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

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.