Strategic facility location: A review - Zoomin

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

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

*

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

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

or a city planner selecting locations for ®re stations, strategic planners are often challenged by dicult spatial resource allocation decisions. As populations shift, market trends evolve, and other environmental factors change, the need to relocate, expand, and adapt facilities ensures the evolution of new planning challenges. The development and acquisition of a new facility is typically a costly, time-sensitive project. Before a facility can be purchased or constructed, good locations must be identi®ed, appropriate facility capacity speci®cations must be determined, and large amounts of capital must be allocated. While the objectives driving a facility location decision depend on the ®rm or government agency, the high costs associated with this process make almost any location project a long-term investment. Thus, facilities which are located today are expected to remain in operation for an extended time. Environmental changes during the facility's lifetime can drastically alter the appeal of a particular site, turning today's optimal location into tomorrow's investment blunder. Determining the best locations for new facilities is thus an important strategic challenge. A vast literature has developed out of the broadly based interest in meeting this challenge. Operations research practitioners have developed a number of mathematical programming models to represent a wide range of location problems. Several dierent objective functions have been formulated to make such models amenable to numerous applications. Unfortunately, the resulting models can be extremely dicult to solve to optimality (most problems are classi®ed as NP-hard); many of the problems require integer programming formulations. The computational hurdle posed by complex facility location formulations has, until recently, limited most 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 decision values. While such problems can provide planners with insight about general location selection, they are not able 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 problem of input data uncertainty. Speci®cally, such research attempts to quantify the eect of a change in parameter values on the optimal objective function value (for example, see Labbe et al. [58]). While such results help in evaluating the robustness of a solution after a model is solved, they do nothing to incorporate uncertainty into models proactively. Both stochastic programming and scenario planning approaches move away from reactive analyses of solution sensitivity toward models which formalize the complexity and uncertainty inherent in real-world problem instances. Similarly, dynamic formulations transform snapshot models of one time decisions into extended horizon models which capture the temporal aspects of real-world problems. In this review, we will see how these proactive approaches have been applied to problems of facility location. More speci®cally, in the following sections we will detail some of the literature which addresses the strategic nature of facility location problems by modeling either stochastic or dynamic problem characteristics. Our goal in writing this paper is to provide an overview of facility location research which, through the consideration of time and uncertainty, has helped to move us toward solving more realistic problem instances. Due to space limitations, we have chosen to focus on the qualitative contributions of the research cited. For a more detailed discussion of stochastic modeling issues, we recommend the recent survey by Louveaux [62]. Those seeking a more general overview of facility location research can refer to one of the many 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 deterministic facility location research as a background for the review. Section 3 highlights contributions in dynamic model formulations which focus on the timing issues involved in locating a facility over an extended horizon. Section 4 then details research which incorporates stochasticity, capturing the uncertainty in forecasting problem input parameters. Finally, we conclude with a discussion of future research directions.


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2. Static and deterministic location problems The study of location theory formally began in 1909 when Alfred Weber considered how to position a single warehouse so as to minimize the total distance between it and several customers [94]. Following this initial investigation, location theory was driven by a few applications which inspired researchers from a range of ®elds. Location theory gained renewed interest in 1964 with a publication by Hakimi [53], who sought 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 as to minimize the total distance between customers and their closest facility or to minimize the maximum such distance. Since the mid-1960s, the study of location theory has ¯ourished. The most basic facility location problem 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 solution will 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 locating obnoxious facilities, have examined multi-objective extensions of these basic models. (For a more detailed review of multi-objective facility location models, see Ref. [31].) In this section, some of the fundamental static and deterministic location problems will be reviewed. Our presentation will be structured around the dierent objective functions required by common applications, and will include a discussion of signi®cant research relating to each problem class.

2.1. Median problems As noted by Church and ReVelle [30], one important way to measure the eectiveness of a facility location is by determining the average distance traveled by those who visit it. (Note that throughout this paper, travel time and travel distance will be used interchangeably to represent the ``cost'' of traveling from one location to another.) As average travel distance increases, facility accessibility decreases, and thus the location's eectiveness decreases. This relationship holds for facilities such as libraries, schools, and emergency service centers, to which proximity is desirable. (To some extent, `ùndesirable'' facilities such as land®lls or nuclear power plants exhibit increases in location eectiveness in response to an increase in average travel distance; models concerning such facilities are discussed in Section 2.4 below.) An equivalent way to measure location eectiveness when demands are not sensitive to the level of service is to weight the distance between demand nodes and facilities by the associated demand quantity and calculate the total weighted travel distance between demands and facilities [75]. The P-median problem (introduced by Hakimi [53]) uses this measure of eectiveness, and is stated as follows: Find the location of P facilities so as to minimize the total demand-weighted travel distance between demands and facilities. To formulate this problem mathematically, the following notation is necessary: Inputs: i index of demand node j index of potential facility site hi demand at node i dij distance between demand node i and potential facility site j P number of facilities to be located

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Decision variables: 8 > < 1 if we locate at potential facility site j; Xj > : 0 if not:

Yij

8 > :0

if not:

Using these de®nitions, the P -median problem can be written as the following integer linear program: XX hi dij Yij 1 Minimize i

subject to: X Yij 1

j

X Xj P ;

2

j

8i;

3

j

Yij ÿ Xj 6 0;

8i; j;

Xj 2 f0; 1g 8j; Yij 2 f0; 1g 8i; j:

