Competitive facility location model with concave demand

European Journal of Operational Research 181 (2007) 598–619

www.elsevier.com/locate/ejor

Discrete Optimization

Competitive facility location model with concave demand

Robert Aboolian a, Oded Berman b, Dmitry Krass b,*

a College of Business Administration, California State University San Marcos, San Marcos, CA 92096, USAb Rotman School of Management, University of Toronto, 105 St. George Street, Toronto, Ont., Canada M5S 3E6

Received 1 March 2004; accepted 6 October 2005Available online 18 September 2006

Abstract

We consider a spatial interaction model for locating a set of new facilities that compete for customer demand with eachother, as well as with some pre-existing facilities to capture the ‘‘market expansion’’ and the ‘‘market cannibalization’’effects. Customer demand is assumed to be a concave non-decreasing function of the total utility derived by each customerfrom the service offered by the facilities. The problem is formulated as a non-linear Knapsack problem, for which wedevelop a novel solution approach based on constructing an efficient piecewise linear approximation scheme for the objec-tive function. This allows us to develop exact and a-optimal solution approaches capable of dealing with relatively large-scale instances of the model. We also develop a fast Heuristic Algorithm for which a tight worst-case error bound isestablished.� 2006 Elsevier B.V. All rights reserved.

Keywords: Location; Integer programming; Competitive facility location models; Non-linear Knapsack problem; Alpha-optimalsolutions; Greedy heuristics; Worst-case bounds

1. Introduction

The current paper has a dual objective: to introduce an interesting and important competitive locationmodel, and to develop an efficient solution technique, which is applicable to a class of non-linear non-separa-ble Knapsack models—which includes our competitive location model.

Consider the problem of locating m new retail or service facilities controlled by the same entity (‘‘com-pany’’) in an area where some other facilities providing comparable product or service may already exist.The new facilities, along with the pre-existing ones (some of which may be owned by the competitors, whileothers may belong to the company) will compete for customer demand with each other. We would like to findan optimal set of locations for the new facilities so that the total customers demand seen at all of the facilitiescontrolled by the company is maximized. We assume that customer demand is concentrated at a set of n dis-crete points and there is a finite set of potential locations for the new facilities.

0377-2217/$ - see front matter � 2006 Elsevier B.V. All rights reserved.

doi:10.1016/j.ejor.2005.10.075

* Corresponding author. Fax: +1 416 978 5433.E-mail address: [email protected] (D. Krass).

mailto:[email protected]

R. Aboolian et al. / European Journal of Operational Research 181 (2007) 598–619 599

In order to be able to determine the total demand seen at a particular facility, it is first necessary to makecertain modeling assumptions with respect to customer behavior, namely how do customers decide which ofthe available facilities to patronize and how much can they be expected to spend there. We adopt the generalframework of ‘‘spatial interaction models’’, which originated with the work of Huff [18] (sometimes referredto as the ‘‘gravity’’ or ‘‘Huff-type’’ models in the location literature, and ‘‘brand choice’’ or ‘‘market shareattraction’’ models in the marketing literature). Under this framework it is assumed that customer choice rulesare probabilistic, i.e., that the customers split their demand between several (possibly all) of the available facil-ities, with the frequency (probability) of a visit to a particular facility decreasing with the distance from thefacility and increasing with the ‘‘attractiveness’’ of the facility. We refer the reader to Berman and Krass[6,5], and Ghosh, McLafferty and Craig [15] for a detailed discussion of this class of competitive locationmodels.

The ‘‘standard’’ spatial interaction model assumes that customers’ expenditures are constant and known inadvance—and are thus independent of the number and location of service facilities. This implies that in thepresence of some existing facilities, the only effect of locating new facilities is the ‘‘cannibalization’’ of someof the market share by the new facilities from the existing ones. However, this assumption ignores the fact thatin many locational decisions one of the key considerations is ‘‘market expansion’’—i.e., attracting new demandfrom customers that were ‘‘underserviced’’ by the existing facilities. This effect may cause the total customerdemand in a given area to increase when new facilities are located there.

In order to capture both the ‘‘cannibalization’’ and the ‘‘market expansion’’ effects, we follow the approachof Berman and Krass [5], who introduced the concept of ‘‘spatial interaction model with variable expendi-tures’’. Under this framework, the total expenditure (demand) of each customer is a function of the total util-ity derived by the customer from all of the available facilities (both new and pre-existing). While this approachdoes capture both the ‘‘cannibalization’’ and ‘‘market expansion’’ effects, it runs into severe technical difficul-ties, due to the complexity of the resulting optimization models when variable expenditures are introduced (infact, even the constant expenditure case considered in [6] is computationally challenging). For the ‘‘general’’form of the expenditure function considered in [5] (which assumed a ‘‘step function’’ form that can be used toapproximate an arbitrary function), the only case that could be analyzed to any degree was that of the‘‘monopolistic competition’’ (where the company owns all of the available facilities); even in that case theresults were quite limited.

In this paper we assume that the total demand functions (i.e., demand seen at each facility) are concave andnon-decreasing functions of the total utility derived by each customer from the service offered by the facilities.We note that this appears to be a stronger assumption than simply assuming that the customer expenditurefunctions are concave and non-decreasing—since the demand observed at a particular facility also dependson the market share captured by this facility, which is itself a non-linear function of the utility derived by eachcustomer. However, we show that it is, in fact, sufficient to assume that the customer expenditure functions areconcave and non-decreasing functions of the utility derived by each customer. Concave non-decreasingdemand assumption appears to be quite mild—it is a standard assumption in much of the marketing and eco-nomics literature—see, for example, Buzzell [7], Lodish, Montgomery and Webster [21], Little and Lodish [20].However, we show that this assumption does come at a price—in some sense, it implies that the market expan-sion effect can never overtake the cannibalization effect.

The concavity of the demand function allows us to formulate our model as a non-linear integer Knapsackproblem with a particular form of the objective function. Non-linear Knapsack models are known to be dif-ficult even with a separable objective function—see Hochbaum [16]; Bretthauer and Shetty [8] provide a goodrecent review.

Unfortunately the Competitive Facility Location Model with concave demand falls within the sub-categoryof non-separable convex Knapsack problems, for which results are much more limited—see Pardalos et al.[27], Caprara et al. [9], and Dussault et al. [14]. An interesting related work was done by Klastorin [19]who considered a non-separable Knapsack model in the context of the media planning; in fact his problemis a special case of the class of problems for which we develop a general solution approach described below.The solution methodology developed in [19] is heavily dependent on the special functional form of his objec-tive function and is not directly extendable to the significantly more general form considered in the currentpaper.

600 R. Aboolian et al. / European Journal of Operational Research 181 (2007) 598–619

Due to complexity of the objective function, we are not aware of any existing methods capable of solvingmoderate or large instances of our problem. This motivated us to develop a new approach that can be appli-cable to other problems as well. Our approach—which we call the Tangent Line Approximation (TLA)—isbased on over-approximating the non-linear objective function with a piecewise linear function and is appli-cable to problems with any number of constraints (i.e., it contains Knapsack problems as a special case).

While piecewise linear approximations are well-studied in the context of separable Knapsack problems, wecould find no evidence of previous application of this approach for non-separable Knapsack problems. Thereare important differences between the TLA procedure and the standard linearization techniques in the opti-mization and numerical analysis literature. The standard approach, outlined in Bazaraa et al. [3] is to use acoarse grid initially and then to use a finer grid around the best solution obtained with the coarse grid—thisprocedure continues until the desired error has been approached (a variation of this approach is used in Meyer[22,23] where new grid points are generated only when necessary). This approach can lead to a large number ofgrid points—which increases the complexity of the approximating problem; moreover several approximationmodels may be required. In contrast, the TLA procedure employs an ‘‘adaptive grid’’ which is optimized tohave the minimum number of grid points required to achieve a certain pre-specified level of the relative error;only one approximating model has to be solved.

The use of the relative error is another distinguishing characteristic of the TLA—the standard numericalanalysis approaches (e.g., see Cheney [10] or Nurenberger [26]) focus on defining an approximation schemethat achieves a certain level of the absolute error—which is of limited usefulness in our context (and, in fact,in most other optimization contexts, as absolute tolerance levels are hard to set a priori). The TLA procedureallows us to provide exact and a-optimal solutions to realistic-size instances of our competitive location model(by ‘‘a-optimal’’ we mean that the maximum allowable deviation from optimality a may be specified inadvance by the user). We believe that the TLA procedure may have applications far beyond the specific loca-tion problem considered in the current paper.

