Analytical solution and optimisation of serial supply chains with multiple nodes, lost sales, continuous review replenishment policies, stochastic lead times and external demand

This work analyses a serial supply chain with an arbitrary number of nodes (retailer, wholesaler, manufacturer, supplier, etc.). Every node faces supply uncertainty in both replenishment lead time and quantity delivered (except for the most upstream node, assumed saturated). The most downstream node (retailer) faces external Poisson demand with rate λ. Each node follows a continuous review (si, Si), i = 1, 2, …, K - 1 replenishment policy. If a node is insufficiently stocked, then the order is only partially satisfied. The system parameters that influence the performance of the supply chain are the number of nodes, the upper level of inventories, the reorder levels, the replenishment times, and the demand rate. The supply network is modelled as a continuous time Markov process with discrete states. We explore the structure of the transition matrices and develop a computational algorithm to generate the stationary probability distributions for any combination of the system parameters. We also use the proposed algorithm as a design and optimisation tool to determine the optimal inventory policies (i.e., reorder level and order-up-to-level) to maximise the fill rate (FR) or minimise the work-in-process (WIP) for multi-echelon serial supply chains.