Store brand introduction in a two-echelon logistics system with a risk-averse retailer

Transportation Research Part E xxx (2015) xxx–xxx

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Transportation Research Part E journal homepage: www.elsevier.com/locate/tre

Store brand introduction in a two-echelon logistics system with a risk-averse retailer Qinquan Cui a, Chun-Hung Chiu a,⇑, Xin Dai b, Zhongfei Li a,* a b

Sun Yat-sen Business School, Sun Yat-sen University, Guangzhou, PR China Xinhua College of Sun Yat-sen University, Guangzhou, PR China

a r t i c l e

i n f o

Article history: Received 28 August 2015 Received in revised form 29 September 2015 Accepted 13 October 2015 Available online xxxx Keywords: Risk analysis Mean–variance efficient frontier Capital constraint Development cost Substitution factor Store brand product

a b s t r a c t We study a risk-averse retailer’s optimal decision of introducing her store brand product by using the mean–variance formulation. The effects of the substitution factor, the capital constraint, and the development cost are examined. Taking the product quantities as the decision variables, the risk deducted surplus of the store brand product and the substitution factor play a vital role in the retailer’s optimal policies. Both the capital constraint and the development cost reduce the mean–variance efficient solution set of the retailer and hence distort the risk management of the retailer. Some meaningful insights are generated. Ó 2015 Elsevier Ltd. All rights reserved.

1. Introduction Store brand of the retailer can date back to several decades ago (Deveny, 1993).1 Nowadays, it has been a common practice that retailers, such as Tesco, Kroger, Woolworths, Walmart, etc., introduce their store brand business and use it as a weapon to compete with upstream manufacturers. The introduction of retailers’ store brands has been a notable trend in the evolving area of retailing. Retailers’ store brands account for about 16% market shares in the U.S. while about 30% in Europe in the food industry (Zimmerman et al., 2007), and account for at least 30% of all products sold in 15 countries, the greatest number ever (see Nielsen data compiled for PLMAs 2014 International Private Label Yearbook). Advantages of store brand have been broadly proved by the practitioners and academics. An appropriate store brand business cannot only directly improve the retailer’s profitability, but also lead to a better bargaining power with the upstream members (Mills, 1995; Morton and Zettelmeyer, 2004). Moreover, the store brand can also build customers’ store loyalty, which is particularly important to retailers (Corstjens and Lal, 2000). However, different from the traditional status, the store brand retailers play a more determinant role in the failure or achievement of their own brand, and have to take all the risk of the development of store brand product. For example, manufacturers’ national brand commodities are considered more popular than the retailer’s store brand product because of their stability and reliability (Pinedo et al., 2008), even though the quality of retailers’ store brand product might be higher than that of the national brand (Richardson et al., 1994; ⇑ Corresponding authors. E-mail addresses: [email protected] (Q.Q. Cui), [email protected], [email protected] (C.-H. Chiu), [email protected] (X. Dai), [email protected] (Z.L. Li). 1 In this study, the term ‘‘store brand” is the same as the term ‘‘private label”. Literatures related to the retail market usually use ‘‘store brand” or ‘‘private label” to describe retailers’ own brand. Please refer to Raju et al. (1995) and Dhar and Hoch (1997) for the typical definitions of ‘‘store brand” and ‘‘private label”. http://dx.doi.org/10.1016/j.tre.2015.10.005 1366-5545/Ó 2015 Elsevier Ltd. All rights reserved.

Please cite this article in press as: Cui, Q., et al. Store brand introduction in a two-echelon logistics system with a risk-averse retailer. Transport. Res. Part E (2015), http://dx.doi.org/10.1016/j.tre.2015.10.005

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Q. Cui et al. / Transportation Research Part E xxx (2015) xxx–xxx

Chintagunta et al., 2002). Moreover, the retailer needs to consider the risk arising from the uncertainty of introducing a new store brand, and to take her risk-averse attitude into consideration (Eeckhoudt et al., 1995; Chen et al., 2007). In addition, capital constraint is a common phenomenon in the retailers’ operations (Buzacott and Zhang, 2004), and the potential market entry retailers should decide how to use the constrained capital efficiently. Motivated by these messages, we examine our research issues: First, what is the optimal stocking quantities of the store brand retailer under the risk-averse situation; Second, how the substitution factor affects the risk-averse retailer’s optimal decision and the desirability to introduce the store brand product; Third, what is the optimal product portfolio when the retailer is faced with the capital constraint and development cost. In the literature, mostly a profit maximizing (riskneural) retailer and channel conflicts are considered in the analysis. In this study, we focus on the risk analysis of the retailer who plan to develop her store brand. This study aims to provide insights for the retailer who is risk sensitive and is going to introduce her store brand product. By formulating the problem as a mean–variance optimization problem (Choi and Chiu, 2012b; Chiu and Choi, 2013), the optimal decisions of the risk-averse retailer for different situations are derived. Then management insights are obtained from the optimal decisions. We find that the retailer’s incentive to develop the store brand business depends on the risk deducted surplus (of selling the store brand product) and the product’s substitution factor. A small risk deducted surplus or a large substitution factor inhibits the retailer to change her strategy to the store brand business. Interestingly, as long as the substitution factor is not sufficiently large and the risk deducted surplus is not extremely small or large, the retailer tends to retail both products concurrently, while it is always optimal for the retailer to give up the risk-free decision and to introduce the store brand product, if the risk deducted surplus is small regardless of the substitution factor. Moreover, we reveal that the retailer is still willing to develop her store brand product even though the unit production cost of the store brand product is higher than the wholesale price of the manufacturer’s product but the retail price of the store brand product is lower than that of the manufacturer’s product. Furthermore, we reveal situations where the risk is unimportant in the retailer’s decision making. These situations include, the profit of selling the store brand product is too low, the profit of selling the manufacturer’s product is very high, and the substitution factor is big (i.e., the differentiation between the two products is small). Both the capital constraint and the development cost may distort the risk management of the retailer. A retailer originally preferring not to develop her store brand product will become preferring to develop her store brand product, and switches from a risk-free decision to a risky decision, just because she has insufficient capital. The development cost causes the polarization of risk level of retailer’s decision, under which only the risk-free solution or extremely high risk solutions are mean–variance efficient to the retailer. This study is organized as follows. We first review the related literature in Section 2, and then provide a stylized model and discuss the assumptions in Section 3. Analytical results of the basic model are presented in Section 4. Later, we extend our model to include the capital constraint and the development cost, and investigate their impacts in Section 5. Section 6 concludes the major findings of this study. All the proofs are presented in the Appendix A.

2. Literature review This study is closely related to two research areas, the area of retailer store brand business and the area of supply chain risk management. There is profound literature on the retailer store brand business, and the related literature can be divided into two streams. The first stream of research is that on incentives of retailers’ market entry decisions. Mills (1995) shows that a well developed store brand program cannot only contribute directly to retailers’ profitability, but also have positive indirect impacts such as better bargaining power with the upstream members. To further explain what makes the store brand entry so conducive for retailers, Raju et al. (1995) present an analytical framework and find that the introduction of store brand could lead to an improvement of the retailer’s profit if the cross-price sensitivity among national brands is low and among the store brand is high. Narasimhan and Wilcox (1998) study the strategic role the store brand plays. They argue that, the private-label development enhances the retailer’s channel power and makes it available to elicit a lower wholesale price by competing with the national brand manufacturer. From the perspective of customer loyalty, Corstjens and Lal (2000) extend the previous studies and find that a retailer can increase her profits by marketing such store brands regardless of whether there exists a cost advantage over the national brand and the retailer is able to obtain lower procurement price from the national brand or not. Chintagunta et al. (2002) provide evidence of both demand and supply effects of store-brand entry. Erdem et al. (2004) empirically study consumer choice behavior with respect to store brands in three different countries. Generally, they find that when consumers are uncertain about the product quality, they may develop expectations about product quality and justify the product by brands, although the quality may be higher for the store brand than for the national brand. However, they do not consider the risk of the retailer of introducing the store brand. Different from the foregoing works, this study focuses on the role of risk which plays in the retailer’s market entry decision when both the stocking quantities of national brand and store brand products are endogenously decided. We focus on the risk analysis and adopt the mean–variance formulation to explore risk-averse retailers’ market entry incentives of store brand business. A second stream of related literature is that on the product position of the store brand. Richardson et al. (1994) examine the relative importance of extrinsic versus intrinsic cues in determining perceptions of store brand quality in an experiment, Please cite this article in press as: Cui, Q., et al. Store brand introduction in a two-echelon logistics system with a risk-averse retailer. Transport. Res. Part E (2015), http://dx.doi.org/10.1016/j.tre.2015.10.005

