The Trangportation-Location Problem

Leon Cooper

Southern Methodist University, Dallas, Texas

(Received January 8, 1971)

This paper defines a problem type, called the transportation-location prob-lem, that can be considered a generalization of the Hitchcock transportationproblem in which, in addition to seeking the amounts to be shipped fromorigins to destinations, it is also necessary to find, at the same time, the opti-mal locations of these sources with respect to a fixed and known set of destina-tions. This new problem is characterized mathematically, and exact andapproximate methods are presented for its solution.

T N A P R E V I O U S paper,'*' I formulated a problem that was a generalization ofboth the Hitchcock 'transportation problem' (see HADLKY'") and a problem

that is sometimes referred to as the 'location-allocation' problem with unlimitedcapacities (see COOPER.'^''') Here I will consider both that problem and an evenmore general formulation in much greater detail.

We are given n fixed locations, to be called destinations, whose positions inEuclidean space are known. Their coordinates will be given by (xDj,yD,), j =1, • • •, n. We are also given a set of known requirements r,,j=l, • • , n, for somecommodity or product at each of the n destinations. We wish to locate m sources,where TO is a given number, from which the product is to be shipped. These sourcesare supposed to have certain limitations on their capacity to ship the product.These numbers are known and will be designated c, i=l, , m. Finally, theremay be 'weights' relating to destination requirements, e.g., multiplicity of trips ina time period, or other possible weights. These will be designated fij,j = i, • • •, n.We shall designate by (xi, y,),i=\, • • • ,m, the locations that are to be determinedfor the sources, and by to,,, the amounts to be 'shipped' from source i to destinationj , which are also to be determined. Finally, we will define a set of 'cost functions'that depend on the relative locations of the sources with respect to the destinations.These will be designated:

^(XD,, 2/D,;x,,!/,) = cost of supplying a unit quantity to the jth destinationfrom the tth source.

We assume that the ^ functions are continuous. Our object is to minimize totalcost subject to capacity and requirement constraints. This problem may beformulated as follows:

min z = £ ; i r S J - I " ISJW.JV'CXcj, yo,; x., 2/.) subject to:

E l - r w.ygc., i=\,--,m; (1)

In what follows, let us designate the set TF as all tii = \wj} satisfying the constraintsof (1).

The relation of the problem given in (1) to the transportation problem and the

The Trantporfation-Locatiem Problem

location-allocation problem is readily seen. If the 4'(XD,, yo,; Xi, y.) =o,,, a set ofconstant costs, and if a,,;3, is designated as y,,, then (1) can be written as:

min 21= £ : : ? J2',-i y./^.,, weW, (2)

which is a usual form of the transportation problem. Hence, for a fixed set of costs,problem (1) becomes a transportation problem.

Returning to equations (1), we may also note that (a) if the requirements arestated in terms of the P, and (b) the capacities associated with each source are un-limited, then the set of m-\-n constraints of (1) are no longer explicitly required and(1) now becomes:

2 T

This is the location-allocation problem discussed in references 2 and 3.The particular form of the functions ^ that we shall be most interested i

throughout this paper is

which is the usual Euclidean distance in a two-dimensional space. However, weshall give some consideration to the more general form given in (1).

CHARACTERISTICS OF THE GENERAL TRANSPORTATION-

LOCATION PROBLEM

W E SHALL REFER to the problem stated in equations (1) as the general transporta-

tion-location problem,. There are a number of important observations that can

be made concerning this problem.

T H E O R E M 1. A necessary and sufficient cmdition for (1) to have a feasible set of w^

is that-£',-'r^^Zl^Ci.

Proof. This result is the same as for a transportation problem (see reference 8).In what follows and throughout this paper, it can be shown that the functions

f(xD,, ym; *., yi) are such that (1) has a finite minimum. The next theorem is animportant result.THEOREM 2. For the problem

min 2= E ; r r E',-r /3, .,iKx«y, y^,-, x,, y,), W€W,

an optimal solution will occur at an extreme point of the convex set W of feasible solu-

tions to (1).

