Solving 0–1 Quadratic Knapsack Problems with a Population-based Artificial Fish Swarm Algorithm

where the coefficients pi, ai(i = 1, 2, . . . , n) and pij(i = 1, 2, . . . , n − 1, j = i + 1, . . . , n) are positive integers and b is an integer such that max{ai : i = 1, 2, . . . , n} ≤ b < ∑n i=1 ai. Here pi is a profit achieved if item i is selected and pij is a profit achieved if both items i and j (j > i) are selected. The goal is to find a subset of n items that yields maximum profit f without exceeding capacity b. The QKP arises in a variety of real world applications including finance, VLSI design, compiler construction, telecommunication, flexible manufacturing systems, locations, hydrological studies. Classical graph and hypergraph partitioning problems can also be formulated as the QKP. Several deterministic [1, 2] as well as stochastic solution methods [4, 8] have been proposed to solve (1). Recently, a population-based artificial fish swarm algorithm that simulates the behavior of the fish swarm inside water was proposed [3, 6]. Applying to the optimization problem, generally a ‘fish’ represents an individual point in a population. Fishes desire to stay close to the swarm, to protect themselves from predators and to look for food, and to avoid collisions within the group. In this paper, we propose a binary version of the artificial fish swarm algorithm for solving (1).