14

For the iron mine example the b; values are:

Card 1:
Word 1: b, Word 5: Ts
Word 2: Gb, Word 6: Q
Word 3: J, Word 7: T,
Word 4: Tp Word 8: zeros

3. The c; values are coded in floating point with one value per word and
eight values per card. All requirements for the b; cards apply to the cj
cards.

For the iron mine example, the cj values are:

Card 1:
Word 1: ¢ Word 5: C,
Word 2: C, Word 6: zeros
Word 3: Cz Word 7: zeros
Word 4: @ Word 8: zeros

4. The non-zero matrix elements are coded in floating point with four
values per card. Words one, three, five, and seven contain the values of
{i, jl for the matrix element values posted in words two, four, six and eight,
respectively. The order of coding the elements is immaterial for this data,
because the subscripted variables [i, j] are used to index the storage of the
matrix elements. The value of i is coded in columns one, two, and three of
the index word, and the value of j is coded in columns four, five, and six of
the index word. Columns seven through ten of the word contain zeros.

The word immediately following the index word contains the floating-point
value of the element ajj- The excess words of the last matrix card must be
filled with zeros. If the number of matrix elements were a multiple of four,
another card containing all zeros would be punched as the final card.

For the iron mine example, the aij elements are:

Card 1:
Word 1: 0010010000 (matrix element location [i, jl)
Word 2: 1000000051 (Element value coded in floating point)
Word 3: 0010020000
Word 4: 1000000051
Word 5: 0010030000
Word 6: 1000000051
Word 7: 0010040000
Word 8: 1000000051

Card 2:
Word 1: 0010050000 Word 5: 0020020000
Word 2: 1000000051 Word 6: Fe,
Word 3: 0020010000 Word 7: 0020030000
Word 4: Fe, Word 8: Fe,


35

TABLE 8. - Values determined by Monte Carlo methods for
bauxite-deposit silica content

Coded random number, u, Silica value from
(two-digit random number | Poisson distribution
divided by 100)
0.06 0.38
17 1.30
+09 -67
+79 4.53
-65 3.67
-88 5.33
+30 2.01
+29 1.95
-80 4.60
-41 2.54

Monte Carlo Model of a Mine-Haulage System

Every mine has the problem of designing an adequate, efficient, and well
balanced haulage system. The numerous system variables are complex and inter-
related, and each mine is faced with a different set of variables. Most haul-
age systems do have one feature in common, the lack of adequate accurate basic
information concerning the system.

Industrial engineering activity in mine plants is providing an increased
quantity of reliable data. However, collection of the data is time consuming
and costly, and much of the data results in calculations that show only aver-
age effects; specific occurrences cannot be accurately predicted. Monte Carlo
methods provide a means by which the effect of random occurrences, such as
delays, breakdowns, and interference, can be simulated.

As an illustration, assume a mine-haulage transfer point underground from
which a main-haulage motor has the capacity to take 5 load units; the load
unit might be any specific number, but the number is not pertinent in this
problem so long as the load units are equal. The transfer point has a capac-
ity of 10 loaded units and 10 empty units, and a loaded unit must be exchanged
for an empty unit. The main-haulage track is available for ore handling for
400 minutes each shift, and the round trip time is 20 minutes. There are 10
production points planned on the level, and each production point is designed
to load 10 units each shift; minimum loading and haulage time to the transfer
point is 20 minutes. Therefore, the productive capacity of the mine is 100
units, and the mainline haulage capacity is 100 units. Figure 13 is an analog
of the mine-haulage system.

The problem is that of determining the minimum-cost arrangement of haul-
age equipment and transfer-point size that is required in order to attain a
production of 100 load units each day.

It is highly improbable that the loaded units will arrive at the transfer
point in orderly sequence or that the haulage motor will always have 5 units


AREA OF THE CIRCLE 7

3.1716)

3.1616)

3.1516]

3.1416

3.1316)

3.1216]

BE

NUMBER OF POINTS

FIGURE 11. - Estimation of Area of a Circle Using a Monte Carlo Method.

Area of Circle Pee ga Pe oO
ny |
)
Ld \ L : ! a 1 ! \ j
jooo 200 3000 4000

$000

Te


40

In order to express the requirements of the system in mathematical models,
symbolic designation of items in table 10 will be required and are defined as
follows:

Ug = number of loaded units arriving at transfer point (arrivals).

Us = number of loaded units on transfer point during any period
(on station).

Ut = number of loaded units hauled from transfer point (taken).
U, = number of loaded units remaining on transfer point (left).
U, = number of units at production points.

The symbols will be modified by expressions of minimum (min) or
maximum (max).

Let ZU represent the total number of units required by the system, then:
ZU = Ure(max) + Us(max) + Up. (1)

Let TP; represent the capacity of the transfer point for loaded units, then:
TPy, = Ug(max). (2)

Let TPp represent the capacity of the transfer point for empty units, then:
TP, = XU - [u,(min) + vp]. (3)

Analysis of Case 1, utilizing table 10, is as follows:
xu = 5 + 15 + 10 = 30,

TP, = 15, and

TPy = 30 - (1 + 10) = 19.

The plan represented by case 1 requires 30 haulage units, expansion of
the transfer point to accommodate 5 additional load units and 9 additional
empty units, and payment of overtime for a haulage crew.

Analysis of the Problem--Case 2

A second solution to the problem would be to not have an overtime crew
and only supply extra load-carrying units and expand the transfer-point capac-
ity, if required. The same transfer-point-arrival data as in case 1 are used,
and calculations in table 11 indicate the situation that might exist in case 2.
After the third day the haulage unit is always operated at capacity. The maxi-
mum number of units on the transfer point is 21, and 10 loaded units are left
at the transfer point at the end of each shift after the third day. The 10
loaded units remaining is obviously a result of lack of haulage capacity on
the first, second, and third day; after the third day the loads produced and
the loads removed are equal. However, as few as 4 units are on the transfer
point during any one 20-minute period after the third day.


The inequalities must be expressed as equalities; this is accomplished by
adding or subtracting positive quantities in the left side of the inequality
statements. The positive quantities thus introduced are "slack" variables;
introducing them provides a method of representing mathematically the choice
of using some or all of the tons and ore grades available (3, p. 13). Less-
than-or-equal-to inequalities are made into equalities by adding slack varia-
bles; greater-than-or-equal-to inequalities are made into equalities by sub-
tracting slack variables. The new expressions are as follows:

-0.025 & + 0.015 X - 1X, = 0, (1b)
0.015 X, - 0.005%,-1X% = 0, (2b)
1.000 X, + 1X, = 40, (3a)
1.000 X, + 1X, = 60, and (4a)
1.000 X, + 1.000 X, = 80. (5)

The preference function:
4X, +6 X = C (min). (6)

Because of a restriction in the simplex procedure (that an initial basis can-
not be negative) it is required that artificial variables be added into the
equations in which a negative slack variable exists; equality statements in
which no slack variable exists also require an artificial variable but for a
different reason.

Iteration time and computational effort can be saved in this problem by
replacing the negative coefficients of slack variables with positive coeffi-
cients (thus avoiding introduction of artificial variables); this can be done
by the simple expedient of multiplying formulas (1b) and (2b) by -1. However,
the procedure would not be followed if the constant values were not 0 or a
negative quantity. Formulas (1b) and (2b) are changed as follows:

0.025 X - 0.015 X + 1 X,
-0.015 X, + 0.005 Xo +1%&

a

0, (1c)
0. (2c)

An artificial variable is required in equation (5):
1X, +1X,+1Q = 80. (5a)

The simplex procedure automatically eliminates solutions which make no
sense from an economic or processing point of view.

The coefficient of an artificial variable is assigned a huge value, desig-
nated by M. In a minimization problem such as this one, the M coefficient is
given a positive sign; in a maximization problem, it is given a negative sign.
The new preference function is:

4X + 6X) +MQ =C (min). (6a)


50

0048
0049
0050
0051
0052
0053
0054
0055
0056
0057
0058
0059
0060
0061
0062
0063
0064
0065
0066
0067
0068
0069
0070
0071
0072
0073
0074
0075
0076
0077
0078
0079
0080
0081
0082
0083
0084
0085
0086
0087
0088
0089
9090
0091
0092
0093

1583
1655
1395
1963
1364
1476
9050
0198
1466
1445
1656
1385
1574
1393
1427
1576
1616
1346
1048
1495
1487
1334
1509
1590
1977
1967
1585
1308
1408
1997
1508
1629
0349
1980
1658
1545
1986
0399
1638
1588
0449
1198
0398
0598
1426
nooo

5980021591
8360001462
4401991252
6132471302
6590071623
9000000050
0000010000
2490521556
4619701620
2490531402
6300001452
8380021343
6912781982
8880021552
6780031435
6990011983
6760001457
1780011507
4416201380
1590541557
4413801492
6919521657
6994281617
4015431444
4515801484
8801001577
4416411991
8139991528
8200481578
2413001458
2490001579
2490101985
2490021558
8900081536
5100011984
6590021609
8201001980
8900081987
6932471988
5000011644
2190011496
4413631446
4804480498
2194180648
6012830748
000008000

APPENDIX A. - COMPUTER PROGRAM LISTING FOR A LINEAR PROGRAMING ROUTINE (Con.)

1591
1462
1252
1302
1562
1490
1322
1556
1970
1402
1452
1343
1982
1552
1435
1983
1457
1507
1620
1557
1999
1657
1617
1543
1580
1484
1641
1528
1578
1458
1975
1985
1558
1536
1984
1609
1636
1987
1988
1644
1496
1363
0448
0648
0748
2000

4813941345
6580021573
5800020199
3990551506
6890071623
0090000047
2490031328
2190511464
4414741620
1090521459
3000041514
6052471502
2490511148
6077601516
6080021443
2412891442
20 12631378
1590501966
6590521477
2090521556
6019511607
2490581614
2476611964
4215091577
3500031640
8200481990
2420001258
5019991534
5000021358
2015221978
6580001584
6914131979
6590591630
6956611634
4015871438
4515861636
6913001660
6020001488
8139991494
5300011994
4215461596
5000021546
7190100498
5200010698
0180031492
o0nn000000

1394
1573
0199
1506
1623
1416
1328
1464
1474
1459
1514
1502
1148
1516
1443
1442
1378
1966
1477
1540
1607
1614
1964
1444
1640
1990
1258
1534
1358
1978
1584
1979
1630
1634
1587
1586
1660
1488
1494
1994
1596
1546
0498
0698
1998
0900

8380021605
1414761437
5800011456
2152471352
8359991480
0000000623
6912831386
3500031973
6912781632
6913001606
5980021524
3436601610
2412711624
4419741526
2190011602
6680060249
6590531485
2090531626
1690591535
6012711976
8080011564
8800001577
5200011627
6090581652
8080030248
8900081396
6996291478
5000011542
5800021628
2014131250
4615371250
2490111486
3500031592
2496181642
4815361992
2190021643
24900 11637
4415381588
5019991544
6090011645
8200481646
4819871996
6099000548
4205981436
0000000002
9900 non0n

1605
1437
1456
1352
1480
1414
1386
1973
1632
1606
1524
1610
1624
1526
1602
0249
1485
1626
1535
1976
1564
1577
1627
1652
0248
1396
1478
1542
1628
1492
1537
1486
1592
1642
1992
1643
1637
1538
1544
1645
1646
1996
0548
1430
1426
0000

