Channel selection in a supply chain with a multi-channel retailer: The role of channel operating costs

Int. J. Production Economics 173 (2016) 54–65

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Channel selection in a supply chain with a multi-channel retailer: The role of channel operating costs Wei Wang a,1, Gang Li a,n, T.C.E. Cheng b,2 a School of Management, Xi'an Jiaotong University, The State Key Lab for Manufacturing Systems Engineering, The Key Lab of the Ministry of Education for Process Control & Efficiency Engineering, Xi'an 710049, China b Department of Logistics and Maritime Studies, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong SAR, China

art ic l e i nf o

a b s t r a c t

Article history: Received 12 February 2015 Accepted 1 December 2015 Available online 15 December 2015

In this paper we establish a linear demand model to explore the channel selection and pricing strategy in a supply chain that comprises a dominant multi-channel retailer, and a manufacturer that sells two horizontally differentiated products through its own direct channel and the retail channel, respectively. We find that the gap between the online and offline channels' operating costs is critical to the retailer's choice of its channel selection strategy. Multi-channel selling is the best choice for the retailer only when the cost-gap is narrow enough. Conversely, the retailer should only select the low-cost channel if the cost-gap is too large. In addition, we find that small product differentiation is more favorable to the manufacturer in a retailer-led supply chain as the retailer is forced to reduce its margin and retail price. Meanwhile, the manufacturer can benefit from a rise in the wholesale price and increasing demand in the retail channel. Finally, we consider the case where the manufacturer, instead of the retailer, acts as the decision maker for channel selection. We find that in theory the manufacturer will adopt the same channel selection strategy as the retailer in this case. & 2015 Elsevier B.V. All rights reserved.

Keywords: Multi-channel retailer Channel selection Operating costs Product differentiation Game theory

1. Introduction E-commerce has been developing rapidly in recent years. B2C e-commerce sales worldwide reached US$1.471 trillion in 2014, a jump of almost 20% over 2013 (eMarketer, 2014). In China, the growth in e-commerce sales is phenomenal, up to 103.7% and 94.1% in 2011 and 2012, respectively (eMarketer, 2013). Against this backdrop, many traditional retailers have built their online arms, evolving into large multi-channel retailers. For instance, most of the top global retailers like War-Mart, Tesco, Metro, Costco Wholesale etc. have built online channels (NRF, 2014). Besides the traditional retailers, some large electronic retailers such as Bonobos, Warby Parker, Fab.com, Gilt.com, JD.com etc. have also transformed themselves into multi-channel retailers by building their own physical stores or cooperating with other bricks-andmortar retailers. The rise of large multi-channel retailers has a profound impact on supply chains. They dominate the market with a big share and wield increasingly strong pricing power in the supply chain. To counter the increasing power of retailers in n

Corresponding author. Tel.: þ 86 13096915852; fax: þ 86 29 8266 4643. E-mail addresses: [email protected] (W. Wang), [email protected] (G. Li), [email protected] (T.C.E. Cheng). 1 Tel.: þ86 15229262173; fax: þ 86 29 8266 4643. 2 Tel.: þ852 27665215; fax: þ852 27665215. http://dx.doi.org/10.1016/j.ijpe.2015.12.004 0925-5273/& 2015 Elsevier B.V. All rights reserved.

supply chains, many manufacturers such as Apple, Haier, Nike, Coca Cola etc. open their direct online channels in droves with a view to improving the performance of the whole supply chain (Chiang et al., 2003). The competition between a manufacturer's direct channel and a retailer's multi-channels has become an important issue to address in supply chain management. Although it has become a general trend for a single-channel retailer to transform itself into a multi-channel retailer, many fundamental questions remain to be answered. Addressing such questions will help firms to adopt the most fitting channel selection strategies. Some giant retailers still focus exclusively on single-channel selling. For instance, 7-eleven does not provide online purchase options for consumers; Amazon.com has not opened a physical store yet. Some retailers have failed in their efforts of building multi-channels. For instance, Suning.com suffered a great drop in net profit when it adopted a new multichannel strategy in 2013, largely due to a sharp increase in the online marketing cost. Likewise, Carrefour was forced to exit the emarket of Tsingdao in 2010, due to the unaffordable online delivery cost. In a nutshell, one of the main reasons for such failures is that retailers underestimate or fail to recognize the impacts of their channel operating costs, i.e., the total cost of marketing, delivering, warehousing, and other costs incurred to sell a product through a specific channel. In view of the great importance of channel operating costs to channel decisions (Park and Keh, 2003;

W. Wang et al. / Int. J. Production Economics 173 (2016) 54–65

Khouja et al., 2010), we carry out this study to explore the role of channel operating costs in retailers' choice of channel selection strategies. To the best of our knowledge, there is a lack of research to address the issues discussed above in the literature. There are studies on the competition between a direct channel and a retail channel, e.g., Chiang et al. (2003), Tsay and Agrawal (2004), which focus on the channel entry decision of a manufacturer, rather than the channel selection of a retailer. There is research that analyzes the channel decisions of multi-channel retailers from a horizontal competition perspective, e.g., King et al. (2004), and Liu et al. (2006), which does not consider vertical competition in a supply chain. Choi (1991) and Pan et al. (2010) considered a retailer-led supply chain, but they only analyzed their problems from a pure vertical competition perspective without involving the manufacturer's direct channel. Moreover, little research has explicitly considered the importance of the operating cost structure to channel decisions. We study the multi-channel selection strategy of a supply chain that comprises a manufacturer with a direct online channel and a dominant multi-channel retailer. Supposing the latter is the leader of the supply chain in pricing, we take the heterogeneous operating costs of different channels into consideration in performing our analysis. Specifically, we address the following fundamental research issues: a) What are the impacts of operating costs on the multi-channel retailer's choice of its channel selection strategy? b) What are the pricing equilibria of the multi-channel retailer and the manufacturer? c) How does the market environment influence the equilibria? We make three contributions to the literature on the multichannel selection strategy in a retailer-led supply chain. First, we find that multi-channel selling is the best choice for the retailer only when the cost-gap is narrow enough. Conversely, the retailer should only select the low-cost channel if the cost-gap is too large. Second, we obtain the counter-intuitive finding that the manufacturer tends to reduce the product differentiation in a retailerled supply chain. In fact, small product differentiation forces the retailer to reduce its margin and retail price, while benefiting the manufacturer by a rise in the wholesale price and increasing demand in the retail channel. Finally, we consider the case where the manufacturer, instead of the retailer, has the power to control the distribution channel. We find that in theory the manufacturer will adopt the same channel selection strategy as the retailer in this case. We organize the rest of the paper as follows: In Section 2 we review the related literature. We introduce the notation and assumptions of the model in Section 3. In Section 4 we present the equilibrium analysis of the model starting with pricing equilibrium, followed by channel selection, and then comparative statics analysis. In Section 5 we discuss the case where the manufacturer acts as the decision maker for channel selection. In Section 6 we conclude the paper and discuss the managerial implications of the research results. We provide all the proofs in the appendices.

2. Literature review Our paper is related to the vast literature on channel competition and selection. A stream of the literature examines the channel decisions of retailers from a horizontal competition perspective. Balasubramanian (1998) presented the seminal work on modeling the horizontal competition between a direct marketer and conventional retailers, showing the importance of information and market coverage in channel competition. Lal and Sarvary

