Fitness-Based Acceleration Coefficients Binary Particle Swarm Optimization (FACBPSO) to Solve the Discounted Knapsack Problem

Abstract

The discounted {0-1} knapsack problem (D{0-1}KP) is a multi-constrained optimization and an extended form of the 0-1 knapsack problem. The DKP is composed of a set of item batches where each batch has three items and the objective is to maximize profit by selecting at most one item from each batch. Therefore, the D{0-1}KP is complex and has found many applications in real economic problems and other areas where the concept of promotional discounts exists. As DKP belongs to a binary class problem, so the novel binary particle swarm optimization variant with modifications is proposed in this paper. The acceleration coefficients are important parameters of the particle swarm optimization algorithm that keep the balance between exploration and exploitation. In conventional binary particle swarm optimization (BPSO), the acceleration coefficients of each particle remain the same in iteration, whereas in the proposed variant, fitness-based acceleration coefficient binary particle swarm optimization (FACBPSO), values of acceleration coefficients are based on the fitness of each particle. This modification enforces the least fit particles to move fast and best fit accordingly, which accelerates the convergence speed and reduces the computing time. Experiments were conducted on four instances of DKP having 10 datasets of each instance and the results of FACBPSO were compared with conventional BPSO and the new exact algorithm using a greedy repair strategy. The results demonstrate that the proposed algorithm outperforms PSO-GRDKP and the new exact algorithm in solving four instances of D{0-1}KP, with improved convergence speed and feasible solution time.