Multiple Facilities Location in the Plane Using the Gravity Model

Transcription

1 Geographical Analysis ISSN Multiple Facilities Location in the Plane Using the Gravity Model Tammy Drezner, Zvi Drezner College of Business and Economics, California State University-Fullerton, Fullerton, CA Two problems are considered in this article. Both problems seek the location of p facilities. The first problem is the p median where the total distance traveled by customers is minimized. The second problem focuses on equalizing demand across facilities by minimizing the variance of total demand attracted to each facility. These models are unique in that the gravity rule is used for the allocation of demand among facilities rather than assuming that each customer selects the closest facility. In addition, we also consider a multiobjective approach, which combines the two objectives. We propose heuristic solution procedures for the problem in the plane. Extensive computational results are presented. Introduction The gravity model, first proposed by Reilly (1931) and later suggested for modeling market share in competitive situations by Huff (1964, 1966), is a spatial distribution model. When customers face a patronage choice among several facilities, it is commonly assumed (such as in location allocation models) that customers select the closest facility (Hotelling 1929). It can be argued that this distribution rule does not reflect reality unless there is central control that forces the selection of the closest facility. Customers may not have full information about distance to all facilities or may not behave in a rational manner. Furthermore, the proximity rule is discontinuous when the distances to the two closest facilities are comparable. In addition, if each demand point represents an area, the closest facility to some customers may be different than the closest facility to other customers in the same area. This phenomenon is termed as source C in aggregation error by Hillsman and Rhoda (1978). For a review of aggregation issues see Francis et al. (2004a, b, in press). According to the proximity rule, a facility is selected according to distances measured to the center of the area representing demand. Thus we can safely assume that not all customers at any location patronize the same facility. Correspondence: Zvi Drezner, College of Business and Economics, California State University-Fullerton, Fullerton, CA zdrezner@fullerton.edu Submitted: March 17, Revised version accepted: May 25, Geographical Analysis 38 (2006) r 2006 The Ohio State University 391

2 Geographical Analysis It is therefore more realistic to assume a spatial distribution of demand according to some probabilistic rule such as the gravity rule. This means that instead of assigning all customers at a demand point to the same facility (the closest one), customers are allocated to the competing facilities in inverse proportion to a nondecreasing function of the distance. The distance decay function can be a power of the distance (as suggested by Huff 1964, 1966), an exponential decay (Wilson 1976; Hodgson 1981), or other functional relationships (Bell, Ho, and Tang 1998). Drezner (1994a, b, 1995a) and Drezner and Drezner (1996) suggested alternative distributions of demand governed by a utility function, a random utility function and the gravity model. Spatial interaction and spatial choice models are described in detail with a comprehensive literature review in Fotheringham and O Kelly (1989) (see also Yamashita 1993). Such models are based on site attributes (independent of location), situation attributes (relative distances to alternative choices), and separation attributes (the distances between customers and facilities). Probabilistic approaches include the random utility maximization and the Logit model (Fotheringham and O Kelly 1989; Drezner and Drezner 1996; Drezner, Wesolowsky, and Drezner 1998). Fotheringham (1983, 1986) combined features of the random utility model and the gravity model. In the gravity model it is assumed that the probability of patronizing a facility (facility utility) is proportional to the attractiveness of the facility and inversely proportional to a function of the distance. Fotheringham calculated a similar expression using the random utility function. Both models, then, calculate the probability of patronage of a given facility by dividing the utility value calculated for that facility by the sum of the utility values for all competing facilities. The gravity rule was applied to hub selection (Drezner and Drezner 2001), to the p-median problem on a network (Drezner and Drezner in press), and to location in discrete space (Hodgson 1978; O Kelly and Storbeck 1984). Gravity-based models are suitable for situations where there is no central control and customers freely select a facility to patronize. When there is central control, such as in public schools, customers are assigned to the closest facility, thus satisfying the objective of minimizing the total distances traveled. In such cases, the proximity rule is the appropriate objective. When there is no central control, customers do not necessarily select the closest facility and a probabilistic rule such as the gravity-based model is the appropriate one. In public service-oriented applications such as post office or Department of Motor Vehicles branches, the gravity p-median model seeks to provide the best service in terms of driving distances while the gravity equity model seeks to evenly distribute demand, or equalize facility load, so that customers do not have to wait in long lines because one facility is busier than another. The p-median problem is well researched in the literature (see, e.g., on the network: Daskin 1995; Current, Daskin, and Schilling 2002, and in the plane: Love, Morris, and Wesolowsky 1988). The objective is to minimize the total travel distance (or the average distance) for all customers using the proximity rule. 392

