EVALUATION OF ALTERNATIVE PLANT LOCATIONS

RICHARD A. KING ROGER A. DAHLGRAN

ANN A. McDERMED and

DAVID L. McPETERS

ECONOMICS SPECIAL REPORT NO. 52 DEPARTMENT OF ECONOMICS AND BUSINESS

NORTH CAROLINA STATE UNIVERSITY RALEIGH. NORTH CAROLINA

JUNE 1979

EVALUATION OF ALTERNATIVE PLANT LOCATIONS

Richard A. King M. G. Mann Professor

Roger A. Dah1gran Graduate Research Assistant

Ann M. McDermed Programmer

and

David L. McPeters Programmer

Economics Special Report No. 52 Department of Economics and Business

North Carolina State University Raleigh, North Carolina

June 1979

ABSTRACT

Exact and approximate procedures for the selection of plant

locations are reviewed . An improved version of an approximate

procedure developed by Hardy is described that makes use of the SAS

computer package. This procedure for site selection introduces a

check for dominance among the set of alternative sites. It provides

a complete printout of the selection process including optimum sites

for systems ranging from ' one plant to any desired maximum number of

plants, product allocation for each system, and the cost of selecting

non-optimum combinations.

A method of searching for superior site combinations is suggested

that is potentially useful for checking systems with small numbers of

plants. It is s hown that the Hardy algorithm, which relies on a

combination improvement check at each step, may fail to find the

optimum site set in situations where two sites s hould be deleted from

a previous combination of locations.

Instructions for use of the program are ' included and sample prob­

lems provided which illustrate potential applications to pricing and

allocation questions.

ACKNO\olLEDGEMENTS

This report is a contribution to the work of the Southern Region

Dairy Marketing Research Committee. Financial support provided by the

Dairy Section, Animal Products Branch, National Economics Division of

the Economics, Statistics, and Cooperatives Service, U. S. Department

of Agriculture is gratefully acknowledged.

2

TABLE OF CONTENTS

I. INTRODUCTION... . • • •

II. LOCATION-ALLOCATION MODELS

III. EXACT SOLUTION METHODS .

A. Complete Enumeration B. Inte"ger " Programming. C. Network Formulation. D. Branch and Bound ..

IV. APPROXIMATE SOLUTION METHODS

A. Preselection of Plant Numbers. B. One-Point Moves •....•.• C. Combination Improvement Check.

V. SELECTION OF PLANT SITES

A. The Hardy Procedure. B. TWelve-site Test Problem C. Small System Optimality Check. D. Extensions of Hardy Procedure.

VI. IMPLEMENTATIONS AND APPLICATIONS .

A. Modified Hardy Computer Program. B. Applications of PROC HARDY

Page 5

6

8

8 10 10 11

11

11 12 12

13

13 18 22 23

24

24 31

VII. SUMMARY. . 42

VIII. REFERENCES 43

IX. APPENDIX A. SUMMARY NOTES ON SOLUTION PROCEDURES FOR MODELS A THROUGH G . . . 45

X. APPENDIX B. COMPUTER OUTPUT FOR l2-SITE PLANT LOCATION TEST PROBLEM • • . • . • . • • . . . . . . . . • • .. 49

3

EVALUATION OF ALTERNATIVE PLANT LOCATIONS

I. INTRODUCTION

The choice of number, size and location of plants is a decision

that is faced by a wide variety of private firms and public agencies.

Conceptually, the issue is quite straightforward, but the computational

problems encountered when the number of options is large can tax the

capacity of the largest computers.

Procedures that have been employed in solving plant location

problems consist of exact methods that lead to a solution which can be

proven to be least-cost and approximate methods that lead to an

acceptable solution which can be shown to be better than others but

which cannot be shown to represent the minimum system cost. Approximate

methods are often necessary when the empirical problem under study is

very large.

Information provided by various solution procedures differs in

important respects. All methods produce information as to the sites

selected and the volume to be handled at each site. Few methods

provide measures of the added costs associated with near-optimal

solutions. In many situations this information can be quite helpful

to the user. Nonquantifiable considerations may lead to the selection

of a system that adds little to total cost if a measure of those added

costs is available.

5

Methods differ with respect to the ease of problem formulation,

the cost of solution, and the effort required for interpretation of

the results. The method described here in some detail is an approxi­

mate solution procedure that offers a number of advantages. A computer

package that is generally available (SAS) is used to minimize set-up

time and input required by the user. Solution time is low and output

is provided in a form that facilitates comparison of alternative

solutions.

II. LOCATION-ALLOCATION MODELS

A wide variety of location-allocation models are in use. Several

of these are sketched out in Figure 1, using the classification system

suggested by Miller and King (16). These models differ with respect

to:

1. Number of transportation stages

2. Number of processing stages

3. Fixed or variable final demands

4 . Plant volumes restricted or unrestricted

5. Location influence on plant costs

6. Economies of scale in processing

7. Distribution costs that vary among plants

8. Number of final destinations

Model A consists of two transportation stages, a single processing

stage, fixed final demands, unrestricted plant volumes, average plant

costs that may vary among plants, no economies of scale in processing,

distribution costs that may vary among plants and one or more final

destinations . This model may be solved by straightforward application

of the transportation problem after preselecting minimum route cost

values from each of i origins to each of j destinations through one of

the ~ plants.

This model may be extended readily to N transportation stages and

N-l processing stages by restricting plant capacities (Model B).

Again, the transportation problem provides a satisfactory solution

method.

6

Characteristic Alternative model assumptions

l. Transportation stages (no. )

2. Processing stages (no. )

3. Final demands

4. Plant volume restricted

5. Average plant cost may vary with location

6. Economies of size in processing

7. Dist. costs may vary among plants

8. Final destinations (number)

Model identifica tion a A C D E F G H B

Figure 1. Characteristics of selected location-allocation models

aClassification used by Miller and King (16).

bHoldS for each of N-l processing stages.

b

b

b

7

Model C is an extension of Model A in that economies of scale in

processing are introduced. However, it is assumed that distribution

costs from each plant to the single destination are equal. Model D is

identical to Model C except that distribution costs from each potential

plant site to the single destination are allowed to vary among sites.

Model E is an extension of Model D by introducing multiple destinations

and varying distribution costs. It is equivalent to Model A except

that the linear total plant costs may have positive intercepts.

Model F introduces variable product demands but assumes constant

average total cost at each plant. The latter assumption is relaxed in

Model G. Two processing stages are included in Model H and economies

of size in processing are allowed, but otherwise the assumptions of

Model A hold. Further comment on alternative models is provided in

Appendix A and in a paper by Revelle, ~ al. (17).

III. EXACT SOLUTION METHODS

A number of exact solution procedures have been used in recently

published location-allocation research. An excellent review of these

methods is provided by Scott (18). Briefly, these include complete

enumeration methods, integer programming and tree-searching procedures.

An overvielv of these provides a base for consideration of alternative

approximation methods.

A. Complete Enumeration

The most widely used complete enumeration method for analyzing

plant location problems was introduced by Stollsteimer (20) in 1963.

His model is that described above as Model C. Plant cost functions

are taken as linear with a positive intercept. All locations are

regarded as equi-distant from a single destination for the final

product of the plants.

The solution to this problem calls for the complete enumeration

of all possible combinations of locations as the number of plants in

the system is allowed to vary from 1 to a maximum of L. For any given

number of plants in the system, say J, there exist k different location

combinations where k is defined as k = (L!)/(J! (L-J!». The results of

8

evaluating all plant combinations for L sites as J varies from 1 to L

can be displayed graphically as in Figure 2 in which L is taken as 4.

Assembly cost is shown for each of the four combinations of 1 and 3

plants, for the six combinations of 2 plants and the single 4-plant

system. The lowest assembly cost for each value of J, shown in boxes,

decreases more slowly as J approaches L.

For problems of moderate size, this method is ideal. All possible

combinations can be explored and comparisons among close alternatives

made readily. However, as the number of possible plant sites increases,

the number of combinations increases so rapidly that even the largest

computers do not have the capacity to make this a feasible procedure .

• •

I

• •

2 3 4

Plants in system (J)

Figure 2. Relation between assembly cost and plant location combinations in a four-location problem

9

B. Integer Programming

Integer programming may be an attractive alternative to complete

enumeration. Existing computer packages perform satisfactorily for

what are described as "well-behaved" problems but experience has shm-m

that some large problems do not converge. Thus, an exact solution is

not guaranteed.

Hilger, McCarl and Uhrig (9) reported the use of a mixed integer

programming model to solve a grain subterminal elevator problem

consisting of 124 grain producing origins, 19 potential subterminals,

105 country elevators and 13 destinations with production specified in

each of twelve months. This problem was too large to be accommodated

on existing software, they found. As an alternative, they used a

technique kno\~ as Benders' decomposition to solve a reduced problem

in two phases. "The iterative procedure is: (a) choose a set of

subterminals, (b) solve the grain distribution and storage problem,

(c) form an equation (representing) the predicted cost of the sub­

problem -- and add to subterminal selection problem, (d) solve the

subterminal selection problem, and (e) take the new set of sub terminals

and go to (b). This procedure continues until the predicted and actual

outcomes converge to a degree acceptable to the researcher" (9, pp.

677-678) •

In the Hilger application the t,~o problems created were an integer

programming problem having 19 zero-one variables and a variable number

of constraints and a linear programming problem having 31,656 variables

and 3,588 constraints. By analyzing the twelve time periods indepen­

dently, the analysis was completed at a computer cost of some $4,000.

The authors concluded that a network formulation could have substan­

tially reduced the computer cost, although they regard the Benders

decomposition procedure of interlocking standard mixed integer and

linear programming routines as superior to heuristic procedures.

C. Network Formulation

Fuller, Randolph and Klingman (5) examined a cotton ginning system

in the Rio Grande Valley with 139 production locations, 14 processing

plants and 16 production weeks. They estimate that, with standard

mixed integer LP computer codes, some 800 hours of computer time would

10

be required. The authors report that special purpose codes available

for solving minimum-cost-flow network problems are 100 to 150 times

faster than the best available LP codes. Since both are exact methods

and are mathematically equivalent, they selected the network format.

