European Journal of Operational Research 234 (2014) 819–829

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European Journal of Operational Research

journal homepage: www.elsevier .com/locate /e jor

Innovative Applications of O.R.

Public facility location using dispersion, population, and equity criteria

0377-2217/$ - see front matter � 2013 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.ejor.2013.10.032

⇑ Corresponding author. Tel.: +1 7166450972; fax: +1 7166453302.E-mail address: batta@buffalo.edu (R. Batta).

Rajan Batta a,⇑, Miguel Lejeune b, Srinivas Prasad b

a Department of Industrial and Systems Engineering, University at Buffalo (SUNY), Buffalo, NY 14260, United Statesb Department of Decision Sciences, School of Business, The George Washington University, Washington, DC 20052, United States

a r t i c l e i n f o

Article history:Received 9 January 2013Accepted 14 October 2013Available online 24 October 2013

Keywords:LocationOR in societal problem analysisHealth carePublic policy

a b s t r a c t

From a practical perspective, the paper demonstrates that the appropriate use of dispersion, population,and equity criteria can lead to fairly good solutions with respect to the p-median objective. The only stip-ulation is that the decision maker verifies (through simple constraint checks) that the chosen locationsmeet the dispersion, population, and equity criteria. An empirical investigation is conducted to obtainappropriate values for these parameters. From a location science perspective, a new location model thataccounts for equity and efficiency simultaneously is studied and analyzed. Specifically, the p-maxianproblem with side constraints on dispersion, population, and equity is developed, its NP-completenessestablished, and valid inequalities and bounds derived. Computational tests show encouraging results.

� 2013 Elsevier B.V. All rights reserved.

1. Introduction

In the OR/MS literature, several researchers have developedmodels and solution methodologies for locating public facilitiessuch as emergency vehicles, libraries, hospitals, and fire stations.The focus usually is on minimizing a single objective function,but there also has been a focus on obtaining efficient solutions sat-isficing several objectives. The objectives used are usually somefunction of the response time; typical functions are the average re-sponse time and the maximum response time. There are many re-view articles for location theory and application papers. Theseinclude those by Daskin (2008), ReVelle and Eiselt (2005), Haleand Maberg (2003), Avella, Benati, Martinez, and Dalby (1998),Owen and Daskin (1998), and Brandeau and Chiu (1989). For agood review of recent trends specifically in the context of publicfacility location modeling, see Serra and Marianov (2004). Despitethe impressive list of citations in these papers, there is a significantgap between model development and analysis and usage of thesemodels to make actual locational decisions. There are, of course,some notable exceptions. Dell et al. (2009) discuss location ofnew schools combined with closing of existing schools as a strat-egy to improve transportation access to school children in Chile.Their work is helping shape the school access debate in the ChileanEducation Ministry. Antunes (1999) developed a model for locatingwaste management facilities in Portugal. The model he developedwas a variant of existing location models and provided a goodstarting point for policy makers in Portugal. Mehrez, Sinuany-Stern, Arad-Geva, and Binyamin (1996) discussed the location of

a hospital in a rural area of Israel. They tried various models thoughnone fit exactly to the specifications of the problem at hand. Final-ly, they ended up using a screening method to suggest alternatives.A well documented success story is Larson’s Hypercube queuingmodel which is used by several police departments to suggest loca-tions and deployment strategies for vehicles (Larson, 1974). An-other successful application of public facility locational analysisis due to Eaton, Daskin, Simmons, Bulloch, and Jansma (1985)who employed the Maximal Covering Location problem due toChurch and ReVelle (1974) to deploy ambulances in Austin, Texas.

We now focus on practical public facility location studies tomotivate our work. The paper by Price and Turcotte (1986) dis-cusses their work as consultants for locating a blood bank in Que-bec City. They adopted an intuitive approach, using a simplisticgravity location model to make a comparison between the fivesites that they had shortlisted. The reasons that they cited for notusing a scientific approach were the lack of data availability andtime for the analysis. The work of Carson and Batta (1990) is a goodexample of a failed locational project. They adopted a scientific ap-proach, which led to an interesting model for locating a campusambulance but required considerable effort in terms of data collec-tion. Despite the fact that the model was empirically validated, theresults were not implemented because of a change of administra-tion and reluctance on part of the students to wait in the cold forso few calls. The project suffered from its inability to capture‘‘political’’ constraints and from the ‘‘vanishing advocate’’ phenom-enon discussed in Larson and Odoni (1981). In a recent talk(Berman & Krass, 2010) the median problem with unreliable facil-ities was presented. An interesting result of the analysis was thatthe facilities tend to colocate (or draw together) when the reliabil-ity level drops. At the same time, facilities tend to be dispersed

820 R. Batta et al. / European Journal of Operational Research 234 (2014) 819–829

when in less populated areas. The authors examined empirical dataof hospital locations in the City of Toronto, Canada, and found thatboth dispersion and population criteria are present when unreli-able public facilities are in question.

In this paper, we attempt to answer the following question inthe context of the p-median problem on a network: How bad cana locational choice be provided that the decision makers use disper-sion, population, and equity criteria. Six other papers that theauthors are aware of which address similar questions are: (1)Larson and Stevenson (1972), who performed calculations withspatially homogeneous demands which suggest that the mean tra-vel time resulting from totally random distribution of facilities inthe region served is only reduced by 25% when the facilities areoptimally located; (2) Larson and Odoni (1981), who performedcalculations to show that ignoring the size of a city block in com-puting travel distances induces an average error of no greater than1/3 the size of the block; (3) Batta and Leifer (1988), who demon-strated that restricting facility location to demand points and bar-rier vertices gives good results for the p-median problem on aManhattan plane both with and without travel barriers; (4) Loveand Walker (1994), who show that using a combination of normsyields an accurate approximation to actual road distances undera variety of circumstances; and (5) Francis, Lowe, and Tamir(2004) who study the loss in accuracy when demand points areaggregated so as to make the problem more computationally trac-table, and provide a summary of the work of several researchers inthis area; and finally (6) Burkey, Bhadury, and Eiselt (2012) whodemonstrated that actual locations of hospitals in four states inthe US yielded median and equity values very close to the optimalp-median value and equity value implied by the optimal p-mediansolution, respectively. This last paper provides the motivation forour study by suggesting that typical criteria used by decision mak-ers in practice may in fact yield optimal solutions that would besuggested by traditional median based location models.

