Journal of the Operational Research Society (2015) 66, 1341–1351
© 2015 Operational Research Society Ltd. All rights reserved. 0160-5682/15 www.palgrave-journals.com/jors/
A joint dynamic pricing and advertising model of perishable products Lin Feng1, Jianxiong Zhang1,2* and Wansheng Tang1 1
Institute of Systems Engineering, Tianjin University, Tianjin, China; and 2School of Management, The University of Texas at Dallas, Richardson, TX, USA Advertising and dynamic pricing play key roles in maximizing profit of a firm. In this paper a joint dynamic pricing and advertising problem for perishable products is investigated, where the time-varying demand rate is decreasing in sales price and increasing in goodwill. A dynamic optimization model is proposed to maximize total profit by setting a joint pricing and advertising policy under the constraint of a limited advertising capacity. By solving the dynamic optimization problem on the basis of Pontryagin’s maximum principle, the analytical solutions of the optimal joint dynamic pricing and advertising policy are obtained. Additionally, to highlight the advantage of the joint dynamic strategy, the case of the optimal advertising with static pricing policy is considered. Numerical examples are presented to illustrate the validness of the theoretical results, and some managerial implications for the pricing and advertising of the perishable products are provided. Journal of the Operational Research Society (2015) 66(8), 1341–1351. doi:10.1057/jors.2014.89 Published online 12 November 2014 Keywords: perishable products; advertising; dynamic pricing; pontryagin’s maximum principle
1. Introduction Advertisements have been widely adopted by firms to lure customers and boost sales. There are two types of advertising, that is, generic advertising that expands the market for the entire product category and brand equity (reputation or goodwill) advertising that increases the firm’s market share (Bass et al, 2005). Managers may pay to distribute advertising messages via various mass media such as the well-known print or electronic media. In 2012, spending on advertising was estimated at $143 billion in the United States and $467 billion worldwide. Another fundamental component of the daily operations of companies is pricing. Because consumers are very price-sensitive, price can be manipulated by managers to influence demand in the short run. It is also a tool that helps to regulate inventory and production pressures (Bitran and Caldentey, 2003). McKinsey’s study indicated that a 1% improvement in pricing can lead to an 8% improvement in profits for a typical S&P 1500 company (Marn et al, 2003). Thus determining the level of advertising expenditure and setting price are two of the key marketing-mix variables. They interact continuously over time and should be taken in conjunction with each other in controlling sales and overall profits (MacDonald and Rasmussen, 2009). Since the pioneering work of Dorfman and Steiner (1954) that incorporated price and advertising decisions in revenue *Correspondence: Jianxiong Zhang, Naveen Jindal School of Management, The University of Texas at Dallas, 800 W. Campbell Road, Richardson, Texas 75080, USA. E-mail:
[email protected]
management model, several efforts have been made to study systematically the joint effects on profit. The relationship between pricing and advertising decisions is reported by Shankar and Bolton (2004), Bagwell (2007) and Srinivasan and Kwon (2012). They showed that advertising could result into lower or higher prices, which depended on the role of advertising. In a monopolistic framework, Dockner and Jørgensen (1988) studied the optimal advertising strategies for a new product using optimal control theory. Thompson and Teng (1984) and Sethi et al (2008) extended this model by considering pricing policy. Fruchter (2009) investigated the roles of price and advertising expenditure that served as the quality indicators in a dynamic framework, assuming that the price was used both as a monetary constraint and as a quality signal, but the advertising expenditure was used only as a signalling device. He et al (2009) modelled a continuous-time dynamic model of co-op advertising and pricing decisions with uncertainty, and derived feedback Stackelberg solutions of the optimal policies for the manufacturer and retailer. In the face of potential competitors, Gupta and Di Benedetto (2007) developed a model to find an optimal price-advertising frontier that maximized the firm’s total discounted profits. Chutani and Sethi (2012) considered a dynamic durable goods duopoly with a manufacturer and two independent and competing retailers. Yue et al (2013) studied the pricing and advertising in a manufacturer-retailer supply by considering both the manufacturer and retailer offering price discounts. For review works on pricing and inventory policies, readers can refer to Huang et al (2012).
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A common characteristic to the aforementioned articles is that deterioration of items is not considered. However, most physical goods undergo deterioration if they lose their value over time, examples being medicine, tickets, volatile liquids, fashion goods, fruits and others. An excellent review of the literature for perishable products has been reported by Goyal and Gunasekaran (1995) and Bakker et al (2012). For the products with high deteriorating rate, managers usually mark down the selling price or invest in advertising to sell them over a finite selling horizon. Airlines selling tickets before planes depart, hotels renting rooms before midnight, and retailers selling seasonal or fashionable goods such as fashion apparel before the end of the season are good examples where combined advertising and pricing activities can be observed almost every day. These perishables have significant impact on profitability, and cannot be decoupled from management decisions (Chung and Li, 2014). Owing to the analytical complexity of solving the decision-making problem for such products, the theory of how to set optimal advertising rates together with optimal prices is yet still in its infancy. The analysis of joint pricing and advertising for perishable items involves different concepts of deterioration. The first concept is the situation in which all items remaining in inventory become simultaneously obsolete at the end of the planning horizon. The second concept is that the items deteriorate throughout their planning horizon (Raafat, 1991). To focus on the perishable products such as airplane tickets, MacDonald and Rasmussen (2009) considered the temporal pricing and advertising strategy of a monopolist under stochastic demand. Helmes and Schlosser (2013) analysed a dynamic pricing and advertising model with isoelastic demand when a firm was selling a finite amount of product in both stochastic and deterministic environment. Considering a continuous decaying inventory, Goyal and Gunasekaran (1995) developed an integrated production-inventory-marketing model for determining the economic production quantity (EPQ) and economic order quantity (EOQ) where the demand depended upon known marketing parameters such as price and the frequency of advertisement. Shah et al (2013) discussed the profit maximization problem in which price and the frequency of advertisement in the electronic or print media were regarded as decision variables. The optimal policies of existing literature on continuous decaying perishable products have confined themselves to onetime pricing and generic advertising, where the demand is directly affected by the frequency of advertisement. However, considering the effects of advertising on brand equity, a firm can contribute to the accumulation of goodwill by advertising his brand towards the consumer and thus achieves its long-term market share and profitability goals. By virtue of the dynamic character of actual market operation, dynamic pricing and advertising practices are particularly useful to regulate inventory pressures, especially in the presence of perishable products. Therefore, developing continuous time joint decision-making models on dynamic pricing and advertising strategies for perishable products is clearly a very interesting and challenging topic.
