# No Refund or Full Refund: When Should a Fashion Brand Offer Full Refund Consumer Return Service for Mass Customization Products?

Authors: Tsan-Ming Choi; Na Liu; Shuyun Ren; Chi-Leung Hui
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

We analytically explore in this paper the consumer return policy under fashion mass customization (MC) program. To be specific, we model the stochastic fashion MC program with the consideration of consumer demand uncertainty. If a consumer return policy is implemented, we further consider return uncertainty. By modeling the optimization objective of the risk averse MC fashion brand via a mean-variance approach, we derive the closed-form optimal solution under each case. We then conduct both analytical and numerical sensitivity analyses. For the scenario with full refund and return, we reveal the analytical conditions under which the optimal retail price and the optimal number of options available for customization (called the “optimal modularity level”) vary monotonically with respect to the salvage value and the return service charge. For the scenario when there is no refund and return, we show that the optimal retail price and the optimal modularity level are decreasing in the MC fashion brand's degree of risk aversion, the demand uncertainty, and the price-demand sensitivity coefficient. In addition, our numerical analysis indicates that whether the risk averse MC fashion brand would prefer offering consumer return with full refund to no return depends heavily on the demand-return correlation (DRC) parameter.


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## Body

## 1. Introduction

Mass customization (MC) is an industrial practice which helps to manufacture customized products to satisfy consumers’ needs with a cost close to mass production [1–7] in a timely manner [8–10]. In the fashion industry, as driven by the consumer needs [11–13], the advances of production and information technology [14–21] and the increasingly competitive market situation [22–25], MC has developed to be a very popular industrial practice [17, 26] and a lot of fashion retail brands and fashion retailers are implementing it. In fact, we can find MC in nearly all kinds of well-established fashion brands, from high-end luxury brands (Hermes, LV, etc.) to mass-market labels (Adidas, Nike, Puma, Brooks Brothers, etc.) (see [8, 27] for more cases and empirical details). However, fashion brands offering MC services face some challenges, and it is known that big name retailers/brands like Levi’s and Lands’ End have already terminated their MC programs for different reasons.In recent years, there are more and more analytical studies on fashion MC and some of them concern about consumer returns. In particular, Liu et al. [28] develop a three-dimensional optimization problem and analytically derive the optimal decisions on retail price [29], modularity level [30–32], and refund rate (under consumer return [33]). From both analytical and numerical analyses, they obtain several important findings and insights. Among others, they argue that the degree of risk aversion of the fashion MC service provider is an influential factor affecting the respective optimal decisions. In addition, Choi [34] studies the optimal return service charging policy for fashion retailers offering MC. He develops the closed-form analytical conditions under which it is optimal for the fashion MC service provider to provide free return service charge to consumers.In the fashion industry, the adoption of consumer return under the MC service program is very polarized. Actually, we find that the fashion MC brands either simply do not allow any consumer return or they would allow full refund in the return policy. For example, the international brand Nike is implementing MC and it allows consumer return with full refund for its NIKEiD products. (p.s.: Nike makes it clear on its website that under its NIKEiD MC program, any consumer can return the MC products (with any reasons) within 30 days of purchase if they want to.) However, many other similar sporty fashion brands do not allow return for MC products (except for those with manufacturing defects or quality problems).In light of the above industrial observation and based on the prior studies in the literature, we examine in this paper the optimal retail pricing and modularity level decisions for a risk averse MC fashion brand (p.s.: an MC fashion brand is a fashion brand which is offering MC.) for both scenarios with and without consumer return. We study the optimal pricing and modularity level decisions under each scenario. We analytically reveal that a linear relationship exists between the optimal price and the optimal level of modularity under both scenarios. We also show analytically how the optimal pricing and level of modularity decisions relate to some important model parameters. After that, we explore the factors which will affect the risk averse MC fashion brand’s decision on when to offer consumer return with full refund. We find that whether it is optimal for the risk averse MC fashion brand to offer consumer return with full refund or no return depends critically on the demand return correlation coefficient.The rest of this paper is organized as follows. We review the related literature concisely in Section2. We present the analytical models in Section 3. We derive the optimal decisions in Section 4. We conduct numerical sensitivity analysis in Section 5. We conclude and discuss future research directions in Section 6.


