Analytical and managerial implications of integrating product substitutability in the joint pricing and procurement problem

Abstract

This paper studies the analytical and managerial implications of product substitutability on the joint pricing and procurement decisions. We consider a single-period model with two products: an existing product and an improved new product that can substitute the demand for the existing product in case of a shortage. Demand for each product follows a general distribution with an expected value that is a linear function of the price of the new product. While the price of the existing product is determined by the market, it is necessary to determine the new product’s price and the procurement quantities of both products so as to maximize the profits. We analytically show that the expected profit function is unimodal and in the existence of substitution: the expected total profit is higher; the optimal price and the safety stock of the new product are higher; and the optimal safety stock of the existing product is less. Using these properties an efficient algorithm is developed. We also provide a numerical analysis to demonstrate that considering substitution in advance could increase the profitability by 58% and the new product price by 5% while decreasing the total procurement quantity by 15%.

Introduction

This paper considers the joint pricing and procurement problem of a company that introduces a new product that can be used as a substitute for an existing product. As companies try to increase their market shares, they expand their product lines by introducing improved versions of their existing products which have already been cloned by other companies. Intel introducing a new generation CPU, Nokia introducing the color screen cellular phones, HP introducing a faster color printer, are some of the many examples. Depending on pricing and product availability, these new products have the potential to cannibalize sales from the existing products. If Motorola prices its newest innovative broadband enabled phone very close to the price of its existing base model phones, customers, who only need a basic mobile phone, will likely prefer the broadband enabled phone even though Motorola has both products in stock (more value for the buck!). On the other hand, customers, who need (or want) a broadband enabled phone, would buy the broadband enabled phone even though its price is twice the price of a basic phone. It is clear that the demand for an existing product is not only influenced by its inherent characteristics, but also by the price and the inventory level of the new product. If substitutability of products is not considered as an integral part of the inventory control and pricing strategies, it is challenging to generate accurate demand forecasts and to determine the right production quantities and price that maximize the profits. Survey results reported at the Harvard/Wharton Merchandizing Effectiveness Project (Fisher and Raman, 1998) have shown that the demand forecasts tend to have an average error of 50%. New York Times (June 2, 1994) reported a survey conducted by a major retailer stating that 50% of customers who visited a store did not purchase any product, and of these 40% stated that they did not buy anything because they did not find what they specifically wanted. Due to these errors in demand forecasts, retailers often understock some products losing sales and overstock some others whose prices then are marked down at the end of the season. Frazier (1986) estimates that these inventory related costs in the US retail industry alone are about $25 billion a year, which sums up to be 25% of total sales. Besides these inventory related costs, AMR research (2001) suggests that if pricing decisions are incorporated with capacity and production related decisions, $90 billion of incremental operating margin is expected to be realized in the US manufacturing industry alone.

Our main objective is to address these demand, price, product substitutability, and inventory related issues. We intend to reveal the benefits of considering substitution as an integral part of the new product introduction strategy rather than a last minute resolution to avoid shortages. We consider two products, a new and an existing product. Demand for each product has a sure part that varies with the new product price and an uncertain component. The objective is to find the optimal procurement quantities and the new product’s price that maximize the firm profits in a single period. We first analytically show that the expected profit, the optimal new product price and stocking factor, which is defined as the amount procured on top of the sure demand, are higher if substitution is allowed; and the stocking factor of the existing product is less in the existence of substitution. Then, we prove the unimodality of the objective function with respect to the decision variables. Using the unimodality of the objective function and properties of the optimal solution, we are able to propose an efficient algorithm that finds the optimal decisions in less than 5 CPUs. A numerical study is conducted to demonstrate the magnitude and the sensitivity of the benefits that can be achieved by considering pricing and substitution issues in advance. We also present some comparative statics and report on the efficiency of the proposed algorithm. Our main conclusion is that considering substitution in joint pricing and procurement problems is very important. It results in an average profit increase of 30% (up to 58%). From the limited number of problem instances that we have solved, we see that, as compared to substitutability, centralizing the pricing decision seems to have a smaller effect. That is, deciding the price in advance without considering the substitutability of the products and then deciding the procurement quantities with this a priori price does not seem to cost the firm much (less than 0.5%). This result could lead to very effective heuristic approaches to the analysis of more realistic problems with multiple products and periods.

