# A Hierarchical Innovation-Related Crowdsourcing Decision in Fast Fashion Industry

Authors: Jizi Li; Yueqing Bian; Chunling Liu; Doudou Wu
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

Due to scarcity of designers in fast fashion industry and proliferation of the Internet, small- and medium-sized garment makers have gradually turned to external designers to enhance their innovation efficiency via crowdsourcing initiative. However, few have investigated the issue of fast fashion customized-design matching decision in the crowdsourcing context. Different from previous works, we split crowdsourcing matching decision process into three hierarchical submodels in terms of three key factors, namely, surplus, due date, and goodwill. From a dynamic perspective, we first develop a two-sided matching model where garment makers and online designers select one another by maximizing their total surpluses with an aim to reach robust final pairs and derive the corresponding conditions under which the optimal pairs can be obtained. Then, the extensions of the matching model are conducted by incorporating the critical factors of due date and garment makers’ goodwill, respectively. Followed by that, an improved Gale–Shapley algorithm is devised to solve the crowdsourcing matching problems. The results illustrate when garment makers exceed online designers in number, crowdsourcing design tasks without due-date constraint are more attractive for designers’ participation than those with due-date constraint, and garment makers intend to share the incremental surpluses with designers to maximize the total surpluses. By contrast, when online designers surpass garment makers in number, designers prefer due-date design tasks to those without it. In addition, regardless of whether under the irregular or regular case, the model with goodwill concern always outperforms the two others. Moreover, celebrated garment makers are invited to post design tasks, thus enabling to entice more designers’ engagement in crowdsourcing activities, which in turn facilitating to transfer myopic designers to strategic ones. Finally, sensitivity analysis further verifies the models are stable and robust.


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## Body

## 1. Introduction

The fast fashion industry is a highly competitive business area, where new designs shift quickly and product life cycle is dramatically short as well [1]. In a fast fashion market, customers tend to purchase only what matches their anticipation and preferences; if garment makers do not offer the option of tailored design for them, which in turn makes consumers not to patronize and even give up their purchase intention accordingly [2]. It means that firms’ design capabilities and innovation play a key role in a fierce market competition. Therefore, fashion designers are given a top priority in fast fashion industry.As observed in practice, successful fast fashion firms such as H&M, Zara, Uniqlo, and among others generally possess strong innovation and design capabilities in response to market uncertainties. Conversely, the fashion firms facing woes primarily resulted from neglecting the importance of designers’ roles. As a result, competitive garment makers invariably keep or recruit more apparel designers for sharpening their edges [3]. More importantly, a majority of fast fashion makers belong to small- and medium-sized enterprises (SMEs) and usually lack sufficient resources to meet customers’ tailored needs, and only leveraging interfirm’s professional designers may lead to the decrease in customer satisfaction. With the advent of Internet, garment makers have an access to other innovation/design alternatives by adopting crowdsourcing paradigm, which mitigates the shortage of offline design capacities [4, 5]. With the support from crowdsourcing, SMEs could integrate external innovations into internal ones, thus compensating for self-innovation capacity deficits and enhancing their design flexibilities as well.Crowdsourcing is a term first coined by Howe in theWired Magazine in 2006. It refers to the innovation of mass participation based on the sweep ubiquity of Internet [6], wherein a firm or an organization distributes assignments once conducted by internal staff to undefined network of people. Crowdsourcing, theoretically, can be key catalysts in attracting a group of creative and enthusiastic people capable of providing better ideas than those attained from conventional patterns [7]. Realizing its potential business values and promising prospect, either giants or startups have adopted crowdsourcing to enhance innovation and design flexibilities and competencies [8]. For example, in two Chinese popular online platforms Zhubaijie and Taobao, crowdsourcing services are offered to a plethora of garment makers and designers. For the garment makers, they post customized-design tasks on crowdsourcing platforms to solicit novel solutions from online dress designers and then chose those with superior qualities upon specific criteria, and only the winners whose solutions are selected are rewarded by garment makers. The crowdsourcing designers are heterogeneous in innovation and design style, and they usually first browse crowdsourcing websites, searching for their favorable design tasks and then submit their solutions according to garment makers’ requirements such as completion duration, quality, and compensation for design tasks in a competitive format. A report from AliResearch revealed that the amounts of deals in the dress-design crowdsourcing sector reached to 7.28 million CNY in 2019, an increase of about 45% compared to the year of 2018, which obviously showing the upward trend.Although the application of crowdsourcing has numerous benefits, there still exist three problems which perplex decision-makers in practice: (i) how to appropriately and efficiently match between garment makers (crowdsourcers) and crowdsourcing designers (crowdsourcees) in process of executing crowdsourcing activities? (ii) which condition can the optimal and robust matching outcomes be obtained? (iii) for a crowdsourcing platform, what is the differences in crowdsourcing design matching in terms of three critical factors, namely, surplus, due date, and goodwill concern? As we have known, most of such matching nowadays run through a primitive individual-selection method on crowdsourcing platforms, thus leading to a lower matching rate, but solving the above problems could facilitate the shape of target and automatic matching and recommend both garment makers (crowdsourcers) and crowdsourcing designers (crowdsourcees) in an efficient way.Motivated by such practices, this paper characterizes a two-sided matching model in a fast fashion market consisting of crowdsourcing designers and garment makers. Our work is similar to [9, 10] those addressing the matching problems; among them, Peng et al. [9] studied a vessel-cargo matching with price game mechanism and mimicked the price bid in the process of matching preferred objects. The literature [10] examines a supply-demand matching model where three settings are taken into account including oversupply, overdemand, and supply-demand equilibrium. Unlike their work addressing one matching model statically, our model breakdowns matching decision into three hierarchical submodels in terms of three key factors and dynamically analyzes and optimizes the robust paring with respect to garment makers and designers.Therefore, the contributions of this paper are threefold. (i) We split a designer-maker matching process into three hierarchical submodels from a dynamic perspective in the crowdsourcing context. These models capture the dynamic process of crowdsourcing decision-making, and the equilibrium state of the stable two-sided matching between designers and garment makers is analyzed. (ii) From a holistic view, we grouped crowdsourcing designers into two categories, namely, myopic and strategic ones, each of whose attributes is depicted in the proposed corresponding matching models to examine its impact on the matching outcomes. (iii) The third contribution is to integrate the practical factors of due date and garment makers’ goodwill into the two-sided matching problem, which is seldom addressed in prior research.The remainder of this paper is configured as follows.Section 2 briefly reviews the related literature. In Section 3, we present main notations used in this paper and problem description. A basic model characterizing a two-sided matching process between crowdsourcing designers and garment makers and the extensions with consideration of due date and garment makers’ goodwill are proposed in Sections 4–6. The algorithm procedure is devised in Section 7. Section 8 presents computational experiments. Finally, we offer concluding remarks along with future research directions in Section 9.


## 2. Literature Review

The work related to the current study is primarily in the following three facets, i.e., fast fashion, crowdsourcing innovation, and matching model.


### 2.1. Research on Fast Fashion Industry

Fast fashion industry is characterized by seasonal demand, mass variety, short shelf life, and diverse consumer preference [2]. In such a context, fast fashion firms will encounter more market variation, hence increasing design uncertainty and volatility. Therefore, fast fashion industry has attracted much attention from scholars. Nonetheless, it is noteworthy that prior literature on this topic mainly underlines fast fashion supply chain coordination [1, 11] and pricing decision [12, 13], as well as quick response issue [14, 15]. Although how to design and evaluate new products is crucial to garment makers, there is less research related to this issue except the literature [3, 4, 16, 17].Among them, Dou et al. [16] proposed an approach of fast fashion product collaborative customization to improve the design process, use design iteration, and configure knowledge to seek the optimal design. Hirscher et al. [17] discussed a new value creation in the fashion sector through social manufacturing, namely, do-it-yourself, do-it-together, and participatory design strategies, and analyzed the different values in social manufacturing. We differ from their work by employing a crowdsourcing design channel to match designers and garment makers.In addition, Dai et al. [3] considered a designer platform service, offering the crowdsourcing mechanism in the fashion market; the price and minimum production quantity decisions are made with consideration of the entrant designer’s objective and customer demand structures. Li et al. [4] proposed multiperiod order models of crowdsourcing supply chain system, where a garment maker chooses the best one from solutions submitted by online crowdsources and utilizes wholesale price, buyback, and profit-sharing scheme to design a mixed contract with an aim to coordinate the whole supply chain. Unlike the above studies, we investigate the crowdsourcing design problem by leveraging a hierarchical matching method.


