Production, Manufacturing and Logistics
The impact of dynamic pricing on the economic order decision

https://doi.org/10.1016/j.ejor.2008.10.011Get rights and content

Abstract

This paper analyzes the impact of dynamic pricing on the single product economic order decision of a monopolist retailer. Items are procured from an external supplier according to the economic order quantity (EOQ) model and are sold to customers on a single market without competition following the simple monopolist pricing problem. Coordinated decision making of optimal pricing and ordering is influenced by operating costs – including ordering and inventory holding costs – and the demand rate obtained from a price response function. The retailer is allowed to vary the selling price, either in a fixed number of discrete points in time or continuously. While constant and continuous pricing have received much attention in the literature, problems with a limited number of price changes are rather rare. This paper illustrates the benefit of dynamically changing prices to achieve operational efficiency in the EOQ model, that is to trigger high demand rates when inventories are high. We provide structural properties of the optimal time instants when the price should be changed. Taking into account costs for changes in price, it provides numerical guidance on number, timing, and size of price changes during an order cycle. Numerical examples show that the benefits of dynamic pricing in an EOQ framework can be achieved with only a few price changes and that products being unprofitable under static pricing may become profitable under dynamic pricing.

Introduction

The application of innovative pricing techniques to improve supply chain performance is receiving growing attention in many industries. In particular, the coordination of dynamic pricing and operations decisions is still at an early stage and offers significant opportunities for improving supply chain performance. In this context, dynamic pricing means the use of intertemporal price discrimination to improve inventory and capacity management. This kind of price discrimination over time is already standard practice in many industries. In service industries like airline, hotel, and travel industries, dynamic pricing or revenue management has been introduced successfully. The Internet and the latest information technology enable us to adjust prices at minimal cost to satisfy demand. Fleischmann et al. (2004) review the linkage between pricing and operational decisions where they indicate different drivers for dynamic pricing strategies like promotions and other marketing-related instruments and operations-driven pricing strategies known from supply chain coordination.

In this paper, we investigate an EOQ model that considers coordinated pricing and lot-sizing decisions. A retailer procures a single product from an external supplier and sells it on a single market without competition. Customer reaction to prices is modeled by a price response function and the ordering process is subject to variable procurement cost and a fixed ordering cost. Inventories are subject to holding costs. The objective is to maximize average profit by choosing an optimal lot-size and pricing strategy where the retailer is allowed to vary the selling price over time. The contribution is to analyze the EOQ model with a limited number of price changes. The main motivation for considering a limited number of price changes are organizational costs associated with each price change. As we will show, the majority of benefits can be captured by very few different price levels and therefore, a discrete number of changes balances benefits and costs of price changes. The retailer has to decide how to set the price for the product in each time interval where the price is fixed and when to switch from one price to another. Implicit results are the optimal cycle length and the optimal order quantity. We assume that the retailer has perfect information on the demand process. Often, deterministic models are used to gain stylized structural properties that can be used to develop heuristic policies in more complex problems. In particular, models with perfect information provide valuable insights on how optimal policies depend on various model parameters. Furthermore, we assume that customers do not act strategically. Customers are myopic if they do not anticipate future price changes or if they do not have the possibility to forward buy, e.g., holding inventory at the customer is prohibited or too expensive. According to the classification of Elmaghraby and Keskinocak (2003), this model can be grouped into the class of R-I-M models (Inventory Replenishment, Independent Demand, and Myopic Customers). From conducted interviews with directors of leading dynamic pricing optimization solution providers (DPOSP) they found that the current offerings are primarily to R-I-M and NR-I-M (No Inventory Replenishment, Independent Demand, and Myopic Customers) markets. If this assumption is relaxed, the main benefit of changing prices in an EOQ framework, that is, achieving operational efficiency by increasing demand when inventories are high, even increases from the perspective of a retailer because inventories are reduced even earlier. We first present two models as benchmarks for a lower and an upper bound on the profit, namely the problem when only one selling price is charged over the whole order cycle (constant pricing) and when price adjustments are made continuously. These extreme cases imply infinite and zero cost associated with a price change, respectively. The general model is illustrated by analytical results obtained for specific price response functions as well as by numerical investigations. Closed-form solutions for pricing and timing strategies are obtained and numerical algorithms are introduced. We consider an example to illustrate the properties of the optimal pricing and timing policies.

