Dynamic lot-sizing models with pricing for new products

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Production, Manufacturing and Logistics

Dynamic lot-sizing models with pricing for new products Xiang Wu a, Yeming (Yale) Gong b,∗, Haoxuan Xu c, Chengbin Chu d, Jinlong Zhang a a

School of Management, Huazhong University of Science and Technology, 1037 Luoyu road, Wuhan 430074, China EMLYON Business School, 23 avenue Guy de Collogue, 69130 Ecully Cedex, France c School of Business Administration, Zhongnan University of Economics and Law, 182 Nanhu avenue, Wuhan 430073, China d Laboratoire Genie Industriel, CentraleSupélec, Université Paris-Saclay, Grande Voie des Vignes, 92290 Chatenay-Malabry, France b

a r t i c l e

i n f o

Article history: Received 4 July 2015 Accepted 2 December 2016 Available online xxx Keywords: Supply chain management Coordinated production and pricing Dynamic lot-sizing New product diffusion

a b s t r a c t While previous dynamic lot-sizing (DLS) models mainly consider mature products, this study analyzes production planning decisions for new products. The demand dynamics caused by new product diffusion complicate production decisions for new products. We integrate DLS and discrete Bass models to provide optimal decisions for pricing and production planning problems. Moreover, we study the joint influence of product diffusion and pricing parameters on the DLS decisions. This leads to the following insights. First, coordinated production-pricing and dynamic pricing improve profitability. Second, the optimal pricing strategy is affected by market conditions. The penetration pricing strategy outperforms the skimming pricing strategy when consumers are less sensitive to relative price changes than to the introductory price, when the product diffuses slowly, or when the consumer initiative level is low. Otherwise, the latter outperforms the former. Finally, pricing strategies and product diffusion patterns reshape the cost structure of a firm. Coordinated decisions and dynamic pricing strategy substantially reduce the cost-revenue ratio, whereas an increase in the consumer initiative level or product diffusion speed can improve cost efficiency. © 2016 Elsevier B.V. All rights reserved.

1. Introduction New products are crucial for modern firms, with sales derived from new products accounting for an average of 28% of firm sales in the United States (Cooper, 2011; Crawford & Di Benedetto, 2008). Our research interest is motivated by production planning problems in a biochemical company in France, which faces dynamic demand when introducing a new patented product, with a monopoly in the period of patent protection. This firm is sufficiently capable in ensuring the supply of the new patented product, that is, the production capacity is assumed to be unlimited. The new product management (NPM) division of this company has exerted efforts to promote the product. However, determining the production time and quantities is difficult because of the dynamics and complexity of the new product diffusion process. Another example is a fitness equipment retailer in China that needs to manage inventory replenishment problems when promoting patented fitness equipment. On the one hand, replenishment causes high fixed cost, and storage requires additional space and maintenance

∗

Corresponding author. E-mail addresses: [email protected] (X. Wu), [email protected], [email protected] (Y. (Yale) Gong), [email protected] (H. Xu), [email protected] (C. Chu), [email protected] (J. Zhang).

cost. On the other hand, stockout may result in the loss of consumers. Thus, replenishment decision becomes essential and challenging for new products. These firms hope to design an appropriate production policy to handle the complex dynamics of new product diffusion, as well as coordinate production planning and NPM. Such complexity lies in two aspects. First, the demand for new products is dynamic and depends on the diffusion pattern. Second, the NPM division constantly uses specific pricing strategies to promote a new product or earn profits. For example, the NPM division may adopt penetration pricing to build a large sales volume, or skimming pricing to make use of consumers’ insensitivity to initial high prices (Noble & Gruca, 1999). Therefore, a well-designed production policy calls for a thorough understanding of the new product diffusion pattern and the effects of pricing strategies. The sales and production sections in the NPM division should cooperate closely, as well as simultaneously consider production and pricing decisions (Bajwa, Sox, & Ishfaq, 2016; Gong, 2013; Hausman, Montgomery, & Roth, 2002). To handle demand dynamics in new product diffusion, we adopt the dynamic lot-sizing (DLS) models to optimize the production planning, since most enterprises implement enterprise resource planning systems comprising a manufacturing resource planning (MRP) modular in making production decisions

http://dx.doi.org/10.1016/j.ejor.2016.12.008 0377-2217/© 2016 Elsevier B.V. All rights reserved.

