Initialization strategies to enhancing the performance of genetic algorithms for the p-median problem
Highlights
► We initialize population of a genetic algorithm (GA) from a spatial perspective. ► Optimal medians cluster at most cases for p-median problem. ► Well-organized initial population outperforms a random one in a genetic algorithm.
Introduction
The goal of solving a p-median problem (PMP) is to locate p facilities in a network of n demand nodes (p < n) so that the total distance between each demand node to its nearest facility is minimized. Hakimi (1964) proved that at least one optimal solution exists when all facilities are located on the nodes. Solving the p-median problem exactly is computationally intensive (Kariv & Hakimi, 1979) and a number of meta-heuristics methods have been developed in the literature in order to find high quality (i.e., optimal or near optimal) solutions efficiently. These approaches include vertex exchange (Teitz & Bart, 1968), Lagrangean relaxation (Beasley, 1985, Christofides and Beasley, 1981, Cornueojols et al., 1977), dual formulations (Gilmore and Gomory, 1961, Hanjoul and Peeters, 1985), simulated annealing (Chiyoshi & Galvão, 2000), tabu search (Brimberg & Mladenovic, 1996), multistart (Resende & Werneck, 2004), and genetic algorithms (Alp et al., 2003, Hosage and Goodchild, 1986, Li and Yeh, 2005). Although each meta-heuristic method has its strength, as a general problem solving framework, the genetic algorithm (GA) approach has shown much potential in the recent literature and has been successfully used in many applications (such as Mladenovi et al., 2007, Reese, 2006). Major research efforts in developing hybrid GAs for the p-median problem have been focused on encoding strategies to represent solutions and operations that can be used to generate new solutions.
The standard GA literature generally suggests the use of random solutions as the initial GA population. Random solutions play a critical role in encouraging convergence as a way of maintaining diversity in the population (Goldberg, 1989). While using random initial solutions is important in a standard GA framework, recent research has shown that the combination of good solutions will generally lead to optimal or near-optimal solutions. Xiao (2008) and Bennett, Xiao, and Armstrong (2004) demonstrated that the use of solutions generated based on domain knowledge (i.e., non-random solutions) in the initial population of a hybrid GA can significantly improve the GA performance. For p-median problems, Rosing and ReVelle (1997), for example, have argued that the meta-heuristic concentration can effectively find optimal or near-optimal solutions by combining high quality solutions generated a priori.
The purpose of this paper is to explore the effect of different initialization strategies in enhancing the overall performance of a hybrid GA in finding optimal or near optimal solutions. We develop an algorithm that can be used to generate a high quality solution; the solution is subsequently used to form the initial population for a hybrid GA. After an examination of existing GA-based methods, we choose a hybrid GA called ADE developed by Alp et al. (2003) to demonstrate our idea. According to their experiments, the ADE algorithm outperforms several other GA-based methods (Bozkaya et al., 2002, Rosing et al., 1999) and a simulated annealing approach (Chiyoshi and Galvão, 2000) in terms of either computational time or the optimal solutions found. We note that the initialization strategies developed in this paper is general and does not depend on a specific hybrid GA.
The reminder of the paper is organized as follows. Section 2 introduces related background including the formulation of the PMP, the principle of GA, and the ADE algorithm. Section 3 proposes a procedure of generating a well-organized initial population. Section 4 defines three different scenarios and compares their performances with a series of experiments. Section 5 concludes the paper.
Section snippets
p-Median problem (PMP)
Let i and j be the indices of any two vertices on a network of n vertices, dij be the distance of the shortest network path between vertexes i and j, decision variable xjj be 1 if a facility is located at vertex j and 0 otherwise, and decision variable xij be 1 if the demand at vertex i is assigned to the facility at vertex j and 0 otherwise. With the above notation, an unweighted and uncapacitated p-median problem can be formulated as follows (ReVelle & Swain, 1970):
Generating the initial GA population
The proposed methodology aims at optimize the generation of initial population to improve the performance of a hybrid GA. In this section, we first described a greedy algorithm that can be used to quickly generate a high-quality solution. We then discuss a local search method that can be used to systematically generate a set of new solutions through local substitution of the vertices in the solution found by the greedy algorithm. A new formula for determining population size is also proposed.
Problems and results
The forty p-median problems with known optimal solutions from the OR-Library (Beasley, 1990) are employed for our experiments. The numbers of vertexes (n) range from 100 to 900, while the p value from 5 to 200. A C# program was developed to implement the proposed approach and ADE. The program is executed in a computer with a 2.4 GHz Intel Core 2 Duo CPU and 2 GB main memory. Three scenarios are used for comparison with different population sizes and methods of generate an initial population.
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ADE:
Conclusion
The research developed in this paper introduces an approach to the optimization of initial population in a GA-based algorithm for the PMP. First, a greedy search algorithm is employed to find an initial near optimal solution. For the 40 PMPs provided by OR Library, the fitness values of the initial solutions are produced by the greedy search algorithm. Then, the initial solution is treated as a seed to produce other solutions. All solutions have the same number of sorted medians and medians at
Acknowledgments
This research was sponsored by National Natural Science Foundation of China (No. 40730526), Scientific Research Starting Foundation for Returned Overseas Chinese Scholars (Ministry of Education, China), Director Grant of Key Lab of Geographical Information Science (No. KLGIS2011C01), and Shanghai Natural Science Foundation (No. 11ZR1410100). The authors are also grateful to the anonymous referees for their helpful comments and suggestions.
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