strategic facility location: a review - mdhzoomin.idt.mdh.se/course/kpp319/ht2010/strategic facility...

Invited Review

Strategic facility location: A review

Susan Hesse Owen *, Mark S. Daskin

Department of Industrial Engineering and Management Sciences, Northwestern University, Evanston, IL 60208-3119, USA

Accepted 1 April 1998

Abstract

Facility location decisions are a critical element in strategic planning for a wide range of private and public ®rms. The

rami®cations of siting facilities are broadly based and long-lasting, impacting numerous operational and logistical

decisions. High costs associated with property acquisition and facility construction make facility location or relocation

projects long-term investments. To make such undertakings pro®table, ®rms plan for new facilities to remain in place

and in operation for an extended time period. Thus, decision makers must select sites that will not simply perform well

according to the current system state, but that will continue to be pro®table for the facility's lifetime, even as envi-

ronmental factors change, populations shift, and market trends evolve. Finding robust facility locations is thus a

di�cult task, demanding that decision makers account for uncertain future events. The complexity of this problem has

limited much of the facility location literature to simpli®ed static and deterministic models. Although a few researchers

initiated the study of stochastic and dynamic aspects of facility location many years ago, most of the research dedicated

to these issues has been published in recent years. In this review, we report on literature which explicitly addresses the

strategic nature of facility location problems by considering either stochastic or dynamic problem characteristics.

Dynamic formulations focus on the di�cult timing issues involved in locating a facility (or facilities) over an extended

horizon. Stochastic formulations attempt to capture the uncertainty in problem input parameters such as forecast

demand or distance values. The stochastic literature is divided into two classes: that which explicitly considers the

probability distribution of uncertain parameters, and that which captures uncertainty through scenario planning. A

wide range of model formulations and solution approaches are discussed, with applications ranging across numerous

industries. Ó 1998 Elsevier Science B.V. All rights reserved.

Keywords: Location; Strategic planning

1. Introduction

Facility location is a critical aspect of strategic planning for a broad spectrum of public and private®rms. Whether a retail chain siting a new outlet, a manufacturer choosing where to position a warehouse,

European Journal of Operational Research 111 (1998) 423±447

* Corresponding author.

0377-2217/98/$19.00 Ó 1998 Elsevier Science B.V. All rights reserved.

PII S 0 3 7 7 - 2 2 1 7 ( 9 8 ) 0 0 1 8 6 - 6

or a city planner selecting locations for ®re stations, strategic planners are often challenged by di�cultspatial resource allocation decisions. As populations shift, market trends evolve, and other environmentalfactors change, the need to relocate, expand, and adapt facilities ensures the evolution of new planningchallenges.

The development and acquisition of a new facility is typically a costly, time-sensitive project. Before afacility can be purchased or constructed, good locations must be identi®ed, appropriate facility capacityspeci®cations must be determined, and large amounts of capital must be allocated. While the objectivesdriving a facility location decision depend on the ®rm or government agency, the high costs associated withthis process make almost any location project a long-term investment. Thus, facilities which are locatedtoday are expected to remain in operation for an extended time. Environmental changes during the facility'slifetime can drastically alter the appeal of a particular site, turning today's optimal location into tomor-row's investment blunder. Determining the best locations for new facilities is thus an important strategicchallenge.

A vast literature has developed out of the broadly based interest in meeting this challenge. Operationsresearch practitioners have developed a number of mathematical programming models to represent a widerange of location problems. Several di�erent objective functions have been formulated to make such modelsamenable to numerous applications. Unfortunately, the resulting models can be extremely di�cult to solveto optimality (most problems are classi®ed as NP-hard); many of the problems require integer program-ming formulations.

The computational hurdle posed by complex facility location formulations has, until recently, limitedmost research in this area to static, deterministic problems. In these problems, all inputs (such as demands,distances, and travel times) are taken as known quantities and outputs are speci®ed as one-time decisionvalues. While such problems can provide planners with insight about general location selection, they are notable to adequately model the uncertainties inherent in making real-world strategic decisions.

As noted by Averbakh and Berman [6], research in the area of sensitivity analysis addresses the problemof input data uncertainty. Speci®cally, such research attempts to quantify the e�ect of a change in pa-rameter values on the optimal objective function value (for example, see Labb�e et al. [58]). While suchresults help in evaluating the robustness of a solution after a model is solved, they do nothing to incor-porate uncertainty into models proactively. Both stochastic programming and scenario planning ap-proaches move away from reactive analyses of solution sensitivity toward models which formalize thecomplexity and uncertainty inherent in real-world problem instances. Similarly, dynamic formulationstransform snapshot models of one time decisions into extended horizon models which capture the temporalaspects of real-world problems. In this review, we will see how these proactive approaches have been ap-plied to problems of facility location.

More speci®cally, in the following sections we will detail some of the literature which addresses thestrategic nature of facility location problems by modeling either stochastic or dynamic problem charac-teristics. Our goal in writing this paper is to provide an overview of facility location research which, throughthe consideration of time and uncertainty, has helped to move us toward solving more realistic probleminstances. Due to space limitations, we have chosen to focus on the qualitative contributions of the researchcited. For a more detailed discussion of stochastic modeling issues, we recommend the recent survey byLouveaux [62]. Those seeking a more general overview of facility location research can refer to one of themany published review articles or texts, including Refs. [1,36,46,57,60,74,86,87].

In the next section, we provide a brief introduction to the general problem areas of static and deter-ministic facility location research as a background for the review. Section 3 highlights contributions indynamic model formulations which focus on the timing issues involved in locating a facility over an ex-tended horizon. Section 4 then details research which incorporates stochasticity, capturing the uncertaintyin forecasting problem input parameters. Finally, we conclude with a discussion of future research direc-tions.

424 S.H. Owen, M.S. Daskin / European Journal of Operational Research 111 (1998) 423±447

2. Static and deterministic location problems

The study of location theory formally began in 1909 when Alfred Weber considered how to position asingle warehouse so as to minimize the total distance between it and several customers [94]. Following thisinitial investigation, location theory was driven by a few applications which inspired researchers from arange of ®elds. Location theory gained renewed interest in 1964 with a publication by Hakimi [53], whosought to locate switching centers in a communications network and police stations in a highway system.To do so, Hakimi considered the more general problem of locating one or more facilities on a network so asto minimize the total distance between customers and their closest facility or to minimize the maximumsuch distance.

Since the mid-1960s, the study of location theory has ¯ourished. The most basic facility locationproblem formulations can be characterized as both static and deterministic. These problems take constant,known quantities as inputs and derive a single solution to be implemented at one point in time. The solutionwill be chosen according to one of many possible criteria (or objectives), as selected by the decision maker.A number of researchers, particularly those working with applied problems and those interested in locatingobnoxious facilities, have examined multi-objective extensions of these basic models. (For a more detailedreview of multi-objective facility location models, see Ref. [31].) In this section, some of the fundamentalstatic and deterministic location problems will be reviewed. Our presentation will be structured around thedi�erent objective functions required by common applications, and will include a discussion of signi®cantresearch relating to each problem class.

2.1. Median problems

As noted by Church and ReVelle [30], one important way to measure the e�ectiveness of a facilitylocation is by determining the average distance traveled by those who visit it. (Note that throughoutthis paper, travel time and travel distance will be used interchangeably to represent the ``cost'' oftraveling from one location to another.) As average travel distance increases, facility accessibility de-creases, and thus the location's e�ectiveness decreases. This relationship holds for facilities such as li-braries, schools, and emergency service centers, to which proximity is desirable. (To some extent,``undesirable'' facilities such as land®lls or nuclear power plants exhibit increases in location e�ective-ness in response to an increase in average travel distance; models concerning such facilities are discussedin Section 2.4 below.)

