A branch-and-price-and-cut for the manpower allocation and vehicle routing problem with staff qualifications and time windows

This paper addresses a manpower allocation and vehicle routing problem with staff qualifications and time windows (MAVRP-SQTW), which is an extension of the manpower allocation and vehicle routing problem. This problem focuses on developing an optimal schedule for integrated staff assignment and vehicle routing in transportation systems to minimize outsourcing costs and travel costs. We first formulate this problem as an arc-flow model. Then, we design a branch-and-price-and-cut (BPC) algorithm based on a set-packing model for MAVRP-SQTW. To accelerate the column generation process, we apply the bounded bidirectional search and the decremental state–space relaxation. Meanwhile, we utilize subset-row inequalities to tighten the master model. Numerical results based on the modified Solomon's instances show that our set-packing model solved by the BPC significantly outperforms the arc-flow model solved by CPLEX. Sensitivity analyses show that the numbers of vehicles and staff members as well as the number of staff types have great effects on the performance of the BPC. Results also suggest that the BPC can be able to optimally solve some instances with 70 requests, but solving large-scale instances is still the limitation of the BPC. Thus, we further develop a primal heuristic to improve the efficiency of solving large-scale instances and obtain relatively good solutions.