A joint replenishment production-inventory model as an MMAP[K]/PH[K]/1 queue

In this paper we analyze a continuous review finite capacity production-inventory system with two products in inventory. With stochastic order quantities and time between orders, the model reflects a supply chain that operates in an environment with high levels of volatility. The inventory is replenished using an independent order-up-to (s, S) policy or a can-order (s, c, S) joint replenishment policy in which the endogenously determined lead times drive the parameters of the replenishment policy. The production facility is modeled as a multi-type MMAP[K]/PH[K]/1 queue in which there are K possible inventory positions when the order is placed and the age process of the busy queue has matrix-exponential distribution. We characterize the system and determine the steady state distribution using matrix analytic methods. Using numerical methods we obtain the inventory parameters that minimize the total ordering and inventory related costs. We present numerical comparisons of independent and joint replenishment policies with varying lead times, order quantities, and cost reductions. We further demonstrate the interplay between the two products in terms of lead times, order quantities and costs.