The plan for the remainder of the paper is as follows: we start by formally introducing the spatial interac-tion models with concave demand function in Section 2. Some structural results relating to model behavior,and an illustrative example of the model are presented there. The model is formulated as a non-linear IntegerProgramming model with concave objective function. In Section 3 we outline an efficient algorithmic approach(the ‘‘Tangent Line Approximation’’) for a general class of concave integer programming problems, whichincludes our problem as a special case (many technical details of the algorithm are relegated to the Appendix).In Section 4, we apply the TLA procedure to develop efficient exact and a-optimal solution approaches for ourcompetitive location model. An alternative heuristic procedure is suggested in Section 5. While the maximumerror for the heuristic cannot be chosen in advance, we do derive a tight worst case error bound. The results ofa series of computational experiments aimed at testing the performance of the exact, a-optimal and the heu-ristics procedures are presented in Section 6. Some concluding remarks are presented in Section 7.

To close the current section, we mention briefly some of the related work in competitive facility location.The problem of finding an optimal location for a single new facility in continuous space in the framework of(constant-expenditure) spatial interaction models is discussed in Drezner [13] and Plastria [28]. A correspond-ing multi-facility location model is investigated in continuous space using spatial interaction models in Drez-ner [12]. Following up on the initial work by Huff [18], discrete multi-facility location models of this kind havebeen at the heart of research in the retail sector (see e.g., [2,15]). In these papers the problem is modeled as anon-linear integer-programming problem, for which explicit enumeration schemes are proposed—these, ofcourse, are applicable only to small-scale problems. Berman and Krass [6] developed a Branch and Boundscheme with a tight upper bound to optimally solve the problem efficiently for small size problems; they alsoprovide an efficient heuristic approach for the constant-expenditure multi-facility case. We note that the TLAprocedure can be applied to the constant-expenditure case as well, providing a significant improvement on theresults in [6]—these will be reported in a separate paper.

2. Spatial interaction models with concave demand functions

We assume that customer demand is concentrated at discrete points N = {1,2, . . . ,n} with point i contain-ing wi customers. All customers residing at point i are assumed to be ‘‘identical’’ with respect to their facility


choice and expenditure decisions—thus, each point can be thought of as containing a separate customer ‘‘seg-ment’’ (type).

Let P � N be the set of potential facility locations—some of which are already occupied by the existingfacilities. We assume that each location is capable of supporting at most one facility. This assumption is madewithout loss of generality, since a ‘‘replicate’’ of a location in P can always be added (e.g., if the actual physicallocation is capable of supporting two facilities, we can simply add a second copy of this location to P).

We assume N to be a metric space, equipped with distance dij, i, j 2 N (e.g., N might consist of the nodes of anetwork, with d being the shortest path distance, or it may be embedded in a plane, with d being the l2 norm, etc.).

Let E � P be the set of existing (competitive) facilities, and E ¼ P � E be the set of locations available fornew facilities. To avoid indefinite forms later, we will assume that there is always some minimal travel distance� > 0 between any customer location and any facility, in other words wj = 0 for j 2 P. We will use S � E,jSj = m to denote the set of locations for m new facilities—the objective of our problem is to find S that opti-mizes the objective function introduced below.

As mentioned in the introduction, we will use the general framework of gravity-type spatial interactionmodels, which were first introduced by Huff [18] to represent customer choice decisions. These models belongto the general class of stochastic utility models—also known as the ‘‘brand share models’’ in the marketingliterature (see Naret and Weverbergh [25], and Nakanishi and Cooper [24]).

A basic assumption in these models is that for a customer at point i 2 N, the utility of a facility at locationj 2 S can be written as a ratio of a non-decreasing function of the ‘‘attractiveness’’ Aj of the facility, and a non-decreasing function of the travel distance dij,

uij ¼F ðAjÞHðdijÞ

: ð1Þ

A typical form of uij is [18]:

uij ¼Aa

j

dbij

a; b > 0: ð2Þ

Another popular functional form for uij involves the use of exponential functions instead of power functions in(2) (see Beaumont [4]). In some applications it might be useful to define utilities so that uij = 0 unless j is withincertain maximum distance from i, since a very attractive facility far away is unlikely to affect consumer’s behav-ior (in marketing terms, this maximum distance represents the radius of the ‘‘consideration set’’ for point i).

Note that the utilities uij are independent of the locations of facilities other than j, and thus can be estimated apriori (i.e., before locational decisions are made)—data collection and estimation techniques are fairly well-developed for this class of models (see Nakanishi and Cooper [24]). For our purposes, the exact functional formof uij is not important—we simply treat these quantities as input parameters to our models. However, in thecomputational experiments reported in Section 5 below, we use the form (2) for generating values for uij.

Let MSij be the frequency (probability) of customer at i using facility at j 2 S, where we assume, for themoment, that the location set S of new facilities is fixed. The quantity MSij can be interpreted as the shareof the market available at demand point i that is captured by the facility at j. A fundamental assumptionof spatial interaction models is that this frequency equals to the relative utility of facility at j compared toother facilities available on the network:

MSij ¼uijP

k2E[Suik: ð3Þ

‘‘Standard’’ spatial interaction models assume that the demand (i.e., total expenditure) of each customer forthe service provided by the facilities is constant and known in advance. Let gi represent the demand (expen-ditures) of a customer at location i 2 N. Then, the total market size available at i is given by wigi. The problemof locating m new facilities can now be written as:

maxS�EjSj¼m

Xi2N

Xj2S

wigiMSij: ð4Þ


This problem was analyzed in detail in Berman and Krass [6]. As mentioned in the introduction, a keyweakness of the model described above is the assumption that expenditures gi are constant and independentof the number and locations of the facilities—thus, this model captures the ‘‘cannibalization’’, but not the‘‘market expansion’’ effect.

To overcome this limitation, Berman and Krass [5] generalized the model above by allowing the demand ofcustomers at location i 2 N to be non-constant; it is assumed that this demand is an increasing function of thetotal utility obtained by a customer at i 2 N from all of the available facilities. Specifically, for i 2 N we definethe demand (expenditure) function giðU 0iðSÞÞ to be a non-negative function of U 0iðSÞ, where

U 0iðSÞ ¼X

j2E[S

uij

is the total utility derived by customers at i from all of the facilities in E [ S. Without loss of generality, we canassume that 0 6 giðU 0iðSÞÞ 6 1, and thus the total expenditure of customers at i 2 N for a particular locationset S is given by wigiðU 0iðSÞÞ.

The problem of locating m new facilities with variable demand function can now be re-written as:

maxS�EjSj¼m

ZðSÞ ¼Xi2N

Xj2S

wigiðU 0iðSÞÞMSij: ð5Þ

Berman and Krass [5] discussed the linear, bounded linear and step-wise forms of the expenditure function.In general, the resulting optimization models are very difficult, due to inherent non-linearities of the objectivefunction. For the general form of the model, only small instances can be solved to optimality.

In this paper we will focus on the special case of model (5), where the demand function satisfies certain con-cavity assumptions. Specifically, we make the following definitions. For i 2 N, let:

H i ¼Xj2E

uij; and UiðSÞ ¼Xj2S

uij ¼ U 0iðSÞ � H i: ð6Þ

Note that these expressions represent the total utility which customers at i derive from all pre-existing facil-ities, and the total utility which customers at i derive from all new facilities, respectively. Note also that Hi areconstant in our model for i 2 N. We also define

MSiðSÞ ¼Xj2S

MSij ¼ 1� Hi

Hi þ U iðSÞ; ð7Þ

which represents the share of market i captured by all facilities in S (the new facilities). For U > 0 andeU ¼ U þ H i, let

V ið eU Þ ¼ gið eU Þ 1� H i

U þ H i

� �: ð8Þ

For eU ¼ U iðSÞ þ Hi, the quantity V ið eU Þ represents the total demand of a customer in market i 2 N capturedby the facilities in S. Note that the expression for the objective function can now be rewritten as follows:

ZðSÞ ¼Xi2N

wiV ið eU iðSÞÞ: ð9Þ

We make the following assumption which will remain in force for the remainder of the paper:

Assumption 1. The total demand function V ið eU Þ is a concave, non-decreasing function of eU 2 ½Hi;1� for alli 2 N.

The following result shows that this assumption is automatically satisfied when the expenditure functiongið eU Þ is concave, non-decreasing and vanishes at 0.

Theorem 1. Suppose that the expenditure function gið eU Þ is a twice differentiable, non-decreasing concave

function such that gi(0) = 0 for some i 2 N. Then, V ið eU Þ satisfies Assumption 1.


Proof. To simplify the notation in this proof we suppress the index i. By the definition of V(Æ) we have:

V 0ð eU Þ ¼ g0ð eU Þ 1� HeU� �

þ gð eU Þ H

ð eU Þ2 P 0;

where the inequality holds since both terms are non-negative. Also,

V 00ð eU Þ ¼ g00ð eU Þ 1� HeU� �

þ g0ð eU Þ 2H

ð eU Þ2 � gð eU Þ 2H

ð eU Þ3 :
The first term is clearly non-positive since g00( ) 6 0. Taking out the positive factor 2H=ð eU Þ3 from the last twoterms, we obtain g0ð eU Þ eU � gð eU Þ. However, this last quantity is clearly non-positive—this can be verified byobserving that its derivative is g00ð eU Þ eU 6 0 and it vanishes at 0. This implies that V00( ) 6 0, which completesthe proof. h
A convenient functional form for the expenditure function that satisfies the conditions of the previous resultis

giðU 0iðSÞÞ ¼ 1� expð�kiU 0iðSÞÞ; ð10Þ

where ki > 0 is the exponential decay parameter (this parameter can be interpreted as the rate at which thetotal demand of a customer in market i reaches its saturation point wi). We will use this functional form inthe computational experiments reported later in the paper.