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suggesting that consumers’ evaluations of store brand grocery items are driven primarily by the extrinsic cues that these products display rather than intrinsic characteristics. Moreover, Palmeira and Thomas (2011) propose that when a retailer carries a single store brand, consumers expect it to be of relatively lower quality even when it is described as a premium brand. On the other hand, when two store brands are presented, consumers infer a superior quality to the one positioned as premium. Specifically, Sayman et al. (2002) study the competition between the private label and national brand by considering the positioning strategy of store brands, and they find that the optimal strategy for the retailer is to position the store brand to imitate the leading and stronger national brand. Morton and Zettelmeyer (2004) also argue that, it is critical for the retailer to consider the national brand’s market share and incremental contribution when replacing the national brand with the store brand. Hansen et al. (2006) develop a multi-category brand choice model by using unique frequentshopper data, and find strong evidence of correlation in household preferences for store brands across categories. Following these results, we consider the scenario where the retailer’s store brand product is an inferior alternative to the manufacturer’s national product, and incorporate the substitution factor and the retailer’s strategic market entry decision in this study. Some joint effects of the substitution factor and the risk are revealed, and management insights are generated. Similarly, there are plenty of literatures on risk analysis and risk measurement in the fields of supply chain management (Zsidisin and Ritchie, 2008; Leng and Parlar, 2010; Sodhi et al., 2012; Choi and Chiu, 2012b) and finance. The well known mean–variance (MV) model is pioneered by Markowitz (1952), then the MV formulation has been discussed and applied broadly (Levy and Markowitz, 1979). The MV criterion is also widely adopted in supply chain management. Lau (1980) first incorporates the mean-standard deviation tradeoff into the analysis of newsvendor problem, then in the recent years, mean– variance has drawn more attention of scholars in the management area, for example, Van Mieghem (2007) studies how a newsvendor can migrate the risk, Choi (2013) applys the mean–variance analysis to study the quick response program, Choi et al. (2008) and Wei and Choi (2010) apply it to the risk management of supply contracts, Liu et al. (2012) apply it to the risk analysis of mass customization, and Shen et al. (2013) and Chiu et al. (2015) study the supply chain coordination problem with mean–variance agents. Besides the mean–variance formulation in the analysis of supply chain risk management, other technological methods are also discussed, such as utility function (Eeckhoudt et al., 1995; Chen et al., 2007), value at risk (VaR) and conditional value at risk (CVaR) criterion (Özler et al., 2009; Chen et al., 2015), experiment (Cantor et al., 2014), downside risk (Choi and Chiu, 2012a), loss-averson (Lee et al., 2015), operational statistics (Lu et al., 2015), etc. We adopt the mean–variance criterion as the measure of the tradeoff between risk and expected profit in this paper because of its intuitive and tractability nature. Moreover, we also focus on the efficiency of capital use under the situation of capital constraint and development cost of the store brand product, which causes the managers to pay more attention to the product strategies. 3. Model and preliminary results In a conventional distribution channel, the manufacturer offers his national brand products to the retailer, who then sells them to consumers with a positive profit margin. However, as is illustrated in Fig. 1, the retailer may want to develop her store brand product, which is a substitution of the manufacturer’s product, to enhance her brand name (Raju et al., 1995; Morton and Zettelmeyer, 2004; Heese, 2010). For brevity, if there is no confusion, we call the retailer’s store brand product as product 1 and the manufacturer’s national product as product 2. Moreover, we use subscript 1 and 2 for their products and the associated variables, respectively. Let q1 P 0 and q2 P 0, respectively, be the stocking quantities of retailer’s store brand product, and the manufacturer’s national brand product. Both q1 and q2 are decision variables of the retailer. The market prices of the products are jointly determined by their respective stocking quantities, via two inverse demand functions. For tractability, this study adopts the commonly used linear inverse demand function (Farahat and Perakis, 2011) for the store brand and the national products. The price-quantity (demand) functions of the two products are

p1 ðq1 ; q2 Þ ¼ a  q1  q2 þ e1 ;

ð1Þ

p2 ðq1 ; q2 Þ ¼ a  bq1  q2 ;

ð2Þ

where a > 0 represents the market potential, b represents the substitution factor of product 1 to product 2, e1 is a random variable with mean 0, and standard deviation r1 P 0, and a; b, and r1 are exogenously given. We consider unified pricequantity relationships except that 0 < b < 1 in (1) and (2) because the retailer usually positions her store brand product as an alternative and inferior product to that of the manufacturer (Heese, 2010), and the manufacturer’s product is more popular than the retailer’s store brand product. Moreover, in the national brand market, consumers usually consider the stability and reliability of brands more than other factors (Pinedo et al., 2008), and hence they tend to prefer the national brand product over that of the retailer.2 Note that E½p1 ðq1 ; q2 Þ < p2 ðq1 ; q2 Þ for any given q1 and q2 .3 Moreover, we consider that the manufacturer’s product is a well known product with a long history. Therefore, the price-demand relationship is stable and predictable. 2 As linear price-quantity relationships are considered, the unified setting does not affect the analysis of the problem. However, different results could be obtained when different coefficients of q1 and q2 in (1) and (2) are considered. We consider identical and unified coefficients of q1 and q2 in (1) and (2), except that 0 < b < 1 in (2), to obtain a simple model, which effectively captures the major characteristic of the store brand market. 3 For b ¼ 1, E½p1 ðq1 ; q2 Þ ¼ p2 ðq1 ; q2 Þ and product 1 perfectly substitutes product 2 with no price difference. For 0 < b < 1; E½p1 ðq1 ; q2 Þ < p2 ðq1 ; q2 Þ and product 1 is a substitution of product 2 if there is a price deduction.

Please cite this article in press as: Cui, Q., et al. Store brand introduction in a two-echelon logistics system with a risk-averse retailer. Transport. Res. Part E (2015), http://dx.doi.org/10.1016/j.tre.2015.10.005

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Retailer’ s Product 1 Manufacturer’ s Product 2

Product 2

Retailer

Manufacturer c2

Market p1 + p2

Fig. 1. The logistics system with the store brand product (product 1) and the national product (product 2).

On the other hand, the retailer’s store brand product is new. Therefore, the response of customers is unknown and highly uncertain, and we include the random variable e1 into (1) to represent the uncertainty of developing the new store brand product. The unit manufacturing cost of product 1 is c1 > 0, and the unit wholesale price of product 2 is c2 > 0. To develop the new store brand product, the retailer incurs a development cost C P 0. The total cost of the retailer is C T ¼ c1 q1 þ c2 q2 þ C when q1 > 0 and C T ¼ c2 q2 when q1 ¼ 0. Moreover, we consider that the retailer has an initial capital k > 0, and there is an ideal bank who provides a risk-free rate g  1 > 0. The ideal bank provides identical rates for saving and borrowing. The retailer will put the excess capital to the ideal bank and gain the interest when k > C T , and the retailer takes a loan from the ideal bank when k < C T . 4 The profit function (or the terminal wealth function) of the retailer is

p1 ðq1 ; q2 Þ ¼ ða  q1  q2 þ e1 Þq1 þ ða  q2  bq1 Þq2 þ gðk  C T Þ  k;

ð3Þ

where the first term is the revenue of the retailer by selling product 1, the second term is the revenue of the retailer by selling product 2, and the last term includes all the costs and the financial income/expense. We explore the optimal stocking quantities of the risk-averse retailer. We consider a mean–variance objective function of the retailer as follow (Lau, 1980; Chiu and Choi, 2013)

max Lðq1 ; q2 Þ ¼ E½p1   kS½p1 ;

q1 P0;q2 P0

ð4Þ

where E½p1  ¼ ða  q1  q2 Þq1 þ ða  q2  bq1 Þq2 þ gðk  C T Þ  k and S½p1  ¼ q1 r1 are the expected profit and the standard deviation of the retailer’s profit, respectively, and k P 0 is the retailer’s risk aversion coefficient. The retailer is riskneutral when k ¼ 0 and is risk-averse when k > 0. 3.1. Preliminary results Let q1 and q2 be the optimal stocking quantities of product 1 and product 2, respectively, of the retailer. Let p1 ; E½p1  and S½p1 , respectively, be the profit, the expected profit, and the standard deviation of profit of the retailer under the optimal solution. Proposition 1. q2 > 0 only if a  gc2 > 0, and q1 > 0 only if a  gc1  kr1 > 0.

Note that gci is the return of the retailer by putting ci in the bank, and a is the maximum (attainable) retail price of the two products. So, a  gci is the maximum (attainable) surplus of selling product i. Therefore, Proposition 1 implies that the retailer has no incentive to sell the product when the maximum surplus is negative. Moreover, in addition to the maximum surplus of selling the product, the retailer needs to deduct the cost of risk, which is reflected by kr1 , from the surplus of product 1 due to the uncertainty of developing product 1. The cost of risk contains the exogenous standard deviation of price of product 1 (r1 ) and the endogenous risk aversion of the retailer (k). As is shown in the later part of this study, k and r1 always appear in the form of kr1 in the optimal solution of the retailer and other important conditions. Therefore, to evaluate the effects of risk on the optimal decision, the retailer must consider the joint effects of kr1 instead of individual effects of k and r1 . For instance, q1 ¼ 0 if kr1 P ða  gc1 Þ, namely, a retailer does not develop her store brand product, if the cost of risk is too high under which both k and r1 must be sufficiently big. Only a big k or only a big r1 does not necessarily lead to q1 ¼ 0. Assumption 1. Assume that a  gci > 0, for all i ¼ 1; 2. In this study, we consider the situation where the retailer has already sold product 2 for a long time, and now considers to develop her store brand product. With Assumption 1, selling the two products are profitable for the retailer. The retailer then turns to take the risk, the substitution factor, the capital constraint, and the development cost into account in deciding q1 and q2 . 4 For a casual bank, the rate of borrowing is usually higher than that of saving. To simplify the analysis and hence obtain more management insights, we follow the common ideal bank setting adopted in literature (Buzacott and Zhang, 2004), and assume the rates of borrowing and saving are identical.