3 Tammy Drezner and Zvi Drezner Multiple Facilities Location in the Plane In the equity model proposed in this article, equity is defined as equitable load distribution across facilities taking the facilities perspective rather than equidistance for customers. Minimizing the variance of loads is achieved by calculating the average load, which is independent of the locations of the facilities (it is the total demand divided by the number of facilities), and minimizing the sum of squares of deviations from this average. Attempting to find a distribution of loads with small deviations from the average was suggested in Church and Murray (1993). The equity model is reminiscent of the voting districting problem (Hess et al. 1965; Garfinkel and Nemhauser 1970; Meholtra, Johnson, and Nemhauser 1998) and to the sales territories delineation problem (Kalcsics, Melo, and Nickel 2002). In the voting districting problem districts need to be delineated so as to equalize the number of voters across districts. In the sales territories problem the territories need to be of comparable buying power or sales potential. In both the voting districting and the sales territories problems the districts need to be connected and form simple shapes. The gravity model is inappropriate for modeling these two problems because there is a central control determining where each voter votes or which customers are assigned to each territory. Models involving minimization of the variance of distances traveled by customers are addressed on the network in Maimon (1986) and in the plane in Carrizosa (1997, 1999) and Drezner, Drezner, and Salhi (2006). The equity model is closely related to the capacitated p-median (van Roy 1986) or p-center problems (Mirchandani and Francis 1990; Drezner 1995b). The equity model is equivalent to finding the lowest capacity, which yields a feasible solution to the capacitated problem. For a discussion of equity models see Marsh and Schilling (1994), Eiselt and Laporte (1995), Berman, Drezner, and Wesolowsky (2003), and Drezner (2004). In the current article, we consider location of multiple facilities in the plane using Euclidean distances. Three problems are analyzed: (a) the p-median (location allocation) problem in the plane using the gravity model, (b) the equity model in the plane using the gravity model (analyzed in a network environment in Berman, Drezner, and Wesolowsky 2003) where the objective is to distribute demand as uniformly as possible among the facilities, and (c) the multiobjective model which incorporates both objectives. The multiobjective approach finds a solution close to the p-median solution (which has the minimal possible cost), while increasing the cost by a relatively small amount to achieve a more equitable solution. Another approach to satisfying both objectives in consolidation of public schools was suggested by Diamond and Wright (1987) who suggested a multiobjective model and Church and Murray (1993) who pointed out that the Diamond and Wright (1987) model tends to recommend closing small schools which is inequitable and propose an improved algorithm to eliminate this bias. In one of their models they attempt to minimize the deviations of the individual utilization values from the average utilization, which is the same concept employed by the variance minimization model. 393

