Research on the Evaluation of Chinese Prefabricated Building Strategic Partners Based on Cloud Model and Improved Evidence Theory

Abstract

In recent years, prefabricated buildings have developed rapidly in China. Compared with traditional buildings, prefabricated buildings require higher capabilities from partners in various aspects. However, due to the early stage of development of prefabricated buildings in China, the level of various enterprises varies greatly. How to evaluate partners scientifically and objectively is a realistic problem that needs to be solved urgently. In order to achieve economies of scale and promote the sustainable development of prefabricated buildings, this study proposes a novel evaluation model for strategic partner selection based on the cloud model and improved Dempster–Shafer (D-S) evidence theory. First, using a literature review and field research method, a strategic partner selection index system is developed that can reflect the characteristics of prefabricated buildings. To address the fuzziness and randomness of the traditional membership function, the cloud model is applied to calculate the membership value between the test samples and the benchmark cloud, which is subsequently transformed into basic probability distribution in the evidence theory. Furthermore, to mitigate the paradox of evidence fusion often encountered in traditional evidence theory, this model combines both the subjective and objective weights of evidence by game theory, and the conflicting evidence is corrected and fused according to the combination weight. Additionally, the Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) method is used to further optimize the strategic partners of prefabricated buildings. Finally, the optimal order obtained from the case analysis is S₁ > S₂ > S₄ > S₃ > S₅, and the evaluation results are consistent with the actual situation, which verifies the effectiveness and superiority of the proposed model in resolving the evidence conflict and selecting strategic partners. The research results have certain reference significance for optimizing the selection mechanism of prefabricated building strategic partners and guiding partners to establish long-term and stable cooperative relations.

1. Introduction

Traditional construction methods, characterized by high energy consumption, significant pollution, low operational efficiency, and extensive production practices fall short of meeting the requirements for industrialization and sustainable urbanization [1]. In 2022, the China Building Energy Conservation Association released a report on carbon emissions in the construction industry, which shows that by 2020, the total energy consumption of the whole construction process has reached 2270 Mtce, representing 45.5% of the country’s total energy consumption [2]. Additionally, total CO₂ emissions were reported to be as high as 5.08 billion tons, accounting for 50.9% of the nation’s total carbon emissions. As energy consumption in the construction sector continues to rise, the industry faces formidable energy-saving challenges. Additionally, China’s demographic dividend is gradually diminishing; the aging workforce phenomenon is intensifying, with fewer young people entering technical roles. This generational shift, combined with the declining interest in traditional construction jobs among the new generation of migrant workers, has contributed to rising labor costs and exacerbating labor shortages in the construction industry.

Prefabricated buildings, as an innovative production model, offer several advantages, including high production efficiency, enhanced quality control, optimized resource utilization, and reduced waste discharge [3]. Since the 18th National Congress of the Communist Party of China emphasized “new industrialization and informatization”, government entities at various levels have intensively introduced several policies aimed at promoting the rapid development of prefabricated buildings. As a result, prefabricated buildings have gradually become a key means for transforming and upgrading the construction industry, driving industrial modernization and catalyzing significant shifts within the sector.

However, there are numerous challenges in the application of prefabricated buildings in China, with high construction costs being one of the important factors hindering the development [4]. Research indicates that economies of scale can effectively reduce construction costs by identifying optimal project management models, so as to enable mass production of prefabricated components [5]. At present, the majority of prefabricated construction projects in China continue to follow the traditional management model, which emphasizes individual responsibility, risks, and interests of each participant. This model often overlooks the need for resource integration and exchange of information between organizations, with cooperative relationships often ending at the completion of the project. Due to such single-use or short-term cooperation, production molds for prefabricated components are frequently used only once, leading to high initial investment costs and limiting opportunities for standardization and mass production, thus inhibiting the benefits of economy of scale [6].

In order to achieve lean production and economies of scale, real estate developers strive to seek strategic partnerships with general contractors with strong comprehensive strength [7]. Through strategic cooperation, enterprises can achieve resource complementarity, information sharing, and risk sharing. However, as prefabricated buildings in China are in the early stages of development, the level of various enterprises is uneven [8]. How to evaluate and select them scientifically and objectively is a realistic issue that needs to be resolved. In addition, considering the technical characteristics and management particularity of prefabricated buildings, the existing evaluation system is not fully applicable to the optimization of prefabricated building contractors [9]. In terms of evaluation methods, most of the existing studies still adopt the traditional single evaluation method, which does not consider the importance of the evidence itself and cannot distinguish the advantages and disadvantages of the evaluation objects at the same level. This paper constructs an index system for optimizing contractors and adopts a cloud model and improved evidence theory to select and sort them.

2. Literature Review

2.1. Research on Partnership of Prefabricated Building

Girmscheid et al. [10] argued that complex processes hinder the development of prefabricated buildings. By establishing cooperative relationships among multiple stakeholders, the tendency for opportunistic behavior can be minimized. The study further proposes that such cooperative relationships can improve the prefabrication levels and provide solutions for the implementation of mass customization. Mohammad et al. [11] concluded that the obstacles to effective cooperation in prefabricated buildings include lack of communication mechanisms at the executive level, lack of common goals, significant cultural differences between enterprises, limited engagement of design units, inadequate industry standards and norms, and lack of trust. Halil et al. [12] explored the concept of the prefabricated building partnership and suggested that long-term cooperation, common vision, mutual trust, and continuous improvement are key elements of this relationship. Mostafa et al. [13] emphasized that in order to reduce construction costs, prefabricated building stakeholders should establish long-term cooperative relationships, collaboratively develop industry standards, and pursue mass production. Based on the cooperative game theory, Zhao et al. [14] used the Shapley value method of the prefabricated building chain model to distribute benefits, and their research made up for the theoretical gap in the distribution of benefits in the prefabricated building industry chain. Yang et al. [15] developed an evolutionary game model to study the collaborative behavior among core enterprises of prefabricated buildings under government intervention and used system dynamics to analyze the influence of initial strategies and key parameters on the strategic evolution pathways, filling the research gap in the relationship between collaborative behavior and production performance of core enterprises.

2.2. Research on Evaluation Index of Strategic Partners

To model the construction partnership, Cheng et al. [16] developed a process model called the procedural mapping model (PMM) by integrating various aspects of the system and process, including success factor detection, interactive process description, and target matrix evaluation. Liu [17] proposed establishing an integrated industrial system with contractors as the core entity, extending upstream to cover process design and downstream to include the main body of research and development of prefabricated components. Li et al. [18] established an information platform that includes design and development, production, transportation, and construction. By facilitating information sharing and resource complementarity, this platform enhanced communication frequency, and mutual trust among partners, thereby significantly improving the project performance. In another study, Yue et al. [19] reported that a strategic alliance of prefabricated buildings covering design institutions, sales teams, transportation companies, and operation service companies should be established, with prefabricated component manufacturers being the core. Liu et al. [20] applied an artificial neural network for partner selection and evaluation within a supply chain environment, enhancing decision-making accuracy. Xia et al. [21] explored the factors that affect green innovation within the integrated supply chain and found that the choice of technology innovation partners is a key factor. Zhao et al. [22] analyzed the importance of supplier selection from the perspective of owner procurement, constructed a supplier selection model for prefabricated components in prefabricated buildings based on hesitant fuzzy sets and prospect theory, and verified the effectiveness and feasibility of the model through numerical examples. Hua et al. [23] investigated the relationship between Building Information Modeling (BIM) technology and prefabricated building supply chain resilience. The study showed that BIM resources and capabilities can enhance supply chain resilience by influencing both participant and partnership dynamics.

2.3. Research on Evaluation Method of Strategic Partners

Considering the ambiguity and uncertainty of partner information, Zhang et al. [24] proposed a combined evaluation method based on the Fuzzy Analysis Network Process (FANP) and TOPSIS to solve the evaluation problem of science and technology collaborative innovation partners. Ayadi et al. [25] proposed a fuzzy collaborative evaluation method for partner trust assessment in horizontal collaborative networks. A standard relationship analysis method based on the Decision-making Trial and Evaluation Laboratory (DEMATEL) and triangular fuzzy numbers (TFNs) was proposed by Li et al. and applied to a real case of sustainable recycling partner selection [26]. Song et al. [27] proposed a supplier selection evaluation model based on mechanism equation modeling and intuitionistic fuzzy analytic hierarchy process. The research results showed that quality, long-term cooperation, after-sales service, and transportation are the key factors affecting supplier selection. Ji et al. innovatively proposed the evaluation and prediction system based on the Driving Stress State Impact Response (DPSIR), TOPSIS, and GM (1,1) models, which could be used for evaluating development performance between social and environmental factors among various cause-effect relationships [28]. Zhao et al. [29] established a collaborative innovation intellectual property partner selection model by combining the maximum entropy model with the grey correlation method. Based on the theory of combination weighting and interval approximation, Liao et al. [30] established a risk analysis and evaluation model for prefabricated building construction.

2.4. Existing Research Gaps

Currently, research on partner selection indicators and partner selection evaluation methods has achieved notable advancement, laying a foundation for our study. However, there is a significant gap in research regarding the selection of strategic partners for prefabricated buildings, which are summarized as follows:

1.

