A distribution free approach to newsvendor problem

4OR-Q J Oper Res DOI 10.1007/s10288-013-0249-9 RESEARCH PAPER

A distribution free approach to newsvendor problem with pricing Syed Asif Raza

Received: 2 October 2012 / Revised: 19 September 2013 © Springer-Verlag Berlin Heidelberg 2013

Abstract The newsvendor problem with pricing is an important tool that provides the relationship between the operational and marketing issues at the firm’s level. This paper investigates the use of the distribution free approach to solve the standard newsvendor problem with pricing and its extension to the holding and shortage cost case. The approach is vital for the situations in which a firm may be missing demand distribution information or historical demand data may not fit any standard probability distributions. The approach develops deterministic approximations by establishing lower bounds on the expected revenues of the standard newsvendor problem with pricing and its extension. The lower bounds are shown jointly concave in price and order quantity. It is also demonstrated that the use of this approach can yield closed form expressions for jointly optimal pricing and order quantities which can find many practical managerial applications in several industries facing yield management decisions. Numerical experimentation demonstrates the performance of the distribution free approach under various demand situations. Keywords Newsvendor problem · Pricing · Stochastic inventory control · Distribution free approach · Revenue management Mathematics Subject Classification

90B05

1 Introduction The newsvendor problem is an essential problem in stochastic inventory control. The analysis of this problem uses some fundamental techniques for probabilistic decision making which finds applications in many other business related problems. In standard S. A. Raza (B) College of Business and Economics, Qatar University, Doha, Qatar e-mail: [email protected]

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newsvendor problem, the price is assumed to be fixed or exogenously given parameter. Although both standard newsvendor problem and pricing problem have well developed results in operations management and micro-economics, yet there is a need to further study an integrated framework which can consider a simultaneous decision making on pricing and inventory (quantity) allocation. In literature, price dependent stochastic demand modeling has been addressed using different approaches. Finding an appropriate modeling and solution approach for newsvendor problem with pricing should make meaningful contributions towards its implications into the related fields. In addition, all such modeling frameworks need essential information about historical data on demand such as demand distribution. Often the demand pattern is required to follow standard distributions which may not be the case in many real life applications. Moreover, many single period sales of items such as fashion products do not have any historical data. Being motivated by the current research status in the area, this paper addresses the three main concerns: (i) How to obtain a deterministic approximation of the newsvendor problem with pricing? (ii) How does this deterministic approximation enables the joint control of pricing and order quantity for a firm? and (iii) How well does this approximation perform computationally for determining the optimal price and order quantity for a firm, and how close these control parameters generate revenue compared to the situation when the firm is fully aware of the demand distribution? For the first question, the stochastic price dependent model can be approximated using a distribution free approach. The approach uses Scarf (1958)’s rule which transforms the newsvendor problem with stochastic price-dependent demand into its deterministic approximation. This approximation will not assume any distribution, however, mean and variance of the stochastic price-dependent demand are needed to model the problem. This deterministic approximation in this situation will develop a lower bound estimate on revenue (profit) which a firm can gain, given the demand distribution is unknown. The lower bound guarantees a revenue for all demand distributions including worst possible demand distribution. In addressing the second concern, this paper studies the structural properties of the lower bound and determines the joint concavity of the lower bound in price and order quantity. Besides this, it outlines a efficient solution procedure to determine a joint optimal control of pricing and order quantity that yield global optimal revenues. For the last concern, the analysis of the lower bound must yield a simplified procedure to determine the optimal price and order quantity, and later the performance of this lower bound must be compared with the original problem assuming known distributions. A single period newsvendor problem is a building block in stochastic inventory control. It incorporates the fundamental techniques of stochastic decision-making and can be applied to a much broader scope. Whitin (1955) was the first to discuss the pricing issues in the inventory control theory. Mills (1959) extended Whitin’s work by modeling the uncertainty of the price sensitive demand. He suggested an additive modeling approach. Later, Karlin and Carr (1962) presented a multiplicative modeling approach to the problem. Historically, both additive and multiplicative models became more vital in pricing research with stochastic inventory control (see Khouja 1999; Petruzzi and Dada 1999; Yao et al. 2006 for more details). In the recent years, there has been a growing interest in addressing the optimal pricing problem in newsvendor setting. Xu et al. (2011) discussed the unimodality of the price-setting newsvendor’s objective

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function with multiplicative demand and its applications. Murray et al. (2012) studied the price-setting newsvendor problem with resource and capacity constraints. Wu et al. (2012) addressed the problem of optimal incentive contracts for multiple advertisers. Jammernegg and Kischka (2013) extended the existing research by considering the service and loss constraints into the newsvendor problem with pricing. Recently, Yu et al. (2013)’s work developed a price-setting newsvendor problem with fuzzy price-dependent demand. Gallego and Moon (1993) studied the distribution free model on the newsvendor problem and its several extensions. They proved the optimality of Scarf (1958)’s ordering rule to the problem. The use of the distribution free approach finds optimal order quantity that maximizes the expected profit to the firm when stochastic demand distribution is unknown. AlFares and Elmorra (2005) extended their work by incorporating the shortage penalty cost. There has been a growing interest towards development of robust optimal models and solution methodologies for fixed prices, to which the use of the distribution free approach makes an excellent fit. Liao et al. (2011) extended Gallego and Moon (1993)’s work for balking and lost sales penalty. Wei et al. (2011) developed robust optimal policies of production and inventory with uncertain returns and demand. Lee and Hsu (2011) studied the effect of advertising on the distribution free newsboy problem. Taleizadeh et al. (2011) also addressed the constrained single period problem under demand uncertainty using distribution free approach. The remainder of this paper is organized as follows. In Sect. 2, a brief analysis of standard newsvendor problem with pricing using both additive and multiplicative modeling approaches is presented. Later, the use of the distribution free approach to solve the standard newsvendor problem with pricing is demonstrated, and in an analysis it is shown how efficiently the distribution free approach can be utilized to develop a deterministic approximation of the stochastic problem by establishing a lower bound. The structural properties of the bound are also investigated and it is shown that the lower bound on the revenue is jointly concave in price and order quantity. Similarly in Sect. 3, the newsvendor problem with pricing is extended to a commonly observed shortage penalty case and holding cost case (Petruzzi and Dada 1999; Yao 2002). Section 4 studies the performance of the distribution free approach on randomly generated problems. Finally, in Sect. 5, the findings of this research are summarized and the directions for future works are identified.

2 Standard newsvendor problem In this section, the standard newsvendor problem with pricing in a monopolistic situation presented in Yao et al. (2006) is briefly discussed. We referred this problem with P, and in the problem, the commodity offered by a firm has a constant cost c and there is no fixed cost associated with the product. The selling price p and the quantity (inventory) q are determined jointly, such that the total revenue is maximized for problem, P. This can be regarded as an extension to the case of price setting since both price p and quantity q are decision controls unlike many previous studies that consider price as an exogenous variable. Consider a firm that faces price dependent stochastic demand D = D( p, ξ ), for brevity D, which has contributions from two sources, the price

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dependent deterministic demand y( p) and the stochastic price independent demand factor, ξ . We assume that ξ has a probability distribution function f (ξ ) and a cumulative probability distribution function F(ξ ). Moreover, ξ is assumed to be bounded such that ξ ∈ [ξ , ξ ] and to follow the Generalized Strictly Increasing Failure Rate (GSIFR). The price dependent deterministic demand y( p) is assumed to be continuous, nonzero positive, twice differentiable and defined between [c, p], where p is the maximum admissible value of p. It is also assumed that y( p), for brevity y = y( p), has increasing price elasticity (IPE) without, however, being restricted to a strictly p . Thus, IPE characteristic. The price elasticity function η is defined as η = − p ∂ y/∂ y

the IPE assumption results in ∂∂ηp ≥ 0, and thus, ∂∂ py ≤ 0. Assuming no shortage and holding costs, the profit (revenue) function of a firm in monopoly is written as follows: P :π ( p, q) = E ξ [ p min{q, D} − c q] = E ξ p q − p [q − D]+ − c q = ( p − c) q − p E ξ [q − D]+

