A New Linear Programming Approach to the Cutting Stock Problem

Abstract

A new approach to the one-dimensional cutting stock problem is described and compared to the classical model for which Gilmore and Gomory have developed a special column-generation technique. The new model is characterized by a dynamic use of simply structured cutting patterns. Nevertheless, it enables the representation of complex combinations of cuts. It can be advantageous in practical applications where many different stock lengths or a relatively large number of order lengths have to be dealt with. The new approach is applied to a real problem where the “trim loss” is not valueless, since it can be used for further demands arising in later planning periods.

Mathematical models and decomposition algorithms for cutting and packing problems

“…In Section 2.4.2 we address the oldest such model, independently developed by Rao [234] in 1976 and by Dyckhoff [111] few years later. The most relevant approach of this kind (somehow anticipated by Wolsey [293] in 1977) was presented in 1999 by Valério de Carvalho [278] for the CSP.…”

“…The idea behind the Rao [234] and Dyckhoff [111] model for the CSP is to simulate the physical cutting process, by first dividing an ideal bin into two pieces (left and right), where the left piece is an item that has been cut, while the right piece is either a residual that can be re-used to produce other items or it is another item. The process is iterated on cutting residuals or new bins, until all demands are fulfilled.…”

“…-ONECUT, C++ implementation of the one-cut model by Rao [234], Dyckhoff [111], and Stadtler [266], solving the resulting ILP through CPLEX 12. -VPSOLVER, C++ implementation by Brandão and Pedroso [45], which uses Gurobi 5.6 as inner routine, available at https://code.google.com/p/vpsolver.…”

“…• ONECUT: C++ implementation of the one-cut CSP model independently defined in the Seventies by Rao [234], and Dyckhoff [111]; • VPSOLVER: link to the C++ implementation by Brandão and Pedroso [45] of their CSP algorithm. This is currently the most effective pseudo-polynomial approach, and its performance is often competitive with that of BELOV.…”

“…For example, through the last decades, no less than six different MILP models have been proposed for the classical bin packing problem: in chronological order, the textbook model by Martello and Toth [214] derived from the work of Kantorovich [167], the set covering formulation by Gilmore and Gomory [134,135], the one-cut by Dyckoff [111], the arc-flow by Valério de Carvalho [278], the DP-flow by Cambazard and O'Sullivan [52], and the general arc-flow with graph compression by Brandão and Pedroso [45]. A seventh, more powerful, formulation is even proposed in this thesis (Chapter 4).…”