# Two-Sided Matching Model of Shared Manufacturing Resources considering Psychological Behavior of Agents

Authors: Peng Liu; Ying Zou
Journal: Discrete Dynamics in Nature and Society
 (2021)

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## Abstract

Sharing manufacturing is a new manufacturing mode based on sharing economy, which is one of the pillars of intelligent manufacturing. This paper proposes a two-sided matching model of shared manufacturing resources considering the psychological behavior of agents. We describe the definition of two-sided matching and introduce the concept of the cloud model. The preference information of agents is transformed to values according to the cloud model. We combine prospect theory and grey relational analysis to calculate the prospect values. Furthermore, an optimization model which aims to maximize the overall satisfaction degree of matching agents is established. A numerical example for the matching of providers and demanders is provided to verify the feasibility and effectiveness of the model.


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## Body

## 1. Introduction

In recent years, driven by the booming mobile Internet technology, an economic model called “sharing economy” is emerging. As a new economic model, it can realize sharing of ownership of resources through the peer-to-peer market platform. The sharing economy has achieved a win-win situation among enterprises through online service and has attracted significant attention in many industries, such as tourism, trade, finance, and medical care. Especially in the transportation and hotel industries, companies such as Uber and Airbnb have achieved great success. In the era of rapid development of sharing economy, shared manufacturing has quietly emerged. The sharing of manufacturing resources which is a combination of sharing economy and manufacturing industry has become a new highlight of development.Shared manufacturing is a new mode of social manufacturing based on the principles of sharing economy. Ellen [1] proposed the concept of “shared manufacturing” in 1990. Ari Samadhi and Hoang [2] proposed the sharing of computer-integrated manufacturing system in the complex manufacturing environment. Sheikhzadeh et al. [3] considered two types of machine sharing configuration in manufacturing systems: total flexibility and chaining. The global manufacturing industry is in a period of transformation from production-oriented manufacturing to service-oriented manufacturing. The rapid changes of the market and the diversified needs of users make manufacturers often be faced with high-efficiency and high-intensity orders, which cannot be met by their manufacturing capabilities. Simultaneously, unbalanced regional development and insufficient market demand often result in idle resources and wasteful capacity. The shared manufacturing platform is the core carrier of the shared manufacturing mode, which provides users with a platform for exchanging information, trading goods, and selling services. And, it provides manufacturing services for bilateral markets. The demanders and providers of manufacturing resources can publish capabilities and requirements on the platform at any time and place. The shared manufacturing platform acting as the interface matches the qualified demanders and providers precisely.The shared manufacturing platform contains a large amount of manufacturing resource information. Among them, there is a large amount of shared manufacturing service with the same or similar functions but different quality of service. The shared manufacturing service can be added and withdrawn dynamically, and the quality of service and requirements are also variable. Through aforementioned analysis, how to match demanders and providers properly in shared manufacturing becomes a key problem. On the one hand, it is vital to choose appropriate method to deal with evaluation information in shared manufacturing platform. On the other hand, in order to improve the accuracy of resource matching in the shared manufacturing platform, more psychological behavior of agents should be considered. Therefore, the problem of two-sided matching should be resolved to facilitate transactions of resources. This paper proposes a two-sided matching model of shared manufacturing resources considering psychological behavior of agents. The cloud model is used to quantify the qualitative language and deal with the quantification of preference information more objectively to embody the fuzziness and uncertainty of the cognitive degree of two agents. At the same time, based on prospect theory, the grey relational analysis method is introduced to deal with the satisfaction of matching agents as a whole, which reflects the agent’s psychological attitude.The rest of this paper is organized as follows: In Section2, the literature review is provided. In Section 3, it introduces the problem description and preliminaries. We propose the construction and solution of the two-sided matching model in Section 4. Numerical example using MATLAB is given in Section 5. Finally, Section 6 provides conclusions.