An explicit enumeration procedure was adopted that reduced the

dimensions of the problem substantially and provided solutions to

p~oblems with 45 production locations, 15 plants and 14 weeks in

approximately one minute on a CDC 6600 using a proprietary code from

the University of Texas (5, p. 432). Both regular and overtime

operating costs and the annual fixed charge were allowed to vary among

locations.

D. Branch and Bound

Daberkow and King (3) encountered a large location-allocation

problem in their study of the location of emergency medical facilities

in northern California. The problem consisted of 72 demand points and

32 potential sites for emergency facilities. A modified branch and

bound algorithm, proposed by Efroymson and Ray (4) and improved by

Khumawala (11), was employed to solve the problem with alternative

sets of constraints using a CDC 7600. This procedure involves solving

a sequence of linear programming problems that do not necessarily meet

the integer restriction but give progressively improved lower bounds

on the objective function of the mixed integer problem, terminating

when the lower value for an integer solution is reached.

IV. APPROXIMATE SOLUTION METHODS

A. Preselection of Plant Numbers

Since the number of plant combinations to be considered can be

very large in many practical problems, a number of approximate methods

have been proposed. It has been suggested by Honeycutt (10) that it

would be useful to estimate the number of plants that will be found in

the least-cost system in order to reduce the range of plant number!

location combinations that must be completely evaluated. To do this

it is necessary to compare the reduction in the transfer cost-system

size relation with the added plant costs associated with increasing by

11

one the number of plants in the system. A formula for calculating

the shape of the transfer cost/plant number relation illustrated in

Figure 2 was developed by Honeycutt, but it is sensitive to the

particular clustering of potential plant sites.

B. One-Point Moves

One early application of an approximate solution method was

reported by Warrack and Fletcher (21). They compare an iterative

eliminations approach with an iterative expansions approach in a study

of the Iowa feed manufacturing industry. The first procedure begins

with the inclusion of all possible plants and tests whether costs

would be lower if one plant were eliminated. If so, the plant that

would reduce cost by the largest amount is removed. The process is

repeated until no further plant eliminations would reduce total system

costs.

The second procedure begins with no plants in operation and seeks

that plant l~hich l~ould result in lowest system cost. If the addition

of a second plant would reduce cost, that plant is added. The process

continues until adding another plant will no longer lower system costs.

Warrack and Fletcher conclude that the latter procedure is preferred

on intuitive and computational grounds. Both are related to steepest

ascent, one-point move algorithms (SAOPMA) discussed by Manne (14).

C. Combination Improvement Check

One-point move procedures suffer from the fact that, once included

(excluded), a plant will not be eliminated from (added to) the system.

It is frequently the case that the best site for a single plant will

not be included in the best pair of sites (see Mathia and King (15),

for example). This deficiency can be remedied by introducing an

intermediate step in which it is possible to make substitutions in

the set of plants selected for any given system size, J. A computer

program developed by Hardy (7) incorporates a combination improvement

check as plant numbers are increased, using an algorithm developed by

Shannon and Ignizio (19). IVhile representing an improvement over

one-point move algorithms, it will be shown that this step still does

not guarantee that minimum cost location combinations have been iden­

tified for every system size.

12

V. SELECTION OF PLANT SITES

A. The Hardy Procedure

The method developed by Hardy can accommodate three types of

problems in which plant capacity is unrestricted:

~ - Best locations selected for specified number of plants such that total assembly costs are minimized.

~ - Best locations for branch facilities around a specified central location such that total travel cost is minimized.

~ - Extension of Type 2 permitting two levels of branch plant locations, using the first level location as the center for second level branch plants.

Of these, the first type is of particular interest in this context.

An application of the method can be found in Hardy and Grissom (8).

The procedure is outlined below and the sample problem given in

(7) is reproduced. Given the ease with which the procedure can be

applied, only a desk calculator is needed for modest-sized problems.

The Hardy program can be improved by the addition of a prelim­

inary step in which the set of possible plant locations is screened

to eliminate any site that is dominated by one or more other locations.

Three such sites are found in the sample problem provided by Hardy.

The desirability of including this step is also suggested by the

problem described by Ladd and Halvorson (13) in which 184 turkey

processing plant sites are examined while only 116 production locations

are considered.

Step 1. Check transfer cost matrix for dominance. Remove any

plant location (column) for which, when compared with

each other column in turn, no route is less costly

than the comparable route to that alternative plant

(see Table 1).

Step 2. Calculate weighted cost matrix assuming each location i n

turn serves every demand (or supply) pOint (see Table 2).

Step 3. Sum each column and select that plant for which total

cost is a minimum. Record plant ID, minimum weighted

cost vector and total cost (last column, Table 3).

13

Table 1. Unit transfer costs and demand quantities used in Hardy examplea

Demand centers

Dl 5 3 5 4 1 3 6 50

D2 7 2 3 3 2 5 5 40

D3 8 6 2 1 3 6 4 25

D4 6 5 4 2 1 2 1 36

D5 3 4 7 4 4 1 2 18

D6 7 6 6 5 3 1 2 84

aSource: Hardy (7, p. 5).

bSince Ll is dominated by L6

, ~r by L5 and L~ by L4 , these three

locations are deleted from the set potential s tes.

Table 2. Weighted cost matrix used to select best plant location

Demand centers L4 L7

Dl 200a 50 150 300

D2 120 80 200 200

D3 25 75 150 100

D4 72 36 72 36

D5 72 72 18 36

D6 420 252 84 168

Total cost 909 565b 674 840

aEath entry is the product of unit transfer cost and demand quantity shown in Table 1-

bPlant to be selected.

14

Table 3. Weighted cost matrix used to select second plant location

Locations available Cost of current

I \

solution

Demand centers L4 L6 L7 (L5

)

(dollars)

Dl 200 150 300 50

D2 120 200 200 80

D3 25 150 100 75

D4 72 72 36 36

D5 72 18 36 72

D6 420 84 168 252

Savings if added .222a [565]b (S.LA.,) 50 120

aplant to be added.

bTotal cost.

Step 4. Calculate which of the remaining locations would permit

the greatest reduction in total cost (savings if added).

This is done by comparing each weighted route cost for

a plant not yet selected with the current solution cost

for that route and summing over all routes any savings

that will result from adding that plant (last row,

Table 3). Add this plant ID, revise minimum cost vector

for two plants selected thus far using the lowest cost

routes available and calculate total cost (last column,

Table 4).

Step 5. Select best third plant location, given the first two

selections, repeating Step 4 (last row, Table 4).

15

Table 4. Weighted cost matrix used to select third plant location

Cost of current .Locations available solution

Demand centers L4 ~ L7 (L5

, L6

)

(dollars)

200 300 50

120 200 80

25 100 75

72 36 36

72 36 18

420 168 84

Savings if added (S.LA.) a

16

ap1ant to be added.

bTota1 cost.

Step 6. Perform "combination improvement check" to see which of

the plants selected would add least to total cost if it

was deleted (C.l.D.). Cost if deleted is the sum of the

increase in cost that would be incurred for each route

if the plant in question was removed from the set being

evaluated (Table 5). Remove plant that has smallest

added cost, unless this is last plant selected. Record

remaining plant lD's.

Step 7. Repeat Steps 4 and 6 until no further reduction in cost

is possible by adding more plants (Table 6) or until the

largest system size desired has been reached.

Table 5. Weighted cost matrix used in combination improvement checka

Least cost Demand centers route

Dl 200 50 150 50

D2 120 80 200 80

D3 25 75 150 25

D4 72 36 721 36

D5 72 72 18 18

D6 420 252 84 84

Cost if deleted (C . LD.) 50 176 222

~emove plant with smallest cost if deleted, unless this is latest plant selected as in this example.

Table 6. Test for plant four

Location available Cost of current solution

Demand centers L7 (L4

, L5

, L6

)

(dollars)

Dl 300 50

D2 200 80

D3 100 25

D4 36 36

D5 36 18

D6 168 84

Savings if added [293]a (S. LA.) 0

aTotal cost.

17

Step 8. Identify cost minimizing flows from each source to one of

selected plants and sum appropriate quantities to obtain

volume at each plant (Table 7). The effect of plant

numbers on transfer cost can be summarized as in Table 8.

B. Twelve-site Test Problem

Use of the combination improvement check as provided in the Hardy

procedure makes it possible to improve a plant site mix by considering

the removal of one of the plants selected at each step in the analysis.

However, it is possible that, especially when plant numbers selected

are small, two or more plants must be removed in order to find the

optimum site mix. For example, if none of the plants in a triad

selected by the Hardy procedure are contained in the optimal pair,

that pair may not be found because only one plant can be deleted at a

time.

A numerical example will serve to illustrate the difficulty.

Figure 3 represents a region 12 miles square with one unit of product

supplied in each of 36 cells. Twelve plant sites have been chosen in

such a way that the best single site is not included in the best pair

and the best pair of sites is not included in the best triad. This

makes it necessary to consider branches of the combination tree other

than those evaluated under the Hardy procedure.

One alternative would be to modify the computer program to make

additional comparisons. However, experience with the test problem

suggests that a simpler procedure is to delete sites that have been

selected and rerun the program as presently written for small values

of plant numbers, say one through four. For larger numbers of active

plants, the combination improvement check procedure is likely to

provide the optimum location mix.

System costs for alternative runs of the test problem are summa­

rized in Table 9. Consideration of all sites (Run 1) provides the

minimum cost combination for the one-plant system and for systems with

four or more plants. However, minimum cost two- and three-plant sites

are found only after deleting plants that were selected as the best

single and the best pair of sites. The justification for doing this

is that, in the case of two-plant systems, all pairs that include

18

Table 7. Allocation for optimum number and location of plants

Demand Plants selected Weighted L4 L5 L6 transfer

centers cost (quantity shipped) (dollars)

Dl 0 50 0 50

D2 0 40 0 80

D3 25 0 0 25

D4 0 36 0 36

D5 0 0 18 18

D6 0 0 84 84

Total 25 126 102 293

Table 8. Effect of plant numbers and locations on cost

Least cost Number of locations cost

dollars) One 5 565

Two 5 and 6 343

Three 4, 5 and 6 293

Four (same as three) 293

19

6

( 1 4

02 D 3 04

2

~ 6 ~7 ~ ~ o

8 9 2

10 11 12 0 D 0

4

6

6 4 2 o 2 4 6

Figure 3. Twelve-site plant location test proble m

Note: Plant numbers correspond to those found in Table 9.

