A Group Intuitionistic Fuzzy Exponential TODIM Method Considering Attribute Interactions Applied to Green Building Material Supplier Selection

Abstract

Green building, driven by the goal of sustainable development, has prompted extensive attention to be paid to the environmental impact of its materials. However, some of the traditional methods of evaluating building material suppliers and attribute systems are not able to adapt to the new issues arising from the green context. This paper aims to provide a new solution for selecting green building material suppliers to enhance the green efficiency of buildings. Specifically, this paper presents a framework for evaluating and selecting suppliers of green building materials that meet the criteria of environmental friendliness and sustainability. A comprehensive evaluation attribute system is established, encompassing cost, quality, service level, delivery capability, and green and sustainable ability. Additionally, a group decision-making method based on the exponential TODIM (an acronym in Portuguese for Interactive and Multi-attribute Decision Making) and intuitionistic fuzzy numbers is developed to integrate expert opinions from diverse domains. Intuitionistic fuzzy numbers represent an extension of traditional fuzzy sets, offering a means of more fully and accurately responding to the inherent vagueness and hesitancy of human thinking. They can often prove invaluable when faced with problems containing uncertainty. Moreover, to obtain more precise attribute weights, the $\lambda$-fuzzy measure, Choquet integral, and Shapley value are employed to consider attribute interactions. Subsequently, a selection case involving six timber suppliers was proposed. Subsystem analysis was employed to ascertain the relative strengths and weaknesses of the various suppliers, with a view to facilitating future improvements. The findings indicated that green and sustainability capability attributes exert a considerable influence on the selection of green building material suppliers. Consequently, suppliers distinguished under this standard may encounter challenges in attaining exemplary rankings. Comparative analysis and robustness analysis have demonstrated the efficacy, superiority, and stability of the proposed framework. The findings of this paper can provide a reference for companies engaged in or planning to develop green buildings and help them choose green building material suppliers, which can help them achieve the expected green building efficiency and promote the sustainable development of the industry.

1. Introduction

Green Supply Chain Management (GSCM) has garnered extensive recognition and research in both businesses and academia due to the growing awareness of the significance of environmental protection [1], involving the entire supply chain process [2]. GSCM is centered around the integration of environmental protection measures and cost control throughout the entire supply chain process [3]. Implementing GSCM can enhance social productivity and effectively drive sustainable socio-economic development [4]. Green building material supplier selection stands out as a crucial element that directly impacts the management efficiency, economic benefits, and environmental friendliness of construction projects [5]. Meticulously selecting green building material suppliers enables enterprises to mitigate environmental impact while enhancing management efficiency and economic benefits, thereby contributing to the creation of a cleaner and more sustainable future [6].

The selection of suppliers in the construction industry is a topic that has been widely studied by scholars in the field, who have offered a range of insights and solutions. Nevertheless, some research gaps pertaining to the selection of green building material suppliers in the context of the new era environment remain unresolved.

Research gap 1: Traditionally, the selection of building material suppliers has been primarily concerned with integrating and coordinating various elements to enhance organizational performance and profitability [7,8]. However, the initial definition failed to adequately consider the significance of environmental factors. Presently, government regulations have placed an imperative on enterprises to comply with environmental standards, and the growing demand for green products has given rise to the necessity for the selection of green building material suppliers [9,10,11]. These developments have introduced new criteria for the selection of green building material suppliers. The selection of traditional building material suppliers may be primarily driven by considerations such as cost, quality, and time [12]. Nevertheless, in certain instances, the enhancement of suppliers’ green performance may conflict with these standards [13]. In light of these considerations, a comprehensive and integrated assessment of a green building material supplier should not only adhere to traditional evaluation attributes but also demand green performance [14].

Research gap 2: The selection of building material suppliers involves multiple stakeholders, such as customers, designers, main contractors, and subcontractors [15]. Green building material suppliers are more likely to involve government environmental departments or green public welfare organizations [16]. In this process, decision making often involves the opinions and suggestions of experts from different fields [17]. Therefore, how to integrate the opinions of different experts to form a consistent decision-making result when evaluating green building material suppliers has become an important research topic [18,19]. In addition, the evaluation process of experts in multiple fields is limited by the differences in their professional fields, educational backgrounds, and occupations, making it difficult for all experts to clearly express their opinions on all evaluation items. This condition requires a tolerance for uncertainty and ambiguity in the evaluation system.

In order to address the aforementioned research gaps, this paper proposes a framework for the evaluation of green building material suppliers. This framework comprises an attribute system and an innovative MAGDM (multi-attribute group decision-making) method. This attribute system incorporates traditional attributes such as cost, time, and service while introducing novel attributes to assess the green performance of alternatives. These include the green business philosophy, application of clean technology, recycling and reuse efficiency, pollutant emission situation, and green energy utilization rate. The new attribute system addresses the limitations of the traditional evaluation system for building material suppliers, which is unable to adapt to the modern green demand context.

On the other hand, the proposed MAGDM method incorporates a range of techniques, including fuzzy set representation [20], expert weight determination [21,22], and the consideration of attribute interaction [23]. The combination of IFNs (intuitionistic fuzzy numbers) and the exponential TODIM (an acronym in Portuguese for Interactive and Multi-attribute Decision Making) method, extended to group decision making, is referred to as the Group-IF-ExpTODIM method. This approach effectively addresses the evaluation differences that may arise due to the involvement of experts from diverse fields. At the same time, the utilization of IFNs alleviates the burden of evaluation and enhances the communicative capacity of experts. Additionally, attribute interaction denotes the interconnection of disparate attributes within an attribute system. This may be exemplified by the intrinsic correlation between a supplier’s service level and cost. In general, the provision of high-level service entails greater expenditure. The presence of attribute interaction can influence the fairness and accuracy of the ultimate evaluation outcomes. To address this issue, the Choquet integrals and Shapley values (two concepts in the field of game theory) introduced in this paper can assist in calculating attribute weights under attribute interactions by reducing reliance on specific weight allocation strategies.

Overall, this paper focuses on various issues related to the selection of green building material suppliers and their impact on the sustainable development of buildings. The main findings and contributions are as follows:

(1) This paper puts forth an MAGDM approach for the selection of suppliers of adaptive green building materials, encompassing IFNs, ExpTODIM, Choquet integral, and Shapley value. After that, an empirically validated case study substantiates the practicality and efficacy of the proposed method in the context of green building material supplier selection. This method offers a promising avenue for addressing the challenge of identifying suitable green building material suppliers.

(2) A system of attributes for evaluating suppliers of green building materials has been established that is distinct from traditional supplier selection, taking into account the green and sustainable capabilities of suppliers. The results of the case evaluation demonstrate that the incorporation of green and sustainable attributes has a considerable impact on the assessment of suppliers. The new attribute system presented in this paper offers a more comprehensive means of characterizing both the basic performance and green performance of suppliers.

The structure of this paper is as follows. Section 2 is a literature review of relevant fields. Section 3 describes the evaluation attributes of green building material supplier selection. The preliminary knowledge and the specific process of the methods used in this paper are introduced in Section 4. Section 5 shows a case study of selecting a timber supplier and verifies the effectiveness and stability of the framework. Section 6 provides comments on managerial and practical implications, the conclusions, and discusses future research directions.

2. Literature Review

2.1. Evaluation Attribute System

In the field of supplier selection, cost and quality are considered to be the two fundamental evaluation factors [24,25]. Due to the particularity of the building material supplier selection problem, that is, the overall building requirements must be completed within a predetermined planning time [26], the service level and delivery capability of the building material supplier have become important considerations to ensure that the building project proceeds as planned [27,28]. Katsaliaki et al. [29] discussed supply chain disruptions and resilience, emphasizing the importance of a supplier’s service level and delivery capability in dealing with supply chain disruptions. In recent years, the concept of environmental friendliness has become increasingly prominent in the construction industry. Significant contributions to this field were made by Khoshnava et al. [30] and Nath et al. [31], who emphasized the importance of sustainability and environmental impact in the selection of building material suppliers. Then, it was further developed into the concept of green supplier selection [20]. Ma et al. [32] combined large-scale group decision making with green suppliers and proposed a comprehensive set of indicators to evaluate supplier environmental performance. Jiang et al. [33] identified key factors affecting green building supply chain resilience and offered strategies to boost it.