(1)

There are two types of modeling approaches utilized to incorporate the stochastic behavior to the price dependent deterministic demand: (i) additive; and (ii) multiplicative. Recent discussions on these modeling approaches can be found in Petruzzi and Dada (1999) and Yao et al. (2006). In the case of additive modeling approach, the price dependent stochastic demand D is the sum of deterministic price dependent demand y, and random factor ξ such that D = y + ξ . Whereas for multiplicative approach, D is the product of deterministic price dependent demand, y and the random factor ξ , and thus, D = yξ . Petruzzi and Dada (1999) has suggested that for an additive approach, a more convenient deterministic demand curve is linear, y = α − β p, where α, β ≥ 0. Furthermore, for multiplicative approach, it is suggested to use a constant elasticity demand curve, y = αp −β , where α ≥ 0, and β > 1. This paper has followed this suggestion from Petruzzi and Dada (1999). For both additive and multiplicative modeling approaches, ξ has an expected mean of E[ξ ] = μ and a standard deviation of σ > 0. In Table 1, a list of notations is presented. Several accents are used to distinguish the problems of optimizing revenue, depending upon the characterization of price dependent demand and the parameters associated to these problems, which are outlined in the following: P : Demand is price dependent stochastic demand with known distribution. P˜ : Demand is price dependent stochastic demand with unknown distribution and distribution free approach is used. Pˇ : Demand is price dependent stochastic demand with unknown distribution and a worst case analysis for distribution free approach is used. In the forthcoming sections of this paper, in each of the problems declared above, all the control (decision) variables will be customized using the corresponding accent defined for each of the above mentioned problems. For instance, when the price dependent stochastic demand is known, which means problem, P, the decision controls would ˜ be p, and q. However, when the price dependent stochastic demand is unknown, P,

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Newsvendor problem with pricing Table 1 Notations Parameters c

Cost per unit sold

α

Maximum perceived demand

β

Price sensitivity for deterministic price dependent demand

η

Demand price elasticity for the price dependent deterministic demand

g

Holding cost per unit of an excess inventory

s

Shortage cost per unit of an unmet inventory

ξ

Stochastic demand factor

μ

Mean of stochastic demand factor, ξ

σ

Standard deviation of stochastic demand factor, ξ

cv

σ Coefficient of variation, cv = μ

E[·]

Expectation of the stochastic parameter

f (ξ )

Probability distribution function for stochastic demand factor ξ

F(ξ )

Cumulative probability distribution for stochastic demand factor, ξ

y = y( p)

Price dependent deterministic demand

D = D( p, ξ )

Price dependent stochastic demand

z = z( p, μ)

Expected price dependent stochastic demand

π = π( p, q)

Total expected revenue to the firm

Accents x˜

Value of parameter x in distribution free approach, x ∈ R, where R is a real set

xˇ

Value of parameter x in an approximate analysis of distribution free approach, x ∈ R

x

Lower bound of a parameter x ∈ R

x

Upper bound of a parameter x ∈ R

Other scripts x∗

Optimal value of a parameter x ∈ R

xa

Value of parameter x ∈ R in additive modeling approach

xm

Value of parameter x ∈ R in multiplicative modeling approach

Decision variables p

Price per unit sold

q

Quantity (inventory)

˜ the controls would be, p, ˜ and q. ˜ Similarly the worst case analysis of the problem, P, ˇ The controls for problem, Pˇ will be p, is distinguished by a new problem, P. ˇ and q. ˇ 2.1 Basic analysis From earlier studies on newsvendor problem with pricing (see Petruzzi and Dada 1999; Yao 2002; Yao et al. 2006 for details), we have used the following expression:

+

q−y

E ξ [q − D] =

(q − y − ξ ) f (ξ )d ξ

(2)

ξ

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As suggested in Yao (2002), we can use integration by parts to simplify Eq. 2 as below: +

q−y

E ξ [q − D] =

F(ξ )d ξ

(3)

ξ

We can then simplify the revenue function established in Eq. 1 by substituting expression for E ξ [q − D]+ obtained in Eq. 3. Thus, while using an additive approach, the revenue function, π in Eq. 1 can be simplified and written as: ∗

∗

π (p , q )

q−y

= max p,q

( p − c) q − p

F(ξ ) d ξ

(4)

ξ

Similarly for multiplicative approach, we can also obtain, E ξ [q − D]+ using earlier studies (see Yao 2002; Yao et al. 2006) in the following: q

E ξ [q − D]+ =

y F(ξ )d ξ

(5)

ξ

This yields a simplified revenue function for multiplicative model upon substituting the value of E ξ [q − D]+ obtained in Eq. 5, resulting: ∗

∗

π (p , q )

q/y = max p,q

( p − c) q − p y

F(ξ ) d ξ

(6)

ξ

In Eqs. 4 and 6, p ∗ and q ∗ are the corresponding optimal price and order quantity that maximizes the profit using additive and multiplicative approaches, respectively. As mentioned in Petruzzi and Dada (1999), it is possible to reduce π ( p, q) to an optimization problem over the single variable p, and then substituting the result back into the revenue function. This method was first suggested in Whitin (1955) and has resulted a fractile rule for determining the optimal order quantities, q ∗ = y + F −1 (ρ), and q ∗ = y F −1 (ρ) for additive and multiplicative approaches respectively, where, ρ = p−c p . Yao et al. (2006) have also used this method to address the problem. An alternative approach was primarily suggested in Zabel (1970) which first determines the optimal price p ∗ for a given q, and searches for an optimal order quantity q ∗ over the trajectory to maximize π ( p ∗ , q). While using Zabel (1970)’s approach, Petruzzi and Dada (1999) stated conditions which determines a unique optimal order quantity q ∗ which guarantee the global maximum of π ( p, q) besides other conditions such as an Increasing Failure Rate (IFR) property of the price dependent stochastic demand, D. Following Petruzzi and Dada (1999)’s approach, we assume that the random factor ξ takes its mean value μ. It can be realistically assumed that the expected demand,

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z = E[D] is nonnegative for both additive and multiplicative modeling situations. In the case of additive model, z = y + μ ≥ 0, and for multiplicative case, z = yμ ≥ 0. The revenue (profit) function in Eq. 6 is shown jointly concave in price, p, and order quantity q (Petruzzi and Dada 1999; Yao et al. 2006). However, the scope of this work is to highlight the use of the proposed approach, and optimality analysis of the problem, when distribution is unknown. In this pursuit we first consider a deterministic analysis. In this case, the deterministic revenue function would be an upper bound on the expected revenue function, therefore, we obtain: π( p) = ( p − c) z

(7)

where in Eq. 7, z = y( p) + μ, and for brevity, π = π( p). In order to continue swiftly with forthcoming distribution free approach analysis, in this paper we highlight some important results from deterministic revenue function, π in Lemma 1. Lemma 1 For the deterministic revenue function, π the followings hold: 1. π is quasi-concave in p, and p ∗ is the unique price which satisfies the first order optimality condition, ∂π ∂ p = 0. 2. For an additive model with deterministic demand function y = α − β p, p ∗ = pa , where pa = α+βc+μ . For a multiplicative model with deterministic demand 2β function y = αp −β , p ∗ = p m , where p m =

Proof See “Appendix”.

βc β−1 .