20

Table 9. Effect of site deletion on cost of systems with 4 plants or less, l2-site plant location test problema

Run 1 Run 2 Run 3 System all ,sites b delete c delete

l2d size considered 2 and 12 2, 6, -and

1 plant 163.65* 163.65* 187.24 (6) (6) (7)

2 plants 135.25 135.25 125.81* (2 - 12) (4 - 10) (5 - n e

3 plants 106.25 101.19* f 104.59 (2 - 10 - 12) (3 - 9 - 10) (1 - 7 - 10)

4 plants 77.25* 89.53 89.53 (2 - 4 - 10 - 12) (3 - 4 - 9 - 10) (4 - 5 - 9 - 10)

aCos t shown in total miles. Sites selected shown in parentheses. See Appendix B for complete computer output.

bSolution using Hardy procedure.

cDeletion of best pair found by Hardy procedure.

dDeletion of best single site and best pair found by Hardy procedure.

eAlternate solution is (3 -11).

fAlternate solution is (3 - 8 -12).

* Indicates ndnimum cost sites'.

21

either the best site (6) or one of the two sites (2-12) have been

fully evaluated under the Hardy procedure. Only pairs that include

none of the three thus remain to be checked. Run 2, with only 2 and

12 deleted, leads to a better triad than either Run 1 with no deletions

or Run 3 with 2, 6 and 12 deleted. However, Run 3 leads to a less

costly two-plant system.

C. Small System Optimality Check

It may be useful to summarize the steps to be followed in per­

forming the optimality check for small systems (2, 3, or 4 plants)

suggested by the test problem described in the preceeding section.

The first four steps correspond with steps 1-6 in Section V.

Step 1. Find the best site for one plant.

Select as Location I the site <lith minimum total weighted

cost. Compare "added cost if chosen" values to identify

close substitutes for the optimum site.

Step 2. Find the best pair of sites that includes Location I.

Select as Location II that site with the largest "savings

if added" value. Compare "savings if added" values for

unused sites to identify equally good or close substitutes

for Location II.

Step 3. Find the best triad of sites that includes Locations I

and II. Select as Location III the site with the largest

"savings if added" value.

Step 4. Compare pair II-III with pair I-II using combination 1 improvement check. Is the "cost if deleted" value for

Location I smaller than that for Location III? If true,

replace pair I-II with pair II-III and repeat steps 3

and 4. If not true, continue.

lNote that it was established in step 2 that pair I-III is not better than I-II.

22

Step 5. Search for a better pair of sites than either I-II or

11-111.2

Delete sites found in Step 4 from the location

matrix and repeat steps I through 4. 3 Is the system

cost found using the reduced site set larger than that

of the full site set? If true, begin search for best

triad. If not true, delete newly selected sites and

repeat steps I through 5 until true.

Having identified the optimum pair of plants by incorporating t he

additional search procedure of step 5, the plant selection process

continues with the search for the best triad, except that now all three

plants may be deleted to test for less costly sets of sites. The

absence of any clumping of volume over the supply (demand) region

explains why the test problem requires the more detailed procedure for

two- and three-plant systems. In many practical applications it is

unlikely that an improvement in system costs would be encountered

because of the attraction of high-volume locations. However, it is

useful to have a method for investigating the possibility that such

solutions exist.

As noted earlier, this small system optimality check need not be

incorporated into the computer program. Instead, it is done by solv:lng

one or more new problems from which the appropriate locations have been

removed. The maximum number of plants can be set at 4 for these new

problems in order to avoid unnecessary calculations since the deletion

of two or more plants at once is unlikely to be necessary in systems

with more than 3 plants.

D. Extensions of Hardy Procedure

The ease of solution of the Hardy algorithm and the wealth of

information provided at each step have been described. Several

2 Recall that steps 2 through 4 do not guarantee that a better pair (or an equally good pair) does not exist since the procedure allows elimination of only one plant at a time.

3An alternative would be to delete site I only to ensure a new start.

23

additional features have been built into the computer program. These

features make it possible to solve problems of Type 1 that include:

(1) variable processing costs that differ among plants, (2) differences

in plant location relative to markets as measured by shadow prices,

reflecting implicit market value of plant output, and (3) location

differences in fixed or capital costs per period.

The input data for these extensions are summarized in Table 10

and the modified matrix of weighted plant costs is outlined in Table 11.

Selection of plant combinations proceeds as outlined earlier except

that comparisons are made using composite TC. values in the last row J

of Table 11 in place of weighted transfer costs alone. The use of

reactive programming [see King and Ho (12)) for calculating implicit

market values of plant outputs is sketched in Table 12. The net effect

of these modifications is to approach the model developed by Boehm (2)

in his GTSS program but with additional output information provided

using the suggestions of Hardy.

VI. IMPLEMENTATIONS AND APPLICATIONS

A. Modified Hardy Computer Program

The computer program described here is a Statistical Analysis

System (SAS) implementation of the program developed by Hardy (7) for

solving facility location problems. A guide to SAS is provided by Barr

(1). The program can be used to solve three types of problems mentioned

in Section V above.

Type 1. Select the best locations for a specified number of facilities such that given costs are minimized.

Type 2. Select the best locations for branch facilities around a specified central location such that given costs are minimized.

Type 3. Select a second level of branch facilities around each best location found in a Type 2 analysis.

The procedure seeks the best one-facility system, the best two­

facility system, and so on up to and including a maximum system size

specified by the user. Each successive solution is based on the

previous solution. That is, the procedure selects the best n+l plant

24

Table 10. Types of input data that may be encountered in modified Hardy Program

Plant location Ori,gin 1 2 3 L Quantities

Unit transEortation a

costs

1 Sl

2 S2 .

Cij

's

M S m

Average variable Erocessing costs a

All origins P. J

ImElicit market value of Elant outEutb

All origins U. J

Total fixed Erocessing costsa

All origins TFC. J

aComputations on ·a desk ·calculator may be simplified by subtracting the smallest entry from all other entries.

bFor Model C thes.e are set equal to zero. For Model D these entries are distribution costs to single market with negative sign. For Model E entries are initially set equal to zero but new values are calculated in Stage II transportation problem for next run of Hardy program (Stage I). ~ndels F and G require reactive programming at Stage II rather than the transportation LP algorithm.

25

Table 11. Plant costs adjusted for differences in average variable costs, total fixed costs and implicit market value of plant output

Origin (i)

1

2

M

Subtotal

Fixed plant costs

Total costs

1 Plant location (j)

TVC . . ~J

TC. J

2

S. (C .. + P. - U.) ~ ~J J J

l: TVC .. i ~J

TFCi

l: TVC .. + TFC. i l.J J

L

Table 12. Reactive programming sub-problem relating plant sites to marketsa

Market (k) Fixed b

Implicit price Plant (j) 1 2 N supplies differentialsc

1

2

Cjk S. U.

J J

L

Demand functions Dk l:S. X X

J

Market Prices Vk X X X X

a Full discussion of a reactive programming algorithm is found King and Ho (12).

dCalculated in Stage I Hardy program for J plants.

CEntered in Table 11 for next run of Hardy program.

26

in

system, given the previous n-plant solution. For each system, the

volume, total weighted cost, and product allocations are printed for

each location in the solution. The costs of selecting non-optimal

locations are also printed. The user may optionally request a print­

out of the cost matrix and the weighted cost matrix.

Using SAS, it is necessary to define variables (columns) and

observations (rows). The data set passed to the procedure must con t ain

a variable name for each potential facility or plant location. Each

observation is a demand or production center. For Type 3 problems, the

demand centers and locations must be identical, and they must appear

in the same order. This means that, for Type 3 problems, variable one

must be in the same place as observation one, variable two the same as

observation two, etc. The value of each location variable may be

dollar transfer cost, miles traveled, time traveled or any other

measure that is to be minimized in satisfying demand for the product

or service. If fixed costs are specified, then an extra observation

(row) must be included in the data set. The value of each location

variable for this extra observation is the fixed cost associated with

that location for the appropriate time period. The data set must

contain two additional variables; one is the total demand (supply) for

each center and the other is a center identification variable.

The SAS Users' Guide provides general information needed to

implement this program. The specific definitions and peculiarities

of the program are discussed below. Dashes (-) are used to indicate

number of characters unless otherwise specified.

The Procedure Hardy (PROC HARDY)

ROWS=# NODOM PRINT FIXC=value FIXN=#

PER2=# LARGE=I!

PROC HARDY ·DATA=data set OUT=data set OUTSOLN ROWID=var name ROWTOT=var name DEC=# CENT=var name NFA=# NFAA=# PER=#

NOPRINT NDIG=#

BEST;

The parameters and options that may appear in the PROC HARDY

statement are the following:

DATA=data set name

The DATA parameter gives the name of the data set to be used by the procedure. If it is omitted, the last data set created will be used.

27

28

OUT=data set name

The name of the data set to be created by PROC Hardy. See the OUTSOLN option. If omitted the data set DATA will be created.

M;

This parameter gives HARDY an upper bound for the number of demand (supply)centers. The actual number may be smaller but it cannot be larger. The default value is 30.

NODOM

The procedure performs a preliminary dominance check to eliminate locations that are dominated by other locations. This option is used to eliminate the dominance check if desired. This option does not apply to Type 3 problems.

PRINT

This option is used to obtain a printout of the unit cost matrix and the weighted cost matrix.

