The combined cutting stock and lot-sizing problem in industrial processes

Abstract

Despite its great applicability in several industries, the combined cutting stock and lot-sizing problem has not been sufficiently studied because of its great complexity. This paper analyses the trade-off that arises when we solve the cutting stock problem by taking into account the production planning for various periods. An optimal solution for the combined problem probably contains non-optimal solutions for the cutting stock and lot-sizing problems considered separately. The goal here is to minimize the trim loss, the storage and setup costs. With a view to this, we formulate a mathematical model of the combined cutting stock and lot-sizing problem and propose a solution method based on an analogy with the network shortest path problem. Some computational results comparing the combined problem solutions with those obtained by the method generally used in industry—first solve the lot-sizing problem and then solve the cutting stock problem—are presented. These results demonstrate that by combining the problems it is possible to obtain benefits of up to 28% profit. Finally, for small instances we analyze the quality of the solutions obtained by the network shortest path approach compared to the optimal solutions obtained by the commercial package AMPL.

Introduction

Consider a production process where rectangular plates have to be cut in order to produce smaller pieces, needed for the assembly of final products, as shown in Fig. 1.

The production process consists of three stages. The first one receives the customers’ orders, providing the quantity of each type of final product and the respective due dates. The second stage converts this demand for final products into a demand for pieces. Finally, the third stage consists of solving the combined cutting stock and lot-sizing problem, that is, deciding how many pieces of each type must be produced over each period of the planning horizon, while minimizing the costs associated with the storage of the pieces, the production setup, and the trim loss.

In this paper we are interested in the third stage. There frequently exists a material loss when the plates are cut in smaller pieces. This waste, which is an important part of the cost structure, is generally smaller if the demand for pieces grows. There are also costs associated with the production setup of each period. There is an economic pressure to produce some pieces in advance of their demand period, in order to minimize the waste and the setup costs. However, the storage costs associated with such anticipated production bring an opposite pressure: to discard such anticipation. This immediately poses a trade-off between the anticipation or not of the production of some lots of pieces (generating better cutting patterns and decreasing the quantity of setups, but increasing storage costs). This trade-off leads us to combine two complex combinatorial optimization problems well known in the production planning literature: cutting stock and lot sizing.

In real situations, some industries, such as furniture, steel bars, and others, usually solve these two problems independently, as follows. At first, for each period of the planning horizon, the quantities of each piece type (the lot size) to be produced are determined. Using this lot-sizing information, the best cutting patterns are generated for each period. However, dealing with these two problems separately can lead to unacceptable global costs in practice, especially if the cutting stage is economically relevant in the process.

The literature for the combined cutting stock and production planning problem is not vast. The paper, presented by Hendry et al. (1996), proposed a two-stage solution procedure to solve this problem for the copper industry. In the first stage they find the best cutting patterns to minimize the waste; these patterns are given as input to the second stage, which provides the daily production planning. Nonas and Thorstenson (2000), studied the combined cutting stock and lot-sizing problem taking into account the cut of steel plates in a company of off-road trucks. They considered the continuous demand, with no capacity restrictions whatsoever, and the cutting patterns were given a priori. Reinders (1992) studied the process of cutting tree trunks to assortments and boards for various markets.

The literature presents some papers concerning the cutting stock problem with setup costs. Haessler (1975) proposed a formulation of the one dimensional cutting stock problem where a setup cost for pattern changes is incurred. Haessler (1988) discussed some approaches for solving the one dimensional roll trim problem. Lefrançois and Gascon (1995), presented an evaluation of four different approaches for solving the cutting stock problem where the trim loss and pattern changes costs are of importance. But, in all those approaches, no storage costs are taken into account.

In this paper, we present a new approach to the combined cutting stock and lot-sizing problem. Our model of the combined problem does not use a previous set of cutting patterns generated a priori. We assume a single type of plate of a single standard size, and the plates are available in sufficient quantities. We consider the restrictions of the cutting capacity where the periods are dealt with as work shifts in an industry and the setup cost is incurred only per period of production. Our approach thus investigates the possibility of anticipating the production of some lots of pieces, increasing the storage costs, but getting an overall advantageous cost decrease through a better composition of the cutting patterns and reduction of production setup costs.

We formulated a new mathematical model and solved it heuristically, using a network shortest path problem representation. In this representation, each arc of the network is associated with a capacitated cutting stock problem that is solved by the classical simplex algorithm with column generation, proposed by Gilmore and Gomory (1965). Finally, after solving all the arcs, the remaining problem consists only of solving a straightforward minimum path problem by one of the readily available methods (Ahuja et al., 1993).

For example, each arc (i, j) contains the costs (trim loss + production setup + storage) associated with the anticipation of the production of all periods i + 1, i + 2, …, j − 1, to period i. After we solve all the arcs, the minimum path will represent the overall cost obtained. For example, if the minimum path includes arc (1, 4), this means that it is advantageous to anticipate the production of periods 2 and 3 to period 1: the additional storage costs are compensated by the overall reduced setup and trim loss costs.

The rest of the paper is organized as follows. First, the new mathematical model of the combined problem is proposed. Then, we represent this problem as a network shortest path problem, and introduce our heuristic solution method. After that, we present the classical approach to each separate problem, to show the convenience of solving them both in conjunction. Next, we present some computational tests, comparing the results of the combined problem (solved by the shortest path) with the solutions to the separate problems. We also compare the results of the combined problem solved by the shortest path approach with that of the optimal solution, obtained by the solver AMPL for some small test cases. Finally, we present some concluding remarks.