Based on the above, this paper establishes a comprehensive attribute evaluation system from the perspective of green building material suppliers, including cost, quality, service level, delivery capability, and green and sustainable ability, to conduct a more comprehensive evaluation of green material suppliers.

Table 1. Comparison of the proposed evaluation attribute system with previous literature

Literature	Cost	Quality	Service Level	Delivery Capability	Green and Sustainable Ability
Noorizadeh et al. (2019) [34]	✓	✓		✓
Liao et al. (2020) [35]	✓	✓		✓	✓
Wan et al. (2020) [36]	✓	✓	✓		✓
Olawumi and Chan (2020) [37]	✓	✓			✓
Xu and Deng (2022) [38]		✓	✓	✓
Kang et al. (2022) [20]	✓	✓	✓		✓
Ma et al. (2023) [32]	✓	✓	✓	✓
This paper	✓	✓	✓	✓	✓

shows a comparison of the evaluation attribute system between this paper and the previous relevant literature.

2.2. Supplier Selection Evaluation Method

MADM is the most commonly used method in the field of supplier selection [39]. Akcan and Taş [40] adopted the SWARA-TOPSIS integrated method to construct a green supply chain evaluation model to find the best suppliers. Yu and Pa [41] attempted to apply principal path analysis to detect evolutionary trajectories in TOPSIS domains to improve the performance of TOPSIS on large-scale data. Kuo [42] applied the Vise Kriterijumska Optimizacija Kompromisno Resenje (VIKOR) method to the selection of material suppliers in industrial sewage systems, which will effectively reduce wastewater generation. With the development of fuzzy theory, research on fuzzy MCDM has received increasing attention [43]. Fallahpour et al. [44] developed a hyper-hybrid model for a multi-criteria decision system, including fuzzy decision making trial and evaluation laboratory (FDEMATEL), fuzzy best worst method (FBWM), fuzzy analytical network process (FANP), and fuzzy inference system (FIS), and applied this to sustainable supplier selection. The TODIM method was initially proposed by Gomes and Lima [45] and has been used in multiple fields such as suppliers [46], healthcare [47], water resource [48], and transportation [49]. Leoneti and Gomes [50] proposed the exponential TODIM (ExpTODIM) method based on the traditional TODIM method, which has better performance than the original method and solves two paradoxes of the TODIM method in extreme situations. In this paper, the ExpTODIM method is improved using IFNs to extend to MAGDM problems (named Group-IF-ExpTODIM). Yu et al. [51] argue that IFNs theory has had a profound impact on dealing with hesitation and ambiguity and is expanding. At present, IFNs are widely used in real-world evaluation problems to describe data that cannot be directly obtained or calculated [52,53]. Consequently, this paper proposes the Group-IF-ExpTODIM method for evaluating green building material suppliers, aiming to improve the accuracy of evaluation results.

The determination of attribute weights is a key issue in the field of MADM, which directly affects the accuracy of the final result [54]. When facing MADM problems in the real world, it is difficult to ensure complete independence between various attributes, and the applicability of additive measurement methods is ineffective [55]. Shen [56] researched the independence of financial stock evaluation attributes, elucidating the interrelationships between said attributes through the use of if–then decision rules. The method of determining attribute weights considering the interaction effects of attributes is generally believed to possess higher accuracy and greater practical significance compared to conventional methods of determining attribute weights [57]. This paper calculates the property weights considering attribute interactions using a Choquet integral based on the $\lambda$-fuzzy measure and Shapley value. The fuzzy measure is a subjective measurement scale for fuzzy objects, and its principle is to transform the probability theory of measuring general things’ emotions into possibility theory [58]. The $\lambda$-fuzzy measure proposed by Sugeno significantly reduces the complexity of solving and calculating fuzzy measures [59]. A Choquet integral is a type of fuzzy integral [60]. Due to its nonlinearity, a fuzzy integral does not assume independence among attributes [61]. A Choquet integral is widely used in MADM problems [62,63,64]. The Shapley value provides a solution for cooperative game theory by calculating the marginal contributions of players [65]. In 1992, Marichal first proposed the application of the Shapley value to MADM as a means to represent the importance coefficients of experts. Subsequently, the Shapley value is also used to calculate attribute weights [66]. In summary, this paper calculates the attribute weights considering attribute interactions using the $\lambda$-fuzzy measure, Choquet integral, and Shapley value.

3. Evaluation Attributes of Green Building Material Supplier Selection

The determination of attributes is one of the core issues in MADM problems. In the field of supplier selection, “Cost” and “Quality” are the two main attributes that are most prioritized to evaluate supplier performance [67,68]. Over time, many scholars have incorporated “Supplier service levels” and “Delivery ability” into the evaluation attributes of building material supplier selection [38]. However, the selection of green building material suppliers has special requirements for reducing energy consumption and protecting the environment. The traditional evaluation attributes of building material suppliers can no longer meet the requirements. For example, a supplier with a higher price has a lower priority in traditional supplier evaluation systems. However, when the supplier uses the least energy to produce products, from the perspective of green building material supplier selection, the priority of this supplier should be raised. Therefore, this paper improves the traditional building supplier evaluation system by adding “Green and sustainable capabilities” to evaluate suppliers’ energy consumption and environmental protection capabilities in the production process. The specific attributes describing the supplier’s green and sustainable capabilities are as follows:

(1)

Green Business Philosophy: This attribute measures the supplier’s business philosophy and values in the green building industry [40]. Suppliers should consciously incorporate environmental protection and sustainable development into their core values. This includes setting clear environmental policies and goals, such as reducing resource consumption, lowering carbon emissions, and protecting ecosystems. Additionally, suppliers can demonstrate their efforts and achievements in green business by obtaining relevant environmental certifications and awards [69].

(2)

Application of Clean Technology: This attribute assesses the extent to which suppliers employ clean technology and environmentally friendly equipment during the construction process. Suppliers should actively adopt low-carbon materials, such as renewable energy sources (e.g., solar, wind energy) and recyclable materials, to minimize their negative impact on the environment [29]. Moreover, they should utilize eco-friendly construction processes and equipment, such as energy-efficient devices and water-saving systems, to improve energy utilization efficiency and reduce natural resource consumption.

(3)

Recycling and Reuse Efficiency: This attribute measures the supplier’s waste recycling and reuse practices during the construction process. Suppliers should implement waste classification and sorting to facilitate effective recycling and reuse [70]. They can take measures to reduce waste generation, such as optimizing material usage and implementing circular economy principles. By enhancing recycling and reuse efficiency, the demand for natural resources can be reduced, thus mitigating the environmental burden [71].

(4)

Pollutant Emission Situation: This attribute evaluates the supplier’s emission of pollutants to the air, water bodies, and soil during the construction process [72]. Suppliers should monitor and document pollutant emissions, including atmospheric emissions and wastewater discharges. They need to ensure that their emissions comply with relevant national and local standards and regulations [73]. By controlling and reducing pollutant emissions, ecological environments can be protected, environmental pollution can be prevented, and public health can be safeguarded [8].

(5)

Green Energy Utilization Rate: This attribute measures the proportion of renewable energy used by suppliers during the construction process and their energy efficiency [9]. Suppliers should strive to increase the share of renewable energy sources, such as solar and wind energy, to reduce dependence on conventional energy sources. Additionally, they should focus on energy consumption efficiency, such as utilizing energy-saving devices and optimizing building design, to lower energy consumption and improve energy utilization efficiency [74]. By adopting green energy sources and enhancing energy efficiency, greenhouse gas emissions can be reduced, promoting sustainable development.

This paper establishes an evaluation attribute system of green building material suppliers, which includes 18 attributes in five dimensions: “Quality”, “Cost”, “Service level”, “Delivery ability”, and “Green and sustainable ability”.

Table 2. Evaluation attribute system of green building material suppliers.