4 5 6

The objective (1), as mentioned above, is to minimize the total demand-weighted distance between customers and facilities. Constraint (2) requires that exactly P facilities be located. Constraint (3) ensures that every demand is assigned to some facility site, while constraint (4) allows assignment only to sites at which facilities have been located. Constraints (5) and (6) are binary requirements for the problem variables. Since demands will naturally be assigned entirely to the nearest facility in this uncapacitated problem (assuming hi dij 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 sites represent the nodes of a network. While one might imagine locating a facility at any point along an edge of the network, Hakimi [53] proves that for any number of facilities P , there is at least one optimal solution to the P -median problem which locates only at network nodes. Thus, the simpli®ed formulation includes only nodes 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 in the presence of competing ®rms. The objective in this retail environment is to locate facilities to maximize the number of new customers captured or to maximize the retailer's added market share. For this maximum capture problem formulation, the author assumes that all ®rms in the area supply the same product and that customers patronize the nearest ®rm. This modi®cation illustrates how the P -median problem can be applied in a strategic decision making context. When applied to a general network, the P -median problem can be dicult to solve to optimality (this class of problems is NP-complete). Limiting potential facility locations to network nodes, however, reduces the number of possible location con®gurations to N N! P !N ÿ P ! P where N represents the number of nodes in the network. Thus, for a ®xed value of P , the P -median problem can be solved in polynomial time. Nevertheless, a total enumeration approach would be computationally


427

prohibitive for reasonable values of N (hundreds to thousands of nodes) and P (tens of locations sited). For variable P , the problem is NP-hard (see Garey and Johnson [47]). Such complexity issues have led to the development 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 moderately sized problems in a reasonable time, several ecient heuristics have also been developed for solving median problems. (See Ref. [36] for an overview of heuristic methods, Refs. [75,89] for more detail on speci®c solution methods.) 2.2. Covering problems The P -median problem described above can be used to locate a wide range of public and private facilities. For some facilities, however, selecting locations which minimize the average distance traveled may 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 `àcceptable'' travel distance or time. Such facilities will thus require a dierent measure of location eciency. To locate such facilities, the key issue is ``coverage''. A demand is said to be covered if it can be served within a speci®ed time. The literature on covering problems is divided into two major segments, that in which coverage is required and that in which it is optimized. Two covering problems which illustrate the distinction are the location set covering problem and the maximal covering problem. We will introduce both problem classes and discuss their relationship to the P -median problem. For a more complete review of covering problems, see Refs. [79,97]. In the set covering problem, the objective is to minimize the cost of facility location such that a speci®ed level of coverage is obtained. The mathematical formulation of this problem requires the following additional notation: Inputs: cj ®xed cost of siting a facility at node j S maximum acceptable service distance (or time) Ni set of facility sites j within acceptable distance of node i i:e:; Ni fjjdij 6 Sg The set covering problem can thus be represented by the following integer program: X cj X j Minimize

7

j

subject to:

X

Xj P 1 8i;

8

j2Ni

Xj 2 f0; 1g

8j:

9

The objective function (7) minimizes the cost of facility location. In many cases, the costs cj are assumed to be equal for all potential facility sites j, implying an objective equivalent to minimizing the number of facilities located. Constraint (8) requires that all demands i have at least one facility located within the acceptable 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 the speci®ed distance, regardless of cost. If the coverage distance S is small, relative to the spacing of demand nodes, the coverage restriction can lead to a large number of facilities being located. Additionally,

428


if an outlying node has a small demand, the cost/bene®t ratio of covering that demand can be extremely high. As stated, the set covering problem allows us to examine how many facilities are needed to guarantee a certain level of coverage to all customers. In many practical applications, decision makers ®nd that their allocated resources are not sucient to build the facilities dictated by the desired level of coverage. (The goal 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 possible the 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 the acceptable service distance S by locating a ®xed number of facilities. The formulation of this problem requires 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 the maximal covering problem: X hi Zi 10 Maximize i

subject to: Zi 6 X Xj 6 P ;

X

Xj

8i;

11

j2Ni

12

j

Xj 2 f0; 1g 8j; Zi 2 f0; 1g 8i:

13 14

The objective (10) is to maximize the amount of demand covered. Constraint (11) determines which demand 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 some j 2 Ni ). If no such facility is located, the right hand side of constraint (11) will be zero, thus forcing Zi to zero. Constraint (12) limits the number of facilities to be located, to account for limited resources. Constraints (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 of potential facility sites. Typically the set of potential sites consists of some (if not all) of the demand nodes of the underlying network. Research extensions to these models have shown that even if facilities are allowed anywhere on the network, the problem can be reduced to one with ®nite choices for facility location (see Ref. [28]). The number of potential sites required to ensure optimality is generally much larger than the number 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 the size of the demand present) so that the objective is simply to maximize the number of demand nodes covered [97]. Another common variant acknowledges that those demands not covered within the desired service distance S should be covered by a less stringent distance standard, T (T > S). This maximal covering with mandatory closeness problem [29] insures that no demand is beyond T units from its nearest facility, while making as many demands as possible within S. All covering models discussed to this point implicitly assume that if a demand is covered by a facility then that facility will be available to serve the demand. In Ref. [41], Daskin and Stern examine siting EMS


429

vehicles to satisfy a speci®ed service requirement. For such an application, the availability assumption is problematic, as EMS vehicles already responding to a call for service will not be available to answer additional demands. Applications where facilities experience busy or inoperative periods have inspired a set of models [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 and Stern establishes a hierarchical objective function which ®rst minimizes the number of vehicles needed to satisfy the service requirement and then locates those vehicles to maximize the multiple coverage of demand nodes. Batta and Mannur [10] also examine models for determining the deployment of multiple EMS vehicles in environments where high demand rates cause frequent unit busy periods. The authors recognize that demands which require a larger response team are typically more critical, and thus should have a tighter coverage level. They formulate generalized deterministic set covering and maximal covering models which incorporate multiple response units and demand-dependent coverage requirements. Solution strategies for each problem class are discussed, including branch and bound algorithms applied to binary representations of reduced problem formulations. In Refs. [30,97], the relationship between the P -median (or central facilities location) and maximal covering problems is examined. The authors show that through a transformation of distances the maximal covering 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: dij0

0

if dij 6 S;

1

if dij > S:

Solving the P -median problem with modi®ed distances dij0 minimizes the amount of demand not served within coverage distance S. It can be shown that this is equivalent to maximizing the amount of demand served within S, and thus the transformed version of the P -median problem is exactly a maximal covering problem. Daskin [36] uses this transformation to develop a multi-objective model that trades o minimizing the 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 NPcomplete for general networks. 2.3. Center problems The set covering problem described above determines the minimum number of facilities needed to cover all demands using an exogenously speci®ed coverage distance. The potential infeasibility of such an approach in many practical contexts led us to examine the maximal coverage problem. As described, this formulation considers the resources available (in terms of the number of facilities we are able to locate) and determines the maximum demand coverage possible. Another problem class which avoids the set covering problem's potential infeasibility is the class of Pcenter problems. In such problems, we require coverage of all demands, but we seek to locate a given number of facilities in such a way that minimizes coverage distance. Rather than taking an input coverage distance S, this model determines endogenously the minimal coverage distance associated with locating P facilities. The P -center problem is also known as the minimax problem, as we seek to minimize the maximum distance between any demand and its nearest facility. If facility locations are restricted to the nodes of the network, the problem is a vertex center problem. Center problems which allow facilities to be located anywhere on the network are absolute center problems. As in the set covering problem, Hakimi's [53] result

430


does not generally hold; the solution to the absolute center problem is often better (i.e., has a lower associated 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 X Xj P ; subject to:

X Yij 1

15 16

j

8i;

17

j

Yij ÿ Xj 6 0 8i; j; X dij Yij 8i; DP

18 19

j

Xj 2 f0; 1g

8j;

Yij 2 f0; 1g 8i; j:

20 21

The objective function (15) is simply to minimize the maximum distance between any demand node and its nearest 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 decision variables Yij are allowed to be fractional, one demand node might be served by multiple facilities. Since the facilities in this simple case are uncapacitated, the solution will assign each demand node to the closest open facility. Thus, any solution which assigns a demand to more than one facility has an alternate optimum in which all Yij are integral. If the input value of P is ®xed, both the vertex center and absolute center problems can be solved in polynomial time. For the vertex center problem, we can ®nd a polynomial algorithm for evaluating all possible locations of the P facilities. The absolute center problem can be reformulated with an augmented network so that its solution will locate on a subset of the original nodes and augmented points. This process can also be completed in polynomial time. (See Ref. [36] for a more detailed discussion on center problem solution algorithms.) If the value of P is variable, however, both types of the P -center problem are NPcomplete. 2.4. Additional problem formulations The P -median, covering, and P -center problems discussed above provide a strong foundation for much of the location theory research done to date. In this ®nal section on static and deterministic models, we will brie¯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 for operating costs once a facility is located. Although we acknowledge that limited resources might dictate the number of facilities sited, in only one model (set covering) did we explicitly consider location costs. The set of ®xed charge facility location problems includes problem instances which have a ®xed charge associated with locating at each potential facility site.


431

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 ®xed cost to the P -median objective function and removing the constraint that dictates the number of facilities to be located. The result is a problem which determines endogenously the number of facilities to locate and sites them so as to minimize total (construction plus travel) costs. The close relationship between the two 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/or construction costs. Similar alterations can extend basic models to incorporate facility capacities. In such models, capacities are input as limits to the number of demands each facility can serve. By adding a set of constraints to the problem formulation, we require that the sum of the demands assigned to each facility not exceed the input capacity. Sankaran and Raghavan [76] extend the classical capacitated ®xed charge facility 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 locating warehouses. Any ®rm deciding where to site a new warehouse must also consider how to best ship products between its facilities and its customers. The set of location-allocation problems builds upon a basic location problem formulation (such as those presented above) to simultaneously locate facilities and dictate ¯ows between facilities and demands. These problems (as reviewed by Scott [80]) combine a standard transportation problem for allocating ¯ow between facilities with a location problem (usually a P -median problem or a ®xed charge problem) for siting the facilities. Just as warehouse applications require us to consider issues of both location and allocation, practical applications often introduce more involved objectives than the simple minimization of cost or maximization of coverage. A class of multi-objective location models have been developed to re¯ect the complexity inherent in many location problem applications. The hierarchical set covering model discussed above is an example 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 factors to be considered in locating new facilities. Finally, note that the models and applications presented thus far focus on locating facilities to make them accessible to customers. Alternatively, several important real-world applications deal with locating facilities 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 detailed above. These applications have, in fact, spawned a special area of research for locating `òbnoxious'' or ``noxious'' facilities. Problems which address these situations include the antimedian problem, which locates 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 such problems are useful in formulating undesirable facility location problems, the political rami®cations involved in locating such facilities often force decision makers to use multi-objective models. A more detailed review of such problems can be found in works by Brandeau and Chiu [19], Daskin [36], Erkut and Neuman [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, their applications and extensions. As noted, many of these problems can be extremely dicult to solve. Thus, it is not surprising that so much work has focused on static and deterministic problem formulations. While such

432


formulations are reasonable research topics, they do not capture many of the characteristics of real-world location problems. The strategic nature of facility location problems requires that any reasonable model consider some aspect of future uncertainty. Since the investment required by locating or relocating facilities is usually large, facilities are expected to remain operable for an extended time period. Thus, the problem of facility location truly involves an extended planning horizon. Decision makers must not only select robust locations which will eectively serve changing demands over time, but must also consider the timing of facility expansions and relocations over the long term. In the next two sections, we will present research which deals explicitly with the uncertainties inherent in facility location. For organizational purposes, a distinction is made between uncertainty related to planning for future conditions and uncertainty due to limited knowledge of model input parameters. We will ®rst address the former, looking at dynamic deterministic facility location models. Section 4 will then examine static 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 was published 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 dynamic problem. For each period in the speci®ed horizon, he solves for the optimal warehouse location, establishing a set of potential ``good'' location sites. Dynamic programming is then used to determine the best schedule for opening a subset of these sites as an `òptimal'' location and relocation strategy for the planning period. This approach was later found to be sub-optimal by Sweeney and Tatham [85] who improve on Ballou's solutions 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' decomposition. The number of solutions (Rt ) varies by period and is found through bounding the overall optimal solution value. The expanded set of potential location sites for each period is then used in a dynamic program to determine an optimal location and relocation strategy. Note that both of these papers allow for frequent facility relocation, but that neither considers construction time or cost in the objective function. Wesolowsky [95] examines another, unconstrained, version of the single facility location problem over a ®nite planning horizon with explicit facility relocation costs. A binary 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 predictable population shifts (i.e., demands change over time but in a deterministic manner). Their objective is to ®nd a single facility location which minimizes the expected cost over the given horizon. The authors also examine the possibility of relocating the facility several times during the horizon. In this case they seek not only 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 facilities are located one at a time at discrete, equally spaced time epochs. Once located, facilities must remain in operation at the speci®ed site. A sub-optimal myopic approach to solving the problem is described, as is a standard dynamic programming approach.