Inspired by the aforementioned studies, we present dynamic pricing and advertising policy for perishable products. To the best of our knowledge, this is the first attempt to deal with the joint dynamic policy for deteriorating inventory. The model we studied incorporates the following features: (1) Perishable products; (2) A continuous time, dynamic environment; (3) Price-sensitive and goodwill-dependent demand rate, where the goodwill is built up by investing in advertising; (4) Two controls at each time: price and advertising effort. For maximizing the total profit, a dynamic optimization model is proposed to allocate a limited investment capacity and set a suitable sales price. The necessary and sufficiency conditions for optimality are obtained by Pontryagin’s maximum principle. Both static and dynamic pricing policies are investigated and compared with each other. The remainder of the paper is organized as follows. In the next section, we present a description of the joint pricing and advertising system for perishable products. The optimal policies are obtained by solving the corresponding dynamic optimization problem in the subsequent section. Then two numerical examples are given in the penultimate section to illustrate the solution procedures. Finally, the paper is concluded with a short summary in the last section.
2. Problem formulation We consider a monopolistic firm that needs to decide its pricing and advertising strategies for a certain branded perishable product, over time. Based on the relationship between adverting and sales, two main models have been proposed to deal with the advertising problems: Vidale-Wolfe model (Vidale and Wolfe, 1957) and Nerlove-Arrow model (Nerlove and Arrow, 1962). The former believes advertising exerts a direct positive impact on potential customers who haven’t yet purchased in the planning cycle, and the latter assumes the sales depend on the stock of goodwill, which is accumulated by investing in advertisement. The concept of goodwill in Nerlove-Arrow model summarizes the effects of advertising effort on all costumers including those who have already purchased. Nerlove and Arrow model is better suited to the perishable products setting where goods are consumed quickly and purchased regularly. Let G(t) be the brand’s goodwill, whose initial value G(0) is given as G0 > 0. According to the Nerlove and Arrow model, the evolution of goodwill can be described by the following dynamic equation: G_ ðt Þ ¼ uðt Þ - ρGðt Þ; (1) where u(t) is the firm’s advertising effort at time t, and ρ > 0 is the decay rate or forgetting effect of goodwill. In the presence of limited resources, the advertising effort is non-negative and upper bounded by the maximum investment capacity U, that is, 0 ⩽ uðt Þ ⩽ U:
(2)
It is shown from (1) and (2) that the goodwill G(t) ⩾ 0 for all t ⩾ 0.
Lin Feng et al—Joint dynamic pricing and advertising model of perishable products
The monopolistic firm also controls the sales price p(t) of its products to customers. And customer demand rate D(t), which increases with the goodwill G(t) but decreases with the price p (t), is determined by
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the total profit. The optimization problem can be formulated as ZT max J ¼ pðtÞðαðt Þ - βðt ÞpðtÞ + γGðtÞÞ - h1 I ðt Þ
pðÞ; uðÞ
0
DðGðt Þ; pðt ÞÞ ¼ αðtÞ - βðt Þpðt Þ + γGðt Þ;
where α(t) is the basic market potential, and β(t) represents the elasticity of the demand with respect to price. Both of them are positive, continuous functions of the time. The parameter γ is a positive parameter capturing the effect on current sales of goodwill. Note that this linear demand function is a regular assumption in the economics literature such as De Giovanni (2011) and Erickson (2012). The inventory system for the firm involving perishable products is considered along a time horizon [0, T], where T is the end of the sales period to be determined. The inventory level at time t is denoted by I(t) with its initial value I0 and terminal value 0. Once the item is in the inventory, deterioration occurs. The deterioration rate is θI, where θ is a non-negative coefficient. This assumption suggests that the deterioration of inventory is proportional to the value of inventory level. Then the inventory variances can be described by the following equation I_ ðtÞ ¼ - θI ðt Þ - DðGðt Þ; pðtÞÞ;
I ð0Þ ¼ I0 :
(4)
To maximize the profit, we set that I(T) = 0. Thus it is shown from (4) that the inventory level remains non-negative at all times, that is, I(t) ⩾ 0 for all t ∈ [0, T], and no backorder occurs in the whole planning period. The firm’s total profit is comprised of the earnings of sales subtracting holding and advertising costs. It is assumed that holding cost is linear. Let h1 be the positive holding cost coefficient per unit item. Advertising cost function may be represented by means of a convex function that takes the following quadratic form, denoted by h2u2(t)/2, where h2 > 0 implies increasing marginal cost of advertising. The selling cost depending on the length of selling cycle, such as sales employee wages, building lease, etc is denoted as ψ(T). It is assumed to be continuously differentiable, increasing and convex with respect to T, which means the marginal cost of the sales cycle is increasing. Accordingly, the profit can be described as ZT J¼
s:t:
I_ ðtÞ ¼ - θI ðtÞ - αðt Þ + βðt ÞpðtÞ - γGðtÞ; I ð0Þ ¼ I0 ; I ðT Þ ¼ 0; G_ ðt Þ ¼ - ρGðtÞ + uðtÞ; Gð0Þ ¼ G0 ; 0 ⩽ uðtÞ ⩽ U:
ð6Þ
3. Solution method In this section, Pontryagin’s maximum principle proposed in Sethi and Thompson (2000) is used to solve the optimization problem (6).
