Efficient Formulations and a Branch-and-Cut Algorithm for a Production-Routing Problem

TLDR: A basic mixed integer linear programming formulation and several strong reformulations of the production-routing problem are introduced and two families of valid inequalities, 2-matching and generalized comb inequalities, are introduced to strengthen these formulations, and they are used within a branch-and-cut algorithm.

Abstract: The production-routing problem can be seen as a combination of two well known combinatorial optimization problems: the lotsizing and the vehicle routing problems. The variant considered in this paper consists in designing a production schedule for an uncapacitated plant, replenishment schedules for multiple customers, and a set of routes for a single uncapacitated vehicle starting and ending at the plant. The aim of the problem is to fulfill the demand of the customers over a finite horizon such that the total cost of distribution, setups, and inventories is minimized. This paper introduces a basic mixed integer linear programming formulation and several strong reformulations of the problem. Two families of valid inequalities, 2-matching and generalized comb inequalities, are introduced to strengthen these formulations, and they are used within a branch-and-cut algorithm. Computational results on a large set of randomly generated instances are presented. Instances with up to 40 customers and 15 time periods or with 80 customers and 8 periods have been solved to optimality within a two-hour limit. The tests clearly indicate the effectiveness of the new formulations and of the valid inequalities. In addition, an uncoordinated approach is considered to demonstrate the benefits of the simultaneous optimization of production and distribution planning. The total cost increases on average by 47% when employing such an uncoordinated approach. Finally, a heuristic algorithm, based on defining an a priori tour for the vehicle routing part, is investigated. The heuristic algorithm shows an excellent performance. The average CPU time is less than 1% of the CPU time for the optimal solution, whereas the average cost increase is only 0.33%.