The effect of market segmentation with demand leakage between market segments on a firm’s price and inventory decisions

European Journal of Operational Research 182 (2007) 738–754 www.elsevier.com/locate/ejor

Production, Manufacturing and Logistics

The effect of market segmentation with demand leakage between market segments on a firm’s price and inventory decisions Michael Zhang *, Peter C. Bell Richard Ivey School of Business, London, Canada N6H 4A9 Received 15 December 2004; accepted 18 September 2006 Available online 15 November 2006

Abstract In this paper, we address the simultaneous determination of price and inventory replenishment in a newsvendor setting when the firm faces demand from two or more market segments in which the firm can set different prices. We allow for demand leakage from higher-priced segments to lower-priced segments and assume that unsatisfied demand can be backlogged. We examine the case where the demands occur concurrently without priority and are met from a single inventory. We consider customer’s buy-down behavior explicitly by modeling demand leakage as a function of segment price differentiation, and characterize the structure of optimal inventory and pricing policies.  2006 Elsevier B.V. All rights reserved. Keywords: Revenue management; Dynamic pricing; Newsvendor problem with price effects; Fences; Market segmentation

1. Introduction The single period stochastic inventory problem (or ‘‘newsvendor’’ problem) is to determine the quantity of product to order by a vendor facing stochastic demand over a short selling season (Nahmias, 1996). If the quantity ordered exceeds actual demand there is an overage cost associated with the unsold inventory, while if demand exceeds the available inventory the vendor pays an underage cost associated with the unsatisfied demand. The optimal quantity to order is well understood under a variety of conditions. For example, Ismail and Louderback (1979) introduced alternative object functions, such as maximizing expected utility, in place of expected profit; Kabak and Weinberg (1972) considered uncertainty in supply; Eppen (1979) and Chang and Lin (1991) investigated the multi-location problem; Bassok et al. (1999) studied a multi-product model with substitution and Khouja (2000) addressed the issue of optimal discounting to sell excess inventory. Revenue management (RM) emerged in the newly deregulated world of the airline industry in the mid1980s, and has since found its way into all sectors of the travel industry, as well as freight, media, utilities *

Corresponding author. E-mail addresses: [email protected] (M. Zhang), [email protected] (P.C. Bell).

0377-2217/$ - see front matter  2006 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2006.09.034

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and retail trade (see Weatherford and Bodily, 1992 for a review). RM has been credited with revenue increases of 2–8% without increasing the supply of product (Smith et al., 1992). A key part of RM is setting product prices, and particularly using aggressive pricing strategies to generate more revenue from the marketplace while selling the same amount of product, but also of interest is the question of whether the RM firm should increase the supply of product. The newsvendor problem with the addition of product pricing decisions has provided a useful model for examining such joint pricing and inventory decision making (Whitin, 1955; Bell and Zhang (2006)). One primary strategy for RM is to segment a single market into multiple sub-markets and to potentially set different prices in each market. For example, Bell and Chen (2006) considered a case of market segmentation based on delivery time preference and showed that differential pricing based on delivery time can greatly enhance revenues for the seller. A second example occurs with the airlines which routinely sell seats at six to ten different prices on a single scheduled flight, ranging from ‘‘supersaver’’ cut price fares to full-fare economy and then business class fares. Such pricing increases revenue to the airline by exploiting the fact that business travelers will pay a high fare in return for convenience and flexibility (and also a larger seat and better service!) while leisure travelers can be segmented into a number of different submarkets based largely on when they buy their tickets. Market segmentation is achieved by attaching a set of conditions and restrictions to each fare category. Devices such as advanced purchase and refund penalties ‘‘fence’’ customers into different market segments and make it difficult or time consuming for the customer to migrate from one market segment to another. As one example, a customer who bought a ticket early but then observed that the fare had gone down as a result of low sales for that flight would want to purchase a ticket at the new lower price and return the old ticket in order to save the fare difference. A non-refundable ticket or a fee to change flight arrangements could be used by the airline to limit ticket switching and so preserve revenues. A ‘‘fence’’ is a device that is designed to preserve market segmentation and limit spillover between market segments; however, most fences are not perfect and allow some degree of demand ‘‘leakage’’ from high priced market segments to low priced segments. The degree of leakage between segments over an imperfect fence can be expected to increase as the price difference increases. In this article we examine the newsvendor problem with price as a decision variable when excess demand can be backlogged. We extend this problem for the case where product is sold to different demand classes at different prices when the demands are realized simultaneously, and examine the impact of demand dependences created by the ‘‘leakage’’ of customers from high price segments to low price segments. We model the leakage across market segments as a function of the price difference between the segments, and investigate the impact when a fence does not perfectly isolate the customer segments and characterize the structure of optimal inventory and prices when demands are stochastic and a function of prices.

2. Previous research The newsvendor problem has been researched extensively (see Khouja, 1999 for a review).Whitin (1955) first examined the newsvendor problem with price effects where selling price and stocking quantity are set simultaneously, and derived a closed-form expression for the case of uniformly distributed demand. Lau and Lau (1988) presented solution procedures for various objectives for normally distributed demand and demand having a distribution constructed using a combination of statistical data analysis and expert’s subjective estimates. Petruzzi and Dada (1999) generalized existing newsboy results for both additive and multiplicative demand cases. A second body of research involves models where demands for different product classes occur concurrently, and inventory is available to multiple classes simultaneously. Gerchak et al. (1985) provided an early example of how revenue management principles can be applied across different businesses. They formulated a dynamic program to determine if a bagel shop should sell a limited supply of bagels as individual items (at a low contribution) or wait for the lunch crowd and sell them as a part of higher contribution combinations (for example, as part of a sandwich). Gerchak et al. noted that this problem was equivalent to the two-fare airline model, with single arrivals and constant arrival probabilities for each type of customers. The work of Gerchak

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et al. (1985) was extended by Lee and Hersh (1993) to multiple fare situations. However, these works do not consider demand dependencies. A number of papers have modeled dependent demands. Pfeifer (1989) partitioned demand into two subsets, price-sensitive and price-insensitive customers, where both subsets were functions of realized demand. Brumelle et al. (1990) treated two-class dependent demands via a bivariate normal distribution. Belobaba and Weatherford (1996) introduced a probability to represent the migration of customers between market segments, and incorporated customers willing to pay full fare or a restricted discount fare, into a static decision rule. Sen and Zhang (1999) considered the newsboy problem with multiple demand classes, where demands were realized sequentially, and modeled the demand dependency created by the returning of a fraction of the unsatisfied demand in a price class to another demand class. Talluri and van Ryzin (2004) specified the probability of purchase for each fare product as a function of the set of fare products offered. Markets segments that were not perfectly isolated leading to leakage between segments appeared in Lovell and Wertz (1981) and La Croix (1983) who considered a monopolist selling a homogeneous product in two markets. Leakage between markets is modeled by assuming ad hoc demand functions for the good in each market that depend on the prices in both markets. In addition, Gerstner and Holthausen (1986) addressed an optimal pricing strategy when considering demand leakage from a transaction cost perspective. 3. The newsvendor problem with backlogged demand We consider a single period inventory problem with a single order and stochastic demand. Define the following notation: p – the price (a decision variable), c – the variable production cost (c < p), I – the quantity available for sale (which may be a decision variable), h – overage cost per unit of inventory unsold at the end of the period, s – underage cost per unit of unsatisfied demand, q – the quantity sold, a random variable with density function f(q). Let: q = D(p, n) be the demand function defined continuously over the interval [p‘, pu]. Following Petruzzi and Dada (1999) we consider a linear demand curve with additive error (n) or q ¼ DðpÞ þ n ¼ a  bp þ n