5880021963
6080031149
6980071512
6090101409
5219991336
0000000001
2412891392
8080031582
24900 31338
8880011612
4813771378
2190001567
65905 31633
6012731974
3312891566
2012511407
1690581493
6080021402
4513881540
4412791530
3000031527
8300081384
5800011434
3000031962
3500041559
7090201428
8120001635
5300010298
4814281990
6680000299
2490001579
7190101593
8080031608
5800010348
7190101980
3000041659
8000001993
2196181595
2496191594
1114141995
1119981199
7190100399
4612509999
0000000001
0000000000
9000000000

1345
1149
1512
1409
1336
9054
1392
1582
1338
1612
1377
1567
1633
1974
1566
1407
1493
1049
1388
1530
1527
1384
1434
1962
1559
1428
1635
0298
1991
0299
1436
1593
1608
0348
1438
1659
1993
1595
1594
1995
1199
1446
1579
1380
0748

8880021655
1114901395
2413001963
4615621364
2040001482
0010000000
6590540198
6036601466
6090501445
2160001656
5880021385
4613781574
3000041393
3990001427
4615761378
8880021616
4513461049
6090511048
1780011495
6090031487
2090591334
7090201509
5100011590
6580021977
2190581967
6096281585
2432471308
4213581408
6980051997
4615089999
6915221629
6990580349
8200001980
5200011658
7190101545
8080021986
8200480399
8120001638
5800021588
4404491446
2190011198
5800080398
8300080598
6014140748
0000000000
0000000000

0104


48

BIBLIOGRAPHY

CARSON, GORDON B. Production Handbook. The Ronald Press Co., New York,
N.Y., 2d ed., 1959, 1648 pp.

FLAGLE, CHARLES D., HUGGINS, WILLIAM H., AND ROY, ROBERT H. Operations
Research and Systems Engineering. The Johns Hopkins Press, Baltimore,
Md., 1960, 889 pp.

GREENWALD, DAKOTA ULRICH. Linear Programming. The Ronald Press Co.,
New York, N.Y., 1957, 75 pp.

HEADY, EARL O., AND CNADLER, WILFRED. Linear Programming Methods. The
Iowa State Univ. Press, Ames, Iowa, 1958, 597 pp.

KING, GILBERT W. The Monte Carlo Method as a Natural Mode of Expression
in Operations Research. Jour. Op. Res. Soc. Am., vol. 1, No. 2, 1953,
pp. 46-51.

McCLOSKEY, JOSEPH F., AND TREFETHEN, FLORENCE N. Operations Research for
Management. Vol. 1. The Johns Hopkins Press, Baltimore, Md., 1956,
409 pp.

G. & C. MERRIAM CO. Webster's New Collegiate Dictionary. The Riverside
Press, Cambridge, Mass., 2d ed., 1956, 1174 pp.




46

Geologic data:

a. Structure.

b. Lithology.

ce. Alteration.

d. Mineralogy.

e. Deposit geometry.

Sampling and testing results:
a. Assay.

b. Metallurgical.

ce. Physical properties.

Historical records:

a. Production.
b. Maps.

Equipment data includes all information concerning equipment. Detailed
and accurate information concerning equipment is requisite to a comprehensive
mine-system analysis; compilation and collation of such data is difficult.
Principal categories are:

1.
2.
3.
4.

Requirements data.
Specifications.

Operating characteristics.
Time standards.

Cost data are interrelated with all other basic data; cost is frequently
important in restrictions data; cost of exploration is likely to affect the
quantity and quality of mineral-deposit data, and equipment-selection deci-
sions usually are made after calculating an optimal balance between utility

and cost.

1.

Cost categories are:
Fixed costs:

a. Overhead
b. Insurance.
c. Taxes.

Operational costs:
a. Exploration.
b. Development.
c. Production.
Equipment costs:
a. Initial.

b. Use.
ce. Amortization.


18

n
+ Alumi iction: , > .
3. Alumina restriction (al, 05 x); 248b,
4. Tonnage restrictions: xj <bj+z, j= 1-1.

Transportation costs per ton are calculated for each deposit using the
following:

Cost j = [ (truck mileage); ($0.12) + (railroad mileage) j (0.007) | .

The problem will be solved by minimizing the cost function (F):

Ms

cx.
jj

n
F=2 (Cost xT), =
j j

uu

The problem was solved for 10 deposits. Their location with respect to
each other and the rail and truck distances are shown in figure 3.

Table 1 shows the values used for gross tonnage, alumina and silica con-
tent, and transportation costs in the 10 bauxite deposits.

~~95
7
/ EXPLANATION
/ log”
40° ---- Truck haulage
+HHtH Railroad haulage
(0) 72 I
SCALE, MILE To Port

FIGURE 3. - Locations of 10 Bauxite Deposits.


36

Transfer point

FIGURE 13, - Analog of Mine-Haulage System.

on each trip. The motor is expected to make a trip each 20 minutes unless
there are no loads on the transfer point.

A method of handling the problem is to divide the shift into 20 periods
of 20 minutes each and schedule unit arrivals at random at the transfer point
from each production point; the arrivals can be scheduled by selecting numbers
from a random number table. Results of a 1-day simulated production schedule
are shown in table 9. Arrivals from each production point were scheduled as
follows: (1) A production point was chosen, (2) a table of random digits was
entered at a randomly selected place, (3) 2-digit numbers from 01 to 20 were
used, and all other 2-digit numbers were ignored. As soon as a number in the
range of 01 to 20 was intercepted, a checkmark was entered in the box occur-
ring at the intersection of the production-point row and the period-number
column of table 9 corresponding to the random number (this indicated a load-
unit arrival at the transfer point); nine other boxes in the production-point
row were filled in the same way. Should a duplicate number be selected from
the random-digit table, it would be ignored. Arrivals from each production
point were scheduled by the same procedure. The process was repeated for
1,000 units, representing 10 days of operation during 200 20-minute periods.

A frequency distribution of arrivals at the transfer point is shown in
figure 14. Note that the frequency distribution approximates a normal dis-
tribution. A minimum of 1 car and a maximum of 9 cars arrived at the transfer
point in any 20-minute period; zero and 10 probably would be encountered if
enough samples were drawn.

The requirements of the system demand a least-cost solution to production
of 100 load units each day.

The mining system is designed for an output of 100 load units each day.
The existing transfer point has a capacity of 10 loaded units and 10 empty
units, and the haulage capacity is 100 units each day.


CONTENTS

Introduction and Summary.....eeeeccececc cece eceeceecee cease esse tee esees
Models....ccceccce secs sccce tesco senses cesses see seeesceeseseees
Construction and methods of solution of mathematical models..
Constructing the model...
Methods of solution.....
Linear programing.....

The simplex method......cesececeseccesseeceace

Digital computer programS.......seeccesecceeeee

General description of algorithm.......

Program description

Input data format.

Output format

Computer information and restrictions. .

Examples of application...

Minimum-cost model. see eee eee

Constant-cost model sone eee eereeees

Monte Carlo methods... seeeeee tee e ee ence ee eeeee
Mathematical application. beeen eee
Examples of simulation. bee e cence eee

Probability distributions. beeen eee eee e ences
Deposit tonnages.......... ove eens

Alumina values...
Silica values..
Monte Carlo model of a mine-haulage system..
Analysis of the problem--case 1l.......
Analysis of the problem--case 2.........
Analysis of the problem--case 3...

Basic Data....

Bibliography see cece eee ee eees see eeee debe cece ences bee e eens

Appendix A. - Computer program listing for a linear programing routine...

Appendix B. - Listing of data in matrix form....... Aone e cee aeecerecreece
ILLUSTRATIONS

Fig.

1. Generalized diagram of a block-caving method....
2. Simplex tableau iterations........
3. Locations of 10 bauxite deposits..
4. Data in matrix form for 10 bauxite deposits...

5. Relationship between transportation cost and tonnage............

6. Relationship between transportation cost and tonnage for two
average Silica CONTENTS. ... ec eee cece cece eee renee ete neeneevens

7. Relationship between cost per ton and tonnage for various produc-
tion rates in millions of tons......sceeeeseeee

8. Four areas of a pit being expanded. No scale.

9. Constant-cost matrix form......seeeeeee deen eeees oe

10
18
20
22

24


29

Solution of the model for tonnage per area (1-4) per period (1-2) is as
follows:

Period Area A Area B Area C Area D
1,810,000 28,307 | 1,319,579 | 412,864
- 1,524,693 450,421 | 1,595,636

Total.... | 1,810,000 | 1,553,000 | 1,770,000 | 2,008,500

Determined tonnage per area for both planning periods corresponds to the
values used (table 5).

Tonnage mined for each planning period is 3,570,750, and this corresponds
to the value of M in the model; the constant tonnage requirement is satisfied.
Other constants determined by solution of the model are:

L
N

1,237,762
2,532,688

ou

Determined total cost was 2,475,524, which is 2 L as expressed in the model as
Cj. Average grade per planning period (N/M) is 0.709 percent.

Equipment and maintenance schedule requirements can be determined more
factually using the results of this solved model.

Monte Carlo Methods

Monte Carlo methods are effective for use when some probabilistic prop-
erty of a system is desired. The methods consist of randomly sampling a known
or assumed probability distribution; the sample statistics provide an estimate
of the performance of the system.

Probability models were used extensively by research personnel during
World War II to solve problems in mathematics; the models were designated by

the code name Monte Carlo (1, sec. 18, p. 19).

Mathematical Application

Several interesting applications of Monte Carlo methods exist; they are
typified in determining the value of 7 by counting the number of times that a
needle, tossed onto a checkerboard, falls across a line (5, p. 47).

As a simple example, the area of a circle will be determined by a Monte
Carlo method. Obviously, it is more expedient to calculate the area by mathe-
matics, but the example effectively illustrates a mathematical application of
Monte Carlo.

Random numbers will be scaled into a square 2 inches on a side. The coor-
dinates of a point, designated by a pair of random numbers, will be tested to
determine if the point is inside or outside a circle inscribed in the square
(fig. 10). The results of the test will be used to count the frequency of a
point falling inside or outside the circle. Basic mathematics are:


Card 3:
Word
Word
Word
Word

Pon

Card 4:
Word 1
Word 2
Word 3:
Word 4:

Card 5:
Word 1
Word 2
Word 3
Word 4

Card 6:
Word 1
Word 2
Word 3:
Word 4

Output Format. = The output format is identical with the input data

0020040000
Fe
0020050000
Fe,

0040020000
1000000051
0050030000
1000000051

0030070000
1000000051
0040080000
1000000051

0070110000
1000000051
0000000000
0000000000

Word
Word
Word
Word

Word
Word
Word
Word

Word
Word
Word
Word

Word
Word
Word
Word

WOIAUG

OIA OIdHAUH

OuUAU

format with the following exceptions:

( a.

0020060000
1000000051
0030010000
1000000051

0060040000
1000000051
0070050000
1000000051

0050090000
1000000051
0060100000
1000000051

0000000000
0000000000
0000000000
0000000000

nations required; this replaces the value of m.

b.

(2) The unused words of the final bj, c,, or aj;
titions of the corresponding words in the preteding cards.