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(Gunasekaran, Marri, McGaughey, & Nebhwani, 2002). One assumption of the lot-sizing decision in MRP is that the demand is deterministic (De Bodt, Gelders, & Van Wassenhove, 1984). Although the uncertainty of demand appears in the diffusion process of new products, such uncertainty can be handled using a rolling scheme or using the safety stock while applying the DLS techniques (De Bodt, Van Wassenhove, & Gelders, 1982). DLS problems have been proven to be an efficient model in handling dynamic demand (Wagner & Whitin, 1958). Researchers have been extending the DLS problems in various settings. For reviews that focus on model category, algorithms, and demand patterns, see Bahl, Ritzman, and Gupta (1987), Karimi, Fatemi Ghomi, and Wilson (2003), Brahimi, Dauzere-Peres, Najid, and Nordli (2006), and Jans and Degraeve (2007). Most lot-sizing models are used to solve production or inventory planning problems for mature products (De Bodt et al., 1984). Our research follows this stream by applying an uncapacitated DLS model in dealing with the production planning problem in a new product environment, which has seldom been considered before. Moreover, we consider the Bass model in specifying the demand in the new product diffusion. We investigate the production planning problem for new products with pricing decisions. In particular, we combine the DLS model with the generalized Bass model that incorporates price effects. This approach enables us to analyze the coordinated production and pricing decisions for new products, thereby providing beneficial guidelines for the firm. Accordingly, we attempt to answer the following questions: How do we make optimal production decisions considering the dynamics of new product diffusion? How do we coordinate and make pricing and production decisions considering the dynamics of new product diffusion? Our contributions are as follows. First, we provide new insights into coordinated production and pricing, as well as optimize production planning according to different new product diffusion patterns. Second, we consider the DLS problem in the context of new products. Only a few studies combine the Bass model with production decisions. Finally, we attempt to coordinate pricing strategies and the DLS concerning different product diffusion patterns. Our research closely relates the growing research stream on coordinated pricing and production management in various settings (see, e.g., Yano and Gilbert, 2004, and Chen & Simchi-Levi, 2012), but focuses on new product situations. 2. Literature review Our study mainly relates to three research domains: coordinated production and pricing models, DLS models with pricing, and Bass-based product diffusion models. This research is categorized in the growing research stream on coordinated pricing and production or inventory management (see review papers of Eliashberg and Steinberg, 1993, Yano and Gilbert, 2004, and Chen & Simchi-Levi, 2012). Recent studies investigate coordinated decisions in various settings such as multiitem cases (Bajwa et al., 2016), perishable or substitutable products (Sainathan, 2013), backordering (Bernstein, Li, & Shang, 2016), random yield (Eskandarzadeh, Eshghi, & Bahramgiri, 2016), free shipping (Hua, Wang, & Cheng, 2012), cost learning (Li, Sethi, & He, 2015), and reference price effect (Güler, Bilgiç, & Güllü, 2015; Wu, Liu, & Zhang, 2015). Most of these studies consider continuoustime models or stochastic models. By contrast, our study uses discrete-time models with price-dependent deterministic demands, as well as incorporates the reference price effect. Since the seminal work of Wagner and Whitin (1958), the DLS problems have been vastly studied. We can easily find several reviews on the related research topics, such as modeling approaches and algorithms (e.g., Bahl et al., 1987; Karimi et al., 2003; Brahimi et al., 2006; Jans & Degraeve, 2007). Researchers have extended

lot-sizing problems in various settings, such as product returns (Teunter, Bayindir, & van den Heuvel, 2006), fixed carbon emissions (Absi, Dauzère-Pérès, Kedad-Sidhoum, Penz, & Rapine, 2016), capacity reservation contract (Akbalik, Hadj-Alouane, Sauer, & Ghribi, 2017), and online retailers (Xu, Gong, Chu, & Zhang, 2017). Our study is most relevant to the integrated DLS problem with pricing, which often appears in the interface of operations and marketing management. Based on the results of Wagner and Whitin (1958), Thomas (1970) studies the first discrete-time lot-sizing model in a new setting by regarding prices of each period as decision variables. Then Kunreuther and Schrage (1973) consider a similar problem in which the price is assumed to be constant over the entire planning horizon and develop a heuristic solution approach. Gilbert (1999) and Van den Heuvel and Wagelmans (2006) investigate the same problem as Kunreuther and Schrage (1973) and provide polynomial-time methods under the assumption of stationary costs and time-varying costs alternatively. While the demand in a given period is assumed to be independent of the prices offered in adjacent periods in most relevant papers, Ahn, Gümüs, and Kaminsky (2007) consider a much realistic scenario where the demand in each period depends on prices of multiple periods. Scholars have also addressed issues of coordinating DLS and pricing decisions with other factors, such as perishability of products (Bhattacharjee & Ramesh, 20 0 0), restriction of production capacity (Geunes, Merzifonluog˘ lu, & Romeijn, 2009), and competition from multiple products (Bajwa et al., 2016). We consider integrated lot-sizing and pricing decisions in the context of new products, analyze both constant and dynamic prices under different product diffusion patterns, and evaluate their effects on production decisions. The Bass model has been extensively used to model new product diffusion and forecast demand (Bass, 1969; Peres, Muller, & Mahajan, 2010). Furthermore, a firm can stimulate product diffusion via pricing, advertising, or channel disintegration (Bass, Krishnan, & Jain, 1994; Ramanan & Bhargava, 2013). Generalized versions of the Bass model consider the effects of prices (e.g., Robinson & Lakhani, 1975) or their changes (e.g., Bass et al., 1994). Our study adopts the discrete form of the Bass model and incorporates the effects of prices and their changes. Incorporating production decisions enables our research to provide additional insights into new product pricing. Only a few studies integrate the Bass model with production or inventory decisions. For example, Shen, Duenyas, and Kapuscinski (2013) examine the optimal inventory decisions for new products by focusing on the effect of supply constraints. Bilginer and Erhun (2015) analyze when to launch a new product concerning inventory-related costs and product diffusion pattern. A similar work by Bhattacharya, Guide, and Van Wassenhove (2006) examines production policy for new products, but without considering the diffusion pattern. By contrast, we combine the DLS and the Bass models, as well as focus on the strategic choice among constant, skimming, and penetration pricing strategies. 3. Dynamic lot-sizing models for new products 3.1. Problem description We consider a firm that introduces a new product into the market. Consumers gradually learn the existence and quality information of the new product. The product diffuses via two types of communication channels, namely, mass media and word of mouth (Bass, 1969). The consumer purchases at most one unit of the product over the entire planning horizon. Therefore, the demand at period t equals the number of new adopters n(t), that is, d (t ) = n(t ). This relationship holds for many products, such as smartphones and cameras. Our model can easily consider multiitem or repeat purchases. If every adopter purchases w units of