An equivalent way to measure location e�ectiveness when demands are not sensitive to the level ofservice is to weight the distance between demand nodes and facilities by the associated demand quantityand calculate the total weighted travel distance between demands and facilities [75]. The P-median problem(introduced by Hakimi [53]) uses this measure of e�ectiveness, and is stated as follows: Find the location ofP facilities so as to minimize the total demand-weighted travel distance between demands and facilities. Toformulate this problem mathematically, the following notation is necessary:

Inputs:i� index of demand nodej� index of potential facility sitehi� demand at node idij� distance between demand node i and potential facility site jP� number of facilities to be located

S.H. Owen, M.S. Daskin / European Journal of Operational Research 111 (1998) 423±447 425

Decision variables:

Xj �1 if we locate at potential facility site j;

0 if not:

8><>:Yij �

1 if demands at node i are served by a facility at node j;

0 if not:

8><>:Using these de®nitions, the P -median problem can be written as the following integer linear program:

MinimizeX

i

Xj

hidijYij �1�

subject to:X

j

Xj � P ; �2�Xj

Yij � 1 8i; �3�

Yij ÿ Xj6 0; 8i; j; �4�Xj 2 f0; 1g 8j; �5�Yij 2 f0; 1g 8i; j: �6�

The objective (1), as mentioned above, is to minimize the total demand-weighted distance between cus-tomers and facilities. Constraint (2) requires that exactly P facilities be located. Constraint (3) ensures thatevery demand is assigned to some facility site, while constraint (4) allows assignment only to sites at whichfacilities have been located. Constraints (5) and (6) are binary requirements for the problem variables. Sincedemands will naturally be assigned entirely to the nearest facility in this uncapacitated problem (assuminghidij P 0 8i; j), constraint (6) can be relaxed to a simple non-negativity constraint (Yij P 0).

Note that this formulation only allows facilities to be located at a ®nite set of potential sites. These sitesrepresent the nodes of a network. While one might imagine locating a facility at any point along an edge ofthe network, Hakimi [53] proves that for any number of facilities P , there is at least one optimal solution tothe P -median problem which locates only at network nodes. Thus, the simpli®ed formulation includes onlynodes as potential facility sites and yet does not penalize the objective function value.

A modi®ed version of the P -median problem is presented by ReVelle [72] for locating retail facilities inthe presence of competing ®rms. The objective in this retail environment is to locate facilities to maximizethe number of new customers captured or to maximize the retailer's added market share. For this maximumcapture problem formulation, the author assumes that all ®rms in the area supply the same product and thatcustomers patronize the nearest ®rm. This modi®cation illustrates how the P -median problem can be ap-plied in a strategic decision making context.

When applied to a general network, the P -median problem can be di�cult to solve to optimality (thisclass of problems is NP-complete). Limiting potential facility locations to network nodes, however, reducesthe number of possible location con®gurations to

N

P

� �� N !

P !�N ÿ P �!where N represents the number of nodes in the network. Thus, for a ®xed value of P , the P -median problemcan be solved in polynomial time. Nevertheless, a total enumeration approach would be computationally


prohibitive for reasonable values of N (hundreds to thousands of nodes) and P (tens of locations sited). Forvariable P , the problem is NP-hard (see Garey and Johnson [47]). Such complexity issues have led to thedevelopment of sophisticated algorithms for solving this problem.

The formulation presented above suggests the use of integer programming techniques for solving P -median problems. While these techniques are often able to reach integer optimal solutions for moderatelysized problems in a reasonable time, several e�cient heuristics have also been developed for solving medianproblems. (See Ref. [36] for an overview of heuristic methods, Refs. [75,89] for more detail on speci®csolution methods.)

2.2. Covering problems

The P -median problem described above can be used to locate a wide range of public and privatefacilities. For some facilities, however, selecting locations which minimize the average distance traveledmay not be appropriate. Suppose, for example, that a city is locating emergency service facilities such as®re stations or ambulances. The critical nature of demands for service will dictate a maximum ``ac-ceptable'' travel distance or time. Such facilities will thus require a di�erent measure of location e�-ciency.

To locate such facilities, the key issue is ``coverage''. A demand is said to be covered if it can be servedwithin a speci®ed time. The literature on covering problems is divided into two major segments, that inwhich coverage is required and that in which it is optimized. Two covering problems which illustrate thedistinction are the location set covering problem and the maximal covering problem. We will introduce bothproblem classes and discuss their relationship to the P -median problem. For a more complete review ofcovering problems, see Refs. [79,97].

In the set covering problem, the objective is to minimize the cost of facility location such that a speci®edlevel of coverage is obtained. The mathematical formulation of this problem requires the following addi-tional notation:

The set covering problem can thus be represented by the following integer program:

MinimizeX

j

cjXj �7�

subject to:Xj2Ni

Xj P 1 8i; �8�

Xj 2 f0; 1g 8j: �9�The objective function (7) minimizes the cost of facility location. In many cases, the costs cj are assumed tobe equal for all potential facility sites j, implying an objective equivalent to minimizing the number offacilities located. Constraint (8) requires that all demands i have at least one facility located within theacceptable service distance. The remaining constraints (9) require integrality for the decision variables.

Note that this formulation makes no distinction between nodes based on demand size. Each node,whether it contains a single customer or a large portion of the total demand, must be covered within thespeci®ed distance, regardless of cost. If the coverage distance S is small, relative to the spacing of de-mand nodes, the coverage restriction can lead to a large number of facilities being located. Additionally,

Inputs:cj� ®xed cost of siting a facility at node jS�maximum acceptable service distance (or time)Ni� set of facility sites j within acceptable distance of node i �i:e:; Ni � fjjdij6 Sg�


if an outlying node has a small demand, the cost/bene®t ratio of covering that demand can be extremelyhigh.

As stated, the set covering problem allows us to examine how many facilities are needed to guarantee acertain level of coverage to all customers. In many practical applications, decision makers ®nd that theirallocated resources are not su�cient to build the facilities dictated by the desired level of coverage. (Thegoal of coverage within distance S may be infeasible with respect to construction resources.) In such cases,location goals must be shifted so that the available resources are used to give as many customers as possiblethe desired level of coverage. This new objective is that of the maximal covering problem [29].

Speci®cally, the maximal covering problem seeks to maximize the amount of demand covered within theacceptable service distance S by locating a ®xed number of facilities. The formulation of this problemrequires the following additional set of decision variables:

Zi �1 if node i is covered;

0 if not:

�Combining these variables with the notation de®ned above, we derive the following formulation of themaximal covering problem:

MaximizeX

i

hiZi �10�

subject to: Zi6Xj2Ni

Xj 8i; �11�Xj

Xj6 P ; �12�

Xj 2 f0; 1g 8j; �13�Zi 2 f0; 1g 8i: �14�

The objective (10) is to maximize the amount of demand covered. Constraint (11) determines which de-mand nodes are covered within the acceptable service distance. Each node i can only be considered covered(with Zi � 1) if there is a facility located at some site j which is within S of node i (i.e., if Xj � 1 for somej 2 Ni). If no such facility is located, the right hand side of constraint (11) will be zero, thus forcing Zi tozero. Constraint (12) limits the number of facilities to be located, to account for limited resources. Con-straints (13) and (14) are integrality constraints for the decision variables.

Note that both the set covering and the maximal covering problem formulations assume a ®nite set ofpotential facility sites. Typically the set of potential sites consists of some (if not all) of the demand nodes ofthe underlying network. Research extensions to these models have shown that even if facilities are allowedanywhere on the network, the problem can be reduced to one with ®nite choices for facility location (seeRef. [28]). The number of potential sites required to ensure optimality is generally much larger than thenumber of demand nodes, however, and augmented networks are often used to formulate such problems.

One variant of the maximal covering problem weights all demand points equally (without regard to thesize of the demand present) so that the objective is simply to maximize the number of demand nodescovered [97].

Another common variant acknowledges that those demands not covered within the desired servicedistance S should be covered by a less stringent distance standard, T (T > S). This maximal covering withmandatory closeness problem [29] insures that no demand is beyond T units from its nearest facility, whilemaking as many demands as possible within S.

All covering models discussed to this point implicitly assume that if a demand is covered by a facilitythen that facility will be available to serve the demand. In Ref. [41], Daskin and Stern examine siting EMS


vehicles to satisfy a speci®ed service requirement. For such an application, the availability assumption isproblematic, as EMS vehicles already responding to a call for service will not be available to answer ad-ditional demands. Applications where facilities experience busy or inoperative periods have inspired a set ofmodels [39,13] which attempt to provide multiple coverage to demand nodes so that if one facility is busy,others will be within the acceptable range to serve incoming demands. The model derived by Daskin andStern establishes a hierarchical objective function which ®rst minimizes the number of vehicles needed tosatisfy the service requirement and then locates those vehicles to maximize the multiple coverage of demandnodes.

