Annals of Operations Research 123, 125–141, 2003 2003 Kluwer Academic Publishers. Manufactured in The Netherlands. 1
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Location of Multiple-Server Congestible Facilities for Maximizing Expected Demand, when Services are Non-Essential
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[email protected] Department of Electrical Engineering Pontificia Universidad Católica de Chile, Vicuña Mackenna 4860, Santiago 6904411, Chile VLADIMIR MARIANOV
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Abstract. We formulate a model for locating multiple-server, congestible facilities. Locations of these facilities maximize total expected demand attended over the region. The effective demand at each node is elastic to the travel time to the facility, and to the congestion at that facility. The facilities to be located are fixed, so customers travel to them in order to receive service or goods, and the demand curves at each demand node (which depend on the travel time and the queue length at the facility), are known. We propose a heuristic for the resulting integer, nonlinear formulation, and provide computational experience.
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Keywords: location, congestion, elastic demand
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Introduction
Some economical activities require the location of several, geographically distributed, service centers. In some cases, these centers receive clients who have to wait in a line in order to be served, as it happens for example with banks, private clinics or inoculation centers, distributed ticket selling facilities, gas stations and distribution centers. When services are essential, customers will stand in line until attended. On the contrary, if the services are not essential, some of the arriving customers will choose not to wait if there is a line, and may or may not come back. In this case, the demand is said to be elastic to the effects of congestion. Demand could also be elastic to other parameters, as price, travel distance or time to the facility, or quality of service. When the demand is elastic, providers capture clients offering better prices, closeness to the client, or a better service. A better service means larger parking space, better qualified attendants, shorter waiting times, a nicer decoration and, in general, all those features that make attractive to the clients visiting the center. The location of the centers and the allocation of an adequate number of attendants to them, have both influence on the waiting time of the clients, and the length of the queue they form. In fact, if a center has more attendants (at a higher cost), the waiting time (or queue length) of the clients is reduced. Also, by assuming that clients go to their closest centers, the location of these centers relative to the clients and to other centers is what defines fundamentally how many clients receive attention from them. As more clients patronize a center, it becomes more congested, and the length of the line or lines that are formed at that center increases, which makes the
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place less attractive. Examples of this kind of externalities include traffic delays in transportation systems, congestion or blocking probabilities in telecommunications systems, congestion in banks, supermarkets, post offices and in many other service networks. If the services are essential and there is a line, the customers either leave to come back later, or stand in a line with other customers waiting to be served. Many location models addressing congestion at the facilities have been developed for systems with inelastic demand, particularly emergency services. These services include health, fire fighting or police, in which facilities are mobile, as opposed to fixed (Jamil, Baveja and Batta, 1999). In the case of fixed facilities, most of the models include static capacity constraints, which force the demand at each center to be smaller than some value. A few formulations have been proposed with probabilistically developed constraints, as (Marianov and Serra, 1998, 2000). The models that address congestion under inelastic demand can be classified in several types. In the first place, there are the models that prescribe the location of facilities and servers, which are based on optimization. Most of these are derived from the Location Set Covering (Toregas et al., 1971), the p-median (Hakimi, 1964; ReVelle and Swain, 1970), the Maximal Covering Location Problems (Church and ReVelle, 1974) and the p-center problems. These models are in general linear. To deal with congestion, they may require each demand to have backup servers or facilities, in a deterministic approach (Daskin and Stern, 1981; Daskin, Hogan and ReVelle, 1988; Hogan and ReVelle, 1986), through added or modified constraints. This deterministic approach works well when congestion is not expected to be severe. As congestion becomes more severe, deterministic approaches do not provide efficient solutions, and an explicit probabilistic optimization approach is better. This approach is based on a probabilistic model of the system, which leads to the development of objectives or constraints that reflect this probabilistic nature. Several authors, as ReVelle and Marianov (1991), Marianov and ReVelle (1992), have presented optimization formulations including explicit probabilistic models of the systems. For a complete review up to 1995, see (Marianov and ReVelle, 1995). Most models include simple probabilistic distributions, and a few include queuing formulations (Batta, Dolan and Krishnamurthy, 1989; Marianov and ReVelle, 1994, 1996). For fixed facilities, Marianov and Serra (1998, 2000), present models that upper bound either the queue-length or the waiting time of the customers who seek service at the facilities. This is an approach that in some sense allows including the quality of service when designing the system. Other models use heuristics rather than formal integer programming optimization formulations. These models are mostly oriented to emergency systems, in which facilities are mobile, as opposed to fixed, although some of them can be applied to systems with fixed facilities. Some of them assume a single server, as Berman, Larson and Chiu (1985) Stochastic Queue Median (SQM), and the models by Batta (1988, 1989), Batta, Larson and Odoni (1988) and Jamil, Baveja and Batta (1999). Models assuming single servers are sometimes used as sub-algorithms for algorithms that locate more than one server. Based on the SQM, and on the Mean Time Calibration (Larson and Odoni, 1981), Berman, Larson and Parkan (1987) develops two heuristics for locating p facili-
