Inventory management with overlapping shrinkages and demands

This article analyzes a discrete time lost sales inventory system with partially observed demand and unobserved shrinkages which happen both before and after the demand realization. When the demand exceeds the remaining inventory, the unmet demand is lost and unobserved. This problem in general has a nonlinear state evolution, and we use unmoralized probability to linearize the system. Despite this, the problem still has a complex-dimensional state space, and we therefore focus on two special cases where the shrinkage either happens before or after the demand realization. With dynamic programming and unnormalized probability, we formulate the Bellman equation. We obtain a lower bound on the cost analytically via the formulation of a fictitious inventory problem. We also develop an iterative algorithm, and compare its solution to a myopic solution as well as the lower bound. This comparison reveals that the solution obtained using the iterative algorithm performs significantly better than the myopic solution in many cases and, moreover, the achieved cost is close to the lower bound, thereby highlighting the value for our algorithm.