Formulations and branch-and-cut algorithms for multi-product multi-vehicle production routing problems with startup cost

In multi-product multi-vehicle production routing problems (MMPRPs), it is necessary to accurately model the capacity utilization of multiple products to obtain feasible production plans. Thus, start-up variables are required to model the capacity consumed or cost incurred when a machine starts a production batch, or when a machine switches from one product to another. In this work, we present a mixed integer linear programming model of the MMPRP with the startup cost (MMPRPSC), which is an extension of the single-item multi-vehicle production routing problem. It also generalizes the multi-item multi-vehicle inventory routing problems by incorporating production decisions. This formulation is tightened with three families of valid inequalities in which the generalized (l, S) inequalities are used in these problem settings for the first time. Using the formulation and valid inequalities, we implement a branch-and-cut algorithm for the solution of the MMPRPSC. Computational experiments also confirm the effectiveness of the valid inequalities. The computational results of a case study show a 15% percent decrease in the total cost.