This paper makes two types of contributions. The first is practi-cal (intended for policy makers with little knowledge or belief inquantitative solution methods for public facility location prob-lems) and the second is methodological (intended for location sci-entists desiring the development and analysis of a location modelwhich simultaneously accounts for efficiency and equity).

Practical aspects: From a practical perspective, the main aim ofthis paper is to demonstrate that appropriate use of dispersion,population, and equity criteria can lead to fairly good solutionswith respect to the p-median objective. Thus a decision-makerwho has little knowledge or belief in quantitative solution methodscan select the location of the p facilities in any manner they wish.The only stipulation is that they verify (through simple constraintchecks) that the chosen locations meet the dispersion, population,and equity criteria. To facilitate this, an empirical investigation isconducted to obtain appropriate parameter values for thesecriteria.

Methodological aspects: The methodological contributions re-sult from the worst-case analysis used to validate the practical ap-proach. Specifically, the p-maxian problem with side constraints ondispersion, population, and equity is formulated and analyzed.After demonstration that the model when location is restricted toa finite set is NP-complete, valid inequalities are developed andused to improve its solvability. Effort is also devoted to the devel-opment of bounds. Thus a location problem that accounts for effi-ciency and equity simultaneously is studied and analyzed,resulting in the identification of parameter values that decisionmakers can then use to generate good quality solutions.

Lastly, some comments are in order regarding the choice of thethree criteria that we focus on in our analysis, and a limitation ofour study. The dispersion and population criteria are proxies formeasures of efficiency, that are typically focused on minimizing

the costs of providing the service. On the other hand, the equity cri-terion accounts for fairness of the service, and addresses an impor-tant consideration that arises often in the location of publicfacilities. The importance of examining the tradeoffs between effi-ciency and equity objectives in public facility location has beenwell established in both theory and practice dating back toMcAllister (1976), and more recently by Burkey et al. (2012) specif-ically in the context of hospital locations. The main aim of thispaper is to demonstrate that appropriate use of dispersion, popula-tion, and equity criteria can lead to fairly good solutions withrespect to a widely used efficiency measure in the location litera-ture, namely the p-median objective. We are not attempting toestablish that these three criteria are the three most important cri-teria in a given location problem. Clearly, other criteria (such as acovering criterion) may be relevant in a given context, and it wouldbe interesting to study the quality of solutions generated by theproposed criteria with respect to such criteria. We suggest this asa future area of study.

The rest of this paper is organized as follows. Notations areintroduced in Section 2, and in Section 3, we develop formulationsto study the worst-case performance of location problems that em-ploy the population, dispersion, and equity criteria with respect tothe median performance measure. In Section 4, we discuss ourcomputational results and present a sensitivity analysis with re-spect to a range of values of the parameters used in these three cri-teria. In Section 5, we present our conclusions.

2. Notation

Consider a network G = (N,L) in which N is the set of nodes and Lis the set of links. Let jNj = n and the elements of N be labeled1,2, . . . ,n. Let l(i, j) denote the length of a link (i, j) 2 L and d(x,y) de-note the length of a shortest length path between two points x andy in G. On any link (i, j) we define a node k breakpoint as a point xk

such that d(xk,k) through node i and node j are exactly equal. Notethat a link may have at most (n � 2) breakpoints. The total arrivalrate of calls into the system is k, with the arrival from node i beingki. Thus the relative frequency of calls from node i is wi = ki/k. Thereare p facilities to be located on the network. Facility locations canbe made at nodes and at any point along a link, and facilities areassumed to be of infinitesimal size. Response units are housed inthese facilities.

There is a natural tendency to favor choices that spread or dis-perse emergency facilities in the geographical region that they ser-vice. To capture this dispersion criterion mathematically, we assumethat the distance between any pair of facilities in a candidate solu-tion is greater than or equal to a threshold k, or d(xi,xk) P k for i,k = 1, . . . , p, i – k. Dispersion alone is probably not the only criterionthat decision makers would use when making public facility loca-tion decisions. If the population is not evenly distributed, decisionmakers are likely to assign more facilities to heavily populatedareas. To capture this population criterion, we assume that the dis-tance of the closest facility to a population center is less than orequal to a threshold, where the threshold itself is inversely propor-tional to the population center’s weight. Mathematically, if bs is thedistance of the closest facility to a node s, then we impose the con-straint bs 6 A/ws, where A is a factor of proportionality. A high valueof A implies that the decision makers are not too concerned bychoosing locations that do not account for population variationsin the geographical region. A low value of A implies much concernfor population variations in facility location decisions. A third crite-rion that is often relevant in the context of public facility location isfair or equitable access to the facilities, where decision makers haveto ensure that no demand point is beyond an established thresholddistance (B) from its closest facility. This is especially true in

R. Batta et al. / European Journal of Operational Research 234 (2014) 819–829 821

contexts that involve emergency response (see, for example, theOklahoma City Fire Department Fire Station Location Study,2006). Mathematically, we impose the constraint bs 6 B, where bs

is as defined earlier. For a discussion of the use of these criteria inthe context of undesirable facility location, see Murray, Church,Gerrard, and Tsui (1999).

3. Analysis

The objective of the p-median model is to locate p facilities so asto minimize the average travel time to a call, under the assump-tions that (i) a call is always answered by a response unit at theclosest facility, and (ii) such a response is always available for ser-vice, i.e., it is not busy servicing another call. Obviously, the modelis applicable in the situation when k ? 0 or when the number ofresponse units available at each facility are large.

