A hybrid approach for solving mixed binary integer programming problems

Abstract

Optimisation appears in many aspects of day to day life and more often, involves integer optimisation of very large scales. Although technology advancements have enabled many combinatorial optimisation problems to be solved exactly, this is only true for small and some of the medium instances. For large instances, they require high computational times and worse, fail to be solved due to the massive usage of the machine’s memory. In this research, we aim to develop a hybrid technique, focusing on solving the MBIP problem rather than finding the best solution for individual problem’s application. Therefore, we proposed a general framework of a hybrid technique that may need minor adjustment when applied to various optimisation problems, in particular to the mixed binary integer programming (MBIP) problems. The hybrid approach proposed in this research is the collaborative combination of the linear programming (LP) relaxation with variable neighbourhood search (VNS). We use LP relaxation solutions to generate initial solutions and use VNS to improve the solutions obtained. To illustrate the flexibility of the proposed method, we implement the proposed method on two similar MBIP problems; the constrained index tracking problem (CITP) and the gas supply chain problem. The proposed hybrid technique generates satisfactory solutions within significantly shorter amount of computational time. For the CITP problem, we compare the obtained solutions with the solutions provided by the CPLEX solver (with time and solution limit imposed) and a genetic algorithm (GA) approach. For most of the instances, our proposed hybrid technique gives better solutions with significant reduction of the computational time compared to the time taken by the CPLEX solver and the GA approach. For the gas supply chain problem, the proposed hybrid technique manage to replicate the solutions generated by the CPLEX solver (with time and solution limit imposed) within a shorter computational time. When we decrease the number of locations that were allowed to supply gas to a specific location, the proposed hybrid technique generated better solutions with lower total costs than the solutions given by the CPLEX solver. The proposed hybrid technique was successfully implemented for both problems by adjusting the optimal LP solutions of the decision variables that are used to guide the search process. Satisfactory solutions were obtained for both problems within a relatively shorter computational time.