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the product, then we have d (t ) = wn(t ). If adopters purchase a unit of the product at every period since their first adoption, then the demand at period t equals the cumulative number of adopters:  d (t ) = N (t ) = ti=1 n(t ). The firm faces dynamic demand during the product diffusion process. When prior data are lacking, the firm can use management judgments or the historical data of analogous products to estimate the diffusion parameters (Mahajan, Muller, & Bass, 1990). For example, the firm can use a weighted sum of the parameters estimated from historical diffusion data of analogous products, where weights represent the similarity between the analogous products and the new product (Mahajan et al., 1990). The firm can also forecast demand by using conjoint analysis and Bayesian updating (Lee, Lee, & Lee, 2012). Thus, we assume that the diffusion parameters are known, and the demand is deterministic. Consider that the firm reviews inventory level x(t) and produces products periodically. Backordering is not allowed, that is, the demand at period t must be satisfied either by production or by inventory held at the end of period t − 1. The firm should decide the production amount z(t) at the beginning of period t over the planning horizon T (1 ≤ t ≤ T). Each production incurs a fixed cost k(t) plus a linear variable production cost c(t)z(t), where c(t) is the unit production cost. Besides, holding a unit of product from period t − 1 to period t incurs a fixed cost h(t). The firm minimizes the total cost over the planning horizon by searching for the optimal production decisions.

and

3.2. Model formulation

Every combination of p and q characterizes a product diffusion pattern and informs us how demand evolves (Bass, 1969). Therefore, we solve the problem using an integrated procedure by first calculating d(t) given coefficients p and q and then using the Wagner–Whitin algorithm to solve the problem. Once the product diffusion parameters p and q are obtained, we can determine the optimal total cost TC∗ . Lawrence and Lawton (1981) report that p + q ranges from 0.3 to 0.7 for most cases. Furthermore, Sultan, Farley, and Lehmann (1990) find p = 0.03, q = 0.38 on average, but both p and q vary significantly. We plot the optimal total cost as a function of product diffusion parameters, that is, T C ∗ = T C ∗ ( p + q, q/p) (see Fig. 1). p + q represents the diffusion speed, whereas q/p characterizes the consumer initiative level. When the sum of the coefficient of imitation q and the coefficient of innovation p is large, the product diffuses rapidly. When the ratio of q to p is large, consumers in the market exhibit a low consumer initiative level. Thus, Fig. 1 shows how the total cost varies with diffusion speed and consumer initiative level. We present the 2-D plots (see Fig. 2) to provide a clear illustration. We observe a nonmonotonic relationship between optimal total cost and product diffusion parameters. Fig. 2(a) shows how the optimal total cost TC∗ varies with diffusion speed p + q. When the consumer initiative level is low (e.g., q/p = 60 or q/p = 80), the total demand increases with diffusion speed p + q, and so does the optimal total cost TC∗ . However, the optimal total cost TC∗ decreases with diffusion speed p + q when the consumer initiative level is high (e.g., q/p = 20 or q/p = 40), in which case the total demand reaches the market potential quickly. The firm enjoys large demands during the first several periods, the sum of which equals the market potential, thereby reducing the setup costs. Fig. 2(b) shows that the optimal total cost TC∗ increases initially but decreases eventually with q/p. When the consumer initiative level decreases but remains relatively high (e.g., from q/p = 1 to q/p = 40), the total demand does not decrease considerably but no longer concentrates merely in the first few periods. Consequently, the optimal total cost TC∗ increases because of setup costs. When the consumer initiative level decreases continuously (e.g., from q/p = 60 to q/p = 100), the total demand decreases substantially, and so does the optimal total cost.

The firm’s objective is to minimize the cost:

TC =

T 

[k(t ) · v(t ) + c (t ) · z(t ) + h(t ) · x(t )].

t=1

The total cost at period t includes the total production cost k(t ) · v(t ) + c(t ) · z(t ), and the inventory holding cost h(t) · x(t). The binary indicator variable v(t ) equals 1 if production occurs at period t and 0 otherwise. The dynamics of inventory and demand are given by

x(t ) = x(t − 1 ) + z(t ) − d (t ), t = 1, . . . , T . n(t) represents the number of new adopters at period t, N (t ) = t τ =1 n (τ ) the cumulative number of adopters at the end of period t, and m the total potential adopters in the market called “market potential.” At any period t, new adopters come from two segments of the remaining nonadopters with the population of m − N (t − 1 ): (1) innovators, whose adoption probability can be characterized by “coefficient of innovation” p, and (2) imitators, whose adoption probability depends on “coefficient of imitation” q and the cumulative proportion of adopters N (t − 1 )/m. Thus, the following is derived (Mahajan et al., 1990):

n(t ) N (t − 1 ) = p+q . m − N (t − 1 ) m Given that every consumer only purchases a unit of product, demand d(t) equals n(t). Thus, the demand at period t is



d (t ) = p +



q N (t − 1 ) · [m − N (t − 1 )], t = 1, . . . , T . m

(3.1)