3.1. Necessary and sufficiency conditions for optimality The necessary conditions for the joint optimal policy (p*, u*) are obtained as follows. Associate adjoint variables λ1 and λ2 with the objective function to form Hamiltonian function as H ðp; u; I; G; λ1 ; λ2 ; t Þ ¼ pðαðtÞ - βðt Þp + γGÞ 1 - h1 I ðt Þ - h2 u2 ðtÞ + λ1 ð - θI ðtÞ - αðt Þ + βðt Þp 2 - γGÞ + λ2 ð - ρG + uÞ:
ð7Þ
The state trajectory (I*, G*) satisfies I_ * ðt Þ ¼ - θI * ðt Þ - αðt Þ + βðtÞp* ðtÞ; I * ð0Þ ¼ I0 ;
I * ðT Þ ¼ 0;
G_ * ðt Þ ¼ - ρG* ðtÞ + u* ðt Þ;
I * ðtÞ ⩾ 0;
G* ð0Þ ¼ G0 ;
G* ðtÞ ⩾ 0:
(5)
0
ð8Þ ð9Þ
The adjoint variables λ1 and λ2 must satisfy the following adjoint equations: ∂H ¼ h1 + λ1 θ; λ_ 1 ¼ (10) ∂I ∂H ¼ ρλ2 + γλ1 - γp; λ_ 2 ¼ ∂G
1 pðtÞDðt Þ - h1 I ðt Þ - h2 u2 ðtÞ dt - ψ ðT Þ: 2
1 - h2 u2 ðt Þ dt - ψ ðT Þ 2
(3)
(11)
as well as transversality condition λ2 ðT Þ ¼ 0:
(12)
The unspecified terminal time T should satisfy Our objective is to find a joint dynamic pricing and advertising policy and determine the optimal sales cycle that maximizes
H ðp; u; I; G; λ1 ; λ2 ; t Þjt¼T - ψ_ ðT Þ ¼ 0:
(13)
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The optimal control policy has to maximize the Hamiltonian function at all points, that is, H p* ; u* ; I * ; G* ; λ1 ; λ2 ; t ⩾ H p; u; I * ; G* ; λ1 ; λ2 ; t : (14) The sufficiency conditions for the optimality can be stated in terms of the maximum principle described in Hartl et al (1995) as the following lemma.
marginal value of a unit inventory level. At the optimal solution, it is natural to set λ1s(0) ⩾ 0 in advance. Then the optimal advertising policy is presented by the following theorem. Theorem 2 For a constant price ps within one selling cycle, the optimal advertising effort is 8 U; > > < u*s ðt Þ ¼ ρΔ1 + 2ρ2 Δ2 eρt + ðθ + ρÞΔ3 eθt ; > > : 0;
Lemma 1 (Hartl et al (1995)) Let p*(t), u*(t) and the corresponding I*(t), G*(t) and λ1(t), λ2(t) satisfy the necessary conditions (8)–(14) for all t ∈ [0, T]. Then, (p*,u*) is an optimal control if H(p, u, I, G, λ1, λ2, t) is concave in (I, G) for each t, and ψ in (5) is convex in T. The sufficiency conditions of the optimality for the optimal control problem can be guaranteed by the following theorem. Theorem 1 The solution (p*, u*, I*, G*) to problem (6) is optimal when (p*, u*, I*, G*, λ1, λ2) satisfies the necessary conditions (8)–(14) Proof Note that the advertising cost function is strictly convex. So the Hamiltonian function H(p, u, I, G, λ1, λ2, t) is concave in (I, G) and the selling cost function ψ(T) is convex in T. Thus according to Lemma 1, the solution obtained by the necessary conditions (8)–(14) is optimal. □
3.2. Static pricing and dynamic advertising policy In this section, we first consider the dynamic advertising policy under a given sales price denoted as ps. The corresponding advertising policy is denoted as us(t). Then an algorithm is designed to obtain the optimal pricing and advertising policy in the whole selling cycle. The corresponding adjoint variables are denoted as λ1s and λ2s. By integrating the adjoint equations (10)–(11) and then substituting the transversality condition (12), the adjoint variables can be obtained as h1 h1 + λ1s ð0Þ eθt - ; (15) λ1s ðtÞ ¼ θ θ γ h1 γ h1 ρðt - T Þ ps + λ1s ð0Þ + λ2s ðtÞ ¼ + 1-e ρ θ-ρ θ θ ð16Þ ´ eθt - eðθ - ρÞT + ρt ; where λ1s(0) is to be determined. Without loss of generality, we consider the parameter θ≠ρ in the following analysis. For the case θ = ρ, it should be mentioned that all the results obtained in this paper also hold when we just take the limit in mathematics such that θ approaches to ρ for the corresponding terms. Let t1 satisfy λ2s(t1) = h2U, and t2 be the minimum positive number satisfying equation λ2s(t2) = 0. Note that λ1s reflects the
0 ⩽ t ⩽ t1 ; t1
where γ h1 1 γ h1 - ρT ps + ps + Δ1 ¼ 2 ; Δ2 ¼ - 2 e ρ h2 2ρ h2 ρ θ θ γ h1 ðθ - ρÞT λ1s ð0Þ + + ; e θ-ρ θ γ h1 Δ3 ¼ 2 2 λ1s ð0Þ + : θ θ - ρ h2
Proof It is obvious that H is a strictly concave function in u. To maximize H, one can obtain the advertising effort is
u*s ðtÞ ¼
8 U; > > < > > :
λ2s ðt Þ ⩾ h2 U;
λ2s ðt Þ h2 ;
0 ⩽ λ2s ðtÞ
0;
otherwise:
(18)
It is shown from (15) that λ1s(t) is a positive and increasing function by virtue of λ1s(0) ⩾ 0. In addition, it can be verified from (11) and (12) that λ2s(t) is a monotonically decreasing function when λ2s(t) > 0. Thus the equation λ2s(t) = h2U has at most a unique solution t = t1, and λ2s(t) > h2U for any t < t1. Furthermore, it can be seen that 0 ⩽ λ2s(t) < h2U implies t1 < t ⩽ t2. Therefore, substituting (16) in (18), the result of Theorem 2 follows. The proof is complete. □ Then solving the state equation (1) under the optimal gives rise to advertising effort u*(t) s 8 U U - ρt > > + G0 e ; > >ρ ρ > > > > < ρðt1 - t Þ + ρΔ2 eρt - eρð2t1 - tÞ Gs ðtÞ ¼ Δ1 + ðGs ðt1 Þ - Δ1 Þe > > > > > + Δ3 eθt - eðρ + θÞt1 - ρt ; > > > : Gs ðt2 Þe - ρðt - t2 Þ ;