The result of solving a problem in linear programing is a matrix from
which may be read the optimum solution in x;.
zero values for an optimum solution may be selected in the following way:

Select a column with a zero c

value 1.0 in an i row of this j codumn
non-zero element in the j column, then the value of the variable xj is given

by the by.

Word two of the first card punched contains the floating-point
value of the function F; this replaces the value of n.

If there is a single coefficient of
» and if that coefficient is the only

15

Word one of the first card punched contains the number of elimi-

cards may contain repe-

Those variables that have non-


21

c. Equipment--assumed constant for various production rates.
d. Labor--assumed constant for various production rates.
e. Transportation--assumed to depend on distance.

It is advisable to emphasize here that all of the above assumptions are
based on estimates or guesses for explanation only and that the collecting of
the true data and computing of more exact costs is a separate problem. How-
ever, the solution of the problem just discussed is presented to illustrate
what can be done with basic data utilizing linear programing.

The minimum transportation costs for any tonnage mined is shown in figure
5. Notice the increasing cost for mining a greater tonnage. This illustrates
the manner in which transportation costs are minimized in this solution; the
deposits having requisite silica-alumina content and minimum haulage cost were
mined first.

The mining sequence for the 10 bauxite deposits is shown in table 2. The
mining sequence as computed for each production rate can be used to determine
the number and capacities of equipment required to meet the computed produc-
tion schedules for the various production rates.

TABLE 2. - Mining sequence for 10 bauxite deposits (by number)
for different production rates per period

Period Production (tonnage x 10°) per period

0.5 1.0 1.5 2.0 2.5 3.0
-|9, 10 5,9, 10] 5, 9, 10 |5, 9, 10]5, 9, 10] 5, 9, 10
~| 2,9 9,7 7,9 3,5 3,5, 8 | 2, 3, 8

-|2,9 5 5 3, 8 2,3, 8/2, 5
2,9 5, 3 3, 5, 8 {1, 2, 8 |2,7 1, 6,7

9,7 3, 8 3, 8 1,7 7,1 -

5 8,2 2, 8 6,7 - -

5 2,7 1, 2,7 - - -

5 2,7 - - - -

~|5 1,7 - - - -

5, 3, 8 - - - - -

-| 3, 8 - - - - -

3, 8 - - - - -

-| 3, 8 - - - - -

-| 3, 8 - - - - -

-| 2, 8 - - - - -

~| 2,7 - - - - -

2,7 - - - - -

~| 2, 7 - - - - -

. ~{| 2, 7 - - - - -
Wessesceceeeeeeeee | 1, 7 - - - - -
ZLesssssoees -| 1,7 - - - - -
22... ~| 1,7 - - - - -
23.04. 7 - - - - -
We rvsesccecscssceee | 7, 6 - - : - -



17

Operation instructions are:

Insert 80-80 board in the 533.

Press computer reset and program start.

Read in program deck.

Read in data sets (1), (2), (3), and (4) in that order.
Press punch on 533.

UPONFH

A plus sign on the sotrage entry switches will cause the computer to punch
only the final matrix. A minus sign will cause punching of the matrix after
each elimination.

Examples of Application

Minimum-Cost Model. - Much of the bauxite in Jamaica is in individual
deposits of varying tonnages, grades, and locations. The operation considered
is one using power-shovel loading and truck-to-train-to-ore-ship haulage. The
bauxite must be mined so that the silica and alumina content of the ore are
within prescribed limits when delivered to the alumina plant. These restric-
tions necessitate selective mining not only within each deposit but also for
all deposits as a group, so that the bauxite delivered to the ore ships will
provide an ore mixture that can be blended to an acceptable silica-alumina
content.

In setting up this problem for solution by linear programing, the above
requirements take the form of restrictions. The alumina content of the ore
must average above a specified grade; therefore, a minimum restriction can be
placed on the alumina content of the bauxite. In addition, the silica content
must have a maximum restriction to insure that it does not exceed some speci-
fied amount. The final objective of the linear programing solution is to
minimize the transportation costs.

A list of variables and their ranges used for this example are:

1. Number of deposits....... bee eee eee eeeee 10.

2. Tonnage in individual deposits --0.5 to 2.5 million tons.
3. Range in grade of Al,0, within deposits.40 to 58 percent.

4. Range in grade of Si0, within deposits..1 to 12 percent.

5. Rail haul distance...... -14 to 20 miles.

6. Truck haul distance «1/2 to 2 miles.

7. Required shipment grade of Si0,... «Less than 2 percent.

8. Required shipment grade of A1,0,.. ..More than 48 percent.

The four general equations necessary to express the mathematical model of
this problem are:

n
1. Production: x xj =b.

n
2. Silica restriction: 2 Wsio, x)j 2b, .
j


30

where:

Nc = number of points inside the circle.
nt = total number of points.

Areacg = area of circle.

Area, = area of square.

This problem is formulated for a digital computer in the following steps:

1. Scale two random numbers into the square; one is the x coordinate,
and the other is the y coordinate of the point.

2. Test and see if the point is inside the circle.
Test:

CP +9 - 1)

(-)
(+)

3. Cumulate the count of number of points inside the circle and total
points tested.

where;

inside circle, and
outside circle.

4. Compute the area of the circle.

The area of the circle is computed by:

Re
Area, = (=) Area,.
oT
In this example, the area of the
square is (2)° or 4 square inches,
(2,0) (2,2) and the area of the circle is Tr?
or 1(1)® or 7. Repeated trials
(looping) with the computer result
in a larger number of points to be
» used in computing the area of the
& circle and, therefore, increased
accuracy in the estimate of area.
Approximation of the area of the
circle (m7) is shown in figure 11.

>

Also, in other areas of
research, random sampling schemes
and Monte Carlo methods often are
used to supply simulated data for
designed experiments; the solution
(0,0) (0,2) is used as data for a mathematical
model. A digital computer makes

FIGURE 10. - A Circle Inside a Square Used for possible, ” a short Period of
Computing Area of the Circle bya | time» development of simulates
Monte Carlo Method operation data representing years
on . of real-life operation.



12

2. Select the element aij which gives the smallest value of this quan-
tity. This is the optimum element to eliminate.

3. When there are no more ayj's such that (1) is satisfied, search those
rows having bj <0 and ajj <0, regardless of its solving status. Select the

ais cs
element in rows having the smallest value of =. This is the optimum
element for elimination. *

4. If there are no rows having bj <0, the problem is solved.
Program Description

Use of this algorithm makes possible a simple data format and rapid com-
putational time. The program requires that only nonzero matrix elements are
stored in the computer's memory, thus making possible the solution of larger
problems.

A simple problem will illustrate the matrix form required by the computer
program. Consider five small iron mines, each with a different tonnage (T, ,
Ta, Tg, 14, and T,). Each mine also has a different grade (Fe, , Fe,, Fes,
Fe,, and Fe,). The mining costs at each mine also are different (C,, Cy, Cy,
G@, and C,). The objective used to solve this problem will be to minimize the
total mining cost from all mines with a production restriction of b, tons per
day and a daily grade restriction equal to or greater than (>) G. The equa-
tions or inequalities represented by a mathematical model, are:

1. T=b or (Lh +h +t +R +1 = b, J.

1

Tmo

J

2. § (FeT) > Gb, or
jel
((FeT), + (FeT), + (FeT), + (FeT), + (FeT); > Gb, ].

3. TyS dye Gels.


Sulfide ore
Oxidized moterial
Conglomerate

EXPLANATION
Caved ore

PER

oie

A= Break in Section

i Pigs ls 5
CUNT
5 | Oey
oP fpf. >> Cee

MOET EY i] [efor n tee
SN

i
IL. EERAS
Noe

4

APPROXIMATE SCALE, FEET

Undercut block

FIGURE 1, - Generalized Diagram of a Block Caving Method.


27

deposit was represented by a mean silica, alumina, and total tonnage value, to
developing schedules for each deposit. Mining schedules for each deposit
would be determined so that the required production per period would corre-
spond to the mean silica and alumina value restrictions for the deposits.

Each working area of each pit could also be considered for all deposits if it
is desirable.

Constant-Cost Model. - The previous model, developed for a production
scheduling problem, had a minimum cost for any tonnage mined. For contrast, a
model was developed for the following production restrictions:

1. Constant costs over the life of the mine.

2. Constant tonnage per planning (mining) period.

3. Constant grade per planning period.

Utilization of this constant-cost model will be illustrated for a portion
of a bench in an open-pit mine. Figure 8 shows four areas of a pit being
expanded. It is desirable to mine the ore out of the four areas of the bench
in 2-year planning periods. The model will be developed for four mining areas
(i) and two mine planning periods (j). The mathematical model is expressed

by the following equations:

1. Constant cost per planning period: & T (i, j) Cy - L=0.
1

2. Constant tonnage per planning period: 2 T (i, j) -M=0.
1

3. Constant grade per planning period: © Gj T (i, j) - N=0.
i
4. Tonnage inventory: 5 T (i, j) < (total tonnage);.

5. Minimization: (Number of planning periods) x (L) to be minimum or
total cost per period,
where:
= area,
planning period,
cost,
grade, percent copper,
tonnage,
constant cost slack variable,
constant tonnage slack variable,
constant grade slack variable.

ZEBrwAQaAGe

This model, expressed in complete equation form for four deposits and two
planning periods, is:

QT, +G& %, +; Ty +G& G, -L=0 )

G@ het p+ Cs Typ +QG Go- Lb Equation 1.

a
°
w


TOTAL TRANSPORTATION COST X10§, dollars

3.0F

2.0

High silico
(av. = 2.302 percent Si0g)——_____

“=— Low silica (av. = 1.472 percent SiO.)

TONNAGE X 10°
FIGURE 6, - Relationship Between Transportation Cost and Tonnage for Two Average Silica Contents.

Via


DESIGN AND APPLICATIONS OF SOME MATHEMATICAL
MODELS FOR MINE-SYSTEMS ANALYSIS!

by

Richard F. Hewlett? and D. E. Redmon?

INTRODUCTION AND SUMMARY

The mining industry utilizes a vast store of knowledge, wisdom, and engi-
neering talent in designing mining systems. However, the complexity of the
required systems and the presence of numerous variables frequently necessi-
tates making important decisions without having adequate accurately detailed
information. That many of the decisions made in the past were nearly correct,
in the long run, is substantiated by the number of successful mining enter-
prises that exist in the world today.

Subsequent to 1940, the disciplines of operations research and systems
engineering were developed. The disciplines have produced methods of provid-
ing better information to persons responsible for making decisions.

A mining system is a composite of all components within the organiza-
tional entity. Mine-systems analysis comprises the scientific and engineering
knowledge applied in the study of existing mining systems or in the design of
new mining systems. The most sifnificant point is that mine-systems analysis
is concerned with the component parts of a mining system individually and in
the relationship of the components to the rest of the system.

Analysis of mine-system components, groups of related components, and
aggregated components provides the analyst and management with information
regarding the effect on the system of procedural or operational changes made
in the components. A model of a mining system can be designed in mathematical
form, and using a computer to solve the mathematical model of the system can
provide, in a short time, an analog of the response of the system to con-
trolled minor changes in the components.

TwWork on manuscript completed March 1962.
? Former mining methods research engineer, Mining Research Center, Bureau of
Mines, Denver, Colo.

SMining methods research engineer, Mining Research Center, Bureau of Mines,
Denver, Colo.


ii

Fig.