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(1999) explored the impact of the Internet on competition in different shopping and distribution contexts, showing that the Internet channel only increases firm profits under certain conditions. King et al. (2004) and Bernstein et al. (2008) supported Lal and Sarvary (1999), highlighting the strategic necessity of the Internet channel. Liu et al. (2006) examined the entry-deterrence strategies of a multi-channel retailer based on the previous literature. Yan (2010) introduced product differentiation into research on horizontal channel competition. Following this stream of research, we explore the channel selection strategy of a multichannel retailer that has both the online and offline channels in the context of a supply chain. Similar to Lal and Sarvary (1999), we find that channel operating costs are a critical factor in the retailer's channel selection strategy. The multi-channel strategy should be adopted only when the gap between the online and offline channel operating costs is narrow enough. Although many studies have highlighted the importance of the operating cost structure to channel decisions, e.g., Park and Keh (2003), and Khouja et al. (2010), there is no research that addresses the issue of how to make the optimal channel selection decision given the channel operating costs. Recently, Hsiao and Chen (2014) studied the channel selection of a supply chain with multi-channel retailers, but they ignored costs in their model. We endeavor to fill the research gap by exploring the role of channel operating costs in retailers' choice of channel selection strategies. Specifically, we provide the critical conditions related to the channel operating costs in a retailer's channel decisions. Moreover, we consider product differentiation in our model and find that small differentiation is profitable to the retailer only when it has a price advantage over the manufacturer. Another stream of research examines the channel strategy of the manufacturer in a supply chain. Choi (1991, 1996) presented the seminal papers examining vertical competition in a supply chain. In particular, Choi (1996) took product differentiation into consideration, finding that product differentiation helps manufacturers but hurts retailers. Recent papers pay more attention to the direct channel entry strategy of the manufacturer in a supply chain. Chiang et al. (2003) constructed a pricing game between a manufacturer and an independent retailer. They argued that the manufacturer's direct channel can be used to constrain the retailer's pricing behavior, thus increasing the performance of the supply chain. Tsay and Agrawal (2004) reached similar conclusions, establishing that the addition of the direct channel can make both the manufacturer and retailer better-off. Wang et al. (2011), however, found that the introduction of the direct channel can be detrimental to manufacturers when they compete along dimensions that differ from prices. Other studies such as Park and Keh (2003), Kumar and Ruan (2006), Guo and Liu (2008), Khouja et al. (2010), and Li et al. (2015) reached the general conclusion that whether a hybrid channel structure benefits the manufacturer and the retailer depends on specific circumstances. However, most of the studies cited above use a manufacturer-Stackelberg game to analyze channel selection decisions. In this paper we extend the research on channel selection by involving a multi-channel retailer that is the leader of the supply chain in pricing. We highlight the strategic necessity of the manufacturer's direct channel in a retailer-led supply chain because it helps to counter the retailer's pricing power. Specifically, relying on the direct channel, the manufacturer can force the retailer to cut price when product differentiation is small. Counter-intuitively, small product differentiation is more favorable to the manufacturer in the retailer-led case, which challenges the consensus of the extant literature.

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3. Notation and assumptions In this study we consider a supply chain that comprises a manufacturer and a multi-channel retailer. The manufacturer has a direct online channel, while the retailer has both a conventional physical store and an online channel. The retailer is the leader of the supply chain in pricing. The manufacturer sells two horizontally differentiated products A and B, where the former is sold through its direct online channel while the latter is sold through the multi-channel retailer. This setup reflects the common phenomenon that manufacturers adopt the product differentiation strategy to alleviate channel conflicts in the real world. The retailer decides the channel(s) to sell B. Obviously, there exist three channel selection strategies for the retailer (see Fig. 1): (I) through the offline channel only, (II) through the online channel only, and (III) through multi-channels (i.e., both online and offline). To avoid channel conflicts, the retailer charges the same price in the online and offline channels if it adopts Strategy III. We develop a linear demand model that is much in the spirit of Cai et al. (2009). One major difference is that we model both consumers' channel preference (through the parameter λ) and product preference at the same time. To build a linear demand model, we first divide consumers into two segments according to their channel preference: (1) the segment preferring shopping in store (denoted as PS consumers) and (2) the segment preferring shopping on the Internet (denoted as PN consumers). This setup is popularly used in many studies, e.g., Cai et al. (2009) and Khouja et al. (2010). Meanwhile, some PS consumers prefer product A while the others prefer product B, so do the PN consumers. We assume that consumers' channel preference and product

preference are independent, yielding four sub-segments as follows: the segment preferring buying A in store, the segment preferring buying A on the Internet, the segment preferring Table 1 Symbols and notation. Symbols and notation

Explanation

αps αpn

The number of the consumers who prefer shopping in store; The number of the consumers who prefer shopping on the Internet; The price sensitivity of consumers; The price-gap sensitivity of consumers (product differentiation); The discount coefficient of demand due to consumers' channel mismatch (the degree of consumers' acceptance of their less preferred channels); The initial proportion of consumers who prefer product A; Retail price; Direct selling price of the manufacturer; Wholesale price; Unit operating cost of the retailer's offline channel; Unit operating cost of the retailer's online channel; Unit operating cost of the manufacturer's online channel; The retailer's margin; Demand; The manufacturer; The retailer; The numbers of channel selection strategies (Offline channel only; online channel only; multi-channels); The online channel of the retailer; The offline channel of the retailer.

β γ λ

θ pr pm w cs crn cmn ur D m r Ι; ΙΙ; ΙΙΙ rn rs

Fig. 1. Supply chain structures in different channel selection strategies.

W. Wang et al. / Int. J. Production Economics 173 (2016) 54–65

buying B in store, and the segment preferring buying B on the Internet. Table 1 summarizes the main symbols and notation used in this paper. Consumers' purchase decision is determined by three factors: product preference, channel preference, and price. Particularly, product preference dominates channel preference, which means that a consumer would rather buy a preferred product from a less preferred channel than buy a less preferred product from a preferred channel. If the price is consistent, i.e., pr ¼ pm , the initial proportion of consumers who prefer product A is θ. Correspondingly, the initial proportion of consumers who prefer product B is 1  θ. If the price is inconsistent, i.e., pr a pm , some consumers might switch from one channel to another due to the price gap characterized by the price-sensitivity parameter γ , which indicates the degree to which the two products are differentiated, whereby the smaller the value of γ is, the higher the degree of product differentiation is (Choi, 1991, 1996; Yan, 2010; Pan et al., 2010). We also assume that the demand function for the percent market size has a linear form with a slope of β . In addition, we introduce the parameter λ A ð0; 1Þ to model the extent to which consumers' purchase decisions are impacted by their channel preferences. We define λ as the discount coefficient of demand due to consumers' channel mismatch. For instance, if pr ¼ pm ¼ 0, the proportion of the PS consumers who will buy A online is θλ o θ, which means that a proposition ð1  λÞ of the PS consumers will give up buying A since it is sold in the channel they dislike (we call it channel mismatch), even though they prefer A and the price is zero. In addition, we assume that αps ¼ α 4 0, αpn ¼ 1, and β ¼ 1 for simplification purposes; the unit operating costs (UOCs) of the channels differ from one another and satisfy crn ; cmn ; cs A ð0; 1Þ to ensure the meaningfulness of the demand functions. Finally, we assume that both the fixed cost of changing the channel selection strategy and the marginal production cost of the manufacturer are zero. In real-life practice, firms can estimate all the parameters used in our model. The channel operating costs (crn , cmn , and cs ) are the basic operating cost elements of firms. Usually, firms can obtain information on them from their financial statements. For example, they can calculate the total cost (or expense) incurred in marketing, delivering, warehousing, and other processes related to selling their products, and then allocate it to online and offline channels according to their respective sales. α and θ are proportional coefficients that reflect the characteristics of consumer groups. α can be obtained from third-party reports on consumer channel preference. For example, many firms, e.g., IBM, Alibaba, and comScore, release their research reports on consumer channel preference annually. As for θ, firms can easily estimate it by conducting a sampling survey of consumer preference of their products. Price sensitivity γ is a parameter in classical economics. Firms can easily estimate it by observing fluctuations in sales volume when price changes. They may also estimate it by employing sophisticated estimating methods proposed by scholars such as Morris and Joyce (1988), Tellis (1988), Hoch et al. (1995) etc. λ essentially indicates consumers' acceptance (or matchness) of their less preferred channels. The larger λ is, the higher the acceptance is, and the less likely for consumers to give up purchasing due to channel mismatch. It can be estimated by various predictors, such as consumers' perceptions of channel transaction cost (Liang and Huang, 1998), shopping experience (Bellman et al., 1999; Zinkhan et al., 2002), risk averseness (Gupta et al., 2004), and perceptions of the performance of channels (Kacen et al., 2013). Besides the most commonly used regression models, neural network models (Chiang et al., 2006) and social network models (Verbraken et al., 2014) are viable options to predict consumers' acceptance of shopping channels.

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In the game, the retailer is the Stackelberg leader and the manufacturer is the follower. The game sequence is described as follows: 1) The retailer selects the channel(s) to sell product B. 2) The retailer decides its margin ur , where ur ¼ pr w (Choi, 1991, 1996; Pan et al., 2010; Wu et al., 2012). 3) The manufacturer decides direct selling price pm and wholesale price w simultaneously.