Please cite this article in press as: Cui, Q., et al. Store brand introduction in a two-echelon logistics system with a risk-averse retailer. Transport. Res. Part E (2015), http://dx.doi.org/10.1016/j.tre.2015.10.005

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Let A1 ¼ a  gc1  kr1 be the risk deducted surplus of selling product 1 (i.e., the surplus is deducted by the cost of risk), and A2 ¼ a  gc2 be the surplus of selling the manufacturer’s product. Proposition 2. Suppose that q1 ¼ 0 is fixed. (a) When borrowing is allowed, q2 ¼ A2 =2; p1 ¼ A22 =4 þ ðg  1Þk and S½p1  ¼ 0. (b) When borrowing is prohibited, i.e., when the constraint C T 6 k is considered, if

k > K 1 :¼ A2 c2 =2;

ð5Þ

then q2 ¼ A2 =2; p1 ¼ þ ðg  1Þk and S½p1  ¼ 0. Otherwise, q2 ¼ k=c2 ; p1 ¼ kðA2 c2  kÞ=c22 þ ðg  1Þk and S½p1  ¼ 0. A22 =4

As is shown in Proposition 2, selling product 2 only is always risk-free for the retailer (S½p1  ¼ 0). Moreover, q2 c2 ¼ k when k 6 K 1 . Thus, K 1 given in (5) represents the ‘‘sufficient capital threshold” of the retailer. If k P K 1 , then the retailer has sufficient capital and makes the decision as there is no capital constraint. However, if k < K 1 , then the retailer has insufficient capital, and the decision of the retailer is limited by the capital constraint. Note that, the sufficient capital threshold of the retailer varies from different situations. 4. Effects of capital constraint In this study, we explore how the substitution factor b, the risk factor, which includes k and r1 , the capital constraint k, and the development cost C affect the optimal decision and the desirability to develop the self-owned store brand product of the retailer. As is shown later in this study, the effects of the substitution factor b and the risk are mixed up with other factors and they are not easy to be separated out from other factors, while the effects of the capital constraint k and the effects of the development cost C are relatively clear and can be separated out easily. To isolate the effects caused by the capital constraint k, and the effects caused by the development cost C, we consider the two factors separately. In this section, we consider C ¼ 0 and investigate the effects caused by the capital constraint k. 4.1. Borrowing is allowed The retailer has three mutually exclusive product development policies: (i) do not develop the store brand product (q1 ¼ 0 and q2 > 0); (ii) concurrently develop the store brand product and sell the manufacturer’s product (q1 > 0 and q2 > 0); and (iii) develop the store brand product and stop selling the manufacturer’s product (q1 > 0 and q2 ¼ 0). Proposition 3 shows the necessary and sufficient conditions of each product development policy. Proposition 3. Suppose that borrowing is allowed. (a) q1 ¼ 0 and q2 ¼ A2 =2 > 0 if

A1 =A2 6 ð1 þ bÞ=2: (b)

q1 ¼

2A1  ð1 þ bÞA2 4  ð1 þ bÞ

2

ð6Þ

> 0 and q2 ¼

2A2  ð1 þ bÞA1 4  ð1 þ bÞ

2

>0

if

ð1 þ bÞ=2 < A1 =A2 < 2=ð1 þ bÞ:

ð7Þ

(c) q1 ¼ A1 =2 > 0 and q2 ¼ 0 if

A1 =A2 P 2=ð1 þ bÞ:

ð8Þ

As the substitution factor of product 2 is 1, 2=ð1 þ bÞ is the ratio of the substitution factors of product 2 to product 1 (ð1 þ bÞ=2 is the ratio of substitution factors of product 1 to product 2). To decide which product development policy to be adopted, the retailer needs to examine the ratio of surplus of selling products A1 =A2 , and compare it to the ratios of substitution factors. The ratio A1 =A2 reflects the profitability difference between selling product 1 and selling product 2. A big [small] A1 =A2 means that selling the retailer’s product is more [less] profitable than selling manufacturer’s product. Therefore, q1 is increasing with A1 =A2 and q2 is decreasing with A1 =A2 . Moreover, the retailer has no intention to develop product 1 when A1 =A2 6 ð1 þ bÞ=2 because the profit of selling product 1 after deducting the cost of risk is marginal. When A1 =A2 > ð1 þ bÞ=2, the profit of selling product 1 becomes sufficiently attractive for the retailer to develop product 1 even Please cite this article in press as: Cui, Q., et al. Store brand introduction in a two-echelon logistics system with a risk-averse retailer. Transport. Res. Part E (2015), http://dx.doi.org/10.1016/j.tre.2015.10.005

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if the cost of risk is included. On the other hand, when A1 =A2 P 2=ð1 þ bÞ, the gain from selling product 1 significantly outweighs that from the retail of product 2. Therefore, the retailer can stop selling product 2 and focus on the store brand business. According to Proposition 3, the retailer prefers to sell both product 1 and product 2 concurrently when b is relatively small, while it is more preferable for the retailer to sell either product 1 or product 2 when b is close to 1. When b is close to 1, from the perspective of customers, the two products have no big difference and the customers will only purchase one of them. Therefore, it is better for the retailer to sell the product that can generate a more considerable profit only. On the other hand, when b is small, from the perspective of customers, product 1 and product 2 are different, and they may buy both products due to the product differentiation. Therefore, the retailer benefits from selling more products to the customer in total. When the retailer sells only one of the two products, no substitution is included, and thus q1 and q2 are independent of b. On the contrary, when the retailer sells both products, the substitution factor takes place and hence both q1 and q2 depend on b. A1 is strictly decreasing in the cost of risk kr1 . Thus, q1 is decreasing in kr1 while q2 is increasing in kr1 . This result is intuitive because selling product 1 is risky while selling product 2 is risk-free. Therefore, a retailer having a bigger [smaller] cost of risk prefers to sell less [more] product 1 and more [less] product 2. Proposition 3 shows that the cost of risk, which is reflected by kr1 , is critical for the optimal solutions and the optimal product development policy of the retailer. For instance, the retailer prefers to develop her store brand product only if she has a small cost of risk. Proposition 4.

(a) The product development policy ðq1 ¼ 0; q2 ¼ A2 =2 > 0Þ is always optimal for the retailer regardless of her risk aversion if and only if

a  gc1 1 þ b : 6 2 A2

ð9Þ

(b) The product development policy ðq1 > 0; q2 ¼ 0Þ is worthy of consideration for the retailer only if

a  gc1 2 > : 1þb A2

ð10Þ

Proposition 4 indicates that (9) is the condition for the risk being critical in the retailer’s decision making. If (9) holds, then the retailer can ignore the risk because selling product 2 only gives her a comparatively big profit,5 and it is optimal for her not to develop product 1. However, if (9) fails, then the risk becomes important for the retailer because developing her store brand product could give her a significant profit. However, developing the store brand is risky for the retailer. Moreover, the right hand side of condition (9) is less than 1. Therefore, the retailer still prefers to develop the store brand product even though c1 > c2 , and the different between c1 and c2 could be big when b is small. This result suggests that the retailer should position the store product as differentiated from the manufacturer’s product if c1 > c2 . On the other hand, if (10) holds, then the expected profit of selling product 1 is sufficiently big such that stop selling product 2 is worthy of consideration for the retailer. Otherwise, the retailer still has to sell product 2 even though she develops the store brand product. Table 1 summaries the expected profits and the standard deviation of the retailer’s profit. To better understand the risk premium and the capital efficiency, we derive the mean–variance (s.d.) efficient frontier in Fig. 2 to illustrate the results in Table 1. All the three product development policies are considered by the retailers with different risk-aversions in Fig. 2(a) because (10) holds.6 In Fig. 2, k increases from right to left, so a retailer having a higher risk aversion tends to select an efficient solution on the left part of the efficient frontier. Consistent with the investment theory, the high expected profit-high risk phenomenon holds, namely, the risk-free solution q1 ¼ 0 and q2 ¼ A2 =2 gives the least profit, and the most risky risk-neutral solution q1 ¼ A1 =2 and q2 ¼ 0 with k ¼ 0 gives the most expected profit, and the retailer with a higher risk aversion prefers an efficient solution with a lower risk level and a lower expected profit. Moreover, from the risk perspective, the policy q1 > 0 and q2 > 0 stretches over the main risk interval, and it is critical for the retailer to focus on the capital efficiency if the policy q1 > 0 and q2 > 0 is adopted. A similar mean–variance efficient frontier is shown in Fig. 2(b) except that there is no efficient solution in the region q1 > 0 and q2 ¼ 0 because (10) does not hold. Finally, Fig. 2(c) shows the case when (9) holds. The risk-free solution is the only optimal solution of the retailer for all different risk aversions, and hence the risk is unimportant in this case.

5

Condition (9) holds when the surplus of selling product 1 (exclude the cost of risk) is small, the surplus of selling product 2 is big, or b is big. In the numerical example of Fig. 2, we take a ¼ 10; b ¼ 0:8; g ¼ 1:05; r1 ¼ 1:5; c2 ¼ 2, and k ¼ 100, respectively. c1 ¼ 1; 3 and 1.5 in Fig. 2(a), (c) and (b), respectively. The setting satisfies (10) for c1 ¼ 1, satisfies (9) for c1 ¼ 3, and does not satisfy (9) and (10) for c1 ¼ 1:5. We aim to highlight the important features of the mean–variance efficient frontiers shown in Fig. 2 and other figures in this paper. We can take other values for the parameters, and similar results can be obtained. 6

Please cite this article in press as: Cui, Q., et al. Store brand introduction in a two-echelon logistics system with a risk-averse retailer. Transport. Res. Part E (2015), http://dx.doi.org/10.1016/j.tre.2015.10.005

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Table 1 Expected profit and s.d. of profit of the retailer (borrowing is allowed). Policy (conditions)

E½p1  and S½p1 

q ¼ 0; q2 > 0
1  A1 1þb A2 6 2

p1 ¼ A22 =4 þ ðg  1Þk and S½p1  ¼ 0

q1 > 0; q2 > 0
 A1 1þb 2 2 < A2 < 1þb

E½p1  ¼ kðg  1Þ þ

q1 > 0; q2 ¼ 0
 A1 2 A2 P 1þb

E½p1  ¼ A21 =4 þ ðg  1Þk þ kr1 A1 =2 and S½p1  ¼ A1 r1 =2.