Batta and Mannur [10] also examine models for determining the deployment of multiple EMS vehicles inenvironments where high demand rates cause frequent unit busy periods. The authors recognize that de-mands which require a larger response team are typically more critical, and thus should have a tightercoverage level. They formulate generalized deterministic set covering and maximal covering models whichincorporate multiple response units and demand-dependent coverage requirements. Solution strategies foreach problem class are discussed, including branch and bound algorithms applied to binary representationsof reduced problem formulations.

In Refs. [30,97], the relationship between the P -median (or central facilities location) and maximalcovering problems is examined. The authors show that through a transformation of distances the maximalcovering problem can be viewed as a special case of the P -median problem. Speci®cally, we consider a P -median problem on a network where the distances dij are transformed as follows:

d 0ij �0 if dij6 S;

1 if dij > S:

�Solving the P -median problem with modi®ed distances d 0ij minimizes the amount of demand not servedwithin coverage distance S. It can be shown that this is equivalent to maximizing the amount of demandserved within S, and thus the transformed version of the P -median problem is exactly a maximal coveringproblem. Daskin [36] uses this transformation to develop a multi-objective model that trades o� minimizingthe demand weighted total distance with maximizing the covered demand.

Similar to the P -median problem above, both the set covering and maximal covering problems are NP-complete for general networks.

2.3. Center problems

The set covering problem described above determines the minimum number of facilities needed to coverall demands using an exogenously speci®ed coverage distance. The potential infeasibility of such an ap-proach in many practical contexts led us to examine the maximal coverage problem. As described, thisformulation considers the resources available (in terms of the number of facilities we are able to locate) anddetermines the maximum demand coverage possible.

Another problem class which avoids the set covering problem's potential infeasibility is the class of P-center problems. In such problems, we require coverage of all demands, but we seek to locate a givennumber of facilities in such a way that minimizes coverage distance. Rather than taking an input coveragedistance S, this model determines endogenously the minimal coverage distance associated with locating Pfacilities.

The P -center problem is also known as the minimax problem, as we seek to minimize the maximumdistance between any demand and its nearest facility. If facility locations are restricted to the nodes of thenetwork, the problem is a vertex center problem. Center problems which allow facilities to be locatedanywhere on the network are absolute center problems. As in the set covering problem, Hakimi's [53] result


does not generally hold; the solution to the absolute center problem is often better (i.e., has a lower as-sociated objective function value) than that for the vertex center problem.

The following additional decision variable is needed in order to formulate the vertex P -center problem:D�maximum distance between a demand node and the nearest facility.

The resulting integer programming formulation of the vertex P -center problem follows.

Minimize D �15�subject to:

Xj

Xj � P ; �16�X

j

Yij � 1 8i; �17�

Yij ÿ Xj6 0 8i; j; �18�D P

Xj

dijYij 8i; �19�

Xj 2 f0; 1g 8j; �20�Yij 2 f0; 1g 8i; j: �21�

The objective function (15) is simply to minimize the maximum distance between any demand node and itsnearest facility. Constraints (16±18) are identical to (2)±(4) of the P -median problem. Constraint (19) de-®nes the maximum distance between any demand node i and the nearest facility j. Finally, constraints (20)and (21) are integrality constraints for the decision variables.

Note that here again constraints (21) can be relaxed to simple non-negativity constraints. If decisionvariables Yij are allowed to be fractional, one demand node might be served by multiple facilities. Since thefacilities in this simple case are uncapacitated, the solution will assign each demand node to the closest openfacility. Thus, any solution which assigns a demand to more than one facility has an alternate optimum inwhich all Yij are integral.

If the input value of P is ®xed, both the vertex center and absolute center problems can be solved inpolynomial time. For the vertex center problem, we can ®nd a polynomial algorithm for evaluating allpossible locations of the P facilities. The absolute center problem can be reformulated with an augmentednetwork so that its solution will locate on a subset of the original nodes and augmented points. This processcan also be completed in polynomial time. (See Ref. [36] for a more detailed discussion on center problemsolution algorithms.) If the value of P is variable, however, both types of the P -center problem are NP-complete.

2.4. Additional problem formulations

The P -median, covering, and P -center problems discussed above provide a strong foundation for muchof the location theory research done to date. In this ®nal section on static and deterministic models, we willbrie¯y describe some of the additional problem formulations found in the literature.

In most of the models discussed thus far, we have focused on travel distance or time as a surrogate foroperating costs once a facility is located. Although we acknowledge that limited resources might dictate thenumber of facilities sited, in only one model (set covering) did we explicitly consider location costs. The setof ®xed charge facility location problems includes problem instances which have a ®xed charge associatedwith locating at each potential facility site.


One model in this set is the uncapacitated ®xed charge facility location problem, a close relative to the P -median problem presented above. The uncapacitated ®xed charge problem is formulated by adding a ®xedcost to the P -median objective function and removing the constraint that dictates the number of facilitiesto be located. The result is a problem which determines endogenously the number of facilities to locateand sites them so as to minimize total (construction plus travel) costs. The close relationship between thetwo problem formulations results in a large degree of similarity between the algorithms used to solve them[36].

Simple formulation changes thus extend basic location models to account for ®xed acquisition and/orconstruction costs. Similar alterations can extend basic models to incorporate facility capacities. In suchmodels, capacities are input as limits to the number of demands each facility can serve. By adding a set ofconstraints to the problem formulation, we require that the sum of the demands assigned to each facilitynot exceed the input capacity. Sankaran and Raghavan [76] extend the classical capacitated ®xed chargefacility location model to incorporate the endogenous selection of facility sizes. Mukundan and Daskin [69]consider a similar problem in a pro®t maximization context.

As mentioned above, one of the earliest applications of facility location modeling considered locatingwarehouses. Any ®rm deciding where to site a new warehouse must also consider how to best ship productsbetween its facilities and its customers. The set of location-allocation problems builds upon a basic locationproblem formulation (such as those presented above) to simultaneously locate facilities and dictate ¯owsbetween facilities and demands. These problems (as reviewed by Scott [80]) combine a standard trans-portation problem for allocating ¯ow between facilities with a location problem (usually a P -medianproblem or a ®xed charge problem) for siting the facilities.

Just as warehouse applications require us to consider issues of both location and allocation, practicalapplications often introduce more involved objectives than the simple minimization of cost or maximiza-tion of coverage. A class of multi-objective location models have been developed to re¯ect the complexityinherent in many location problem applications. The hierarchical set covering model discussed above is anexample of how multiple objectives are used to simultaneously optimize along multiple criteria. In Ref. [31],Current, Min and Schilling review a variety of multi-objective formulations, illustrating the range of factorsto be considered in locating new facilities.

Finally, note that the models and applications presented thus far focus on locating facilities to makethem accessible to customers. Alternatively, several important real-world applications deal with locatingfacilities which are undesirable to nearby populations. For example, if a city locates a waste disposal plant,a water treatment center or even an airport, the objectives for optimal location are contrary to those de-tailed above. These applications have, in fact, spawned a special area of research for locating ``obnoxious''or ``noxious'' facilities. Problems which address these situations include the antimedian problem, whichlocates a server to maximize average distance between server and demand points; the anticenter problem,which maximizes the minimum distance between server and demand points; and the p-dispersion problem,which locates facilities to maximize the minimum distance between any pair of facilities. While suchproblems are useful in formulating undesirable facility location problems, the political rami®cations in-volved in locating such facilities often force decision makers to use multi-objective models. A more detailedreview of such problems can be found in works by Brandeau and Chiu [19], Daskin [36], Erkut andNeuman [44], and Shilling et al. [79].

3. Dynamic location problems

Much of the research published on location theory is drawn from the models described above, theirapplications and extensions. As noted, many of these problems can be extremely di�cult to solve. Thus, it isnot surprising that so much work has focused on static and deterministic problem formulations. While such


formulations are reasonable research topics, they do not capture many of the characteristics of real-worldlocation problems.