In Lemma 1, an optimal solution to the deterministic version of the problem is presented. pa and p m are the corresponding optimal prices which are the largest prices a firm can set under no risk, using linear deterministic price dependent demand function in the additive model, and the constant elasticity price dependent deterministic demand function in the multiplicative model. Later, in this paper it will be shown that these prices are the maximum allowable prices that a firm can set when information about demand distribution is unknown and the distribution free approach is used. In the next section, the use of distribution free approach is presented to develop a deterministic approximation of the stochastic problem. The optimization of this approximation would also yield optimal revenue against worst possible demand distribution a firm might have experienced. 2.2 Distribution free analysis The distribution free approach to the standard newsvendor problem with pricing is now investigated. The proposed distribution free approach uses the max-min scheme suggested in Scarf (1958) to develop a lower bound estimate on the revenue function, π˜ = π˜ ( p, ˜ q). ˜ One can obtained following simplified relationship established earlier in AlFares and Elmorra (2005).

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min{q, D} = q − E ξ [q − D]+

E ξ [q − D]+ = q − E ξ (D) + E ξ [D − q]+ E ξ [D − q]+ = E ξ (D) − q + E ξ [q − D]+ It is important to notice here that E ξ (D) = z. Now, introducing the Scarf (1958)’s rule to the above relationships, the following expression is obtained: 1/2 σ 2 + (q˜ − z˜ )2 + (q˜ − z˜ ) E ξ [q − D] ≤ 2

+

(8)

Now, the lower bound estimate on the profit function π˜ ( p, ˜ q) ˜ using Scarf (1958)’s rule would actually substitute, E ξ [q − D]+ with its value obtained in Eq. 8 as follows: (σ 2 + (q˜ − z˜ )2 )1/2 + (q˜ − z˜ ) P˜ : π˜ ( p, ˜ q) ˜ = ( p˜ − c) q˜ − p˜ 2

(9)

˜ is a lower bound estimate. In Notice here that the revenue function in problem, P, ˜ the expected (mean) demand would be z˜ = y˜ +μ, and z˜ = y˜ μ for additive problem P, and multiplicative modeling approaches, respectively. Given a mean μ, the maximum expected demand z a firm can experience is at z˜ p=c . As discussed earlier, following Petruzzi and Dada (1999), in the case of additive approach the linear deterministic demand function is y˜ = α − β p. ˜ For multiplicative approach, a constant elastic −β deterministic demand, y˜ = α p˜ is used. The demand elasticity is, η˜ = − p˜ ∂ z˜z˜/∂ p˜ , due to increasing price elasticity (IPE) property, however in non-strict sense, it is very obvious to notice that ∂∂ zp˜˜ ≤ 0 , and ∂∂ ηp˜˜ ≥ 0. Let q˜ ∗ be the optimal quantity and ˜ q), ˜ thus optimal revenue would be p˜ ∗ , the optimal pricing policy that maximize π˜ ( p, π˜ ( p˜ ∗ , q˜ ∗ ) by following Eq. 10. (σ 2 + (q˜ − z˜ )2 )1/2 + (q˜ − z˜ ) max P˜ : π˜ ( p˜ ∗ , q˜ ∗ ) = p, ˜ q˜ ( p˜ − c) q˜ − p˜ 2

(10)

It is important to notify that y˜ ≥ 0, and often μ ≥ 0 for most realistic applications, and thus z˜ ≥ 0 for both additive and multiplicative modeling approaches. In rest of the paper, for brevity, we may refer π˜ ( p, ˜ q) ˜ and π˜ ( p˜ ∗ , q˜ ∗ ) by π˜ and π˜ ∗ , respectively. Next, we investigate the behavior of π˜ ( p, ˜ q), ˜ and ideally looking for joint concavity of π˜ in the two decision controls, p˜ and q. ˜ In this research, Yao et al. (2006)’s approach is followed via Proposition 1. The approach first determines an optimal order quantity, q˜ ∗ for any given price p˜ , and then substitutes this expression of optimal order quantity ˜ q). ˜ This yields a revenue function π˜ ( p, ˜ q˜ ∗ ) in which the price, p˜ is the q˜ ∗ into π˜ ( p, only control variable. Later, the behavior of π˜ ( p, ˜ q˜ ∗ ) is investigated to prove that it is quasi-concave (unimodal) in p, ˜ and a simple search can yield, p ∗ . ˜ the followings hold: Proposition 1 For revenue function, π˜ ( p, ˜ q), ˜ in problem, P, 1. π˜ ( p, ˜ q) ˜ is quasi-concave in q˜ for any given price, p. ˜ For any given price, p, ˜ an p−c ˜ σ (2ρ−1) ˜ ∗ √ optimal order quantity, q˜ = z˜ + ,where ρ˜ = p˜ . 2

123

ρ(1− ˜ ρ) ˜


2. There exists a unique price, p˜ ∗ that satisfies the first order optimality condition ˜ q˜ ∗ ) , ∂ π˜ (∂p, = 0. p˜ 3. π˜ ( p, ˜ q˜ ∗ ) is quasi-concave in p˜ such that p˜ ∈ pa , pa using additive approach and p˜ ∈ p m , p m using multiplicative approach respectively, where, p a = α+2βc−2βs , 3β

2β and p m = c 2β−1 .


Proposition 1 proves the joint concavity of π( ˜ p, ˜ q) ˜ in p˜ and q, ˜ and it also outlines a sequential process to achieve the joint optimization of π˜ ( p, ˜ q). ˜ It can be noticed from ˜ q) ˜ σ c Proposition 1 that the first order optimality condition ∂ π˜ ∂( p, = z ˜ (1− ρ ˜ η)− ˜ = p 2 p−c ˜

0 (see “Appendix”), of course, there is no closed-form solution to determine p˜ ∗ , however, p˜ ∗ can be very efficiently determined using numerical methods. In the following, a procedure is outlined to determine p˜ ∗ , and q˜ ∗ c 1. Solve, z˜ (1 − ρ˜ η) ˜ − σ2 p−c = 0, and determine the optimal price, p˜ ∗ ˜ 2. Determine ρ˜ ∗ =

p˜ ∗ −c p˜ ∗

using the value of p˜ ∗

∗ −1) , ρ˜ (1−ρ˜ ∗ )

σ (2ρ˜ 3. Determine the optimal order quantity, q˜ ∗ = z˜ + √ ∗ 2

where, z˜ = z˜ p= p˜ ∗

Notice from Proposition 1, if pa ≤ p˜ ∗ ≤ pa , then it also guarantees the global optimality. This condition on lower bound on the price, p˜ established in Proposition 1 becomes redundant, if pa ≤ c for additive and p m ≤ c for multiplicative approach, respectively. When ρ˜ ∗ ≤ 21 , a firm’s optimal stocking decision, q˜ ∗ ≤ z˜ p= p˜ ∗ , which means a firm stocks less than expected deterministic demand at an optimal price, p˜ ∗ . Otherwise the firm stocks more than deterministic demand, z˜ p= p˜ ∗ . Although the solution procedure outlined using Proposition 1 has yielded simplified search for optimal price, p˜ ∗ , and order quantity, q˜ ∗ , yet there is no closed-form expressions for pricing and order quantities established. 2.3 A worst case analysis to distribution free approach In this section, we present a worst case analysis for expected revenue to a firm when demand distribution is unknown and distribution free approach is used. In the forthcoming analysis it will be shown that this worst case analysis would actually enable determining a closed-form or a more simplified solution for optimal pricing and order quantity decisions. In Proposition 2, a worst case analysis on π˜ is proposed. ˜ π˜ ( p, Proposition 2 For revenue function, π˜ ( p, ˜ q), ˜ in problem, P, ˜ q) ˜ ≥ ( p˜ − c) z˜ − p˜ σ2 , where, z˜ = α − β p˜ for additive model, and z˜ = α p˜ −β for multiplicative model. Using Proposition 2, one can establish a more simplified lower bound on revenue developed earlier using distribution free approach in Eq. 10. The advantage is that unlike previously established results in Proposition 1, we can obtain closedform expression for optimal pricing and order quantity for an additive model.