OUTSOLN

When this option appears on the PROC HARDY statement, the procedure creates a new data set containing the solution for each system. The data set will contain these variables:

BY variables, if any

NP

ROWID

TC

LEVEL

CENT

plant location variables

the number of plants in the solution, i.e. 1;1 plant system, 2=2 plant system

demand center identification

total cost of serving that demand center

level designation for type 3 problems, l=primary level, 2=secondary level

name of the central location or satellite location depending on _LEVEL_

the volume supplied by that location to that demand center

NOPRINT

This option is used to suppress the printing of each solution step. It can be used when only an output data set is desired.

ROWID~ ----

This parameter is the name of the variable that identifies demand centers. This variable may be numeric or contain up to eight characters. This parameter is required.

ROWTOT=

This parameter is the name of the variable that contains the quantity demanded at each demand center. This parameter is required.

FIXC~

This parameter is the character value of the ROWID variable that is to be considered the fixed cost row. This parameter is required if ROWID is a character variable and fixed costs are specified.

DEC~

This parameter specifies the number of the places to the right of the decimal point. If this parameter is omitted, then E notation will be used.

FIXN~

This parameter is the numeric value of the ROWID variable that is to be considered the fixed cost row. This parameter is used only if ROWID is a numeric variable and fixed costs are specified.

NDIG~

This parameter specifies the number of significant digits that are to be printed. The default value is 4.

CENT~

This parameter is the variable name of the central facility for type 2 and type 3 problems.

NFA=

This parameter is required. It is the maximum number of facilities that are to be located.

NFAA~

This parameter specifies the number of secondary branch facilities that are to be located in Type 3 problems.

29

PER=

This parameter is the percent of the original demand that is to be referred from primary satellite facilities to the central facility in Type 2 problems.

PER2=

This parameter is the percent of total demand that is to be referred from the secondary to the primary satellites in Type 3 problems.

LARGE=

This parameter is used to block locations out of the solution. If a possible location variable has a cost=LARGE, then the cost is replaced with a large number so that it will not enter the solution. The default is a missing value C. ).

BEST

This option tells Hardy to print only the best system. Otherwise the solution for each system size from 1 to NP, unless the best system is reached earlier, will be printed.

Procedure Information Statements

VARIABLES statement

VARIABLES list_of_variables;

Only the variables listed in the VARIABLES statement will be

considered by the Hardy procedure. They will be inserted in the cost

matrix in the order that they are listed regardless of position in the

data set. If no VARIABLES statement is given, then all of the variables

in the data set except the BY variables will be considered. The

procedure assumes that all variables specified, except the ROWID and

ROWTOT variables, are possible plant locations. Therefore, to perform

a small system optimization check, it is necessary to list the locations

to be included.

BY statement

If a BY statement is included, the procedure will compute a solu­

tion for each BY group in the data set. The data set must be sorted

by the subsets of variables listed in the BY statement.

30

NOPRINT

This option is used to suppress the printing of each solution step. It can be used when only an output data set is desired.

ROWID=

This parameter is the name of the variable that identifies demand centers. This variable may be numeric or contain up to eight characters. This parameter is required.

ROWTOT=

This parameter is the name of the variable that contains the quantity demanded at each demand center. This parameter is required.

FIXC=

This parameter is the character value of the ROWID variable that is to be considered the fixed cost row. This parameter is required if ROWID is a character variable and fixed costs are specified.

DEC=

This parameter specifies the number of the places to the right of the decimal pOint. If this parameter is omitted, then E notation will be used.

FIXN=

This parameter is the numeric value of the ROWID variable that is to be considered the fixed cost row. This parameter is used only if ROWID is a numeric variable and fixed costs are specified.

NDIG= __ _

This parameter specifies the number of significant digits that are to be printed. The default value is 4.

CENT= __ _

This parameter is the variable name of the central facility for type 2 and type 3 problems.

NFA=

This parameter is required. It is the maximum number of facilities that are to be located.

NFAA= __ _

This parameter specifies the number of secondary branch facilities ~hat are to be located in Type 3 problems.

29

This parameter is the percent of the original demand that is to be referred from primary satellite facilities to the central facility in Type 2 problems.

This parameter is the percent of total demand that is to be referred from the secondary to the primary satellites in Type 3 problems.

This parameter is used to block locations out of the solution. If a possible location variable has a cost=LARGE, then the cost is replaced with a large number so that it will not enter the solution. The default is a missing value (. ).

BEST

This option tells Hardy to print only the best system. Otherwise the solution for each system size from 1 to NP, unless the best system is reached earlier, will be printed.

Procedure Information Statements

VARIABLES statement

VARIABLES list_of_variables;

Only the variables listed in the VARIABLES statement will be

considered by the Hardy procedure. They will be inserted in the cost

matrix in the order that they are listed regardless of position in the

data set. If no VARIABLES statement is given, then all of the variables

in the data set except the BY variables will be considered. The

procedure assumes that all variables specified, except the ROWID and

ROWTOT variables, are possible plant locations. Therefore, to perform

a small system optimization check, it is necessary to list the locations

to be included.

BY statement

If a BY statement is included, the procedure will compute a solu­

tion for each BY group in the data set. The data set must be sorted

by the subsets of variables listed in the BY statement.

30

Treatment of Missing Values

A missing value in the ROWTOT variable will cause an error exit.

Missing values in the location variab~es will be replaced with a large

value. If the LARGE parameter has been specified, then missing values

in the location variables will be treated as zeros.

B. Applications of PROC HARDY

Milk Processing

An illustration of the use of the Hardy algorithm is provided

using milk production and consumption data for subs tate regions in

North Carolina and South Carolina. The data, developed by the Southern

Region Dairy Marketing Committee, were aggregated to form ten production

centers and seven potential dairy plant locations (Table 13). The

unit cost matrix and weighted cost matrix are shown in Table 14.

In this example, the SAS data set (Table 15) contains nine varl­

ab1es. The PROD variable contains the names of the production centers.

The variable OUTPUT is the quantity available at each production center.

The remaining variables are potential plant locations. They contain

the transfer cost from each production center to that location. The

PROC HARDY invokes the HARDY program. The NODOM and PRINT options are

used and the best seven-plant system is to be determined. NDIG

specifies seven significant digits and DEC requests two places to the

right of the decimal point. The dat~ set will be created containing

the seven solutions.

The portion of the computer output provided by the modified Hardy

procedure is shown in Table 16. The first section identifies the best

location for a sirig1e plant. The first column names each possible

plant location. The next column is the total weighted cost (T.W. Cost)

for each possible location. The third column gives the added cost if

chosen (A.C.I.C.) for each alternative to the least-cost location. In

this problem, RALEIGH is the best location for a single plant. The

last column shows the total volume moving to the best location.

Selection of the best pair of locations is shown in the second

section of Table 16. The second column is the savings if added (S.I . A.)

when each location is added to the RALEIGH location to form a 2-plant

31

~ Table 13. Aggregation of S-40 regions for three-state example

Production Market

Production center Areas Volume a

Plant location Areas Volume a

Roanoke VA 1.1, 1.2 374 Staunton, VA 12.1, 1.1, 1.4 411

Warrenton VA 1.3, 1.4 588 Norfolk, VA 1. 2, 1.3 450

Norfolk VA 1.5, 1.6 284 Bristol, TN 1.5 136

Asheville NC 2.5, 2.6 254 Charlotte, NC 2.2, 2.5 471

Greensboro NC 2.1, 2.2 618 Raleigh, NC 2.3, 2.4 344

Fayetteville NC 2.3, 2.4, 2.7 332 Greenville, SC 2.1, 3.1 254

Anderson SC 3.1, 4.2 129 Charleston, SC 3.2, 33, 3.4, 4.2 347

Newberry SC 3.2, 3.3, 3.4, 166 Subtotal 2413

Orangeburg SC 3.5, 3.6 148 Excess 678

Jonesboro TN 7.9 198 Total 3091

Total 3091

aExpressed in millions of pounds.

Source: E. A. Stennis, V. G. Hurt and B. J. Smith. Levels and Locations of Fluid Milk Production, Processing, and Corisumption in the South, 1965 and 1975. Southern Cooperative Series Bulletin No. 163, January 1971. Table 1, pp. 7-9 and Table 3, pp. 14-16.

Table 14. Three-state milk processing problem: Unit cost matrix and weighted cost matrix

THNEE-STATE DAIRY EXAMPLE

COST MATRIX

PRODa STAUNTON NORFOLK BRISTOL CHARLOTT RALE I GH GREENVIL CHARLEST OUTPuT

ROANOKE 0.45 0.60 ~ollars fber cwt)

0.4 .51 0.48 0.64 0.73 (mil. l8s~

370. 0 WARRENTN 0.3<;; 0.48 0.75 0.67 0.56 0.77 0.86 590.00 NORFOLK 0.58 0.31 0.82 0.71 0.53 0.84 0.81 280.00 ASHEVILE 0.77 0.83 0.44 0.46 0.63 0.40 0.65 250. 0 0 GREENSBO 0.47 0.55 0.47 0.37 0.35 0.49 0.57 620.00 FAYETTEV 0.63 0.60 0~65 0.47 0.40 0.60 0.56 ~30.00

ANDERSON 0.82 0.86 0.55 0.46 0.65 0.36 0.58 130.00 NEWBERRY 0.76 0.80 0.58 0.42 0.58 0.3<;; 0.49 170.00 ORANGEBG 0.80 o.eo 0.66 0.47 0.60 0.49 0.41 150.00 JONESBOR 0.68 0.79 0.34 0.45 0.60 0.44 0.67 200.00

FIX COST 0.00 0.00 0.00 0.00 0.00 0.00 0.00

'IIEIGHTED COST MATRIX

PROD STAUNTON NORFOLK ~ollars 0000 omittedt

~RI TOL CHARLOTT RALLIGH GREENVIL CHARLEST

ROANOKE 166.50 222.00 199.80 188.70 177.60 236.80 270010 'IIARRENTN 230.10 283.20 442.50 395.30 :1.30.40 454.30 507.40 NORFOLK 162.40 86.130 229.60 198.80 148.40 235.20 22£>. 8 0 ASHEVILE 192.50 207.50 110.00 115.00 157.50 .100.00 162.50

GRE2:NSBO 2<;;1.40 341.00 291.40 22<;;.40 217.00 303.80 353.40

FAYETTEV 207.90 1<;8.00 214.50 155.10 132.00 198.00 184.'30

ANDERSON 1 C6. 60 111.80 71.50 59.80 84.50 46.80 75.40

NEWBERRY 129.20 136.00 98.60 71.40 <;;8.60 66.30 8 3.30

CPANGEBG 1 2 0.00 120.00 99.00 70.50 90.00 73.50 61.50 J i) NE SBOR 136.00 158.00 68.00 90.00 120.00 88.00 134.00

w aSee Table 13 for aggregation of S-40 production and consumption regions. w

"" "'"

Table 15. Program statements for three-state milk processing problem

STATISTICAL ANALYSIS SYSTEM

NOTE: THE JOB HARDYI HAS BEEN RUN UNDER RELEASE 76.60 OF SAS AT TRIANGLE UNIV~RSITIES COMPUTATION CENTER.

