Positions and covering: A two-stage methodology to obtain optimal solutions for the 2d-bin packing problem

Abstract

We present a two-stage methodology called Positions and Covering (P&C) to solve the twodimensional bin packing problem (2D-BPP). The objective of this classical combinatorial NP-hard problem is to pack a set of items (small rectangles) in the minimum number of bins (larger rectangles). The first stage is the key-point of the Positions and Covering, where for each item, it is generated in a pseudo-polynomial way a set of valid positions that indicate the possible ways of packing the item into the bin. In the second stage, a new set-covering formulation, strengthen with three sets of valid inequalities, is used to select the optimal non-overlapping configuration of items for each bin. Experimental results for the P&C method are presented and compared with some of the best algorithms in the literature for small and medium size instances. Furthermore, we are considering both cases of the 2D-BPP, with and without rotations of the items by 90˚. To the best of our knowledge, this is one of the first exact approaches to obtain optimal solutions for the rotation case. OPEN ACCESSCitation: Cid-Garcia NM, Rios-Solis YA (2020) Positions and covering: A two-stage methodology to obtain optimal solutions for the 2d-bin packing problem. PLoS ONE 15(4): e0229358. https://doi.

Exact solutions for the 2d-strip packing problem using the positions-and-covering methodology

“…In this section, we present the materials and methods used to solve the 2SP. First, we show the adaptation of the P&C proposed by [ 10 ] to solve the 2d-bin packing problem. Then, we describe each part of the methodology for the 2SP.…”

“…Else, the H -value is increased by one or in a dichotomous way, the new set of valid positions for the items is generated, and the covering model is solved again. These last three steps represent the main difference with the P&C for the 2D-BPP (see [ 10 ]). The procedure ends when P&C finds a feasible solution of the D-2SP( H ), and the current height becomes the optimal one.…”

“…Besides valid inequalities ( 5 ) and ( 6 ), other valid inequalities could enhance the ILP, but these proved in preliminary results to be the more efficient for the 2SP (and also for the 2BPP, see [ 10 ]).…”

“…In this study, we propose an adaptation of the Positions and Covering (P&C) methodology, used by [ 10 ] to obtain optimal solutions for the two-dimensional bin packing problem. Considering the similarity of the 2SP with other packing problems, some potential applications for the P&C methodology can be found in agricultural fields to delineate rectangular and homogeneous management zones [ 11 – 13 ], in scheduling problems applied to environmental, automotive, ferry, and manufacturing industries to ensure maximum use of materials [ 14 – 16 ], in healthcare applied to operating rooms schedules [ 17 – 19 ], in cloud computing where a set of jobs must be processed on virtual machines of physical servers with the aim of improves the energy efficiency [ 20 – 22 ], and in optimal deliveries in e-commerce where the objective is optimizing the total number of containers to integrate into several delivery trips [ 23 ].…”

A Systematic Review of Hyper-Heuristics on Combinatorial Optimization Problems

“…Traveling Salesman [92] Traveling Salesman [93] Multiple Traveling Salesman [94] Bottleneck Traveling Salesman [95] Cutting Stock [96] Cutting Stock [97] 2D Cutting [98] Packing [99] Packing [100] 2D Packing [101] Bin Packing [102] Knapsack [103] Knapsack [104] Subset Sum [105] Unbounded Knapsack [105] Bounded Knapsack [106] Multiple Knapsack [107] Quadratic Knapsack [108] Scheduling [109] Scheduling [110] Production Scheduling [111] Workforce Scheduling [112] Job-Shop Scheduling [113] Precedence Constrained Scheduling [114] Educational Timetabling [115] Educational Timetabling [116] Facility Location [117] Assignment [118] Quadratic Assignment [119] Spanning Tree [120] Maximum Leaf Spanning Tree [121] Degree Constrained Spanning Tree [122] Minimum Spanning Tree [123] Boolean Satisfiability [124] Boolean Satisfiability [125] Covering [126] Minimum Vertex Cover [127] Set Cover [128] Exact Cover [129] Minimum Edge Cover [130] Vehicle Routing [131] Vehicle Routing…”