- Email: [email protected]
On the optimality conditions of a price-setting newsvendor problem
Accepted Manuscript On the optimality conditions of a price-setting newsvendor problem Sirong Luo, Suresh P. Sethi, Ruixia Shi PII: DOI: Reference:
S0167-6377(16)30083-9 http://dx.doi.org/10.1016/j.orl.2016.08.005 OPERES 6136
To appear in:
Operations Research Letters
Received date: 7 June 2016 Revised date: 22 August 2016 Accepted date: 23 August 2016 Please cite this article as: S. Luo, S.P. Sethi, R. Shi, On the optimality conditions of a price-setting newsvendor problem, Operations Research Letters (2016), http://dx.doi.org/10.1016/j.orl.2016.08.005 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
*Manuscript Click here to view linked References
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
On the Optimality Conditions of a Price-Setting Newsvendor Problem Sirong Luo School of Statistics and Management, Shanghai University of Finance and Economics, Shanghai, 200433, China. [email protected]
Suresh P. Sethi Naveen Jindal School of Management, The University of Texas at Dallas, Richardson, TX, 75083, USA. [email protected]
Ruixia Shi* School of Business, University of San Diego, San Diego, CA, 92110, USA. [email protected]
We analyze a price-setting newsvendor problem with an additive-multiplicative demand. We show that the unimodality of the newsvendor profit function holds when the underlying random term has an increasing failure rate and the demand functions satisfy certain concavity conditions. Further, we show that the optimal price decreases in the order quantity. Finally, we compare our optimality conditions with those existing in the literature. Key words : Inventory Control, Price-Setting Newsvendor, Unimodality, Elasticity
1.
Introduction
Integrating pricing and inventory replenishment decisions under demand uncertainty has proved to be a successful operations strategy of many firms including Amazon, Dell, Walmart and J. C. Penney [4, 6]. By adopting proactive pricing, firms can better match supply with demand, which leads to significant profit increases. The benefits of jointly deciding pricing and inventory replenishment level have also been well documented in various academic studies [5, 8, 9]. The building block for joint inventory and pricing decisions research is the newsvendor model with pricing. The major difficulty in studying this problem is to establish concavity or unimodality of the profit function. Although many analytical results on the optimality conditions have been * Corresponding author. Email: [email protected]. Address: School of Business, University of San Diego, San Diego, CA, 92110, USA. 1
2
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
developed in recent years [10, 12, 13, 17], the existing literature often makes strong assumptions on the demand function forms and demand uncertainty distributions, such as one parameter additive or multiplicative only demand model and specific uncertainty distributions. These assumptions simplify analysis, but their limitations in capturing reality limit their applicability in practice. To address this issue, we use a general additive-multiplicative demand model to analyze the problem of joint inventory and pricing decisions. To derive the optimality conditions, we make two assumptions: 1) the random term in the demand has an increasing failure rate (IFR); 2) the demand function satisfies certain concavity conditions. [12] studies the same problem by first solving for the optimal order quantity and then finding the optimal price. It identifies three conditions to be met for establishing the optimality of the profit function: 1)the riskless profit function is log-concave; 2) the coefficient of variation is log-convex and 3) the distribution of the random term satisfies a specific condition. [10] introduces the concept of lost-sale elasticity. Assuming the random term has IFR distribution, it shows that when the lost-sale elasticity satisfies certain conditions, the concavity or unimodality of the profit function is implied. [14] conducts a similar analysis of the problem from the price elasticity view point. It shows that when both price elasticities of the location and scale parameters in demand are increasing in price, and the elasticity of the location parameter increases faster than the price elasticity of the scale parameter, the unimodality of the profit function is obtained. [1] analyzes a risk-averse price-dependent newsvendor and shows the concavity of the profit functions for additive and multiplicative demands. A focus of the paper is to elaborate the difference in pricing and ordering behaviors of the risk averse and risk neutral newsvendors. [17] obtains the unimodality when the random term has log-concave distribution and the demand functions satisfy certain conditions. Compared with the aforementioned studies, our proof is new and the resulting optimality conditions are different. As our analysis unfolds, we show that our optimality conditions and the conditions obtained in the existing literature do not imply each other. In fact, our result complements the existing results. The rest of this paper is organized as follows. Section §2 presents the newsvendor pricing model and our analytical results. We discuss our results by comparing them with the existing literature
3
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
in Section §3.
2.
Model and Analysis
In this section, we present the price-setting newsvendor model and derive conditions for the optimality of the firm’s expected profit function. 2.1.
Model
A risk neutral firm buys a product at a unit cost c and sells the product to customers at a retail price p ∈ [p, p] over a single selling season. The demand during the selling season depends on the retail price p and is random. Let D(p, ϵ) denote the price-dependent demand, where ϵ is a random variable. The firm simultaneously decides the retail price p and the order quantity y at the beginning of the selling season before observing the demand. After the demand materializes, the firm satisfies the demand with the product’s available stock. If the firm does not have enough stock, that is, y ≤ D(p, ϵ), the unsatisfied demand is lost with no penalty for lost sales. The unsold inventory, if any, is salvaged at zero value. Note that our analysis can be extended easily to the cases of nonzero penalty and nonzero salvage value. As in [7, 8, 12, 17], we consider the additive-multiplicative demand model D(p, ϵ) = µ(p) + σ(p)ϵ,
(1)
where µ(·) ≥ 0 and σ(·) ≥ 0 are deterministic functions of price p, and ϵ is a nonnegative random variable with mean µ ˜ and standard deviation σ ˜ . This demand model generalizes the widely used multiplicative or additive only demand models. In our generalized model, the retail price can have different effects on the location and scale parameters of the demand. Whereas, in the multiplicative only models, the price has the same effects on the demand mean and standard deviation, i.e., the coefficient of variation is independent of price. Let f (·) and F (·) denote the density and cumulative distribution functions of ϵ, respectively. In addition, let F¯ (·) = 1 − F (·). We define G(·|p) as the conditional distribution of D(p, ϵ) for a given ) ( . The failure rate of ϵ is defined as h(·) = f (·)/F¯ (·). retail price p. Thus, G(x|p) = F x−µ(p) σ(p)
4
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
For a given retail price p and order quantity y, the firm’s expected profit (profit, hereafter) is π(y, p) = p E min{y, µ(p) + σ(p)ϵ} − cy.
(2)
The firm determines y ∈ [0, ∞) and p ∈ [p, p] to maximize its profit, that is maxy,p π(y, p) = p E min{y, µ(p) + σ(p)ϵ} − cy. Unfortunately, the concavity of the profit function does not hold under general conditions. Thus, researchers have tried to establish conditions under which the profit function is quasi-concave (i.e., unimodal) or log-concave (since the log-concavity guarantees unimodality [12]). 2.2.
Analysis of the optimality conditions
To derive the optimality conditions, we make two assumptions: i) the random term in the demand has an increasing failure rate (IFR); ii) the demand function satisfies certain concavity conditions. Assumption i) the distribution of ϵ has an increasing failure rate (IFR), that is, f (·)/F¯ (·) is an increasing function. Assumption ii) the functions µ(p) and σ(p) are twice continuously differentiable and strictly decreasing in price p. In addition, pµ(p) and pσ(p) are concave in p. Most distributions commonly used in the operations management literature, such as normal, gamma, uniform and logistic distributions have IFR; see [3, 10, 11, 15]. As our analysis unfolds, we show that the IFR distribution assumption leads to a new set of optimality conditions for the unimodality of the profit function. The assumption of the demand to be decreasing in price is standard. The concavity assumption implies that the revenue function is concave in price, e.g., marginal revenue is decreasing in retail price. It is also a widely adopted assumption in the operations management literature [10, 18]. To facilitate analysis, we rewrite the profit function as π(y, p) = pE min{y, µ(p) + σ(p)ϵ} − cy = pσ(p)E min{y/σ(p), κ(p) + ϵ} − cy = pσ(p)S(y, p) − cy,
(3)
5
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
where κ(p) = µ(p)/σ(p) and S(y, p) = E min{y/σ(p), κ(p) + ϵ}. For ease of exposition, define Θ(x) = ∫∞ (u − x)f (u)du. The profit function can then be written as x { } π(y, p) = pσ(p) κ(p) + [˜ µ − Θ(y/σ(p) − κ(p))] − cy.