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ties on a congested network, suitable for systems with only a few facilities. All of these models are nonlinear. Some of the models have considered the problem of districting. Berman and Larson (1985) solve this problem for a two-server network in the presence of queuing, with a nonlinear model and a heuristic algorithm. Each district behaves as a M/G/1 system. The average response time is minimized. No interaction exists between districts. Later, Berman and Mandowsky (1986), use the SQM, combined with this 2-server districting algorithm, to develop a general location – districting iterative algorithm for n-nodes, m-facilities networks. All of the models are nonlinear, heuristic, and they minimize the expected response time of servers that travel to the site of an emergency. Also, all of these models use approximations in order to obtain tractable representations of the system. Some authors have addressed elasticity. For example, Wagner and Falkson (1975), study the effects of price elasticity, and Labbé and Hakimi (1991) analyze elasticity to price and distance. ReVelle and Church (1977), formulate a model that addresses elasticity to the size of the facilities. The formulation maximizes utilization of the facilities by customers, when the utilization depends on the scale of the facilities as well as on the distance. However, elasticity to congestion has been seldom included in location– allocation models, because of the difficulty of its treatment. Lee and Cohen (1985) characterize user-choice equilibrium under elastic demand. Berman and Kaplan (1987), locate a facility on a network when demand depends on service delay, while Cadwallader (1975) study the relationship between congestion and attractivity. Several models for congestion-elastic demand, as well as a good review of the literature on models for both mobile and fixed facilities, with elastic and inelastic demand, can be found in (Brandeau et al., 1995). The models they present cover fixed and elastic demand conditions, in both dictatorial (central planning) and user-choice environments. We propose a model for location of service centers, which will locate multiple-server, fixed facilities on a network, and allocate demands to facilities, when demand is elastic and demand nodes are assigned to facilities by central planning. The public sector provides some examples of this type of systems, as preventive care or child inoculation services. Public polling or voting places are another example of this type of facilities. The model maximizes total demand served. As opposed to previous works, we consider location of several facilities with multiple servers in each one, (instead of just one server at each facility); use a practical or experimental demand curve that may be obtained from polls and focus groups, which does not necessarily have an analytical form (instead of an analytical demand function); and use a demand curve that is demandnode-dependent, as opposed to a common curve for all demand nodes. We also propose a special heuristic for solving the resulting integer, nonlinear optimization model. The model builds on a spatial queuing system with one or more servers at each service center. The demand curves at each demand node are considered known.
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1. Basic assumptions of the model Each node of the network is either a demand node, representing concentrations of population with similar characteristics, or a potential location for a center (facility), or
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Figure 1. Demand surface for each node i.
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both. At each demand node, customers have a known demand, which is elastic to both travel time and congestion. This demand curve is actually a surface, as the one shown in figure 1. Note that the demand needs not to have the same curve at all demand nodes. Through measuring congestion by the number of customers in the system, we achieve two goals. In the first place, we allow measuring the demand elasticity by asking customers in terms of travel time and queue length, so the shape of this surface can be found from polls or behavioral studies of customers. Note that the queue length is equal to the number of customers in the system that exceeds the number m of servers at the facility. Secondly, a curve of demand versus queue length (including queue length zero, corresponding to a number of customers in the system of at most m) can be drawn for each facility, since travel time to that facility from the demand node is known (assuming it is deterministic). Hence, if customers at a node i ∈ I patronize facility j ∈ J , and the number of customers n at that facility is known, the demand level corresponds to a known point on the surface. If f¯i is the maximum demand at node i, we can represent this point (the portion of the demand from i that patronizes facility j ) as fij n = βij n f¯i . The factor βij n ∈ [0, 1] accounts for the decrease of the demand respect to the maximum, due to both travel time and the number of customers in the system, and 1 − βij n customers decide not to obtain service at j , given that there are n other customers in line. Analytic equations for these curves are not required, but they can be used if available. Instead, the curve is just a set of points. This demand function is strictly non-increasing with distance and strictly non-increasing with the number of customers, unless there are crowd-lovers among the customers, which is not the case of interest.
