An iterative pseudo-gap enumeration approach for the Multidimensional Multiple-choice Knapsack Problem

Chao GAO, Guanzhou LU, Xin YAO, Jinlong LI

Research output: Journal PublicationsJournal Article (refereed)peer-review

20 Citations (Scopus)


The Multidimensional Multiple-choice Knapsack Problem (MMKP) is an important NP-hard combinatorial optimization problem with many applications. We propose a new iterative pseudo-gap enumeration approach to solving MMKPs. The core of our algorithm is a family of additional cuts derived from the reduced costs constraint of the nonbasic variables by reference to a pseudo-gap. We then introduce a strategy to enumerate the pseudo-gap values. Joint with CPLEX, we evaluate our approach on two sets of benchmark instances and compare our results with the best solutions reported by other heuristics in the literature. It discovers 10 new better lower bounds on 37 well-known benchmark instances with a time limit of 1 hour for each instance. We further give direct comparison between our algorithm and one state-of-the-art “reduce and solve” approach on the same machine with the same CPLEX, experimental results show that our algorithm is very competitive, outperforming “reduce and solve” on 18 cases out of 37. © 2016 Elsevier B.V.
Original languageEnglish
Pages (from-to)1-11
Number of pages11
JournalEuropean Journal of Operational Research
Issue number1
Early online date29 Nov 2016
Publication statusPublished - Jul 2017
Externally publishedYes

Bibliographical note

We thank Yuning Chen and Jin-Kao Hao for answering questions about their solvers PEGF and PERC. We are grateful to the anonymous reviewers for their constructive comments, which have been very helpful for us in improving the quality and presentation of this paper. This research work was supported by the National Natural Science Foundation of China under Grants 61573328 and 61329302. Xin Yao was supported by a Royal Society Wolfson Research Merit Award.


  • Heuristics
  • Integer programming
  • Multidimensional Multiple-choice Knapsack
  • Reduced cost constraint


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