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A Tabu Search Approach Based on Strategic Vibration for Competitive Facility Location Problems with Random Demands

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A Tabu Search Approach Based on Strategic Vibration for Competitive Facility Location Problems with Random Demands Proceedings of the International MultiConference of Engineers and Computer Scientists 2009 Vol II IMECS 2009, March 18 - 20, 2009, Hong Kong A Tabu Search Approach based on Strategic Vibration for Competitive Facility Location Problems with Random Demands Takeshi Uno∗, Hideki Katagiriy, and Kosuke Katoz Abstract| This paper proposes a new location Based upon the CFLP proposed by Wendell and McK- problem of competitive facilities, e.g. shops and elvey, Hakimi [5] considered CFLPs under the condi- stores, with uncertain demands in the plain. By rep- tions that the DM locates her/his facilities on a network resenting the demands for facilities as random vari- that other competitive facilities were already located. ables, the location problem is formulated to a stochas- Drezner [3] extended Hakimi's CFLPs to the CFLP on tic programming problem, and it is reformulated to the plain that there are DPs and competitive facilities. three deterministic programming problems: expec- As extension of their CFLPs, CFLPs with quality or size tation maximizing problem, probability maximizing problem, and satisfying level maximizing problem. of facilities are considered by Uno et al. [16], Fern´andez After showing that one of their optimal solutions can et al. [4], Bruno and Improta [2], and Zhang and Rush- be found by solving 0-1 programming problems, their ton [21], CFLPs with fuzziness are considered by Moreno solution method is proposed by improving the tabu P´erezet al [10], and CFLPs based on maximal covering search algorithm based on strategic vibration. The are considered by Plastria and Vanhaverbeke [13]. efficiency of the solution method is shown by apply- ing it to numerical examples of the facility location In the above studies of CFLPs, the demands of customers problems. for facilities is represented as definite values. Wagnera et al. [19] considered facility location models with random Keywords: facility location, competitiveness, stochastic demands in a noncompetitive environment. For the de- programming, 0-1 programming, tabu search tails of location models with random demands, the reader can refer to the study of Berman and Krass [1]. 1 Introduction In this paper, we proposes a new competitive facility Competitive facility location problem (CFLP) is one of location problem with random demands by extending optimal location problems for commercial facilities, e.g. Drezner's location model that introduces Huff's attrac- shops and stores, and the objective of a decision maker tive function [8]. Then, the location problem can be for- (DM) for the CFLP is mainly to obtain as many demands mulated as a stochastic programming problem. The prob- for her/his facilities as possible. Mathematical studies on lem is reformulated to the three deterministic program- CFLPs are originated by Hotelling [7]. He considered the ming problems: expectation maximizing problem, proba- CFLP under the conditions that (i) customers are uni- bility maximizing problem, and satisfying level maximiz- formly distributed on a line segment, (ii) each of DMs can ing problem. Because the problems are nonlinear pro- locate and move her/his own facility at any times, and gramming problems, it is difficult to find a strict optimal (iii) all customers only use the nearest facility. CFLPs solution of the problems directly. We show that the prob- on the plain were studied by Okabe and Suzuki [12], etc. lems can be reformulated as 0-1 programming problems, As an extension of Hotelling's CFLP, Wendell and McK- and propose their solution method improving the tabu elvey [20] assumed that there exist customers on a finite search algorithm with strategic vibration, which Hanafi number of points, called demand points (DPs), and they and Freville [6] proposed for multidimensional knapsack considered the CFLP on a network whose nodes are DPs. problems. For details of the tabu search algorithms, the readers can refer to the book of Reeves [14]. We ap- ∗Manuscript received December 24, 2008. Faculty of Inte- ply it to numerical examples of the CFLPs with random grated Arts and Sciences, the University of Tokushima, 1-1, demands, and show its efficiency by comparing to other Minamijosanjima-cho, Tokushima-shi, Tokushima, 770-8502 Japan (Email: [email protected]) solution algorithms. yGraduate School of Engineering, Hiroshima University, 1-4- 1, Kagamiyama, Higashihiroshima-shi, Hiroshima, 739-8527 Japan The remaining structure of this article is organized as fol- (Email: [email protected]) lows. In Section 2, we formulate the CFLP with random zGraduate School of Engineering, Hiroshima University, 1-4- demand as a stochastic programming