The strategic nature of facility location problems requires that any reasonable model consider someaspect of future uncertainty. Since the investment required by locating or relocating facilities is usuallylarge, facilities are expected to remain operable for an extended time period. Thus, the problem of facilitylocation truly involves an extended planning horizon. Decision makers must not only select robust loca-tions which will e�ectively serve changing demands over time, but must also consider the timing of facilityexpansions and relocations over the long term.

In the next two sections, we will present research which deals explicitly with the uncertainties inherent infacility location. For organizational purposes, a distinction is made between uncertainty related to planningfor future conditions and uncertainty due to limited knowledge of model input parameters. We will ®rstaddress the former, looking at dynamic deterministic facility location models. Section 4 will then examinestatic stochastic models, which attempt to locate facilities under incomplete or imperfect information.

3.1. Dynamic single facility location models

The ®rst paper which recognized the limited application of static and deterministic location models waspublished by Ballou in 1968 [7]. Attempting to locate a single warehouse so as to maximize pro®ts over a®nite planning horizon, Ballou uses a series of static deterministic optimal solutions to solve the dynamicproblem. For each period in the speci®ed horizon, he solves for the optimal warehouse location, estab-lishing a set of potential ``good'' location sites. Dynamic programming is then used to determine the bestschedule for opening a subset of these sites as an ``optimal'' location and relocation strategy for theplanning period.

This approach was later found to be sub-optimal by Sweeney and Tatham [85] who improve on Ballou'ssolutions by extending the set of potential location sites. Their method ®nds the Rt best (rank ordered)solutions in each period t through an iterative procedure of solving integer programs with Benders' de-composition. The number of solutions (Rt) varies by period and is found through bounding the overalloptimal solution value. The expanded set of potential location sites for each period is then used in a dy-namic program to determine an optimal location and relocation strategy.

Note that both of these papers allow for frequent facility relocation, but that neither considers con-struction time or cost in the objective function. Wesolowsky [95] examines another, unconstrained, versionof the single facility location problem over a ®nite planning horizon with explicit facility relocation costs. Abinary integer programming formulation of the objective function is given, and enumeration procedures(including a branch and bound method) are suggested for solving it to optimality.

More recently, Drezner and Wesolowsky [43] consider locating a facility in a growing city with pre-dictable population shifts (i.e., demands change over time but in a deterministic manner). Their objective isto ®nd a single facility location which minimizes the expected cost over the given horizon. The authors alsoexamine the possibility of relocating the facility several times during the horizon. In this case they seek notonly the locations for the facility but also the times at which changes in location should occur.

3.2. Dynamic multiple facility location models

Scott [81] examines dynamic extensions of the location-allocation problem in which multiple facilitiesare located one at a time at discrete, equally spaced time epochs. Once located, facilities must remain inoperation at the speci®ed site. A sub-optimal myopic approach to solving the problem is described, as is astandard dynamic programming approach.


Wesolowsky and Truscott [96] extend the analysis of multi-period node location-allocation problems,allowing facilities to be relocated in response to predicted changes in demand. An integer programmingmodel is presented, with a constraint restricting the number of location changes in each period. A dynamicprogramming formulation is also presented.

Tapiero [88] further extends the dynamic location-allocation problem to include possible facility ca-pacities and shipping costs. The optimal solution to this transportation-location-allocation problem willprovide the facility locations (as located in the Euclidean plane), allocations of demands to sources (withincapacity restrictions), and the quantities to be shipped between facilities and demand points. In this for-mulation, supply and demand values are known and are given in aggregate terms for the horizon. A dy-namic programming formulation is given, and optimality conditions are de®ned.

Dynamic multiple facility problem formulations are not limited to the location-allocation problem class.Sheppard [84] seeks to extend a wide range of basic facility location models so that they include both spatialand temporal aspects of real-world problems. The author presents a variety of models which determine notonly the location of multiple facilities, but also the size of the facilities and the timing of plant constructionor expansion. While Sheppard's models capture many aspects of the true location problems faced in in-dustry or the public sector, the majority of his formulations are nonlinear, integer, and dynamic, and thuscomputationally intractable.

Drezner [42] formulates the progressive P -median problem, which locates P facilities over a planninghorizon of T periods, without relocation. Inputs to the progressive P -median problem include time-de-pendent (known) demands and times at which the facilities are to be located. The objective is to ®nd thefacility locations which minimize total transport cost (or distance) over the horizon. Since the general formof the problem is nonlinear, a heuristic solution procedure for ®nding local minima is presented.

The computational complexity of most facility location problems has inspired a number of heuristicprocedures for determining near-optimal solutions. In an attempt to evaluate the relative merits of somesuch procedures, Erlenkotter [45] compares the performance of several heuristic solution approaches on asingle problem formulation. He examines a dynamic, ®xed charge, capacitated, cost minimization problemwith discrete time intervals, a special case of which is the static simple plant location problem. While limitedin scope, Erlenkotter's computational study suggests that combining heuristic approaches in a multiplephase solution process may prove most e�ective.

VanRoy and Erlenkotter [91] later study a dynamic uncapacitated facility location problem in whichgoods are shipped from facilities to meet known customer demands. New facilities are allowed to be openedand initially existing facilities are allowed to be closed over the time horizon. The objective is to minimizetotal discounted costs, including facility location and operating costs as well as production and distributioncosts for goods shipped. For this problem, a branch and bound solution procedure is proposed with lowerbounds obtained through solving LP-relaxations with a heuristic dual ascent method.

Driven by an application to freight carrier transportation terminals, Campbell [25] seeks simple strat-egies for locating and relocating facilities. Speci®cally, he examines the e�ectiveness of myopic approachesfor ®nding near-optimal location solutions. The author develops a general continuous distribution modelwhich includes linehaul transportation and economies of scale. The model considers trade-o�s betweentransportation, location, and relocation costs, with the objective of overall cost minimization. Campbelldevelops bounds on the optimal objective value using myopic strategies which ®rst ignore relocation costs(providing a lower bound) and then disallow relocations altogether (deriving an upper bound). Campbellshows that a myopic strategy with limited relocation is nearly optimal for locating terminals in both oneand two dimensions, unless relocation costs are high. Campbell thus suggests that extensive relocations maynot be necessary to obtain near-optimal distribution costs.

Gunawardane [52] moves away from private sector applications to consider location problems withinthe public sector. Speci®cally, he examines several covering problems in which public facilities are located(and possibly relocated) over a planning horizon. Both the set covering and the maximal covering problem


formulations are extended to account for a T period planning horizon. Decreasing weights wt (indexed overall planning periods t) are used as coe�cients to the location variables Xjt (indexed over potential facilitysites j and planning periods t) in the dynamic set covering objective function to encourage postponingfacility locations until they are required. Another model formulation discourages frequent changes in lo-cations by charging against each facility opening and closing. Computational results are highlighted; theauthor reports that most LP-relaxations return integer optimal solutions.

3.3. Alternative dynamic approaches

All of the dynamic deterministic problems discussed thus far seek an optimal or near-optimal solution toa single objective function. Schilling [77] considers an alternate approach to solving facility locationproblems, inspired by the public sector need to locate EMS facilities. Speci®cally, he considers a multi-objective maximal cover problem formulation and seeks a set of good solutions from which the decisionmaker can select one for implementation. The model formulation requires the following notation:

Decision variables:

Xjt �1 if a facility is operating at site j in period t;

0 otherwise;

8><>:Yit �

1 if node i is covered in period t;

0 otherwise:

�The mathematical model formulation is given by the following:

MaximizeX

i

hitYit 8t � 1; . . . ; T �22�

subject to:Xj2Nit

Xjt P Yit 8i; t; �23�Xj

Xjt � Pt 8t � 1; . . . ; T ; �24�

Xjt P Xj;tÿ1 8j; t � 2; . . . ; T ; �25�Xjt 2 f0; 1g 8j; t � 1; . . . ; T ; �26�Yit 2 f0; 1g 8i; t � 1; . . . ; T : �27�

This model combines T maximal covering problems, one for each period in the time horizon. The objectivefunction (22) is actually a vector of T individual period objectives which will not, in general, have a uniqueoptimum. The model assumes in Eq. (25) that once a facility is opened it remains open for all future pe-riods. The author discusses multi-objective approaches for generating a set of ``e�cient'' solutions for thedecision maker to choose between.

