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IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS: SYSTEMS, VOL. 47, NO. 12, DECEMBER 2017 3189

Optimal Shipping Strategy and Return Service Charge Under No-Reason Return Policy

in Online Retailing Zhongsheng Hua, Huijun Hou, and Yiwen Bian

Abstract—Under no-reason return policy in an online setting, online retailers must determine whether to offer free shipping services when delivering products to consumers, and how to charge consumers for returns. To address such decision-making challenges, we develop a theoretic model to derive the optimal shipping strategy and return service charge (RSC) for an online retailer under two decision scenarios, in which decisions on ship- ping strategy and RSC are made either jointly or separately. We find that the retailer is better off in joint decision scenario than in separate decision scenario; a shipping free strategy is usually accompanied by a higher RSC, while a shipping fee strategy is typically accompanied by a lower RSC. We also find that, market parameters (e.g., the base return quantity, prod- uct price, consumers’ sensitivities of shipping fee on demand, RSC on demand and return quantity) have important effects on the retailer’s decisions on shipping strategy and RSC. Our find- ings suggest that the retailer can benefit from taking positive actions toward influencing the market to determine the favor- able shipping strategy and RSC. Furthermore, our results can provide theoretical explanations for widely used shipping strate- gies and RSCs within the context of no-reason return policies in online settings. In particular, our analytical results explain why some real-world online retailers offer both free shipping and free return services to certain consumers.

Index Terms—No-reason return, online retailer, return service charge (RSC), shipping strategy.

I. INTRODUCTION

R APID advances in information technology, e.g., onlinecommunication and information exchange platforms, have significantly boosted the development of electronic com- merce. According to a National Bureau of Statistics of China report, online retail sales has reached approximately $407.63 billion in China during the first three quarters of 2015,

Manuscript received November 28, 2015; revised March 5, 2016; accepted April 3, 2016. Date of publication June 1, 2016; date of current version November 16, 2017. This work was supported by the National Natural Science Foundation of China (NSFC) Major International (regional) Joint Research Program under Grant 71320107004. The work of Y. Bian was supported by the NSFC Program under Grant 71571115. This paper was recommended by Associate Editor T.-M. Choi. (Corresponding author: Yiwen Bian.)

Z. Hua is with the School of Management, University of Science and Technology of China, Hefei 230026, China, and with the School of Management, Zhejiang University, Hangzhou 310058, China.

H. Hou is with the School of Management, University of Science and Technology of China, Hefei 230026, China.

Y. Bian is with the SHU-UTS SILC (Sydney Institute of Language and Commerce) Business School, Shanghai University, Shanghai 201899, China (e-mail: [email protected]).

Digital Object Identifier 10.1109/TSMC.2016.2564920

which has increased by approximately 36.2% from that in 2014. In the U.S., during the second quarter of 2008, online retail sales increased by almost 9.5% from that in 2007 and reached roughly $35 billion [1]. A key feature of online retail- ing for physical products is the existence of a spatial separation between buyers and sellers. In this context, on the one hand, buyers’ orders are mainly fulfilled via shipping. Hence, ship- ping strategy is an important decision-making issue for online retailers. On the other hand, online markets face barriers with regard to physical experience products that cannot easily be described through the Internet interface [2]. Thus, consumers cannot check and physically touch ordered products prior to shipment, and this potentially results in product returns or refunds [3]. Product returns may incur return costs, includ- ing shipping, repackaging and handling costs, for retailers. More specifically, in the case of no-reason or unconditional returns, online retailers must decide whether to accept con- sumer returns for free or whether to partly or fully charge for return costs. The primary goal of this paper is to inves- tigate how online retailers determine whether to offer free shipping services when delivering products to consumers, and how to charge consumers for returns under no-reason return policies.

In order to address the two issues described above, it is important to understand that shipping strategy and return service charge (RSC) can indeed be substantial for online transactions. To boost sales and to attract and retain customers, many online retailers offer free shipping services. Many com- panies on their websites, e.g., Amazon.com, Nordstrom.com, and JD.com, advertise that they offer free shipping services for some or all products irrespective of what or how much is ordered [4]. Some retailers offer free shipping services when an order value exceeds a certain threshold amount. For example, Natural Sense, a Canadian aromatherapy prod- ucts company, offers free shipping for online orders over C$300 [5]. Free shipping service has proven to serve as an effective online promotion tool, which incentivizes customers to buy more products. However, such strategy also significantly increases costs. For instance, because of shipping and related costs, Amazon.com incurred losses of $630M in 2008 and $849M in 2009 [1]. In terms of absolute shipping costs under a shipping free strategy (SFRS), per order values reach roughly $15 for prescription drugs and $26 for toys [6]. Thus, to reduce losses related to shipping costs, some retailers charge con- sumers for shipping costs in addition to the product price rather

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than offering free shipping services. Note that, any charge for shipping costs can have significantly negative effects on consumer decisions in online transactions. Consumers consis- tently respond that high shipping costs are the main deterrent to buying more products online [7]. Market research also shows that shipping fee is a main complaint related to online retailing, and more than 60% of consumers abandon their orders once shipping fees are added [8]. Therefore, shipping strategy can substantially affect consumers’ online purchasing decisions.

RSC generally corresponds to return policy that specifies conditions under which retailers can accept consumer returns for full or partial refund. Since consumers in online purchas- ing settings may not have access to experience the product or have ability to assess the product quality, a clear return policy has been regarded as an effective tool that increases customer confidence, and thus encourages more customers to buy products [9]. Survey evidence shows that over 70% of consumers are at least somewhat likely to consider return poli- cies before making purchases [10]. Accordingly, many online retailers, e.g., Amazon.com, Gap.com, and JD.com, offer their return policies. In particular, Nike allows unconditional returns with refund within 30 days of purchase from any consumers dissatisfied with their NIKEiD products [3]. Notably, some online retailers offer different return policies among coun- tries. For example, in the U.S., Amazon accepts returns on most items within 30 days, and pays for return shipping costs only for its own mistakes [9]. However, for Amazon in China, the time limit for no-reason returns is seven or 30 days, which is determined according to the product cat- egory (Amazon.cn). Similarly, in China and the U.S., GAP permits no-reason returns without any charge within 10 and 45 days, respectively, after a product is received (GAP.cn and GAP.com). One can refer to [11] for more return poli- cies. A viable and clear return policy might increase the probability that an order is completed, and in turn, might increase product return quantity. Return rate in online retail- ing generally exceeds 18% of total sales, which is much larger than that for offline retailing (5%–10%); particularly, for fashion products such as fashion apparel, the rate even reaches 74% [12]. Product returns might typically result from quality defects, delivery mistakes, or product dam- ages due to shipping. In such cases, when consumers return their products for these reasons, online retailers such as JD.com and GAP.com typically bear all return costs (JD.com and GAP.com, in China), or third-party shippers may bear related costs due to damages resulting from the shipping pro- cess. Thus, a larger number of returns may directly increase the return costs of online retailing operations. Indeed, the total cost of returned products from online stores reached roughly $11 billion by 2002, which resulted in a loss of roughly $1.5–$2.5 billion in revenue [9]. Furthermore, many of these returned products are resold at a significant dis- count if not at a scrap value. Thus, as the profit for an online retailer increases when a clear return policy is applied but decreases owing to increasing return costs, return pol- icy constitutes an important decision variable for any online retailer.

No-reason return policy is an unconditional return policy that allows consumers return purchased products without any reason within a specified time period from online purchase. More recently, with improvements in product quality, most product returns are no longer due to functional quality but for other consumer behavior-related reasons, e.g., consumers who are not satisfied with the purchased products, who do not understand how to use the products, or who regret an impulsive purchase [13]. Such returns are also referred to as false-failure returns. In practice, many online retailers, e.g., NIKEiD.com, Amazon.com, and JD.com, offer their no-reason return policies. Particularly, Taobao.com as a platform also grants full refunds within seven days for new clothes and appliances [13]. Note that, by offering no-reason return ser- vices, online retailers can foster consumer trust, attract more consumers, and retain consumers; however, such services may increase great return costs. In such a case, online retailers specify their own no-reason return policies with particular con- ditions, whereby service providers may grant customers a full refund minus an RSC. In this way, RSC can somewhat avoid the abuse of the return right while subsidizing return costs and losses from returns [3]. As noted above, Amazon.com pays for return shipping costs only for its own mistakes. JD.com offers a seven-day no-reason return policy for most of its items, with the exception of fresh and perishable products, unpacked audiovisual products and software, and newspapers, journals, and some magazines (see JD.com). Nike offers two different return policies to its consumers, i.e., nonmembers must pay a service charge while members do not [3].

Clearly, RSC specifies an online retailer’s return policy and is reflective of whether the retailer offers free return service. Higher RSC can decrease the number of consumer returns for subjective reasons, but can also decrease the retailer’s sales, as consumers consider refunds before making purchases [14].

The above mentioned findings show that both shipping strat- egy and return policy have significant effects on consumer demands, retailer profits, and related costs. These two issues have thus greatly shaped online retailer product decisions. When employing no-reason return policy, online retailers may make their practical decisions by jointly considering both ship- ping strategy and RSC. Accordingly, online retailers often resort to two distinct strategies.

1) Retailers may offer free shipping services when deliv- ering products, but when consumers return products, retailers may require consumers to pay for no, partial or full shipping and return costs.

2) Retailers do not offer free shipping services when delivering products, necessitating that consumers bear shipping costs; however, when consumers return prod- ucts, retailers may require consumers to pay for no or partial return costs. Similar strategies can be widely observed on online retailer websites such as JD.com.

These two distinct strategies raise the following research questions.

1) How do online retailers determine whether to offer free shipping services when delivering products? Note that, when retailers do not offer free shipping services, all related costs will be paid by consumers.

HUA et al.: OPTIMAL SHIPPING STRATEGY AND RSC UNDER NO-REASON RETURN POLICY IN ONLINE RETAILING 3191

2) In considering shipping strategy with the presence of no-reason returns, how do online retailers charge for incurred shipping and return costs?

Despite the importance of shipping strategy and return pol- icy to online retailing, the prior studies have not well examined the above issues. The primary motivation of this paper is to address this gap. To this end, we develop a theoretical model to investigate the optimal shipping strategy, i.e., SFRS or ship- ping fee strategy (SFES), and RSC for an online retailer by maximizing his profit. First, we examine the retailer’s optimal shipping strategy by fixing RSC, and the optimal RSC by fix- ing shipping strategy. Second, we jointly explore the optimal shipping strategy and RSC, and provide several key theoreti- cal conclusions. Properties of the retailer’s optimal profit are also characterized. The results show that the retailer is better off when determining the optimal shipping strategy and RSC jointly. Some important findings and management insights are obtained.