### 2.2. Research on Crowdsourcing Innovation

Crowdsourcing refers to an online, distributed, problem-solving, and production model that uses the collective intelligence of networked communities for specific purposes [7]. Although several researchers question the role of crowdsourcing and highlight its negative effects, such as participants’ repeatedly submitted similar solutions, sloppiness, or cheating [18], crowdsourcing is effective in enhancing design capabilities and innovative competencies for sharpening firms’ competitive edges [19].Most studies have focused on how to facilitate crowdsourcing innovation activities using empirical models from two lenses, namely, crowdsourcer (task poster) and crowdsourcee (task solver). From the crowdsourcer lens, researchers have investigated the task-specific factors regarding crowdsourcing development and found the significant impacts of task type, task specificity, monetary reward, and duration on crowdsourcing performance [8]. Specifically, task complexity has significant negative impact on the number of crowdsourcing solutions submitted, while duration for the tasks and monetary reward have significant positive effect. In addition, brand strength of the crowdsourcer also has a significant effect on number of submissions [7].From the crowdsourcee lens, crowdsourcee individuals’ extrinsic, intrinsic, and social motives are effective predictors of individuals’ participation and performance in the crowdsourcing context [20]. Prior studies have also explored that competitors’ experience, intrinsic motivation, and extrinsic motivation with immediate payoffs have a significant positive impact on the quantity of crowdsourcing solutions. While earning reputations, learning, improving skills, getting employed as a freelancer, and self-marketing have an effect on average quality of submissions [21]. Additionally, the significant effect of number of competitors and superstars are influencing factors on crowdsourcing performance [22].Overall, the aforementioned papers study the influencing factors of crowdsourcing by utilizing the empirical study. But in this paper, we use an analytical research method to examine the decision process of crowdsourcing design for matching crowdsourcers and crowdsourcees.


### 2.3. Research on Matching Model

Matching models are kind of analytical tool which has been applied in many domains: hospital-resident matching [23], stable marriage problem [24], and vessel-cargo matching [9], as well as supply-demand matching problem [10, 25].But the application of matching model in crowdsourcing design is rare, especially in the fast fashion sector. Some authors used matching models to capture two-sided matching problems in the loan market [26]. Some other authors utilized matching models to characterize supply and demand matching management. For instance, Boysen et al. [27] focused on deterministic matching problems and provided a classification scheme for the resulting optimization problems occurring in different areas of the sharing economy [27], whereas Yin and Li [28] proposed a decision-making method for matching management of supply and demand based on fuzzy sets and considered interactions among aspiration criteria, and Peng et al. [23] proposed a model of stable vessel-cargo matching with price game mechanism in the dry bulk shipping market. Similar to the latter paper, in this paper we employed the matching model to depict the dynamic hierarchy matching problem for fast fashion customized design in a crowdsourcing environment.All in all, different from previous work addressing one matching model statically or using the empirical method to examine crowdsourcing innovation, our model splits design crowdsourcing matching decision into three hierarchical submodels in terms of three key factors and dynamically analyzed and optimized the robust paring with respect to garment makers and designers.


## 2.1. Research on Fast Fashion Industry

Fast fashion industry is characterized by seasonal demand, mass variety, short shelf life, and diverse consumer preference [2]. In such a context, fast fashion firms will encounter more market variation, hence increasing design uncertainty and volatility. Therefore, fast fashion industry has attracted much attention from scholars. Nonetheless, it is noteworthy that prior literature on this topic mainly underlines fast fashion supply chain coordination [1, 11] and pricing decision [12, 13], as well as quick response issue [14, 15]. Although how to design and evaluate new products is crucial to garment makers, there is less research related to this issue except the literature [3, 4, 16, 17].Among them, Dou et al. [16] proposed an approach of fast fashion product collaborative customization to improve the design process, use design iteration, and configure knowledge to seek the optimal design. Hirscher et al. [17] discussed a new value creation in the fashion sector through social manufacturing, namely, do-it-yourself, do-it-together, and participatory design strategies, and analyzed the different values in social manufacturing. We differ from their work by employing a crowdsourcing design channel to match designers and garment makers.In addition, Dai et al. [3] considered a designer platform service, offering the crowdsourcing mechanism in the fashion market; the price and minimum production quantity decisions are made with consideration of the entrant designer’s objective and customer demand structures. Li et al. [4] proposed multiperiod order models of crowdsourcing supply chain system, where a garment maker chooses the best one from solutions submitted by online crowdsources and utilizes wholesale price, buyback, and profit-sharing scheme to design a mixed contract with an aim to coordinate the whole supply chain. Unlike the above studies, we investigate the crowdsourcing design problem by leveraging a hierarchical matching method.


## 2.2. Research on Crowdsourcing Innovation

Crowdsourcing refers to an online, distributed, problem-solving, and production model that uses the collective intelligence of networked communities for specific purposes [7]. Although several researchers question the role of crowdsourcing and highlight its negative effects, such as participants’ repeatedly submitted similar solutions, sloppiness, or cheating [18], crowdsourcing is effective in enhancing design capabilities and innovative competencies for sharpening firms’ competitive edges [19].Most studies have focused on how to facilitate crowdsourcing innovation activities using empirical models from two lenses, namely, crowdsourcer (task poster) and crowdsourcee (task solver). From the crowdsourcer lens, researchers have investigated the task-specific factors regarding crowdsourcing development and found the significant impacts of task type, task specificity, monetary reward, and duration on crowdsourcing performance [8]. Specifically, task complexity has significant negative impact on the number of crowdsourcing solutions submitted, while duration for the tasks and monetary reward have significant positive effect. In addition, brand strength of the crowdsourcer also has a significant effect on number of submissions [7].From the crowdsourcee lens, crowdsourcee individuals’ extrinsic, intrinsic, and social motives are effective predictors of individuals’ participation and performance in the crowdsourcing context [20]. Prior studies have also explored that competitors’ experience, intrinsic motivation, and extrinsic motivation with immediate payoffs have a significant positive impact on the quantity of crowdsourcing solutions. While earning reputations, learning, improving skills, getting employed as a freelancer, and self-marketing have an effect on average quality of submissions [21]. Additionally, the significant effect of number of competitors and superstars are influencing factors on crowdsourcing performance [22].Overall, the aforementioned papers study the influencing factors of crowdsourcing by utilizing the empirical study. But in this paper, we use an analytical research method to examine the decision process of crowdsourcing design for matching crowdsourcers and crowdsourcees.


## 2.3. Research on Matching Model

Matching models are kind of analytical tool which has been applied in many domains: hospital-resident matching [23], stable marriage problem [24], and vessel-cargo matching [9], as well as supply-demand matching problem [10, 25].But the application of matching model in crowdsourcing design is rare, especially in the fast fashion sector. Some authors used matching models to capture two-sided matching problems in the loan market [26]. Some other authors utilized matching models to characterize supply and demand matching management. For instance, Boysen et al. [27] focused on deterministic matching problems and provided a classification scheme for the resulting optimization problems occurring in different areas of the sharing economy [27], whereas Yin and Li [28] proposed a decision-making method for matching management of supply and demand based on fuzzy sets and considered interactions among aspiration criteria, and Peng et al. [23] proposed a model of stable vessel-cargo matching with price game mechanism in the dry bulk shipping market. Similar to the latter paper, in this paper we employed the matching model to depict the dynamic hierarchy matching problem for fast fashion customized design in a crowdsourcing environment.All in all, different from previous work addressing one matching model statically or using the empirical method to examine crowdsourcing innovation, our model splits design crowdsourcing matching decision into three hierarchical submodels in terms of three key factors and dynamically analyzed and optimized the robust paring with respect to garment makers and designers.