I 2 3 4

CATA A; INPUT PROD $ OUTPUT ST_UNTON NORFOLK BRISTOL CH_RLOTT RALEIGH GREEtjVIL CHARLEST;

OUTPUT=CUTPUT*IO; CARDS;

NOTE: DATA SET WORK.A HAS 10 OBSERVATIONS AND 9 VARIABLES. 171 CBS/TRK. NOTE: THE DATA STATEMENT USED 0.14 SECONDS AND 104K.

15 16

OROC ORINT; TITLE 'THREE-STATE CAIRY EXAMPLE';

NOTE: THE PROCEDURE PRINT USED 0.21 SECCNDS AN0 104K _NO RRINTEC PAGE I.

17 18

PROC HARDY ROWID=PROD RGWTUT=OUTPUT NFA=7 NODOM PRINT NDIG=7 DEC=2 OUTSOLN:

NOTE: DATA SET WORK.OATAI HAS 60 OBSERVATIONS AND 12 VARIABLES. 130 OBS/TRK. NOTE: THE PROCEDURE hARDY USED 0.74 SECO~~S AND 200K AND P~INTED PAGES 2 TU 14.

19 RROC PR I NT;

NOTE: THE PROCEDURE PRINT USED 0.3S SECONDS AND 106K AND PRINTED PAGES 15 TU 16.

Table 16. Computer output for three- s tate milk processing problem

- PLANT SYSTEM

T ..... COST A.C.I.C VOLUM E

S TAUNTUI~ 174 2 . 6 0 136 . 60 NORFOL K 1 864.30 308.30 ClR ISTLlL 1824. 9 0 268 .90 CHARLCTT 1574.00 1 g .00 RALEIGH 1556 .00 0.0 3090.00 GRcEN V!L 1'102.7 'J 246.70 CHARLEST 2059 . 20 50.3.20

TOT CCS T 1556.00 0.00 3090 .00

2 PLANT SYSTE .'1

S .I.A. S .I.A. C.l.DEL. T.iI. COS T VOLUM E

STAUNTON 111.,,0 111.40 1 1 1.4 0 NORFCLK 108.80 10 8 .80 BRISTOL 11 2 . SO 20.00 CHAJ;LOTT 143. 90 3.00 RALt;IGH 154 .40 100 5 .40 2190.00 GREEN VIL 17 6 .0 0 176.00 J 7 ', . 60 900 .00 CH AJ;L ES T 52.90 12. 00

TOTAL 0.00 0.00 0 .00 13 80 .00 3090.00

3 PLANT S Y S TC: M

S .I.A. S .I.A. C.I.DEL. T.W. CO S T VOLUME

S TAUNTCN l1l. "0 64.20 396.60 960.00 NORFOLK 108. 80 6 1 .60 6 1. 60 BR I STOL 20 . 00 20.00 CHAFLOTT 3.00 3 .uO RALEIGH 140.40 497.40 1230.00 GkEENVIL 176.00 374.60 900.00 CHA RL ES T 12. 00 12.00

TOTAL 0.00 0.00 0 .00 1 2&d . 60 3090.00

4 - PLANT SY ST=M

S .I. A. S .I.A. C . 1 • DEL. T.w. COS T VOLUr-IE

STAUNTON 6 4.20 396. 60 960.00 NOFFCLK 61.60 6 1.60 86.80 280.00 BR IST UL 20.00 20.00 20 .00 CH AJ; LOTT 3.00 3.00 RALEIGH 140.40 349.00 950 .00 GREENVIL 83.50 374.60 900.00 CHAFLEST 12.00 12 .00

TOTAL 0 .. 00 0.00 0.00 1207.00 3090.00

35

Table 16 (continued)

5 - PLANT SYST2:M

S.I.A. S.I.A. C. I • DEL. T.w.COST VOLUME

STAUNTGN 64.20 396.60 960.00 NORFOLK 61.60 86.80 280.00 ORI ST OL 20.00 20.00 68.00 200.00 CHAFLOTT 3.00 3.00 RALEIGH 127.20 349.00 950.00 GRE'=.NVIL 51.70 286.60 700.00 CHAFLEST 12.00 12.00 12.00

TOTAL 0.00 0.00 0.00 1187.00 3090.00

6 - PLANT SYSTEM

S.I.A. T.III.COST VOLUME

STAUNTON 390.60 960.00 NORFOLK 86.'30 280.00 BRISTOL . 68.00 200. J 0 CHARLOTT 3.00 RALE IGH 349.00 950.00 GREENVIL 213.10 550.00 CHARLE ST 12.00 6 1.50 150.00

TOTAL 0.00 1175.00 3090.00

T~E SOLUTION WILL NOT BE IMPRUVED BY TH~ ADDITION OF MOPE THAN

(, LOCATIONS.

PPODUCT ALLOCATION AMONG PLANTS

PAL:,I GH GR~ENVIL STAUNTON NORFOLK BRISTOL CHARLEST

ROANUKE 370.00 WARRENTI\ 5<;0.00 NORFOLK 280.00 ASHEVILE 250.00 GREENS80 620.00 FA YETTEV 330.00 ANDEf<SCN · 13.0 ~ J 0 NEW8Ff'FY 170.00 ORANGEElG 150.00 JONESBOR 200.00

TOT VOL 950.00 550.00 <;60.00 280,00 200.00 150.00

36

system. Since GREENVIL has the largest S.I.A., it is selected next.

The third column is the savings if added to RALEIGH-GREENVIL to form a

3-plant system. In this case, STAUNTON has the greatest S.I.A. The

next column shows the cost if deleted (C.I.DEL) for each of the 3 plants

selected thus far. STAUNTON has the least C.I.DEL. Since this was the

last location added, the best 2-plant system ts RALEIGH-GREENVIL. The

last two columns give the total weighted cost and volume for the two

locations selected.

Optimum sites for 3~, 4-, 5-, and 6-plant systems follow this

same format. The last section of the table summarizes product flows

for the least-cost system. Minimizing transfer costs alone, it can be

shown that there is no economic incentive to locate a plant at Charlotte

since there is no savings possible over the six-plant system. Introduc­

tion of a fixed cost component would likely reduce the number of plants

that would minimize total system costs.

Sweet Potato Processing

A second illustration is drawn from a sweet potato processing

plant location study by Mathia and King (15). The unit cost matrix

with and without variable processing costs are provided in Table 17.

The SAS data set (Table 18) contains the variable PROD to identify

production centers and OUTPUT for quantity produced. The remaining

variables are again potential locations. The last observation in the

data set represents fixed processing costs at each location. The first

PROC HARDY statement requests that the last observation considered be

observation 15. Therefore fixed costs will not be included in the

solution. A four-plant system is requested. A second input matrix is

constructed in statement 25. This adds a $2 average variable processing

cost to the transportation costs at each location. The problem is re­

run using fixed costs this time. Again a four-plant system is requested.

The parameter FIXN tells HARDY that the fixed costs are located in the

16th row. Solutions for the two problems are shown in Tables 19 and 20.

37

w 00

Table 17. Input data for sl<eet potato processing problems

Witnout fixed costs CJ5T MATR I X

PI<GD d[N$GNa OETHEL CHAOODUR ELI Z CTY F~I~DN (asembly cost in cent s per cwt

fAY",TTE" O'U TPUT 1 00 cwt)

7.600E+OI 2 . 900E +OI 1.140 E +0 2 t.. . 900E +OI 6 .200E+OI 9 . 6uO[+O I 1. 200E +OI 2 e . ~OOE + O I J . 600E +OI 1.<!60E+0 2 5 . eOOE +OI 7.40010+01 1.0 70 t:: +0 2 6 .UOOE+OO ~ 9.10 'l <': +01 4 .9JO E +OI 1019010+0 2 701 OOE +O I 7.200[+01 1.0 70E +0 2 4 .000E+00 4 7.9 00Eh)! 2 . 600E+O I 1.200E+02 6 • .300,, +0 I b.700E+OI 9 . 900E +OI I.OOO E +OI 5 1.03UE+02 4.600E+OI 1.520 [=+02 J.4i.lOE+OI 9.700E+UI 1. 260E +0 2 4 .()00E+00 6 90100E+OI 3 .7 00E +OI 1.43UE+0 2 4.800[':+01 8.800E+OI 1.140[+U2 I.OOO E +OO 7 8 . 4JOE+O I 3.300<':+01 I. 380E+ 02 5. ,,00E+O I 8.2UOE+OI 1.080E +0 2 I.JOOE+OO e 9.900[+01 4 . 200E+O I 1.470E+02 3 .700[ +0 I 9 . £'00E +OI 1. 2£'01:::+02 I.OOOE+O() ~ 9 .0 00::;+0 1 3 .3 00E + OI 1.380E+-02 4.60010+01 8 . 300 [+01 1.130<':+02 4.000E+00