A21 þA22 ð1þbÞA1 A2 32bb2

2 þ kr1 2A1 ð1þbÞA 2

32bb

and S½p1  ¼ ½2A1 ð1þbÞA22 r1 4ð1þbÞ

Fig. 2. The mean–variance efficient frontier when C ¼ 0 and borrowing is allowed.

4.2. Borrowing is prohibited In Section 4.1, we discuss the optimal decisions for the retailer as a potential market entrant on the premise of sufficient capital. However, capital constraint is a common phenomenon in the process of firms’ operating. Therefore, it is necessary to explore the optimal decisions and related conditions when the retailer has insufficient capital. We first find the sufficient capital threshold for different cases when borrowing is prohibited, and then investigate the possible stocking strategies under different conditions.

Please cite this article in press as: Cui, Q., et al. Store brand introduction in a two-echelon logistics system with a risk-averse retailer. Transport. Res. Part E (2015), http://dx.doi.org/10.1016/j.tre.2015.10.005

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Proposition 5. Suppose that borrowing is prohibited. The sufficient capital threshold of the retailer is given by

(a)

K 1 :¼ A2 c2 =2; for (6) holds. (b)

K 3 :¼

c1 ½2A1  ð1 þ bÞA2  þ c2 ½2A2  ð1 þ bÞA1  2

4  ð1 þ bÞ

;

ð11Þ

for (7) holds. (c)

K 2 :¼ A1 c1 =2;

ð12Þ

for (8) holds. For q1 ¼ 0 and q2 > 0 the sufficient capital threshold is identical to the case where the retailer has no intention to develop her store brand product. The risk and the substitution factor have no effect in this case because q1 ¼ 0. For q1 > 0 and q2 ¼ 0 the sufficient capital threshold K 2 is decreasing in kr1 , or equivalently, a retailer who has a bigger cost of risk is more likely to have sufficient capital, because the retailer usually has less inventory of product 1 and hence requires less capital. For q1 > 0 and q2 > 0, unlike K 2 , the sufficient capital threshold K 3 is strictly increasing in kr1 if c1 =c2 < ð1 þ bÞ=2, remains constant if c1 =c2 ¼ ð1 þ bÞ=2, and is strictly decreasing in kr1 if c1 =c2 > ð1 þ bÞ=2. A retailer with a bigger [smaller] cost of risk has less [more] inventory of product 1 but more [less] inventory of product 2, and the inventory levels of the two products depend on the costs of products and the substitution factor. Therefore, the influence of risk on the sufficient capital threshold is mixed up with the costs of products and the substitution factor too. Proposition 6. Suppose that borrowing is prohibited, and k is smaller than the sufficient capital threshold. Then c1 q1 þ c2 q2 ¼ k, and

(a)

q1 ¼ 0 and q2 ¼ k=c2 ;

ð13Þ

if k 6 K 1 and

½c1 =c2  ð1 þ bÞ=2k 6

c 2 A2 ðc1 =c2  A1 =A2 Þ: 2

ð14Þ

(b)

q1 ¼ k=c1

and q2 ¼ 0;

ð15Þ

if k 6 K 2 and

½2=ð1 þ bÞ  c1 =c2 k 6

c 1 A2 ðA1 =A2  c1 =c2 Þ: 1þb

ð16Þ

(c)

q1 ¼

½2c1  ð1 þ bÞc2 k  c1 c2 A2 þ c22 A1 2½c21 þ c22  ð1 þ bÞc1 c2 

and q2 ¼

½2c2  ð1 þ bÞc1 k  c1 c2 A1 þ c21 A2 ; 2½c21 þ c22  ð1 þ bÞc1 c2 

ð17Þ

if k 6 K 3 , and (14) and (16) fail. According to (14) and (16), in addition to the ratio of surplus of the two products (A1 =A2 ), and the ratios of substitution factors of the two products (2=ð1 þ bÞ and ð1 þ bÞ=2), the feasible ranges of k for each optimal solution pair also depend on the ratio of costs of the two products (c1 =c2 ). The ratio of costs of the products is included because the efficiency of the use of Please cite this article in press as: Cui, Q., et al. Store brand introduction in a two-echelon logistics system with a risk-averse retailer. Transport. Res. Part E (2015), http://dx.doi.org/10.1016/j.tre.2015.10.005

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Q. Cui et al. / Transportation Research Part E xxx (2015) xxx–xxx Table 2 Feasible range of k of the optimal solution pairs. Conditions nðq1 ; q2 Þ

Given by (17)

Given by (15)

(a) Case A1 =A2 6 ð1 þ bÞ=2 c1 A1 1 6 k 6 K1 K c 2 < A2 c1 A1 k 6 K1 c 2 P A2

Given by (13)

2 < k < K 1 K Not optimal

2 k6K Not optimal

c1 c2 A1 A2 c1 c2

(b) Case ð1 þ bÞ=2 < A1 =A2 < 2=ð1 þ bÞ < AA12 Not optimal ¼ cc12 Not optimal A1 1 > A2 k6K

 2 < k 6 K3 K k 6 K3  1 < k 6 K3 K

2 k6K Not optimal Not optimal

(c) Case A1 =A2 P 2=ð1 þ bÞ: k 6 K 3 Not optimal 6 AA12 Not optimal 1 k6K

Not optimal Not optimal 1 < k < K 2 K

k 6 K2 k 6 K2  2 6 k 6 K2 K

c1 2 c2 6 1þb 2 < cc12 1þb c1 A1 c 2 > A2

Table 3 Expected profit and s.d. of profit of the retailer (borrowing is prohibited). E½p1  and S½p1 

Policy q1 q1

¼ 0,

q2

>0

> 0,

q2

>0

p1 ¼ A2 k=c2 þ ðg  1Þk  k2 =c22 and S½p1  ¼ 0 2 2 2 ½2c ð1þbÞc2 kc1 c2 A2 þc22 A1 2 ÞþA2 ð2c 2 ð1þbÞc 1 Þð32bb Þk þ ðg  1Þk þ kr1 1 2 c2 ð1þbÞc E½p1  ¼ ðA2 c1 A1 c2 Þ þ2k½A1 ð2c14ð1þbÞc 2 ½c2 ð1þbÞc1 c2 þc2  ½ 1 c 2 þc  1

2

1

2

½2c ð1þbÞc kc c A þc2 A

1 2 2 2 1 and S½p1  ¼ 1 2½c2 þc22ð1þbÞc r1 1 c2 
1 2  k k E½p1  ¼ A1  c1 c1 þ ðg  1Þk þ kr1 ck1 and S½p1  ¼ kcr11

q1 > 0; q2 ¼ 0

capital becomes important when borrowing is prohibited, and the costs of products directly affect the efficiency of the use of capital. Let 2  1 ¼ c1 c2 A2  c2 A1 K 2c1  ð1 þ bÞc2

2  2 ¼ c 1 c 2 A1  c 1 A2 : and K 2c2  ð1 þ bÞc1

Table 2 shows the feasible range of k of the optimal solution pairs for different situations which are classified according to the value of c1 =c2 . We obtain three implications from Table 2. First, the feasible range of k varies for different situations. For instance, q1 > 0 and q2 ¼ 0 is optimal for a small k when A1 A2

A1 A2

6 1þb . However, q1 > 0 and q2 ¼ 0 is optimal for a big k when 2

P 1þb . Therefore, the optimal solutions and the optimal product development policy of the retailer could vary for different 2

situations even though k remains unchanged. Second, unlike the case where the retailer has sufficient capital and only one product development policy is optimal, i.e., q1 ¼ 0 and q2 > 0 for

A1 A2

6 1þb ; q1 > 0 and q2 > 0 for 2

1þb 2

2 < AA12 6 1þb , and q1 > 0 and q2 ¼ 0 for

A1 A2

2 P 1þb , when the retailer has

insufficient capital, all the three product development policies can be optimal for all cases. In general, (i) the policy q1 ¼ 0 and q2 > 0 becomes mean–variance efficient for the retailer when c1 =c2 is big, because only selling product 2 leads to a more efficient use of capital; (ii) the policy q1 > 0 and q2 ¼ 0 becomes efficient when c1 =c2 is small, because only selling product 1 leads to a more efficient use of capital; and (iii) the policy q1 > 0 and q2 > 0 becomes efficient when c1 =c2 is medium. As all the capital is used to purchase the products, the retailer can improve the efficiency of use of capital by reducing the quantity of the higher cost product and then using the saved capital to increase the inventory of the cheaper product. Therefore, when c1 =c2 is big, i.e., c1 is relatively big or c2 is relatively small, the retailer could reduce the stocking level of the more expensive product 1 and replace them with the cheaper product 2 to improve the efficiency of use of capital, and vice versa. Third, similar to the scenario where borrowing is allowed, the cost of risk kr1 is also critical for the optimal solutions and the optimal product development policy when borrowing is prohibited. However, as the feasible range of k varies with different situations, kr1 has no monotone relationship with the optimal solutions. For instance, the policy q1 > 0 and q2 ¼ 0 is optimal for both A1 =A2 P 2=ð1 þ bÞ P c1 =c2 , i.e., A1 =A2 (or kr1 ) is big [small],7 and for A1 =A2 6 ð1 þ bÞ=2, i.e., A1 =A2 (or kr1 ) is small [big]. This finding is quite different from the case when borrowing is allowed under which the retailer prefers to reduce the stocking level of product 1 or would rather not develop product 1 when kr1 increases. According to the conditions discussed in Propositions 5 and 6, we can obtain the retailer’s expected profits and the corresponding standard deviations in Table 3. Intuitively, capital constraint k becomes another factor that should be considered when the retailer determines her product policies. 7

A1 =A2 is strictly decreasing in kr1 .