## 3. Problem Description

There is a fast fashion crowdsourcing platform community consisting of a population of garment makers and designers with different requirements and competencies, respectively. Assume that both designers and garment makers are rational, and a designer only chooses and undertakes one design task, while one task is only completed by one crowdsourcing designer. In addition, regardless of whether a submitted solution is selected or not, the solution is not allowed to be repeatedly submitted again. In this setup, garment makers distribute their crowdsourcing tasks and announce the corresponding financial rewards on the platform. However, incumbent or entrant designers choose crowdsourcing tasks based on their competencies and design styles and then submit their design proposals. To this end, the matching between garment makers and designers is proceeded.In general, the designers and the garment makers select their matchers/partners considering two parts, namely, present utilities and future expected benefits. The former is the behavior of myopic designers who only concern about present utilities, which commonly occurs under two cases, i.e., crowdsourcing without and with consideration of due-date factor. In the case of the absence of due-date factor, the designers and the garment makers merely focus on cost, reward, and design style as well and believe that the time issue is not a problem because of their confidence in being capable of on-time completion of crowdsourcing tasks. By contrast, in the presence of due-date constraint, both sides not only paid attention to the abovementioned factors, but they also took due date into account. The latter part, i.e., future expected benefits, refers to the act of strategic designers who care about next-round gains, which is strongly associated with the garment makers’ current goodwill and reputation; for instance, given two different garment makers, the higher goodwill the garment maker has, the more expected benefits the designers will obtain in the next round. In this regard, we thus breakdown matching decision process into three levels, namely, basic two-sided matching without due-date constraint, two-sided matching with due-date constraint, and two-sided matching with due-date and goodwill concern. The dynamic hierarchy matching decision process is shown in Figure1.Figure 1
Criteria of hierarchical matching between garment makers and designers.As shown in Figure1, designers first filter out design tasks which they are unable to punctually complete or which are beyond their capabilities. Most importantly, preannounced reward by garment makers is key driver for crowdsourcing designers to determine the choice of matching with crowdsourcing tasks; if online apparel designers perceive the amount of task rewards below their expectations, they tend to be likely to select other alternatives with higher monetary ones, which in turn forces garment makers to raise their subsequent rewards for attracting more and better-qualified participants. This can serve as a proxy of pricing mechanism.Furthermore, when online designers are far-sighted for future gains, then garment maker’s goodwill is taken into account by crowdsourcees, especially for strategic designers, because the garment makers’ reputation and goodwill have a significant impact on the likelihood of designers’ success of being selected in the future. To this end, the anticipated benefits will be transformed into designers’ utilities. Since designers are inclined to choose the tasks with higher utility, each individual designer has a sequence list of preferences for design tasks.Following the same logit as designers, garment makers also first exclude the designers whose professional capabilities cannot serve their requirements well. Generally, the garment makers’ utilities are determined by the quality of designer’s solution and marginal cost. Likewise, with observing prior crowdsourcing participants, garment makers will determine the current amount of crowdsourcing rewards through tradeoff between transaction cost and payoff. In other words, the simpler the tasks or the more competitive crowdsourcing is, the less the rewards are offered by garment makers. As a result, each garment maker has the sequence list of preferences with respect to designers. To this end, the designers and garment makers dynamically make decision, and the optimal match between both could be finalized until mutual gratifications are obtained.


## 4. Two-Sided Matching Model

In this section, we consider the case where crowdsourcing designers are myopic and only concern about present utility, rather than focus on long-term gains, i.e., myopic crowdsourcees underline the current benefits and do not care about the next-round benefits. To match both sides, the main notations used in this paper are illustrated in Table1. The designer’s and the garment maker’s utilities are derived out as follows.Table 1
Main notations used in this paper.
Type
Notation
Definition
Sets
I
Set of garment makers or tasks,i∈I
J
Set of designers or design solutions,j∈J
Θ
j
Set of designer’s (j) completion tasks
μ
Set of stable parings,μ=i,jZij=1,∀i∈I,∀j∈J
Parameters
C
i
j
Cost of completion of taski by designer j
P
i
j
,
next
Expected payoff for the next task if designerj completes the present design task i
T
i
j
Duration of taski completed by designer j
c
i
j
Unit time cost of taski completed by designer j
C
T
i
j
Cost of completion of taskI within the time limits by designer j
P
N
−
L
Anticipated payoff for the case in which a normal-goodwill garment maker will offer a low reward in the near future
P
N
−
H
Anticipated payoff for the case in which a normal-goodwill garment maker will offer a high reward in the near future
P
B
−
L
Anticipated payoff for the case in which a better-goodwill garment maker will offer a low reward in the near future
P
B
−
H
Anticipated payoff for the case in which a better-goodwill garment maker will offer a high reward in the near future
B
i
j
Benefit of garment makeri from designer’s (j) solution
S
i
j
Payment to the crowdsourcing platform by garment makeri
α
,
β
Estimated parameters
t
j
E
Earliest start time of designerj
t
i
L
Latest submission time required by garment makeri
U
¯
i
j
M
Minimum utility obtained when garment makeri chooses designer j
U
¯
i
j
D
Minimum utility obtained when designerj accomplishes task i
Decision variables
Z
i
j
Z
i
j
=
1 if garment maker i and designer j match as a pairing, Zij=0, otherwise
R
i
j
,
present
Reward for design solutioni is selected by garment maker j
Auxiliary variables
U
i
j
D
Total utility of designerj taking on task i
U
i
j
M
Total utility of garment makeri selecting the designer’s (j) solution
V
C
D
i
,
j
Total surpluses of designerj if he matches with garment maker i
V
G
M
i
,
j
Total surpluses of garment makeri if he matches with designer j


### 4.1. Designer’ s Utility for Ranking Garment Makers

In the myopic designer setting, assume that each designerj, j∈J, (J is the set of designers or design solutions) only submits one design solution on the crowdsourcing platform and independently makes decision in choosing tasks i, i∈I, (I is the set of garment makers or tasks) by maximizing the utilities. Meanwhile, we deposit that the crowdsourcing platform does not charge any fees from myopic designers with an aim to encourage their participation. Considering that the designers j are short-sighted, it is reasonable if we assume each designer assesses his payoffs based on the current-round tasks i and the corresponding costs. Given a crowdsourcing task i, designer’s utility is determined by the difference between payoffs Rij,present and incurred costs Cij. Where the myopic designer j payoff Rij,present is the reward from the garment maker i, while the corresponding costs Cij include time cost or invest cost. Thus, designer’s (j) utility is expressed by(1)UijD=Rij,present−Cij,where Rij,present is the reward of designer j for completion of task I and Cij represents the cost of completion of task i by designer j.


### 4.2. Garment Maker’s Utility for Ranking Designers

As the same logical way as myopic designers, garment makers,i∈I, (I is the set of garment makers or tasks) are rational and individually determine their crowdsourcing tasks being assigned to the desirable designer j, j∈J, (J is the set of designers or design solutions) by using a utility-maximization method. Also, assume that each task only matches one solution. Given a crowdsourcing solution j, garment maker’s utility is evaluated with the gap between the benefit Bij from crowdsourcing solution j and the compensation-for-designer cost Rij,present.Additionally, consider that the crowdsourcing platform serves as an interface between online designers and garment makers, such that the garment makers are charged at a platform service feeSij, thus implying the garment maker’s (i) cost includes two parts Rij,present and Sij. Therefore, the garment maker’s utility is defined as follows:(2)UijM=Bij−Rij,present−Sij.The first term of the right side of equation (2) Bij indicates the benefit of the garment maker i from design solution j, the second term demonstrates the garment maker’s (i) payment (or reward) to designer j, and the third term Sij denotes the service fee charged by the crowdsourcing platform only if the garment maker i and designer j match successfully.


### 4.3. Matching Model

To obtain a robust garment maker-designer matching equilibrium, we consider a decision problem in which the garment makers and the designers act as a single entity to make decision with a goal of the overall surplus maximization. Conversely, if designers and garment makers separately make decision on their own interests rather than jointly as a whole. As a sequence, crowdsourcing design cannot be sustainable and even end up with platform’s operational inefficiency or disclosure. Parallelly, from the crowdsourcing platform perspective, the optimal match both sides from the whole systematic way is aligned with platform’s purpose. Hence, we formulate the model by maximizing the surplus of both garment makers and designers as follows:(3)max∑i∑jVCDi,j+VGMi,j×Zij,pairi,j,which is subject to(4)VGMi,j=UijM−U¯ijM≥0,∀i∈I,∀j∈J,(5)VCDi,j=UijD−U¯ijD≥0,∀i∈I,∀j∈J,Equation (3) aims to maximize the total surpluses of garment makers i and designers j. Equations (4) and (5) represent the surplus of garment makers i and designers j, respectively, and guarantee that the surpluses of both sides are nonnegative.(6)∑i∈j∪IZij=1,∀j∈J,(7)∑j∈i∪JZij=1,∀i∈I,(8)Zij∈0，1,∀i∈I,∀j∈J.Equation (6) guarantees that one designer j undertakes no more than one task. Equation (7) guarantees that one task is completed by only one designer j. Equation (8) denotes that Zij is a 0-1 variable, and if garment maker i and designer j successfully match as a pair, Zij=1; otherwise,Zij=0.It is worthy to note that the abovementioned analytical model could derive out the total surplus, yet the corresponding pairing is probably not stable due to that there exists double marginalization effect in some pairs (i, j). In such a context, we give the following equilibrium conditions under which garment makers and designers can match with stable final pairing:(9)VCDi∗,j∗≥VCDi,j∗+VGMi,j∗−VGMi,j,(10)VCDi,j≥VCDi∗,j+VGMi∗,j−VGMi∗,j∗,∀i,j,i∗,j∗∈μ,μ=i,jZij=1,∀i∈I,∀j∈J.Equations (9) and (10) ensure that the final pairings are stable. In detail, when garment i matches with designer j∗, garment i can obtain the surplus of VGMi,j∗−VGMi,j, more than current matching with designer j. Assuming garment maker i wants to change the current matching, garment maker i is willing to at most transfer VGMi,j∗−VGMi,j to crowdsourcing designer j∗ to change his rank. But right now, designer j∗ is assigned to garment maker i∗, and VCDi∗,j∗≥VCDi,j∗+VGMi,j∗−VGMi,j implies that there is no incentives for the designer j∗ to change the current matching. Following the same logic, garment maker i∗ has no motivation to change the present pairing.