Proof. Let {£*, y*, iv*\ be an optimal solution to (1), where x=(xi, x^, • • •,

Xm), y=(yi,yu •••,«/«.), and w=(wu,wn, •••,wi,,,wn, •• •,vhn, •• ^ l o^n ) . If we

fix (x, y) so that (x, y) = (x*, §*), we obtain a resulting problem

min2= E ; ^ H'.-i 0/^',4'(xoi,yDi)X*,yi*), weW,

which is a transportation problem. Hence, it has a solution at an extreme pointWgtW. From this it follows that (x*, y*, Wg) is an optimal solution to the originalproblem (1).

96 Leon Cooper

THE TRANSPORTATION-LOCATION PBOBLEM

LET US NOW consider the problem of equations (1) for a particular form of the

function 4/:^(xo,, yv,; X., y,) = [(xD,-x,)'+(y:>i-y,)']"\ (4)

Hence, what we shall now call the transportation-location problem is the following:

We shall now characterize the transportation-location problem. The constraint

set is clearly a convex set. What of the objective function? Theorem 3 has some

less-than-comforting information.

THEOREM 3. The function 2 = E ™ T.',-" ^/u}>A{XD,-x.)^+(yD,-y.)^f'' is

neither a convex function nor a concave function of the vector (x, y, w) for all values of

X, y, and w.

Proof. Let us represent the objective function

as z(x, y, iv). Consider the special case where all w,,=0 except Wu- Further,

let all Xi, y, be fixed and known for i>l and let 2/1 = 2/01 and xm>xi. Hence z(x,

y, w) =zi(xi, Wn)+K, where K is the sum of all the constant terms. If 2, is neither

convex nor concave over all of the space defined by (x, y, w), then the same will be

true of 2. Hence, we examine 2i(xi, w u) =/SIM'U(XDI-XI). It is well known'"

that a function such as Zi will be neither convex nor concave if the Hessian matrix

associated with 21 is neither positive nor negative semidefinite. The Hessian matrix

for z is

(8)

which is indefinite. Hence 21 is neither convex nor concave in the portion of the

space defined by xi and «;„. Therefore, z(x, y, w) is neither convex nor concave

over all of the space defined hy(x,y,iv).

This proof shows that the objective function 2 is not a convex or a concave func-

tion over all possible values of x, y, iv. A more pertinent question is whether or

not the objective function is convex or concave over values of x, y, and w limited

by the constraint set of the transportation-location problem. The answer to this

is given in the following result.

THEOREM 4. The function 2= E S ^ Ei-i" ;8,w.,[(xBy-x.)'+(2//>,-j/.)T « rtei-

ther a convex nor a concave function of the vector (x,y,w) over the constraint set of (7).

Proof. Consider the following problem:

subject to (9)

where D.y=l(xBy-x.)'+(2/B-2/i)'r-In order to simplify the argument but allow sufficient variation over the con-

strMnt set, we assume that (xj, j/,) = (x«, j/«) and further that v,=vm = Wn. and

The Transportation-LocaHon Problem

XDI> XI, xta> Xl. This simplifies the objective function to:

2(Xi, Wu, toil, UI-i) ='Pl'Wll(xDl—Xi)+^Wu(Xm—Xi

where KD^XDI—XM-

Let us consider the Hessian matrix of 2(xi, Wn, Wn, Wn):

0

-/3j 0

0 0

0 0

0 0

Since the Hessian matrix is indefinite for the problem given by (9), we see that, ingeneral, the objective function is neither convex nor concave over the constraintset of the transportation-location problem.

A consequence of Theorem 4 is that the transportation-location problem, whichis given by equations (7), is a nonconvex nonlinear programming problem in whichwe are minimizing a nonconvex objective function over a convex set. However,because of Theorem 2, we can state the equivalent result for the transportation-location problem.T H E O R E M 5. For the problem

I extreme point

r, w,W, (7)

X set of feasible solution!some optimal solution wiU (x

to (7).

Proof. The problem above is a special case of the general formulation provenin Theorem 2.

The importance of this result is that it makes it unnecessary to consider anybut basic feasible solutions to the constraints of (7). This will be important whenwe consider computational approaches in the next section of this paper.

AN EXACT ALGORITHM FOR THE TRANSPORTATION-LOCATION PROBLEM

THE FIRST algorithm we shall consider is exact and relatively simple in concept.However, its use will be limited, as will be evident, to relatively small problems.