Inputs:dijt shortest distance or time from node i to node j in period tNit � fjjdijt6 Sg� set of sites which can cover node i in period thit � demand weight on node i in period tPt� number of facilities operational in period t


A multi-objective approach is also examined by Min [65], who considers expanding and relocating publiclibraries in the Columbus metropolitan area. The criteria considered in choosing library locations includecoverage of population, proximity to each community, proximity to facilities being closed, and accessibilityto transportation routes or parking lots. A discrete location model based on ``fuzzy'' goal programming isformulated as a mixed integer program. The model is solved multiple times, using inputs from the decisionmaker to obtain a number of potential siting con®gurations and to illustrate trade-o�s between objectives.

Another unique approach to locating facilities over time was proposed by Daskin, Hopp and Medina[40]. The authors acknowledge that the di�culty in solving dynamic facility location problems arises fromthe uncertainty surrounding future conditions. Even establishing an appropriate horizon length is a non-trivial problem which is ignored in most formulations. They argue that the best way to manage uncertaintyis to postpone decision making as long as possible, collecting information and improving forecasts as timeadvances. Since the ®rst period decisions are the only ones to be implemented immediately, the authorsclaim that the goal of dynamic location planning should not be to determine locations and/or relocationsfor the entire horizon, but to ®nd an optimal or near-optimal ®rst period solution for the problem over anin®nite horizon. Their approach ®nds an endogenously determined forecast horizon length, T �, and aninitial decision such that all horizons with length T P T � have an optimal or near-optimal policy whichbegins with the speci®ed initial decision.

4. Stochastic location problems

The dynamic models described in the previous section attempt to locate facilities over a speci®ed timehorizon in an optimal or near-optimal manner. While capturing more of the complexity inherent in real-world problem instances than static and deterministic formulations, these models assume that input pa-rameters are known values or that they vary deterministically over time. In this section, we will reviewresearch which addresses the stochastic nature of real-world systems.

Research on stochastic location problems can be broken down into two primary approaches, referred tohere as the probabilistic approach and the scenario planning approach. In both cases, any number of systemparameters might be taken as uncertain, including travel times, construction costs, demand locations, anddemand quantities. The objective is to determine robust facility locations which will perform well (ac-cording to the de®ned criteria) under a number of possible parameter realizations. Probabilistic modelsexplicitly consider the probability distributions of the modeled random variables, while scenario planningmodels consider a generated set of possible future variable values. (For a more general discussion on theadvantages and disadvantages of scenario planning versus stochastic programming, we recommend thepaper by Mulvey et al. [71].)

4.1. Probabilistic models

In this section we will examine models which capture the stochastic aspects of facility location throughexplicit consideration of the probability distributions associated with modeled random quantities. Someauthors incorporate these distributions into standard mathematical programs, while others use them withina queueing framework.

4.1.1. Standard formulationsIn 1961, Manne [63] published one of the earliest papers to consider stochastic problem inputs. In this

paper, he examines the problem of capacity expansion over an in®nite horizon, with the objective of se-lecting expansion sizes which minimize the sum of discounted installation costs. Manne models demand


probabilistically and allows backordering of unsatis®ed demands. The addition of probabilistic demandsdoes not greatly a�ect the model used, but the additional uncertainty does increase the desired level ofexcess capacity. A modi®ed discount rate is found to capture information about the magnitude of uncer-tainty. As demand variance increases, the e�ective discount rate decreases, and thus the optimal level ofexpansion increases. In the backorder case, Manne shows that optimal cost levels will decrease with in-creasing, low levels of variance.

Bean et al. [11] revisit the capacity expansion problem, with stochastically growing demand and anin®nite horizon. Relaxing a number of Manne's original assumptions, they allow for nonstationary demandprocesses which are either discrete or continuous and for general cost structures. No backorders are allowedand it is assumed that capacity is added only when existing capacity has been exhausted. A deterministicequivalent is found for demands which follow nonlinear Brownian motion or non-Markovian birth anddeath processes. The e�ect of demand uncertainty is again seen as a drop in the e�ective interest rate.

Stochastic problem inputs have been studied in a number of other problem classes as well. Carbone [26]considers locating public facilities on a network when demand values are unknown. The author refor-mulates a deterministic p-median problem as a chance-constrained program, incorporating uncertainty indemand. Assuming that demands have a multi-variate normal distribution, Carbone utilizes analyticalresults on multi-variate statistics in formulating a nonlinear deterministic equivalent to the chance-con-strained program. A computational procedure for solving the nonlinear deterministic problem is detailed.

Mirchandani and Odoni [67] further extend Hakimi's early results on network median problems toinclude random length arcs with known discrete probability distributions. The authors prove that Hakimi'sresult on the existence of an optimal solution which locates facilities only at the nodes of the network can begeneralized to stochastic networks. Speci®cally, they determine that at least one set of expected optimal k-medians exists on the nodes in a non-oriented stochastic network if the utility function for travel time isconvex and non-increasing. Another version of this result is given by Hurter and Martinich [55], whoconsider integrated production and location problems under uncertainty.

Mirchandani and Odoni's generalization of Hakimi's result should have greatly simpli®ed solutionsearches. At that time, however, no computational results or solution procedures were presented. Weaverand Church [93] later attempt to ®ll these computational gaps by outlining solution procedures for the P -median problem on a stochastic network where the travel time on any arc may be a discrete randomvariable. An integer linear program is formulated and a solution method involving Lagrangian relaxationand an exchange heuristic is examined.

Berman and Odoni [18] and Berman and LeBlanc [17] extend the analysis of Mirchandani and Odoni toincorporate the possibility of relocating one or more of the P facilities in reaction to changes in link traveltimes. Network states are de®ned such that each state di�ers from all others by a change in the travel timealong at least one network link. In both papers, a Markov transition matrix is assumed to govern tran-sitions between states. Berman and Odoni present a substitution-based heuristic for determining the op-timal strategy for locating/relocating one facility in this multi-state environment. When the number offacilities is greater than one, the relocation decision must also consider changes in the assignment of de-mands to facilities. Berman and LeBlanc address this complication, developing a heuristic for the multi-facility, multi-state problem. Assuming a given steady-state probability vector, their heuristic attempts tominimize the weighted sum of long-term expected travel time per unit time and the expected relocation costof all facilities per unit time. (Note that these models determine a static optimal strategy which dictates thebest facility locations associated with each state, thus we classify them as stochastic, as opposed to dy-namic.)

Mirchandani [66] further examines the P -median problem and the uncapacitated warehouse locationproblem when travel characteristics and supply and demand patterns are stochastic. Looking speci®cally atthe application of locating ®re-®ghting units on a transportation network, the author also considers thecase of service-congested environments, in which a facility may not be available to service a demand. With


assumptions regarding the distributions of demands, service times, and travel times, the system is modeledas a Markov process with states de®ned according to unit availability status and the number of demands inthe queue. Problem formulations and solution issues are discussed, including proofs of the applicability ofHakimi's result for a number of problem variants.

The issue of facility availability has been the focus of a number of papers in the literature. Daskin [33,34]extends deterministic maximal covering models used to site EMS vehicles to account for the probabilitythat vehicles may be busy when demands arrive. The result is a maximum expected covering problem. Thisproblem assumes that the probability of ®nding one vehicle busy is independent of the probability ofanother vehicle being busy. Daskin further assumes that the busy probabilities are the same for all vehicles.These assumptions allow the author to compute the incremental expected coverage that results from havingthe kth vehicle able to cover each node. Daskin also presents a single node substitution-based heuristic forsolving the maximum expected covering problem. The algorithm begins with the case in which all vehiclesare busy virtually all the time. Under these conditions, Daskin argues that all vehicles should be located atthe node that covers the most demand. His algorithm then uses single node substitutions to ®nd values ofthe system-wide busy probabilities at which locations should change.

Additional work on vehicle availability is reviewed by Daskin et al. [39]. All models presented explicitlyaccount for the possibility that vehicles may be busy when demands for them arise. Some models do so bylocating additional coverage without consideration of the likelihood that a certain number of vehicles willbe busy. Other models, like the maximum expected covering problem above, actually account for thedistribution of the number of busy vehicles.

