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

Pattern Generation for Three Dimensional Cutting Stock Problem

We consider the problem of three-dimensional cutting of a large block that is to be cut into some small block pieces, each with a specific size and request. Pattern generation is an algorithm that has been used to determine cutting patterns in one-dimensional and two-dimensional problems. The purpose of this study is to modify the pattern generation algorithm so that it can be used in three-dimensional problems, and can determine the cutting pattern with the minimum possible cutting residue. The large block will be cut based on the length, width, and height. The rest of the cuts will be cut back if possible to minimize the rest. For three-dimensional problems, we consider the variant in which orthogonal rotation is allowed. By allowing the remainder of the initial cut to be rotated, the dimensions will have six permutations. The result of the calculation using the pattern generation algorithm for three-dimensional problems is that all possible cutting patterns are obtained but there are repetitive patterns because they suggest the same number of cuts.

Penyelesaian Algortima Pattern Generation dengan Model Arc-Flow pada Cutting Stock Problem (CSP) Satu Dimensi

Permasalahan optimasi dalam kasus pengkombinasian pola pemotongan yang hanya memperhatikan salah satu sisi pemotongan dikenal dengan Cutting Stock Problem (CSP) satu dimensi. Penelitian ini menggunakan algoritma pattern Generation dan model arc-flow untuk menyelesaikan permasalahan pola pemotongan kayu. Berdasarkan hasil dan pembahasan didapatkan bahwa algoritma pattern generation menghasilkan pola-pola pemotongan yang optimal tanpa adanya trim loss. Pola-pola tersebut selanjutnya dimodelkan ke dalam model arc-Flow. Model yang terbentuk hanya menggunakan kendala pemenuhan permintaan dan kendala non negatif, sedangkan kendala yang berkaitan dengan konservasi flow tidak digunakan

A new heuristic algorithm for two-dimensional defective stock guillotine cutting stock problem with multiple stock sizes

U radu se uglavnom raspravlja o problemu rezanja giljotinom dvodimenzijske oštećene robe raspoložive u različitim veličinama. Za raspravu o problemu predlaže se novi heuristički algoritam u obliku stabla. Takav se algoritam sastoji od dva dijela: prvi dio je početno rješenje problema rezanja robe kad ne postoje oštećenja robe; drugi dio je konačno rješenje optimizacije utemeljeno na prvom dijelu uz razmatranje oštećenja. U radu se također ocjenjuju rezultati predloženog algoritma. Eksperimentalnim se rezultatima demonstrira učinkovitost algoritma za problem rezanja dvodimenzijske oštećene robe i pokazuje da se algoritmom može poboljšati ne samo stopa iskoristivosti robe već i stopa ponovne uporabe ostataka smanjenjem fragmentacije ostataka.This paper mainly addresses a two-dimensional defective stocks guillotine cutting stock problem where stock of different sizes is available. Herein a new heuristic algorithm which is based on tree is proposed to discuss this problem. In particular, such an algorithm consists of two parts: the first part is an initial solution of the cutting stock problem where there are no defects on the stocks; the second part is the final optimization solution which is set up on the basis of the first part and takes the defects into consideration. This paper also evaluates the performance of the proposed algorithm. The experimental results demonstrate the effectiveness of the algorithm for the two-dimensional defective stocks cutting stock problem and show that the algorithm can improve not only the utilization rate of stocks, but also the reuse rate of remainders by reducing the fragmentation of remainders

Models and Solutions of Resource Allocation Problems based on Integer Linear and Nonlinear Programming

In this thesis we deal with two problems of resource allocation solved through a Mixed-Integer Linear Programming approach and a Mixed-Integer Nonlinear Chance Constraint Programming approach. In the first part we propose a framework to model general guillotine restrictions in two dimensional cutting problems formulated as Mixed-Integer Linear Programs (MILP). The modeling framework requires a pseudo-polynomial number of variables and constraints, which can be effectively enumerated for medium-size instances. Our modeling of general guillotine cuts is the first one that, once it is implemented within a state of-the-art MIP solver, can tackle instances of challenging size. Our objective is to propose a way of modeling general guillotine cuts via Mixed Integer Linear Programs (MILP), i.e., we do not limit the number of stages (restriction (ii)), nor impose the cuts to be restricted (restriction (iii)). We only ask the cuts to be guillotine ones (restriction (i)). We mainly concentrate our analysis on the Guillotine Two Dimensional Knapsack Problem (G2KP), for which a model, and an exact procedure able to significantly improve the computational performance, are given. In the second part we present a Branch-and-Cut algorithm for a class of Nonlinear Chance Constrained Mathematical Optimization Problems with a finite number of scenarios. This class corresponds to the problems that can be reformulated as Deterministic Convex Mixed-Integer Nonlinear Programming problems, but the size of the reformulation is large and quickly becomes impractical as the number of scenarios grows. We apply the Branch-and-Cut algorithm to the Mid-Term Hydro Scheduling Problem, for which we propose a chance-constrained formulation. A computational study using data from ten hydro plants in Greece shows that the proposed methodology solves instances orders of magnitude faster than applying a general-purpose solver for Convex Mixed-Integer Nonlinear Problems to the deterministic reformulation, and scales much better with the number of scenarios

