Coordinated Pricing and Inventory Control With Batch Production and Erlang Leadtimes

Abstract

This paper addresses a joint pricing and inventory control problem for a batch production system with random leadtimes. Assume that demand arrives according to a Poisson process with a price-dependent arrival rate. Each replenishment order contains a single batch of a fixed lot size. The replenishment leadtime follows an Erlang distribution, with the number of completed phases recording the delivery state of outstanding orders. The objective is to determine an optimal inventory-pricing policy that maximizes total expected discounted profit or long-run average profit. We first show that when there is at most one order outstanding at any point in time and that excess demand is lost, the optimal reorder policy can be characterized by a critical stock level and the optimal pricing decision is decreasing in the inventory level and delivery state. We then extend the analysis to mixed-Erlang leadtime distribution which can be used to approximate any random leadtime to any degree of accuracy. We further extend the analysis to allowing three outstanding orders where the optimal reorder point becomes state-dependent: the closer an outstanding order is to its arrival or the more orders are outstanding, the lower selling price is charged and the lower reorder point is chosen. Finally, we address the backlog case and show that the monotone pricing structure may not be true when the optimal reorder point is negative. 530Z. Pang and F.Y. Chen challenges intrinsic to problems with dynamically controllable prices (Federgruen and Heching [13]). In standard inventory models with fully backordering and exogenous prices (hence exogenous random demand), the impact of current inventory positions (= inventory levels + outstanding orders − backorders) is evaluated on the basis of the present value of expected inventory cost one order leadtime later. However, in the presence of positive leadtime, with dynamically controllable prices, these future costs depend on the pricing decisions of future periods during the leadtime, and they are therefore unspecified in the current period. An exact formulation requires a multi-dimensional state space to record the inventory levels on hand and on order, and is therefore very difficult to analyze. Pang, Chen, and Feng [31] represent the early effort to provide some partial characterization of the optimal policy structure for periodic-review systems. Our paper attempts to continue their effort to address the continuous-review systems.Furthermore, the literature on inventory-pricing control has also confined itself to models with arbitrary ordering sizes (i.e., lot-for-lot sizes), whereas in supply chains, materials often flow in fixed batch sizes. For example, consumer packaged goods typically arrive at stores in cartons or case packs, finished goods may be transported in full containers or truckloads, and work-in-process (WIP) is usually processed in some convenient lot sizes (Kapuscinski et al. [22]). In addition, a batch production system can serve as an approximation of a capacitated inventory s...

Inventory Rationing in a Make‐to‐Stock System with Batch Production and Lost Sales

“…(2006, 2009) address the joint pricing and production control problem for the unit and batch (exponential) production system, respectively. Pang and Chen (2010) generalize Chen et al. (2009) to the case with Erlangian lead times and two outstanding orders.…”

“…2012). In the continuous‐review setting, Pang and Chen (2010) present some preliminary analysis for these properties when there are at most three orders outstanding (i.e., k = 3). However, it is not easy to further extend their analysis to the general case where k can be any positive integer.…”