The assumption that demand is concave (equivalent to our Assumption 1) is quite standard in much of theeconomic literature. It will be seen below that Assumption 1 allows us to develop efficient exact and algorith-mic approaches to our problem. However we stress that this assumption does come at a price. In particular,the following result, which follows directly from Theorem 1 in [5] (which is based on a result from [1]), showsthat the ‘‘market expansion’’ effect is limited when Assumption 1 holds.

Theorem 2. Let S 2 P be the set of company’s facilities that have already been located. For a facility j 2 S let

DjðSÞ ¼P

i2N wigið eU iðSÞÞMSij be the demand currently seen by facility j. Suppose Assumption 1 holds. Then for

any k 2 P � S � E,

DjðSÞP DjðS [ fkgÞ;
i.e., if a new facility is located at k, the demand seen by all existing facilities cannot increase.
This results indicates that, from the point of view of individual facilities, the market expansion resultingfrom location of any new facilities will never overcome the cannibalization of existing demand by these facil-ities. We stress that this does not mean that the total demand seen by all of the facilities owned by the com-pany (both new and already located) will not increase—indeed it can, and from the ‘‘company-wide’’ point ofview the location of additional facilities may well be warranted.

In spite of the previous result, it should be recognized that the location model described above, even whenAssumption 1 holds, does provide a lot of flexibility to the decision-maker and can be applied to many real-life situations. For example, when the distance sensitivity parameter b in (2) is small, the facilities are likely tobe centrally located and in close proximity to each other (under the optimal solution)—particularly if thedemand elasticity is high. In practice, this may lead the decision maker to aggregate some of the facilities intoa smaller number of larger units—‘‘big box’’ retail stores provide a good example of this situation. On theother hand, if the distance sensitivity b is high, the optimal solution will tend to spread out facilities resultingin a ‘‘service network’’ type of solutions—seen in practice in the case of, for example, bank machines. Thefollowing example illustrates how the demand cannibalization and market expansion effects are captured withour model.

Example 1. Consider two customer demand nodes located at the endpoints of a unit line segment. We assumethat 50 units of demand are available at each node (i.e., w1 = w2 = 50). We will use the exponential form of theexpenditure function (10), set the distance sensitivity b = 2 and attractiveness Aj = 1 for all j in (2), andexamine model behavior under the different values of the demand elasticity parameter k (here, the higher thevalue of k, the less elastic the demand).


We will locate two facilities sequentially in this market: first pick the optimal location for facility 1, andthen find the optimal location for facility 2 assuming the location of facility 1 is already fixed. We are inter-ested in finding location patterns and measuring cannibalization and expansion effects. We will designate thelocation of facilities 1 and 2 by y1, y2—here yi represents the distance from the first demand point, and 1 � yi—from the second, for i = 1,2.

First note that if the expenditures are constant (or, equivalently, k is high), then we have a ‘‘classic’’ Hotell-ing problem (see [17]), with the optimal location for the first facility at the midpoint (y1 = 0.5), and one facilitycapturing 100% (i.e., all 100 units) of the available demand. When the second facility is brought in, its optimallocation is to co-locate with facility 1 (y2 = 0.5), it captures 50 units of demand and the total amount of cap-tured demand stays at 100 units. Thus there is a 50% cannibalization effect (representing the decrease in thedemand seen by facility 1 when the second facility is opened) and 0% market expansion effect (since the twofacilities together intercept the same amount of demand as the first facility by itself). Note that these resultswill hold for any distance sensitivity parameter b—since the expenditures do not react to increased distancesensitivity. It can be verified numerically that the situation described above holds for the parameter valuesdefined above when k > 1.

However, the results are quite different for low values of k (corresponding to elastic demand). For example,for k = 0.1, in the one—facility case the location y1 = 0.5 is no longer desirable since the relatively high dis-tance sensitivity coupled with large demand elasticity mean that the customers are spending just under 33% oftheir potential expenditures, with the total intercepted demand of about 33 units. We can increase this to 55units by moving the facility very close to either endpoint (y1 = 0.86 is one optimal location; y1 = 0.14 isanother)—this location receives 99.3% of the available potential demand from the ‘‘nearby’’ node, and only12.6% from the ‘‘distant’’ one, for a total intercepted demand of 56 units. Suppose y1 = 0.86 is selected.The optimal location for the second facility is y2 = 0.06—i.e., this facility is ‘‘specialized’’ to node 1. Thetwo facilities together intercept 99.8 units of demand—the market expansion is 78% in this case. However,the demand seen by facility 1 declines to 49.3 units—representing the cannibalization effect of 12%. Thus,in the case of elastic demand, the facilities are forced to focus on local markets—instead of a ‘‘shopping cen-tre’’ solution in the constant expenditure case (with both facilities located in close proximity to each other), thenew solution resembles a ‘‘corner store’’ pattern.

Table 1 illustrates these effects for a range of the values of k (columns 2 and 3 contain the facility locations,while the next two columns contain cannibalization and expansion effects). It can be seen that as k gets smaller(i.e., as the demand sensitivity increases), the market expansion effect grows, the cannibalization effectdecreases, and the distance between the two facilities grows. On the other and, for larger values of k weobserve the same location pattern as in the constant expenditure case.

We also note that an obvious extension of the model (5) is to make the constraint on the number of newfacilities to be located flexible (by changing the constraint to jSj 6 m) and changing the objective to the profitmaximization (rather than revenue maximization).

We now turn our attention to formulating the location model defined above as a mathematical pro-gram. Let xj be a binary variable that is equal to 1 if locating a facility at node j and 0 otherwise. With thisdecision variables, we can formulate the location problem defined above as the following optimizationproblem.

Table 1Facility locations, cannibalization and expansion effects in Example 1

k y1 y2 Cannibalization (%) Expansion (%)

0.05 0.92 0.06 6 890.1 0.86 0.09 12 780.15 0.82 0.12 18 680.2 0.78 0.15 24 570.25 0.73 0.17 30 470.3 0.68 0.21 36 380.4 0.5 0.5 40 20


maxXi2N

wigi

Xj2E

uij þXj2E

uijxj

0@ 1A 1�P

j2EuijPj2Euij þ

Pj2Euijxj

!ðPOÞ

s:t:Xj2E

xj ¼ m;

xj 2 f0; 1g; j 2 E:

We will refer to problem PO as the ‘‘original model’’. This model is a non-linear IP (where the non-linearitycomes from both, the expenditure function gi( ) and the ratio term in the objective) that contains jEj binaryvariables with one Knapsack-type constraint. By relaxing the integrality condition xj 2 {0,1} to xj 2 [0, 1] weobtain continuous relaxation P R

O.Let x̂ ¼ ðx1; x2; . . . ; xjEjÞ be a feasible solution of PO. Clearly, there is a one-to-one correspondence between

a feasible solution x̂ and a feasible location set S for the new facilities. Thus we will use the notations Ui(S),Z(S) and Uiðx̂Þ, Zðx̂Þ interchangeably. Note that

Uiðx̂Þ ¼Xj2E

uijxj; ð11Þ

is a linear function of the decision variables.From (9) and Assumption 1 it follows that each term in the objective function of PO is a composition of a

one-dimensional concave non-decreasing function and a linear function. In the following section we developan efficient a—optimal procedure for integer programs with this type of objective function.

3. TLA procedure for a class of concave integer programming models

Consider the following problem:

max xðxÞ ¼Xn

i¼1

xið/iðxÞÞ ðProblem DOÞ

s:t: AðxÞ 6 b

where x non-negative, integer-valued vector in Rm, where for i = 1, . . . ,n; xi(y) is a twice differentiable, non-decreasing, concave function of y 2 R+, with xi(0) = 0; /iðxÞ ¼ cT

i x; i ¼ 1; . . . ; n is a linear functional withnon-negative coefficient vector ci P 0, and A(x) 6 b is a system of linear constraints. Moreover, we assumethat 0 6 /iðxÞ 6 �/i, for each i, where �/i is a constant. Note that one possible sufficient condition under whichthis assumption holds is that the feasible region of DO is bounded.

We will denote the continuous relaxation DO by DRO. Observe that each term in the objective function x(x) is

a composition of a concave non-decreasing function with a linear functional. As noted at the end of Section 2above, the competitive facility location model PO defined in the previous section is a special case of DO.

