the simple plant-location problem under uncertainty

The Simple Plant-Location Problem under UncertaintyAuthor(s): James V. Jucker and Robert C. CarlsonSource: Operations Research, Vol. 24, No. 6 (Nov. - Dec., 1976), pp. 1045-1055Published by: INFORMSStable URL: http://www.jstor.org/stable/169977 .

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OPERATIONS RESEARCH, Vol. 24, No. 6, November-December 1976

The Simple Plant-Location Problem under Uncertainty

JAMES V. JUCKER and ROBERT C. CARLSON

Stanford University, Stanford, California

(Received original November 1, 1974; final, February 12, 1976)

There is a class of simple plant-location problems under uncertainty that can be decomposed into two simpler problems that can be solved sequentially with surprising ease. The salient characteristic of this class of problems is that for any demand generating region, one plant dominates all others as a supply source. We develop and discuss the conditions required for this dominance. The risk-averse nature of the firm is incorporated into this model with a mean-variance formulation of the objective function. The general formulation is particu- larized to four paradigms of firms characterized by a range of behavioral and market assumptions.

W E START with the simple, uncapacitated plant-location problem that Efroymson and Ray [3, p. 366] described as follows:

In its simplest form, plant location can be posed as a transportation problem with no constraint on the amount shipped from any source. However, there is a cost associated with each sourcet (plant). This cost (called a fixed cost or fixed charge) is zero if nothing is shipped from the plant, i.e., the plant is "closed." It is positive and independent of the amount shipped if any shipment from the plant takes place, i.e., the plant is "open."

It should also be noted that this is a one-product, one-period, static problem with plants restricted to m feasible sites and the total market divided into n distinct regions.

VTe will use the following notation: tij = the unit transportation cost from plant i to region j, dj = the demand in region j (in units of product), xij = the fraction of the demand in region j supplied from plant i, and fi = the fixed cost associated with plant i. In addition, we will need ui = the per unit cost of producing the (single) product at plant i, pj = the per unit selling price of the product in region j, and rij = pj - ui- tj= the marginal profit (contribution to profit and fixed costs) derived from selling a unit of product from plant i in region j.

Given the usual assumption of certainty, this simple plant-location problem can be formulated as a mixed-integer programming problem [3, p. 362]. We choose to state the problem as one of maximizing total profit, 11, rather than minimizing costs for reasons that will become clear as we proceed. The

1045



1046 James V. Jucker and Robert C. Carison

problem is to

maxII Ej E rijdjxi - Ei-fiy

Es xij l, j- l,= n 0 xi< _ i = l, ,m;j-1 ,n

yi c 0 l, i = , * m.

This problem, in its cost minimization form, has received a substantial amount of attention in the literature [3, 4,1 0, 16]; and, in the sense that many problems of a practical size can be economically solved using current algorithms, the problem itself can be considered "solved." It is our purpose to consider extensions of this problem that permit uncertainty in either price or demand. We will show that for certain cases of practical significance the plant-location problem under uncertainty can be solved almost as easily as can the corresponding deterministic problem.

Relaxing the assumption of certainty makes this problem not only more realistic, but more interesting as well. Once it is assumed that there is either price or demand uncertainty, we are faced with the necessity of identifying some behavioral characteristics of the firm whose decision problem we are modeling. We must know something of the firm's attitude toward risk and the extent to which the firm waits to see the uncertainty resolved before making decisions.

We will start with a discussion of the assumed behavioral characteristics of the firm. Then we will describe a simple dominance condition that leads to an efficient solution procedure for the problems under consideration. Finally, examples illustrating some of the characteristics of simple plant- location problems under uncertainty will be given.

1. RISK AVERSITY

In the plant-location decision, a substantial capital commitment is required, and long-term constraints on production and distributionl patterns are implied by the decision. The uncertainties that exist with regard to both total future demand and the spatial distribution of future demand are in most instances significant. Thus, we would argue that the plant-location decision is certainly one of those decisions for which Sharpe [15, p. 27] reports, "A large body of evidence indicates that almost everyone is a risk averter when making important decisions. Clear counter-examples are rarely found."