4 Geographical Analysis The solution to the equity model presented in this article tends to be dispersed in the region, which is similar in many ways to the p-dispersion or obnoxious facilities models. In the p-dispersion model (Kuby 1987) we wish to locate p facilities in a bounded region such that the minimum distance between any two facilities is maximized. In obnoxious facility models (Church and Garfinkel 1978; Hansen, Peeters, and Thisse 1981) we wish to locate one or more facilities that are as far as possible from a given set of demand points. The article is organized as follows: in the next section, we present the basic formulas for the three models. In the following section, the models are formulated and in section that follows, solution procedures are presented. In the penultimate section computational experiments are reported and we conclude in the last section. Notation and basic formulas For flexibility in the modeling we define for each facility an attractiveness level, and a capacity (or factor to determine equity). For most applications both the attractiveness level and the capacity factor are assumed to be equal to one. In the classical p-median model, for example, all facilities are equally attractive and have equal capacity. Equal attractiveness is implicitly assumed in the proximity rule. Let: n be the number of demand points p be the number of facilities (a i, b i ) be the location of demand point i for i 5 1,..., n (x j, y j ) be the unknown location of facility j for j 5 1,...,p X be the vector of locations (x i, y i ), for j 5 1,..., p d ij be the distance between demand point i and facility j for i 5 1,..., n; j 5 1,..., p w i be the demand generated at demand point i for i 5 1,..., n A j be the attractiveness of facility j for j 5 1,..., p (normally equal to 1) a j be the factor (inverse of capacity) of facility j for j 5 1,..., p (normally equal to 1) l be the power to which the distance is raised By the gravity rule: The proportion of the demand generated at demand point i that patronizes facility k is A k d r¼1 A rd ir ð1þ 394

5 Tammy Drezner and Zvi Drezner Multiple Facilities Location in the Plane The average distance traveled by a customer residing at demand point i is A k d P k¼1 p d r¼1 A r d ir P ¼ Xp p A k d P k¼1 p r¼1 A r d ir k¼1 The total demand served by facility k is w i A k d r¼1 A rd ir A k d r¼1 A rdir d ¼ k¼1 A kd 1 k¼1 A kd ð2þ ð3þ The models Three models are analyzed: The p-median model, the minimum variance model, and a multiobjective model, which incorporates these two models. The location allocation (p-median) model In the location allocation (p-median) model customers total travel distance is minimized. By equation (2) the total distance T(X) traveled is P T ðxþ ¼ Xn p k¼1 w A kd 1 i k¼1 A ð4þ kd The gradient of T(X) is calculated in Appendix A. In Drezner and Drezner (in press) several properties of the gravity p-median model on the network were proven. The same properties are also valid for the gravity p-median model in the plane. The proofs are based on the property that the weighted mean of distances is greater than or equal to the minimum distance with equality holding for p 5 1. The properties are: Property 1. The value of the objective function of the standard p-median problem is not higher than the value of the objective function for the gravity p-median problem. Property 2. The gravity 1-median problem is identical to the standard 1-median problem. Property 1 demonstrates that central control leads to a better value of the objective function. Property 2 implies that, when a location of one facility is sought, the two models result in the same location solution. Minimum variance model The total demand patronizing all facilities is constant: P n w i. If all loads are proportional to the factor a i, the ideal demand allocated to facility k is a k = r¼1 a r P n w i (or 1=p P n w i for equal factors). Minimizing the sum of squares of 395

