Optimization of a stochastic joint replenishment inventory system with service level constraints

An inventory system with stochastic demands is considered, where multiple items are jointly replenished. The joint replenishment is controlled by a P(s,S) policy, where s defines the aggregate reorder level of all items, and the order-up-to level of each item is defined by the vector S. The cost of this system includes major ordering costs, minor ordering costs, and holding costs, and the service level of each item is required to achieve a certain given level. The objective is to optimize the parameters of this policy so that the expected total cost of this system for one period is minimized subject to the service level constraints. After deriving analytically the cost function and the service levels of this system based on probability analysis of the time between two orders, we develop an efficient algorithm for this optimization. Numerical experiments of this algorithm on a large number of instances demonstrate its efficiency. Furthermore, a parameter sensitivity analysis of the system provides some managerial insights for its inventory management.