Note that [˜ µ − Θ(y/σ(p) − κ(p))] =
∫ y/σ(p)−κ(p) 0
(4)
F¯ (t)dt ≥ 0. Since ϵ is a nonnegative random variable,
for any given p and y, we have y ≥ µ(p). Thus, y/σ(p) − κ(p) ≥ 0. Define V (z) =
∫z
0 tf (t)dt [˜ µ−Θ(z)]
and U (z) =
F (z) . [˜ µ−Θ(z)]
To establish the optimality conditions for the profit
function, we need the monotone property of V (z) and U (z) stated in the following Lemma. Lemma 1 (Monotone Property). Under Assumptions i) and ii), both V (z) and U (z) are nondecreasing functions of z and 0 ≤ V (z) < 1. Proof of Lemma 1.
Under Assumption i), the distribution of ϵ has an IFR, so it also has
an increasing generalized failure rate (IGFR). It follows that V (z) is nondecreasing in z, which is proved in [15]. Since V ′ (z) ≥ 0 and limz→∞ V (z) =
∫∞ tf (t)dt ∫0∞ ¯ 0 F (t)dt
= 1, we have 0 ≤ V (z) < 1.
Next we prove the monotone property of U (z). Taking the first derivative of U (z), we have f (z)[˜ µ − Θ(z)] − F (z)F¯ (z) [˜ µ − Θ(z)]2 ∫z f (z) 0 F¯ (t)dt − F (z)F¯ (z) = [˜ µ − Θ(z)]2 ∫z h(z) F¯ (t)dt − F (z) 0 = [F¯ (z)] . [˜ µ − Θ(z)]2
U ′ (z) =
Define △(z) = h(z)
∫z 0
F¯ (t)dt − F (z). We have limz→0 △(z) = 0. Further △′ (z) = h′ (z)
(5) ∫z 0
F¯ (t)dt ≥ 0,
and therefore, when the distribution of ϵ has an IFR, we have U ′ (z) ≥ 0. This completes the proof. We are now ready to present the optimality conditions. Theorem 1 (Optimality Conditions). Under Assumptions i) and ii), there exists a unique maximizer (ˆ y , pˆ) which maximizes the firm’s expected profit function. Furthermore, the optimal price pˆ(y) for any given order quantity y is decreasing in y. Proof of Theorem 1.
We follow [17] to first consider the choice of p(y) to maximize π(y, p)
for a fixed y. We then show that π(y, p(y)) is concave in y.
6
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
For a fixed y, take the first derivative of the profit function π(y, p) with respect to p to obtain ∂π(y, p) = pσ ′ (p) ∂p
∫
∫
y/σ(p)−κ(p)
y/σ(p)−κ(p)
[κ(p) + t]f (t)dt + κ′ (p)f (t)dt + σ(p)S(y, p) 0 0 { ∫ y/σ(p)−κ(p) tf (t)dt F (y/σ(p) − κ(p)) ′ ′ = [˜ µ − Θ(y/σ(p) − κ(p))] pµ (p) + pσ (p) 0 µ ˜ − Θ(y/σ(p) − κ(p)) µ ˜ − Θ(y/σ(p) − κ(p)) } µ(p) + + σ(p) . (6) µ ˜ − Θ(y/σ(p) − κ(p))
The first equality follows from S(y, p) = F¯ (y/σ(p) − κ(p))y/σ(p) + ′
′
(p) (p) second equality follows from κ′ (p) = κ(p)[ µµ(p) − σσ(p) ].
∫ y/σ(p)−κ(p) 0
[κ(p) + t]f (t)dt. The
To facilitate the exposition, we define z(y, p) = y/σ(p) − κ(p) ≥ 0. Since µ ˜ − Θ(y/σ(p) − κ(p)) ̸= 0, the first-order condition of π(y, p) with respect to p is: µ(p) + σ(p) µ ˜ − Θ(z(y, p)) µ(p)F¯ (z(y, p)) = [pµ′ (p) + µ(p)]U (z(y, p)) + [pσ ′ (p) + σ(p)]V (z(y, p)) + + σ(p)[1 − V (z(y, p))] µ ˜ − Θ(z(y, p))
W (p, y) = pµ′ (p)U (z(y, p)) + pσ ′ (p)V (z(y, p)) +
(7)
= 0.
To further simplify the notation, let z = z(y, p). Then, [ ′ ] dz(y, p) 1 −yσ ′ (p) dp µ (p) σ ′ (p) dp = + 2 − κ(p) − dy σ(p) σ (p) dy µ(p) σ(p) dy [ ′ ] σ (p) µ′ (p) dp 1 − z+ . = σ(p) σ(p) σ(p) dy
(8)
By taking derivative of W (p, y) with respect to y, we see that the optimal price must satisfy the following equation: } dp µ′ (p)F¯ (z) ′ + σ (p)[1 − V (z)] [pµ (p) + 2µ (p)]U (z) + [pσ (p) + 2σ (p)]V (z) + µ ˜ − Θ(z) dy { } { } ′ ′ ¯ µ(p)F (z) 1 σ (p) µ (p) dp + pµ′ (p)U ′ (z) + pσ ′ (p)V ′ (z) − −[ z+ ] · = 0. [˜ µ − Θ(z)]2 σ(p) σ(p) σ(p) dy
{
′′
′
′′
′
(9)
According to Lemma 1, U ′ (z) ≥ 0, V ′ (z) ≥ 0, and 1 − V (z) ≥ 0. Moreover, under Assumption [ ′ ] ′ (p) (p) dp ii), µ′ (p) ≤ 0, and thus σσ(p) z + µσ(p) ≤ 0. Therefore, we have dy ≤ 0 if pµ′′ (p) + 2µ′ (p) ≤ 0 and pσ ′′ (p) + 2σ ′ (p) ≤ 0. Furthermore, according to Equation (9), we also have
dz(y,p) dy
≥ 0 if
dp dy
≤ 0.
7
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
By taking the first derivative of π(y, p(y)) with respect to y and using ∂π(y, p(y))/∂p = 0, we obtain ∂π(y, p(y)) ∂π(y, p(y)) dp(y) = + p(y)F¯ (z(y, p(y))) − c ∂y ∂p dy = p(y)F¯ (z(y, p(y))) − c. Note that when y → 0, p(y) → p¯, µ(¯ p) → 0,
∂π(y,p(y)) ∂y
(10)
→ p¯ − c > 0; and when y → ∞, z(y, p(y)) =
y/σ(p(y)) − κ(p(y)) → ∞ because µ(p(y)) and σ(p(y)) are bounded. Thus,
∂π(y,p(y)) ∂y
→ −c < 0.
