Mathematical model and solution approaches for integrated lot-sizing, scheduling and cutting stock problems

Abstract

In this paper, we address a two-stage integrated lot-sizing, scheduling and cutting stock problem with sequence-dependent setup times and setup costs. In production stage one, a cutting machine is used to cut large objects into smaller pieces, in which cutting patterns are generated and used to cut the pieces, and should be sequenced in order to obtain a complete cutting plan for the problem. The cut pieces, from production stage one, are used to assemble final products in production stage two, where the final products are scheduled in order to meet the client’s demands. To solve the two-stage integrated problem, we present solution methods based on a price-and-branch approach, in which a column generation procedure is proposed to generate columns and the integer problem is solved by decomposition solution approaches. A computational study is conducted using randomly generated data and an analysis showing the impact of the solution approaches in the two-stage integrated problem is presented. In addition, the performance and benefits of the integrated approach are compared to an empirical simulation of the common practice (sequential approach).

Appendices

Appendix A: General overview of the computational results

Tables 14, 15 and 16 show, for each class and heuristic approach, an overview of the average values for the gap and the computational time considering all the instances and data variation in terms of the number of final products and pieces, the length of pieces and the capacity parameters, respectively.

Table 14 Overview: number of final products and pieces

Table 15 Overview: length of pieces

Appendix B: Performance profile

An analysis of all the results presented in Sect. 5.3 is also performed using the performance profile technique (see Dolan and Moré 2002 for more details). This technique provides a tool which facilitates the comparison among approaches by taking into account all the instances of a computational study, even the ones that have not been solved by an approach. Figures 4 and 5 show the gap profile and the computational time profile, respectively. The starting point, on the left hand side, for each approach corresponds to the percentage of the instances in which it found the shortest values for the gap and computational time over all other approaches, and the point reached on the right hand side, by each approach, corresponds to the number of instances in which the method is able to find a feasible solution for the two-stage integrated problem. The gap profile shows that the best results are found by the P&B_S1S2 approach in around 30% of the instances, whereas P&B_F and P&B_T provided the lowest quality of the gaps in the computational study. In Fig. 5, we can see that the P&B_S1S2 and P&B_T present a good overall performance compared to the remaining solution approaches, since their performance profile dominates all the other solution methods. The P&B_S1S2 and P&B_T found the shortest computational time in around 50% and 40% of the instances, respectively. However, after some increments of the best values ($\tau > 2$), the P&B_S1S2 is dominated by the P&B_T heuristic. It is worth mentioning that the P&B_S1S2 heuristic dominates considerably the other stage decomposition, succeeding in more than 30% of the best results. The rest of the approaches have quite similar behavior to the performance profile, in terms of best results in less than 10% of the instances, in which the worst performance profile is generated by the MIP approach. Therefore, the P&B_S1S2 and P&B_T have shown to be the outstanding approaches in terms of gap and computational time, when dealing with the bottleneck more efficiently and solving a small version of the whole problem, respectively. The P&B_T approach has an advantage of finding a feasible solution to more than 98% of the instances, when compared to 95% of feasible instances with the P&B_S1S2 heuristic.

Table 16 Overview: capacity values

Fig. 4

Gap profile

Fig. 5

Computational time profile