## 4.1. Designer’ s Utility for Ranking Garment Makers

In the myopic designer setting, assume that each designerj, j∈J, (J is the set of designers or design solutions) only submits one design solution on the crowdsourcing platform and independently makes decision in choosing tasks i, i∈I, (I is the set of garment makers or tasks) by maximizing the utilities. Meanwhile, we deposit that the crowdsourcing platform does not charge any fees from myopic designers with an aim to encourage their participation. Considering that the designers j are short-sighted, it is reasonable if we assume each designer assesses his payoffs based on the current-round tasks i and the corresponding costs. Given a crowdsourcing task i, designer’s utility is determined by the difference between payoffs Rij,present and incurred costs Cij. Where the myopic designer j payoff Rij,present is the reward from the garment maker i, while the corresponding costs Cij include time cost or invest cost. Thus, designer’s (j) utility is expressed by(1)UijD=Rij,present−Cij,where Rij,present is the reward of designer j for completion of task I and Cij represents the cost of completion of task i by designer j.


## 4.2. Garment Maker’s Utility for Ranking Designers

As the same logical way as myopic designers, garment makers,i∈I, (I is the set of garment makers or tasks) are rational and individually determine their crowdsourcing tasks being assigned to the desirable designer j, j∈J, (J is the set of designers or design solutions) by using a utility-maximization method. Also, assume that each task only matches one solution. Given a crowdsourcing solution j, garment maker’s utility is evaluated with the gap between the benefit Bij from crowdsourcing solution j and the compensation-for-designer cost Rij,present.Additionally, consider that the crowdsourcing platform serves as an interface between online designers and garment makers, such that the garment makers are charged at a platform service feeSij, thus implying the garment maker’s (i) cost includes two parts Rij,present and Sij. Therefore, the garment maker’s utility is defined as follows:(2)UijM=Bij−Rij,present−Sij.The first term of the right side of equation (2) Bij indicates the benefit of the garment maker i from design solution j, the second term demonstrates the garment maker’s (i) payment (or reward) to designer j, and the third term Sij denotes the service fee charged by the crowdsourcing platform only if the garment maker i and designer j match successfully.


## 4.3. Matching Model

To obtain a robust garment maker-designer matching equilibrium, we consider a decision problem in which the garment makers and the designers act as a single entity to make decision with a goal of the overall surplus maximization. Conversely, if designers and garment makers separately make decision on their own interests rather than jointly as a whole. As a sequence, crowdsourcing design cannot be sustainable and even end up with platform’s operational inefficiency or disclosure. Parallelly, from the crowdsourcing platform perspective, the optimal match both sides from the whole systematic way is aligned with platform’s purpose. Hence, we formulate the model by maximizing the surplus of both garment makers and designers as follows:(3)max∑i∑jVCDi,j+VGMi,j×Zij,pairi,j,which is subject to(4)VGMi,j=UijM−U¯ijM≥0,∀i∈I,∀j∈J,(5)VCDi,j=UijD−U¯ijD≥0,∀i∈I,∀j∈J,Equation (3) aims to maximize the total surpluses of garment makers i and designers j. Equations (4) and (5) represent the surplus of garment makers i and designers j, respectively, and guarantee that the surpluses of both sides are nonnegative.(6)∑i∈j∪IZij=1,∀j∈J,(7)∑j∈i∪JZij=1,∀i∈I,(8)Zij∈0，1,∀i∈I,∀j∈J.Equation (6) guarantees that one designer j undertakes no more than one task. Equation (7) guarantees that one task is completed by only one designer j. Equation (8) denotes that Zij is a 0-1 variable, and if garment maker i and designer j successfully match as a pair, Zij=1; otherwise,Zij=0.It is worthy to note that the abovementioned analytical model could derive out the total surplus, yet the corresponding pairing is probably not stable due to that there exists double marginalization effect in some pairs (i, j). In such a context, we give the following equilibrium conditions under which garment makers and designers can match with stable final pairing:(9)VCDi∗,j∗≥VCDi,j∗+VGMi,j∗−VGMi,j,(10)VCDi,j≥VCDi∗,j+VGMi∗,j−VGMi∗,j∗,∀i,j,i∗,j∗∈μ,μ=i,jZij=1,∀i∈I,∀j∈J.Equations (9) and (10) ensure that the final pairings are stable. In detail, when garment i matches with designer j∗, garment i can obtain the surplus of VGMi,j∗−VGMi,j, more than current matching with designer j. Assuming garment maker i wants to change the current matching, garment maker i is willing to at most transfer VGMi,j∗−VGMi,j to crowdsourcing designer j∗ to change his rank. But right now, designer j∗ is assigned to garment maker i∗, and VCDi∗,j∗≥VCDi,j∗+VGMi,j∗−VGMi,j implies that there is no incentives for the designer j∗ to change the current matching. Following the same logic, garment maker i∗ has no motivation to change the present pairing.


## 5. Extended Model with Due-Date Constraint

As we know that fashion products such as dresses, bags, footwears, and among others belong to the fast consumption industry, in which product design renews swiftly. For instance, both H&M and Zara, two famous multinational fashion companies, require that its each chain store launches new product styles twice a week and that all items on sales must be off shelf and totally replaced by new ones a month later. Therefore, in the crowdsourcing context, crowdsourcing tasks usually have deadline request or due-date constraint for solution submission; herein the due date is denoted byTij, thus warranting garment makers in a quick response to consumers’ diverse needs. In addition, myopic designer’s cost is directly associated with the length of time they expend such that we assume that designer’s cost is measured or quantified by task’s due date Tij and unit time cost cij. Hence, the utility of myopic designer with due-date constraint is represented as(11)UijD−II=Rij,present−CTij,CTij=cij×Tij,whereas the utility for garment makers is the same as before.With the aforementioned analysis, we incorporate the factor of due-date constraint into the abovementioned matching model and obtain the extended one as follows:(12)max∑i∑jVCDIIi,j+VGMIIi,j×Zij,pairi,j,which is subject to (6)–(10), and(13)VGMIIi,j=UijM−IIU¯ijM≥0,∀i∈I,∀j∈J,(14)VCDIIi,j=UijD−II−U¯ijD≥0,∀i∈I,∀j∈J,(15)CTij=cij×Tij≥0,Tij≥0,(16)tjE+Tij×Zij≤tiL,∀i∈I,∀j∈J,(17)VCDIIi∗,j∗≥VCDIIi,j∗+VGMIIi,j∗−VGMIIi,j,(18)VCDIIi,j≥VCDIIi∗,j+VGMIIi∗,j−VGMIIi∗,j∗,∀i,j,i∗,j∗∈μ，μ=i,jZij=1,∀i∈I,∀j∈J.Equation (12) aims to maximize the total surpluses of the whole matching system between garment makers and myopic designers under due-date constrain. Equations (13) and (14) represent the surplus of garment makers and designers, respectively. Equation (15) characterizes designer’s expenditure cost. Equation (16) guarantees that designers punctually submit their design solution. Equations (17) and (18) are the conditions under which the robust and stable final parings can be obtained.