We set the initial conditions as follows:

x ( 0 ) = 0, N ( 0 ) = 0. No inventory exists, and no demand occurs at the beginning of the planning horizon. Moreover, both inventory level and production size are nonnegative,

x(t ) ≥ 0, t = 1, . . . , T

z(t ) ≥ 0, t = 1, . . . , T . Therefore, we obtain the DLS model for a new product as follows: Minimize

TC =

T  [k(t ) · v(t ) + c (t ) · z(t ) + h(t ) · x(t )]

(3.2)

t=1

Subject to

x (0 ) = 0 N (0 ) = 0 x(t ) = x(t − 1 ) + z(t ) − d (t ), t = 1, . . . , T q d (t ) = N (t ) − N (t − 1 ) = [ p + N (t − 1 )] · [m m − N (t − 1 )], t = 1, . . . , T x(t ) ≥ 0, t = 1, . . . , T z(t ) ≥ 0, t = 1, . . . , T The decision variable of this model is the lot size z(t) at each period t. 3.3. Numerical experiments

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900 800

optimal total cost

700 600 500 400 300 200 100 0 100 80

1 60

0.8 0.6

40 0.4

20

0.2 0

q/p

0

p+q

Fig. 1. How optimal total cost evolves with ( p + q ) and q/p. Note: We set market potential m = 10 0 0, number of periods T = 12, setup cost k(t ) = 72, unit production cost c (t ) = 0, and unit holding cost h(t ) = 1.

Fig. 2. How optimal total cost varies with ( p + q ) and q/p.

To conclude, the optimal total cost TC∗ varies significantly with the diffusion pattern characterized by diffusion speed and consumer initiative level. In the next section, we discuss how to maximize profits by influencing the diffusion process via pricing decisions. 4. Dynamic lot-sizing models with pricing for new products In this section, we consider the DLS problem with the effect of prices. Noble and Gruca (1999) identify three strategies for new product pricing situation: (1) price skimming, which means firms set a high introductory price and systematically reduce prices over time; (2) penetration pricing, which means firms set a low introductory price to accelerate product adoption; and (3)

experience curve pricing, which means firms set a low introductory price to build a volume based on which they can reduce costs through accumulated experience. Our model retains the cost constant and does not consider experience curve pricing. We consider three specific pricing strategies for new products (Kunreuther & Schrage, 1973; Noble & Gruca, 1999), namely, constant, skimming, and penetration pricing strategies. We let P0 be the introductory price, that is, P (0 ) = P0 . For both skimming and penetration pricing strategies, we assume that the price varies with a constant rate: [P (t + 1 ) − P (t )]/P (t ) = η (Kunreuther & Schrage, 1973). That is,

P (t ) = P0 (1 + η )t , t = 0, . . . , T .

(4.1)

The rate of price change η = 0, η < 0, and η > 0 represents constant, skimming, and penetration pricing strategy, respectively.

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4.1. Model formulation

4.2. Algorithm

We maintain the assumption that all adopters will purchase a unit of product, but different from the model described in the preceding section from two aspects: (1) price decisions influence product diffusion process and (2) the firm aims to maximize its profit rather than to minimize its cost. The objective is to maximize the profit:

We first consider the sequential and coordinated decision models with constant price, in which cases g(t ) = 1 + μlnP, and give an upper and lower bound for the optimal price P∗ by the following lemma given that c (t ) = c.

π=

T 

e

[P (t ) · d (t ) − k(t ) · v(t ) − c (t ) · z(t ) − h(t ) · x(t )].

t=1

The original Bass model does not consider marketing decisions. Accordingly, Bass et al. (1994) develop a generalized version of the Bass model, known as the generalized Bass model (GBM), which involves marketing decisions (e.g., price) represented as follows:

dN (t ) = [ p + qN (t )] · [1 − N (t )] · g(t ), dt

(4.2)

where g(t), called “current marketing effort”, reflects the effect of marketing variables on the diffusion rate. Furthermore, Krishnan, Bass, and Jain (1999) modify the GBM by incorporating introductory price into g(t), where

g(t ) = 1 + μlnP0 + γ

P (t ) − P (t − 1 ) . P (t − 1 )

(4.3)

Krishnan et al. (1999) term γ as the diffusion price parameter because it characterizes how price changes accelerate and decelerate the product diffusion process. Introductory price also influences the diffusion process, which is described by μ. That is, the product diffusion process jointly depends on the diffusion and price-related parameters. Using Eq. (4.1), we can analyze three kinds of pricing strategies in the GBM and rewrite Eq. (4.3) as follows:

g(t ) = g(P0 , η ) = 1 + μlnP0 + γ η.