## 2. Literature Review

Two streams of the literature are relevant to this study, i.e., development of shared manufacturing and two-sided matching.The shared manufacturing model is the application practice of using mobile Internet technology to solve the imbalance of social manufacturing resources. Yu et al. [4] presented the concept and definition of shared manufacturing on the basis of a service hierarchy and analyzed the service operations of shared manufacturing along with the trends of two significant manufacturing evolutions. He et al. [5] introduced the research status and significance of shared manufacturing in the Chinese manufacturing industry and classified the main shared manufacturing contents in detail. Yu et al. [6] proposed the blockchain-based shared manufacturing (BSM) framework in support of the application of Cyber-Physical Systems (CPS). Rozman et al. [7] presented a scalable framework for blockchain-based on shared manufacturing. Wang et al. [8] proposed a digital twin-driven service model and optimal allocation of manufacturing resources in shared manufacturing. Li et al. [9] considered the pricing strategies for the shared manufacturing model based on the cloud platform. Ayala et al. [10] studied the dynamic application of knowledge sharing in manufacturing companies depended on the type of collaboration and service object. Zhao and Du [11] analyzed the impact of information sharing through the whole supply chain on the strategies of manufacturing capability sharing under the context of Internet of Things. Bao and Cai [12] studied the intelligent production sharing and user experience based on the case of Shenyang Machine Tool. Dai et al. [13] considered the realization mechanism of sharing pattern innovation in service manufacturing enterprises by using a grounded theory analysis based on service leading logic.The study of two-sided matching first appeared in Gale and Shapley [14] on college enrollment and marital stability, which immediately aroused widespread concern in academia. Li et al. [15] selected service by calculating the similarity of function and quality and proposed an intelligent service searching and matching method for cloud manufacturing service based on formal description of cloud manufacturing service. Dong and Guo [16] obtained the quality of service acquisition method of cloud manufacturing through a mutual evaluation mechanism and established the trust evaluation model for cloud manufacturing service. Zhao and Wang [17] proposed a two-sided matching model for cloud service based on quality of service. Some scholars also studied the matching problem in practical situations from the perspective of preference information. Fan et al. [18] proposed a decision analysis method for two-sided satisfied matching considering stable matching conditions. Kamiyama [19] proposed an acceptable matching problem with two-sided preference lists and matroid constraints. Zhang et al. [20] developed a new method of two-sided matching decision with the fuzzy preference relation with self-confidence on the basis of the extended logarithmic least squares method and the proposed consistency improvement algorithm. Zhang et al. [21] developed a two-sided matching decision-making method with multigranularity hesitant fuzzy linguistic term sets and constructed optimization models to determine the standard weights of matching objects. Zhang et al. [22] proposed a new method of stable two-sided matching decision-making with incomplete fuzzy preference relation based on the disappointment theory. Li et al. [23] proposed a bilateral matching model based on double hesitation and ambiguity in cloud manufacturing platform. Li et al. [24] put forward a two-sided matching decision model with bidirectional projection under uncertain and fuzzy preference information. Zhang and Su [25] studied a combined fuzzy DEMATEL and TOPSIS method to estimate participants. Besides, some scholars also pay attention to the psychological behavior of the agent in two-sided matching. Zhang et al. [26] proposed a two-sided matching method considering regret aversion psychological behavior and matching aspiration of the agents. Zhao and Li [27] determined the preference of the matching agent based on the prospect theory and designed a “many-to-many” matching algorithm to obtain a stable matching result. Zhang et al. [28] considered the decision model and application of two-sided matching of uncertain information based on grey relational analysis. Bi et al. [29] studied the decision method for two-sided matching with linguistic evaluation based on cloud model and prospect theory.Although there have studies that have solved various forms of two-sided matching problems in cloud manufacturing, few studies consider the two-sided matching in the shared manufacturing environment. At the same time, the preference information of most matching problem is based on an order value. However, most of the preference information given by the demanders and providers on the shared manufacturing platform is based on the linguistic evaluation. Meanwhile, the cloud model can use linguistic values to express the uncertainty transformation between qualitative and quantitative concepts. Also, the matching agent is usually abstract. The agent’s psychological behavior in the matching process is ignored. Therefore, this paper combines prospect theory with the grey relational analysis method. According to this proposed method, we can holistically judge the degree of influence between the demanders and providers under all indicators. Then, a new calculating method of prospect value is obtained, which better reflects the psychological behavior of agents.


## 3. Problem Description and Preliminaries

### 3.1. Problem Description

With the development of the sharing economy and the popularization of mobile Internet technology, manufacturing enterprises tend to share production materials selectively. At this time, a third-party platform is used to integrate and release this production information and to operate and maintain the entire sharing mechanism. This paper is devoted to examining such a shared manufacturing platform that provides an accurate matching between a large number of orders and providers and helps the demanders quickly find the providers of qualified manufacturing resources and service capabilities. Also, the platform provides matching and trading services forM demanders and N providers. In the process of matching shared manufacturing resources with two-sided demand, the shared manufacturing platform provides a reasonable matching plan based on the expected level and the actual measured level of quality of service under the condition of comprehensively considering the satisfaction of both agents. The two-sided matching model of shared manufacturing resources considering psychological behavior of agents can be described in Figure 1.Figure 1
Two-sided matching model of shared manufacturing resources considering psychological behavior of agents.The basic description of the problem about two-sided matching is set out. Let the resource provider side beS=s1,s2,…sn, and sj means the jthj=1,2,…,n provider in the S. The resource demander side is D=d1,d2,…,dm, and di means the ithi=1,2,…,m demander in the D. The resource provider sj (demander di) hopes to choose a satisfied demander di (provider sj).The satisfaction of the agent of two-sided matching is calculated based on the prospect theory and grey relational analysis method. LetC=c1,c2,…cf be the set of satisfaction evaluation indicators on the quality of service from the demander di to the provider sj. ck is the kthk=1,2,…,f indicator, and each indicator is separate. The set of satisfaction evaluation C includes the following indicators: the price (c1), technical service level c2, reputation c3, and delivery date c4 (the number of days required for processing and transportation). A=a1,a2,…,ag is the set of indicators on the quality of service evaluated from the provider sj to the demander di. at is the ttht=1,2,…,g indicator, and each indicator is separate. The set of satisfaction evaluation A includes the following indicators: the technical difficulty a1 (“difficult” means that it can process all tasks below “difficult” technical requirements), reputation a2, and payment speed a3 (how many days do you want to pay 95% of the full amount after the order task is placed, and 5% is the warranty deposit). ωk is the weight corresponding to each indicator, which satisfies ωk>0and∑k=1f/gωk=1. Subsequently, the demander di gives the expected value eik and the actual measured value rijk of each indicator of the provider sj. The provider’s expected value and actual measured value of the demander’s indicators can be obtained in the same way.Combined with reality, indicators can be described as numerical values or evaluation phrases. One type is an indicator that can be measured with actual numbers such as price, and the corresponding subset is denoted asCRAR. And, the other type is an indicator that can only be evaluated by language, such as reputation. The corresponding subset is represented as CFAF and C=CR∪CFA=AR∪AF. In this paper, the cloud model is used to convert the indicators of language evaluation into quantitative standard values.


### 3.2. Preliminaries

#### 3.2.1. Definition of the Two-Sided Matching

Suppose there be two agents; they are the demander and the provider. And, the set of provider isS=s1,s2,…,sn,n≥2, where sj is the jth provider, j=1,2,…,n. The set of demander is D=d1,d2,…,dm, m≥2, where di represents the ith demander, i=1,2,…,m. There is a hypothesis that m≤n. The basic model of two-sided matching is shown in Figure 2.Figure 2
Basic model of two-sided matching.Definition 1 (see [30]).
For a one-to-one mapping relationshipμ: D∪S⟶D∪S, meanwhile, ∀di∈D and ∀sj∈S satisfy the following conditions:(1)
μdi∈S∪di, if μdi=di, there is no matching agent in di(2)
μsj∈D∪sj, if μsj=sj, there is no matching agent in sj(3)
μdi=sj, if and only if μsj=di, μ satisfies two-sided matching
Among them,μdi=sj means di is matched with sj in μ, which can be represented as di,sj and μsj=sj means sj is matched with itself, that is, it does not be matched.