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Finally, this worst case analysis yields a revised lower bound on revenue defined in ˇ problem, P: σ Pˇ : πˇ = ( pˇ − c) zˇ − pˇ 2

(11)

In this situation, pˇ ∗ and qˇ ∗ are the corresponding optimal price and order quantity for ˇ In Proposition 3, Proposition 2 is followed revenue function established in problem, P. and a closed-form solution expression to determine pˇ ∗ and qˇ ∗ is obtained. However, due to demand function, the closed expressions can only be obtained for linear demand curve. ˇ is concave in p. Proposition 3 The revenue function, πˇ , in problem P, ˇ The optimal ∗ price, pˇ is determined as follows: 1. For additive model, pˇ ∗ = 2(α+βc+μ)−σ . For multiplicative model, pˇ ∗ is determined 4β ˇ − σ2 = 0. by solving, αμ pˇ −β−1 (βc − β pˇ + p) σ (2ρˇ ∗ −1) 2. For an optimal price, pˇ ∗ , an optimal order quantity, qˇ ∗ = zˇ ∗ + √ , where ∗ ∗ ρˇ ∗

=

pˇ ∗ −c pˇ ∗ .


2

ρˇ (1−ρˇ )

Proposition 3 proves the concavity of πˇ in price, p, ˇ and following the results established in Proposition 2, Proposition 3 outlines a sequential process to achieve the joint optimization of πˇ ( p, ˇ q). ˇ It can be noticed from Proposition 3 that using the first order optimality condition ∂∂ πpˇˇ = zˇ (1 − ρˇ η) ˇ − pˇ σ2 = 0 (see appendix), there is a ∗ closed-form solution to determine pˇ for additive model, however, pˇ ∗ can be only be determined numerically for multiplicative model. 3 Extension to shortage penalty and holding cost case In this section, the standard newsvendor problem with pricing discussed in Sect. 2 is extended to consider the shortage and holding cost, and we refer this as an Extended Problem(P ). Likewise standard newsvendor problem with pricing, P, the extended problem P has the variants addressed in the following: P : Demand is price dependent stochastic demand with known distribution. P˜ : Demand is price dependent stochastic demand with unknown distribution and distribution free approach is used. Pˇ : Demand is price dependent stochastic demand with unknown distribution and a worst case analysis for distribution free approach is used. Let g be the holding cost per unit for an excess inventory, and s be the shortage cost per unit associated with an unmet demand. The revenue function to the firm in this situation would be established in Eq. 12, (see Yao 2002; AlFares and Elmorra 2005; Yao et al. 2006 for more details).

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P : π ( p, q) = E ξ p min{q, D} − c q − g [q − D]+ − s [D − q]+ = p(q − E ξ [q − D]+ ) − c q − g E ξ [q − D]+ − s E ξ [D] − q + E ξ [q − D]+ = ( p + s − c) q − s z − ( p + s + g) E ξ [q − D]+

(12)

3.1 Basic analysis Following a basic analysis previously presented for standard newsvendor problem, the revenue function for the firm when demand distribution is known for an additive modeling framework would be: q−y

F(ξ )d ξ

π ( p, q) = ( p + s − c) q − s z − ( p + s + g)

(13)

ξ

Similarly for a multiplicative modeling framework, the revenue function would be: q

y F(ξ )d ξ

π ( p, q) = ( p + s − c) q − s z − ( p + s + g)y

(14)

ξ

The standard newsvendor problem with pricing extended to shortage and holding cost for a known demand distribution has been addressed in Petruzzi and Dada (1999) and Yao (2002). It is shown that the revenue function is quasi-concave for a wide range of demand distributions. Now, the use of the distribution free approach is discussed for the problem in the forthcoming. 3.2 Distribution free analysis The lower bound on the revenue function of problem, P , using the distribution free approach is established in Eq. 15. The extended problem is referred as P˜ in the following: (σ 2 + (q˜ − z˜ )2 )1/2 + (q˜ − z˜ ) ˜ q) ˜ = ( p˜ + s − c) q˜ − s z˜ − ( p˜ + s + g) P˜ : π˜ ( p, 2 (15) In Proposition 4, the proof of joint concavity of π˜ ( p, ˜ q) ˜ established in Eq. 15 is presented. The findings in this proposition apparently resemble to that of Proposition 1, however, the optimal price, p˜ ∗ , and order quantity q˜ ∗ in this case would vary because the first order optimality conditions are distinct in each of these extensions. Proposition 4 For revenue function, π˜ ( p, ˜ q), ˜ in problem, P˜ , the followings hold:

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1. π˜ ( p, ˜ q) ˜ is quasi-concave in q˜ for any given price, p. ˜ For any given price, p, ˜ an p+s−c ˜ κ−1) ˜ optimal order quantity, q˜ ∗ = z˜ + 2σ√(2κ(1− , where κ ˜ = . p+s+g ˜ ˜ κ) ˜ 2. There exists a unique price, p˜ ∗ that satisfies the first order optimality condition, ∂ π˜ ( p, ˜ q˜ ∗ ) = 0. ∂ p˜ 3. π˜ ( p, ˜ q˜ ∗ ) is quasi-concave in p˜ such that p˜ ∈ pa − 23 s, pa using additive β approach and p˜ ∈ p m − 2 2β−1 s, p m using multiplicative approach respectively.


Proposition 4 proves the joint concavity of π˜ ( p, ˜ q) ˜ in p˜ and q. ˜ It also outlines a sequential procedure to achieve the joint optimization of π˜ ( p, ˜ q). ˜ It can ˜ q˜ ∗ ) be noticed from Proposition 4 that the first order optimality condition ∂ π˜ (∂p, = p c+g = 0 (see “Appendix”, proof of Proposition 4), there is no z˜ (1 − κ˜ η) ˜ − σ2 p+s−c ˜ closed-form solution to determine p˜ ∗ , however, p˜ ∗ can be very efficiently determined using numerical methods for finding roots. In the following, a procedure is outlined to determine p˜ ∗ , and q˜ ∗ . c+g = 0, and determine the optimal price, p˜ ∗ . 1. Solve, z˜ (1 − κ˜ η) ˜ − σ2 p+s−c ˜ 2. Determine κ˜ ∗ =

p˜ ∗ +s−c p˜ ∗ +s+g ,

using the value of p˜ ∗ .

3. Determine the optimal order quantity, q˜ ∗ = z˜ +

σ (2κ˜ ∗ −1) √ , 2 κ˜ ∗ (1−κ˜ ∗ )

where, z˜ = z˜ p= p˜ ∗ .

Similar to an earlier interpretation presented in Proposition 1, for an extended newsvendor problem with pricing, it is intuitive to notice that if pa ≤ p˜ ∗ ≤ pa , then it will also guarantee the global optimality. This condition on lower bound of the price, p˜ established in the Proposition 4 becomes redundant if pa − 23 s ≤ c for additive and

β s ≤ c for multiplicative approaches respectively. Also when κ˜ ∗ ≤ 21 , a p m − 2 2β−1 firm’s optimal stocking decision is, q˜ ∗ ≤ z˜ p= p˜ ∗ , which means a firm stocks less than expected riskless demand, z˜ p= p˜ ∗ , at an optimal price, p˜ ∗ . Otherwise, the firm stocks more than expected deterministic demand.

3.3 A worst case analysis to distribution free approach Following Proposition 2, Proposition 5 suggests a worst case analysis for a firm. A new lower bounding scheme is outlined in Proposition 5. ˜ q) ˜ ≥ ( p˜ − c)˜z − Proposition 5 For revenue function, π˜ ( p, ˜ q), ˜ in problem, P˜ , π˜ ( p, σ ( p˜ +s +g) 2 , where, z˜ = α −β p˜ for additive model, and z˜ = α p˜ −β for multiplicative model. Proof See “Appendix”.