10 5 .3 00E +OI 9 . 6001::: +01 3 .100[+0 1 1. 530':+02 4.::'00E+OI 3 . 70010 +0 1 I.OOOE+OJ II 4 . 700E+Ol 9 . 400(0 +0 1 3.500E+Ol 1. 530E+0£' 4.400[+01 2 . 9JOE+O l I.OOOE+OO 1 2 7 . 500E +OI 1.0101:::+02 4.200[=+01 1. 520E+02 5 . 700[ +01 6 . 500[ +0 1 3.000E+00 13 9 .1 00E+O I 1. 260E +02 3.t..00E+Ol 1.78 C.E +0': 7.800E+Ol 7 . ::;00E + O l 1. 000E +Ol 14 9 . 500E + OI 1.180E+02 5 . 300E+ 1' 1 1. 65f)E +02 7.6000;+01 0 . 400[ +01 1.000E+00 1 = 8 . 000E+O I 1.050E + 02 4.500E+(L 1. 56 ,)[+0£' 6.200E+Ol 7 .1 001::+01 1.000 1::: +01

FIX COST O . 000t:+ 00 0 .000[ + 00 O. OOOE+OO 0 . 00 ' )£::+00 0.000E+00 o . 000E +J O

WITH fiXE D COS T S

CU~T MATRIX

PRGD d[NSc'Na BE TH EL CHAI)BuUf< ELI Z CTY FAIS GN FAYET Tt:" OUTPUT (assenbly and variable processing cos t in cents per cwt) (100 cwt)

1 7.00;)::;+01 3 .1 00r.: + 0 1 1.160E+02 7.1 00E +Ol 6.400E+Ol 9 . 800E+O I 1.200E+Jl 2 C; ")OOE+O I 3.800E+OI 1. 260E+02 6 . 000::: + 0 1 7.600C+JI 1.0 90£:: + 02 6.000[+00 3 9 .3 00E+OI 5 .1 uOE + OI I' L I OE + 02 7 . 300L +-Ol 7.400E+OI 1.()90E+02 4.000E+00 4 8 .1 00E+O I 2 . 800<': + 01 1.220E+02 6 . 500t:: + 0 1 6 . 900E + 0 1 1.010 E +02 I.OO OE +OI :; I. U,,0t: +02 4.8JOC+ul 1.;'40[+0£, 3 . 600E +0 I 9 . 900E + UI 1. £'80E +02 4.000E+00 6 9 . 300E+O I 3 . 900E + OI 1.450:+02 5 . 000E + OI 9.000E+OI 1.160E+02 l. uOOE + OO 7 8 . 6 0 0E+O I 3 . 5uO £:: +01 1. 400E + 02 5.7UO": +01 8 .40 0E +01 1.IJOE+02 1.00 0[= +00 8 1.010 E +0 2 4 .400 2: + 0 1 1.4 90E+02 J . 9UO::: +O I 9 . 400E + OI 1.£'40E+02 1. 00oE + 00 9 9 . 2 00E+OI 3 . 500E + OI 1. 400E+C2 4 . 8 00E+OI 8 . 500E + OI 1.150E+02 4.00 0E + 00

1 0 ;, . 500E + OI .. . 800E + OI .J .J OtJ E+ Ol I. S"OE +02 4 .70()E+O I .3 . <)OOE +OI I. JOOE + OO 1 I 4.900E+OI 9 . (:00E + OI 3 . 700E + OI 1.5c;OE+02 4.000E+OI 3 .1 00E +() 1 I.JO OE +OO 1 2 7.700t:+Ol 1.030E+02 4.400,,+01 1. 5400: +0 2 5 . <;;0() 0:+01 <> . 700<: + :1 1 3 .000[ + 00 I J 9 .3 00£:: + 0 1 1.280E+02 3 . I:lOOE + OI l. eOO[ +02 >3 . 000E +Ol 7.700~+01 1.0001:+01 14 9 .7 00[ + 01 1. 2()OE +0 2 5 . 50 0E+ OI 1.670£::+02 7./:j00E+OI 8 . 600[: +01 1.000E+00 15 0 . 200E +OI 1.070 E + 02 4.70JE+Ol 1. 580E +0 2 0.400E+OI 7.300E+OI I.OOOE+OI

FIX CJ S Tb 1. 00uE, + 02 1.000E+02 I.OO OE +O " I. DOOE +0 2 1.000 E+0 2 I.OOO E +02

aBenso n is dominated by Faison.

aDol lar s per period. Processing cost fu'nct ion is C hundredweight,

$100. 00 + $0, 0 2 Q IJhere Q is expressed in

W \D

Table 18 . Pr ogr am s t a t ement s f or swee t pota t o processing probl ems

S TAT i S TIC A L AI-I"LYSIS SYSTE~

~OTE: THE JOB HARDY2 HAS BEEI-I RUN UI-IOER RELEASE 76.60 OF S"S AT TRIAI-IG~ E UI-IIVERSITIES :~~~JTATI~~ :E~TER.

1 2 3

DATA LAST; II-IPUT PROD BEI-ISON BETHEL CHADBOUR ELIZ CTY FAISON FAYETTEV OUTPUT; -

CARDS;

I-IOTE: DATA SET WORK. LAST HilS 16 OBSERVATIONS AND 8 VARIABLES. 191 JBS/TR<. ~OTE: THE OATA STATE~E~T USED 0.16 SECONDS AI-ID 104K.

20 21 22 23

TIT~EI 'SWEET POTATO PRO:ESSING PLANT EXAMPLE·; TITLE2 'WITHOUT FIXED COSTS': PROC HARDY DATA=LAST(OBS=IS) ROWIO=PROD ROWTOT=OUTPUT N~)O~ NFA=4

PRINT OUTSOLI-I;

NOTE: DATA SET WORK.DATAI HAS 60 OBSERVATIONS AND 11 VARIA3LES. 141 OBS'TRK. NOTE: THE PROCEDURE HARDY USED 0.72 SECONDS AN~ 200K A~D P~INTED P"GES I TO 9.

2. PROC PRINT:

NOTE: THE PROCEDURE PRINT USED 0.39 SECONDS AN) IIOK AND P~I~TED P"GES to T~ II.

2S 26 27 28 29 30 31 32 33 34 35 36

DATA; SET LAST; • PROCESSING COST = 100 + 2 • Q • LAST R~W. _N_=16, CONTAI~S FIXED COSTS • FOR ALL Q; IF _N_=16 THEN RETURN; BENSON = BENSO~ +2 BETHEL BETHEL +2 CHADBOUR C~ADBQUR +2 ELIZ_CTY = EL(Z_CTY +2 FAISON = FAISON +2 FAYETTEV FAYETTEV +2 TITLE2 'WITH FIXED COS S';

100

NOTE: DATA SET WORK.DATII2 HAS 16 OBSERVATIONS "NO 8 VARIABLES. 191 OBS'TRK. ~OTE: THE DATA STATEMENT USED 0.11 SECONDS AND 104K.

37 PROC HARDY ROWID=PROD ROWTOT=OUTPUT NODOM NFA=4 FIXN=16 >~INT OUTSOLN;

~OTE: DATA SET WORK.DATA3 HAS 30 OBSERVATIONS AND II VARIABLES. 141 OBS/TRK. NOTE: THE PROCEDURE HARDY USED 0.47 SECONDS AN) 200K AND P~INTED P'GES 12 TO 16.

38 PROC PRII-IT:

NOTE: THE PROCEDURE PRINT ~SED 0.29 SECONDS AND 110K AND P~INTED P'GE 17.

Table 19. Computer output for sweet potato processing problem without fixed costs

1 - PLANT SYSTEM

T • \II. CO S T A.C.I.C VOLUME

BENSON 5.770E+03 1.401E+0~ . BETHEL 4 • .369E+0.3 0.0 6.900E+Ol CHADBOUR 6.443E+03 2.074E+0.3 ELIZ eTY 6.817E+03 2.448E+03 FAISON 4.864E+0~ 4.950E+02 • FAYETTEV 6.317E+03 1.948E+03

TOT COST 4.369E+03 O.OOOE+OO 6.900 E+O 1

2 PLAI';T SYSTEM

S.l.A. S.I.A. C. I .DEL. T.\III.COST VOLUME

BENSUN 7.910E+02 0.0 . . • BETHEL . 1.155E+03 1.448E+03 4.300E+Ol CHADE30UR 1.866E+0-3 1.866E+03 1.055E+0.3 2.600E+Ol ELIZ CTY 5.300E+Ol 5.300E+Ol 5.300E+Ol • • FAISON 1.185E+03 0.0 FAYETTEV 1.116E+03 6.000E+00 •

TOTAL O.OOOE+OO O.OOOE+OO O.OOOE+OO 2.503E+03 6.900E+Ol

3 PLANT SYSTEM

S.I.A. S.I.A. C. I .DEL. T.W.COST VOLUME

BENSON 0.0 0.0 • • • EETHEL 1.IS5E+03 1.222E+03 .3.800E+Ol CHADBOUR . 1.560E+02 1.055E+03 2.600E+Ol ELIZ_ CTY 5.300E+Ol • 5.300E+Ol 1.730E+02 5.000E+00 FAISON 0.0 0.0 FAYETTEV 6.000E+00 6.000E+00 6.000E+OO • •

TOTAL O.OOOE+OO O.OOOE+OO O.OOOE+OO 2.450E+03 6.900E+Ol

4 PLANT SYSTEM

S.I.A. T.W.COST VOLUME

BENSON 0.0 . . BETHEL • 1.222E+03 3.800E+Ol CHADBOUR 1.020E+03 2.500E+Ol ELIZ CTY . 1.730E+02 5.000E+00 FAISON 0.0 . . FAYETTEV 6.000E+OO 2.900E+Ol 1.000E+OO

TOTAL O.OOOE+OO 2.444E+03 6.9 OOE+ 0 1

40

Table 20. Computer output for sweet potato processing problem with fixed costs

BENSON BETHEL CHAOBGUR ELIZ CTY FAISON FAYETTEV

TOT COST

BEr-SON BETHEL CHAD80UR ELIZ_CTY FAISON FAYETTEV

TOTAL

1 - PLANT SYSTEM

T.W.COST

6.008Ei-03 4.607Ei-03 6.681Ei-03 7.055E-i-03 5.102Ei-03 6.555Ei-03

4.6 07Ei- 03

A.C.I.C VOLUME

1.401Ei-0.3 • 0.0 6.900Ei-Ol

2.074Ei-0.3 2.448Ei-03 4.950Ei-02 1.948Ei-03

O.OOOE+OO 0.900Ei-01

2 PLA~T SYSTEM

S.I.A.