The remainder of this paper is organized as follows. In the following section, we review the most relevant litera- ture. Section III presents the basic model. In Section IV, the optimal shipping strategies and RSCs are analytically exam- ined, and properties of the retailer’s optimal profit are also characterized. A numerical study is conducted to explore the impact of some return related market parameters on condi- tions associated with the optimal decisions, and illustrate the advantage of joint decision-making in Section V. Section VI presents concluding remarks. All proofs are provided in the Appendix.

II. LITERATURE REVIEW

Although the specific research questions of this paper have not been well addressed in the literature, there are certain streams of prior research studies on shipping strategy and return policy. As this paper is related to shipping strategy and RSC policy in online retailing, we review only the most relevant studies.

A. Shipping Strategy

In recent years, an increasing number of studies have explored shipping strategies, i.e., SFRS or SFES. Some empir- ical studies have shown how consumers react to shipping strategy in online retailing. Lewis et al. [6] empirically showed that consumers are highly sensitive to shipping charges, and that shipping fees affect purchase incidences and order sizes. Lewis [15] further indicated that customer acquisition is more sensitive to order size incentives, whereas retention is more heavily influenced by base shipping fee levels. Melnik and Richardson [16] empirically tested the impact of shipping fees on auction outcomes on eBay. Yao and Zhang [17] examined the price portioning decisions of product base price, shipping and handling fees, and find that sellers can strategically determine product base and shipping prices to maximize their profits.

Another stream of studies focus on examining the opti- mal product prices, order quantities and inventory levels for online retailers under SFRS. Leng and Parlar [5] and

Leng and Becerril-Arreola [18] used theoretical models to investigate the optimal free shipping threshold (i.e., free ship- ping for orders over a threshold quantity) and price decisions related to B2B and B2C transactions, respectively. They find that free shipping cutoff levels have significant effects on consumer order sizes. Hua et al. [19] examined the optimal order quantity and pricing decisions related to newsvendor problem with SFRS by considering a free shipping thresh- old, and find that even though the retailer faces uncertain demand, SFRS can effectively increase the number of orders. Becerril-Arreola et al. [20] investigated the promotional prices, free-shipping thresholds and inventory levels for online retail- ers by using a simulation analysis. Their results show that variations in a positive finite free shipping threshold affect both the average value and standard deviation of order sizes. These two studies mainly explore the optimal prices, inventory levels or order quantities by taking free shipping threshold into account. Similar studies include [4] and [21].

Note that, the above mentioned two streams of studies have mainly examined the effects of shipping fees and free shipping services on consumer orders, and the optimal free shipping threshold as well as related pricing, order quan- tity, and inventory level decisions. Unlike the above studies, Frischmann et al. [22] empirically showed that online retail- ers pursue two shipping cost strategies, i.e., free shipping and the sum of the net product price and shipping costs, and they find that these strategies conflict, and target different consumer segments. Gümüş et al. [1] applied game-theoretic model and empirical approach to examine how online retailers choose the optimal shipping strategy (i.e., SFRS or SFES), and therefore determine an equilibrium market structure in terms of the pro- portion of retailers that apply SFRS and SFES. Their results show that shipping fee retailers charge lower product prices than shipping free retailers do, but that the total price (i.e., the sum of product price and shipping related costs) charged is higher for shipping fee retailers.

We use the insights from all three literature streams to develop our demand framework. To this end, shipping strategy, i.e., SFRS or SFES, is directly incorporated into the demand function. This paper differs from the existing studies in the following three respects.

1) While the first two streams of literature are mainly based on investigations of the consumer behavior or reac- tions, we explore the optimal shipping strategy for online retailers at the firm level.

2) The second stream of literature focuses on examining the impact of free shipping thresholds on consumer orders and attempts to determine the optimal free ship- ping thresholds and related variables’ decisions such as pricing decisions. We assume that shipping strat- egy is imposed on one unit product without considering order size.

3) While at the firm level, Gümüş et al. [1] studied two price partitioning strategies (i.e., shipping fees or ship- ping free) for online retailers, we assume in our model that shipping fee is constant, exogenous, and cannot be partitioned. As a possible example of this assump- tion, shipping can be carried out by third-party logistics

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companies whose shipping fees are typically known. In such a case, online retailers cannot determine shipping fee regardless of whether they offer free shipping services.

B. Return Policy

There is a rich body of literature on return policy in the context of online retailing. The most relevant studies can be grouped into two streams, i.e., RSC policy and no- reason return policy. Regarding RSC, related studies have focused on determining how to refund consumers or how to allocate return costs when accepting consumer returns. Mukhopadhyay and Setaputra [23] examined full and par- tial refund policies based on an assumption that demand and return functions linearly depend on selling price and refund amount in an Internet supply chain. Furthermore, Mukhopadhyay and Setaputra [9] developed a profit maxi- mization model to study the trade-off between sales and return quantities, and obtain the optimal refund policy and product price. These studies nevertheless do not explicitly model indi- vidual consumer choices. In a recent study, Shulman et al. [24] proposed an analytical model to derive the optimal pricing and restocking fee policy by taking consumer preferences into consideration. Su [25] developed a model to examine both full and partial refund policies and pricing strategy in a sim- ilar setting by investigating the impact of return policies on supply chain contracts. Li et al. [26] presented several the- oretical models to analyze the relationships between return policy, product quality, and pricing strategy based on consumer purchase and return behaviors, and they also examine a full refund policy. Their results show that return policy decisions are mutual and complementary to product quality and pric- ing strategies. By incorporating both strategic retailer options and consumer trials into their proposed analytical optimiza- tion models, Li et al. [13] identified the retailer’s optimal pricing and refund policies for advance-selling fashionable products. In their work, full and partial refund policies are both examined. Similar work related to return policy on fast fash- ion products are also found in [27]. Following [28], Choi [3] explored RSC with mass customized fashion products, and derives the optimal conditions for offering free return services. The results show that retailers can offer differentiated RSC schemes to members and nonmembers.

The aforementioned studies focus on examining the opti- mal return policy or RSC policy under which retailers offer full or partial refunds to consumers, but do not clearly discriminate whether returns are caused by objective or sub- jective reasons. By contrast, in this paper, we assume that returns are driven not by objective quality issues but by other consumer behavior-related issues [13]. Unlike other return policies, under no-reason return policy, returned products can be resold as regular products. Hence, RSC policy focuses only on how to deal with return costs.

With respect to no-reason return policy, Ferguson et al. [29] are pioneers on examining false failure returns in supply chain systems, in which false-failure returns are derived from differential pricing between two sellers. Similar studies on supply chain practices can also be found in [30] and [31].

In online selling practice, most no-reason product returns are made because products cannot cater to consumer tastes or other consumer behavior-related needs [13], [25]. However, only a few academic studies have examined the impact of no- reason returns on consumer behaviors and rights protection regarding this issue. For example, Inderst and Ottaviani [32] studied contract cancellations and product return policies in markets by considering full rational and credulous consumers, and find that competition policy effectively reduces contrac- tual inefficiencies for rational customers, but not for credulous customers. Sparks et al. [33] conducted a qualitative study to examine purchase rescinding patterns and develop a concep- tual model using a timeshare context. Their results suggest that rescission is related to a mismatch between product features and personal circumstances, post-purchase concerns regarding product value, financial capability reassessments, reflections on sales presentations, and cautionary reference group effects. Note that, these studies focus mainly on the effects of relevant no-reason return policies on consumer protections or behav- iors. No academic study related to RSC under no-reason return policy can be found in the literature.

This paper attempts to identify the optimal shipping strategy and RSC under no-reason return policy for online retailers. To the best of our knowledge, this is the first study to address this issue in the context of online retailing. It is also the first piece of analytical research that specifically examines dif- ferentiated RSC under no-reason return policy with different shipping strategies. Hence, some new managerial insights are obtained.

III. BASIC MODEL

We consider an online retailer who sells a particular prod- uct online to a group of consumers. For ease of analysis, we assume that each consumer buys only one product online. In our model, the retailer makes two decisions, i.e., shipping strat- egy and RSC. When delivering products to consumers, the retailer decides to choose SFRS or SFES. When the retailer chooses SFES, consumers must pay for all shipping cost. Such consumers are allowed to return products within a specified time limit under a no-reason return policy when they are not satisfied with the purchased products. In this case, the retailer determines how to charge consumers for return costs. Notably, under the specified no-reason return policy, we assume that products are returned not for any quality or function related reasons but for some consumer behavior-related reasons (e.g., dissatisfaction with the purchased products, lack of under- standing how to use the purchased products, or regretting for an impulsive purchase) [13]. Thus, after being repacked, the returned products can be resold at the same market price as the regular products. To examine the optimal shipping strategy and RSC, we assume that the retailer is rational and self-interested, and that his aim is thus to maximize his own profit.

Before we present the basic theoretical model, we first present the notations used in this paper, as summarized in Table I.

In online retailing, the retailer presents product price p (p > c), shipping strategy x and RSC l on his website. In

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TABLE I SUMMARY OF NOTATIONS USED

our model, x and l are two decision variables, while the prod- uct price, shipping fee, return cost, and per unit product cost are assumed to be exogenous. This assumption allows us to focus on the main decisions considered in this paper, while retaining analytical tractability. Note that, when x = 0, the retailer offers SFRS for delivering products; however, when x = 1, the retailer employs SFES, and consumers pay ship- ping fee s. We assume that consumers make their purchase decisions after completely understanding the shipping strategy and RSC employed. This assumption reflects typical practice as widely observed in online retailing. Upon receiving the ordered product, if accepting, the consumer confirms the deal, and the retailer obtains a net profit p − c −(1 − x)s. Otherwise, if the consumer is not satisfied with the product, he returns the product with refund p − l, which equals the full refund minus the RSC. In this case, the retailer suffers an associated loss (1 − x)s + ω − l. We also note that, the return cost per product ω includes shipping, repackaging and handling costs for the returned product.

Product demand is generally assumed to be linearly depen- dent on product price and return refund [3], [26]. We extend this assumption by considering the retailer’s shipping strategy, and we model the demand function as

D(x, l) = α − βp + γ(p − l) − λxs. (1)

To render product demand more practical, without loss of generality, we assume that the product has positive demand, and that α > 0, β > 0, γ > 0, and λ > 0. Note that, the spe- cific linear demand function indicates that demand increases in return refund p − l, while decreases in retail price and ship- ping strategy, especially when SFES is offered. The effect of shipping strategy on customer demand is intuitive and can be commonly observed in practice.

