Optimal (z, Z)-type contracts for vendor-managed inventory

We examine (z, Z)-type contracts for vendor-managed inventory (VMI) between a supplier and a retailer from the retailer's perspective. A (z, Z) VMI contract specifies minimum and maximum inventory levels and their corresponding under- and over-stocking penalties. The retailer chooses the contract parameter values and the supplier decides whether to accept the contract and, if accepts, manages the inventory at the retailer and makes replenishment decisions. The supplier accepts the contract if his cost under VMI is no larger than his reservation cost. We provide the optimal (z, Z) VMI contract for the retailer and the corresponding optimal replenishment decisions for the supplier and show that the optimal (z, Z) VMI contract can coordinate the supply chain under mild conditions. We also examine a VMI contract with stockout penalty and holding-cost sharing, which is a special type of (z, Z) contract, and find that it may perform well compared with the optimal (z, Z) VMI contract, although its performance may depend on several factors such as the supplier's reservation cost.