A Complete Algebraic Transformational Solution for the Optimal Dynamic Policy in Inventory Rationing across Two Demand Classes
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In this paper, we apply the sensitivity-based optimization to propose and develop a complete algebraic transformational solution for the optimal dynamic rationing policy in inventory rationing across two demand classes. Our results provide a unified framework to set up a new transformational threshold type structure for the optimal dynamic rationing policy. Based on this, we can provide a complete description that the optimal dynamic rationing policy is either of critical rationing level (i.e. threshold type or a static rationing policy) or of no critical rationing level. Also, two basic classifications can be described by means of our algebraic transformational solution. To this end, we first establish a policy-based birth-death process and set up a more general reward (or cost) function with respect to both states and policies of the birth-death process, hence this gives our policy optimal problem. Then we set up a policy-based Poisson equation, which, together with a performance difference equation, characterizes monotonicity and optimality of the long-run average profit of the rationing inventory system. Finally, we apply the sensitivity-based optimization to construct a threshold type policy to further study the rationing inventory system. Furthermore, we use some numerical experiments to verify our theoretic results and computational validity, and specifically, compare the optimal dynamic rationing policy with the optimal threshold type rationing policy from two different policy spaces. We hope that the methodology and results developed in this paper can shed light to the study of rationing inventory systems, and will open a series of potentially promising research by means of the sensitivity-based optimization and our algebraic transformational solution.
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