First-Level Attribute	Second-Level Attribute	Data Type	Explanation	Source
Quality	Product qualification rate ($C_1$)	Real	The percentage of products that meet the quality standards	Liao et al. [35]
	Production standardization ($C_2$)	IFNs	The degree of conformity to the production standards	Kang et al. [20]
	Product quality management ability ($C_3$)	IFNs	The ability to control and improve the product quality	Kang et al. [20]
Cost	Transportation cost ($C_4$)	Real *	The cost of transporting the products from the supplier to the customer	Kang et al. [20]
	Storage cost ($C_5$)	Real *	The cost of storing the products in warehouses or other facilities	Mahato et al. [75]
	Product purchase price ($C_6$)	Real *	The price of buying the products from the supplier	Kang et al. [20]
Service level	Customer satisfaction ($C_7$)	IFNs	The degree of satisfaction of the customers with the products and services	Ghadge et al. [76]
	Product research and development ability ($C_8$)	IFNs	The ability to innovate and develop new products	Kang et al. [20]
	Enterprise financial condition ($C_9$)	IFNs	The financial performance and stability of the supplier	Liao et al. [35]
	After-sales ability ($C_{10}$)	IFNs	The ability to provide after-sales service and support to the customers	Owida et al. [77]
Delivery ability	Average delivery time ($C_{11}$)	Real *	The average time of delivering the products from the supplier to the customer	Kang et al. [20]
	Timely delivery rate ($C_{12}$)	Real	The percentage of products that are delivered on time or earlier	Kang et al. [20]
	Supply flexibility ($C_{13}$)	IFNs	The ability to adapt to changes in demand and supply conditions	Owida et al. [77]
Green and sustainable ability	Green business philosophy ($C_{14}$)	IFNs	The degree of commitment to green and sustainable practices	Akcan and Taş [40]
	Application of clean technology ($C_{15}$)	IFNs	The degree of adoption of clean technology in production and operation	Katsaliaki et al. [29]
	Recycling and reuse efficiency ($C_{16}$)	Real	The percentage of materials that are recycled or reused in production	Kamble et al. [70]
	Pollutant emission situation ($C_{17}$)	IFNs *	The degree of pollution caused by production and operation activities	Zheng et al. [72]
	Green energy utilization rate ($C_{18}$)	Real	The percentage of energy used that comes from renewable sources	Yildizbasi and Arioz [9]

* indicates a cost-type attribute; higher values are less favorable. Unmarked attributes are benefit-type.

shows the details of the evaluation attribute system for green building material suppliers. Qualitative indicators are evaluated by experts using IFNs. The clear and standardized definitions of quantitative attributes are as follows.

• Product qualification rate ($C_1$): qualified product quantity divided by total production quantity.
• Transportation cost ($C_4$): the transportation cost per unit of product from the supplier to the target location.
• Storage cost ($C_5$): the amount spent by the decision-maker’s company on storing each unit of product from the supplier.
• Product purchase price ($C_6$): the amount spent on purchasing each unit of product from the supplier.
• Average delivery time ($C_{11}$): total product delivery time divided by quantity of product delivery units.
• Timely delivery rate ($C_{12}$): on time delivery times divided by total delivery times.
• Recycling and reuse efficiency ($C_{16}$): the amount of recycled materials used divided by the total amount of materials used.
• Green energy utilization rate ($C_{18}$): green energy used (e.g., wind, solar) divided by total energy consumption.

4. Evaluation Methodology for Green Building Material Suppliers

This section mainly introduces the IFNs, expert weights determination method, and attribute weights determination method, and combines them with the Group-IF-ExpTODIM method to provide solutions for the selection of green building material suppliers in group decision making, and the main processes of the proposed method are shown in Figure 1.

Decision makers, often management personnel within construction firms, are tasked with assessing and selecting suppliers based on multiple attributes, necessitating the collaboration of relevant experts. This scenario forms an MAGDM problem. The supplier evaluation attributes consist of two data types: real numbers, which are directly collected and measured, and IFNs, derived from expert assessments. Invited experts rate qualitative attributes using IFNs, and the hesitation degree within these numbers can be utilized to calculate expert weights. These weights facilitate the aggregation of all experts’ opinions into a singular, complete scoring matrix. Subsequently, attribute weights are determined using the Choquet integral and Shapley value methods. The ExpTODIM method’s $Phi$ function is then applied to ascertain the preference order of all suppliers, a higher degree of preference indicates a better ranking.

4.1. Preliminary Knowledge

4.1.1. Intuitionistic Fuzzy Numbers

IFNs are an extension of fuzzy set theory that takes into account both membership and non-membership degrees [78]. An IFN has three components: membership degree, non-membership degree, and trust degree. Assuming X is a non-empty universe, the intuitionistic fuzzy set on X is defined as [79]:

$$\begin{array}{c}\hfill A=\{(x,{\mu }_A\left(x\right),v_A\left(x\right))\mid x\in X\}\end{array}$$

(1)

Among them: ${\mu }_A\left(x\right):X\to [0,1]$, $v_A\left(x\right):X\to [0,1]$ and $\forall x\in X$, $0\le {\mu }_A\left(x\right)+v_A\left(x\right)\le 1$. ${\mu }_A\left(x\right)$ and $v_A\left(x\right)$ represent the membership and non-membership of element x in X that belongs to A. Moreover, ${\pi }_A\left(x\right)=1-{\mu }_A\left(x\right)-v_A\left(x\right)$ represents the hesitation degree of element x in X belonging to A, in particular, if ${\pi }_A\left(x\right)=0$, then A degenerates into a conventional fuzzy set, and the set of intuitionistic fuzzy sets on X is denoted by $IFNs\left(X\right)$. For the intuitionistic fuzzy number $\alpha =(0.6,0.1)$, which means ${\mu }_{\alpha }\left(x\right)=0.6,v_{\alpha }\left(x\right)=0.1,{\pi }_{\alpha }\left(x\right)=0.3$, it can be explained by the voting model as follows: in a vote, there are 10 participants, among which 6 vote for, 1 votes against, and 3 abstain.

Let $s\left(A\right)$ and $s\left(B\right)$ be the scoring functions of intuitionistic fuzzy numbers $A=({\mu }_{\alpha },v_{\alpha })$ and $B=({\mu }_{\beta },v_{\beta })$, then the calculation formula for the scoring function of intuitionistic fuzzy numbers is as follows [80]:

$$\begin{array}{c}\hfill s\left(X\right)={\mu }_x-v_x+\left({\displaystyle \frac{2+3({\mu }_x-v_x)}{6}}\right){\pi }_x\end{array}$$

(2)

There are: If $s\left(A\right)>s\left(B\right)$, then $A\succ B$; if $s\left(A\right)<s\left(B\right)$, then $A\prec B$; if $s\left(A\right)=s\left(B\right)$, then ${\pi }_A>{\pi }_B$, $A\prec B$, when ${\pi }_A={\pi }_B$, $A=B$. The scoring function considers the influence of approval, disapproval, and abstention. Considering the conformity psychology of the abstainers, they may vote for, against, or still abstain. Therefore, the abstainers are divided into three categories: inclined to approve, inclined to disapprove, and still inclined to abstain. Assuming that the number of people in the three categories of abstention is equal, that is, assigning a weight value of $1/3$ to ${\pi }_x$, finally, use half of the difference between the approval and disapproval votes to correct this weight value.

4.1.2. $\lambda$-Fuzzy Measures, Choquet Integrals, and Shapley Value

Let $C=\left\{c_1,c_2,\cdots ,c_n\right\}$ be a finite set of attributes in an MADM problem, and $P\left(C\right)$ is the power set of C. For the given $\lambda \in (-1,\infty )$, $g_{\lambda }:P\left(C\right)\to [0,1]$, the following conditions are met [81].

(1)

$g_{\lambda }(⌀)=0,g_{\lambda }\left(C\right)=1$;

(2)

$\forall M,N\in P\left(C\right)$ and $M\cap N=\varphi$, then $g_{\lambda }(M\cup N)=g_{\lambda }\left(M\right)+g_{\lambda }\left(N\right)+\lambda g_{\lambda }\left(M\right)g_{\lambda }\left(N\right)$;

(3)

$g_{\lambda }$ is a continuous function.