6 Geographical Analysis deviations of the actual demand from the desired demand is (and defining F(X) and F k (X)): FðXÞ ¼ Xp k¼1 w i A k d r¼1 A rdir a k r¼1 a r w i! 2 ¼ Xp k¼1 F k ðxþ ð5þ Note that the variance of the demand loads at the facilities, for equal factors, is F(X)/p or F(X)/(p 1) for sample variance. The gradient of F(X) is calculated in Appendix A. For the minimum variance problem there are two types of optimum with a zero variance: Type 1. If all facilities are located at the same point, demand is evenly distributed among all facilities and thus the variance of the allocated demand is zero. Type 2. If all facilities are located at infinity, demand is also evenly distributed among the facilities with a zero variance. We experimented extensively with the gradient search to solve the minimum variance problem and repeatedly obtained a zero variance solution. Examining the solutions obtained we found that in all cases, the solution was a Type 2 one, that is, the facilities were located at a great distance from the set of demand points. The gradient appears to push the facilities away from the cluster of demand points rather than attract them to a common location. Such a solution is a poor location solution in terms of the quality of service provided by the facilities. The average distance customers have to travel to the facilities is unnecessarily excessive. Similar results for the Lorenz equity measure were obtained by Drezner (2004) for the location of casualty collection points. Note that the Type 1 optimal solution where all facilities are located at one point is not a good location solution either. The best Type 1 solution is to locate all facilities at the 1-median solution. If the 1-median location solution is used, the advantage of having multiple facilities is lost, and one might as well utilize only one facility. It follows that, in order to obtain equity, all customers receive poor service. A multiobjective approach As the minimum variance problem is not an interesting one by itself, we investigated the multiobjective of the p median combined with the minimum variance. We define the objective function of H(X) 5 T(X) 5 yf(x) for some y40. The gradient of H(X) is simply HHðXÞ ¼HT ðxþþyhf ðxþ 396

7 Tammy Drezner and Zvi Drezner Multiple Facilities Location in the Plane Objective Function diagonal axis Distance from Center Figure 1. The value of the objective function for the square example. Solution procedure We solve the three models by using the gradient search method, a technique commonly used for the optimization of an objective function with no constraints in a continuous space (see Appendix A for details). Running the gradient search on the p-median (or the multiobjective) model we observed cases where the number of iterations was vary large, occasionally exceeding 100,000, compared with about 100 iterations for most problems. Examining these special cases we found that at least one facility was located at a great distance from the cluster of demand points and its progress according to the gradient was very slow. As all facilities were generated in the convex hull of the demand points, we were puzzled as to the reason for this phenomenon. To better understand the reasons for this phenomenon, we analyzed a case of locating one facility to serve a square of side 2 centered at the origin with uniform demand and a second facility located at the origin. The objective function is the p median. Details of the analysis are given in Appendix B. An examination of Fig. 1 reveals that the value of the p-median objective function decreases when the second facility is located farther than approximately 2.5 units from the center of the square. It is plausible that under different configurations, the decrease in the value of the objective function starts inside the square. This analysis provides an explanation to the rare occurrences in which one facility is moved away from the square to infinity causing the excessive number of iterations, which in turn, leads to a lengthy computational effort and inferior solutions. Note that this phenomenon is not an issue in the standard p-median problem where the value of the objective function cannot decrease when a facility is located farther away from the convex hull of demand points. In fact, once a facility is farther 397

8 Geographical Analysis from the boundary of the convex hull than the diameter of the convex hull, no demand point is using its services and the objective function remains constant beyond that point. As a result of this analysis, we implemented the following modification to the gradient search: in every iteration it is checked whether a facility is located outside the convex hull. If this is encountered: (i) (ii) (iii) the present solution is discarded, all the facilities are randomly generated inside the convex hull (yielding a completely new starting solution), and the iterations continue. With this modification, we did not experience any instances of excessively long run times. Generating dispersed starting solutions As there is no real advantage in locating two facilities next to one another, we expect the solution to both the p-median and the multiobjective models to be dispersed. Therefore, we first experimented with generating dispersed starting solutions following the approach used in Drezner, Drezner, and Salhi (2002). A parameter K is used. The location for the first facility is randomly generated. We repeat the following for the location of the rest of the facilities: K candidate locations are randomly generated; the candidate location, which is farthest from the already selected locations is chosen as the starting location for the facility. Note that K 5 1 is the standard way of generating starting solutions. In Table 1 we report the results for K 5 1,..., 10, with each instance of K solved 10 times from 10 randomly generated starting solutions. The best solution found for each problem for all 10 K s is the best found (BF) solution. This BF solution may be improved in subsequent experiments to the best known (BK) solution. Table 1 depicts: (i) (ii) (iii) (iv) the number of problems for which the BF solution was found at least once, the average percent of the BF solution for each K over the BF for the 44 problems, the average percent of the average solution for each K over the BF for all 44 problems, and the total time in hours it took to solve all 44 problems 10 times each. We selected for the experiments K 5 4 which seems to perform best (see Table 1). Computational experiments Fortran programs were coded, compiled by Microsoft Fortran PowerStation 4.0, and were run on a 2.8 MHz Pentium IV computer with 256 Mb RAM. Problems were randomly generated in a unit square, with w i 5 1, i 5 1,..., n, A j 5 1, j 5 1,...,p, and a j 5 1, j 5 1,...,p. Problems were generated for n 5 10, 50, 100, 1000, 398