12.

A circ
by a
Estima’

ILLUSTRATIONS (Con.)

le inside a square used for computing area of the circle
Monte Carlo method.........ecseeeeeeeeeees
tion of area of a circle using a Monte Carlo method..

Relative cumulative frequency for a Poisson distribution with

he
Analog

Frequency distribution of load arrivals at transfer point....

Basic
prod

of mine-haulage system.

data; constituents and relationship to system design and
luction scheduling...

TABLES

Values assigned to the 10 bauxite deposits....... deen cece ence eeeaee

Mining sequence for 10 bauxite deposits (by number) for different
production rates per Period......ceecee cece ee ceeeeeecenceeseeeens

Transportation costs and tonnages possible to mine for the

diff
Sequen
rate!
Values

erent production rates per period.......cececeecreeceecesvece
ice of mining out deposits by number for different production
S PCL Period... .cceecsecscceeceeeee beeen eee eee e eee enee
used for a mathematical model with constant costs per

Planning Period. ..ecccsesccccrece ce cene ee eernee renee eeeneeeenens

Values
tonn:
Values

alumina content

Values
sili
One-da:

Case 1;
Case 2;
Case 3;

determined by Monte Carlo methods for bauxite-deposit
ALCS eee eee cence ee ee eee eee ee ee eee Eee EEE EEE Eas
determined by Monte Carlo methods for bauxite-deposit

determined by Monte Carlo methods for bauxite-deposit
ca content......... .
yy production; simulated by Monte Carlo methods...
10-day production, 5-unit haulage, overtime crew.
10-day production, 5-unit haulage, no overtime crew.
10-day production, 6-unit haulage, no overtime crew..

19

21

23

26


13

The mathematical model is described by the following matrix and vectors:

jo n by column
1 2 3 4 5 6 7 8 9 10 1
1 1 1 1 1 1 b,
2 Fe, Fe, Fe, Fea Fe, -1 Gyb,
3 1 1 Tt,
4 1 1 tT
5 1 1 Ty
i
L 6 1 1 ae
m
7 1 1 Ts

c j row q Cg Cy G, Cy

Row 1 contains the five coefficients (each equal to 1) of the tonnage for each
deposit (equation 1). Row 2 contains the average iron assay values for each
of the five mines, elements [2,1] through [2,5]; element [2,6] is a minus one
which makes the inequality (greater than G,b,) an equality. Some value will
be added to the summation of the product of tonnage times grade for each mine,
so that the result _is greater than G,b, (equation 2). The coefficients [3,1],
(4,2], [5,3], [6,4], and [7,5] are used to designate mining a_tonnage equal to
T,, Tz, Ts, %, and Ty. The slack variable coefficients [3,7], [4,8], [5,9],
[6,10], and {7,11] are used to indicate that a tonnage less than the maximum
can be mined from any or all five deposits (equation 3).

Input Data Format. - Four sets of data cards are required to describe the
problem for this linear programing routine as used with the IBM 650:

1. One card is required to give the number of rows (m) and the number of
columns (n). The value of m is coded in word one, and the value of n is coded
in word two in fixed point with leading zeros. Words 3 - 8 may be blank.

For the iron example, m= 7 and n = 11.

2. The by values are coded in floating point with one value per word and
eight values per card. Signs need to be punched only if they are minus. The
values of b; are coded consecutively from i= 1toi-=m. Any unused words of
the last b; card should be punched with zeros. It is essential that the
values be coded consecutively and with no missing values.


These definitions are satisfactory when used with familiar mining models
such as maps, flowsheets, and three-dimensional models. However, the defini-
tions are not adequately definitive when used with less familiar abstract
models. Attempts to establish standard categories for models have been
largely unsuccessful--primarily because of the existence of multitudinous
varieties of models, complex relationships between various types of models,
and semantic problems encountered in definitions.

As used in this report, the description of a model is: "Anything that
represents all or part of a system." Specific models are designated by
descriptive modifiers. No attempt will be made to rigorously justify use of
each modifier; however, description will be adequate for clarity.

Each model is developed for explanatory purposes and is used in the study
of part or all of a system. Four types of models might logically be distin-
guished as (1) pictorial, (2) iconic, (3) analog, and (4) symbolic.

Pictorial models, as the modifier implies, are photographs, paintings, or
ground plans which are in a two-dimensional medium and pictorially illustrate
the item that they represent. Isometric drawings, such as the one illustrated
in figure 1, probably should be included in the pictorial-model category.

Iconic models have the same appearance as the items that the models repre-
sent and consist of three-dimensional mine models, topographic models, and
scale models. Pictorial and iconic models are useful for representing a thing
at a specific moment in time; the model describes but does not explain or
interpret a physical situation.

Analog models show analogy; analogy is a likeness between two things or
of one thing to another, the resemblance consisting not of or between the
things themselves but rather of or between two or more attributes, circum-
stances, or effects. Graphs are simple analog models; distances along desig-
nated axis are analogous to properties such as grade, tonnage, cost, percent,
and frequency. Company organization charts, computer flow charts, laboratory
setups, and pilot plants are typical of effective analog models.

Symbolic models are abstract descriptions of an operation or system.
Symbols are employed to represent the components of a system and the relation-
ships between the components. The symbols might be logical or mathematical,
depending upon the system being studied and the data available.

Symbolic logic is a shorthand representation of thought processes such as
might lead to a decision (2, p. 133). In symbolic logic, symbols are used in
place of word propositions, classes of things, or functional statements; rela-
tionships between propositions, classes, and functions also are concisely
expressed by means of symbols (6, p. 188).

In analysis of mining systems, mathematical symbols and models are used
more frequently than symbolic logic. Mathematical models are concise forms of
symbolic models, in which relationships between components are expressed by
symbols of mathematics and the symbolically designated components can be


19

TABLE 1. - Values assigned to the 10 bauxite deposits

Tonnage, Average | Average Transportation
Deposit No. x Al, 0, , SiO, , cost per ton
10° percent percent
1. 1.1570 47.544 2.192 0.367
2. 1.5530 52.516 2.070 -361
3. 1.1810 46.695 +746 +319
4. +0440 52.706 2.224 -415
5. 2.0085 48.813 1.045 +298
6.. 1.5660 52.192 2.489 +373
Tee 2.1085 52.070 -515 -367
8. 1.7700 49.254 +923 «322
9. . 1.3025 47.776 1.148 +241
lO... eee eeee cere eeece - 1395 48.955 1.364 +226

The matrix form in fixed point for this problem using 10 deposits is
shown in figure 4. A listing of the data in matrix form for a 5-million-ton
production is shown in appendix B.

Using basic data from each of the 10 deposits, the following questions
can be answered with linear programing solutions:

1. Minimum transportation costs for any tonnage mined.
2. Mining sequence for any production rate.

3. Effect of assay or ore reserve computational errors on the transpor-
tation costs for any tonnage.

4. Total transportation costs for any desired production rate for any
total tonnage.

5. Equipment requirements and scheduling for various production rates.
Also determined from a study of these results would be the economic production
rate and the economic total tonnage to be mined (life of the operation from
these 10 deposits).

It is realized at this point that a linear programing solution to this
problem, or similar problems, depends on the following basic data and
assumptions:

1. Ore reserve data:

a. Tonnage in each deposit.
b. Grade of SiO, in each deposit.
c. Grade of Al,O, in each deposit.

2. Cost data:

a. Mining--assumed constant for each deposit.
b. Milling--assumed constant for various production rates.


b f mi
FE | report of investigations OLOS

DESIGN AND APPLICATIONS
OF SOME MATHEMATICAL MODELS
FOR MINE-SYSTEMS ANALYSIS

By Richard F. Hewlett and D. E. Redmon

metadc38701 |

!
UNITED STATES DEPARTMENT OF THE INTERIOR
BUREAU OF MINES

1963


42

The system is in balance after the third day, so analysis of case 2 uti-
lizes information from table 11 after the third day, as follows:

xU = 5 + 21 + 10 = 36,
TP; = 21, and
TPE = 36 -(9 + 10) = 17.

The plan represented by case 2 requires 36 haulage units and expansion of
the transfer point to accommodate 11 additional load units and 7 additional
empty units.

Analysis of the Problem--Case 3

If the dump capacity and miin-haulage capacity could be increased, it
might provide the cheapest and most efficient operation. The same transfer-
point-arrival data as in case 1 and case 2 are used in the construction of
table 12. It is assumed that the haulage capacity is now 6 units instead of
5 and that no extra haulage crew is used. According to data presented in
table 12, three loaded units are left on the transfer point on the third day;
the maximum number of units on the transfer point is 10, and a minimum of 1
unit is on the transfer point during any period.

Analysis of case 3, utilizing table 12, is as follows:

zu = 6 + 10 + 10 = 26,

u

TP, 10, and

Pg

25 -(1 + 10) = 15.

The plan represented by case 3 requires 26 haulage units, expansion of
the transfer point to accommodate 5 additional empty units, and expansion of
the dump and haulage-motor capacity.

The foregoing example of an application of the Monte Carlo technique to
a mine-haulage problem is obviously oversimplified, but the example illus-
trates a plausible use of the technique in solving problems for which basic
data are not available. The technique is especially appropriate for solving
some of the problems encountered in designing a new mine plant as well as for
examining the effect of proposed changes in an existing plant.


are profits, costs, production, and other measures of effectiveness. The
optimum solution is subject to certain restraining conditions, such as avail-
ability of various raw materials, government or legal restrictions, marketing
or storage restrictions, mill and other production requirements, and so forth.
To use linear programing, the objective must be represented by a linear equa-
tion, or the variables must be able to be made linear by a suitable transfor-
mation. Restriction must be represented or expressed by a system of linear
equalities or inequalities. A digital computer is ideal for solving problems
using a linear programing technique because the method involves many itera-
tions to find a better or optimum solution.

The Simplex Method

The simplex method provides a systematic iterative computational proce-
dure by which linear-programing problems are solved. Each successive itera-
tion provides a solution which is no worse (and which is probably better) than
the previous solution. The iterative (successive solution) procedure ends,
after a finite number of steps, with an optimal (best) solution. The simplex
method is a simple and effective aid to decision making and can be highly use-
ful to anyone that needs to know the most desirable course of action in a sit-
uation governed by many known variables and conditions which can be

quantified (3, p. 4).

A linear-programing problem has three quantitative components: (1) An
objective, (2) alternative methods or processes for attaining the objective
and, (3) resource or other restrictions (4, p. 2).

If a problem exists in which linear programing might be used, that is,
an objective represented by a linear equation and restrictions represented by
a system of linear equalities or inequalities, the simplex method provides a
systematic manual procedure by which the set of basic feasible solutions can
be examined and an optimal solution determined.

The simplex method probably can be described best by an example of a
problem that can be solved fairly easily by algebraic computation.

Assume two stopes in a mine producing a lead-zinc ore.

Basic information:
Production limit, Cost, man-hours

Zn, percent Pb, percent tons per day per_ton
Stope l.. 4 6 40 4
Stope 2.. 8 4 60 6
Requirements:

1. Eighty tons of ore per day.

2. Ore to contain an average of not less than 6.5 percent zinc and not
less than 4.5 percent lead.


quantified. A familiar formula used in determining pounds of metal in a short
ton of rock is: P = 2,000 g, where P is pounds metal and g is proportion of
metal expressed as a decimal. The formula constitutes a simple mathematical
model.

