Abstract
The discounted {0-1} knapsack problem (DKP01) is a kind of knapsack problem with group structure and discount relationships among items. It is more challenging than the classical 0-1 knapsack problem. In this paper, we study binary particle swarm optimization (PSO) algorithms with different transfer functions and a new encoding scheme for DKP01. An effective binary vector with shorter length is used to represent a solution for new binary PSO algorithms. Eight transfer functions are used to design binary PSO algorithms for DKP01. A new repair operator is developed to handle isolation solution while improving its quality. Finally, we conducted extensive experiments on four groups of 40 instances using our proposed approaches. The experience results show that the proposed algorithms outperform the previous algorithms named FirEGA and SecEGA . Overall, the proposed algorithms with a new encoding scheme represent a potential approach for solving the DKP01.
Highlights
Related WorksSuppose that the search space is D-dimensional, and the position of the ith particle of the swarm can be described using a D-dimensional vector, xi
We conducted extensive experiments on four groups of 40 instances using our proposed approaches. e experience results demonstrate that the proposed algorithms outperform the genetic algorithm and original binary PSO in solving 40 DKP01 instances. e main contributions of this work can be listed as follows: (i) Binary particle swarm optimization algorithms with difference binary transfer functions and new solution presentation are proposed to solve the discounted {0-1} knapsack problem
Eight new algorithms have been proposed based on the binary particle swarm optimization with a new repair operator to solve discounted 0-1 knapsack problem efficiently
Summary
Suppose that the search space is D-dimensional, and the position of the ith particle of the swarm can be described using a D-dimensional vector, xi E binary particle swarm optimization algorithm was introduced by Bansal and Deep to allow the PSO algorithm to operate in binary problem spaces [21,22,23]. It uses the concept of velocity as a probability that a bit (position) takes on one or zero. We propose 8 binary algorithms based on BSO named BPSO1 to BPSO8. e algorithm BPSOx uses transfer function Sx (where x is integer in [1, 8]), and BPSO1–BPSO4 use formula (7), while BPSO5–BPSO8 use formula (8) to calculate binary vector X
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