## 6. Extended Model with Goodwill Concern

Different from the two aforementioned matching models where designers are near-sighted. In this section, we assume that designers are strategic rather than myopic ones, it means that strategic designers underline both the current utilities and the next-round expected utilities as well.Considering that the market fluctuates with the changes in time and demand, garment makers anticipate the better-qualified designers to participate; the practical way of assessing qualified designers is to observe which company designers worked before. To this end, the designers who employed by sound goodwill firms are apt to be regarded as more capable candidates, which elicits designers not only to focus on the present utilities but also care about present selected garment makers’ goodwill, thus in turn help the designers to probably win next-round tasks, since goodwill refers to an intangible asset taken into account in reflecting its commercial reputation, customer connections, etc. Therefore, strategic designers prefer the tasks from makers with a higher goodwill.Based on the model with due-date constraint and goodwill concerns, the strategic designer’s utility is the sum of current task utilityRij,present−C and expected task utilityPij,next, which is illustrated as(19)UijD−III=Rij,present−CTij+Pij,next.For the expected taskutilityPij,next, assume that the incumbent garment makers’ goodwill can be grouped into two parts, namely, normal and better ones, and suppose that the strategic designer facing next-round tasks exist in two available utilities: low or high. In such a setup, low utility with normal or better goodwill is denoted by PN−L and PB−L, while high utility with normal or better goodwill is set by PN−H and PB−H. In addition, the parameters αandβ indicate the probabilities of strategic designers’ taking on tasks with a normal and better goodwill, respectively. Thereby, the strategic designer’s expected task utilityPij,next can be demonstrated as αPN−H+1−αPN−L+βPB−H+1−βPB−L.For the garment makers’ utility, it remains the same as before.In the context of goodwill concern, the extended model is formulated as follows:(20)max∑i∑jVCDIIIi,j+VGMIIIi,j×Zij,pairi,j,which is subject to (6)–(8), (15), and (16),(21)VGMIIIi,j=UijM−III−U¯ijM≥0,∀i∈I,∀j∈J,(22)VCDIIIi,j=UijD−III−U¯ijD≥0,∀i∈I,∀j∈J,(23)Pij,next=αPN−H+1−αPN−L+βPB−H+1−βPB−L,(24)VCDIIIi∗,j∗≥VCDIIIi,j∗+VGMIIIi,j∗−VGMIIIi,j,(25)VCDIIIi,j≥VCDIIIi∗,j+VGMIIIi∗,j−VGMIIIi∗,j∗,∀i,j,i∗,j∗∈μ,μ=i,jZij=1,∀i∈I,∀j∈J.Equation (20) aims to maximize the total surpluses of designers and garment makers. Equations (21) and (22) represent the surplus of garment makers and designers, respectively. Equation (23) is the strategic designer’s expected utility regarding the next-round task. Equations (24) and (25) are the conditions under which the robust and stable final parings can be obtained.


## 7. Algorithm Procedure

The classical matching problem can be solved by using Gale–Shapley algorithm. However, the proposed matching models of this paper characterize both the crowdsourcing natures and the dynamic hierarchy attributes; thus, we design an improved Gale–Shapley algorithm to adjust our garment maker-designer matching problem under the crowdsourcing environment. The algorithm procedure for the present models is descripted as follows (Algorithm1).Algorithm 1: The solution procedure of the matching models.
(1)
μ
0
⟵
Φ; UijM0⟵EU¯ijD,U¯ijM; n ⟵ 0
(2)
do
(3)
n ⟵ n + 1
(4)
UijMn⟵EU¯ijD,U¯ijM; xj ⟵ rank VCDk,j in descending order;
(5)
a ⟵ 0; b ⟵ 0;
(6)
while (a < n) do
(7)
a ⟵ a + 1; b ⟵ 0;
(8)
while (b < Hj) do
(9)
b ⟵ b + 1; j ⟵ a; y ⟵ b; i∗ ⟵ xjy
(10)
if (not (VCDi∗,j∗≥VCDi,j∗+VGMi,j∗−VGMi,j)) then
(11)
if (VGMi,j∗−VGMi,j>VCDi∗,j∗−VCDi,j∗option) then
(12)
U
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∗
M
n
+
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∗
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(13)
else
(14)
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i
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∗
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n
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1
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−
V
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(15)
end if
(16)
end if
(17)
end while
(18)
end while
(19)
μn+1 ⟵ GS algorithm;
(20)
while (not (μn−1==μnandUijMn−1==UijMn))
(21)
Output results


## 8. Numerical Study

In this section, we present an application of garment maker-designer matching model related to the dynamical hierarchical method in the crowdsourcing context. The relevant data come fromzbj.com, one of the most famous online-crowdsourcing platforms in China, providing a wide variety of crowdsourcing services including customized design for fast fashion firms. We suppose α=0.3,β=0.7,PN−H=PB−H=10000,PN−L=PB−L=8000. In addition, two scenarios are taken into account based on the number of designers and garment makers. (i) Scenario I refers to a regular crowdsourcing case in which designers surpass garment makers in number (i.e., n>m), thus implying the garment maker dominates in the crowdsourcing community, whereas (ii) scenario II stands for an irregular one in which garment makers exceed designers in number (i.e., n<m), indicating the designers have bargaining power over garment makers. The specific parameters and attributes of garment makers and designers in two different scenarios are presented in Tables 2–5.Table 2
The specific attributes of garment makers in scenario I.
Garment maker
Task type
T
ij (day)
S
ij (CNY)
Goodwill
Reward (CNY)
U
¯
i
j
M (CNY)
1
Casual
9
1200
Normal
8000
45000
2
Casual
10
1200
Better
9600
50000
3
Casual
12
1200
Better
10000
48000
4
Sporty
10
1200
Better
12000
60000
5
Sporty
14
1200
Normal
13000
58000
6
Retro
18
1200
Better
12000
70000Table 3
The relevant parameters of designers in scenario I.
Designer
Style type
C
ij
c
ij (CNY/day)
B
i
j (CNY)
1
Casual
3600
300
62000
2
Casual
3100
250
65000
3
Casual
3300
260
64500
4
Casual
3250
320
60000
5
Casual
3350
280
61500
6
Casual
3500
300
60500
7
Casual
3200
310
61000
8
Sporty
3700
380
73000
9
Sporty
3800
400
69000
10
Sporty
3650
390
74500
11
Sporty
3850
370
75000
12
Sporty
3760
380
71000
13
Retro
4000
350
83000
14
Retro
4200
330
79000
15
Retro
4100
360
80000Table 4
The specific attributes of garment makers in scenario II.
Garment maker
Task type
T
ij (day)
S
ij (CNY)
Goodwill
Reward (CNY)
U
¯
i
j
M (CNY)
1
Casual
9
1200
Normal
8000
45000
2
Casual
10
1200
Better
9600
50000
3
Casual
12
1200
Better
10000
48000
4
Casual
13
1200
Normal
10500
53000
5
Casual
11
1200
Normal
8500
52000
6
Casual
15
1200
Better
11000
53000
7
Sporty
10
1200
Better
12000
60000
8
Sporty
14
1200
Normal
13000
58000
9
Sporty
12
1200
Normal
15000
52000
10
Sporty
13
1200
Better
14500
50000
11
Retro
18
1200
Better
12000
62000
12
Retro
16
1200
Normal
15000
60000Table 5
The relevant parameters of designers in scenario II.
Designer
Style type
C
ij
c
ij (CNY/day)
B
i
j
a (CNY)
1
Casual
3600
300
62000
2
Casual
3100
250
65000
3
Casual
3300
260
64500
4
Casual
3250
320
60000
5
Casual
3350
280
61500
6
Sporty
3700
380
73000
7
Sporty
3800
400
69000
8
Sporty
3650
390
74500
9
Retro
4000
350
83000
10
Retro
4200
330
79000