The current research on project partners mainly focuses on the analysis of traditional cooperative relationships, with an emphasis on the selection and evaluation of partners from a single buyer’s perspective. This single-sided approach lacks the reciprocal perspective, which is essential for assessing mutual benefits between both parties [31]. Unlike traditional buildings, the development of prefabricated buildings requires developers and general contractors to establish a long-term and stable cooperative relationship. Effective management of this partnership is key to the success of the project [32].

2.

Most studies on strategic cooperation of prefabricated buildings mainly focus on supply chain cooperation, specifically among contractors, subcontractors, and suppliers [33]. However, there is a lack of research on the selection of general contractors who play a strategic core role. Additionally, current evaluation indicators or models are often inadequate, as they do not fully consider the unique characteristics of the prefabricated construction industry or the specific attributes of the enterprises involved.

3.

The existing selection and evaluation methods of strategic partners including the analytic hierarchy process, fuzzy comprehensive evaluation method, group decision-making, matter-element extension method, and artificial neural network (ANN) method, do not encompass a composite combination evaluation approach. Consequently, a standardized and systematic evaluation system has yet to be developed. Since the selection of strategic partners for prefabricated buildings is characterized by multi-attribute, complexity, and diversification, most of the existing literature only makes a simple evaluation of partners, often ignoring the importance of the evidence itself. In addition, these studies can only assess the level of the research object, and it is difficult to compare the objects of the same level [34].

Therefore, establishing an evaluation index system for general contractors as strategic partners and seeking scientific and objective evaluation methods are the focus and difficulty of current research.

3. Methodology

The following section aims to establish a strategic partner selection evaluation model for prefabricated buildings and mainly includes the following aspects: (i) a combination of literature research and project investigation interviews is used to establish a strategic partner index system for prefabricated buildings; (ii) considering the multi-attribute, complexity and diversification of the selection of strategic partners for prefabricated buildings, the cloud model is used to generate the membership degree corresponding to each index, and subsequently transformed into the basic probability distribution in the evidence theory; (iii) given the conflicting evidence problems related to “combination contradiction”, “one-vote veto” and “robustness” in traditional evidence theory, the static and dynamic weights of evidence are combined by game theory, and the conflict correction and fusion are carried out according to the combination weights [35]. Simultaneously, referring to the idea of TOPSIS average closeness degree, the preferred order of partners is determined by calculating the basic probability distribution of each potential partner and the closeness degree of positive and negative ideal solutions.

3.1. Establishment of Strategic Partner Optimization Index System

3.1.1. Influence Factor Identification and Analysis

This present study identifies and summarizes the general requirements for evaluating enterprise partner selection through an extensive review of existing literature, focusing on common indicators that can reflect the basic requirements and conditions of the enterprise to be evaluated [36]. Given that the development of prefabricated buildings in China is still in its early stages, there is relatively little literature on strategic partnership research [37,38]. To address this gap, based on the identified common indicators, this study conducts field research on representative prefabricated building projects, refines the personality indicators, and obtains the final evaluation index system after comprehensive research and judgment. In summary, the selection indicators of prefabricated building strategic partners are divided into four categories: interests, capabilities, costs, and compatibility [39,40,41].

1.

Benefits

The primary motivation for establishing a strategic partnership between owners and contractors in prefabricated construction is to achieve mutual, enhanced benefits [42]. To strengthen market competitiveness and realize core business values, enterprises often enter into formal cooperative relationships through various contracts.

(i)

Experience in prefabricated construction projects: Prefabricated building technology has distinct characteristics that require high capabilities and experience from the partnering units. When the selected partner lacks relevant experience, it can introduce significant risks to safety and quality management, and also construction timeline control [43].

(ii)

Innovation level of prefabricated building technology: Technological innovation is a critical competency for enterprises, with a prominent role in the field of prefabricated construction. It is difficult to continue cooperation between partners if they do not meet the requirements of the technological innovation capabilities. Therefore, technological innovation capability is an important indicator for selecting strategic partners for prefabricated construction.

(iii)

Supply level of prefabricated buildings: This indicator measures a company’s ability to produce and supply prefabricated components or has a stable supply channel for prefabricated components [44]. Contractors with good, prefabricated building supply capabilities not only ensure the quality of prefabricated components but also cooperate with product delivery plans for continuous supply, facilitating timely delivery.

(iv)

Financial competency of the enterprise: Currently, the capital turnover of the real estate and construction industries has significantly decreased, increasing difficulty in enterprise operation. In China’s developing prefabricated buildings sector, economies of scale have yet to materialize, making prefabricated construction methods relatively more costly than traditional methods. Therefore, when choosing strategic partners, full attention should be paid to the financial management capabilities of potential partners.

(v)

Reserve of prefabricated construction talents: Shortage of skilled professionals is a key factor hindering the rapid development of prefabricated buildings. Skilled management and technical personnel with a systematic understanding of the various processes and concepts of prefabricated buildings can ensure the smooth implementation of projects.

2.

Ability

In the selection of strategic partners, enterprises should assess not only whether the resources owned by potential partners are complementary, but also whether potential partners possess the necessary capabilities required by the enterprise. A partnership can only be effective and sustainable if the partners demonstrate relevant capabilities [45].

(i)

Prefabricated building quality management system: The characteristics of factory production and on-site assembly construction of prefabricated components or parts require a fundamental transformation of the traditional quality management system. The quality management system should be developed around the production, transportation, lifting, and assembly of prefabricated components, and the standards for control should be detailed and clear.

(ii)

Specialized organizational management level of prefabricated buildings: This indicator evaluates the organizational ability of enterprises in prefabricated construction. Compared to traditional construction methods, prefabricated buildings require meticulous attention to construction organization including setting of on-site stacking areas for prefabricated components, logistics routes for transport vehicles both within and outside the site, the selection of lifting equipment, synchronization of production and construction plans for prefabricated components, lifting scheme for prefabricated components, and the support system for prefabricated components [46].

(iii)

The ability to respond quickly to market demand: The smooth production of prefabricated components determines whether the project schedule can be implemented on time and is closely related to mold preparation and the supply of raw materials such as steel bars and concrete. In practice, delays in the prefabricated components production due to an untimely supply of raw materials or delays in construction due to shortages of transportation equipment occur occasionally [47].

(iv)

Adaptability to new technologies: The ability to adapt to new technologies reflects a company’s ability to new technological awareness, enabling it to identify the emergence of new technologies and equipment in the market. This adaptability is essential for responding quickly and continuously improving its comprehensive strength through learning.

3.

Cost

Currently, prefabricated buildings are still in the initial stage of development in China, and the relatively high cost has become an important factor affecting the development of this sector [48]. In this study, the cost evaluation index is divided into the following four aspects: (a) prefabricated component product price (this covers many components, including the cost of the main material, auxiliary material, embedded material, mold, depreciation, labor, manufacturing, factory inspection, management, profit, and transportation); (b) purchasing cost of prefabricated components; (c) transportation cost of prefabricated components; and (d) repair cost of prefabricated components including both the repair expenditure and the repair rate of components, as well as the construction delay loss [49].

4.

Compatibility

Compatibility is the most important prerequisite in the selection of strategic partners. Previous studies have shown that in case of low compatibility, even highly capable cooperative units struggle to synchronize effectively, often leading to friction and, ultimately, undermining the benefits of cooperation [50].

(i)

Whether the strategic objectives are compatible: Compatibility of strategic objectives is the first criterion for selecting strategic partners. Although owners and contractors operate in different corporate cultures and systems, they must work towards a shared goal, otherwise, it can lead to a breakdown of cooperative relationships.

(ii)

Whether the organizational structure is compatible: Compatible organizational structure helps to form a counterpart relationship between partners around business development and improves communication across different levels. Such compatibility blurs the boundary between organizational structure, which enhances the rapid flow of information, material, and capital during collaboration, and ultimately improves the cooperation performance.

(iii)

Whether the culture is compatible: Corporate culture, a reflection of the enterprise values, behavior style, and concept varies significantly across enterprises due to differences in development environments, management systems, and leadership styles. As each enterprise introduces its own corporate culture into the cooperative relationship, the compatibility of these cultures will critically affect the effectiveness of cooperation.

(iv)

Whether the willingness to cooperate is compatible: A mutual willingness to carry out long-term, stable cooperation will enable both parties to measure the benefits of cooperation from a long-term perspective. This will reduce the occurrence of opportunistic behavior, avoid conflicts of interest in short-term cooperation, facilitate information and risk sharing, and achieve the unity of goals and interests in specific project cooperation [51].

3.1.2. Establishment of the Index System

Scientific index evaluation system is the basis of an accurate and reasonable assessment [52]. Based on the analysis of the partner selection index as detailed in Section 3.1.1, this study develops an evaluation index system specifically for the selection of assembly building strategic partners, as shown in

Table 1. Evaluation index system for selecting strategic partners in prefabricated construction.