ða > 0; b > 0Þ:

The error term (n) is assumed to be price independent with density function f(n), mean E(n) and standard deviation rn. As is common in the single period stochastic inventory (or ‘‘newsvendor’’) problem, we assume that the parameter values are such that the probability of negative demand is essentially zero and can be ignored. We assume that unsatisfied demand is backlogged at a cost, as would be the case for a firm that could purchase extra product from a supplier and meet demand by expedited shipping. The contribution, p, depends on the random demand q, distributed f(q)  pðp; IÞ ¼

qp  cI  hðI  qÞ q < I; qp  cI  sðq  IÞ q P I:

The overage cost (h) in this formulation requires further interpretation. The firm establishes a market price (p) and prefers to hold inventory rather than sell at a price below p. We therefore conclude that unsold units carry a capital cost that is proportional to p and we define the overall overage cost as h = rp + H where H is the cost of carrying the inventory (which could be negative if the product is sold for salvage) and r is the periodic interest rate. We will pay particular interest to the special case when r = 0. The vendor chooses values for the decision variables (p and I) based on expected contribution, E(p), which depends on the forecast of the random demand q.

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741

Expected contribution is EðpÞ ¼ pEðqÞ  cI  ðrp þ H Þ

Z

I

ðI  qÞf ðqÞ dQ  s

Z

0

1

ðq  IÞf ðqÞ dQ:

ð1Þ

I

For notational convenience, we write E(q) as D. For the case of linear demand, q ¼ D þ n and f ðqÞ ¼ f ðn ¼ q  DÞ and (1) becomes Z ID Z 1 EðpÞ ¼ pD  cI  ðrp þ H Þ ½I  ðD þ nÞf ðnÞ dn  s ½D þ n  If ðnÞ dn: 1

ID

The vendor might have fixed inventory (I) and must choose a price (p) to maximize E(p), may have a fixed price and must determine inventory to maximize E(p), or may need to choose both I and p. The necessary condition for a maximum for E(p) w.r.t. p is that oE(p)/op = 0 or " # Z Z ID

D  bp þ b s  ðs þ rp þ H Þ

ID

f ðnÞ dn  r

1

½I  ½D þ nÞf ðnÞ dn ¼ 0

ð2Þ

1

or Z

ID

f ðnÞ dn ¼

1

r

R ID

nf ðnÞ dn þ D  bp þ bs : rðI  DÞ þ bðs þ rp þ H Þ 1

ð2aÞ

The necessary condition for a maximum for E(p) w.r.t. I is that oE(p)/oI = 0 which provides a result analogous to the familiar ‘‘newsvendor’’ problem, or Z ID sc f ðnÞ dn ¼ s þ rp þ H 1 or F ½I  D ¼

sc s þ rp þ H

or

I ¼ D þ F 1



 sc ; s þ rp þ H

where F ½I  D is the probability of having sufficient inventory to meet forecast demand. The second partial derivatives of E(p) taken with respect to p and I are " # Z ID o2 EðpÞ ¼ 2b 1 þ r f ðnÞ dn  b2 ðrp þ H þ sÞf ½I  D < 0; op2 1 o2 EðpÞ ¼ ðrp þ H þ sÞf ½I  D < 0: oI 2

ð3Þ

ð4Þ ð5Þ

The optimization problem can be solved by a unidimensional search by first solving for the optimal value of I as a function of p using (3), and then substituting the result back into (2). If the overage cost does not depend on the sales price (i.e. r = 0) (2) becomes " # Z ID D  bp þ b s  ðs þ H Þ f ðnÞ dn ¼ 0 1

or Z

ID

1

f ðnÞ dn ¼

D=b  p þ s : sþH

The necessary condition for a maximum for E(p) w.r.t. I is the familiar ‘‘newsvendor’’ result Z ID sc f ðnÞ dn ¼ s þH 1

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or D=b  p þ s sc ¼ sþH sþH

or

p ¼ D=b þ c

or

p ¼ ða þ cbÞ=2b:

We conclude that the for the case of r = 0, the firm sets the deterministic optimum price [found by choosing p to maximize (p  c) (a  bp)] and then sets the order quantity based on the classic ‘‘newsvendor’’ solution. 4. Two demand classes Now we consider the model with two demand classes, when both demands are realized simultaneously. This is a common situation in the practice of RM, where, for example, airlines sell the same seat at different fares to different customer segments at the same time, or a firm may price products offered on its website differently from the same products offered in a retail store. The firm using multiple purchase channels offering different prices will use a number of different ‘‘fencing’’ devices to try to prevent customers who would normally buy from the high priced channel from purchasing from a lower priced channel. Most of these fences are imperfect in that a determined customer can migrate across segments; that is, demand leakage occurs from high priced to low priced segments. In order to capture the intuition that more customers are willing to cross a fence as the price difference increases, we model the leakage as a linear function of the difference between the prices. Specifically, we set the indices such that p1 > p2, and use the linear function c(p1p2) with c P 0 to denote the demand that leaks from the high priced market to the low priced market. c = 0 represents the case of the perfect fence where demand leakage is zero. The demand functions in the two market segments are given by D1 ðp1 ; p2 Þ ¼ a1  b1 p1  cðp1  p2 Þ; D2 ðp1 ; p2 Þ ¼ a2  b2 p2 þ cðp1  p2 Þ with actual market demand qi ¼ Di þ ni ¼ ai  bpi þ ni

i ¼ 1; 2:

We assume that the errors, (ni, i = 1, 2) are independent, normally distributed random variables with means 0 and standard deviations rni. We define the random variable X X X qi ¼ Eðqi Þ þ ni : QN ¼ N

N

N

P Letffiffiffiffiffiffiffiffiffiffiffiffiffi n ¼ ffi N ni where n is the aggregate error that is normal distributed with mean 0 and standard deviation q P 2 N rni . For notational convenience, we write E(qi) as Di . The overage cost is a function of the average selling price hðp1 ; p2 Þ ¼ r

p1 D1 þ p2 D2 þH D1 þ D2

and EðpÞ ¼ p1 D1 þ p2 D2  cI  h

Z

I

ðI  QN Þf ðQN Þ dQN  s 0

Z

1

ðQN  IÞf ðQN Þ dQN :

ð6Þ

I

4.1. General existence conditions The linear demand functions are only valid for ai  bipi P 0 requiring pi 2 [0, ai/bi] for i = 1, 2. The demand loss from the high price market c(p1  p2) cannot exceed the market demand in that market (a1  b1p1), requiring c 6 (a1  b1p1)/(p1  p2). We restrict attention to p1 > p2 > c.