(4.4)

Therefore, we obtain the DLS model for a new product under different pricing strategies as follows: Maximize

π=

T 

Lemma 1. For both sequential and coordinated decision models under constant pricing strategy, the optimal price P∗ satisfies c < P ∗ <

[P0 (1 + η )t · d (t ) − k(t ) · v(t ) − c (t ) · z(t ) − h(t ) · x(t )]

t=1

(4.5) Subject to

x (0 ) = 0 x(t ) = x(t − 1 ) + z(t ) − d (t ), t = 1, . . . , T d (t ) = N (t ) − N (t − 1 )   q = p + N (t − 1 ) · [m − N (t − 1 )] · (1 m + μlnP0 + γ η ), t = 1, . . . , T x(t ) ≥ 0, t = 1, . . . , T z(t ) ≥ 0, t = 1, . . . , T The decision variables of this model are the introductory price P0 , the rate of price change η, and the production amount z(t) at each period t. To evaluate the performance of this coordinated pricing and production model, we also analyze another strategy that the firm makes decisions in the following sequence. The sales section de∗ to maximize its revenue; then the termines a constant price Pseq production section produces products to minimize the inventory∗ ) afterward related costs given the known demand d (t ) = d (t, Pseq (see model formulation for the sequential decision model in Appendix A). We call this strategy “sequential decision strategy” and use its performance as a baseline for the comparison with our coordinated pricing and production strategies.

1−mT μmT

.

Proof. The firm charges a price to make a net profit. On the one hand, the price must cover all costs including the unit production cost, that is, P∗ > c. On the other hand, the price should ensure the existence of demand, that is, the number of cumulative adopters T t=1 d (t ) ≥ 1. Since T 

d (t ) =

t=1

T   t=1

<

T 

p+



q N (t − 1 ) · [m − N (t − 1 )] · (1 + μlnP ) m

m(1 + μlnP )

t=1

= mT (1 + μlnP ), we have mT (1 + μlnP ) > 1−mT μmT

T

t=1

d (t ) ≥ 1; thus, we obtain P ∗ < 1−mT

e . Therefore, the optimal price P∗ satisfies c < P ∗ < e μmT for both sequential and coordinated decision models with constant price. Considering the complexity of new product pricing problems (Krishnan et al., 1999), we develop heuristic algorithms for both models.  We first solve the problem under the constant pricing strategy and denote the optimal constant price for the coordinated decision ∗ . Our algorithm deals with the DLS problem with a model as Pcons T ∗ constant price; thus, we further assume T1 t=0 P0 (1 + η )t = Pcons to obtain the optimal rate of price change η for the coordinated decision model with skimming and penetration pricing strategy: ∗ P0 = ηT Pcons /[(1 + η )T +1 − 1].

(4.6)

Our method is inspired by Kunreuther and Schrage (1973), which consists of iteratively solving the pricing and lot-sizing problems. As long as η and P0 are known, the demand is known for each period of the planning horizon, and the problem reduces to a classical DLS problem that can be solved by the Wagner– Whitin algorithm. Note that the zero inventory property holds, as in the Wagner–Whitin algorithm, in this reduced problem. We denote v = {v(1 ), . . . , v(T )} to be the binary indicator vector, and z = {z(1 ), . . . , z(T )} to be the production amount vector over the planning horizon. Consequently, one can derive the value of v from that of z and vice versa. Thus, we only consider v in the reminder of the discussion. Moreover, as long as P0 and η are given, we can compute the corresponding optimal value of v. For a given value of v, we know the production quantity of each period, which is a function of P0 and η. The inventory level of each period can be calculated accordingly. Consequently, as long as v is known, the profit, which is denoted as π (P0 , η) and a function of P0 and η, can be computed. The maximum value of π (P0 , η) can be calculated by solving the nonlinear program with two continuous variables, namely, P0 and η. In summary, for given values of P0 and η, the problem reduces to a DLS model; and the optimal production plan, that is, the value of v(t ), can be calculated using the Wagner–Whitin algorithm. On the other hand, when the value of v is known, the problem reduces to a pricing problem involving two continuous variables, namely, P0 and η. This case is a nonlinear program. The algorithm consists of iteratively solving these two problems. The algorithm can be specifically described as follows:

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X. Wu et al. / European Journal of Operational Research 000 (2016) 1–12 Table 5.1 Cumulative sales. Time N(t)

04/2011 100

03/2012 200

08/2012 300

10/2012 350

02/2013 400

08/2013 500

02/2014 600

06/2014 700

11/2014 800

Note: We set November 2008 as t = 0, and the cumulative sales reached 100 at t = 2.5 etc.

Step 1: k = 0, set initial values P0 and η to P00 and η0 , respectively. Step 2: Compute the corresponding production plan using the Wagner–Whitin algorithm, and denote the solution as vk . Step 3: If k > 0 and vk = vk+1 , then the algorithm converges, STOP. Step 4: Compute the optimal values of P0 and η, which are denoted as P0k+1 and ηk+1 respectively, by solving the pricing problem given the production plan defined by vk . Step 5: k ← k + 1, and go to Step 2. The convergence of this algorithm is guaranteed by the following relationship, which holds by definition:

π (P0k−1 , ηk−1 , vk−1 ) ≤ π (P0k , ηk , vk−1 ) ≤ π (P0k , ηk , vk ),

(4.7)

where π represents the objective function. 5. Numerical experiments and case study In this section, we first use data from a newly patented automated plate streaker system in a biochemical company to test our model, as well as conduct a sensitivity analysis to test the robustness of our results in Sections 5.1 and 5.2. Then we analyze the coordination between pricing strategies and new product diffusion patterns in Section 5.3. Finally, we assess in Sections 5.4 and 5.5 how the pricing strategy and the outcomes, such as profit and cost efficiency, vary with the new product diffusion patterns. 5.1. Case background and parameter estimation We consider a revolutionary automated plate streaker system for microbiology specimen processing in the automated microbiological streaking instrument market. This patented robotic system standardizes inoculum quantity and improves workflow by managing the major steps in processing liquid microbiology specimen samples. The company produces this automated plate streaker system with a constant unit production cost. The company pays royalty provision annually and renegotiate it every three years. If the company chooses to produce the patented automated plate streaker systems and retain them as inventories, then each product incurs a constant annual holding cost. We estimate the diffusion parameters given a price Porg = 110, 0 0 0 US dollars using the following equation:

d (t ) = N (t ) − N (t − 1 ) = [ p + qN (t )] · [m − N (t )] · [1 + μln(P/Porg ) + γ η].