4. Equilibrium analysis 4.1. Pricing game The demand functions of each strategy are described as follows: a) Strategy I: offline channel only

 DΙm ¼ αps θλð1  βpm Þ  γ ðpm  pr Þ þ αpn θð1  β pm Þ  γ ðpm pr Þ ; ð1Þ

 DΙrs ¼ αps ð1  θ Þð1  β pr Þ  γ ðpr  pm Þ þ αpn ð1  θ Þλð1  βpr Þ  γ ðpr  pm Þ :

ð2Þ b) Strategy II: online channel only

 DΙΙ m ¼ αps θλð1  β pm Þ  γ ðpm  pr Þ þ αpn θ ð1  β pm Þ  γ ðpm pr Þ ; ð3Þ DΙΙ rn ¼ αps





 λð1  θÞð1  βpr Þ  γ ðpr  pm Þ þ αpn ð1  θÞð1  βpr Þ  γ ðpr  pm Þ :

ð4Þ c) Strategy III: multi-channels

 DΙΙΙ m ¼ αps θλð1  β pm Þ  γ ðpm  pr Þ þ α pn θ ð1  β pm Þ  γ ðpm  pr Þ ; ð5Þ
 DΙΙΙ rs ¼ αps ð1  θ Þð1  β pr Þ  γ ðpr pm Þ ;

ð6Þ


 DΙΙΙ rn ¼ αpn ð1  θ Þð1  β pr Þ  γ ðpr pm Þ :

ð7Þ

In Eq. (1), θλð1  βpm Þ  γ ðpm  pr Þ is the percentage of PS consumers who buy product A from the direct online channel. Correspondingly, θð1  βpm Þ  γ ðpm  pr Þ is the percentage of PN consumers who buy product A from the direct online channel. We can interpret Eqs. (2)–(7) in a similar way. We work out the profit functions of each strategy as follows: a) Strategy I: offline channel only

π Ιm ¼ DΙm ðpm cmn Þ þ DΙrs w;

ð8Þ

π Ιr ¼ DΙrs ður  cs Þ:

ð9Þ

b) Strategy II: online channel only

π ΙΙm ¼ DΙΙm ðpm cmn Þ þ DΙΙrn w;

ð10Þ

π ΙΙr ¼ DΙΙrn ður  crn Þ:

ð11Þ

c) Strategy III: multi-channels ΙΙΙ ΙΙΙ ΙΙΙ π ΙΙΙ m ¼ Dm ðpm  cmn Þ þ ðDrs þ Drn Þw:

ð12Þ

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Table 2 The equilibria of Strategies I–III.

uor wo pom π or Dor π om

Strategy I

Strategy II

Strategy III

t 1 þ cmn t 0 cs 2 þ 2ðt 0 þ t 1 Þ ð1  cmn Þt 0 1  cs 4 þ 4ðt 0 þ t 1 Þ 1 þ cmn 2 2 1 ½ð1  cs Þt 1 þ ðcmn  cs Þt 0  t0 þ t1 8

t 2 þ cmn t 0 crn 2 þ 2ðt 0 þ t 2 Þ ð1  cmn Þt 0 1  crn 4 þ 4ðt 0 þ t 2 Þ 1 þ cmn 2 2 1 ½ð1  crn Þt 2 þ ðcmn  crn Þt 0  t0 þ t2 8

ð1  cs Þt 1 þ ðcmn  cs Þt 0 4
4ð1  cmn Þ2 ðα2  α2 θ þ αλ þ θ  αt 1 Þ

ð1  crn Þt 2 þ ðcmn  crn Þt 0 4 4ð1  cmn Þ2 ½ðαλ þ 1Þðt 2 þ t 0 Þ  t 22  þ ½ðcmn  crn Þt 0 þ ð1  crn Þt 2 2

t 3 þ cmn t 0 c0 2 þ 2ðt 0 þ t 3 Þ ð1  cmn Þt 0 1  c0 4 þ 4ðt 0 þ t 3 Þ 1 þ cmn 2 2 1 ½ð1  c Þt þ ðc s 3 mn  cs Þt 0 þ ðcs  crn ÞtÞ 8 t0 þ t3 ð1  cs Þt 3 þ ðcmn  cs Þt 0 þ ðcs  crn Þt 4 4ð1 cmn Þ2 ½ðλα  t 3 Þðt 3 þ t 0 Þ þ t 3 t

ðt 1 þ t 0 Þ þ t 0 t 1  þ ½ðcmn  cs Þt 0 þ ð1 cs Þt 1 2 16ðt 1 þ t 0 Þ Else

16ðt 2 þ t 0 Þ

þ ½ð1 cs Þt 3 þ ðcmn  cs Þt 0 þ ðcs  crn ÞtÞ2 16ðt 3 þ t 0 Þ

t3  t þ crn t0 þt t3 t 0 ¼ ð1 þ αÞγ, t 1 ¼ ðα þ λÞð1  θÞ, t 2 ¼ ðαλ þ 1Þð1 θÞ, t 3 ¼ ðα þ 1Þð1  θÞ, t ¼ 1  θ þ γ, c0 ¼ cs t 0tþ 0 þ t3

ΙΙΙ ΙΙΙ π ΙΙΙ r ¼ Drs ður  cs Þ þ Drn ður  crn Þ:

ð13Þ

Both the retailer and the manufacturer maximize their own profits in the game. Based on backward induction, we obtain the equilibria of each strategy shown in Table 2 (we provide the details in Appendix A). Definition 1. : Since the retailer charges the same price online and offline when adopting Strategy III, the multi-channels can be treated as one “unified” channel (i.e., the retail channel), and we define c0 as the retail channel's average UOC. t3  t α þc 1 0 þ crn t 0 þt t3 ¼ cs α þ Note that c0 ¼ cs t 0tþ rn α þ 1, so c is the 1 0 þ t3 weighted average of each channel's UOC, and the weighting coefficients are the proportions of the consumers who prefer the corresponding channels. It is apparent that c0 falls between cs and crn , i.e., min fcn ; crs g o c0 o max fcn ; crs g. Compared with only selecting the low-cost channel, the retailer has to bear a highercost when selecting multi-channels.

4.2. Channel selection Obviously, the retailer will choose a channel strategy from the three strategies to maximize its profit. Lemma 1. : To compare the profits of the three strategies, we only o need to compare π rΙΙΙo and π rΙo and π rΙΙΙo and π ΙΙ respectively. r o Comparing π rΙo and π ΙΙ is unnecessary. r Proof. : See Appendix B. ■ A multi-channel retailer never sells products only through the high-cost channel. The trade-off that a retailer encounters when selecting selling channels, as Lemma 1 shows, is actually between “multi-channels” and the “low-cost channel”. Multi-channels cover a larger market than the low-cost channel because they can fit consumers' heterogeneous channel preference, but incur a higher operating cost than the low-cost channel, i.e., c0 4 min fcn ; crs g as aforementioned. As to the high-cost channel, it is never an alternative since it is inferior to multi-channels both in terms of cost and market coverage. Proposition 1. : For the retailer, 1) If 0 o cs o A13 crn þ B13 , the optimal channel selection strategy is Strategy I; 2) If A13 crn þ B13 o cs o A23 crn þ B23 , the optimal channel selection strategy is Strategy III; 3) If A23 crn þ B23 o cs ocs , the optimal channel selection strategy is Strategy II.