Under no-reason return policy, following [3] and [9], we characterize the return quantity as a linear function of the return refund, i.e.,

R(l) = φ + ψ(p − l). (2) Similar to the demand function, we assume that the return

quantity is positive, and thus φ > 0 and ψ > 0. Obviously, the return quantity increases in refund p − l.

Based on (1) and (2), we can easily determine the net product demand for the product by using the following function:

N(x, l) = D(x, l) − R(l) = α − φ − βp + (γ − ψ)(p − l) − λxs. (3)

Based on the practice in online retailing with respect to no- reason return policy, the retailer’s total profit depends on two factors: 1) profit obtained from net sold products (i.e., the quan- tity of products sold minus that of returns) and 2) loss resulting from returns. Thus, the retailer’s profit can be expressed as

π(x, l) = N(x, l)(p − c − (1 − x)s) − R(l)[(1 − x)s + ω − l]. (4)

Note that, N(x, l)(p − c − (1 − x)s) is the retailer’s profit derived from the products sold, and that R(l)[(1−x)s+ω−l] is the loss incurred from consumer returns. Similar profit func- tions can also be found in [3] and [9]. Note that, we assume that returned products can be resold at the same product price p as regular products.

Notably, the retailer’s maximum total operational cost asso- ciated with shipping and return is ω + s(1 − x). This cost can be rationalized as follows. First, when the retailer offers free shipping services (i.e., x = 0), the maximum total cost is ω + s, which is equal to the sum of shipping fee and return cost. Second, when the retailer does not offer free shipping services (i.e., x = 1), the maximum total cost is ω, which is only the return cost. Thus, it is intuitive that the retailer cannot set an RSC that is larger than the maximum cost, i.e.,

l ≤ ω + s(1 − x). (5) Hence, the retailer attempts to maximize his profit by consid- ering the constraint formulated in (5). Therefore, we directly obtain the following model:

max l,x

π(x, l)

s.t. x ∈ {0, 1} l ≤ ω + s(1 − x) l ≥ 0. (6)

In what follows, we will examine the optimal shipping strategy and RSC under no-reason return policy based on model (6).

IV. THEORETICAL ANALYSIS

To identify the optimal shipping strategy and RSC under no-reason return policy in online retailing, in this section, we consider two scenarios: 1) separate and 2) joint decisions. In separate decision scenario, we first examine the retailer’s opti- mal shipping strategy for given fixed RSC, and then investigate

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the optimal RSC for given fixed shipping strategy. In joint decision scenario, we attempt to explore the optimal shipping strategy and RSC jointly.

A. Separate Decision

For a given fixed RSC l, we can easily identify the optimal shipping strategy, which is characterized by Proposition 1.

Proposition 1: For a given RSC l(l ∈ [0,ω]), the retailer offers free shipping service when l > [α − βp + γ p − λ(p − c)]/γ .

Proposition 1 shows that the retailer provides free ship- ping service when delivering products, when RSC l is larger than the threshold [α − βp + γ p − λ(p − c)]/γ . In other words, under the specified no-reason return policy, if l > [α − βp + γ p − λ(p − c)]/γ , the retailer might adopt SFRS for delivering products; otherwise, he would not offer free shipping service but would instead charge shipping fees. In this context, the retailer should set the shipping strategy and RSC threshold before he sells products.

Note that, the RSC threshold l > [α − βp + γ p − λ(p − c)]/γ can be transformed into its equivalent form λ > D(0, l)/(p − c). Thus, the choice of the optimal ship- ping strategy is independent of shipping fee, which contradicts the common wisdom that shipping strategy decision may depend on shipping fee to some extent. This finding further suggests that whether the retailer chooses free shipping ser- vice completely hinges on the given RSC threshold employed under no-reason return policy. One HRC Advisory study indi- cates that online returns constitute a major problem for 95% of retailers and as much as 30% of orders are sent back. After free shipping is implemented, returns will likely become even worse [34]. This in turn indicates that, when a retailer offers SFRS, a certain RSC is required. This reasoning partly supports the conclusion of Proposition 1.

Note that, it is assumed that the given RSC l ∈ [0,ω] for Proposition 1. The rationale for this is that, when l ∈ [0,ω], we can derive the condition under which whether the retailer chooses SFRS for a given RSC. However, when l ∈ (ω,ω+s], according to model (6), the retailer will definitively choose SFRS.

As for given fixed shipping strategy, the optimal RSC can also be obtained based on model (6), which is shown in Proposition 2.

Proposition 2: When offering SFRS, the retailer’s optimal RSC is

l∗0 =

⎧ ⎪⎪⎨

⎪⎪⎩

s + ω, φ ≥ γ(p − c − s) − ψ(2p − 2s − c − ω) 0 φ < γ(p − c − s) − ψ(2p − c + ω) (φ − γ(p − s − c)

+ ψ(2p − c + ω))/2ψ, otherwise. When choosing SFES, the retailer’s optimal RSC is

l∗1 =

⎧ ⎪⎪⎨

⎪⎪⎩

ω, φ ≥ γ(p − c) − ψ(2p − c − ω) 0, φ < γ(p − c) − ψ(2p − c + ω) (φ − γ(p − c)

+ ψ(2p − c + ω))/2ψ, otherwise. Note that, l∗0 and l

∗ 1 refer to the optimal RSCs for given the

SFRS and SFES, respectively. Proposition 2 characterizes

the retailer’s optimal RSCs and related conditions. Obviously, the retailer’s optimal RSC is partly dependent on shipping fee s under SFRS, while independent of s under SFES.

Proposition 2 indicates that, when the base return quantity is sufficiently large, the retailer might charge the maximum fee for returns. Otherwise, the retailer might charge less. Indeed, large base return quantity is related to more returns received, and the retailer is better off to choose a larger charge for returns in such a case to reduce the customers’ willingness of returning products. In contrast, when the base return quantity is relatively small, the retailer might charge less for product returns, which in turn may increase product demand. In par- ticular, when the base return quantity is sufficiently low, the retailer may offer free return service for consumer returns.

It is noted that, γ describes the sensitivity of sales with respect to return policy and represents the rate of sales increase as return policy becomes more clear and generous. When consumers are more sensitive to return policy, the retailer might charge less for return service, which may in turn stim- ulate product demand, and compensate for the cost increase due to product returns. Thus, a higher γ is desirable since it can increase sales and thus profitability. Note also that, ψ is the refund-return quantity sensitivity coefficient, which reflects consumers’ willingness of returning products or the rate of increase in the return quantity when RSC decreases, i.e., return policy becomes more clear and generous. When consumers are less sensitive to the RSC, a clear and gener- ous return policy will not make much difference from that with an increase in γ . In turn, a decrease in ψ is just like an increase in γ . These insights are similar to those presented by Mukhopadhyay and Setaputra [9]. As they suggested, the retailer can take some possible efforts to influence the market to increase γ while decrease ψ. For example, retailers can use advertising or promotional means to make consumers aware of clear return policy, which can be regarded as an increase in γ , and also can show good faith in consumers, which can be seen as a decrease in ψ. Similar strategies can be found in eBay.com.

Note that in net product demand as shown in (3), γ −ψ can be used to reflect the sensitivity of net demand with respect to the return policy. Such sensitivity represents the rate of increase in the net demand from the base demand as the return policy becomes more clear and generous. Therefore, online retailers might expect a sufficiently large γ − ψ to reduce the effect of ψ on return quantity. This expectation can be sup- ported by practical operations in real settings. For example, online retailers might highlight return policies or other pro- motions to prevent consumers from returning products, and such efforts can in turn increase sales [3], [9], [35]. Hence, a large (γ − ψ)/ψ is desirable in practice.

Coincidentally, based on Proposition 2, we find that the ratio (γ − ψ)/ψ has a vital impact on the retailer’s RSC decisions. When (γ − ψ)/ψ = 1, the retailer’s optimal RSC is inde- pendent of the product price; and when (γ − ψ)/ψ < 1, the retailer’s optimal RSC increases in the product price. As noted in the above discussion, these two cases are not desirable for online retailers, who will try their best to guard against these two cases. It is practical that, when (γ − ψ)/ψ > 1,

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the retailer’s optimal RSC decreases in the product price. This is intuitive, since a higher product price can directly decrease product demand, while such a decrease can be partly offset by a lower RSC, which can effectively increase net demand. Similar considerations are also presented in relevant studies (see [3], [9]). Neiman Marcus.com mainly sells lux- ury products online and states on its website that it offers free shipping on all orders and will pay for return shipping fee within 15 days from purchase. This practice partly supports this conclusion.

Since p > c, when (γ − ψ)/ψ ≤ 1, it is easy to verify that φ < γ(p − c − s) − ψ(2p − c + ω) and φ < γ(p − c) − ψ(2p − c + ω) do not hold. Hence, we can conclude that, the retailer has no incentive to offer free return service regard- less of whether offering free shipping service. That is, the retailer always charges fees for consumer returns under no- reason return policy in this case. However, more practically, when (γ − ψ)/ψ > 1, the two conditions may hold, and thus the retailer may offer free return service. For instance, when φ < γ(p − c − s) − ψ(2p − c + ω), the retailer may offer free return service under SFRS; otherwise, he may charge for return service.

Notably, Proposition 1 characterizes the retailer’s optimal shipping strategy for a given RSC, and we find that the RSC has a significant effect on the retailer’s shipping strategy deci- sions; Proposition 2 specifies the retailer’s optimal RSCs by fixing shipping strategies, and shows that shipping strategies significantly affect the retailer’s decision on RSC. As such, it may be preferable for the retailer to determine the optimal shipping strategy and RSC jointly, which is examined in the following section.

B. Joint Decision

The retailer’s optimal shipping strategy and RSC are denoted by x∗ and l∗, respectively. By solving model (6), the retailer’s optimal decisions can be determined as shown in Theorem 1.

Theorem 1: When λ > λ̄, x∗ = 0, and l∗ = l∗0 ; otherwise, x∗ = 1 and l∗ = l∗1 where

λ̄ = α − βp + γ ( p − l∗0

)

p − c + γ(p − c) − φ − ψ

( 2p − c + ω − l∗1 − l∗0

)

p − c · l∗0 − l∗1

s .

Theorem 1 characterizes the optimal shipping strategy and RSC as well as the condition related to the threshold of ship- ping fee-demand sensitivity λ. Note that, the optimal decisions made in the joint decision scenario are associated with the optimal RSCs (l∗0 , l

∗ 1 ) for given a fixed shipping strategy in

the separate decision scenario, and clearly, λ̄ is piecewise. We can find that the retailer offers SFRS when the shipping fee- demand sensitivity λ is higher than a particular threshold λ̄; in this case, the optimal RSC is l∗0 . Otherwise, the retailer may employ SFES, and charge l∗1 for returns. Theorem 1 indi- cates that, when customers are sufficiently sensitive to shipping fees, the retailer may offer free shipping service for delivering products. Offering such service can increase product demand,

which may compensate for shipping costs. By contrast, when customers are not sufficiently sensitive to shipping fees, the retailer will charge for shipping costs, which may generate extra profits.