Then, $g_{\lambda }$ is called a $\lambda$-fuzzy measure defined on $P\left(C\right)$ [82], where $\lambda$ represents the interaction of attributes in the Choquet integral. In MADM problems, ${\bigcup }_{i=1}^n{c}_i=C$, the $\lambda$ fuzzy measure definition of $P\left(C\right)$ is as follows [83].

$$g_{\lambda }\left(C\right)=\left\{\begin{array}{cc}{\displaystyle \frac{1}{\lambda }}\left({\displaystyle \prod _{i=1}^n}\left(1+\lambda g_{\lambda }\left(c_i\right)\right)-1\right),\hfill & \lambda \ne 0\hfill \\ \sum _{i=1}^n{g}_{\lambda }\left(c_i\right),\hfill & \lambda =0\hfill \end{array}\right.$$

(3)

When $g_{\lambda }\left(C\right)=1$, it can be obtained that

$$\begin{array}{c}\hfill \lambda =\prod _{i=1}^n\left(1+\lambda g_{\lambda }\left(c_i\right)\right)-1\end{array}$$

(4)

Let $g_{\lambda }$ be a fuzzy measure of $(C,P(C\left)\right)$, then the discrete Choquet integral function is defined as follows:

$$\int fd{g}_{\lambda }=\sum _{i=1}^n\left(g_{\lambda }\left(F_{\left(i\right)}\right)-g_{\lambda }\left(F_{(i+1)}\right)\right)f\left(c_{\left(i\right)}\right)$$

(5)

where $\left(i\right)$ is a permutation of $f\left(c_{\left(i\right)}\right)$, and $F_{(n+1)}=0,F_{\left(i\right)}=\left\{c_{\left(i\right)},c_{(i+1)},\cdots ,c_{\left(n\right)}\right\}$, $0\le f\left(c_{\left(1\right)}\right)\le f\left(c_{\left(2\right)}\right)\le \cdots \le f\left(c_{\left(n\right)}\right).$

The Shapley value is a relatively fair and interpretable method for profit allocation in cooperative games, used to calculate the contribution of each participant to the entire profit [84]. In MADM problems, the Shapley value can help decision makers understand the interaction between attributes and the contribution of each attribute to the overall population. The calculation method for the Shapley value is as follows:

$${\phi }_i(g,C)=\sum _{S\subseteq C\setminus c_i}{\displaystyle \frac{(n-s-1)!}{n!}}[g(S\cup c_i)-g\left(S\right)]$$

(6)

where n and s indicate the number of elements in C and S, and ${\phi }_i\left(g\right)⩾0$, ${\sum }_{i=1}^n{\phi }_i\left(g\right)=1$. ${\phi }_i\left(g\right)$ is the ith attribute weight.

4.2. General Framework

The overall structure of the stakeholders involved in this research method includes an opinion evaluation group composed of t experts, n evaluation attributes for green building material suppliers, and m alternative suppliers. The proposed method is divided into four parts: determining expert weights, determining attribute weights, obtaining descriptions of alternatives under every attribute, and ranking suppliers, which are usually the basic components of an MAGDM problem. In order to improve the comprehensibility of the method proposed in this paper, the detailed quotation of symbols is shown in

Table 3. Quotation.

Symbol	Interpretation	Symbol	Interpretation
i	Index of attributes, $i=1,2,\cdots ,n$	j	Index of alternatives, $j=1,2,\cdots ,m$
k	Index of experts, $k=1,2,\cdots ,t$	$C_i$	ith attribute
$A_j$	jth alternative	$E_k$	kth expert
$w_i$	Weight of $C_i$	$w_k^e$	Weight of $E_k$
$g_{\lambda }\left(c_i\right)$	$\lambda$-fuzzy measure of attribute $C_i$	${\phi }_i\left(g\right)$	Shapley value of attribute $A_i$, ${\phi }_i\left(g\right)=w_i$
${\phi }_i\left(A_p,A_q\right)$	Superiority of $A_p$ for $A_q$ under $C_i$	$\Phi \left(A_j\right)$	Overall performance of $A_j$

.

4.3. Group-IF-ExpTODIM Method with Choquet Integrals and Shapley Value to Determine Attribute Weights

This paper introduces IFNs into an improved TODIM method (i.e., the ExpTODIM method) and extends it to the field of group decision making. It proposes the Group-IF-ExpTODIM method to rank green building material suppliers and uses Choquet integrals and Shapley value to calculate attribute weights.

4.3.1. Problem Definition

Step 1. Define the goal, the alternatives (i.e., suppliers) $A_j,j=1,2,\cdots ,m$, the attributes $C_i,i=1,2,\cdots ,n$, and the experts $E_k,k=1,2,\cdots ,t$ with respect to the problem. This is the preparation procedure before multi-criteria group decision making, and the usually defined goal is to select suitable suppliers. $C_i$ refers to each attribute, n is the total number of attributes in the attribute system, $E_k$ refers to each expert, and t is the total number of experts participating in the evaluation. The types of attributes are classified into two categories: real number type and IFNs type. This distinction allows for the presentation of attribute data in two distinct forms. In particular, real data are called $C_r,r=1,2,\cdots ,n_r$; these kind of data are obtained through data collection. The intuitionistic fuzzy data types are called $C_f,f=1,2,\cdots ,n_f$, obtained through experts’ evaluation, and $n=n_r+n_f$.

Step 2. The acquisition of data pertaining to attribute $C_r$ is typically accomplished through direct calculation, measurement, collection, and other methodologies. The data pertaining to $C_r$ are quantitative in nature. In contrast, the data pertaining to $C_f$ are qualitative in nature and cannot be directly measured. Accordingly, experts are invited to evaluate the IFNs type attributes of each alternative solution in order to obtain the data $C_f$. After that, obtain the initial real number evaluation matrix $R_r=a_{ri}$ and the initial intuitionistic fuzzy evaluation matrix $R_f^k=a_{fi}^k$. The size of $R_r$ is $n_r\ast m$, where $n_r$ is the number of attributes represented by real numbers in the attribute system and m is the number of alternatives. The value of $C_f$ is represented by matrix $R_f^k$, which has a size of $n_f\ast m\ast t$. Here, $n_f$ is the number of attributes represented by IFNs in the attribute system, and m is the number of alternatives, k is the number of experts, and each expert forms a matrix composed of IFNs.

$$\begin{array}{c}\hfill \begin{array}{cc}{R}_r=\left[\begin{array}{cccc}{a}_{11}& a_{12}& \cdots & a_{1m}\\ a_{21}& a_{22}& \cdots & a_{2m}\\ ⋮& ⋮& \ddots & ⋮\\ a_{n_{r}1}& a_{n_{r}2}& \cdots & a_{n_{r}m}\end{array}\right]& R_f^k=\left[\begin{array}{cccc}{a}_{11}^k& a_{12}^k& \cdots & a_{1m}^k\\ a_{21}^k& a_{22}^k& \cdots & a_{2m}^k\\ ⋮& ⋮& \ddots & ⋮\\ a_{n_{f}1}^k& a_{n_{f}2}^k& \cdots & a_{n_{f}m}^k\end{array}\right]\end{array}\end{array}$$

4.3.2. Weight Determination

Step 3. Determine the expert weights. The weight of experts is calculated from the IFNs in the matrix $R_f^k=a_{fi}^k$. The fundamental concept is that IFNs are comprised of three components: membership degree, non-membership degree, and hesitation degree. In general, experts with a higher hesitation degree may lack a comprehensive understanding of the problem and should be assigned lower weights. Specifically, let $a_{fj}^k=({\mu }_{fj}^k,v_{fj}^k,{\pi }_{fj}^k)$ be an IFN for rating of the fth attribute for $C_f$ the jth alternative of the kth expert. Then, the weight of the $k_{th}$ expert can be obtained as [85]:

$$\begin{array}{c}\hfill w_k^e={\displaystyle \frac{{\displaystyle \sum _{f=1}^{nf}\sum _{j=1}^m}\left({\mu }_{fj}^k+{\pi }_{fj}^k\left(\frac{{\mu }_{fj}^k}{{\mu }_{fj}^k+v_{fj}^k}\right)\right)}{{\displaystyle \sum _{k=1}^t\sum _{f=1}^{nf}\sum _{j=1}^m}\left({\mu }_{fj}^k+{\pi }_{fj}^k\left(\frac{{\mu }_{fj}^k}{{\mu }_{fj}^k+v_{fj}^k}\right)\right)}}\end{array}$$

(7)

where ${\sum }_{k=1}^t{w}_k^e=1$.

