Multidimensional Assortment Problemwith an Application

Ashish Tripathy,1 Haldun Sural,2 Yigal Gerchak2

1 USAir, Arlington, Virginia

2 Department of Management Sciences, University of Waterloo, Waterloo, Ontario,Canada N2L 3G1

Received 18 November 1997; accepted 27 August 1998

Abstract: This paper addresses the discrete multidimensional assortment problem. Assortmentissues arise frequently in practice as an important design and inventory problem which simulta-neously seeks the answers to two related questions: (a) Which items (or sizes of a product) to stock?(b) How much of each to stock? Its discrete multidimensional version concerns itself with choosingsizes from among a discrete set of possible ones with each size being characterized by more than onedimension. Our research is motivated by an application of the problem in the distribution center of aglobal manufacturer of telecommunications equipment where the goal was to standardize the sizesof three-dimensional crates used to package finished items by selecting a few from among all cratesizes. The main contributions of this research are (1) modeling the assortment problem as a facilitylocation problem, (2) devising a heuristic procedure that generates a good solution to the problem aswell as a bound on the optimal solution, and (3) implementing the heuristic procedure on a PC so asto obtain solutions for actual large-scale instances of a three-dimensional problem. © 1999 John Wiley& Sons, Inc. Networks 33: 239 –245, 1999

1. INTRODUCTION

An important part of inventory management concerns thechoice of end-items to stock. The stocking decision, whichis based on objectives such as profit maximization (or costminimization) and customer satisfaction, has to take intoaccount interdependencies that are often present among thevarious items involved. Our research is directed towardfinding optimal stocking policies for inventory problemswhere the end-items are related throughsubstitution,that is,an item can be substituted for by modification of another

one at a certain cost. This problem is often referred to as theassortment problem.The assortment problem essentiallyseeks to determine a subset of sizes to be stocked fromamong a continuum or a large discrete set of sizes. If ademanded size is not stocked, it will be substituted for by a“feasible” stocked size (i.e., a stocked size which is capableof substituting for the unstocked size), thus incurring anadaptation cost.

The principal objective of this research was to develop amathematical model for the discrete multidimensional as-sortment problem and a solution technique that can effi-ciently solve large problems encountered in practice. Itsmotivation is an actual problem faced by the distributionCorrespondence to:Y. Gerchak; e-mail: ygerchak@engmail.uwaterloo.ca

© 1999 John Wiley & Sons, Inc. CCC 0028-3045/99/030239-07

239

center (DC) of a global telecommunications equipmentmanufacturer based in Ontario, Canada.

The DC receives finished goods from various manufac-turing facilities based in North America, packages them intocrates, and dispatches orders to customers all over the globevia land, water, and air. The packaging operation involvesmaking wooden crates of various sizes. Standardization ofsome or all crate sizes can smooth production and reduceexcess capacity as well as simplify production control.However, if a larger-volume crate substitutes for a smallerone, extra costs of material and shipping are incurred.Having a large set of historical data, the company wishes toselect a few crate sizes from among all sizes actuallyshipped from the DC in the past. With crate sizes beingthree-dimensional, the problem is construed as a specialcase of the multidimensional assortment problem. Thisproblem has received limited attention in the past; mostresearch has focused on specific solution techniques forone- and two-dimensional problems, and solution tech-niques for the multidimensional problem are still elusive.

The classic motivation in the one-dimensional case is thedetermination of lengths of structural steel beams to stock,in which a longer-than-demanded beam can be cut down toa shorter length, incurring a substitution cost consisting ofthe cutting cost and the cost of scrap. There is a demand fora large number of lengths. The larger the subset of lengthsstocked, the smaller the steel loss. However, due to setupcosts associated with the production of each length, it isdesirable to produce a small subset of lengths. The problemis to determine the optimum subset of lengths to stock so asto minimize the sum of all relevant costs. In early articles,Sadowski [12] and Wolfson [15] addressed this problem inthe steel industry for continuous and discrete cases. Sad-owski restricted his analysis to a fixed machine-installationcost for each size and linear variable cost for the quantityproduced of a particular size and used dynamic program-ming to solve the problem. Pentico [8, 9] extended thediscrete problem to consider (general) concave production-inventory and substitution costs and probabilistic demands.For different types of concave cost functions, he developedoptimum stocking and substitution policies and provedsome properties which simplify computational proceduresin his dynamic programming algorithms. Tryfos [13] ana-lyzed a continuous version of the problem which arises inthe apparel industry. Since each size offered involves addi-tional setup and design costs and complicates inventorymanagement, there is a strong incentive for attempting to fita population with as few sizes as possible. But fewer sizesmeans higher discomfort for customers or lower profit. Theproblem, thus, is to find how many and which sizes toproduce and stock. Jones et al. [6] showed that a specialcase of the one-dimensional deterministic assortment prob-lem can be formulated as a specially structured uncapaci-tated location problem.