## 2. Literature Review

MC is an important make-to-order/build-to-order [32, 35] business strategy widely adopted in fashion retailing. In fact, MC helps consumers to develop customized products [36] and participate in the codesign process [37] which is a known means of cocreating value [38, 39]. It is also a popular topic in operations management (see [40] for a relatively recent review) and both empirical and analytical studies are reported in recent years. For example, Peters and Saidin [16] explore via a case study approach the challenges of MC implementation. They study the critical factors which drive the service provider to adopt MC. They discuss the challenges behind the MC implementation in the service sector. They further propose a novel framework which can help the service provider to offer MC. Gu et al. [41] develop an optimization model to describe the implementation of MC as a gradual process. They derive two optimization methods to find the optimal customization quantity. They indicate that their proposed methods can successfully reduce the amount of customized items in every part of the overall production process and hence enhances the efficiency of MC. Helms et al. [42] examine e-commerce and knowledge management and identify how they can support MC. They find that e-commerce provides capabilities for the MC service provider to reach global market and helps in learning about customer preferences. They reveal that knowledge management can help manage the intellectual capital. They argue that both e-commerce and knowledge management can significantly improve the implementation of MC. Bock [43] studies the pros and cons on employing off-shore and near-shore production strategies under MC. He proposes a mixed-model assembly line balancing approach to provide a direct comparison of the estimated variable manufacturing costs by generating a location-based line layout for all competing locations. He runs several tests with various location configurations and identifies the factors which affect the optimal sourcing and production strategy for the MC company. Brun and Zorzini [44] investigate the relationships between postponement and modularization under MC. They examine the industrial cases in Italy with a case study approach. They reveal that product/process customization and product/process complexity are critical factors which affect the postponement and modularization relationship. Yao and Liu [45] study an interesting problem on how companies handle the probable contradiction between scale production effect and customized demand under MC. They develop a dynamic multiobjective optimization model and derive effective algorithm to solve it. They test their proposed algorithm by simulation and show that their proposed algorithm can enhance the scheduling efficiency under the MC program. Most recently, motivated by a typical MC production company that has an inefficient scheduling operations problem, Zhong et al. [46] examine the applicability of an RFID-enabled manufacturing execution system (MES) for MC. In their proposed MES, the RFID-based devices are deployed on the shop floor to keep track of the production data. As a result, they find that both operational planning and scheduling decisions become enhanced. They conduct a case study and conclude that their proposed MES is effective.In the domain of fashion apparel, MC has also been extensively studied. For example, Dong et al. [1] discuss how MC can be implemented and integrated with the mass production line to achieve high quality and low price fashion products. Ulrich et al. [37] study the consumer-to-design scheme of fashion products under MC and generate a number of interesting insights. Yeung and Choi [27] explore the implementation of MC in the Hong Kong apparel industry. They identify a few barriers which hinder the development of fashion MC in markets like Hong Kong. Choi et al. [47] conduct a mean-variance analysis of the fashion MC program with consumer return. Based on the assumption that the consumer demand function is independent of the return service charging policy, they derive a couple of counter-intuitive findings. Liu et al. [28] study the optimal retail pricing, refund, and modularity decision making problem for a fashion mass customization system. They obtain several important insights via extensive sensitivity analysis.Based on the above reviewed literature, this paper conducts an analytical research on MC program with a risk averse fashion brand under the mean-variance framework [48, 49]. This paper is related to [28, 32, 47] because they all consider optimal decisions related to retail pricing and modularity. However, this paper is different from [28, 32] in which the demand and return uncertainty models are different and this paper focuses on examining the polarized strategic decisions on allowing consumer return with full refund or disallowing return at all whereas [28, 32] focus on the optimal operational decisions on price, modularity, and refund rate. This paper is also different from [47] because we argue that demand must relate to the return service charge whereas [47] assumes consumer demand independent of return service charge. Finally, this paper is also different from [34] because these two papers have totally different optimal decisions and the research objectives are also different.


## 3. Analytical Models

In this paper, we consider a risk averse MC fashion brand which sells an MC fashion product directly to consumers via an online platform (e.g., the MC retailing website). The whole process is a make-to-order type and the fashion brand will start producing the product after receiving the order from the consumer. In this paper, we consider the situation that the MC fashion brand makes a few important decisions, namely the retail price of the MC product (p), the number of options available for customization (we call it the level of modularity of the product (m)) (notice that this product modularity is different from the production process modularity concept commonly explored in the literature [50]), and whether to allow consumer return with full refund or no return. Notice that the “consumer return scheme” here refers to the product return from consumers who are unhappy with the purchase with any reason (probably within some reasonable time, such as one month), but it is not related to the situations with manufacturing mistakes, defects, or other product quality problems. As widely observed from the industrial practice in the fashion apparel industry, the MC fashion brands are adopting very polarized measure on consumer return: they either disallow return (i.e., no return, no refund) or accept return with full refund. Based on this industrial practice, we first represent the amount of refund under the consumer return policy by  r. Thus, a full refund under the consumer return scheme implies r=p, and the no return scheme has r=0.Following [28, 32], we construct the following analytical models in two separate scenarios, namely“full refund under consumer return” and “no consumer return.”The notation employed in this paper basically follows the literature [28, 32, 34].


### 3.1. Full Refund under Consumer Return

Demand function:D~1=α-βp+γr+δm+ε~D,(β>γ); (p.s.: notice that this price dependent function is very commonly employed in the literature, see, e.g., [25, 51].)
Return quantity function:R~=ϕ+ψr+ε~R;
Salvage value:S(m)=s+vm;
Modularity cost:C(m)=(1/2)θm2.Since under the consumer return scheme, we assume a full refund will be granted for each return, we have the rate of refundr=p. Thus
(1)D~1=α-βp+γ(p-l)+δm+ε~D,(β>γ);
(p.s.: we use subscripts 1 and 2 to represent the scenarios “with consumer return and full refund” and “no return,” respectively.)  R~=ϕ+ψ(p-l)+ε~R, l<p is the service charge.The profit function is hence expressed as follows:(2)P1=p(α-βp+γ(p-l)+δm+ε~D)+(S(m)-l)(ϕ+ψ(p-l)+ε~R)-12θm2.
We consider the scenario when the demand function and return quantity are correlated with a coefficient of correlation of ρ. Thus, the expected profit and the variance of profit are given as follows:
(3)E[P1]=p[α-βp+γ(p-l)+δm]+(s+vm-l)[ϕ+ψ(p-l)]-12θm2,V[P1]=p2σD2+(s+vm-l)2σR2+2ρσRσDp(s+vm-l).Define in the following the mean-variance (MV) objective function [28, 52–57] of the MC fashion brand, where k represents the degree of risk aversion for the MC fashion brand:
(4)U1=E[P1]-kV[P1].
In order to examine whether U1 is concave, we construct the Hessian matrix as follows:
(5)HU1=|∂2U1∂m2∂2U1∂m∂p∂2U1∂p∂m∂2U1∂p2|=|-θ-2kv2σR2δ+vψ-2kvρσDσRδ+vψ-2kvρσDσR-2β+2γ-2kσD2|=(2β-2γ+2kσD2)(θ+2kv2σR2)-(δ+vψ-2kvρσDσR)2=2θ(β-γ)+2kθσD2+4kρσDσR(vδ+v2ψ)+4kv2σR2(β-γ)+4k2v2σD2σR2(1-ρ2)-(δ+vψ)2>0.
By checking the Hessian matrix, we can see that U1 will be a concave function of p and m if and only if (5) holds.


