An exact algorithm for generating homogenous T-shape cutting patterns

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Abstract

Both the material usage and the complexity of the cutting process should be considered in generating cutting patterns. This paper presents an exact algorithm for constrained two-dimensional guillotine-cutting problems of rectangles. It uses homogenous T-shape patterns to simplify the cutting process. Only homogenous strips are allowed, each of which contains rectangular blanks of the same size and direction. The sheet is divided into two segments. Each segment consists of strips of the same length and direction. The strip directions of the two segments are perpendicular to each other. The algorithm is based on branch and bound procedure combined with dynamic programming techniques. It is a bottom-up tree-search approach that searches the solution tree from the branches to the root. Tighter bounds are established to shorten the searching space. The computational results indicate that the algorithm is efficient both in computation time and in material usage.

Introduction

The constrained two-dimensional cutting (CTDC) problem discussed is as follows: m types of rectangular blanks are to be cut from a rectangular sheet of size L×W, where any cuts that are made are restricted to be guillotine cuts. The ith type has size li×wi, value ci and demand di, i=1,,m. Assume that pattern A includes zi pieces of type i. The mathematical model for the CTDC problem is maxi=1mcizi;A is a feasible pattern;zi integer,0zidi,i=1,,m.

This paper applies the cutting patterns shown in Fig. 1, where the numbers indicate the blank types. A cut denoted by the arrow divides the sheet into two segments. Each segment consists of strips in the same direction. The strip directions of the two segments are perpendicular to each other. Only homogenous strips consisting of blanks of the same type are allowed. These patterns are referred to as homogenous T-shape patterns. The pattern in Fig. 1a is a TX-pattern where the dividing cut is vertical, and that in Fig. 1b is a TY-pattern where the dividing cut is horizontal. Here and in the following sections one should only consider the TX-pattern, and note that the TY-pattern appears by rotating the sheet and all items by 90.

Homogenous T-shape patterns are a subset of three-stage cutting patterns. They are not a superset of two-stage cutting patterns, as two-stage cutting patterns allow general strips, each of which may include blanks of different types.

Homogenous T-shape patterns are interesting to OR practice for the reasons below. The first reason is that in the sheet metal industry, the sheet is often divided into blanks in two phases. First a guillotine shear cuts the sheet into strips. Then a stamping press punches out the blanks from the strips. The stamping phase requires that each strip include only blanks of the same type. The second reason is to simplify the cutting process. In developing countries, non-numerical controlled machines are widely used to divide the sheet into blanks. The cutting process of the T-shape patterns is simple, for that each strip consists of only blanks of the same type.

Many authors have investigated the CTDC problem. There exist two exact approaches: the top–down approach [1] and the bottom-up one [2], [3], [4], [5], [6], [7]. The top–down approach uses a tree-search procedure. All possible cuts that can be made on the stock sheet are enumerated by means of a tree in which branching corresponds to guillotine cuts and nodes represent sub-rectangles obtained through the guillotine cut. The bottom–up approach is based on the observation that any pattern satisfying the guillotine constraint can be obtained through horizontal and vertical builds of rectangles. All possible combinations of smaller rectangles are generated to obtain larger rectangles until no more guillotine patterns can be obtained. Both approaches can use upper and lower bounds to discard some non-promising branches.

Both homogenous T-shape patterns and two-stage patterns are the subsets of three-stage patterns. Algorithms for constrained two-stage patterns have been presented in the literature [8], [9], [10]. The author has not seen any report of the constrained homogenous T-shape patterns.

This paper presents an algorithm for CTDC problems. It can generate the optimal homogenous T-shape pattern. The computational results indicate that the algorithm is efficient both in material usage and in computation time.

Section snippets

Some basic concepts

To facilitate presentation, it is assumed that all blank types have fixed direction, and the sizes of the blanks and the stock sheet are integers.

Definition 1

X-strips, Y-strips, X-segments, and Y-segments

An X-strip is a horizontal strip, and a Y-strip is a vertical one. A TX-pattern includes two segments (Fig. 1a). The left segment consisting of X-strips is an X-segment, and the right segment consisting of Y-strips is a Y-segment. The value of a strip is the sum of the values of the blanks included. The value of a segment is the sum of the values of

The approach

The approach for generating the optimal homogenous T-shape patterns consists of the following steps: (1) Generate all possible X-strips and Y-strips; (2) Obtain all possible X-segments through vertical build of X-strips; (3) Obtain all possible Y-segments through horizontal build of Y-strips. (4) Generate all possible T-shape patterns and select the best one as the optimal solution. Each T-shape pattern is generated through the horizontal build of an X-segment and a Y-segment.

The computational results

The computations were performed on a computer with Pentium 4 CPU, clock rate 2.8 GHz, and main memory 512 MB. Fifty test problems were randomly generated. There are 20 blank types for each problem. The value of each blank is equal to its area. The variable ranges are: sheet length in (2000, 2600), sheet width in (1000, 1300), blank length and width both in (50, 450), blank demand in (1, 10). The variables are uniformly distributed in the ranges. Once the paper is published, these problems will be

Conclusions

Because of the simplicity of the cutting process, cutting patterns of homogenous strips have been widely used in practice. They are especially appropriate for the case where non-numerical controlled guillotine shears are used, or the shearing and punching process is applied.

Other exact algorithms exist for the CTDC problems. Usually the patterns generated do not consist of strips, and are not adequate for the shearing and punching process. The staged patterns of general or uniform strips are

Acknowledgements

This paper is part of the project supported by Guangxi Science Foundation (Grant 0236017). The authors wish to express their appreciation to the supporter.

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