Notably, the threshold λ̄ can vary with market parameters. To elaborate on this point, we take the following three spe- cial cases based on the ratio (γ − ψ)/ψ as examples in what follows. We provide the following three corollaries.

Corollary 1: When (γ − ψ)/ψ > 1 and φ ≤ γ(p − c − s) − ψ(2p − c + ω), l∗0 = l∗1 = 0, where λ̄ = A/(p − c) and A = α − βp + γ p; otherwise, l∗0 > l∗1 .

The condition (γ − ψ)/ψ > 1 is desirable, as discussed in the previous section. In this case, when φ ≤ γ(p − c − s) − ψ(2p − c + ω), which means the base return quantity is lower than a particular threshold, the retailer may offer free return service regardless of whether he pro- vides free shipping service. This result is intuitive, as the retailer is better off to provide free return service when the base return quantity is relatively small. Providing such service, in turn, can increase product demand, and thus can generate more profits to cover return costs. Note that, φ ≤ γ(p − c − s)−ψ(2p − c + ω) can be expressed as γ ≥ [φ+ψ(2p − c + ω)]/(p − c − s), which means that when γ ≥ max{[φ+ψ(2p − c + ω)]/(p − c − s), 2ψ} and λ > λ̄, the retailer will offer both free shipping and free return services. Many online retailers, e.g., Nordstrom.com, Zappos.com, Diapers.com, and Neiman Marcus.com, declare on their websites that free shipping and free return ser- vices are offered for online purchases with or without order value thresholds. Kyle [36] lists 28 online stores that offer both free shipping and return services for some or all of their online orders. It is reported in [36] that, most of the stores mainly sell fashion products such as shoes, apparel and household products (e.g., kitchen, bath, and dining room products). For shoes and apparel, such stores may strive to increase γ to be sufficiently large to meet the condition φ ≤ γ(p − c − s) − ψ(2p − c + ω). For durable goods such as luxury and household products, the base return quantity can be small, and thus, the condition can also be satisfied. These evidences sufficiently support the reasonability of this conclusion.

In practice, some online retailers such as Nike and JD.com offer free or relatively low charge return services to registered or prime members rather than to all members (or consumers). These practices can be partly explained by Corollary 1. Note that, φ ≤ γ(p − c − s)−ψ(2p − c + ω) can be trans- formed into p ≥ [φ + (γ − ψ)(c + s) + ψ(ω + s)]/(γ − 2ψ). This expression partially indicates that, when the corre- sponding revenues that retailers can earn from registered members are sufficiently high, free return service may also be offered to these members. Revenues in this case can also be some additional business values offered by regis- tered members in addition to product prices, e.g., more repeat purchases, higher degrees of loyalty, and more trustworthy and meaningful comments or feedback. Similar insights are presented in [3].

Motivated by these findings, we further examine Corollary 1 and present the following conclusion.

3196 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS: SYSTEMS, VOL. 47, NO. 12, DECEMBER 2017

Fig. 1. RSC with γ = 12.

Remark 1: The retailer may offer free shipping and free return services, if and only if

λ > γ − β,(γ − ψ)/ψ > 1 and

p ≥ max {

α + λc λ + β − γ ,

φ + (γ − ψ)(c + s) + ψ(ω + s) γ − 2ψ

}

.

Remark 1 theoretically characterizes the conditions under which that the retailer offers both free shipping and free return services.

However, according to Corollary 1, when φ > γ(p − c − s)−ψ(2p − c +ω), the retailer may charge for product returns. The rationale for this is that, the retailer may use the RSC to prevent the abuse of return policy in order to decrease returns. Note that, when (γ − ψ)/ψ > 1 is not satisfied, which is not desirable for the retailer, as in the case of φ > γ(p − c − s) − ψ(2p − c + ω), the retailer may use the RSC to decrease the return quantity.

Interestingly, we find in Corollary 1 that, λ̄ = A/(p − c) is independent of the shipping fee s, because we have the same RSC (i.e., l∗0 = l∗1 = 0) under SFRS and SFES when φ ≤ γ(p−c−s)−ψ(2p−c+ω). As l∗0 = l∗1 , according to λ̄ defined in Theorem 1, λ̄ = A/(p − c). Coincidentally, similar result is also found in Proposition 1, again owing to the same fixed RSC l (l ∈ [0,ω]). This finding implies that the optimal shipping strategy choice is made independent of shipping fee in this case. However, when the base return quantity increases to be a sufficiently large value, i.e., φ > γ(p−c−s)−ψ(2p−c+ω), decision regarding the optimal shipping strategy will depend on shipping fee to some extent.

To better illustrate Corollary 1, we apply a numerical exam- ple in what follows. We set α = 1000, β = 10, p = 140, c = 90, s = 10, ω = 25, and ψ = 2. According to Corollary 1, the results regarding the RSC with respect to φ and γ are shown in Figs. 1 and 2, respectively.

Fig. 1 shows that, when φ ≤ γ(p − c − s) − ψ(2p − c + ω) = 50, the retailer chooses free return service regardless of whether he offers free shipping service; otherwise, he charges more under SFRS strategy than he does under SFES strategy.

Fig. 2 shows that, when γ ≥ max{[φ + ψ(2p − c + ω)]/ (p − c − s), 2ψ} = 11.75, the retailer offers free return service even when offering free shipping service; otherwise, when γ < 11.75, he charges more for return service under SFRS than he does under SFES. Note that in these two figures, SFRS is chosen when λ > λ̄ while SFES is chosen when λ ≤ λ̄.

Fig. 2. RSC under Corollary 1 with φ = 40.

Fig. 3. RSC with γ = 8.

Corollary 2: When (γ − ψ)/ψ > 1 and φ ≥ γ(p − c) − ψ(2p−c−ω), or (γ − ψ)/ψ ≤ 1 and φ ≥ γ(p−c−s)−ψ(2p− 2s − c − ω), l∗0 = s + ω, and l∗1 = ω, where λ̄ = C/(p − c), C = A + γ(p − ω − c − s) − φ − ψ(2p − ω − c − s).

Corollary 2 indicates that, when the base return quantity is sufficiently high, the retailer may charge the maximum fee for return service, i.e., l∗0 = s + ω under free shipping strategy and l∗1 = ω under SFES, respectively, regardless of whether (γ − ψ)/ψ > 1 or (γ − ψ)/ψ ≤ 1. Indeed, when the base return quantity is sufficiently high, the retailer will choose the highest RSC to restrict product returns to compensate for return losses. A numerical example based on the same data as used in Corollary 1 is applied to show the optimal RSC for the retailer in this case. The RSC results regarding φ are depicted in Fig. 3.

Fig. 3 shows that, when φ ≥ γ(p − c) − ψ(2p − c − ω) = 70, the retailer will charge the maximum fee for return service regardless of whether he offers free ship- ping service. In this case, the optimal RSC under SFRS is also larger than that under SFES. Note that, the condition φ ≥ γ(p − c) − ψ(2p − c − ω) = 70 can be transformed into γ ≤ [φ + ψ(2p − c − ω)]/(p − c). The impact of γ on RSC is shown in Fig. 4.

Fig. 4 shows that, when γ ≤ [φ +ψ(2p − c −ω)]/(p − c) = 7.40, the retailer will charge the highest fee for product returns.

Notably, λ > λ̄ can be reformulated as s > s̄1, where s̄1 = [α − φ − βp + (γ − ψ)(2p − ω − c) − λ(p − c)]/(γ −ψ), based on Corollary 2, we can derive the following conclusion.

Remark 2: When γ − ψ > 0, if s > s̄1, the retailer offers SFRS; otherwise, the retailer offers SFES.

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Fig. 4. RSC under corollary 2 with φ = 40.

Remark 2 shows that, when shipping fee is sufficiently high, i.e., s > s̄1, the retailer may offer free shipping service for delivering products and charge the maximum fee for returns, i.e., s + ω; otherwise, the retailer employs SFES and charges the maximum fee for returns, i.e., ω. Intuitively, the retailer would like to offer free shipping service when shipping fee is relatively low, and would not do when shipping fee is relatively high. However, when shipping costs are high, the retailer may set a higher RSC in order to decrease the return quantity under the conditions shown in Corollary 2.

Corollaries 1 and 2 characterize RSCs under the conditions of a sufficiently low base return quantity or a sufficiently large base return quantity, respectively. However, when the base return quantity falls into a specified interval, the following conclusion can be made.

Corollary 3: When (γ − ψ)/ψ ≥ 1 + 2ω/s and γ(p − c − s) − ψ(2p − 2s − c − ω) ≤ φ ≤ γ(p − c)−ψ(2p − c + ω), l∗0 = s + ω and l∗1 = 0, where λ̄ = G/(p − c) and G = C+[γ(p − c) − φ − ψ(2p − c)]ω/s.

Corollary 3 presents an interesting case in which the retailer provides free return service under SFES while charges the highest return service fee under SFRS. Such a strategy is rea- sonable as the retailer is better off by trading off the RSC and shipping cost to balance incurred costs even though the effect of the RSC on the refund-net demand is more signifi- cant than that on the return quantity, owing to the relatively high base return quantity. In other words, when offering SFRS, the retailer charges the maximum RSC to decease the return quantity; when offering SFES, it is preferable for the retailer to offer free return service. This case can be illustrated through a numerical example based on the same data used in the previous section, which is shown in Fig. 5.

Fig. 5 shows that, when γ ≥ ψ + ψ(1 + 2ω/s) = 14 and 430 ≤ φ ≤ 470, the retailer charges maximum fee for prod- uct returns under SFRS strategy and offers free return service under SFES strategy.

Figs. 1–5 clearly show that the retailer may set a higher RSC when the base return quantity is substantially higher, and a lower charge when the base return quantity is lower regardless of whether he offers free shipping service. Such an approach is used because when the base return quantity is higher, which means that more returns will be received, it makes sense for the retailer to increase the RSC to lower con- sumers’ willingness of returning products, and thus to decrease

Fig. 5. RSC with γ = 18.

the level of potential product returns. By contrast, the retailer may have no incentive to charge more for returns, as doing so may decrease product demand. Furthermore, the retailer charges higher fee for return service under SFRS than under SFES. Such a strategy is intuitive in that the retailer pays for shipping fees when delivering products. Consequently, the retailer will charge a higher compensation fee to cover the loss. One exception is the case of (γ − ψ)/ψ > 1 and φ ≤ γ(p − c − s) − ψ(2p − c + ω) as shown in Corollary 1. Such a strategy is reasonable in that, when the base return quantity is relatively low, consumers may not be sensitive to the RSC, and in this case, free return service may help to increase product demand.