It will be assumed throughout that the firm is risk-averse and that (a) the objective of the firm is to maximize its expected utility for end-of-period



Plant Location under Uncertainty 1047

profit, and (b) this objective can be attained through the maximization of z = E - XV, where E is the expected value of end-of-period profit, V is the variance of the end-of-period profit, and X is a non-negative parameter that represents the rate at which the firm will substitute variance for expected value. It is known (see references 8, 12, and 13) that the use of the mean- variance criterion is approximately consistent with the principle of maximizing expected utility if (a) the firm's utility function can be represented by a quadratic function of end-of-period profit, or (b) the subjective probability distribution of end-of-period profit is a two-parameter distribution, such as the normal distribution.

A crucial aspect of any multicriteria formulation is the determination of X. Past work has proposed and applied one or more of four techniques. In the first, a single value of X is selected. The sources of this estimate of X range from management judgment to the analysis of historical cost data, as suggested by Ijiri [9]. This approach does not account for the fact that X may depend on the values of E and V. A second technique involves assum- ing the functional form of the utility function and then using known solutions (possibly historical values of E and V) to determine the values of the function's parameters (e.g., Briskin [2] uses an exponential form of the utility function). Both of the above techniques require rather arbitrary se- lections. In a third technique, solutions are generated for a number of values of X within its acceptable range. This set of efficient solutions is then dis- played to the decision maker, who selects the one he prefers. This selection implicitly identifies the appropriate value of X but presents a more difficult choice to the decision maker. Sharpe [15] advocates this approach for portfolio-analysis problems. In the plant-location problem the generation of a large set of efficient solutions will be computationally infeasible. Therefore, a fourth technique involving the simultaneous determination of the pre- ferred solution and the value of X by an interactive process using methods such as those suggested by Baum and Carlson [1] and Geoffrion, Dyer, and Feinberg [5] appears to hold the most promise for this problem.

This difficult problem of determining X is far from solved. In a survey article Roy [14] points out problems in more detail. Furthermore, Wallenius [17] reports experiments showing that managers do not feel at ease making the trade-off decisions required to obtain a value for X. Nevertheless, even the crudest technique is probably better than forcing the problem into a single criterion formulation.

2. FOUR TYPES OF FIRMS

We have selected four types of firms (see Table I) exhibiting four very different control behaviors for specific analysis. Although these four firms



1048 James V. Jucker and Robert C Carlson

TABLE I CHARACTERISTICS OF FOUR TYPES OF FIRMS

I MONOPOLY PUBLIC UTILITY NEWSBOY AGRIuSInESS Price-setting firm Price-taking firm Price-taking firm Quantity setting firm Prpducing producing producing a to order to order perishable good

1. qi set ex ante 1. pi set ex ante 1. pi set exoge- 1. pi set exogenously nously

2. pj's are func- 2. di's are func- 2. di's are random 2. di's are set exog- tions of qj and tions of pi and variables with enously as ran- uncertainty uncertainty known means dom variables

and variances with known den- and are set sity functions exogenously

3. Firm sells qi at 3. Firm sets qj ex 3. Firm sets qi ex 3 qi set ex ante Pj post to meet post to meet

demand, di demand, di pi = vi - wiqi + Ej qi = si = d= ai qi = sj = di = aj di = a; + Ei

-bip2 + Ej + 'e

do not represent all possible control behaviors, they are comumonly en- countered in many economies; and they demonstrate the generality of the model developed here. In addition to the definitions and the terminology already introduced, we will follow Leland [11] in using: ex ante decisions: decisions made before demand is known, ex post decisions: decisions mnade after demand is known, qj = the quantity of product produced for sale in region j, sj = the quantity of product sold in region j, Cij(qj) the variable cost of supplying qj units to region j from plant i, - qj) =j contribution to profit and fixed costs from region j sales given that region j is sup- pliedfrom plant i, irij(qj) =p jSj - Ci(qj) , and ej a normally distributed

2 random variable with mean zero and variance ai .