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The number of servers at each facility is fixed and equal to m. Previous models consider either only one facility, or several facilities with only one server per facility. Requests for service at each demand node i, appear according to an independently distributed Poisson process with maximum intensity f¯i . The requests for service at each facility j are the sum of the requests at the demand nodes that are allocated to it, delayed each one by the corresponding travel time. Consequently, they also appear according to a Poisson process, with intensity λj , equal to the sum of the intensities of the processes at the nodes that assign to the facility. This assignment (node i to facility j ) is given by the variable xij 1. Then, if the demand was not elastic, the total portion of the demand at i that would go to j for service is f¯i xij . Since the demand is elastic, the percentage of the demand that goes from i to j and stays in line until served, given that there are n other customers in line, is λij n = βij n f¯i xij . Finally, the expected value of the demand from i that is served at j , over all possible numbers of customers in the system, is βij n f¯i xij Pn (λj ), (1) λ¯ ij =
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λj =
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λ¯ ij =
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βij n Pn (λj )f¯i xij .
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If we assume that the interarrival time (time between requests for service) at each demand node is exponentially distributed, the interarrival time at each center is also exponentially distributed, because it is the sum of the exponentially distributed processes at the demand nodes, delayed each one by the corresponding travel time. Actually, since customers can refuse the service, and the refusal is somehow related to the expected occupation state of the facilities, the processes might not be exactly exponentially distributed. However, there is no better tractable approximation. We assume that customers assign a different value to travel time and service time. Thus, we treat both values separately in the demand surface. Consequently, travel time is not added to service time, and service time at each center is exponentially distributed for the type of services we are dealing with. The average service rate of each server in center j is µj . If we assume steady state, we can use the known results for a M/M/m queuing system for each center and its allocated users. Notice that typically, demand nodes represent aggregated population, rather than single call sources. Then, it is safe to assume that each time a call for service appears at a node, the probability of a new call appearing at the same
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where Pn (λj ) is the probability of a number n of customers at the facility, which is, in turn, a function of the total average demand at the facility λj . Furthermore, the number of customers at each facility is assumed to be limited to K, due to the limited space at each facility and to elasticity (no customer is willing to stay in a line that is longer than K − m). This is a system with finite queues. However, infinite queues could also be assumed, with zero probabilities for all queues longer than K − m. Thus, the total demand at facility j is:
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node remains practically unchanged. For this reason, we use an infinite source queuing system. The equations for the probabilities of each state in an M/M/m queuing system with a finite number K of customers in the system are (Hillier and Lieberman, 1986):
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n−m −1 K ρ ρ + , n! m! n=m+1 m
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ρn P n! 0 Pn = ρn P0 m!mn−m 0
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for n m,
for m n K, for n K,
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where ρ = λ/µ 1. One set of similar equations can be written for each facility j . The probabilities Pn at each facility j are a function of ρj , which is, in turn, a function of λj . Let the right-hand side of equation (2) be (λj ). Then, equation (2) can be written as λj = (λj ). For this equation to make sense, there must exist demand equilibrium at each facility. In other words, if demand is elastic, there must exist an equilibrium demand. We prove the existence and uniqueness of such demand equilibrium in appendix. Note that for this proof to hold, the assignments of demands to facilities must be given, which is always true if there is a central planner. A central planner makes assignment of demands to facilities. Demand nodes are assigned to facilities in such a way that the total demand is maximized. Note that the planner assigns demand nodes to those facilities in which the attractiveness is maximized for users. Thus, the objective also preserves users’ goal (minimum travel and congestion costs), except for the fact that assigning complete demand nodes to facilities is suboptimal as compared to the case in which a demand node could be assigned to more than one facility (i.e., using a zero-one variable xij is suboptimal as compared with using a real variable xij ∈ [0, 1]). Except for this fact, the objective works in the same way as the objective in the p-median problem, in which both the planner and the users seek to minimize travel distance or time.
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Development of the model for m servers per center
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The problem can be stated as: “Locate p facilities with m servers each, and assign customers to them so to maximize the demand, under elastic demand conditions.”