(5.1)

We have η = 0 and current marketing effort g(t ) = 1 because the firm charges a constant price P (t ) = Porg . Therefore, we estimate the coefficient of innovation p, coefficient of imitation q, and market potential m under the given prices P (t ) = Porg . Similarly, the 1−mT

upper bound of price decision should be Porg e μmT in Lemma 1. We estimate the diffusion parameters with the data from the sales record. We note that the new patented product was launched in November 2008 and show cumulative sales data in Table 5.1. We recalculate the data using the continuous Bass model, present the cumulative sales as (Srinivasan & Mason, 1986)

N (t ) = mF (t ) =

m[1 − e−( p+q )t ] , 1 + qp e−( p+q )t

(5.2)

Table 5.2 Nonlinear least squares estimations.

p q m Note:

∗

Estimate

Std. error

t value

Pr(> |t|)

.013 .774 1185

1.95e−03 1.09e−01 1.79e+02

6.79 7.11 6.62

.0 0 050∗ ∗ ∗ .0 0 039∗ ∗ ∗ .0 0 057∗ ∗ ∗

-significant at .05,

∗∗

-significant at .01,

∗∗∗

-significant at .001

and estimate diffusion parameters. To overcome the shortcomings such as multicollinearity in the ordinary least squares approach proposed by Bass (1969), maximum likelihood and nonlinear least squares (NLS) estimations are preferred (Srinivasan & Mason, 1986). We adopt the NLS approach and show the results in Table 5.2. The estimated parameters in the continuous Bass model are as follows: coefficient of innovation p = 0.013, coefficient of imitation q = 0.774, market potential m = 1185, and R2 = 0.995. Fig. 3 shows the predicted and real cumulative sales.

5.2. Pricing and production decisions under different strategies Using the biochemical company as a background, we test our model and present the optimal pricing and production decisions under four strategies in Table 5.3. The firm charges the optimal ∗ = 147, 0 0 0 and earns profit π = 96, 357, 0 0 0 US dollars price Pseq over the planning horizon under the sequential decision strategy, ∗ and charges the optimal price Pcons = 149, 0 0 0 and earns profit π = 96, 806, 0 0 0 US dollars over the planning horizon under the constant pricing strategy. The coordinated decision model evaluates all costs and revenues, thereby obtaining an improved method to price and produce new products. Since the sequential decision model maximizes the gross profit, the sales section tends to charge a lower price to increase the sales volume. We find that the demands under the constant pricing strategy are lower than those under the sequential decision model. However, the increase in sales volume does not ensure profitability. Therefore, the cooperation between the sales and production sections is valuable. When dynamic pricing is allowed, the penetration pricing strategy is better than the constant pricing strategy, whereas skimming pricing cannot bring extra profit. By setting a low introductory price P0∗ = 102, 0 0 0 US dollars and raising prices continuously at a constant rate η∗ = 5.65%, the company’s using the penetration pricing strategy can obtain additional profits (π = 100, 554, 000 > 96, 806, 0 0 0). Table 5.3 shows that coordinated production-pricing decision or dynamic prices can improve profits. To test the robustness of our models, we conduct a sensitivity analysis by setting h(t ) = h ∼ N (10, 32 ), μ ∼ N (−1, .12 ), and γ ∼ N (−8, 12 ). We randomly generate 1, 0 0 0 groups of (h, μ, γ ) and test our model for different strategies, and find our results are robust. The coordinated pricing and production strategy increases 1.44% profit on average compared with the sequential decision strategy. The dynamic pricing strategy earns 15.62% more than the constant pricing strategy. Therefore, we present the first observation as follows: Observation 1. Coordinated production-pricing decision and dynamic pricing improve profitability.

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7

1000 cumulative sales predicted cumulative sales 900

800

700

600

500

400

300

200

100

0 2.5

3

3.5

4

4.5 year

5

5.5

6

6.5

Fig. 3. Bass model estimated by NLS. Table 5.3 Pricing and production decisions under different strategies. Period

t=1

2

3

4

= 147, 0 0 0, π = 96, 357, 0 0 0 Sequential decision strategy: d(t) 11 17 26 39 z(t) 28 0 65 0 ∗ = 149, 0 0 0, π = 96, 806, 0 0 0 Constant pricing strategy: Pcons d(t) 11 17 25 38 z(t) 28 0 63 0 ∗ = 102, 0 0 0, η∗ = 5.65%, Best pricing strategy (penetration): Ppene d(t) 10 14 21 30 z(t) 45 0 0 73

5

6

7

8

9

10

11

12

57 57

82 82

111 111

142 142

164 164

166 166

143 143

103 103

107 107

136 136

159 159

164 164

146 146

108 108

81 81

105 105

128 128

144 144

146 146

131 131

∗ Pseq

55 78 55 78 π = 100, 554, 000 43 60 0 60

Note: Unit production cost c (t ) = 50, 0 0 0 US dollars, setup cost k(t ) = 60 0, 0 0 0 US dollars, unit holding cost h(t ) = 10, 0 0 0 US dollars, number of periods T = 12, coefficient of innovation p = .013, coefficient of imitation q = .774, market potential m = 1185, baseline price parameter μ = −1, and diffusion price parameter γ = −8.