We may note, according to Theorems 2 and 5, that an optimal solution t» thetransportation-location probleni_will occur at an extreme point of the convex set ofsolutions, W=\iv,,\Aw^b,w^O\. We know that the extreme points of W cor-respond to basic feasible solutions of Aw^b, w^O. If the number of basic feasiblesolutions is designate<l as Ners, for an m-source, n-destination problem, NarsS( T^— )• "^^ follows from the basic theory of the transportation problem.'"

98 leon Cooper

( \1 is the number of basic solutions, most of which are infeasible.

TO+n-1/Hence, it is usually the case that NBrs«.("^^_^\ For example, in the simple

example we shall present, with m = 2, n = 4, f _ ^ _ j j = U j = 8!/5!3! = 56.

However, there are only nine different nondegenerate basic feasible solutions.According to results of DEMUTH'" AND DOIG,'" the minimum number of basic

feasible solutions, for mgn, to a transportation problem of order mXn is w!/(n—m+1)! Form = 2, n=4, wehave4!/3! = 4. Hence, the number of bases we havehad to examine, 9, is much closer to 4, the minimum number, than to the unrealisticupper bound, 56. This is somewhat encouraging.

In any case, Nara is a finite number. Suppose we generate all basic feasiblesolutions. Let us designate any such solution t6={i»ol- (We consider, subse-quently, how to do this.) For each such solution, we can then solve the problem:

by considering this problem as a set of problems of the form:

We can do this, since the {iJ,,| are simply known nonnegative constants or weightsAn iterative technique for solving problems of the form given by equations (13)

was given by the author in references 2 and 3, and is repeated here for convenience.The initial estimates of (z,, j/,) are given by

and the general iteration equations are:

where the superscript on the z, and y, is the iteration parameter and

D\i = {(xu,-x^f^(y,>-y^ff\ (16)

If we now designate the minimum value of z for the 2th basic feasible solution as z*,

then it is obvious that the optimal value of the objective function 2* for the trans-portation-location problem will be:

/ = minV,2i*. (17)

If the minimum is taken on at l=s, then the optimal values of the variables are(X,,, Vi.), t = l , •••,m, and it>,y., t = l, •••,m,j=l, •••,n, where the designation(xi,, 2/..) indicates (x,, y.) for the sth basic feasible solution and T».y. are the set of?»,j for this solution.

Let us now return to a point we g l o s ^ over, earlier, viz., generating all thebasic feasible solutions to the constraints of the transportation-location problem,i.e., the constraints:

i:;:r...c., .=i,. . . , .; Ei-»o=r. ,=i,...,«; ^ ^w.,gO, 1=1, ,m, j = l, ,n.

The Tronsporfation-LocaHon Problem 99

As has already been mentioned, these are simply the constraints of the standardtransportation problem. We can make use of the transportation problem tableau(see reference 8) to advantage in order to generate only basic feasible solutions.The alternative would be to find all basic solutions and discard the infeasible solu-

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tions. This may require orders of magnitude more work. The method describedbelow is more efficient.

As an example, consider a problem with m =2, n =3. There wiU be at most 2-1-3-1 =4noniero values of w.y in a basic feaaibk solution. Suppose c, =80, c = 120, r, =70, r, =90,r. =40. An initial tableau is shown in Tableau 1 of Table I. The blank squares bave zerovalues of u>,y, i.e., wwO, Wsi=O. It is a simple matter to find all basic feasible solutions

100 Leon Cooper

from this initial Tableau 1. We can use the standard 'loop' method for allowing a zerovariable to become positive and still remain feasible (see reference 8). This is merely theapplication of the simplex method to the special case of the transportation problem. FromTableau 1, we can generate two new tableaux. These are given in Tableaux 2 and 3. FromTableau 2 we can generate one new tableau given in Tableau 4. From Tableau 3 we cangenerate Tableau 5. From Tableau 4 we can generate Tableau 5, and from Tableau 5 backto Tableau 4.

We have now generated all the basic feasible solutions. The relations between them forthis example can be represented as the directed graph shown in Fig. 1.

It can be seen that there are five basic feasible solutions. If all basic solutions had been

found, we would have had to solve four simultaneous equations in four variables, ( 1 = 15

times. The above procedure is simpler and very much less work.