We know from Hakimi (1964) that an optimum solution to thep-median model can be found among the nodes of the network.Therefore, the optimum p-median value can be written as

f ¼ minx1 ;...;xp2N

Xs2N

ws min½dðx1; sÞ; . . . ;dðxp; sÞ�: ð1Þ

By definition, the worst-case value is

g ¼ maxx1 ;...;xp2G

Xs2N

ws min½dðx1; sÞ; . . . ; dðxp; sÞ�: ð2Þ

Other than to compute how inefficient a solution can be interms of the p-median objective, there are several other inherentapplications of a solution to (2). For example, the optimizationproblem on the right side of Eq. (2) is referred to in the literatureas the p-maxian problem (see, e.g., Erkut, Baptie, & Hohenbalken,1990; Erkut & Neuman, 1989), and this has applications in thelocation of obnoxious facilities. Church and Garfinkel (1978) showthat the p-maxian is equivalent to the 1-maxian problem and pro-pose a method to solve it. Minieka (1983) and Ting (1984) providesmore efficient methods for solving this problem. All these methodstake advantage of the fact that the search for the 1-maxian can berestricted to a finite set of points. For more recent work in this area,see the paper by Cheng and Kang (2010), who study the p-maxianon an interval graph.

Recall from the introduction that our objective is to study howbad the p-median value can be for certain parameter values for thedispersion, population and equity criteria. In other words, wewould like to demonstrate that appropriate use of dispersion, pop-ulation, and equity criteria can in fact lead to fairly good solutionswith respect to the p-median objective. With that as our goal, theproblem of interest can then be formulated as:

ðPÞ MaximizeXs2N

ws bs ð3Þ

Subject to the constraints:

dðxj; xlÞP k for j ¼ 1; . . . ;p� 1; l ¼ jþ 1; . . . ; p ð4ÞMinjfdðs; xjÞgP bs for j ¼ 1; . . . ;p; s 2 N ð5ÞMinjfdðs; xjÞg 6 A=ws for s 2 N ð6Þxj 2 G for j ¼ 1; . . . ;p ð7Þbs 6 B for s 2 N ð8Þ

The objective function (3) is to maximize the weighted travel dis-tance from nodes to their closest facility. Constraint (4) ensures thatdistances between facilities are at least k. Constraint (5) ensuresthat the distance between a node and its closest facility is computedcorrectly. Constraint (6) ensures that the distance between a nodeand its closest facility is less than or equal to a threshold, wherethe threshold itself is inversely proportional to the node’s weight.

Constraint (7) says that a facility must be located on the network it-self. Finally, constraint (8) ensures that there is a facility within dis-tance B of each demand node.

The above formulation belongs to a class of problems referred toin the literature as network location problems with distance con-straints. One of the earliest reference to this class of problems isToregas and ReVelle (1972). Other relevant papers are those byChurch and Meadows (1979), Church (1984), and an early surveyof related articles by Moon and Chaudhry (1984). More recently,Berman and Huang (2008) studied the minimum weighted cover-ing problem in such a context, and establish that even checkingfor the feasibility of a given solution with respect to the distanceparameters is NP hard, and developed heuristics for solving theproblem. Network location problems with distance constraintsare typically very hard to solve, and efficient procedures are usu-ally possible for only the simplest cases. This is due to the non-con-vexity of the solution space that the distance constraints introduce.Convexity is retained, however, for a class of problems with upperbounded constraints and when the underlying network structure isa tree. This enables the development of efficient procedures such asin Dearing, Francis, and Lowe (1976), Francis, Lowe, and Ratliff(1978). Erkut, Francis, and Lowe (1988) develop a methodologythat could be used to convert a continuous tree network locationproblem with distance constraints to mathematical programmingproblems. Erkut, Francis, and Tamir (1992) use this methodologyin the context of distance constrained multifacility minimax loca-tion problems on tree networks. All these articles deal with thelocation of ‘desirable’ facilities, for which the distance betweenfacilities and demand points is, in some sense, minimized. Anotherclass of problems, that is relevant to our problem, is the location ofobnoxious facilities. These problems attempt to maximize the dis-tance between facilities and demand points, which is the objectiveof our problem. These are also referred to as multifacility maxisumproblems by Erkut and Neuman (1989). Kuby (1987) and Erkutet al. (1988) consider discrete versions of this problem. A recent re-view article by Farahani, Seifi, and Asgari (2010) covers multiplecriteria facility location problems in general, and includes a gooddiscussion of obnoxious facility location.

Problem (P) is hard to solve, especially since the location offacilities is unrestricted. Our approach will be to restrict locationto a finite set of sites, and use this to derive a valid upper boundon the optimal value of (P). We will start, however, by analyzingthe complexity of (P) for a few simple cases.

3.1. Unrestricted facility location

The first case that we consider is the 1-maxian with distanceconstraints. Constraint set (4) is not relevant in this case. Along alink, the 1-maxian achieves a maximum at either of the end pointsof the link or at one of the finitely many break points along the link.Since feasibility of a break point with respect to the distance con-straints can be easily verified, the optimal solution can be deter-mined by a straightforward procedure, whose order is polynomialin the number of links and vertices of the underlying graph.

The next simple case is when the underlying network structureis a chain. In this case, problem (P) can be reformulated as a linearprogram. This is possible because all distances can be computedwith respect to a common reference point, which could be eitherend of the chain. The decision variables can be defined as xi,i = 1, . . . , p, where xi is the distance of facility i from the referencepoint. Constraint (4) in (P) can then be written as jxi � xjjP k forall i, j = 1, . . . , p, i – j. Constraint (5) can be written as jds � xijP bs

for i = 1, . . . , p, s 2 N, where ds is distance of node s from the refer-ence point. Thus, (P) can be solved in polynomial time on a chainnetwork, as the number of variables and constraints in the result-ing linear program are polynomial.