5.3. Rate of price change η∗ Fig. 4 shows how the optimal rate of price changes η∗ varies with diffusion price parameter γ by maintaining other parameters constant. The optimal rate of price changes η∗ > 0 holds when diffusion price parameter |γ | < 8.5, whereas η∗ < 0 holds when |γ | > 8.5. Therefore, firms should adopt the penetration pricing strategy when consumers are less sensitive to relative price changes (i.e., γ /μ is small), and choose the skimming pricing strategy when consumers care more about relative price changes (i.e., γ /μ is large). Fig. 5 shows how the optimal rate of price changes η∗ varies with diffusion speed p + q and consumer initiative level p/q given diffusion price parameter γ = −8.5. The rate of price changes η shows a negative relationship with p + q (see Fig. 5(b)). When the diffusion speed increases, η∗ shows slightly decreasing trends. Similarly, the rate of price changes η∗ > 0 when the consumer initiative level p/q is low, whereas η∗ < 0 holds when the consumer

initiative level is high. Therefore, firms can adopt the penetration pricing strategy when the product diffuses slowly or the consumer initiative level is low, and choose the skimming pricing strategy when the product diffuses rapidly or the consumer initiative level is high. Note that the skimming pricing strategy combines high introductory price and markdowns. On the one hand, high introductory price slows product diffusion during the early periods. On the other hand, continuous markdowns stimulate product diffusion during the later periods in which the number of cumulative adopters is relatively large. When the consumer initiative level is high, the product diffuses rapidly during the early periods but slowly during the later periods. Consequently, the skimming pricing strategy should be a preferable tool to improve profits by promoting product diffusion during the later periods at the expense of slowing down product diffusion during the early periods. Besides, whether to use the skimming pricing strategy also depends on consumers’ response to prices. If consumers are considerably

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Fig. 4. How rate of price changes varies with γ .

Fig. 5. How rate of price changes varies with diffusion pattern (when γ = −8.5).

sensitive to relative price changes but care less about the introductory price (i.e., γ /μ is large), then the skimming pricing strategy should also be a suitable option. Similarly, the penetration pricing strategy is better than the skimming pricing strategy when the consumer initiative level is low (i.e., p/q is small) or consumers are less sensitive to relative price changes than to the introductory price (i.e., γ /μ is small). Thus, we present the following observation:

Observation 2. The penetration pricing strategy outperforms the skimming pricing strategy when consumers are less sensitive to relative price changes than to the introductory price, when the product diffuses slowly, or when the consumer initiative level is low; otherwise, the firm can choose the skimming pricing strategy.

5.4. Profit analysis Since only one between skimming and penetration pricing strategies outperforms the constant pricing strategy, we select the better one and call it the “dynamic pricing strategy.” We further compare profits under three strategies, namely, sequential decision, constant pricing, and dynamic pricing strategies. Fig. 6 shows how profits vary with different strategies and diffusion patterns. We keep consumer initiative level p/q constant, vary diffusion speed p + q from 0.2 to 1 (Krishnan et al., 1999; Sultan et al., 1990), and show the results in Fig. 6(a) and (b). The penetration pricing outperforms the skimming pricing strategy when γ = −8 (i.e., consumers are not that sensitive to price changes), whereas the skimming pricing strategy dominates when γ = −9 (i.e., consumers are sensitive to price changes), as discussed before

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Fig. 6. How optimal profit varies with diffusion pattern.

(see Fig. 4). Similarly, we maintain diffusion speed p + q constant, vary consumer initiative level p/q from 0.01 to 0.1, and show the results in Fig. 6(c) and (d). From Fig. 6(a) to (d), we observe that the constant pricing strategy makes more profits than the sequential decision strategy, thereby advocating the adoption of coordinated pricing and production decision. Furthermore, the dynamic pricing strategy shows its advantage over the constant pricing strategy, suggesting that the firm should adjust its prices dynamically. Moreover, profits increase with diffusion speed and consumer initiative level for all strategies. 5.5. Cost structure analysis How will product diffusion patterns and pricing strategies influence the DLS decisions? We answer this question by performing a cost structure analysis. Since total demands vary with product diffusion parameters and pricing strategies, we define two ratios that are independent of the sales volume:

ρ1 =

T

t=1 [k

and

ρ2 = T

(t ) · v(t ) + c(t ) · z(t ) + h(t ) · x(t )] T t=1 P (t ) · d (t )

T

t=1 [k

(t ) · v(t ) + h(t ) · x(t )]

t=1 [k (t ) · v (t ) + c (t ) · z (t ) + h (t ) · x (t )]

(5.3)

,

(5.4)

where ρ 1 represents the cost efficiency to earn a unit profit, and ρ 2 characterizes the cost efficiency to satisfy a unit demand. These two ratios enable us to obtain insights into how product diffusion patterns and pricing strategies jointly influence the cost structure.