Each of the four types of firms is characterized in Table I as to: 1. Which variables are determined exogenously and which are decision

variables under the control of the decision maker, and 2. Which of the decision variables are set ex ante and which are set

ex post. We will assume that the price-demand relations can be approximated by the linear functions given in Table I for each type of firm.

3. DOMINANCE

For a set K1 of open plants Efroymson and Ray 13] showed how the deterministic plant-location problem can be decomposed into n easily solved




subproblems. The condition required for decomposability is called dominance, and it is equally computationally advantageous in our stochastic problem. Dominance occurs when for every feasible K, there exists an m(j) C Kl such that 7m(j),j(qj) > ij(qj), j = 1, 2, ** *, n; i E K1, over the relevant ranges of the qj's. From the definition of irij(qj), this is equiva- lent to requiring that Cm(j),j(q) = miniEK Ci,(qj), j = 1, 2, , n, over the relevant ranges of the qj's.

A sufficient condition for dominance to be present is the linearity of the variable supply cost function, that is, Cij(qj) = cijqj, where cij = ui + tij. In this case the contribution function and the dominance relation can be written as

7rij(qj) = pjsj - cijqj and (2)

Cm(j),j = miniE(K ci1, j = 1, *.* , n. (3)

Thus for any realization of pj, sj, and qj, m(j) is the plant in K1 with the lowest unit variable supply cost; and either m(j) will be selected to meet the demand in region j or the demand in region j will not be met. The man- ner and extent to which the demand in region j is met are determined by an optimization procedure described in the next section. The important point here is that

_ SO or 1 = m(j), j = 1, ,n 02 i On t (j),) j ly 1 , #n.

That the xij are 0-1 variables when there are dominant plants permits significant simplifications in the formulation and solution of the simple plant- location problem under uncertainty.

Although we will follow Efroymson and Ray [3] in making the linearity assumption, it is useful to point out a case in which the costs are nonlinear and dominance holds. This occurs when the marginal costs of production are constant and identical at all plants (that is, ui = u, i = 1, ***, m), even though the transportation cost functions, tij(qj), are subject to price breaks. If Cij(qj) = tij(qj) + uqj and if the transportation cost functions are such that the same percentage price breaks occur at the same values of q (such as at car-load lots) for all plants, then the closest plant in K1 will dominate all others for each region j. Figure 1 illustrates such a situation.

4. SPLPUU: STATEMENT AND SOLUTION OF THE PROBLEM

Given a firm with the objective max z = E[II] - XV[II], given that a dominant plant exists for each region j in every feasible set of open plants K1, and given that all random variables are independently distributed, the simple plant-location problem under uncertainty (SPLPUU) can be stated



1050 James V. Jucker and Robert C. Carlson

t.j(qj)

tej qj)

tfj (qj)

/r~~~~~~~~~~M X tmA ), (qj'

' 9" qj

Fig. 1. Example of transportation cost functions with price breaks at q' and q" and such that for e, f, m(j) e K1, plant m(j) is the dominant plant in region .

as follows:

max z = E[E Ej irijxij - fy] - X Ei Ej V[7rj]xj = Ei Ej (E[7rij] -V[7rij])xij- Eif

Eixii < 1, j- 1, *l , n

s; < dj, sj _ qjy j n(4

fj(pi, dj, Ei) = 0, j = 1, , n

xij, yi = 0, 1, i= 1, ***,m; j =1,.., n

0 < xij _ yi _ 1, i .. 1,* m; j = ... *, n,

where lrij = irij(qj) for notational convenience, and f,(pi, di, Ej) = 0 represents the implicit function form of either the demand function or the in- verse demand function.

