In this paper, we propose a multiperiod single-sourcing problem (MPSSP), which takes both transportation and inventory into consideration, suitable for evaluating the performance of a logistics distribution network in a dynamic environment. We reformulate the MPSSP as a Generalized Assignment Problem (GAP) with a convex objective function. We then extend a branch-and-price algorithm that was developed for the GAP to this problem. The pricing problem is a so-called Penalized Knapsack Problem (PKP), which is a knapsack problem where the objective function includes an additional convex term penalizing the total use of capacity of the knapsack. The optimal solution of the relaxation of the integrality constraints in the PKP shows a similar structure to the optimal solution of the knapsack problem, that allows for an efficient solution procedure for the pricing problem. We perform an extensive numerical study of the branch-and-price algorithm.
Freling, R., Romeijn, H. E., Romero Morales, D., & Wagelmans, A. P. M. (2003). A Branch-and-Price Algorithm for the Multiperiod Single-Sourcing Problem. Operations Research, 51(6), 922-939. https://doi.org/10.1287/opre.51.6.922.24914