An integrated manufacturer-buyer chain with bounded production cycle length

A mathematical formulation is developed for modeling a joint two-echelon supply chain consisting of a single manufacturer (i.e., vendor) who continuously produces a production lot of size Q items with constant production rate P≥D, where D is the given demand rate set by the retailer (i.e., buyer). The unit production cost c_V(P) is assumed to be a linearly decreasing function of the production rate P. This assumption may hold as long as economies of scale are in effect and production rate does not exceed very high level which entails heavy wear associated with expensive equipment. The optimization problem of minimizing the joint total cost is solved analytically. The proposed formulation avoids scenarios under which the production rate approaches the demand rate (i.e., producing almost without breaks) and scenarios under which the production cycle length (at the manufacturer's warehouse) is unbounded. Very long production cycle lengths inevitably mean very large lots, which are associated with immense and inefficient warehouses. Moreover, they might be infeasible due to scheduled maintenance, holidays or end weeks. We prove that the optimal solution can consist of no more than two switching points along the shipment-size axis n.