With respect to the impact of the base return quantity on the threshold of shipping fee-demand sensitivity λ, we present the following conclusion.

Proposition 3: ∂λ̄/∂φ = 0, when φ ≤ γ(p − c − s)−ψ(2p − c + ω); otherwise, ∂λ̄/∂φ < 0.

Proposition 3 indicates that, when the base return quan- tity is relatively low, the threshold of λ̄ does not vary with changes in φ. However, when the base quantity is suffi- ciently high, a greater base return quantity φ will lead to a smaller threshold λ̄. Consequently, according to Theorem 1, the retailer will more likely provide free shipping service. Hence, the base return quantity may have a significant impact on the shipping strategy choices. Notably, the condition φ ≤ γ(p−c−s)−ψ(2p−c+ω) never holds when (γ − ψ)/ψ ≤ 1, as discussed in the previous section. The effects of γ and ψ on the threshold λ̄ will be examined in the following section.

As in Proposition 2, product price may also have a signifi- cant impact on the retailer’s joint decision of shipping strategy and RSC; thus, based on Theorem 1, we present the following interesting conclusion.

Corollary 4: When (γ − ψ)/ψ > 1, the optimal RSC is constant or decreasing in product price; when (γ − ψ)/ψ < 1, the optimal RSC is increasing or constant in product price. An exception occurs when the optimal shipping strategy changes from SFES to SFRS, as this causes the optimal RSC to increase considerably.

To better illustrate Corollary 4, two numerical examples are considered. First, we set α = 1000, β = 10, c = 90, s = 10, ω = 25, λ = 12, γ = 8, ψ = 2, and φ = 40, and increase product price p from 110 to 190 at a step size of 5. The optimal RSC results are depicted in Fig. 6.

3198 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS: SYSTEMS, VOL. 47, NO. 12, DECEMBER 2017

Fig. 6. Jointly optimal RSC when (γ − ψ)/ψ > 1.

Fig. 7. Jointly optimal RSC when (γ − ψ)/ψ < 1.

Fig. 6 shows the RSC results for (γ − ψ)/ψ=3 > 1. We find that, when p < 129, the optimal shipping strategy is SFES, and the optimal RSC is constant (25) with an increasing product price. However, when p = 129, the optimal shipping strategy changes into SFRS, and the optimal RSC increases to 35 and remains constant until p = 142.5; then the optimal RSC is decreasing in the product price. Note that, when p = 177.5, the RSC decreases to zero and then remains constant with an increasing product price.

Second, we set α = 1000, β = 10, c = 20, s = 10, ω = 25, λ = 12, γ = 3.9, ψ = 2, and φ = 5, and let the product price p increase from 30 to 110 at a step size of 5. The optimal RSC results are depicted in Fig. 7.

Fig. 7 shows changes in the RSC with an increasing product price when (γ − ψ)/ψ = 0.95 < 1 and illustrates the ratio- nale behind the conclusion in this case for Corollary 4. Note that, when p = 62.11, the optimal shipping strategy changes from SFES to SFRS, and there is a jump in the RSC.

Corollary 4 shows the important management insights required for the online retailer to jointly determine the opti- mal shipping strategy and RSC. Within a particular product price range, the retailer’s optimal shipping strategy remains unchanged. In this case, when (γ − ψ)/ψ > 1, the retailer may charge less for return service when the product price is higher, and such a strategy may significantly boost prod- uct net-demand, and vice versa. This conclusion can also be reached from Proposition 2. As noted in the previous section, return policy employed by Neiman Marcus.com partly reflects this conclusion. When (γ − ψ)/ψ < 1, which is not desirable, the online retailer might charge more to decrease returns. With an increasing product price, the optimal shipping strategy may

shift from SFES to SFRS, and a dramatic jump occurs in the optimal RSC, which is used to compensate for the increased shipping costs resulting from free shipping service.

C. Profit Analysis

To examine the properties of the retailer’s optimal profit, without loss of generality, we assume that the product price satisfies the condition p > max{c + s,ω + s}. This condi- tion indicates that the price should be sufficiently large to cover the total cost incurred by SFRS and no-reason returns. In this case, properties of the optimal profit associated with shipping fee-demand sensitivity λ, refund-demand sensitivity γ and refund return-quantity sensitivity ψ are shown in the following proposition.

Proposition 4: The monotonicity of the online retailer’s optimal profit π(x∗, l∗) is determined by the following rules.

1) π(x∗, l∗) decreases in λ when λ ≤ λ̄, while is indepen- dent of λ when λ > λ̄.

2) π(x∗, l∗) increases in γ when γ > [φ +ψ(ω − c)]/(p − c − s).

3) π(x∗, l∗) decreases in ψ when ψ > max{[γ(p − c) − φ]/(p − ω − s), [γ(p − c) − φ]/(p − c − s)}.

Proposition 4 1) indicates that, when parameter λ is not greater than the threshold λ̄, the retailer may charge shipping fees for delivering products. As λ increases, product demand goes down, which leads to a lower profit for the retailer. However, when λ is larger than λ̄, the retailer’s optimal ship- ping strategy is to offer SFRS, and in this case, the retailer’s optimal profit is independent of λ.

Proposition 4 2) shows that, when parameter γ is larger than a specific threshold [φ + ψ(ω − c)]/(p − c − s), the retailer may charge less for returns. In this context, as γ increases, product demand goes up, which directly benefits the retailer. Proposition 4 3) suggests that, when the refund return- quantity sensitivity coefficient ψ is sufficiently large, i.e., ψ > max{[γ(p − c) − φ]/(p − ω − s), [γ(p − c) − φ]/(p − c − s)}, the retailer may charge more for returns. Charging more in this case decreases product demand, which may directly lead to a lower profit for the retailer. These insights can serve as useful guidelines for the retailer to market his products by choosing favorable shipping strategies and RSCs.

V. NUMERICAL STUDY

In this section, we present a numerical study to explore the impact of refund-demand sensitivity γ and refund return- quantity sensitivity ψ on the threshold (λ̄) of shipping fee- demand sensitivity. We then compare profits obtained under different shipping strategies and RSCs. To this end, we use the data presented in the previous section as initial values, i.e., α = 1000, β = 10, λ = 12, s = 10, γ = 8, c = 90, ψ = 2, ω = 25 and p = 140, and set φ = 40.

A. Sensitivity Analysis

We first examine the impact of parameter γ on the thresh- old λ̄. To this end, we vary γ from 6 to 10 while keeping the other parameters constant. The results are displayed in Fig. 8.

HUA et al.: OPTIMAL SHIPPING STRATEGY AND RSC UNDER NO-REASON RETURN POLICY IN ONLINE RETAILING 3199

Fig. 8. Sensitivity results of γ .

Fig. 9. Sensitivity results of ψ.

Fig. 8 shows that the threshold λ̄ increases in γ . Hence, when the effect of return policy on product demand increases, λ̄ increases. In this case, the retailer may lower RSC to encour- age more consumers to make purchases, which may result in some return loss. To compensate for this possible loss, accord- ing to Theorem 1, the retailer may have less incentive to provide SFRS. By contrast, the retailer may increase the RSC and offer SFRS.

To examine the impact of parameter ψ on the threshold λ̄, we let ψ increase from 1 to 2.5 at a step size of 0.1 while keep- ing the other parameters unchanged. The results are shown in Fig. 9.

Fig. 9 shows that, when ψ ≤ 1.30, λ̄ is relatively large and remains unchanged as ψ increases; when ψ > 1.30, the threshold λ̄ decreases in ψ. These results imply that, when ψ is sufficiently small, owing to a relatively high λ̄, the retailer is less able to offer SFRS. As γ remains unchanged, a lower ψ results in a higher γ − ψ. In this case, the retailer may set a lower RSC in order to boost sales. Such a strategy, in turn, can generate more profit and thus cover part of the return loss. By contrast, when ψ > 1.30, as ψ increases, the threshold λ̄ decreases, and as a result, the retailer tends to offer SFRS and bears shipping costs. In this context, the effect of return policy on return quantity becomes more pronounced, and thus, the retailer may charge more to deter consumer returns. Note that, a higher RSC can decrease product demand, however, which can be counteracted by an increase in demand caused by SFRS.

B. Profit Comparison

To illustrate the advantages of joint shipping strategy and RSC decisions, we first compare the retailer’s profits obtained

Fig. 10. Profits under (a), (c), and (e) SJD, SS1, and SS2, and (b), (d), and (f) SJD, SRR, and SRE.

based on the optimal joint shipping strategy and RSC decisions (x = x∗, l = l∗) with those obtained through the separate decisions, i.e., the optimal shipping strategy with fixed RSC and the optimal RSC with fixed shipping strategy. As it is difficult for us to obtain the conditions under which to compare profits of the optimal joint decisions and separate decisions, we use this numerical study to compare profits for the two decision scenarios. To this end, we set two fixed RSCs as l = ω and l = 10 in the separate decision scenarios. For ease of notation, we denote the optimal joint decision as SJD, separate shipping strategy decisions by fixing the RSC with l = ω and l = 10 as SS1 and SS2, and separate decisions on the RSC by fixing SFRS and SFES as SRR and SRE, respectively. We conduct the analysis by increasing the product price from 110 to 190, γ from 6.4 to 10, and ψ from 1 to 2.5, respectively, while keeping the other parameters constant. The results are depicted in Fig. 10.

Fig. 10(a) clearly shows that, when the product price increases from 110 to 190, profits obtained from the opti- mal joint decision are greater than or equal to those obtained from the optimal shipping strategy decision by fixing RSC regardless of whether l = ω or l = 10. In particular, the profit line achieved under the optimal joint decision and those under separate decision by fixing l = ω or l = 10

3200 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS: SYSTEMS, VOL. 47, NO. 12, DECEMBER 2017

overlap for certain product prices. Similar results are shown in Fig. 10(b). These results indicate that the retailer is bet- ter off jointly determining the optimal shipping strategy and RSC rather than separately determining when the product price changes.

Fig. 10(c)–(f) clearly shows that, the retailer’s profits from joint decision scenario are greater than or equal to those from separate decision scenarios when γ and ψ increase from 6.4 to 10, and from 1 to 2.5, respectively. These observations further indicate that the retailer can benefit more from deter- mining the optimal shipping strategy and RSC with a joint decision rather than with separate decisions.