Now we consider ∂ 2 π(y, p(y)) dp(y) dz(y, p(y)) = F¯ (z(y, p(y))) − p(y)f (z(y, p(y))) . 2 ∂y dy dy Since
dp(y) dy
≤ 0 and
dz(y,p(y)) dy
≥ 0, we have
∂ 2 π(y,p(y)) ∂y 2
(11)
≤ 0. Therefore, π(y, p(y)) is concave in y. Thus,
there is a unique root for p(y)F¯ (z(y, p(y))) − c = 0. This completes the proof. The examples below illustrate the demand functions that satisfy Assumption ii). As an important observation from these examples, the unimodality holds when the demand function is highly nonlinear and convex in response to price changes. Examples satisfying Assumption ii): D(p, ϵ) = σ(p)ϵ for p ∈ [p, p] where 1. Linear-power function: σ(p) = (α − βp)γ , α > 0, β > 0 and γ ≤ −1. 2. Exponential function: σ(p) = αe−βp , α > 0, β > 0 and βp ≤ 2. 3. Iso-elastic function: σ(p) = αp−β , α > 0 and 0 < β < 1. From these examples, we see that our result applies to some concave (e.g., linear-power function with γ < −1) and convex (e.g., exponential function, or Iso-elastic function with β > 0) demands commonly studied in the literature. Although we use multiplicative demands in these illustrative examples, our result applies to general additive-multiplicative demands as long as the parameters defining σ(p) are as specified above.
3.
Comparison with Existing Results
In this section, we compare our optimality conditions with the existing results. By discussing the similarities and differences, we show that our optimality conditions complement the existing ones.
8
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
From a practical perspective, decision makers are often interested in using the price elasticity of demand to facilitate their decisions. In the literature of the price-setting newsvendor, the optimality conditions are also often related to the price elasticity of demand [16, 14]. Definition 1 (Price Elasticities of Demand). The price elasticities of µ(p) and σ(p) are defined, respectively, as δµ (p) = −pµ′ (p)/µ(p) and δσ (p) = −pσ ′ (p)/σ(p). [10] studies a similar price-setting newsvendor as ours. To establish the unimodality of the profit function, it assumes that i) the random term ϵ in the demand function D(p, ϵ) exhibits IFR; ii) the revenue function pD(p, ϵ) is concave in p for any realization of ϵ. Besides these assumptions, it also introduces the concept of the lost-sale elasticity ε(p, x) =
pHp (p,x) , 1−H(p,x)
where H(p, x) is the probability
of no lost sale and Hp (p, x) is its first derivative with respect to price p. The unimodality of the profit function and the optimality conditions are established under assumptions i), ii) and the values of the lost-sale elasticity. In particular, when ε(p, x) ≥ 0.5, the profit function is jointly concave in price and order quantity. Further, it defines ε∗ (x) = ε(p∗ (x), x), ∀x and ε∗ (p) = ε(p, x∗ (p)), ∀p, where p∗ (x) is the optimal price p for a given inventory level x and x∗ (p) is the optimal inventory level for a given price p. They show that when ε∗ (x) ≥ 0.5, ∀x, or ε∗ (p) ≥ 0.5, ∀p, the newsvendor pricing model has a unique optimal solution. Note that under the same demand model with the nonnegative random term ϵ, the revenue function pD(p, ϵ) is concave in p for any realization of ϵ if both pµ(p) and pσ(p) are concave in p. Thus, our optimality conditions imply assumption i) of [10]. Since the condition ε(p, x) ≥ 0.5, depends on the inventory level, it is not comparable to our conditions. Moreover, there are cases that violate the conditions ε∗ (x) ≥ 0.5, ∀x and ε∗ (p) ≥ 0.5, ∀p, whereas our coditions hold regardless. We show this by an example. Example Consider the price-dependent demand D(p, ϵ) = σ(p)ϵ for p ∈ [p, p], where ϵ has the uniform distribution with support [a,b]. It can be shown that the lost-sale elasticity ε(p, x) = −
1 pσ ′ (p)x f (x/σ(p)) x · . = δσ (p) · 2 ¯ [σ(p)] F (x/σ(p)) σ(p) b − x/σ(p)
9
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
The profit function π(x, p) = pσ(p)E min{x/σ(p), ϵ} − cx = pσ(p)
[2b − l(x, p)]l(x, p) − a(2b − a) − cx, 2(b − a)
where l(x, p) = x/σ(p). The optimal order quantity x = σ(p)[(b − a)(1 − c/p) + a]. Thus, we have [
] bp ε (p) = δσ (p) · −1 . (b − a)c ∗
On the other hand, for a given order quantity x, the optimal price p satisfies a[δσ (p) − 1](2b − a) + 2b l(x, p) − [1 + δσ (p)]l2 (x, p) = 0.
(12)
Once the optimal l(x, p) is obtained from the above equation, we can calculate the lost-sale elasticity ε∗ (x) = δσ (p) ·
l(x, p) . b − l(x, p)
(13)
Now, consider the iso-elastic demand with σ(p) = 100p−0.25 . Clearly, pσ(p) is concave and δσ (p) = 0.25. Assume a=0.2 and b=1.8. When p ≤ 2.6c, ε∗ (p) ≤ 0.5. On the other hand, for any given x, from Equations (12) and (13), we have ε∗ (x) = 0.02, implying ε∗ (x) ≤ 0.5. Therefore, this example violates the optimalily conditions in [10]. However, the example satisfies our optimality conditions. [12] identifies a set of new optimal conditions based on the log-concavity of the profit function. Following are its optimality conditions: i) (p − c)µ(p) is log-concave, ii) the coefficient of variation σ(p)/µ(p) is log-convex ∫ F −1 (1− pc ) p iii) w(p) = − p−c xf (x)dx is log-convex in p. ∞
The first condition requires that the riskless profit function is log-concave. Since a concave function is log-concave, the concavity of pµ(p) implies that (p − c)µ(p) is log-concave. Moreover, [12] points out that if µ(p) is log-concave, then µ(p) has an increasing price elasticity (IPE). For the second
10
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
condition, from [14], we know that it implies δµ′ (p) ≥ δσ′ (p) ≥ 0, whenever δµ (p) ≥ δσ (p) (assumption iii in [14]). Thus, it is a slightly stronger condition. Moreover, combining conditions i) and ii), [12] shows that σ(p) needs to be log-convex if µ(p) is log-concave. Thus, it loses some generality. For the third condition, [12] demonstrates that it can be satisfied by many commonly used distributions, such as normal, uniform, logistic and gamma. Comparing our optimality conditions with those established in [12], we see that the two sets of conditions do not imply each other. Indeed, they complement each other. [14] illustrates a relationship between the price elasticity of demand and the unimodality of the profit function. It identifies the following optimality conditions: i) ϵ has IFR distribution, and both µ(p) and σ(p) are decreasing in p, ii) both µ(p) and σ(p) are IPE, i.e., δµ′ (p) ≥ 0 and δσ′ (p) ≥ 0, iii) δµ′ (p) ≥ δσ′ (p) ≥ 0 whenever δµ (p) ≥ δσ (p). Clearly, our concavity conditions for pµ(p) and pσ(p) are different from the above IPE conditions. Note that the first derivative of the price elasticity can be written as δµ′ (p) = −
} 1 { ′′ pµ (p) + 2µ′ (p) + µ′ (p)[δµ (p) − 1] . µ(p)
Clearly, if the demand function is price inelastic, i.e., δµ (p) ≤ 1, the IPE condition δµ′ (p) ≥ 0 implies the concavity of the demand function, i.e., pµ′′ (p) + 2µ′ (p) ≤ 0. Thus, the concavity condition is weaker than the IPE condition. On the other hand, if the demand function is price elastic, i.e., δµ (p) ≥ 1, the concavity condition implies that δµ′ (p) ≥ 0. In other words, the IPE condition is weaker. In the following Theorem we generalize our optimality conditions using the IPE properties of demand functions. Theorem 2 (Optimality Conditions with IPE Demand). Under Assumptions i) and ii), there exists a unique (ˆ y , pˆ) which maximizes the profit. Moreover, the optimal price pˆ(y) for a given y is decreasing in y, if i) δµ (p) ≥ δσ (p) > 1, σ(p) is IPE, and pµ(p) is concave in p; or
