Abstract
The two-dimensional heterogeneous vector bin packing problem (2DHet-VBPP) consists of packing the set of items into the set of various type bins, respecting their two resource limits. The problem is to minimize the total cost of all bins. The problem, known to be NP-hard, can be formulated as a pure integer linear program, but optimal solutions can be obtained by the CPLEX Optimizer engine only for small instances. This paper proposes a metaheuristic approach to the 2DHet-VBPP, based on Reduced variable neighborhood search (RVNS). All RVNS elements are adapted to the considered problem and many procedures are designed to improve efficiency of the method. As the Two-dimensional Homogeneous-VBPP (2DHom-VBPP) is more often treated, we considered also a special version of the RVNS algorithm to solve the 2DHom-VBPP. The results obtained and compared to both CPLEX results and results on benchmark instances from literature, justify the use of the RVNS algorithm to solve large instances of these optimization problems.
Highlights
The two-dimensional heterogeneous vector bin packing problem (2DHet-VBPP) can be stated as follows: given the N pairs ( 2 -dimensional items) and the finite number of bin types, characterized by the capacity and the cost, the problem is to select bins and pack all items into these bins, so that the total cost is minimized and the resource constraints are met.The problem 2DHet-VBPP is a special case of the Vector bin packing problem (VBPP), where bins and items are vectors of dimension N
Turky et al [36] developed Hyper-heuristic framework based on automatically selecting local search algorithm and the internal operators to solve 2DHom-VBPP denoted as Multi-Capacity Bin Packing Problem (MCBPP)
The newly generated 2DHet-VBPP instances and the 2CBP set of 2DHom-VBPP benchmark instances provided by Caprara and Toth [8] are used to evaluate the Reduced variable neighborhood search (RVNS) algorithm
Summary
The two-dimensional heterogeneous vector bin packing problem (2DHet-VBPP) can be stated as follows (see Han et al [13], where the name M2BP — multitype two-dimensional bin packing problem, was used): given the N pairs ( 2 -dimensional items) and the finite number of bin types, characterized by the capacity and the cost, the problem is to select bins and pack all items into these bins, so that the total cost is minimized and the resource constraints are met. Turky et al [36] developed Hyper-heuristic framework based on automatically selecting local search algorithm and the internal operators to solve 2DHom-VBPP denoted as Multi-Capacity Bin Packing Problem (MCBPP). A consistent neighborhood search was developed for solving the one-dimensional bin packing problem and applied to the 2DHom-VBPP by Buljubašić and Vasquez [7]. Brandao and Pedroso [6] presented an exact method based on an arc-flow formulation with side constraints for bin packing, including 2DHom-VBPP, and cutting stock problems In their method, all the patterns formed a very compact graph to which a graph compression algorithm was applied in order to reduce the size of a graph without weakening the model. As the neighbourhood of a 2DHet-VBPP feasible solution is very large and complicated, the Reduced VNS (RVNS, see Hansen et al [15], [18]) was used, the variant obtained from the basic VNS by omitting the Local search step
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