Fig. 2. General network example.

822 R. Batta et al. / European Journal of Operational Research 234 (2014) 819–829

The above argument holds even in the case of a tree network(not necessarily a chain), as long as the facility locations are re-stricted to lie on a path. The tree network can be represented ona rectangular coordinate system and the distances can be com-puted correctly. This is illustrated in Fig. 1 in which facility locationis restricted to be on the path 1 � 2 � 5. Using node 1 as the origin,the distances between facilities and nodes can be computed cor-rectly. This, however, is not possible if we allow location at anypoint on the tree. Then the distance between xi on link 1 � 3 andxj on 2 � 4 cannot be computed correctly for the given rectangularcoordinate transformation. For any given rectangular coordinatetransformation, there will exist at least one pair of such links forwhich the distance cannot be computed correctly.

Nevertheless, for a specific assignment of facilities to links onthe network, the optimal locations of the facilities on the respec-tive links can be found by formulating a linear program. The deci-sion variables can be defined as xi, i = 1, . . . , p, where xi is thedistance of facility i from either of the two end nodes of the linkto which facility i is assigned. The number of such linear programsis, however, exponential in n and p. Therefore, this is not a compu-tationally feasible approach.

We now consider the case of a general network. As in the case ofa tree network, the optimal locations of facilities can be deter-mined by solving a linear program for a given assignment of facil-ities to links. As an example, consider the network shown in Fig. 2.Assume that facilities 1 and 2 are to be located on the network, onlinks 1 � 2 and 1 � 3 respectively. Let x1 and x2 represent the dis-tances of facilities 1 and 2 from nodes 2 and 3 respectively. Then,the distance between facilities 1 and 2 can be written as

dðx1; x2Þ ¼ Min 2þ x1 þ x2;7� x1 � x2f g:

Such a constraint can be incorporated within a linear program andthus the problem is solvable polynomially for a given assignment offacilities to links. As mentioned earlier, such an approach is notcomputationally feasible, since there are an exponential numberof possible assignments of facilities to links.

It may be possible, however, to develop efficient procedures tosolve (P) on a general network for certain values of the parametersk, A and B. For example, Problem (P) for large values of A and B andrelatively small values of k, reduces to the p-maxian (which isequivalent to the 1-maxian), and can therefore be solved polyno-mially. It is not clear, however, that the problem is polynomiallysolvable for general values of A, B, and k. In fact there is consider-able evidence that leads us to conjecture that problem (P) isNP-complete. For example, for a certain version of the p-maxianproblem where the objective is to maximize the sum of thedistances between facilities and demand points and the distancesbetween the facilities, Hansen and Moon (1988) have shown thatwhen location is restricted to a discrete set of points theproblem is strongly NP-complete; and Tamir (1991) proved NP-completeness for the more general case of unrestricted location.

1

2 3

4 5

1 2 5

3 4

Fig. 1. Tree network example.

3.2. Location restricted to a finite set

We now consider a discrete version of problem (P), where loca-tion is restricted to a finite set of sites. We first note that the dis-crete version could essentially solve the unrestricted version ofthe problem, if it can be shown that the solution to the unrestrictedlocation could be restricted to a Finite Dominating Set (FDS) oflocations. In fact, the identification of FDSs has been studied exten-sively in the literature - see for example, the definition of NetworkIntersection Point Set or NIPS (Church & Meadows, 1979), the CicleIntersect Point Set or CIPS that was defined in the context of planarlocation poblems (Church, 1984), and the Multiple Constraint Dis-tance Intersect Point Set or MIPS (Berman & Huang, 2008). Sincethe size of the FDS could be exponentially large, especially in thepresence of dispersion constraints (Berman & Huang, 2008), otherforms of discretization have been considered in the literature – forexample, Kuby, Lim, and Upchurch (2005) consider the added nodedispersion problem for dispersing additional discrete sites alongarcs of the network, and Schobel, Hamacher, Liebers, and Wagner(2009) consider the continuous stop location problem as a discretelocation problem by identifying a finite dominating set. Althoughthe discrete version of the problem is hard to solve (as establishedby the NP-completeness result a little later), we can at least formu-late it in a concise manner unlike the unrestricted version. We pro-pose several equivalent formulations for the discrete locationproblem, examine their features, and suggest a way to discretizelocations that helps us develop an upper bound for the problemof interest.

Let Z represent the set of sites available for facility location.Furthermore, let dij denote the length of a shortest distance pathbetween sites i and j, and aj be a 0–1 decision variable so thataj = 1 if a facility is at site j and 0 otherwise, for j = 1, . . . , z.Let the sites be labeled 1, . . . , z. We define the index setsHs = {j:dsj 6min(A/ws,B), j = 1, . . . , z}, s = 1, . . . , n. Each set Hs con-tains the indices j of the possible locations which are withinacceptable distance (i.e., minimum between A/ws and B) of nodes. With this notation in place, the problem (P) can be reformulatedas the following bilinear programming problem:

ðPZ1 Þ MaximizeXn

s¼1

ws bs ð9Þ

Subject to the constraints:

Xz

j¼1

aj ¼ p ð10Þ

ð1�aj alÞMþaj al djl P k for j¼ 1; . . . ;z�1; l¼ jþ1; . . . ;z ð11Þð1�ajÞ Dsþaj dsj P bs for s¼ 1; . . . ;n; j2Hs ð12ÞXj2Hs

a½j�P 1; for s¼ 1; . . . ;n ð13Þ

aj ¼ 0 or 1 for j¼ 1; . . . ;z ð14Þ

R. Batta et al. / European Journal of Operational Research 234 (2014) 819–829 823

The decision variables in the above formulation are the aj’s whichare binary variables (14): aj = 1 indicates that a facility is locatedat site j and aj = 0 otherwise. Constraint (10) ensures that only pfacilities are located among the z sites. Constraints (11)–(13) areanalogous to constraints (4)–(6), respectively. Constraints (11)and (12) are big-M constraints. The role of the constraints (12) isto calculate the distance bs between a node s and the closest openfacility. Since a facility must be opened within an acceptable dis-tance (i.e., at most equal to the minimum of A/[w[s] and B) of anynode s, we have a number jHsj(and not jZj) of constraints of form(12) for each node s. Indeed, Hs is the set of sites that are withinmin(A/w[s],B) from s and constraint (13) ensures that at least onefacility will be opened at one of the sites j 2 Hs. The values of the po-sitive constants M and Ds must be chosen such that they do not ex-clude any admissible solution. From a computational perspective, itis important to set them equal to the smallest possible value allow-ing for the above objective to be attained. Assigning a very large va-lue (say 105) to M and Ds would produce a valid (integer) model.However, its continuous relaxation would be very loose and wouldlikely generate an excessive number of branching and increase thecomputational time. It is straightforward to see that k and min(A/ws,B) are the smallest values that M and Ds can respectively taketo guard us from rejecting any admissible solution.

Theorem 1. Problem ðPZ1 Þ is NP-complete.

Proof. We show the above when the set Z is restricted to the set ofdemand nodes N. This can be shown by reducing a known NP-com-plete problem (in this case the dominating set problem) to thegiven problem. The decision version of the dominating set problemcan be stated as follows: Given a graph G = (N,L), and a positiveinteger K 6 jNj, is there a dominating set of size K for G, i.e., a subsetN0 # N with jN0j = K such that for all i 2 N � N0 there is a j 2 N0 forwhich (i, j) 2 L? Let G0 = (N,L) be derived from G such that l(i, j) = 1for (i, j) 2 L, and ws = 1 for all s 2 N. Consider the problem PN overG0 with the parameters (A, B and k) all set to unity, and p = K. It iseasy to verify that the graph G has a dominating set of size K ifand only if the solution to PN over G0 is equal to n � K. h

If the set N0 is required to be both a dominating and indepen-dent set of size K (which is also NP-complete), then lettingA = 1,B = 1 and k = 2 in the solution of PN over G0 will also yieldthe same result.

We introduce the following notations: Mjl 2 R+, j = 1, . . . , (z � 1),l = j + 1, . . . , z is a positive parameter, djl is the known distance be-tween the potential facility sites j and l. We can now rewrite thenonlinear constraint (11) in a linear form.

Proposition 1. Let Mjl ¼ max k;kþdjl

2

� �. The linear constraint

Mjlð2� aj � alÞ þ ðaj þ al � 1Þdjl P k: ð15Þ

is equivalent to

ð1� aj alÞ kþ aj al djl P k: ð16Þ

Constraint (16) is obtained by setting M in (11) to its smallestadmissible value k.

Proof. The goal is to find the smallest value for Mjl that does notexclude any feasible solution for (16). Each pair (aj,al) of binarydecision variables in (15) and (16) can take values {(1,1); (1,0);(0,1); (0,0)}: (1,1) does not impose any restriction on Mjl. However,setting (aj,al) to (0,1) or (1,0) in (15) requires Mjl P k. Similarly,(0,0) imposes that Mjl P kþdjl

2 . This provides the result that we setout to prove. h

Consider two locations j and l such that djl > k. Setting Mjl = kwould implicitly force the opening of at least one facility at sitesj or at l. Indeed, the option of not opening any facility (aj,al) = (0,0)at j and l results into inequality (15) being rewritten as2Mjl � djl P k which is violated if djl > k and Mjl = k. Clearly, thiswould cut a feasible solution (which could be optimal) and alterthe feasible set defined by (16).

The reformulation of the bilinear problem ðPZ1 Þ as the mixed-integer programming (MIP) problem ðPZ2 Þ is a direct consequenceof Proposition 1:

ðPZ2 Þ MaximizeXn

s¼1

ws bs

Subject to the constraints:(10), (12)–(14)

Mjlð2� aj � alÞ þ ðaj þ al � 1Þdjl P k; j ¼ 1; . . . ; ðz� 1Þ;l ¼ jþ 1; . . . ; z ð17Þ

We shall now derive another MIP formulation ðPZ3 Þ equivalent toðPZ1 Þ and ðPZ2 Þ. We call the tuple set T2 the set of interdicted jointfacility locations and define it as:

T2 ¼ ðj; lÞ : djl < k; j ¼ 1; . . . ; ðz� 1Þ; l ¼ jþ 1; . . . ; z� �

:

Clearly each tuple (j, l) 2 T2 refers to a pair of locations wheretwo facilities are not allowed to operate concurrently.

Proposition 2 can now be derived.

Proposition 2. The inequalities

Mjlð2� aj � alÞ þ ðaj þ al � 1Þdjl P k; j ¼ 1; . . . ; ðz� 1Þ;l ¼ jþ 1; . . . ; z

can be subsumed by the set of jT2jcover inequalities

aj þ al 6 1; ðj; lÞ 2 T2: ð18ÞIt is evident that the set of constraints (18) prevents from open-

ing (aj = 1,al = 1) a facility at both locations j and l if the distanceseparating them is not at least equal to k.

It follows from Proposition 2 that the MIP problem ðPZ3 Þ isequivalent to problem ðPZ2 Þ

ðPZ3 Þ MaximizeXn

s¼1

ws bs

Subject to the constraints:(10), (12)–(14), and (18)We can strengthen the formulation of problem ðPZ3 Þ through

the introduction of valid cover inequalities. Let Tk denote the setof k-length tuples (v1, . . . , vk) defined as:

Tk ¼ ðv1; . . . ;vkÞ : dvtv t0 < k; t ¼ 1; . . . ; ðk� 1Þ; t0 ¼ t þ 1; . . . ; k� �

:

The set Tk contains the indices of the groups of k facilities amongwhich at most one can be opened. We construct all the sets Tk fork = 3, . . . , (z � p � 1). If the set Tk is non-empty for k P z � p � 1,then the problem is infeasible. Clearly, the larger the value of k,the smaller the cardinality of the set Tk. It is therefore straightfor-ward that:

Proposition 3. The inequalities

Xj2Tk

aj 6 1; k ¼ 3; . . . ; ðz� p� 1Þ: ð19Þ

are valid for ðPZ3 Þ.