Fig. 2. Distribution of the optimal channel selection strategies. t0 In Proposition 1, cs ¼ t1tþ1 þcmn t 0 , and the expressions of A13 , A23 , B13 , and B23 are given in Appendix B (see Appendix B for the proof). In Fig. 2, we use a numerical example to illustrate Proposition 1 by setting α ¼ 0:5; λ ¼ 0:3; θ ¼ 0:6; γ ¼ 0:01; cmn ¼ 0:04. When 0 o cs o A13 crn þ B13 (region Ι), the UOC of the offline channel is much lower than that of the online channel, so the retailer only selects the offline channel to sell product B. When A23 crn þ B23 o cs o cs (region ΙΙ), the UOC of the offline channel is much higher than that of the online channel, so the retailer only selects the online channel to sell product B. When A13 crn þB13 o cs oA23 crn þ B23 (region ΙΙΙ), the gap between online and offline UOCs is medium and there exists no significant difference between the online and offline channels in terms of channel operating costs, so the retailer tends to select multi-channels to take advantage of their broad cover of the market in spite of a relatively high-cost. Proposition 1 reveals that multi-channel selling is not always the best choice for the retailer, at least in the case where the price is consistent. This finding provides important managerial insights. Although firms tend to adopt the multi-channel strategy, it is counter-intuitive that multi-channel selling has a significant hidden hazard caused by the unbalanced operating costs between the online and offline channels. Therefore, the multi-channel strategy should not be adopted if there is a wide gap between the two channels' operating costs. This finding is consistent with business practice in the real world. For instance, multi-channel retailers generally do not sell all product categories through their multichannels, especially when they charge a consistent price online

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Fig. 3. The sensitivity of each region's size w.r.t the parameters.

and offline. For example, in China, Walmart sells furniture only in its retail stores because of the high shipping cost, while Suning sells pedometers only in its online shops because of the high displaying cost. Also, many retailers only build multi-channels in some specific cities or areas. For example, Carrefour launches online stores only in several countries such as France and Romania; JD.com only provides offline purchase options in Taiyuan. One of the reasons for these phenomena is that the cost-structure, i.e., the coordinate of ðcrn ; cs Þ in Fig. 2, varies when retailers sell different product categories or locate in different areas. Therefore, they adopt differentiated channel selection strategies for different categories and areas. Selecting the right channel for the dominant retailer is not a straight-forward decision, which is widely affected by many factors, including consumers' population size α, product preference θ, price sensitivity γ , acceptance of less preferred channels λ, and the operating cost of the manufacturer's direct channel cmn . To succeed in the market, the dominant retailer should take all these factors into consideration to craft its optimal channel selection strategy according to the critical conditions given in Proposition 1. To generate insightful implications from the analytical results, we conduct a numerical sensitivity analysis to dig some qualitative and deeper insights implied by Proposition 1. The size of each region in Fig. 2 reflects the likelihood that the corresponding channel selection strategy is chosen. We explore how a retailer should adjust its channel selection strategy as the market environment changes by analyzing the sensitivity of each region's size with respect to various model parameters. Fig. 3 shows the results of the sensitivity analysis and Appendix C provides the calculation details.

Remember that a retailer's competitive edge stems from its owning multi-channels that cover a wider range of the market than a single channel. So a retailer should exploit the edge by favouring multi-channel selling in the situation where price competition intensifies (i.e., γ increases, shown in Fig. 3(c)), or when it is disadvantaged in terms of channel operating costs (i.e., cmn decreases, shown in Fig. 3(a)) or product popularity (i.e., θ increases, shown in Fig. 3(b)). Particularly, it should avoid selling products through the online channel (the same channel type as the manufacturer's direct channel) when its product is less preferred by consumers. Fig. 3(e) reveals that consumers' acceptance of their less preferred channels (i.e., λ) has the most significant impact on the dominant retailer's channel selection. Specifically, the retailer should more favor single-channel selling if consumers have high acceptance of their less preferred channels (i.e., λ is larger, shown in Fig. 3(e)) because consumers' channel mismatch now has less influence on their purchase and, as a result, multichannels lose their advantage over a single channel in market coverage but retain their disadvantage in cost. Besides, our results also suggest that a retailer should less favor multi-channels if more consumers prefer the offline channel (i.e., α is larger, shown in Fig. 3(d)). 4.3. Comparative statics analysis To gain further insights on how the market environment influences the market equilibrium, we make a comparative statics analysis here. Table 3 shows the effects of parameter changes on the equilibrium prices (we provide the proof in Appendix D). The results in Table 3 provide a number of insights. The manufacturer's direct selling price is independent of the market

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Table 3 The effects of parameter changes on equilibrium prices.

pr ur w

θ ↑

γ ↑

λ ↑

α↑

↓ ↓ ↑

↓ ↓ ↑

↑ðΙ; ΙΙÞ ↑ðΙ; ΙΙÞ ↓ðΙ; ΙΙÞ

↑ðΙÞ; ↓ðΙΙÞ; signðcs  crn Þ ðΙΙΙÞ ↑ðΙÞ; ↓ðΙΙÞ; signðcs  crn Þ ðΙΙΙÞ ↓ðΙÞ; ↑ðΙΙÞ; signðcrn  cs Þ ðΙΙΙÞ

Explanation of symbols: ↑ stands for “increase”, ↓ stands for “decrease” ↑ðΙÞ means “increase” in Strategy I and so forth.

environment, e.g., θ, γ , α, and λ, which is consistent with the result of Cai et al. (2009).3 However, the retailer seems to be more sensitive to changes in market conditions. An increase in θ increases the wholesale price and decreases the retail price, which suggests that a retailer should reduce the price when its product is less popular. The reason is straightforward. With more consumers preferring the manufacturer's product, the retailer is on the losing side in product competition. As a result, it should enhance its advantage in price competition by reducing the retail price. It is noted that the marginal increment of pr w.r.t θ is twice as that of ur , which means that a retailer has to squeeze its margin greatly to ensure that it can eventually lower the retail price; otherwise, the manufacturer may prevent it from reducing the retail price by raising the wholesale price. When γ increases, the price competition becomes fiercer because consumers are more sensitive to price, so they are more likely to switch from the high-price channel to the low-price channel. Consequently, a retailer should cut price to guarantee an advantageous position in price competition. λ influences a retailer's pricing in a way that is opposite to θ and γ , which means that a retailer can raise the retail price to earn a higher margin when λ increases, i.e., consumers are less sensitive to channel mismatch. At the same time, those consumers with channel mismatch are less likely to exit the market, so it is less necessary for a retailer to suppress price to induce demand. Similarly, when α increases (i.e., the number of PS consumers increases), a retailer will be less stressful to suppress price to attract consumers if it sells its product through the offline channel, so it can raise price to obtain a higher margin; conversely, it will have a stronger motivation to attract consumers by cutting price if it sells its product through the online channel. If it sells its product through both online and offline channels with a consistent price, changing price with α is unnecessary when cs ¼ crn . But when cs acrn , it should change price accordingly because the average UOC of multi-channels changes with α. Proposition 2. : Reducing product differentiation (i.e., a higher value of γ ) benefits the manufacture regardless of the strategy that the retailer adopts. However, the retailer can benefit from reducing product differentiation only when it has a price advantage over the manufacturer. Proof. : see Appendix B. ■ As aforementioned, a higher value of γ indicates smaller product differentiation (i.e., higher product substitutability) and more intense price competition. Proposition 2 states that the manufacturer, whether advantaged in price or not, always expects small product differentiation. However, the retailer is reluctant to take part in fierce competition caused by small product differentiation if he has no price advantage. The result is intuitive for the retailer but somewhat counter-intuitive for the manufacturer. Hence, we provide further explanations below. 3 In Cai et al. (2009), the direct selling price of the supplier in the dual-channel supplier-Stackelberg game is ð1 þ βcs Þ=2β.

∂π io

We decompose the partial derivative ∂γm into two terms, h i ∂ Dio ðpio cmn Þ þ Dio wio m m r ∂π io 1 cmn ∂Dio m m1 ¼ ¼ ∂γ ∂γ 2 ∂γ |fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl} 

 1  cmn ∂wio ∂Dio r þwio þ þ ; 2 ∂γ ∂γ |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

1

i ¼ Ι; ΙΙ; ΙΙΙ

ð14Þ

2

Dio m

io Dio m1 þ Dr

¼ and Dio where m1 is the demand in the direct channel. The manufacturer's profit comes from both its direct channel and the retail channel. The increments of the two profit streams are measured by term 1 and term 2, respectively. The direct selling price keeps unchanged while the retail price decreases when γ increases (see Table 3), consequently, some consumers will switch from the direct channel to the retail channel, which implies ∂Dio r