The two-dimensional assortment problem is motivatedby numerous practical applications like selection of length

and strength of steel beams, length and width of glass sheets(Chambers and Dyson [2]; Diegel and Bocker [4]), andrectangular steel bars (Page [7]) as well as composition andquality of chemical compounds. The most general analysisof the discrete two-dimensional problem so far was per-formed by Pentico [10], who formulated the problem as asingle-source multiple-destination network flow problem,characterized optimal stocking patterns, and presented heu-ristic solution procedures for a concave stocking cost and alinear substitution cost. Tryfos [14] suggested ap-medianlocation model for the general multidimensional determin-istic assortment problem.

In this paper, we first define the problem and some of itsbasic properties in Section 2. An uncapacitated facilitylocation problem formulation of our problem and its solu-tion procedure are discussed in Section 3. In Section 4, wepresent computational results for randomly generated testproblems and for real data obtained from the company.Section 5 outlines our concluding remarks.

2. THE PROBLEM AND SOME BASICPROPERTIES

We are given a discrete set of sizes of a certain item andtheir demands. It may be preferable to stock a subset ofthese sizes because of some or all of the following:

1. Limitations in production and storage.

2. The costs associated with producing different sizes andwith holding them in stock.

3. Risk-pooling over uncertain demands.

Demand for an unstocked size can then be satisfied from thestock of a “feasible” size with an associated substitutioncost. The problem is to decide on the particular subset ofsizes, and amounts thereof, to stock and the substitutionrules to follow so as to minimize the sum of all relevantcosts. In short, the problem is to find an optimal stockingand substitution policy.

Let (i1, i2, . . . , i r) denoter dimensions of sizei , wherei1 $ i2 $ . . . $ i r and i 5 1, . . . , n. In our setting, sizei can substitute for sizej only if i q $ j q for each q5 1, . . . , r . While we use the term “dimensions,” in someapplications, these may actually be attributes like color,quality, etc. If sizei can substitute for the demand of sizej ,then sizei will be referred to as “larger” than sizej . Withoutloss of generality, we may assume that sizes are not iden-tical.

There are two types of costs that need to be considered inour problem: A stocking cost is incurred to maintain a stockof sizei and is assumed to have a fixed componentfi and alinear variable componentvi. It may include the fixed costsof production and acquiring inventory and the variable costsof production and holding inventory. A substitution cost is

240 TRIPATHY, SURAL, AND GERCHAK

incurred if a larger sizei is used to satisfy the demand of asmaller sizej . This substitution cost is assumed to have afixed componentgij representing the cost of buying andsetting up equipment for the transformations needed for thesubstitution and a linear variable costwij including the costof extra material used in production and the cost of addi-tional space needed for storage. Note that the variablecomponent may also include the cost of transformingi intoj (e.g., cutting a steel beam).

Feasible substitutions between sizes in a discrete assort-ment problem can be represented by a graph in which nodei corresponds to sizei and an arc (i , j ) represents a feasiblesubstitution between sizesi andj . Such a graph is called thesubstitution graphof the assortment problem. In our setting,the substitution graph is directed and acyclic. Considering adirected graphG 5 (V,A) with a node setV and an arc setA, wheren 5 uVu, Ahuja et al. [1] showed thatG is acyclicif and only if there exists an ordering of the nodesv1,v2, . . . , vn such that for every directed arc (vi, vj) [ A wehave i , j . Such an ordering of the nodes is called atopological ordering. It implies that for the discrete assort-ment problem we can always number all sizes in such a waythat if a size numbered asi can substitute for the demand ofa size numbered asj then i , j . Here is a simple example:

Example 1. Consider a two-dimensional assortment prob-lem with sizes {(13, 12), (11, 10), (9, 8), (14, 9)}. We firstillustrate its substitution graph with the given order of sizesin Figure 1(a) and next show the graph with the topologicalordering of sizes in Figure 1(b). In Figure 1(c), we illustratethe problem where additional arcs are introduced to repre-sent the stocking option of sizes. Here, an arc (s, i ) fromsource nodes to size nodei denotes the stocking of sizei .Unless otherwise stated, in this study, we assume that allsizes are numbered in the topological order.