Step 4. Determine the attribute weights. It is the responsibility of experts to use IFNs to evaluate the $\lambda$-fuzzy measure value of each attribute. This should be performed based on the belief that the greater the impact of the attribute on the evaluation objective, the higher the evaluation value should be. These IFNs will be transformed through score function and the expert weights obtained in the previous step. Subsequently, using Equation (3) to calculate the Choquet integral for different attribute combinations, and then using Equation (6) to calculate the Shapley value for each attribute, the attribute weight can be calculated. In order to reduce the calculation steps, it is possible to perform hierarchical calculations based on first-level and second-level attributes.

Step 5. Gather the opinions of experts through expert weights and convert IFNs into score function. The opinions of experts are only focused on attributes expressed by IFNs. The aggregation formula of intuitionistic fuzzy numbers is as follows:

$$\begin{array}{cc}\hfill a_{fj}=& w_1^e{a}_{fj}^1\oplus w_2^e{a}_{fj}^k\oplus \cdots \oplus w_k^e{a}_{fj}^k\oplus \cdots \oplus w_t^e{a}_{fj}^t\hfill \\ \hfill =& \left(1-\prod _{k=1}^t{\left(1-{\mu }_{fj}^k\right)}^{w_k^e},\prod _{k=1}^t{\left(v_{fj}^k\right)}^{w_k^e},\prod _{k=1}^t{\left(1-{\mu }_{fj}^k\right)}^{w_k^e}-\prod _{k=1}^t{\left(v_{fj}^k\right)}^{w_k^e}\right)\hfill \end{array}$$

(8)

After gathering expert opinions, the IFNs are transformed into a score function through Equation (2), recorded as ${\widehat{a}}_{fj}\in (0,1)$, and in order to unify these, it is necessary to standardize the real number attributes $C_r$; the standardize function is

$$\begin{array}{c}\hfill {\widehat{a}}_{rj}={\displaystyle \frac{a_{rj}}{max{a}_r}}\end{array}$$

(9)

Hence, a complete scoring matrix $R=R_f+R_r={\widehat{a}}_{ij}$, ${\widehat{a}}_{ij}\in (0,1)$ is obtained.

$$R=\left[\begin{array}{cccc}{\widehat{a}}_{11}& {\widehat{a}}_{12}& \cdots & {\widehat{a}}_{1m}\\ {\widehat{a}}_{21}& {\widehat{a}}_{22}& \cdots & {\widehat{a}}_{2m}\\ ⋮& ⋮& \ddots & ⋮\\ {\widehat{a}}_{n1}& {\widehat{a}}_{n2}& \cdots & {\widehat{a}}_{nm}\end{array}\right]$$

4.3.3. Preference Calculation

Step 6. Attributes are divided into benefit attributes and cost attributes. The higher the value of benefit attributes, the more advantageous the evaluation object is, while cost attributes are the opposite.

$${\overline{a}}_{ij}=\left\{\begin{array}{cc}{\widehat{a}}_{ij}/{\displaystyle \sum _{j=1}^m}{\widehat{a}}_{ij},\hfill & \mathrm{if} C_i \mathrm{is} \mathrm{benefit} \mathrm{criteria};\hfill \\ {\displaystyle \frac{1}{{\widehat{a}}_{ij}}}/{\displaystyle \sum _{j=1}^m}{\displaystyle \frac{1}{{\widehat{a}}_{ij}}},\hfill & \mathrm{if} C_i \mathrm{is} \mathrm{cost} \mathrm{criteria}\hfill \end{array}\right.$$

(10)

Step 7. Paired comparison of alternative for calculating the $Phi$ function; the $Phi$ function is as follows:

$${\phi }_i\left(A_p,A_q\right)=\left\{\begin{array}{cc}{w}_i\left(1-{10}^{-\rho \left|a_{ip}-a_{iq}\right|}\right),\hfill & \mathrm{if}\left(a_{ip}-a_{iq}\right)>0\hfill \\ 0,\hfill & \mathrm{if}\left(a_{ip}-a_{iq}\right)=0\hfill \\ -w_i\lambda \left(1-{10}^{-\rho \left|a_{ip}-a_{iq}\right|}\right),\hfill & \mathrm{if}\left(a_{ip}-a_{iq}\right)<0\hfill \end{array}\right.$$

(11)

where $A_p,A_q\in A_j$ and $A_p\ne A_q$, $\lambda$ and $\rho$ are two related parameters.

The determination of $\lambda$ typically falls under the purview of decision makers, enabling them to express their preferences for viable alternatives. The value of $\lambda$ reflects the decision maker’s preferences. A lower value of $\lambda$ indicates a leaning towards alternatives with more advantages, while a higher value suggests a preference for alternatives with fewer disadvantages. Additionally, the parameter $\rho \in N^{*}$ signifies the significance of the decision based on the decision maker’s perception and exhibits the robustness of the $Phi$ function to changes in its parameters. A higher $\rho$ implies that even slight parameter alterations can significantly impact the outcome of the $Phi$ function. Conversely, a lower $\rho$ suggests that minor parameter adjustments bear only a marginal influence on the ultimate result. In line with the research findings of Leoneti and Gomes (2021) [50], the combination of $\lambda$ = 2.25 and $\rho$ = 3 is regarded as an efficacious one, conducive to optimal decision making [86].

Step 8. Calculate the overall performance by comparing the sum of each alternative to the superiority of the other alternatives for all attributes. The calculation formula is as follows:

$$\begin{array}{c}\hfill \Phi \left(A_j\right)=\sum _{j\ne p,p=1}^m\sum _{i=1}^n{\phi }_i\left(A_j,A_p\right)\end{array}$$

(12)

Step 9. Rank according to the $\Phi$ value from highest to lowest, and the higher the $\Phi$ value, the higher the ranking obtained by the alternative.

5. Case Study

Based on the established attribute system and evaluation model for selecting green building material suppliers, this paper offers a solution to choose suppliers in the domain of sustainable construction materials effectively. This section presents a case study examining the procurement of timber in the context of green building projects to demonstrate the proposed model’s robustness and applicability. Furthermore, an in-depth analysis of the performance and effectiveness of the framework across multiple dimensions is conducted, thereby contributing to the existing body of knowledge in the field of green building material supplier selection.

5.1. Case Description

Green buildings involve various sustainable and environmentally friendly building materials, including timber, concrete, metal, recyclable materials, and high-performance insulation materials. In the construction phase of green buildings, the green degree, quality, and price of different materials vary depending on the supplier. The selection of green building material suppliers will directly affect the greenness of the final building. Timber is one of the most common building materials in the construction process of green buildings. It is commonly used in structural framing, exterior wall decoration, interior decoration, and furniture production, giving the building a natural and warm appearance and adding comfort to the space. Timber plays an important role in green building construction. It not only meets the requirements of sustainability and environmental protection but also has good energy efficiency and health performance. Therefore, this paper takes the selection of timber suppliers as an example.

Z Company is a real estate development company for commercial residential buildings established in Wuhan, China. The business scope includes architecture, decoration engineering design and construction, real estate development, and commercial housing sales. Its developed real estate projects have obtained China’s green building certification multiple times. In order to improve the green level and quality of residential products delivered and reduce construction costs through supply chain management, the developer management committee adopts the MAGDM method for supplier decision making. Six timber suppliers were selected from the list of suppliers for a case study.