#### 3.2.2. Concept of the Cloud Model

For the indicators that can only be evaluated in language, this paper adopts the cloud model to transform. The cloud model (Li et al. [31]) is an uncertain transformation that deals with qualitative concepts and quantitative numerical descriptions. The overall characteristics of the concepts described by the cloud model can be represented by three digital attributes of the cloud, expectation Ex, entropy En, and hyper-entropy He, called the eigenvectors of the cloud which are denoted by: CEx,En,He. Ex is the central position of the indicator concept in the universe of discourse and is the value that best represents the qualitative concept; En is the measure of the randomness of the qualitative concept, reflecting the degree of dispersion of the qualitative concept of cloud drops; and He is a measure of the uncertainty of entropy, which is determined by the randomness and ambiguity of entropy. Due to the general applicability of normal cloud, this paper expresses the language variables in the language evaluation set as the normal cloud model, that is, the digital features of cloud model satisfy(1)y=e−x−Ex2/2En2.In the process of matching shared manufacturing resources, reviews are generally divided into five levels. Therefore, the phrase evaluation of each indicator is also divided into five groups which, respectively, represent “very satisfied/very difficult, satisfied/difficult, basically satisfied/fair, dissatisfied/easy, very dissatisfied/very easy.” According to the right domainxmin,xmax specified by experts, five clouds can be generated. The middle cloud is C0Ex0,En0,He0. The left and right neighboring clouds are, respectively, C−1Ex−1,En−1,He−1, C+1Ex+1,En+1,He+1, C−2Ex−2,En−2,He−2, and C+2Ex+2,En+2,He+2. And, the golden section can be used to generate the following digital features:(2a)Ex0=xmin+xmax2,Ex−2=xmin,Ex+2=xmax,Ex−1=Ex0−0.382xmin+xmax2,Ex+1=Ex0+0.382xmin+xmax2,En−1=En+1=0.382xmax−xmin6,En=0.618En+1,(2b)En−2=En+2=En+10.618,(2c)He−1=He+1=He00.618,He−2=He+2=He+10.618.Let the effective universe interval be [0, 1] and the value ofHe0 is generally 0.005. “Very satisfied/very hard” corresponds to the cloud model (1, 0.104, 0.013), and “satisfied/difficult” corresponds to the cloud model (0.691, 0.064, 0.008); “basically satisfied/average” corresponds to the cloud model (0.5, 0.039, 0.005) and “unsatisfied/easy” corresponds to the cloud model (0.309, 0.064, 0.008). And, “very unsatisfied/very easy” corresponds to the cloud model (0, 0.104, 0.013). The cloud model can effectively represent qualitative language variables. The expected value of the cloud model of each language variable is used as the satisfaction value of the matching agents to construct the preference matrix, respectively.


## 3.1. Problem Description

With the development of the sharing economy and the popularization of mobile Internet technology, manufacturing enterprises tend to share production materials selectively. At this time, a third-party platform is used to integrate and release this production information and to operate and maintain the entire sharing mechanism. This paper is devoted to examining such a shared manufacturing platform that provides an accurate matching between a large number of orders and providers and helps the demanders quickly find the providers of qualified manufacturing resources and service capabilities. Also, the platform provides matching and trading services forM demanders and N providers. In the process of matching shared manufacturing resources with two-sided demand, the shared manufacturing platform provides a reasonable matching plan based on the expected level and the actual measured level of quality of service under the condition of comprehensively considering the satisfaction of both agents. The two-sided matching model of shared manufacturing resources considering psychological behavior of agents can be described in Figure 1.Figure 1
Two-sided matching model of shared manufacturing resources considering psychological behavior of agents.The basic description of the problem about two-sided matching is set out. Let the resource provider side beS=s1,s2,…sn, and sj means the jthj=1,2,…,n provider in the S. The resource demander side is D=d1,d2,…,dm, and di means the ithi=1,2,…,m demander in the D. The resource provider sj (demander di) hopes to choose a satisfied demander di (provider sj).The satisfaction of the agent of two-sided matching is calculated based on the prospect theory and grey relational analysis method. LetC=c1,c2,…cf be the set of satisfaction evaluation indicators on the quality of service from the demander di to the provider sj. ck is the kthk=1,2,…,f indicator, and each indicator is separate. The set of satisfaction evaluation C includes the following indicators: the price (c1), technical service level c2, reputation c3, and delivery date c4 (the number of days required for processing and transportation). A=a1,a2,…,ag is the set of indicators on the quality of service evaluated from the provider sj to the demander di. at is the ttht=1,2,…,g indicator, and each indicator is separate. The set of satisfaction evaluation A includes the following indicators: the technical difficulty a1 (“difficult” means that it can process all tasks below “difficult” technical requirements), reputation a2, and payment speed a3 (how many days do you want to pay 95% of the full amount after the order task is placed, and 5% is the warranty deposit). ωk is the weight corresponding to each indicator, which satisfies ωk>0and∑k=1f/gωk=1. Subsequently, the demander di gives the expected value eik and the actual measured value rijk of each indicator of the provider sj. The provider’s expected value and actual measured value of the demander’s indicators can be obtained in the same way.Combined with reality, indicators can be described as numerical values or evaluation phrases. One type is an indicator that can be measured with actual numbers such as price, and the corresponding subset is denoted asCRAR. And, the other type is an indicator that can only be evaluated by language, such as reputation. The corresponding subset is represented as CFAF and C=CR∪CFA=AR∪AF. In this paper, the cloud model is used to convert the indicators of language evaluation into quantitative standard values.