Similar to Proposition 3, Proposition 5 can be utilized for revised lower bound on revenue function developed using the distribution free approach in Eq. 15. Again the

123


advantage of this lower bound would be to obtain a closed-form expression for an optimal price and order quantity. Thus, the revised lower bound in this case would be: σ ˇ = ( pˇ − c)ˇz − ( pˇ + s + g) Pˇ : πˇ ( p) 2

(16)

where in Eq. 16, πˇ = πˇ ( p). ˇ Following the results established in Proposition 3, it is very intuitive to show that the revenue function, πˇ established in Eq. 16 is concave (unimodal) in p. ˇ Furthermore, a closed-form solution for the optimal price, pˇ ∗ , is ˇ For obtained which is the same as the optimal price expression obtained for problem P. an additive model one can obtain an expression for optimal price, pˇ ∗ = 2(α+βc+μ)−σ 4β which matches the optimal price for standard newsvendor problem. However, for ˇ − σ2 = 0. multiplicative model, pˇ ∗ is determined by solving, αμ pˇ −β−1 (βc − β pˇ + p) An optimal order quantity given that pˇ is already calculated, would be, qˇ ∗ = zˇ ∗ + ∗ +s−c σ (2κˇ ∗ −1) √ , where κˇ ∗ = ppˇˇ∗ +s+g . This finding reveals that a firm under this strategy ∗ ∗ 2

κˇ (1−κˇ )

sets an optimal price pˇ ∗ which will be indifferent to standard newsvendor problem and its extension. Nevertheless, the optimal order quantity, qˇ ∗ would be distinct as it can be noticed from the previously presented analysis. 4 Numerical experimentation

In this section a numerical experimentation is presented to calibrate the performance of the proposed distribution free approach on newsvendor problem with extension to shortage and holding costs. It can be clearly noticed here that the two modeling frameworks, additive and multiplicative are substantially different and therefore, each of the two models have different analysis and have been experimented separately. Nevertheless, for both models three price dependent stochastic demand distributions, uniform, normal, and log normal are considered. As suggested in Mostard et al. (2005), √ √ the random factor, ξ is bounded such that, ξ ∈ [μ − 3σ, μ + 3σ ]. Additionally, for the numerical experimentation with lognormal distribution the mean, μ and standard σ 2 ) ), deviation, σ are re-scaled, so that the modified mean is, μln = ln μ−0.5 ln (1 + ( μ σ 2 and the standard deviation, σln = ln (1 + ( μ ) ). This transformation is often used for comparative purposes (see Bain and Engelhardt 1992; Mostard et al. 2005 for details). Gallego and Moon (1993) suggested a framework to calibrate the effectiveness of the distribution free approach on newsvendor problem with exogenous price. A performance measure, expected value of additional information (EVAI) is introduced, which is determined by taking the difference of the optimal revenue when demand distribution is perfectly known and the optimal revenue generated using the distribution free approach, i.e., EVAI = π( p ∗ , q ∗ ) − π( pˇ ∗ , qˇ ∗ ). A very similar performance pˇ ∗ ,qˇ ∗ ) measure often used is, Ratio = π( π( p ∗ ,q ∗ ) . It is the relative measure of revenue that a firm will yield operating at distribution free control parameters, pˇ ∗ , qˇ ∗ to the corresponding revenue at optimal control parameters p ∗ and q ∗ , when demand distribution is perfectly known. This performance measure was first introduced in Gallego and Moon (1993) and has been used in many related studies (see Liao et al. 2011) to calibrate the

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S. A. Raza Table 2 Data set for additive model

Parameter values c ∼ U[5, 13] α ∼ U[1,000, 100,000] β ∼ U[5, 50] μ ∼ U[500, 2,000] σ ∼ U[0.1, 0.9] × μ s ∼ U [1.2, 1.4] × c g ∼ U [0.2, 0.4] × c

Table 3 Performance analysis of distribution free approach for additive model

Mean

Uniform

Normal

Lognormal

0.9990

0.9931

0.9399

Min.

0.9982

0.9923

0.6166

Max.

0.9993

0.9941

0.9682

SE

0.0002

0.0006

0.0545

performance of the distribution free approach, which is also used in this numerical study. 4.1 Additive model A linear riskless demand curve, y( p) = α − β p, is assumed for additive model. Following an earlier work in Smith et al. (2007), Table 2 outlines the data generation criteria for problem related parameters. Using Table 2, 100 problem instances are generated following uniform distribution. All such problems were solved with distribution free approach, and later a situation is considered in which the unknown demand distribution may have followed uniform, normal and lognormal distributions, respectively. Table 3 summarizes the results of numerical experimentation with additive model. While knowing the demand distribution, a firm can achieve an additional revenue gain which is about 0.10, 0.69, and 6.01 %, if unknown demand distribution may have followed uniform, normal and lognormal distributions respectively, relative to the corresponding optimal revenues when demand distribution is perfectly known. Next, the minimum ratio, reveals the maximum revenue gain a firm can yield by the knowledge of demand behavior that has followed uniform, normal, and lognormal distribution. These gains are, 0.18, 0.77, and 38.34 %, respectively. 4.2 Multiplicative model Unlike the additive model, for the multiplicative model, constant elasticity riskless demand function y( p) = αp −β is used. Table 4 outlines the details of the problem related parameters that are used in numerical experimentation. These values have been used after some customizations from a related numerical experimentation in Smith et

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Newsvendor problem with pricing Table 4 Data set for multiplicative model

Parameter values c ∼ U[5, 8] α ∼ U[50,000, 100,000] β ∼ U[1.5, 2.7] μ ∼ U[1, 2] σ ∼ U[0.1, 0.2] × μ s ∼ U [1.2, 1.4] × c g ∼ U [0.2, 0.4] × c

Table 5 Performance analysis of distribution free approach for multiplicative model

Uniform

Normal

Lognormal

Mean

0.9171

0.9362

0.1033

Min.

0.7976

0.8254

0.0071

Max.

0.9745

0.9867

0.1293

SE

0.0543

0.0471

0.0351

al. (2007). It is important to mention here that, due to the multiplicative nature of the random factor, ξ , an increase in cv, could result infeasible (negative) revenue even if the distribution is perfectly known. Therefore, in Table 4, cv is kept up to 0.2. Despite cv is small, yet due to multiplicative nature, this demand variability is still quite substantial for the present numerical experimentation. The problem instances are generated following uniform distribution as outlined in Table 4. The performance pˇ ∗ ,qˇ ∗ ) measure was Ratio = π( π( p ∗ ,q ∗ ) . A total of 100 problems instances are generated randomly and tested for demand situations when the price dependent stochastic demand follows uniform, normal,and lognormal distributions, respectively. There were some problem instances noticed to result a negative π( p ∗ , q ∗ ), and thus Ratio was found negative. This situation was mainly observed with lognormal distribution, as remedy, these instances were identified as outliers, and were removed from further analysis. Using the mean ratio in Table 5, it can be noticed that, by knowing the price dependent stochastic demand distribution, a firm can yield superior revenue gains relative to the situation when it does not know the demand distribution. These gains are 8, 6, and 90 % compared to the case when the firm uses distribution free approach and the unknown price dependent stochastic demand may have followed uniform, normal, and lognormal distributions, respectively. The gains could augment to a maximum of 20 %, 17 % and 99 % for uniform, normal, and lognormal distribution, as can be noticed from Table 5. To author’s knowledge only Mostard et al. (2005) have used lognormal distribution for distribution analysis of newsvendor problem with exogenous price. The study has reported that the performance of distribution free approach in the case of lognormal distribution can fall as low as 74 % compared to the corresponding optimal revenue, when the distribution is known. Whereas, in the same study with uniform and normal distribution, it has been reported that the deviation was to maximum of 5 % in the same context. However, this indeed requires further analysis, but at present this is not the focus of this paper.