5.2. Initial Data of Suppliers

The six alternative timber suppliers are listed as $A_1$ to $A_6$. The data obtained through the calculation and investigation are divided into two parts: attributes expressed in real numbers (i.e., $C_1$, $C_4$, $C_5$, $C_6$, $C_{11}$, $C_{12}$, $C_{16}$, $C_{18}$) and attributes expressed in IFNs (i.e., $C_2$, $C_3$, $C_7$, $C_8$, $C_9$, $C_{10}$, $C_{13}$, $C_{14}$, $C_{15}$, $C_{17}$). The data corresponding to the former are called $D_r$, and the data corresponding to the latter are called $D_f$. The details in $D_r$ are shown in

Table 4. Original data $D_r$

Real Attribute	${\mathit{A}}_1$	${\mathit{A}}_2$	${\mathit{A}}_3$	${\mathit{A}}_4$	${\mathit{A}}_5$	${\mathit{A}}_6$
$C_1$	0.92	0.88	0.79	0.95	0.82	0.93
$C_4$ (1000 CNY)	50.36	68.14	59.33	31.39	62.03	46.27
$C_5$ (1000 CNY)	16.24	19.30	20.80	15.69	14.22	16.54
$C_6$ (CNY/T)	555	482	375	448	531	496
$C_{11}$ (day)	2.5	3.8	1.6	2.7	2.6	3.1
$C_{12}$	0.67	0.89	0.86	0.88	0.85	0.87
$C_{16}$	0.28	0.26	0.17	0.20	0.12	0.16
$C_{18}$	0.48	0.36	0.27	0.15	0.33	0.31

.

The attributes represented by IFNs were evaluated by experts. In this case, four experts were invited, and their detailed information can be found in

Table 5. Basic information of the four experts

Expert	Field of Expertise	Years of Experience	Occupation	Education
$E_1$	Environmental science	8–10 years	Real estate company	Master
$E_2$	Architectural planning management	16–20 years	University associate professor	Master
$E_3$	Decision science	6–8 years	University associate professor	Doctor
$E_4$	Business management	10–12 years	Real estate company	Doctor

.

The evaluation results of the IFNs’ attributes of these four experts on six different timber suppliers are shown in

Table 6. Data from experts evaluating qualitative attributes (i.e., data $D_f$)

		${\mathit{E}}_1$	${\mathit{E}}_2$	${\mathit{E}}_3$	${\mathit{E}}_4$		${\mathit{E}}_1$	${\mathit{E}}_2$	${\mathit{E}}_3$	${\mathit{E}}_4$
$A_1$	$C_2$	(0.5, 0.3)	(0.6, 0.15)	(0.75, 0.2)	(0.7, 0.2)	$C_{10}$	(0.3, 0.2)	(0.5, 0.3)	(0.7, 0.1)	(0.45, 0.4)
$A_2$		(0.65, 0.1)	(0.65, 0.15)	(0.75, 0.1)	(0.65, 0.15)		(0.75, 0.1)	(0.5, 0.4)	(0.7, 0.2)	(0.45, 0.5)
$A_3$		(0.85, 0.05)	(0.65, 0.1)	(0.9, 0.0)	(0.6, 0.15)		(0.25, 0.05)	(0.45, 0.55)	(0.75, 0.1)	(0.55, 0.1)
$A_4$		(0.55, 0.15)	(0.7, 0.15)	(0.85, 0.05)	(0.8, 0.1)		(0.35, 0.5)	(0.25, 0.35)	(0.6, 0.25)	(0.6, 0.25)
$A_5$		(0.75, 0.1)	(0.6, 0.1)	(0.6, 0.2)	(0.9, 0.1)		(0.4, 0.5)	(0.5, 0.2)	(0.5, 0.15)	(0.3, 0.4)
$A_6$		(0.6, 0.2)	(0.65, 0.25)	(0.75, 0.05)	(0.65, 0.2)		(0.15, 0.7)	(0.6, 0.35)	(0.6, 0.05)	(0.3, 0.6)
$A_1$	$C_3$	(0.4, 0.5)	(0.35, 0.3)	(0.5, 0.2)	(0.5, 0.35)	$C_{13}$	(0.15, 0.8)	(0.3, 0.6)	(0.25, 0.6)	(0.3, 0.5)
$A_2$		(0.65, 0.2)	(0.45, 0.4)	(0.7, 0.1)	(0.5, 0.35)		(0.05, 0.9)	(0.05, 0.7)	(0.15, 0.6)	(0.3, 0.5)
$A_3$		(0.35, 0.25)	(0.5, 0.1)	(0.55, 0.4)	(0.65, 0.25)		(0.25, 0.6)	(0.3, 0.5)	(0.4, 0.55)	(0.2, 0.65)
$A_4$		(0.55, 0.15)	(0.55, 0.2)	(0.5, 0.2)	(0.5, 0.3)		(0.15, 0.75)	(0.25, 0.55)	(0.3, 0.35)	(0.3, 0.5)
$A_5$		(0.4, 0.2)	(0.35, 0.2)	(0.5, 0.3)	(0.55, 0.4)		(0.4, 0.55)	(0.5, 0.35)	(0.55, 0.3)	(0.2, 0.75)
$A_6$		(0.6, 0.15)	(0.5, 0.4)	(0.65, 0.15)	(0.4, 0.2)		(0.15, 0.65)	(0.3, 0.45)	(0.25, 0.35)	(0.3, 0.6)
$A_1$	$C_7$	(0.7, 0.1)	(0.5, 0.2)	(0.6, 0.35)	(0.55, 0.25)	$C_{14}$	(0.65, 0.2)	(0.85, 0.1)	(0.9, 0.05)	(0.4, 0.6)
$A_2$		(0.75, 0.15)	(0.5, 0.3)	(0.8, 0.1)	(0.75, 0.2)		(0.3, 0.6)	(0.55, 0.2)	(0.45, 0.4)	(0.4, 0.5)
$A_3$		(0.5, 0.3)	(0.4, 0.2)	(0.85, 0.05)	(0.85, 0.1)		(0.7, 0.15)	(0.35, 0.3)	(0.6, 0.25)	(0.7, 0.1)
$A_4$		(0.65, 0.2)	(0.9, 0.05)	(0.65, 0.1)	(0.5, 0.4)		(0.85, 0.05)	(0.65, 0.2)	(0.55, 0.3)	(0.3, 0.4)
$A_5$		(0.45, 0.3)	(0.65, 0.2)	(0.7, 0.2)	(0.8, 0.2)		(0.85, 0.15)	(0.45, 0.5)	(0.6, 0.2)	(0.7, 0.1)
$A_6$		(0.3, 0.5)	(0.75, 0.1)	(0.65, 0.3)	(0.4, 0.3)		(0.4, 0.55)	(0.85, 0.1)	(0.75, 0.1)	(0.7, 0.25)
$A_1$	$C_8$	(0.3, 0.4)	(0.25, 0.4)	(0.2, 0.45)	(0.5, 0.3)	$C_{15}$	(0.4, 0.25)	(0.45, 0.25)	(0.5, 0.25)	(0.7, 0.2)
$A_2$		(0.1, 0.5)	(0.45, 0.2)	(0.4, 0.4)	(0.5, 0.35)		(0.5, 0.2)	(0.6, 0.2)	(0.4, 0.2)	(0.5, 0.3)
$A_3$		(0.3, 0.4)	(0.25, 0.35)	(0.5, 0.2)	(0.35, 0.45)		(0.2, 0.25)	(0.25, 0.4)	(0.35, 0.2)	(0.45, 0.4)
$A_4$		(0.6, 0.2)	(0.4, 0.35)	(0.6, 0.15)	(0.45, 0.4)		(0.1, 0.25)	(0.35, 0.5)	(0.25, 0.6)	(0.3, 0.5)
$A_5$		(0.55, 0.15)	(0.5, 0.1)	(0.45, 0.2)	(0.5, 0.25)		(0.05, 0.9)	(0.5, 0.2)	(0.15, 0.7)	(0.2, 0.6)
$A_6$		(0.35, 0.3)	(0.6, 0.2)	(0.5, 0.4)	(0.35, 0.55)		(0.05, 0.9)	(0.45, 0.3)	(0.1, 0.65)	(0.1, 0.85)
$A_1$	$C_9$	(0.55, 0.4)	(0.6, 0.3)	(0.7, 0.2)	(0.8, 0.1)	$C_{17}$	(0.25, 0.7)	(0.45, 0.3)	(0.6, 0.25)	(0.7, 0.15)
$A_2$		(0.65, 0.25)	(0.75, 0.15)	(0.8, 0.1)	(0.45, 0.5)		(0.35, 0.4)	(0.45, 0.4)	(0.1, 0.65)	(0.1, 0.85)
$A_3$		(0.7, 0.1)	(0.45, 0.25)	(0.75, 0.1)	(0.65, 0.15)		(0.5, 0.1)	(0.35, 0.5)	(0.5, 0.25)	(0.4, 0.15)
$A_4$		(0.85, 0.1)	(0.5, 0.3)	(0.55, 0.25)	(0.8, 0.05)		(0.65, 0.3)	(0.45, 0.4)	(0.25, 0.4)	(0.35, 0.4)
$A_5$		(0.45, 0.2)	(0.6, 0.35)	(0.6, 0.1)	(0.7, 0.15)		(0.45, 0.5)	(0.6, 0.15)	(0.15, 0.65)	(0.35, 0.25)
$A_6$		(0.75, 0.1)	(0.5, 0.2)	(0.65, 0.2)	(0.5, 0.25)		(0.55, 0.4)	(0.45, 0.3)	(0.15, 0.65)	(0.1, 0.85)