## 3.2. Preliminaries

### 3.2.1. Definition of the Two-Sided Matching

Suppose there be two agents; they are the demander and the provider. And, the set of provider isS=s1,s2,…,sn,n≥2, where sj is the jth provider, j=1,2,…,n. The set of demander is D=d1,d2,…,dm, m≥2, where di represents the ith demander, i=1,2,…,m. There is a hypothesis that m≤n. The basic model of two-sided matching is shown in Figure 2.Figure 2
Basic model of two-sided matching.Definition 1 (see [30]).
For a one-to-one mapping relationshipμ: D∪S⟶D∪S, meanwhile, ∀di∈D and ∀sj∈S satisfy the following conditions:(1)
μdi∈S∪di, if μdi=di, there is no matching agent in di(2)
μsj∈D∪sj, if μsj=sj, there is no matching agent in sj(3)
μdi=sj, if and only if μsj=di, μ satisfies two-sided matching
Among them,μdi=sj means di is matched with sj in μ, which can be represented as di,sj and μsj=sj means sj is matched with itself, that is, it does not be matched.


### 3.2.2. Concept of the Cloud Model

For the indicators that can only be evaluated in language, this paper adopts the cloud model to transform. The cloud model (Li et al. [31]) is an uncertain transformation that deals with qualitative concepts and quantitative numerical descriptions. The overall characteristics of the concepts described by the cloud model can be represented by three digital attributes of the cloud, expectation Ex, entropy En, and hyper-entropy He, called the eigenvectors of the cloud which are denoted by: CEx,En,He. Ex is the central position of the indicator concept in the universe of discourse and is the value that best represents the qualitative concept; En is the measure of the randomness of the qualitative concept, reflecting the degree of dispersion of the qualitative concept of cloud drops; and He is a measure of the uncertainty of entropy, which is determined by the randomness and ambiguity of entropy. Due to the general applicability of normal cloud, this paper expresses the language variables in the language evaluation set as the normal cloud model, that is, the digital features of cloud model satisfy(1)y=e−x−Ex2/2En2.In the process of matching shared manufacturing resources, reviews are generally divided into five levels. Therefore, the phrase evaluation of each indicator is also divided into five groups which, respectively, represent “very satisfied/very difficult, satisfied/difficult, basically satisfied/fair, dissatisfied/easy, very dissatisfied/very easy.” According to the right domainxmin,xmax specified by experts, five clouds can be generated. The middle cloud is C0Ex0,En0,He0. The left and right neighboring clouds are, respectively, C−1Ex−1,En−1,He−1, C+1Ex+1,En+1,He+1, C−2Ex−2,En−2,He−2, and C+2Ex+2,En+2,He+2. And, the golden section can be used to generate the following digital features:(2a)Ex0=xmin+xmax2,Ex−2=xmin,Ex+2=xmax,Ex−1=Ex0−0.382xmin+xmax2,Ex+1=Ex0+0.382xmin+xmax2,En−1=En+1=0.382xmax−xmin6,En=0.618En+1,(2b)En−2=En+2=En+10.618,(2c)He−1=He+1=He00.618,He−2=He+2=He+10.618.Let the effective universe interval be [0, 1] and the value ofHe0 is generally 0.005. “Very satisfied/very hard” corresponds to the cloud model (1, 0.104, 0.013), and “satisfied/difficult” corresponds to the cloud model (0.691, 0.064, 0.008); “basically satisfied/average” corresponds to the cloud model (0.5, 0.039, 0.005) and “unsatisfied/easy” corresponds to the cloud model (0.309, 0.064, 0.008). And, “very unsatisfied/very easy” corresponds to the cloud model (0, 0.104, 0.013). The cloud model can effectively represent qualitative language variables. The expected value of the cloud model of each language variable is used as the satisfaction value of the matching agents to construct the preference matrix, respectively.


## 3.2.1. Definition of the Two-Sided Matching

Suppose there be two agents; they are the demander and the provider. And, the set of provider isS=s1,s2,…,sn,n≥2, where sj is the jth provider, j=1,2,…,n. The set of demander is D=d1,d2,…,dm, m≥2, where di represents the ith demander, i=1,2,…,m. There is a hypothesis that m≤n. The basic model of two-sided matching is shown in Figure 2.Figure 2
Basic model of two-sided matching.Definition 1 (see [30]).
For a one-to-one mapping relationshipμ: D∪S⟶D∪S, meanwhile, ∀di∈D and ∀sj∈S satisfy the following conditions:(1)
μdi∈S∪di, if μdi=di, there is no matching agent in di(2)
μsj∈D∪sj, if μsj=sj, there is no matching agent in sj(3)
μdi=sj, if and only if μsj=di, μ satisfies two-sided matching
Among them,μdi=sj means di is matched with sj in μ, which can be represented as di,sj and μsj=sj means sj is matched with itself, that is, it does not be matched.