6.910Ei-02 . 1.766Ei-0.3

-4.700Ei-Ol 1.085Ei-03 1.016Ei-03

O.OOOEi-OO

T.W.COST

. 1.634Ei-03 1.207Ei-03

•

2.841Ei-03

VOLUME

• 4.300Ei-Ol 2.600Ei-Ol

•

6.900Ei-Ol

T~E SOLUTION WILL NCT 8E IMPROVED BY THE ACOITIDN OF MORE THAN

2 LOCAT LJNS.

PRODUCT ALLOCATION AMONG PLANTS

BETHEL

1 1.200Ei-Ol 2 6.00UEi-00 3 4.0UOEi-00 4 1.000Ei-Ol 5 4.000Ei-00 6 1.000Ei-00 7 1.000Ei-00 8 1.0UOEi-00 9 4.000Ei-00

10 1 1 • 12 13 14 15

TOT VGL 4.300Ei-Ol

ChAOiJOUR

•

• · 1.000Ei-00 1.000Ei-00 3.000Ei-00 1.000Ei-Ol 1.000Ei-00 1.000Ei-Ol

2.600Ei-Ol

41

VII. SUMMARY

This report describes an approximate method of selecting plant

locations. Small problems may be solved easily on a desk calculator.

A computer program using SAS, a widely available general purpose

computer package, minimizes the effort required to solve larger

problems and produces detailed output for each system size. Sample

problems are provided to illustrate the data requirements and output

format.

Procedures are suggested for incorporating variable costs that

differ across alternative sites, differences in value of output

associated with distance to final markets and fixed costs that vary

among locations.

Exact methods for solving plant location problems are reviewed.

Although leading to exact solutions in theory, computer capacity

limitations may be encountered in reaching solutions to large practical

problems. The algorithm described here represents an improvement over

several other approximate methods that have been used. An advantage

of the modified Hardy algorithm is that it provides detailed information

on changes in the system as plant numbers are increased and measures

of the added cost of selecting non-optimal sites. Such informati on is

often lost when using algorithms that are designed to locate a s i ngle

least-cost solution.

42

VIII. REFERENCES

1. Barr, A. J., et al. A User's Guide to SAS 76, (Raleigh, N.C.: Sparks Pres~ 1976).

2. Boehm, WID. T. Generalized Transportation Solution System (GTSS). AE 19, Dept. of Agr. Econ., VPI & SU, Blacksburg, VA 24061, February 1976, 71 pp.

3. Daberkow, S. G., and G. A. King. "Response Time and the Location of Emergency Medical Facilities in Rural Areas: A Case Study." Amer. J. Agr. Econ. (1977):466-477.

4. Efroymson, J., and T. Ray. "A Branch and Bound Algorithm for Plant Location." Oper. Res. 14 (1966):361-368.

5. Fuller, Stephen W., Paul Randolph and Darwin Klingman. "Optimizing Subindustry Marketing Organizations: A Network Analysis Approach." Amer. J. Agr. Econ. (1976):425-436.

6. Galler, B. A., and P. S. Dwyer. "Translating the Method of Reduced Matrices to Machines." Naval Research Logistics Quarterly, March 1957, pp. 55-71.

7. Hardy, William E. A Computer Program for Locating Economic Facilities. Agr. Econ. Series No. 24, Agr. Exp. Station of Auburn University, Auburn, Alabama, March 1973.

8. Hardy, W., and C. Grissom. "An Economic Analysis of a Regionalized Rural Solid Waste Management System." Amer. J. Agr. Econ. 58 (1976):179-85.

9. Hilger, D., B. McCarl and J. Uhrig. "Facilities Location: The Case of Grain Subterminals." Amer. J. Agr. Econ. 59 (1977): 674-682.

10. Honeycutt, T. L. "A General Location Equilibrium and Planning System (LEAPS)." Proceedings of the Summer Computer Simulation Conference, Society for Computer Simulation, Chicago, Illinois, July 1977.

11. Khumawala, B. "An Efficient Branch and Bound Algorithm for the Warehouse Location Problem." Manage. Sci. 18 (1972) :618-639.

12. King, R. A., and F.-S. Ho. Reactive Programming: A Market Simulating Spatial Equilibrium Algorithm. Economics Research Report No. 21, Department of Economics and Business, North Carolina State University, Raleigh, April 1972.

13. Ladd, G., and M. Halvorson. "Parametric Solutions to the StollsteimerModel." Amer. J. Agr. Econ. 52 (1970):578-580.

43

14. Manne, A. "Plant Location Under Economies-of-Scale: Decentralization and Computation." Manage. Sci. 11 (1964): 213-235.

15. Mathia, G. A., and R. A. King. Planning Data for the Sweet Potato Industry: 3. Selection of the Optimum Number, Size and Location of Processing Plants in Eastern N. C. AE Information Series No. 97, Dept. of Agr. Econ. N. C. State University, Raleigh, December 1962.

16.

17.

Miller, B., and R. King. Regional System."

"Location Models in the Context of a S. Econ. J. 38 (1971):59-68.

Revelle, C., D. Marks and J. C. Liebman. and Public Sector Location Models." 692-707.

"An Analysis of Private Manage. Sci.16 (1970):

18. Scott, A. Combinatorial Programming, Spatial Analysis and Planning. (London: Methuen and Co., Ltd., 1971.)

19. Shannon, Robert E., and James P. Ignizio. Algorithm for Warehouse Location." (1970):334-339.

"A Heuristic Programming AIlE Transactions II

20. Stollsteimer, J. "A Working Model for Plant Numbers and Location." J. Farm Econ. 45 (1963):631-645.

21. Warrack, A. A., and L. B. Fletcher. "Plant Location Model Suboptimization for Large Problems." Amer. J. Agr. Econ. 52 (1970): 587-590.

44

IX. APPENDIX A SUMMARY NOTES ON SOLUTION PROCEDURES FOR MODELS

A THROUGH G

45

Summary Notes on Solution Procedures for Plant Location Models

Model A. 2-Stage Transportation LP

1. P 1 . CS f h . rese ect m1n ij or eac 1, j pair

where i 1, 2, ... , m origins

j 1, 2, ... , n final demands

s = 1, .•. , S < L plant sites

2. s

Include average plant costs in Cij if these differ by site, s.

Model B. N-Stage Transportation LP

Processing capacity limited at each processing stage

[see Galler and Dwyer (6)].

2. Do not preselect minimum C~. because plant capacity constraints 1J

may require use of more costly routes.

3. Include average plant costs where these differ by site.

Model C. Stollsteimer Model

46

1. Complete enumeration of all plant combinations, Jk

,

L for J = 1, 2, ... , L; k = 1, 2, .•. , (J)'

2. Assumes single destination for output of all plants wi t h identical unit distribution costs from each plant and positive intercepts for linear total plant cost functions.

3. Average variable plant costs in Cij

matrix may differ if appropriate.

4. Include differences in fixed costs among sites before selecting each minimum total cost combination, J

k.

5. Shannon-Ignizio [see Hardy (7)] heuristic algorithm is attractive alternative to complete enumeration used by Stolls'teimer (20). Hardy program must be modified by including average processing costs in Cij values and by adding fixed plant cost to total variable processing costs at each site before selection of any least-cost site combination.

Model D. Modified Stollsteimer I

1. Identical to Model C except that distribution costs from each plant to single destination are allowed to vary.

2. Cost matrix in Model C is modified by adding to each assembly cost element, Cij , the appropriate distribution cost from

plant j to the market.

3. Hardy procedure preferred algorithm.

Model E. Modified Stollsteimer IT

1. Extension of Model D allowing mUltiple destinations and vary­ing distribution costs. Equivalent to Model A except that linear total plant cost functions may have positive intercepts.

2. One option is to solve as Model A transportation problem, assuming uniform fixed costs at all plant sites, L, to identify S plant locations. Then

systematically, evaluate the S plants thus identified, allowing J to vary from 1 to S.

3. Preferred option is modified Hardy algorithm.

Hodel F. Variable Demands, Constant ATC

1. Preselect minimum C~j for each i, j pair as in Model A.

Include average plant costs in C~o where these differ by site, ~J

2.

3.

Model G.

1.

2.

3.

4.

s.

Solve with reactive programming or quadratic programming algorithm.

Variable Demands with Economies of Size

Define average total revenue ATRo = a - bJoQJo J j

Define net total revenue, NTRi 0 as total revenue J - total SJ

plant cost s - total transfer cost i o' sJ

Define average total cost for plant s as as - bsQs

Define average net revenue as

NIRo 0

~SJ

47

48

Subject to

and

Supply i smaller than plant s Plant s smaller than mkt. j Assures pos. price possible

Demand fn slope steeper than average cost fn slope.