0 ⩽ t ⩽ t1 ; t1
(19)
Lin Feng et al—Joint dynamic pricing and advertising model of perishable products
With the demand function (3) and the goodwill (19), it can be calculated from (4) that
8 Rt > γ U > ðαðτÞ - βðτÞps Þeθτ dτ e - θt I0 + γU > > ρθ + θ - ρ G0 - ρ > > 0 > 0 ⩽ t ⩽ t1 ; > > > > γU γ U - ρt > G ; e > 0 > ρθ θ ρ ρ > > > ! > > > Rt > > θt1 θτ > I ð t Þe ð α ð τ Þ β ð τ Þp Þe dτ e - θt > s 1 s > > t1 > > > > > > γ ðGs ðt1 Þ - Δ1 Þ ρðt1 - tÞ θðt1 - tÞ γΔ1 > > > e -e 1 - eθðt1 - tÞ > > θ-ρ θ > < t1
- γρΔ2 e e + e > 2 > θ+ρ θ-ρ θ - ρ2 > > > > > > 1 1 θ + ρ θð2t1 - tÞ > > ; eθt eðθ + ρÞt1 - ρt + e > - γΔ3 > > 2θ θ-ρ 2θðθ - ρÞ > > > 0 1 > > > Zt > > > γGs ðt2 Þeθt2 B C > θt2 > I ð t Þe + ðαðτÞ - βðτÞps Þeθτ dτAe - θt @ > s 2 > θ-ρ > > > t t2 2 > > > > > γGs ðt2 Þ ρðt2 - tÞ > > : e ; θ-ρ
(20)
Then λ1s(0) and the final time T can be determined by the terminal inventory level I(T) = 0 and the terminal time condition (13). With λ1s(0) and the final time T calculated, the optimal advertising effort can be obtained from (17). From the perspective of economics, the adjoint variables λ1s(t) and λ2s(t) can be interpreted as marginal value or shadow price of a unit inventory level and goodwill. It can be seen from (17) that the advertising effort will reach its maximum value U when the marginal value of goodwill is comparatively higher (ie, λ2s(t) ⩾ h2U). Thus t1 represents the time point before which large additional profits can be produced by advertising. For a comparatively lower marginal value of goodwill (ie, λ2s(t) < 0), the advertising effort is 0. Thus t2 is the time point ever since which advertising is not worth implementing. Otherwise, the optimal adverting effort depends on the marginal value of goodwill. The total profit Js under the given price ps can be obtained by substituting the optimal advertising decision (17), the corresponding goodwill (19) and inventory level (20) into (5). Now we have to find the optimal price that maximizes the total profit. We assume that the sales price is upper bounded by pmax. In reality, setting the highest allowed price is very common in many industries. An algorithm for searching the optimal policy (p*, s u*) s is designed as follows.
Algorithm Step 1: Initialize ps = 0, p*s = 0, J*s = 0. Set a sufficiently small iterative step-size δ. Step 2: Compute us from (17) and Js from (5). If Js > Js*, set Js* = Js, ps* = ps and us* = us. Let ps = ps + δ. Step 3: If ps < pmax, go back to Step 2, otherwise compute us from (17) and Js from (5) with ps = pmax. If Js > Js*, set Js* = Js, ps* = ps and us* = us. Output the optimal price ps*, advertising effort us* and profit Js*.
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3.3. Dynamic pricing and advertising policy In this section, the joint dynamic strategy for the optimization problem (6) is considered, where the price sensitive factor β is assumed to be time-invariant for simplicity. Because H is a strictly concave function in p and u, it follows from the necessary condition (14) that the optimal policies are 8 0; γGðt Þ + βλ1 ðt Þ< - αðt Þ; > > > > > > < αðtÞ + γGðtÞ + βλ1 ðtÞ ; - αðt Þ ⩽ γGðt Þ + βλ1 ðt Þ<αðtÞ; p* ðt Þ ¼ 2β > > > > αðtÞ + γGðtÞ > > ; otherwise; : β (21) 8 0; λ2 ðtÞ<0; > > > < λ 2 ðt Þ u * ðt Þ ¼ (22) ; 0 ⩽ λ2 ðtÞ h2 > > : U; otherwise: Adjoint variable λ1(t) is the marginal value of inventory level. It can be seen from (21) that the price is zero to promote sales and regulate inventory pressures if the firm has a low reputation or high holding cost (ie, γG + βλ1(t) < − α(t)). The firm can’t earn from selling in this case since the products are free. In contrast, the price reaches its maximum value (α(t) + γG(t))/β if the firm has a well-deserved reputation or low holding cost (ie, γG + βλ1(t) ⩾ α(t)). However, the demand rate is 0 that no customer wants to buy the products in this case. It is obvious that the above two cases go against from making profits. Thus in the following, we only investigate the joint strategy when − α(t) ⩽ γG(t) + βλ1(t) < α(t), that is, the firm’s pricing strategy depends positively on the goodwill and the marginal value of inventory level. To study the joint policy, we introduce the following matrix function 02 31 " # - ρ h1 φ11 ðtÞ φ12 ðtÞ 2 5t A :¼ φðt Þ ¼ ; (23) exp @4 γ2 φ21 ðtÞ φ22 ðtÞ - 2β ρ where the elements of φ(t) are defined as follows. If ρ2 − γ2/ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (2βh2) ⩾ 0, let m ¼ ρ2 - γ 2 =ð2βh2 Þ: Denote φ11(t) = ((ρ + m) e − mt − (ρ − m)emt)/(2m), φ12(t) = (emt − e − mt)/(2h2m), φ21(t) = (γ2e − mt − γ2emt)/(4βm), φ22(t) = ((ρ + m)emt − (ρ − m)e − mt)/(2m). pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Otherwise, let m ¼ - ρ2 + γ 2 =ð2βh2 Þ: Denote φ11(t) = cos(mt) − (ρsin(mt))/m, φ12(t) = (sin(mt))/(h2m), φ21(t) = − (γ2sin(mt))/(2βm), φ22(t) = cos(mt) + (ρ sin(mt))/m. Denote G1 = G(t1), G2 = G(t2), I1 = I(t1) and I2 = I(t2). Let t1 satisfy λ2(t1) = h2U, and t2 be the minimum positive number satisfying equation λ2(t2) = 0. In practice, the goodwill is often built up by investing in advertisement with its maximum effort U at the beginning of sales cycle. This is often occurring due to large stocks in the
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early stage, which will lead to a large amount of deterioration of products. Advertising with its maximum effort as earlier as possible is better for stimulating demands and further reducing the loss of deterioration. Thus, to simplify the analysis, in the following we only consider the case λ2(t) ⩾ h2U is equivalent to t ⩽ t1, 0 ⩽ λ2(t) < h2U is equivalent to t1 < t ⩽ t2, and λ2(t) < 0 is equivalent to t2 < t ⩽ T. Under this case, along with the general assumption − α(t) ⩽ γG(t) + βλ1(t) < α(t), the joint dynamic policy can be obtained by the following theorem. Theorem 3 The optimal pricing and advertising policies are p * ðt Þ ¼
αðtÞ + γGðtÞ + βλ1 ðtÞ ; 2β
8 U; > > > < λ 2 ðt Þ u* ðt Þ ¼ ; > h2 > > : 0;
(24)
t1
Zt γh1 1 - eρðt - t2 Þ γ ρt e e - ρτ αðτÞdτ + 2β 2θρ
(25)
0 ⩽ t ⩽ t1 ;
t1
t2
(26) h1 h1 + λ1 ð0Þ eθt - ; θ θ
8 > γ hθ1 + λ1 ð0Þ eθt - eρt > γh1 ð1 - eρt Þ > > + > > > 2θρ 2ðθ - ρÞ > > > > > t Z > > > γ ρt γ2 U > > e e - ρτ αðτÞdτ + ðe - ρt - eρt Þ G0 > > 2β ρ 4βρ > > > 0 > > > > > > γ 2 U ð1 - eρt Þ ρt > > + + e λ2 ð0Þ; > > 2βρ2 > > > > < φ ðt - t ÞG + φ ðt - t Þh U 1 1 1 2 21 22 λ2 ðtÞ ¼ > > Zt > > > > + γ φ22 ðt - τÞ h1 + λ1 ð0Þ eθτ - αðτÞ - h1 dτ; > > > 2 β θ θ > > t1 > > > > > > > γ hθ1 + λ1 ð0Þ eθt - eρt + ðθ - ρÞt2 > γh1 1 - eρðt - t2 Þ > > > + > > 2ðθ - ρÞ 2θρ > > > > > t > Z > 2 > > > - γ eρt e - ρτ αðτÞdτ + γ G2 e - ρðt - t2 Þ - eρðt - t2 Þ ; > > : 2β 4βρ
γ G2 - ρðt - t2 Þ ρðt - t2 Þ ; -e e + 4βρ 2
t2
8 U - ρt U > > G e + ; > 0 > > ρ ρ > > > > > > ð t t ÞG + φ φ 1 1 12 ðt - t1 Þh2 U > < 11 t Z G ðt Þ ¼ γ h1 αðτÞ h1 > > > + λ1 ð0Þ eθτ φ12 ðt - τÞ + dτ; > > θ θ 2 β > > > t1 > > > > : - ρð t - t 2 Þ G2 ; e
λ 1 ðt Þ ¼
When λ2(t) < 0, that is, t2 < t ⩽ T, the goodwill can be calculated from (1) under zero advertising effort that GðtÞ ¼ e - ρðt - t2 Þ G2 . By virtue of the condition λ2(t2) = 0 and adjoint equation (11), we get γ hθ1 + λ1 ð0Þ eθt - eρt + ðθ - ρÞt2 λ 2 ðt Þ ¼ 2 ðθ - ρ Þ
0
0 ⩽ t ⩽ t1 ;
where
initial value λ1(0), the adjoint equation (10) gives rise to h1 h1 λ 1 ðt Þ ¼ + λ1 ð0Þ eθt - : θ θ
(27)
t2
When λ2(t) ⩾ h2U, that is, 0 ⩽ t ⩽ t1, the advertising effort u reach its maximum value, it follows from the state equation (1) that U - ρt U e + : Gðt Þ ¼ G0 ρ ρ With the initial value λ2(0), the corresponding adjoint variable can be obtained from (11) that γ hθ1 + λ1 ð0Þ eθt - eρt γh1 ð1 - eρt Þ λ2 ðtÞ ¼ + 2θρ 2ðθ - ρÞ γ ρt e 2β
Zt
e - ρτ αðτÞdτ
0
+ 0 ⩽ t ⩽ t1 ;
t1
t2
0
(28)
γ U γ 2 U ð1 - eρt Þ G0 ðe - ρt - eρt Þ + ρ 2βρ2 4βρ 2
+ eρt λ2 ð0Þ;
t ⩽ t1 :
When 0 ⩽ λ2(t) < h2U, that is, t1 < t ⩽ t2, let x(t) = [G(t)λ2(t)]T. It can be seen from the differential equations (1) and (11) that x_ ðtÞ ¼ Axðt Þ + Bðt Þ; (29) " #
- ρ h12 0 where A ¼ , BðtÞ ¼ γ ðβλ1 ðtÞαðtÞÞ . It follows γ2 - 2β ρ 2β from the differential equation (29) and the definition of φ(t) in (23) that Zt eAðt - τÞ BðτÞdτ ¼ φðtÞxðt1 Þ xðtÞ ¼eAt xðt1 Þ + 0
Zt Proof Note from (21) and (22) that the optimal policies depend on the goodwill and adjoint variables. With the
φðt - τÞBðτÞdτ:
+ 0
ð30Þ
Lin Feng et al—Joint dynamic pricing and advertising model of perishable products
Note that x(t1) = [G1 h2U]T. Then it is shown from (30) that Gðt Þ ¼φ11 ðt - t1 ÞG1 + φ12 ðt - t1 Þh2 U γ + 2
Zt φ12 ðt - τÞ t1
h1 αðτÞ h1 θτ + λ 1 ð0 Þ e dτ; β θ θ
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4. Numerical examples In this section, we present two numerical examples to illustrate the theoretical results. Example 1 offers the optimal advertising strategy with the static sales price, while Example 2 provides the optimal advertising effort and the dynamic optimal sales price. Example 1 Consider the following parameters setting:
λ2 ðt Þ ¼φ21 ðt - t1 ÞG1 + φ22 ðt - t1 Þh2 U γ + 2
Zt φ22 ðt - τÞ t1
αðt Þ ¼ 4 + t - 0:1t2 ; β ¼ 0:3; ψ ðT Þ ¼ 0:5T 2 ;
h1 αðτÞ h1 θτ + λ1 ð0Þ e dτ; β θ θ
t1
The proof is complete. □ With the demand function (3) and the goodwill (26), it can be calculated from (4) that