Exact Approaches for Higher-Dimensional Orthogonal Packing and Related Problems

NP-hard problems of higher-dimensional orthogonal packing are considered. We look closer at their logical structure and show that they can be decomposed into problems of a smaller dimension with a special contiguous structure. This decomposition influences the modeling of the packing process, which results in three new solution approaches. Keeping this decomposition in mind, we model the smaller-dimensional problems in a single position-indexed formulation with non-overlapping inequalities serving as binding constraints. Thus, we come up with a new integer linear programming model, which we subject to polyhedral analysis. Furthermore, we establish general non-overlapping and density inequalities and prove under appropriate assumptions their facet-defining property for the convex hull of the integer solutions. Based on the proposed model and the strong inequalities, we develop a new branch-and-cut algorithm. Being a relaxation of the higher-dimensional problem, each of the smaller-dimensional problems is also relevant for different areas, e.g. for scheduling. To tackle any of these smaller-dimensional problems, we use a Gilmore-Gomory model, which is a Dantzig-Wolfe decomposition of the position-indexed formulation. In order to obtain a contiguous structure for the optimal solution, its basis matrix must have a consecutive 1's property. For construction of such matrices, we develop new branch-and-price algorithms which are distinguished by various strategies for the enumeration of partial solutions. We also prove some characteristics of partial solutions, which tighten the slave problem of column generation. For a nonlinear modeling of the higher-dimensional packing problems, we investigate state-of-the-art constraint programming approaches, modify them, and propose new dichotomy and intersection branching strategies. To tighten the constraint propagation, we introduce new pruning rules. For that, we apply 1D relaxation with intervals and forbidden pairs, an advanced bar relaxation, 2D slice relaxation, and 1D slice-bar relaxation with forbidden pairs. The new rules are based on the relaxation by the smaller-dimensional problems which, in turn, are replaced by a linear programming relaxation of the Gilmore-Gomory model. We conclude with a discussion of implementation issues and numerical studies of all proposed approaches.Es werden NP-schwere höherdimensionale orthogonale Packungsprobleme betrachtet. Wir untersuchen ihre logische Struktur genauer und zeigen, dass sie sich in Probleme kleinerer Dimension mit einer speziellen Nachbarschaftsstruktur zerlegen lassen. Dies beeinflusst die Modellierung des Packungsprozesses, die ihreseits zu drei neuen Lösungsansätzen führt. Unter Beachtung dieser Zerlegung modellieren wir die Probleme kleinerer Dimension in einer einzigen positionsindizierten Formulierung mit Nichtüberlappungsungleichungen, die als Bindungsbedingungen dienen. Damit entwickeln wir ein neues Modell der ganzzahligen linearen Optimierung und unterziehen dies einer Polyederanalyse. Weiterhin geben wir allgemeine Nichtüberlappungs- und Dichtheitsungleichungen an und beweisen unter geeigneten Annahmen ihre facettendefinierende Eigenschaft für die konvexe Hülle der ganzzahligen Lösungen. Basierend auf dem vorgeschlagenen Modell und den starken Ungleichungen entwickeln wir einen neuen Branch-and-Cut-Algorithmus. Jedes Problem kleinerer Dimension ist eine Relaxation des höherdimensionalen Problems. Darüber hinaus besitzt es Anwendungen in verschiedenen Bereichen, wie zum Beispiel im Scheduling. Für die Behandlung der Probleme kleinerer Dimension setzen wir das Gilmore-Gomory-Modell ein, das eine Dantzig-Wolfe-Dekomposition der positionsindizierten Formulierung ist. Um eine Nachbarschaftsstruktur zu erhalten, muss die Basismatrix der optimalen Lösung die consecutive-1’s-Eigenschaft erfüllen. Für die Konstruktion solcher Matrizen entwickeln wir neue Branch-and-Price-Algorithmen, die sich durch Strategien zur Enumeration von partiellen Lösungen unterscheiden. Wir beweisen auch einige Charakteristiken von partiellen Lösungen, die das Hilfsproblem der Spaltengenerierung verschärfen. Für die nichtlineare Modellierung der höherdimensionalen Packungsprobleme untersuchen wir moderne Ansätze des Constraint Programming, modifizieren diese und schlagen neue Dichotomie- und Überschneidungsstrategien für die Verzweigung vor. Für die Verstärkung der Constraint Propagation stellen wir neue Ablehnungskriterien vor. Wir nutzen dabei 1D Relaxationen mit Intervallen und verbotenen Paaren, erweiterte Streifen-Relaxation, 2D Scheiben-Relaxation und 1D Scheiben-Streifen-Relaxation mit verbotenen Paaren. Alle vorgestellten Kriterien basieren auf Relaxationen durch Probleme kleinerer Dimension, die wir weiter durch die LP-Relaxation des Gilmore-Gomory-Modells abschwächen. Wir schließen mit Umsetzungsfragen und numerischen Experimenten aller vorgeschlagenen Ansätze

Un algoritmo FFD-Eficiente para resolver el problema de corte de guillotina con demanda no unitaria de requerimientos sobre stock de tamaño variado

Resuelve el problema Guillotine Cutting Stock Problem with Demand on Varied Stock (GCSP-DVS) a través de un algoritmo FFD-Eficiente variado (FFD-E 2DGV). Además, demuestra la capacidad del algoritmo propuesto para incidir en el ahorro significativo a través del reúso de materia prima reciclable para el proceso industrial de corte bidimensional. Asimismo, compendia los resultados del algoritmo propuesto aplicado al GCSP-DVS y los resultados comparativos entre el FFD y el FFD-E aplicado al GCSP-D; generando un banco inédito para instancias de cortes 2 dimensiones de tipo guillotina sobre stock de tamaño variado y otra de demostraciones numéricas comparativas del FFD-E respecto al FFD, respectivamente. Finalmente, implementa un sistema computacional parametrizable que ejecute el algoritmo propuesto y arroje reportes de solución del citado problema GCSP con demanda sobre stock variado (GCSP-DVS).Tesi