### 8.1. Scenario I: Garment Makers’ Domination in Crowdsourcing

In this subsection, considering a regular case in which the number of designers exceeds that of garment makers, we compare the pairs under three matching submodels from the hierarchical perspective, i.e., the two-sided matching model without due-date constraint (the first-level matching), that with due-date constraint (the second-level matching), and that with due-date constraint and goodwill concern (the third-level matching).The utilities regarding the garment makers and designers are calculated according to equations (1) and (2), respectively, followed by ranking their candidates based on the values of utilities. Then, the match between garment makers and designers proceeds by maximizing the total surplus. The first matching result of garment makers and designers can be obtained by the improved GS algorithm, and it is illustrated in Table 6, which shows that task 1 is assigned to designer 1, task 2 is assigned to designer 3, and task 3 is assigned to designer 2; while garment maker 5 chooses designer 11, and garment maker 4 chooses designer 10. Interestingly, we find that garment maker 6 does not choose anyone although three of the designers have a strong desire to match with garment maker 6, when the surpluses the designers create are under the 6th garment maker’s minimum anticipation.Table 6
Scenario I: the first matching pairs in the matching submodel without due-date constraint.
GM1
GM2
GM3
GM4
GM5
GM6
CD1
(4400, 7800)∗
(6000, 1200)
(6400, 2800)
—
—
—
CD2
(4900, 10800)
(6500, 4200)
(6900, 5800)∗
—
—
—
CD3
(4700, 10300)
(6300, 3700)∗
(6700, 5300)
—
—
—
CD4
(4750, 5800)
(6350, −800)
(6750,800)
—
—
—
CD5
(4650, 7300)
(6250, 700)
(6650, 2300)
—
—
—
CD6
(4500, 6300)
(6100, −300)
(6500, 1300)
—
—
—
CD7
(4800, 6800)
(6400, 200)
(6800, 1800)
—
—
—
CD8
—
—
—
(8300, −200)
(9300, 800)
—
CD9
—
—
—
(8200, −4200)
(9200, −3200)
—
CD10
—
—
—
(8350, 1300)∗
(9350, 2300)
—
CD11
—
—
—
(8150, 1800)
(9150, 2800)∗
—
CD12
—
—
—
(8240, −2200)
(9240, −1200)
—
CD13
—
—
—
—
—
(8000, −200)
CD14
—
—
—
—
—
(7800, −4200)
CD15
—
—
—
—
—
(7900, −3200)
∗Indicates successful matching ones.However, garment maker 2 finds that matching with designer 2 can help him get,VGM2,2−VGM2,3 = 500, more surpluses. Therefore, with intention to change his ranking, garment maker 2 transfers no more than 500 surpluses to designer 2. But designer 2 prefers garment maker 3, such that designer 2 may lose, VCD3,2−VCD2,2=400, surpluses if he matches with designer 2. Thereby, only if garment maker 2 compensates designer 2 at least 400 surpluses, can he change the ranking. Meanwhile, garment maker 3, which is now assigned to designer 2 will give up at most, VGM3,2−VGM3,3 = 500, to resist garment maker 2’s matching behavior, avoiding being assigned to designer 3 at the loss of 500 surpluses. So VCD3,2−VCD2,2<VGM2,2−VGM2,3<VCD3,2−VCD2,2+VGM3,2−VGM3,3 indicates that garment maker 3 can retain his status by transferring the surplus of VGM2,2−VGM2,3−VCD3,2−VCD2,2 to garment maker 2. Therefore, we can obtain the following observation.Observation 1.
(necessary condition for crowdsourcing participation). IfVCDi∗,j∗−VCDi,j∗<VGMi,j∗−VGMi,j<VCDi∗,j∗−VCDi,j∗+VGMi∗,j∗−VGMi∗,j,∀i,j,i∗,j∗∈μ holds, a garment maker i∗ has to increase his reward to a designer j∗ to eliminate the possibility of the designer j∗ being assigned to a garment maker i.
Observation1 implies if the rewards announced by garment makers are below a certain threshold, thus leading to crowdsourcing matching failure. Even worse, the crowdsourcing platform cannot be sustainable to operate due to being unable to attract sufficient designers’ participation. The proof of Observation 1 is showed in Appendix.
Meanwhile, garment maker 1 matching with designer 1 also prefers designer 2, withVGM1,2−VGM1,1 = 3000, whereas designer 2 whose best choice is garment maker 3 will reduce his surpluses of VCD′3,2−VCD1,2=2100. To keep his surpluses, garment maker 3 is willing to transfer no more than VGM′3,2−VGM3,3 = 400 to designer 2, thus eliminating the possibility of being assigned to designer 3. However, VGM1,2−VGM1,1>VCD′3,2−VCD2,2+VGM′3,2−VGM3,3 (i.e., garment maker 1 has more surpluses to compensate designer 2), garment maker 1 can change his ranking by transferring VCD′3,2−VCD2,2+VGM′3,2−VGM3,3 to designer 2. Additionally, garment maker 4 would match with designer 11 rather than designer 10, but VCD5,11≥VCD4,11+VGM4,11−VGM4,10 satisfies the stable constraint, thus garment maker 4 does not intend to alter the current matching. Therefore, we have the following observation shown as follows.Observation 2.
(sufficient condition for crowdsourcing participation). IfVGMi,j∗−VGMi,j>VCDi∗,j∗−VCDi,j∗+VGMi∗,j∗−VGMi∗,j,∀i,j,i∗,j∗∈ holds, a garment maker i has adequate surpluses to defeat a garment maker i∗ and match with a designer j∗.
Observation2 demonstrates that although the crowdsourcing platform attracts tremendous designers’ participation, garment makers who compete against the others to win customized design solutions rely on both expected utilities and their goodwill.
In line with the stable matching rules, we summarize the results of matching shown in Tables7 and 8. Tables 7 and 8 illustrate that the final stable pairs regarding three submodels are attained until the 3rd, 5th, and 5th matching, respectively.
Tables7 and 8 reveal that the overall surpluses of garment makers and designers under three submodels slightly rise in the scenario I. Nonetheless, from each model aspect, the surplus of designers shows upward trend, whereas that of garment makers declines under the first two submodels. Surprisingly, for the last submodel, the surplus of designers almost remains unchanged as that of garment makers do, which implies that the final pairs of the last submodel are more stable and robust than those of the first two ones. In addition, an increase of designers’ surplus in the matching process also further verifies designers’ domination in a competitive fast fashion market.Table 7
Scenario I: the multimatching results in the submodel without due-date constraint.
Matching process
Garment makers
Designers’ surplus
Garment makers’ surplus
Total surpluses
1
2
3
4
5
6
Designers
1st matching
1
3
2
10
11
—
35100
21400
56500
2nd matching
2
1
3
10
11
—
37600
18900
56500
3rd matching
1
2
3
10
11
—
37000
19500
56500Table 8
Scenario I: The multimatching results of two extended submodels.
Matching process
Garment makers
Designers’ surplus
Garment makers’ surplus
Total surpluses
1
2
3
4
5
6
Designers
Extended submodel with due-date constraint
1st matching
1
2
3
11
10
—
35120
21400
56520
2nd matching
2
3
1
11
10
—
36860
19530
56390
3rd matching
1
3
2
11
10
—
36910
19600
56510
4th matching
1
2
3
11
10
—
36790
19730
56520
5th matching
1
3
2
11
10
—
38080
18460
56540
Extended submodel with goodwill concern
1st matching
1
2
3
11
10
—
80520
21400
101920
2nd matching
3
1
2
11
10
—
83040
18750
101790
3rd matching
1
3
2
11
10
—
81490
20450
101940
4th matching
1
2
3
11
10
—
81370
20550
101920
5th matching
1
3
2
11
10
—
81860
20080
101940


### 8.2. Scenario II: Designers’ Domination in Crowdsourcing

In this subsection, we examine an irregular case in which the designer size is smaller. Following with the same logic as scenario I, we can achieve the matching outcomes of three submodels, which are shown in Table9. The results show that garment maker 4, garment maker 5, and garment maker 6 still remain unmatched with any crowdsourcing designers in the 1st matching, and for garment maker 7 and garment maker 8, they fail to match in the upcoming competition because they do not have enough surpluses to adjust their rewards with an aim to attract designers. Interestingly, there exist unmatchings for both garment makers and designers in such case, which to some extent reflects the real state of crowdsourcing market under the small scale of designers.Table 9
Scenario II: multimatching results of three submodels.
Matching process
Garment makers
Designers’ surplus
Garment makers’ surplus
Total surpluses
1
2
3
4
5
6
7
8
9
10
11
12
Designers
Two-sided matching model
1st matching
5
1
3
2
—
—
—
—
8
6
10
9
65700
37200
102900
2nd matching
5
2
3
—
1
—
—
—
8
6
10
9
67600
36300
103900
3rd matching
5
2
3
—
1
—
—
—
8
6
10
9
68600
35300
103900
Extended model with due-date constraint
1st matching
5
3
1
2
—
—
—
—
8
6
10
9
61470
37200
98670
2nd matching
2
3
1
—
—
—
—
—
8
6
10
9
57250
37640
94890
3rd matching
1
2
3
—
—
—
—
—
8
6
10
9
58460
36560
95020
4th matching
1
2
3
—
—
—
—
—
8
6
10
9
58490
36530
95020
Extended model with goodwill concern
1st matching
1
2
3
—
—
—
—
—
8
6
10
9
118020
41500
159520
2nd matching
2
3
1
—
—
—
—
—
8
6
10
9
123050
36340
159390
3rd matching
1
3
2
—
—
—
—
—
8
6
10
9
121500
38040
159540
4th matching
1
2
3
—
—
—
—
—
8
6
10
9
121380
38140
159520
5th matching
1
3
2
—
—
—
—
—
8
6
10
9
121870
37670
159540Table9 shows that under scenario II the overall surpluses of garment makers and designers illustrate the same trend as scenario I. Parallelly, in scenario II, the surpluses of designers under the first two submodels increase in the matching process, whereas the surplus of garment makers drops. By contract, the surplus of designers under the last submodel in scenario II descend, while that of garment makers climb up, and it implies that, under an irregular case, the final pairs of the last submodel are more sensitive than the two others, which further confirms the robustness of the last submodel.The abovementioned analysis focuses on each scenario; the following discussion will highlight the comparison of two scenarios. Comparing the first submodel (two-side matching model) to the second submodel (extended model with due-date constraint), it is easy to witness that the myopic designers’ surpluses of the latter model are higher than those of the former in scenario I, whereas in scenario II the consequence is opposite (as shown in Figure2). Meanwhile, the surpluses of the third submodel (extended model with goodwill concern) outperforms those of the two others in terms of garment makers and designers. This implies under the irregular case, i.e., when garment makers exceed online designers in number, crowdsourcing design tasks without due-date constraint are more attractive for designers’ participation than those with due-date constraint. Particularly, in matching process, garment makers intend to share the incremental surpluses with designers so as to maximize the total surpluses of both sides. By contrast, under the regular case, i.e., when online designers surpass garment makers in number, designers are prone to choose crowdsourcing design tasks with due-date constraint than those without due date.Figure 2
The comparison of three matching submodels in scenarios I and II. (a) Scenario I (regular case); (b) scenario II (irregular case).
(a)
(b)In addition, we find that regardless of whether under the irregular case or the regular case, the surpluses of strategic designers are obviously higher than those of myopic ones. It reminds that both crowdsourcing platforms and garment makers should emphasize the issue of goodwill concerns, for example, if the crowdsourcing platform attracts more well-known garment makers to post their name-labeled tasks online, thus making myopic designers transfer to be strategic ones, which in turn helps entice more designers’ engagement in crowdsourcing activities due to the incremental surplus.


