A Memetic Algorithm Based on Probability Learning for Solving the Multidimensional Knapsack Problem

The multidimensional knapsack problem (MKP) is a well-known combinatorial optimization problem with many real-life applications. In this article, a memetic algorithm based on probability learning (MA/PL) is proposed to solve MKP. The main highlights of this article are two-fold: 1) problem-dependent heuristics for MKP and 2) a novel framework of MA/PL. For the problem-dependent heuristics, we first propose two kinds of logarithmic utility functions (LUFs) based on the special structure of MKP, in which the profit value and weight vector of each item are considered simultaneously. Then, LUFs are applied to effectively guide the repair operator for infeasible solutions and the local search operator. For the framework of MA/PL, we propose two problem-dependent probability distributions to extract the special knowledge of MKP, that is, the marginal probability distribution (MPD) of each item and the joint probability distribution (JPD) of two conjoint items. Next, learning rules for MPD and JPD, which borrow ideas from competitive learning and binary Markov chain, are proposed. Thereafter, we generate MA/PL's offspring by integrating MPD and JPD, such that the univariate probability information of each item as well as the dependency of conjoint items can be sufficiently used. Results of experiments on 179 benchmark instances and a real-life case study demonstrate the effectiveness and practical values of the proposed MKP.