11
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
ii) δσ (p) ≥ δµ (p) > 1, µ(p) is IPE, and pσ(p) is concave in p. We use a proof similar to that of Theorem 1. It is sufficient to show
Proof of Theorem 2. that
dz(y,p) dy
≥ 0 and
(7). Let m(p) =
≤ 0. Consider the first-order condition W (p, y) = 0 as defined in Equation
dp dy
pµ′ (p) . σ(p)
Then the first-order condition can be written as
W (p, y)/σ(p) =
{
} κ(p) m(p)U (z(y, p)) − δσ (p)V (z(y, p)) + + 1 = 0. µ ˜ − Θ(z(y, p))
(14)
Similarly, by taking the first derivative with respect to y, we have { } κ′ (p) dp ′ ′ m (p)U (z) − δσ (p)V (z) + . µ ˜ − Θ(z) dy } { { } κ(p)F¯ (z) 1 σ ′ (p) µ′ (p) dp · + m(p)U ′ (z) − δσ (p)V ′ (z) − − [ z + ] = 0. [˜ µ − Θ(z)]2 σ(p) σ(p) σ(p) dy
(15)
dp Since m(p) ≤ 0, U ′ (z) ≥ 0, V ′ (z) ≥ 0, if δσ′ (p) ≥ 0, m′ (p) ≤ 0 and κ′ (p) ≤ 0, we have dy ≤ 0 and [ ′ ] ′ (p) dz(y,p) (p) ≥ 0. When κ′ (p) = κ(p) µµ(p) − σσ(p) ≤ 0, it implies δµ (p) ≥ δσ (p). On the other hand, we dy
have
m′ (p) =
pµ′′ (p) + 2µ′ (p) + µ′ (p)(δσ (p) − 1) . σ(p)
Therefore, when δµ (p) ≥ δσ (p) > 1 and pµ(p) is concave in p, we have m′ (p) ≤ 0 and κ′ (p) ≤ 0. This proves part i). Similarly, let n(p) = W (p, y)/µ(p) =
pσ ′ (p) . µ(p)
{
The the first-order condition W (p, y) = 0 can be written as
} 1 + 1/κ(p) = 0. −δµ (p)U (z(y, p)) + n(p)V (z(y, p)) + µ ˜ − Θ(z(y, p))
(16)
It can be shown that when κ′ (p) ≥ 0 and n′ (p) ≤ 0, i.e. when δσ (p) ≥ δµ (p) ≥ 1 and pσ(p) is concave in p, we have
dz(y,p) dy
≥ 0 and
dp dy
≤ 0. This proves part ii).
Comparing our results with those in [14], we see that although our optimality conditions require the demand functions to be price elastic, our results complement findings in [14] in two aspects. First, we consider both the cases of δµ (p) ≥ δσ (p) and δµ (p) ≤ δσ (p). Second, we do not require δµ′ (p) ≥ δσ′ (p) ≥ 0, which is a stronger condition. It is valuable to point out that if we can further relax the concavity in both conditions to the IPE condition, the results are better. However, we have not been able to obtain such more general conditions.
12
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
[17] studies the joint inventory and pricing problem using the same demand model as ours. It shows that when ϵ has log-concave distribution and the demand functions satisfy several conditions, i.e., pθ′ (p)+θ(p) ≤ 0 for large p (where θ(p) = µ(p)+σ(p)) and the concavity conditions, there exists a unique optimal solution. We extend its results from log-concave distribution to IFR distributions and relax the conditions on the demand function. Although log-concave distributions have IFR, there are IFR distributions that are not log-concave; see [2]. Therefore, our optimality conditions include a broader range of demand uncertainty distributions. Both [7, 8] use the same demand model to analyze the dynamic pricing and inventory control problem. Although their demand model is similar to ours, they assume the unmet demand to be backordered while we assume it to be lost. Also, the optimality conditions derived in [7, 8] are different from ours. For example, [7] requires the demand function to be concave in price, and [8] needs the second derivatives of the two demand functions discussed there to satisfy some complex conditions; see Theorem 3 in [8]. Thus, their analysis is different from ours and their results do not imply ours.
Acknowledgement Sirong Luo’s research is supported by National Natural Science Foundation of China [NSFC71471107] and Shanghai Pujiang Program (12PJC051), and in part by the State Key Program in the Major Research Plan of National Natural Science Foundation of China [grant number 91546202] and Program for Changjiang Scholars and Innovative Research Team in SUFE [PCSIRT-IRT13077].
References [1] V. Agrawal and S. Seshadri. Impact of uncertainty and risk aversion on price and order quantity in the newsvendor problem. Manufacturing & Service Operations Management, 2: 410–423, 2000. [2] M. Bagnoli and T. Bergstrom. Log-concave probability and its applications. Economic Theory, 26(2):445–469, 2005. [3] M. Banciu and P. Mirchandani. Technical note - new results concerning probability distributions with increasing generalized failure rate. Operations Research, 61:925–931, 2013.
13
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
[4] L. M. A. Chan, Z. J. M. Shen, D. Simchi-Levi, and J. Swann. Coordination of pricing and inventory decisions: A survey and classification. In S. D. Wu and Z. J. M. Shen, editors, Handbook of Quantitative Supply Chain Analysis: Modeling in the E-Business Era., pages 335–392. Kluwer Academic Publishers., 2005. [5] X. Chen and D. Simchi-Levi. Coordinating inventory control and pricing strategies with random demand and fixed ordering cost: the finite horizon case. Operations Research, 52(6):887– 896, 2004. [6] W. Elmaghraby and P. Keskinocak. Dynamic pricing in the presence of inventory considerations: research overview, current practices, and future directions. Management Science, 49:1287–1309, 2003. [7] A. Federgruen and A. Heching. Combining pricing and inventory control under uncertainty. Operations Research, 47:454–475, 1999. [8] Q. Feng, S. Luo, and D. Zhang. Dynamic inventory-pricing control under backorder: Demand estimation and policy optimization.
Manufacturing & Service Operations Management,
16:149–160, 2014. [9] L. Gimpl-Heersin, C. Rudlof, M. Fleischmann, and A. Taudes. Integrating pricing and inventory control: Is it worth the effort? Business Research, 1(1):106–123, 2008. [10] A. Kocabıyıkoˇglu and I. Popescu. An elasticity approach to the newsvendor with price-sensitive demand. Operations Research, 59(2):301–312, 2011. [11] M. A. Lariviere. A note on probability distributions with increasing generalized failure rate. Operations Research, 54:602–604, 2006. [12] Y. Lu and D. Simchi-Levi. On the unimodality of the profit function of the pricing newsvendor. Production and Operations Management, 22:615–625, 2013. [13] N. Petruzzi and C. M. Dada. Pricing and the newsvendor problem: A review with extensions. Operations Research, 47:183–194, 1999. [14] G. Roels. Risk premiums in the price-setting newsvendor model. Working paper, UCLA
14
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
Anderson School of Management, Los Angeles, CA., 2010. [15] Y. Song, S. Ray, and T. Boyaci. Optimal dynamic joint inventory-pricing control for multiplicative demand with fixed order costs and lost sales. Operations Research, 57:245–250, 2009. [16] L. Yao, Y. F. Chen, and H. Yan. The newsvendor problem with pricing. International Journal of Management Science Engineering Management, 1(1):3–16, 2006. [17] L. Young. Price, inventory and the structure of uncertain demand. New Zealand Operations Research, 6:157–177, 1978. [18] S. Ziya, H. Ayhan, and R. D. Foley. Relationships among three assumptions in revenue management. Operations Research, 52:804–809, 2004.