Evaluation index system for selecting strategic partners in prefabricated construction (S)	Primary index	Secondary index
	Benefits (C1)	Experience in prefabricated construction projects (C11)
		Innovation level of prefabricated building technology (C12)
		Supply level of prefabricated buildings (C13)
		Financial condition of the enterprise (C14)
		Reserve of prefabricated construction talents (C15)
	Ability (C2)	Prefabricated building quality management system (C21)
		Specialized organizational management level of prefabricated buildings (C22)
		The ability to respond quickly to market demand (C23)
		Adaptability to new technologies (C24)
	Costs (C3)	Prefabricated component product price (C31)
		Purchasing cost of prefabricated components (C32)
		The transportation cost of prefabricated components (C33)
		Repair cost of prefabricated components (C34)
	Compatibility (C4)	Whether the strategic objectives are compatible (C41)
		Whether the organizational structure is compatible (C42)
		Whether the culture is compatible (C43)
		Whether the willingness to cooperate is compatible (C44)

.

3.2. Combination Weighting Based on G1-Improved CRITIC Method

3.2.1. G1 Analysis Method

The G1 analysis method is a subjective weighting method. Contrary to the analytic hierarchy process, this method does not require the construction of a judgment matrix, let alone a consistency test. It has the advantage of simple calculation and no requirement for the number of indicators [53].

1.

Determination of order relation. Firstly, the evaluation index set $D=\left\{D_1,D_2,\dots ,D_n\right\}$ is determined, and the most important index is selected according to the evaluation criteria, which is recorded as $D_1^{\ast }$. Then, among the remaining indicators, the most important indicator is selected and recorded as $D_2^{\ast }$. This process is repeated iteratively, selecting the most significant indicator from the remaining set at each step, until the final indicator $D_n^{\ast }$ is reached.

2.

Determine the relative importance between adjacent indicators: Calculate the ratio of importance between the evaluation index $D_{k-1}^{\ast }$ and $D_k^{\ast }$ using Equation (1).

$$f_k={\displaystyle \frac{w_{k-1}^{\ast }}{w_k^{\ast }}}$$

(1)

where $w_{k-1}^{\ast }$ and $w_k^{\ast }$ are the weight coefficients of $D_{k-1}^{\ast }$ and $D_k^{\ast }$, respectively.

3.

Determine the weight coefficient: The subjective weight coefficient can be calculated according to Equations (1) and (2):

$$w_n^{\ast }={\left[1+{\displaystyle \sum _{k=2}^n\left({\displaystyle \prod _{j=k}^n{f}_j}\right)}\right]}^{-1}$$

(2)

3.2.2. Improved CRITIC Weighting Method

The Criteria Importance Through Intercriteria Correlation (CRITIC) method is a well-established objective weighting analysis method. In this approach, the objective weight of the index is mainly determined by the discrimination and conflict of the index to be tested. Discriminability represents the degree of difference of an indicator in different sample values. If the difference is greater, the discrimination is stronger. There are two inherent problems in the conventional CRITIC weighting method. First, the dimension and order of magnitude of each index are different, and the standard deviation with dimension cannot directly compare the discrimination of the index. Second, the coefficient reflecting the conflict between indicators may be negative. It is generally believed that whether it is a positive or negative correlation, similar absolute values indicate that the correlation is the same. Therefore, this study effectively improves the CRITIC weighting method using the following steps [54]:

1.

The coefficient of variation $v_j$ was calculated according to Equation (3).

$$v_j={\displaystyle \frac{s_j}{{\overline{x}}_j}}\hspace{1em}j=1,2,\dots ,n$$

(3)

where ${\overline{x}}_j={\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n{x}_{ij}}$ and $s_j=\sqrt{{\displaystyle \frac{{\displaystyle \sum _{i=1}^n{\left(x_{ij}-{\overline{x}}_j\right)}^2}}{n-1}}}$.

2.

Calculate the conflict coefficient using Equation (4).

$$R_j={\displaystyle \sum _{i=1}^n\left(1-\left|r_{ij}\right|\right)}\hspace{1em}j=1,2,\dots ,n$$

(4)

where $r_{ij}$ is the correlation coefficient between the ith index and the jth index.

3.

Calculate the amount of information using Equation (5).

$$C_j=v_j\cdot R_{j}j=1,2,\dots ,n$$

(5)

3.2.3. Improved Game Theory Combination Weighting

Game theory combination weighting is essentially a compromise between the weights obtained by different methods, to minimize the deviation between the optimal weight and the basic weight value [55]. In this paper, a combination weighting method based on game theory is proposed. The static weight and dynamic weight of evidence are integrated to obtain the combination weight. Based on this, the conflict evidence is determined and its basic probability distribution function is modified. This method not only considers the differences between evidence but also retains the original information as much as possible to improve the accuracy of evidence fusion.

Assuming that the number of indicators to be evaluated is n and there are P methods for calculating and analyzing the index weight. Then the number of basic weight sets is P, and a weight set can be expressed as $w_p=\left\{w_{p1},w_{p2},\dots ,w_{pn}\right\}$. Therefore, any linear combination of basic weight sets can be expressed as follows:

$$w={\displaystyle \sum _{p=1}^p{\alpha }_p{w}_p^T}$$

(6)

The improved game theory combination weighting objective function and constraint conditions are:

$$\begin{gathered} \underset{{\alpha }_1,{\alpha }_2,\dots ,{\alpha }_p}{\min }f={\displaystyle \sum _{t=1}^p\left|\left({\displaystyle \sum _{t=1}^p{\alpha }_p{w}_t{w}_p^T}\right)-w_t{w}_p^T\right|} \\ \mathrm{s}.\mathrm{t}.{\displaystyle \sum _{p=1}^p{\alpha }_p^2}=1 \end{gathered}$$

(7)

where ${\alpha }_p>0$, $p=1,2,\dots ,P$.

By solving the model, the optimal combination weight considering subjective human factors and objective data rules can be obtained. The problem is to solve the minimum value under the equality constraint condition, and the Lagrangian function is constructed as follows [56]:

$$L({\alpha }_p,\lambda )={\displaystyle \sum _{t=1}^p\left|\left({\displaystyle \sum _{t=1}^p{\alpha }_p{w}_t{w}_p^T}\right)-w_t{w}_p^T\right|}+{\displaystyle \frac{\lambda }{2}}\left({\displaystyle \sum _{p=1}^p{\alpha }_p^2-1}\right)$$

(8)

The combination coefficient solution can be expressed as:

$${\alpha }_p={\displaystyle \frac{{\displaystyle \sum _{t=1}^p{w}_t{w}_p^T}}{\sqrt{{\displaystyle \sum _{p=1}^p{\left({\displaystyle \sum _{t=1}^p{w}_t{w}_p^T}\right)}^2}}}}$$

(9)

3.3. Evaluation Model Based on Cloud Model and D-S Evidence Theory

The selection of strategic partners for prefabricated buildings is essentially a multi-attribute evaluation and decision-making problem. The key aspect of this process is to calculate the membership degree of each index across various evaluation levels and integrate this degree with multi-source information to yield a comprehensive membership degree to determine the final evaluation result. Traditional methods of determining membership degree are usually unable to take into account the randomness and fuzziness in multi-attribute decision-making. The cloud model has advantages in solving the problems of fuzziness and randomness and can transform the index data value into a qualitative description belonging to a certain risk level. As one of the most established methods for information fusion, D-S theory is an effective tool for dealing with uncertain and fuzzy information. Therefore, to ensure the evaluation results are more reliable, this study combines the combination weighting, cloud model, and improved D-S theory to construct a strategic partner selection evaluation model for prefabricated buildings.

3.3.1. Cloud Model Theory

1.

Introduction to the theory

The cloud model is a theoretical framework that facilitates the conversion between qualitative and quantitative concepts [57]. It can not only explain natural language through classical probability theory and fuzzy mathematics but also reflect the relationship between randomness and fuzziness. The three digital features of expectation $Ex$, entropy $En$ and hyper entropy $He$ of cloud model can describe fuzziness, randomness, and their correlation. $Ex$ can represent the corresponding fuzzy information in the center value of quantitative evaluation. $En$ reflects the dispersion degree and fluctuation range of cloud droplets. $He$ signifies the stability of cloud droplets and represents the uncertainty of entropy.

2.