. It contains 10 attributes that need to be expressed using IFNs in the attribute system. Each row within the criterion matrix represents a supplier, while the columns correspond to the various experts. The intersecting values denote the evaluation score given by the expert for the supplier on that particular attribute. These scores are represented by IFNs, with the parentheses indicating the degree of membership and the degree of non-membership, respectively.

5.3. Determination of Expert Weights and Attribute Weights

The calculation of weights is divided into two parts: expert weights and attribute weights. Firstly, using the evaluation results in Table 4 and Equation (7), the weights of the four experts can be obtained as (0.235, 0.257, 0.270, 0.238). Secondly, through Step 4 in Section 4.3.2, four experts need to perform fuzzy measures on each attribute using IFNs, and then use the expert weights obtained above to aggregate expert opinions; the score is obtained from Equation (2). The specific situation is shown in

Table 7. Expert fuzzy measure and score.

Attribute	${\mathit{E}}_1$	${\mathit{E}}_2$	${\mathit{E}}_3$	${\mathit{E}}_4$	Aggregation	Score
$C_1$	(0.75, 0.1)	(0.7, 0.2)	(0.55, 0.15)	(0.65, 0.2)	(0.67, 0.16)	0.61
$C_2$	(0.5, 0.2)	(0.15, 0.6)	(0.25, 0.6)	(0.3, 0.3)	(0.31, 0.39)	0.28
$C_3$	(0.5, 0.3)	(0.7, 0.2)	(0.8, 0.1)	(0.9, 0.05)	(0.77, 0.13)	0.71
$C_4$	(0.6, 0.3)	(0.75, 0.1)	(0.4, 0.5)	(0.6, 0.05)	(0.60, 0.17)	0.56
$C_5$	(0.5, 0.4)	(0.75, 0.1)	(0.6, 0.2)	(0.6, 0.1)	(0.63, 0.17)	0.58
$C_6$	(0.7, 0.1)	(0.7, 0.2)	(0.45, 0.5)	(0.7, 0.05)	(0.65, 0.15)	0.61
$C_7$	(0.45, 0.3)	(0.25, 0.6)	(0.3, 0.5)	(0.65, 0.15)	(0.43, 0.34)	0.17
$C_8$	(0.4, 0.25)	(0.65, 0.1)	(0.3, 0.2)	(0.7, 0.1)	(0.54, 0.15)	0.56
$C_9$	(0.7, 0.1)	(0.35, 0.25)	(0.45, 0.5)	(0.7, 0.05)	(0.57, 0.16)	0.55
$C_{10}$	(0.8, 0.05)	(0.75, 0.2)	(0.4, 0.2)	(0.55, 0.2)	(0.65, 0.15)	0.63
$C_{11}$	(0.5, 0.1)	(0.2, 0.6)	(0.5, 0.3)	(0.85, 0.05)	(0.58, 0.18)	0.53
$C_{12}$	(0.75, 0.05)	(0.55, 0.1)	(0.45, 0.2)	(0.75, 0.1)	(0.64, 0.10)	0.69
$C_{13}$	(0.3, 0.45)	(0.4, 0.25)	(0.45, 0.5)	(0.8, 0.15)	(0.54, 0.30)	0.30
$C_{14}$	(0.45, 0.35)	(0.5, 0.15)	(0.35, 0.35)	(0.85, 0.05)	(0.59, 0.17)	0.54
$C_{15}$	(0.65, 0.05)	(0.2, 0.7)	(0.65, 0.2)	(0.7, 0.1)	(0.59, 0.17)	0.55
$C_{16}$	(0.3, 0.4)	(0.6, 0.2)	(0.7, 0.15)	(0.45, 0.25)	(0.54, 0.23)	0.43
$C_{17}$	(0.6, 0.1)	(0.8, 0.15)	(0.55, 0.25)	(0.75, 0.1)	(0.69, 0.14)	0.65
$C_{18}$	(0.3, 0.45)	(0.4, 0.55)	(0.45, 0.4)	(0.8, 0.1)	(0.54, 0.32)	0.29

. Four experts used IFNs to provide fuzzy measures for each attribute and aggregated expert opinions through expert weights. The score value of each attribute is the $\lambda$-fuzzy measure and indicates the importance of the attribute in the evaluation process. The higher the value, the higher the importance.

By Equations (3), (4), and (6), it can be concluded that the attribute weights are (0.096, 0.053,0.124, 0.047, 0.049, 0.052, 0.048, 0.303, 0.300, 0.349, 0.224, 0.316, 0.114, 0.110, 0.113, 0.082, 0.138, 0.052).

5.4. Ranking Using Group-IF-ExpTODIM Method

Using Equations (8) and (9) to obtain the complete score matrix R, as shown in

Table 8. Complete score matrix R.

	${\mathit{A}}_1$	${\mathit{A}}_2$	${\mathit{A}}_3$	${\mathit{A}}_4$	${\mathit{A}}_5$	${\mathit{A}}_6$
$C_1$	0.968	0.926	0.832	1.000	0.863	0.979
$C_2$	0.764	0.840	0.971	0.874	0.855	0.817
$C_3$	0.612	0.729	0.711	0.726	0.657	0.733
$C_4$	0.739	1.000	0.871	0.461	0.910	0.679
$C_5$	0.781	0.928	1.000	0.754	0.684	0.795
$C_6$	1.000	0.868	0.676	0.807	0.957	0.894
$C_7$	0.744	0.807	0.843	0.837	0.757	0.701
$C_8$	0.511	0.569	0.570	0.688	0.751	0.603
$C_9$	0.756	0.776	0.818	0.827	0.770	0.774
$C_{10}$	0.715	0.717	0.794	0.618	0.643	0.646
$C_{11}$	0.658	1.000	0.421	0.711	0.684	0.816
$C_{12}$	0.753	1.000	0.966	0.989	0.955	0.978
$C_{13}$	0.330	0.250	0.376	0.395	0.512	0.409
$C_{14}$	0.837	0.555	0.765	0.772	0.773	0.795
$C_{15}$	0.701	0.707	0.577	0.441	0.388	0.304
$C_{16}$	1.000	0.929	0.607	0.714	0.429	0.571
$C_{17}$	0.657	0.378	0.691	0.567	0.584	0.434
$C_{18}$	1.000	0.750	0.563	0.313	0.688	0.646

, its presentation integrates both qualitative and quantitative data, and by leveraging expert weights, it consolidates and standardizes the opinions of four experts. Each value in the complete score matrix lies within the range of 0 to 1, corresponding to the standardized performance of each supplier under the attribute system. Afterward, the dominance degree of alternatives is compared by the $Phi$ function. Finally, the alternative suppliers were ranked using Equations (10)–(12). The results are shown in Figure 2. Suppliers with greater overall superiority receive higher rankings, and the ranking order of the six suppliers is “$A_4>A_3>A_5>A_2>A_6>A_1$”; therefore, $A_4$ should be the best supplier in the selection of timber.