Please cite this article in press as: Cui, Q., et al. Store brand introduction in a two-echelon logistics system with a risk-averse retailer. Transport. Res. Part E (2015), http://dx.doi.org/10.1016/j.tre.2015.10.005

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Fig. 3. The efficient frontier when C ¼ 0 and borrowing is prohibited.

We trace out the efficient frontier in Fig. 3.8 Although the efficient frontier is similar to the previous cases traced out in Fig. 2 when the risk aversion factor k is relatively large, the policy q1 > 0 and q2 ¼ 0 when k is small is quite different from that of in Fig. 2. It can be found that the efficient frontier is discontinuous when the profit level is high (i.e., when k is small). In specific, if the retailer chooses the policy (q1 > 0; q2 ¼ 0) in order to obtain more expected profit than what the (q1 > 0; q2 > 0) policy can achieve, she needs to bear a big jump in the risk level and sticks the risk level at S½p1  ¼ kr1 =c1 , regardless of her risk attitude, while the improvement of expected profit is very small. This indicates that the capital constraint distorts the risk management of the retailer. 5. Effects of development cost In this section, we investigate the effect of the development cost C > 0. To isolate the effect of C > 0 and the capital constraint, we consider that borrowing is allowed first. Then we investigate the joint effect of the development cost and the capital constraint afterward. 5.1. Borrowing is allowed  2 ¼ ðA2  A2 Þ=4g, and C  3 ¼ ½2A1  ð1 þ bÞA2 2 =f4g½4  ð1 þ bÞ2 g. Let C 1 2 Proposition 7. Suppose borrowing is allowed and C > 0.  3 , or (8) holds and C P C 2. (a) q1 ¼ 0 and q2 ¼ A2 =2 > 0 if (6) holds, or (7) holds and C P C (b)

q1 ¼

2A1  ð1 þ bÞA2 4  ð1 þ bÞ

2

> 0 and q2 ¼

2A2  ð1 þ bÞA1 2

4  ð1 þ bÞ

> 0;

3. if (7) holds and C 6 C 2. (c) q1 ¼ A1 =2 > 0 and q2 ¼ 0 if (8) holds and C 6 C  2 are the upper limits of C for the retailer to develop her store brand product, for ð1 þ bÞ=2 < A1 =A2 < 2=ð1 þ bÞ  3 and C C  2 and C  3 are strictly decreasing in kr1 . Therefore, the risk reduces the intention of and A1 =A2 P 2=ð1 þ bÞ, respectively. Both C the retailer to develop product 1.  2 is independent of b because the retailer sells product 1 only under the policy q > 0 and q ¼ 0, and hence the substiC 1 2  3 is strictly decreasing in b. C  3 is considered when the retailer adopts the tution factor can be ignored. On the other hand, C

8

In Fig. 3, a ¼ 20; b ¼ 0:8; g ¼ 1:05;

r1 ¼ 1:5; c1 ¼ 1; c2 ¼ 2, and k ¼ 10, respectively.

Please cite this article in press as: Cui, Q., et al. Store brand introduction in a two-echelon logistics system with a risk-averse retailer. Transport. Res. Part E (2015), http://dx.doi.org/10.1016/j.tre.2015.10.005

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policy q1 > 0 and q2 > 0. The two products are very similar to the customers when b is close to 1, and a cut-throat competition between the two products can be expected if the retailer sells both products concurrently. Therefore, it is unworthy for the retailer to develop the store brand product if the develop cost is too high. However, the two products are different to the customers when b is small, and it gives space for the co-existing of the two products. Therefore, it is still worthy for the retailer to develop the store brand product even thought the development cost is high. This finding shows that product differentiation may be good for the retailer even though the product differentiation causes a lower expected retail price of the store brand product (i.e., E½p1  < p2 is always true for b < 1). Figs. 4 and 5 show that the efficient frontier varies with C. The solid lines show the mean–variance efficient frontier for C > 0, and the dash lines show the infeasible part of the mean–variance efficient frontier due to a big C > 0. First of all, there is a drop in the expected profit for q1 ! 0 because C > 0 incurs for all q1 > 0 and it is excluded for q1 ¼ 0. Moreover, a bigger C > 0 leads to a bigger drop. As is shown in Fig. 4(a), when the development cost C is sufficiently small, all the three product policies could be mean-variance efficient for the retailer, but her intension could be inhibited by the positive development  3 when her risk aversion is high. Therefore, the efficient frontier is discontinuous from the risk-free point to the cost C P C risky part, and there are jumps in both the expected profit and the risk on the efficient frontier. As a result, for a retailer with a middle k, her risk management is distorted by the development cost because she can only take a lower risk solution and earns a lower profit like a highly risk-averse retailer with a big k, or take a high level risk solution like a risk-neutral retailer

Fig. 4. C = 0.3 and C = 5.

Please cite this article in press as: Cui, Q., et al. Store brand introduction in a two-echelon logistics system with a risk-averse retailer. Transport. Res. Part E (2015), http://dx.doi.org/10.1016/j.tre.2015.10.005

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Fig. 5. C ¼ 3:486 and C ¼ 3:894.

with a small k, but can not be in the middle when C > 0. To cover the development cost, the retailer will introduce her store brand only when the store brand product generates a significant profit. According to the high expected profit-high risk phenomenon, the retailer needs to face a high risk too. Therefore, the development cost causes the gap on the efficient frontier. Fig. 5(a) and (b) further indicates that the gaps between the risk-free solution and the lowest risky solution are increasing with C. For an extreme case with a relatively large C, only the policies q1 ¼ 0; q2 > 0 and q1 > 0; q2 ¼ 0 are mean–variance  2 and C  3 are decreasing in k, C P C  3 usually holds for the retailer whose efficient. As the thresholds of the development cost C k is big, and she would not develop her store brand business under this situation. However, for the same development cost  2 usually holds for the retailer with a small k, and she would rather develop her store brand product. Therefore, a C; C < C polarization of mean–variance efficient frontier is caused by a big C under which only the polarized solutions, the riskfree solution and extremely high risk solutions, are mean–variance efficient to the retailer.

5.2. Borrowing is prohibited Next, we explore the integrated effect of non zero development cost and the capital constraint. The optimal policy and corresponding conditions are summarized in Proposition 8. Please cite this article in press as: Cui, Q., et al. Store brand introduction in a two-echelon logistics system with a risk-averse retailer. Transport. Res. Part E (2015), http://dx.doi.org/10.1016/j.tre.2015.10.005

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Proposition 8. Suppose that borrowing is prohibited and C > 0. C T ¼ k, and

(a) q1 ¼ 0 and q2 ¼ k=c2 for the following cases: 1. k 6 K 1 and (14) holds. ^ 3 and 2. k 6 minfK 1 ; K 0 g; C P C 3

c 2 A2 ðc1 =c2  A1 =A2 Þ; 2 c1 A2 ½2=ð1 þ bÞ  c1 =c2 ðk  CÞ > ðA1 =A2  c1 =c2 Þ: 1þb

½c1 =c2  ð1 þ bÞ=2ðk  CÞ >

3.

k6

minfK 1 ; K 02 g; C

ð18Þ ð19Þ

^ 2 and PC

½2=ð1 þ bÞ  c1 =c2 ðk  CÞ 6

c 1 A2 ðA1 =A2  c1 =c2 Þ: 1þb

ð20Þ

(b) q1 ¼ k=c1 and q2 ¼ 0 for the following cases: ^ 2 , and (20) holds, 1. k 6 minfK 1 ; K 0 g; C < C 2

2.

~ 2 , and (20) holds. K 1 < k < K 02 ; C < C

(c)

q1 ¼

½2c1  ð1 þ bÞc2 ðk  CÞ  c1 c2 A2 þ c22 A1 ; 2½c21 þ c22  ð1 þ bÞc1 c2 

q2 ¼

½2c2  ð1 þ bÞc1 ðk  CÞ  c1 c2 A1 þ c21 A2 ; 2½c21 þ c22  ð1 þ bÞc1 c2 

and

for the following cases: 1.

^3, (18) and (19) hold, k 6 minfK 1 ; K 03 g and C < C 0 ~ (18) and (19) hold, K 1 < k < K , and C < C 3 ,

2. where K 02 ¼ K 2 þ C; K 03 ¼ K 3 þ C,

3

 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ^ 2 ¼ 1 c2 ð2k  A1 c1  c2 gÞ þ c1 c2 ðA1 þ c1 gÞ2 þ 4C2 ; C 1 2 2c2  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ~ 2 ¼ 1 2k  c1 ðA1 þ c1 gÞ þ c1 ðA1 þ c1 gÞ2  ðA2 þ 4gkÞ ; C 2 2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h i 2/1=2 c22 A21 þ A22  ð1 þ bÞA1 A2 þ g C1 þ g 2 / þ ð1  bÞð3 þ bÞC2 A c þ A c þ 2g/ 2 2 ^3 ¼ k  1 1 þ ; C ð1  bÞð3 þ bÞ c2 ð1  bÞð3 þ bÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 /1=2 ½2A1  ð1 þ bÞA2  þ 4g C1 þ 4g 2 /  4ð1  bÞð3 þ bÞk ~ 3 ¼ k  A1 c1 þ A2 c2 þ 2g/ þ ; C 2 2 3  2b  b 3  2b  b

c1 ¼ 2c1  ð1 þ bÞc2 ; c2 ¼ 2c2  ð1 þ bÞc1 ; / ¼ c21  ð1 þ bÞc1 c2 þ c22 ; and C1 ¼ A1 c1 þ A2 c2 ; C2 ¼ kðk  A2 c2  gc22 Þ:

Table 4 Expected profit and Lðq1 ; q2 Þ of the retailer (borrowing is allowed). E½p1  and Lðq1 ; q2 Þ