The remainder of this paper is organized as follows. After a literature review in Section 2, we introduce the assumptions and notation as well as the model for a given number of price changes in Section 3. Sections 4 Constant pricing, 5 Continuous price adjustments characterize two models that determine a lower and an upper bound for the average profit when the retailer coordinates the decisions on pricing and order quantity. To gain more structural insights, Sections 6 Linear price response, 7 Exponential price response analyze linear and exponential price response functions. In Section 8, we present a numerical example. The paper ends with conclusions and an outlook on further research in Section 9.

Section snippets

Literature review

A variety of aspects of joint pricing and manufacturing has been analyzed in operations management and marketing literature. Eliashberg and Steinberg (1993) provide a comprehensive review of problems at the interface between marketing and operations management. Bitran and Caldentey (2003) review research results of dynamic pricing policies and their relation to revenue management. Elmaghraby and Keskinocak (2003) provide a review of literature and current practices in dynamic (intertemporal)

Model description

Consider a retailer who is selling a single product on a market without competition over an infinite planning horizon. We assume that customer demand follows a function of the selling price P and arrives continuously at a rate of D(P) which is a differentiable, strictly decreasing, and convex function in P with D(P)0, D(P)<0, and D(P)0. We assume that there exists a critical price P¯(0,) such that D(P)>0 for P[0,P¯) and D(P)=0 for P[P¯,). Furthermore, we assume that D(P) satisfies 2-D(P

Constant pricing

A lower bound for the profit is given by a model of simultaneous decision of pricing and inventory when the retailer charges a constant selling price over the whole order cycle (t1=T). The decision maker maximizes the average profit by the simultaneous setting of the order quantity Q and a single sales price P. The profit function can be obtained from (7)Π(T,P)=P-c-h2TD(P)-FT,subject to PP¯. The optimal order interval and the optimal price result from analyzing the Lagrange function L(T,P,μ)=Π(

Continuous price adjustments

Assume that the retailer is allowed to vary the selling price continuously. At each time t, the retailer is allowed to charge a different price P(t). The objective is to maximize the average profit by determining the optimal price trajectory P(t) and the optimal cycle length T. This yields an upper bound for the profit. The problem can be solved by a two-stage hierarchical optimization problem. At the second-stage, we determine the optimal price trajectory P(t) given a fixed T and without

Linear price response

Assume that at any point in time t the market potential is denoted by a and an amount of bP customers decide that the price is too high and do not buy. In particular, the reservation price P¯=ab denotes the price where the demand rate drops to zero.D(P)=a-bP:0Pab,0:P>ab.A linear price response function is often used in economics, marketing, and operations management literature. In the following, we analyze constant pricing, discrete-time price changes, and continuous price adjustment for the

Exponential price response

A second class of demand functions assumes a non-linear reaction to price variations. In the following we assume an exponential relationshipD(P)=ae-bP.The limiting value limPD(P)=0, i.e. the reservation price is infinite Simon (1989). For non-linear price response functions, in general, we do not obtain closed-form solutions for the optimal prices and the optimal cycle length. By inserting (41) into (8), (12), the optimal static price and the optimal cycle length areT=2Fhae-bPandP=c+1b+h22F

Numerical study

Consider a linear price response function D(P)=a-bP with a=500 and b=20.5. The setup cost is F=900, purchasing cost c=15 per unit, and inventory holding cost h=1.5 per unit and time unit. Furthermore, we set the menu cost κ(N)=0. This assumption provides an upper bound for the optimal profit if price adjustments occur continuously (N=) (see Fig. 4).

The results of Table 1 and the behavior of the optimal average profit illustrated in Fig. 3 indicate that potential for improvement exists even

Conclusions

This paper analyzes a problem of jointly determining the profit-maximizing pricing and lot-sizing policy with intertemporal price discrimination in an EOQ framework. Besides providing further evidence for the benefits of dynamic pricing (which admittedly has already been pointed out in more complex environments), we especially show its impact on operational (order quantity and order cycle) decisions. In the general dynamic pricing model, we could not derive closed-form solutions for the optimal

Acknowledgements

The authors would like to thank Professor Peter Kelle from the Louisiana State University, USA and the referees of this paper for their very constructive remarks and feedback. We especially acknowledge that Proposition 4 was suggested by a referee.

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