Out of Stock?

FENG, Youyi | XU, Jianjun | ZHENG, Shaohui

Downward substitution is common in many industries and occurs in supply chain management when a firm used a higher value item to satisfy the demand of a lower value item that is out of stock. This type of substitution is driven by the supplier in order to increase customer service quality and / or revenue. For example, a hotel might choose to fill its reservations with more expensive rooms when the more affordable rooms are not available.

Although there is extensive research covering single-period inventory problems with substitution and stochastic demands, only a very limited number of papers have looked into multiperiod inventory problems with substitution.

Previous studies have shown that the optimal solution is to first use any available inventory to satisfy same-class demand, and then to upgrade customer demands until the inventory reaches protection limit. Additionally, more recent studies have worked towards addressing the issue of joint replenishment and pricing decisions for substitutable products. The other stream of research that is closely related to the inventory control of substitutable products is on inventory systems with transshipment. This field of research investigates the optimal joint control of inventory and transshipment for a system that produces has inventories of a product in multiple locations and has demand uncertainty.  

A recent study by Youyi Feng, Jianjun Xu, and Shaohui Zheng considers a periodic-review inventory system that has stochastic demand for two products. Product 1 is of better quality or has more functionality and can be used as a substitute for product 2 if needed. In this instance, downward substitution is also allowed. The researchers assume that the replenishment lead time for each product is positive and that there is a fixed replenishment/production capacity for each product. The aim of their study was to characterize the structures of the optimal replenishment and substitution policy and to find useful insights from the structural properties.

The authors explain, “Our model differs from the research in the literature on the inventory systems of substitutable products in two respects. First, substitution is a key decision to be made in our model, which interacts with the replenishment decision in each period, whereas the dominant research in the related literature assumes that substitution follows a predetermined rule and is not a decision. Second, we assume a positive replenishment lead time for each product, while all of the studies mentioned above assume a zero replenishment lead time.” A positive replenishment lead time is more practical but will make the system much more difficult to analyze. 

Based on the results of their model, the researchers found that under simple conditions, the optimal ordering and substitution policies have monotone structures, the properties which are important for deriving the policies. However, for large replenishment lead times, they explained that it was very difficult to compute the optimal polices because of the curse of dimensions. As such, the authors mentioned that they may need to rely on heuristics in such a case. For example, they suggested, “One possible heuristic [among others] is to assume that substitution follows a predetermined rule and is not a decision. With this simplification, we will only have one decision on the replenishment quantities in each period”.

ZHENG, Shaohui

Head, Chair Professor
Information Systems, Business Statistics & Operations Management