∂Dio m1 ∂γ

o0 and ∂γ 40. Besides, ∂w ∂γ is positive as shown Table 3. Thus, term 1 is negative and term 2 is positive, indicating that even though the manufacturer suffers a loss in its direct channel as competition intensifies, it can get compensated and eventually obtains more profit from the retail channel instead. Especially if γ is quite large (i.e., small product differentiation) that the demand in the direct channel Dio m1 equals zero, the manufacturer can still get “indirect benefits” from the direct channel, which is consistent with the finding in Chiang et al. (2003). Proposition 2 tells a manufacturer that building a direct channel and reducing the product differentiation between the retail channel and direct channel can be an effective way to compete with dominant retailers that wield strong power in pricing. Reducing product differentiation can force the retailer to decrease its margin and retail price (see Table 3). As a result, the manufacturer can benefit from a rise in the wholesale price and an increase in the demand in the retail channel. In this sense, the direct channel enhances the manufacturer's competitiveness in competition and can act as a counterweight to the powerful retailer. io

5. Extension Although multi-channel retailers have strong power in pricing, some manufacturers retain their power to determine the channel selection strategies for their products. To gain a better understanding of the optimal channel strategy under various supply chain structures, we analyze the case where the manufacturer has the power to decide the distribution channel(s) for product B. Proposition 3. : In a market where the manufacturer can decide the distribution channel(s) for its product sold through the retailer, π ko r lo ko lo Z π lo (or π ko to π ko Z π lo r r o π r ) is identical m m (or π m o π m ) with   respect to ðcrn ; cs Þ, where k; l A Ι; ΙΙ; ΙΙΙ and k a l, i.e., the manufacturer will select the same channel selection strategy as the retailer. Proof. : see Appendix B. ■ We perform numerical studies to illustrate Proposition 3 by assuming α ¼ 0:5, λ ¼ 0:3, θ ¼ 0:6, γ ¼ 0:01, and cmn ¼ 0:04. We show the results in Fig. 4. We see from Fig. 4 that the intersection lo of the curved surface of π ko r  π r and the zero plane, and the lo intersection of the curved surface of π ko m  π m and the zero plane ko ko lo overlap. Besides, the curved surfaces of π r  π lo r and π m  π m are above (or below) the zero plane with one accord. Proposition 3 states that the optimal decision does not change when the decision maker for channel selection changes from the retailer to the manufacturer. Theoretically speaking (i.e., the costinformation of both firms is transparent to each other), the retailer

W. Wang et al. / Int. J. Production Economics 173 (2016) 54–65

61

Fig. 4. The profit-gaps of the retailer and the manufacturer in selecting different strategies.

and the manufacturer will select the same channel(s) to sell product B, no matter who controls the distribution channel. In practice, retailers usually desire to decide the selling channel of specific products by themselves, instead of being dictated by manufacturers. However, our results show that the decision-making power over channel selection is somewhat unimportant under certain circumstances, and a retailer and a manufacturer need not be entangled in the battle for decision-making power over channel selection because their optimal channel selection strategies are essentially consistent, at least from the perspective of channel operating costs. Besides, Proposition 3 also reminds manufacturers that it is not always the best choice to ask their retailers to sell products through different types of channel from their own direct channel to alleviate channel conflicts (we call such a practice “differentiation of distribution channel” and denote it by DDC). For instance, the online retail channel should be selected if A23 crn þ B23 ocs o cs , even though the manufacturer also owns an online channel. Manufacturers should select the optimal channel (s) based on operating costs rather than adopting DDC blindly. They should keep in mind that although DDC alleviates channel conflicts, it may increase costs as well.

6. Conclusion Multi-channel retailers play an increasingly important role in supply chains. Multi-channels can cover a larger market, which improves the multi-channel retailer's competitiveness in the market compared with the traditional retailer and pure electronic retailer. However, multi-channel retailers face the challenge of finding the optimal channel selection and pricing strategies to distribute products. Poor integration of the online and offline channels will render multi-channel retailers less competitive. We develop a linear demand model to analyze the optimal channel selection decisions in a supply chain that comprises a dominant multi-channel retailer, and a manufacturer that sells two horizontally differentiated products through its own direct channel and the retail channel, respectively. We highlight the key role of channel operating costs in the multi-channel retailer's choice of its channel selection strategy. For the retailer, multichannel selling is optimal only when the gap between its online and offline channels' operating costs is narrow enough. Otherwise, it should only select the low-cost channel. It should be noted that although multi-channel selling can help the retailer cover a wider range of the market, it has a significant hidden hazard caused by

the unbalanced operating costs between the online and offline channels. Our findings shed light on a multi-channel retailer's decisions on integration of the online and offline channels in the contemporary market. As the O2O (i.e., offline to online or online to offline) mode has gained popularity in recent years, an increasing number of single-channel retailers are transforming themselves into multi-channel retailers. They often over-estimate the advantage of multi-channels while over-looking the potential hazards. Our findings remind retailers that, even if they are powerful in pricing, they should be cautious about adopting the multi-channel strategy. Specifically, retailers should decide their channel selection strategies according to regional characteristics, product categories, and other factors that affect the cost-structure of multichannels. A “one size fits all” viewpoint is inadvisable. Besides, our findings indicate that reducing product differentiation is an effective means for the manufacturer to enhance its power in a retailer-led supply chain. A reduction in product differentiation drives the retailer to cut its margin and retail price. On the contrary, the manufacturer has room to raise the wholesale price while keeping the direct selling price stable when competition intensifies. As a result, reducing product differentiation will benefit the manufacturer by increasing the wholesale price and increasing the demand in the retail channel. Essentially, adding the direct channel is a strategic necessity for the manufacturer, especially in a retailer-led supply chain. Finally, we find that no matter whether the retailer or the manufacturer controls the distribution channel, in theory they will make the same channel selection decision for the product sold through the retailer. This finding indicates that adopting the DDC strategy in today's market does not always benefit the manufacturer from the perspective of operating costs. DDC can be so costly that it offsets the benefit of alleviating channel conflicts. The main limitation of this paper concerns the assumption that the retailer charges the same price online and offline when adopting multi-channel selling, which is based on the observation that establishing a consistent pricing scheme is a popular practice. However, it is not clear whether, or under which conditions, a consistent pricing scheme is better than an inconsistent pricing scheme in the supply chain we consider. On the one hand, a consistent pricing scheme can reduce channel conflicts, thus benefiting the retailer (Cattani et al., 2006; Cai et al., 2009). On the other hand, it reduces the retailer's flexibility in pricing, which is a down side when the gap between the online and offline costs is

62

W. Wang et al. / Int. J. Production Economics 173 (2016) 54–65

wide. The optimal pricing scheme for a multi-channel retailer deserves further research.

Acknowledgments This research was supported by the Natural Science Foundation of China under Grant numbers 61174171 and 71571140, and the Fundamental Research Funds for the Central Universities under Grant number SK2014035.

Then we obtain the reaction function of the manufacturer through the first-order condition as follows: 8 Ι 8  1 þ cmn ∂π m > > > < ∂pm ¼ 0 < pm ¼ 2 ðA15Þ ) > > > : w  ¼ 1  ur : ∂π Ιm ¼ 0 2 ∂w Finally, substituting the reaction function Eq. (A15) into the ∂π Ι retailer's profit function and using first-order condition ∂urr ¼ 0 again, we obtain the equilibrium solutions as follows: urΙo ¼

cs t 1 þ cmn t 0 þ ; 2 2ðt 0 þ t 1 Þ

ðA16Þ

1  cs ð1  cmn Þt 0 þ ; 4 4ðt 0 þ t 1 Þ

ðA17Þ

Appendix A. : Analysis of game equilibria In Strategy I, the demand functions for the manufacturer and the retailer are

wΙo ¼

DΙm ¼ ðαθλ þ θÞð1 pm Þ  ðα þ 1Þγ ðpm  w  ur Þ;

ðA1Þ

1 cs t 1 þ cmn t 0 prΙo ¼ þ þ ; 2 4 4ðt 0 þ t 1 Þ

DΙrs ¼ ðα þ λÞð1  θÞð1  w  ur Þ  ðα þ1Þγ ðw þ ur  pm Þ;

ðA2Þ

Ιo ¼ pm

1 þ cmn ; 2

ðA19Þ

1 ½ð1  cs Þt 1 þ ðcmn  cs Þt 0 2 ; 8 t0 þ t1

ðA20Þ

and the profit functions for the manufacturer and the retailer are

π Ιm ¼ DΙm ðpm  cmn Þ þ DΙrs w;

ðA3Þ

π rΙo ¼

π Ιr ¼ DΙrs ður cs Þ:

ðA4Þ

π Ιmo ¼

ðA18Þ


 4ð1  cmn Þ2 ðα2  α2 θ þ αλ þ θ  αt 1 Þðt 1 þ t 0 Þ þ t 0 t 1 þ ½ðcmn  cs Þt 0 þ ð1 cs Þt 1 2 ; 16ðt 1 þ t 0 Þ

In Strategy II, the demand functions for the manufacturer and the retailer are DΙΙ m ¼ ðαθλ þ θ Þð1 pm Þ  ðα þ 1Þγ ðpm  w  ur Þ;

ðA5Þ

DΙΙ rn ¼ ðαλ þ 1Þð1  θ Þð1  w  ur Þ  ðα þ 1Þγ ðw þ ur  pm Þ;

ðA6Þ

ðA21Þ Ιo ¼ Drs

ð1  cs Þt 1 þ ðcmn cs Þt 0 ; 4

where t 0 ¼ ðα þ 1Þγ ;

ðA22Þ

t 1 ¼ ðα þ λÞð1  θÞ. ■

and the profit functions for the manufacturer and the retailer are

π ΙΙm ¼ DΙΙm ðpm  cmn Þ þ DΙΙrn w;

ðA7Þ

π ΙΙr ¼ DΙΙrn ður  crn Þ:

ðA8Þ

In Strategy III, the demand functions for the manufacturer and the retailer are DΙΙΙ m ¼ ðαθλ þ θ Þð1  pm Þ  ðα þ 1Þγ ðpm  w  ur Þ;

ðA9Þ


 DΙΙΙ rs ¼ α ð1  θ Þð1  pr Þ  γ ðw þ ur  pm Þ ;

ðA10Þ

DΙΙΙ rn ¼ ð1  θ Þð1 pr Þ  γ ðw þ ur pm Þ;

ðA11Þ

Proof of Lemma 1. : Rewrite t 1 ¼ ðα þ λÞð1  θÞ, t 2 ¼ ðαλ þ 1Þð1  θÞ, and t 3 ¼ ðα þ1Þ ð1  θÞ. From t 1 t 3 ¼ ðλ  1Þð1  θÞ o 0, we get t 1 o t 3 ; similarly, we get t 2 o t 3 from t 2  t 3 ¼ αðλ 1Þð1  θÞ o 0. Also, it can be proven that t 0 þt 3  t ¼ αð1  θ þ γ Þ 4 0. First, we consider the case where crn r cs . From DΙrs ¼ ð1  cs Þt1 þ4ðcmn cs Þt0 Z0, we obtain ð1  cs Þt 1 þ ðcmn  cs Þ t0 t 0 Z 0, i.e., cs rcs ¼ t1tþ1 þcmn t 0 . Since t 3 4 t 1 , ð1  cs Þt 3 þ ðcmn  cs Þt 0 4 0. 2

and the profit functions for the manufacturer and the retailer are ΙΙΙ ΙΙΙ ΙΙΙ π ΙΙΙ m ¼ Dm ðpm  cmn Þ þ ðDrs þDrn Þw;

Appendix B. : Proofs of the Lemmas and Propositions

ðA12Þ

Define the function f ðxÞ ¼ 18 ½ð1  cs Þx þt0ðcþmnx  cs Þt0  , where cs A t0 ð0; t 1tþ1 þcmn t 0 Þ. Since ð1  cs Þx þ ðcmn  cs Þt 0 Z 0, f ðxÞ is monotonically increasing with x. Recalling that t 3 4 t 1 , we obtain:

π rΙo ¼ f ðt 1 Þ o f ðt 3 Þ ¼ 18 ½ð1  cs Þt3tþ0 þðctmn3  cs Þt0 

2

ðB1Þ

o 18 ½ð1  cs Þt 3 þ ðcmnt0þcts 3Þt0 þ ðcs  crn Þt ¼ π rΙΙΙo 2

ΙΙΙ ΙΙΙ π ΙΙΙ r ¼ Drs ður  cs Þ þ Drn ður  crn Þ:

ðA13Þ

Since the analysis of the three strategies' equilibria are similar, we only present the analysis of Strategy I for simplification purposes. First, the following results establish that the Hessian matrix is negative definite, so the profit function is concave. 8 2 Ι ∂ πm > ¼ 2½αλθ þ θ þ ðα þ 1Þγ  o0 > ∂p2m > > > > 2 Ι > ∂ π > 2m ¼ 2½ðα þ λÞð1  θÞ þðα þ 1Þγ  o 0 < ∂w ðA14Þ ∂2 π Ιm > > ∂w∂pm ¼ 2ðα þ 1Þγ > > > 2 Ι 2 > 2 Ι 2 Ι > > : ∂ πm ∂ πm  ∂ πm 40 ∂p2m ∂w2

∂w∂pm

Second, we consider the case where crn 4 cs . ð1  crn Þt 2 þ ðcmn crn Þt 0 From DΙΙ Z 0, we obtain ð1 crn Þt 2 þ ðcmn  rn ¼ 4 t0 crn Þ t 0 Z 0, i.e., crn r crn ¼ t2tþ2 þcmn t 0 . Since t 3 4 t 2 , ð1  crn Þt 3 þ 2

ðcmn  crn Þt 0 4 0. Define the function gðxÞ ¼ 18 ½ð1  crn Þx þt 0ðcþmnx  crn Þt0  , where crn A ð0; t 2tþ2 þcmnt0t0 Þ. Since ð1  crn Þx þ ðcmn crn Þt 0 Z 0, gðxÞ is monotonically increasing with x. Recalling that t 3 4 t 2 and t 3 þ t 0  t 4 0, we obtain:

π ΙΙr o ¼ gðt 2 Þ ogðt 3 Þ ¼ 18 ½ð1  crn Þt3tþ0 þðctmn3  crn Þt0  ¼

2

1 ½t 3  crn ðt 3 þ t 0  tÞ  crn t þ cmn t 0 2 1 ½t 3  cs ðt 3 þ t 0  tÞ  crn t þ cmn t 0 2 o 8 8 t0 þ t3 t0 þ t3

W. Wang et al. / Int. J. Production Economics 173 (2016) 54–65

¼

1 ½ð1  cs Þt 3 þ ðcmn  cs Þt 0 þðcs  crn Þt 2 ¼ π rΙΙΙ o 8 t0 þ t3

ðB2Þ

Combing Eq. (B1) and Eq. (B2), we see that π rΙΙΙo ranks second at least. Since we seek to find the largest of the three profits, we only o and π rΙΙΙo respectively, and need to compare π rΙo and π rΙΙΙo and π ΙΙ r o comparing π rΙo and π ΙΙ is unnecessary. ■ r

63

( o ðcmn  crn Þt 0  ð1 þ crn Þt 2 þ 2cmn t 2 4 0 3 prΙΙo o pΙΙ m Noting that , ΙΙ o ΙΙ ðcmn  crn Þt 0  ð1 þ crn Þt 2 þ 2cmn t 2 o 0 3 pr 4 pmo ( IIo 4 0; if pIIo r o pm ∂π ΙΙo we obtain ∂γr . IIo o 0; if pr 4 pIIo m _

∂π rΙΙΙo ½ðcmn  cs Þt 0 þ ðcs  crn Þt þ 2cmn t 3  ðcs þ 1Þt 3 ½ð1  cs Þt 3 þ ðcmn  cs Þt 0 þ ðcs  crn Þt  ¼ 8ðt 0 þ t 3 Þt ∂γ

ðB16Þ

Proof of Proposition 1. : What is critical in comparing π rΙΙΙo and π rΙo and π rΙΙΙo and π rΙΙo is o to find the indifferent points where π rΙΙΙo ¼ π rΙo and π rΙΙΙo ¼ π ΙΙ r . From π rΙΙΙo ¼ π rΙo , we obtain that cs ¼ A13 crn þ B13 , where pffiffiffiffiffiffiffiffiffiffiffiffiffi t t þt pffiffiffiffiffiffiffiffiffiffiffiffiffi 1 0 pffiffiffiffiffiffiffiffiffiffiffiffiffi A13 ¼ ðB3Þ ðt 0 þ t 1 Þ t 0 þt 3  ðt 3 þ t 0  tÞ t 0 þ t 1 pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi ðcmn t 0 þ t 1 Þ t 0 þ t 3  ðcmn t 0 þ t 3 Þ t 0 þ t 1 pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi B13 ¼ ðt 0 þ t 1 Þ t 0 þ t 3  ðt 3 þ t 0 tÞ t 0 þ t 1