## 3.2.2. Concept of the Cloud Model

For the indicators that can only be evaluated in language, this paper adopts the cloud model to transform. The cloud model (Li et al. [31]) is an uncertain transformation that deals with qualitative concepts and quantitative numerical descriptions. The overall characteristics of the concepts described by the cloud model can be represented by three digital attributes of the cloud, expectation Ex, entropy En, and hyper-entropy He, called the eigenvectors of the cloud which are denoted by: CEx,En,He. Ex is the central position of the indicator concept in the universe of discourse and is the value that best represents the qualitative concept; En is the measure of the randomness of the qualitative concept, reflecting the degree of dispersion of the qualitative concept of cloud drops; and He is a measure of the uncertainty of entropy, which is determined by the randomness and ambiguity of entropy. Due to the general applicability of normal cloud, this paper expresses the language variables in the language evaluation set as the normal cloud model, that is, the digital features of cloud model satisfy(1)y=e−x−Ex2/2En2.In the process of matching shared manufacturing resources, reviews are generally divided into five levels. Therefore, the phrase evaluation of each indicator is also divided into five groups which, respectively, represent “very satisfied/very difficult, satisfied/difficult, basically satisfied/fair, dissatisfied/easy, very dissatisfied/very easy.” According to the right domainxmin,xmax specified by experts, five clouds can be generated. The middle cloud is C0Ex0,En0,He0. The left and right neighboring clouds are, respectively, C−1Ex−1,En−1,He−1, C+1Ex+1,En+1,He+1, C−2Ex−2,En−2,He−2, and C+2Ex+2,En+2,He+2. And, the golden section can be used to generate the following digital features:(2a)Ex0=xmin+xmax2,Ex−2=xmin,Ex+2=xmax,Ex−1=Ex0−0.382xmin+xmax2,Ex+1=Ex0+0.382xmin+xmax2,En−1=En+1=0.382xmax−xmin6,En=0.618En+1,(2b)En−2=En+2=En+10.618,(2c)He−1=He+1=He00.618,He−2=He+2=He+10.618.Let the effective universe interval be [0, 1] and the value ofHe0 is generally 0.005. “Very satisfied/very hard” corresponds to the cloud model (1, 0.104, 0.013), and “satisfied/difficult” corresponds to the cloud model (0.691, 0.064, 0.008); “basically satisfied/average” corresponds to the cloud model (0.5, 0.039, 0.005) and “unsatisfied/easy” corresponds to the cloud model (0.309, 0.064, 0.008). And, “very unsatisfied/very easy” corresponds to the cloud model (0, 0.104, 0.013). The cloud model can effectively represent qualitative language variables. The expected value of the cloud model of each language variable is used as the satisfaction value of the matching agents to construct the preference matrix, respectively.


## 4. Construction and Solution of the Two-Sided Matching Model

### 4.1. Calculation of Satisfaction Based on Prospect Theory and Grey Relational Analysis

The prospect theory uses the gap between the actual measured value and the reference point as the basis for measuring gains and losses and evaluating the quality. In this paper, taking the demander side as an example, the demander’s expected valueeik of the provider’s indicators is used as a reference. When the demander’s actual measured value rijk of the provider’s indicators is obtained, the relationship between the two agents is compared. Meanwhile, the gain or loss value of rijk to eik is obtained. When making decisions in prospect theory, people care more about the gap between the actual measured value and the reference point. Therefore, this paper introduces the grey relational analysis method. The grey relational coefficient is used to judge the closeness of expected value eik from the demander side to the provider side to the actual measured value rijk under each indicator, so as to improve the value function in the prospect theory. The provider side uses the same calculation method for the demander side.The distance between the actual measured value and the reference point is denoted as(3)dijk=rijk−eik,i=1,2,…,m;j=1,2,…,n.On this basis, the gain and loss value ofrijk relative to eik is described as(4)Bijk=dijk,rijk≥eik−dijk,rijk<eik,i=1,2,…,m;j=1,2,…,n.WhenBijk is positive, it is called gain and when it is negative, it is called loss. On the basis of the gain and loss value, the grey relational coefficient between the demander di and the provider sj under the indicator ck is defined as(5)ζijk=miniminjBijk+ρmaximaxjBijkBijk+ρmaximaxjBijk,i=1,2,…,m;j=1,2,…,n.Here, ρ is the resolution coefficient and ρ∈0,1, which is generally 0.5.Taking into account the different risk attitudes of the agent to gains and losses, the grey relational coefficient is substituted into the utility value function and the indicator prospect value is denoted as(6)Vijk=1−ζijkα,rijk≥eik−λ−ζijk−1β,rijk<eik,i=1,2,…,m;j=1,2,…,n;k=1,2,…,f.Among them,0<α<1, 0<β<1, and λ>1. The coefficient λ reflects the agent’s psychological performance of being more sensitive to losses relative to gains. The larger the λ, the greater the degree of loss aversion. Referring to He and Zhou [32], it is taken as α=β=0.88andλ=2.25.Before the weighted summation, the foreground value of the indicator needs to be de-dimensionalized to truly reflect the weight relationship of each indicator. The prospect value is normalized as(7)V¯ijk=VijkmaxVijkj=1,2.…,n,i=1,2,…,m;j=1,2,…,n;k=1,2,…,f.The satisfaction prospect value of the demanderdi to the provider sj is denoted as(8)Pij=∑k=1fωkV¯ijk,i=1,2,…,m;j=1,2,…,n;k=1,2,…,f.Correspondingly, the satisfaction prospect matrix of the demanderdi to the provider sj is VD=Pijm×n, and in the same way, the satisfaction prospect value P˜ij of the provider si to the demander di can be obtained. The satisfaction prospect matrix of the provider sj to the demander di is VS=P˜ijm×n.


### 4.2. Construction of the Model

In the process of matching the agents in shared manufacturing, there are one-to-one, one-to-many, and many-to-many situations. And, only one-to-one situation is considered here. Letxij represent a 0-1 variable where xij=0 represents that di and sj do not match in μ. And, xij=1 represents that di and sj match in μ. The multiobjective matching model is established by the prospect matrixes VD=Pijm×n and VS=P˜ijm×n as follows:(9a)maxGD=∑i=1m∑j=1nPijxij,maxGS=∑i=1m∑j=1nP˜ijxij,(9b)s.t.∑j=1nxij=1,i=1,2,…,m,(9c)∑i=1mxij≤1,j=1,2,…,n,(9d)∑i=1m∑j=1nxij=minm,n,xij∈0,1,i=1,2,…,m,j=1,2,…,n.The meaning of the objective functions is to maximize the sum of the prospect value of the demander side to the provider side and to maximize the sum of the prospect value of the provider side to the demander side. The meaning of equation (9b) is that each demander must be matched with only one provider. And, equation (9c) indicates that each provider can only be matched with at most one demander. Equation (9d) is the constraint on the number of two-sided matching, which guarantees the maximum number of successful matches and the platform’s interests.