### 3.2. No Consumer Return

For the scenario without consumer return, the model is much simpler because there will be no return and no salvage value of the returned product. As such, the demand function, the modularity cost, and the profit function are derived as follows.Demand function:D~2=α-βp+δm+ε~D.Modularity cost:C=(1/2)θm2.The profit function is:(6)P2=p(α-βp+δm+ε~D)-12θm2.
The expected profit and variance of profit are hence given as follows:
(7)E[P2]=p(α-βp+δm)-12θm2;V[P2]=p2σD2.
Similarly, we construct the MV objective function as follows:
(8)U2=E[P2]-kV[P2].
By checking the Hessian matrix, we can see that U2 is a concave function of p and m if and only if (9) is satisfied as follows:
(9)HU2=|∂2U2∂m2∂2U2∂m∂p∂2U2∂p∂m∂2U2∂p2|=|-θδδ-2β-2kσD2|=2θ(β+kσD2)-δ2>0.


## 3.1. Full Refund under Consumer Return

Demand function:D~1=α-βp+γr+δm+ε~D,(β>γ); (p.s.: notice that this price dependent function is very commonly employed in the literature, see, e.g., [25, 51].)
Return quantity function:R~=ϕ+ψr+ε~R;
Salvage value:S(m)=s+vm;
Modularity cost:C(m)=(1/2)θm2.Since under the consumer return scheme, we assume a full refund will be granted for each return, we have the rate of refundr=p. Thus
(1)D~1=α-βp+γ(p-l)+δm+ε~D,(β>γ);
(p.s.: we use subscripts 1 and 2 to represent the scenarios “with consumer return and full refund” and “no return,” respectively.)  R~=ϕ+ψ(p-l)+ε~R, l<p is the service charge.The profit function is hence expressed as follows:(2)P1=p(α-βp+γ(p-l)+δm+ε~D)+(S(m)-l)(ϕ+ψ(p-l)+ε~R)-12θm2.
We consider the scenario when the demand function and return quantity are correlated with a coefficient of correlation of ρ. Thus, the expected profit and the variance of profit are given as follows:
(3)E[P1]=p[α-βp+γ(p-l)+δm]+(s+vm-l)[ϕ+ψ(p-l)]-12θm2,V[P1]=p2σD2+(s+vm-l)2σR2+2ρσRσDp(s+vm-l).Define in the following the mean-variance (MV) objective function [28, 52–57] of the MC fashion brand, where k represents the degree of risk aversion for the MC fashion brand:
(4)U1=E[P1]-kV[P1].
In order to examine whether U1 is concave, we construct the Hessian matrix as follows:
(5)HU1=|∂2U1∂m2∂2U1∂m∂p∂2U1∂p∂m∂2U1∂p2|=|-θ-2kv2σR2δ+vψ-2kvρσDσRδ+vψ-2kvρσDσR-2β+2γ-2kσD2|=(2β-2γ+2kσD2)(θ+2kv2σR2)-(δ+vψ-2kvρσDσR)2=2θ(β-γ)+2kθσD2+4kρσDσR(vδ+v2ψ)+4kv2σR2(β-γ)+4k2v2σD2σR2(1-ρ2)-(δ+vψ)2>0.
By checking the Hessian matrix, we can see that U1 will be a concave function of p and m if and only if (5) holds.


## 3.2. No Consumer Return

For the scenario without consumer return, the model is much simpler because there will be no return and no salvage value of the returned product. As such, the demand function, the modularity cost, and the profit function are derived as follows.Demand function:D~2=α-βp+δm+ε~D.Modularity cost:C=(1/2)θm2.The profit function is:(6)P2=p(α-βp+δm+ε~D)-12θm2.
The expected profit and variance of profit are hence given as follows:
(7)E[P2]=p(α-βp+δm)-12θm2;V[P2]=p2σD2.
Similarly, we construct the MV objective function as follows:
(8)U2=E[P2]-kV[P2].
By checking the Hessian matrix, we can see that U2 is a concave function of p and m if and only if (9) is satisfied as follows:
(9)HU2=|∂2U2∂m2∂2U2∂m∂p∂2U2∂p∂m∂2U2∂p2|=|-θδδ-2β-2kσD2|=2θ(β+kσD2)-δ2>0.