S0167-6377(16)30083-9 http://dx.doi.org/10.1016/j.orl.2016.08.005 OPERES 6136
To appear in:
Operations Research Letters
Received date: 7 June 2016 Revised date: 22 August 2016 Accepted date: 23 August 2016 Please cite this article as: S. Luo, S.P. Sethi, R. Shi, On the optimality conditions of a price-setting newsvendor problem, Operations Research Letters (2016), http://dx.doi.org/10.1016/j.orl.2016.08.005 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
*Manuscript Click here to view linked References
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
On the Optimality Conditions of a Price-Setting Newsvendor Problem Sirong Luo School of Statistics and Management, Shanghai University of Finance and Economics, Shanghai, 200433, China. [email protected]
Suresh P. Sethi Naveen Jindal School of Management, The University of Texas at Dallas, Richardson, TX, 75083, USA. [email protected]
Ruixia Shi* School of Business, University of San Diego, San Diego, CA, 92110, USA. [email protected]
We analyze a price-setting newsvendor problem with an additive-multiplicative demand. We show that the unimodality of the newsvendor profit function holds when the underlying random term has an increasing failure rate and the demand functions satisfy certain concavity conditions. Further, we show that the optimal price decreases in the order quantity. Finally, we compare our optimality conditions with those existing in the literature. Key words : Inventory Control, Price-Setting Newsvendor, Unimodality, Elasticity
1.
Introduction
Integrating pricing and inventory replenishment decisions under demand uncertainty has proved to be a successful operations strategy of many firms including Amazon, Dell, Walmart and J. C. Penney [4, 6]. By adopting proactive pricing, firms can better match supply with demand, which leads to significant profit increases. The benefits of jointly deciding pricing and inventory replenishment level have also been well documented in various academic studies [5, 8, 9]. The building block for joint inventory and pricing decisions research is the newsvendor model with pricing. The major difficulty in studying this problem is to establish concavity or unimodality of the profit function. Although many analytical results on the optimality conditions have been * Corresponding author. Email: [email protected]. Address: School of Business, University of San Diego, San Diego, CA, 92110, USA. 1
2
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
developed in recent years [10, 12, 13, 17], the existing literature often makes strong assumptions on the demand function forms and demand uncertainty distributions, such as one parameter additive or multiplicative only demand model and specific uncertainty distributions. These assumptions simplify analysis, but their limitations in capturing reality limit their applicability in practice. To address this issue, we use a general additive-multiplicative demand model to analyze the problem of joint inventory and pricing decisions. To derive the optimality conditions, we make two assumptions: 1) the random term in the demand has an increasing failure rate (IFR); 2) the demand function satisfies certain concavity conditions. [12] studies the same problem by first solving for the optimal order quantity and then finding the optimal price. It identifies three conditions to be met for establishing the optimality of the profit function: 1)the riskless profit function is log-concave; 2) the coefficient of variation is log-convex and 3) the distribution of the random term satisfies a specific condition. [10] introduces the concept of lost-sale elasticity. Assuming the random term has IFR distribution, it shows that when the lost-sale elasticity satisfies certain conditions, the concavity or unimodality of the profit function is implied. [14] conducts a similar analysis of the problem from the price elasticity view point. It shows that when both price elasticities of the location and scale parameters in demand are increasing in price, and the elasticity of the location parameter increases faster than the price elasticity of the scale parameter, the unimodality of the profit function is obtained. [1] analyzes a risk-averse price-dependent newsvendor and shows the concavity of the profit functions for additive and multiplicative demands. A focus of the paper is to elaborate the difference in pricing and ordering behaviors of the risk averse and risk neutral newsvendors. [17] obtains the unimodality when the random term has log-concave distribution and the demand functions satisfy certain conditions. Compared with the aforementioned studies, our proof is new and the resulting optimality conditions are different. As our analysis unfolds, we show that our optimality conditions and the conditions obtained in the existing literature do not imply each other. In fact, our result complements the existing results. The rest of this paper is organized as follows. Section §2 presents the newsvendor pricing model and our analytical results. We discuss our results by comparing them with the existing literature
3
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
in Section §3.
2.
Model and Analysis
In this section, we present the price-setting newsvendor model and derive conditions for the optimality of the firm’s expected profit function. 2.1.
Model
A risk neutral firm buys a product at a unit cost c and sells the product to customers at a retail price p ∈ [p, p] over a single selling season. The demand during the selling season depends on the retail price p and is random. Let D(p, ϵ) denote the price-dependent demand, where ϵ is a random variable. The firm simultaneously decides the retail price p and the order quantity y at the beginning of the selling season before observing the demand. After the demand materializes, the firm satisfies the demand with the product’s available stock. If the firm does not have enough stock, that is, y ≤ D(p, ϵ), the unsatisfied demand is lost with no penalty for lost sales. The unsold inventory, if any, is salvaged at zero value. Note that our analysis can be extended easily to the cases of nonzero penalty and nonzero salvage value. As in [7, 8, 12, 17], we consider the additive-multiplicative demand model D(p, ϵ) = µ(p) + σ(p)ϵ,
(1)
where µ(·) ≥ 0 and σ(·) ≥ 0 are deterministic functions of price p, and ϵ is a nonnegative random variable with mean µ ˜ and standard deviation σ ˜ . This demand model generalizes the widely used multiplicative or additive only demand models. In our generalized model, the retail price can have different effects on the location and scale parameters of the demand. Whereas, in the multiplicative only models, the price has the same effects on the demand mean and standard deviation, i.e., the coefficient of variation is independent of price. Let f (·) and F (·) denote the density and cumulative distribution functions of ϵ, respectively. In addition, let F¯ (·) = 1 − F (·). We define G(·|p) as the conditional distribution of D(p, ϵ) for a given ) ( . The failure rate of ϵ is defined as h(·) = f (·)/F¯ (·). retail price p. Thus, G(x|p) = F x−µ(p) σ(p)
4
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
For a given retail price p and order quantity y, the firm’s expected profit (profit, hereafter) is π(y, p) = p E min{y, µ(p) + σ(p)ϵ} − cy.
(2)
The firm determines y ∈ [0, ∞) and p ∈ [p, p] to maximize its profit, that is maxy,p π(y, p) = p E min{y, µ(p) + σ(p)ϵ} − cy. Unfortunately, the concavity of the profit function does not hold under general conditions. Thus, researchers have tried to establish conditions under which the profit function is quasi-concave (i.e., unimodal) or log-concave (since the log-concavity guarantees unimodality [12]). 2.2.
Analysis of the optimality conditions
To derive the optimality conditions, we make two assumptions: i) the random term in the demand has an increasing failure rate (IFR); ii) the demand function satisfies certain concavity conditions. Assumption i) the distribution of ϵ has an increasing failure rate (IFR), that is, f (·)/F¯ (·) is an increasing function. Assumption ii) the functions µ(p) and σ(p) are twice continuously differentiable and strictly decreasing in price p. In addition, pµ(p) and pσ(p) are concave in p. Most distributions commonly used in the operations management literature, such as normal, gamma, uniform and logistic distributions have IFR; see [3, 10, 11, 15]. As our analysis unfolds, we show that the IFR distribution assumption leads to a new set of optimality conditions for the unimodality of the profit function. The assumption of the demand to be decreasing in price is standard. The concavity assumption implies that the revenue function is concave in price, e.g., marginal revenue is decreasing in retail price. It is also a widely adopted assumption in the operations management literature [10, 18]. To facilitate analysis, we rewrite the profit function as π(y, p) = pE min{y, µ(p) + σ(p)ϵ} − cy = pσ(p)E min{y/σ(p), κ(p) + ϵ} − cy = pσ(p)S(y, p) − cy,
(3)
5
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
where κ(p) = µ(p)/σ(p) and S(y, p) = E min{y/σ(p), κ(p) + ϵ}. For ease of exposition, define Θ(x) = ∫∞ (u − x)f (u)du. The profit function can then be written as x { } π(y, p) = pσ(p) κ(p) + [˜ µ − Θ(y/σ(p) − κ(p))] − cy.