### 4.3. Solution to the Model

In order to solve the multiobjective optimization model, letϕD and ϕS denote the weights of GD and GS, respectively, and satisfy 0<ϕD,ϕS<1 and ϕD+ϕS=1. The linear weighting method is used to convert the multiobjective model into the single-objective model:(10a)maxG=ϕD∑i=1m∑j=1nPijxij+ϕS∑i=1m∑j=1nP˜ijxij=∑i=1m∑j=1nTijxij,(10b)s.t.∑j=1nxij=1,i=1,2,…,m,(10c)∑i=1mxij=1,j=1,2,…,n,(10d)∑i=1m∑j=1nxij=minm,n,x∈0,1,i=1,2,…,m,j=1,2,…,n.Here,Tij=ϕDPij+ϕSP˜ij and ϕD and ϕS reflect the importance of two agents. Generally, the fairness of the two agents is considered, and let ϕD =  ϕS = 0.5.The process of two-sided matching of shared manufacturing resources is depicted in in Figure3 and is described as follows:Step 1: by using the numerical values and evaluation phases, the expected values and actual measured values of the indicator of the demanderdi to the provider sj and the provider sj to the demander di are obtainedStep 2: through the cloud model, the evaluation phrases are converted into numerical valuesStep 3: the grey relational coefficients between expected values and actual measured values of the demanderdi to the provider sj and the provider sj to the demander di are calculatedStep 4: by applying equations (3)–(8), the satisfaction prospect matrix Pij of di to sj and P˜ij of sj to di is obtainedStep 5: the multiobjective model in (9a)–(9d) is establishedStep 6: by using linear weighting, the multiobjective model is transformed into the single-objective one (10a)–(10d)Step 7: by solving the single-objective model, the matching results are obtainedFigure 3
Two-sided matching process of shared manufacturing resources.


## 4.1. Calculation of Satisfaction Based on Prospect Theory and Grey Relational Analysis

The prospect theory uses the gap between the actual measured value and the reference point as the basis for measuring gains and losses and evaluating the quality. In this paper, taking the demander side as an example, the demander’s expected valueeik of the provider’s indicators is used as a reference. When the demander’s actual measured value rijk of the provider’s indicators is obtained, the relationship between the two agents is compared. Meanwhile, the gain or loss value of rijk to eik is obtained. When making decisions in prospect theory, people care more about the gap between the actual measured value and the reference point. Therefore, this paper introduces the grey relational analysis method. The grey relational coefficient is used to judge the closeness of expected value eik from the demander side to the provider side to the actual measured value rijk under each indicator, so as to improve the value function in the prospect theory. The provider side uses the same calculation method for the demander side.The distance between the actual measured value and the reference point is denoted as(3)dijk=rijk−eik,i=1,2,…,m;j=1,2,…,n.On this basis, the gain and loss value ofrijk relative to eik is described as(4)Bijk=dijk,rijk≥eik−dijk,rijk<eik,i=1,2,…,m;j=1,2,…,n.WhenBijk is positive, it is called gain and when it is negative, it is called loss. On the basis of the gain and loss value, the grey relational coefficient between the demander di and the provider sj under the indicator ck is defined as(5)ζijk=miniminjBijk+ρmaximaxjBijkBijk+ρmaximaxjBijk,i=1,2,…,m;j=1,2,…,n.Here, ρ is the resolution coefficient and ρ∈0,1, which is generally 0.5.Taking into account the different risk attitudes of the agent to gains and losses, the grey relational coefficient is substituted into the utility value function and the indicator prospect value is denoted as(6)Vijk=1−ζijkα,rijk≥eik−λ−ζijk−1β,rijk<eik,i=1,2,…,m;j=1,2,…,n;k=1,2,…,f.Among them,0<α<1, 0<β<1, and λ>1. The coefficient λ reflects the agent’s psychological performance of being more sensitive to losses relative to gains. The larger the λ, the greater the degree of loss aversion. Referring to He and Zhou [32], it is taken as α=β=0.88andλ=2.25.Before the weighted summation, the foreground value of the indicator needs to be de-dimensionalized to truly reflect the weight relationship of each indicator. The prospect value is normalized as(7)V¯ijk=VijkmaxVijkj=1,2.…,n,i=1,2,…,m;j=1,2,…,n;k=1,2,…,f.The satisfaction prospect value of the demanderdi to the provider sj is denoted as(8)Pij=∑k=1fωkV¯ijk,i=1,2,…,m;j=1,2,…,n;k=1,2,…,f.Correspondingly, the satisfaction prospect matrix of the demanderdi to the provider sj is VD=Pijm×n, and in the same way, the satisfaction prospect value P˜ij of the provider si to the demander di can be obtained. The satisfaction prospect matrix of the provider sj to the demander di is VS=P˜ijm×n.


## 4.2. Construction of the Model

In the process of matching the agents in shared manufacturing, there are one-to-one, one-to-many, and many-to-many situations. And, only one-to-one situation is considered here. Letxij represent a 0-1 variable where xij=0 represents that di and sj do not match in μ. And, xij=1 represents that di and sj match in μ. The multiobjective matching model is established by the prospect matrixes VD=Pijm×n and VS=P˜ijm×n as follows:(9a)maxGD=∑i=1m∑j=1nPijxij,maxGS=∑i=1m∑j=1nP˜ijxij,(9b)s.t.∑j=1nxij=1,i=1,2,…,m,(9c)∑i=1mxij≤1,j=1,2,…,n,(9d)∑i=1m∑j=1nxij=minm,n,xij∈0,1,i=1,2,…,m,j=1,2,…,n.The meaning of the objective functions is to maximize the sum of the prospect value of the demander side to the provider side and to maximize the sum of the prospect value of the provider side to the demander side. The meaning of equation (9b) is that each demander must be matched with only one provider. And, equation (9c) indicates that each provider can only be matched with at most one demander. Equation (9d) is the constraint on the number of two-sided matching, which guarantees the maximum number of successful matches and the platform’s interests.