M. Zhang, P.C. Bell / European Journal of Operational Research 182 (2007) 738–754

Proposition 1. When n ¼

P

N ni

743

¼0, the deterministic solution ðp01 ; p02 ; I 0 Þ satisfies

ðb2 þ cÞða1 þ b1 cÞ þ cða2 þ cb2 Þ ; 2½b1 b2 þ cðb1 þ b2 Þ ðb1 þ cÞða2 þ b2 cÞ þ cða1 þ cb1 Þ p02 ¼ ; 2½b1 b2 þ cðb1 þ b2 Þ 1 I 0 ¼ ða1  cb1 þ a2  cb2 Þ: 2

p01 ¼

Proof. When n = 0, p ¼ p1 D1 þ p2 D2  cðD1 þ D2 Þ ¼ ðp1  cÞD1 þ ðp2  cÞD2 , the first-order conditions op ¼ D1  ðb1 þ cÞðp1  cÞ þ cðp2  cÞ ¼ a1  2b1 p1  2cðp1  p2 Þ þ b1 c ¼ 0; op1 op ¼ ðp1  cÞc þ D2  ðp2  cÞðb2 þ cÞ ¼ a2  2b2 p2 þ 2cðp1  p2 Þ þ b2 c ¼ 0; op2 lead to (Appendix A) ðb2 þ cÞða1 þ b1 cÞ þ cða2 þ cb2 Þ ; 2½b1 b2 þ cðb1 þ b2 Þ ðb1 þ cÞða2 þ b2 cÞ þ cða1 þ cb1 Þ p02 ¼ 2½b1 b2 þ cðb1 þ b2 Þ p01 ¼

and I 0 ¼ D1 þ D2 .

h

Proposition 2. The optimal inventory level (I*) satisfies F ½I   ðD1 þ D2 Þ ¼

sc s þ hðp1 ; p2 Þ

or

I  ¼ D1 þ D2 þ F 1



 sc : s þ hðp1 ; p2 Þ

ð7Þ

Proof. From (6), oE(p)/oI = 0 leads to c þ s  ½s þ hðp1 ; p2 Þ

Z

I  ðD1 þD2 Þ

f ðnÞ dn ¼ 0; 1

which rearranges to (7) directly. We note that the optimal inventory level given by (7) is analogous to the classical newsvendor solution when demand is aggregated and overage cost/unit based on average selling price. h Proposition 3. The difference between the optimal prices in the two markets depends only on the market parameters p1  p2 ¼

a1 b2  a 2 b1 : 2½b1 b2 þ cðb1 þ b2 Þ

2 a2 b1 Since (8) is valid for all rn, this implies: p01  p02 ¼ 2½b1ab12bþcðb . 1 þb2 Þ

Proof. Using (6) (An extended derivation is presented in Appendix A) Z oEðpÞ ohðp1 ; p2 Þ IðD1 þD2 Þ ¼ D1  ðb1 þ cÞp1 þ cp2 þ b1 ½s  ðs þ hÞP  ½I  ðD1 þ D2 þ nÞf ðnÞ dn; op1 op1 1 Z oEðpÞ ohðp1 ; p2 Þ IðD1 þD2 Þ ¼ D2  ðb2 þ cÞp2 þ cp1 þ b2 ½s  ðs þ hÞP  ½I  ðD1 þ D2 þ nÞf ðnÞ dn; op2 op2 1 R IðD þD Þ where P ¼ 1 1 2 f ðnÞ dn.

ð8Þ

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M. Zhang, P.C. Bell / European Journal of Operational Research 182 (2007) 738–754

2 p2 Expanding: hðp1 ; p2 Þ ¼ r D1Dp1 þD þ H and simplifying leads to þD 1

2

2D1  a1 þ b1 c 2D2  a2 þ b2 c ¼ ; ð2D1  a1 ÞðD1 þ D2 Þ þ b1 ðD1 p1 þ D2 p2 Þ ð2D2  a2 ÞðD1 þ D2 Þ þ b2 ðD1 p1 þ D2 p2 Þ which leads to the necessary condition a1 b2  a2 b1  2b1 b2 ðp1  p2 Þ  ð2b1 þ 2b2 Þcðp1  p2 Þ ¼ 0; which rearranges to (8). h Proposition 4. The optimal price when rn > 0 is strictly less than the deterministic case, or pi < p0i :

ð9Þ

(Proof in Appendix A). Proposition 5 (1) E(p) is concave in I for a given p1 (or p2); (2) E(p) is concave in p1 (or p2) for a given I. Proofs are in Appendix A. Theorem 1. p1 and p2 exist "c P 0. Proof. By definition, p1 > p2 hence from (8) we require: a1b2 > a2b1. From Proposition 2 p1 > p2 and combined with Proposition 4, the optimums exist. Proposition 3 specifies the affiliation of optimal prices, while Proposition 4 provides a useful upper bound for the optimal prices. Propositions 4 and 5 provide the basis for an iterative procedure to find the optimal pricing policy. The three-variable optimization problem can be reduced to a search procedure for a twovariable optimization problem. For the case of r = 0, the first-order conditions for optimum prices reduce to the deterministic solutions 2D1  a1 þ b1 c ¼ a1  2b1 p1  2c1 ðp1  p2 Þ þ b1 c ¼ 0; 2D2  a2 þ b2 c ¼ a2  2b2 p2 þ 2c1 ðp1  p2 Þ þ b2 c ¼ 0; or ðb2 þ cÞða1 þ b1 cÞ þ cða2 þ b2 cÞ ; 2½b1 b2 þ cðb1 þ b2 Þ ðb1 þ cÞða2 þ b2 cÞ þ cða1 þ b1 cÞ p2 ¼ : 2½b1 b2 þ cðb1 þ b2 Þ p1 ¼



We now consider a numerical example to illustrate the case of two demand classes, before extending the model to the general case of n demand classes. 5. A numerical example for two demand classes We provide an example to illustrate the model and results. Consider a firm attempting to determine inventory and selling prices for a product in two market segments. Let: c = 5, s = 6, H = 1 and r = 0.1 with demand parameters: a1 = 80, a2 = 180, b1 = 2, and b2 = 8. Assume the uncertainty in demand in each segment is independent and normally distributed with mean 0 and standard deviation rq i = 1, 2. Therefore, the aggregated demand uncertainty is a normal distribution with mean ni where ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 and r ¼ r2n1 þ r2n2 .

