A matheuristic algorithm for the one-dimensional cutting stock and scheduling problem with heterogeneous orders

Abstract

Cutting stock and bin packing problems are common in several industrial sectors, such as paper, glass, furniture, steel industry, construction, transportation, among others. The classical cutting stock problem (CSP) ignores the production planning and scheduling of multiple customer orders. Nevertheless, in real industrial settings customer orders have to be planned and scheduled over time so as to meet demand and required due dates. We propose an integer linear programming model for the one-dimensional cutting stock and scheduling problem with heterogeneous orders. As this problem is a generalization of the classical single-period one-dimensional CSP, which is known to be NP-hard, it is difficult to solve real-sized instances to optimality. Thus, we propose a novel matheuristic algorithm based on a fix-and-optimize strategy hybridized with a random local search. The proposed matheuristic was tested on a set of 160 synthetic problem instances based on a real-world problem and compared with CPLEX solver. In larger instances, the proposed matheuristic performed better than CPLEX, with average relative percentage deviation (RPD) regarding objective values as high as 72%. On the other hand, in small instances CPLEX showed a marginal advantage, with best average RPD of 18% with relation to the matheuristic. We also performed a paired t-test with significance level 0.05 and null hypothesis: no difference between the proposed matheuristic and CPLEX. In small test instances, the performance of the proposed matheuristic was statistically indistinguishable from CPLEX, while in larger instances the matheuristic outperformed CPLEX in most cases with $p\,\text {value} < 0.05$.