problem. Since it is 1, Kagamiyama, Higashihiroshima-shi, Hiroshima, 739-8527 Japan difficult to solve the formulated problem directly, we show (Email: [email protected]) ISBN: 978-988-17012-7-5 IMECS 2009 Proceedings of the International MultiConference of Engineers and Computer Scientists 2009 Vol II IMECS 2009, March 18 - 20, 2009, Hong Kong that one of its optimal solutions can be found by solv- For finding an optimal solution of (3), we consider the ing a 0-1 programming problem in Section 3. In Section following three deterministic programming problems: (i) 4, we propose an efficient solution method based upon expectation maximizing problem tabu search algorithms by utilizing characteristics of the } maximize E[f(x)] CFLPs. We show the efficiency of the solution method (4) by applying to numerical examples of the CFLPs with subject to x 2 R2n random demands in Section 5. Finally, in Section 6, con- cluding comments and future extensions are summarized. (ii) probability maximizing problem } maximize P r[f(x) ≥ f ] 2 Formulation of CFLP with random de- 0 (5) subject to x 2 R2n mands where f0 means a given satisfying level of obtaining BP, In the proposed CFLPs, we assume that all customers 2 and (iii) satisfying level maximizing problem only exist on DPs in plain R . For convenience sake, by 9 aggregating all customers on the same DP, we regard one maximize f0 = DP as one customer. ≥ ≥ subject to P r[f(x) f0] α ; (6) x 2 R2n There are n DPs in R2, and let D = f1; ··· ; ng be the set of indices of the DPs. Let m be the number of new where α is a given satisfying level of probability that the facilities that the DM locates, and k be the number of DM can obtain BP level f . competitive facilities which have been already located 0 in R2. The sets of indices of the new facilities and the Problems (4), (5), and (6) are nonconvex nonlinear pro- competitive facilities are denoted by F = f1; : : : ; mg and gramming problems and we need to find at least one opti- FC = fm + 1; ··· ; m + kg, respectively. mal solution for each of the problems. However, for most CFLPs [3, 16], it is difficult to find an optimal solution Let u 2 R2 be the site of DP i 2 D, and x 2 R2 and i j by using general analytic solution methods with derived q > 0 be the site and quality of facility j 2 F [ F , j C functions of the objective function, Kuhn-Tucker condi- respectively. Then, attractive power of facility j for DP tions, etc. Moreover, Uno and Katagiri [17] and Uno et i is represented as the following function introduced by al. [18] show that it is also difficult to find an optimal so- Huff [8]: 8 lution of such CFLPs by using heuristic solution methods < qj jj − jj for nonlinear programming problems, e.g. genetic algo- 2 ; if ui xj > "; ≡ jjui − xjjj rithm for numerical optimization for constrained problem ai(xj; qj) q (1) : j jj − jj ≤ (GENOCOP) [9]. In the next section, we show that the 2 ; if ui xj "; " above three problems can be reformulated as 0-1 pro- where " > 0 is an upper limit of the distance that cus- gramming problems. tomers can move without any trouble. It is assumed that all customers only use one facility with the largest at- tractive power, and if the two or more attractive powers 3 Reformulation to combinatorial opti- are the same, they use the facility in reverse numerical mization problems order of the indices of facilities; that is, in the order of competitive facilities and new facilities. In the location model introduced in the previous section, if the new facilities are located, then the values of (2) for Let x = (x1;:::; xm) be the location of the new facilities. all facilities and DPs are given. On the other hand, we Then we use the following 0-1 variable for representing propose the following solution method: whether DP i uses new facility j 2 F : { 1; if DP i uses the new facility j; 1. Decide the set of DPs that she/he wants to obtain 'j(x) = (2) i 0; otherwise: their BPs preferentially by giving the values of (2) for all facilities and DPs, and Letw ¯i be the random variable meaning the buying power 2 (BP) of DP i. New facility j F can obtain the BPw ¯i 2. Find the location of the new facilities such that (2) j if 'i (x) = 1. The objective of the DM is maximizing the for each facility and DP is equal to or more than the sum of BP that all the new facilities obtain. Then, the given value. CFLP with random demand is formulated as the follow- ing stochastic programming problem: 9 For DP i 2 D, the largest attractive power among all Xn Xm => competitive facilities is denoted as follows: maximize f(x) = w¯ 'j(x) i i (3) i=1 j=1 ;> C ≡ f g 2 2m ai min ai(xj; qj) : (7) subject to x R j2FC ISBN: 978-988-17012-7-5 IMECS 2009 Proceedings of the International MultiConference of Engineers and Computer Scientists 2009 Vol II IMECS 2009, March 18 - 20, 2009, Hong Kong From (1), the set of DPs that new facility j 2 F cannot Proof: Let x∗ be an optimal solution of (4).
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