Note that this problem statement is valid only when there are dominant plants (when the xij are restricted to 0-1 values) since each 7rij is, in general, a nonlinear function of the fraction of the region served.

As a first step toward solving this problem we rewrite the objective function to emphasize the fact that there are three levels of optimization.

z = maxyi {maxx i [i Ej maxp,,q, (E[rij] - -Ei fiyi]. If we are given a set of open plants, K1, the remaining problem has two parts. First, the dominant plant, m (j), must be selected for each region j. Next, the remaining optimization subproblems associated with each region must be solved. It is most efficient to solve each of these m X n subproblems as a pre-optimization step and then use the resulting value of the ob-




jective function, zij, in the branch-and-bound algorithm used to solve (4). Each subproblem has the form

zij = maxp,,q, (E[iJj]- XV[7r'j]), for all i and j

f(pj, d,) = 0, j- 1, *.., n (5)

sj _dj, sj <qj) j-1 , n.

Now the SPLPUU becomes

max z E jz .ijxij - Ei fyi

I,ixij _ 1, 1, 1, * j+, n (6)

zXii Yi = O,1, i= 1, *-m; j ,**,n

O _!! zx j < y i < 1,. i-1, *-,; j=1 -n.

This problem can be solved directly by the branch-and-bound methods of Efroymson and Ray [3] or Khumawala [10] with but one minor modifica- tion. For any zij < 0 the corresponding xij will be set equal to zero. Thus it is possible that some regions will not be served. To allow this, the first constraint in problems (4) and (6) is an inequality rather than the equality constraint of problem (1).

Problem (4) has now been decomposed into two simpler problems: 1. Determine the values of zij by solving the m X n optimization prob-

lems defined in (5). In many cases these problems can be easily solved by traditional methods.

2. Solve problem (6) using one of the more efficient methods available for this problem [3, 10].

Since the methods for solving (6) are very efficient and since the determination of the zij's will often be straightforward, it should be possible to solve many problems of practical significance. We will now illustrate what is involved in the solution of problem (4) for the four types of firms that are characterized in Table I.

5. EXAMPLES

Example 1. The Quantity-Setting Firm

We assume that the firm establishes supply quantities ex ante for each region. Then market-clearing prices that are linear functions of the quantities available in each region and uncertainty are established ex post. The resulting equilibrium between the markets and the firm implies that s, =

dj = qj. We also assume that the marginal cost of supply is constant and that e -- N(O, o-2) and COY ki, Ek] = 0 for all j and k 5 j. For this type of




firm ri j = (vj - wjqj + Ej)qj - cijqj and E[7rii] = (vj - wjqj - cij)qj, V[7rij] = qj2 o2. Problem (5) can now be written as

Zij = max., [(vj - wjq1 - cij)qj - Xq2j o2], for all i, j.

The first set of constraints in (5) have here been substituted into the objective function, and the other constraints are satisfied as equalities because of the equilibrium assumption. The resulting unconstrained maximization problems produce qj*(i), the optimal supply quantity for each region j given that the region is served by plant i: qj*(i) = (vj - cij)/2(wj + Xoi 2), for all i, j. These conditional supply quantities imply that zij = (Vj - Cij)2/

4(wj + j2), for all i, j. Now these .,ij's can be used in problem (6) to solve the SPLPUU by a branch-and-bound algorithm.

The impact of increasing risk-aversion (increasing X) on the firm's behavior is now apparent. As the firm becomes increasingly risk-averse, it will decrease the supply quantities. This will, in turn, effect reductions in both the mean and the variance of the contribution from each region. The firm, becoming more risk-averse, will be willing to sacrifice some expected contribution in order to reduce its risk.

