An integer programming model for two- and three-stage two-dimensional cutting stock problems (original) (raw)
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Contribution to Solving a Two-Dimensional Cutting Stock Problem with Two Objectives
ECONOMIC COMPUTATION AND ECONOMIC CYBERNETICS STUDIES AND RESEARCH
In this paper, we propose a technique for solving a twodimensional cutting stock problem with two-objective. It's about cutting a number of rectangular pieces from a set of raw material plates, themselves identical. These are available in unlimited quantities, where we try to minimize the total area lost and the number of setups to be carried out. This technique is made up of two stages, the first of which consists in generating all the feasible cutting patterns and the second makes it possible to construct cutting plans, satisfying the demands, thanks to a subset of these patterns. These different cutting plans represent all the feasible solutions, each of which is characterized by a number of setups and the total quantity of falls.
Models for the two-dimensional two-stage cutting stock problem with multiple stock size
Computers & Operations Research, 2013
We consider a Two-Dimensional Cutting Stock Problem (2DCSP) where stock of different sizes is available, and a set of rectangular items has to be obtained through two-stage guillotine cuts. We propose and computationally compare three Mixed-Integer Programming models for the 2DCSP developing formulations from the literature. The first two models have a polynomial and pseudo-polynomial number of variables, respectively, and can be solved with a general-purpose MIP solver. The third model, having an exponential number of variables, is solved via branch-and-price techniques. We conclude the paper describing the results of extensive computational experiments on a set of benchmark instances from the literature.
An exact algorithm for the N-sheet two dimensional single stock-size cutting stock problem
ORiON, 2015
The method introduced in this paper extends the trim-loss problem or also known as 2D rectangular SLOPP to the multiple sheet situation where N same size two-dimensional sheets have to be cut optimally producing demand items that partially or totally satisfy the requirements of a given order. The cutting methodology is constrained to be of the guillotine type and rotation of pieces is allowed. Sets of patterns are generated in a sequential way. For each set found, an integer program is solved to produce a feasible or sometimes optimal solution to the N-sheet problem if possible. If a feasible solution cannot be identified, the waste acceptance tolerance is relaxed somewhat until solutions are obtained. Sets of cutting patterns consisting of N cutting patterns, one for each of the N sheets, is then analysed for optimality using criteria developed here. This process continues until an optimal solution is identified. Finally, it is indicated how a given order of demand items can be totally satisfied in an optimal way by identifying the smallest N and associated cutting patterns to minimize wastage. Empirical results are reported on a set of 120 problem instances based on well known problems from the literature. The results reported for this data set of problems suggest the feasibility of this approach to optimize the cutting stock problem over more than one same size stock sheet. The main contribution of this research shows the details of an extension of the Wang methodology to obtain and prove exact solutions for the multiple same size stock sheet case.
Algorithms for the one-dimensional two-stage cutting stock problem
European Journal of Operational Research, 2018
In this paper, we consider a two-stage extension of one-dimensional cutting stock problem which arises when technical requirements inhibit cutting large stock rolls to demanded widths of finished rolls directly. Therefore, demands on finished rolls are fulfilled through two subsequent cutting processes, in which rolls produced in the former are used as input for the latter, while the number of stock rolls used is minimized. We tackle the pattern-based formulation of this problem which typically has a very large number of columns and constraints. The special structure of this formulation induces both a column-wise and a row-wise increase when solved by column generation. We design an exact simultaneous column-and-row generation algorithm whose novel element is a row-generating subproblem that generates a set of columns and rows. For this subproblem, which is modeled as an unbounded knapsack problem, we propose three algorithms: implicit enumeration, column generation which renders the overall methodology nested column generation, and a hybrid algorithm. The latter two are integrated in a well-known knapsack algorithm which forges a novel branch-and-price algorithm for the row-generating subproblem. Extensive computational experiments are conducted, and performances of the three algorithms are compared.