ðB4Þ

o From π rΙΙΙo ¼ π ΙΙ r , we obtain that cs ¼ A23 crn þB23 , where pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi ðt 0 þ t 2 Þ t 3 þt 0  t t 2 þ t 0 A23 ¼ ðt 0 þ t 2 Þðt 3 þ t 0  tÞ

ðB5Þ

pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi ðcmn t 0 þ t 3 Þ t 0 þ t 2  ðcmn t 0 þ t 2 Þ t 0 þ t 3 B23 ¼ ðB6Þ ðt 0 þ t 2 Þðt 3 þt 0  tÞ   o ΙΙΙo ¼ π Ιo if 0 o c o A Then, it is obvious that max π rΙo ; π ΙΙ s 13 r ; πr r  Ιo ΙΙo ΙΙΙo  t0 ΙΙ o crn þ B13 ; max π r ; π r ; π r ¼ π r if A23 crn þB23 o cs o t1tþ1 þcmn t0 ;  Ιo ΙΙo ΙΙΙo  ¼ π rΙΙΙo if A13 crn þ B13 o cs o A23 crn þ B23 . ■ max π r ; π r ; π r Proof of Proposition 2. : h i Ιo ðt 0 þt 1 Þ2 ðcmn  cs Þ2 þ 3t 21 ð1  cmn Þ2 ðα þ 1Þ ∂π m ¼ 40 ∂γ 16ðt 0 þt 1 Þ2 ΙΙo ∂π m ¼ ∂γ

h

i ðt 0 þ t 2 Þ2 ðcmn  crn Þ2 þ 3t 22 ð1  cmn Þ2 ðα þ1Þ 16ðt 0 þ t 2 Þ

2

∂π ΙΙΙo

ðcmn t 0 þ cmn t 3 crn t þ cs t  cs t 0  cs t 3 Þ m ¼ ∂γ 16ðt 0 þt 3 Þt

2

40

þ 3ð1  cmn Þ2 t 23

ðB7Þ

_

r

∂γ

¼

½ðcmn  cs Þt 0  ð1 þ cs Þt 1 þ 2cmn t 1 ½ð1 cs Þt 1 þ ðcmn  cs Þt 0 ðα þ 1Þ 8ðt 0 þ t 1 Þ2

Ιo 40 _ð1 cs Þt 1 þ ðcmn  cs Þt 0 ¼ 4Drs

ðB8Þ

40

ðB10Þ

ðB11Þ

_

r

∂γ

¼

½ðcmn  crn Þt 0  ð1 þ crn Þt 2 þ 2cmn t 2 ½ð1  crn Þt 2 þ ðcmn  crn Þt 0 ðα þ 1Þ 2

8ðt 0 þ t 2 Þ

ðB13Þ o _ð1 crn Þt 2 þ ðcmn  crn Þt 0 ¼ 4DΙΙ rn 4 0

‘ sign

o ∂π ΙΙ r ¼ sign½ðcmn crn Þt 0  ð1 þ crn Þt 2 þ 2cmn t 2  ∂γ

ðB14Þ ðB15Þ

ðB17Þ

∂π rΙΙΙo ¼ sign½ðcmn  cs Þt 0 þ ðcs  crn Þt þ 2cmn t 3  ðcs þ 1Þt 3  ∂γ ðB18Þ

( Noting that ΙΙΙo ðcmn cs Þt 0 þ ðcs  crn Þt þ 2cmn t 3  ðcs þ 1Þt 3 4 0 3 prΙΙΙo o pm ΙΙΙ o ΙΙΙo , we ðcmn cs Þt 0 þ ðcs  crn Þt þ 2cmn t 3  ðcs þ 1Þt 3 o 0 3 pr 4 pm ( ΙΙΙo 4 0; if prΙΙΙo o pm ∂π ΙΙΙo obtain ∂rγ ¼ ΙΙΙo . ■ o 0; if prΙΙΙo 4 pm Proof of Proposition 3. : The direct way to prove Proposition 3 is to calculate the lo ko lo indifferent points of π ko r ¼ π r and π m ¼ π m , then check whether lo lo ¼ π is the same as π ko the indifferent point of π ko r r m ¼ π m . However, the mathematical expressions of the indifferent points are too complex to compare. So we prove Proposition 3 in an indirect way. Ιo ¼ π ΙΙΙo . First, we prove π rΙo ¼ π rΙΙΙo 3 π m m _

Ιo Ιo π rΙo ¼ Drs ðpr  wΙo  cs Þ

ðB19Þ

_

ΙΙΙo ΙΙΙo ΙΙΙo ΙΙΙo π rΙΙΙo ¼ Drs ðpr  wΙΙΙo cs Þ þ Drn ðpr  wΙΙΙo  crn Þ

ðB20Þ

‘

π rΙo ¼ π rΙΙΙo

3

ΙΙΙo ΙΙΙo ΙΙΙo Ιo Ιo ðDrs þDrn Þw Drs w

ΙΙΙo ΙΙΙo Ιo Ιo ¼ Drs ðprΙΙΙo  cs Þ þ Drn ðpr crn Þ  Drs ðpr  cs Þ

∂ π Ιo ðB12Þ ‘ sign r ¼ sign½ðcmn  cs Þt 0  ð1 þ cs Þt 1 þ 2cmn t 1  ∂γ ( Ιo ðcmn  cs Þt 0  ð1 þ cs Þt 1 þ 2cmn t 1 4 0 3 prΙo o pm Noting that Ι o Ιo , ðcmn  cs Þt 0  ð1 þ cs Þt 1 þ 2cmn t 1 o 0 3 pr 4 pm 8 Ιo < 4 0; if prΙo o pm ∂π Ι o we obtain ∂γr Ι o Ιo . : o 0; if pr 4 pm ∂π ΙΙo

‘ sign

ΙΙΙo

ðB9Þ ∂π Ιo

ΙΙΙo ΙΙΙo þ 4Drn 40 _ ð1  cs Þt 3 þ ðcmn  cs Þt 0 þ ðcs  crn Þt ¼ 4Drs

ðB21Þ

_

Ιo ¼ DΙo ðpo c Þ þ DΙo wΙo πm mn m m rs

ðB22Þ

_

ΙΙΙo ¼ DΙΙΙo ðpo  c Þ þ ðDΙΙΙo þ DΙΙΙo ÞwΙΙΙo πm mn m m rs rn

ðB23Þ

‘

ΙΙΙo π Ιmo ¼ π m

3

ΙΙΙo ΙΙΙo ΙΙΙo Ιo Ιo ΙΙΙo þDrn Þw Drs w ¼ ðDΙmo  Dm Þðpom  cmn Þ ðDrs

ðB24Þ

Ιo ¼ pΙΙo ¼ pΙΙΙo ¼ 1  cmn . where ¼ pm m m 2 Ιo ¼ π ΙΙΙo is equivalent to proving Hence, to prove π rΙo ¼ π rΙΙΙo 3 π m m

pom

ΙΙΙo ΙΙΙo ΙΙΙo ΙΙΙo Ιo Ιo Ιo ΙΙΙo ðpr cs Þ þ Drs ðpr  crn Þ  Drs ðpr  cs Þ ¼ ðDm  Dm Þðpom  cmn Þ: Drs

ðB25Þ Eq. (B25) can be easily proven by substituting the specific expressions presented in Table 2 into each term. Ιo ¼ π ΙΙo and π ΙΙo In the same way, we can prove π rΙo ¼ π rΙΙo 3 π m m r ΙΙo ¼ π ΙΙΙo by proving the following equations, respec¼ π rΙΙΙo 3 π m m tively: ΙΙo ΙΙo Ιo Ιo Ιo o o Drn ðpr  crn Þ  Drs ðpr  cs Þ ¼ ðDm  DΙΙ m Þðpm  cmn Þ

ðB26Þ

ΙΙΙo ΙΙΙo ΙΙΙo ΙΙΙo o ΙΙo ðpr cs Þ þ Drn ðpr  crn Þ  DΙΙ Drs rs ðpr  crn Þ ΙΙo ΙΙΙo  Dm Þðpom  cmn Þ ¼ ðDm