9 Tammy Drezner and Zvi Drezner Multiple Facilities Location in the Plane Table 1 Results for Various Values of K Property K 5 1 K 5 2 K 5 3 K 5 4 K 5 5 K 5 6 K 5 7 K 5 8 K 5 9 K 5 10 The p-median problem # BF (of 44) % over BF % of average over BF Total runtime (h) The multiobjective problem # BF (of 44) % over BF % of average over BF Total runtime (h)

10 Geographical Analysis Table 2 Best-Known Results N p p-median Multiobject N p p-median Multiobject , , , , , , , , , and 10,000 demand points. The number of facilities to be located is p 5 2, 3, 4, 5, 10, 20, 50, 100 with the stipulation that 5p n, yielding 44 different problems. We did not experiment with the location of one facility because the 1-median in the plane is a convex problem. Note that one set of demand points was generated for each n and was used for all p s with the same n. If during the iterations a facility is located outside the square, the solution is abandoned and a new random starting solution is generated. For the multiobjective model we normalized the variance objective by dividing F(X) by the mean weight n/p, that is, using y 5 p/n. This normalization borrows from the formula for the w 2 test. We ran each problem 100 times. In Table 2 we report the BK solution for the gravity p-median and the multiobjective models for all 44 problems incorporating the best results found in Table 1. Table 3 depicts the results for each of the 44 problems, each run 100 times using K 5 4. For each of the two objectives, the p-median and the multiobjective, we report: (a) the number of times the BK solution was found, (b) the percentage of the BF solution over the BK solution, (c) the average percentage over the BK 400

11 Tammy Drezner and Zvi Drezner Multiple Facilities Location in the Plane Table 3 Results for both objectives on the test problems N p The p-median The multiobjective # best % best % average Time (s) # best % best % average Time (s) , , , , , , , , , , , , , , , ,

12 Geographical Analysis solution, and (d) the total run time in seconds for running all 100 replications for each problem. Results For the gravity p-median problem, the BK solution was found for 41 of 44 problems. The average of the BF solution over the BK solution is %. The average solution was 0.172% over the BK solution. Total time for all 44 problems, each solved 100 times, is h. For the multiobjective problem, the BK solution was found for 43 of 44 problems. The average of the BF solution is practically the same as the BK solution. The average solution was 0.088% over the BK solution. Total time for all 44 problems, each solved 100 times, is h. In conclusion, applying K 5 4 for generating dispersed starting solutions performed very well. Run times for gravity multiobjective problems are about four to five times slower than those required for the solution of gravity p-median problems. The average of the solutions did not exceed 2% above the BK solution for any of the problems. The gravity multiobjective model provided average results, which are about twice as close to the BK solution than the gravity p-median results. Conclusions In this article we considered the p-median, minimum variance of loads, and the combined multiobjective location models. The objective of the p-median model is to minimize the total cost for customers. The objective of the minimum load variance is to provide an equitable solution for the facilities. Both of these objectives are important and therefore we proposed the multiobjective model. The goal of the multiobjective approach is to consider both objectives and find a solution where the total cost for customers is a bit higher than the minimum possible cost (obtained at the p-median solution) while improving the equity by reducing the variance from its value at the p-median solution. Allocation of demand to facilities is estimated by the gravity rule. In many applications customers do not necessarily use the services of the closest facility to them. The gravity rule divides the customers located at each demand point among all facilities. The gravity rule allocates more customers to closer facilities and only a few of them to far away facilities. A gradient search was constructed for the solution of these problems with very good results. We also incorporated the construction of dispersed starting solutions and analyzed the effect of the construction of the starting solution on the final result of the algorithm. As future research we suggest to solve the multiobjective model proposed in this model in a network environment. 402

