Mathematical models and decomposition algorithms for makespan minimization in plastic rolls production

Nesello, Vitor; Delorme, Maxence; Iori, Manuel; Subramanian, Anand

doi:10.1057/s41274-017-0221-8

Cited by 12 publications

(6 citation statements)

References 23 publications

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“…Nesello et al [75] proposed an arc flow model, similar to the one by Macedo et al [61], to solve a three-stage two-dimensional strip packing problem where a limit is imposed on the number of shelves and setup times between items must be taken into account. Delorme et al [32] adapted the classical arc flow model for the CSP to solve the one-dimensional contiguous bin packing problem, which often appears as a sub-problem in two-dimensional cutting and packing solution methods.…”

Section: Multi-dimensional Problemsmentioning

confidence: 99%

Arc Flow Formulations Based on Dynamic Programming: Theoretical Foundations and Applications

de Lima,

Alves,

Clautiaux

et al. 2020

Preprint

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Network flow formulations are among the most successful tools to solve optimization problems. One of such formulations is the arc flow, where variables represent flows on individual arcs of the network. For NP-hard problems, polynomial-sized arc flow models typically provide weak linear relaxations and may have too much symmetry to be efficient in practice. Instead, arc flow models with a pseudo-polynomial size usually provide strong relaxations and are efficient in practice. The interest in pseudo-polynomial arc flow formulations has grown considerably in the last twenty years, in which they have been used to solve many open instances of hard problems. A remarkable advantage of arc flow models is the possibility to solve practical-sized instances directly by a Mixed Integer Linear Programming solver, avoiding the implementation of complex methods based on column generation.In this survey, we present theoretical foundations of pseudo-polynomial arc flow formulations, by showing a relation between their network and Dynamic Programming (DP). This relation allows a better understanding of the strength of these formulations, through a link with models obtained by Dantzig-Wolfe decomposition. The relation with DP also allows a new perspective to relate state-space relaxation methods for DP with arc flow models. We also present a dual point of view to contrast the linear relaxation of arc flow models with that of models based on paths and cycles. To conclude, we review the main solution methods and applications of arc flow models based on DP in several domains such as cutting, packing, scheduling, and routing.

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Section: Multi-dimensional Problemsmentioning

confidence: 99%

Arc Flow Formulations Based on Dynamic Programming: Theoretical Foundations and Applications

de Lima,

Alves,

Clautiaux

et al. 2020

Preprint

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“…In some specific applications, cutting and packing problems are combined with other problems, such as combined cutting and packing problems [15] or routing problems with loading constraints [8]. Packing problems also appear in many other fields such as telecommunications [12,14], newspaper pagination [17], manufacturing [16], scheduling [10], and maritime logistics [19]. Most cutting and packing problems are NP-hard and are highly challenging in practice, necessitating sophisticated and effective solution methods [8].…”

Section: Introductionmentioning

confidence: 99%

A Comparative Study of Mathematical Models for Two-Dimensional Cutting Stock Problems and Solving Algorithms

Almani,

Gholamnazhad

2024

Preprint

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Two-dimensional cutting problem involves cutting a set of small pieces from large sheets in a way that optimizes one or more objectives. The objective of this problem can be minimizing the number of sheets used, waste, cost, and so forth. These problems have various applications in different industries such as glass, paper, wood, plastic, steel, etc. This paper studies one of the recent exact polynomial models proposed for these problems. Although the presented model, named Original(a), is quite promising for finding a solution, since these problems belong to the mixed integer non-linear programming model (MINLP) group, solving them will be difficult. Therefore, a linearization technique by multiplying continuous variables for this model would be used. The goal is to find an optimum solution. The studied models are implemented in the Matlab environment and the results of numerical experiments are presented.

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“…In addition, for the standard one-dimensional Bin Packing Problem (1D-BPP) we refer to the surveys by Valério de Carvalho (2002) and Delorme et al (2016. For interesting problem generalizations we refer, among others, to the works by Bortfeldt and Wäscher (2013) and Crainic et al (2012) (multi-dimensional packing), by Martello (2010, 2013) (integrated routing and packing problems), by Castro and Oliveira (2011) and Nesello et al (2018) (scheduling and packing problems), by Melega et al (2018) (lot-sizing and cutting stock problems), and by Trivella and Pisinger (2016) (load balancing and packing).…”

Section: Introductionmentioning

confidence: 99%

Combinatorial Benders Decomposition for the Two-Dimensional Bin Packing Problem

Côté

Haouari

Iori

2021

INFORMS Journal on Computing

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The two-dimensional bin packing problem calls for packing a set of rectangular items into a minimal set of larger rectangular bins. Items must be packed with their edges parallel to the borders of the bins, cannot be rotated, and cannot overlap among them. The problem is of interest because it models many real-world applications, including production, warehouse management, and transportation. It is, unfortunately, very difficult, and instances with just 40 items are unsolved to proven optimality, despite many attempts, since the 1990s. In this paper, we solve the problem with a combinatorial Benders decomposition that is based on a simple model in which the two-dimensional items and bins are just represented by their areas, and infeasible packings are imposed by means of exponentially many no-good cuts. The basic decomposition scheme is quite naive, but we enrich it with a number of preprocessing techniques, valid inequalities, lower bounding methods, and enhanced algorithms to produce the strongest possible cuts. The resulting algorithm behaved very well on the benchmark sets of instances, improving on average on previous algorithms from the literature and solving for the first time a number of open instances. Summary of Contribution: We address the two-dimensional bin packing problem (2D-BPP), which calls for packing a set of rectangular items into a minimal set of larger rectangular bins. The 2D-BPP is a very difficult generalization of the standard one-dimensional bin packing problem, and it has been widely studied in the past because it models many real-world applications, including production, warehouse management, and transportation. We solve the 2D-BPP with a combinatorial Benders decomposition that is based on a model in which the two-dimensional items and bins are represented by their areas, and infeasible packings are imposed by means of exponentially many no-good cuts. The basic decomposition scheme is quite naive, but it is enriched with a number of preprocessing techniques, valid inequalities, lower bounding methods, and enhanced algorithms to produce the strongest possible cuts. The algorithm we developed has been extensively tested on the most well-known benchmark set from the literature, which contains 500 instances. It behaved very well, improving on average upon previous algorithms, and solving for the first time a number of open instances. We analyzed in detail several configurations before obtaining the best one and discussed several insights from this analysis in the manuscript.

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Mathematical models and decomposition algorithms for makespan minimization in plastic rolls production

Cited by 12 publications

References 23 publications

Arc Flow Formulations Based on Dynamic Programming: Theoretical Foundations and Applications

Arc Flow Formulations Based on Dynamic Programming: Theoretical Foundations and Applications

A Comparative Study of Mathematical Models for Two-Dimensional Cutting Stock Problems and Solving Algorithms

Combinatorial Benders Decomposition for the Two-Dimensional Bin Packing Problem

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