ReVelle and Hogan [73] later develop two new models which capture the problem of vehicle availabilitywithin a location set covering context. In the maximum expected covering problem described above, prepresents the average fraction of time that a vehicle spends servicing demands. ReVelle and Hogan uselocalized estimates of this value to derive expressions for the probability that one or more servers within thecoverage distance is free to take a call for service from a given demand node. This probability is thenconstrained to be greater than or equal to a, a level of reliability that must be met for all nodes. Theirmodels attempt to balance the number of facilities located, the reliability of vehicle availability, and thecoverage level. The ®rst formulation is for the a-reliable P -centerproblem, which locates P facilities so as tominimize the maximum time within which service is available with a reliability. While this formulation takesa ®xed a and returns an optimal coverage level, the maximum reliability location problem takes a ®xedcoverage level S and locates P facilities which provide service within S time units and which maximize theminimum reliability of service.

Further work on locating EMS vehicles is presented in Ref. [37]. Here, Daskin and Haghani considercases in which multiple vehicles are simultaneously dispatched to an emergency scene. Recognizing theimportance of a rapid response to such demands, they develop a model to estimate the distribution of thearrival time of the ®rst vehicle at the scene. Travel times on each link are stochastic, and are assumed to benormally distributed (with variances proportional to the associated means). Analysis shows that decreasingthe expected common travel time between two paths (i.e., the travel time two responding vehicles spend oncommon links) will increase the probability of the ®rst vehicle reaching the scene within a speci®ed time.The result is shown to hold even if implementing such a decrease causes the mean travel time for somevehicles to increase slightly.

Daskin [35] reviews and presents additional research on deploying EMS vehicles on stochastic traveltime networks. A combined location, dispatching and routing model is presented, for which multiple,nonlinear objectives are de®ned. Solution issues are discussed for the large integer program which results.

Clearly the problem of locating emergency service vehicles has inspired a signi®cant amount of researchin stochastic location theory. In fact, researchers examining a wide range of applications have contributedmethods for solving stochastic location problems. Ghosh and Craig [49] discuss a method for multiple retailsite location by a retailer operating in a duopoly with ®xed market potential. The retailer's objective is to


maximize pro®t, and a nonlinear multiple store location model is developed. The market environmentrequires that the competitor relocate in response to decisions made by the retailer. A competitive equi-librium is thus sought through an iterative procedure which solves a multiple store location problem in-stance at each step. The computational intensity of this procedure leads to the development of a heuristicwhich utilizes the Tietz and Bart exchange approach [89].

Belardo et al. [12] examine the problem of locating response resources for maritime oil spills. If an oilspill occurs, the response equipment dispatched to the scene is responsible for containing the spill andremoving oil from the water. The objective behind these operations is to minimize the amount of oil thatwashes onto the shore, where clean-up is more di�cult and very expensive. Drawing from the literature onlocating ®re stations, the authors develop a partial set covering model for locating six types of responseequipment. The model considers spills of various oil types, occurring in a number of weather conditions,and incorporates assessments of relative probabilities of spills occurring in di�erent regions and of theenvironmental impact of various types of spills. A multi-objective formulation allows decision makers todiscriminate between coverage of likely spills and coverage of potentially high impact spills. The problem oflocating oil spill response resources on Long Island Sound is examined.

In yet another approach to facility location, Hanink [54] uses portfolio theory to solve a class of multi-plant location problems. Drawing from ®nancial economics, he casts such problems as geographical al-locations of assets. As portfolio models generally recommend diversi®cation of holdings to reduce portfoliorisk, the author considers large (multi-plant) ®rms as having a geographically diverse portfolio of plants. Abinary quadratic program is developed to maximize the ®rm's expected return, depending largely on therisk-aversion of the ®rm's management.

Gregg et al. [51] present a method for siting public libraries in the Queens borough of New York City.They attempt to model uncertain future demands for service through a stochastic programming approach.The model developed incorporates expected overage and underage cost curves as penalties in a nonlinearmathematical program. These penalties represent the expected cost of the realized supply being over orunder the realized demand (making use of the probability distribution of demand). The authors show howsuch a model was employed interactively using sensitivity analysis and exogenously speci®ed location al-ternatives so that decision makers could in¯uence solution values to account for unmodeled politicalfactors.

Also drawing from stochastic programming, Louveaux [61] introduces two-stage stochastic programswith recourse for solving simple plant location problems and P -median problems. In his models, the ®rststage decisions determine the location and size of facilities to be built, while the second stage speci®es theallocation of production resources to meet the most pro®table demands. The author considers uncertaintiesin demand, production and transportation costs, and selling prices. The relationship between the simpleplant and P -median problem is explored, but solution methods are not discussed in this paper.

4.1.2. Queueing modelsThe models and methodologies described in Section 4.1.1 incorporate a range of stochastic problem

parameters. In this section, we will see how the probability distributions associated with these parametershave been combined with results from queueing theory to examine additional aspects of facility location.

Larson's hypercube model [59] was the ®rst to embed queueing theory in facility location problems. Inthe original model, Larson examines problems relating to vehicle location and response district design foremergency service organizations. Considering interdistrict (as well as intradistrict) responses, probabilisticcall arrivals, and variable service times, Larson models the emergency service system as a multi-serverqueueing system with distinguishable servers. Speci®cally, the author assumes a Poisson call arrival processand exponential service times. Given a geographical description of the region and a dispatch criterion,Larson then develops an iterative method for generating the transition matrix for the associated continuoustime Markov process. The state space of the process is depicted as the vertices of an N-dimensional unit


hypercube in the positive orthant (where N is the number of servers), each vertex representing somecombination of service unit availabilities. This descriptive model is then used to generate a number ofperformance measures which characterize the system's behavior. As described below, Larson's hypercubemodel is embedded in a number of heuristic procedures for solving a range of queueing-based locationmodels.

Batta et al. [9] use Larson's hypercube results in addressing the following three assumptions of Daskin'smaximum expected covering model [34] (described above):1. servers operate independently,2. servers have equal busy probabilities (p), and3. server busy probabilities are invariant with respect to server locations.The authors recognize that server cooperation is a common practice, driven by the need for quick responsesto emergency calls. Using an elementary queueing system, they prove that such cooperation disquali®es the®rst (independence) assumption. Simultaneously relaxing all three assumptions, the authors embed ahypercube queueing model in a single node substitution heuristic procedure to determine a near-optimal setof server locations. To do this, the authors assume that the service system is in steady state and that theassumptions of the hypercube model are valid (e.g., they assume a ®rst come ®rst served (FCFS) queueingdiscipline, a Poisson call arrival process, and exponential service times). Computational results show somedisagreement between the expected coverage predicted by the hypercube procedure and that reported byDaskin's model. The two approaches are largely in agreement, however, on the locations selected. Theauthors also suggest adjustments to Daskin's model and heuristic solution procedure which allow for therelaxation of the independence assumption for server busy probabilities.

Berman et al. [15] further examine the problem of locating a vehicle in a congested network by explicitlyconsidering the arrival process of customer calls for service. The authors note that when the server is busyand customers are queued, the mean time spent in the queue may be much greater than the mean traveltime, and thus they consider queueing delay in formulating the problem objective. Two models are for-mulated as extensions to Hakimi's original P -median problem, both of which assume that demands ariseaccording to a homogeneous Poisson process. In the ®rst model, demands which ®nd the server busy arelost. The objective of this stochastic loss median problem is to minimize a weighted sum of mean travel timeand rejection costs. In the second model, demands which ®nd the server busy are entered into a queue whichis depleted according to a ®rst in, ®rst out (FIFO) manner. The objective in this stochastic queue mediancase is to minimize the sum of the mean in-queue delay and the mean travel time. Both systems are modeledas an M/G/1 queue, operating in steady state, with zero or in®nite queue capacity, respectively. Berman,Larson and Chiu show that the solution to the loss case reduces to the standard Hakimi median, while thein®nite capacity case has a nonlinear objective function which can lead to an optimal location at any pointof the network. The authors develop an exact, ®nite procedure for ®nding the optimal location and exploreproperties of the optimal location as a function of the demand rate.

