The cutting stock problem with mixed objectives: Two heuristics based on dynamic programming

Abstract

The purpose of this paper is to provide efficient methods for solving a large spectrum of industrial cutting stock problems. We propose two methods. Both are based on dynamic programming. In both methods, we reduce the computation burden by keeping, at each stage of the dynamic programming process, only the states which seem to be the most promising in terms of cost. One of the methods favours the computation time at the expense of the quality of the solution; this method is used in the sale department, where the goal is to propose a “good” solution in less than one minute. The second method favours the quality of the solution. Industrial applications are proposed.

Introduction

Cutting problems are NP-hard (see 5, 7, 8). Thus, only small size problems can be solved optimally. These problems are solved using either integer linear programming (see 6, 9) or dynamic programming, or branch-and-bound, depending on the type of problem.

But most of the cutting problems use heuristic algorithms. The next fit (NF) heuristic (see 1, 2) consists in selecting the stored bars in a random order and of assigning the ordered bars also in a random order. If the next ordered bar cannot be assigned to the stored bar under consideration, a new ordered bar is selected and the unused part of the previous ordered bar is considered as scrap. The first fit (FF) heuristic (see 1, 2, 3) is similar to NF, except that the unused part of a stored bar is considered as a stored bar and can be used later. The first fit decreasing (FFD) algorithm (see 1, 2) is the heuristic NF in which the ordered bars are selected in the decreasing order of their length. Numerous other heuristic algorithms are available as, for instance, best fit (BF) (see 1, 2, 3), worst fit (WF) (see 1, 2, 3), almost worst fit (AWF) (see 1, 2) to quote only a few. All these approaches are similar, except in the choice of the stored and ordered bars, and in the way the unused parts of the stored bars are processed.

The heuristic algorithms presented hereafter are based on a dynamic programming approach in which, at each step of the process, only some of the states are kept for further computations. Each of these states is supposed to be among the most promising (in terms of cost). The smaller the number of states kept for further consideration, the faster the computation, but the more likely is the solution obtained far from the optimal one.

Two algorithms are proposed. The first one, called Heuristic Constraining the Utilisation of the Stored Bars (HCUS) aims at reducing as much as possible the computation time while reaching a reasonable cost. It is used by the salesmen to respond in “real time” to the customers' demands. The second one, called Heuristic Constraining the Utilisation of Stored and Ordered bars (HCUSO) aims at reaching a near-optimal solution: reducing the computation burden is no more the main goal in this case, since the computation is performed during the night to prepare the production of the next day.

In Section 2, we present the dynamic programming formulation and introduce the notations. Section 3is devoted to HCUS and Section 4to HCUSO. Section 5proposes comparative results extracted from an industrial application. Section 6is the conclusion.