M. Zhang, P.C. Bell / European Journal of Operational Research 182 (2007) 738–754

745

We focus on the impact of demand leakage and aggregated uncertainty on optimal solutions and expected profit. Table 1 summarizes p1 and p2 as a function of c for r = 0, 5, 10, 15 and 20. The numerical results illustrate the analytical relationship that the difference between the optimal prices does not depend on c or r (Proposition 3), and also that optimal prices are notably unaffected by the aggregated demand uncertainty: a change in r from 0 to 20 results in a decline in p1 of less than 0.01%. The numerical results do show that prices are quite sensitive to the value of the leakage parameter c: an increase in c from 0 to 20 leads to almost a 30% decrease in p1 providing vivid illustration of the expectation that the ability to maintain a price difference between the two market segments erodes as the leakage parameter increases. Fig. 1 summarizes the effect of increasing c on expected profit and emphasizes the point that a firm wishing to enhance revenues through market segmentation and differential pricing to each market segment must have fences in place to limit demand leakage from high priced to low priced market segments. Table 1 also shows I as a function of c for various values for the aggregated demand uncertainty. These results suggest that demand leakage has only a slight impact on I: I increases no more than 0.2% asc increases from 0 to 50. This insensitivity is expected since total demand does not depend on the quantity of sales that leaks across segments. I is quite sensitive to the value of aggregated errors as would be expected from the newsvendor problem: I decreases about 32.3% when rchanges from 0 to 20. Overall, these results suggest that the firm would not deviate much from optimality if it chose to disregard demand leakage in calculating the optimal inventory. Fig. 1 graphically illustrates the loss of expected profit caused by demand leakage and uncertainty: for example, an increase in c from 0 to 20 leads to a decline in expected profit of about 10.0%. In Table 2, we report these values and also show the potential gains from market segmentation and demand-splitting. Using aggregated demand given by D ¼ 260  10p ¼ D1 ðpÞ þ D2 ðpÞ, the results show that splitting demand always enhances expected profits, even when there is a large amount of demand leakage from the high priced to the lower priced market. The maximum gain occurs when there is no leakage (c = 0) and amounts to an increase of 11.4%. The gain decreases to about 2.8% if c = 5. At c = 100 there is still a gain, but it is now very small. Table 1 The impact of variation in c and r on optimal prices and inventory c=

0

r

p1

p2

I*

p1

2 p2

I*

p1

5 p2

I*

p1

20 p2

I*

p1

50 p2

I*

0 5 10 15 20

22.50 22.50 22.50 22.50 22.49

13.75 13.75 13.75 13.75 13.74

105 99.0 93.1 87.1 81.1

18.61 18.61 18.61 18.61 18.61

14.72 14.72 14.72 14.72 14.72

105 99.1 93.1 87.1 81.2

17.20 17.20 17.19 17.19 17.19

15.08 15.07 15.07 15.07 15.07

105 99.1 93.1 87.2 81.2

16.02 16.02 16.02 16.01 16.01

15.37 15.37 15.37 15.37 15.36

105 99.1 93.1 87.2 81.3

15.72 15.72 15.71 15.71 15.71

15.45 15.44 15.44 15.44 15.44

105 99.1 93.1 87.2 81.3

E(π) 1250

1200 Aggregated demand γ=0

1150

γ=5 γ=20

1100

1050 0

5

10

σ

15

Fig. 1. Sensitivity of expected profit to aggregate demand uncertainty.

20

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M. Zhang, P.C. Bell / European Journal of Operational Research 182 (2007) 738–754

Table 2 Expected profit gain through market segmentation E(p)

2-market segment model

r=0 r=5 r = 10 r = 15 r = 20

1-segment model

c=0

c=2

c=5

c = 20

c = 50

c = 100

1225.0 1216.6 1208.1 1199.7 1191.3

1156.9 1148.5 1140.1 1131.7 1123.3

1132.2 1123.8 1115.4 1107.0 1098.6

1111.6 1103.2 1094.8 1086.4 1078.0

1106.3 1097.9 1089.5 1081.1 1072.7

1104.4 1096.0 1087.6 1079.2 1070.8

1102.5 1094.1 1085.7 1077.3 1068.9

6. n demand classes We examine two possible models of demand curves incorporating leakage. We index the demand segments such that p1 > p2 > ..pi >    pn > c. In the first demand model, leakage only occurs between adjacent demand classes; that is D1 ðp1 ; p2 Þ ¼ a1  b1 p1  c1 ðp1  p2 Þ; Di ðpi1 ; pi ; piþ1 Þ ¼ ai  bi pi þ ci1 ðpi1  pi Þ  ci ðpi  piþ1 Þ Dn ðpn1 ; pn Þ ¼ an  bn pn þ cn ðpn1  pn Þ:

for i ¼ 2; . . . ; ðn  1Þ;

ð10Þ

In the second demand model we assume that leakage only occurs from each demand class to the lowest price segment, or  Di ðpi ; pn Þ ¼ ai  bi pi  ci ðpi  pn Þ for i ¼ 1; 2; . . . ; ðn  1Þ; ð11Þ Dn ðp1 ; . . . ; pi ; . . . ; pn Þ ¼ an  bn pn þ c1 ðp1  pn Þ þ    þ cn1 ðpn1  pn Þ: We note the following properties of demand for both demand models (10) and (11): 1. 2.

P

n Di

P

o

n

¼

pi Di

opi

P

n ðai

 bi pi Þ ¼ Qn that is total demand is independent of leakage.

¼ 2Di  ai .

Again, we relate the overage cost to the average selling price, or hðp1 ; . . . ; pN Þ ¼ r

RDi pi þ H: RDi

The expected profit is given by EðpÞ ¼

X

pi Di  cI  h

n

Z

I

ðI  Qn Þf ðQn Þ dQn  s 0

Z

1

ðQn  IÞf ðQn Þ dQn : I

Proposition 6. For both demand models (10) and (11), the optimal I satisfies   X sc Di þ F 1 I ¼ : s þ hðp1 ; . . . ; pn Þ n Proposition 7. For both demand models (10) and (11), the optimal pi satisfy 2D1  a1 2Di  ai 2Dn  an ¼  ¼ ¼  ¼ : b1 bi bn When r = 0, this becomes

M. Zhang, P.C. Bell / European Journal of Operational Research 182 (2007) 738–754

747

2D1  a1 2Di  ai 2Dn  an ¼  ¼ ¼  ¼ ¼ c; b1 bi bn which can be solved to determine p1, p2, . . . , pn (Proofs are in Appendix A). 7. A numerical example for three market segments For this example, we divide market 2 from the two-segment model ðD2 ¼ 180  8p2 Þ into two sub-segments ðD2 ¼ 100  3p2 and D3 ¼ 80  5p3 Þ which, with segment 1 ðD1 ¼ 80  2p1 Þ satisfy the condition DðpÞ ¼ D1 ðpÞ þ D2 ðpÞ þ D3 ðpÞ for all p. We use the same cost parameters as for the two-segment model [c = 5, s = 6, H = 1, and r = 0.1] and use (10) as the demand function. For simplicity, we fix c2 = 5. Table 3 illustrates the increase in expected profit that arises from the additional market segmentation these increases are substantial, reflecting the fairly effective fence that limits leakage from segment 2 to segment 3 (c2 = 5). Table 4 summarizes optimum prices and inventory for this three-segment case. The figures illustrate the insensitivity of prices to the level of uncertainty, and the insensitivity of the optimum inventory to the demand leakage between segments. These results suggest that a heuristic that sets inventory based on aggregated demand and then sets prices based on the optimum prices for the deterministic model will be near optimum. Finally we show some results for the three-market segment model with demand model (11): that is all demand leakage that occurs does so to the lowest price segment. As expected this change of demand assumption reduces expected profit markedly when the fencing is less than perfect (Table 5). Table 6 summarizes Table 3 Expected profit for three market segments with demand model (10) 3-price model

r=0 r=5 r = 10 r = 15 r = 20

c1 = 0

c1 = 5

c1 = 20

c1 = 50

1291.3 1255.4 1247.5 1239.5 1231.6

1265.5 1195.5 1187.5 1179.5 1171.6

1249.4 1183.2 1175.2 1167.3 1159.0

1244.9 1180.1 1172.1 1164.2 1156.2

Table 4 Optimum prices and inventory for 3-market segment model with demand model (10) c1=