We now state the algorithm we have been discussing for solving the transporta-tion-location problem and then present a numerical example.

Enumeration Algorithm for the Transportation-Location Prohlem

1. Using the transportation-problem tableau, starting with any basic feasible solution,generate the connected graph of all basic feasible solutions. Designate each such solution,I ui,,i 1,2 = 1, • , T, where there are T basic feasible solutions.

2. For each such solution, solve the set of location problems:

The Transportafion-Locolion Problem

with w*,, and (i* , yf,) being

and set 2,*= X i i ? Z.I.

3. The optimal solution is found by z* =min;_,.the corresponding values of the variables.

A sample problem. Let (xm, J/ci)=(O, 0); (x«, ym)=(O, I ) ; (xm, VDZ) =(1, 1); (xw,yot) = (1, 0); m =2, n =4; c, =50, Cj = 100; n =20, r, =40, r, =60, r, =30. A basic feasiblesolution is given in Tableau 1 of Table II . From Tableau 1 we can generate Tableaux 2, 3and 4. From Tableau 2 we can generate Tableaux 5 and 6. From 3 we generate 7 and 4.From 4 we generate 8 and 4. From 5 we generate 6 and 7. From 6 we generate 5 and 8.

TABLE II

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20

1

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1

20

20

30

10

40

60

60

30

30

Tableau 1

2 3 4

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40

30

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60

30

30

Tableau 3

2 3 4

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50

60

30

30

Tableau 5

50

100

c.

50

100

c.

50

2

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1

2

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1

1

2

r.

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rablea

3

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Tablea

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Tablea

30

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4

30

30

i4

4

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50

100

c.

50

100

50

100

leon Cooper

TABLE n-Cont.

1 2 3

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100 20

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Tableau 7 Tableau 8

1 2 3 4c,

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Tableau 9

From 7 we generate 9 and 5. From 8 we generate 6 and 9. From 9 we generate 4 and 8.Hence we are done. All of these tableaux are shown in Table II. Figure 2 gives the graphof the nine basic feasible solutions.

For each of these nine basic feasible solutions to the constraints of the transportation-location problem, two location problems were solved with the results shown in Table III.It can be seen that the minimum occurs for solution 2. Hence the solution to our problem is:U)u = 10, «'is=40, wn=O, u)n=O; Wn = 10, teM=O, u>23=60, Wj» = 30; (xi, i/i) =(0, 1), (xj, 1/2) =( l , l ) ; z* =54.143.

Fig. 2. Graph of solutions to the sample problem.

TJie Transportation-Location Problem 103

We have made use, in the above algorithm, of the usual transportation-problem

method. A worthwhile point to investigate is whethei ~ '

t h a n this ' loop' method might no1

solutions. MoTZKiN, R A I F ,

all the vertices of a convex polyhedra

W E T S " " have also examined this problem. One of the most comprehens

ies, and perhaps the most promising, ii

general problem, bu t can be used to find all e" ' . . . . ^^^ invpsfi(Tfl+inn thai lT pithpr nt Tnpisp mernon.s « .*^^«

simplified

It be found to generate all possible basic feasible

i THRALL""' have pr^ented a method for finding

ledral set, as has BALTNSKI."' WITZGALL AND

" o f the most comprehensive stud-

,_ KOHLER;"' it considers a more

gdld'tii |jruuit;iiij uut U£LJ1 UC uisci-* I*J iiim an cXtrGIXlG-pOUlt SOlUtlOHS. It IS C|Ult6possible, and this is under investigation, that, if either of these methods is applied

to the specific polyhedral convex set of the transportation problem.

TABLE III

Basic feasiblesolution

1

2

5

9

(x<, y.)

(0,1)(1,1)

(0,1)(1,1)

(1, 1)(0.641, 0.792)

(1,0)(1,1)

(0,1)(1,1)

(0, 1)(1,1)

(1, 1)(0.287, 0.596)

(1,0)(1,1)

(1,0)(0.173,0.924)

Z,i

2040

1044.143

28.28665.490

2040

1058.284

14.14248.286

069.126

28.28648.286

2059.579

60

54.143

93.776

60

68.284

62.428

69.126

76.572

79.579

method for finding all the basic feasible solutions will result t ha t is more efficient

than the usual ' loop' method. This would greatly improve the algorithm presented

here.