An optimal stocking and substitution policy,for thediscrete multidimensional assortment problem, consists offinding subset(s) of sizes, their amounts that are actuallystocked both (either) for themselves and(or) for demands ofunstocked sizes, and the substitution rules to follow, so as tominimize the sum of the stocking and substitution costs.

When the stocking costs are concave, the substitution costsare linear, andwij 5 wik 1 wkj for all sizesk wherei , k , j,the optimal policy can be shown to be such that if sizeisubstitutes for sizej then sizei also substitutes for all interme-diate sizeskwherei , k, j (Pentico [9]). Such policy is calleda segmented stocking and substitution policy.

It is important to note that in a multidimensional problemi , j does not necessarily imply that sizei can substitute forthe demand of sizej . For instance, in Figure 1(b), neithersize 1 nor size 2 can substitute for size 3. Since the substi-tution relation is only a partial order, optimal policies in themultidimensional case are rarely segmented. We, however,can state a general result about an optimal policy.

Proposition 1. If the stocking and substitution cost func-tions have a fixed and a linear variable cost component,then there is always an optimal solution in which each sizewill be either stocked or supplied entirely from the stock ofa single larger size.

Fig. 1. Graph representation of Example 1.

MULTIDIMENSIONAL ASSORTMENT PROBLEM 241

Proof. Consider the graph representation of the assort-ment problem where the flow along an arc from sources tosize i represents the amount to be stocked of sizei and theflow along an arc from sizei to sizej represents the amountof sizei to be used to satisfy the demand of sizej . Supposethat nodes denotes a supply node with¥i di and the nodes1 throughn denote the demand nodes withdi, the demandfor sizei . This representation is equalivalent to a minimumconcave cost flow problem, whose optimal solution isknown to consist of flows with only one flow into each node(Zangwill [16]). ■

The above proposition implies that at the optimum stock-ing and substitution policy the set of all sizes can be clus-tered into subsets with only the largest size in each subsetbeing stocked to supply its own demand and the totaldemand of all other sizes in the subset.

Let f 5 ( f1, . . . , fn), v 5 (v1, . . . , vn), and d5 (d1, . . . , dn) be the vectors of fixed stocking costs,variable stocking costs, and demands, respectively.

Example 2. Consider a two-dimensional assortment prob-lem with sizes {(8, 8), (7, 7), (5, 5), (4, 4), (2, 2)}, wheref5 (10, 15, 23, 4, 40),v 5 (7, 10, 5, 8, 6), andd 5 (13,19, 4, 9, 5). The values of the pairs (gij , wij ) are indicatedon the associated arcs in Figure 2.

The optimal policy calls for stocking sizes 1, 2, and 4 toentirely supply their own demands. Size 3 is entirely substi-tuted for by size 1. Size 5 is entirely substituted for by size 2.Thus, the sizes are clustered into subsets {1, 3}, {2, 5}, and{4}. The largest size in each subset is stocked to supply all thedemands in that subset. Note that size 1 substitutes for size 3but not for size 2. Also, size 2 substitutes for size 5 but not forsize 4. Hence, the optimal policy is not segmented. In practice,it is likely that if size i can substitute for sizej and sizej cansubstitute for sizek, then gik $ gij and wik $ wij for allintermediate sizesj wherei , j , k, that is, the substitutioncosts are usually nondecreasing in the difference in size. How-ever, one can show that even in such cases the optimal policymight not be segmented.

3. UNCAPACITATED FACILITY LOCATIONMODEL FOR THE ASSORTMENT PROBLEM

The uncapacitated facility location(UFL) problem can bedefined as follows: Given a set of potential sitesI5 {1, . . . , n}, a set of clientsJ 5 {1, . . . , m}, andrelevant cost data, the aim is to find a minimum-cost planfor the number of facilities to open, their locations, and anallocation of each client to an open facility. In general, thereare fixed costs for opening the facilities and production andtransportation costs for distributing the commodities be-tween facilities and clients.

Let xi be a binary variable indicating whether facilityi isopened or not andyij be a binary variable indicatingwhether facilityi satisfies the demand of clientj . A costcij

is incurred to satisfy the demand of clientj from a facilityi , and the cost of opening a facility at sitei is fi. The 0–1integer programming model is

Constraint (1) states that each client should be served byone of the facilities. Constraint (2) states that if no facilityis located at sitei then none of the clients’ demands can beserved from that site, that is, each client can be served onlyby an open facility.

