Planning and managing replenishment policies of items plays an important role in supply chain management. In this paper, a new mathematical model is developed to optimize replenishment policies of a two-echelon inventory system under demand uncertainty. The system consists of one factory warehouse at the upper echelon and three supermarkets at the lower echelon. A special case of this model is where sales price and scheduled inventory replenishment periods are uniformly fixed over all echelons. Demand at the supermarkets is stochastic and stationary. Adopting a Markov decision process approach, the states of a Markov chain represent possible states of demand for milk powder product. The objective is to determine in each echelon of the planning horizon an optimal replenishment policy so that the long run sales revenue is maximized for a given state of demand. Using weekly equal intervals, the decisions of when to replenish additional units are made using dynamic programming over a finite period planning horizon. A numerical example demonstrates the existence of an optimal state-dependent replenishment policy and sales revenue over the echelons.

This paper presents a finite horizon Markov decision process model for determining the optimal production lot size (PLS) of multiple items with demand uncertainty. The model is formulated using states of a Markov chain that represent possible states of demand for items. Using weekly equal intervals, the decision of whether or not to produce additional units is made using dynamic programming over a finite period planning horizon. The proposed model demonstrates the existence of an optimal state-dependent production lot size as well as the corresponding production-inventory costs for items. A numerical example is taken to illustrate the solution procedure of the developed model.