Constrained two-dimensional cutting stock problems a best-first branch-and-bound algorithm
International Transactions in Operational Research, 2000
In this paper, we develop a new version of the algorithm proposed in Hi® (Computers and Operations Research 24/8 (1997) 727±736) for solving exactly some variants of (un)weighted constrained twodimensional cutting stock problems. Performance of branch-and-bound procedure depends highly on particular implementation of that algorithm. Programs of this kind are often accelerated drastically by employing sophisticated techniques. In the new version of the algorithm, we start by enhancing the initial lower bound to limit initially the space search. This initial lower bound has already been used in (Journal of the Operational Research Society, 49, 1270±1277), as a heuristic for solving the constrained and unconstrained cutting stock problems. Also, we try to improve the upper bound at each internal node of the developed tree, by applying some simple and ecient combinations. Finally, we introduce some new symmetric-strategies used for neglecting some unnecessary duplicate patterns. The performance of our algorithm is evaluated on some problem instances of the literature and other hard-randomly generated problem instances. 7 (M. Hi®). J (NV-pattern but not a NH-pattern) horizontal combination of the patterns A and
A Solution of the Rectangular Cutting-Stock Problem
IEEE Transactions on Systems, Man, and Cybernetics, 1976
A method of solving a version of the two-dimensional cutting-stock problem is presented. In this version of the problem one is given a number of rectangular sheets and an order for a specified number of each of certain types of rectangular shapes. The goal is to cut the shapes out of the sheets in such a way as to minimize the waste. However, in many practical applications computation time is also an important economic consideration. For such applications the goal may be to obtain the best solution possible without using excessive amounts of computation time. The method of solution described here avoids exhaustive search procedures by employing an approach utilizing a constrained dynamic programming algorithm to lay out groups of rectangles called strips. This paper also describes the results obtained when the algorithm was tested with some typical rectangular layout problems.
Journal of Combinatorial Optimization, 2001
In this paper we propose two algorithms for solving both unweighted and weighted constrained two-dimensional two-staged cutting stock problems. The problem is called two-staged cutting problem because each produced (sub)optimal cutting pattern is realized by using two cut-phases. In the first cut-phase, the current stock rectangle is slit down its width (resp. length) into a set of vertical (resp. horizontal) strips and, in the second cut-phase, each of these strips is taken individually and chopped across its length (resp. width). First, we develop an approximate algorithm for the problem. The original problem is reduced to a series of single bounded knapsack problems and solved by applying a dynamic programming procedure. Second, we propose an exact algorithm tailored especially for the constrained two-staged cutting problem. The algorithm starts with an initial (feasible) lower bound computed by applying the proposed approximate algorithm. Then, by exploiting dynamic programming properties, we obtain good lower and upper bounds which lead to significant branching cuts. Extensive computational testing on problem instances from the literature shows the effectiveness of the proposed approximate and exact approaches.
Applications of Cutting Stock Problem
Communications - Scientific letters of the University of Zilina
We present one heuristic solution for the well-known cutting stock problem which was formulated by Kantorovich in 1939. It is the problem of filling an order at minimum cost for specified numbers of lengths of material to be cut to given stock lengths of given cost. When expressed as an integer programming problem the large number of variables involved generally makes computation infeasible. The same difficulty persists when only an approximate solution is being sought by linear programming.
International Transactions in Operational Research, 2014
We address a 1-dimensional cutting stock problem where, in addition to trimloss minimization, cutting patterns must be sequenced so that no more than s different part types are in production at any time. We propose a new integer linear programming formulation whose constraints grow quadratically with the number of distinct part types and whose linear relaxation can be solved by a standard column generation procedure. The formulation allowed us to solve problems with 20 part types for which an optimal solution was unknown.
A software for the one-dimensional cutting stock problem
In this paper, one-dimensional cutting stock problem is taken into consideration and a new heuristic algorithm is proposed to solve the problem. In this proposed algorithm, a new dynamic programming algorithm is applied for packing each of the bins. The algorithm is coded with Delphi and then by computational experiments with the real-life constraint optimization problems , and the obtained results are compared with the other one-dimensional cutting stock commercial packages. The computational experiments show the efficiency of the algorithm.