Calculation of cloud digital features and membership degree

Firstly, the cloud eigenvalue $\left(Ex,En,He\right)$ of each evaluation index grade is determined according to the evaluation grade classification standard. Assuming that a data point is expressed as $x_{ij}$, where $i$ represents the evaluation index and $j$ denotes the evaluation level of the data matching, the upper and lower boundary values of data $x_{ij}$ are denoted as $x_{ij}^1$ and $x_{ij}^2$, respectively. Then the total number of cloud models can be obtained as $i\times j$ and the expected value can be expressed as:

$$E{x}_{ij}=\left(x_{ij}^1+x_{ij}^2\right)/2$$

(10)

Considering the boundary of each level belongs to both the previous levels and the next levels, then the membership degree $\mu =\exp (-{\displaystyle \frac{{(x_{ij}^1-x_{ij}^2)}^2}{8E_{n_{ij}}^2}})=0.5$ can be obtained. The entropy value ($En$) can be obtained as:

$$E{n}_{ij}=\left(x_{ij}^2-x_{ij}^1\right)/2.355$$

(11)

According to the calculated cloud feature parameter values and the actual evaluation index data after screening, the membership degree of each index matching each grade is obtained by using the X-conditional cloud generator algorithm. The membership degree matrix $U={[{\mu }_{ij}]}_{m\times n}$ is formed by the membership degree of each index, and the membership degree of the actual index data $x_0$ is expressed as follows [58]:

$$\begin{gathered} {\mu }_0=\exp \left[-{\displaystyle \frac{{(x_0-Ex)}^2}{2{\left(Enn\right)}^2}}\right] \\ Enn=En+He\ast Randbetween(-1,1) \end{gathered}$$

(12)

where $Enn$ is a normal random distribution with $En$ as expectation and $He$ as standard deviation. $He$ is usually based on experience, which is set to $En/10$ for this study.

3.3.2. D-S Evidence Theory and Evidence Fusion Process

1.

The connotation of D-S evidence theory

Evidence theory is a method for uncertain reasoning, primarily applied to the fusion of uncertain information [59]. Under the recognition framework $\mathrm{\Theta }$, the set of all subsets is represented as $2^{\mathrm{\Theta }}$, and the function $m:2^{\mathrm{\Theta }}$ is called the basic probability assignment function, which satisfies the following condition:

$$\left\{\begin{array}{c}m(\phi )=0\\ {\displaystyle \sum _{A\subseteq \mathrm{\Theta }}m(A)=1}\end{array}\right.$$

(13)

2.

Evidence combination rules of D-S evidence theory

To effectively use multiple items of information, it is essential to apply fusion rules to process evidence. Specifically, D-S synthesis rules are employed for this purpose, which is as follows:

$$m(A)=\left\{\begin{array}{c}{\displaystyle \frac{{\displaystyle \sum _{A_i\cap B_j\cap C_l\dots =A}{m}_1(A_i)m_2(B_j)m_3(C_l)\dots }}{1-k}}\\ 0,A=\varnothing \end{array}\right.,A\ne \varnothing$$

(14)

where $k={\displaystyle \sum _{A_i\cap B_j\cap C_l\dots =\varnothing }{m}_1(A_i)m_2(B_j)m_3(C_l)\dots }$, k reflects the degree of conflict between the evidence.

3.

Improved D-S evidence theory fusion process

Evidence theory is one of the well-established methods of information fusion, particularly advantageous in solving uncertain reasoning. However, the traditional evidence theory has several limitations such as “combination contradiction”, “one-vote veto”, and “robustness” in the face of conflicting evidence. To address these challenges, this paper improves the D-S evidence theory to enhance the fusion of conflicting evidence. The specific implementation process is outlined as follows [60]:

(i)

The cloud model is used to transform the value of each index into the membership degree corresponding to each grade, and then the membership degree is processed by the normalization method.

$$\left\{\begin{array}{c}{\theta }_i=1-\max ({\mu }_{i1},{\mu }_{i2},\dots ,{\mu }_{iq})\\ m_i(\mathrm{\Theta })={\theta }_i\\ m(R_j)=(1-{\theta }_i){\mu }_{ij}/{\displaystyle \sum _{j=1}^h{\mu }_{ij}}\end{array}\right.$$

(15)

(ii)

Calculate the weight of evidence

Determine the average basic probability value of $n$ sets of evidence for each proposition $A_k$.

$$\overline{m}(A_k)={\displaystyle \frac{1}{n}}{\displaystyle \sum _{k=1}^n{m}_i(A_k)}k=1,2,\dots ,n$$

(16)

Calculate the degree of trust in turn.

$$d_i={\displaystyle \sum _{k=1}^n\left|m_i(A_k)-\overline{m}(A_k)\right|} k=1,2,\dots ,n$$

(17)

The weight ($w_i$) of the evidence is assigned according to the size of $d_i$.

$${\varphi }_i={\displaystyle \frac{1}{d_i}}\hspace{1em}{w}_i={\displaystyle \frac{{\varphi }_i}{sum{\varphi }_i}}$$

(18)

(iii)

Correct the conflict evidence

The formula for calculating the correction coefficient of conflict evidence $i$ is as follows:

$$discount={\displaystyle \frac{w_i}{\max (w_i)}}$$

(19)

After the correction coefficient is determined, it is corrected according to the following formula:

$$m_i^{\prime }(A)=discount\cdot m_i(A),A\subset \mathrm{\Theta }$$

(20)

$$m_i^{\prime }(\mathrm{\Theta })=discount\cdot m_i(\mathrm{\Theta })+1-discount$$

(21)

(iv)

Evidence fusion

According to the Dempster rule, the modified conflict evidence is synthesized with the remaining original evidence, and the decision is made according to the final fusion result.

3.3.3. Optimization Based on TOPSIS Average Closeness Degree Thought

Evidence synthesis involves integrating multiple pieces of evidence to form a comprehensive membership degree. The level of attribution of the object to be decided can then be determined by applying the principle of maximum membership degree. However, when multiple decision objects exist and belong to the same level, these decision objects cannot be further compared. To address this, the present study incorporates the basic idea of TOPSIS [56], where the preferred order of partners is determined by calculating the closeness between the basic probability assignment function obtained by the evidence fusion of each potential partner and the basic probability assignment function corresponding to the positive and negative ideal solutions. The steps are as follows:

1.

Calculate the positive and negative ideal cloud probability assignment function: Combined with the index attributes, the optimal and worst values corresponding to each index across all potential partners are selected, and the membership matrix of positive ideal cloud and negative ideal cloud is calculated. Using the above evidence theory fusion method, the probability assignment of positive and negative ideal clouds after evidence fusion is obtained, which represents the most ideal and the least ideal solution, respectively.

2.

Calculate the closeness of each potential partner to the ideal cloud: The basic probability assignment function, derived through evidence fusion, is calculated using the basic probability assignment function of positive and negative ideal clouds, and the average value of the fit degree of potential partners is calculated as:

$$R^{+}=\left[\begin{array}{cccc}{r}_{11}^{+}& r_{12}^{+}& \dots & r_{1n}^{+}\\ r_{21}^{+}& r_{22}^{+}& \dots & r_{2n}^{+}\\ ⋮& ⋮& \ddots & ⋮\\ r_{p1}^{+}& r_{p2}^{+}& \dots & r_{pn}^{+}\end{array}\right]$$

(22)

$$\mathrm{where}{r}_{ij}^{+}=\left|mas{s^{\prime }}_i({{\mu }^{\prime }}_j)-mas{s}_i^{+}\right|$$

(23)

Similarly, the closeness matrix $R^{-}$ of the negative ideal scheme can be obtained as:

$$\overline{R}=\left[\begin{array}{cc}{\overline{r}}_1^{+}& {\overline{r}}_1^{-}\\ {\overline{r}}_2^{+}& {\overline{r}}_2^{-}\\ ⋮& ⋮\\ {\overline{r}}_p^{+}& {\overline{r}}_p^{-}\end{array}\right]$$

(24)

$$\mathrm{where}{\overline{r}}_i^{+}={\displaystyle \frac{1}{n}}{\displaystyle \sum _{j=1}^n{r}_{ij}^{+}};{\overline{r}}_i^{-}={\displaystyle \frac{1}{n}}{\displaystyle \sum _{j=1}^n{r}_{ij}^{-}}$$

(25)

3.

Calculate the average fit difference: The average closeness degree is used to calculate the difference between the average fit degree of each potential partner with both the maximum and minimum average fit degree. The larger the difference, the closer the contractor is to the ideal solution, and the better the potential partners are sorted.

$$\mathrm{\Delta }{\overline{r}}_i^{+}=\max ({\overline{r}}_1^{+},{\overline{r}}_2^{+},\dots ,{\overline{r}}_p^{+})-{\overline{r}}_i^{+}$$

(26)

$$\mathrm{\Delta }{\overline{r}}_i^{-}={\overline{r}}_i^{-}-\min ({\overline{r}}_1^{-},{\overline{r}}_2^{-},\dots ,{\overline{r}}_p^{-})$$

(27)

3.4. Implementation Steps of the Model

First of all, according to Section 3.1, a strategic partner selection index system is established. Secondly, the benchmark cloud is constructed according to the index grade range, and the membership degree of each index is obtained. Thirdly, the conflict evidence is corrected by game combination weighting, and the evidence fusion of each membership degree is carried out. Finally, the TOPSIS idea is used to optimize the partners. The overall process of selection evaluation is shown in Figure 1.

4. Case Study

4.1. Project Introduction

In order to verify the validity and feasibility of the model, this study takes a prefabricated building project in Xi’an as an example to conduct an empirical analysis of the established evaluation and optimization model. The basic information of the project is as follows: A nationally listed real estate company wants to develop and construct a prefabricated residential project consisting of three high-rise residential buildings, each with 34 floors. The building height will be 98.835 m, with a floor height of 2.9 m, and a total construction area of 118,700 m². In order to ensure construction quality and process safety, enhance its competitive position in the local market, and obtain government subsidy incentives for prefabricated buildings, the developer plans to find suitable contractors as strategic cooperation units in the region. Therefore, the owner initially selected five highly qualified candidate partners from the contractors who had cooperated previously and set up an expert review group to analyze, evaluate, and select the above five contractors.