CONSTRUCTION AND METHODS OF SOLUTION OF MATHEMATICAL MODELS

Construction and solution of a mathematical model are intimately asso-
ciated with each other. Construction involves logic; an orderly sequence of
steps is followed to describe a situation.

After the model has been constructed, selection of a method of solution
can be made. Mathematical models are solved by analytical, numerical, and
Monte Carlo methods.

Constructing the Model

The first phase in constructing a mathematical model is formulating the
problem. An analysis of the system to be studied must be made, and the prin-
cipal phases of the system requiring study must be isolated. Methods of sys-
tem control must be considered. Measurements of system effectiveness are to
be determined, and policies affecting the system must be evaluated.

The second phase of model construction is preparing a flow chart of the
portion of the system being studied. Control variables must be formulated
into equations that will express the effectiveness of the system. The flow
chart will be either a pictorial or an analog model. In general, the varia-
bles of the system will be of two types. One type can be controlled by manage-
ment decision, and the other type is the uncontrollable aspect of the system.

The third and final phase of constructing the model requires listing the
components that contribute to the efficiency or inefficiency of the system.
Cost items commonly are used because they are a good measure of efficiency or
profitability. Analysis of the individual components of the system is impor-
tant in order that a realistic and practical solution can be obtained from the
constructed mathematical model; the results are only as good as the data and
the method of solution.

Methods of Solution

Linear programing and Monte Carlo are two methods of solving mathematical
models that are described in this report and illustrated with examples. A
general digital computer linear-programing routine is listed as an example of
a method to solve linear programing problems. Other techniques used for
linear programing and Monte Carlo methods of solution are included.

Linear Programing
Linear programing refers to methods used for solving a general class of

problems that have many variables and few equations relating the variables in
which optimization is essential. Some of the objectives that are optimized


38

TABLE 9. - One-day production; simulated by Monte Carlo methods

Place Period Range

1/2/3141 5] 6] 7] 819 [10] 11] 12/13 114 [15 |16/17 118/19] 20

x X} [X} | X/X] xX X] X X| xX

XX} |X} X)X}X x x x x

x X/X/X} |X| X] X} xX x x

x X/X/X} X| X} X x X| xX

X|X|X|X x x] X x X| xX

x X/X|X x X| X| X} X x

x X| [xX xX] X X| X| X x x

X|X X|X/X x X| xX x x

X} |X|X}] | X/X) [xX] x xX] X x

X] |X] |X X| X| X X| X| xX x
Arrivals...... | 6/4/5/3/6|5)/7/4]7] 6} 7) 5) 3} 4] 7] 3] 4] 6] 6] 2]}2- 7
On station. 6] 5}5}3) 6/6) 8} 7}9}10/12)/12)10}] 9}11} 9} 8) 9/10] 7] 3 - 12
Taken... 5}5]5})315/515)5/5] 5] 5) 5) 5} 5] 5] 5] 5) 5) 5) 5/3 - 5
Left.... 1]/0j0/0} 1/1) 3)2)4) 5) 7] 7) 5] 4] 6] 4) 3) 4) 5) 2} 0- 7

The analysis, in this oversimplified example, must be concerned with:
(1) Capacity of transfer point and number of load-carrying units required to
satisfy the system requirements, and (2) cost of modifications, equipment, and
operations.

Analysis of the Problem--Case 1

The first solution is to make modifications in the transfer-point capac-
ity, if required, and to retain the present main-line haulage. The simulated
10-day-production is recapitulated in table 10 which shows: (1) Number of
units arriving at the station in each 20-minute period, (2) number of units on
the station before the haulage trip is made, (3) number of units taken by the
haulage trip, and (4) number of units left on the station. Minimum and maxi-
mum occurrences are listed in the range colum. Calculations in table 10
illustrate the situation that might exist if the transfer point were not
limited in capacity and a haulage crew worked overtime to remove loads remain-
ing on the transfer point at the end of shift.


26

TABLE 4. - Sequence of mining out deposits by number for

different production rates per period

Order of being Production rate (tons x 10°) per period

0.5 1.0 1.5 2.0 2.5 3.0
First...........| 10 10 10 10&9 |10&9 10 & 9
Second... 9 9 9 5 5 3&8
Third.... 5 5 5 3 3&8 5&2
Fourth..........] 3 3 3 8&2) 2 7&1
Fifth...........] 8 8 8 1 7 -
Sixth.........0. 2 2 7 2 7 - -
Seventh.... «{ 1 7 - - - -
Eighth..... . 7 - - - - -
Deposits left...| (4 & 6)|(1& 4 & 6)}/(1& 4 & 6)/(4 & 6)|/(1 & 4 & 6)/(4 & 6)

faz _——— Present pit limits

/

NYS \ /
4
Ve

D Leexpansion limits

FIGURE 8. - Four Areas of a Pit Being Expanded.

No Scaie.

Use of this technique
to solve scheduling problems
in existing ore bodies or
for a number of ore bodies
is simpler than for the hypo-
thetical situation just
described. The restriction
for costs, numbers of work-
ing mines or areas, and
grades of ore can be defined.
Construction of the mathemat-
ical model can be done
according to the objectives
of the study. Available
equipment determines the
minimum and maximum number
of sites that can be mined
in any given period.
Required production is
usually determined by the
mill, as is grade. It would
require only simple changes
in the model to study mill-
ing problems. A study of
both the mining and milling
operation would result in an
optimum production rate and
recovery. This would give
maximum deposit utilization
at a minimum cost.

Use of this production
scheduling application can
be extended from the entire
operation, where each


43

- Case 3; 10-day production, 6-unit haulage, no overtime crew

TABLE 12.

Range

Period

Item

Day

1234 56789 10 11 12 13 14 15 16 17 18 19 20

noon

pres
NNNO

ANNNO

wood

wowed

ttst0

atto

nnoea

t4tt0

otto

nnwod

nROON

onond

nnoea
tnNNO
nnoa
ninne
wows

aANMNS
ninne
ttto
wows

On station

Taken

Left

Arrivals

1

anon

prog
NNNO

mninINnd

wove

wowvsd

ttt°o

TONS

nnod

tattoo

nnne

wows

wowed

ninNne

+t4t0
mnNNnS
tinine
trond
AHO”

wowed
ANNNO
AMNS
mnNne

On station

Taken

Arrivals
Left

2

anoM

pros
aan

naon

mOON

trond

wnon

naon

anon

wowed

minNned

mnnNnS

aAMNS

NAMM

nnod
mniNnNne
ammo
ninne
wowd

nnne
ANNO
mnnne
aoaHo

On station

Arrivals
Taken
Left

3

ARON

prog
AMMO

AIMS

wowed

tatsto

wowd

wove

tattoo

mnnne

wttto

+tNWNO

nnod

NDON

AnOM
minNne
ANNO
mnNnNne
tiNNO

nnod
nooo
wnon
wnon

On station
Taken
Left

Arrivals

4

anon

prued
HNNO

nowosd

owned

onwen

nrod

tast0

manne

NOON

anon

ANNNO

wows

amMnNe

moon
NNNO
minned
wood
otto

wonwond
onwon
nnod
ANN

On station

Arrivals
Taken
Left

5

woot
a

preg
ANN

otto
onwend
onwen
nnod
tatt0
ttt0
AMM
minined
atto
wnon
nnod

ttsto°o

wood

MOosd

naAoM

a~oort
a

wmnmnon
mninNne
aAMNe
tattoo

On station
Taken

Left

Arrivals

nnod

hare
aAMNS

+ttt0

Novos

ono

wnond

nnod

AMMO

wowd

minNne

aAmMnNo

On station

Arrivals
Taken
Left

ROON

read
dane

wows

wowed

mnne

tattoo

wowed

atto

nnor

wood

tattoo

wooed

ANNO

Hans
mnnNne
AMMO
minNne
tooo

noon
onwd
nnod
wood

On station

Taken

Arrivals
Left

8

WOON

pres
NNNO

nnne

+tNNS

wonwon

nnod

t4tt0

anne

nnne

nnn?

nowd

nnod

wowed

tNNS
nnod
woon
WOON
wowed

NANO
amo
tattoo
minnoe

On station

Arrivals
Taken
Left

7

3°95 5 2-

5374 24646 75764 64 7

Arrivals

10

non

mae
NNO

nino

nno

tt0

nor

tt0

wod

nn oO

nod

nod

wwod

nod

wood
tt)
wood
tt)
NNO

nino
nod
Amo

On station

Taken
Left



CUMULATIVE TRANSPORTATION COST PER TON, cents

26

1 1 1 L 1 L L 1 L

3 4 5 6 7 8 9 10 in
TONNAGE x 10°

FIGURE 7. - Relationship Between Cost per Ton and Tonnage for Various Production Rates in Millions of Tons.

St


The reliability of results obtained by solving a mathematical model is
obviously dependent upon the reliability of the model and of the input data.
Correct design of a mathematical model usually follows preliminary analysis;
the solution of the model, utilizing appropriate data, constitutes terminal
analysis.

Basic data consists of all data pertinent to the system or component
being studied. The types of data required are determined from the nature and
scope of the problem and from the model that is used to describe the system or
component.

Mine-systems analysis is used in some degree by all operating mining com-
panies. Level of use depends entirely upon the desires, requirements, and
attitude of individual company management.

The purpose of this publication is to show possible application, within
the field of mining, of some innovations to methods of analysis that have been
developed and improved in recent years. An effort has been made to provide
examples of methods that are of interest to systems analysts, regardless of
the extent to which systems analysis is practiced within their mining
operation.

A short discussion of models in general is followed by discussions of the
construction and solution of mathematical models.

Linear programing methods and applications are described and illustrated
by examples of a manual method and a digital computer routine.

Monte Carlo methods are useful in determining a desired probabilistic
property of a system. Illustrative examples, in which Jamaican bauxite depos-
its and an underground mine-haulage system are simulated, are included in this
report.

Lengthy discussions concerning details of technique are avoided, because
numerous books are available in which the theories and computational proce-
dures are thoroughly discussed.

It is hoped that the examples of model design and solutions as presented
in this report will be of assistance in solution of similar or analogous
problems that are present in all mining systems.

MODELS

The term "model" has several definitions; the most common are: A set of
plans for a building to be constructed, or drawings to scale, or a ground plan;
a copy; a miniature representation or facsimile; any thing (or person) that
might serye as an example for imitation; and, a pattern of something to be
made (7).

Underlined numbers in parentheses refer to items in the bibliography at the
end of this report. Page references refer to pages in the items and not in
this report.


16

Consider this example:

1 1 1 0 0 0 2
aj 2 0 1 1 1 0 7 by
3 0 0 0.3 0 1 3
5

The solution is: x, = 2, x, = 0, x, = 0, % =7, x, = 3. Thus, all variables
not accounted for by the method just presented are zero for the optimum
solution.

Computer Information and Restrictions

The possibility of exceeding the capacity of the computer exists. First,
the programer must be sure that the data does not exceed:

n <200, and
m S100.

Number of non-zero elements < 599.

If the number of non-zero elements exceeds 599 during the course of the
computations, the computer halts with an identifiable error condition. In
general, a safety factor of two (2) is necessary for the number of non-zero
elements in the data.