### 8.3. Sensitivity Analysis

In this subsection, we will examine the robustness of our approach of the current study. Tables10 and 11 show the variance of matching results in scenario I and scenario II with the change of α and β, respectively. When α ranges from 0.3 to 0.9 and β ranges from 0.1 to 0.7, the matching pairs keep stable, which means our proposed method applied in hierarchical matching decision process on crowdsourcing design in a fast fashion market has strong robustness.Table 10
The matching results and surplus at differentα and β in scenario I.Table 11
The matching results and surplus at differentα and β in scenario II.In addition, no matter what the irregular case or regular case is, the areas with respect toα and β are relatively bigger, which verifies that the extended submodel with due-date constraint and goodwill concern is more stable and more practical than the two others.


## 8.1. Scenario I: Garment Makers’ Domination in Crowdsourcing

In this subsection, considering a regular case in which the number of designers exceeds that of garment makers, we compare the pairs under three matching submodels from the hierarchical perspective, i.e., the two-sided matching model without due-date constraint (the first-level matching), that with due-date constraint (the second-level matching), and that with due-date constraint and goodwill concern (the third-level matching).The utilities regarding the garment makers and designers are calculated according to equations (1) and (2), respectively, followed by ranking their candidates based on the values of utilities. Then, the match between garment makers and designers proceeds by maximizing the total surplus. The first matching result of garment makers and designers can be obtained by the improved GS algorithm, and it is illustrated in Table 6, which shows that task 1 is assigned to designer 1, task 2 is assigned to designer 3, and task 3 is assigned to designer 2; while garment maker 5 chooses designer 11, and garment maker 4 chooses designer 10. Interestingly, we find that garment maker 6 does not choose anyone although three of the designers have a strong desire to match with garment maker 6, when the surpluses the designers create are under the 6th garment maker’s minimum anticipation.Table 6
Scenario I: the first matching pairs in the matching submodel without due-date constraint.
GM1
GM2
GM3
GM4
GM5
GM6
CD1
(4400, 7800)∗
(6000, 1200)
(6400, 2800)
—
—
—
CD2
(4900, 10800)
(6500, 4200)
(6900, 5800)∗
—
—
—
CD3
(4700, 10300)
(6300, 3700)∗
(6700, 5300)
—
—
—
CD4
(4750, 5800)
(6350, −800)
(6750,800)
—
—
—
CD5
(4650, 7300)
(6250, 700)
(6650, 2300)
—
—
—
CD6
(4500, 6300)
(6100, −300)
(6500, 1300)
—
—
—
CD7
(4800, 6800)
(6400, 200)
(6800, 1800)
—
—
—
CD8
—
—
—
(8300, −200)
(9300, 800)
—
CD9
—
—
—
(8200, −4200)
(9200, −3200)
—
CD10
—
—
—
(8350, 1300)∗
(9350, 2300)
—
CD11
—
—
—
(8150, 1800)
(9150, 2800)∗
—
CD12
—
—
—
(8240, −2200)
(9240, −1200)
—
CD13
—
—
—
—
—
(8000, −200)
CD14
—
—
—
—
—
(7800, −4200)
CD15
—
—
—
—
—
(7900, −3200)
∗Indicates successful matching ones.However, garment maker 2 finds that matching with designer 2 can help him get,VGM2,2−VGM2,3 = 500, more surpluses. Therefore, with intention to change his ranking, garment maker 2 transfers no more than 500 surpluses to designer 2. But designer 2 prefers garment maker 3, such that designer 2 may lose, VCD3,2−VCD2,2=400, surpluses if he matches with designer 2. Thereby, only if garment maker 2 compensates designer 2 at least 400 surpluses, can he change the ranking. Meanwhile, garment maker 3, which is now assigned to designer 2 will give up at most, VGM3,2−VGM3,3 = 500, to resist garment maker 2’s matching behavior, avoiding being assigned to designer 3 at the loss of 500 surpluses. So VCD3,2−VCD2,2<VGM2,2−VGM2,3<VCD3,2−VCD2,2+VGM3,2−VGM3,3 indicates that garment maker 3 can retain his status by transferring the surplus of VGM2,2−VGM2,3−VCD3,2−VCD2,2 to garment maker 2. Therefore, we can obtain the following observation.Observation 1.
(necessary condition for crowdsourcing participation). IfVCDi∗,j∗−VCDi,j∗<VGMi,j∗−VGMi,j<VCDi∗,j∗−VCDi,j∗+VGMi∗,j∗−VGMi∗,j,∀i,j,i∗,j∗∈μ holds, a garment maker i∗ has to increase his reward to a designer j∗ to eliminate the possibility of the designer j∗ being assigned to a garment maker i.
Observation1 implies if the rewards announced by garment makers are below a certain threshold, thus leading to crowdsourcing matching failure. Even worse, the crowdsourcing platform cannot be sustainable to operate due to being unable to attract sufficient designers’ participation. The proof of Observation 1 is showed in Appendix.
Meanwhile, garment maker 1 matching with designer 1 also prefers designer 2, withVGM1,2−VGM1,1 = 3000, whereas designer 2 whose best choice is garment maker 3 will reduce his surpluses of VCD′3,2−VCD1,2=2100. To keep his surpluses, garment maker 3 is willing to transfer no more than VGM′3,2−VGM3,3 = 400 to designer 2, thus eliminating the possibility of being assigned to designer 3. However, VGM1,2−VGM1,1>VCD′3,2−VCD2,2+VGM′3,2−VGM3,3 (i.e., garment maker 1 has more surpluses to compensate designer 2), garment maker 1 can change his ranking by transferring VCD′3,2−VCD2,2+VGM′3,2−VGM3,3 to designer 2. Additionally, garment maker 4 would match with designer 11 rather than designer 10, but VCD5,11≥VCD4,11+VGM4,11−VGM4,10 satisfies the stable constraint, thus garment maker 4 does not intend to alter the current matching. Therefore, we have the following observation shown as follows.Observation 2.
(sufficient condition for crowdsourcing participation). IfVGMi,j∗−VGMi,j>VCDi∗,j∗−VCDi,j∗+VGMi∗,j∗−VGMi∗,j,∀i,j,i∗,j∗∈ holds, a garment maker i has adequate surpluses to defeat a garment maker i∗ and match with a designer j∗.
Observation2 demonstrates that although the crowdsourcing platform attracts tremendous designers’ participation, garment makers who compete against the others to win customized design solutions rely on both expected utilities and their goodwill.
In line with the stable matching rules, we summarize the results of matching shown in Tables7 and 8. Tables 7 and 8 illustrate that the final stable pairs regarding three submodels are attained until the 3rd, 5th, and 5th matching, respectively.
Tables7 and 8 reveal that the overall surpluses of garment makers and designers under three submodels slightly rise in the scenario I. Nonetheless, from each model aspect, the surplus of designers shows upward trend, whereas that of garment makers declines under the first two submodels. Surprisingly, for the last submodel, the surplus of designers almost remains unchanged as that of garment makers do, which implies that the final pairs of the last submodel are more stable and robust than those of the first two ones. In addition, an increase of designers’ surplus in the matching process also further verifies designers’ domination in a competitive fast fashion market.Table 7
Scenario I: the multimatching results in the submodel without due-date constraint.
Matching process
Garment makers
Designers’ surplus
Garment makers’ surplus
Total surpluses
1
2
3
4
5
6
Designers
1st matching
1
3
2
10
11
—
35100
21400
56500
2nd matching
2
1
3
10
11
—
37600
18900
56500
3rd matching
1
2
3
10
11
—
37000
19500
56500Table 8
Scenario I: The multimatching results of two extended submodels.
Matching process
Garment makers
Designers’ surplus
Garment makers’ surplus
Total surpluses
1
2
3
4
5
6
Designers
Extended submodel with due-date constraint
1st matching
1
2
3
11
10
—
35120
21400
56520
2nd matching
2
3
1
11
10
—
36860
19530
56390
3rd matching
1
3
2
11
10
—
36910
19600
56510
4th matching
1
2
3
11
10
—
36790
19730
56520
5th matching
1
3
2
11
10
—
38080
18460
56540
Extended submodel with goodwill concern
1st matching
1
2
3
11
10
—
80520
21400
101920
2nd matching
3
1
2
11
10
—
83040
18750
101790
3rd matching
1
3
2
11
10
—
81490
20450
101940
4th matching
1
2
3
11
10
—
81370
20550
101920
5th matching
1
3
2
11
10
—
81860
20080
101940