433

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 programming model is presented, with a constraint restricting the number of location changes in each period. A dynamic programming formulation is also presented. Tapiero [88] further extends the dynamic location-allocation problem to include possible facility capacities and shipping costs. The optimal solution to this transportation-location-allocation problem will provide the facility locations (as located in the Euclidean plane), allocations of demands to sources (within capacity restrictions), and the quantities to be shipped between facilities and demand points. In this formulation, supply and demand values are known and are given in aggregate terms for the horizon. A dynamic 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 spatial and temporal aspects of real-world problems. The author presents a variety of models which determine not only the location of multiple facilities, but also the size of the facilities and the timing of plant construction or expansion. While Sheppard's models capture many aspects of the true location problems faced in industry or the public sector, the majority of his formulations are nonlinear, integer, and dynamic, and thus computationally intractable. Drezner [42] formulates the progressive P -median problem, which locates P facilities over a planning horizon of T periods, without relocation. Inputs to the progressive P -median problem include time-dependent (known) demands and times at which the facilities are to be located. The objective is to ®nd the facility locations which minimize total transport cost (or distance) over the horizon. Since the general form of 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 heuristic procedures for determining near-optimal solutions. In an attempt to evaluate the relative merits of some such procedures, Erlenkotter [45] compares the performance of several heuristic solution approaches on a single problem formulation. He examines a dynamic, ®xed charge, capacitated, cost minimization problem with discrete time intervals, a special case of which is the static simple plant location problem. While limited in scope, Erlenkotter's computational study suggests that combining heuristic approaches in a multiple phase solution process may prove most eective. VanRoy and Erlenkotter [91] later study a dynamic uncapacitated facility location problem in which goods are shipped from facilities to meet known customer demands. New facilities are allowed to be opened and initially existing facilities are allowed to be closed over the time horizon. The objective is to minimize total discounted costs, including facility location and operating costs as well as production and distribution costs for goods shipped. For this problem, a branch and bound solution procedure is proposed with lower bounds obtained through solving LP-relaxations with a heuristic dual ascent method. Driven by an application to freight carrier transportation terminals, Campbell [25] seeks simple strategies for locating and relocating facilities. Speci®cally, he examines the eectiveness of myopic approaches for ®nding near-optimal location solutions. The author develops a general continuous distribution model which includes linehaul transportation and economies of scale. The model considers trade-os between transportation, location, and relocation costs, with the objective of overall cost minimization. Campbell develops 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). Campbell shows that a myopic strategy with limited relocation is nearly optimal for locating terminals in both one and two dimensions, unless relocation costs are high. Campbell thus suggests that extensive relocations may not be necessary to obtain near-optimal distribution costs. Gunawardane [52] moves away from private sector applications to consider location problems within the 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

434


formulations are extended to account for a T period planning horizon. Decreasing weights wt (indexed over all planning periods t) are used as coecients to the location variables Xjt (indexed over potential facility sites j and planning periods t) in the dynamic set covering objective function to encourage postponing facility locations until they are required. Another model formulation discourages frequent changes in locations by charging against each facility opening and closing. Computational results are highlighted; the author 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 to a single objective function. Schilling [77] considers an alternate approach to solving facility location problems, inspired by the public sector need to locate EMS facilities. Speci®cally, he considers a multiobjective maximal cover problem formulation and seeks a set of good solutions from which the decision maker can select one for implementation. The model formulation requires the following notation: Inputs: dijt shortest distance or time from node i to node j in period t Nit fjjdijt 6 Sg set of sites which can cover node i in period t hit demand weight on node i in period t Pt number of facilities operational in period t Decision variables: 8 > < 1 if a facility is operating at site j in period t; Xjt > : 0 otherwise; Yit

1

if node i is covered in period t;

0

otherwise:

The mathematical model formulation is given by the following: X hit Yit 8t 1; . . . ; T Maximize

22

i

subject to: X Xjt Pt

X

Xjt P Yit

8i; t;

23

j2Nit

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

j

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


435

A multi-objective approach is also examined by Min [65], who considers expanding and relocating public libraries in the Columbus metropolitan area. The criteria considered in choosing library locations include coverage of population, proximity to each community, proximity to facilities being closed, and accessibility to transportation routes or parking lots. A discrete location model based on ``fuzzy'' goal programming is formulated as a mixed integer program. The model is solved multiple times, using inputs from the decision maker to obtain a number of potential siting con®gurations and to illustrate trade-os between objectives. Another unique approach to locating facilities over time was proposed by Daskin, Hopp and Medina [40]. The authors acknowledge that the diculty in solving dynamic facility location problems arises from the uncertainty surrounding future conditions. Even establishing an appropriate horizon length is a nontrivial problem which is ignored in most formulations. They argue that the best way to manage uncertainty is to postpone decision making as long as possible, collecting information and improving forecasts as time advances. Since the ®rst period decisions are the only ones to be implemented immediately, the authors claim that the goal of dynamic location planning should not be to determine locations and/or relocations for the entire horizon, but to ®nd an optimal or near-optimal ®rst period solution for the problem over an in®nite horizon. Their approach ®nds an endogenously determined forecast horizon length, T , and an initial decision such that all horizons with length T P T have an optimal or near-optimal policy which begins 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 time horizon in an optimal or near-optimal manner. While capturing more of the complexity inherent in realworld problem instances than static and deterministic formulations, these models assume that input parameters are known values or that they vary deterministically over time. In this section, we will review research which addresses the stochastic nature of real-world systems. Research on stochastic location problems can be broken down into two primary approaches, referred to here as the probabilistic approach and the scenario planning approach. In both cases, any number of system parameters might be taken as uncertain, including travel times, construction costs, demand locations, and demand quantities. The objective is to determine robust facility locations which will perform well (according to the de®ned criteria) under a number of possible parameter realizations. Probabilistic models explicitly consider the probability distributions of the modeled random variables, while scenario planning models consider a generated set of possible future variable values. (For a more general discussion on the advantages and disadvantages of scenario planning versus stochastic programming, we recommend the paper by Mulvey et al. [71].) 4.1. Probabilistic models In this section we will examine models which capture the stochastic aspects of facility location through explicit consideration of the probability distributions associated with modeled random quantities. Some authors incorporate these distributions into standard mathematical programs, while others use them within a queueing framework. 4.1.1. Standard formulations In 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 selecting expansion sizes which minimize the sum of discounted installation costs. Manne models demand