I ðt Þ ¼
8 > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > < > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > :
β 2θ
θt h1 e h1 γ U θe - ρt U G0 + λ1 ð0Þ + 2θ ρ θ-ρ ρ θ 2 θ
+ e - θt I0 -
e - θt 2
Zt
αðτÞeθτ dτ -
0
0 ⩽ t ⩽ t1 ; βe - θt h1 γe - θt ðθG0 - U Þ; λ1 ð0Þ + 4θ θ 2θðθ - ρÞ
2h β h1 1 + λ1 ð0Þ eθt - eθð2t1 - tÞ 1 - eθðt1 - tÞ θ θ 4θ + I1 eθðt1 - tÞ -
e - θt 2
Zt
t1
t1
β 4θ
2h h1 1 + λ1 ð0Þ eθt - eθð2t2 - tÞ 1 - eθðt2 - tÞ θ θ
+ I2 eθðt2 - tÞ -
e - θt 2
Zt
αðτÞeθτ dτ +
γG2 ρt2 - ðρ + θÞt - e - θt ; e 2ρ
t2
t2
ð31Þ
Then λ1(0), λ2(0) and the final time T can be further determined by the terminal inventory level I(T) = 0, the transversality condition (12) and the terminal time condition (13). With λ1(0), λ2(0) and the final time T calculated, the optimal pricing and advertising policies can be obtained from (24) and (25) respectively. Theorem 3 shows the optimal dynamic pricing and advertising policy when the goodwill is built up by investing in advertisement with its maximum effort U at the beginning of sales cycle, which is common in practice. It should be pointed out that, to better match demand, it may be possible to postpone the advertising with its maximum effort to some time in selling period, especially when the peak of demand occurs close to the end of the sales period. In this case, the advertising with the maximum effort U may occur in the middle or close to the end of the sales period, and the optimal dynamic pricing and advertising policy can be correspondingly obtained by applying a similar method deriving Theorem 3.
and γ = 0.5, h1 = 0.13, h2 = 0.6, θ = 0.2, ρ = 0.25, U = 1, G0 = 2, I0 = 30. It should be mentioned that the quadratic form of the basic market potential, which has a maximum during the selling period and can reflect the seasonal varying of the demand for the product, has been widely applied in existing literature (Gaimon, 1988; Adida and Perakis, 2007). The other parameters about the demand parameters and goodwill parameters are similarly selected according to previous studies in marketing and operations management, for example, De Giovanni (2011) and Erickson (2012). We assume the sales price ps is a static variable to be determined. According to Theorem 2 and applying the Algorithm, the optimal advertising effort can be obtained from (17) as ( u*s ðt Þ ¼
1;
0 ⩽ t ⩽ 5:6540;
112:3333 + 60:4958e0:25t - 115:8733e0:2t ; 5:6540
where t1 = 5.6540, t2 = T*s = 5.8644 λ1s(0) = 2.8262, and the optimal sales price p*s = 16.20. The corresponding total profit J*s = 198.8308. The blue curve us ðtÞ in Figure 1 shows the optimal advertising effort. In the first stage of the planning horizon, the advertising capacity is utilized to increase the goodwill level. And at the end of planning horizon, the advertising effort is reduced gradually to zero to save advertising cost. With the optimal sales price and advertising effort, the corresponding optimal goodwill level can be calculated and is depicted in the blue curve Gs ðtÞ of Figure 2. It is shown that the goodwill decreases at the end of planning horizon due to the gradually reduced advertising effort. The optimal inventory level can be obtained as the blue curve I s in Figure 3. Example 2 In this example we study the optimal advertising and dynamic pricing problem. Consider the same parameters with that in Example 1 except p being a timevarying decision variable. For the given parameters, noted that ρ2 − γ2/(2βh2) < 0. It is calculated that φ11(t) = cos(0.6319t) − 0.3956 sin(0.6319t), φ12(t) = 2.6374 sin(0.6319t), φ21(t) = − 0.6593 sin(0.6319t), φ22(t) = cos(0.6319t) + 0.3956 sin(0.6319t). According to Theorem 3, one can get the optimal advertising effort and dynamic price as that of (24) and (25) with λ1(0) = 2.8672, λ2(0) = 13.3331, t1 = 6.262, t2 = T* = 6.5051, which are shown in the red curves u ðtÞ; p ðtÞ respectively in
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1.4
22
1.2
20
1
18
0.8
16
0.6
14
0.4
12
0.2
10
0
0
1
2
3
4
5
6
7
Figure 1 The optimal advertising effort.
8
0
1
2
3
4
5
6
7
Figure 4 The optimal sales prices.
Figures 1 and 4. With the optimal sales price and advertising effort, the corresponding optimal goodwill level, optimal inventory level can be calculated, and are depicted respectively in the red curves G ðtÞ; I ðtÞ of Figures 2 and 3. And the total profit is calculated as J* = 223.9680. In addition, it can be verified that the condition − α(t) ⩽ γG(t) + βλ1(t) < α (t) for all t ⩽ T* is satisfied.
3.6 3.4 3.2 3 2.8 2.6 2.4 2.2 2
0
1
2
3
4
5
6
7
6
7
Figure 2 The optimal goodwill levels.
30
25
20
15
10
5
0
0
1
2
3
4
5
Figure 3 The optimal inventory levels.