Further analysis of the stochastic queue median as a function of the customer demand rate is presentedby Brandeau and Chiu [20] for the case of a planar region with rectilinear distances. Through convexityanalysis, the authors determine a necessary and su�cient ratio condition for ®nding the optimal location.The stochastic queue median trajectory, as a function of the systemwide demand rate (k), is characterizedand an algorithm for ®nding the optimal trajectory is detailed. Finally, the authors discuss the extension oftheir results to problems with stochastic travel times.

Berman [14] considers vehicle availability in a combined probabilistic and scenario planning model (seeSection 4.2 below) which captures uncertainty in both link travel times and service demands. This paperattempts to combine the work of Berman et al. [15] (capturing customer demand rates) with that of Mi-rchandani and Odoni [67] (concerning uncertain travel times), both of which have been detailed above. Theauthor attempts to locate a single service vehicle anywhere on a given network, assuming that servicedemands constitute a homogeneous Poisson process and that link lengths are discrete random variables. As


in Ref. [15], two cases are examined, alternately disallowing and allowing queueing of demands (resulting inthe stochastic expected loss median problem and the stochastic expected queue median problem, respectively).The objective in both models is to minimize the expected cost of response. The stochastic expected lossmedian is found to reduce to the expected median. A heuristic for the stochastic expected queue median isdeveloped and the behavior of the optimal location is considered as a function of the demand arrival rate.Results for the special case of a tree network are also presented.

E�orts to extend the stochastic median analysis to the location of P service units are complicated by theabsence of a closed form expression for the expected waiting time in queue for the M/G/P system. Bermanet al. [16] analyze this problem and propose two heuristics for ®nding the locations of P units. The ®rstheuristic takes an initial set of locations and uses the hypercube model to provide information to eachserver on the likelihood of being dispatched to calls for service from each demand point. This information isthen used in a 1-median problem to improve each server's location. The entire process is continued until nofurther improvement is found. The second heuristic described by the authors is similar to the ®rst, exceptthat a stochastic queue median problem is substituted for the 1-median problem. These heuristics areconceptually similar to the neighborhood search algorithm originally proposed by Maranzana [64], andboth versions are shown to perform well through computational testing. The ®rst heuristic requires lesscomputational e�ort than the second and the authors recommended it for almost all values of k.

Batta [8] examines the stochastic queue median problem with the added restriction that potential facilitylocations are limited to a ®nite set of points. Trying to ®nd the location of a single server which minimizesthe average server response time, the author develops an algorithm which solves for the optimal siteparametrically in k. Batta also presents a worst case analysis for the the stochastic queue median problem inwhich a facility can be located at any point in the network.

As mentioned in Section 2.3 above, the expected (or average) service level is not an appropriate objectivefor all applications. The P -center problem addresses situations in which service inequities are more im-portant than average performance. Brandeau and Chiu [22] examine how this model can be extended tocongested systems in their stochastic queue center problem. Their objective is to locate a facility whichminimizes the maximum expected response time, the total of expected time in queue and expected traveltime. Using analyses similar to Berman et al. [15], Brandeau and Chiu develop a ®nite step algorithm for®nding the optimal location on a general network. The special case of locating on a tree network is alsoconsidered, and extensions involving probabilistic travel times or demand distributions are formulated.

Brandeau and Chiu [21] attempt to unify the stochastic queue center problem, the stochastic queuemedian problem, and several other single-server facility location problems in a general class of models.Explicitly considering both queueing and travel delays, the authors model the system as an M/G/1 queue.The objective of the queueing location problem is to minimize response time to customers, using an Lp norm-based cost function to measure system performance. (Here the parameter p speci®es a power cost functionfor response time to calls.) The family of models is parameterized on both the customer call arrival rate (k)and the Lp cost parameter (p). Brandeau and Chiu establish convexity properties of the objective functionand discuss methods for ®nding the optimal facility location for cases when k and p are ®xed and when theyare unknown.

Queueing and network congestion are also factors in considering consumer choice and facility utiliza-tion. In the Ghosh and Craig model discussed above, customers select which facility to patronize based ontravel cost (or distance) and facility attributes such as cleanliness or size. This is an extension of much of thefacility location literature which typically assumes that customers patronize the closest facility. In all ofthese models, however, each customer's choice is considered to be independent of the actions of othercustomers. Brandeau and Chiu [23] recognize that this independence assumption is not necessarily a goodone for modeling practical situations. They highlight many instances in which a public facility's totalmarket share may a�ect customer choice, speci®cally considering the impact that crowds or congestionmight have on a customer's patronage. Brandeau and Chiu's model attempts to ®nd an equilibrium in


which facilities are located and customers select the facility that minimizes their total cost of (one way)travel and externalities. The example externality function used by the authors considers waiting time due toqueue formation at each facility. Demands are assumed to be generated according to a Poisson process andthe special case of a tree network with customers at nodes and two facilities is considered. An enumerativealgorithm for ®nding the optimal locations for this special case is detailed.

Whereas the above model seeks to ®nd a user equilibrium for public facilities, in Ref. [24], Brandeauand Chiu consider a competitive facility location problem. In this instance, the facility owners are re-sponsible for site selection, with the objective of maximizing equilibrium market share. Customers havethe choice of which facility to patronize, and these decisions continue to be a�ected by market exter-nalities as well as travel costs. Analysis is again focused on a special tree network with nodal demands,and externalities are captured through customer delay in queue. The authors characterize the problem inthis environment as a three stage sequential game in which the leader locates ®rst, then the followerlocates, and ®nally a customer choice game is played by the users, determining a customer choiceequilibrium.

4.2. Scenario planning models

Scenario planning is a method in which decision makers capture uncertainty by specifying a number ofpossible future states. The objective is to ®nd solutions which perform well under all scenarios. In someapplications, scenario planning replaces forecasting as a way to evaluate trends and potential changes in thebusiness environment [68]. Firms can thus develop strategic responses to a range of environmental changes,more adequately preparing themselves for the uncertain future. Under such circumstances, scenarios arequalitative descriptions of plausible future states, derived from the present state with consideration ofpotential major industry events. In other applications, scenario planning is used as a tool for formulatingand solving speci®c operational problems [70]. While scenarios here also depict a range of future states, theydo so through a quantitative characterization of the various values that problem input parameters mayrealize. As detailed below, the use of scenario planning for facility location follows the latter, morequantitative approach. Vanston et al. [92] discuss the use of scenario planning techniques and present a 12-step procedure for generating a set of appropriate scenarios. Additional works by Amara and Lipinski [2],Georgantzas and Acar [48], and van der Heijden [90] provide a more general overview of scenario planningtechniques.

A recent text by Kouvelis and Yu [56] discusses the use of a robustness approach to decision making inenvironments characterized by uncertain data values. Their approach develops decisions which hedgeagainst worst case scenarios (thus eliminating the need for assigning probabilities to parameter scenarios).Focusing on discrete optimization problems, Kouvelis and Yu develop a framework for ®nding robustsolutions and detail complexity results for a range of problem classes. Included in this volume is an analysisof the robust 1-median location problem on a tree. The authors examine a range of problem instances,beginning with those having node demands and edge lengths changing linearly with time and moving tomodels having uncertain node demands and edge lengths.

A number of additional researchers have used scenario planning techniques to solve a broad spectrum offacility location problems. The references detailed in this section illustrate the strategic nature of this ap-proach and the advantages of its use. Before discussing scenario planning research contributions, we brie¯youtline some of the core concepts of the approach.

As stated above, the goal of scenario planning is to specify a set of scenarios which represent the possiblerealizations of unknown problem parameters and to consider the range of scenarios in determining acompromise (robust) location solution. There are at least three approaches to incorporating scenarioplanning into location modeling.


1. optimizing the expected performance over all scenarios,2. optimizing the worst-case performance, and3. minimizing the expected or worst-case regret across all scenarios.In what follows, we focus on regret-based approaches.

The regret associated with a scenario is calculated by comparing the performance of the optimal lo-cations for the scenario (had planners known for certain that the scenario would be realized) with theperformance of the compromise locations when the scenario is realized. Using a regret-based objective thusallows us to evaluate robust solution alternatives with respect to the optimal solution obtained under datacertainty. Note that some papers use this measure in objectives which require the assessment of scenariorealization probabilities, while others implicitly assume that all scenarios are equally likely. The Kouvelisand Yu criteria of hedging against the worst case outcome is an example of the latter, in which the authorsseek to minimize the maximum regret. Another commonly used decision criterion which requires scenarioprobabilities is minimizing the expected regret.