## 4. Optimal Pricing and Modularity Level Decisions

For a notational purpose, we define the following:(10)A1=θ(α-lγ)-θψ(l-s)+v(δ+vψ)(ϕ-lψ),B1=2ρσDσR(lθ-sθ-v2ϕ+lv2ψ)+2σR2(v2α-lv2γ+lvδ-svδ),G1=(δ+vψ)(α-lγ)-ψ(δ+vψ)(l-s)+2v(ϕ-lψ)(β-γ),I1=2vσD2(ϕ-lψ)+4vσR2(β-γ)(l-s)-2ρσDσR(v(α-lγ)-(δ+2vψ)(l-s)),J1=4vσD2σR2(l-s)(1-ρ2),K1=2θ(β-γ)-(δ+vψ)2,L1=2θσD2+4vρσDσR(δ+vψ)+4v2σR2(β-γ),F1=4v2σD2σR2(1-ρ2).In order to have more analytical closed-form results, we assume in the rest of this paper that (5) and (9) hold which implies that  U1 and U2 are concave functions of p and m. Thus, we can easily derive Lemma 1.Lemma 1.
Under the scenario with full refund under consumer return, (a) the optimal pricing and modularity level decisions are given as follows:(11)p1*=A1+B1kK1+L1k+F1k2,m1*=G1+I1k+J1k2K1+L1k+F1k2.
(b) The optimal pricing and modularity level decisions are linearly related to one another:  p1*=ξm1*, where ξ=(A1+B1k)/(G1+I1k+J1k2).Lemma1 shows the closed-form expression of the optimal decisions for the MC fashion brand when it offers full refund consumer return. Notice from Lemma 1(b) that it is very interesting to reveal analytically that the optimal pricing decision and the optimal modularity level decisions are directly proportional to one another if ξ>0. Despite being complicated, we can further derive Theorem 2 by checking the respective first-order partial derivatives of the optimal decisions.Theorem 2.
Under the consumer return policy with full refund, (a) the optimal retail price is a strictly decreasing (increasing) function of the service chargel if -γθ-vδψ+2kvσR2(δ-vγ)+(2kρσDσR-ψ)(θ+v2ψ)<0 (sufficient conditions are δ-vγ<0 and 2kρσDσR-ψ<0). (b) The optimal retail price is a strictly increasing (increasing) function of the salvage value s if θ>2kvδσR2/(ψ-2kρσDσR).In parallel to the scenario with consumer return, we can derive Lemma3 by the same logic.Lemma 3.
Under the no consumer return scenario, (a) the optimal pricing and modularity level decisions are given as followsp2*=αθ/(2θ(β+kσD2)-δ2) and m2*=αδ/(2θ(β+kσD2)-δ2). (b) The relationship between the optimal pricing and the optimal modularity level is given by p2*=(θ/δ)m2*.Lemma3 shows the closed-form expression of the optimal decisions for the MC fashion brand when there is no consumer return. Similar to the case with consumer return, it is very interesting to note that the resulting optimal retail price is directly proportional to the optimal modularity level. In other words, Lemmas 1(b) and 3(b) both reveal that it is optimal for the MC fashion brand to charge a higher price if more options are available to consumers (as quantified by the level of modularity in the model) no matter whether the “consumer return with full refund” (p.s.: for the case with consumer return and full refund, this result holds whenever  ξ is positive)or “no return” strategy is imposed. This supplements the common belief that fashion brands can charge a premium by offering MC (compared to mass production) because of having more variety of options for consumers. With the result in Lemma 3, we can further derive Theorem 4.Theorem 4.
Under the no consumer return scenario, (a) If k increases, bothp2* and m2* will decrease. (b) If β or σD increases, both p2* and m2* will decrease. (c) If α increases, both p2* and m2* will increase. (d)  If θ increases, m2* will decrease. (e) If δ increases, p2* will decrease.From Theorem2, we can see that, for the scenario when the MC fashion brand offers consumer return, the optimal decisions on retail price and modularity level depend highly on the salvage value and the return service charge. From Theorem 4, we can observe that for the scenario when the MC fashion brand does not offer consumer return, the optimal decisions on retail price and modularity level depend highly on different market parameters such as the primary demand, the price-demand sensitivity coefficient, the modularity-demand sensitivity coefficient, and the modularity cost sensitivity coefficient; the corresponding relationships are shown in closed-form as summarized by Theorem 4.After we have analyzed the separate scenarios, we now examine a critical research question of this paper on whether the MC fashion brand should offer consumer return with full refund or no return. To address this question, we employ the analytical results above and defineΔU≡U1(p1*,m1*)-U2(p2*,m2*). Obviously, whenever ΔU is strictly positive, we have U1(p1*,m1*)>U2(p2*,m2*) which means that it is better to implement the consumer return policy with full refund, compared to the strategy of offering no return. As such, ΔU represents the amount of benefit that can be brought by employing consumer return with full refund (as compared to the no return no refund scenario) in MV domain.