ðB27Þ

Having proven π ¼ π 3 π ¼ π we can easily verify that lo ko lo ko lo ko lo π ko (or r 4 πr  π r oπ r ) is identical to π m 4 π m (or π m o π m ), where k; l A Ι; ΙΙ; ΙΙΙ and k a l.■ ko r

lo r

ko m

lo m,

Appendix C. : The numerical sensitivity analysis in Section 4.2 2

 B23 Þ Þ, S2 ¼ ðcs 2A , and The sizes of regions Ι–ΙΙΙ are S1 ¼ A213 ðcs þ AB13 13 23

S3 ¼ c2s  S1  S2 , respectively (see Fig. C1). Since cs changes with

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W. Wang et al. / Int. J. Production Economics 173 (2016) 54–65

∂pio m ¼0 ∂θ

i ¼ Ι; ΙΙ; ΙΙΙ

ðD8Þ

The effects of changes in γ are as follows: ∂pΙo r

∂γ

Fig. C1. A diagrammatic sketch of the retailer's optimal channel selection.

the parameters, we examine the sensitivity of the relative sizes of the three regions, namely s1 ¼ S1 =c2s , s2 ¼ S2 =c2s , and s3 ¼ 1  s1  s2 , with respect to the parameters cmn , θ, γ , α, and λ. When examining the sensitivity w.r.t cmn , we set α ¼ 0:5, θ ¼ 0:6, γ ¼ 0:01, λ ¼ 0:3, and the range of cmn is ð0; 1Þ. When examining the sensitivity w.r.t θ, we set α ¼ 0:5, γ ¼ 0:01, λ ¼ 0:3, cmn ¼ 0:04, and the range of θ is ð0; 1Þ. When examining the sensitivity w.r.t γ , we set α ¼ 0:5, θ ¼ 0:6, λ ¼ 0:3, cmn ¼ 0:04, and the range of γ is ð0; 0:1Þ. When examining the sensitivity w.r.t α, we set θ ¼ 0:6, γ ¼ 0:01, λ ¼ 0:3, cmn ¼ 0:04, and the range of α is ð0; 2Þ. When examining the sensitivity w.r.t λ, we set α ¼ 0:5, θ ¼ 0:6, γ ¼ 0:01, cmn ¼ 0:04, and the range of λ is ð0; 1Þ.

Appendix D. : Analysis of the comparative statics The effects of changes in θ are as follows:

¼

ð1  θÞðα þ 1Þð1  cmn Þðα þ λÞ 4ðαγ  αθ  λθ þ α þ γ þ λÞ2

ðD10Þ

∂prΙΙΙo ð1  θÞð1  cmn Þ ¼ o0 ∂γ 4ð1  θ þ γ Þ2

ðD11Þ

∂wΙo ð1  cmn Þð1  θÞðα þ 1Þðα þ λÞ ¼
2 4 0 ∂γ 4 ðα þ λÞð1  θÞ þ ðα þ 1Þγ

ðD12Þ

∂wΙΙo ð1 cmn Þð1  θÞðα þ 1Þðαλ þ 1Þ ¼ 40 ∂γ 4ð  αλθ þ αγ þ αλ þ 1  θ þ γ Þ2

ðD13Þ

∂wΙΙΙo ð1 cmn Þð1  θÞ ¼ 40 ∂γ 4ð1  θ þ γ Þ2

ðD14Þ

∂uio ∂pio ∂wio r ¼ r  o0 ∂γ ∂γ ∂γ

ðD15Þ

∂pio m ¼0 ∂γ

∂prΙo ð1  θÞðα þ 1Þð1  cmn Þγ ¼ 40 ∂λ 4ðαγ  αθ  λθ þ α þ γ þ λÞ2

ðD17Þ

o ∂pΙΙ αð1  θÞðα þ1Þð1  cmn Þγ r ¼ 40 ∂λ 4ð  αγθ þ αγ þ αγ þ γ  θ þ 1Þ2

ðD18Þ

∂prΙΙΙo ¼0 ∂λ

ðD19Þ

∂wΙo ð1  cmn Þðα þ 1Þð1  θÞγ ¼
2 o 0 ∂λ 4 ðα þ λÞð1  θÞ þðα þ 1Þγ

ðD20Þ

∂wΙΙo αð1  cmn Þðα þ 1Þð1  θÞγ ¼ o0 ∂λ 4ð  αλθ þ αγ þ αλ þ1  θ þ γ Þ2

ðD21Þ

ðD22Þ

o ∂pΙΙ ðαλ þ1Þðα þ 1Þðcmn  1Þγ r ¼ o0 ∂θ 4ð  αγθ þ αγ þ αγ þ γ  θ þ 1Þ2

ðD2Þ

∂uio ∂pio ∂wio r ¼ r  40 ∂λ ∂λ ∂λ

ðD4Þ

∂wΙΙo ð1  cmn Þðα þ 1Þðαλ þ 1Þγ ¼ 40 ∂θ 4ð  αλθ þ αγ þ αλ þ 1  θ þ γ Þ2

ðD5Þ

∂wΙΙΙo ð1  cmn Þγ ¼ 40 ∂θ 4ð1  θ þ γ Þ2

ðD6Þ

∂uio ∂pio ∂wio r ¼ r  o0 ∂θ ∂θ ∂θ

i ¼ Ι; ΙΙ; ΙΙΙ

ðD7Þ

ðD16Þ

The effects of changes in λ are as follows:

∂wΙΙΙo ¼0 ∂λ

∂wΙo ð1  cmn Þðα þ 1Þðα þ λÞγ ¼
2 4 0 ∂θ 4 ðα þ λÞð1  θÞ þ ðα þ1Þγ

i ¼ Ι; ΙΙ; ΙΙΙ

i ¼ Ι; ΙΙ; ΙΙΙ

ðD1Þ

ðD3Þ

ðD9Þ

o ∂pΙΙ ðαλ þ 1Þð1  θÞðα þ 1Þð1  cmn Þ r ¼ o0 ∂γ 4ð  αγθ þ αγ þ αγ þ γ  θ þ 1Þ2

∂prΙo ðα þ λÞðα þ 1Þðcmn 1Þγ ¼ o0 ∂θ 4ðαγ  αθ  λθ þ α þ γ þ λÞ2

∂prΙΙΙo ðcmn  1Þγ ¼ o0 ∂θ 4ð1  θ þ γ Þ2

o0

i ¼ Ι; ΙΙ

ðD23Þ

o ∂uΙΙΙ ∂pΙΙΙo ∂wΙΙΙo r ¼ r  ¼0 ∂λ ∂λ ∂λ

ðD24Þ

∂pio m ¼0 ∂λ

ðD25Þ

i ¼ Ι; ΙΙ; ΙΙΙ

The effects of changes in α are as follows: ∂pΙo r

∂α

¼

ð1  θÞð1  λÞð1  cmn Þγ 4ðt 0 þt 1 Þ2

40

o ∂pΙΙ ð1  θÞð1  λÞð1  cmn Þγ r ¼ o0 ∂α 4ðt 0 þ t 2 Þ2

∂prΙΙΙo cs  crn ¼ ∂α 4ðα þ 1Þ2

(

40 r0

if cs 4 crn if cs r crn

ðD26Þ

ðD27Þ

ðD28Þ

W. Wang et al. / Int. J. Production Economics 173 (2016) 54–65

∂wΙo ð1  θÞð1  λÞð1  cmn Þγ ¼ o0 ∂α 4ðt 0 þ t 1 Þ2

ðD29Þ

∂wΙΙo ð1  θÞð1  λÞð1  cmn Þγ ¼ 40 ∂α 4ðt 0 þ t 2 Þ2

ðD30Þ

∂wΙΙΙo crn  cs ¼ ∂α 4ðα þ 1Þ2

(

40

if crn 4 cs

r0

if crn r cs

∂urΙo ∂prΙo ∂wΙo ¼  40 ∂α ∂α ∂α o ∂uΙΙ ∂pΙΙo ∂wΙΙo r ¼ r  o0 ∂α ∂α ∂α ( ∂urΙΙΙo ∂prΙΙΙo ∂wΙΙΙo 40 ¼  r0 ∂α ∂α ∂α

∂pio m ¼0 ∂α

i ¼ Ι; ΙΙ; ΙΙΙ

ðD31Þ

ðD32Þ ðD33Þ if cs 4 crn if cs r crn

ðD34Þ

ðD35Þ

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