An optimal solution algorithm for the constant lot-size model with equal and unequal sized batch shipments for the single product multi-stage production system

This paper presents a new heuristic solution procedure for the constant lot-size model for the production of a single product requiring processing through a fixed sequence of manufacturing stages. There is a single set-up at each production stage followed by continuous production of the whole lot. However, the lot may be transferred to subsequent stages in partial lots, a set of possibly unequal batches, which may vary in size between production stages. Previous models have used a heuristic solution procedure based on the concept of differentiation of the cost function, the sum of the costs of set-up, transportation and inventory. This approach has drawbacks when many of the parameters have to be integer. It also implicitly assumes the cost is a convex function of the lot size. In this situation it can be shown that the function may often be non-convex. Furthermore, the heuristic does not provide a solution directly when the production rates of machines in adjacent stages are equal, and is also unable to consider zero transportation cost. By formulating the constraint that the largest batch size at any stage does not exceed the transport equipment capacity in a different way, a number of properties that the optimal solution should satisfy are developed. An algorithm giving the optimal solution is then derived based on these properties. This is illustrated by numerical examples, which indicate further cost reductions on the most recent models proposed are possible. This modified model and solution enables the sensitivity of the total cost to variations in lot size around the optimal value to be investigated.