## 4.3. Solution to the Model

In order to solve the multiobjective optimization model, letϕD and ϕS denote the weights of GD and GS, respectively, and satisfy 0<ϕD,ϕS<1 and ϕD+ϕS=1. The linear weighting method is used to convert the multiobjective model into the single-objective model:(10a)maxG=ϕD∑i=1m∑j=1nPijxij+ϕS∑i=1m∑j=1nP˜ijxij=∑i=1m∑j=1nTijxij,(10b)s.t.∑j=1nxij=1,i=1,2,…,m,(10c)∑i=1mxij=1,j=1,2,…,n,(10d)∑i=1m∑j=1nxij=minm,n,x∈0,1,i=1,2,…,m,j=1,2,…,n.Here,Tij=ϕDPij+ϕSP˜ij and ϕD and ϕS reflect the importance of two agents. Generally, the fairness of the two agents is considered, and let ϕD =  ϕS = 0.5.The process of two-sided matching of shared manufacturing resources is depicted in in Figure3 and is described as follows:Step 1: by using the numerical values and evaluation phases, the expected values and actual measured values of the indicator of the demanderdi to the provider sj and the provider sj to the demander di are obtainedStep 2: through the cloud model, the evaluation phrases are converted into numerical valuesStep 3: the grey relational coefficients between expected values and actual measured values of the demanderdi to the provider sj and the provider sj to the demander di are calculatedStep 4: by applying equations (3)–(8), the satisfaction prospect matrix Pij of di to sj and P˜ij of sj to di is obtainedStep 5: the multiobjective model in (9a)–(9d) is establishedStep 6: by using linear weighting, the multiobjective model is transformed into the single-objective one (10a)–(10d)Step 7: by solving the single-objective model, the matching results are obtainedFigure 3
Two-sided matching process of shared manufacturing resources.