## 5. Numerical Sensitivity Analysis

Owing to the complexity of the analytical expression ofΔU, we have to employ extensive numerical analysis to reveal under what conditions the strategy of “implementing consumer return with full refund” will outperform the strategy of “no return and no refund.” In determining the set of numerical values in conducting the analysis, we make reference to the data set as employed in [28, 32]. To be specific, we set the parameters as in Table 1.Table 1
Parameter
α
β
γ
θ
δ
ϕ
φ
v
s
l
k
ρ
σ
D
σ
R
Value
2000
30
2
30
10
20
8
0.2
1
0.8
0.02
0.3
100
20We focus on analyzing how the degree of risk aversionk, the return uncertainty σR, and the demand uncertainty σD affect the value of ΔU (which in turn determines the optimal strategy) because they are the most critical parameters related to the consumer return policy under MC. We also explore the problem with “positive, zero, and negative ρ” because a different sign of  ρ can significantly affect the finding. The results are shown in Tables 2 to 4 and also plotted in Figures 1 to 3, respectively. Notice that the numerical range of each parameter is set with reference to the conditions with which the mean-variance objective function remains concave and the optimal pricing and modularity level decisions are all feasible.Table 2
(a) Sensitivity analysis on the effect brought byk  (with ρ=-1). (b) Sensitivity analysis on the effect brought by k (with ρ=-0.3). (c) Sensitivity analysis on the effect brought by k (with ρ=0). (d) Sensitivity analysis on the effect brought by k (with ρ=0.3). (e) Sensitivity analysis on the effect brought by k (with ρ=1).
(a)
k
Δ
U
(
ρ
=
-
1
)
Optimal strategy
0
3573.464
Consumer return with full refund
0.0025
1346.646
Consumer return with full refund
0.005
828.6819
Consumer return with full refund
0.0075
618.2397
Consumer return with full refund
0.01
508.4378
Consumer return with full refund
0.0125
442.3226
Consumer return with full refund
0.015
398.6414
Consumer return with full refund
0.0175
367.8473
Consumer return with full refund
0.02
345.0745
Consumer return with full refund
0.0225
327.6048
Consumer return with full refund
0.025
313.8096
Consumer return with full refund
0.0275
302.6579
Consumer return with full refund
0.03
293.4675
Consumer return with full refund
0.0325
285.7699
Consumer return with full refund
0.035
279.2335
Consumer return with full refund
0.0375
273.617
Consumer return with full refund
0.04
268.7412
Consumer return with full refund
0.0425
264.4703
Consumer return with full refund
0.045
260.6994
Consumer return with full refund
0.0475
257.3464
Consumer return with full refund
0.05
254.3461
Consumer return with full refund
(b)
k
Δ
U
(
ρ
=
-
0.3
)
Optimal strategy
0
3573.464
Consumer return with full refund
0.0025
1077.694
Consumer return with full refund
0.005
550.2467
Consumer return with full refund
0.0075
351.1763
Consumer return with full refund
0.01
253.4773
Consumer return with full refund
0.0125
197.6467
Consumer return with full refund
0.015
162.3939
Consumer return with full refund
0.0175
138.5062
Consumer return with full refund
0.02
121.4449
Consumer return with full refund
0.0225
108.7525
Consumer return with full refund
0.025
98.99825
Consumer return with full refund
0.0275
91.30056
Consumer return with full refund
0.03
85.09028
Consumer return with full refund
0.0325
79.98568
Consumer return with full refund
0.035
75.72236
Consumer return with full refund
0.0375
72.11208
Consumer return with full refund
0.04
69.01759
Consumer return with full refund
0.0425
66.33674
Consumer return with full refund
0.045
63.99209
Consumer return with full refund
0.0475
61.924
Consumer return with full refund
0.05
60.08586
Consumer return with full refund
(c)
k
Δ
U
(
ρ
=
0
)
Optimal strategy
0
3573.464
Consumer return with full refund
0.0025
972.1282
Consumer return with full refund
0.005
447.7589
Consumer return with full refund
0.0075
257.8076
Consumer return with full refund
0.01
168.0751
Consumer return with full refund
0.0125
118.62
Consumer return with full refund
0.015
88.45607
Consumer return with full refund
0.0175
68.6862
Consumer return with full refund
0.02
55.01266
Consumer return with full refund
0.0225
45.15184
Consumer return with full refund
0.025
37.79863
Consumer return with full refund
0.0275
32.16292
Consumer return with full refund
0.03
27.74351
Consumer return with full refund
0.0325
24.2099
Consumer return with full refund
0.035
21.33693
Consumer return with full refund
0.0375
18.96689
Consumer return with full refund
0.04
16.98659
Consumer return with full refund
0.0425
15.31309
Consumer return with full refund
0.045
13.88447
Consumer return with full refund
0.0475
12.65375
Consumer return with full refund
0.05
11.58473
Consumer return with full refund
(d)
k
Δ
U
(
ρ
=
0.3
)
Optimal strategy
0
3573.464
Consumer return with full refund
0.0025
872.1531
Consumer return with full refund
0.005
354.8006
Consumer return with full refund
0.0075
176.2229
Consumer return with full refund
0.01
95.8788
Consumer return with full refund
0.0125
53.7666
Consumer return with full refund
0.015
29.38607
Consumer return with full refund
0.0175
14.25304
Consumer return with full refund
0.02
4.366616
Consumer return with full refund
0.0225
−2.34816
No return
0.025
−7.0484
No return
0.0275
−10.4174
No return
0.03
−12.8778
No return
0.0325
−14.7011
No return
0.035
−16.0677
No return
0.0375
−17.1003
No return
0.04
−17.8848
No return
0.0425
−18.4824
No return
0.045
−18.9374
No return
0.0475
−19.2824
No return
0.05
−19.542
No return
(e)
k
Δ
U
(
ρ
=
1
)
Optimal strategy
0
3573.464
Consumer return with full refund
0.0005
2374.058
Consumer return with full refund
0.001
1653.531
Consumer return with full refund
0.0015
1190.932
Consumer return with full refund
0.002
878.7702
Consumer return with full refund
0.0025
659.8301
Consumer return with full refund
0.003
501.4762
Consumer return with full refund
0.0035
384.0431
Consumer return with full refund
0.004
295.1391
Consumer return with full refund
0.0045
226.6607
Consumer return with full refund
0.005
173.1403
Consumer return with full refund
0.0055
130.7888
Consumer return with full refund
0.006
96.91849
Consumer return with full refund
0.0065
69.58424
Consumer return with full refund
0.007
47.35295
Consumer return with full refund
0.0075
29.15195
Consumer return with full refund
0.008
14.16698
Consumer return with full refund
0.0085
1.772005
Consumer return with full refund
0.009
−8.51977
No return
0.0095
−17.0909
No return
0.01
−24.2446
No return
0.0105
−30.2239
No return
0.011
−35.2243
No return
0.0115
−39.4048
No return
0.012
−42.8955