436


probabilistically and allows backordering of unsatis®ed demands. The addition of probabilistic demands does not greatly aect the model used, but the additional uncertainty does increase the desired level of excess capacity. A modi®ed discount rate is found to capture information about the magnitude of uncertainty. As demand variance increases, the eective discount rate decreases, and thus the optimal level of expansion increases. In the backorder case, Manne shows that optimal cost levels will decrease with increasing, low levels of variance. Bean et al. [11] revisit the capacity expansion problem, with stochastically growing demand and an in®nite horizon. Relaxing a number of Manne's original assumptions, they allow for nonstationary demand processes which are either discrete or continuous and for general cost structures. No backorders are allowed and it is assumed that capacity is added only when existing capacity has been exhausted. A deterministic equivalent is found for demands which follow nonlinear Brownian motion or non-Markovian birth and death processes. The eect of demand uncertainty is again seen as a drop in the eective 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 reformulates a deterministic p-median problem as a chance-constrained program, incorporating uncertainty in demand. Assuming that demands have a multi-variate normal distribution, Carbone utilizes analytical results on multi-variate statistics in formulating a nonlinear deterministic equivalent to the chance-constrained 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 to include random length arcs with known discrete probability distributions. The authors prove that Hakimi's result on the existence of an optimal solution which locates facilities only at the nodes of the network can be generalized to stochastic networks. Speci®cally, they determine that at least one set of expected optimal kmedians exists on the nodes in a non-oriented stochastic network if the utility function for travel time is convex and non-increasing. Another version of this result is given by Hurter and Martinich [55], who consider integrated production and location problems under uncertainty. Mirchandani and Odoni's generalization of Hakimi's result should have greatly simpli®ed solution searches. At that time, however, no computational results or solution procedures were presented. Weaver and 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 random variable. An integer linear program is formulated and a solution method involving Lagrangian relaxation and an exchange heuristic is examined. Berman and Odoni [18] and Berman and LeBlanc [17] extend the analysis of Mirchandani and Odoni to incorporate the possibility of relocating one or more of the P facilities in reaction to changes in link travel times. Network states are de®ned such that each state diers from all others by a change in the travel time along at least one network link. In both papers, a Markov transition matrix is assumed to govern transitions between states. Berman and Odoni present a substitution-based heuristic for determining the optimal strategy for locating/relocating one facility in this multi-state environment. When the number of facilities is greater than one, the relocation decision must also consider changes in the assignment of demands to facilities. Berman and LeBlanc address this complication, developing a heuristic for the multifacility, multi-state problem. Assuming a given steady-state probability vector, their heuristic attempts to minimize the weighted sum of long-term expected travel time per unit time and the expected relocation cost of all facilities per unit time. (Note that these models determine a static optimal strategy which dictates the best facility locations associated with each state, thus we classify them as stochastic, as opposed to dynamic.) Mirchandani [66] further examines the P -median problem and the uncapacitated warehouse location problem when travel characteristics and supply and demand patterns are stochastic. Looking speci®cally at the application of locating ®re-®ghting units on a transportation network, the author also considers the case of service-congested environments, in which a facility may not be available to service a demand. With


437

assumptions regarding the distributions of demands, service times, and travel times, the system is modeled as a Markov process with states de®ned according to unit availability status and the number of demands in the queue. Problem formulations and solution issues are discussed, including proofs of the applicability of Hakimi'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 probability that vehicles may be busy when demands arrive. The result is a maximum expected covering problem. This problem assumes that the probability of ®nding one vehicle busy is independent of the probability of another 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 having the kth vehicle able to cover each node. Daskin also presents a single node substitution-based heuristic for solving the maximum expected covering problem. The algorithm begins with the case in which all vehicles are busy virtually all the time. Under these conditions, Daskin argues that all vehicles should be located at the node that covers the most demand. His algorithm then uses single node substitutions to ®nd values of the system-wide busy probabilities at which locations should change. Additional work on vehicle availability is reviewed by Daskin et al. [39]. All models presented explicitly account for the possibility that vehicles may be busy when demands for them arise. Some models do so by locating additional coverage without consideration of the likelihood that a certain number of vehicles will be busy. Other models, like the maximum expected covering problem above, actually account for the distribution of the number of busy vehicles. ReVelle and Hogan [73] later develop two new models which capture the problem of vehicle availability within a location set covering context. In the maximum expected covering problem described above, p represents the average fraction of time that a vehicle spends servicing demands. ReVelle and Hogan use localized estimates of this value to derive expressions for the probability that one or more servers within the coverage distance is free to take a call for service from a given demand node. This probability is then constrained to be greater than or equal to a, a level of reliability that must be met for all nodes. Their models attempt to balance the number of facilities located, the reliability of vehicle availability, and the coverage level. The ®rst formulation is for the a-reliable P -centerproblem, which locates P facilities so as to minimize the maximum time within which service is available with a reliability. While this formulation takes a ®xed a and returns an optimal coverage level, the maximum reliability location problem takes a ®xed coverage level S and locates P facilities which provide service within S time units and which maximize the minimum reliability of service. Further work on locating EMS vehicles is presented in Ref. [37]. Here, Daskin and Haghani consider cases in which multiple vehicles are simultaneously dispatched to an emergency scene. Recognizing the importance of a rapid response to such demands, they develop a model to estimate the distribution of the arrival time of the ®rst vehicle at the scene. Travel times on each link are stochastic, and are assumed to be normally distributed (with variances proportional to the associated means). Analysis shows that decreasing the expected common travel time between two paths (i.e., the travel time two responding vehicles spend on common 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 some vehicles to increase slightly. Daskin [35] reviews and presents additional research on deploying EMS vehicles on stochastic travel time 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 research in stochastic location theory. In fact, researchers examining a wide range of applications have contributed methods for solving stochastic location problems. Ghosh and Craig [49] discuss a method for multiple retail site location by a retailer operating in a duopoly with ®xed market potential. The retailer's objective is to