Policy q1 q1

¼ 0,

q2

>0

Lo ðÞ ¼ p1 ¼ A22 =4 þ kðg  1Þ

> 0,

q2

>0

E½p1  ¼ kðg  1Þ  Cg þ o

L ðÞ ¼ kðg  1Þ  Cg þ q1 > 0, q2 ¼ 0

A21 þA22 ð1þbÞA1 A2 2 32bb A21 þA22 ð1þbÞA1 A2 2 32bb

2 þ kr1 2A1 ð1þbÞA 2

32bb

E½p1  ¼ A21 =4 þ kðg  1Þ  Cg þ kr1 A1 =2 Lo ðÞ ¼ A21 =4 þ kðg  1Þ  Cg

Please cite this article in press as: Cui, Q., et al. Store brand introduction in a two-echelon logistics system with a risk-averse retailer. Transport. Res. Part E (2015), http://dx.doi.org/10.1016/j.tre.2015.10.005

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Table 5 Expected profits and Lðq1 ; q2 Þ of the retailer (borrowing is prohibited). Policy

E½p1  and Lðq1 ; q2 Þ

q1 ¼ 0, q2 > 0

p1 ¼ Lc ðÞ ¼ ðA2  k=c2 Þk=c2 þ ðg  1Þk

q1 > 0, q2 > 0

E½p1  ¼ ðA2 c1 A1 c2 Þ c

q1 > 0, q2 ¼ 0

2

2

2

2

þ2ðkCÞ½A1 ð2c1 ð1þbÞc2 ÞþA2 ð2c2 ð1þbÞc1 Þð32bb ÞðkCÞ 4½c21 ð1þbÞc1 c2 þc22  2

2

ðA2 c1 A1 c2 Þ þ2ðkCÞ½A1 ð2c1 ð1þbÞc2 ÞþA2 ð2c2 ð1þbÞc1 Þð32bb ÞðkCÞ 4½c21 ð1þbÞc1 c2 þc22 

L ðÞ ¼
 kC kC E½p1  ¼ A1  kC c1 c1 þ ðg  1Þk  Cg þ kr1 c1 ,
 c kC kC L ðÞ ¼ A1  c1 c1 þ ðg  1Þk  Cg

þ ðg  1Þk  Cg þ kr1

½2c1 ð1þbÞc2 ðkCÞc1 c2 A2 þc22 A1 2½c21 ð1þbÞc1 c2 þc22 

,

þ ðg  1Þk  Cg

In contrast with the scenarios where only the development cost exists, or only borrowing is prohibited, more complex and restrictive limitations of C and of the capital are found, when both capital constraint and development cost of store brand business co-exist. Moreover, Proposition 8 indicates that the retailer should focus on the capital surplus k  C instead of k in the scenario where q1 > 0 because only the part of k  C can generate profits. 6. Conclusion In this study, we consider a conventional retailer’s potential market entry incentives. Besides the traditional retail business, the retailer may also develop her store brand product in order to improve the profitability. We consider a stochastic price-demand relationship, and apply the mean–variance formulation to analyze the risk premium of the development of the store brand business. We consider the quantities of the two products as decisions of the retailer, and focus on the risk analysis and the effects of the product substitution, the capital constraint, and the development cost. Our analysis shows that it is critical for the retailer to jointly consider the internal risk aversion level and the external price-demand uncertainty, which we call it the cost of risk. A retailer with high cost of risk never introduces her store brand product, while a retailer with a smaller cost of risk is more willing to introduce her store brand product. When borrowing is allowed and the development cost is zero, we find that the retailer’s optimal strategies depend on the ratios of risk involved profitability A1 =A2 and the ratios of substitution factors of the store brand product and the national brand product. Three interesting results that are related to the substitution factor b, are obtained, and two of them are related to the risk. First, the retailer prefers to sell both the store brand product and the national brand product concurrently when b is small and the cost of risk kr1 is not polarized. Second, the retailer is still willing to develop her store brand product even though the unit production cost of the store brand product is bigger than the wholesale price of the manufacturer’s product, i.e., c1 > c2 . We argue that customers are willing to buy both products because of the product differentiation when b is small. Thus, the retailer is able to generate more total demand and benefits from selling both products. Third, the risk becomes unimportant if b is close to 1 and A2 is not too small. When b is close to 1, the product differentiation is small. The customers are not willing to buy both products. With the addition of selling the manufacturer’s product is attractive (A2 is not too small), the retailer prefers to sell the manufacturer’s product only to avoid any uncertainty that are caused by introducing the store brand product. When borrowing is prohibited, we show that the retailer can improve the efficiency of capital by reducing the stocking level of more expensive product and replacing them with the cheaper one. Moreover, the capital constraint distorts the risk management of the retailer. A retailer originally preferring the risk-free decision of not developing her store brand product will become willing to take the risk and to develop her store brand product just because she has insufficient capital. We also reveal that the development cost of the store brand product reduces the choices of efficient solution of the retailer too. The retailer can only choose the risk-free solution or extremely high risk solutions. These results show that the effects of the capital constraint and the development cost on the risk management are different. Our research aims to provide some insights to the retailer’s store brand business management in reality and better understandings on the impact of risk factors. There are still many interesting issues worth further studying in such a setting. For instance, the interactions between the retailer and the manufacturer during the introduction of store brand of the retailer, when both agents are risk-averse or loss-averse. Furthermore, the quality of the store brand product can also be the decision variable of the retailer, although some conflicts with the upstream members might be induced. Acknowledgements We sincerely thank the editor, the guest editor of the special issue ‘‘risk management”, and the anonymous reviewers for their many helpful suggestions and kind advices. Chun-Hung Chiu is partially supported by the National Natural Science Foundation of China with the Grant No. of 71371197. Zhongfei Li is partially supported by the National Natural Science Foundation of China with the Grant No. of 71231008, and the Natural Science Foundation of Guangdong Province of China with the Grant No. of 2014A030312003. Xin Dai is partially supported by the Guangdong Soft Science Research Project with the Grant No. of 2013B070206029.

Please cite this article in press as: Cui, Q., et al. Store brand introduction in a two-echelon logistics system with a risk-averse retailer. Transport. Res. Part E (2015), http://dx.doi.org/10.1016/j.tre.2015.10.005

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15

Appendix A. Proofs and technical results

Proof of Proposition 1. If the retailer has sufficient capital, her profit function is

p1 ðq1 ; q2 Þ ¼ ða  q1  q2 þ e1 Þq1 þ ða  q2  bq1 Þq2 þ gðk  C T Þ  k; therefore, the mean–variance objective function is given by

max Lðq1 ; q2 Þ ¼ E½p1   kS½p1 ;

q1 P0;q2 P0

where E½p1  ¼ ða  q1  q2 Þq1 þ ða  q2  bq1 Þq2 þ gðk  C T Þ  k and S½p1  ¼ q1 r1 . By considering the K–T conditions, we have

@Lðq1 ; q2 Þ @Lðq1 ; q2 Þ ¼ a  2q1  ð1 þ bÞq2  c1 g  kr1 6 0; q1 ¼ 0; @q1 @q1 @Lðq1 ; q2 Þ @Lðq1 ; q2 Þ ¼ a  2q2  ð1 þ bÞq1  c2 g 6 0; q2 ¼ 0: @q2 @q2 Because all the parameters are nonnegative, if q2 > 0, there must exist a  gc2 > 0. Similarly, a  gc1  kr1 > 0 is necessary for q1 > 0. On the contrary, if kr1 P ða  gc1 Þ; q1 ¼ 0. h Proof of Proposition 2. Note that, we assume the rates of borrowing and saving (g) are identical for the retailer and define A1 ¼ a  gc1  kr1 ; A2 ¼ a  gc2 . Suppose that q1 ¼ 0 is fixed, then the retailer’s decision is risk-free. When borrowing is allowed, the retailer’s objective function can be written as

L1 ðq2 Þ ¼ ða  q2 Þq2 þ gðk  c2 q2 Þ  k: We have @L1 ðq2 Þ=@q2 ¼ a  gc2  2q2 and @ 2 L1 ðq2 Þ=@q22 ¼ 2 < 0. Therefore, L1 ðq2 Þ is concave in q2 , and the existence and uniqueness are guaranteed. Consider the K–T condition, we have

@L1 ðq2 Þ ¼ a  2q2  c2 g 6 0; @q2

q2

@L1 ðq2 Þ ¼ 0: @q2

We assume a  gc2 > 0, then q2 > 0 can be easily obtained, thus @L1 ðq2 Þ=@q2 ¼ a  gc2  2q2 ¼ 0 and q2 ¼ A2 =2,

p ¼ A22 =4 þ ðg  1Þk, and S½p1  ¼ 0.  1

When borrowing is prohibited, the retailer’s optimization problem becomes

8 max Lðq2 Þ ¼ ða  q2 Þq2 þ ðk  c2 q2 Þg  k > < q2 s:t: q2 P 0; > : k  c2 q2 P 0: Formulating the Lagrangian function yields

Gðq2 ; k0 Þ ¼ ða  q2 Þq2 þ ðk  c2 q2 Þg  k þ k0 ðk  c2 q2 Þ; and according to the K–T condition, we have

@G @G ¼ a  2q2  c2 g  c2 k0 6 0; q ¼ 0; @q2 @q2 2 @G @G 0 ¼ k  c2 q2 P 0; k ¼ 0: @k0 @k0 If k  c2 q2 P 0, the proof is the same as the case ‘‘borrowing is allowed”. Otherwise, if the capital constraint exists, i.e. k ¼ c2 q2 , thus q2 ¼ k=c2 , p1 ¼ kðA2 c2  kÞ=c22 þ ðg  1Þk and S½p1  ¼ 0. h Proof of Proposition 3. We consider C ¼ 0 and investigate the effects caused by the capital constraint. Suppose that borrowing is allowed, the retailer’s profit function is

p1 ðq1 ; q2 Þ ¼ ða  q1  q2 þ e1 Þq1 þ ða  q2  bq1 Þq2 þ gðk  C T Þ  k; therefore, the mean–variance objective function is given by

max Lðq1 ; q2 Þ ¼ E½p1   kS½p1 ;

q1 P0;q2 P0

where E½p1  ¼ ða  q1  q2 Þq1 þ ða  q2  bq1 Þq2 þ gðk  C T Þ  k and S½p1  ¼ q1 r1 . By applying the K–T conditions, we have