No return
0.0125
−45.8034
No return
0.013
−48.217
No return
0.0135
−50.2098
No return
0.014
−51.8436
No return
0.0145
−53.17
No return
0.015
−54.2326
No return(a) Sensitivity analysis on the effect brought byk (with ρ=-1). (b) Sensitivity analysis on the effect brought by k (with ρ=-0.3). (c) Sensitivity analysis on the effect brought by k (with ρ=0). (d)  Sensitivity analysis on the effect brought by k (with ρ=0.3). (e) Sensitivity analysis on the effect brought by k (with ρ=1).
(a)
(b)
(c)
(d)
(e)From Tables2(a) to 2(e), it is interesting to note that the no return scenario would outperform the consumer return scenario with full refund only when ρ and k are both big enough. In other words, when ρ≤0, consumer return with full refund is more preferred to the MC fashion brand irrespective of the degree of risk aversion. Only under the case when ρ≥0.3, for a sufficiently risk averse MC fashion brand (i.e., with a sufficiently big k), having no return is better than the case with consumer return with full refund. This result is partially intuitive as it indicates that a sufficiently risk averse MC fashion brand will prefer no return to having consumer return because the corresponding risk from consumer return is too high to bear. However, our result also indicates the important role played by the “very strong mediating factor” ρ which can affect the result very significantly. In addition, as observed from Figures 1(a) to 1(e), for all cases, when k increases, ΔU decreases which implies that the amount of benefit that can be brought by employing consumer return with full refund (as compared to the no return no refund scenario) in MV domain is decreasing with the MC fashion brand’s degree of risk aversion.From Tables3(a) to 3(e), we can observe that when ρ is small (i.e., ρ≤0.3), the consumer return scenario with full refund is the preferred choice for all values of σR under study. When ρ=1, (i) we observe that the consumer return scenario with full refund is still the preferred choice when  σR is small. (ii) If σR is sufficiently high, then the no return scenario outperforms the full refund scenario. Again, similar to our prior findings, ρ is a very important “mediating factor” which can substantially affect the optimal choice of whether to adopt no return or return with full refund when the return uncertainty varies. From Figures 2(a) to 2(e), we can observe that, (i) for ρ<0, when σR increases, ΔU increases which means that the respective amount of benefit that can be brought by employing consumer return with full refund (as compared to the no return no refund scenario) in MV domain is increasing with the return uncertainty. (ii) On the contrary, for ρ≥0, when σR increases, ΔU decreases which indicates that the effect as brought by increasing σR on  ΔU is just reverted when ρ≥0 (compared to the case when ρ<0).Table 3
(a) Sensitivity analysis on the effect brought byσR (with ρ=-1). (b) Sensitivity analysis on the effect brought by σR (with ρ=-0.3). (c) Sensitivity analysis on the effect brought by σR (with ρ=0). (d) Sensitivity analysis on the effect brought by σR (with ρ=0.3). (e) Sensitivity analysis on the effect brought by σR (with ρ=1).
(a)
σ
R
Δ
U
(
ρ
=
-
1
)
Optimal strategy
0
57.47614
Consumer return with full refund
5
112.6601
Consumer return with full refund
10
178.8924
Consumer return with full refund
15
256.321
Consumer return with full refund
20
345.0745
Consumer return with full refund
25
445.2616
Consumer return with full refund
30
556.9698
Consumer return with full refund
35
680.2648
Consumer return with full refund
40
815.1897
Consumer return with full refund
45
961.7641
Consumer return with full refund
50
1119.983
Consumer return with full refund
55
1289.818
Consumer return with full refund
60
1471.215
Consumer return with full refund
65
1664.095
Consumer return with full refund
70
1868.354
Consumer return with full refund
75
2083.861
Consumer return with full refund
80
2310.462
Consumer return with full refund
85
2547.977
Consumer return with full refund
90
2796.201
Consumer return with full refund
95
3054.907
Consumer return with full refund
100
3323.843
Consumer return with full refund
(b)
σ
R
Δ
U
(
ρ
=
-
0.3
)
Optimal strategy
0
57.47614
Consumer return with full refund
5
72.71861
Consumer return with full refund
10
88.54183
Consumer return with full refund
15
104.8257
Consumer return with full refund
20
121.4449
Consumer return with full refund
25
138.2716
Consumer return with full refund
30
155.1791
Consumer return with full refund
35
172.0444
Consumer return with full refund
40
188.7511
Consumer return with full refund
45
205.1917
Consumer return with full refund
50
221.269
Consumer return with full refund
55
236.8976
Consumer return with full refund
60
252.0048
Consumer return with full refund
65
266.5306
Consumer return with full refund
70
280.4274
Consumer return with full refund
75
293.6601
Consumer return with full refund
80
306.2044
Consumer return with full refund
85
318.0468
Consumer return with full refund
90
329.1825
Consumer return with full refund
95
339.615
Consumer return with full refund
100
349.3545
Consumer return with full refund
(c)
σ
R
Δ
U
(
ρ
=
0
)
Optimal strategy
0
57.47614
Consumer return with full refund
5
57.31906
Consumer return with full refund
10
56.85037
Consumer return with full refund
15
56.07753
Consumer return with full refund
20
55.01266
Consumer return with full refund
25
53.67209
Consumer return with full refund
30
52.0757
Consumer return with full refund
35
50.24625
Consumer return with full refund
40
48.20862
Consumer return with full refund
45
45.98904
Consumer return with full refund
50
43.61431
Consumer return with full refund
55
41.11118
Consumer return with full refund
60
38.5057
Consumer return with full refund
65
35.82278
Consumer return with full refund
70
33.08575
Consumer return with full refund
75
30.31611
Consumer return with full refund
80
27.53335
Consumer return with full refund
85
24.75483
Consumer return with full refund
90
21.99579
Consumer return with full refund
95
19.26937
Consumer return with full refund
100
16.58673
Consumer return with full refund
(d)
σ
R
Δ
U
(
ρ
=
0.3
)
Optimal strategy
0
57.47614
Consumer return with full refund
1
54.50733
Consumer return with full refund
2
51.56711
Consumer return with full refund
3
48.65629
Consumer return with full refund
4
45.77567
Consumer return with full refund
5
42.92605
Consumer return with full refund
6
40.10819
Consumer return with full refund
7
37.32283
Consumer return with full refund
8
34.57069
Consumer return with full refund
9
31.85248
Consumer return with full refund
