A Metaheuristic Approach for a Two-dimensional Fuzzy Version of the Variable Size and Cost Bin Packing Problem

The Variable Size and Cost Bin Packing Problem (VSCBPP) focuses on minimizing the overall cost of containers used to pack a specified set of items. This problem has significant applications across various fields, including energy, cargo transport, and informatics, among others. Most research conducted on this problem has concentrated on enhancing solution methodologies. Recently, some studies have investigated the use of fuzzy approaches to VSCBPP, which allow for the relaxation of certain constraints. In this paper, we introduce a metaheuristic method for solving the fuzzy version of VSCBPP, facilitating the simultaneous relaxation of two constraints: the overloading of containers and the exclusion of specific items from the packing process. Consequently, this two-dimensional fuzzy relaxation of the VSCBPP enables us to derive a range of solutions that present varying trade-offs between cost and the satisfaction levels of the original constraints. We employ mechanisms from the multi-objective metaheuristic approach to maximize the degrees of relaxation while minimizing the original cost function. To demonstrate the efficacy of our proposed solution, we utilized two well-known multi-objective evolutionary P-metaheuristics (Multi-Objective Genetic Algorithm and NSGA-II) and two S-metaheuristics (Multi-Objective Local Search and Ulungu Multi-Objective Simulated Annealing) specifically tailored for the fuzzy version of the VSCBPP. Computational experiments were conducted on 39 instances to validate the effectiveness of this approach.