A comparative study of the arcflow model and the one-cut model for one-dimensional cutting stock problems

This study aims to optimize the one-dimensional cutting process of aluminum bars for the production of aluminum doors. Reducing the use of bars and the amount of material that becomes scrap is a key factor in process efficiency, reducing the need for raw material procurement. The mathematical model used considers the size of the bar, the number and size of cuts, the size of the leftovers that can be used and the size of the leftovers that are considered scrap. Based on real data from a company in the aluminum frame segment, the mathematical model was used to simulate three different scenarios. Three different objective functions were used in the simulations, and the results obtained in each scenario were described in order to indicate the advantages and disadvantages of using each objective function. For the instance sizes studied, the model is able to obtain optimal solutions with little computational time.

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“…Inventory Utilization Rate , corresponds to the percentage of material used in stock, given by Eq. (14).…”

Section: Resultsmentioning

confidence: 99%

“…The problem of one-dimensional cutting is making an object available for production by making cuts on it in order to meet a specific dimension demand [4,13]. The cutting process refers to obtaining smaller items from cutting a larger matrix using cutting patterns or decision elements [5,14].…”

Section: Theoretical Backgroundmentioning

confidence: 99%

Aluminum bar cutting optimization for door and window manufacturing

Machado

Zayatz

Silva

et al. 2020

DYNA

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“…Its current name goes back to Zak [14], who presented a first pattern-based modelling approach for arbitrary frequencies. Although this model is conjectured to possess a very tight LP relaxation, similiar to the cutting context [6,10,12,13], the number of variables is exponential with respect to the number of items. Consequently, pseudo-polynomial models for the SSP have been established in literature [9] and were shown to exhibit an equally good relaxation.…”

Section: Introductionmentioning

confidence: 99%

“…The reflect graph for the instance E =(3, (18, 16, 8), 20,(10,10,10)). An optimal solution with z RE = 15 is given by ξ 0,2,r = ξ 0,4,r = ξ 0,8,s = ξ 2,4,l = ξ 8,10,l = 10, ξ 4,8,l = 20, and ξ 10,10,r = −5.…”

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confidence: 99%

An Improved Arcflow Model for the Skiving Stock Problem

Martinovic

Delorme

Iori

et al. 2019

Operations Research Proceedings

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Because of the sharp development of (commercial) MILP software and hardware components, pseudo-polynomial formulations have been established as a viable tool for solving cutting and packing problems in recent years. Constituting a natural (but independent) counterpart of the well-known cutting stock problem, the one-dimensional skiving stock problem (SSP) asks for the maximal number of large objects (specified by some threshold length) that can be obtained by recomposing a given inventory of smaller items. In this paper, we introduce a new arcflow formulation for the SSP applying the idea of reflected arcs. In particular, this new model is shown to possess significantly fewer variables as well as a better numerical performance compared to the standard arcflow formulation.

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“…In general, the skiving stock problem can be considered as a natural counterpart of the extensively studied one-dimensional cutting stock problem (CSP) [8,12,16,27,37,40]. Indeed, the latter consists of cutting larger objects into smaller ones, whereas the SSP asks for the reverse direction.…”

Section: Introductionmentioning

confidence: 99%

New Theoretical Investigations on the Gap of the Skiving Stock Problem

Martinovic

Scheithauer

2019

Pesqui. Oper.

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The one-dimensional skiving stock problem is a combinatorial optimization problem being of high relevance whenever an efficient and sustainable utilization of given resources is intended. In the classical formulation, a given supply of (small) item lengths has to be used to build as many large objects (specified by some target length) as possible. For this N P-hard (discrete) optimization problem, we investigate the quality of the continuous relaxation by considering the additive integrality gap, i.e., the difference between the optimal values of the integer problem and its LP relaxation. In a first step, we derive an improved upper bound for the gap by focusing on the concept of residual instances. Moreover, we show how further upper bounds can be obtained if all problem-specific input data are considered. Additionally, we constructively prove the integer round-down property for two new classes of instances, and introduce several construction principles to obtain gaps greater than or equal to one.

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A comparative study of the arcflow model and the one-cut model for one-dimensional cutting stock problems

Cited by 42 publications

References 36 publications

Aluminum bar cutting optimization for door and window manufacturing

Aluminum bar cutting optimization for door and window manufacturing

An Improved Arcflow Model for the Skiving Stock Problem

New Theoretical Investigations on the Gap of the Skiving Stock Problem

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