12

3

45

7

6

8

9 10

104

192

119

130

161

160

151

168

177

143

170

206

147

81

123

98

Fig. 3. New York network.

824 R. Batta et al. / European Journal of Operational Research 234 (2014) 819–829

3.3. An Upper Bound on (P) via (PZ)

Our focus is to develop a method to obtain a good upper boundfor (P). A good upper bound suffices, since our motive is in deter-mining how bad things can get. We attempt to do this by establish-ing a relationship between the optimal values of (P) and (PZ). Weassume for now that we have chosen a finite set of candidate sitesZ; later we will develop a sensible choice for the set Z. An upperbound on (P) can be obtained based on the following theorem:

Theorem 2. Let dZ = maxx2G[min(d(x,1), . . . , d(x,z))] be the maxi-mum distance from a point on the network to its closest facility site inthe set Z and let mk,A,B(P) denote the optimum objective function valueof a problem P when a distance standard k, a proportionality standardA, and an equity standard B are used. Then, mk;A;BðPÞ 6 mk0 ;A0 ;B0 ðP

LZÞ þ dZ,

where k0 = max[0,k � 2dZ] and A0 = A + wsdZ and B0 = B + dZ are definedfor each node s.

Table 1Population densities on network.

Township Node (i) Weight (wi)

Knox 1 0.0072Guilderland 2 0.0833Colonie 3 0.2741Berne 4 0.0080New Scotland 5 0.0362Albany City 6 0.4591Bethlehem 7 0.0929Rensselaerville 8 0.0061Westerlo 9 0.0089Coeymans 10 0.0266

Proof. Let x�1; . . . ; x�p be the facility locations on the network corre-sponding to an optimal solution to (P). Associated with x�1; . . . ; x�p,let v�1; . . . ;v�p be the closest sites (in terms of distance) chosen fromthe set Z, respectively. The points x�1; . . . ; x�p certainly satisfy the dis-tance constraints (10) and (12) but the points v�1; . . . ;v�p may notwhen the same standards k,A,B is used. However, the pointsv�1; . . . ;v�p will satisfy the constraints when a distance standard ofk0 = max[0,k � 2dZ], a proportionality standard A0 = A + wsdZ andan equity standard B0 = B + dZ for each node s is used; the 2 in theexpression max[0,k � 2dZ] stems from the fact that two facilitiesmay be reassigned to sites in a manner that makes both of themmove dZ units towards one another. Similarly, the constraint bs -6 A/ws needs to be modified to bs 6 A0/ws, and bs 6 B needs to bemodified to bs 6 B0 since a facility that was closest to a node s couldhave moved dZ units away. Let V denote the objective functionvalue at the points v�1; . . . ;v�p. Then V P mk,A,B(P) � dZ, since themaximum reduction in the objective function value is dZ, due tothe fact that the nodes can each move closer to their nearest facil-ity by at most dZ and that

Ps2Nws ¼ 1 by assumption. Also,

mk0 ;A0 ;B0 ðPLZÞP V , since the points v�1; . . . ;v�p are feasible to PL

Z whendistance standard k0, proportionality standard A0, and equity stan-dard B0 are used. Combining these two inequalities yieldsmk;A;BðPÞ 6 mk0 ;A0 ;B0 ðPL

ZÞ þ dZ . The theorem follows. h

Clearly, it is desirable to keep dZ small in relation to the value ofmk,A,B(P). For a chosen value of dZ, the sites can be chosen in the fol-lowing manner: select max½b lij

2dZc;1� sites at equally spaced intervals

on the interior of a link (i, j), where lij is the length of a link (i, j), andby having nodes as sites. For example, a link (i, j) of length 20 and adZ value of 3 is handled by choosing max½b20

6 c;1� ¼ 3 sites at pointson link (i, j) that are 5, 10, and 15 units away from node i, respec-tively. (There is a possibility of having some unnecessary sites iflinks of length less than 2dZ exist. Such links have one site at thecenter of the link. It is a simple matter to check if every point onsuch a link (i, j) is within dZ distance of a site on another link — thiscan be done by checking if the length of a shortest path from themidpoint of link (i, j) to the nearest site on an adjacent link is lessthan or equal to dZ. If such a check is affirmative, the site is re-moved from consideration.) What remains then is to obtainmk0 ;A0 ;B0 ðPZ3 Þ, using which an upper bound can be obtained on mk,A,B(P)based on the result from Theorem 2.

Some observations are now made regarding the choice of dZ.The smaller dZ is, the better the approximation thus resulting in atighter upper bound. Ideally, dZ should be as small as possible,but this results in a harder problem to solve since there will be alarge number of sites. As can be seen from the formulationPZ3 , there are order Oðz2Þ number of constraints. Therefore, as z

increases, the problem quickly becomes harder to solve. The larg-est value of dZ that can be chosen is given by dZ = max(i,j)lij/2 and re-sults in Z = N. When dZ is chosen this way, the value of mk0 ;A0 ;B0 ðPNÞwhich represents an upper bound on mk,A,B(PN), tells us how badthe p-median objective could be if locations are restricted to theset of nodes.

