A hybrid genetic algorithm for the two-dimensional single large object placement problem

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Abstract

In the two-dimensional single large object placement problem, we are given a rectangular master surface which has to be cut into a set of smaller rectangular items, with the aim of maximizing the total value of the pieces cut. We consider the special case in which the items cannot be rotated and must be cut with their edges always parallel to the edges of the surface. We present new greedy algorithms and a hybrid genetic approach with elitist theory, immigration rate, heuristics on-line and tailored crossover operators. Extensive computational results for a large number of small and large benchmark test problems are presented. The results show that our approach outperforms existing heuristic algorithms.

Introduction

This paper considers the problem of cutting a plane rectangle into smaller rectangular items, each of a given size and value, with the objective of maximizing the total value of the items cut. This problem arises in many industrial applications; for example, the cutting of steel, glass, wood or paper and the placement of adverts in newspapers and magazines. The related problem of minimizing the amount of wasted material produced by cuts can be converted into the value maximization problem by simply taking the value of each item to be proportional to its area.

The two-dimensional cutting problem can also be viewed as the problem of packing smaller items into a larger stock rectangle. Packing problems are common in the optimization of stock areas in industry layouts. In this paper, the terms cutting and packing will be used interchangeably.

The above problem studied in this paper is known in the literature as the two-dimensional single large object placement problem (2SLOPP). The name of the problem is derived from the typology developed by Wäscher et al. [52]. Note that, in the same typology, the related problem in which there is a strongly heterogeneous assortment of items is defined as the two-dimensional knapsack problem (2KP).

We consider the constrained two-dimensional non-guillotine cutting problem in which each item must be cut with its edges always parallel to the edges of the master surface (i.e., orthogonal cuts) and it must have a fixed orientation (i.e., no rotation is allowed). Cutting problems in which the orientation of the items is fixed arise for materials that are not isomorphic, as wood or corrugated sheet. The optimal cutting patterns are not restricted to be of the guillotine type (a guillotine cut on a rectangle runs from one edge of the rectangle to the opposite edge, parallel to the two remaining edges). Moreover, the constrained form of this problem imposes restrictions on the maximum number of items of each type (i.e., same size and value) required to be cut.

In this paper, we present a simple but very effective heuristic approach for solving the 2SLOPP. In a preliminary phase, simple upper bounds are computed and initial solutions are obtained through greedy algorithms. Then a genetic search, which uses parent selections, elitist theory, immigration and different crossover operators, is performed. The genetic approach is hybridized with an on-line heuristic. The computational results reported show that the algorithm finds the best solution in a fast computational time for most of the cases and, hence, outperforms the other metaheuristics in the literature.

The rest of the paper is organized as follows: In Section 2, we survey the main results in the literature. In Section 3, we briefly describe the problem and its main characteristics. In Section 4, we present a simple upper bound from the literature and propose new greedy algorithms for the computation of lower bounds. Section 5 presents our hybrid genetic approach and Section 6 reports extensive computational results of this approach for a wide range of problem instances.

Section snippets

Literature review

Many surveys are available for multidimensional cutting and packing problems. We refer to Dyckoff [17], Haessler and Sweeney [28], Sweeney and Paternoster [51], Dowsland and Dowsland [16] and Lodi et al. [41].

After the seminal work of Gilmore and Gomory [24], a number of authors have addressed the two-dimensional cutting problem in its different forms. The unconstrained non-guillotine problem has been considered by a few authors, namely, Arenales and Morabito [2] who proposed an approach based

Problem description

The 2SLOPP can be defined as follows: A large rectangular master surface, here defined as S, of width W0 and length L0 has to be cut into a number of smaller rectangular items chosen from a set of m available types. Each item type j is defined by width wj, length lj and value vj, for j = 1,  , m. The objective is to construct a cutting pattern for S with the highest possible total value.

The constrained 2SLOPP is considered, in which the number of replicates of each item produced should not exceed a

A one-dimensional knapsack relaxation

Consider a natural relaxation of the 2SLOPP, given by the one-dimensional knapsack (1KP), in which each item has an associated value vj and a weight dj equal to the area of the item j in the 2SLOPP (dj = wjlj) for j = 1,  , M, and the knapsack capacity D is equal to the area of S in the 2SLOPP (D = W0L0). The 1KP can be formally stated asmaxz=j=1Mvjxj,j=1MdjxjD,xj{0,1}(j=1,,M).

For a 2SLOPP instance I, let z(I) denote the optimal solution value of I and zUB the upper bound on z(I) corresponding to

Genetic algorithm

In this section we will outline a hybrid genetic algorithm, referred to as GA2SLOPP, which we have developed for the 2SLOPP. The following elements of GA2SLOPP will be discussed:

  • Representation: How a solution to the 2SLOPP is represented;

  • Fitness: Assessing the value of a solution;

  • Evolution process: Elitist criteria, reproduction and immigration;

  • Crossover: Having a child from parents;

  • On-line heuristic: Finding a solution.

Computational results

The heuristic algorithm GA2SLOPP was coded in FORTRAN 90 and run on a Pentium IV 1700 MHz (Windows 2000 operating system), using a set of both small and large-size test problems and a particular problem of packing squares.

Conclusions

In this paper we have presented a new hybrid genetic algorithm for the two-dimensional knapsack problem. The algorithm was evaluated on a very large number of test problems taken from the literature of varying size and complexity. The reported computational results indicate that as problem size increases, our heuristic performs better, getting results that are (empirically) very close to optimality in small computational times. Indeed, for the largest problems solved, our results outperform

Acknowledgements

We thank the Ministero dell’Istruzione, dell’Università e della Ricerca (MIUR) and the Consiglio Nazionale delle Ricerche (CNR), Italy, for the support given to this project. The computational experiments have been executed at the Laboratory of Operations Research (LabOR) of the University of Bologna.

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