HEURISTIC ALGORITHM NO. 1 FOR THE TRANSPORTATION-

LOCATION PROBLEM

THE HEURISTIC algorithm described in this section was a first attempt to devise a

rapid suboptimal method for solving transportation-location problems. It was

suggested by a heuristic method previously developed for the pure location-alloca-tion problem and described in reference 3. This method, called the 'alternatelocation-allocation method,' is as follows:

1. Select some subset consisting of m of the n destinations that are given and considerthese as source locations.

2. Allocate each of the remaining n -m destinations to the closest of the m sources givenin Step 1.

3. Within each of the TO sets of destinations determined in Step 2, use the iteration methodgiven in references 2 and 3 [also used in equations (14) and (15) of this paper] to find theexact location of the optimal source location.

4. Determine for each destination whether or not it is closer to another of the sourceslocated in Step 3 than the one to which it is allocated. This defines a new grouping of msubsets of destinations.

5. Repeat Steps 3 and 4 until no further changes are possible.

It is shown in reference 3 that this algorithm is a moderately successful one, butby no means the best of the several heuristics tested for the pure location-allocationproblem. A modification of this method for the transportation-location problemcan be made as follows.

Alternating Transportation-Location Heuristic

1. Select arbitrarily m of the (XD,, yoi) and let these be the m initial source locations.This then yields a set of distances between each of the destinations and the assumed sources.

2. Using these distances as cost coefficients 7,y [as in equations (2)], we can solve anordinary transportation problem to find a set of [w,,\.

3. Using the |te,j| from Step 2, we can solve a location problem using equations (14)and (15) and find a new set of source locations.

4. We now iterate Steps 2 and 3 until no further changes, to within some tolerance, areobtained in two successive cycles.

It can be seen that the general notion behind this approximate method is tolocate sources alternately given a pattern of allocations and to determine an alloca-tion given a set of source locations. The location-allocation problem method andthe usual transportation-problem method are alternately applied to perform thecalculations. It can readily be seen that this iteration method yields a convergentmonotone nonincreasing sequence of values for z. However, there is no guaranteethat it will converge to the global maximum we seek. Still, what experience wehave with this and similar algorithms indicates that the result, when not optimal,lies within -^lO percent and usually within 2-3 percent of the optinval solution.

Table IV indicates the results with this heuristic method for the first sevenproblems given in reference 2. These are location-allocation problems with m=2,n = 7. They were solved as transportation-location problems by using a set ofcapacities and requirements such that each destination was supplied by only onesource. The starting points in each of these problems were completely random.For problems 3 and 4 the optimal solutions were obtained. In the others, resultsof varying degrees of closeness to optimality are obtained.

The Transportation-location Problem

Problem

1

2

*

5

6

7

ALTEKNATING TRANS

Allocationsobtained

(1, 2, 3, 4, 5)(6,7)

(1, 3, 4, 6, 7)(2,5)

(1,2,3,4)(5, 6, 7)

(1. 2, 3, 7)(4, 5, 6)

(1, 2, 3, 4, 5)(6,7)

(1, 2,3)(4, 5, 6, 7)

(1,5,6,7)(2, 3, 4)

PORTATION-LOCATION HEUI

allocations

(1,2,4,5)(3, 6, 7)

(1,3,4,6)(2, 5, 7)

(1, 2,3,4)(5, 6, 7)

(1,2,3,7)(4, 5, 6)

(1, 2,3,5)(4, 6, 7)

(1, 2,3, 4)(5,6,7)

(1 ,3 ,4 ,5 ,6 ,7)(2)

isTic RESULTS

Zobuined

52.118

81.764

38.323

48.850

38.560

44.564

61.935

Optimal

50.450

72.000

38.323

48.850

38.033

36.175

59.716

In order to study how the results obtained by the heuristic are related to

initial choice of the sources, 17 different (randomly selected) starting values w

chosen and the heuristic then applied for Problem 1 of Table IV. Table V ii

cates the results obtained. It will be noted, from Table IV, that, for problem ni

Allocationobtained

(1, 2,3,4,5)(6,7)

(1, 2, 4, 5)(3, 6, 7)

(1, 2, 3, 4)(5, 6, 7)

(1,3,4,5,6,7)(2)

(1, 2, 3, 7)(4, 5, 6)

(1,2,4,5,6)(3,7)

obt4ied

52.118

50.450

52.001

59.705

60.128

57.672

R PROBLKM 1

No. of times (out of 17)this solution occurs

3

3 (optimal solution)

3

2

2

106 Leon Cooper

ber 1, the optimal allocations are (1, 2, 4, 5), (3, 6, 7) and the optimal value of z

is .50.450.