To illustrate how such objectives are formulated, we will examine the P -median problem under thescenario planning approach. To do so, we introduce the following additional notation:

Decision variables:

Xj �1 if we locate at potential facility site j ;

0 if not;

8><>:Yijk �

1 if demand node i is assigned to facility j under scenario k;

0 if not:

8><>:The regret associated with the scenario k is thus given by Rk � Vk ÿ V̂k, where Vk is the value of the demandweighted total distance (i.e., the P -median objective value) under the compromise locations(Vk �

Pi

Pj hikdijkYijk). The expected regret problem can thus be formulated as follows:

MinimizeX

k

qkRk �28�

subject to:X

j

Xj � P ; �29�Xj

Yijk � 1 8i; k; �30�

Yijk ÿ Xj6 0 8i; j; k; �31�

Rk ÿX

i

Xj

hikdijkYijk ÿ V̂k

!� 0 8k; �32�

Xj 2 f0; 1g 8j; �33�Yijk 2 f0; 1g 8i; j; k: �34�

Inputs:k� index of possible scenarioshik � demand at node i under scenario kdijk � distance from node i to facility site j under scenario kV̂k � optimal P-median solution value for scenario k


The objective function (28) minimizes the expected regret, with regret de®ned in constraint (32). The re-maining constraints are the scenario planning equivalents to the standard P -median constraints detailed inSection 2.1. Note that the locations are common to all scenarios and must be determined before knowingwhich scenario is realized. The demand assignments, however, are scenario-speci®c. In essence, they are theresult of optimizing the assignments conditional on the chosen sites, but after we know which scenario isrealized. This formulation requires the decision maker to input probability values qk for each scenario,values which typically must be estimated. To avoid making such estimates, we can instead minimize themaximum regret across scenarios. This objective is more conservative, and is formulated with the con-straints as above, but with the following objective function:

Minimize maxkfRkg:

Recent work by Averbakh and Berman [6] extends the work of Kouvelis and Yu, and looks at the robust1-median location problem on a general network (as formulated above with P � 1). The authors present apolynomial algorithm for this problem and develop improved algorithms (in terms of computationalcomplexity) for the tree network problem.

Additional research by Averbakh and Berman considers algorithms for robust center problems. Firstfocusing on the weighted 1-center problem, the authors examine problem instances with uncertainty in bothdemand values and edge lengths [4]. Having previously shown [3] that this problem is strongly NP-hard ona general network, they present a polynomial algorithm for the 1-center problem on a tree. The algorithm'shigh order of complexity (O(n6)) illustrates the increased problem di�culty associated with uncertainty inedge lengths, even on tree networks. In a later paper [5], Averbakh and Berman study the P -center problemon a network with uncertain demand values. They seek the minimax regret solution when interval estimatesof the demand values are given. The authors present an algorithm which involves solving n� 1 regularweighted P -center problems (where n represents the number of nodes in the network). Complexity resultsfor both tree and general networks are detailed for the P � 1 case.

Ghosh and McLa�erty [50] use scenario planning concepts to make retail location decisions in an un-certain environment. In this problem, a retail chain seeks to locate stores in such a way that market share ismaximized. The scenarios generated describe possible future marketing environments. An exchange heu-ristic is used to identify non-inferior strategies which perform well under all scenarios. When the number ofnon-inferior strategies is small, the authors leave the ultimate location selection to the decision maker.When the number of strategies is too large for such subjective discrimination, they propose a regret-minimization method for selecting the optimal solution.

Reminiscent of the dynamic problems detailed in Section 3 above, Schilling [78] uses scenario planningto analyze the problem of locating a number of facilities over time. Individual scenarios are used to identifya set of good locational con®gurations, each of which can be seen as a contingency plan. Schilling proposesbuilding facilities which are common to all contingencies early in the horizon, leaving decisions whichdiscriminate between contingencies for a later time, at which point more information about the future willbe available. (Daskin et al. [40] show that following this strategy can, in some circumstances, lead to poorlocation decisions.)

Recent research in scenario planning has broadened its applicability to a wider range of problem classes.Serra et al. [83] extend the maximum capture problem to a setting with uncertain demands by generating aset of possible demand scenarios. In this model, two objectives are analyzed: maximizing the minimumdemand captured and minimizing the maximum regret. A solution procedure is developed which involves®nding an initial solution and then improving upon it with an exchange heuristic. The initial solution isfound by determining the optimal locations associated with each scenario independently (as if it were astatic problem) and then choosing the ``best'' starting point from among those, based on the objective beingused.


Serra and Marianov [82] utilize an equivalent solution method to site ®re stations in Barcelona, withuncertainty in both the demand at the network nodes and the travel times along the network arcs (whichvary depending on time of day or day of the week). Scenarios are used to capture di�erent demand patternsand/or travel times. Over these scenarios, facilities are sited with the objectives of minimizing the maximumaverage travel time and minimizing the maximum regret.

On a smaller scale, Carson and Batta [27] use a similar scenario planning approach to site a singleambulance on the Amherst campus of the State University of New York at Bu�alo. The authors use four(unequal duration) states to capture the movement of campus populations during the 24 hour day. Tominimize system-wide average response times, they formulate a model for determining the optimal am-bulance position under each state. Evaluating the resultant optimal strategy with historical data, the au-thors predict a 30% reduction in average response time. However, a test implementation of this strategyshowed only a 6% reduction. This di�erence was attributed to the short operating range of the campusambulance, an unmodeled factor which makes response times relatively insensitive to travel distances.

Taking a rather di�erent approach, Current et al. [32] examine uncertainty in the number of facilities tobe located over time. Speci®cally, they consider the problem of locating an initial number of facilities (p0)when the total or ®nal number of facilities to be located (pF ) is unknown. Two decision criteria are con-sidered within a P -median context: minimizing the expected opportunity loss and minimizing the maximumregret.

Daskin et al. [38] generalize the minimax regret objective in an e�ort to make location decisions morerealistic by making them less conservative. The authors recognize that the standard minimax regret ob-jective is sensitive to potentially ``bad'' scenarios, even if the likelihood of such a scenario evolving is verysmall. Thus, solutions for a minimax regret location problem can be driven by a single, unlikely scenario.Daskin, Hesse, and ReVelle develop the a-reliable P -minimax regret problem which takes a user-input re-liability level, a, which captures the risk aversion of the decision maker. The model endogenously selects asubset of scenarios (the reliability set) over which the minimax regret solution is found. The probability ofrealizing a scenario which is not in the reliability set must be at most 1ÿ a. Thus if a � 0:95, the decisionmaker will hedge against at least 95% of the possible future outcomes when selecting robust facility sites.Taking a � 1:0 forces all scenarios into the reliability set and is thus equivalent to solving the standardminimax regret problem. Using an 88-node problem with 9 scenarios, they show that scenarios may movein and out of the reliability set as the level of reliability increases. This suggests that identifying an extremescenario in a network planning context is more complex than identifying an extreme scenario in problemswithout network interactions.

5. Conclusions

In this review, we have attempted to provide an overview of facility location literature dedicated tocapturing the complex time and uncertainty characteristics of most real-world problem instances. Advancesin integer programming, dynamic programming, stochastic programming, and scenario planning tech-niques have clearly increased our capacity for analyzing, modeling, and solving important strategic facilitylocation problems.

We expect future research to continue in this direction. Speci®cally, we look for improved heuristics tosupport the solution of larger, more complex and more realistic problem instances. The increased use ofscenario planning techniques will drive such solution advances, as scenario-based models grow rapidly withthe number of scenarios generated. Also, we look for the development of tractable models which considerboth the stochastic and dynamic aspects of facility location. Recent developments in multiple stage sto-chastic programming with recourse might be employed to capture the complexities of such locationproblems.


Acknowledgements

This work was supported by NSF grant DMI-9634750 and the GE Faculty of the Future fellowshipprogram. This support is gratefully acknowledged.

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