438


maximize pro®t, and a nonlinear multiple store location model is developed. The market environment requires that the competitor relocate in response to decisions made by the retailer. A competitive equilibrium is thus sought through an iterative procedure which solves a multiple store location problem instance at each step. The computational intensity of this procedure leads to the development of a heuristic which 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 oil spill occurs, the response equipment dispatched to the scene is responsible for containing the spill and removing oil from the water. The objective behind these operations is to minimize the amount of oil that washes onto the shore, where clean-up is more dicult and very expensive. Drawing from the literature on locating ®re stations, the authors develop a partial set covering model for locating six types of response equipment. 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 dierent regions and of the environmental impact of various types of spills. A multi-objective formulation allows decision makers to discriminate between coverage of likely spills and coverage of potentially high impact spills. The problem of locating 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 multiplant location problems. Drawing from ®nancial economics, he casts such problems as geographical allocations of assets. As portfolio models generally recommend diversi®cation of holdings to reduce portfolio risk, the author considers large (multi-plant) ®rms as having a geographically diverse portfolio of plants. A binary quadratic program is developed to maximize the ®rm's expected return, depending largely on the risk-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 nonlinear mathematical program. These penalties represent the expected cost of the realized supply being over or under the realized demand (making use of the probability distribution of demand). The authors show how such a model was employed interactively using sensitivity analysis and exogenously speci®ed location alternatives so that decision makers could in¯uence solution values to account for unmodeled political factors. Also drawing from stochastic programming, Louveaux [61] introduces two-stage stochastic programs with recourse for solving simple plant location problems and P -median problems. In his models, the ®rst stage decisions determine the location and size of facilities to be built, while the second stage speci®es the allocation of production resources to meet the most pro®table demands. The author considers uncertainties in demand, production and transportation costs, and selling prices. The relationship between the simple plant and P -median problem is explored, but solution methods are not discussed in this paper. 4.1.2. Queueing models The 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 parameters have 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. In the original model, Larson examines problems relating to vehicle location and response district design for emergency service organizations. Considering interdistrict (as well as intradistrict) responses, probabilistic call arrivals, and variable service times, Larson models the emergency service system as a multi-server queueing system with distinguishable servers. Speci®cally, the author assumes a Poisson call arrival process and 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 continuous time Markov process. The state space of the process is depicted as the vertices of an N-dimensional unit


439

hypercube in the positive orthant (where N is the number of servers), each vertex representing some combination of service unit availabilities. This descriptive model is then used to generate a number of performance measures which characterize the system's behavior. As described below, Larson's hypercube model is embedded in a number of heuristic procedures for solving a range of queueing-based location models. Batta et al. [9] use Larson's hypercube results in addressing the following three assumptions of Daskin's maximum expected covering model [34] (described above): 1. servers operate independently, 2. servers have equal busy probabilities (p), and 3. 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 responses to 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 a hypercube queueing model in a single node substitution heuristic procedure to determine a near-optimal set of server locations. To do this, the authors assume that the service system is in steady state and that the assumptions of the hypercube model are valid (e.g., they assume a ®rst come ®rst served (FCFS) queueing discipline, a Poisson call arrival process, and exponential service times). Computational results show some disagreement between the expected coverage predicted by the hypercube procedure and that reported by Daskin's model. The two approaches are largely in agreement, however, on the locations selected. The authors also suggest adjustments to Daskin's model and heuristic solution procedure which allow for the relaxation 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 explicitly considering the arrival process of customer calls for service. The authors note that when the server is busy and customers are queued, the mean time spent in the queue may be much greater than the mean travel time, and thus they consider queueing delay in formulating the problem objective. Two models are formulated as extensions to Hakimi's original P -median problem, both of which assume that demands arise according to a homogeneous Poisson process. In the ®rst model, demands which ®nd the server busy are lost. The objective of this stochastic loss median problem is to minimize a weighted sum of mean travel time and rejection costs. In the second model, demands which ®nd the server busy are entered into a queue which is depleted according to a ®rst in, ®rst out (FIFO) manner. The objective in this stochastic queue median case is to minimize the sum of the mean in-queue delay and the mean travel time. Both systems are modeled as 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 the in®nite capacity case has a nonlinear objective function which can lead to an optimal location at any point of the network. The authors develop an exact, ®nite procedure for ®nding the optimal location and explore properties 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 presented by Brandeau and Chiu [20] for the case of a planar region with rectilinear distances. Through convexity analysis, the authors determine a necessary and sucient ratio condition for ®nding the optimal location. The stochastic queue median trajectory, as a function of the systemwide demand rate (k), is characterized and an algorithm for ®nding the optimal trajectory is detailed. Finally, the authors discuss the extension of their results to problems with stochastic travel times. Berman [14] considers vehicle availability in a combined probabilistic and scenario planning model (see Section 4.2 below) which captures uncertainty in both link travel times and service demands. This paper attempts to combine the work of Berman et al. [15] (capturing customer demand rates) with that of Mirchandani and Odoni [67] (concerning uncertain travel times), both of which have been detailed above. The author attempts to locate a single service vehicle anywhere on a given network, assuming that service demands constitute a homogeneous Poisson process and that link lengths are discrete random variables. As