Note that [˜ µ − Θ(y/σ(p) − κ(p))] =
∫ y/σ(p)−κ(p) 0
(4)
F¯ (t)dt ≥ 0. Since ϵ is a nonnegative random variable,
for any given p and y, we have y ≥ µ(p). Thus, y/σ(p) − κ(p) ≥ 0. Define V (z) =
∫z
0 tf (t)dt [˜ µ−Θ(z)]
and U (z) =
F (z) . [˜ µ−Θ(z)]
To establish the optimality conditions for the profit
function, we need the monotone property of V (z) and U (z) stated in the following Lemma. Lemma 1 (Monotone Property). Under Assumptions i) and ii), both V (z) and U (z) are nondecreasing functions of z and 0 ≤ V (z) < 1. Proof of Lemma 1.
Under Assumption i), the distribution of ϵ has an IFR, so it also has
an increasing generalized failure rate (IGFR). It follows that V (z) is nondecreasing in z, which is proved in [15]. Since V ′ (z) ≥ 0 and limz→∞ V (z) =
∫∞ tf (t)dt ∫0∞ ¯ 0 F (t)dt
= 1, we have 0 ≤ V (z) < 1.
Next we prove the monotone property of U (z). Taking the first derivative of U (z), we have f (z)[˜ µ − Θ(z)] − F (z)F¯ (z) [˜ µ − Θ(z)]2 ∫z f (z) 0 F¯ (t)dt − F (z)F¯ (z) = [˜ µ − Θ(z)]2 ∫z h(z) F¯ (t)dt − F (z) 0 = [F¯ (z)] . [˜ µ − Θ(z)]2
U ′ (z) =
Define △(z) = h(z)
∫z 0
F¯ (t)dt − F (z). We have limz→0 △(z) = 0. Further △′ (z) = h′ (z)
(5) ∫z 0
F¯ (t)dt ≥ 0,
and therefore, when the distribution of ϵ has an IFR, we have U ′ (z) ≥ 0. This completes the proof. We are now ready to present the optimality conditions. Theorem 1 (Optimality Conditions). Under Assumptions i) and ii), there exists a unique maximizer (ˆ y , pˆ) which maximizes the firm’s expected profit function. Furthermore, the optimal price pˆ(y) for any given order quantity y is decreasing in y. Proof of Theorem 1.
We follow [17] to first consider the choice of p(y) to maximize π(y, p)
for a fixed y. We then show that π(y, p(y)) is concave in y.
6
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
For a fixed y, take the first derivative of the profit function π(y, p) with respect to p to obtain ∂π(y, p) = pσ ′ (p) ∂p
∫
∫
y/σ(p)−κ(p)
y/σ(p)−κ(p)
[κ(p) + t]f (t)dt + κ′ (p)f (t)dt + σ(p)S(y, p) 0 0 { ∫ y/σ(p)−κ(p) tf (t)dt F (y/σ(p) − κ(p)) ′ ′ = [˜ µ − Θ(y/σ(p) − κ(p))] pµ (p) + pσ (p) 0 µ ˜ − Θ(y/σ(p) − κ(p)) µ ˜ − Θ(y/σ(p) − κ(p)) } µ(p) + + σ(p) . (6) µ ˜ − Θ(y/σ(p) − κ(p))
The first equality follows from S(y, p) = F¯ (y/σ(p) − κ(p))y/σ(p) + ′
′
(p) (p) second equality follows from κ′ (p) = κ(p)[ µµ(p) − σσ(p) ].
∫ y/σ(p)−κ(p) 0
[κ(p) + t]f (t)dt. The
To facilitate the exposition, we define z(y, p) = y/σ(p) − κ(p) ≥ 0. Since µ ˜ − Θ(y/σ(p) − κ(p)) ̸= 0, the first-order condition of π(y, p) with respect to p is: µ(p) + σ(p) µ ˜ − Θ(z(y, p)) µ(p)F¯ (z(y, p)) = [pµ′ (p) + µ(p)]U (z(y, p)) + [pσ ′ (p) + σ(p)]V (z(y, p)) + + σ(p)[1 − V (z(y, p))] µ ˜ − Θ(z(y, p))
W (p, y) = pµ′ (p)U (z(y, p)) + pσ ′ (p)V (z(y, p)) +
(7)
= 0.
To further simplify the notation, let z = z(y, p). Then, [ ′ ] dz(y, p) 1 −yσ ′ (p) dp µ (p) σ ′ (p) dp = + 2 − κ(p) − dy σ(p) σ (p) dy µ(p) σ(p) dy [ ′ ] σ (p) µ′ (p) dp 1 − z+ . = σ(p) σ(p) σ(p) dy
(8)
By taking derivative of W (p, y) with respect to y, we see that the optimal price must satisfy the following equation: } dp µ′ (p)F¯ (z) ′ + σ (p)[1 − V (z)] [pµ (p) + 2µ (p)]U (z) + [pσ (p) + 2σ (p)]V (z) + µ ˜ − Θ(z) dy { } { } ′ ′ ¯ µ(p)F (z) 1 σ (p) µ (p) dp + pµ′ (p)U ′ (z) + pσ ′ (p)V ′ (z) − −[ z+ ] · = 0. [˜ µ − Θ(z)]2 σ(p) σ(p) σ(p) dy
{
′′
′
′′
′
(9)
According to Lemma 1, U ′ (z) ≥ 0, V ′ (z) ≥ 0, and 1 − V (z) ≥ 0. Moreover, under Assumption [ ′ ] ′ (p) (p) dp ii), µ′ (p) ≤ 0, and thus σσ(p) z + µσ(p) ≤ 0. Therefore, we have dy ≤ 0 if pµ′′ (p) + 2µ′ (p) ≤ 0 and pσ ′′ (p) + 2σ ′ (p) ≤ 0. Furthermore, according to Equation (9), we also have
dz(y,p) dy
≥ 0 if
dp dy
≤ 0.
7
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
By taking the first derivative of π(y, p(y)) with respect to y and using ∂π(y, p(y))/∂p = 0, we obtain ∂π(y, p(y)) ∂π(y, p(y)) dp(y) = + p(y)F¯ (z(y, p(y))) − c ∂y ∂p dy = p(y)F¯ (z(y, p(y))) − c. Note that when y → 0, p(y) → p¯, µ(¯ p) → 0,
∂π(y,p(y)) ∂y
(10)
→ p¯ − c > 0; and when y → ∞, z(y, p(y)) =
y/σ(p(y)) − κ(p(y)) → ∞ because µ(p(y)) and σ(p(y)) are bounded. Thus,
∂π(y,p(y)) ∂y
→ −c < 0.
