An Improved Arcflow Model for the Skiving Stock Problem

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

Improved flow-based formulations for the skiving stock problem

Thanks to the rapidly advancing development of (commercial) MILP software and hardware components, pseudo-polynomial formulations have been established as a powerful tool for solving cutting and packing problems in recent years. In this paper, we focus on the one-dimensional skiving stock problem (SSP), where a given inventory of small items has to be recomposed to obtain a maximum number of larger objects, each satisfying a minimum threshold length. In the literature, different modeling approaches for the SSP have been proposed, and the standard flow-based formulation has turned out to lead to the best trade-off between efficiency and solution time. However, especially for instances of practically meaningful sizes, the resulting models involve very large numbers of variables and constraints, so that appropriate reduction techniques are required to decrease the numerical efforts. For that reason, this paper introduces two improved flow-based formulations for the skiving stock problem that are able to cope with much larger problem sizes. By means of extensive experiments, these new models are shown to possess significantly fewer variables as well as an average better computational performance compared to the standard arcflow formulation

Arc flow formulations based on dynamic programming: Theoretical foundations and applications

Network flow formulations are among the most successful tools to solve optimization problems. Such formulations correspond to determining an optimal flow in a network. One particular class of network flow 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 pseudo-polynomial 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

Reel and sheet cutting at a paper mill

This work describes a real-world industrial problem of production planning and cutting optimization of reels and sheets, occurring at a Portuguese paper mill. It will focus on a particular module of the global problem, which is concerned with the determination of the width combinations of the items involved in the planning process: the main goal consists in satisfying an order set of reels and sheets that must be cut from master reels. The width combination process will determine the quantity/weight of the master reels to be produced and their cutting patterns, in order to minimize waste, while satisfying production orders. A two-phase approach has been devised, naturally dependent on the technological process involved. Details of the models and solution methods are presented. Moreover some illustrative computational results are included

ONE-DIMENSIONAL CUTTING STOCK PROBLEM THAT MINIMIZES THE NUMBER OF DIFFERENT PATTERNS

Cutting stock problem (CSP) is a problem of cutting an object into several smaller objects to fulfill the existing demand with a minimum unused object remaining. Besides minimizing the remaining of the object, sometimes there is another additional problem in CSP, namely minimizing the number of different cutting patterns. This happens because there is a setup cost for each pattern. This study shows a way to obtain a minimum number of different patterns in the cutting stock problem (CSP). An example problem is modeled in linear programming and then solved by a column generation algorithm using the Lingo 18.0 software

PENYELESAIAN MASALAH CUTTING STOCK DENGAN PENGELASAN MENGGUNAKAN MODEL ARC-FLOW DAN ALGORITMA PATTERN GENERATION

Masalah cutting stock dengan pengelasan adalah masalah penentuan pola pemotongan dan pengelasan bahan baku untuk memenuhi permintaan dengan bahan baku yang sedikit mungkin. Penelitian ini menggunakan model Arc-Flow dan algoritma Pattern Generation untuk menyelesaikan permasalahan pemotongan pipa. Model Arc-Flow merepresentasikan permasalahan cutting stock dengan pengelasan dalam bentuk graf berarah asiklik. Model ini bertujuan untuk menentukan aliran minimum (flow) dari simpul awal ke simpul akhir pada graf. Sedangkan, algoritma Pattern Generation menghasilkan pola-pola pemotongan yang feasible dengan menggunakan pohon pencarian. Pola pemotongan yang telah diperoleh dapat dipilih kembali agar diperoleh pola pemotongan optimal. Hasil implementasi menunjukan bahwa model Arc-Flow dan algoritma Pattern Generation dapat menyelesaikan masalah cutting stock dengan pengelasan. Berdasarkan hasil pengujian diperoleh bahwa solusi yang dihasilkan model Arc-Flow lebih optimal jika dibandingkan dengan solusi hasil implementasi algoritma Pattern Generation. Cutting stock with welding problem is a problem to find the patterns of cutting and welding raw materials to meet demand with as few raw materials as possible. In this research, we use Arc-Flow model and Pattern Generation algorithm to solve the problem. The Arc-Flow Model represents the problem using acyclic directed graphs. Then, we should determine the minimum flowfrom the initial node to the end node on the graph. On the other hands, the Pattern Generation algorithm produces feasible cutting patterns using a search tree. The cutting patterns that have been obtained can be reselected to an optimal level. Then, we should choose the optimal patterns. The computational results show that the Arc-Flow model and Pattern Generation algorithm can be implemented to solve the cutting stock with welding problem. According to the test data, we can conclude that the solutions of the Arc-Flow model are more optimal than the solutions of the Pattern Generation algorithm

OPTYMALIZACJA WIELOKRYTERIALNA W PROCESIE PRODUKCJI MEBLI

This paper concerns the use of integer linear programming in a multi-criteria optimization. The aim of the research was to develop a model of the decision support system allowing simultaneous minimization of the intermediate products stocks level and waste generated in the process of cutting. The goal of controller was to select the appropriate cutting program, including production orders, the current inventory and limits on permissible stocks levels.Opracowanie dotyczy zastosowania programowania liniowego całkowitoliczbowego w optymalizacji wielokryterialnej. Celem badań było opracowanie modelu sterownika decyzyjnego umożliwiającego jednoczesną minimalizację poziomu zapasów półfabrykatów wygenerowanych w procesie cięcia, jak i odpadów po rozkroju. Zadaniem sterownika było dobranie odpowiedniego programu rozkroju z uwzględnieniem zamówień produkcyjnych, bieżących zapasów półfabrykatów i ograniczeń odnośnie dopuszczalnych poziomów zapasów