A Fixed-Length Subset Genetic Algorithm for the p-Median Problem

In this paper, we review some classical recombination operations and devise new heuristic recombinations for the fixed-length subset. Our experimental results on the classical p-median problem indicate that our method is superior and very close to the optimal solution. 1 Fixed-Length Subset Recombinations We study the Fixed Length Subset Genetic Algorithm (FLS-GA), whose candidate solutions are represented by the fixed-length subset (FLS ), which can be defined as any subset with a fixed size for a given set. In FLS-GA, we adopt a subset encoding [CHWS97], which uses a list of elements to represent the candidate FLS. [Rad93] studies two pure recombinations for FLS, which are Random Respectful Recombination (RRR) and Random Assorting Recombination (RAR). We extend them to heuristic recombinations. 1. Construct candidate set S′, and inherited pattern s0, from FLSs A and B; 2. Choose sub-optimal FLS from S′ using the heuristic procedure H(S′, s0). Let the result of H(S′, s0) be the child of recombinations; For FLS-GA, we could use RRR and RAR respectively to construct candidate set S′, and inherited pattern s0, leading the following two heuristic recombinations. – Heuristic RRR (H-RRR) The inherited pattern s0 = A∩B and the candidate set S′ = A∪B−A∩B. Thus the size of the inherited pattern |s0| is equal to |A ∩B|, while the size of the candidate set |S′| is |A ∪B| − |A ∩B|; – Heuristic RAR (H-RAR) Each element of the inherited pattern s0, is chosen from s0 with probability q. And each element of the candidate set S′, is chosen from A∩B− s0 with probability p0, or from A∪B−A∩B with probability p1, or from S−A∪B with probability p2. Moreover, to balance the diversity and pattern, an adaptive heuristic recombinations can be designed as follows. E. Cantú-Paz et al. (Eds.): GECCO 2003, LNCS 2724, pp. 1596–1597, 2003. c © Springer-Verlag Berlin Heidelberg 2003 A Fixed-Length Subset Genetic Algorithm for the p-Median Problem 1597 – Threshold H-RAR (T-H-RAR) Different heuristic recombinations are used in different situations of diversity. 1. If the diversity of the current population is less than a threshold H, we adopt H-RAR with a smaller q and bigger p2 to increase the diversity; 2. If the diversity of the current population is larger than a threshold H, we adopt H-RAR with a bigger q and smaller p2 to decrease the diversity; 2 FLS-GA Application to P-Median Problem (PMP) PMP [ODE03] can be formulated as a FLS optimization problem. – Instance: 1. A set S = {v1, .., vm} with m vertices; 2. A positive number p ≤ m; 3. A distance matrix Dm×m, where Dij represents the distance between vertex vi and vertex vj ; 4. A function F (s) = ∑ ∀vi∈S ( min ∀vj∈s Dij), ∀s ⊆ S; – Constraint: |s| = m – Output: A fixed length subset s∗, where s∗ ⊆ S and |s∗| = m. – Objective: To minimize the value of F (s∗); Two heuristic procedures are adoptted for when we applying FLS-GA to PMP. – Decreasing Heuristics (DH ) For a given candidate set S′ and an inherited pattern s0, firstly add all elements of S to the result FLS. Then select the rest (p−|s0|) elements from S′−s0 one by one. In each round, the element, which decreases the objective function most, is added to the result FLS. – Increasing Heuristics (IH ) [ODE03] For a given candidate set S′ and an inherited pattern s0, delete (|S′| − p) elements from S′ one by one. In each round, the element, which is excluded from s0 and increases the objective function the least, is deleted from S′. The experimental results have shown that the FLS-GAs with heuristic recombinations over-perform the GA with pure genetic operations, and that the hybrid heuristic genetic recombinations exhibit the best results over all. For most problems, FLS-GAs with T-H-RAR can obtain solutions that are very close to the optimal solution (the gap < 0.1%).