We note that if there is only one linear constraint (as is the case in PO), DO is belongs to the class of non-linear integer Knapsack problem with non-separable objective function—which are notoriously difficult tosolve, as discussed in the introduction. While it may appear that the problem can be transformed to a sepa-rable non-linear Knapsack problem by a simple variable substitution (defining new decision variablesyi = /i(x)), this approach does not work since the argument of the objective function is no longer integerand it may be impossible to express the constraint in terms of the new variables, which means that the trans-formed model no longer has the Knapsack structure (see [8] for a nice discussion). The method we developbelow allows us to solve DO directly in an efficient manner and with arbitrary number of linear constraints.

In our opinion, the importance of the optimization models satisfying the same properties as DO extendsbeyond the competitive facility location model PO defined in the previous section. For example, the ‘‘classical’’spatial interactive model analyzed in [6] also belongs to this class. Applications outside the field of locationalanalysis may also exist.

Our ultimate goal in this section is to develop an approximate Linear Program DRa whose optimal solution

lies within a pre-specified error bound a from the solution of the relaxed non-linear problem DRO. We will then


show how this approximating LP can be used to find exact and a-optimal solutions to DO within a standardBranch-and-Bound scheme.

Suppose the objective function x(x) is replaced by a concave piecewise linear function xA(x), which boundsx(x) from above. More specifically, for i 2 N, let xA

i ð/Þ be a concave piecewise-linear function such that

xAi ð/ÞP xið/ÞP 0 for 0 6 / 6 �/i; ð12Þ

where xi(Æ) is the ith term in x. Note that the existence of xAi ð/Þ follows from the fact that xi(/) is bounded on

½0; �/i�. The following result now follows.

Lemma 1. For i = 1, . . . , n let

ai ¼ max06/i6

�/i

xAi ð/iÞ � xið/iÞ

xið/iÞ

� �ð13Þ

be the maximum relative error for approximating xi(/i) by xAi ð/iÞ. Let

a0 ¼ maxi2Nfaig: ð14Þ

Then, for any feasible solution x of DO (or DRO),

xðxÞ 6 xAðxÞ 6 ð1þ a0ÞxðxÞ ð15Þ

Proof. Follows directly from (12) and (14). h

Suppose we want to make sure that the maximum relative error from such an approximation does notexceed some pre-specified level a > 0. Then this requirement can be met by constructing concave piecewise lin-ear functions xa

i for i = 1, . . . ,n that satisfy (12) and for which ai < a holds in (13). The procedure for con-structing such a piecewise linear function will be developed in Section 3.1 below.

Suppose xai ð/iÞ with the property described above has been constructed for i = 1, . . . ,n, and /i 2 ½0; �/�. For

x 2 Rm. Let

xaðxÞ ¼Xn

i¼1

xai ð/iðxÞÞ;

and define mathematical programs DA and DRA that have xa as the objective function and the same constraint

sets as DO and DRO, respectively. Note that since xa is piecewise linear and concave by construction, DA (DR

A)can be re-formulated as a linear integer program (linear program) using standard formulation techniques forproblems with concave piecewise linear objectives. The following results follows immediately from the preced-ing lemma.

Lemma 2. For any a 2 [0,1], the optimal solution of DA is a-optimal for DO and the optimal solution of DRA is a-

optimal in DRO.

It follows that by applying the preceding result to our original problem PO, it will be possible to obtain a-optimal solutions by solving a linear integer problem instead of a non-linear one (provided the concave piece-wise linear over-approximating functions xa

i can be efficiently constructed, of course). We will come back tothis issue in Section 4.

We also note that in constructing our piecewise linear approximator, it is desirable to minimize the numberof linear segments (for a given accuracy a). This is because each linear segment requires a separate decisionvariable in the LP representation of a piecewise linear function. It will be shown that the TLA proceduredeveloped below is ‘‘optimal’’ in this regard—the piecewise linear approximator it constructs has the smallestpossible number of linear segments for a given relative error a.

3.1. Construction of piecewise linear over-approximator for a given relative error a

In this section we develop the fundamental results allowing us to construct the piecewise linear functionsxa

i ð/iÞ possessing the properties described above for a given a 2 (0, 1) for i = 1, . . . ,n. To simplify the notation


in this section, we will drop the subscript i. For easy reference, we re-state the basic problem addressed in thissection.

Piecewise linear over-approximator problem. Let x(/) be a concave, non-decreasing function for / 2 ½0; �/�.Assume that x(/) is twice-differentiable and that x(0) = 0. For a given a 2 (0,1) we would like to construct aconcave, non-decreasing, piecewise linear function xa(/) such that

xað/ÞP xð/Þ for / 2 ½0; �/�; ð16Þ

max/

xað/Þ � xð/Þxð/Þ j0 < / < �/

� �6 a; ð17Þ

and xa(0) = 0.The next result, stated in Theorem 3, shows that for any piecewise linear function that satisfies (16), in order

for the property (17) to be satisfied it is sufficient that it be satisfied at the ‘‘breakpoints’’. The Proof of The-orem 3 can be found in the Appendix.

Theorem 3. Let xA(/) be a non-decreasing concave piecewise linear concave function on ½0; �/� such that

xAð/ÞP xð/Þ for / 2 ½0; �/�.Let �(/) = [xA(/) � x(/)]/x(/) be the relative error (distance) between xA(Æ) and x(Æ). Then �(/) is

maximized at the beginning of one of the line segments of the approximating function xA(/).

Theorem 3 implies that if we ensure that �(/) 6 a holds at the starting points of all line segments definingour piecewise linear over-approximator, then the relative error will indeed be equal to a. In fact, it will beshown with that in order to minimize the number of line segments used it is advantageous to set �(/) = aat each starting point, except for the first one, where �(0) can be set to 0. This observation forms the basisof the TLA procedure.

In defining our piecewise linear function xa, the following notational conventions will be used: (1) we willindex each line segment by l, with l = 1, . . . ,L, (2) bl will be used to represent the slope of segment l, and (3) thestarting point of segment l will be represented by cl, with cLþ1 ¼ �/. Thus, segment l is given by [cl,cl+1] forl = 1, . . . ,L.

Intuitively, the TLA procedure operates as follows (the technical details of how different steps are per-formed, and the correctness of the procedure are provided in the Appendix):

Step A: We start by setting xa(0) = 0 and use the point 0 as the starting point of the first segment—i.e., setc1 = 0. Note that the relative error at 0 is set to 0 by convention (since x(0) = 0). We set the slope ofthe first segment equal to the first derivative of x(/) at 0, i.e., set b1 = x 0(0). Thus, the segment istangent to the graph of x(Æ) at 0.

Step B: We find the endpoint of the current segment by locating a point wa(c2) on the ray originating at 0 andwith slope b1 such that the relative error �(c2) = a. This provides the starting point for the nextsegment.

Step C: To find the slope of segment l for l = 2, . . . ,L, we find the slope of the ray originating at (cl,wa(cl))that is tangent to the graph of x(Æ) on ½cl; �/�.

Step D: To find the endpoint of segment l for l = 2, . . . ,L, we repeat step B.

The procedure continues until the xa(/) function has been defined for all points in ½0; �/�. The followingresult establishes the convergence of the TLA procedure and provides an upper bound on the number of seg-ments in the resulting piecewise linear approximator (which is the same as the number of major iterations ofthe TLA). The proof can be found in the Appendix.

Theorem 4

1. For any a > 0, the TLA procedures converges in finitely many steps to a piecewise linear non-decreasing con-

cave function xa(/) such that 1 6 xað/Þxð/Þ 6 1þ a for all / 2 ½0; �/�.

2. Let � > 0 be the length of the first linear segment in xa(/). Then the total number of segments (which is the

same as the number of major iterations in the TLA) is bounded above by 1þ d�/�e.


We note that the upper bound on the maximum number of ‘‘major iterations’’ in the TLA procedurederived in the preceding result is very rough. In our experience, the procedure tends to require only asmall number of iterations even for small values of a. This will be illustrated through a set of computa-tional experiments below. However, we first discuss an interesting ‘‘optimality’’ property of the TLA pro-cedure.

Recall that our intent is to use the concave piecewise linear over-approximator xa in place of x in the math-ematical programming formulations. Since each segment in xa will require a separate linearization (i.e., severalnew decision variables will have to be added to the formulation), it is desirable that xa have as few segments aspossible. The next result shows that the function constructed by the TLA procedure is, in fact, optimal in thisregard.

Theorem 5. For a given a > 0, the TLA procedure constructs a concave piecewise linear function, which

approximates x(/) from above within relative error a with the minimum number of line segments.

