A Scatter Search Method for the Bi-Criteria Multi-dimensional {0,1}-Knapsack Problem using Surrogate Relaxation

Abstract

Abstract. This paper presents a scatter search (SS) based method for the bi-criteria multi-dimensional knapsack problem. The method is organized according to the usual structure of SS: (1) diversification, (2) improvement, (3) reference set update, (4) subset generation, and (5) solution combination. Surrogate relaxation is used to convert the multi-constraint problem into a single constraint one, which is used in the diversification method and to evaluate the quality of the solutions. The definition of the appropriate surrogate multiplier vector is also discussed. Tests on several sets of large size instances show that the results are of high quality and an accurate description of the entire set of the non-dominated solutions can be obtained within reasonable computational time.Comparisons with other meta-heuristics are also presented. In the tested instances the obtained set of potentially non-dominated solutions dominates the set found with those meta-heuristics. (2000): 90C27. Mathematics Subject Classification

The multiobjective multidimensional knapsack problem: a survey and a new approach

“…We have obtained a large number of results. It is unfortunate that we are not able to obtain the results of Gomes da Silva et al ().…”

“…The solution is then improved by a local search that also uses the linear sum. They compared their method with SPEA2 and MOGLS and showed that they obtained better results. Gomes da Silva et al () adapted their SS for the MOKP (Gomes da Silva et al, ) to also tackle the MOMKP. Surrogate relaxation was used to convert the MOMKP into an MOKP by generating adequate surrogate multipliers.…”

Real Estate Property Maintenance Optimization Based on Multiobjective Multidimensional Knapsack Problem

“…The multidimensional knapsack, as many other combinatorial optimization problems, was intensively investigated within metaheuristics at the end of the 1980s to the beginning of the 1990s (Fréville, ). These last decades, in addition to the classically used evolutionary algorithms (GA, NSGA2, and SPEA, see Lust and Teghem, ), several metaheuristics are used to solve the multidimensional knapsack problem: human learning optimization (Wang et al., ), firefly (Baykasoğlu and Ozsoydan, ), scatter search (Gomes da Silva, ), ant colony optimization (Kong et al., ), fruit fly optimization (Wang et al., ), PSO (Kong and Tian, ; Hembecker et al., ; Labed et al., ; Bansal and Deep, ; Bonyadi and Michalewicz, ; Chih, ), etc. In addition, based on the “no free lunch theorem” (Wolpert and Macready, ), stating that any algorithm is not better than all the others for any problem of optimization, the PSO that is easy to tune has been used as search algorithm.…”

A scatter search method for the multidimensional knapsack problem with generalized upper bound constraints

“…Gholipour-Kanani et al [7] tackled a multicriteria group scheduling problem for a cellular manufacturing system by scatter search. Our paper differs from the previous scatter search knapsack applications [17,23] by (1) utilizing GRASP to populate reference-set solutions, (2) introducing different diversity measurement to capture the GUB structure, and (3) incorporating the benefit-cost ratio (referred to the B/C ratio hereafter) when making crossover combination.…”