4.2. Assessment Process

4.2.1. The Determination of Evaluation Index Weight Based on Game Theory

1.

Due to the fact that this case is for the evaluation and comparative analysis of five candidate partners, which involves strong professionalism, in order to obtain objective and reasonable results, expert members must have a deep understanding and knowledge of prefabricated buildings, and their sources must be extensive. Therefore, this article invites an expert who has been engaged in prefabricated construction-related work for a long time from the company’s engineering management department, technology development department, financial management department, cost contract department, and legal production department. In addition, the general manager is also invited, with a total of six expert members. First of all, six experts assigned the relative importance of each index in the index system, as shown in

Table 2. The relative importance of strategic partners to select primary and secondary indicators.

Primary Index	Relative Importance						Secondary Index	Relative Importance
	Scoring by Six Experts							Scoring by Six Experts
C1	1	1	1	1	1	1	C11	1.2	1.3	1.2	1.2	1.3	1.1
							C12	1.1	1.2	1.1	1.1	1.1	1.2
							C13	1.1	1.1	1.1	1.1	1.1	1.1
							C14	1.1	1.1	1.1	1.2	1.2	1.2
							C15	—	—	—	—	—	—
C2	1.1	1	1	1.1	1	1	C21	1.2	1.3	1.2	1.3	1.2	1.3
							C22	1.1	1.1	1.1	1.1	1.1	1.2
							C23	1.1	1.1	0.9	1.1	1.1	1.1
							C24	—	—	—	—	—	—
C3	1.1	1.1	1.1	1.1	1.1	1.1	C31	1.1	1.1	1.1	1.1	1.1	1
							C32	1.1	1.2	1.1	1.1	1.2	1.2
							C33	1.1	1.2	1.1	1.2	1.1	1.1
							C34	—	—	—	—	—	—
C4	—	—	—	—	—	—	C41	1.1	1.2	1.1	1.1	1.2	1.2
							C42	1.1	1.1	1.2	1.1	1.1	1.1
							C43	1.3	1.2	1.2	1.3	1.2	1.1
							C44	—	—	—	—	—	—

.

According to Equations (1) and (2), the subjective weights of the first- and second-level indicators are calculated.

2.

Experts evaluate each contractor based on the actual situation and experience (the total score is 10 points; the score is accurate to 0.1). According to Equations (3)–(5), the objective weight of the index is calculated sequentially, as shown in

Table 3. Objective weight calculation table based on improved CRITIC method.

Index	Confliction	Variability	Information Content	Objective Weights
C11	6.37	0.16	1.03	0.063
C12	5.43	0.10	0.54	0.033
C13	4.77	0.07	0.31	0.019
C14	4.83	0.26	1.26	0.077
C15	4.78	0.10	0.45	0.028
C21	5.81	0.12	0.67	0.041
C22	11.27	0.17	1.86	0.114
C23	6.32	0.19	1.22	0.075
C24	11.80	0.16	1.94	0.119
C31	5.23	0.08	0.43	0.026
C32	6.93	0.07	0.49	0.030
C33	4.64	0.08	0.37	0.023
C34	10.78	0.14	1.50	0.092
C41	4.77	0.23	1.11	0.068
C42	6.98	0.12	0.87	0.053
C43	8.85	0.13	1.17	0.072
C44	4.55	0.24	1.09	0.067

.

3.

Based on the idea of improved game theory combination weighting, according to Equations (6)–(10), the weight distribution coefficients of the evaluation index are calculated as 0.473 and 0.527, respectively, and the comprehensive weight of the evaluation index is calculated. The specific results are shown in

Table 4. Combination weight table of evaluation index.

Evaluation index system for selecting strategic partners in prefabricated construction S	Primary index	Weights	Secondary index	Weights	Combination weights
	Benefits (C1)	0.235	Experience in prefabricated construction projects (C11)	0.256	0.060
			Innovation level of prefabricated building technology (C12)	0.187	0.044
			Supply level of prefabricated buildings (C13)	0.147	0.035
			Financial condition of the enterprise (C14)	0.269	0.063
			Reserve of prefabricated construction talents (C15)	0.139	0.033
	Ability (C2)	0.297	Prefabricated building quality management system (C21)	0.172	0.051
			Specialized organizational management level of prefabricated buildings (C22)	0.315	0.094
			The ability to respond quickly to market demand (C23)	0.216	0.064
			Adaptability to new technologies (C24)	0.294	0.087
	Costs (C3)	0.208	Prefabricated component product price (C31)	0.220	0.046
			Purchasing cost of prefabricated components (C32)	0.221	0.046
			The transportation cost of prefabricated components (C33)	0.176	0.037
			Repair cost of prefabricated components (C34)	0.386	0.081
	Compatibility (C4)	0.262	Whether the strategic objectives are compatible (C41)	0.282	0.073
			Whether the organizational structure is compatible (C42)	0.200	0.052
			Whether the culture is compatible (C43)	0.261	0.068
			Whether the willingness to cooperate is compatible (C44)	0.260	0.067

, and the index weight line is shown in Figure 2.

4.2.2. Determination of Basic Probability Assignment

1.

Dividing the evaluation grade

In this study, each index is scored by experts, and a specific value is obtained. The full score is set to 10 points. According to the grade, the scoring interval is divided into {0–2,2–4,4–6,6–8,8–10} (

Table 5. Classification standard of evaluation grade of prefabricated building contractors to be evaluated.

Grade	Range of Value	Description
Grade I	[8,10)	Good
Grade II	[6,8)	Better
Grade III	[4,6)	General
Grade IV	[2,4)	Medium
Grade V	[0,2)	Poor

).

2.

Score of each index of alternative units

Six experts scored the evaluation indicators, as shown in

Table 6. The index value of prefabricated building contractors to be evaluated.

Index	Contractor S1						Contractor S2						…	Contractor S5
C11	9.4	8.8	9.2	7.2	9.2	9.1	8.8	8.7	9.2	7.8	8.7	8.9	…	6.2	5.8	4.4	6.1	6.3	6.2
C12	9.6	9.0	9.5	8.5	7.0	9.4	7.6	8.6	8.7	8.6	8.5	9.4	…	8.1	7.0	7.3	7.5	7.7	6.4
C13	8.8	8.9	9.3	7.6	8.7	9.4	8.4	9.0	8.5	8.4	7.3	8.6	…	7.8	6.6	7.8	7.6	8	7.7
C14	9.5	9.6	8.8	9.5	9.4	8.2	7.9	7.9	8.0	8.4	8.0	6.8	…	5.4	3.8	5.6	5.2	5.3	5.4
C15	8.9	9.1	8.1	8.5	9.1	9.2	8.3	6.5	8.6	8.4	8.7	8.3	…	8.0	7.2	7.3	6.0	7.4	7.5
C21	9.5	9.3	9.2	8.2	9.2	9.3	8.5	8.7	8.8	8.4	9.0	7.4	…	6.7	7.1	6.8	5.9	7	6.2
C22	9.1	9.7	9.5	9.2	7.7	9.4	6.2	6.4	6.5	4.8	5.9	6.2	…	7.7	7.9	7.6	7.8	6	7.8
C23	8.9	8.8	8.9	9.3	8.8	9.1	8.0	6.5	8.2	8.1	8.3	8.2	…	5.2	5.3	5.4	5.1	4	5.8
C24	7.2	7.4	6.3	7.0	7.1	7.3	6.6	7.7	7.9	7.8	7.8	8.4	…	7.9	7.8	6.7	7.8	8.2	8.0
C31	9.6	9.7	8.3	9.5	9.5	9.5	8.0	6.9	8.2	8.1	8.2	8.5	…	7.8	8.1	7.8	8.0	8.1	6.9
C32	7.2	7.4	7.6	6.8	6.5	5.4	6.8	8.3	9.0	8.5	8.3	8.4	…	8.1	8.1	8.2	6.9	8.0	8.5
C33	8.4	9.1	9.2	9.1	9.1	8.5	7.3	8.3	8.5	8.4	8.4	8.5	…	7.9	7.8	6.6	7.2	7.7	7.8
C34	8.8	9.4	9.1	8.0	8.7	8.6	9.0	8.5	8.6	8.3	8.7	8.8	…	6.7	8.0	8.2	7.9	8.1	8.8
C41	9.3	9.8	9.7	9.6	9.4	8.2	7.8	7.8	8.1	8.3	7.7	8.2	…	4.2	5.6	5.7	5.8	6.2	6.1
C42	4.8	6.2	6.5	6.8	6.7	6.9	6.7	6.4	5.3	6.5	6.6	6.7	…	8.2	8.1	7.8	6.7	7.9	8.0
C43	7.6	9.1	8.7	8.9	8.8	8.8	6.5	6.7	4.6	5.9	7.2	6.4	…	5.8	7.1	7.5	7.4	7.2	7.6
C44	9.3	9.4	9.5	8.2	9.3	8.6	8.0	6.5	8.3	8.1	8.0	8.1	…	3.4	4.6	5.7	5.3	5.4	5.0

.

