Abstract
The multidemand multidimensional knapsack problem (MDMKP) is a significant generalization of the popular multidimensional knapsack problem with relevant applications. In this work we investigate for the first time how solution-based tabu search can be used to solve this computationally challenging problem. For this purpose, we propose a two-stage search algorithm, where the first stage aims to locate a promising hyperplane within the whole search space and the second stage tries to find improved solutions by exploring the reduced subspace defined by the hyperplane. Computational experiments on 156 benchmark instances commonly used in the literature show that the proposed algorithm competes favorably with the state-of-the-art results. We analyze several key components of the algorithm to highlight their impacts on the performance of the algorithm.
Highlights
Given a set V = {1, 2, . . . , n} of n items, a set R = {r1, r2, . . . , rm} of m resources with a capacity upper limit bi for resource ri (1 ≤ i ≤ m), where each item j of V is associated with a profit cj and consumes a given quantity aij for each resource ri (i ∈ {1, 2, . . . , m}), the popular NP-hard 0–1 multidimensional knapsack problem (MKP) involves selecting a subset of items from V such that the resource consummation of the selected items does not exceed the given capacity upper limit for each resource in R, while maximizing the total profit of the selected items
We employed four sets of benchmark instances to assess the performance of our two-stage solution-based tabu search (TSTS) algorithm, where the first two sets of benchmark instances are available at http://www.optsicom.es/binaryss, and the third and fourth sets of benchmark instances are available in OR-Library 1
We provide in the Appendix the computational results of TSTS on the third and fourth sets of benchmark instances for which no detailed results are available for existing algorithms in the literature
Summary
Rm} of m resources with a capacity upper limit bi for resource ri (1 ≤ i ≤ m), where each item j of V is associated with a profit cj and consumes a given quantity aij for each resource ri M}), the popular NP-hard 0–1 multidimensional knapsack problem (MKP) involves selecting a subset of items from V such that the resource consummation of the selected items does not exceed the given capacity upper limit for each resource in R (knapsack constraints), while maximizing the total profit of the selected items. The MKP can be written as follows: n (M KP ) Maximize z = cjxj (1) j=1 n s.t. aijxj ≤ bi, ∀i ∈ {1, 2, . The MDMKP can be formulated as follows [1,5]:
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