r >c (V cv — r-. —

tn — »vf^3- — c^MOC'C-ccjco one c. r' — — r. c- o o o

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c c c c. o r c c c c - c

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c c o o r t.-- o C' o c C' c r o a

o — — — — c- ooooo c O O O rj O O O Ci O C D C D o rroooc>c;'or'Ocoociro

o o o o o o ci o o m o o c> o o — — «-^3-l/l^^— r^oxLO^r^o— fN*'^3-

C' CT'OOC'C' — — — —— — —— rsjfv(\isrv

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1

APPENDIX B: Listing of PAROBJ
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in CO IT K; c IT vDm (\l 3- d- c c O 1^ r- IT 3 KV

vr rr l<- ^ C If 3- <\ ir

— c c 3 K 3 ^ K", c (V cv o lO d- f; 3- c CO c o c c o o c c o

CJ c tM o c O o c o C' ^ fV OJ c (V c c o c o o c c c o c o O o c o o o c o o c: c o o o o cr o o o c o o o o c c o c c c o o o o o c c o o o o c o o o o o

c; a.a o o o o c c o c c c o o c c CJ o o o o c o o o o o c o c o o c c o c o o o c o o o o o o c o c o o o c c o o o c o o c o c o c o c o o o c o o o o c o o o o o o c c o o o o o o c o o o o

_J li. L. tiJ 2

CO r- in _I

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cr l-H »-l l-l cr ir o o «-l c ^ o c o c c c c o c c o c o o c o o c coo o c c c o c c c o c c o o o o o c o o o o o c o coo o c o o o o o o o c c c o o o o c o o o o o o o o o o o o o o o o c o o o c o o o o o o c o o o o O .H CJ to 3- c c c c c

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APPENDIX D: Timing Considerations
APPENDIX D

TIMING CONSIDERATIONS

RANGES SAMPLES: Time Per Right-Hand Side

and Objective Function Coefficient

PROBLEM SIZE

*i ELAPSED

TIME

RANGES

(SEC.)

TIME

PER

R.H.S.

TIME

PER

COEFFICIENT

m n

10 20 .1388 .0031 .0054

15 20 .167A .0031 .0060

20 20 .2216 .0035 .0075

20 30 .2750 .0032 .0071

20 80 .9258 .0033 .0108

50 150 4.0446 .0039 .0257

D-1
APPENDIX D (continued)

PAROBJ SAMPLES: Time Per Critical Value

FROBLM SIZE ELAPSED NO. OF TIME

TIME COMPUTED PER

PAROBJ CRITICAL CRITICAL

m n (SEC.) VALUES VALUE

10 20 .7608 3 .2536

15 20 .0554 1 .0554

20 20 14.1665 5 2.8333

20 30 12.5100 9 1.3900

20 80 9.4508 8 1.1814

50 150 190.2708 23 8.2726

j

D-3
h2.

APPENDIX D (continued)

PARRHS SAMPLES: Time Per Critical Value

PROBLEM SIZE ELAPSED

TIME

PARRHS

(SEC.)

NO. OF

COMPUTED

CRITICAL

VALUES

TIME

PER

CRITICAL

m n VALUE

10 20 .7930 2 .3965

15 20 1.8704 3 .6235

20 20 4.3180 10 .3925

20 30 1.9914 1 1.9914

20 80 4.7322 1 4,7322

50 150 73.2580 2 36.6290

USCOMM NliS DC D-2
NBS-1 14A (REV. 7-73)

U.S. DEPT. OF COMM.

BIBLIOGRAPHIC DATA

SHEET

1. PUBLICATION OR REPORT NO.

NBSIR 75-666

2. Gov't Accession

No.

3, Recipient's Accession No.

4. TITLE AND SUBTITLE

"Post Optimality and Parametric Analysis with

the National Bureau of Standards' Linear Prograrnming

Subroutine"

5. Publication Date February 1975

6. Performing Organization Code

7. AUTHOR(S)

Tyrone B. Ayers

8. Performing Organ. Report No.

9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

10. Project/Task/Work Unit No.

11. Contract/Grant No.

12. Sponsoring Organization Name and Complete Address (Street, City, State, ZIP)

Same as No, 9

13. Type of Report & Period

Covered

Final

14. Sponsoring Agency Code

15. SUPPLEMENTARY NOTES

16. ABSTRACT (A 200-word or less factual summary of most si^ificant information. If document includes a significant

bibliography or literature survey, mention it here.)

This report is a sequel to NBS Report IO695 (February 1972), "The National

Bureau of Standards' Linear and Quadratic Programming Subroutines," which

documented one phase of an effort to provide users, of the facility operated

by the National Bureau of Standards' Computer Services Division, with reliable,

clearly -de scribed solution algorithms for selected frequently -arising classes

of special mathematical problems. The present report presents subroutines

which perform post -optimality analysis and parametric programming studies on linear programming problems solved by the National Bureau of Standards' RVSMPX

subroutine. (The present versions of these codes use internal storage only.

)

17. KEY WORDS (six to twelve entries; alphabetical order; capitalize only the first letter of the first key word unless a proper name; separated by semicolons

)

Algorithms; linear programming; parametric programming; post -optimality analysis.

18. AVAILABILITY [X] Unlimited 19. SECURITY CLASS

(THIS REPORT)

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