Now we consider ∂ 2 π(y, p(y)) dp(y) dz(y, p(y)) = F¯ (z(y, p(y))) − p(y)f (z(y, p(y))) . 2 ∂y dy dy Since
dp(y) dy
≤ 0 and
dz(y,p(y)) dy
≥ 0, we have
∂ 2 π(y,p(y)) ∂y 2
(11)
≤ 0. Therefore, π(y, p(y)) is concave in y. Thus,
there is a unique root for p(y)F¯ (z(y, p(y))) − c = 0. This completes the proof. The examples below illustrate the demand functions that satisfy Assumption ii). As an important observation from these examples, the unimodality holds when the demand function is highly nonlinear and convex in response to price changes. Examples satisfying Assumption ii): D(p, ϵ) = σ(p)ϵ for p ∈ [p, p] where 1. Linear-power function: σ(p) = (α − βp)γ , α > 0, β > 0 and γ ≤ −1. 2. Exponential function: σ(p) = αe−βp , α > 0, β > 0 and βp ≤ 2. 3. Iso-elastic function: σ(p) = αp−β , α > 0 and 0 < β < 1. From these examples, we see that our result applies to some concave (e.g., linear-power function with γ < −1) and convex (e.g., exponential function, or Iso-elastic function with β > 0) demands commonly studied in the literature. Although we use multiplicative demands in these illustrative examples, our result applies to general additive-multiplicative demands as long as the parameters defining σ(p) are as specified above.
3.
Comparison with Existing Results
In this section, we compare our optimality conditions with the existing results. By discussing the similarities and differences, we show that our optimality conditions complement the existing ones.
8
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
From a practical perspective, decision makers are often interested in using the price elasticity of demand to facilitate their decisions. In the literature of the price-setting newsvendor, the optimality conditions are also often related to the price elasticity of demand [16, 14]. Definition 1 (Price Elasticities of Demand). The price elasticities of µ(p) and σ(p) are defined, respectively, as δµ (p) = −pµ′ (p)/µ(p) and δσ (p) = −pσ ′ (p)/σ(p). [10] studies a similar price-setting newsvendor as ours. To establish the unimodality of the profit function, it assumes that i) the random term ϵ in the demand function D(p, ϵ) exhibits IFR; ii) the revenue function pD(p, ϵ) is concave in p for any realization of ϵ. Besides these assumptions, it also introduces the concept of the lost-sale elasticity ε(p, x) =
pHp (p,x) , 1−H(p,x)
where H(p, x) is the probability
of no lost sale and Hp (p, x) is its first derivative with respect to price p. The unimodality of the profit function and the optimality conditions are established under assumptions i), ii) and the values of the lost-sale elasticity. In particular, when ε(p, x) ≥ 0.5, the profit function is jointly concave in price and order quantity. Further, it defines ε∗ (x) = ε(p∗ (x), x), ∀x and ε∗ (p) = ε(p, x∗ (p)), ∀p, where p∗ (x) is the optimal price p for a given inventory level x and x∗ (p) is the optimal inventory level for a given price p. They show that when ε∗ (x) ≥ 0.5, ∀x, or ε∗ (p) ≥ 0.5, ∀p, the newsvendor pricing model has a unique optimal solution. Note that under the same demand model with the nonnegative random term ϵ, the revenue function pD(p, ϵ) is concave in p for any realization of ϵ if both pµ(p) and pσ(p) are concave in p. Thus, our optimality conditions imply assumption i) of [10]. Since the condition ε(p, x) ≥ 0.5, depends on the inventory level, it is not comparable to our conditions. Moreover, there are cases that violate the conditions ε∗ (x) ≥ 0.5, ∀x and ε∗ (p) ≥ 0.5, ∀p, whereas our coditions hold regardless. We show this by an example. Example Consider the price-dependent demand D(p, ϵ) = σ(p)ϵ for p ∈ [p, p], where ϵ has the uniform distribution with support [a,b]. It can be shown that the lost-sale elasticity ε(p, x) = −
1 pσ ′ (p)x f (x/σ(p)) x · . = δσ (p) · 2 ¯ [σ(p)] F (x/σ(p)) σ(p) b − x/σ(p)
9
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
The profit function π(x, p) = pσ(p)E min{x/σ(p), ϵ} − cx = pσ(p)
[2b − l(x, p)]l(x, p) − a(2b − a) − cx, 2(b − a)
where l(x, p) = x/σ(p). The optimal order quantity x = σ(p)[(b − a)(1 − c/p) + a]. Thus, we have [
] bp ε (p) = δσ (p) · −1 . (b − a)c ∗
On the other hand, for a given order quantity x, the optimal price p satisfies a[δσ (p) − 1](2b − a) + 2b l(x, p) − [1 + δσ (p)]l2 (x, p) = 0.
(12)
Once the optimal l(x, p) is obtained from the above equation, we can calculate the lost-sale elasticity ε∗ (x) = δσ (p) ·
l(x, p) . b − l(x, p)
(13)
Now, consider the iso-elastic demand with σ(p) = 100p−0.25 . Clearly, pσ(p) is concave and δσ (p) = 0.25. Assume a=0.2 and b=1.8. When p ≤ 2.6c, ε∗ (p) ≤ 0.5. On the other hand, for any given x, from Equations (12) and (13), we have ε∗ (x) = 0.02, implying ε∗ (x) ≤ 0.5. Therefore, this example violates the optimalily conditions in [10]. However, the example satisfies our optimality conditions. [12] identifies a set of new optimal conditions based on the log-concavity of the profit function. Following are its optimality conditions: i) (p − c)µ(p) is log-concave, ii) the coefficient of variation σ(p)/µ(p) is log-convex ∫ F −1 (1− pc ) p iii) w(p) = − p−c xf (x)dx is log-convex in p. ∞
The first condition requires that the riskless profit function is log-concave. Since a concave function is log-concave, the concavity of pµ(p) implies that (p − c)µ(p) is log-concave. Moreover, [12] points out that if µ(p) is log-concave, then µ(p) has an increasing price elasticity (IPE). For the second
10
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
condition, from [14], we know that it implies δµ′ (p) ≥ δσ′ (p) ≥ 0, whenever δµ (p) ≥ δσ (p) (assumption iii in [14]). Thus, it is a slightly stronger condition. Moreover, combining conditions i) and ii), [12] shows that σ(p) needs to be log-convex if µ(p) is log-concave. Thus, it loses some generality. For the third condition, [12] demonstrates that it can be satisfied by many commonly used distributions, such as normal, uniform, logistic and gamma. Comparing our optimality conditions with those established in [12], we see that the two sets of conditions do not imply each other. Indeed, they complement each other. [14] illustrates a relationship between the price elasticity of demand and the unimodality of the profit function. It identifies the following optimality conditions: i) ϵ has IFR distribution, and both µ(p) and σ(p) are decreasing in p, ii) both µ(p) and σ(p) are IPE, i.e., δµ′ (p) ≥ 0 and δσ′ (p) ≥ 0, iii) δµ′ (p) ≥ δσ′ (p) ≥ 0 whenever δµ (p) ≥ δσ (p). Clearly, our concavity conditions for pµ(p) and pσ(p) are different from the above IPE conditions. Note that the first derivative of the price elasticity can be written as δµ′ (p) = −
} 1 { ′′ pµ (p) + 2µ′ (p) + µ′ (p)[δµ (p) − 1] . µ(p)
Clearly, if the demand function is price inelastic, i.e., δµ (p) ≤ 1, the IPE condition δµ′ (p) ≥ 0 implies the concavity of the demand function, i.e., pµ′′ (p) + 2µ′ (p) ≤ 0. Thus, the concavity condition is weaker than the IPE condition. On the other hand, if the demand function is price elastic, i.e., δµ (p) ≥ 1, the concavity condition implies that δµ′ (p) ≥ 0. In other words, the IPE condition is weaker. In the following Theorem we generalize our optimality conditions using the IPE properties of demand functions. Theorem 2 (Optimality Conditions with IPE Demand). Under Assumptions i) and ii), there exists a unique (ˆ y , pˆ) which maximizes the profit. Moreover, the optimal price pˆ(y) for a given y is decreasing in y, if i) δµ (p) ≥ δσ (p) > 1, σ(p) is IPE, and pµ(p) is concave in p; or
