Production, Manufacturing and LogisticsOptimal inventory policy for two substitutable products with customer service objectives
Introduction
Product substitution is important to many firms. For example, manufacturers of components used in consumer electronics (CE) routinely design new generations of their devices in a way that allows more than one model of a generation to work in the end-product of a CE manufacturer. If one such model being supplied to a CE manufacturer for use in an end-product stocks out, then the component manufacturer is often able to still meet the CE manufacturer’s demand by providing, in lieu of the model that has stocked out, a superior model. This is an example of downward, supplier-driven substitution. Downward refers to a supplier meeting customer demand for a product j by providing a product i that has better quality and/or more functionality than product j. Supplier-driven indicates it is the supplier of the product that reacts to the stockout and takes action. This is unlike the situation in a restaurant for instance where after a customer learns that his first choice is unavailable, he picks another item from the menu, which is an example of customer-driven substitution.
The earliest study of substitution was undertaken several decades ago by McGillivray and Silver (1978). There now exists a stream of research on this topic. Interestingly, none of it considers customer service objectives, e.g. fill rates or in-stock probabilities (see Cachon & Terwiesch, 2012 for a thorough discussion of these measures). Juxtaposed with this reality are (1) surveys such as the one by Aberdeen (Kay, 2005), which reported that 70 percent of companies feel that providing a high level of service is critical to their business operations, and (2) the fact that in the more general operations and supply chain management literature, there is a steady flow of research with a service level aspect (e.g. see Alptekinoglu, Banerjee, Paul, Jain, 2013, Bensoussan, Feng, Sethi, 2011, Chen, Shen, 2012, Wang, Xiao, Yang, 2014). The lacuna in the substitution literature with respect to service levels, combined with the increasing importance of service levels to business, motivates our study of a substitution problem with customer service objectives.
More specifically, we consider a firm that faces stochastic demand for two products where an in-stock probability service level objective has been set for each product. The in-stock probability of a product is the probability the firm has inventory to meet every customer demand for the product. We designate the two products “1” and “2” with product 1 as good as or superior to product 2 in every respect, hence product 1 can substitute for product 2. As is common practice in the CE industry, we assume that the substitution is supplier-driven. Furthermore, we also assume both products are perishable or prone to obsolescence, i.e. both products have short life cycles. The firm under consideration therefore faces a single period problem, which is to determine in advance of observing demand the profit maximizing inventory/stocking levels of both products that will meet given customer service objectives. Note that throughout the paper we speak of inventory or stocking levels, however the products may be either goods or services and in the latter case we are interested in determining capacities to put in place. Also, we use increasing to mean non-decreasing and decreasing to mean non-increasing.
The rest of the paper is organized as follows. In Section 2, we review the relevant substitution literature. In Section 3, we begin our study of the problem setting without customer service objectives. We develop a general model of inventory policies for the two products with downward, supplier-driven substitution. In Section 4, we introduce into the analysis service level constraints of the in-stock variety and identify the optimal inventory policy for the firm. In Section 5, we present the results of a numerical study that reveal the benefits obtained from substitution and show the impact of customer service objectives on optimal inventory levels. Section 6 summarizes our findings and highlights future research opportunities.
Section snippets
Literature review
As already noted, the earliest investigation of inventory management involving substitution was undertaken by McGillivray and Silver (1978). In their study inventory was managed using an order-up-to policy with a fixed review period. Some analytical results were established for limiting cases including complete-substitutability and no-substitutability. For cases in-between these two extremes, a heuristic approach was developed and tested for establishing order-up-to levels.
Research undertaken
Model description
We consider a firm that faces stochastic demand for two products, designated “1” and “2”, where product 1 can downward substitute for product 2. Furthermore, we assume that the substitution is supplier-driven. In addition to the substitutability aspect, we assume that both products are perishable or prone to obsolescence, hence the firm faces a single period problem. The objective of the firm is to determine in advance of observing demand the profit maximizing inventory levels of both products.
Optimal ordering policies with a service commitment
In this section, we consider the situation where the firm commits to providing a minimum level of service, which is defined in terms of an in-stock probability. For product i, the target in-stock probability is given by αi, which is a number that lies between 0 and 1. The selection of a value αi indicates that the firm wants to stock-out of product i with probability no greater than 1 − αi. Hence, when the stocking level for product i is Qi, for i = 1, 2, the service commitment can be
Numerical study
In this section, we present the results of a numerical study in which we investigated (1) how optimal inventory levels are affected by service level requirements specified as target in-stock probabilities, and (2) how profitability is affected by substitution. In the following we let q1(α1, α2) and q2(α1, α2) denote the optimal inventory levels of products 1 and 2, respectively, when their respective target in-stock probabilities are α1 and α2.
Lemma 5 For fixed α2, as α1 increases, q1(α1, α2), the
Conclusions and suggestions for further research
In this paper, we considered a single period problem in which a firm seeks to determine the profit maximizing inventory levels for two products in advance of observing random demand. In making its stocking decisions, the firm wants to take into account downward, supplier-orientated substitution that it can employ and the firm needs to take into account customer service commitments that it has made. For the firm with this problem setting, we identified the optimal inventory policy with and
Acknowledgments
This work was funded by a Marie Curie International Incoming Fellowship within the 7th Framework Programme of the European Commission (PIIF-GA-2009-253720), National Natural Science Foundation of China, China (nos. 71272128, 71432003), Program for New Century Excellent Talents in University, China (no. NCET-12-0087), and Youth Foundation for Humanities and Social Sciences of Ministry of Education of China (no. 11YJC630022). The authors are grateful to the editor and the two referees, whose
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