0

r

p1

p2

p3

I*

p1

5 p2

p3

I*

p1

20 p2

p3

I*

p1

50 p2

p3

I*

0 5 10 15 20

22.50 22.50 22.50 22.49 22.49

15.23 15.23 15.22 15.22 15.22

12.86 12.86 12.86 12.86 12.86

105.0 99.01 93.02 87.04 81.05

18.38 18.37 18.37 18.37 18.37

16.73 16.72 16.72 16.72 16.72

13.61 13.61 13.61 13.61 13.61

105.0 99.03 93.06 87.09 81.12

17.53 17.53 17.53 17.53 17.52

17.03 17.03 17.03 17.03 17.03

13.77 13.77 13.76 13.76 13.76

105.0 99.03 93.07 87.10 81.14

17.32 17.32 17.32 17.31 17.31

17.11 17.11 17.11 17.11 17.10

13.81 13.80 13.80 13.80 13.80

105.0 99.04 93.07 87.11 81.14

Table 5 Expected profit for three market segments with demand model (11) 3-price model

r=0 r=5 r = 10 r = 15 r = 20

c1 = 0

c1 = 5

c1 = 20

c1 = 50

1291.3 1255.4 1247.5 1239.5 1231.6

1194.7 1145.6 1137.7 1129.7 1121.7

1152.4 1121.9 1114.0 1106.0 1098.1

1140.4 1115.9 1108.0 1100.0 1092.0

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M. Zhang, P.C. Bell / European Journal of Operational Research 182 (2007) 738–754

Table 6 Optimum prices and inventory for 3-market segment model with demand model (11) c1=

0

r

p1

p2

p3

I*

p1

5 p2

p3

I*

p1

20 p2

p3

I*

p1

50 p2

p3

I*

0 5 10 15 20

22.50 22.50 22.50 22.50 22.49

15.23 15.23 15.23 15.22 15.22

12.86 12.86 12.86 12.86 12.86

105.00 99.01 93.02 87.04 81.05

16.80 16.80 16.80 16.80 16.80

16.26 16.26 16.26 16.26 16.26

14.52 14.52 14.52 14.52 14.52

105.00 99.05 93.09 87.14 81.19

15.57 15.57 15.57 15.57 15.57

16.49 16.49 16.48 16.48 16.48

14.88 14.88 14.88 14.88 14.87

105.00 99.05 93.11 87.16 21.22

15.26 15.26 15.26 15.26 15.25

16.54 16.54 16.54 16.54 16.54

14.97 14.97 14.97 14.97 14.96

105.00 99.06 93.11 87.17 81.22

optimum pricing and inventory decisions for this case. We first note that optimum inventory is hardly affected by this change in allowable demand leakage, confirming the view that optimum inventory depends on aggregate demand conditions and is not much affected by the segment in which the demand materializes. Second, we note again that prices are insensitive to demand uncertainty. Finally, we note that the prices in segments 1 and 3 converge as the fencing between these segments becomes less effective. We conclude that effective revenue enhancement through market segmentation and differential pricing requires effective fencing to prevent demand leaking from higher priced market segments to lower priced segments. 8. Conclusions This paper addresses the newsvendor problem with backlogged demand, in which initial inventory and selling price are determined simultaneously. We extend this problem to the case where the single product can be sold to different demand classes at different prices. We note that if such market segmentation is to lead to enhanced revenues and profits, customers from the higher priced market must be incentivized to not place their business in the lower priced market. Devices that provide such an incentive are called ‘‘fences’’, but most fences are not perfect in that a certain amount of demand will leak from a high priced segment to a lower priced segment. We model the imperfection of a fence as a leakage function between market segments with the amount of leakage increasing as the price difference increases. We find a straightforward relationship between the optimal prices for different demand class, which together with a newsvendor-like optimal inventory expression provides an efficient search procedure for finding the optimal prices and inventory levels for this problem. We illustrate the results using numerical examples. We note the near optimality of an heuristic that sets the optimum inventory based on the aggregate demand distribution for all segments while ignoring demand leakage between segments, and ignores demand uncertainty in computing optimum prices for each segment. In the practice of revenue management, the leakage parameter (c) is related to the effort invested in fences. Our results show dramatically that the price difference between the multiple segments and the consequent gain in expected profit can be improved by decreasing c: that is by designing more impermeable fences. Increasing the effort invested in designing and operating effective fences can reduce the value of c and improve revenue and profit gains. Many research issues remain, including how much cost should be devoted to fences, what is the empirical relationship between the cost of a fence and its ‘‘c’’ value, and how to design low cost effective fences. Appendix A Proof of Proposition 1 op ¼ D1  ðb1 þ cÞðp1  cÞ þ cðp2  cÞ ¼ a1  2b1 p1  2cðp1  p2 Þ þ b1 c ¼ 0; op1 op ¼ ðp1  cÞc þ D2  ðp2  cÞðb2 þ cÞ ¼ a2  2b2 p2 þ 2cðp1  p2 Þ þ b2 c ¼ 0: op2

ðA:1Þ ðA:2Þ

M. Zhang, P.C. Bell / European Journal of Operational Research 182 (2007) 738–754

749

2 þb1 c (A.1) leads to: p1 ¼ a1 þ2cp . 2b1 þ2c

2 þb2 cÞþcða1 þb1 cÞ Substituting into (A.2) and rearranging leads to: p2 ¼ ðb1 þcÞða . 2½b1 b2 þcðb1 þb2 Þ

I 0 ¼ a1 þ a2 

b1 ½ðb2 þ cÞða1 þ b1 cÞ þ cða2 þ cb2 Þ b2 ½ðb1 þ cÞða2 þ b2 cÞ þ cða1 þ cb1 Þ  2½b1 b2 þ cðb1 þ b2 Þ 2½b1 b2 þ cðb1 þ b2 Þ

 b1 b2 a1  ca1 b1  b1 b1 cb2  b1 b1 cc  b1 a2 c  b1 cb2 c  b1 b2 a2  ca2 b2  b2 cb1 b2  b2 ccb2  ca1 b2  cb1 cb2 1 cb2 Simplifies to: I 0 ¼ a1 þa2 cb . h 2