440


in Ref. [15], two cases are examined, alternately disallowing and allowing queueing of demands (resulting in the 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 loss median is found to reduce to the expected median. A heuristic for the stochastic expected queue median is developed 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. Eorts to extend the stochastic median analysis to the location of P service units are complicated by the absence of a closed form expression for the expected waiting time in queue for the M/G/P system. Berman et al. [16] analyze this problem and propose two heuristics for ®nding the locations of P units. The ®rst heuristic takes an initial set of locations and uses the hypercube model to provide information to each server on the likelihood of being dispatched to calls for service from each demand point. This information is then used in a 1-median problem to improve each server's location. The entire process is continued until no further improvement is found. The second heuristic described by the authors is similar to the ®rst, except that a stochastic queue median problem is substituted for the 1-median problem. These heuristics are conceptually similar to the neighborhood search algorithm originally proposed by Maranzana [64], and both versions are shown to perform well through computational testing. The ®rst heuristic requires less computational eort 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 facility locations are limited to a ®nite set of points. Trying to ®nd the location of a single server which minimizes the average server response time, the author develops an algorithm which solves for the optimal site parametrically in k. Batta also presents a worst case analysis for the the stochastic queue median problem in which 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 objective for all applications. The P -center problem addresses situations in which service inequities are more important than average performance. Brandeau and Chiu [22] examine how this model can be extended to congested systems in their stochastic queue center problem. Their objective is to locate a facility which minimizes the maximum expected response time, the total of expected time in queue and expected travel time. 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 also considered, 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 queue median 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 normbased cost function to measure system performance. (Here the parameter p speci®es a power cost function for 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 function and discuss methods for ®nding the optimal facility location for cases when k and p are ®xed and when they are unknown. Queueing and network congestion are also factors in considering consumer choice and facility utilization. In the Ghosh and Craig model discussed above, customers select which facility to patronize based on travel cost (or distance) and facility attributes such as cleanliness or size. This is an extension of much of the facility location literature which typically assumes that customers patronize the closest facility. In all of these models, however, each customer's choice is considered to be independent of the actions of other customers. Brandeau and Chiu [23] recognize that this independence assumption is not necessarily a good one for modeling practical situations. They highlight many instances in which a public facility's total market share may aect customer choice, speci®cally considering the impact that crowds or congestion might have on a customer's patronage. Brandeau and Chiu's model attempts to ®nd an equilibrium in


441

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 to queue formation at each facility. Demands are assumed to be generated according to a Poisson process and the special case of a tree network with customers at nodes and two facilities is considered. An enumerative algorithm 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], Brandeau and Chiu consider a competitive facility location problem. In this instance, the facility owners are responsible for site selection, with the objective of maximizing equilibrium market share. Customers have the choice of which facility to patronize, and these decisions continue to be aected by market externalities 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 in this environment as a three stage sequential game in which the leader locates ®rst, then the follower locates, and ®nally a customer choice game is played by the users, determining a customer choice equilibrium. 4.2. Scenario planning models Scenario planning is a method in which decision makers capture uncertainty by specifying a number of possible future states. The objective is to ®nd solutions which perform well under all scenarios. In some applications, scenario planning replaces forecasting as a way to evaluate trends and potential changes in the business 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 are qualitative descriptions of plausible future states, derived from the present state with consideration of potential major industry events. In other applications, scenario planning is used as a tool for formulating and solving speci®c operational problems [70]. While scenarios here also depict a range of future states, they do so through a quantitative characterization of the various values that problem input parameters may realize. As detailed below, the use of scenario planning for facility location follows the latter, more quantitative approach. Vanston et al. [92] discuss the use of scenario planning techniques and present a 12step 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 planning techniques. A recent text by Kouvelis and Yu [56] discusses the use of a robustness approach to decision making in environments characterized by uncertain data values. Their approach develops decisions which hedge against 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 robust solutions and detail complexity results for a range of problem classes. Included in this volume is an analysis of 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 to models having uncertain node demands and edge lengths. A number of additional researchers have used scenario planning techniques to solve a broad spectrum of facility location problems. The references detailed in this section illustrate the strategic nature of this approach and the advantages of its use. Before discussing scenario planning research contributions, we brie¯y outline 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 possible realizations of unknown problem parameters and to consider the range of scenarios in determining a compromise (robust) location solution. There are at least three approaches to incorporating scenario planning into location modeling.

442


1. 2. 3. In

optimizing the expected performance over all scenarios, optimizing the worst-case performance, and minimizing the expected or worst-case regret across all scenarios. what follows, we focus on regret-based approaches. The regret associated with a scenario is calculated by comparing the performance of the optimal locations for the scenario (had planners known for certain that the scenario would be realized) with the performance of the compromise locations when the scenario is realized. Using a regret-based objective thus allows us to evaluate robust solution alternatives with respect to the optimal solution obtained under data certainty. Note that some papers use this measure in objectives which require the assessment of scenario realization probabilities, while others implicitly assume that all scenarios are equally likely. The Kouvelis and Yu criteria of hedging against the worst case outcome is an example of the latter, in which the authors seek to minimize the maximum regret. Another commonly used decision criterion which requires scenario probabilities is minimizing the expected regret. To illustrate how such objectives are formulated, we will examine the P -median problem under the scenario planning approach. To do so, we introduce the following additional notation: Inputs: k index of possible scenarios hik demand at node i under scenario k dijk distance from node i to facility site j under scenario k V^k optimal P-median solution value for scenario k Decision variables: 8 > < 1 if we locate at potential facility site j ; Xj > : 0 if not; 8 > < 1 if demand node i is assigned to facility j under scenario k; Yijk > : 0 if not:

The regret associated with the scenario k is thus given by Rk Vk ÿ V^k , where Vk is the value of the demand weighted P Ptotal distance (i.e., the P -median objective value) under the compromise locations (Vk i j hik dijk Yijk ). The expected regret problem can thus be formulated as follows: X qk R k 28 Minimize k

X

subject to:

j

X Yijk 1

Xj P ;

29

8i; k;

30

j

Yijk ÿ Xj 6 0 8i; j; k; Rk ÿ

XX hik dijk Yijk ÿ V^k i

31

! 0 8k;

32

j

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

33 34


443

The objective function (28) minimizes the expected regret, with regret de®ned in constraint (32). The remaining constraints are the scenario planning equivalents to the standard P -median constraints detailed in Section 2.1. Note that the locations are common to all scenarios and must be determined before knowing which scenario is realized. The demand assignments, however, are scenario-speci®c. In essence, they are the result of optimizing the assignments conditional on the chosen sites, but after we know which scenario is realized. 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 the maximum regret across scenarios. This objective is more conservative, and is formulated with the constraints as above, but with the following objective function: Minimize

maxk fRk g:

Recent work by Averbakh and Berman [6] extends the work of Kouvelis and Yu, and looks at the robust 1-median location problem on a general network (as formulated above with P 1). The authors present a polynomial algorithm for this problem and develop improved algorithms (in terms of computational complexity) for the tree network problem. Additional research by Averbakh and Berman considers algorithms for robust center problems. First focusing on the weighted 1-center problem, the authors examine problem instances with uncertainty in both demand values and edge lengths [4]. Having previously shown [3] that this problem is strongly NP-hard on a general network, they present a polynomial algorithm for the 1-center problem on a tree. The algorithm's high order of complexity (O(n6 )) illustrates the increased problem diculty associated with uncertainty in edge lengths, even on tree networks. In a later paper [5], Averbakh and Berman study the P -center problem on a network with uncertain demand values. They seek the minimax regret solution when interval estimates of the demand values are given. The authors present an algorithm which involves solving n 1 regular weighted P -center problems (where n represents the number of nodes in the network). Complexity results for both tree and general networks are detailed for the P 1 case. Ghosh and McLaerty [50] use scenario planning concepts to make retail location decisions in an uncertain environment. In this problem, a retail chain seeks to locate stores in such a way that market share is maximized. The scenarios generated describe possible future marketing environments. An exchange heuristic is used to identify non-inferior strategies which perform well under all scenarios. When the number of non-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 regretminimization method for selecting the optimal solution. Reminiscent of the dynamic problems detailed in Section 3 above, Schilling [78] uses scenario planning to analyze the problem of locating a number of facilities over time. Individual scenarios are used to identify a set of good locational con®gurations, each of which can be seen as a contingency plan. Schilling proposes building facilities which are common to all contingencies early in the horizon, leaving decisions which discriminate between contingencies for a later time, at which point more information about the future will be available. (Daskin et al. [40] show that following this strategy can, in some circumstances, lead to poor location 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 a set of possible demand scenarios. In this model, two objectives are analyzed: maximizing the minimum demand 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 is found by determining the optimal locations associated with each scenario independently (as if it were a static problem) and then choosing the ``best'' starting point from among those, based on the objective being used.

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Serra and Marianov [82] utilize an equivalent solution method to site ®re stations in Barcelona, with uncertainty in both the demand at the network nodes and the travel times along the network arcs (which vary depending on time of day or day of the week). Scenarios are used to capture dierent demand patterns and/or travel times. Over these scenarios, facilities are sited with the objectives of minimizing the maximum average travel time and minimizing the maximum regret. On a smaller scale, Carson and Batta [27] use a similar scenario planning approach to site a single ambulance on the Amherst campus of the State University of New York at Bualo. The authors use four (unequal duration) states to capture the movement of campus populations during the 24 hour day. To minimize system-wide average response times, they formulate a model for determining the optimal ambulance position under each state. Evaluating the resultant optimal strategy with historical data, the authors predict a 30% reduction in average response time. However, a test implementation of this strategy showed only a 6% reduction. This dierence was attributed to the short operating range of the campus ambulance, an unmodeled factor which makes response times relatively insensitive to travel distances. Taking a rather dierent approach, Current et al. [32] examine uncertainty in the number of facilities to be 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 considered within a P -median context: minimizing the expected opportunity loss and minimizing the maximum regret. Daskin et al. [38] generalize the minimax regret objective in an eort to make location decisions more realistic by making them less conservative. The authors recognize that the standard minimax regret objective is sensitive to potentially ``bad'' scenarios, even if the likelihood of such a scenario evolving is very small. 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 reliability level, a, which captures the risk aversion of the decision maker. The model endogenously selects a subset of scenarios (the reliability set) over which the minimax regret solution is found. The probability of realizing a scenario which is not in the reliability set must be at most 1 ÿ a. Thus if a 0:95, the decision maker 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 standard minimax regret problem. Using an 88-node problem with 9 scenarios, they show that scenarios may move in and out of the reliability set as the level of reliability increases. This suggests that identifying an extreme scenario in a network planning context is more complex than identifying an extreme scenario in problems without network interactions. 5. Conclusions In this review, we have attempted to provide an overview of facility location literature dedicated to capturing the complex time and uncertainty characteristics of most real-world problem instances. Advances in integer programming, dynamic programming, stochastic programming, and scenario planning techniques have clearly increased our capacity for analyzing, modeling, and solving important strategic facility location problems. We expect future research to continue in this direction. Speci®cally, we look for improved heuristics to support the solution of larger, more complex and more realistic problem instances. The increased use of scenario planning techniques will drive such solution advances, as scenario-based models grow rapidly with the number of scenarios generated. Also, we look for the development of tractable models which consider both the stochastic and dynamic aspects of facility location. Recent developments in multiple stage stochastic programming with recourse might be employed to capture the complexities of such location problems.


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