We retain the diffusion speed p + q, vary the consumer initiative level p/q from 0.01 to 0.1, and show the results in Fig. 7. We attempt to further understand how to improve profits by selecting among different pricing strategies. Fig. 7 shows that the constant pricing strategy always has a smaller cost-revenue ratio ρ 1 than that of the sequential decision strategy. However, ρ 2 shows the opposite result (see Fig. 7(b) and (d)). Hence, coordinated decisions assist in refining the cost efficiency to earn a unit profit rather than facilitate the improvement of the cost efficiency to satisfy a unit demand. The dynamic pricing strategy reduces ρ 1 compared with the constant pricing strategy. Despite a possible increase in total demand, dynamic pricing refines cost effectiveness to earn a unit profit. We further find that this strategy also significantly reduces ρ 2 , thereby suggesting that the pricing strategy could reshape the cost structure in the DLS problems. Moreover, ρ 1 and ρ 2 show decreasing trends with consumer initiative level p/q when diffusion speed p + q remains constant. Therefore, an increase in the consumer initiative level helps improve cost effectiveness. Subsequently, we maintain consumer initiative level p/q constant, vary p + q from 0.2 to 1, and show the results in Fig. 8. Similarly, we find that the constant pricing strategy always has a smaller cost-revenue ratio ρ 1 than that of the sequential decision strategy, whereas ρ 2 shows the opposite result. Furthermore, the dynamic pricing strategy helps reduce ρ 1 considerably. ρ 1 and ρ 2 significantly decrease with diffusion speed p + q when consumer initiative level p/q remains constant. Thus, an increase in product diffusion speed facilitates the improvement of cost effectiveness. To conclude, we present the following observation:

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Fig. 7. How cost structure varies with consumer initiative level p/q.

Observation 3. Both pricing strategies and product diffusion pattern reshape the firm’s cost structure. Coordinated decisions and dynamic pricing strategy substantially reduce the cost-revenue ratio, whereas an increase in the consumer initiative level or product diffusion speed can improve cost efficiency. 6. Concluding remarks Our research complements the existing research stream in coordinating production and pricing management. Different from previous studies on DLS models by mainly considering mature products, this study considers production planning decisions for new products. Observing new product diffusion will lead to dynamics of demand; thus, we integrate DLS and discrete Bass models to provide optimal decisions for production planning problems. We further analyze coordinated pricing and DLS decisions for new products when both pricing strategies and new product diffusion influence the dynamics of demand. We study the joint influence of product diffusion and pricing parameters on the DLS decisions, as well as show that coordinated production-pricing decision can improve profitability. This observation suggests that the production and sales sections can improve the cooperation in crossfunctional decisions, which is consistent with previous findings (Hausman et al., 2002). We also provide new insights into the strategic choice among constant, penetration, and skimming pricing strategies when coordinating with production planning. A firm can choose the penetration pricing strategy when consumers are less sensitive to relative price changes than to the introductory price, when the product diffuses slowly, or when the consumer initiative level is low. By contrast, the skimming pricing strategy fits the opposite situations. A strategic choice among constant,

penetration, and skimming pricing strategies improves profitability. Hence, a firm should vary its pricing strategy to fit different product diffusion patterns. Finally, both product diffusion patterns and pricing strategies reshape the cost structure of a firm. An increase in diffusion speed or consumer initiative level assists in refining cost efficiency. Coordinated decision or dynamic pricing strategy can substantially reduce the cost-revenue ratio. Therefore, a firm can refine its cost structure by stimulating new product diffusion or varying its pricing strategy. In future work, our models can be extended by considering capacity constraints. During the new product diffusion process, the peak demand might exceed the firm’s production capacity. Further, we can extend our model by considering cost learning during the product life cycle. For new products, marginal production costs decrease with cumulative product sales. Therefore, it is interesting to incorporate cost learning in marginal production costs and setups in future research. Acknowledgments We gratefully acknowledge the editor and the referees for their valuable comments. This research is supported by National Natural Science Foundation of China (Nos. 71531009; 71271095; 71620107002), National Social Science Foundation of China (No. 16ZDA013), and Chutian Scholarship. Appendix A. Sequential decision model In this section, we formulate the sequential decision model in which the firm makes decisions in the following sequence: the ∗ sales section determines a constant price Pseq to maximize its

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Fig. 8. How cost structure varies with diffusion speed p + q.

revenue (see Model A.1), then the production section produces products to minimize the inventory-related costs given the known ∗ ) afterwards (see Model A.2). We evaluate demand d (t ) = d (t, Pseq the outcomes such as profit and cost structure under sequential decision strategy and coordinated decision strategy. In the first stage, the sales section optimizes the price P to maximize the revenue: Maximize

R=

T 

P · d (t )

(A.1)

t=1

Subject to

P > c, t = 0, . . . , T d (t ) = N (t ) − N (t − 1 ) q = [ p + N (t − 1 )] · [m − N (t − 1 )] · [1 + μlnP ], t = 1, . . . , T m ∗ , and then the Denote the optimal price in Model (A.1) to be Pseq ∗ demand is d (t, Pseq ). In the second stage, the production section minimizes the total cost by searching for an optimal production decision: Minimize

TC =

T 

[k(t ) · v(t ) + c (t ) · z(t ) + h(t ) · x(t )]

t=1

Subject to

x (0 ) = 0 x(t ) = x(t − 1 ) + z(t ) − d (t ), t = 1, . . . , T ∗ d (t ) = d (t, Pseq ), t = 1, . . . , T

(A.2)

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