Proof. Let [cl,cl+1] for l = 1, . . . ,L be the lth segment in xa(/), with c1 = 0 and cLþ1 ¼ �/.Let xA(/) be another concave piecewise linear function such that

xð/Þ 6 xAð/Þ 6 ð1þ aÞxð/Þ for / 2 ½0; �/� ð18Þ
Assume xA(/) consists of LA segments, with segment l given by ½cA
l ; cAlþ1�. To prove the result we need to show

that L 6 LA.Consider l 6 L � 1 and let cT 2 [cl,cl+1] be the point where xa(/) is tangent to x(/), i.e., xa(cT) = x(cT) and

bl = x 0(cT), where bl is the slope of segment l in xa(/). Consider segment l in xA(/) with endpoints cAl ; c

Alþ1 and

slope bAl . First we establish the following preliminary result:

cAl 6 cl implies that cA

lþ1 6 clþ1:

Suppose, to the contrary, that cAl 6 cl but cA

lþ1 > clþ1. Then ½cl; clþ1� � ½cAl ; c

Alþ1�, and cT 2 ½cA

l ; cAlþ1Þ. By

assumption, xA(cT) P x(cT) = xa(cT). We distinguish between two cases:

Case 1: xA(cT) > xa(cT).
By linearity of xA(/) we have
xAðcT Þ ¼ xAðclÞ þ bAl ðcT � clÞ > xaðcT Þ ¼ xaðclÞ þ x0ðcT ÞðcT � clÞ: ð19Þ

However, by construction of xa we have

xaðclÞ ¼ ð1þ aÞxðclÞP xAðclÞ;which contradicts (19) if bA

l 6 x0ðcT Þ. It follows that bAl > x0ðcT Þ must hold in this case. Now,

xAðclþ1Þ ¼ xAðcT Þ þ bAl ðclþ1 � cT Þ > xaðcT Þ þ x0ðcT Þðclþ1 � cT Þ ¼ xaðclþ1Þ ¼ ð1þ aÞxðclþ1Þ;

which contradicts (18).
Case 2: xA(cT) = x(cT) = xa(cT).
By linearity of xA(/) on ½cAl ; c

Alþ1�, it follows that bA

l ¼ x0ðcT Þ ¼ bl. Thus, xa(/) = xA(/) for all /2 [cl,cl+1].Recall that by the TLA procedure and Lemma 4 (see Appendix), the point cl+1 is the unique root of

xðcT Þ þ x0ð/� cT Þ ¼ ð1þ aÞxð/Þ;on ½cT ; �/�, and that (1 + a)x(/) < x(cT) + x 0(cT)(/ � cT) for / > cl+1. But, since cA

lþ1 > clþ1, thisimplies that

xAðcAlþ1Þ ¼ xAðcT Þ þ bA

l ðcAlþ1 � cT Þ ¼ xðcT Þ þ x0ðcT ÞðcA

lþ1 � cT Þ > ð1þ aÞxðcAlþ1Þ;

which contradicts (18).

It follows that cAl 6 cl implies that cA

lþ1 6 clþ1 for all l = 1, . . . ,L � 1. Since c1 ¼ cA1 ¼ 0 this immediately

implies that cAL 6 cL. It follows that LA P L must hold, thus establishing the result. h

Table 2Number of line segments for approximation of each customer node found by TLP

ki 0.1 0.1 0.1 0.1 0.5 0.5 0.5 0.5 1 1 1 1 10 10 10 10

Hi 1 2 5 10 1 2 5 10 1 2 5 10 1 2 5 10

ai = 5% 4 4 4 2 5 5 5 4 5 5 5 4 5 5 5 4

ai = 2% 7 7 6 4 8 8 8 6 9 9 8 6 9 9 9 7

ai = 1% 9 8 8 6 12 12 11 8 12 12 11 8 12 12 12 9

ai = 0.1% 24 23 22 15 37 35 32 23 37 36 33 25 38 37 33 26


As mentioned earlier, the usefulness of the TLA procedure for the purposes of solving the optimizationproblem DO depends critically on having a relatively small number of linear segments in the approximatingfunction xa(/). To evaluate the efficiency of the TLA procedure in this regard we conducted a set of compu-tational experiments, the results of which are presented in Table 2.

For x(/) we used the exponential expenditure function (10) in the competitive location model describedin the previous section (we looked at the component corresponding to customer i 2 N, with i chosen arbi-trarily). We examined how the number of segments in the approximating function xa(/) depends on therequired relative error rate a for a = 5%, 2%, 1% and 0.1%, as well as on several other model parameters:the exponential decay factor k = ki and the total utility derived from pre-existing facilities H = Hi—as definedin (6). The upper limit of the relevant interval �/ was set to �/ ¼

Pj 62Euij � H i—this is the upper limit on Ui

in (8).Table 2 shows that the most critical parameter in determining the required number of segments is the rel-

ative error rate a, with the number increasing sharply as a becomes smaller. The required number of segmentsalso tends to increase with the exponential decay parameter k, but this effect appears to be weaker than that ofthe relative error rate. This effect is likely due to the fact that the exponential function becomes steeper as kincreases. Finally, the required number of segments decreases with H—this is likely due to the fact H is inver-sely related to �/.

Overall, Table 2 shows that the number of segments required to achieve a reasonable level of accuracy isnot large—for example, 12 segments are sufficient to reduce the relative error to below 1%. It appears that theTLA procedure provides a practical approach for solving the competitive location problem P0 defined in (11).This approach is explained in greater detail in the following section.

4. Applying TLA to the competitive location model P0

We begin by developing a linear ‘‘a-approximating’’ model for our original model P0 given in (11).For i 2 N, let Vi(U), given in (8), be the ith term of the objective function of P0, and let xa

i ðUÞ be thea-approximator of Vi(U) constructed by the TLA procedure, with U 2 ½0;Ui� and U i ¼

Pj 62Euij.

Let Li be the number of linear segments in xai ðUÞ, with the endpoints of segment l given by ci

l and cilþ1 for

l 2 {1, . . . ,Li}. Let bli be the slope of segment l, and ai

l ¼ cilþ1 � ci

l be the length of this segment (projected ontothe X axis). The function xa

i ðUÞ can be represented as follows for U 2 ½0;U i�:

xai ðUÞ ¼

XLiðUÞ

l¼1

ailb

ily

il;

where

LiðUÞ ¼ maxfl : cl 6 Ug

(note that Li ¼ LiðUÞ), and

yli ¼

1 if l < LiðUÞU�ci

lai

lif l ¼ LiðUÞ:

(


We now introduce the following linear integer program P a0:

Za ¼ maxXi2N

XLi

l¼1

wiailb

ily

il ð20Þ

s:t:Xj2E

uijxj ¼XLi

l¼1

aily

il for i 2 N ;

Xj2E

xj ¼ m;

0 6 yil 6 1 for i 2 N ; l ¼ 1; . . . ; Li;

xj 2 f0; 1g for j 2 E:

We will denote the linear relaxation of P a0 by P a

R and use Za(x) to represent the objective value of P a0 for a

particular feasible location vector x (with the y variables chosen optimally for this x).It should be clear that for any feasible location vector x,

ZaðxÞ ¼Xi2N

xaðUiðxÞÞ;

where UiðxÞ ¼P

j2Euijxj, as in (11).The following result now follows from (9), Lemma 1, Theorem 4, and the fact that any location vector x is

feasible in problem P0 if and only if it is feasible in problem P a0.

Theorem 6. Let x* be the optimal location vector in problem P0 (problem PR). For some a > 0, let xa be the

optimal location vector in problem P a0 (problem P a

R). Then

Zðx�Þ 6 ZaðxaÞ 6 ð1þ aÞZðx�Þ and1

1þ aZðx�Þ 6 ZðxaÞ 6 Zðx�Þ:

The preceding result suggests two different ways to apply the TLA procedure to the competitive locationmodel P0:

Exact procedure: We apply a ‘‘regular’’ branch-and-bound algorithm, using the relaxed approximatorproblem P a

R to obtain upper bounds, and the objective function of the origin non-linear Integer ProgramP0 to obtain lower bounds. We call this approach the ‘‘exact procedure’’ because the resulting solution is guar-anteed to be optimal for the original problem P0—no matter what value of a is used (here smaller values of aresult in tighter upper bounds at the cost of increasing the dimensionality of the LPs that have to be solved toobtain these bounds).

a-Optimal approximate procedure: Under this procedure we solve the approximator problem P a0 directly and

then use the resulting optimal location vector xa as an a-optimal solution in P0. The main advantage of thisapproach is that the problem P a

0, which is a linear Integer Program, can be solved by any of the ‘‘off-the-shelf’’IP solvers—it is not necessary to develop a special-purpose solution algorithm. The obvious disadvantage isthat the resulting solution is only guaranteed to be a-optimal for the original model.

Both of the procedures described above will be explored further in our computational experimentsdescribed in Section 6 below. However, we point out that for practical problems both procedures involve solv-ing large-scale Integer Programs, which may not be possible if the problem dimensionality is sufficiently large.Thus there is a need for an efficient heuristic procedure capable of providing good solutions quickly. Such aprocedure is developed in the following section.

