The continuous (S,s,S_e) inventory model with dual sourcing and emergency orders

This paper examines an inventory system of a retailer who uses a continuous-review base-stock policy to manage inventory. We consider two types of demands: Markov-modulated fluid demands with linear rates and batch-size demands that arrive according to a Markov arrival process (MAP). In addition, we consider two types of supply: (i) a regular supply that follows an (S,s) policy with stochastic lead times under a lost-sales assumption and (ii) an additional emergency supply that brings the inventory up to level 0<S_e≤S when stockout. We assume that such an emergency supply has a negligible lead time but incurs an extra cost. Hence, we analyze four models that are each adequate for a different demand-supply pattern. Our cost structure includes variable and fixed costs for each type of order, together with holding and lost-sales costs. By integrating a multidimensional martingale technique and hitting-time transforms for fluid models and MAPs, we derive explicit formulas for the cost components that aim to minimize the expected discounted total cost. A numerical study provides insights into the best parameters and how they depend on different factors. For example, we show that the MAP-type arrivals yield higher control values; increasing the lead time (average and variance) and the lost cost increases these values, and eventually leads to the (S,s=S−1) policy. We further show that increasing the discount factor decreases s and the fluid models are less sensitive to the timing. By comparing the models, we provide managers with a framework for assessing when dual sourcing is more beneficial than single sourcing.