10
29.16887
Consumer return with full refund
11
26.52051
Consumer return with full refund
12
23.90802
Consumer return with full refund
13
21.33201
Consumer return with full refund
14
18.79306
Consumer return with full refund
15
16.29172
Consumer return with full refund
(e)
σ
R
Δ
U
(
ρ
=
1
)
Optimal strategy
0
57.47614
Consumer return with full refund
1
47.75089
Consumer return with full refund
2
38.4596
Consumer return with full refund
3
29.60083
Consumer return with full refund
4
21.17313
Consumer return with full refund
5
13.175
Consumer return with full refund
6
5.604925
Consumer return with full refund
7
−1.53862
No return
8
−8.25721
No return
9
−14.5524
No return
10
−20.4259
No return
11
−25.8792
No return
12
−30.914
No return
13
−35.532
No return
14
−39.735
No return
15
−43.5246
No returnTable 4
(a) Sensitivity analysis on the effect brought byσD (with ρ=-1). (b) Sensitivity analysis on the effect brought by σD (with ρ=-0.3). (c) Sensitivity analysis on the effect brought by σD (with ρ=0). (d) Sensitivity analysis on the effect brought by σD (with ρ=0.3). (e) Sensitivity analysis on the effect brought by σD (with ρ=1).
(a)
σ
D
Δ
U
(
ρ
=
-
1
)
Optimal strategy
20
3449.186
Consumer return with full refund
30
2594.183
Consumer return with full refund
40
1844.719
Consumer return with full refund
50
1307.854
Consumer return with full refund
60
946.619
Consumer return with full refund
70
705.1343
Consumer return with full refund
80
541.0528
Consumer return with full refund
90
426.7885
Consumer return with full refund
100
345.0745
Consumer return with full refund
110
285.1112
Consumer return with full refund
120
240.0424
Consumer return with full refund
130
205.4228
Consumer return with full refund
140
178.3037
Consumer return with full refund
150
156.6838
Consumer return with full refund
160
139.1748
Consumer return with full refund
170
124.7939
Consumer return with full refund
180
112.8319
Consumer return with full refund
190
102.7681
Consumer return with full refund
200
94.2139
Consumer return with full refund
(b)
σ
D
Δ
U
(
ρ
=
-
0.3
)
Optimal strategy
20
2450.895
Consumer return with full refund
30
1608.23
Consumer return with full refund
40
1016.281
Consumer return with full refund
50
650.045
Consumer return with full refund
60
430.0031
Consumer return with full refund
70
296.1204
Consumer return with full refund
80
212.2569
Consumer return with full refund
90
157.9012
Consumer return with full refund
100
121.4449
Consumer return with full refund
110
96.19273
Consumer return with full refund
120
78.1769
Consumer return with full refund
130
64.97559
Consumer return with full refund
140
55.06682
Consumer return with full refund
150
47.46713
Consumer return with full refund
160
41.52439
Consumer return with full refund
170
36.7957
Consumer return with full refund
180
32.9736
Consumer return with full refund
190
29.84028
Consumer return with full refund
200
27.23856
Consumer return with full refund
(c)
σ
D
Δ
U
(
ρ
=
0
)
Optimal strategy
20
2084.977
Consumer return with full refund
30
1266.59
Consumer return with full refund
40
741.2984
Consumer return with full refund
50
438.7885
Consumer return with full refund
60
268.2584
Consumer return with full refund
70
170.537
Consumer return with full refund
80
112.777
Consumer return with full refund
90
77.41918
Consumer return with full refund
100
55.01266
Consumer return with full refund
110
40.34647
Consumer return with full refund
120
30.4589
Consumer return with full refund
130
23.61273
Consumer return with full refund
140
18.75729
Consumer return with full refund
150
15.23858
Consumer return with full refund
160
12.63853
Consumer return with full refund
170
10.68328
Consumer return with full refund
180
9.189367
Consumer return with full refund
190
8.031324
Consumer return with full refund
200
7.121739
Consumer return with full refund
(d)
σ
D
Δ
U
(
ρ
=
0.3
)
Optimal strategy
20
1752.936
Consumer return with full refund
30
968.2728
Consumer return with full refund
40
508.8139
Consumer return with full refund
50
264.978
Consumer return with full refund
60
138.1811
Consumer return with full refund
70
71.41194
Consumer return with full refund
80
35.41802
Consumer return with full refund
90
15.54633
Consumer return with full refund
100
4.366616
Consumer return with full refund
110
−1.98807
No return
120
−5.59179
No return
130
−7.59094
No return
140
−8.63718
No return
150
−9.1104
No return
160
−9.237
No return
170
−9.15444
No return
180
−8.9475
No return
190
−8.66903
No return
200
−8.35226
No return
(e)
σ
D
Δ
U
(
ρ
=
1
)
Optimal strategy
20
1099.709
Consumer return with full refund
25
704.3688
Consumer return with full refund
30
425.6345
Consumer return with full refund
35
238.2991
Consumer return with full refund
40
116.6536
Consumer return with full refund
45
39.76602
Consumer return with full refund
50
−7.6261
No return
55
−35.9901
No return
60
−52.2552
No return
65
−60.9155
No return
70
−64.8518
No return(a) Sensitivity analysis on the effect brought byσR (with ρ=-1). (b) Sensitivity analysis on the effect brought by σR (with ρ=-0.3). (c) Sensitivity analysis on the effect brought by σR (with ρ=0). (d) Sensitivity analysis on the effect brought by σR (with ρ=0.3). (e) Sensitivity analysis on the effect brought by σR (with ρ=1).
(a)
(b)
(c)
(d)
(e)(a) Sensitivity analysis on the effect brought byσD (with ρ=-1). (b) Sensitivity analysis on the effect brought by σD (with ρ=-0.3). (c) Sensitivity analysis on the effect brought by σD (with ρ=0). (d) Sensitivity analysis on the effect brought by σD (with ρ=0.3). (e) Sensitivity analysis on the effect brought by σD (with ρ=1).
(a)
(b)
(c)
(d)
(e)From Tables4(a) to 4(e), it is interesting to note that the impact on the optimal choice of return policy (no return versus return with full refund) as brought by the demand uncertainty is rather similar to the return uncertainty. The consumer return with full refund scenario is the preferred choice for all demand uncertainty (σD) values under study when ρ≤0. When ρ≥0.3, we find that, (i) the consumer return scenario with full refund is still the preferred choice when  σD is small. (ii) If σD is sufficiently big, the no return scenario outperforms the full refund scenario. As observed from Figures 3(a) to 3(e), we notice that different from the case with σR, ΔU is always a decreasing function of σD irrespective of whether ρ is positive or negative. In other words, the amount of benefit that can be brought by employing consumer return with full refund (as compared to the no return no refund scenario) in MV domain is decreasing with the demand uncertainty for all ρ.