It is noted that the optimal total profit is much greater than that without the optimal dynamic pricing in Example 1; thus the dynamic pricing policy can improve the profit of the system compared with the static pricing policy. However, it is worthwhile to mention that, although the dynamic pricing strategy always dominates the static one in the framework of this paper because we have omitted the costs associated with the dynamic pricing setting, a constant sales price obtained from the Algorithm may be a better choice for those firms with very high transaction costs to implement a dynamic pricing strategy. Note that T* > Ts*, which implies the selling period under the case of dynamic pricing is longer than that of static pricing. This is because the dynamic pricing policy can better utilize the varying of the goodwill caused by advertising effort to maximize the profit. The firm should set a high price when the corresponding goodwill is great, which, indicated in Figure 4, finally extends the selling period. It can be seen from Figure 4 that in the first stage of the planning horizon the firm should set a lower selling price (red curve p ðtÞ) than the static price (blue curve ps ), and at the end of planning horizon the firm should set a higher selling price than the static price. This is because in the initial stage the goodwill is relatively low, as shown in Figure 2, a low selling price can stimulate the demand and simultaneously reduce the inventory cost and the loss caused by deterioration of the perishable products. In the later stage of the selling period, Figure 2 shows that the goodwill is high which allows the firm to set a high price to maximize the selling profit while maintaining a suitable demand rate. On the other hand, with the dynamic price adjusted with the varying of the
Lin Feng et al—Joint dynamic pricing and advertising model of perishable products
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Table 1 Computational results for different values of θ θ T* Ts* t1* t1s* ps* J* Js* (J* − J*s)/Js*
0.16
0.17
0.18
0.19
0.20
0.21
0.22
7.1051 6.3970 6.8743 6.2550 16.2910 255.6125 221.1031 0.1561
6.9392 6.2540 6.7024 6.1002 16.2763 246.9121 215.1514 0.1476
6.7848 6.1121 6.5442 5.9453 16.2351 238.7709 209.4886 0.1398
6.6390 5.9859 6.3961 5.8004 16.2230 231.1048 204.0228 0.1327
6.5051 5.8644 6.2620 5.6540 16.2000 223.9680 198.8308 0.1264
6.3842 5.7500 6.1404 5.5101 16.1750 217.4727 193.9836 0.1211
6.2683 5.6406 6.0183 5.3605 16.1467 211.2083 189.3067 0.1157
goodwill, the corresponding inventory level shown in Figure 3 (red curve I ðtÞ) decreases faster than that with static price (blue curve I ðtÞ) in the first stage of the planning horizon due to the large demand incurred by low price, and declines more slowly in the later stage of the selling period due to the high price. And the firm can benefit from the high levels of goodwill and sales price in the later stage of the planning horizon. In addition, the dynamic pricing policy can enhance the effectiveness of advertising effort in maximizing the system profit. The firm should bear more advertising efforts in the case of dynamic pricing than that in the case of static pricing, which is also indicated from Figure 1 that the optimal advertising effort (red curve u ðtÞ) is always above that (blue curve us ðtÞ) in the case of static pricing. And as shown in Figure 2, the corresponding goodwill level (red curve Gs ðtÞ) is greater than that (blue curve G ðtÞ) of the case with static pricing. When the deterioration coefficient θ takes the values from [0.16, 0.22] with the step size 0.1 and other parameter values are fixed, the main optimal computational results under dynamic pricing and advertising scenario, and static pricing and advertising scenario respectively are presented in Table 1. The optimal switching point t1 under static pricing scenario is denoted as t1s*. It is calculated that the optimal switching point t2 = T* under dynamic pricing policy, and t2 = Ts* under static pricing policy for all the above parameters. Additionally, it can be verified that the condition − α(t) ⩽ γG(t) + βλ1(t) < α(t) for all t ⩽ T* is satisfied in the case of joint dynamic policy. It is shown from Table 1 that both the selling periods under dynamic pricing policy and static pricing policy are decreasing with the deterioration coefficient θ, which means that at the optimal solution, the highly perishable items imply a short selling period. That is intuitive in practice. In addition, due to the greater freedom of decision-making under joint dynamic policy compared with the static pricing policy, for a given deterioration coefficient θ, the selling period under dynamic pricing policy is always greater than that under static pricing policy. Table 1 shows that, under both the dynamic pricing policy and the static pricing policy, the greater the deterioration coefficient θ is, the smaller the switching points t1 are, which implies that the highly perishable items (ie, items with a high
deterioration coefficient θ) bring more inertness in advertising investment of the firm. Additionally, for a given deterioration coefficient θ, the switching points t1 under dynamic pricing policy are always larger than that under static pricing policy, which implies more advertising investment in the dynamic pricing scenario. This is because, compared with the static pricing policy, the dynamic pricing policy enhances the effectiveness of advertising effort in maximizing the system profit, which induces more advertising efforts in the case of dynamic pricing than that in the case of static pricing. Furthermore, it is shown that the optimal static selling price p*s slightly decreases with the increase of the deterioration coefficient θ, which illustrates that in the case of static pricing policy, the firm should reduce the selling price for the highly perishable products. Finally, it can be seen from Table 1 that both the profits under dynamic pricing policy and under static pricing policy are decreasing with the deterioration coefficient θ. In particular, when the deterioration coefficient θ decreases from 0.22 (associated with profits J* = 211.2083 and Js* = 189.3067) to 0.16 (associated with profits J* = 255.6125 and Js* = 221.1031), the percentages of profit growth are respectively calculated as 21.02% in the dynamic pricing policy and 16.80% in the static pricing policy, which means that the perishable characteristic of the product has a significant negative impact on the profits. In addition, for a given θ the profit under dynamic pricing policy is always greater than that under static pricing policy. Specifically, the percentages of profit growth in the case of dynamic policy compared with the static pricing policy are always above 10% (the maximum 15.61% and the minimum 11.57%) for the deterioration coefficient θ ∈ [0.16, 0.22]. Thus it is concluded that, compared with the static pricing and advertising policy, the dynamic pricing and advertising policy presented in this paper can greatly improve the profitability for the firms with perishable products.
