Genotype-phenotype heuristic approaches for a cutting stock problem with circular patterns

The cutting stock problem has been studied in the context of different industrial applications inducing NP-hard problems in most instances. However, the application in sawmill has not received the same attention. In this paper, we deal with the problem of determining the number of logs to cut over a period of several days and the geometry of sawmill patterns in order to satisfy the demand while minimizing the loss of material. First, the problem is formulated as an integer programming problem of the form of a constrained set covering problem where the knowledge of a priori cutting patterns is necessary to generate its columns. In our implementation, these patterns are obtained using a genetic algorithm (GA) or a simulated annealing method (SA). Then, two different approaches are introduced to solve the problem. The first approach includes two methods that combine a metaheuristic to generate the number of logs and a constructive heuristic to generate the cutting patterns for each of the logs. In the second approach, we use an exact procedure CPLEX to solve the integer programming model where the cutting patterns are generated with the GA method (GA+CPLEX) or the SA method (SA+CPLEX). These four methods are compared numerically on 11 semi-randomly generated problems similar to those found in real life. The best results for the loss are obtained with the two-stage GA+CPLEX approach that finds the best values for 7 problems.