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and its formulation is the following: λj Max
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and
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λj =
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or
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βij γn Pn (λj )f¯i xij ,
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where the new variable yj is one if a facility is opened at site j , and zero otherwise. The objective (5) maximizes the total demand in the system. Constraint (6) is the demand equilibrium constraint. Constraint (7) states that each demand is assigned to at most one facility. Constraint (8) forces demand to be assigned to open facilities, and constraint (9) limits the number of facilities to p. Note that constraint (6) is a nonlinear function of λj . There are conditions under which the problem can be solved as a convex, integer optimization problem. In fact, if the elasticity factors βij n can be factorized as βij n = βij · γn , the equilibrium equation (6), at equilibrium, can be rewritten as:
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G(λj ) =
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βij f¯i xij ,
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where
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If the denominator of G(λj ) is concave (which happens for certain choices of demand curves), then G(λj ) is convex (which is proved easily by differentiating twice) and the objective is a sum of concave functions (also concave). Then, the formulation is a concave objective, integer formulation, which can be solved using enumerative techniques (Brandeau et al., 1995). However, if the elasticity factors can not be factorized, and/or the denominator of G(λj ) is not concave, a different approach must be used. We choose to relax the nonlinear equilibrium equation, which leaves a linear problem. In order to assure the demand equilibrium, we use an iterative procedure that converges to the solution. We also apply Lagrangean relaxation.
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Lagrangean relaxation and iterative heuristic
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3.
In order to solve the problem, keeping the equilibrium, we use traditional Lagrangean relaxation, plus an iterative procedure. The procedure can be summarized as follows. Procedure:
Use constraint (8) to formulate a relaxed version of the problem. Outer loop: fix the values of the Lagrangean multipliers. Inner loop: For fixed values of the multipliers, find the optimal values for the variables xij and yj (optimal locations and allocations). The problem is separable in a problem for the location variables yj and a problem for the assignment variables xij .
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∀i, j, ∀j.
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Then, the equilibrium constraint is λj = G−1 ( i∈I βij f¯i xij ), and this value of λj can be replaced in the objective. The new formulation becomes: −1 ¯ G βij fi xij Max
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Iterative procedure for assignment variables: find the equilibrium value of λj . To this equilibrium value, corresponds also a best value of the allocation variables xij .
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Find the upper bound (unfeasible solution) that corresponds to the values of the location and assignment variables.
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Find a lower bound (feasible solution): use the solution obtained for the location variables yj , and solve the iterative assignment problem.
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Save the inner loop solution (lower bound).
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End of the Procedure
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Min Max ω
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where ωij are the Lagrange multipliers. For fixed values of ωij , the problem separates in two, (P1) and (P2). ωij yj (P1) Max
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A more detailed description of the procedure follows. Replacing the demands λj by the right-hand side of the equilibrium equation in the objective, the relaxed version of the problem can be stated as:
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Choose the best solution found and determine its quality by comparing upper and lower bounds.
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Go back to the outer loop. Find new values of the Lagrange multipliers using subgradient optimization, until the optimal values are found.
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Problem (P1) can be solved optimally using
a greedy approach. It consists in finding the p values of yj with the largest factors i ωij . The problem (P2) in the allocation variables can be stated as:
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Pn (λ(k) j ).
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3. For the assignments xij(k) found in step 2, compute
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is the maximum. If Kj 0, then set xij(k) = 0. If Kj > 0, then set xij(k) = 1. Save the value of the objective.
K
(k)
(k) ¯ βij n Pn λ(k) λj = j fi xij . n=0 i∈I
4. Compute a new value
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1 (k) 4λj + λ(k) . j 5 Note that, at each step, this value is most probably smaller than the equilibrium value of λj . Thus, we assure convergence by obtaining nondecreasing values of λ(k) j . If the value at some iteration decreases, we keep the preceding one.
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5. Set k = k + 1. If the value of the objective is better than the value at the last iteration, go to step 2. Else, go to step 3 until convergence. Convergence is guaranteed because the objective cannot increase indefinitely (because we keep decreasing the distance between λj and its equilibrium value). Thus, at some step it decreases. Once it does,
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2. For each demand i, find the j for which the coefficient K
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xij 1 (i assigned to one facility);
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1. Set k = 0 and = 0 ∀j . Compute Once computed, the problem becomes separable, and a subproblem is written for each demand i: K
(k) βij n Pn λj f¯i − ωij xij (P2i ) Max
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For problem (P2) we use an iterative procedure that finds a good (not necessarily optimal) solution and the equilibrium values of the demand and allocation variables. The iterative procedure has the following steps:
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the assignments stay fixed, and the iterations only are used for finding the demand equilibrium for those assignments.