## 6. Conclusion

In this paper, we have explored the stochastic fashion MC service programme with the consideration of consumer demand uncertainty and risk aversion of the company. We have obtained a number of important findings and insights. For example, for the scenario with full refund and return, we have derived the closed-form analytical optimal decisions in retail pricing and modularity level. We have also revealed the analytical conditions under which the optimal retail price and the optimal modularity level would vary monotonically with respect to the return service charge and the salvage value.  For the scenario when there is no refund and no return, we have obtained the neat closed-form expressions of the optimal pricing and modularity decisions. Our analysis has also indicated that the optimal retail price and the optimal modularity level are decreasing in the MC fashion brand’s degree of risk aversion, the demand uncertainty, and the price-sensitivity coefficient. In addition, for both scenarios, we have interestingly found that the optimal retail pricing decision is linearly proportional to the optimal modularity decision. Finally, we have discovered that whether the risk averse MC fashion brand would prefer offering consumer return with full refund to no return depends critically on the demand-return correlation parameterρ.For future research, we believe that it can be done in a number of ways. First, it will be interesting to examine the MC schemes with multiple players under competition in a game-theoretic setting. Second, it will be promising to explore empirically more operational and marketing factors which may affect the demand of MC fashion products. After that, new analytical models based on the new empirical results can be constructed for further in-depth theoretical analysis.



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*Source: 561846-2013-04-11.xml*