Computer halts occur:

1. If the computer stops with a storage selection light after punching
the matrix; this indicates completion of the problem.

2. If the computer stops on a halt code, and the upper accumulator
(8003) reads:

a. 00 0000 0001--no positive solution to the problem exists.

b. 99 9999 9999--the number of non-zero elements has exceeded 599.
If 2a and 2b stops occur, set the storage selection switches to (+) and press
program start for the problem punched out in its present state.

Console settings are:

1. Storage entry: 70 1951 1999" 5. Display: 8003
2. Programed: STOP 6. Overflow: STOP
3. Half cycle: RUN 7. §Error: STOP

4. Control: RUN




33

Alumina Values

Deposit alumina values also were estimated to have a normal frequency
distribution with a mean of 50 and a standard deviation of 2. These values
were drawn from a normal random number table with a mean of zero and then
coded as shown in table 7. The average grade of the alumina values is 50.702,
which is within the limits of expectation for ten assay values.

TABLE 7. - Values determined by Monte Carlo methods for
bauxite-deposit alumina content

Normal random number, u, Alumina value

Value from a population, with (u + 50)
w= 0, o = 2.0

Le sscccccccccee 2.489 52.489

Zeccccccccecces -.515 49.485

-923 50.923

-1.148 48.852

1.364 51.364

.113 50.113

-931 50.931

.378 50.378

3.863 53.863

-1.377 48.623

Silica Values

Deposit silica values were assumed to have a Poisson frequency distribu-
tion with a mean and variance equal to 3.6.5 The plot of the relative cumula-
tive frequency (r.c.f.) of this distribution is shown in figure 12. Two-digit
random numbers were coded by being divided by 100 and were scaled on the ordi-
nate (Y axis); the r.c.f. plot then determined the corresponding silica value
on the abscissa (X axis). Table 8 shows some values determined. The selected
values shown have a mean of 2.70, which is less than the mean of 3.6. These
values were selected because they are closer to a mean of 2.0, which is the
maximum average silica content commonly used in blending. These values are
more typical of the range of silica values within a deposit.

®some text books show this as \ and others as Np-


PROBABILITY DISTRIBUTION AND RANDOM NUMBER SCALE

4 L 1 1 L 1 1

2 3 4 5 6 7 8

CLASS INTERVALS AND SILICA VALUES

FIGURE 12. - Relative Cumulative Frequency for a Poisson Distribution With A = 3.6.

ok

ve


1952
9053
1953
1953
1953
1939
1940
1940
1940
1940
1940
0001
0002
0003
0004
0005
0006
0007
0008
0009
0010
0011
0012
0013
0014
0015
0016
0017
0018
0019
0020
0021
0022
0023
0024
0025
0026
0027
0028
0029
0030
0031
0032
0033
0034
0035
0036
0037
0038
0039
0040
0041
0042
0043
0044
0045
0046
0047

4090
n0n0
7019
0000
1519
6919
6919
6919
2219
6919
2400
1250
1280
1318
0048
1272
1275
1254
1431
1415
1447
1371
0098
1265
1425
1306
1367
1525
1261
1419
1301
1355
1361
1284
1568
1621
1411
1631
1314
0049
1981
1511
1340
1332
1619
1618
0099
1312
1650
1482
1517
1335
1366
1622
1470
1551
1613
1555

0919529054+
0901098000+
6919561953+
6919541953
6519561953
4419531939
2419111940
2419171940
2419231940
2419291940
2419351940
8100021256
6912831286
5000011324
2790001253
3500031281
6912781331
5019991310
3500031389
4413191270
8139991304
3312891465
2412511354
0000000000
4412791380
6913001303
3500031475
6080031433
1090531369
4813721323
3490551504
2020001531
4614541565
6580021293
2136601571
6090571329
5980021469
1590541589
6090591971
8080021357
6090561989
5980021569
1590501597
8180021391
3000041429
6515221277
6940001553
2490531968
3000041961
6590531440
4513201968
8080031344
3990551420
1090521479
5980021529
3990551505
5219991520
630000165)

APPENDIX A. - COMPUTER PROGRAM LISTING FOR A LINEAR PROGRAMING ROUTINE

5980
9000
3819
0000
0280
0880
1819
2419
2919
0319
0019
1256
1286
1324
1253
1281
1331
1310
1389
1270
1304
1465
1354
1273
1279
1303
1475
1433
1369
1372
1504
1531
1565
1293
1571
1329
1469
1589
1971
1357
1989
1569
1597
1391
1429
1277
1553
1968
1961
1440
1432
1344
1420
1479
1529
1505
1520
1651

8000529055+
2790008002+
2419401954+
2419371937
2419381954
1080011954
6519011912
6919061918
6919051924
2400001930
3000041936
8200001262
2412891292
5300471330
0920001339
8880031290
2490511287
4613181276
6580021397
6012731319
5019991360
4614681318
6994001410
1000000027
6912511404
8880011259
3000031383
3000031341
2160001353
5880021481
2132471450
6590531489
4515181454
3500031351
3957601560
1090521337
4814221373
2090521296
1190521379
6937601463
1090531547
4814721423
2090531604
6032471401
8280031288
1514301285
2490101309
6090561326
5980021969
1590501647
8180021441
6590521451
3256601387
2160001653
4815321533
3252471523
2440001482
3000041412

0051
0000
3870
0070
0300
0200
1622
2222
2869
3424
4070
1262
1257
1330
1339
1290
1287
1276
1397
1319
1360
1468
1410
1260
1404
1259
1383
1341
1353
1323
1450
1489
1518
1351
1560
1337
1422
1296
1379
1463
1547
1472
1604
1401
1288
1285
1309
1326
1969
1647
1441
1451
1387
1653
1532
1523
1533
1412

6690569057+
4090569053+
6919571955+
7019511951
6919571955
1019581955
4619021913
2400001919
2400001925
3000041931
6919391937
6912651268
6594001315
4213331292
6180051297
6076601365
8120001294
6912781381
3500031305
3913681418
67800 31267
6913681421
2412631318
6090511317
8180011460
6990501515
2190571291
2090560148
6300001400
8880021531
8139991307
1590501497
6512631467
8280031510
3214131539
2160001403
5880021581
6590561453
4612821621
2490551519
2160001503
5880021332
1690581561
3990551654
6057601316
2015221975
6932471600
1090531483
4815721473
2090531405
6060001455
3500031359
2156601424
6300001501
5880021491
2152471601
8880021541
1614161972

0001
0000
1951
1951
0000
0001
1918
1924
1904
0000
1911
1268
1315
1333
1297
1365
1294
1381
1305
1418
1267
1421
1283
1317
1460
1515
1291
0148
1400
1481
1307
1497
1467
1510
1539
1403
1373
1453
1282
1519
1503
1423
1561
1654
1316
1581
1600
1483
1572
1405
1320
1359
1455
1501
1491
1601
1541
1972

2920009055+
5100509054+
2419391958+
0000000000
2419361937
1580018003
6519081914
3000041920
3000041926
6919341932
2219391938
2412711274
4613181269
5200481339
1013001255
2413681321
3432471347
2412711374
8880031264
8120001325
6080021375
2412891342
9999999999
4414711322
6932471350
2490531521
6512631417
8200001454
3000041311
8180021439
5019991313
2090531554
3500031575
6036601615
2114131266
6300001500
8880021424
1690521461
6590571639
6990501625
6300001550
8880021382
4619811965
2190051611
3390051295
8180021390
2490551356
2160001603
5880021432
1690581513
6590561327
8280031370
3500031563
3000041362
8380020149
4414821406
6012651570
4613761426

9057
0000
1952
1952
0000
0000
1917
1923
1929
1935
1911
1274
1269
1292
1255
1321
1347
1374
1264
1325
1375
1342
1278
1471
1350
1521
1417
1454
1311
1439
1313
1554
1575
1615
1266
1500
1424
1461
1639
1625
1550
1382
1965
1611
1295
1390
1356
1603
1473
1513
1327
1370
1563
1362
0149
1406
1570
1376

49

2192009058+
0000000000+
1119571952+
0000000000
6019561936
6919038002
3500041915
1019071921
6919291927
2219341933
6919021939
2490511280
4512721318
5000020048
4412571260
4412751276
8139991254
6594001431
6077601415
3432471447
2413681371
6980050098
1111111111
6012711425
2490551306
6012631367
3500031525
6090561261
5980021419
6032471301
6820001355
1690581361
8080031284
3490551568
2090521424
3000041411
6590521631
4513141424
3000040049
2490531981
3000041511
6590531340
4516181981
6090531619
2157601382
8280020099
6990501312
6300001650
8880021482
4619681517
3500031335
6136601366
3000031622
2490071470
6132471551
8359991613
2460001555
1780011583


TRANSPORTATION COST PER TON, cents

35,

30

25

3 4 5 6 7
TONNAGE x 10%

FIGURE 5. - Relationship Between Transportation Cost and Tonnage.

(A




This publication has been cataloged as follows:

Hewlett, Richard F

Design and applications of some mathematical models for
mine-systems analysis, by Richard F. Hewlett and D. E.
Redmon. [Washington] U. S. Dept. of the Interior, Bureau of
Mines [1963]

ii, 51 p. illus., tables. 27 cm. (U. S. Bureau of Mines. Report
of investigations, 6158)
Bibliography: p. 48

1 Mining engineering. I. Redmon, D © E _joint author. I.
Title. IL Title: Mine-systems analysis. (Series)

TN23.U7 no. 6158 622.06173

U. S, Dept. of the Int. Library



41
Range

Period
12,3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

- Case 2; 10-day production, 5-unit haulage, no overtime crew

Item

TABLE 11.

Day

nawnn
a
pone
AMMO
anna
wonn
a
want
towne
Mans
nan3e©
a
tant
monn
a
nann
a
nawnn
a
wonn
a
Rant
tna
none
nond
won
anno
minne
tAINS

oond

On station

Taken

Left

1 Arrivals

aonn
=

enn
NANO
monn
a

wonn
=

want
tone
tant
monn
=

tone
nant
want
wane
nnn
tN INN
INoOINM
tone
tans
aonn
=

wond
NANNO

aminine

On station

Taken

Left

2 Arrivals

0 19 1 0
at
pone
aAMO
nono
aoe
nono
a
tan©
a
wnnn
a
rane
=I
Oant
won
nInHS
ninNne
anno
atto
nna
miInINnS
Otto
mond
oond
mniANS
Otto°

nond

1
6
5
1

On station

Taken

3 Arrivals
Left

9
8

5
9-13

3-
14 - 18
3-
5
7-14

5 5

13 13 13 12 11 11 10 11 12 11 12 10
6 6 3

955445 4664 6 3
14 12 12 15 18 18 16 17 16 18 19 17 17 15 14 13 15 16 17 15 12 - 19

18 18 18 17 16 16 15 16 17 16 17 15
55555 55 5 5 5

55555555555 55 55 55 5 5 5
9 7 7 10 13 13 11 12 1113 1412 12 10 9 8 10 11 12 10

4358 853647635344 7

6
6
5
1

l
1

On station

Taken

Left
On station

Taken

Left

On station

Taken

Left

4 Arrivals

5 Arrivals
6 Arrivals

3- 7
12 - 17
5
7 - 12

7665 4
14 15 16 16 15
5 5 5 5 5
7 910 1111 10

3
12
5

5 6
13 14
5 5
89

7 7 In
aa
ninne
aos
n4nnS
aoa
woninea
a
mann
4
tt
=
nono
aoa
sto in 00
a
tN AD
~
tT
=

tond
soa

7
17
5
12

On station

Taken

7 Arrivals
Left

7
8
6
1
7

1
1

5
7 - 16

1-
12 - 21
2-
9

4

2-
10 - 17
5
5-12

3°95 5

6755 53 4764 5
14 16 16 16 16 14 13 15 16 15 15
5 55 55 55 5 5 5 5
911111111 9 8 10 11 10 10