In this formulation, theyij ’s will assume integer values atoptimum even though they were allowed to be continuous,because all clients will be assigned to a single open (unca-pacitated) facility. Recall that by Proposition 1, in the op-timal solution to the assortment problem, each size is eitherwholly stocked or wholly supplied by the stock of one largerstocked size. This single-assignment property thus plays animportant role in using the UFL as a viable model for theassortment problem. The UFL problem itself isNP-hard;however, it has some features which make it a relativelyeasyNP-hard problem (Cornuejols et al. [3]).

Let us redefine the parameters of the UFL formulation ofthe assortment problem withn possible sizes as follows:The setI 5 {1, . . . , n} now represents the set of potentialsizes to be stocked. CallI the set offacility-sizes.The setJ5 {1, . . . , m} now represents the demanded sizes whosedemands need to be satisfied by the facilities. Thus,m 5 uJu5 uI u 5 n. Call J the set ofclient-sizes. fi is the fixed costof stocking facility-sizei . The costcij of satisfying all thedemand of client-sizej from the stock of facility-sizei isdefined as

Fig. 2. Graph representation of Example 2.

242 TRIPATHY, SURAL, AND GERCHAK

cij 5 Hdjvi if i 5 j ,gij 1 dj~vi 1 wij! if i Þ j and i , j ,` if i Þ j and i . j .

Note thati 5 j implies that sizej itself is stocked to supplyits entire own demand, whilei Þ j implies that the entiredemand of sizej is supplied by a different sizei , which ispossible only if sizei is larger than sizej .

Thus, the problem is to select a subset of facility-sizes tostock and to assign each client-size to exactly one stockedfacility-size such that the total cost is minimized. With theredefinition of parameters, the 0–1 integer program of theUFL problem adequately represents the discrete multidi-mensional deterministic assortment problem.

Substantial literature is available on the UFL problem. Adual-descent procedure has been used for solving UFLproblems, although it may perform poorly on certain hardinstances. It is used in a branch-and-bound algorithm de-veloped by Erlenkotter [5], which is one of the most effi-cient programs available for solving UFL problems. Theprocedure attempts to find a good solution to a certaincondensed form of the dual of the linear programmingrelaxation of model IP and then uses the complementaryslackness conditions to find a candidate integer solution tothe problem. The dual solution provides a bound on theoptimal solution to the primal problem. Below, we formal-ize this procedure for solving multidimensional assortmentproblems.

Suppose that the minimization problem [IP] is convertedto a mathematically equivalent maximization problem:

maximizeOi[I

Oj[J

pijyij 2 Oi[I

fixi,

subject to (1)–(4), wherepij 5 2cij . Consider an assort-ment problem withn possible sizes and letP be ann 3 nprofit-matrix whose theij th element contains the profitpij .Additionally, definepj

k as thekth lowest profit in thej thcolumn of matrixP. Let U 5 { u1, . . . , un}, whereuj is adual variable corresponding to client-sizej . Now, let I (U)be the set of facility-sizes eligible to be stocked for a givenset U of feasible dual variables andK(U) be the set offacility-sizes in I (U) that are ultimately stocked.imax

j de-notes the facility-size in setI (U) which can serve thedemand of client-sizej with the maximum profit.

There are three modules in the dual-descent heuristic forthe assortment problem. Module 1 uses the dual-descentprocedure. Starting with an initial set of dual variables, itattempts to improve the solution by maintaining dual feasi-bility. We use a simple rule for improvement: reduceuj tothe lowestpij which does not violate feasibility; otherwise,uj is blocked and we consideruj11. Module 2 uses this setof dual variables to find a set of facility-sizes eligible to bestocked by considering complementarity conditions. Thelast module improves upon the output of module 2 (ifneeded) by deleting some of the eligible facility-sizes andfinds the set of facility-sizes that are actually stocked and,for each stocked facility-size, the client-sizes whose de-mands it supplies. We thus find a minimal subsetK(U) ofI (U), where most profitable facility-sizes are kept for eachclient-size. Let (a)1 5 max{a, 0}.

procedure find U (module 1):input: Poutput: U such that if anyuj is decreased thenU becomesinfeasible.beginU :5 { p1

n, p2n, . . . , pn

n} (initialize)for all j [ J do

for l :5 1 to n 2 1 dowhile (¥ j[J ( pij 2 uj)