There is a significant body of work in the Operations Management, Economics, and Marketing disciplines that addresses the issue of product substitutability. Most of the economics and marketing literature, that we are aware of, ignores the inventory control aspects of these problems (see Guadagni and Little, 1983, Bultez et al., 1989, Anupindi et al., 1998). The operations management literature, although has a vast number of papers that integrate pricing and inventory control decisions (see Chan et al., 2004, Elmaghraby and Keskinocak, 2003), seems to have ignored price as a decision variable when product substitutability is considered. In what follows we discuss and compare our paper with some of the most relevant existing work.

McGillivray and Silver (1978) investigate the effect of full substitutability on inventory control policies and costs in a multi-product model. Full substitutability is when any product can substitute the demand for any other product. The authors acknowledge that there is no known tractable way of analyzing this problem, and for the two-product case they generate order up to levels using simulation and heuristic techniques. Parlar and Goyal, 1984, Pasternack and Drezner, 1991 consider two fully substitutable products and each independently show that for given prices the expected profit function is a concave and submodular function of the production quantities. Bassok et al. (1999) consider inventory control decisions for one-way substitutable products under exogenous prices. They assume that the products are classified into different grades and the higher grade products can be used to substitute lower grade products. They show that the expected profit function is concave and submodular, and characterize the behavior of the optimal solution with respect to the initial inventories. Similar to Bassok et al., 1999, Netessine et al., 2002 consider one-level downward substitution for products with correlated demands that follow multi-variate Normal distribution. Allowing substitution by only one level and considering a specific demand distribution simplify the expressions for optimality conditions, which help the authors calculate the optimal solution in an efficient way. Rajaram and Tang (2001) consider a fully substitutable multiple-product model in which the demand for each product is Normally distributed and are correlated. The authors approximate the effective demand under substitution and develop a heuristic to solve the problem. Netessine and Rudi (2003) consider the problem of determining optimal procurement quantities of multiple substitutable products under competition and exogenous prices. They show that first-order conditions can be used to estimate optimal procurement quantities if the distribution of the effective demand under substitution is available.

While none of the above studies consider price as a decision variable, Birge et al. (1998) address the joint pricing and procurement decisions for two substitutable products each with independent Uniform demand distribution. The authors analyze three scenarios: (1) Price of each product is given; (2) Price of one product and the production capacity of the other product are given; and (3) Production capacity of each product is given. For each scenario authors determine the optimal values of the remaining variables.

In terms of cost structure, planning horizon, and inventory control aspects, our paper is most related with Pasternack and Drezner, 1991, Birge et al., 1998. We extend these earlier works in several ways: first, not only the procurement quantities of the products but also the price of the new product is a decision variable. Second, the demand distribution of the products are assumed to be any general distribution. Third, extensive analytical results along with numerical results are provided.

In the following section, we formulate the problem. Section 3 discusses the analytical challenges that are created by the pricing and substitutability of the products, and the managerial implications of considering substitution as an integral part of the planning stage. We show that the optimal stocking factor and the optimal price of the new product are greater and the stocking factor of the existing product is less in the existence of substitution. Our results in this section are completely independent of the demand distributions. In Section 4, we provide a condition on the demand distribution of the new product that helps us show that the total expected profit function is unimodal. The condition we develop is a weak one and all log-concave distributions satisfy it. Using certain properties of the optimal solution, we provide an efficient algorithm in Section 5 to find the optimal procurement and pricing strategy. We discuss the managerial implications of substitution in Section 6 where we report on a numerical study. In the last section we conclude the paper and provide future research avenues.