From the results of Table V we can see that in three of the 17 trials we obtained

the optimal solution. In roughly 60 percent of the trials we obtained a solution no

worse than about 3 percent of optimal and usually better than this. The maximum

error was about 15 percent. However, it can be seen that repeated use of the

heuristic method, with varying starting values, is apt to give a reasonably good

approximation to the optimal solution, if not the optimal solution itself.

HEURISTIC ALGORITHM NO. 2 FOR THE TRANSPORTATION-

LOCATION PROBLEM

THIS ALGORITHM is based on earlier work reported in reference 4. It seems to be

extremely efficient, as will be seen. The heuristic method that has been developed

is as follows.

Transportat ion-Location Heuristic Method

1. Use one of the heuristic methods reported in reference 3 (the "alternate location-allocation" method referred to in the previous section is one of these) to find a solution withunspecified destination requirements and unlimited source capacities. Let the matrixA =||a,,|| he the allocation matrix for the solution obtained, i.e., a,y = l if source i suppliesdestination j and a,,- =0 otherwise. A is a matrix of zeros and ones such that each columncontains exactly one '1.* There are no restrictions on the rows, since one source may supplymore than one destination. The allocation matrix A is merely a convenient way of indicatingthe subdivision of n destinations into m subsets, i.e., m subsets served by each of the m

2. Replace the nonzero elements of A by their respective ry, thus forming a new matrixW', = l|«.;-, II where

""•'"lo',' o!!=0.'3. Using the original matrix A of allocations, we now derive two new matrices, analogous

to Wl of Step 2 above, as follows. Find the pair of points being served by the same sourcesuch that the distance between them is a maximum. Let the sources be 8,,, u = l, • • •,m,and let Q« be the subsets of destinations. Then we wish to find a pair of points p,, p, asfollows:

Let the subset in which this occurs be Qi, with source »». We then eliminate source 8 andreplace it with points p., p,. We now have a set of m + 1 sources (««, u = l, • • . , m; u^h),p., p,. Therefore, we have one more source than is desired. Let ft = ((««, u = l, , m;U}^h), p., p,\. The source coordinates are (z., ;/«) for u.,ih, (XD,, yo,), (XD,, yot). Fornotational simplicity, we rename the sources as »,. Therefore, R = [sJ\v = \, • • • , T O + 1 1 ,where the first m — 1 sources are «,,(u^h) and where s™ =p, , s«+i =pt. We now wish to findthe pair of sources s. and % that are closest together. Let us designate the set of indicesV, corresponding to »,eK, as V, i.e., V = l»|»,eSl. The pair of sources »„ and »i is now de-

The Transportation-Location PrtMem 107

terminedbyUx.-Xt)*+(y.-yt)']'i'=imiii.,.T-t^,Uxt-x,)'+(yt-y,)']^".

Having found the pair of sources «„,»»that are closest toother, first we eliminate s, and applythe 'alternate' method of successive location and allocation until convergence, as referred tounder step 1 above. This will determine a second solution. Next we restore source ».and eliminate sj, apply the 'alternate' procedure and obtain a third solution. We calculatetwo new matrices Wi and W) corresponding to these solutions, as we did in step 2.

4. Sum the requirements for each subset of destinations served by one source, i.e., calcu-late E . . . . I-/for each subset Si, i = l, , m. We now calculate the differences, c.—E^.-i r,,1 = 1, ,m. A negative difference implies a capacity deficit and a positive differenceindicates a capacity surplus.