3.

Cloud model data feature calculation

According to the evaluation grade division of Table 5, and in accordance with Equations (10) and (11), the data characteristics corresponding to each grade are calculated as follows: Grade I (9,0.849,0.085), Grade II (7,0.849,0.085), Grade III (5,0.849,0.085), Grade IV (3,0.849,0.085), Grade V (1,0.849,0.085). The forward cloud generator algorithm is used to draw the evaluation benchmark cloud, as shown in Figure 3. In the figure (Figure 3), the standard clouds corresponding to grades I to V are represented from right to left, with each grade represented by different colors.

4.

Membership calculation

Due to space limitations, this study focuses only on the evaluation of contractor S₁ as a representative example to explain the specific calculation process. Based on the index score in Table 6, the corresponding cloud digital features are calculated according to Equations (10)–(12). These parameters are then applied to calculate the membership degree of each index belonging to each grade sequentially. Following this, the membership distribution of contractor S₁ is calculated, as shown in

Table 7. The index membership distribution of contractor S₁.

Index	V	IV	III	II	I	Index	V	IV	III	II	I
C11	0.00	0.00	0.00	0.02	0.89	C31	0.00	0.00	0.00	0.01	0.77
	0.00	0.00	0.00	0.11	0.97		0.00	0.00	0.00	0.01	0.71
	0.00	0.00	0.00	0.04	0.97		0.00	0.00	0.00	0.31	0.71
	0.00	0.00	0.04	0.97	0.11		0.00	0.00	0.00	0.02	0.84
	0.00	0.00	0.00	0.04	0.97		0.00	0.00	0.00	0.02	0.84
	0.00	0.00	0.00	0.05	0.99		0.00	0.00	0.00	0.02	0.84
C12	0.00	0.00	0.00	0.01	0.77	C32	0.00	0.00	0.04	0.97	0.11
	0.00	0.00	0.00	0.07	1.00		0.00	0.00	0.02	0.89	0.17
	0.00	0.00	0.00	0.02	0.84		0.00	0.00	0.01	0.77	0.26
	0.00	0.00	0.00	0.21	0.84		0.00	0.00	0.11	0.97	0.04
	0.00	0.00	0.07	1.00	0.07		0.00	0.00	0.21	0.84	0.02
	0.00	0.00	0.00	0.02	0.89		0.00	0.02	0.89	0.17	0.00
C13	0.00	0.00	0.00	0.11	0.97	C33	0.00	0.00	0.00	0.26	0.78
	0.00	0.00	0.00	0.09	0.99		0.00	0.00	0.00	0.05	0.99
	0.00	0.00	0.00	0.03	0.94		0.00	0.00	0.00	0.04	0.97
	0.00	0.00	0.01	0.77	0.26		0.00	0.00	0.00	0.05	0.99
	0.00	0.00	0.00	0.14	0.94		0.00	0.00	0.00	0.05	0.99
	0.00	0.00	0.00	0.02	0.89		0.00	0.00	0.00	0.21	0.84
C14	0.00	0.00	0.00	0.02	0.84	C34	0.00	0.00	0.00	0.11	0.97
	0.00	0.00	0.00	0.01	0.77		0.00	0.00	0.00	0.02	0.89
	0.00	0.00	0.00	0.11	0.97		0.00	0.00	0.00	0.05	0.99
	0.00	0.00	0.00	0.02	0.84		0.00	0.00	0.00	0.50	0.50
	0.00	0.00	0.00	0.02	0.89		0.00	0.00	0.00	0.14	0.94
	0.00	0.00	0.00	0.36	0.64		0.00	0.00	0.00	0.17	0.89
C15	0.00	0.00	0.00	0.09	0.99	C41	0.00	0.00	0.00	0.03	0.94
	0.00	0.00	0.00	0.05	0.99		0.00	0.00	0.00	0.01	0.64
	0.00	0.00	0.00	0.43	0.56		0.00	0.00	0.00	0.01	0.71
	0.00	0.00	0.00	0.21	0.84		0.00	0.00	0.00	0.01	0.77
	0.00	0.00	0.00	0.05	0.99		0.00	0.00	0.00	0.02	0.89
	0.00	0.00	0.00	0.04	0.97		0.00	0.00	0.00	0.37	0.64
C21	0.00	0.00	0.00	0.02	0.84	C42	0.00	0.11	0.97	0.04	0.00
	0.00	0.00	0.00	0.03	0.94		0.00	0.00	0.36	0.64	0.01
	0.00	0.00	0.00	0.04	0.97		0.00	0.00	0.21	0.84	0.02
	0.00	0.00	0.00	0.37	0.64		0.00	0.00	0.11	0.97	0.04
	0.00	0.00	0.00	0.04	0.97		0.00	0.00	0.14	0.94	0.03
	0.00	0.00	0.00	0.03	0.94		0.00	0.00	0.09	0.99	0.05
C22	0.00	0.00	0.00	0.05	0.99	C43	0.00	0.00	0.01	0.77	0.26
	0.00	0.00	0.00	0.01	0.71		0.00	0.00	0.00	0.05	0.99
	0.00	0.00	0.00	0.02	0.84		0.00	0.00	0.00	0.14	0.94
	0.00	0.00	0.00	0.04	0.97		0.00	0.00	0.00	0.09	0.99
	0.00	0.00	0.01	0.71	0.31		0.00	0.00	0.00	0.11	0.97
	0.00	0.00	0.00	0.02	0.89		0.00	0.00	0.00	0.11	0.97
C23	0.00	0.00	0.00	0.09	0.99	C44	0.00	0.00	0.00	0.03	0.94
	0.00	0.00	0.00	0.11	0.97		0.00	0.00	0.00	0.02	0.89
	0.00	0.00	0.00	0.09	0.99		0.00	0.00	0.00	0.02	0.84
	0.00	0.00	0.00	0.03	0.94		0.00	0.00	0.00	0.37	0.64
	0.00	0.00	0.00	0.11	0.97		0.00	0.00	0.00	0.03	0.94
	0.00	0.00	0.00	0.05	0.99		0.00	0.00	0.00	0.17	0.89
C24	0.00	0.00	0.04	0.97	0.11	C24	0.00	0.00	0.07	1.00	0.07
	0.00	0.00	0.02	0.89	0.17		0.00	0.00	0.05	0.99	0.09
	0.00	0.00	0.31	0.71	0.01		0.00	0.00	0.03	0.94	0.14

.

5.

Basic probability assignment calculation of evaluation index

The membership degree of each index in Table 7 is normalized according to Equation (15), and the basic probability distribution of each index is obtained, as shown in

Table 8. The basic probability assignment of contractor S₁.

