Strong cutting planes for the capacitated multi-pickup and delivery problem with time windows

In this paper, we propose an improved 2-index mixed integer linear programming formulation for capacitated multi-pickup and delivery problems with time windows. We develop new families of valid inequalities, present some simple separation heuristics and solve the benchmark problems using a cutting plane approach. Specifically, we propose and demonstrate the effectiveness of incompatible request sets based valid inequalities, exploiting thereby the time window and capacity constraints. Further, we introduce two sub-families of these inequalities: node-based and request-based incompatible request set cuts, along with their separation heuristics. The computational results show that the incompatible request set cuts are highly useful as cutting planes for the benchmark instances. Overall, there is a marked improvement in solution performance vis-à-vis the different formulations and solution approaches used in extant literature. The approach in this paper is able to solve 12%, 37% and 43% more instances to optimality, improve the root gap by 18%, 33% and 36%, and final gap by 36%, 45% and 43% vis-à-vis the asymmetric representative formulation, three-index and the two-index formulations respectively.