A Global Method for a Two-Dimensional Cutting Stock Problem in the Manufacturing Industry

A two-dimensional cutting stock problem (2DCSP) needs to cut a set of given rectangular items from standard-sized rectangular materials with the objective of minimizing the number of materials used. This problem frequently arises in different manufacturing industries such as glass, wood, paper, plastic, etc. However, the current literatures lack a deterministic method for solving the 2DCSP. However, this study proposes a global method to solve the 2DCSP. It aims to reduce the number of binary variables for the proposed model to speed up the solving time and obtain the optimal solution. Our experiments demonstrate that the proposed method is superior to current reference methods for solving the 2DCSP

The use of geometric information in heuristic optimization

The trim-loss, or cutting stock, problem arises whenever material manufactured continuously or in large pieces has to be cut into pieces of sizes ordered by customers. The problem is so to organize the cutting as to minimize the amount of waste (trim-loss) resulting from it. Brown (1971) remarks that no practical solution method has been found for the generalized 2-dimensional trim-loss problem. This thesis discusses the applicability of heuristic search methods as solution techniques for this and other problems. Chapter 2 describes three types of combinatorial search method, state-space search, problem reduction, and branch-and-bound. There is a discussion of the ways in which heuristic information can be incorporated into these methods, and descriptions of the versions of the methods used in the work described in succeeding chapters. In the 1-dimensional trim-loss problem order lengths of some material such as steel bars must be cut from stock lengths held by the supplier. Gilmore and Gomory (1961, 1963) have formulated a mathematical programming solution of this problem, which also arises with the slitting of steel rolls, cutting of metal pipe and slitting of cellophane rolls. Their approach has been developed by Haessler (1971,1975) who is particularly concerned with problems arising in the paper industry. In the 1½-dimensional case the material is manufactured as a continuous sheet of constant width and it is required to minimize the length produced to satisfy orders for rectangular pieces. In the 2-dimensional case the orders are again for rectangular pieces, but here the stock is held as large rectangular sheets. In both cases there may be restrictions as to the way in which the material may be cut; the generalized problem in each case occurs when no such restrictions exist. The 1½-dimensional problem appears to be easier of solution than the 2-dimensional case since in the latter it is necessary not only to determine the relative positions of the required pieces in a cutting pattern, but also to partition the pieces into sets to be cut from separate stock sheets. A solution method for the easier problem might provide some insight into possible methods of solution of the more difficult. In chapter 3, a state-space search method for the solution of generalized 1½-dimensional problems where the number of pieces in the order list is fairly small and the dimensions are small integers is described. This method can be developed to solve 2-dimensional problems in which the order list is fairly small and the size of stock sheets variable but affecting the cost of the material. This development is described in chapter 4. A similarly structured state-space search can be used for finding solutions to optimal network problems. Such searches do not prove the solutions they find to be optimal, so it is of interest also to develop a method for finding solutions to the problems that proves them to be optimal. In chapter 5 the state-space search method is compared with one using branch-and-bound.problems change when large numbers of identical pieces are ordered, so a solution method with a different structure is required. Chapter 6 describes a problem reduction method for generalized 2-dimensional problems in which the order lists are large and the dimensions are small integers. Even when there are restrictions on the way in which the material may be cut, the presence of other constraints may make a mathematical formulation of the 2-dimensional trim-loss problem intractable, so again a heuristic solution method may be desirable. In a problem where there are sequencing constraints on the design of successive cutting patterns, problem reduction is again found to provide a useful solution method. This is described in chapter 7. Some conclusions about the efficacy and potential of the methods used are drawn in chapter 8. The remainder of the present chapter is concerned with setting the work described in this thesis in the context of other work on the same and related problems

