A Joint Production and Delivery Problem in Multi‐Factory Multi‐Distribution‐Center Multiproduct Systems With Limited Capacities

We consider a joint production and delivery problem in multi‐factory multi‐DC (distribution center) multiproduct systems with limited production and delivery capacities over a finite horizon. The objective is to minimize the system's expected total cost. Since the structure of the optimal policy is hard to find, we propose a Lagrangian relaxation heuristic to solve the problem. The proposed heuristic is based on solving a Lagrangian relaxation of the original problem. Although the Lagrangian relaxation problem remains challenging due to the joint production and delivery decisions, we identify a zero‐inventory policy that enables further decomposition into independent single‐product, single‐DC subproblems, each of which can be solved independently. We evaluate the heuristic's performance by deriving a theoretical upper bound on its expected loss. In numerical experiments, we compare the Lagrangian relaxation heuristic with a benchmark myopic heuristic. The results consistently show that the Lagrangian relaxation heuristic achieves a significantly smaller expected relative loss and exhibits greater stability than the myopic heuristic.