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Max
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βij n Pn (λj )f¯i xij
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xij 1
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j
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where the step size is:
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(15)
Here, LB is the best lower bound obtained so far, k n is a parameter that decreases after a number of non-improving iterations (its starting value was set to 2) and Ln is the value of the Lagrangean objective at the nth iteration. The procedure finishes with best values of lower and upper bounds. The best lower bound corresponds to the solution of the problem. Both the choice of the subgradient optimization and the starting value for k n are the usual ones in Lagrangean relaxation (Daskin, 1995). Note that, although we use a Lagrangean relaxation type of procedure, we are not solving to optimality the upper bound part of it. Consequently, the best value obtained for the Lagrangean objective is not a valid upper bound, in the usual sense. However, this best value is sought as an intermediate step for finding feasible solutions. In fact, at each iteration, together with the value of the Lagrangean objective, we find values for the primal variables. These primal variables are in turn used for finding the new lower bound or feasible solution corresponding to that iteration. Evidently, by not finding the optimal value of the upper bound, we are not able to ensure the quality of the solution; however, we share this shortcoming with many other good heuristics.
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k (LB − L ) tn =
n n 2. i j (xij − yj ) n
26
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12
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17
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10
which is solved using the same iterative procedure described previously. Note that the only difference with (P2) is that the locations are restricted to the ones obtained in (P1). If desired, at each step, these locations could be improved using a one-opt procedure. Once the inner loop is complete, a new step of the outer loop must be performed (new value for Lagrange multipliers ωij ). The new value is computed using a subgradient optimization, as follows: (14) ωij(n+1) = max 0, ωij(n+1) + t n (xij − yj ) ,
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5
9
xij = 0, 1; j ∈ Ni , ∀i, for j such that yj = 1,
14
4
8
PR O
8
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2 3
Once the iterative procedure has finished, the solutions of (P1) and (P2) provide an upper bound for the problem. To each upper bound, we match a lower bound (feasible solution), which is found using the values of the location variables yj , obtained from (P1), and solving the assignment problem (P3):
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136 1
4.
MARIANOV
Computational experience
1
2
10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43
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Initial Lagrange multipliers: random. Maximum node demand: node population (between 710 and 10). Service rate at each potential facility, each server: 800. Maximum queue length: 15. Number of servers at each facility: 3. Number of facilities: 1, 2, 3, 4, 5. Number of nodes: 30, 55.
In order to take advantage of its random starting Lagrange multipliers, the heuristic was run ten times for each value of the number of facilities. Except for the case in which there is one facility to be located, several solutions were obtained. The best and worse solutions were always within 1%, except for the case 30 nodes, 5 facilities, in which there was a difference of 1.85%. The heuristic terminates when the difference between upper and lower bound is smaller than 0.1%. Note, however, that the solution to the assignment problems (P2) and (P3) are not optimal. Thus, in some cases, the upper bound was lower than the lower bound. Tables 1 and 2, as well as figure 2, show the results of the runs. The tables show the number p and locations of the facilities, the objective values (obtained as lower bounds), the frequency of each solution in ten runs, the percentage by which the best solution obtained in ten runs is better than the worst solution, the number of locations that contain all located facilities in the 10 runs (“solution set”), and the run time of each run. Note that the heuristic was programmed in Quick Basic, and the routines were not optimized. In the column corresponding to the objective, the bold numbers correspond to the best solution, while the numbers in italic correspond to the worst solution for each instance. Figure 2 shows the decreasing marginal gain in the amount of attended (or “captured”) demand when more facilities are added. This is due to the fact that, as the number p of facilities grows, adding a (p + 1)th facility has a lesser impact both on the reduction of the distance between the demand and its chosen facility (say, the closest), and on the reduction of the queuing effect at the centers. This is easy to see if both elasticity to distance and elasticity to waiting are analyzed separately. Suppose first that the demand is elastic only to distance and assume that all the nodes of the network are on a line. As more facilities are added, the reduction in the distance from the demands to their closest facilities is necessarily smaller. Now assume that the demand is only elastic to congestion, and not to distance, and assume that there are p facilities already located. Then, what can be roughly expected is that the demand becomes evenly distributed among the p facilities (assuming that the demand curves of all customers are similar). When a (p + 1)th facility is added, the demand at each facility is p/(p + 1) times the demand
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• • • • • • •
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2
The heuristic was programmed in Basic, on a Pentium III, 500 MHz machine. Several instances of the problem were solved. The parameters were set at the following values:
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LOCATION OF MULTIPLE-SERVER CONGESTIBLE FACILITIES
Table 1 30 nodes.
