Inventory Management with Multiple Demand Classes and Pricing


Student thesis: Doctoral Thesis

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Awarding Institution
  • Yimin YU (Supervisor)
  • Zhiying Liu (External person) (External Supervisor)
Award date1 Dec 2017


In practice, there are many situations where customers can be classified into different classes. For example, customers may have different service level requirements, profit margins, priorities (e.g., for spare part services, some demands may be more important than other demands), etc. This thesis mainly concentrates on the problem of how to satisfy demands from distinct customer classes, either by inventory rationing or by dynamic pricing. In the former, customer classes are characterized by their backlogging costs. In the latter, customers can be regarded as in the same class if they pay the same price.

Chapter 1 gives a brief introduction to the research background and the problems studied in the following chapters. In Chapter 2, we revisit the classical model in Topkis (1968). Topkis (1968) shows the optimality of the critical rationing level policy for an inventory system with multiple demand classes. However, even with this result, the corresponding dynamic programming problem for the backordering setting is still multidimensional and computationally challenging, which limits its applicability in practice. Under a novel state transformation, we are able to show that the corresponding value functions are decomposable, i.e., each of them can be expressed as the sum of single-variable convex functions. We show that alternatively, the optimal inventory allocation policy can be described by a nested base stock policy under our transformation. Our approach allows us to extend the results to the models with exogenous supply and priority upgrading.

In some cases, customers require fast shipping/logistics in addition to product availability. As a commonly observed practice, order expediting has been used by firms to shorten the lead time and maintain fast response to demands. Therefore in Chapter 3, we focus on how to coordinate inventory ordering, expediting and rationing within the context of multiple demand classes. Through a novel transformation method of state variables, we first prove the L$^{\natural}$-convexity for the value function of the underlying inventory system. Then with a convex expediting cost structure, we show that it is optimal to sequentially fulfill demands with higher unit backordering costs first and expedite inventory from lower leadtime positions first. Based on these results, we then partially characterize the structure of the optimal ordering, expediting and allocation policies. The optimal expediting policy can be described by a state-dependent nested threshold policy. It is optimal to use a state-dependent inventory rationing policy for inventory allocation. For the ordering decision, a state-dependent base stock policy is optimal and the optimal base stock level depends on both the amount of backorders in different demand classes and inventory levels of leadtime positions. Furthermore, we also obtain a series of monotone properties regarding the optimal ordering, expediting, and allocation decisions, which are useful in reducing the computational complexity. We can further simplify the optimal policies under various special settings of our model.

Finally, we consider another way to differentiate customers, pricing. Chapter 4 investigates the problem of coordinating inventory and pricing decisions with general demand functions. We show for the backorder setting that a BSLP policy is optimal if demand functions are decreasing in price and strictly decreasing (increasing) in the realizations of random noises, and their sensitivities in price have the upper-set (lower-set) decreasing property (USDP/LSDP). For the lost-sales setting, a BSLP policy is proved to be optimal if (i) demand functions are decreasing in price and strictly decreasing (increasing) in the realizations of random noises, and their sensitivities in price have the USDP (LSDP), and (ii) the single-period expected profit function is submodular in price and inventory level. Moreover, the proposed concept of USDP/LSDP can be applied to other operations management problems. For example, the backorder/lost-sales inventory system with inventory-dependent demand or quality-dependent demand.