We use SJD and SS1 as an example to illustrate these observations. In scenario SJD, Corollary 1 shows that φ ≤ γ(p − c − s) − ψ(2p − c + ω) can be equivalently trans- formed into p ≥ [φ + (γ − ψ)(c + s) + ψ(ω + s)]/(γ − 2ψ). In this scenario, (γ − ψ)/ψ=3 > 1 and [φ + (γ − ψ) (c + s) + ψ(ω + s)]/(γ − 2ψ) = 177.5. In addition, λ̄ = A/(p − c) = 7.37 < λ = 12, and the value of λ̄ decreases in the product price when p increases from 177.5 to 190. According to Theorem 1 and Corollary 1, the retailer will offer free shipping and return services simultaneously. In this case, a possible decrease in product demand from an increase in product price may be counteracted by offering both free shipping and return services. However, in scenario SS1, for a given fixed l = ω, when the product price increases from 110 to 134.3, the condition l > [α − βp + γ p − λ(p − c)]/γ cannot be satisfied. Thus, according to Proposition 1 in this case, the retailer may charge shipping cost when delivering a product. However, when p > 134.3, free shipping ser- vice is provided. More specifically, when the product price increases from 177.5 to 190, the retailer will provide free shipping service but will charge ω for return service. In com- parison with SJD, this case may decrease product demand to some extent. These results can reasonably explain why the retailer can obtain higher profits from joint decisions than those from separate decisions when the product price increases from 177.5 to 190. Similarly, the increase in the product price from 110 to 177.5 can also be explained by using our theo- retical results, e.g., Proposition 1, Theorem 1, Remark 1, and Corollary 2, and the explanations are thus omitted here.

To further illustrate the advantage of joint shipping strat- egy and RSC decision, we compare the retailer’s profits from the optimal joint decision (x = x∗, l = l∗) with those from other possible shipping strategies and RSCs. We consider the following four scenarios: 1) x = 0 and l = s + ω; 2) x = 0 and l = 0; 3) x = 1 and l = ω; and 4) x = 1 and l = 0. These four scenarios denote four possible combinations of shipping strategies and RSCs. For example, in scenario 1), the retailer chooses SFRS and charges the maximum fee s + ω for a return. For ease of notation, we denote the optimal joint decision as scenario SJD, and the other four scenarios as S1, S2, S3, and S4, respectively. For the sake of simplicity, we conduct an analysis by increasing only the product price from 110 to 190 while keeping the other parameters constant. The results are displayed in Fig. 11.

Fig. 11 shows that the retailer’s profits based on the opti- mal decisions are not lower than those in any of the four

Fig. 11. Profits under (a) SJD, S1, and S2, and (b) SJD, S3, and S4.

scenarios. Importantly, profits from the optimal decisions vary within a relatively small interval, i.e., from 4000 to 8000, while profits in other four scenarios change dramatically, and even are lower than zero. These results indicate that, the retailer’s profits secured based on the optimal decisions are more stable with the product price fluctuations than those based on other choices of shipping strategies and RSCs.

An important finding can be concluded from the above results.

Observation 1: The retailer is better off by determining the optimal shipping strategy and RSC jointly rather than determining them separately.

Changes in any market parameter (i.e., λ, γ , ψ, or p) can directly cause market environment variations, and thus will affect shipping strategy and RSC choices. In such circum- stances, although the retailer can acquire the same profits from separate decision scenario as those from joint deci- sion scenario under certain conditions, the retailer is better off in joint decision scenario rather than in separate decision scenario under other conditions. In other words, the retailer may not acquire the maximum profit by fixing any shipping strategy or RSC, and joint decision is a better choice. This result can be observed in Fig. 10. This observation also partly explains why many online retailers such as Nordstrom.com, Zappos.com, Diapers.com, and Neiman Marcus.com, provide both shipping strategies and RSCs simultaneously. For more examples, one can refer to [36]. This observation importantly highlights that the retailer can adjust his optimal shipping strategy and RSC to obtain a superior profit with market changes.

VI. CONCLUSION

As observed in real online retailing settings, many retailers provide free shipping strategy and allow dissatisfied customers to return products with no reason to boost sales. However, such a strategy, in turn, increases operational costs for retailers. In such a circumstance, it is important for retailers to determine whether to offer free shipping service when delivering prod- ucts, and how to charge consumers for returns. To address these two issues, in this article, we consider that one retailer sells a particular product to a group of consumers online, and we develop a theoretical model to examine the optimal shipping strategy and RSC under no-reason return policy in

HUA et al.: OPTIMAL SHIPPING STRATEGY AND RSC UNDER NO-REASON RETURN POLICY IN ONLINE RETAILING 3201

online retailing. Using the introduced model, we first examine the optimal shipping strategy by fixing RSC, and the optimal RSC by fixing shipping strategy, respectively. We then explore the optimal shipping strategy and RSC jointly. The retailer’s optimal shipping strategies and RSCs as well as the corre- sponding conditions are derived. We then conduct a numerical study to illustrate the advantage of joint decision of the optimal shipping strategy and RSC.

Some key findings and insights are obtained and are sum- marized as follows.

1) The retailer’s optimal shipping strategy and RSC inter- act with each other, and such decisions also depend on market parameters, i.e., λ, γ , ψ, p, and φ. The retailer is better off when determining the optimal shipping strategy and RSC jointly than that determining them sep- arately. Our numerical study supports this finding. This finding importantly shows that the retailer can always set the optimal combinations of shipping strategies and RSCs according to market changes.

2) Our theoretical results show that, the retailer typically uses SFRS accompanied by a higher RSC and SFES accompanied by a lower RSC. However, when con- sumers are sufficiently sensitive to RSC, the retailer will offer free return service regardless of whether providing free shipping service.

3) The retailer may charge a higher fee for return ser- vice when the base return quantity is relatively high, and a lower fee when the base return quantity is rela- tively low. In particular, when the base return quantity is sufficiently high, the retailer may charge the maxi- mum fee for return service. In contrast, when the base return quantity is sufficiently low, the retailer may offer free return service. Interestingly, the base return quantity has a significantly positive effect on decisions regarding SFRS. Specifically, as the base return quantity increases, the retailer has more incentive to offer free shipping service, and vice versa.

4) Surprisingly, when the effect of RSC on net demand is greater than that on return quantity, the retailer charges less for return service when the product price is higher, and vice versa. Conversely, the retailer charges more for return service when the product price is higher, and vice versa. However, when the optimal shipping strategy shifts from SFES to SFRS, the optimal RSC increases considerably.

5) The three market parameters (i.e., λ, γ , and ψ) have significant impacts on the optimal decisions on ship- ping strategy and RSC as well as the retailer’s profits. When λ exceeds a particular threshold, the retailer offers free shipping service and his profit remains constant when λ changes; otherwise, the retailer will charge for shipping service, and in this case his profit decreases in λ. Our numerical study shows that the threshold increases in γ but decreases in ψ. This relationship will influence the retailer’s choice of SFRS or SFES and RSC. Furthermore, the retailer’ optimal profit increases in γ and decreases in ψ, when they exceed their partic- ular thresholds. These results can help to guide online

retailers in managing their shipping and return ser- vices. In other words, to obtain higher profits, managers can strive to influence the market to increase γ while decrease ψ.

This paper presents key findings that shed light on retailers’ joint decisions on the optimal shipping strategies and RSCs that can help online retailers to improve their operational per- formance in terms of shipping and return service management. However, this paper also presents some limitations that may serve as future research topics. First, we develop our model based on the assumption that the demand function is determin- istic. Our model may generate different results when stochastic demand is considered. Second, in online settings, consumers have to pay shipping fees when retailers do not offer free ship- ping services. In such context, shipping fees can be regarded as sunk costs when consumers decide to return products. In fact, such sunk costs may prevent consumers from returning products and thus decrease returns. Consideration of such sunk costs in our model may provide more insights for retailers to better manage their shipping and return services. Third, issues such as seasonal discounts and limited time free shipping ser- vices are also important, and may have significant impact on shipping strategy and RSC decisions. These issues can also be seen as important extensions in future studies.

More important, we consider only one retailer in this paper, and it would be interesting to examine the optimal decisions made under competitive market environments. We can illus- trate this issue by using a simple example. We consider two retailers, A and B, in the same setting. On the one hand, when retailer B adopts SFRS while retailer A does not, some of retailer A’s consumers may move to retailer B. In such a case, retailer A may then also adopt SFRS, which exactly depends on how much of his demand has shifted. Thus, we can characterize conditions under which whether retailer A may choose SFRS. On the other hand, when retailer B adopts more generous return policy, some of retailer A’s consumers may also move to retailer B. We can thus identify condi- tions when retailer A tends to decrease his RSC. Note that, in this case, competition between retailers A and B will lead to an equilibrium of shipping strategies and RSCs. Thus, study on this issue may also help online retailers to better manage their shipping and return services in practice. This issue will constitute an important topic in our future research.

APPENDIX

Proof of Proposition 1: For a fixed RSC l (l ∈ [0,ω]), when offering free shipping service, the retailer’s profit is

π(0, l) = [α − φ − βp + (γ − ψ)(p − l)](p − s − c) − [φ + ψ(p − l)](s + ω − l).

When not offering free shipping service, the retailer’s profit is

π(1, l) = [α − φ − βp − λs + (γ − ψ)(p − l)](p − c) − [φ + ψ(p − l)](ω − l).

We have

π(0, l) − π(1, l) = −[α − βp + γ(p − l) − (p − c)λ]s.

3202 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS: SYSTEMS, VOL. 47, NO. 12, DECEMBER 2017

Thus, the retailer offers free shipping service if and only if π(0, l) − π(1, l) > 0, i.e., l > [α − βp + γ p − λ(p − c)]/γ . This completes the proof.

Proof of Proposition 2: We prove Proposition 2 by using the following two cases.

1) When the retailer offers free shipping service, i.e., x = 0, the retailer’s profit optimization problem is formulated as

max l

π(0, l)

s.t. l ≤ ω + s l ≥ 0.

Note that

π(0, l) = [α − φ − βp + (γ − ψ)(p − l)](p − s − c) − [φ + ψ(p − l)](s + ω − l).

Since

∂π(0, l)/∂l = φ − (γ − ψ)(p − s − c) + ψ(p + s + ω) − 2ψl = 0

l = −(γ − ψ)(p − s − c) + φ + ψ(p + s + ω) 2ψ

.

Since

∂ 2 π(0, l)/∂l2 = −2ψ < 0

when −(γ − ψ)(p − s − c) + φ + ψ(p + s + ω)

2ψ ≥ s + ω

the profit π(0, l) increases in l in the interval [0, s + ω]. When

0 < −(γ − ψ)(p − s − c) + φ + ψ(p + s + ω)

2ψ < s + ω

in the interval [0, s + ω], π(0, l) first increases in l, and reaches the maximum value when

l = −(γ − ψ)(p − s − c) + φ + ψ(p + s + ω) 2ψ

and then decreases. When −(γ − ψ)(p − s − c) + φ + ψ(p + s + ω)

2ψ < 0

π(0, l) decreases in the interval [0, s + ω]. Then, the results in Proposition 2 1) are achieved.