5. Heuristic procedure

In this section we develop a greedy heuristic for our competitive location problem. This algorithm is muchfaster than either the ‘‘exact’’ or the ‘‘a-optimal’’ procedures described in the previous section, but, of course, it


does not provide any direct control over the size of the relative error. However it is possible to obtain a boundon the maximum relative error that can result from the application of this greedy heuristic.

Heuristic algorithm (the greedy heuristic): Starting with the empty set of facilities, find the potential loca-tion that results in the greatest increase in the objective function and locate a new facility there. Proceed untilall m facilities have been located.

Note that the running time for this algorithm is OðN jEjmÞ. The worst case analysis of the algorithm is basedon the following Theorem, whose proof can be found in Appendix.

Theorem 7. The objective function Z(S) of the problem P0 (defined in (9) above) is non-decreasing and sub-modular.

The following corollary now follows immediately by the results of [11].

Corollary 1. Let ZHO be the objective function value produced by the Heuristic Algorithm. Let Z* be the optimal

value of the objective function in original problem, thenZH

OZ� P 1� 1

e � :63.

The computational results in the next section show that the average performance of the Heuristic Algo-rithm above is much better than the performance guaranteed by the worst case bound. In fact, the relativeerrors are typically well below 2%. Thus, the Heuristic Algorithm above provides an efficient way to obtainhigh-quality approximate solutions for our competitive location model.

On the other hand, the bound established in the preceding result is tight, as shown by the following con-struction. Consider an instance where we are to locate m facilities, E = ; (there are no pre-existing facilities),n = 3m, with nodes 1, . . . ,m being the customer nodes (i.e., having positive demand), and nodes m + 1, . . . , 3m

serving as potential facility locations (i.e., jEj ¼ 2m). For convenience, we will use index j = 1, . . . , 2m to referto the potential locations. We assume that the expenditure function gi( ) is exponential—as defined in (10)above. Let wi = ki = 1 for i = 1, . . . ,m. The utility vector uij for i 2 {1, . . . ,m} and j 2 {1, . . . , 2m} is given by

uij ¼� lnð1� 1=mÞ for 1 6 i; j 6 m;

� lnð�Þ for 1 6 i 6 m; j ¼ iþ m;

0 otherwise;

8><>:
where � is chosen so that 0 6 � 6 (1 � 1/m)m.
For example, for the case of m = 4, the utilities are given by:

� lnð0:75Þ � lnð0:75Þ � lnð0:75Þ � lnð0:75Þ � lnð�Þ 0 0 0

� lnð0:75Þ � lnð0:75Þ � lnð0:75Þ � lnð0:75Þ 0 � lnð�Þ 0 0

� lnð0:75Þ � lnð0:75Þ � lnð0:75Þ � lnð0:75Þ 0 0 � lnð�Þ 0

� lnð0:75Þ � lnð0:75Þ � lnð0:75Þ � lnð0:75Þ 0 0 0 � lnð�Þ

2666664

3777775:

Let A = {1, . . . ,m} and B = {m + 1, . . . , 2m}. We make the following observations.

Proposition 1. Let S* represent the optimal location set, and Sh represent the location set selected by the

Heuristic Algorithm above. Then

1. S* = B and z(S*) = m(1 � �).2. Sh = A and z(Sh) = m(1 � (1 � 1/m)m).

Proof

1. Consider an arbitrary solution S, consisting of k nodes from set A and m � k nodes from set B, k 6 m. It iseasy to check that the objective function value of this solution is given by

kð1� exp½lnð1� 1=mÞk�Þ þ ðm� kÞð1� exp½lnð1� 1=mÞk þ lnð�Þ�Þ;


a quantity which is clearly decreasing in k. Thus, the optimal solution must have k = 0 and the first state-ment follows.

2. The proof follows by induction. By the definition of �, it is clear that the first facility will be located in A bythe heuristic. Suppose that the algorithm locates the first i facilities in A. Now, if the i + 1-st facility islocated in B, the resulting objective function value is

ðm� 1Þð1� exp½lnð1� 1=mÞi�1�Þ þ ð1� exp½lnð1� 1=mÞi�1 þ lnð�Þ�Þ;

which increases with �. By setting � to its maximum value (1 � 1/m)m, and substituting above we obtain

ðm� 1Þð1� 1=mÞi�1 þ ð1� 1=mÞmþi�16 mð1� 1=mÞi;

where the quantity on the right can be recognized as the objective function value obtained by locating theith facility in set A. It follows, that at each step of the algorithm, it is beneficial to locate the next facility inset A, which completes the proof. h

From the preceding result it follows that

ZGO

Z�¼ 1� ð1� 1=mÞm

1� � ;

which approaches 1 � (1 � 1/m)m as �! 0. Thus, by selecting a large value of m and a small value of �, we canapproach the bound in Corollary 1 arbitrarily closely.

6. Computational results

We have conducted a series of computational experiments to test the efficiency of the three computationalapproaches developed above, namely the ‘‘exact’’ and ‘‘a-optimal’’ approaches of Section 4, as well as theHeuristic Algorithm of Section 5. Throughout this section, we assume that the expenditure function is of expo-nential form, as defined by (10).

Each problem instance was generated as follows:

• We varied the number of candidate nodes jEj between 20 and 200 (in steps of 10) and the number of newfacilities m between 2 and 21. The number of pre-existing facilities jEj was drawn randomly from{1, . . . , 2m}.

• We set the number of customer (demand) nodes to jEj þ jEj. Thus jN j ¼ 2ðjEj þ jEjÞ, but only the customernodes have positive demand. Weights (amount of total demand) for each node were randomly drawn froma uniform distribution on [1, 10].

• We randomly generated the demand elasticity parameter ki for node i from a uniform distribution on[0.1,1].

• We generated the utility coefficients uij for each customer node i and candidate location j as follows. Weused the form uij ¼ Aj

dbij

, with distance sensitivity b randomly drawn from [1,4], the shortest distances dij

obtained by first randomly generating distances for all links in the graph and then constructing a shortestdistance table (the distances for links were generated from a uniform distribution on [5,100]), and theattractiveness parameter Aj was generated from a uniform distribution on [1, 100]. We note that this formof the utility coefficients corresponds to (2) – which is the ‘‘traditional’’ form for Huff-type models.

For each combination of jEj and m values, we generated five random instances. For each problem, weapplied the ‘‘exact’’ and the ‘‘a-optimal’’ approaches with a = 0.001. The time limit for all algorithms wasset to 60 minutes of CPU time. The resulting optimization models were solved using CPLEX IP solver ona Dell Poweredge 6300 machine running Redhat Linux.

Results of the experiments can be found in Table 3 for moderate size problems, and in Table 4 for largerproblems. The following measures were recorded for each run (and then averaged for each combination of jEjand m values): the relative errors for the objective function values for the a-optimal and the heuristic solutions(columns 3 and 4), and the CPU time in seconds (columns 5–7 in Table 3 and columns 5–6 in Table 4); the

Table 3Results of the experiment for moderate size problems

# Candidate sites (jEj) # Facilities to locate (m) Relative error CPU time (seconds)

Heuristic 0.1%-Optimal Heuristic 0.1%-Optimal Exact

20 4 0.0106 0.0002 0.00 0.93 3.2630 4 0.0116 0.0000 0.01 1.63 32.7840 4 0.0000 0.0000 0.01 1.52 123.4540 5 0.0029 0.0000 0.02 15.40 232.0950 4 0.0000 0.0000 0.02 1.11 218.1250 5 0.0000 0.0000 0.02 1.93 1042.32

100 2 0.0012 0.0000 0.02 2.57 5.36100 3 0.0000 0.0000 0.03 8.28 479.35100 4 0.0033* 0.0010* 0.03 12.35 N/A120 2 0.0000 0.0000 0.02 6.92 9.71120 3 0.0000 0.0000 0.03 9.85 789.02120 4 0.0060* 0.0010* 0.03 15.77 N/A160 2 0.0000 0.0000 0.02 37.59 29.15160 3 0.0017 0.0000 0.03 43.59 1452200 2 0.0000 0.0000 0.03 67.91 77.42200 3 0.0010* 0.0010* 0.03 102.38 N/A

For entries marked with (*) relative error is evaluated versus the best known upper bound.


relative errors were computed as (Z* � Z)/Z, where Z* is the optimal objective function value and Z is thevalue of the candidate solution.

Entries marked ‘‘N/A’’ in the last column of Table 3 mark combinations of parameters for which the ‘‘exactmodel’’ failed to converge within the specified time limit—in this case the relative errors were measured versusthe best available upper bounds; the best available upper bound was computed as the smaller of the followingquantities: Za(xa) (i.e., the objective function of the problem P a

0 corresponding to its optimal solution xa) andthe best intermediate upper bound produced by the exact procedure of Section 4 before the time limit wasreached. Note that since for all problems in Table 4 the ‘‘exact model’’ failed to converge, the relative errorsare measured versus the best available upper bound.