Please cite this article in press as: Cui, Q., et al. Store brand introduction in a two-echelon logistics system with a risk-averse retailer. Transport. Res. Part E (2015), http://dx.doi.org/10.1016/j.tre.2015.10.005

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@Lðq1 ; q2 Þ ¼ A1  2q1  ð1 þ bÞq2 6 0; @q1 @Lðq1 ; q2 Þ ¼ A2  2q2  ð1 þ bÞq1 6 0; @q2

@Lðq1 ; q2 Þ q1 ¼ 0; @q1 @Lðq1 ; q2 Þ q2 ¼ 0: @q2

(a) If q1 ¼ 0 and q2 > 0 hold, then q2 ¼ A2 =2 if feasible. Moreover, it is necessary to guarantee A1  2q1  ð1 þ bÞq2 6 0,

thus we get A1 =A2 6 ð1 þ bÞ=2, E½p1  ¼ A22 =4 þ ðg  1Þk, and S½p1  ¼ 0. (b) If q1 > 0 and q2 > 0 hold, then

@Lðq1 ; q2 Þ ¼ A1  2q1  ð1 þ bÞq2 ¼ 0; @q1 @Lðq1 ; q2 Þ ¼ A2  2q2  ð1 þ bÞq1 ¼ 0; @q2 by solving the above equations, we obtain the retailer’s optimal stocking quantities

q1 ¼

2A1  ð1 þ bÞA2 4  ð1 þ bÞ

2

and q2 ¼

2A2  ð1 þ bÞA1 4  ð1 þ bÞ

2

;

and, 2A1  ð1 þ bÞA2 > 0 and 2A2  ð1 þ bÞA1 > 0 should be satisfied, i.e.,

ð1 þ bÞ=2 < A1 =A2 < 2=ð1 þ bÞ: ð1þbÞA2 r1 Besides, S½p1  ¼ ½2A14ð1þbÞ and 2

E½p1  ¼ kðg  1Þ þ

A21 þ A22  ð1 þ bÞA1 A2 3  2b  b

2

þ kr1

2A1  ð1 þ bÞA2 3  2b  b

2

:

(c) If q1 > 0 and q2 ¼ 0 hold, then q1 ¼ A1 =2 if feasible. Meanwhile, it is necessary to have A2  2q2  ð1 þ bÞq1 6 0, then it can be obtained that A1 =A2 P 2=ð1 þ bÞ; E½p1  ¼ A21 =4 þ ðg  1Þk þ kr1 A1 =2, and S½p1  ¼ A1 r1 =2. h

Proof of Proposition 4. According to the proof of Proposition 3, the product development policy ðq1 ¼ 0; q2 > 0Þ is optimal if

agc1 kr1 A2

q2

1 6 1þb . Therefore, agc 6 1þb is always true such that agcA12kr1 6 1þb because kr1 > 0. In addition, the policy (q1 > 0 and 2 2 2 A2

agc1 kr1 2 2 1 P 1þb , then agc > 1þb becomes mean-variance A2 A2   whether q1 > 0 and q2 ¼ 0 is optimal depends on k 1 . h

¼ 0) is based on the condition

under the condition

agc1 A2

>

2 , 1þb

efficient for the retailer, because

r

Proof of Proposition 5. The sufficient capital threshold varies from different policies, and according to Proposition 3, for (6) holds, K 1 ¼ q2 c2 ¼ A2 c2 =2; for (7) holds,

K 3 ¼ q2 c2 þ q1 c1 ¼

c1 ½2A1  ð1 þ bÞA2  þ c2 ½2A2  ð1 þ bÞA1  2

4  ð1 þ bÞ

;

for (8) holds, K 2 ¼ q1 c1 ¼ A1 c1 =2. h Proof of Proposition 6. Suppose that borrowing is prohibited and k is smaller than the sufficient capital threshold, then the retailer’s objective function can be written as

8 max Lðq1 ; q2 Þ ¼ ða  q1  q2 Þq1 þ ða  q2  bq1 Þq2 þ ðk  C T Þg  k  kq1 r1 > < q1 ;q2 s:t: q1 P 0; q2 P 0; > : k  C T P 0: The corresponding Lagrangian function is

G ¼ ða  q1  q2 Þq1 þ ða  q2  bq1 Þq2 þ ðk  C T Þg  k  kq1 r1 þ k0 ðk  c1 q1  c2 q2 Þ; it can be derived from the K–T conditions that

Please cite this article in press as: Cui, Q., et al. Store brand introduction in a two-echelon logistics system with a risk-averse retailer. Transport. Res. Part E (2015), http://dx.doi.org/10.1016/j.tre.2015.10.005

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@G @G ¼ a  2q1  ð1 þ bÞq2  c1 g  kr1  c1 k0 6 0; q ¼ 0; @q1 @q1 1 @G @G ¼ a  2q2  ð1 þ bÞq1  c2 g  c2 k0 6 0; q ¼ 0; @q2 @q2 2 @G @G 0 ¼ k  c1 q1  c2 q2 P 0; k ¼ 0: @k0 @k0

(a) The condition k 6 K 1 and q1 ¼ 0; q2 > 0 is equivalent to

8 0 > < a  ð1 þ bÞq2  c1 g  kr1  c1 k 6 0 0 a  2q2  c2 g  c2 k ¼ 0 > : k  c2 q2 ¼ 0: Thus we have q1 ¼ 0 and q2 ¼ k=c2 , and

½c1 =c2  ð1 þ bÞ=2k 6

c 2 A2 ðc1 =c2  A1 =A2 Þ: 2

Further, it can be obtained that

p1 ¼ A2 k=c2 þ ðg  1Þk  k2 =c22 and S½p1  ¼ 0.

(b) Similarly, the condition k 6 K 2 and q1 > 0; q2 ¼ 0 is equivalent to

8 0 > < a  2q1  c1 g  kr1  c1 k ¼ 0 0 a  ð1 þ bÞq1  c2 g  c2 k 6 0; > : k  c1 q1 ¼ 0: Then we have q1 ¼ k=c1 and q2 ¼ 0, and

½2=ð1 þ bÞ  c1 =c2 k 6

c 1 A2 ðA1 =A2  c1 =c2 Þ: 1þb


 Also, E½p1  ¼ A1  ck1 ck1 þ ðg  1Þk þ kr1 ck1 .

(c) The condition k 6 K 3 and q1 > 0; q2 > 0 is equivalent to

8 0 > < a  2q1  ð1 þ bÞq2  c1 g  kr1  c1 k ¼ 0; 0 a  2q2  ð1 þ bÞq1  c2 g  c2 k ¼ 0; > : k  c1 q1  c2 q2 ¼ 0: Then we get

q1 ¼

½2c1  ð1 þ bÞc2 k  c1 c2 A2 þ c22 A1 2½c21 þ c22  ð1 þ bÞc1 c2 

and q2 ¼

½2c2  ð1 þ bÞc1 k  c1 c2 A1 þ c21 A2 ; 2½c21 þ c22  ð1 þ bÞc1 c2 

and

c 1 A2 ðA1 =A2  c1 =c2 Þ; 1þb c 2 A2 ½c1 =c2  ð1 þ bÞ=2k > ðc1 =c2  A1 =A2 Þ: 2

½2=ð1 þ bÞ  c1 =c2 k >

Furthermore, we have

E½p1  ¼

2

ðA2 c1  A1 c2 Þ2 þ 2k½A1 ð2c1  ð1 þ bÞc2 Þ þ A2 ð2c2  ð1 þ bÞc1 Þ  ð3  2b  b Þk   4 c21  ð1 þ bÞc1 c2 þ c22 þ kr1

2

½2c1  ð1 þ bÞc2 k  c1 c2 A2 þ c22 A1   þ ðg  1Þk; 2 c21  ð1 þ bÞc1 c2 þ c22

and

S½p1  ¼

½2c1  ð1 þ bÞc2 k  c1 c2 A2 þ c22 A1 r1 : 2½c21 þ c22  ð1 þ bÞc1 c2 



Please cite this article in press as: Cui, Q., et al. Store brand introduction in a two-echelon logistics system with a risk-averse retailer. Transport. Res. Part E (2015), http://dx.doi.org/10.1016/j.tre.2015.10.005

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Q. Cui et al. / Transportation Research Part E xxx (2015) xxx–xxx

Proof of Proposition 7. When borrowing is allowed and C > 0; Lðq1 ; q2 Þ is discontinuous at q1 ¼ 0 and q1 > 0. As consequence, we should consider the fixed cost C on the basis of Proposition 3. To distinguish the different Lðq1 ; q2 Þ in the two cases (borrowing is allowed and, the capital is inefficient and borrowing is prohibited), we use a superscript ‘‘o” for those when borrowing is allowed, and ‘‘c” for those in the scenario where the retailer’s capital is inefficient and borrowing is prohibited. Besides, we define

 2 ¼ ðA2  A2 Þ=4g; C 1 2

2 C 3 ¼ ½2A1  ð1 þ bÞA2 

.n

h io 2 4g 4  ð1 þ bÞ :