2) When the retailer does not provide free shipping service, i.e., x = 1, the retailer’s profit optimization problem is expressed as

max l

π(1, l)

s.t. l ≤ ω l ≥ 0.

Note that

π(1, l) = [α − φ − βp − λs + (γ − ψ)(p − l)](p − c) − [φ + ψ(p − l)](ω − l).

Similar to Proposition 2 1), it is easy to verify Proposition 2 2). This completes the proof.

Proof of Theorem 1: For fixed shipping strategy, the retailer’s optimal RSC is presented in Proposition 2, i.e., l∗0 and l∗1 . When optimizing shipping strategy and RSC jointly, the retailer will choose SFRS when π(0, l∗0) − π(1, l∗1) > 0; otherwise, the retailer will choose SFES. Hence, x∗ = 0 when λ > λ̄, and x∗ = 1 when λ ≤ λ̄, where

λ̄ = α − βp + γ ( p − l∗0

)

p − c + γ(p − c) − φ − ψ

( 2p − c + ω − l∗1 − l∗0

)

p − c l∗0 − l∗1

s .

To elaborate on Theorem 1, for ease of notation, we define the following parameters:

A = α − βp + γ p B = α − βp − γφ/2ψ + γ(c − ω)/2

+ γ 2(p − c)/2ψ − γ 2s/4ψ C = α − φ − βp + (γ − ψ)(2p − ω − c − s) D = B − [γ(p − c) − φ − ψ(2p − c − ω)]2/4sψ E = B − [γ(p − c) − φ − ψ(2p − c + ω)]2/4sψ F = C + [γ(p − c) − φ − ψ(2p − c − ω)]2/4sψ G = C + [γ(p − c) − φ − ψ(2p − c)]ω/s.

Theorem 1 can be specified by using the following three cases.

Case 1: When (γ − ψ)/ψ < 1, φ+(γ −ψ)(c + s)−ψ(ω+ s) < φ + (γ − ψ)c − ψω.

1) When φ ≥ γ(p − c − s)−ψ(2p − 2s − c −ω), l∗0 = s +ω and l∗1 = ω. The retailer offers free shipping service if π(0, l∗0) − π(1, l∗1) > 0, i.e., (γ − ψ)s2 − [α − φ − βp + (γ − ψ)

(2p − ω − c) − λ(p − c)]s > 0. This is equivalent to

λ(p − c) > α − φ − βp + (γ − ψ)(2p − ω − c − s) = C.

Thus, π(0, l∗0) − π(1, l∗1) > 0 when λ > C/(p − c), and π(0, l∗0) − π(1, l∗1) ≤ 0 when λ ≤ C/(p − c). That is, when λ > C/(p − c), x∗ = 0 and l∗ = l∗0 ; otherwise, x∗ = 1 and l∗ = l∗1 .

2) When

γ(p − c) − ψ(2p − c − ω) ≤ φ < γ(p − c − s) − ψ(2p − 2s − c − ω), l∗1 = ω

and

l∗0 = −(γ − ψ)(p − s − c) + φ + ψ(p + s + ω)

2ψ .

Similar to 1), the retailer may offer free shipping service when π(0, l∗0) − π(1, l∗1) > 0, which is equivalent to λ(p − c) > D. Thus, π(0, l∗0) − π(1, l∗1) > 0 when λ > D/(p − c), and π(0, l∗0) − π(1, l∗1) ≤ 0 when λ ≤ D/(p − c). Hence, x∗ = 1

HUA et al.: OPTIMAL SHIPPING STRATEGY AND RSC UNDER NO-REASON RETURN POLICY IN ONLINE RETAILING 3203

and l∗ = l∗1 when λ ≤ D/(p − c); and x∗ = 0 and l∗ = l∗0 when λ > D/(p − c).

3) When

φ < γ(p − c) − ψ(2p − c − ω), l∗0 = −(γ − ψ)(p − s − c) + φ + ψ(p + s + ω)

2ψ

and

l∗1 = −(γ − ψ)(p − c) + φ + ψ(p+ω)

2ψ .

Similarly, the retailer offers free shipping service when π(0, l∗0) − π(1, l∗1) > 0, which is equal to λ(p − c) > B. Thus, π(0, l∗0) − π(1, l∗1) > 0 when λ > B/(p − c), and π(0, l∗0) − π(1, l∗1) ≤ 0 when λ ≤ B/(p − c). Therefore, x∗ = 0 and l∗ = l∗0 when λ > B/(p − c), x∗ = 1 and l∗ = l∗1 when λ ≤ B/(p − c).

Case 2: When 1 + 2ω/s > (γ − ψ)/ψ > 1, it is easy to verify that

φ + (γ − ψ)(c + s) + ψ(ω + s) > φ + (γ − ψ)c + ψω > φ + (γ − ψ)(c + s) − ψ(ω + s) > φ + (γ − ψ)c − ψω.

1) When φ < γ(p − c − s) − ψ(2p − c + ω), l∗0 = 0 and l∗1 = 0. The retailer offers free shipping service when π(0, l∗0)−π(1, l∗1) > 0, which is equivalent to λ(p−c) > A. Then, π(0, l∗0) − π(1, l∗1) > 0 when λ > A/(p − c), and π(0, l∗0)−π(1, l∗1) ≤ 0 when λ ≤ A/(p − c). Hence, x∗ = 0 and l∗ = 0 when λ > A/(p − c), and x∗ = 1 and l∗ = 0 when λ ≤ A/(p − c).

2) When

γ(p − c − s) − ψ(2p − c + ω) ≤ φ < γ(p − c) − ψ(2p − c + ω)

l∗0 = −(γ − ψ)(p − s − c) + φ + ψ(p + s + ω)

2ψ

and l∗1 = 0. The retailer offers free shipping service when π(0, l∗0) − π(1, l∗1) > 0, which is equivalent to λ(p − c) > E. That is, π(0, l∗0) − π(1, l∗1) > 0 when λ > E/(p − c), and π(0, l∗0) − π(1, l∗1) ≤ 0 when λ ≤ E/(p − c). Hence, x∗ = 0 and l∗ = l∗0 when λ > E/(p − c); x∗ = 1 and l∗ = l∗1 when λ ≤ E/(p − c).

3) When

γ(p − c) − ψ(2p − c + ω) ≤ φ < γ(p − c − s) − ψ(2p − 2s − c − ω)

l∗0 = −(γ − ψ)(p − s − c) + φ + ψ(p + s + ω)

2ψ

and

l∗1 = −(γ − ψ)(p − c) + φ + ψ(p+ω)

2ψ .

Similar to 3) in case 1, we can easily obtain the optimal decisions, and thus the proof is omitted here.

4) When

γ(p − c − s) − ψ(2p − 2s − c − ω) ≤ φ < γ(p − c) − ψ(2p − c − ω)

l∗0 = s + ω and l∗1 = −(γ − ψ)(p − c) + φ + ψ(p+ω)

2ψ .

The retailer offers free shipping service when π(0, l∗0)− π(1, l∗1) > 0, which is equivalent to λ(p − c) > F. That is, π(0, l∗0) − π(1, l∗1) > 0 when λ > F/(p − c), and π(0, l∗0) − π(1, l∗1) ≤ 0 when λ ≤ F/(p − c). Hence, x∗ = 0 and l∗ = l∗0 when λ > F/(p − c); x∗ = 1 and l∗ = l∗1 when λ ≤ F/(p − c).

5) When φ ≥ γ(p − c) − ψ(2p − c − ω), l∗0 = s + ω, and l∗1 = ω. Similar to 1) in case 1, we can easily obtain the optimal decisions for the retailer.

Case 3: When

(γ − ψ)/ψ ≥ 1 + 2ω/s, φ + (γ − ψ)(c + s) + ψ(ω + s) > φ + (γ − ψ)(c + s) − ψ(ω + s) ≥ φ + (γ − ψ)c + ψω > φ + (γ − ψ)c − ψω.

1) When φ < γ(p − c − s) − ψ(2p − c + ω), l∗0 = 0 and l∗1 = 0. The proof is similar to that of 1) in case 2, and thus omitted here.

2) When

γ(p − c − s) − ψ(2p − c + ω) ≤ φ < γ(p − c − s) − ψ(2p − 2s − c − ω)

l∗0 = −(γ − ψ)(p − s − c) + φ + ψ(p + s + ω)

2ψ

and l∗1 = 0. The proof is similar to that of 2) in case 2, and thus omitted here.

3) When γ(p − c − s) − ψ(2p − 2s − c − ω) ≤ φ < γ(p − c) − ψ(2p − c + ω), l∗0 = s + ω, and l∗1 = 0. The retailer offers free shipping service when π(0, l∗0)− π(1, l∗1) > 0, which is equivalent to λ(p − c) > G. That is, π(0, l∗0) − π(1, l∗1) > 0 when λ > G/(p − c), and π(0, l∗0) − π(1, l∗1) ≤ 0 when λ ≤ G/(p − c). Hence, x∗ = 0 and l∗ = l∗0 when λ > G/(p − c); x∗ = 1 and l∗ = l∗1 when λ ≤ G/(p − c).

4) When

γ(p − c) − ψ(2p − c + ω) ≤ φ < γ(p − c) − ψ(2p − c − ω)

l∗0 = s + ω and l∗1 = −(γ − ψ)(p − c) + φ + ψ(p+ω)

2ψ .

The proof is similar to that of 4) in case 2, and thus omitted here.

5) When φ ≥ γ(p − c) − ψ(2p − c − ω), l∗0 = s + ω and l∗1 = ω. The proof is similar to that of 5) in case 2, and thus omitted here. This completes the proof.

Proof of Corollary 1: By comparing l∗0 and l ∗ 1 in the proof of

Theorem 1, we can directly obtain the results in Corollary 1, and the proof is thus omitted here.

Proof of Remark 1: According to Theorem 1 and Corollary 1, the retailer may offer free shipping and return services, if and only if (γ − ψ)/ψ > 1, λ ≥ λ̄ = A/(p − c) and φ ≤ γ(p − c − s)−ψ(2p − c +ω), where A = α−βp +γ p.