Index	V	IV	III	II	I	mΘ	Index	V	IV	III	II	I	mΘ
C11	0.00	0.00	0.00	0.02	0.95	0.02	C31	0.00	0.00	0.00	0.02	0.97	0.02
	0.00	0.00	0.00	0.09	0.81	0.10		0.00	0.00	0.00	0.01	0.98	0.01
	0.00	0.00	0.00	0.04	0.92	0.04		0.00	0.00	0.00	0.21	0.48	0.30
	0.00	0.00	0.03	0.75	0.08	0.13		0.00	0.00	0.00	0.02	0.96	0.02
	0.00	0.00	0.00	0.04	0.92	0.04		0.00	0.00	0.00	0.02	0.96	0.02
	0.00	0.00	0.00	0.05	0.90	0.05		0.00	0.00	0.00	0.02	0.96	0.02
C12	0.00	0.00	0.00	0.01	0.97	0.01	C32	0.00	0.00	0.03	0.75	0.09	0.13
	0.00	0.00	0.00	0.06	0.88	0.06		0.00	0.00	0.02	0.67	0.13	0.18
	0.00	0.00	0.00	0.02	0.96	0.02		0.00	0.00	0.01	0.55	0.18	0.26
	0.00	0.00	0.00	0.16	0.64	0.20		0.00	0.00	0.09	0.75	0.03	0.13
	0.00	0.00	0.05	0.78	0.05	0.12		0.00	0.00	0.16	0.62	0.01	0.21
	0.00	0.00	0.00	0.02	0.95	0.02		0.00	0.02	0.67	0.13	0.00	0.18
C13	0.00	0.00	0.00	0.09	0.81	0.10	C33	0.00	0.00	0.00	0.19	0.56	0.25
	0.00	0.00	0.00	0.07	0.85	0.08		0.00	0.00	0.00	0.05	0.90	0.05
	0.00	0.00	0.00	0.03	0.94	0.03		0.00	0.00	0.00	0.04	0.92	0.04
	0.00	0.00	0.01	0.55	0.18	0.26		0.00	0.00	0.00	0.05	0.90	0.05
	0.00	0.00	0.00	0.11	0.76	0.13		0.00	0.00	0.00	0.05	0.90	0.05
	0.00	0.00	0.00	0.02	0.95	0.02		0.00	0.00	0.00	0.16	0.64	0.20
C14	0.00	0.00	0.00	0.02	0.96	0.02	C34	0.00	0.00	0.00	0.09	0.81	0.10
	0.00	0.00	0.00	0.02	0.97	0.02		0.00	0.00	0.00	0.02	0.95	0.02
	0.00	0.00	0.00	0.09	0.81	0.10		0.00	0.00	0.00	0.05	0.90	0.05
	0.00	0.00	0.00	0.02	0.96	0.02		0.00	0.00	0.00	0.25	0.25	0.50
	0.00	0.00	0.00	0.02	0.95	0.02		0.00	0.00	0.00	0.11	0.76	0.13
	0.00	0.00	0.00	0.23	0.40	0.37		0.00	0.00	0.00	0.14	0.70	0.16
C15	0.00	0.00	0.00	0.07	0.85	0.08	C41	0.00	0.00	0.00	0.03	0.94	0.03
	0.00	0.00	0.00	0.05	0.90	0.05		0.00	0.00	0.00	0.01	0.98	0.01
	0.00	0.00	0.00	0.24	0.32	0.43		0.00	0.00	0.00	0.01	0.98	0.01
	0.00	0.00	0.00	0.16	0.64	0.20		0.00	0.00	0.00	0.01	0.97	0.02
	0.00	0.00	0.00	0.05	0.90	0.05		0.00	0.00	0.00	0.02	0.95	0.02
	0.00	0.00	0.00	0.04	0.92	0.04		0.00	0.00	0.00	0.23	0.40	0.37
C21	0.00	0.00	0.00	0.02	0.96	0.02	C42	0.00	0.09	0.75	0.03	0.00	0.13
	0.00	0.00	0.00	0.03	0.94	0.03		0.00	0.00	0.23	0.40	0.00	0.37
	0.00	0.00	0.00	0.04	0.92	0.04		0.00	0.00	0.16	0.62	0.01	0.22
	0.00	0.00	0.00	0.23	0.40	0.37		0.00	0.00	0.08	0.75	0.03	0.13
	0.00	0.00	0.00	0.04	0.92	0.04		0.00	0.00	0.11	0.72	0.02	0.15
	0.00	0.00	0.00	0.03	0.94	0.03		0.00	0.00	0.07	0.77	0.04	0.12
C22	0.00	0.00	0.00	0.05	0.90	0.05	C43	0.00	0.00	0.01	0.55	0.18	0.26
	0.00	0.00	0.00	0.01	0.98	0.01		0.00	0.00	0.00	0.05	0.90	0.05
	0.00	0.00	0.00	0.02	0.96	0.02		0.00	0.00	0.00	0.11	0.76	0.13
	0.00	0.00	0.00	0.04	0.92	0.04		0.00	0.00	0.00	0.07	0.85	0.08
	0.00	0.00	0.01	0.48	0.21	0.31		0.00	0.00	0.00	0.09	0.81	0.10
	0.00	0.00	0.00	0.02	0.95	0.02		0.00	0.00	0.00	0.09	0.81	0.10
C23	0.00	0.00	0.00	0.07	0.85	0.08	C44	0.00	0.00	0.00	0.03	0.94	0.03
	0.00	0.00	0.00	0.09	0.81	0.10		0.00	0.00	0.00	0.02	0.95	0.02
	0.00	0.00	0.00	0.07	0.85	0.08		0.00	0.00	0.00	0.02	0.96	0.02
	0.00	0.00	0.00	0.03	0.94	0.03		0.00	0.00	0.00	0.23	0.40	0.37
	0.00	0.00	0.00	0.09	0.81	0.10		0.00	0.00	0.00	0.03	0.94	0.03
	0.00	0.00	0.00	0.05	0.90	0.05		0.00	0.00	0.00	0.14	0.70	0.16
C24	0.00	0.00	0.03	0.75	0.08	0.13	C24	0.00	0.00	0.05	0.78	0.05	0.12
	0.00	0.00	0.02	0.67	0.13	0.18		0.00	0.00	0.04	0.77	0.07	0.12
	0.00	0.00	0.21	0.48	0.01	0.31		0.00	0.00	0.02	0.72	0.11	0.15

.

4.2.3. Correction and Fusion of Conflict Evidence

In general, evidence conflict does not typically arise from all sources but is often due to specific conflict evidence within the data. Therefore, it is necessary to address the individual evidence to preserve the integrity of the original dataset. This study corrects the conflict evidence according to Equations (16)–(21) to improve the reliability of the decision-making process. The corrected evidence is shown in

Table 9. Revised basic probability assignment for contractor S₁.

Index	V	IV	III	II	I	mΘ	Index	V	IV	III	II	I	mΘ
C11	0.00	0.00	0.00	0.01	0.39	0.60	C31	0.00	0.00	0.00	0.02	0.97	0.02
	0.00	0.00	0.00	0.09	0.81	0.10		0.00	0.00	0.00	0.01	0.98	0.01
	0.00	0.00	0.00	0.02	0.45	0.53		0.00	0.00	0.00	0.04	0.09	0.87
	0.00	0.00	0.00	0.09	0.01	0.90		0.00	0.00	0.00	0.02	0.96	0.02
	0.00	0.00	0.00	0.02	0.45	0.53		0.00	0.00	0.00	0.02	0.96	0.02
	0.00	0.00	0.00	0.05	0.90	0.05		0.00	0.00	0.00	0.02	0.96	0.02
C12	0.00	0.00	0.00	0.01	0.54	0.45	C32	0.00	0.00	0.01	0.29	0.03	0.67
	0.00	0.00	0.00	0.06	0.88	0.06		0.00	0.00	0.02	0.67	0.13	0.18
	0.00	0.00	0.00	0.01	0.56	0.43		0.00	0.00	0.00	0.21	0.07	0.72
	0.00	0.00	0.00	0.16	0.64	0.20		0.00	0.00	0.03	0.31	0.01	0.65
	0.00	0.00	0.01	0.14	0.01	0.84		0.00	0.00	0.16	0.62	0.01	0.21
	0.00	0.00	0.00	0.01	0.58	0.41		0.00	0.00	0.09	0.02	0.00	0.89
C13	0.00	0.00	0.00	0.09	0.81	0.10	C33	0.00	0.00	0.00	0.08	0.23	0.70
	0.00	0.00	0.00	0.03	0.31	0.66		0.00	0.00	0.00	0.05	0.90	0.05
	0.00	0.00	0.00	0.01	0.18	0.82		0.00	0.00	0.00	0.04	0.92	0.04
	0.00	0.00	0.00	0.04	0.01	0.95		0.00	0.00	0.00	0.05	0.90	0.05
	0.00	0.00	0.00	0.11	0.76	0.13		0.00	0.00	0.00	0.05	0.90	0.05
	0.00	0.00	0.00	0.00	0.17	0.83		0.00	0.00	0.00	0.09	0.36	0.54
C14	0.00	0.00	0.00	0.01	0.29	0.71	C34	0.00	0.00	0.00	0.03	0.27	0.70
	0.00	0.00	0.00	0.00	0.27	0.73		0.00	0.00	0.00	0.00	0.11	0.89
	0.00	0.00	0.00	0.09	0.81	0.10		0.00	0.00	0.00	0.01	0.14	0.86
	0.00	0.00	0.00	0.01	0.29	0.71		0.00	0.00	0.00	0.01	0.01	0.97
	0.00	0.00	0.00	0.01	0.31	0.69		0.00	0.00	0.00	0.11	0.76	0.13
	0.00	0.00	0.00	0.02	0.03	0.95		0.00	0.00	0.00	0.14	0.70	0.16
C15	0.00	0.00	0.00	0.07	0.85	0.08	C41	0.00	0.00	0.00	0.03	0.94	0.03
	0.00	0.00	0.00	0.03	0.56	0.42		0.00	0.00	0.00	0.01	0.62	0.37
	0.00	0.00	0.00	0.05	0.07	0.88		0.00	0.00	0.00	0.01	0.65	0.34
	0.00	0.00	0.00	0.16	0.64	0.20		0.00	0.00	0.00	0.01	0.97	0.02
	0.00	0.00	0.00	0.03	0.55	0.42		0.00	0.00	0.00	0.02	0.95	0.02
	0.00	0.00	0.00	0.02	0.50	0.48		0.00	0.00	0.00	0.03	0.06	0.90
C21	0.00	0.00	0.00	0.01	0.63	0.36	C42	0.00	0.01	0.12	0.00	0.00	0.86
	0.00	0.00	0.00	0.03	0.94	0.03		0.00	0.00	0.23	0.40	0.00	0.37
	0.00	0.00	0.00	0.04	0.92	0.04		0.00	0.00	0.16	0.62	0.01	0.22
	0.00	0.00	0.00	0.04	0.07	0.90		0.00	0.00	0.04	0.33	0.01	0.62
	0.00	0.00	0.00	0.04	0.92	0.04		0.00	0.00	0.11	0.72	0.02	0.15
	0.00	0.00	0.00	0.03	0.94	0.03		0.00	0.00	0.03	0.30	0.02	0.66
C22	0.00	0.00	0.00	0.05	0.90	0.05	C43	0.00	0.00	0.00	0.05	0.02	0.93
	0.00	0.00	0.00	0.01	0.52	0.47		0.00	0.00	0.00	0.01	0.25	0.74
	0.00	0.00	0.00	0.01	0.56	0.43		0.00	0.00	0.00	0.11	0.76	0.13
	0.00	0.00	0.00	0.04	0.92	0.04		0.00	0.00	0.00	0.03	0.33	0.64
	0.00	0.00	0.00	0.06	0.03	0.91		0.00	0.00	0.00	0.09	0.81	0.10
	0.00	0.00	0.00	0.02	0.95	0.02		0.00	0.00	0.00	0.09	0.81	0.10
C23	0.00	0.00	0.00	0.07	0.85	0.08	C44	0.00	0.00	0.00	0.03	0.94	0.03
	0.00	0.00	0.00	0.02	0.19	0.79		0.00	0.00	0.00	0.02	0.95	0.02
	0.00	0.00	0.00	0.07	0.85	0.08		0.00	0.00	0.00	0.02	0.75	0.23
	0.00	0.00	0.00	0.00	0.14	0.86		0.00	0.00	0.00	0.06	0.11	0.83
	0.00	0.00	0.00	0.02	0.19	0.79		0.00	0.00	0.00	0.03	0.94	0.03
	0.00	0.00	0.00	0.01	0.24	0.74		0.00	0.00	0.00	0.14	0.70	0.16
C24	0.00	0.00	0.03	0.75	0.08	0.13	C24	0.00	0.00	0.03	0.52	0.03	0.41
	0.00	0.00	0.02	0.67	0.13	0.18		0.00	0.00	0.04	0.77	0.07	0.12
	0.00	0.00	0.04	0.09	0.00	0.87		0.00	0.00	0.02	0.72	0.11	0.15