Proof of Proposition 2. We have (6) EðpÞ ¼ p1 D1 þ p2 D2  cI  h

Z

I

ðI  QN Þf ðQN Þ dQN  s

Z

0

¼ p1 D1 þ p2 D2  cI  hI Z

ðQN  IÞf ðQN Þ dQN I

Z

I

f ðQN Þ dQN þ h

Z

0

s

1

I

QN f ðQN Þ dQN

0

1

QN f ðQN Þ dQN þ sI  sI

Z

I

f ðQN Þ dQN 1

I

or EðpÞ ¼ p1 D1 þ p2 D2 þ ðs  cÞI  ðh þ sÞI

Z

Z

I

f ðQN Þ dQN þ h 0

I

QN f ðQN Þ dQN  s 0

Z

1

QN f ðQN Þ dQN : I

The first-order condition for I gives Z

I

f ðQN Þ dQN ¼

0

sc : sþh

ðA:3Þ

Leading to EðpÞ ¼ p1 D1 þ p2 D2 þ h

Z

I

QN f ðQN Þ dQN  s 0

Z

1

QN f ðQN Þ dQN : I

Rewriting the integrals in terms of n EðpÞ ¼ p1 D1 þ p2 D2 þ h

Z

IðD1 þD2 Þ

ðD1 þ D2 þ nÞf ðnÞ dn  s

Z

IðD1 þD2 Þ

¼ p1 D1 þ p2 D2 þ h 1 Z 1 nf ðnÞ dn  IðD1 þD2 Þ

ðD1 þ D2 þ nÞf ðnÞ dn  sðD1 þ D2 Þ

Z

1

ðD1 þ D2 Þf ðnÞ dn  s IðD1 þD2 Þ

Z

IðD1 þD2 Þ

¼ p1 D1 þ p2 D2 þ h½D1 þ D2  " # 1 Z Z IðD1 þD2 Þ  1 f ðnÞ dn  s 1

ðD1 þ D2 þ nÞf ðnÞ dn

IðD1 þD2 Þ

1

Z

1

f ðnÞ dn þ h

Z

IðD1 þD2 Þ

1 1

IðD1 þD2 Þ

nf ðnÞ dn

nf ðnÞ dn  s½½D1 þ D2 

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M. Zhang, P.C. Bell / European Journal of Operational Research 182 (2007) 738–754

Using (A.3) ¼ p1 D1 þ p2 D2 þ h½D1 þ D2  "  1

Z

Z

#

IðD1 þD2 Þ

IðD1 þD2 Þ

f ðnÞ dn þ h 1

Z

IðD1 þD2 Þ

nf ðnÞ dn  s½½D1 þ D2 

1

Z

f ðnÞ dn  s

1

nf ðnÞ dn ¼ p1 D1 þ p2 D2  c½D1 þ D2  IðD1 þD2 Þ

1

þh

Z

IðD1 þD2 Þ

nf ðnÞ dn  s

Z

1

nf ðnÞ dn

IðD1 þD2 Þ

1

EðpÞ ¼ ðp1  cÞD1 þ ðp2  cÞD2  h

Z

IðD1 þD2 Þ

nf ðnÞ dn  s

Z

1

nf ðnÞ dn:

IðD1 þD2 Þ

1

Hence the first-order conditions are Z IðD1 þD2 Þ oEðpÞ ohðp1 ; p2 Þ p1 ¼ D1  ðb1 þ cÞðp1  cÞ þ cðp2  cÞ  nf ðnÞ dn ¼ 0; op1 o 1 Z ohðp1 ; p2 Þ IðD1 þD2 Þ 2D1  a1 þ b1 c  nf ðnÞ dn ¼ 0; op1 1 Z oEðpÞ ohðp1 ; p2 Þ IðD1 þD2 Þ ¼ ðp1  cÞc þ D2  ðp2  cÞðb2 þ cÞ  nf ðnÞ dn ¼ 0; op2 op2 1 Z ohðp1 ; p2 Þ IðD1 þD2 Þ 2D2  a2 þ b2 c  nf ðnÞ dn ¼ 0: op2 1

ðA:4Þ

ðA:5Þ

Eliminating the integral term between (A.1) and (A.2) provides the necessary condition ½2D1  a1 þ b1 c

ohðp1 ; p2 Þ ohðp1 ; p2 Þ ¼ ½2D2  a2 þ b2 c : op2 op1

ðA:6Þ

2 D2 Using: hðp1 ; p2 Þ ¼ r p1 DD1 þp þ H , we have þD 1

2

ohðp1 ; p2 Þ ðD1 þ D2 ÞðD1 þ p1 ðb1  cÞ þ p2 cÞ  ðD1 p1 þ D2 p2 Þðb1 Þ ¼r 2 op1 ðD1 þ D2 Þ ¼r

ðD1 þ D2 Þð2D1  a1 Þ þ ðD1 p1 þ D2 p2 Þb1 ðD1 þ D2 Þ2

;

ðA:7Þ

ohðp1 ; p2 Þ ðD1 þ D2 ÞðD2 þ p2 ðb2  cÞ þ p1 cÞ  ðD1 p1 þ D2 p2 Þðb2 Þ ¼r 2 op2 ðD1 þ D2 Þ ¼r

ðD1 þ D2 Þð2D2  a2 Þ þ ðD1 p1 þ D2 p2 Þb2 ðD1 þ D2 Þ

2

:

Combining (A.6) and the derivatives (A.7) and (A.8) ½2D1  a1 þ b1 cr

ðD1 þ D2 Þð2D2  a2 Þ þ ðD1 p1 þ D2 p2 Þb2

¼ ½2D2  a2 þ b2 cr

ðD1 þ D2 Þ

2

ðD1 þ D2 Þð2D1  a1 Þ þ ðD1 p1 þ D2 p2 Þb1 ðD1 þ D2 Þ

2

) ½2D1  a1 þ b1 c½ðD1 þ D2 Þð2D2  a2 Þ þ ðD1 p1 þ D2 p2 Þb2   ½2D2  a2 þ b2 c½ðD1 þ D2 Þð2D1  a1 Þ þ ðD1 p1 þ D2 p2 Þb1  ¼ 0:

ðA:8Þ

M. Zhang, P.C. Bell / European Journal of Operational Research 182 (2007) 738–754

751

Expanding 2D2 2D1 ðD1 þ D2 Þ  a2 2D1 ðD1 þ D2 Þ þ 2D1 b2 ðD1 p1 þ D2 p2 Þ  2D2 a1 ðD1 þ D2 Þ þ a2 a1 ðD1 þ D2 Þ  a1 b2 ðD1 p1 þ D2 p2 Þ þ 2D2 b1 cðD1 þ D2 Þ  a2 b1 cðD1 þ D2 Þ þ b1 cðD1 p1 þ D2 p2 Þb2  2D2 2D1 ðD1 þ D2 Þ þ a1 2D2 ðD1 þ D2 Þ  2D2 b1 ðD1 p1 þ D2 p2 Þ þ a2 2D1 ðD1 þ D2 Þ  a1 a2 ðD1 þ D2 Þ þ a2 b1 ðD1 p1 þ D2 p2 Þ  b2 c2D1 ðD1 þ D2 Þ þ a1 b2 cðD1 þ D2 Þ  b2 cb1 ðD1 p1 þ D2 p2 Þ ¼ 0; which becomes 2D1 b2 ðD1 p1 þ D2 p2 Þ  a1 b2 ðD1 p1 þ D2 p2 Þ þ 2D2 b1 cðD1 þ D2 Þ  a2 b1 cðD1 þ D2 Þ  2D2 b1 ðD1 p1 þ D2 p2 Þ þ a2 b1 ðD1 p1 þ D2 p2 Þ  2D1 b2 cðD1 þ D2 Þ þ a1 b2 cðD1 þ D2 Þ ¼ 0 or ½D1 ðp1  cÞ þ D2 ðp2  cÞ½2D1 b2  a1 b2  2D2 b1 þ a2 b1  ¼ 0: Now D1 > 0; p1 > c; D2 > 0 and p2 > c so the necessary condition reduces to ½2D1 b2  a1 b2  2D2 b1 þ a2 b1  ¼ 0; 2a1 b2  2b1 b2 p1  2b2 cðp1  p2 Þ  a1 b2  2b1 a2 þ 2b1 b2 p2  2b1 cðp1  p2 Þ þ a2 b1 ¼ 0; a1 b2  a2 b1  2b1 b2 ðp1  p2 Þ  ð2b1 þ 2b2 Þcðp1  p2 Þ ¼ 0; a1 b2  a2 b1 :  p1  p2 ¼ 2½ðb1 þ b2 Þc þ b1 b2  Proof of Proposition 4. From (A.5) and (A.8) 2D2  a2 þ b2 c ¼ r

ðD1 þ D2 Þð2D2  a2 Þ þ b2 ðD1 p1 þ D2 p2 Þ ðD1 þ D2 Þ

2

Z

IðD1 þD2 Þ

½I  ðD1 þ D2 þ nÞf ðnÞ dn:

ðA:9Þ

1

The sign of the rhs of (A.9) depends on the sign of ðD1 þ D2 Þð2D2  a2 Þ þ b2 ðD1 p1 þ D2 p2 Þ ¼ ðD1 þ D2 Þ½a2  2b2 p2 þ 2cðp1  p2 Þ þ b2 ðD1 p1 þ D2 p2 Þ > ðD1 þ D2 Þ½a2 þ 2cðp1  p2 Þ  2b2 p2 ðD1 þ D2 Þ þ b2 ðD1 p2 þ D2 p2 Þ ¼ ½a2  b2 p2 þ 2cðp1  p2 ÞðD1 þ D2 Þ > 0 since p1 > p2 : Therefore the rhs of (A.9) is greater than zero, hence 2D2  a2 þ b2 c > 0 ) a2  2b2 p2 þ 2cðp1  p2 Þ þ b2 c > 0:

ðA:10Þ

Combining (A.6) and (A.10) a2  2b2 p2 þ

cða1 b2  a2 b1 Þ cða1 b2  a2 b1 Þ þ b2 c > 0 ) 2b2 p2 < a2 þ b2 c þ 2b2 p2 b1 b2 þ cðb1 þ b2 Þ b1 b2 þ cðb1 þ b2 Þ a2 b1 b2 þ b1 b2 b2 c þ ca2 b1 þ cb2 cb1 þ ca2 b2 þ cb2 cb2 þ ca1 b2  ca2 b1 p2 < b1 b2 þ cðb1 þ b2 Þ <

ðb1 þ cÞða2 þ b2 cÞ þ cðcb1 þ a1 Þ ¼ p02 : 2½b1 b2 þ cðb1 þ b2 Þ

We have p1  p2 ¼ p01  p02 [from (8)]. Hence p2 < p02 implies p1 < p01 . h

752

M. Zhang, P.C. Bell / European Journal of Operational Research 182 (2007) 738–754

Proof of Proposition 5. Let H 11 ¼

oh ð2D1  a1 ÞðD1 þ D2 Þ þ b1 ðD1 p1 þ D2 p2 Þ ¼r 2 op1 ðD1 þ D2 Þ

H 12 ¼

oh ð2D2  a2 ÞðD1 þ D2 Þ þ b2 ðD1 p1 þ D2 p2 Þ ¼r op2 ðD1 þ D2 Þ2

H 21 H 22

! o2 h b1 ða2  b2 p2 Þ þ c½a1 þ a2  ðb1 þ b2 Þp2  ¼ 2 ¼ 2r½a1 þ a2  ðb1 þ b2 Þp2  3 op1 ðD1 þ D2 Þ ! o2 h b2 ða1  b1 p1 Þ þ c½a1 þ a2  ðb1 þ b2 Þp1  : ¼ 2 ¼ 2r½a1 þ a2  ðb1 þ b2 Þp1  3 op2 ðD1 þ D2 Þ

We note that H12 > 0 (from Proposition 4), and H21 < 0 [since: a2b2p2 P 0; and [a1 + a2(b1 + b2)p2] > [a1b1p1 + a2b2p2] P 0]. From (A.9), the first-order condition for p2 is Z IðD1 þD2 Þ ½I  ðD1 þ D2 þ nÞf ðnÞ dn ¼ 0: ðA:11Þ ð2D2  a2 þ b2 cÞ  H 12 1

The second derivative w.r.t. p2 is Z IðD1 þD2 Þ o2 EðpÞ ¼ 2ðb2 þ cÞ  H 22 ½I  ðD1 þ D2 þ nÞf ðnÞ dn op22 1 Z IðD1 þD2 Þ f ðnÞ dn  b22 f ðI  D1  D2 Þðh þ sÞ:  2H 12 b2 1

R IðD þD Þ 2H 12 b2 1 1 2 f ðnÞ dn < 0 ðsince H 12 > 0Þ and  b22 f ðI  D1  D2 Þðh þ sÞ < 0, hence if R IðD1 þD2 Þ R IðD1 þD2 Þ 2ðb2 þ cÞ  H 22 1 ½I  ðD1 þ D2 þ nÞf ðnÞ dn < 0, then E(p) is strictly concave in p2.H 22 1 2 þb2 cÞ ½I  ðD1 þ D2 þ nÞf ðnÞ dn ¼ H 22 ð2D2 a [from (A.11)] so we are interested in the sign of H 12 Now:

ð2D2  a2 þ b2 cÞ H 12 2½a1 þ a2  ðb1 þ b2 Þp1 ½b2 ða1  b1 p1 Þ þ c½a1 þ a2  ðb1 þ b2 Þp1  ð2D2  a2 þ b2 cÞ: ¼ 2ðb2 þ cÞ þ ðD1 þ D2 Þ½ð2D2  a2 ÞðD1 þ D2 Þ þ b2 ðD1 p1 þ D2 p2 Þ

 2ðb2 þ cÞ  H 22

We begin with the term: [a1 + a2  (b1 + b2)p1]. If [a1 + a2 (b1 + b2)p1] = 0, then the sign is negative and the second derivative is strictly negative. Consider next the case of [a1 + a2(b1 + b2)p1] < 0. 1 þb1 cÞþcða2 þcb2 Þ (Proposition 4). We have, p1 < p01 ¼ ðb2 þcÞða 2½b1 b2 þcðb1 þb2 Þ