555555555 55 55 555 55 5 5
10 810 9 6 5 6 5 6 8 8 10 11 10 11 10 12 10 10 10

15 13 15 14 11 10 11 10 11 13 13 15 16 15 16 15 17 15 15 15

53742464675 764 64 7

6
6
5
1
5
5
5
0)

1
1
1
L

On station

Taken

Left
On station

Taken

Left

On station

Taken
Left

8 Arrivals
9 Arrivals

10 Arrivals


39

Range

Period
123 4 5 6 7 89 10 11 12 13 14 15 16 17 18 19 20

roduction, 5-unit haulage, overtime crew

10- da:

- Case 1;

TABLE 10.
Item

Day

Rann
=

pene
NNN

annN
wonn
a
want
tone
Mant
noine
a
tant
monn
a
nann
a
Rann
a
wonn
a

nant
tnnNN

none

nond

Arrivals
On station

1

aonn
a

pond
NNNO

nonn
a
wonn
=
want
stone
tant
nonn
a
taonm

nant

Left

On station

Arrivals
Taken

2

anno

aoa
pou

amano

nono
aoa
nano
a
tan0©
a
wann
=
nano
a
Dans
wonrd
mniNnKHe
mnNne
ANNO
atto

nnina
miNniNne

otto

mond

Arrivals
On station

3

aannt

toad
otto

mond

waonn

tRNA

won

wnna

toned

mnnnA

tn

tone

nant

mans

aant
mnnNno

atto

nond

Taken

On station
Left

Arrivals

4

aonn
a

pene
AMMO

nonn
wonn
ant
ann
ond
nnN
ant
ant
ane
ond
tt°

~anm
mine

mnm
~aInm

nnn

376 63 65 28 1629 5347 6

Arrivals

5

aann
a

pane
aNNS

mone
wonn
a
want
nann
trond
th nN
Monn
nonn
=
monn
a
wane
a
Rone
a

tant
wonn
a
Nant
nine
a
DHIN|]
a
~~ Ine

mnnne

43
m 4 3
43
00

On station

Taken

Arrivals
Left

6

mans

pene
otto

to iInem

mans

want

waonn

nnnN

minine

onnan

mond

non

NOAM

nonm

won
otto

toned

nna

Arrivals
On station

7

trond

wonna

mond

Rann

oonn

tNNO

wonnd

ANNs

atto
None

monn

Arrivals

8

Arrivals
On station

9

nawnn
a

poe
aNmMO

nonn
=
nonn
=
nonn
a
Rann
at
tonn
=
wnne
a
tonn
a
wnneo
a
nonn
a
wo in
none

wonn
tN

wonnd
wttt°o

NNN

nRInN
maaAno
niAnNnO

On station

Taken

Arrivals
Left

10



The mathematical model can now be arranged in a form which will fit into
the simplex tableau; also, each variable is entered into each equation:

C= 4.000 X + 6.000 X+OX, +OX% +0X, +0, + MQ, (6b)
0 = 0.025 % - 0.015 xX, +1X%,+0% +0X, +0% + 0Q, (1a)
0 = -0.015 X, + 0.005 X% +0X,+1X% +0X, +0X;, + 0Q, (2d)
40 = 1.000 X, + 0.000 X,+0X, +0X% +1, +0, +00, (3b)
60 = 0.000 X, + 1.000 X»+0X, +0% +0X, +1X;, + 0Q, and (4b)
80 = 1.000 X + 1.000 X, +0X, +0% +0X, +0X, + 1Q. (5b)

Note that the values of the equations are unchanged; the only change is in
arrangement. A simplex tableau (table) can now be entered. Tableau 1 in fig-
ure 2 shows the restatement of the mathematical model. Note that the numeri-
cal entries in the tableau are the same as the coefficients in the mathemati-
cal model. The Cj row contains the coefficients to the variables in the
preference function formula (6b). The Pg row contains the variables which
designate the columns into which the coefficients should be entered. The Py
column contains the constant terms associated with each equation. The two
columns on the left side of the Pg column are known as the stub. The first
column in the stub is the Cj column which contains entries that are the coef-
ficients, from the C; row, of the variable which is being considered. The
second column in the’ stub contains the variable which is being considered.

In two iterations the simplex criterion is satisfied, and the optimum
program is outlined in the stub and Po column of tableau 3 (fig. 2):

1. X, designates tons produced from stope 1; 30 tons at a cost of 4 man-
hours per ton.

2. X& designates the quantity of lead (0.2 ton) produced in excess of
the requirement expressed in formula (2c).

3. X, designates the quantity of ore (10 tons) not mined from stope 1,
formula (3a).

4. Xg, designates the quantity of ore (10 tons) not mined from stope 2,
formula (4a).

5. X, designates tons produced from stope 2; 50 tons at a cost of 6
man-hours per ton.

6. The total minimum production cost is 420 man-hours; 2 - Cy row, Py
column.

7. X, does not enter the final program, indicating that the zine require
ments are met exactly.


0000000013
5000000057
1566000057
3670000050
24&1NNNNN5N
anannannan
lob Malas Ralelale)
aniangannn
9919NMENNAN
9920030090
0070070000
0020110000
0030040090
9030080000
anaanianco
OlaT lal Yalalale)
0170090090
9069150000
9100190090
Nnnnnnanannn

APPENDIX B.

0000000022

1900000058 2400000059
2108500057 1770000057
3610000050 3190009050
2264900050 0009009000
annnnnanng onoaanno00
1900000051 091002N000
yanaannnsy oninnenono
1999000051 0910100000
7460000050 0920040000
5150000050 0020080000
1000000051 0030010000
5270600052 0930050000
4925400052 0030099000
19900000051 0950020000
1nn9900051 0990069000
1900000051 0130190000
1000000051 0970160000
1000000051 0110200000
annnanaann anannnnonnn

- LISTING OF DATA IN MATRIX FORM.

1157000057
1302500057
4150000050
onnn0nn0n0
o9o0nnannnn
1900009051
1anananos1
1900000051
2224000051
9230090050
4754400052
4881300052
4777600052
1990000051
19anennns1
1000000051
1000009051
1000000051
anannnannan

1553000057
1395000056
2981000050
nonnnnnndn
oonnnnnnnn
0010930000
901NNTANDN
90020010000
0020050000
0020090000
0030020000
003006N000
0030100000
9960030000
910007NN000
0040130000
0080170000
0120210000
annnagnann

1181000057
0000000000
3730000050
0000000000
noononnnnn
1000000051
1900000051
2192000051
1045000051
1148000051
5251600052
5219200052
4895500052
1000000051
1000000051
1000000051
1000000051
1000000051
noonnnnnann

LINEAR PROGRAMING PROBLEM, 5-MILLION-TON
PRODUCTION, 2 PERCENT SILICA, 48 PERCENT ALUMINA

4400000055
0000000000
3670000050
9000000000
990000 000
0010040000
0910080000
0020020000
0020060000
0020100000
0030030000
0030070000
0030120000
0070040000
0110080000
0050140000
0090180000
0130220000
9900000000

51

2008500057
0000000000
3221000050
0000000000
9000000000
1000000051
1000000051
2070000051
2489000051
1364000051
4669500052
5207000052
1000000051-
1000000051
1000000051
1000000051
1000000051
1000000051
9000000000

0020

INT. 0BULOF MEMES. POH. .PAL 4772


11

problems will be presented; the example problems are kept relatively simple so
that the value of the method can be recognized. Problem size is limited only
by availability of computational equipment.

Digital Computer Program

A program listing for a linear programing routine is included (appendix A)
for an IBM 650 digital computer with immediate access storage, index registers,
and automatic floating-decimal arithmetic. This program can be used by either
the above computer or by other types of computers utilizing an augmented IBM
650 simulation computer program. This routine is a revised elimination method
of solving a linear programing problem where some measure of productivity is
to be minimized. Computer programs that either maximize or minimize the cost
function are available for many other computers.

General Description of Algorithm

The linear programing problem is described by a set of linear equations
and inequalities, where each variable is assigned a certain cost measure (cj)
per unit. There are more variables than equations in the system; therefore,
there are an infinite number of possible solutions. The object of linear
programing is to solve the system so as to minimize (or maximize) a cost
function, F:

oy
W
Ms

CiXj-
jel
Common notations used for elements in a matrix are i for the rows and j for

the columns. An element a,, would be located in row 1, column 4, with i equal
to 1 and j equal to 4. The system may be described symbolically as:

4,1 % + a pX> + wees + a gXptoceee eee + ar nk = by
Ag, % + agoX + -eeeee Fag GXjt veeeeee + apnX¥n = by
Aijy + AifXo + ceeeee + AUGRGT ceeeeee + ainXn = bi
Am, % + aneX_ + eeeeeet AmjXjt oeeeeee + Amn¥n = bm
CH + Qk t eeeeeet C]X] + ceeeeee + CX

The aij's are the coefficients in the equations; the cj's are the cost coeffi-
cients; and the bi's are the elements of the vector formed by the right side

of the equations. The element of the matrix to be solved for and eliminated
from all other equations is selected in the following way:

Db

1. Calculate for all unsolved equations such that aij # 0 and


DESIGN AND APPLICATIONS
OF SOME MATHEMATICAL MODELS
FOR MINE-SYSTEMS ANALYSIS

By Richard F. Hewlett and D. E. Redmon

so x» » & 4 =» * » » » » report of investigations 6158

UNITED STATES DEPARTMENT OF THE INTERIOR
Stewart L. Udall, Secretary

BUREAU OF MINES
Marling J. Ankeny, Director


32

Examples of Simulation

Probability Distributions

It is often desirable to simulate values of a specific population so that
expensive sampling is not required. The only information that is required
concerning the population is its distribution. This method will be illus-
trated for the bauxite deposits previously considered in the linear programing
section. Values were determined for deposit tonnages, average grades of
silica for each deposit, and the average grades of alumina for each deposit.
The three characteristics that must be determined or estimated for the deposit
tonnages and average grades are:

1. Frequency distribution.
2. Average of the deposit grades and tonnages.
3. Standard deviation of the deposit grades and tonnages.

These characteristics are discussed separately for tonnage, grade of silica,
and grade of alumina. Experience and available information on existing
deposits were used to estimate these characteristics.

Deposit Tonnages

Geology and geometry of the deposits in Jamaica affect the tonnage in
such a way that it was assumed the frequency distribution of all tonnages from
the area of interest follows a normal distribution and would have a standard
deviation of 2.0. Normal random numbers were drawn from a distribution with a
mean of zero and a standard deviation of 2.0. The numbers (u) were coded by
(u + 3)0.5 to transform the values for tonnage into millions of tons, as shown
in table 6. The average tonnage for the ten deposits is about 1,365,150 tons,
which is within expectation of the estimated mean tonnage.