5. NR. . is constantly decreasing if one or both of the lSJ

inequalities b < b. and Q < Q. holds. s - J s - J

Solve by reactive programming using (5) to maximize returns

to the fixed supplies at each origin. An alternative form

of (5) is

NR .. lSJ

(a. - a - Ci

- C . ) - b. Qi

- b /Q J s s sJ J s s

X. APPENDIX B COMPUTER OUTPUT FOR 12-SITE PLANT LOCATION TEST PROBLEM

49

Appendix Table 1. Computer output for 12-site plant location test problem; all sites included

- PLANT SYSTE~""

T.".COST A.Col.C VOLUME

01 21E.2221 52.5724 02 ~0<;.2519 4S.t-022 [l~ 'P·7.2425 ,~3.592:J n4 ?O<;:.2519 45. i. 022: O~ lP'.2lJ.25 23.~(,,2:J 01; 16~.64<:}7 O.J 3&.0000 [l7 187.2425 23.5<;2-3 08 '?' c:: .O93( 51.44.,7 09 ":?1~.O93':" :, 1" 4 437 I).J.() ?O<;.2519 4S.F·022 Dl I )P7.24?5 2~.5<723 rll 2 ~O~ .. 2519 .+5. t. 022

TOT COST If<: .6497 0.0000 36.J 000

2 - PLANT :;YS TE ~

S.! .A. ~" J.A" C. I .OEL. S" I" A" C" r .DEl..." r.w.C0 5 T

[ll 23.35 \ 1 1?9~63 18.S4 5:i , -O:!- -24. (d.,r~7 ?c.2667 44.:5462 '1".141(1

0:: ?2 . C2 ()3 11.97Sf. 20.204'} 1)4 2"'. 2~ c"7 2~.('Il?:<; ~8 .c9I3u

Of. 2? .O2()' t 1 .97SE 20.304'1 06 20.1340 20.1346 07 22.02()~ 22.0?O3 20.3049 01' ?2 .780f: !<;l.4850 25" 76::a,1 O~ 22.781)6 22.7~0~ 8.264:> . I;)\-() ~4-.-""'6 7- ~->-.O~.c;- ·2-8-9980 ,-2-1>_99&0-0 1 ) ?: .~" O?I')] ??O?O:~ 20.3049 1'12 24.26f;7 24.;>667 24.2667 44.:';402 4 q .9')71

TCTAL (l.OOGO 1).0000 c,.OOOO O.OOOu o .JOOu l35.251')9

3 - PLANT SYS T= M

S.I. ~. S. I. A. C. I .DEL. T." .COST VOLl)'1~

01 18.545e 18.5458 ll-:!- . 28.c;:.980 52.8287 I 'i .M)OO C3 ?O .::01).<; 1<;.95~9 Dn "8.o9~O ?F.9geo 2d.c;.<;-8J ()'5 20.2C6.C 1I.71H5 r,,,. 20ol:;4~ \2.8<;57 07 20.304<; 1<;.S559 08 2=.7€31 2.5P.58 0 9 8.2E4= ~.2 (45

~.OOOO -Il-!.-O ~QJ>..Cl 28. 99-':lO 19.3137 1')11 ?f). 3 Cl. C '.71'35 fl' 2 ,'2B .<;<;80 3" .! 104 12.')000

t::J:rAL o.JOoe C.OOOO o.OOOJ 106. 2529 :16.0000

4 - °LAf\T 5Y5 Ti..:M

5 .1.A. S .1.A. C.l.0I::.L. T. W.C CST VOl.U"'~

D~. ! 8.545P. :t . 8 15? . G4 21.759c! 19.31.3-7 + • .l)OOO 0.:' 19.<?':S<; 4.:;694 1':4 2Q.CJ9 A ('I ~1.7592 19.3137 ~.OOOO 05 4.718': 4.3t:.94 Of, 12.8<;57 ~.6569 5.1;569 97 19. 9~5G ~ .36<;4 08 2.se~e 2 .. 5858 C9 a .?t:4= ' .5858

-O+(J 21.7592 1<;;.3137 'l...1~0

ell 4.71'3': ". ](<;4 C!2 21.7592 19 • .3 137 'i. I)").,);)

, T'l-T,A"-- .0....) O.Q 0 G.OOOO 0.0000 77 .~548 3f,. JI)'I) 0

'. ' 50

VOLUME

~1 . • 0000

1=').0000

]f).oono

Appendix Table 2 . Computer output f or l2-site plant location test problem; . sites 2 and 12 deleted

- PLANT sys T :: . ..,

T .w. C':ST A. C.l. C VOL UM E

D I ~ IF. 2221 '32.~7 24

03 1~? 242S 23 . 592-3 ')4 ~oc . ~51~ '1.5. f .. 02,2, ')C '~7 . ?425 ~?59?'o ')1; 1 f:::' ,, 64 ~ 7 O.t> 36 . )000 i)7 1 ~7 . 2 " 2~ 23 . ~,~2O

all ,? 1 ':5.. 09.:!4 51 . 4437 ')0 2. 1 c::" OC:;)4 51 . 4411 01') 'OS . ?5 19 4.5 . I'lO?2: 1) 11 l A7.242'5 2J . 5 c2~

TrT <:C<;, 1 f.:3 " t 4 <;) 7 O. OOOJ .$" . u OUO

2 - ;':lLAN'T 5Y"> rc M

S .I • A . <; . I • A • C . ! . OEL . 5 .!. A. C. I . :> EL . ' . " . C05 T VOL LH",E

[) I 23 . ~511 1 2 . c 563 1"' . 5458 [l3 2 2.0 201 I ! . <; 7 5 € 20 . ~049

04 ":' ~,,2~· 057 ?~ . ;6c7 3(' . lJ145 q6 . 34~8 2 1 . OOO~ r>C ??O?O~ 2 .?0203 20 . ,)044 DE 20 .1.346 20.134 0 07 ?2 . 0203 11.9756 20 • .304Q fl F 22 . 7eOti 22 .7 eOf 8 " 2 645 0<; 22.780t 19.413~ O 25 .7 63 1 25 " 7631 fl l O 24 . 2667 ? 4. 2667 ?4 . 2667 52 . 73JS 41 . ?,)71 15. 0000 0 11 22 . 020) 22 . 0203 20 . 3049

lC TAL o.on,>c n • ) 00 0 0.000') 0 . 0UO) U . j O'JO l3 3 . ~~:1Q ~6.0000

3 - PLAf'.. l 5YSTE~1

S .t. l\ . ~ .J. A . c. J.OEL. S .t. A. c . I . DEL . T.W.C ·)ST V..,LUME

0 1 1 8 . ~a~8 1'l.545€ 4.0 F\49 . .J) ~ ?'O ~ 304C; . , Q . 9<;~C; 1 &:; ·. ·9-559 . 1 9 • <;;{)\>9 -4'1...f"~.~ +E;.-Q.O-(}Q -

fl. 11. 6SS .. II. i 58 .. 11.6584 flc 20 . ~O4<; l ° . c~c:;<; 8 . l.208 f)€ 20 .1 ~4f: 13 . 22<")(; 5 . 249) 07 20 . 30il.&:; 3 . Po 74 2 5 . 8234 O· e . ?€~= !' . 2c45 4. 3J8() . flC 25 .7 fi31 2!:- .4141 25.4 '14 1 23 . 87 3 2 11. 0000 010 :::<; . c237 39.623 7 23 . 7858 10 . 0000

.\l-1l ~O"';>Q.4..'"- 4 .. 6.2RJi ~_4,..t..Qa

T'"'TA.l 0 . :00'> () . OCOO O. OOuO 0.0000 o . 0000 1 f) 1.1 903 3 6 . 0000

4 - ':> L A f\.IT ~YSTEM

S .l. A . S . t • A. C.I. DEL . f.W.t CST V 0 LU '--1 C

01 4.084 C; :' .1 605 · cr~ · llo 4 20d 3l . ~~5 ~O~O-')4 11 . 6~P4 11. t 58<+ 11. b569 "> . 0 .)00 C'o <:} . 1I? I) s: ~ .42 Ot) B .4 <08 Dt) 5 . 2493 <; . 24<;.3 07 ~ . 82) ill < . 9320 08 4 . 1~Bg •• 3 38<;; . C9 25 . 4141 22 .5 486 9 . 0000 010 14.3 246 23 . 7S513 1 1 . 00)0

· Gl .1 7.? 7Q8 7.2793

T'"'.'TAL O . OCOC C. OOOO 0 . 0000 ,J9.5318 l() .O Qf)O

51

Appendix Table J. Computer output for 12-site plant location problem; sites 2, 6, and 12 deleted

Ln N 1 - PLANT SYSTEI-t

r.W.COST A.e.I.e VOLUME

01 216.2221 28. <;796 03 187.2425 O.OODO 04 209.251S 22.0094 05 187.2425 0.01'00 07 187.2425 ( .0 36.0000 08 215.0934- 27.8508 09 215.0934 27.8508 010 209.2519 22.0094 011 187.2425 0.0000

TOT COST 187.2425 0.0000 36.0000

2 - PLANT SYSTEM

S. I ,A, S.I.A. e.I.DEL. T.W.COST VOLUME

01 37.2712 16.7775 16.7775 03 41.3389 14.2982 04 21.1812 15.2376 . .

1S,0000 05 61.4282 40.<;346 62.9071 07 40.9346 62.9071 18.0000 08 55.5958 10.2454-09 11.0120 10.2454-010 53.5553 15.2376 Oil 41.3389 14.2'982

TOTAL 0.0000 0.0000 0.0000 1· 25".6143 36.0000

3 - PLANT SYSTEM

S. (.A. S. I.A. C. I .DEL. S.I.A. e.I.OEL. r.w.COST VOLUME

01 16.7775 . 16.7775 16.7775 34.1886 11.0aoo 03 14.2982 1.6056 3.9782 04 15.2376 7.6434- 7.84,34

10,7935 (;5 10.7935 10.7935 07 34.9909 34.9909 38.1461 13.0000 08 10.2454 10.2454 4.e545 09 10.2454 10.2454 9.3324

15.2376 010 15.2376 15.2376 15.2376 ' 8.4208

J2.2580 12.0000 011 14.2982 14.298,

TOTAL 0.0000 0.0000 0.0000 0.0000 0.0000 104.5926 36.0000

4 - PLANT SYSTEM

S. I .A. S.l.A. C.I.DEL. 5.1. A. C.l.DEL. S. I. A. C.I.DEL. T.w.C;)ST VQLUME

01 . 1.0056

16.7775 9.3833 9.3833 9.3833 03 3.<i782 3.4627 8.4208 04 7.8434 7.8434 11.998e 11. <;988 11.9988 23.7658 10. 0000 05 10.7935 10.7935 10.7935 10.7935 31.5405 11. 0000 C7 7.6805 7.6E05 3.5251 08 4.8545 1.4076 1.4076 1.4076 09 9.3324 <;.3324 9.3324 25.4141 25.4141 22.5486 9.0000 010 . . 14.3246 14.3246 . 14.3246 11.6569 6.0000 011 8.4208 7.8284 6.6874 6.6874

TOTAL 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 89.5318 36.0000