3204 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS: SYSTEMS, VOL. 47, NO. 12, DECEMBER 2017

Note that, λ ≥ λ̄ = A/(p − c) is equivalent to (λ+β−γ)p ≥ α + λc. Since α + λc > 0 and p > 0, this condition can also be expressed as λ > γ − β and p ≥ (α + λc)/(λ + β − γ). In addition, when (γ − ψ)/ψ > 1, the condition φ ≤ γ(p − c − s) − ψ(2p − c + ω) can be transformed into p ≥ [φ + (γ − ψ)(c + s) + ψ(ω + s)]/(γ − 2ψ). Based on these results, Remark 1 can be easily achieved. This completes the proof.

Proof of Corollary 2: According to 1) of case 1, 5) of case 2, and 5) of case 3 in the proof of Theorem 1, we can easily obtain the results in Corollary 2, and thus omitted here.

Proof of Corollary 3: According to subcase of case 3 in the proof of Theorem 1, we can easily obtain the results in Corollary 3.

Proof of Proposition 3: Note that, the threshold λ̄ is continu- ous in φ. For each case and subcase in the proof of Theorem 1, we can easily verify the monotonicity of λ̄ regarding φ, and thus the proof is omitted here.

Proof of Corollary 4: We prove Corollary 4 by using the following two cases.

Case 1) When the retailer’s optimal shipping strategy remains unchanged, according to Proposition 2, it is easy to verify that the optimal RSC is continu- ous regarding the product price, and the optimal RSC is decreasing in the product price when (γ −ψ)/ψ > 1, while is increasing in the product price when (γ − ψ)/ψ < 1.

Case 2) When the retailers’ optimal shipping strategy shifts from SFES to SFRS, the optimal RSC changes from l∗1 to l

∗ 0 accordingly. Since l

∗ 1 ≤ l∗0 , the opti-

mal RSC increases. In particular, when l∗1 < l ∗ 0 ,

there is a jump in the optimal RSC. This completes the proof.

Proof of Proposition 4: The retailer’s optimal profit is π(x∗, l∗), which satisfies π(x∗, l∗) = max{π(0, l∗0),π(1, l∗1)}. π(x∗, l∗) is increasing in γ when both π(0, l∗0) and π(1, l

∗ 1)

are increasing in γ ; and π(x∗, l∗) is decreasing in ψ when both π(0, l∗0) and π(1, l

∗ 1) are decreasing in ψ. Thus, to prove

Propositions 4 2) and 4 3), we only need to verify that both π(0, l∗0) and π(1, l

∗ 1) are increasing in γ but decreasing in ψ.

1) Theorems 1 shows that, for a particular λ̄, x∗ = 0 when λ > λ̄, and x∗ = 1 when λ ≤ λ̄. Note that, the threshold λ̄ varies with changes of market parameters. For exam- ple, if the parameters satisfy the case (1), λ̄ = A/(p − c). When x∗ = 0, π(x∗, l∗) = π(0, l∗0); when x∗ = 1, π(x∗, l∗) = π(1, l∗1). Since l∗0 and l∗1 are independent of λ, π(0, l∗0) and π(1, l

∗ 1) are derivable, i.e.,

∂π ( 0, l∗0

)

∂λ = 0

∂π ( 1, l∗1

)

∂λ = −(p − c)s.

Hence, π(x∗, l∗) is decreasing in λ when λ ≤ λ̄, and π(x∗, l∗) is independent of λ when λ > λ̄.

2) To prove that π(0, l∗0) and π(1, l ∗ 1) are increasing in

γ , we first prove that π(0, l∗0) is increasing in γ . We can easily have π(0, l∗0) = max{π(0, 0),π(0, l∗0), π(0, s + ω)}.

Case 1: When

π ( 0, l∗0

) = π(0, 0), ∂π ( 0, l∗0

)

∂γ = p(p − c − s).

Case 2: When

π ( 0, l∗0

) = π(0, s + ω) ∂π

( 0, l∗0

)

∂γ = (p − s − ω)(p − c − s).

Case 3: When π(0, l∗0) = π(0, l∗0) and

l∗0 = −(γ − ψ)(p − s − c) + φ + ψ(p + s + ω)

2ψ ∂π

( 0, l∗0

)

∂γ = −(p − c − s)[φ + ψ(ω − c) − γ(p − c − s)]

2ψ .

It is assumed that p > max{c + s,ω + s}, p(p − c − s) > 0 and (p − s − ω)(p − c − s) > 0. When (p − c − s)[φ + ψ(ω − c) − γ(p − c − s)]/2ψ < 0, it is clear that π(0, l∗0) is increasing in γ . Hence, π(0, l

∗ 0) is

increasing in γ when γ > [φ + ψ(ω − c)]/(p − c − s). Regarding

π ( 1, l∗1

) ,π

( 1, l∗1

) = max{π(1, 0),π(1, l∗1 ) ,π(1,ω)

} .

Case 1: When

π ( 1, l∗1

) = π(1, 0), ∂π ( 1, l∗1

)

∂γ = p(p − c).

Cases 2: When

π ( 1, l∗1

) = π(1,ω), ∂π ( 1, l∗1

)

∂γ = (p − ω)(p − c).

Case 3: When π(1, l∗1) = π(1, l∗1) and

l∗1 = −(γ − ψ)(p − c) + φ + ψ(p + ω)

2ψ ∂π

( 1, l∗1

)

∂γ = −(p − c)[φ + ψ(ω − c) − γ(p − c)]

2ψ .

It is assumed that p > max{c + s,ω + s}, p(p − c) > 0, and (p − ω)(p − c) > 0. When (p − c)[φ + ψ(ω − c) − γ(p − c)]/2ψ < 0, we can conclude that π(1, l∗1) is increasing in γ . Hence, π(1, l

∗ 1)

is increasing in γ when γ > [φ + ψ(ω − c)]/(p − c). In conclusion, when γ > [φ + ψ(ω − c)]/(p − c − s), γ > [φ + ψ(ω − c)]/(p − c) is always satisfied. Thus, both π(0, l∗0) and π(1, l

∗ 1) are increasing in γ when γ >

[φ + ψ(ω − c)]/(p − c − s). 3) To prove that π(0, l∗0) and π(1, l

∗ 1) are decreasing in ψ,

we first verify that π(0, l∗0) is decreasing in ψ. We have

π(0, l∗0) = max { π(0, 0),π

( 0, l∗0

) ,π(0, s + ω)}.

Case 1: When

π ( 0, l∗0

) = π(0, 0), ∂π ( 0, l∗0

)

∂ψ = −p(p − c + ω).

HUA et al.: OPTIMAL SHIPPING STRATEGY AND RSC UNDER NO-REASON RETURN POLICY IN ONLINE RETAILING 3205

Case 2: When

π ( 0, l∗0

) = π(0, s + ω) ∂π

( 0, l∗0

)

∂ψ = −(p − s − ω)(p − c − s).

Case 3: When π(0, l∗0) = π(0, l∗0) and l∗0 = [ − (γ − ψ)(p − s − c) + φ + ψ(p + s + ω)]/2ψ, ∂π(0, l∗0)/∂ψ = −[2(p−l)(p−l+ω−c)ψ2 +(φ−γ(p− c − s))(2p − 2l +ω− c)ψ +(φ − γ(p − c − s))2]/(2ψ2). We assume that p > max{c+s,ω+s}, −p(p−c+ω) < 0 and −(p − s − ω)(p − c − s) < 0. When [2(p − l)(p − l +ω− c)ψ2 +(φ −γ(p − c − s))(2p − 2l + ω−c)ψ +(φ − γ(p − c − s))2]/(2ψ2) > 0, we find that π(0, l∗0) is decreasing in ψ. Hence, π(0, l

∗ 0) is decreas-

ing in ψ when ψ > max{[γ(p − c) − φ]/(p − ω − s), [γ(p − c) − φ]/(p − c − s)}. With respect to π(1, l∗1), π(1, l∗1) = max{π(1, 0),π(1, l∗1),π(1,ω)}. Case 1: When

π(1, l∗1) = π(1, 0), ∂π(1, l∗1) ∂ψ

= −p(p − c + ω). Case 2: When

π ( 1, l∗1

) = π(1,ω), ∂π ( 1, l∗1

)

∂ψ = −(p − ω)(p − c).

Case 3: When π(1, l∗1) = π(1, l∗1) and l∗1 = [ − (γ − ψ)(p − c) + φ + ψ(p + ω)]/2ψ, ∂π(1, l∗1)/∂ψ = −[2(p − l)(p − l + ω − c)ψ2 + (φ − γ(p − c))(2p − 2l + ω − c)ψ + (φ − γ(p − c))2]/(2ψ2). It is assumed that p > max{c + s,ω + s}, −p(p − c + ω) < 0, and −(p − ω)(p − c) < 0. When [2(p − l)(p − l + ω − c)ψ2 + (φ − γ(p − c))(2p − 2l + ω − c)ψ + (φ − γ(p − c))2]/(2ψ2) > 0 we can conclude that π(1, l∗1) is decreasing in ψ. Hence, π(1, l∗1) is decreasing in ψ when ψ > max{[γ(p − c) − φ]/(p − ω),[γ(p − c) − φ]/(p − c)}.

We conclude that, when

ψ > max {[ γ(p − c) − φ]/(p − ω − s)

[ γ(p − c) − φ]/(p − c − s)}

ψ > max {[ γ(p − c) − φ]/(p − ω)

[ γ(p − c) − φ]/(p − c)}.

Therefore, both π(0, l∗0) and π(1, l ∗ 1) decrease in ψ when ψ >

max{[γ(p − c) − φ]/(p − ω − s), [γ(p − c) − φ]/(p − c − s)}. This completes the proof.

ACKNOWLEDGMENT

The authors would like to thank the editor and the three reviewers for helpful comments and suggestions on earlier versions of the manuscript.

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Zhongsheng Hua received the Ph.D. degree in computer science from the University of Science and Technology of China, Hefei, China.

He is a Professor with the School of Management, Zhejiang University, Hangzhou, China. His current research interests include service science, produc- tion and operations management, and supply chain management.

Prof. Hua’s publications have appeared in Production and Operations Management, the Journal of Operations Management, Marketing

Science, IIE Transactions, the IEEE TRANSACTIONS ON INTELLIGENT TRANSPORTATION SYSTEMS, and the IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS, and Information and Management.

Huijun Hou is currently pursuing the Ph.D. degree with the School of Management, University of Science and Technology of China, Hefei, China.

His research interests include service operations management.

Yiwen Bian received the Ph.D. degree in manage- ment science and engineering from the University of Science and Technology of China, Hefei, China.

He is an Associate Professor with the SHU-UTS SILC Business School, Shanghai University, Shanghai, China. His current research interests include performance evaluation and service opera- tions management.

Dr. Bian’s publications have appeared in the Annals of Operations Research, Omega, Energy Policy, Computers and Operations Research, and

Computers and Industrial Engineering.