The following observations can be made based on experimental results:

1. The ‘‘a-optimal’’ approach appears to perform extremely well even with a = 0.1%. The computationaltimes are quite reasonable and do not appear to grow overly fast with the number of candidate facilitysites. The computation times do increase with the number of facilities to be located, however problemswith up to 20 facilities can be solved in reasonable time. The deviations from the optimal solutionappear to be negligible—in fact, as shown in Table 3, the optimal solution is found for over 90% ofall ‘‘moderate-size’’ instances solved to optimality; the largest observed departure from optimalitywas 0.02%.

2. The ‘‘exact’’ approach appears to be less attractive, with solutions times much larger than for the ‘‘a-opti-mal’’ approach. The exact approach appears to be viable only for small-to-moderate size instances of theproblem.

3. The Heuristic Algorithm performs extremely well. For the moderate-size instances (Table 3), the averagedeviation from the optimal value of the objective function was only 0.19%; the maximum deviationrecorded was 1.3% for the instances that were solved to optimality; the optimal solution was found for47% of these instances. Even for larger size problems (Table 4), the maximum relative error versus theupper bound did not exceed 6.2%. Moreover, there is no evidence that the size of the relative error tendsto increase with the problem size. The Heuristic Algorithm is quite fast, with the average CPU time of only0.02 seconds, increasing to 0.03 seconds for the largest instances solved.

It appears that both a-optimal and the heuristic approaches are viable for dealing with large-scale instancesof the problem. Of course, if the problem can be solved to optimality in a reasonable time using the a-optimal

Table 4Results of the experiment for larger size problems

# Candidatesites (jEj)

# Facilities tolocate (m)

Relative error (versus upper bound) CPU time (seconds)

Heuristic 0.1%-Optimal Heuristic 0.1%-Optimal

100 5 0.0034 0.001 3.56 542.36100 10 0.0142 0.001 3.85 875.31100 15 0.0030 0.001 4.13 1023.60100 20 0.0010 0.001 4.86 953.08120 6 0.0061 0.001 4.02 675.38120 6 0.0010 0.001 4.01 795.51120 8 0.0112 0.001 3.90 738.09120 8 0.0260 0.001 3.48 1164.86120 9 0.0004 0.001 4.33 1325.84120 9 0.0010 0.001 5.13 1369.21120 12 0.0015 0.001 5.63 1865.63120 12 0.0015 0.001 5.32 1653.28120 15 0.0090 0.001 6.25 1135.91120 15 0.0027 0.001 5.89 986.26160 6 0.0079 0.001 6.58 386.94160 6 0.0010 0.001 8.41 628.68160 8 0.0074 0.001 6.55 1212.57160 8 0.0010 0.001 6.37 996.58160 9 0.0011 0.001 7.55 1296.36160 9 0.0010 0.001 7.82 1358.60160 12 0.0245 0.001 8.01 1624.61160 12 0.0037 0.001 8.95 1298.12160 15 0.0269 0.001 9.65 2013.56160 15 0.0010 0.001 9.21 2154.09160 18 0.0619 0.001 10.06 1785.32160 18 0.0033 0.001 9.97 1568.37200 6 0.0010 0.001 15.43 1119.5200 6 0.0059 0.001 15.26 1710.23200 8 0.0010 0.001 12.15 1004.69200 8 0.0033 0.001 16.81 331.21200 9 0.0010 0.001 15.39 1652.09200 9 0.0202 0.001 17.23 1065.83200 12 0.0302 0.001 17.65 2013.84200 12 0.0007 0.001 18.15 1864.66200 15 0.0150 0.001 21.3 2216.2200 15 0.0038 0.001 19.65 2315.65200 18 0.0010 0.001 21.54 1985.32200 18 0.0010 0.001 20.95 2865.54200 21 0.0328 0.001 23.50 2132.14200 21 0.0012 0.001 22.79 2291.07


approach, it should be preferred since it provides a guarantee on the quality of the solution. Where this is notviable due to problem size, the heuristic method can be applied with a reasonable expectation of obtaining anear-optimal solution.

We note that the excellent performance of the Heuristic Algorithm is not surprising—it is our experiencethat this performance is, in fact, to be expected for problems of this type where the sub-modularity-basedbound established in Section 5 holds (see, e.g., [5]). On the other hand, we underscore that the results discussedin this section are based on a limited number of randomly generated instances, and thus may not extend toproblem instances with structure significantly different from the one used above; in particular we do not knowhow sensitive the algorithmic performance is to the form of the utility coefficients and other problem param-eters (we would expect this to be more of an issue for the Heuristic Algorithm than for the a-optimal proce-dure, since the latter employs a generic IP solver).


7. Concluding remarks

In this paper, we presented a model for locating competitive service facilities with concave demand. The keyadvantage of the model is that it captures two key aspects of the underlying problem: the market expansionand market cannibalization effects. Previous modeling attempts in this direction had run into significant com-putational problems due to non-linearity and complexity of the underlying optimization models. By focusingon the concave demand functions we were able to overcome these difficulties through the development of theTLA procedure which allowed us to obtain a-optimal solutions to large-scale models using standard IntegerProgramming software.

In addition, we developed a Heuristic Algorithm with a tight worst-case error bound, which is somewhatunexpected in view of the non-linearity of the underlying model. Our computational experiments show thatthis heuristic has excellent performance both, with respect to solution quality, and the computational times.

In fact, in view of the performance of the heuristic it may be asked whether the a-optimal procedure is reallynecessary. In our opinion, the answer is ‘‘yes’’. While the performance of the heuristic in our computationalexperiments is certainly encouraging, it must be remembered that, in the worst case, the heuristic solution maybe up to 37% worse than the optimal one. Given the strategic nature of the facility location problems, and thelarge capital commitments involved in the locational decisions, it has been our experience that real-worlddecision makers are unwilling to accept this level of potential deviation from optimality, especially whenthe uncertainty regarding the solution quality can be resolved with a few hours (or even days) of computertime. The a-optimal procedure has a distinct advantage in this regard since by setting the a level sufficientlylow one can guarantee that the potential ‘‘error’’ in the solution is below the ‘‘noise’’ level introduced by otherparameters in the model.

In future we plan to concentrate on incorporating facility design decisions into our model, i.e., focusing onproblems where the decision is not just where to locate the new facilities, but what kind of facilities in terms oftheir design (and thus their attractiveness level Aj) they should be, where ‘‘design’’ can include many aspectssuch as facility layout, size, customer service, etc.

Appendix

Proof of Theorem 3. Let the lth segment defining xA(Æ) have endpoints cl, and cl+1 and the slope bl. To provethe result it suffices to show that

max/2½cl;clþ1�

xAð/Þxð/Þ 2

xAðclÞxðclÞ

;xAðclþ1Þxðclþ1Þ

� �:

This holds trivially if x(/) is constant on [cl,cl+1]. For the remainder of the proof, assume that x(/) is increas-ing on this interval.

For / 2 [cl,cl+1] let Sð/Þ ¼ xðclÞ þ ð/� clÞ xðclþ1Þ�xðclÞðclþ1�clÞ be a straight line connecting points (cl,x(cl)) and

(cl+1,x(cl+1)), and let sl ¼ xðclþ1Þ�xðclÞclþ1�cl

be the slope of this line. Since x(/) is assumed to be concave and increas-

ing on [cl,cl+1], it follows that sl > 0 and

Sð/Þ ¼ xð/Þ for / 2 fcl; clþ1g and Sð/Þ < xð/Þ for / 2 ðcl; clþ1Þ: ð21Þ
Thus,
xAð/ÞSð/Þ P

xAð/Þxð/Þ for / 2 ½cl; clþ1�; ð22Þ

with equality holding for / = cl and / = cl+1.Let r ¼ bl

sl. By definition of the relative error �(/), we have �ðclÞ ¼ xAðclÞ�xðclÞ

xðclÞ . Now, for / 2 [cl,cl+1],

xAð/Þ ¼ xAðclÞ þ blð/� clÞ ¼ ð1þ �ðclÞÞxðclÞ þ rslð/� clÞ¼ ð1þ �ðclÞÞ½xðclÞ þ slð/� clÞ� þ ½r � ð1þ �ðclÞÞ�slð/� clÞ:


It follows that

xAð/ÞSð/Þ ¼ ð1þ �ðclÞÞ þ ½r � ð1þ �ðclÞÞ�

slð/� clÞxðclÞ þ slð/� clÞ

:

We conclude that xAð/ÞSð/Þ is non-decreasing is r P 1 + �(cl) and is decreasing otherwise. In either case, xAð/Þ

Sð/Þ ismaximized at one of the endpoints of [cl,cl+1]. The result now follows from (21) and (22).

Detailed algorithm and technical results for the TLA procedure Steps B and C of the informal proceduredescribed in Section 3.1 above are obviously the crucial ones in the procedure described above. They are basedon the two results presented below.