Example 2. Price-Setting Firm Producing to Order

In this example we assume that "producing to order" implies equilibrium between the markets and the firm. Moreover, we assume that this equilibrium is a function of ex ante regional prices as well as random perturbations. Specifically, we assume that qj = = di = ai - bjpj + fj, and, as always, we assume the ej's are N(O, oj2) and independently distributed. Following example 1, we can show that the solution of problem (5) produces an optimal price in each regionj given that the region is served by plant i: pj*(i) =

cij + (ai - bjcij)/2(bj + Xo_2), for all i, j. These conditional prices imply zi; = (aj - bcCij)2/4(bj + Xo_2), for all i, j. These zij's can now be used in (6) to solve the SPLPUU.

The impact of increasing risk-aversion (increasing X) on the firm's behavior is interesting. As the firm becomes more risk averse, it lowers prices. This, of course, will increase regional demands and the quantities supplied. As X -* oo , the regional prices approach the costs of supply.

Example 3. Price-Taking Firm Producing to Order

Once more we assume equilibrium between the markets and the firm. This time, however, we assume that prices are set exogenously and that demand is known as a random variable. Hence qj = si = dj = aj + ej, for all j.




We also assume constant marginal supply costs and we make the standard assumptions concerning the ej's. For this case rirj = (pj- cj) dj = (pj -

cij) (aj + ej), and problem (5) now becomes simply zij = (pj - cj)aj -

X(pi - cii)2oi2, for all i, j. The first set of constraints in (5) is satisfied by assumption: both prices and demands are determined exogenously. The second set of constraints is satisfied by the equilibrium assumption. Problem (5) is thus solved without an optimization procedure and we go directly to problem (6). Although problem (6) can be solved by the methods of references 3 or 10, one difficulty remains. Observe that as X increases (the firm becomes increasingly risk-averse), zij becomes smaller; and for X sufficiently large, all zij will be negative. This implies that, regardless of the values of the (pj - cj), the solution to (6) for a highly risk-averse firm will be nullity. No plants will be built and no markets will be served. An even more dis- turbing observation is that zij will be a decreasing function of (pj - cij) for (pi - cij) > aj/2Xo-,j since zij is a quadratic function of (pj - ci,). But we know from the dominance argument that any risk-averse firm will prefer plant e to plantf in region j if (pj - cej) > (pj - Cf j) . Therefore, it is neces- sary that Zej > Zfj if (pj - Cej) > (pj - Cfj). That is, the solution to problem (6) will give results that are consistent with the firm's preferences only if X is restricted to the interval [0, Xwmax] where Xmax = mini,j {aj/2(pj -

cij) o-j2} for when X is restricted to this range, Z,j is an increasing function of (pj - cij). Also, for X in this restricted range, zij will be negative only for (pj - cij) < 0. Thus, for this type of firm problem (6) is valid only for XA6[0 Xmax].

It is worth emphasizing that because of dominance, the assignment of regions to plants will be the same for any Xe[0, Xmax]. The value of the objective function in (6) will vary with X, however, and thus the set of open plants in the solution to (6) will be a function of X.

Example 4. Price-Taking Firm Producing Perishable Good

In this example prices are set exogenously and demands are known as normally distributed random variables with means aj and variances o-r2. It is assumed that the firm sets the supply quantities ex ante. Since supply quantity has no influence on the quantity demanded, the firm may either over- or undersupply the market. Further assumptions are that the marginal supply cost is constant and that the unsold product in any region has zero value and zero disposal cost. Given t,hese assumptions, the contribution function is

J pjd; - cijqj, dj < qj pjqj -cijqj, dj > qj




where dj - N(aj, o-j2). Since in this case the objective max [E - XV] produces a very messy problem, we will limit the example to the case of X - 0 (risk neutrality). Given this restriction, problem (5) becomes z; - max qjE[7rij], for all i and j. Again the first set of constraints in problem (5) is satisfied by the assumption that there is no relationship between price and quantity demanded. The second set of constraints is incorporated in the expression for E[7rijl. Thus the zij are determined by solving m X n maximization problems of the type that are commonly called "newsboy" or "Christmas tree" problems. The optimal supply quantity for region j, given that it is supplied by plant i, is known to satisfy the relation [7, pp. 297-299] f qj*(i) gj(x) dx = Cij/pj, where g(j) represents the normal probability density function with mean aj and variance fj,2. Thus zij = E[irij(qj*(i))], and these zij's can be used in problem (6) to solve this SPLPUU by branch and bound.

