Closed-Form Solution for Lot-Sizing Problem in Manufacturing Systems with Imperfect Matching

Integrated single-vendor single-buyer problem has received a lot of attention as it is the building block for wider supply chains. This paper considers a manufacturing system in which a single manufacturer procures raw materials from a single supplier, processes them to produce finished products, and delivers to a just-in-time (JIT) customer. The manufacturer's production cycle is imperfect where the production uptime and cycle time are not exact integer multiples of the shipment interval to the customer. The closed-form solution for the optimal batch size is developed to minimize the total cost of procurement and production. It is found that the total cost under imperfect matching is always lower than or equal to that under perfect matching. Different from previous works under perfect matching, it is found that the optimal total cost is a piecewise concave function of the ordering and setup costs, and also a piecewise convex function of the shipment size to the customer