11
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
ii) δσ (p) ≥ δµ (p) > 1, µ(p) is IPE, and pσ(p) is concave in p. We use a proof similar to that of Theorem 1. It is sufficient to show
Proof of Theorem 2. that
dz(y,p) dy
≥ 0 and
(7). Let m(p) =
≤ 0. Consider the first-order condition W (p, y) = 0 as defined in Equation
dp dy
pµ′ (p) . σ(p)
Then the first-order condition can be written as
W (p, y)/σ(p) =
{
} κ(p) m(p)U (z(y, p)) − δσ (p)V (z(y, p)) + + 1 = 0. µ ˜ − Θ(z(y, p))
(14)
Similarly, by taking the first derivative with respect to y, we have { } κ′ (p) dp ′ ′ m (p)U (z) − δσ (p)V (z) + . µ ˜ − Θ(z) dy } { { } κ(p)F¯ (z) 1 σ ′ (p) µ′ (p) dp · + m(p)U ′ (z) − δσ (p)V ′ (z) − − [ z + ] = 0. [˜ µ − Θ(z)]2 σ(p) σ(p) σ(p) dy
(15)
dp Since m(p) ≤ 0, U ′ (z) ≥ 0, V ′ (z) ≥ 0, if δσ′ (p) ≥ 0, m′ (p) ≤ 0 and κ′ (p) ≤ 0, we have dy ≤ 0 and [ ′ ] ′ (p) dz(y,p) (p) ≥ 0. When κ′ (p) = κ(p) µµ(p) − σσ(p) ≤ 0, it implies δµ (p) ≥ δσ (p). On the other hand, we dy
have
m′ (p) =
pµ′′ (p) + 2µ′ (p) + µ′ (p)(δσ (p) − 1) . σ(p)
Therefore, when δµ (p) ≥ δσ (p) > 1 and pµ(p) is concave in p, we have m′ (p) ≤ 0 and κ′ (p) ≤ 0. This proves part i). Similarly, let n(p) = W (p, y)/µ(p) =
pσ ′ (p) . µ(p)
{
The the first-order condition W (p, y) = 0 can be written as
} 1 + 1/κ(p) = 0. −δµ (p)U (z(y, p)) + n(p)V (z(y, p)) + µ ˜ − Θ(z(y, p))
(16)
It can be shown that when κ′ (p) ≥ 0 and n′ (p) ≤ 0, i.e. when δσ (p) ≥ δµ (p) ≥ 1 and pσ(p) is concave in p, we have
dz(y,p) dy
≥ 0 and
dp dy
≤ 0. This proves part ii).
Comparing our results with those in [14], we see that although our optimality conditions require the demand functions to be price elastic, our results complement findings in [14] in two aspects. First, we consider both the cases of δµ (p) ≥ δσ (p) and δµ (p) ≤ δσ (p). Second, we do not require δµ′ (p) ≥ δσ′ (p) ≥ 0, which is a stronger condition. It is valuable to point out that if we can further relax the concavity in both conditions to the IPE condition, the results are better. However, we have not been able to obtain such more general conditions.
12
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
[17] studies the joint inventory and pricing problem using the same demand model as ours. It shows that when ϵ has log-concave distribution and the demand functions satisfy several conditions, i.e., pθ′ (p)+θ(p) ≤ 0 for large p (where θ(p) = µ(p)+σ(p)) and the concavity conditions, there exists a unique optimal solution. We extend its results from log-concave distribution to IFR distributions and relax the conditions on the demand function. Although log-concave distributions have IFR, there are IFR distributions that are not log-concave; see [2]. Therefore, our optimality conditions include a broader range of demand uncertainty distributions. Both [7, 8] use the same demand model to analyze the dynamic pricing and inventory control problem. Although their demand model is similar to ours, they assume the unmet demand to be backordered while we assume it to be lost. Also, the optimality conditions derived in [7, 8] are different from ours. For example, [7] requires the demand function to be concave in price, and [8] needs the second derivatives of the two demand functions discussed there to satisfy some complex conditions; see Theorem 3 in [8]. Thus, their analysis is different from ours and their results do not imply ours.
Acknowledgement Sirong Luo’s research is supported by National Natural Science Foundation of China [NSFC71471107] and Shanghai Pujiang Program (12PJC051), and in part by the State Key Program in the Major Research Plan of National Natural Science Foundation of China [grant number 91546202] and Program for Changjiang Scholars and Innovative Research Team in SUFE [PCSIRT-IRT13077].
References [1] V. Agrawal and S. Seshadri. Impact of uncertainty and risk aversion on price and order quantity in the newsvendor problem. Manufacturing & Service Operations Management, 2: 410–423, 2000. [2] M. Bagnoli and T. Bergstrom. Log-concave probability and its applications. Economic Theory, 26(2):445–469, 2005. [3] M. Banciu and P. Mirchandani. Technical note - new results concerning probability distributions with increasing generalized failure rate. Operations Research, 61:925–931, 2013.
13
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
[4] L. M. A. Chan, Z. J. M. Shen, D. Simchi-Levi, and J. Swann. Coordination of pricing and inventory decisions: A survey and classification. In S. D. Wu and Z. J. M. Shen, editors, Handbook of Quantitative Supply Chain Analysis: Modeling in the E-Business Era., pages 335–392. Kluwer Academic Publishers., 2005. [5] X. Chen and D. Simchi-Levi. Coordinating inventory control and pricing strategies with random demand and fixed ordering cost: the finite horizon case. Operations Research, 52(6):887– 896, 2004. [6] W. Elmaghraby and P. Keskinocak. Dynamic pricing in the presence of inventory considerations: research overview, current practices, and future directions. Management Science, 49:1287–1309, 2003. [7] A. Federgruen and A. Heching. Combining pricing and inventory control under uncertainty. Operations Research, 47:454–475, 1999. [8] Q. Feng, S. Luo, and D. Zhang. Dynamic inventory-pricing control under backorder: Demand estimation and policy optimization.
Manufacturing & Service Operations Management,
16:149–160, 2014. [9] L. Gimpl-Heersin, C. Rudlof, M. Fleischmann, and A. Taudes. Integrating pricing and inventory control: Is it worth the effort? Business Research, 1(1):106–123, 2008. [10] A. Kocabıyıkoˇglu and I. Popescu. An elasticity approach to the newsvendor with price-sensitive demand. Operations Research, 59(2):301–312, 2011. [11] M. A. Lariviere. A note on probability distributions with increasing generalized failure rate. Operations Research, 54:602–604, 2006. [12] Y. Lu and D. Simchi-Levi. On the unimodality of the profit function of the pricing newsvendor. Production and Operations Management, 22:615–625, 2013. [13] N. Petruzzi and C. M. Dada. Pricing and the newsvendor problem: A review with extensions. Operations Research, 47:183–194, 1999. [14] G. Roels. Risk premiums in the price-setting newsvendor model. Working paper, UCLA
14
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
Anderson School of Management, Los Angeles, CA., 2010. [15] Y. Song, S. Ray, and T. Boyaci. Optimal dynamic joint inventory-pricing control for multiplicative demand with fixed order costs and lost sales. Operations Research, 57:245–250, 2009. [16] L. Yao, Y. F. Chen, and H. Yan. The newsvendor problem with pricing. International Journal of Management Science Engineering Management, 1(1):3–16, 2006. [17] L. Young. Price, inventory and the structure of uncertain demand. New Zealand Operations Research, 6:157–177, 1978. [18] S. Ziya, H. Ayhan, and R. D. Foley. Relationships among three assumptions in revenue management. Operations Research, 52:804–809, 2004.