2
Locations
p
Objective
4
Frequency of this solution
1 2
% Best/worst
2
2352.301
10
2
3, 5 3, 4 3, 11 3, 9
3504.085 3496.625 3491.919 3491.375
4 3 2 1
0.36
2, 9, 19 5, 9, 19 4, 5, 30 4, 11, 29 3, 4, 5 2, 9, 30 4, 11, 30
4013.272 4004.393 4003.025 3977.205 4012.744 3998.770 3992.345
1 2 2 1 2 1 1
0.5
1, 2, 6, 23 2, 4, 5, 22 1, 2, 9, 19 2, 9, 11, 17 1, 2, 4, 19 4, 5, 8, 17 8, 9, 11, 17 3, 5, 9, 19 3, 4, 5, 23 1, 3, 9, 22
4286.752 4278.989 4280.821 4283.629 4265.769 4273.240 4268.579 4274.017 4248.925 4248.903
1 1 1 1 1 1 1 1 1 1
0.9
1, 3, 4, 9, 22 3, 5, 9, 22, 29 2, 7, 9, 10, 17 3, 4, 13, 15, 22 4, 5, 7, 10, 12 3, 6, 8, 13, 22 3, 4, 7, 13, 22 2, 5, 6, 8, 17 1, 2, 9, 10, 22 2, 3, 5, 6, 12
4469.077 4421.020 4414.596 4439.536 4422.952 4435.150 4423.355 4462.654 4422.150 4387.700
14 15 16
4
17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43
5
5
2:51–6:01 2:16–3:22 3:01–3:35 3:16
6
9
13
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2:14–3:39
5
TE
12
EC
11
1 1 1 1 1 1 1 1 1 1
3
1
1.85
16
5:21 2:40–3:33 1:33–5:25 3:33 3:44–6:39 3:36 2:22 2:41 3:43 1:17 2:02 3:01 2:28 2:30 4:29 1:32 1:26 1:41 1:23 1:53 1:30 1:46 3:09 1:47 1:46 2:07 1:15
that was attended before the addition of the new facility, so as p grows, the reduction is more negligible. Furthermore, if the demand curves are exponential, and if the distance and queuing effects are combined, the decreasing marginal gains are more notorious. It is interesting to note that the solution set, that is, the number of nodes that contain located facilities, is rather limited. Since the solutions are not necessarily optimal, a second heuristic could be run a posteriori, considering as possible locations only those in the solution set. For example, a heuristic that has shown to provide excellent results is the Heuristic Concentration, by Rosing and ReVelle (1997). If this heuristic were
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Run time
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7
Solution set
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MARIANOV
Table 2 55 nodes.
1 2
Locations
p
Objective
0
2
17:55–31:50 8:05–30:00
5
2
1, 6 6, 45
3855.851 3855.850
9 1
0
3
30:15–126:01 52:05
7
3
9, 31, 45 1, 9, 31 9, 16, 31 9, 20, 45 34, 41, 45
4612.130 4612.130 4609.375 4599.450 4598.993
3 3 2 1 1
0.3
8
37:38–50:05 37:23–50:12 38:09–42:15 34:54 27:50
9
1, 2, 9, 49 2, 9, 45, 49 3, 11, 12, 45 3, 16, 34, 49 1, 9, 18, 21 1, 6, 11, 12 1, 3, 11, 18 16, 34, 49, 51
5011.339 5011.339 4998.295 5002.433 4999.168 4997.139 4998.222 4996.609
1 1 2 1 2 1 1 1
0.3
29:02 31:29 33:53–42:15 44:56 37:57–44:32 24:40 33:25 29:48
13
2, 3, 5, 12, 45 2, 3, 5, 18, 45 1, 5, 6, 12, 21 1, 2, 3, 5, 18 3, 18, 21, 34, 45 1, 2, 3, 5, 12 3, 5, 18, 21, 45 2, 5, 6, 12, 45
5282.451 5282.451 5278.450 5282.440 5279.137 5282.440 5282.385 5276.452
18:41–27:09 36:46 22:42 17:29 29:53 25:28 48:02 19:12
20
4
14 15 16 17 18 19
21 22 23 24 25 26
5
29 30 31 32 33 34 35 36 37 38 39 40 41
0.1
10
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3 1 1 1 1 1 1 1
PR O
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11
20
4
2508.643 2508.643
10
13
6
8
10 11 12
14 15 16 17 18 19
21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41
42 43
3
33 51
8 9
Run time
1
6 7
Solution set
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% Best/worst
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Frequency of this solution
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42
Figure 2. Demand as a function of the number of facilities.