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It is obvious to notice from this numerical experimentation that both modeling framework and demand distribution could be main factors to impact the performance of the distribution free approach to the problem. In summary, the performance of the distribution free can be sensitive to the modeling framework of the newsvendor problem with pricing, and also sensitive to distribution of the unknown demand. The distribution free approach, in general performs very well with the additive modeling framework for newsvendor problem with pricing compared to the multiplicative model. Fortunately, the use of distribution free approach is found very promising in additive model, and in the situation when the unknown behavior of the price dependent stochastic demand may have followed widely observed demand distributions such as uniform and normal. In pricing research, the use of linear demand curve in an additive modeling framework is widely practiced (Zhang et al. 2010). Indeed, numerical experimentations have provided support that the distribution free approach is very robust in this context, and has a very significant implication for real life business decision in revenue management and pricing. 5 Conclusions, limitations, and future research directions This paper proposes the use of the distribution free approach to newsvendor problem with pricing and its extension to shortage and holding cost case. The approach develops lower bounds on the revenue (profit) for the problem and its extension. The lower bounds are shown jointly concave in price and order quantity which resulted the global optimal revenue for a firm that faces stochastic price dependent demand with unknown distribution. However, the firm still needs information on basic demand related parameters, mean and standard error. It is shown that the lower bounds are jointly concave in price and order quantity decisions, and therefore, a global optimal revenue for a firm can be determined. This optimal revenue would be robust with respect to demand behaviors that a firm may experience, including worst possible demand distribution. The research also suggests simplified solution procedures that have resulted closedform expressions of the optimal pricing and order quantity, by using the distribution free approach. These closed-form expressions find many practical applications for a firm exercising yield management decisions. Numerical examples are presented to evaluate the performance of the distribution free approach in randomly generated problems. The experimentations consider a situation in which a firm’s demand distribution is unknown and distribution free approach is used. Later, a comparison is made to situations when the unknown demand distribution follows uniform, normal, and lognormal distributions. The performance of distribution free approach is noticed competitive by using a performance measure, often used in related studies. However, the performance of the distribution free approach being an approximation scheme, is found dependent on the problem’s modeling framework (additive or multiplicative), and on the distribution of the unknown demand. The mean performance of the distribution free in the case of additive model when the price dependent stochastic demand is normal and uniform, is about 0.5 % away from the corresponding optimal revenue when the demand distribution is known. The work presented in this paper has direct implications on the situations where pricing and inventory control decisions are required to maximize the profit on per-

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ishable assets, like in the field of revenue management and pricing. This study also provides an integration of quantity based and price based revenue management. The closed-form expressions for optimal pricing and order quantities developed in this research may find many practical applications for industries practicing yield management decisions. Future work might include the investigation of this approach for industrial applications, i.e. where revenue management is widely practiced (airlines, hotels, car rentals, cruise liners etc.). Other potential extensions of the proposed work include the newsvendor pricing problem with setup cost, random yield, multi-item constraints like budget, resalable returns when a firm faces after sales returns. Moreover, the distribution free approach may also be adapted to the competitive pricing in the context of multiple firms. Acknowledgments

Author would like to thank Mihaela Turiac for her careful editing of the paper.

Appendix Proof of lemma 1 Part i π( p) = ( p − c) z

(17)

For brevity, π( p) is referred by π . The first order optimality condition yields: ∂z ∂π = ( p − c) +z ∂p ∂p = z (1 − ρ η)

(18) p

∂z ∂p

p−c Thus, ∂π ∂ p = 0 ⇒ (1 − ρ η) = 0, where ρ = p , η = − z . Consider the expression (1 − ρ η), which can also be written as 1 = G( p), where G( p) = ρ η. To lim determine, there exist only a unique price, p ∗ , we need to show that p→c G( p) ≤ 0, ∂G( p) ∂G( p) ∂η ∂ρ lim and ∂ p ≥ 0. Thus, here p→c G( p) = 0, and ∂ p = ρ ∂ p + η ∂ p ≥ 0, since ∂η ∂p

≥ 0, and ∂∂ρp = pc2 ≥ 0. This proves the uniqueness of p ∗ . In order to prove that π( p) is quasi-concave in p, all we need to prove that ∂2π | ∂π =0 ≤ 0, which is proven in the following expression: ∂ p2 ∂p

∂ 2π ∂η ∂ρ ∂z − z ρ + η = (1 − ρ η) ∂ p2 ∂p ∂p ∂p It is very obvious to determine that, ∂∂ pπ2 ≤ 0, since 1−ρ η = 0, ∂∂ρp = and z ≥ 0. 2

(19) c p2

≥ 0, ∂∂ηp ≥ 0.

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Part ii Using additive modeling approach, the riskless demand is y = α − β p, z = y + μ, p . Now solving the first order optimality condition, 1 − ρ η = 0 for and η = α−ββ p+μ p, results, optimal price, pa = y=α

p −β , z

α+βc+μ . 2β

Now for multiplicative modeling approach,

= y μ, and η = β, the first optimality condition, yields, p m =

βc β−1 .

Proof of proposition 1 Part i Consider the lower bound on the revenue function: π˜ ( p, ˜ q) ˜ = ( p˜ − c) q˜ − p˜

(σ 2 + (q˜ − z˜ )2 )1/2 + (q˜ − z˜ ) . 2

(20)

The first order derivative w.r.t q˜ yields following expression: ∂ π( ˜ p, ˜ q) ˜ p˜ = ( p˜ − c) − ∂ q˜ 2 Now, the first order optimality condition, q˜ ∗

q˜ − z˜

+1 σ 2 + (q˜ − z˜ )2

∂ π˜ ( p, ˜ q) ˜ ∂ q˜

(21)

= 0 yields an optimal order quantity

σ (2ρ˜ − 1) q˜ ∗ = z˜ + 2 ρ(1 ˜ − ρ) ˜

(22)

˜ where in Eq. 22, ρ˜ = p−c p˜ . ρ˜ is commonly used in pricing research and it is referred as critical fractile (Petruzzi and Dada 1999; Porteus 1990). Now, take the second 2 ˜ q) ˜ p˜ σ 2 = − ˜ ∈ [c, p]. This proves that derivative test, ∂ π˜∂(q˜p, 2 3/2 ≤ 0, ∀ p 2((q−˜ ˜ z )2 +σ 2 ) π˜ ( p, ˜ q) ˜ is concave in q˜ for a given price, p. ˜

Part ii Substituting the expression for q˜ ∗ from Eq. 22 into revenue function expression in Eq. 20 results:

σ (1 − ρ) ˜ π( ˜ p, ˜ q˜ ∗ ) = ( p˜ − c) z˜ − ρ(1 ˜ − ρ) ˜

(23)

Alternatively, the expression in Eq. 23 can also be written as: π˜ ( p, ˜ q˜ ∗ ) = ( p˜ − c)˜z − σ

123

c( p˜ − c)

(24)


Now the first derivative on π˜ ( p, ˜ q˜ ∗ ) w.r.t p˜ results the following expression: ∂ π˜ ( p, ˜ q˜ ∗ ) ∂ z˜ σ = ( p˜ − c) + z˜ − ∂ p˜ ∂ p˜ 2

c p˜ − c

(25)

Applying the first order optimality condition (FOC) for p, ˜ ∂∂ πp˜˜ = 0, gives: σ z˜ (1 − ρ˜ η) ˜ − 2

c =0 p˜ − c

(26)