4. Computational results

We present our results based on three different data sets. Thefirst one is a transportation network in the county of Albany,New York (NY). The second one is a transportation network fromthe South-East (SE) US that has been used in a disaster prepared-ness study (Rawls & Turnquist, 2010). The last set of results arebased on a number of randomly generated networks (RAN) of dif-ferent sizes and weights on nodes. Each problem instance wasmodeled with AMPL and solved with the CPLEX 12.4 solver on a64-bit Dell Precision T5400 Workstation with Quad Core Xeon Pro-cessor X5460 3.16 gigahertz CPU, and 4 � 2 gigabyte of RAM. Allproblem instances were solved to optimality in less than11 second.

Before proceeding with a discussion of the results, some lastcomments are in order regarding the choice of dZ. Recall, that themaximum value of dZ, say dmax

Z , is determined by the length ofthe longest link on the network and is therefore fixed for a givennetwork. Also, this value of dZ results in the set of sites, Z, restrictedto the set of nodes (N).

Also, the values of the parameters A, k and B are varied based onbase starting values Ap, Bp and kp as implied by the p-median solu-tion. Successive values of A, B, and k are obtained using incrementsof 10% of Ap, Bp and kp in both directions of the base values,representing tighter constraints on one side of the base values

Fig. 4. Results for NY network.

Fig. 5. South East network.

Fig. 6. Results for SE network.

R. Batta et al. / European Journal of Operational Research 234 (2014) 819–829 825

and looser constraints on the other. Also, preliminary analysis wasconducted for a range of values of dZ, the largest value for whichthe set Z of potential sites essentially reduces to the set N, theset of nodes. Clearly, the tightest bounds on the objective function(when location is unrestricted on the network) are obtained for thesmallest value of dZ which results in a fairly large number of poten-

tial sites over which the problem is solved. Our findings showedthat the upper bounds are fairly loose even for the smallest valueof dZ suggesting that the worst case values when location is unre-stricted could be quite bad relative to the p median solution. More-over, as discussed in the Introduction section, in actual practice it isquite common that location choices are not unrestricted but

Table 2Median Deviations for Random Networks

No. facilities Network % Deviation

w1 w2 w3 w4

3 r1 23.70 34.90 35.28 24.093 r2 16.42 21.48 23.84 35.233 r3 23.71 32.44 31.89 15.995 r1 40.34 42.12 35.21 49.845 r2 35.00 57.27 46.33 69.595 r3 38.55 36.52 46.41 42.67

10 r1 51.35 71.08 81.98 90.5910 r2 73.93 98.14 86.24 87.3710 r3 70.87 75.84 97.22 89.78

Table 3Equity deviations for random networks.

No. facilities Network % Deviation

w1 w2 w3 w4

3 r1 �7.44 �2.43 �5.73 �4.223 r2 0.00 �1.05 �1.90 �2.903 r3 0.00 0.00 �3.68 0.005 r1 �2.73 �3.33 �4.98 �6.575 r2 �1.05 0.00 �2.11 �1.055 r3 0.00 0.00 0.00 �2.50

10 r1 �2.50 0.00 0.00 �5.0010 r2 �1.25 �1.33 �1.43 �1.2510 r3 0.00 �1.54 0.00 0.00

826 R. Batta et al. / European Journal of Operational Research 234 (2014) 819–829

instead limited to a discrete set of locations which are often the setof demand points (see for example, the study by Carling, Han, &Hakansson (2012)). As such, in our computational study, we lim-ited our analysis to locations that are restricted to the set of de-mand nodes.

4.1. New York data set

The county of Albany in NY state divides into ten townshipswith highly variant population densities and relative locations asshown in Fig. 3. Only the major roadways have been consideredin constructing the network — these are shown in the figure as

Fig. 7. Feasible

links connecting nodes and the numbers alongside the links repre-sent the lengths. Consideration of only major road segments isquite typical when studying public facility location problems,e.g., see the recent paper by Erdemir et al. (2008) for locating aero-medical bases in the State of New Mexico. The population densitiesof the nodes are shown in Table 1.

We solve problems with 3, 5, and 7 facilities on this network,and results are presented in Fig. 4. The Y-axis on the figure showsthe % deviation of the worst-case median value from the optimal p-median value, while the X-axis shows the parameter values. Themiddle value on the X-axis, as shown in the figure as (ABL6) corre-sponds to the parameter values as implied by the p-median solu-tion. The rest of the values on the X-axis represent successive10% changes from this base value.

For tighter values of the parameters (represented in the figureby (ALB1) through (ALB5)), the problem becomes infeasible. Forthe other values of the parameters, as seen in the graph, the worstcase p-median value is very stable and does not change much at all.In fact, for both the 3 and 7 facility case, the deviation from the p-median solution is zero, and about 26% for the 5 facility case. Foreach computed worst case median solution, we also assessed thedeviation with respect to the equity measure from its value as im-plied by the p-median solution, and that was 0 for the 3 and 7 facil-ity case, and improved by about 15% in the 5 facility case. Theseresults, we suspect, are somewhat sensitive to the particular fea-tures of this network which has an extreme variation in the popu-lation densities across the nodes – roughly 80% of the entirepopulation of the network is contained in three nodes of the net-work. As such, the parameter values implied by the p-median solu-tion seem to place rather tight bounds on the correspondingcriteria, thus not allowing for too many feasible solutions.

4.2. South-East (SE) US network

This network was used in a disaster preparedness study byRawls and Turnquist (2010) and consists of 30 nodes with citiesand towns along the gulf from Texas to Florida as shown inFig. 5. We solve problems with 3, 5, 7, and 10 facilities on this net-work, and results are presented in Fig. 6. As in the previous figurefor the NY network, the Y-axis on this figure again shows the %

solutions.

Fig. 8. Infeasible wrt Pop. Para. alone.

Fig. 9. Infeasible wrt Disp. Para,. alone.

R. Batta et al. / European Journal of Operational Research 234 (2014) 819–829 827

deviation of the worst-case median value from the optimal p-med-ian value, while the X-axis shows the parameter values.