1 2 fi # 0 for all i [ I )do uj :5 pj

n2l

endwhileendfor

endforend.

procedure find I (U) (module 2):input: Uoutput: I (U)beginI (U) :5 f (initialize)

for all i [ I doif (¥ j[J ( pij 2 uj)

1 2 fi 5 0)then I (U) :5 I (U) ø { i }

endforend.

procedure find K(U) (module 3):input: I (U), imax

j (@j [ J)

TABLE I. Computational results for Set I

ProblemNo. Potential

Sizes (n)Average Ratio of Cost

to Lower BoundMinimum-Cost

RatioMaximum-Cost

Ratio

1 50 1.0035 1.0023 1.0043

2 100 1.0038 1.0016 1.0059

3 150 1.0030 1.0019 1.0049

4 200 1.0037 1.0028 1.0045

MULTIDIMENSIONAL ASSORTMENT PROBLEM 243

output: K(U)beginK(U) :5 f (initialize)

for all j [ J dofor all i [ I (U) do

if (i 5 imaxj ) then K(U) :5 K(U) ø { i }

endforendfor

end.

A program incorporating all modules was implementedin the C programming language on a PC. The output con-sists of (1) a good stocking and substitution policy, (2) itstotal cost, and (3) a lower bound on the optimal solution.

4. COMPUTATIONAL RESULTS

In this section, the performance of the dual-descent heuristicfor the assortment problem is reported for both randomlygenerated problems and the real application data. We com-pare the solution found by the heuristic with the lowerbound on the optimal solution. Pentico [10] outlined thecomputational results of his heuristics for the discrete two-dimensional assortment problem. However, the problemsets on which he tested the heuristics have two limitations inthe context of our application: (1) The fixed cost of stockingis the same for every size, and (2) the substitution costfunction is linear. The problems that we used have unequalstocking and substitution costs, all having different fixedand variable components. We have three sets of problems intotal. The last set contains problems with real data obtainedfrom the company.

Set I: We chosen 5 50, 100, 150, and 200 andsolved50 test problems for each. All three dimensions of each sizeare assumed to be equal and consecutive integers that startwith 1. Thus, for example, the sizes in a problem withkpotential sizes are{(1, 1, 1), (2, 2, 2), . . . , (k, k, k)}.The demanddi, the fixed stocking costfi, and the unitstocking costvi are randomly distributed on U[20,40],U[50,60], and U[10,30], respectively. The unit cost of sub-stitutionwij is taken as the sum of the difference in volumeand the difference in surface area. For each pair of sizes, thefixed cost of substitutiongij is randomly distributed onU[10,20]. Table I shows the average cost ratios of the

dual-descent heuristic to the lower bound. The minimum-and the maximum-cost ratios are also reported.

Set II: All data are generated in the same way as for SetI, except for the dimensions of sizes. Each dimension of asize is assumed to be a number from 1 ton with equalprobability, and the dimensions of a size are not necessarilyarranged in a nonincreasing order. The results of 50 testproblems for eachn are given in Table II.

Set III: This contains problems with actual data aboutthree-dimensional crates obtained from the company. Thereare five different instances with 200 sizes each and one with320 sizes. Table III summarizes the results.

All problems in Sets I and II were solved to within 0.7%of the lower bound, which constitutes a remarkable accu-racy. However, the dual-descent procedure did not do sowell for the real-life problems. The results are betweenwithin 7% and within 31% of the lower bound. The reasoncould be that (1) in random problems the dimensions havethe same distribution while the same is not true of real data,and (2) the total variable cost (demand times unit variablecost) of real data is comparable in magnitude to the fixedcost while it is much larger than the fixed cost in the case ofrandomly generated data.

The second observation led us to rerun the program onSet I and Set II problems wherefi is randomly distributed onU[430,630], which better corresponds to Set III’s cost struc-ture. The rest of the data has not been changed. The resultsare shown in Table IV.

From the above results, it is clear that the performance ofthe procedure declined significantly with increase in fixedcosts. On average, the solution is within 13–22% of thelower bound.