## 8.2. Scenario II: Designers’ Domination in Crowdsourcing

In this subsection, we examine an irregular case in which the designer size is smaller. Following with the same logic as scenario I, we can achieve the matching outcomes of three submodels, which are shown in Table9. The results show that garment maker 4, garment maker 5, and garment maker 6 still remain unmatched with any crowdsourcing designers in the 1st matching, and for garment maker 7 and garment maker 8, they fail to match in the upcoming competition because they do not have enough surpluses to adjust their rewards with an aim to attract designers. Interestingly, there exist unmatchings for both garment makers and designers in such case, which to some extent reflects the real state of crowdsourcing market under the small scale of designers.Table 9
Scenario II: multimatching results of three submodels.
Matching process
Garment makers
Designers’ surplus
Garment makers’ surplus
Total surpluses
1
2
3
4
5
6
7
8
9
10
11
12
Designers
Two-sided matching model
1st matching
5
1
3
2
—
—
—
—
8
6
10
9
65700
37200
102900
2nd matching
5
2
3
—
1
—
—
—
8
6
10
9
67600
36300
103900
3rd matching
5
2
3
—
1
—
—
—
8
6
10
9
68600
35300
103900
Extended model with due-date constraint
1st matching
5
3
1
2
—
—
—
—
8
6
10
9
61470
37200
98670
2nd matching
2
3
1
—
—
—
—
—
8
6
10
9
57250
37640
94890
3rd matching
1
2
3
—
—
—
—
—
8
6
10
9
58460
36560
95020
4th matching
1
2
3
—
—
—
—
—
8
6
10
9
58490
36530
95020
Extended model with goodwill concern
1st matching
1
2
3
—
—
—
—
—
8
6
10
9
118020
41500
159520
2nd matching
2
3
1
—
—
—
—
—
8
6
10
9
123050
36340
159390
3rd matching
1
3
2
—
—
—
—
—
8
6
10
9
121500
38040
159540
4th matching
1
2
3
—
—
—
—
—
8
6
10
9
121380
38140
159520
5th matching
1
3
2
—
—
—
—
—
8
6
10
9
121870
37670
159540Table9 shows that under scenario II the overall surpluses of garment makers and designers illustrate the same trend as scenario I. Parallelly, in scenario II, the surpluses of designers under the first two submodels increase in the matching process, whereas the surplus of garment makers drops. By contract, the surplus of designers under the last submodel in scenario II descend, while that of garment makers climb up, and it implies that, under an irregular case, the final pairs of the last submodel are more sensitive than the two others, which further confirms the robustness of the last submodel.The abovementioned analysis focuses on each scenario; the following discussion will highlight the comparison of two scenarios. Comparing the first submodel (two-side matching model) to the second submodel (extended model with due-date constraint), it is easy to witness that the myopic designers’ surpluses of the latter model are higher than those of the former in scenario I, whereas in scenario II the consequence is opposite (as shown in Figure2). Meanwhile, the surpluses of the third submodel (extended model with goodwill concern) outperforms those of the two others in terms of garment makers and designers. This implies under the irregular case, i.e., when garment makers exceed online designers in number, crowdsourcing design tasks without due-date constraint are more attractive for designers’ participation than those with due-date constraint. Particularly, in matching process, garment makers intend to share the incremental surpluses with designers so as to maximize the total surpluses of both sides. By contrast, under the regular case, i.e., when online designers surpass garment makers in number, designers are prone to choose crowdsourcing design tasks with due-date constraint than those without due date.Figure 2
The comparison of three matching submodels in scenarios I and II. (a) Scenario I (regular case); (b) scenario II (irregular case).
(a)
(b)In addition, we find that regardless of whether under the irregular case or the regular case, the surpluses of strategic designers are obviously higher than those of myopic ones. It reminds that both crowdsourcing platforms and garment makers should emphasize the issue of goodwill concerns, for example, if the crowdsourcing platform attracts more well-known garment makers to post their name-labeled tasks online, thus making myopic designers transfer to be strategic ones, which in turn helps entice more designers’ engagement in crowdsourcing activities due to the incremental surplus.


## 8.3. Sensitivity Analysis

In this subsection, we will examine the robustness of our approach of the current study. Tables10 and 11 show the variance of matching results in scenario I and scenario II with the change of α and β, respectively. When α ranges from 0.3 to 0.9 and β ranges from 0.1 to 0.7, the matching pairs keep stable, which means our proposed method applied in hierarchical matching decision process on crowdsourcing design in a fast fashion market has strong robustness.Table 10
The matching results and surplus at differentα and β in scenario I.Table 11
The matching results and surplus at differentα and β in scenario II.In addition, no matter what the irregular case or regular case is, the areas with respect toα and β are relatively bigger, which verifies that the extended submodel with due-date constraint and goodwill concern is more stable and more practical than the two others.


## 9. Conclusion

In this paper, we focus on the issue of garment maker-designer matching in the fast fashion crowdsourcing context. To handle such a problem, we propose the matching model using a dynamic hierarchical method and analyze the impact of three different matching submodels on crowdsourcing performance, including the two-sided matching model and the extended model with due-date constraint, as well as one with due-date and garment makers’ goodwill concern. After that, we compare the optimal outcomes and examine the condition under which the garment makers and designers can reach stable equilibriums.The main findings of our paper lie in several facets as follows:(1)
From the hierarchical perspective, we find that only when three key factors including surplus, due date, and goodwill are taken into account as a whole in the crowdsourcing matching context, both garment makers and designers are better off, and the whole matching system obtains the optimal result.(2)
When online designers surpass garment makers in number, dress designers prefer crowdsourcing tasks with due-date constraint to those without it. By contrast, when garment makers exceed online designers in number, crowdsourcing tasks without due-date constraint are more attractive for designers’ participation than those with due-date constraint because garment makers are willing to share the incremental surpluses with designers to maximize the total surpluses.(3)
In addition, regardless of under the irregular or the regular case, more strategic designers necessarily benefit garment makers and crowdsourcing platforms than myopic ones, which implies that crowdsourcing platforms should encourage more well-known garment makers to post their name-labeled tasks, thus making myopic designers transfer to be strategic ones.(4)
Finally, sensitivity analysis reveals that no matter the irregular case or regular case, the extended submodel with goodwill concern is more stable and robust than the two others to achieve the final matching pairs.Some limitations of this paper that should be considered in future research include the following. First, while our assumption of the crowdsourcing matching with an information symmetry could characterize most fast fashion design platforms; in practice, there is indeed a tiny portion of them which are those with an information asymmetry. Thus, future study should relax this condition. Second, for analytical convenience, we assume that the submodels proposed are the same in unit time cost of designer and due date, which could be considered differently later. Third, extension of the present study from only design sector to design-production sector could be an interesting research in the future.



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*Source: 9424132-2020-04-13.xml*