.

Based on the probability distribution of the modified evidence in Table 9, the evidence is fused according to Equation (14), and the result is shown in

Table 10. Evidence fusion results of contractor S₁.

Index	V	IV	III	II	I	m(Θ)
C11	0.00	0.00	0.00	0.00	1.00	0.00
C12	0.00	0.00	0.00	0.00	0.99	0.00
C13	0.00	0.00	0.00	0.02	0.97	0.01
C14	0.00	0.00	0.00	0.03	0.95	0.03
C15	0.00	0.00	0.00	0.01	0.99	0.00
C21	0.00	0.00	0.00	0.00	1.00	0.00
C22	0.00	0.00	0.00	0.00	1.00	0.00
C23	0.00	0.00	0.00	0.01	0.99	0.00
C24	0.00	0.00	0.00	1.00	0.00	0.00
C31	0.00	0.00	0.00	0.00	1.00	0.00
C32	0.00	0.00	0.02	0.94	0.02	0.02
C33	0.00	0.00	0.00	0.00	1.00	0.00
C34	0.00	0.00	0.00	0.04	0.95	0.01
C41	0.00	0.00	0.00	0.00	1.00	0.00
C42	0.00	0.00	0.05	0.94	0.00	0.01
C43	0.00	0.00	0.00	0.01	0.99	0.00
C44	0.00	0.00	0.00	0.00	1.00	0.00

.

4.2.4. Results and Discussion

Table 10 shows the integration result of contractor S₁. Similarly, the integration results of the remaining contractors are calculated. Then, the above results are weighted, and the final fusion results are calculated, as shown in

Table 11. The evidence fusion results of each alternative contractor.

Contractor	V	IV	III	II	I	m(Θ)
S1	0.0000	0.0000	0.0000	0.0007	0.9990	0.0002
S2	0.0000	0.0000	0.0000	0.0013	0.9987	0.0000
S3	0.0000	0.0000	0.0000	1.0000	0.0000	0.0000
S4	0.0000	0.0000	0.0000	0.0015	0.9985	0.0000
S5	0.0000	0.0000	0.0002	0.9998	0.0000	0.0000

. At this point, only the level of each contractor can be obtained, and further processing of the data is required.

The average closeness degree, the average closeness degree difference between each contractor, and the positive and negative ideal clouds are calculated according to Equations (22)–(27), and the values are shown in

Table 12. Average closeness degree and its difference between each contractor and positive and negative ideal clouds.

Contractor	Mean Value of Closeness to Positive Ideal Cloud	Mean Value of Closeness to Negative Ideal Cloud	Average Closeness Difference
S1	0.0003	0.3325	0.232
S2	0.0004	0.3324	0.230
S3	0.3333	0.2048	0.003
S4	0.0005	0.3323	0.229
S5	0.3333	0.2047	0.001

.

According to Table 12, the average closeness differences of the five candidate contractors are 0.232, 0.230, 0.003, 0.229, and 0.001, respectively. According to the principle that a larger average closeness difference indicates greater proximity of the user to the most ideal result, the evaluation ranking of five alternative contractors is determined in descending order: Contractor S₁ > Contractor S₂ > Contractor S₄ > Contractor S₃ > Contractor S₅. Therefore, the contractor S₁ emerges as the preferred strategic partner.

In terms of weight calculation, according to Figure 2, it can be found that the subjective and objective weight values of six indicators C₁₁, C₁₅, C₄₁, C₄₂, C₄₃, and C₄₄ are relatively consistent, while the weight values of other indicators differ greatly. Among them, indicator C₂₄ has the largest weight value difference, reaching 0.091. The objective weight clearly tends towards the four indicators C₁₄, C₂₂, C₂₄, and C₃₄, while the subjective weight is relatively balanced, but cannot highlight the main indicators. Therefore, the combination weighting method based on game theory realizes the organic combination of The G1 method and the improved CRITIC method, which greatly eliminates the limitations of a single method.

In terms of strategic partner selection, the traditional evidence theory is to fuse multiple pieces of evidence to form a comprehensive membership vector, and then determine the level of the research object according to the principle of maximum membership degree, which cannot sort multiple research objects. For example, in this case, if this method is adopted, according to Table 11, the grades of the five contractors are grade I, grade I, grade II, grade I, and grade II, respectively. At this time, further comparison of the contractors cannot be made. Therefore, the optimization method based on the TOPSIS average closeness degree proposed in this paper realizes the comparison of multiple decision objects.

5. Conclusions and Future Work

5.1. Conclusions

As a new type of construction method, prefabricated building has several advantages, such as high production efficiency, good quality control, high resource utilization rate, less environmental pollution, and reduced safety hazards. It is an important measure for the low-carbon cycle and green development of China’s construction industry. However, most of China’s prefabricated construction projects still follow the traditional management mode, and the cooperation relationship often terminates at the end of a project, which is not conducive to the realization of economies of scale. Given that the traditional cooperation model cannot match the development needs of prefabricated buildings, it is very important to choose partners with strong comprehensive strength for the smooth implementation of the project. Therefore, this present study conducts an in-depth analysis of the selection and evaluation of strategic partners for prefabricated buildings. The research conclusions are as follows:

1.

Through a literature survey, this study identifies the common indicators for the selection of enterprise partners. These indicators can reflect the basic requirements and conditions of the enterprise that need to be evaluated. Considering the characteristics of prefabricated buildings, this study conducted field research on prefabricated building projects and summarized individual indicators. Finally, an evaluation index system for selecting strategic partners in prefabricated buildings was constructed from four aspects: benefits, capabilities, costs, and compatibility.

2.

Considering the multi-attribute, complex, and diverse characteristics of strategic partner selection for prefabricated buildings, this study develops an evaluation model for such selection based on the cloud model and improved evidence theory. The membership value between the sample to be tested and the reference cloud is calculated by the cloud model, and it is transformed into basic probability distribution in the evidence theory, which helps to overcome the fuzziness and randomness of the traditional membership function. Furthermore, to address the issues in traditional evidence theory, the weight of evidence is combined by game theory, and the conflict evidence is corrected and fused according to the combination weight. The TOPSIS method is then applied to optimize the strategic partners of prefabricated buildings.

3.

The rationality of the indicator system and the feasibility of the strategic partner selection model were verified. The evaluation model was applied to practical engineering projects, and the optimal order is S₁ > S₂ > S₄ > S₃ > S₅. The analysis showed that the results obtained by the model proposed in this paper were consistent with the actual situation, which further verified the effectiveness and superiority of the model.

5.2. Future Work

Based on the research results of this paper, the following tasks will be the future research direction.

1.

It is important to note that this study mainly focuses on the research of prefabricated construction projects invested by real estate developers and does not involve government investment and infrastructure projects. Whether the proposed index system applies to these government projects is worthy of further research and optimization.

2.

In the face of an increasingly complex and changing market environment, cooperation performance will be affected by many factors in the implementation of prefabricated construction projects. Therefore, finding the key factors affecting cooperation performance and conducting efficiency measurements will be the next research direction.

3.

This study mainly focuses on selecting partners for prefabricated construction in China. Considering the significant differences in economic and cultural backgrounds among countries, the research results have certain limitations. The next step is to expand the evaluation results of strategic partnerships for prefabricated buildings based on a larger international perspective.