Multicriteria 0-1 knapsack problems with k-min objectives

Authors:
Aiying Rong;Kathrin Klamroth;José Rui Figueira
Affiliations:
Cemapre (Center of Applied Mathematics and Economics), ISEG-Technical University of Lisbon, Rua do Quelhas 6, 1200-781 Lisboa, Portugal;Department of Mathematics and Natural Sciences, University of Wuppertal, Gauístr. 20, 42119 Wuppertal, Germany;CEG-IST, Instituto Superior Técnico, Av. Rovisco Pais, 1049-001 Lisboa, Portugal
Venue:
Computers and Operations Research
Year:
2013

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Abstract

This paper deals with the multicriteria 0-1 knapsack problem (KP) with k-min objectives (MkMIN-KP) in which the first objective is of classical sum type and the remaining objectives are k-min objective functions. The k-min objectives are ordinal objectives, aiming at the maximization of the k th smallest objective coefficient in any feasible knapsack solution with at least k items in the knapsack. We develop efficient algorithms for the determination of the complete nondominated set of MkMIN-KP. The MkMIN-KP can be transformed into a series of single objective 0-1 multidimensional KPs (mKPs) with cardinality constraints based on the @e-constraint method. The resulting mKPs are NP-hard. Therefore, it is necessary to solve the mKP efficiently. In this paper, we first develop different variants of hybrid algorithms for the mKP considering the properties of the MkMIN-KP problem and suggest a hybrid two stage solution approach that is the synthesis of various solution techniques such as linear programming relaxation, dynamic programming (DP), bounding techniques, state dominance relations, core concept and greedy principle. Numerical experiments with different types of MkMIN-KP instances with two k-min objectives show that the hybrid algorithm can find the complete nondominated set significantly faster with much less memory requirements than a sequential DP algorithm. We also evaluate the effect of different combinations of k values of the k-min objectives on the cardinality of the nondominated set, solution time and memory requirements of the algorithm.