Abstract
We study pseudo-polynomial formulations for the classical bin packing and cutting stock problems. We first propose an overview of dominance and equivalence relations among the main pattern-based and pseudo-polynomial formulations from the literature. We then introduce reflect, a new formulation that uses just half of the bin capacity to model an instance and needs significantly fewer constraints and variables than the classical models. We propose upper- and lower-bounding techniques that make use of column generation and dual information to compensate reflect weaknesses when bin capacity is too high. We also present nontrivial adaptations of our techniques that solve two interesting problem variants, namely the variable-sized bin packing problem and the bin packing problem with item fragmentation. Extensive computational tests on benchmark instances show that our algorithms achieve state of the art results on all problems, improving on previous algorithms and finding several new proven optimal solutions.
Highlights
The bin packing problem (BPP) requires to pack a set of weighted items into the minimum number of identical capacitated bins
We show how to solve the two main problem variants: (i) minimize the number of bins used for the packing while the total number of fragmentations is at most F; and (ii) minimize the number of fragmentations while the number of bins is at most B
We tested our algorithms on the most-well known and challenging BPP and cutting stock problem (CSP) benchmark sets: (1) Classical: A set of 1615 instances proposed in various articles in the last decades and having variegate characteristics; (2) GI: 4 sets of 60 instances proposed by Gschwind and Irnich (2016) and involving bin capacities up to 1 500 000; (3) AI/ANI: 2 sets of 100 challenging instances proposed by Delorme et al (2016)
Summary
The bin packing problem (BPP) requires to pack a set of weighted items into the minimum number of identical capacitated bins. Gilmore and Gomory (1961) modeled the CSP as a setcovering by using a pattern-based representation, and proposed the well-known column generation algorithm The strength of their model comes from the very high quality of its continuous relaxation value. We extend the previous result and provide a clear picture of the dominance and equivalence relations that exist among the main pattern-based and pseudo-polynomial MILP formulations that have been proposed for the BPP and the CSP;. We perform extensive computational tests and show that our algorithms achieve state of the art results, improving previous algorithms from the literature, finding several new optimal solutions for the BPP and the CSP, and solving to optimality all attempted instances of the variable-sized BPP and the BPP with item fragmentation. We formally describe the BPP and the CSP, give the necessary notation, and present the main formulations developed in the literature for their solution
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