We have p1 > c and 0 < a1b1p1 < a1b1c or a1 > b1c therefore ðb2 þ cÞða1 þ b1 cÞ þ cða2 þ cb2 Þ ¼ cða1 þ a2 Þ þ cðb1 c þ b2 cÞ þ b2 ða1 þ b1 cÞ < cða1 þ a2 Þ þ cða1 þ a2 Þ þ b2 ða1 þ a1 Þ ¼ 2½cða1 þ a2 Þ þ a1 b2 : Hence p1 <

2½cða1 þ a2 Þ þ a1 b2  : 2½b1 b2 þ cðb1 þ b2 Þ

Or cða1 þ a2 Þ þ a1 b2  ½b1 b2 þ cðb1 þ b2 Þp1 > 0; ½b2 ða1  b1 p1 Þ þ c½a1 þ a2  ðb1 þ b2 Þp1  > 0:

ðA:12Þ

M. Zhang, P.C. Bell / European Journal of Operational Research 182 (2007) 738–754

753

Examining the term 2½a1 þ a2  ðb1 þ b2 Þp1 ½b2 ða1  b1 p1 Þ þ c½a1 þ a2  ðb1 þ b2 Þp1  : ðD1 þ D2 Þ½ð2D2  a2 ÞðD1 þ D2 Þ þ b2 ðD1 p1 þ D2 p2 Þ ð2D2  a2 þ b2 cÞ > 0 and ½ð2D2  a2 ÞðD1 þ D2 Þ þ b2 ðD1 p1 þ D2 p2 Þ > 0 (From Proposition 4) and with 2 < 0. (A.12) and [a1 + a2  (b1 + b2)p1] < 0 we conclude that this term is negative leading to o opEðpÞ 2 2 We now examine the case of a1 + a2  (b1 + b2)p1 > 0 2½a1 þ a2  ðb1 þ b2 Þp1 ½b2 ða1  b1 p1 Þ þ c½a1 þ a2  ðb1 þ b2 Þp1  ð2D2  a2 þ b2 CÞ ðD1 þ D2 Þ½ð2D2  a2 ÞðD1 þ D2 Þ þ b2 ðD1 p1 þ D2 p2 Þ   2ðD1 þ D2 Þ b2 ða1  b1 p1 Þ þ cðD1 þ D2 Þ ð2D2  a2 þ b2 cÞ < 2ðb2 þ cÞ þ ðD1 þ D2 Þ½ð2D2  a2 ÞðD1 þ D2 Þ þ b2 ðD1 p1 þ D2 p2 Þ

2ðb2 þ cÞ þ

since a1 þ a2  ðb1 þ b2 Þp1 < a1 þ a2  b1 p1  b2 p2 ¼ D1 þ D2 ¼ 2ðb2 þ cÞ þ ¼

2½b2 ða1  b1 p1 Þ þ cðD1 þ D2 Þð2D2  a2 þ b2 cÞ ð2D2  a2 ÞðD1 þ D2 Þ þ b2 ðD1 p1 þ D2 p2 Þ

2 fb2 ½ð2D2  a2 ÞðD1 D2 Þ þ b2 ðD1 p1 þ D2 p2 Þ ð2D2  a2 ÞðD1 þ D2 Þ þ b2 ðD1 p1 þ D2 p2 Þ  ða1  b1 p1 Þð2D2  a2 þ b2 cÞ þ c½ð2D2  a2 ÞðD1 D2 Þ þ b2 ðD1 p1 þ D2 p2 Þ  ðD1 þ D2 Þð2D2  a2 þ b2 cÞg

¼

2 fb2 ½ð2D2 þ a2 þ b2 cÞD1 þ b2 ðD1 ðp1  cÞ þ D2 ðp2  cÞ ð2D2  a2 ÞðD1 þ D2 Þ þ b2 ðD1 p1 þ D2 p2 Þ þ cb2 ½D1 ðp1  cÞ þ D2 ðp2  cÞg

which is <0. Therefore, we have Similarly,

o2 EðpÞ op22

< 0.

o2 EðpÞ ¼ 2ðb2 þ cÞ  H 22 op22

Z

IðD1 þD2 Þ

½I  ðD1 þ D2 þ nÞf ðnÞdn  2H 12 b2 1

Z

IðD1 þD2 Þ 1

o2 EðpÞ ¼ f ðI  D1  D2 Þðh þ sÞ < 0: oI 2

f ðnÞdn  b22 f ðI  D1  D2 Þðh þ sÞ < 0;



Proof of Proposition 6 & 7. The first-order condition oEðpÞ ¼ c þ s  ðs þ hÞ oI oEðpÞ opi

Z

IQn

1

¼ 2Di  ai þ bi ½s  ðs þ hÞ

hðp1 ; . . . ; pn Þ ¼

r RDQi pi n

f ðnÞ dn ¼ 0; R IQn 1



) I ¼ Qn þ F

1 ;...;pn Þ f ðnÞ dn  ohðpop i

R IQn 1

1



 sc ; s þ hðpÞ

9 ½I  Qn  nf ðnÞ dn =

þH

Z ð2Di  ai ÞQn þ bi RDi pi IQn ½I  Qn  nf ðnÞ dn ¼ 0 Q2n 1 Z IQn 2Di  ai þ bi c 1 ¼ 2 ½I  Qn  nf ðnÞ dn ¼ 0 ) ð2Di  ai ÞQn þ bi RDi pi Qn 1

) 2Di  ai þ bi c 

;

754

M. Zhang, P.C. Bell / European Journal of Operational Research 182 (2007) 738–754

)

2Di  ai þ bi c 2D1  a1 þ b1 c ¼ ð2Di  ai ÞQn þ bi RDi pi ð2D1  a1 ÞQn þ b1 RDi pi

) ð2Di  ai þ bi cÞ½ð2D1  a1 ÞQn þ b1 RDi pi  ¼ ð2D1  a1 þ b1 cÞ½ð2Di  ai ÞQn þ bi RDi pi  ) ð2Di  ai Þð2D1  a1 ÞQn þ bi cð2D1  a1 ÞQn þ ð2Di  ai þ bi cÞb1 RDi pi ¼ ð2D1  a1 Þð2Di  ai ÞQn þ b1 cð2Di  ai ÞQn þ ð2D1  a1 þ b1 cÞbi RDi pi RDi pi RDi pi ¼ b1 cð2Di  ai Þ þ ð2D1  a1 Þbi Qn Qn ðh  H Þ ðh  H Þ ¼ b1 cð2Di  ai Þ þ ð2D1  a1 Þbi ) bi cð2D1  a1 Þ þ ð2Di  ai Þb1 r r   ðh  H Þ ðh  H Þ ) bi ð2D1  a1 Þ c  ¼ b1 ð2Di  ai Þ c  r r ) bi cð2D1  a1 Þ þ ð2Di  ai Þb1

) bi ð2D1  a1 Þ ¼ b1 ð2Di  ai Þ )

2D1  a1 2Di  ai 2Dn  an ¼  ¼ ¼  ¼ : b1 bi bn



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