A new discrete electromagnetism-based meta-heuristic for solving the multidimensional knapsack problem using genetic operators

Abstract

The Standard Electromagnetism-like Mechanism (SEM) is one of the swarm-based optimization methods which is examined in this paper. The SEM works based on the charges in electrons and hence its operators have been especially designed for continuous space problems. Although the SEM was successfully applied to the standard optimization problems, it was not that notable when it came to tackling discrete space problems. This shortcoming was obvious when the SEM was applied to some standard discrete problems such as Travelling Salesman Problem, Nurse Scheduling Problem, etc. In this paper, a modified SEM called Discrete Electromagnetism-like Mechanism is proposed which utilizes Genetic Algorithm (GA) operators to work in discrete spaces. In fact, the vector calculations (which are at the heart of the SEM) in the SEM are replaced by specific types of GA operators to determine the effects that particles have on one another. Also, a new operator based on the principles of quantum mechanics is proposed which further improves the performance of the method. In our experiments, the proposed algorithm is applied to a well-studied discrete space problem called Multidimensional Knapsack Problem (MKP). All tests are done on standard problems of the MKP and the results are reported and compared with several stochastic population-based optimization methods. Experiments showed that the proposed algorithm not only found comparable (and even better in some cases) solutions for the standard problems of the MKP, but also took much less computational time (75% improvement in average in comparison to other methods).

Appendix

Here, the best found solutions by the AMMDEM + ls are reported. Due to the results of the test benches 5–100, and 10–100 were solved to optimality (Chu and Beasley 1998), these results are not reported here. It is worthwhile to note that the AMMDEM + ls could find these values too. Also, the best results of test benches 5–500, 10–500, and 30–500 have been reported in Boussier et al. (2010), Vasquez and Vimont (2005) and are not reported in this appendix. The results of other test benches (5–250, 10–250, 30–100, and 30–250) are reported here. In 10–250 and 30–250 test benches, the AMMDEM + ls could find optimal solutions as they can be find in Boussier et al. (2010). These values are distinguished by star. Also, the results which were better than GA_CB have been bolded - italicized (Table 8).

Table 8 The results of 5–250, 10–250, 30–100, and 30–250, found by AMMDEM + ls