5. Conclusion This paper is concerned with the problem of setting a joint pricing and advertising policy for perishable products in a capacitated dynamic setting. Demand rate is price-sensitive
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and depends on the goodwill. The firm controls the selling price to customers, and invests in advertising in order to build up the brand equity. To maximize the total profit, we establish a dynamic optimization model to set a suitable price and decide how much to advertise over time. The optimal advertising policies under both static and dynamic prices are obtained by using Pontryagin’s maximum principle. Examples are provided for comparison between the two strategies. The results obtained in this paper have the following theoretical contributions and important managerial implications. Firstly, the analytical solutions of the optimal dynamic policies obtained by using Pontryagin’s maximum principle serve as powerful tools to support the managers in making the pricing and advertising decisions. Secondly, it is shown that dynamic pricing policy can enhance the effectiveness of advertising effort and prolong the selling period compared with the static one. The profit of the system is improved by dynamic pricing. Thirdly, for those firms with high transaction costs to implement a dynamic pricing strategy, this paper provides an algorithm to obtain a constant sales price for administrative convenience within a selling cycle. Fourthly, the perishable characteristic of the product has a significant negative impact on the profits. Additionally, the high perishability of the product brings more inertness in advertising investment of the firm. Several extensions to the analysis presented in this article are currently under development. For instance, we have only considered the case of one period selling problem in which the inventory is fixed at the start of the sales horizon, with no replenishment. The case of joint dynamic pricing and advertising for multi-period selling problem with continuous-time production or fixed period replenishment is worthwhile to study. In addition, the impact of competition between different firms or brands with respect to both advertising and pricing should be considered in future extensions to this research. Furthermore, we have omitted the costs associated with the dynamic pricing setting; the optimal dynamic pricing and advertising for perishable products with costly price adjustments is an interesting potential extension. Acknowledgements —This work was supported by the National Nature Science Foundation of China Nos. 61473204, 71371133 and the Program for New Century Excellent Talents in Universities of China (NCET-110377), and China Scholarship Council.
References Adida E and Perakis G (2007). A nonlinear continuous time optimal control model of dynamic pricing and inventory control with no backorders. Naval Research Logistics 54(7): 767–795. Bagwell K (2007). The economic analysis of advertising. In: Armstrong M and Porter R (eds). Handbook of Industrial Organization. Vol. 3. Amsterdam: North-Holland, pp 1701–1844. Bakker M, Riezebos J and Teunter RH (2012). Review of inventory systems with deterioration since 2001. European Journal of Operational Research 221(2): 275–284.
Bass FM, Krishnamoorthy A, Prasad A and Sethi SP (2005). Advertising competition with market expansion for finite horizon firms. Journal of Industrial and Management Optimization 1(1): 1–19. Bitran G and Caldentey R (2003). An overview of pricing models for revenue management. Manufacturing & Service Operations Management 5(3): 203–229. Chung J and Li D (2014). A simulation of the impacts of dynamic price management for perishable foods on retailer performance in the presence of need-driven purchasing consumers. Journal of the Operational Research Society 65(8): 1177–1188. Chutani A and Sethi S (2012). Optimal advertising and pricing in a dynamic durable goods supply chain. Journal of Optimization Theory and Applications 154(2): 615–643. De Giovanni P (2011). Quality improvement vs. advertising support: Which strategy works better for a manufacturer? European Journal of Operational Research 208(2): 119–130. Dockner E and Jørgensen S (1988). Optimal advertising policies for diffusion models of new product innovation in monopolistic situations. Management Science 34(1): 119–130. Dorfman R and Steiner PO (1954). Optimal advertising and optimal quality. The American Economic Review 44(5): 826–836. Erickson GM (2012). Transfer pricing in a dynamic marketingoperations interface. European Journal of Operational Research 216(2): 326–333. Fruchter GE (2009). Signaling quality: Dynamic price-advertising model. Journal of Optimization Theory and Applications 143(3): 479–496. Gaimon C (1988). Simultaneous and dynamic price, production, inventory and capacity decisions. European Journal of Operational Research 35(3): 426–441. Goyal SK and Gunasekaran A (1995). An integrated productioninventory-marketing model for deteriorating items. Computers & Industrial Engineering 28(4): 755–762. Gupta MC and Di Benedetto CA (2007). Optimal pricing and advertising strategy for introducing a new business product with threat of competitive entry. Industrial Marketing Management 36(4): 540–548. Hartl R, Sethi S and Vickson R (1995). A survey of the maximum principles for optimal control problems with state constraints. SIAM Review 37(2): 181–218. He X, Prasad A and Sethi SP (2009). Cooperative advertising and pricing in a dynamic stochastic supply chain: Feedback stackelberg strategies. Production and Operations Management 18(1): 78–94. Helmes KL and Schlosser R (2013). Dynamic advertising and pricing with constant demand elasticities. Journal of Economic Dynamics and Control 37(12): 2814–2832. Huang J, Leng M and Liang L (2012). Recent developments in dynamic advertising research. European Journal of Operational Research 220(3): 591–609. MacDonald L and Rasmussen H (2009). Revenue management with dynamic pricing and advertising. Journal of Revenue & Pricing Management 9(1): 126–136. Marn MV, Roegner EV and Zawada CC (2003). The power of pricing. McKinsey Quarterly 1: 26–39. Nerlove M and Arrow KJ (1962). Optimal advertising policy under dynamic conditions. Economica 29(114): 129–142. Raafat F (1991). Survey of literature on continuously deteriorating inventory models. Journal of the Operational Research Society 42(1): 27–37. Sethi SP and Thompson GL (2000). Optimal Control Theory: Applications to Management Science and Economics. Kluwer: Dordrecht, the Netherlands. Sethi SP, Prasad A and He X (2008). Optimal advertising and pricing in a new-product adoption model. Journal of Optimization Theory and Applications 139(2): 351–360.
Lin Feng et al—Joint dynamic pricing and advertising model of perishable products
Shah NH, Soni HN and Patel KA (2013). Optimizing inventory and marketing policy for non-instantaneous deteriorating items with generalized type deterioration and holding cost rates. Omega 41(2): 421–430. Shankar V and Bolton RN (2004). An empirical analysis of determinants of retailer pricing strategy. Marketing Science 23(1): 28–49. Srinivasan A and Kwon C (2012). Operations of online advertising services and publisher’s option. Journal of the Operational Research Society 63(5): 674–682. Thompson GL and Teng JT (1984). Optimal pricing and advertising policies for new product oligopoly models. Marketing Science 3(2): 148–168.
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Vidale ML and Wolfe HB (1957). An operations-research study of sales response to advertising. Operations Research 5(3): 370–381. Yue JF, Austin J, Huang ZM and Chen BT (2013). Pricing and advertisement in a manufacturer-retailer supply chain. European Journal of Operational Research 231(2): 492–502.
Received 17 March 2014; accepted 4 September 2014 after one revision
