An optimal vendor-buyer cooperative policy under generalized lead-time distribution with penalty cost for delivery lateness

This paper presents an integrated inventory model for a single-vendor and single-buyer where lead-time is a stochastic variable with general distribution function. The vendor delivers goods at a fixed lot size to the buyer who has a constant demand rate. Storage is permitted on both parties, but a penalty cost is assessed when the vendor delivers the shipment late beyond a threshold time. The problem is formulated as a nonlinear cost model which needs to be minimized to arrive at an optimal policy for reorder point, order quantity, and number of shipments from the vendor to buyer, to cooperatively operate the joint contract. The solution procedure involves both closed form solution and iterative search procedure for this multi-dimensional problem. Numerical examples are presented for uniform, exponential and normally distributed lead-times to demonstrate the solution methodology as well as a precursor test for special cases. The model presented here are applicable for numerous two-stage supply chain systems including retail businesses, assembly and production houses, where two parties of the supply chain are tied to their common goal of achieving minimum total cost of inventory operation.