TABLE II. Computational results for Set II

ProblemNo. Potential

Sizes (n)Average Ratio of Cost

to Lower BoundMinimum-Cost

RatioMaximum-Cost

Ratio

1 50 1.0007 1.0003 1.0015

2 100 1.0035 1.0022 1.0070

3 150 1.0031 1.0015 1.0059

4 200 1.0039 1.0025 1.0056

TABLE III. Computational results for Set III

InstanceNo. Potential

SizesRatio of Cost toLower Bound

1 200 1.1183

2 200 1.1218

3 200 1.0769

4 200 1.0641

5 200 1.2196

6 320 1.3063

244 TRIPATHY, SURAL, AND GERCHAK

5. CONCLUDING REMARKS

Our research constitutes a first attempt at an investigation ofthe multidimensional assortment problem. The motivationbehind it is an actual three-dimensional assortment problemwhich arose at a distribution center of a telecommunicationsequipment manufacturing company. The discrete multidi-mensional assortment problem is modeled as a facility lo-cation problem where it allowed us use the well-knowndual-descent procedure that not only finds a good feasiblesolution to the problem but also computes a bound on theoptimal solution. This heuristic approximately solves thedual of the 0–1 integer programming formulation of theuncapacitated facility location model of the assortmentproblem.

It may be possible to improve the dual solution further bymodifying the procedure of findingU. As an anonymousreferee proposed, the way in which theuj are updated can bechanged so that in every step eachuj is decreased, ifpossible, only to the next lowerpij , rather than to the lowestpij . If we insist on obtaining a better bound, then thepossibility of further decreases to the minimum value (thatfeasibility constraints allow) can be checked before theprocedure terminates. For further discussion of this subject,the reader is referred to Cornuejols et al. [3].

In addition to addressing a real-life application of stan-dardizing numerous crate sizes, this paper provides the firstpractical approach for solving multidimensional problemsand allows for fixed, as well as linear, substitution costs andnonequal stocking costs for different sizes. We also believethat our approach of treating the problem as an uncapacitedfacility location problem could be applied efficiently even toproblems of less general cost structure or dimensionality.Modeling and solving a multidimensional assortment prob-lem with probabilistic demands and relaxing the assumptionthat every size can be stocked in unlimited quantities war-rant future research. We would like to note that when thereis a limitation on the amount of a particular size that may bestocked a more difficult plant location problem arises,which so far has received little attention in the literature(Revelle and Laporte [11]).

The authors are grateful to David W. Pentico for his useful

comments on an earlier draft. We also thank anonymous refereesfor their helpful comments.

REFERENCES

[1] R.K. Ahuja, T.L. Magnanti, and J.B. Orlin, Network flows:Theory, algorithms and applications, Prentice-Hall, Engle-wood Cliffs, NJ, 1993.

[2] M.L. Chambers and R.G. Dyson, The cutting stock problemin the flat glass industry: Selection of stock sizes, Oper ResQ 27 (1976), 949–957.

[3] G. Cornuejols, G.L. Nemhauser, and L.A. Wolsey, “Theuncapacitated facility location problem,” Discrete locationtheory, R.L. Francis and P. Mirchandani (Editors) Wiley,Chichester, 1988, pp. 119–171.

[4] A. Diegel and H.J. Bocker, Optimal dimensions of virginstock in cutting glass to order, Decision Sci 15 (1984),260–274.

[5] D. Erlenkotter, A dual-based procedure for uncapacitatedfacility location, Oper Res 26 (1978), 992–1009.

[6] P.C. Jones, T.J. Lowe, G. Muller, N. Xu, Y. Ye, and J.L.Zydiak, Specially structured uncapacitated facility locationproblems, Oper Res 43 (1995), 661–669.

[7] E. Page, A note on a two-dimensional dynamic program-ming problem, Oper Res Q 26 (1975), 321–324.

[8] D.W. Pentico, The assortment problem with probabilisticdemands, Mgmt Sci 21 (1974), 286–290.

[9] D.W. Pentico, The assortment problem with nonlinear costfunctions, Oper Res 24 (1976), 1129–1142.

[10] D.W. Pentico, The discrete two-dimensional assortmentproblem, Oper Res 36 (1988), 324–332.

[11] C.S. Revelle and G. Laporte, The plant location problem:New models and research prospects, Oper Res 44 (1996),864–874.

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TABLE IV. Results for problems with higher fixed costs

ProblemNo. of Potential

Sizes (n)Average Ratio of

Cost to Lower Bound Minimum-Cost Ratio Maximum-Cost Ratio

Set I Set II Set I Set II Set I Set II

1 50 1.13 1.15 1.12 1.13 1.15 1.16

2 100 1.16 1.16 1.15 1.17 1.17 1.19

3 150 1.19 1.21 1.18 1.18 1.20 1.22

4 200 1.21 1.22 1.18 1.21 1.23 1.25

MULTIDIMENSIONAL ASSORTMENT PROBLEM 245