Gonzales [6], considering a variation of this basic newsboy-type firm and approaching the problem from the point of view of stochastic programming, developed the same solution procedure that we have developed here for solving the newsboy location problem. Knowledge of his results helped us see how the newsboy-type firm fits into the SPLPUU framework.

6. CONCLUSIONS

We have shown that: 1. There is a class of SPLPUU's that can accommodate a variety of as-

sumptions about both the behavior and the cost and revenue functions of the firm whose problem is being considered.

2. When the variable cost of supply functions is such that the condition for the existence of dominant plants can be satisfied, then these problems can be decomposed into two simpler problems that can often be solved with surprising ease.

REFERENCES

1. S. BAUM AND R. C. CARLSON, "Multi-goal Optimization in Managerial Science," Omega Int. J. Management Sci. 2, 607-623 (1974).

2. L. E. BRISKIN, "A Method of Unifying Multiple Objective Functions," Man- agement Sci. 12, B406-B416 (1966).

3. M. A. EFROYMSON AND T. L. RAY, "A Branch-Bound Algorithm for Plant Lo- cation," Opns. Res. 14, 361-368 (1966).

4. R. L. FRANCIS AND J. M. GOLDSTEIN, "Location Theory: A Selective Bibliog- raphy," Opns. Res. 22, 400-410 (1974).




5. A. M. GEOFFRION, J. S. DYER, AND A. FEINBERG, "An Interactive Approach for Multi-Criterion Optimization with an Application to the Operation of an Academic Department," Management Sci. 19, 357-368 (1972).

6. F. GONZALES-VTALENZUELA, "Simple and Capacitated Warehouse Location Problems with Stochastic Demand," Ph.D. Dissertation, Stanford Uni- versity, January 1975.

7. G. HADLEY AND T. M. WHITIN, Anlysis of Inventory Systems, Prentice-Hall, Englewood Cliffs, N. J., 1963.

8. G. HANOCH AND H. LEVY, "The Efficiency Analysis of Choices Involving Risk," Rev. Econ. Stud. 36, 335-346 (1969).

9. Y. IJIRI, "A Historical Cost Approach to Aggregation of Multiple Goals," p. 395-405, in J. COCHRANE AND M. ZELENY (eds.), Multiple Criteria Decision Making, University of South Carolina Press, Columbia, S. C., 1973.

10. B. M. KHUMAWALA, "An Efficient Branch-Bound Algorithm for the Ware- house Location Problem," Management Sci. 18, B718-B729 (1972).

11. H. E. LELAND, "Theory of the Firm Facing Uncertain Demand," Am. Econ. Rev. 62, 278-291 (1972).

12. G. C. PHILLIPPATOS AND N. GRESSIS, "Conditions of Equivalence among E-V, SSD, and E-H Portfolio Selection Criteria: The Case for Uniform, Normal, and Lognormal Distributions," Management Sci. 21, 617-625 (1975).

13. R. B. PORTER AND J. E. GAUMNITZ, "Stochastic Dominance vs. Mean Variance Portfolio Selection Analysis: An Empirical Evaluation," Am. Econ. Rev. 62, 438-445 (1972).

14. B. Roy, "Problems and Methods with Multiple Objective Functions," Math. Programming 1, 239-266 (1971).

15. W. F. SHARPE, Portfolio Theory and Capital Markets, McGraw-Hill, New York, 1970.

16. K. SPIELBERG, "Algorithms for the Simple Plant Location Problem with Some Side Conditions," Opns. Res. 17, 85-111 (1969).

17. J. WALLENIUS, "Comparative Evaluation of Some Interactive Approaches to Multicriterion Optimization," Management Sci. 21, 1387-1396 (1975).



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