13 Tammy Drezner and Zvi Drezner Multiple Facilities Location in the Plane Appendix A Gradient search The following is the standard gradient search. To simplify the notation we used H(X) in the description of the algorithm that stands for either T(X) or F(X). 1. Generate p random locations for the facilities: (x j, y j ) for j 5 1,..., p. 2. Calculate the gradient (dh(x)/dx j, dh(x)/dy j for j 5 1,..., p). 3. If the maximum absolute value of the elements of the gradient is less than a prespecified e, stop with the present (x j, y j ) as the solution. 4. Perform a one-dimensional search on t 0 by the Golden Section search (Zangwill 1969). The objective function to be minimized is H(x j tdh(x)/dx j, y j tdh (X)/dy j for j 5 1,...,p). 5. Update (x j, y j ) by the best t and go to step 2. Calculating the gradient of T(X) We calculate the derivative dt(x)/dx j for 1 j p. dtðxþ dx j ¼ Xn ¼ A j w i k¼1 A kd 2 ð1 lþa j dij ( X p X ) p A k d þ la j dij 1 A k d 1 xj a i d k¼1 k¼1 ij ( ) w i dij 2 ðx j a i Þ X p k¼1 A kd 2 ð1 lþd ij A k d þ l Xp A k d 1 k¼1 k¼1 ð6þ and a similar expression holds for dt(x)/dy j. Calculating the gradient of F(X) We first find the derivative F k (X)byx j for some j. This yields different expressions for j 5 k and j6¼k. For j 5 k ¼ 2 k w i ¼ 2lw i A k d 1 A k d r¼1 A rdir w i a k r¼1 a r A k d r¼1 A rd ir! A k d 1 r¼1 w i w A rdir þ la 2 k d 2l 1 i r¼1 A rdir 2 a k r¼1 a r! r¼16¼k w A rdir i r¼1 A rdir 2 403

14 Geographical Analysis For k ij ¼ 2 Xn w i A k d r¼1 A k ðxþ ¼ 2 lw ia j d 1 X ij ij r¼1 A rd 2 8 >< >: r¼16¼k ir A r d ir A k d a k r¼1 a r jj ¼ k jj 6¼ k w i A k d r¼1 A rd ir! A k d w i w d ij 1 i r¼1 A rdir 2! a k r¼1 a r w i ð7þ According to the chain rule: or df k ðxþ dx j ¼ k ðxþ x j a ij d ij dfðxþ dx j ¼ Xp k ðxþ x j a ij d ij ð8þ and similarly dfðxþ dy j ¼ Xp k ðxþ y j b ij d ij Appendix B Location of two facilities in a uniform square demand To analyze this special case, let one facility be located at the center of the square, and the other facility be located at a point (a, b). We considered two locations for (a, b): (d, 0)or(d, d), and used l 5 2. The value of the p-median objective function is by (4): T ða; bþ ¼ Z 1 Z p 1 þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffi 1 x 2 þy 2 ðx aþ 2 þðy bþ 2 dxdy ð9þ 1 1 þ x 2 þy 2 ðx aþ 2 þðy bþ 2 The value of the objective function in (9) was calculated using the Simpson rule for each dimension. The value of the objective function as a function of the distance from the origin for (a, b) 5 (d, 0) (on the axis) and (a, b) 5 (d, d) (on the diagonal) is depicted in Fig

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