## 5. Numerical Example Analysis

In this paper, a numerical example of automobile mold parts is given and the historical data of the last year are obtained from the Mustard Network, a shared manufacturing platform, which includes 1081 orders and 23 providers. This paper extracts four demanders and five providers to match in the platform. The evaluation information is obtained by the preliminary processing and analysis of the related data of the shared manufacturing platform. The two-sided matching and verification processes are as follows.In the two-sided matching of shared manufacturing resources considering psychological behavior of agents, the demander side evaluates the satisfaction to the provider side through four indicators: pricec1, technical service level c2, reputation c3, and delivery date c4. And, the corresponding weights are 0.250, 0.231, 0.512, and 0.007. Meanwhile, the provider side evaluates the satisfaction to the demander side through three indicators: technical difficulty a1, reputation a2, and payment speed a3. And, the corresponding weights are 0.453, 0.504, and 0.043. Among them, c1 and c4∈CR and a3∈AR, where the indicator values c1, c4, and a3 are negative actual numbers. The larger the c1 and c4 are, the lower the price is and the shorter the delivery period is, which is more beneficial for the demander. And, the larger the a3 is, the shorter time the demander pays, which is better for the provider. For the sake of simplification, rounding and removing the dimensions, c2 and c3∈CF and a1 and a2∈AF, this paper uses the cloud model for numerical conversion which is consistent with the description in Section 3.2.2. The expected value and actual measured value of both demanders and providers are shown in Tables 1–4.Table 1
Expected evaluation information of demanders to providers.
DemanderProvider’s indicatorc1c2c3c4d1−6SatisfiedBasically satisfied−13d2−7Basically satisfiedBasically satisfied−8d3−6Basically satisfiedUnsatisfied−19d4−5Very satisfiedUnsatisfied−5Table 2
Actual measured evaluation information of demanders to providers.
DSc1c2c3c4d1s1−6Basically satisfiedUnsatisfied−10s2−8Very satisfiedUnsatisfied−15s3−7SatisfiedBasically satisfied−11s4−5Basically satisfiedSatisfied−18s5−6Basically satisfiedVery satisfied−14d2s1−6Basically satisfiedBasically satisfied−11s2−8SatisfiedUnsatisfied−7s3−7UnsatisfiedBasically satisfied−11s4−5UnsatisfiedVery satisfied−9s5−6SatisfiedBasically satisfied−9d3s1−6UnsatisfiedVery unsatisfied−12s2−8Basically satisfiedUnsatisfied−20s3−7Basically satisfiedSatisfied−13s4−5Very unsatisfiedVery unsatisfied−19s5−6SatisfiedUnsatisfied−17d4s1−6SatisfiedVery satisfied−7s2−8UnsatisfiedBasically satisfied−6s3−7Very unsatisfiedVery unsatisfied−10s4−5Very satisfiedBasically satisfied−5s5−6Basically satisfiedBasically satisfied−9Table 3
Expected evaluation information of providers to demanders.
ProviderDemander’s indicatora1a2a3s1AverageBasically satisfied−5s2AverageBasically satisfied−7s3HardSatisfied−7s4Very hardBasically satisfied−5s5AverageBasically satisfied−5Table 4
Actual measured evaluation information of providers to demanders.
SDa1a2a3s1d1HardVery satisfied−7d2AverageBasically satisfied−4d3HardBasically satisfied−5d4AverageBasically satisfied−6s2d1AverageSatisfied−7d2Very easyUnsatisfied−4d3EasySatisfied−5d4EasyVery satisfied−6s3d1AverageUnsatisfied−7d2HardVery satisfied−4d3Very hardSatisfied−5d4HardBasically satisfied−6s4d1AverageBasically satisfied−7d2Very hardSatisfied−4d3Very hardSatisfied−5d4HardBasically satisfied−6s5d1EasyUnsatisfied−7d2AverageSatisfied−4d3EasySatisfied−5d4Very hardVery satisfied−6According to equations (2a)–(2c), the indicators of the five-level language evaluation in Tables 1–4 are represented by the cloud model and the expected value of the five cloud models is used as the preference values of two matching agents, as shown in Tables 5–8.Table 5
Expected values of demanders to providers.
DemanderProvider’s indicatorc1c2c3c4d1−60.6910.5−13d2−70.50.5−8d3−60.50.309−19d4−510.309−5Table 6
Actual measured values of demanders to providers.
DSc1c2c3c4d1s1−60.50.309−10s2−810.309−15s3−70.6910.5−11s4−50.50.691−18s5−60.51−14d2s1−60.50.5−11s2−80.6910.309−7s3−70.3090.5−11s4−50.3091−9s5−60.6910.5−9d3s1−60.3090−12s2−80.50.309−20s3−70.50.691−13s4−500−19s5−60.6910.309−17d4s1−60.6911−7s2−80.3090.5−6s3−700−10s4−510.5−5s5−60.50.5−9Table 7
Expected values of providers to demanders.
ProviderDemander’s indicatora1a2a3s10.50.5−5s20.50.5−7s30.6910.691−7s410.5−5s50.50.5−5Table 8
Actual measured values of providers to demanders.
SDa1a2a3s1d10.6911−7d20.50.5−4d30.6910.5−5d40.50.5−6s2d10.50.691−7d200.309−4d30.3090.691−5d40.3091−6s3d10.50.309−7d20.6911−4d310.691−5d40.6910.5−6s4d10.50.5−7d210.691−4d310.691−5d40.6910.5−6s5d10.3090.309−7d20.50.691−4d30.3090.691−5d411−6Besides, the grey relational coefficients between the expected value under each indicator and the actual measured value are calculated by equations (3)–(5), which can be shown in Tables 9 and 10.Table 9
Grey relational coefficients between the expected values and the actual measured values of demanders to providers.
DSc1c2c3c4d1s110.72360.64400.5385s20.42860.61800.64400.6364s30.6110.6364s40.60.72360.64400.4118s510.72360.40860.7778d2s10.6110.5385s20.60.72360.64400.7778s310.723610.5385s40.42860.72360.40860.7778s50.60.723610.7778d3s110.72360.52790.3333s20.4286110.7778s30.610.47490.3684s40.60.50.52791s510.723610.6364d4s10.60.61800.33330.6364s20.33330.41980.64400.7778s30.42860.33330.52790.4118s4110.64401s50.60.50.64400.4667Table 10
Grey relational coefficients between the expected values and the actual measured values of providers to demanders.
SDa1a2a3s1d10.56690.33330.4286d2110.6d30.566911d4110.6s2d110.56691d20.33330.56690.3333d30.56690.56690.4286d40.56690.33330.6s3d10.56690.39561d210.44720.3333d30.447210.4286d410.56690.6s4d10.333310.4286d210.56690.6d310.56691d40.447210.6s5d10.56690.56690.4286d210.56690.6d30.56690.56691d40.33330.33330.6The grey relational coefficient is put into the utility value function of prospect theory, and the prospect matricesVD and V˜D are calculated by equations (6)–(8), which can be shown in Table 11.Table 11
Satisfaction prospect matrix.
Matrixs1s2s3s4s5VD−0.5701−0.6662−0.18060.21610.37170.1041−0.6160−0.11210.38740.1688−0.6814−0.22280.0706−0.56040.0632−0.1783−0.2580−1.00000.1775−0.1604V˜D0.33890.1671−0.8139−0.4960−0.72880.0274−0.91400.32070.25140.25140.20130.09400.29260.50400.0510−0.0430−0.0188−0.4849−0.49600.5215From the satisfaction prospect matrix, this paper constructs a multiobjective optimization model, takingϕD = ϕS = 0.5 and converting the multiobjective optimization model to a single-objective optimization model, and the coefficient matrix T=tij4×5 is obtained, which can be shown in Table 12.Table 12
Coefficient matrix.
DemanderProviders1s2s3s4s5d1−0.1156−0.2496−0.4973−0.1400−0.1785d20.0658−0.76500.10430.31940.2101d3−0.2401−0.06440.1816−0.02820.0571d4−0.1107−0.1384−0.7424−0.15920.1805The software package Matlab R2018b can be used to solve the above model, the objective value is 0.5660, and the matching result isμd1,s1,d2,s4,d3,s3,d4,s5; that is, the demander 1 matches the provider 1, the demander 2 matches the provider 4, the demander 3 matches the provider 3, and the demander 4 matches the provider 5. The provider 2 has not obtained the demander.


## 6. Conclusions

Shared manufacturing resources are different from traditional manufacturing resources. The complexity of the transaction process and the characteristics of the manufacturing industry determine the particularity of service matching options. A two-sided model of shared manufacturing resources considering psychological behavior of agents is proposed in this paper. The cloud model is used to realize the conversion from qualitative evaluation to quantitative evaluation, which can solve the fuzziness of the linguistic evaluation. In accordance with preference information extracted by using the cloud model, the satisfaction prospect matrix is constructed through the combination of prospect theory and grey relational analysis. Based on it, the satisfaction prospect values of both agents are comprehensively considered. The model of two-sided matching for maximizing satisfaction prospect values from both agents is constructed. For the sake of obtaining the matching results, the solution is given. Meanwhile, the validity and rationality of the model are verified through a numerical example. Subsequently, the model is applied to solve the matching problem for shared manufacturing resources.The future research of this research is as follows. First, it can be used to consider the intelligent optimization algorithm for shared manufacturing resource allocation. Second, the supply chain capability between agents of shared manufacturing will be investigated. Third, the application of implementing the one-to-many and many-to-many matching for shared manufacturing resources will be another work.



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