Abstract
In this paper, we propose to solve the max-min knapsack problem with multiple scenarios by using an iterative algorithm that uses three main phases: (1) construction phase, (2) improvement phase, and (3) destroying/repairing phase. The first phase yields a (starting) pool of elite solutions for the problem by applying a greedy randomized search. The second phase tries to improve each solution at hand by using an intensification search using path-relinking combined with a look-ahead strategy. The third phase can be viewed as a diversification strategy, where the iterative algorithm tries to avoid premature convergence towards local optima. Finally, the proposed method is evaluated on a set of benchmark instances taken from the literature. Its obtained results are compared to those reached by recent algorithms available in the literature. The computational part shows that the method remains competitive (in term of the quality of solutions achieved), where it is able to provide better bounds than those already published ones.
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Appendix
Appendix
In this part, the bounds reached by IA, for the instances containing more than two-scenarios, are detailed. We recall that instances are composed of two types of instances: weakly and strongly correlated ones and for each type, there are three sub-groups organized following the size of the set I (the number of items), the number of scenarios m and the capacity of the knapsack c. This section presents the best results provided by IA for instances with n varying from 1000 to 20000 and m from 100 to 1000. For each ten instances, (i) the best provided bound (with and without using the path-relinking strategy) is reported according to the parameter \(\beta \) that varies from \(10\%\) to \(20\%\) (as explained in the experimental part, Sect. 4) and (ii) the average value of each ten bounds (corresponding to the ten instances of each sub-group) are displayed on the line labelled “Average” while the line labelled “Total Average” tallies the mean of all achieved bounds for the three sub-groups (Tables 8, 10 and 12, respectively).
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Al-douri, T., Hifi, M. & Zissimopoulos, V. An iterative algorithm for the Max-Min knapsack problem with multiple scenarios. Oper Res Int J 21, 1355–1392 (2021). https://doi.org/10.1007/s12351-019-00463-7
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DOI: https://doi.org/10.1007/s12351-019-00463-7