Then the retailer’s objective function is

(

max Lðq1 ; q2 Þ ¼ ða  q1  q2 Þq1 þ ða  q2  bq1 Þq2 þ ðk  C T Þg  k  kq1 r1 q1 ;q2

q1 P 0;

s:t:

q2 P 0:

The corresponding Lagrangian function is

G ¼ ða  q1  q2 Þq1 þ ða  q2  bq1 Þq2 þ ðk  C T Þg  k  kq1 r1 ; it can be derived from the K–T conditions that

@G @G ¼ a  2q1  ð1 þ bÞq2  c1 g  kr1  c1 k0 6 0; q ¼ 0; @q1 @q1 1 @G @G ¼ a  2q2  ð1 þ bÞq1  c2 g  c2 k0 6 0; q ¼ 0: @q2 @q2 2

(a) q1 ¼ 0 and q2 ¼ A2 =2 if (i) A1 =A2 6 ð1 þ bÞ=2; (ii) ð1 þ bÞ=2 < A1 =A2 < 2=ð1 þ bÞ, and

2

Lo ðq1 > 0; q2 > 0Þ  Lo ðq1 ¼ 0; q2 > 0Þ ¼

2

½2A1  ð1 þ bÞA2   4Cgð3  2b  b Þ 2

4ð3  2b  b Þ

3; i.e. C P C A =A P 2=ð1 þ bÞ, and (iii) 1 2

Lo ðq1 > 0; q2 ¼ 0Þ  Lo ðq1 ¼ 0; q2 > 0Þ ¼ ðA21  A22 Þ=4  Cg 6 0;

6 0;

i:e: C P C 2 ;

(b)

q1 ¼

2A1  ð1 þ bÞA2 4  ð1 þ bÞ

2

> 0 and q2 ¼

2A2  ð1 þ bÞA1 2

4  ð1 þ bÞ

> 0;

3; if ð1 þ bÞ=2 < A1 =A2 < 2=ð1 þ bÞ, and Lo ðq1 > 0; q2 > 0Þ  Lo ðq1 ¼ 0; q2 > 0Þ > 0, i.e., C < C (c) q1 ¼ A1 =2 > 0 and q2 ¼ 0 if A1 =A2 P 2=ð1 þ bÞ, and Lo ðq1 > 0; q2 ¼ 0Þ  Lo ðq1 ¼ 0; q2 > 0Þ ¼ ðA21  A22 Þ=4  Cg > 0, i.e.,  2 . Table 4 summarizes the expected profits and Lðq ; q Þ of the retailer (borrowing is allowed). h C 0; k < K 03 , and k is smaller than the sufficient capital threshold, thus the retailer’s objective function can be written as

8 max Lðq1 ; q2 Þ ¼ ða  q1  q2 Þq1 þ ða  q2  bq1 Þq2 þ ðk  C T Þg  k  kq1 r1 > < q1 ;q2 s:t: q1 P 0; q2 P 0; > : k  C T P 0: where C T ¼ c1 q1 þ c2 q2 þ C when q1 > 0 and C T ¼ c1 q1 þ c2 q2 when q1 ¼ 0. The corresponding Lagrangian function is

G ¼ ða  q1  q2 Þq1 þ ða  q2  bq1 Þq2 þ ðk  C T Þg  k  kq1 r1 þ k0 ðk  C T Þ; @G @G ¼ a  2q1  ð1 þ bÞq2  c1 g  kr1  c1 k0 6 0; q ¼ 0; @q1 @q1 1 @G @G ¼ a  2q2  ð1 þ bÞq1  c2 g  c2 k0 6 0; q ¼ 0; @q2 @q2 2 @G ¼ @k0



k  c1 q1  c2 q2 P 0;

if q1 ¼ 0;

k  c1 q1  c2 q2  C P 0;

if q1 > 0:

Please cite this article in press as: Cui, Q., et al. Store brand introduction in a two-echelon logistics system with a risk-averse retailer. Transport. Res. Part E (2015), http://dx.doi.org/10.1016/j.tre.2015.10.005

Q. Cui et al. / Transportation Research Part E xxx (2015) xxx–xxx

19

Then, the proof is similar to that of Proposition 7. (a) q1 ¼ 0 and q2 > 0 is equivalent to (i) k 6 K 1 and

½c1 =c2  ð1 þ bÞ=2k 6

c 2 A2 ðc1 =c2  A1 =A2 Þ: 2

(ii) k 6 minfK 1 ; K 03 g and

8 a  2q1  ð1 þ bÞq2  c1 g  kr1  c1 k0 ¼ 0; > > > > > > a  2q2  ð1 þ bÞq1  c2 g  c2 k0 ¼ 0; > < k  c1 q1  c2 q2  C ¼ 0; > > > c c > > L ðq1 > 0; q2 > 0Þ  L ðq1 ¼ 0; q2 > 0Þ 6 0; > > : q1 > 0; q2 > 0:

By solving the above equations, we have

c 2 A2 ðc1 =c2  A1 =A2 Þ; 2 c 1 A2 ½2=ð1 þ bÞ  c1 =c2 ðk  CÞ > ðA1 =A2  c1 =c2 Þ; 1þb

½c1 =c2  ð1 þ bÞ=2ðk  CÞ >

^3. and C P C (iii) k 6 minfK 1 ; K 02 g and

8 a  2q1  c1 g  kr1  c1 k0 ¼ 0; > > > > > > a  ð1 þ bÞq1  c2 g  c2 k0 6 0; > < k  c1 q1  C ¼ 0; > > > c c > > L ðq1 > 0; q2 ¼ 0Þ  L ðq1 ¼ 0; q2 > 0Þ 6 0; > > : q1 > 0; q2 ¼ 0: ^2. 1 A2 We have ½2=ð1 þ bÞ  c1 =c2 ðk  CÞ 6 c1þb ðA1 =A2  c1 =c2 Þ and C P C  0 o c  2 P C P C ~ 2 , which is in (iv) K 2 < k < K 1 and L ðq1 > 0; q2 ¼ 0Þ  L ðq1 ¼ 0 ; q2 > 0Þ 6 0, i.e., A1 =A2 P 2=ð1 þ bÞ and C ~  conflict with C 2 < C 2 , i.e. this case is infeasible.

(b) q1 > 0 and q2 ¼ 0 is equivalent to (i) k 6 minfK 1 ; K 02 g and

8 a  2q1  c1 g  kr1  c1 k0 ¼ 0; > > > > > > a  ð1 þ bÞq1  c2 g  c2 k0 ¼ 0; > < k  c1 q1  c2 q2  C ¼ 0; > > > c c > > L ðq1 > 0; q2 ¼ 0Þ  L ðq1 ¼ 0; q2 > 0Þ > 0; > > : q1 > 0; q2 ¼ 0:

^2. 1 A2 We have ½2=ð1 þ bÞ  c1 =c2 ðk  CÞ 6 c1þb ðA1 =A2  c1 =c2 Þ and C < C (ii) K 1 < k < K 02 and

8 a  2q1  c1 g  kr1  c1 k0 ¼ 0; > > > > > > a  ð1 þ bÞq1  c2 g  c2 k0 ¼ 0; > < k  c1 q1  C ¼ 0; > > > > Lc ðq1 > 0; q2 ¼ 0Þ  Lo ðq1 ¼ 0; q2 > 0Þ > 0; > > > : q1 > 0; q2 ¼ 0:

1 A2 ~2. We have ½2=ð1 þ bÞ  c1 =c2 ðk  CÞ 6 c1þb ðA1 =A2  c1 =c2 Þ and C < C   (c) q1 > 0 and q2 > 0 is equivalent to

Please cite this article in press as: Cui, Q., et al. Store brand introduction in a two-echelon logistics system with a risk-averse retailer. Transport. Res. Part E (2015), http://dx.doi.org/10.1016/j.tre.2015.10.005

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Q. Cui et al. / Transportation Research Part E xxx (2015) xxx–xxx

(i) k 6 minfK 1 ; K 3 g and

8 a  2q1  c1 g  kr1  c1 k0 ¼ 0; > > > > 0 > > < a  ð1 þ bÞq1  c2 g  c2 k ¼ 0; k  c1 q1  c2 q2  C ¼ 0; > > > > Lc ðq1 > 0; q2 > 0Þ  Lc ðq1 ¼ 0; q2 > 0Þ > 0; > > : q1 > 0; q2 > 0:

Thus we have

c 2 A2 ðc1 =c2  A1 =A2 Þ; 2 c1 A2 ½2=ð1 þ bÞ  c1 =c2 ðk  CÞ > ðA1 =A2  c1 =c2 Þ; 1þb

½c1 =c2  ð1 þ bÞ=2ðk  CÞ >

^3. and C < C (ii) K 1 < k < K 03 and

8 a  2q1  c1 g  kr1  c1 k0 ¼ 0; > > > > 0 > > < a  ð1 þ bÞq1  c2 g  c2 k ¼ 0; k  c1 q1  c2 q2  C ¼ 0; > > c o   > > L > ðq1 > 0; q2 > 0Þ  L ðq1 ¼ 0; q2 > 0Þ > 0; > : q1 > 0; q2 > 0:

We can get

c 2 A2 ðc1 =c2  A1 =A2 Þ; 2 c1 A2 ½2=ð1 þ bÞ  c1 =c2 ðk  CÞ > ðA1 =A2  c1 =c2 Þ; 1þb

½c1 =c2  ð1 þ bÞ=2ðk  CÞ >

~ 3 . Table 5 summarizes the expected profits and Lðq ; q Þ of the retailer (borrowing is prohibited). h and C < C 1 2

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Please cite this article in press as: Cui, Q., et al. Store brand introduction in a two-echelon logistics system with a risk-averse retailer. Transport. Res. Part E (2015), http://dx.doi.org/10.1016/j.tre.2015.10.005