Lemma 3. Consider c 2 ½0; �/� and d 2 R such that x(c) 6 d. For a point cT 2 ½c; �/�, let bLðcT Þ ¼ xðcT Þ�dcT�c be the

slope of the line from (c,d) to (cT,x(cT)), and let L(/) = d + bL(cT)(/ � c), / 2 [c, cT] be the equation of thatline. Then

1. There exists cT 2 ½c; �/� such that x 0(cT) = bL(cT) (i.e. the slopes of L(/) and x(/) coincide at cT) if and only if

xð�/ÞP x0ð�/Þð�/� cÞ þ d: ð23Þ
2. If condition (23) holds, then cT is unique, and can be found by solving the following equation for / 2 ½c; �/�:
x0ð/Þð/� cÞ þ d � xð/Þ ¼ 0: ð24Þ

Proof. For / 2 ½c; �/�, consider the function T(/) = x 0(/)(/ � c) + d � x(/). Note that T 0(/) = x00(/)(/ � c).Since, by assumption, x(/) is concave, x00(/) < 0, and thus T(/) is non-increasing on ½c; �/� (in fact, it is strictlydecreasing on ½c; �/�).

Recall that d P x(c). If d = x(c), then T(c) = 0 and thus cT = c provides the unique solution to T(/) = 0.Finally, since T(/) is non-increasing, T(c) = 0, and �/ P c, it follows that T ð�/Þ 6 0, which shows thatcondition (23) holds as well.

Assume now that d > x(c). Then T(c) > 0, and T(/) is decreasing on ½c; �/�. It follows that T(/) = 0 has asolution if and only if condition (23) holds. Let cT be the root of T(/) = 0. Then bL(cT) = x(cT) by definitionof bL, thus completing the proof. h

Note that if the solution to (24) exists, then it can be found by a standard line search procedure—e.g., bisec-tion. The convergence of the line search procedure is guaranteed by the monotonicity of the function T(Æ)defined in the proof of Lemma 3 above.

Lemma 4. Consider c 2 ½0; �/�. For / 2 ½c; �/�, let L(/) = x(c) + x 0(c)(/ � c) be the line tangent to x(/) at point

c. Then, for a > 0 there exist cE 2 ½c; �/� such that

LðcEÞ � xðcEÞð1þ aÞ ¼ 0 ð25Þ
if and only if
Lð�/ÞP xð�/Þð1þ aÞ: ð26Þ
Moreover, if a solution to (25) exists, then it is unique.
Proof. Let T(/) = x(/)(1 + a) � L(/) for / 2 ½c; �/�. This function is concave (by the concavity of x(/)), andpositive at c. Thus, it has at most one root on ½c; �/�. If (26) holds, then T ð�/Þ 6 0 and such a root exists. Theconverse also holds. h

We note that if (26) holds, the unique root of (25) can be found by a standard line search technique (e.g.,bisection)—the convergence of the latter is guaranteed by the uniqueness of the root. We can present thedetails of TLA Procedure for constructing the approximating function xa.

The TLA procedure

Step 1: Set l = 1, c = c1 = cT = 0, b1 = x 0(0).Step 2: If condition (26) fails, set clþ1 ¼ �/, L = l and STOP.


Else, use the bisection method to find the unique root of (25) and set cl+1 equal to that root. Ifclþ1 ¼ �/, set L = l and STOP. Else, set c = cl+1 and proceed to Step 3a.

Step 3a: Set l = l + 1.
If condition (23) holds, proceed to Step 3b. Else, set L = l and cL ¼ �/. If
xðclÞð1þ aÞ 6 xð�/Þ; ð27Þthen set bl ¼ xð�/Þ�ð1þaÞ�xðclÞ

�/�cland STOP.

Else, set bl = 0 and STOP.
Step 3b: Use the bisection method to find the unique root of (24), set cT equal to that root and bl = x 0(cT).Step 4: If cT ¼ �/, set cl+1 = cT, L = l and STOP.
Else, repeat Step 2.

Comments

Note that cT is used as a temporary variable in the procedure above—it is the point where segment l is tan-gent to x(/). We note that from the two preceding results it follows that if a stopping point has not beenreached (i.e., if the next iteration is initiated in Step 4), the relative error for the current segment will be aat its right endpoint. Thus, the relative error at the left endpoint is 0 for the first segment and a for every sub-sequent segment. Since we know from Theorem 3 that the maximum relative error for each segment is reachedat one of the endpoints, the overall relative error of a is assured as long as the relative error at the right end-point of the last segment does not exceed a. This is easy to check by analyzing the possible stopping conditionsof the procedure above.

The procedure has several stopping points. In Step 2 it stops if either condition (26) is not satisfied (which—by Lemma 4—implies that the relative error is less than a at �/, and thus the current segment can be extendedto �/), or if the right endpoint of the current segment (computed by Lemma 4) coincides with �/. In either case,the relative error for the last segment cannot exceed a.

In Step 3a, the stopping point is reached if condition (23) fails, indicating that the tangent line to x(/) can-not be constructed on ½cl; �/�. In this case, it is natural to simply extend the current segment to the pointð�/;xð�/ÞÞ, thus ensuring that the relative error for the final segment is 0. However, this extension may leadto the negative slope for the last segment if condition (27) holds—which would violate the requirement thatxa(/) must be non-decreasing. Thus, if condition (27) holds, we simply set the slope of the last segment to 0.Note that in this case xð�/Þ 6 xað�/Þ ¼ xaðclÞ ¼ ð1þ aÞxðclÞ 6 ð1þ aÞxð�/Þ (where we have used the fact thatx(/) is non-decreasing and that the relative error is a at the previous endpoint). This shows that the relativeerror is no more than a for the final segment in this case as well.

Finally, if the stopping point is reached in Step 4, then the tangent point for the last segment has coincidedwith �/, and thus the relative error at �/ is 0.

The discussion above shows that the relative error for the piecewise linear function xa(/) constructed by theTLA procedure does not exceed a (in fact, the relative error is equal to a at the right endpoint of every seg-ment, with the possible exception of the last segment). Moreover, by construction, the slope of every segmentexceeds that of the following segment, since the slope of each segment equals to the slope of x(/) for some /belonging to the segment, and x(/) is concave. This ensures that xa(/) is concave and non-decreasing.

Proof of Theorem 4. The properties of xa(/) constructed by the TLA procedure have already been establishedin the discussion above. It remains to show that the procedure converges in finitely many steps.

Consider segment l in xa(/) for l 2 {1, . . . ,L � 1}. It consists of two sub-segments: [cl,cT] and [cT,cl+1].Note that xa(cT) = x(cT) and xa(cl+1) = (1 + a)x(cl+1). Thus,

clþ1 � cl > clþ1 � cT ¼ð1þ aÞxðclþ1Þ � xðcT Þ

x0ðcT ÞP

axðclþ1Þx0ðcT Þ

;

where the last inequality follows since x(/) is increasing.Since x(/) is increasing and concave, the quantity on the right-hand side above is increasing with l. Thus,

the lengths of the segments in xa(/) are bounded below by an increasing sequence (in particular, the length ofthe sub-segment [cT,cl+1] is strictly increasing with l). Let � > 0 be the length of segment 1. Since for this


segment, cT = c1 = 0, � is the length of the [cT,c2] sub-segment, and thus the length of every other segment is

bounded below by �. Thus, xa(/) has at most bL ¼ 1þ d�/�e segments, and the TLA procedure stops after at

most bL major iterations, which concludes the proof.

Proof of Theorem 7. To prove that ZOðSÞ ¼P

i2N wiV iðU iðSÞÞ the set function in the original problem is non-

decreasing we note the reader to the fact that Vi(Ui(S)) is a non-decreasing function in Ui(S) and since fromdefinition Ui(S) is a non-decreasing function in Ui(S), then Vi(Ui(S)) is a non-decreasing function in S. Nowsince wi P 0 for i 2 N, therefore

Pi2N wiV iðU iðSÞÞ is also a non-decreasing function in S.

Define the following terms: ti ¼P

j2T uij, si ¼P

j2Suij, where S � T � E. To prove that ZO(S) is sub-

modular we have to show that ZO(T [ {k}) � ZO(T) 6 ZO(S [ {k}) � ZO(S) where S � T � E and k 62 T.Since Vi(U) is concave, for 0 6 u1 6 u2 and r P 0 we have

V iðu1 þ rÞ � V iðu1ÞP V iðu2 þ rÞ � V iðu2Þ for i 2 N : ð28Þ
Since uik P 0 and si 6 ti for i 2 N, from (28) we obtain
V iðsi þ uikÞ � V iðsiÞP V iðti þ uikÞ � V iðtiÞ for i 2 N : ð29Þ

From (29) it follows that
Xi2N
wiV iðsi þ uikÞ �Xi2N

wiV iðsiÞPXi2N

wiV iðti þ uikÞ �Xi2N

wiV iðtiÞ: ð30Þ

Thus, ZO(S) is sub-modular, which concludes the proof.

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Competitive facility location model with concave demand

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