Integer Solutions to Cutting Stock Problems

We consider two integer linear programming models for the one-dimensional cutting stock problem that include various difficulties appearing in practical real problems. Our primary goals are the minimiza- tion of the trim loss or the minimization of the number of master rolls needed to satisfy the orders. In particular, we study an approach based on the classical column-generation procedure by Gilmore and Gomory for solving the linear programming (LP) relaxations and an extra column-generation procedure before solving a final integer problem. For our computational tests we use some data sets from the paper industry and some others generated randomly. This work deals with the real-world industrial problem of reel cutting optimization, usually called Cut- ting Stock Problem (CSP) that can be described as follows: find the most economical way of cutting the master rolls (with large width) into rolls with smaller widths previously specified by customers together with the quantity they need. It is well known that several formulations are possible (see for example (1)). Here, we consider two integer linear programming models for the one-dimensional CSP differing in the objective cost function: in the Model I, our primary goal is the minimization of the trim loss and in the Model II, we consider the minimization of the number of master rolls needed to satisfy the orders. This last model has been used more frequently and has some interesting features like the modified integer round-up property. Theoretically, both models are equivalent when the demands must be satisfied exactly. Nevertheless, in practical real problems the demands usually may be over satisfied, because the customer accepts certain deviations of the quantity ordered. These conditions of sale can be agreed upon the client and the seller in the contract, but there exist some standard tolerances (see the "General Conditions of Sale of Paper and Board Manufacturers in EEC", by the European Confederation of Pulp, Paper and Board industries (CEPAC) a document produced in 1991, but still in use). Other practical difficulties usually appearing are: the limitation on the maximum waste allowed in each cutting pattern (i.e. list of items to be cut from a master roll) and a limited number of slitting knives. In particular, we study an approach based on the classic column-generation procedure by Gilmore and Gomory (6) for solving the LP relaxations (without integer variables) of the integer CSP. The LP optimal solution, which generally is non-integer, has associated a small fraction of all possible patterns. To obtain an integer solution we solve a final integer CSP after using an extra column-generation procedure. For our computational tests we use some data sets from the paper industry and others randomly generated with the problem generator CUTGEN1, see (4). In the first case we obtain a very good performance in terms of paper savings when compared with those solutions obtained by the paper mill. The models are compared with respect to solution quality and computing time. The work presented here corresponds to a first stage of a project with a paper company whose last interest was to produce an interactive system able to solve this problem (with several adjustable parameters) for running on a personal computer. The use of a second machine (rewinder) that allows making further cuts on the rolls initially produced on the first machine (winder) will be considered in another paper. On the other hand, the main results of this study are included in the undergraduate academic work (3), where more details can be found. According to the typology of Wascher et al. (11), the problem under consideration is of type SSSCSP (Single Stock Size Cutting Stock Problem).