The results indicate that the deviation from the p-median isagain quite low (about 20–40%) and relatively stable especiallyfor the 10 and 15 facility cases. For the 3 and 5 facility cases, thedeviation is slightly higher, but under 80% for the base values ofthe parameters, and around 80–100% for higher values. The devia-tion with respect to the equity measure from its value as impliedby the p-median solution is generally negative or around 0 forparameter values close to the base values. This suggests that theequity values are not as sensitive to variations in parameter valuesas the median values. This is consistent with the findings in Burkeyet al. (2012) where they determined that actual locations of hospi-tals in four states in the US yielded median and equity values veryclose to the optimal p-median value and equity value implied bythe optimal p-median solution, respectively.

4.3. Randomly (RAN) generated networks

A number of tests were also conducted using randomly gener-ated problem instances. Three network sizes were generated com-prising of 50 nodes, with the number of edges being 125, 175, and230. For each of these three networks, 4 sets of weights were gen-erated ranging from somewhat uniformly populated nodes (w1) tonodes that differed considerably in their population densities (w4).For each such problem, 5 random instances were generated and re-sults were averaged over the 5 instances. The experiment thus in-volved solving a total of 180 problems instances (3 networksizes � 4 sets of weights � 5 problem instances � 3 number offacilities = 180).

The results show that the base parameter values (as implied bythe p median solution) when location is restricted to nodes yieldsolutions that are in the worst case within 100% of the optimal

Fig. 10. Infeasible wrt both.

828 R. Batta et al. / European Journal of Operational Research 234 (2014) 819–829

p-median solution, with even lower deviations for the specificcases of p = 3 and p = 5 (see Table 2). Table 3 shows the deviationsof the worst case solutions in terms of the equity criteria relative toequity values implied by the p median solution. It is evident thatthe equity values are mostly about the same as those implied bythe p-median solution or marginally better. These results againconfirm the findings in Burkey et al. (2012), suggesting that theparameter values as implied by the p-median solution do seemto capture the criteria used by decision makers in practice.

4.4. Summary of results

The results presented above for the three categories of networksdemonstrate that by imposing constraints based on dispersion,population, and equity criteria, the quality of solutions obtainedwith respect to the p-median objective is quite reasonable, whenlocation is restricted to the nodes of the networks. The deviationfrom the optimal p-median values seems to increase with the sizeof the network and the number of facilities to be located – this isevident in comparing the results of the larger SE US network withthose of the New York network, and also in examining the resultsfor the randomly generated networks. We also observe that thedeviation is larger for networks where there is considerable varia-tion in population densities among the nodes. This is somethingthat perhaps would not be encountered typically in practice, sincelocation strategies would typically account for such variation byallocating a larger number of facilities to the more populated areas.We also note that the equity values that result from the worst casep-median solution are generally better than those implied by theoptimal p-median solution, which is a reflection of the implicittradeoffs between efficiency and equity criteria that are likely tobe present. This would suggest that better solutions with respectto the p-median objective may be found for tighter values of theequity parameters, and something that may be worthwhile toinvestigate in future studies. In summary, the results demonstratethat appropriately chosen parameter values for the dispersion,population, and equity criteria can lead to fairly good solutionswith respect to the p-median objective.

4.5. Sensitivity of solution set

Finally, we provide some results that illustrate the effectivenessof the parameters in eliminating poor solutions from consider-ation. In Figs. 7–10, we illustrate for a particular instance of theNY data set, the effect of varying the population and dispersionparameter values on the set of feasible solutions. The parametervalues are plotted on the X-axis, starting with Ap and kp as A1L1,and the successive values A2L2 through A9L9 representing 10%successive relaxations on the starting values which were estab-lished based on the p median solution. Fig. 7 shows that a numberof the poor solutions are indeed eliminated at tighter values of Aand k, and the parameters are indeed quite effective in eliminatingthe worst solutions for a range of values. Figs. 8–10 show solutionsthat are infeasible with respect to the population parameter alone,the dispersion parameter alone, and both, respectively. We can seefrom these graphs, that the dispersion parameter is not as effectiveas the population parameter in eliminating poor solutions – asshown by Figs. 9 and 10, almost all solutions eliminated by the dis-persion parameter are also eliminated by the population parame-ter. This suggests that with a proper choice of the populationparameter, solutions are naturally well dispersed and automati-cally satisfy the dispersion parameter.

5. Conclusions

In conclusion, our analysis suggests that when location is re-stricted to nodes, parameters based on the population, dispersion,and equity criteria seem effective in eliminating most of the poorsolutions. Prescriptions derived in this manner based on the p-median solution may serve as a useful prescriptive aid for decisionmakers to help generate/assess a number of good quality solutionsthat they can choose from. In the context of larger problem in-stances, if the p median itself is hard to solve, a question arisesas to how good values for the parameters can be generated? Preli-minary testing using the NY and SE networks with parameter val-ues implied by solutions that are within a specified small% (25%) ofthe p-median solution suggests that the worst case median value is

R. Batta et al. / European Journal of Operational Research 234 (2014) 819–829 829

often close to the optimal p-median value, and within 100% in al-most all cases. This suggests the use of a heuristic to get a goodsolution based on which the parameter values can be derived.For example, one option is to use values based on a solution foundusing a heuristic such as the 1-opt interchange (Rosing & Hodgson,2002; Rosing & ReVelle, 1997; Teitz & Bart, 1968). If the objective isto actually generate a set of solutions to present to the DMs forconsideration, we could employ a modification of the Teitz andBart heuristic to generate, say, the k best solutions starting withthe best heursitic solution that would satisfy the parameters. Themodification would be to consider the swap in the 1-opt inter-change on the basis of two conditions, (1) that the new solutionsatisfies the parameters, and (2) the new solution results in thesmallest increase with respect to the p-median value. It would beinteresting to assess the average and worst solution among thegenerated solutions relative to the p-median. Another area for fu-ture research would be to extend the analysis in this paper to loca-tions on a plane.

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