A joint production and delivery schedule for a single-vendor single-buyer system over finite horizon

In this paper, we address a joint production and delivery scheduling problem in which a single-vendor supplies goods to a single-buyer over a finite horizon. The vendor's production rate and the buyer's demand rate can vary from period to period and are known in advance. The objective is to determine a production/shipment schedule that minimises the total cost of production setup, shipment, and holding of inventory at both the vendor and the buyer. We approach this problem using a dynamic programming framework, each stage of which constitutes solutions to different types of single-period problems depending on the production phase encountered. We develop effective methods for the solutions of these single-period problems, which are then embedded within the dynamic programming framework. We show that the optimal solution in each period follows a pattern of geometric-then-equal shipment sizes except for the last shipment, which may be larger in size. Furthermore, we show that an optimal solution for the infinite horizon problem can be obtained by using a special case of our finite horizon approach. In addition, we propose two fast heuristic methods for the finite horizon problem, which, as we show, can obtain almost optimal solutions.