5. The iterative calculation begins with this step. Let /_ be any set with a capacitydeficit and let /+ be any set with a capacity surplus. Choose a destination point pt in the.let /_ such that the difference in the distances from pt to the source of /_ and from pt tothe source of 7+ is a minimum. Symbolically, if pt has coordinates {x^t, y,,t) and the sourcelocation of /_ is (a:_, yJ) and that of 7+ is {x+, y+), then we choose p* as the index j from 7_such that

e'=imn,.,JUx,,-x..y-t(yD,-y-)'V"-l(xD,-x+)'-i-(yo,-y^)'V"\.

6. We now reallocate part or all (if possible) of the requin7_ to the source in I+. The amount reallocated depends on tthe surplus at 7+. The four cases are

(a) If the (requirement at pt) S (deficit for 7_) and the (surplus for 7 .) £ (deficit for 7_),then we reallocate the deficit from 7_ by supplying this amount to p* from 7+ instead of 7_.

(b) If the (requirement at p*) fe (deficit for 7_) and the (surplus for 7+) <(deficit for 7_),then we reallocate the amount of the surplus at 7+ by supplying this amount to p* from 7+instead of 7...

(c) If the (requirement at pt) < (deficit for 7_) and the (surplus at 7+) fe (requirement atPk), then we reallocate the entire requirement at pt from 7_ to 7+.

(d) If the (requirement at pt) < (deficit for 7_) and the (surplus at 7+) < (requirement atPt), then we reallocate the amount of the surplus at 7+ by supplying this amount to pt from7+ instead of 7...

7. With the new allocations, new source locations are computed for each subset of desti-nations S, by the use of equations (14) and (15), using the allocations as weights.

8. Each subset of destination points S, is now examined for the points that might becloser to a different source with an excess capacity. If possible, we reallocate part or all ofits requirement, depending on the amount of the surplus at this source.

9. We now repeat steps 7 and 8 (exact source location and subsequent reallocation tosatisfy requirements), until no further change in the allocation matrix occurs.

10. The entire reallocation process (steps 5-9) is now repeated until all capacity deficitsare removed. The value of z for this allocation matrix Wi is computed.

11. The procedure of steps 4-10 is repeated for allocation matrices W, and W,.

12. The minimum of the three values of z obtained is chosen as the solution, together with

its source locations and destination allocations.

This heuristic method was tested on the eight two-source, seven-destinationproblems listed in reference 2. The requirements were generated randomly. Thecapacities were chosen all equal, with their sum 5 percent higher than the sum of the

Leon Cooper

requirements. In order to have exact solutions to compare the heuristic against,

the exact extremal equations (14) and (15) were used to solve for all possible alloca-

tions that, for these test problems, included the optimal solutions. For these eight

problems, the heuristic method produced the optimal solution.

In reference 4 this basic method was also applied to 100 randomly generated

problems with apparently good results, although the correct solutions were not

known in advance.

1. M. BALINSKI, "An Algorithm for Finding All Vertices of Convex Polyhedral Sets," J.Soc. Ind. & Appl. Math 9, 72-88 (1961).

2. L. COOPER, "Location-Allocation Problems," Opns. Res. U, 331-343 (1963).3. — — , "Heuristic Methods for Location-Allocation Problems," SIAM Review 6, 37-52

(1964).4. — , "Solutions of Generalized Locational Equilibrium Models," J. of Regional Science

7, 1-18 (1967).5. 0 . DEMUTH, "A Remark on the Transportation Problem," Casopis pest. mat. 86,103-110

(1961).6. A. DoiG, "The Minimum Number of Basic Feasible Solutions to a Transport Problem,"

Opnat. Res. Quart. U, 387-391 (1963).7. W. H. FLEMING, Functions of Severat Variables, Addison-Wesley, Reading, Mass., 1965.8. G. HADLEY, Linear Programming, Addison-Wesley, Reading, Mass., 1962.9. D. A. KoHLEE, "Projections of Convex Polyhedral Sets," ORC 67-29, University of

California, Berkeley, 1967.10. T. S. MoTZKiN, H. RAIFPA, AND R . M. THRALL, "The Double Description Method,"

in H. W. KuHN AND A. W. TUCKER (eds.), ContribiUions to ihe Theory of Games, Vol.II, Annals of Mathematics Study No. 28, Princeton U. Press, Princeton, 1953.

11. C. WiTZGALL, AND R. J. WETS, Journal of Research of the Naiimal Bureau of Standards71B, 1-7 (1967).