TABLE 6. - Values determined by Monte Carlo methods for
bauxite-deposit tonnages

Normal random number, u, Coded (u + 3)0.5 tonnage

Value from a population, with in millions of tons
uw = 0, 0 = 2.0 (absolute value)

-1.828 0.586

-3.073 0365

-.817 1.0915

-.778 1.1110

1.866 2.4330

+715 1.8575

-802 1.9010

-.363 1.3185

2.878 2.939

72.245 ~3775



1 2 3 4 5 6 7 8 9 10 a 12 13 14 15 16 7 18 19 20 2i 22

1 1 1 | ' t 1 1 ' 1 ! b,

2 | 2.192] 2.070] 0.746|2.224| 1.045 |2.489/0.515 |0.923/1.148 |1.364| | 2b,

3 Lah fe Lao lo? a Loo [ac® a 1 48,

oe ' 1,157,000
5 t ' 1,553,000
6 ' ' 1,181,000
7 ' | 44,000
8 ' 1 2,008,500
9 ' 1 1,566,000
10 \ \ 2,108,500
u \ \ 1,770,000
12 1 i 1,302,500
13 1 139,500

0.367|0.361 {0.319 |0.415 |0.298|0.373 |0.367/0.322 [0.241 |0.226

FIGURE 4, - Data in Matrix Form for 10 Bauxite Deposits.

02




44

BASIC DATA

Basic data comprise all constituent data; their collection, collation,
and processing are an essential part of systems analysis. The types of data
required are determined by the nature and scope of the problem and from the
model that is used to describe the problem.

Constituent data are selected and obtained from many sources such as pub-
lished information, related or unrelated operations, policy decisions, and
experimental results. Constituent data in basic data, which are utilized in
all areas of mine-systems analysis, are shown in figure 15.

Restrictions data consist of items that directly affect operation of the
system but are imposed on the system from outside sources; principal catego-
ries are:

1. Management decisions:
a. Investment capital available.
b. Profit requirements.
c. Delivery dates (mine-plant completion).
2. Political controls and implications:
a. International.
b. National.
c. Local.
d. Tax structure.
3. Demand for the product from the system:
a. Quantity.
b. Quality.
ce. Delivery dates (product).
4. Availability of requisite personnel:
a. Appropriate skills.
b. Willingness to serve.

c. Training facilities.

Many of the restrictions cannot be expressed as an exact quantity but can
be expressed by upper or lower limits or assigned a value by rank correlation.

Mineral deposit data include collectively all information pertaining to
mineral deposits; principal categories are:

1. Geographic data:

a. Location.
b. Regional characteristics.


28

Ti+ Ta + Ts + Ua -M=
Tle + Ton + Top + Wo - M=

nod
oo

1 +G, Ty + Gy Ty) + G1 - N= 0
2+ G The + Gy Ton +& Lg -N=0
TM. +e = Ts
Ti + Toe = Tp
Ts: + Too =T¢
Mi + he = Tp
0+0+0+0- 2L

)
3
)
}
)
j
3
)

Equation

Equation

Equation

Equation

These equations expressed in matrix form are shown in figure 9.

Data used for the constant cost model are shown in table 5.

TABLE 5.

- Values used for _a mathematical model with

constant costs per planning period

Mine area Tonnage Grade, percent | Cost per ton
1,810,000 0.65 $0.2075
1,553,000 55 +327
1,770,000 75 -550
2,008 , 500 85 +308
1 2 3 4 5 6 7 8 9 10 i}

Taf Tz} Tai} Tea] Tail Tse] Tai] Tao] & | M | ON
rfc, c, cy cy “| °
2 Cc, c, cs cyl -1 0
3 1 I if | -I oO
4 | ! 1 1 -I ce)
s |S, 6, 6, S, - 10
6 S| g, 6, 6, “1 °
7 | | Th
8 1 i} Ty
9 \ 1 To
10 ' \ T

lo} lo) oO ° ° ° Le) fo) -2 O °

FIGURE 9. - Constant-Cost Matrix Form,

2.

t Equation

t Equation

Equation

r Equation

} Equation


FREQUENCY

50
48
46
44
42
40
38
36
34
32
30
28
26
24
22
20
18
16
14
12
10

On £ DD

T T T T T T T T

ie} i 2 3 4 5 6 7 8 9
NUMBER OF LOAD ARRIVALS PER PERIOD

FIGURE 14, - Frequency Distribution of Load Arrivals at Transfer Point.

10

37


23

The effect of an error in computing the mean (x) of the silica content
for each deposit was studied by assuming that the estimated silica content was
low by an average of 0.83 percent. The effect of higher silica on total trans-
portation costs is observed in figure 6. This figure shows the additional
cost involved in more rigorous blending of the ore from each deposit to obtain
less than 2 percent silica.

Transportation costs per ton for various production rates and tonnages
are plotted in figure 7. Initial costs can be determined for the various pro-
duction rates from this graph. It is observed that using a low (0.5 x 10°
tons) production rate per planning period results in the lowest overall trans-
portation costs, and a production rate of 1,500,000 tons per planning period
costs more for mining lower total tonnages (up to about 7 million tons). This
transportation cost data for the various production rates can be used for fur-
ther cost analysis required for optimizing the mining system. Final total
transportation cost per ton and total minable tonnage for each production rate
are shown in table 3. For preliminary estimation, the transportation cost per
ton for the maximum possible tonnage using any of the various production rates
is about 32 to 33 cents per ton. It can be seen in table 3 that production
rates of 2 and 3 million tons per period result in optimum (or maximum)
deposit utilization or conservation. The minimum transportation cost for
mining any tonnage is 0.5 million tons per period. The number of deposits to
be mined in any single period for production tonnages of 0.5 to 3.0 million
tons is from nearly 2 to nearly 3. This information can be of help to other
mine planning activities and allow other cost considerations to be used.
Usually a suboptimum solution must be used, so that information obtained by
this type of a study determines the range of combinations possible in the
optimum overall solution.

TABLE 3. - Transportation costs and tonnages possible to mine
for the different production rates per period

Production rate Final cost | Tonnage x 10° Tonnage x 10°
(tons x 10°) per per ton mined unmined
period
$0.32083 11.5026 1.3274
- 31800 8.563 4.2670
«32247 10.500 2.3300
«32842 12.000 +8300
~32247 10.500 2.3300
+ 32842 12.000 -8300

Property optioning or purchase can also be approached by considering the
same variables. A modification of table 3 gives the information. Table 4
shows the order in which the deposits were mined; deposits mined out first are
the more desirable. From table 4 it can be shown that deposits 4 and 6 are
not economical, and that deposit number 1 is near marginal. Value or desira-
bility depends on such factors as purchase price per ton, location, grade of
ore, politics, and other factors difficult to measure quantitatively.


Solve for that balance of production from the stopes which will fulfill
production requirements at a minimum man-hour cost (the objective).

The ore-production problem is expressed as a mathematical statement
(model), in which the restriction are expressed as equalities or inequalities.

For the structural variables, let X, represent tons produced from stope 1

and X, represent tons produced from stope 2. The statement which places the
restriction on zinc is:

0.04 X, + 0.08 X, >0.065 (KX, + Xp). (1)
The restriction on lead is:
0.06 X + 0.04 X, 50.045 (X, + X,). (2)

The inequalities merely state that the total tons of zinc and/or lead produced
will be equal to or greater than the respective tons of metal required.

Similarly, the restrictions on stope-production limits are expressed by:

X, S40, and (3)
X <60. (4)

Equations (3) and (4) merely restate that 40 tons or less can be produced
from stope 1 and that 60 tons or less can be produced from stope 2.

A production of 80 tons is required, and this is expressed as an equality:
X +X = 80 (5)
The mathematical model that has been developed thus far is listed as

follows: (Note that the variables in formulas (1) and (2) have been collected
on the left side of the inequality sign.)

-0.025 % + 0.015 X, >0, (Ja)
0.015 % - 0.005 X, 30, (2a)
Xx S40, (3)

X, $60, and (4)

X + Xp = 80. (5)

The restrictions and requirements have now been expressed except for the
cost-of-production statement which is:

4X, + 6X, =C (min). (6)
This equation is customarily called the functional or preference function and

represents the objective, which is to produce the required quantity and qual-
ity of ore at the minimum man-hour cost; therefore, C is to be a minimum.


RESTRICTIONS DATA

MINERAL DEPOSIT DATA

EQUIPMENT DATA \ / COST DATA

BASIC
DATA

DEPOSIT
VALUATION

SYSTEM DESIGN

Optimization of:
|. Service-point location
2. Mine-plant layout
3. Mine planning
4. Mining methods

5. Equipment selection

PRODUCTION SCHEDULING

Ore
control

Equipment
utilization

Labor
utilization

FIGURE 15. - Basic Data; Constituents and Relationship to System Design

and Producti

ion Scheduling.

45


47

4. Personnel costs:
a. All categories.

Cost data are obtained from numerous sources. A well-organized indus-
trial engineering department is a fruitful source of cost data. .Such data are
usually in the form of standard time, which can be readily converted to cost
elements.

An accounting department can provide numerous cost data; however, balance
sheets and condensed reports for management are frequently insufficiently
detailed and must be carefully analyzed if the reports are to provide data for
use in systems analysis. Records and reports from geology, engineering,
research, and maintenance departments provide important additional data.

The quantity and quality of basic data is related to the size and effi-
ciency of the mining organization involved as well as to the size, value, and
relative development of the mineral deposit under consideration.

Basic data, as illustrated in figure 15, is one requisite for efficient
analysis in mine-system design and production scheduling. A second requisite
is utilization of the data in a representative model or models from which
meaningful results can be obtained.


10

cy 4 6 ° ° ° ° i) Ratio
Po | TM yi %2 | %3 | %a | Xs | %s | %
=| 0 |“; | 0 |.o2s]/-o1is | 1 ’ (min)
o | X4 | 0 |-015 |I+.005 1
o | Xs | 40] 1 1 5
x <
° 6 | 60 \ 1 ct
mM} 2% | so} ijt I #
z som| mM | M M
Z) - ©) | 8om|M-4 | M-6 M-M
—~—|4{%*, / 0 1 | -6 )| 40 Ss NF.
o |X, } 0 -.004]| 6 1 aoaal NEF.
o | Xs | 40 6 ||-40 1 s $2 =667
© | X— | 60 \ fl a 0; 60
=| |, | 80 1.6 J[-40 2 $9 = 50 (min)
7) [eow| « [EFS :
2) -%) | eom| 4-4 | eM | 189 M-M
4 |*, |] 30] 1 25 375
o |X, | 2 5 | 1 0025
o | Xs | 10 -25 ' -.375 °
© |X | 10 25 1 f62s|p i
—| 6 | *, | 50 1 |-25 625 z
zy 420| 4 6 |-50 5.25
7; - ©; |420] 4-4] 6-6 |-50 526 I

FIGURE 2, - Simplex Tableau Iterations.

This simplified application of the simplex algorithm illustrates the
feasibility and practicality of the simplex method. Obviously, the time spent
in computation and chance for error in computation increases rapidly as the
size of the tableau is increased. However, the simplex method provides a
means for solving appropriate problems manually.

The foregoing simple problem does not adequately exemplify the powerful
tool available in linear programing. Very large problems can be solved by the
use of computers. Examples of computer solutions to linear programing