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LOCATION OF MULTIPLE-SERVER CONGESTIBLE FACILITIES 1 2 3 4
139
to be used, the solution set would be what Rosing and ReVelle call the Concentration Set. We remark that at the first step of the iterative heuristic, the value of the objective corresponds to λj = 0 for all j . This means that the assignment is made only by distance.
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A new formulation is presented for the location–allocation problem of congestible facilities. The formulation considers elasticity to congestion and to travel time or distance. For the first time, the location of multiple-server facilities is included in such a model. The formulation also relies on data that can be easily and practically obtained through polls or focus groups; thus, it does not need analytical expressions of the demand curves. A procedure has also been presented for solving the resulting nonlinear, integer problem, which provides a good solution and keeps the required equilibriums. Finally, the procedure is tested through some computational experience. A second heuristic is suggested for possible improvement of the results. Further research should include the search for faster heuristics. The reader must note, however, that classic heuristics are not readily applicable to this problem, because of the equilibrium issue in the model.
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Conclusions
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Appendix
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24
Existence and uniqueness of demand equilibrium for elastic demand and given allocations
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Lemma 1. If the assignments (allocations) xij are given, the function (λj ) is continuous, strictly nonincreasing with λj .
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Proof. Recall that (λj ) = n=0 i∈I βij n Pn (λj )f¯i xij . The probabilities Pn (λj ) are continuous with λj , and so is the function (λj ), a linear combination of these probabilities. (i) Let first βij n = 1 for all i, j, n (no elasticity). As λj increases, the probabilities of higher numbers of customers in the system, i.e., the values
of Pn (λj ) for higher values of n, tend to increase or, at least, not to decrease. Since n Pn (λj ) = 1, the probabilities Pn (λj ) for smaller values of n must decrease in the same amount, so the function (λj ) does not change. (ii) Now, if the demand is congestion averse, the factors βij n 1 and they decrease with n. Then, when λj increases, the decrease in the values of Pn (λj ) for smaller values of n is multiplied by larger factors βij n than the corresponding increase of Pn (λj ) for larger values of n. Thus, function (λj ) must be strictly nonincreasing with λj .
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Theorem 1. If the facilities are located and the assignment of demands to facilities is specified, the demand equilibrium exists and it is unique.
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Note that
3
0 (λj )
4 5 6 7
1
K
f¯i xij
λ∗j
10
13 14 15
2
f¯i xij .
n=0 i∈I
Since function (λj ) is strictly nonincreasing with λj , equation (2) must have exactly one solution λ∗j , with
9
12
0 λj
n=0 i∈I
8
11
and
K
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Proof.
2
=
K
βij n Pn (λ∗j )f¯i xij ,
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n=0 i∈I
D
This research has been possible thanks to a grant by FONDECYT (Project Nr 1000602). References
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Batta, R. (1988). “Single Server Queuing – Location Models with Rejection.” Transportation Science 22, 209–216. Batta, R. (1989). “A Queuing – Location Model with Expected Service Time dependent Queuing Disciplines.” European Journal of Operational Research 39, 192–205. Batta R., J. Dolan, and N. Krishnamurthy (1989). “The Maximal Expected Covering Location Problem: Revisited.” Transportation Science 23, 277–287. Batta, R., R. Larson, and A. Odoni. (1988). “A Single-Server Priority Queueing – Location Model.” Networks 8, 87–103. Berman, O. and E.H. Kaplan. (1987). “Facility Location and Capacity Planning with Delay-Dependent Demand.” International Journal of Production Research 25(12), 1773–1780. Berman, O. and R. Larson. (1985). “Optimal 2-Facility Network Districting in the Presence of Queuing.” Transportation Science 19, 261–277. Berman, O., R. Larson, and S. Chiu. (1985). “Optimal Server Location on a Network Operating as a M/G/1 Queue.” Operations Research 12(4), 746–771. Berman, O., R.Larson, and C. Parkan. (1987). “The Stochastic Queue p-Median Location Problem.” Transportation Science 21, 207–216. Berman, O. and R. Mandowsky. (1986). “Location–Allocation on Congested Networks.” European Journal of Operational Research 26, 238–250. Brandeau, M., S. Chiu, S. Kumar, and T. Grossman. (1995). “Location with Market Externalities.” In Z. Drezner (ed.), Facility Location: A Survey of Applications and Methods. New York: Springer. Cadwallader, M. (1975). “Behavioral Model of Consumer Spatial Decision Making.” Economic Geography 51, 339–349. Church, R. and C. ReVelle. (1974). “The Maximal Covering Location Problem.” Papers of the Regional Science Association 32, 101–118.
