Abstract
The discounted {0–1} knapsack problem may be a kind of backpack issue with gathering structure and rebate connections among things. A moth-flame optimization algorithm has shown good searchability combined with an effective solution presentation designed for the discounted {0-1} knapsack problem. A new encoding scheme used a shorter length binary vector to help reduce the search domain and speed up the computing time. A greedy repair procedure is used to help the algorithm have fast convergence and reduce the gap between the best-found solution and the optimal solution. The experience results of 30 discounted {0-1} knapsack problem instances are used to evaluate the proposed algorithm. The results demonstrate that the proposed algorithm outperforms the two binary PSO algorithms and the genetic algorithm in solving 30 DKP01 instances. The Wilcoxon rank-sum test is used to support the proposed declarations.
Highlights
Academic Editor: Mohammad Yazdi e discounted {0–1} knapsack problem may be a kind of backpack issue with gathering structure and rebate connections among things
A greedy repair procedure is used to help the algorithm have fast convergence and reduce the gap between the best-found solution and the optimal solution. e experience results of 30 discounted {0-1} knapsack problem instances are used to evaluate the proposed algorithm. e results demonstrate that the proposed algorithm outperforms the two binary PSO algorithms and the genetic algorithm in solving 30 DKP01 instances. e Wilcoxon rank-sum test is used to support the proposed declarations
A successful 0-1 vector with 2∗dimensional length is utilized for a solution combined with moth-flame optimization (MFO). is advantageous solution present is first used by Truong [20]. e experience results on 30 discounted {0-1} knapsack problem (DKP01) instances are used to evaluate the proposed algorithm. e results demonstrate that the proposed algorithm outperforms the two binary PSO algorithms and genetic algorithm in solving 30 DKP01 instances: (i) Moth-flame optimization algorithm has shown good searchability combined with an effective solution presentation designed to the discounted {0-1} knapsack problem
Summary
Where vki,d is dth dimension velocity of particle i in cycle k; xki,d is the dth dimension position of particle i in cycle k; pki,d is the dth dimension position of personal best (pbest) of particle i in cycle k; pkg,d is the dth dimension position of global best particle (gbest) in cycle k; w is the inertia weight; c1 is the cognitive weight and c2 is a social weight; and r1 and r2 are two random values uniformly distributed in the range of [0, 1] [23]. When using the MFO algorithm, F can be thought of as M’s best location in the search space. To mathematically model this action, each search agent’s location is modified as follows: Mi SMi, Fj,. Where Mi is the ith search agent and Fj is the jth best position found so far, and S indicates the logarithmic spiral function which is updated as follows:. Where r is a random number in [−1, 1], c is a constant that defines the shape of the logarithmic spiral, and Di factor is the distance of the ith search agent for the jth flame, which is calculated as follows: Di Fj − Mi. Where t is the current iteration number, N is the maximum number of flames, and T is the maximum iteration number
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