In Eq. 26, η˜ = − p˜ ∂ z˜z˜/∂ p˜ , ∂ z˜ /∂ p˜ ≤ 0. Due to IPE property ∂∂ ηp˜˜ ≥ 0. In addition to c this, it can be concluded that ρ˜ η˜ ≤ 1, since σ2 p−c ≥ 0. To show that there exist ˜ a unique p, ˜ p˜ ∗ that solves Eq. 26, we re-write Eq. 26 as, X ( p) ˜ + G( p) ˜ = 0, where, ∂ X ( p) ˜ σ c ∂ z˜ X ( p) ˜ = z˜ (1 − ρ˜ η), ˜ and G( p) ˜ = − 2 p−c ˜ − ˜ . Notice here, ∂ p˜ = ∂ p˜ (1 − ρ˜ η) ∂G( p) ˜ ∂ η˜ ∂ ρ˜ σ c 1 z˜ ρ˜ ∂ p˜ + η˜ ∂ p˜ , and ∂ p˜ = 2 p−c . In order to show that there exists a ˜ 2( p−c) ˜

unique price, p˜ ∗ , we need to prove that

∂G( p) ˜ ∂ p˜

≥ 0, and

∂ X ( p) ˜ ∂ p˜

≤ 0, or vice versa. Now,

˜ ≤ 0, which means, X ( p) ˜ is (strictly) decreasing in p. ˜ Notice here, we prove that ∂ X∂ (p˜p) ∂ η˜ ∂η ∂ ρ˜ c ∂ z˜ ∂ p˜ ≥ 0, likewise ∂ p due to IPE property. Also, ∂ p˜ = p˜ 2 ≥ 0. In addition, ∂ p˜ ≤ 0

due to IPE, and indeed z˜ ≥ 0. Moreover, using FOC condition, 1 − ρ˜ η˜ ≥ 0. Thus, ∂ X ( p) ˜ ∂ p˜ ≤ 0. Q.E.D. Next, it is very intuitive to show that

to this,

lim p→c ˜

∂G( p) ˜ ∂ p˜

is strictly increasing in p. ˜ In addition

X ( p) ˜ → z, where z is the maximum expected price dependent stochastic lim

demand experienced at p˜ = c, and p→c G( p) ˜ → −∞. Since X ( p) ˜ is decreasing in ˜ p, ˜ whereas G( p) ˜ is increasing in p, ˜ thus there will be a unique p˜ ∗ which will satisfy, X ( p) ˜ + G( p) ˜ = 0. This guarantees a unique price, p˜ ∗ . Part iii In order to prove that π˜ ( p) ˜ is quasi-concave in p, ˜ all we need to prove that ∂∂ p˜π˜2 | ∂ π˜ =0 ≤ 2

∂ p˜

0. Thus,

∂ 2 π˜ ∂ ρ˜ 1 c ∂ z˜ ∂ η˜ σ + η ˜ = − ρ ˜ η) ˜ − z ˜ ρ ˜ + (1 ∂ p˜ 2 ∂ p˜ ∂ p˜ ∂ p˜ 2 p˜ − c 2( p˜ − c) In Eq. 27 we replace

σ 2

c p−c ˜

(27)

with z˜ (1 − ρ˜ η), ˜ and after simplification results

∂ η˜ ∂ 2 π˜ z˜ (1 − ρ˜ η) ˜ ∂ ρ˜ + ρ ˜ = − 2 ρ ˜ η) ˜ − z ˜ η ˜ (1 ∂ p˜ 2 2( p˜ − c) ∂ p˜ ∂ p˜

(28)

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S. A. Raza

In Eq. 28,

z˜ (1−ρ˜ η) ˜ 2 ( p−c) ˜

∂ η˜ ∂ p˜

≥ 0, also

≥ 0, and

∂ ρ˜ ∂ p˜

=

c p2

≥ 0. Thus, ∂∂ p˜π˜2 ≤ 0, given 2

ρ˜ η˜ ≥ 1/2. In summary, π˜ ( p) ˜ is quasi-concave in p˜ given that 21 ≤ ρ˜ η˜ ≤ 1. Thus we conclude quasi-concavity of π( ˜ p, ˜ q). ˜ Next, we can use this condition to derive improved bounds on price p˜ which will determine joint concavity of π˜ ( p, ˜ q) ˜ as follows. Using additive approach , z˜ = y˜ +μ. Assuming linear riskless demand function, y˜ = . Similarly, the condition,ρ˜ η˜ ≥ 21 , α−β p. ˜ The condition, ρ˜ η˜ ≤ 1 yields, p˜ ≤ α+βc+μ 2β . Thus, the bound on price, p˜ ∈ pa ≤ p˜ ≤ pa , where, pa = gives p˜ ≥ α+2βc+μ 3β α+2βc+μ 3β

which can also be expressed as c + 3zβ , and pa is referenced from Lemma 1. Next, for multiplicative approach, assuming expected price dependent demand z˜ = 2βc βc ≤ p˜ ≤ β−1 , which can be also written as p m ≤ p˜ ≤ p m , where α p˜ −β μ, 2β−1 pm =

2βc 2β−1 ,

and p m is again referenced from Lemma 1.

Proof of proposition 2 Referring to Eq. 24 as established in proof of Proposition 1 π( ˜ p) ˜ = ( p˜ − c)˜z − σ Recall, ρ˜ = re-written as:

p−c p ,

c( p˜ − c)

(29)

thus, using this notation the revenue function in Eq. 23 can be

π˜ = ( p˜ − c) z˜ − p˜ σ Now consider, the expression, f (ρ) ˜ = of f (ρ) ˜ is:

ρ(1 ˜ − ρ) ˜

(30)

ρ(1 ˜ − ρ) ˜ in Eq. 30. The first order derivative

∂ f (ρ) ˜ 1 − 2ρ˜ = ∂ ρ˜ 2 (1 − ρ) ˜ ρ˜

(31)

˜ = 0 ⇒ ρ˜ ∗ = 21 . It is important to The first order optimality condition yields, ∂ f∂(ρ˜ρ) notice that the second derivative test, as presented in Eq. 32, shows that f (ρ) ˜ is strictly concave for all values of ρ˜ such that 0 ≤ ρ˜ ≤ 1, and f (ρ) ˜ is maximum at ρ˜ ∗ = 21 .

∂ 2 f (ρ) ˜ 1 =− 2 ∂ ρ˜ 4((1 − ρ) ˜ ρ) ˜ 3/2 Next, substituting, ρ˜ = ρ˜ ∗ =

1 2

in Eq. 30, we can obtain the following:

π˜ ≥ ( p˜ − c)˜z − p˜

123

(32)

σ 2

(33)


Proof of proposition 3 Again referring to Eq. 33, we can write a lower bound that can be observed using the distribution free approach as follows: πˇ = ( pˇ − c)ˇz − pˇ

σ 2

(34)

Now, taking the first derivative of πˇ w.r.t pˇ results the following expression: ∂ πˇ ∂ zˇ σ = ( pˇ − c) + zˇ − ∂ pˇ ∂ pˇ 2

(35)

Applying the first order optimality condition for p, ˇ ∂∂ πpˇˇ = 0, gives: z˜ (1 − ρˇ η) ˇ −

σ =0 2

(36)

In Eq. 36, ηˇ = − pˇ ∂ zˇzˇ/∂ pˇ , ∂ zˇ /∂ pˇ ≤ 0. Due to IPE property ∂∂ ηpˇˇ ≥ 0. In addition to this, it can be concluded that ρˇ ηˇ ≤ 1, since σ2 ≥ 0. To show that there exist ˇ + G( p) ˇ = 0, a unique price, pˇ ∗ that solves Eq. 36, we re-write Eq. 36 as, X ( p) where, X ( p) ˇ = zˇ (1 − ρˇ η), ˇ and G( p) ˇ = − σ2 . Following the proof procedure given ˇ p) ˇ ≤ 0, and ∂G( ≤ 0 to prove uniqueness in Proposition 1, we need to show ∂ X∂ (pˇp) ∂ pˇ ˇ = ∂∂ zpˇˇ 1 − ρˇ ηˇ − zˇ ρˇ ∂∂ ηpˇˇ + ηˇ ∂∂ ρpˇˇ , and following a result from of pˇ ∗ . Here, ∂ X∂ (pˇp) Proposition 1,