High Multiplicity Strip Packing Problem With Three Rectangle Types

The two-dimensional strip packing problem (2D-SPP) involves packing a set R = {r1, ..., rn} of n rectangular items into a strip of width 1 and unbounded height, where each rectangular item ri has width 0 \u3c wi ≤ 1 and height 0 \u3c hi ≤ 1. The objective is to find a packing for all these items, without overlaps or rotations, that minimizes the total height of the strip used. 2D-SPP is strongly NP-hard and has practical applications including stock cutting, scheduling, and reducing peak power demand in smart-grids. This thesis considers a special case of 2D-SPP in which the set of rectangular items R has three distinct rectangle sizes or types. We present a new OPT + 5/3 polynomial-time approximation algorithm, where OPT is the value of an optimum solution. This algorithm is an improvement over the previously best OPT + 2 polynomial-time approximation algorithm for the problem

Optimal surface cutting

Surface cutting problems in two dimensions are considered for nonrectangular items. An exact solution method is discussed. Outlines of several possible heuristic algorithms are also presented. For the heuristic methods a first approximation to the optimal solution is obtained by encompassing each item by a rectangle and then using some available strategy for this standard problem. Different approaches are then suggested for more accurate methods

The Two-Dimensional, Rectangular, Guillotineable-Layout Cutting Problem with a Single Defect

In this paper, a two-dimensional cutting problem is considered in which a single plate (large object) has to be cut down into a set of small items of maximal value. As opposed to standard cutting problems, the large object contains a defect, which must not be covered by a small item. The problem is represented by means of an AND/OR-graph, and a Branch & Bound procedure (including heuristic modifications for speeding up the search process) is introduced for its exact solution. The proposed method is evaluated in a series of numerical experiments that are run on problem instances taken from the literature, as well as on randomly generated instances.Two-dimensional cutting, defect, AND/OR-graph, Branch & Bound

Comparing several heuristics for a packing problem

Packing problems are in general NP-hard, even for simple cases. Since now there are no highly efficient algorithms available for solving packing problems. The two-dimensional bin packing problem is about packing all given rectangular items, into a minimum size rectangular bin, without overlapping. The restriction is that the items cannot be rotated. The current paper is comparing a greedy algorithm with a hybrid genetic algorithm in order to see which technique is better for the given problem. The algorithms are tested on different sizes data.Comment: 5 figures, 2 tables; accepted: International Journal of Advanced Intelligence Paradigm

Penerapan Algoritma Kelelawar pada Masalah Pemotongan Bahan

ABSTRAKSI: Masalah pemotongan bahan dua dimensi non-guillotine sering terjadi pada banyak industri tekstil, dimana sebuah stock persegi panjang harus dipotong menjadi potongan kecil dengan ukuran dan jumlah yang berbeda-beda. Masalah ini termasuk masalah kombinatorial dengan ruang solusi yang besar dan sulit untuk diselesaikan.Pada tugas akhir ini digunakan algoritma kelelawar yang merupakan algoritma optimasi metaheuristik dan termasuk kedalam swarm intelligent. Algotirma ini terinspirsi dari peristiwa echolocation pada microbats. Algoritma ini dikembangkan berdasarkan kelebihan dari penentuan posisi dengan menggunakan velocity pada algoritma particle swarm optimization(PSO) dan cooling schedule pada algoritma simulated annealing(SA) sehingga proses pencarian yang dilakukan tidak hanya eksplorasi (global search) tetapi juga eksploitasi (local search).Hasil percobaan pada tiga buah dataset yang berbeda menunjukan algoritma kelelawar dapat menyelesaikan masalah pemotongan bahan dengan optimasi diatas 90%.Kata Kunci : masalah pemotongan bahan, swarm intelegent, algoritma kelelawar, optimasi.ABSTRACT: Two dimensional non-guillotine cutting stock problem often occurs in many textile industries where a rectangular stock must be cut into smaller pieces with different size and number. This problem included into combinatorial problems with large solution space and is hard to resolve.In this final project used bat algorithm which metaheuristics optimization algorithm and included into swarm intelligent. This Algotirma inspired from echolocation in microbats. The algorithm was developed based on advantage of positioning using velocity in particle swarm optimization(PSO) and cooling schedule in simulated annealing(SA) so the search process is done not only exploration (global search) but also exploitation (local search).The experiment results on three datasets show the bat algorithm can solve cutting stock problem with an optimization above 90%.Keyword: cutting stock problem, swarm intelegent, bats algorithm, optimization