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Acknowledgment
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The preceding theorem proves that there exists a unique demand equilibrium under conditions of elasticity to negative externalities.
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which corresponds to the unique demand equilibrium at j . This argument is repeated for each facility.
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Daskin, M. (1995). Networks and Discrete Location: Models, Algorithms and Applications. WileyInterscience Series in Discrete Mathematics and Optimization. New York: Wiley. Daskin, M., K. Hogan, and C. ReVelle. (1988). “Integration of Multiple, Excess, Backup, and Expected Covering Models.” Environment and Planning B: Planning and Design 15, 15–35. Daskin, M. and E. Stern. (1981). “A Hierarchical Objective Set Covering Model for Emergency Medical Service Vehicle Deployment.” Transportation Science 15, 137–152. Hakimi, S.L. (1964). “Optimal Locations of Switching Centers and the Absolute Centers and Medians of a Graph.” Operations Research 12, 450–459. Hillier, F.S. and G.J. Lieberman. (1986). Introduction to Operations Research. Oakland, CA: Holden-Day. Hogan, K. and C. ReVelle. (1986). “Concepts and Applications of Backup Coverage.” Management Science 32, 1434–1444. Jamil, M., A. Baveja, and R. Batta. (1999). “The Stochastic Queue Center Problem.” Computers and Operations Research 26, 1423–1436. Labbé, M. and S.L. Hakimi. (1991). “Market and Locational Equilibrium for Two Competitors.” Operations Research 39, 749–756. Larson, R. and A. Odoni. (1981). Urban Operations Research. Englewood Cliffs, NJ: Prentice Hall. Lee, H.L. and M.A. Cohen. (1985). “Equilibrium Analysis of Disaggregate Facility Choice Systems Subject to Congestion Elastic Demand.” Operations Research 33(2), 293–311. Marianov, V. and C. ReVelle. (1992). “A Probabilistic Fire Protection Siting Model with Joint Vehicle Reliability Requirements.” Papers in Regional Science 71, 217–241. Marianov, V. and C. Revelle. (1994). “The Queuing Probabilistic Location Set Covering Problem and Some Extensions.” Socio-Economic Planning Sciences 28(3), 167–178. Marianov, V. and C. ReVelle. (1995). “Siting of Emergency Services.” In Z. Drezner (ed.), Facility Location: A Survey of Applications and Methods. New York: Springer. Marianov, V. and C. Revelle. (1996). “The Queuing Maximal Availability Location Problem: a Model for the Siting of Emergency Vehicles.” European Journal of Operations Research 93, 110–120. Marianov, V. and D. Serra. (1998). “Probabilistic Maximal Covering Location–Allocation for Congested Systems.” Journal of Regional Science 38 (3), 401–424. Marianov, V. and D. Serra. (2000). “Hierarchical Location–Allocation Models for Congested Systems.” European Journal of Operational Research, forthcoming. ReVelle, C. and R. Church. (1977). “A Spatial Model for the Location Construct of Teitz.” Papers of the Regional Science Association 39, 129–135. ReVelle, C. and V. Marianov. (1991). “A Probabilistic FLEET Model with Individual Vehicle Reliability Requirements .” European Journal of Operational Research 53, 93–105. ReVelle, C. and S. Swain. (1970). “Central Facilities Location.” Geographical Analysis 2, 30–42. Rosing, K. and C. ReVelle. (1997). “Heuristic Concentration: Two Stage Solution Construction.” European Journal of Operational Research 97, 75–86. Toregas, C., R. Swain, C. ReVelle, and L. Bergmann. (1971). “The Location of Emergency Service Facilities.” Operations Research 19, 1363–1373. Wagner, J.L. and L.M. Falkson. (1975). “The Optimal Nodal Location of Public Facilities with PriceSensitive Demand.” Geographical Analysis 7, 69–83.
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