∂ X ( p) ˇ ∂ pˇ

≤ 0 , which means it is decreasing in p, ˇ and

∂G( p) ˇ ∂ pˇ

does not

change w.r.t p. ˇ In addition to this, X ( p) ˇ → z, and G( p) ˇ → − σ2 . Since X ( p) ˇ is decreasing in p, ˇ whereas G( p) ˇ is constant in p, ˇ there will be a unique, pˇ ∗ which will satisfy, X ( p) ˇ + G( p) ˇ = 0. This guarantees a unique price, pˇ ∗ . lim p→c ˇ

lim p→c ˇ

Part iii In order to prove that πˇ ( p) ˇ is quasi-concave in p, ˇ we need to show that Thus,

∂ ηˇ ∂ ρˇ ∂ 2 πˇ ∂ zˇ 1 − ρ ˇ η ˇ − z ˇ ρ ˇ + η ˇ = ∂ pˇ 2 ∂ pˇ ∂ pˇ ∂ pˇ ∂ X ( p) ˇ ≤0 = ∂ pˇ

∂ 2 πˇ ∂ pˇ 2

| ∂ πˇ =0 ≤ 0. ∂ pˇ

(37)

This yields, ∂∂ pˇπˇ2 ≤ 0. In summary, πˇ ( p) ˇ is quasi-concave in p. ˇ ∗ In order to determine optimal price, pˇ , the first order optimality condition is applied, zˇ (1 − ρˇ η) ˇ − σ2 . For additive model, we assume expected price dependent . Next, for demand zˇ = α − β pˇ + μ. The optimal price would be, pˇ ∗ = 2(α+βc+μ)−σ 4β 2

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S. A. Raza

multiplicative approach, assuming expected price dependent demand zˇ = α pˇ −β μ. Again using the first order optimality condition, the optimal price, pˇ ∗ , is determined ˇ − σ2 = 0. by solving, αμ pˇ −β−1 (βc − β pˇ + p) Proof of proposition 4 Part i Consider the lower bound on the revenue function: π˜ ( p, ˜ q) ˜ = ( p˜ + s − c) q˜ − s z˜ − ( p˜ + s + g)

(σ 2 + (q˜ − z˜ )2 )1/2 + (q˜ − z˜ ) 2 (38)

The first order optimality condition, quantity q˜ ∗

∂ π˜ ( p, ˜ q) ˜ ∂ q˜

= 0, yields following an optimal order

σ (2κ˜ − 1) q˜ ∗ = z˜ + √ 2 κ(1 ˜ − κ) ˜ where in Eq. 39, κ˜ 2 ˜ − σ ( p+s+g) 3/2 2((q−˜ ˜ z )2 +σ 2 )

=

p+s−c ˜ . p+s+g ˜

The second derivative,

(39) ∂ 2 π˜ ( p, ˜ q) ˜ ∂ q˜ 2

=

≤ 0, which proves that π˜ ( p, ˜ q) ˜ is quasi-concave in q˜ for any given

price, p. ˜ Part ii

Now, substituting the expression for q˜ ∗ from Eq. 39 into revenue function expression derived in Eq. 38 and algebraic simplification results: π( ˜ p) ˜ = ( p − c) z˜ − σ

(c + g)( p˜ + s − c)

(40)

Again following Proposition 1’s findings, the first order optimality condition, ∂ π˜ ( p, ˜ q˜ ∗ ) = 0, on Eq. 40 gives: ∂ p˜ σ π˜ ( p, ˜ q˜ ∗ ) = z˜ (1 − ρ˜ η) ˜ − ∂ p˜ 2

c+g p˜ + s − c

(41)

Notice here Eq. 41 can be re-written as, X ( p)+ ˜ G( p) ˜ = 0, where, X ( p) ˜ = z˜ (1− ρ˜ η), ˜ c+g p−c ˜ − p˜ ∂ z˜ /∂ p˜ σ and G( p) ˜ = − 2 p+s−c , ρ˜ = p˜ , and η˜ = . It is easy to show here that ˜ z˜ there exist at least one, p˜ that satisfies the first order optimality condition, if ρ˜ η˜ ≤ 1, c+g because σ2 p+s−c ≥ 0, thus ρ˜ η˜ ≤ 1 is the necessary condition from first order ˜ optimality condition.

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Similar to Proposition 1, in order to prove that there exists only a unique p˜ ∗ which ˜ p) ˜ ≤ 0, and, ∂G( also satisfies the first order optimality condition, we can show ∂ X∂ (p˜p) ∂ p˜ ≥ ∗ 0, which will guarantee uniqueness of price, p˜ . Notice here that, lim p→c ˜

G( p) ˜ =

c+g s

≤ 0,

X ( p) ˜ =z≥

∂ X ( p) ˜ ∂ p˜

≤ 0 is (strictly) decreasing in p˜ (see proof of p) ˜ σ c 1 = ≥ 0 is (strictly) Proposition 1 for details). Additionally, ∂G( 2 ∂ p˜ p+s−c ˜ 2( p+s−c) ˜ 0,

− 21 σ

lim p→c

increasing in p. ˜ This proves the uniqueness of p˜ ∗ .

Part iii Now to prove quasi-concavity we apply the second derivative test:

∂ ρ˜ 1 c+g ∂ z˜ ∂ 2 π˜ ∂ η˜ σ + η ˜ = − ρ ˜ η) ˜ − z ˜ ρ ˜ + (1 ∂ p˜ 2 ∂ p˜ ∂ p˜ ∂ p˜ 2 p˜ + s − c 2( p˜ + s − c)

(42)

c+g Using the first order optimality condition, from Eq. 41 replace σ2 p+s−c with z˜ (1 − ˜ ρ˜ η) ˜ in Eq. 43, and after simplification yields following equation: ∂ 2 π˜ z(1 − ρ˜ η) ˜ = 2 ∂ p˜ 2( p˜ + s − c)

( p˜ + s − c) 1 − 2ρ˜ η˜ p˜ − c

∂ ρ˜ ∂ η˜ − z˜ η˜ + ρ˜ ∂ p˜ ∂ p˜

(43)

ρ˜ η) ˜ ≥ 0, also ∂∂ ηp˜˜ ≥ 0, and ∂∂ ρp˜˜ = p˜c2 ≥ 0. Thus, ∂∂ p˜π˜2 ≤ 0, In Eq. 43, 2(z˜ (1− p+s−c) ˜ ≤ 0. In summary, π( ˜ p) ˜ is quasi-concave in p˜ given that given 1 − 2ρ˜ η˜ ( p+s−c) p−c 2

p−c 2( p+s−c)

≤ ρ˜ η˜ ≤ 1. Thus we conclude quasi-concavity of π˜ ( p, ˜ q). ˜

Extending the analysis following Proposition 1, we get, pa − 23 s ≤ p˜ ≤ pa , where

α+2β c for additive modeling approach. For multiplicative modeling approach, 3β 2β βc . In addition, pi , ∀ i = {a, m} are referred p m − 2 β−1 s ≤ p˜ ≤ p m , where, p m = 22β−1

pa =

from Lemma 1.

Proof of proposition 5 Again recalling Eq. 40 which is established in Proposition 4: π( ˜ p) ˜ = ( p˜ − c) z˜ − σ

(c + g)( p˜ + s − c)

(44)

p+s−c ˜ earlier, it has been noticed that κ˜ = p+s+g . For many practical applications of the ˜ problem, κ˜ resembles ρ˜ as we already know, 0 ≤ ρ˜ ≤ 1. Eq. 44 can be re-written as:

π˜ ( p) ˜ = ( p˜ − c) z˜ − ( p˜ + s + g)σ

κ(1 ˜ − κ) ˜

(45)

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S. A. Raza

Likewise√Proposition 2, we can establish a lower bound estimate based on Eq. 45, as ˜ − κ) ˜ which is maximum at κ˜ = 21 . Therefore, we can conclude that: f (κ) ˜ = κ(1 1 π˜ ( p) ˜ ≥ ( p˜ − c) z˜ − ( p˜ + s + g)σ. 2

(46)

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