In Figure 2, the left side illustrates the local superiority of each supplier. In contrast, the right side presents the overall superiority, which is the weighted sum of the former based on attribute weights. Local superiority represents a supplier’s leading position in that attribute; a value greater than zero indicates an advantage over other suppliers, whereas a negative value signifies a disadvantage. Supplier $A_4$, ranking first in overall superiority, holds an advantage in 12 of the 18 attributes (i.e., 12 local superiorities are greater than 0), placing it at the top among the six suppliers. However, the worst-ranked supplier, $A_1$, does not have the fewest positive local superiority. $A_1$ only trails $A_4$ by one, with seven. Nevertheless, $A_1$’s significant disadvantages in attributes $C_8$, $C_{12}$, and $C_{17}$ overshadow its advantages in other attributes, leading to its undesirable ranking.

5.5. Further Analysis

Further analysis will include subsystem analysis, comparative analysis, and robustness analysis to demonstrate the advantages of the methods and models proposed in this paper.

5.5.1. Subsystem Analysis

To further discuss the impact of different subsystems on the ranking of green building material suppliers, Figure 3 shows the performance of six suppliers in the “Quality” subsystem, “Cost” subsystem, “Service level” subsystem, “Delivery ability” subsystem, and “Green and sustainable ability” subsystem. The blue circle indicates that the local superiority of the supplier under the subsystem is less than 0, while the green circle indicates that the local superiority is greater than 0. The larger the area of the circle, the greater the degree of advantage or disadvantage of the supplier under the subsystem. From Figure 4, it can be seen that $A_4$, as the top-ranked supplier, has gained advantages in both quality and cost subsystems, making it the only supplier among the six to gain advantages in two different subsystems. This is one of the reasons why $A_4$ ranks first. However, another main reason that supports $A_4$’s ranking first is that $A_4$ has fewer disadvantages in the service level, delivery ability, and green and sustainable ability subsystems. This is because decision makers choose suppliers who are more inclined to provide fewer disadvantages when setting model parameters (i.e., $\lambda$).

$A_3$ performs well on the service level subsystem, and $A_5$ performs well on the deliver ability subsystem. This has enabled them to achieve good rankings. $A_6$ has also gained advantages in the quality subsystem, but its ranking is not satisfactory, and its poor performance in the green and sustainable ability subsystem provides an obstacle to $A_6$’s ranking. $A_1$, as the worst-ranked supplier, ranks first among all suppliers in terms of sustainable ability subsystem performance. However, it ranks bottom in terms of the quality subsystem, service level subsystem, and delivery capability subsystem, making it the lowest-ranked supplier. The rankings of $A_1$ and $A_6$ reveal that only suppliers who perform well in both traditional supplier evaluation attributes and green capability attributes can achieve better rankings.

5.5.2. Comparative Analysis

In order to demonstrate the superiority of the method proposed in this paper, in the comparative analysis, it was compared with the method proposed in this paper from three different perspectives: the equal expert weights, the equal attribute weights, and the traditional ExpTODIM method. In the equal expert weight, all experts participating in decision making will be given equal weight, and the equal attribute weight is also similar. That is, all attributes will be given equal weight. This paper improves the ExpTODIM method using IFNs. Therefore, in the traditional ExpTODIM method, IFNs will be replaced by real numbers, and experts will be required to provide specific evaluation values during the evaluation process to advance the decision-making process. The details of the comparative analysis are shown in Figure 4.

From Figure 4, it can be seen that the ranking order of equal expert weights is “$A_3>A_4>A_5>A_2>A_6>A_1$”, the ranking order of equal attribute weights is “$A_4>A_3>A_6>A_5>A_2>A_1$”, and the ranking order of traditional ExpTODIM method is “$A_4>A_3>A_6>A_5>A_1>A_2$”. However, the ranking result of the method proposed in this paper is “$A_4>A_3>A_5>A_2>A_6>A_1$”. In different situations, the ranking results of the six suppliers showed slight changes but still maintained certain similarities. For example, suppliers $A_3$ and $A_4$ always become the two best suppliers, while $A_1$, $A_2$, and $A_6$ have disadvantages in the vast majority of cases. Overall, the Group-IF-ExpTODIM method with Choquet integers and a Shapley value used to determine attribute weights, proposed in this paper, considers both attribute weights and expert weights, which can achieve advantages in group decision making and make the evaluation results more realistic. Secondly, this paper improves the ExpTODIM method using IFNs. Compared to traditional ExpTODIM methods, the advantage lies in the fact that experts are allowed to use fuzzy expressions when evaluating. This reduces the possibility of excessive evaluation bias when facing uncertain or inexperienced evaluation objects and improves the credibility and accuracy of the results.

5.5.3. Robustness Analysis

Robustness analysis aims to assess the extent of a system’s response to parameter variations, input data fluctuations, or model structure uncertainties. Monte Carlo simulation employs random sampling techniques to simulate the uncertainties of parameters and input data within the system, thereby enabling system performance evaluation under such uncertainties. In this subsection, Monte Carlo simulation is employed to randomly assign weights to experts in the case study 10,000 times (In the original case, expert weight values were 0.235, 0.257, 0.270, 0.238). However, it still ensured that the sum of expert weights remained at 1.

The results of the robustness analysis are depicted in Figure 5. Notably, the supplier ranked first in the case was $A_4$, maintaining its leading position in 97% of the results even after subjecting the system to 10,000 random expert weights. Moreover, the ranking order of most suppliers in the robust analysis aligns with that observed in the initial case study, except for $A_6$. This divergence can be attributed to the relatively close performance levels of suppliers $A_1$, $A_2$, and $A_6$, resulting in potential shifts in their rankings in response to variations in expert weights. In summary, within a defined range of expert weight adjustments, the stability of supplier rankings remains largely unaffected, indicating the robustness of this methodology.

5.6. Discussion

Green buildings are increasingly favored by governments and construction firms worldwide. The selection of green building material suppliers, as a pivotal component within the green building system, significantly influences the efficacy demonstrated by green buildings. This paper introduces a framework for green building material suppliers’ selection, comprising an MAGDM approach and attribute system applicable to assessing green building material suppliers in research contexts. In comparison to traditional approaches to evaluating building material suppliers [87,88], this paper enhances the quality and validity of evaluations in three significant ways:

Firstly, an MAGDM method based on IFNs is constructed. The evaluation of green building material suppliers is more complex than traditional evaluations [89], involving a broader spectrum of stakeholders. Decision makers in construction enterprises often cannot rely solely on their experience to judge suppliers, as green supplier evaluation encompasses not only construction costs and operational management but also environmental protection issues. Thus, it is necessary to incorporate opinions from experts across various fields. However, this also introduces a new challenge: experts from different domains are required to evaluate the same target, which can be daunting given their area of expertise. For instance, some may be well-versed in production and transportation costs, while others are knowledgeable about green technological innovations. Asking experts to precisely assess areas outside their expertise can create significant pressure, and studies have shown that excessive pressure can lead to biased evaluation outcomes. Therefore, it is crucial to provide experts with an accessible and uncertainty-tolerant expression format during the evaluation process. IFNs, which allow experts to express judgments in terms of membership, non-membership, and hesitation degrees, effectively serve this purpose. The hesitation degree within IFNs represents the experts’ lack of confidence in their evaluations and can be used to adjust expert weights, thereby aggregating the opinions of multiple experts.

Secondly, the construction of an attribute system for the selection of green building material suppliers is addressed. Traditional building material supplier evaluation attribute systems fail to measure a supplier’s green performance [90,91], preventing decision makers from screening suppliers who meet the green requirements of upcoming construction projects through conventional systems. However, traditional concerns such as cost, transportation time, and delivery cycles must still be considered, as construction enterprises are not willing to pursue green efficiency at any cost. Therefore, this paper develops a comprehensive attribute system for the selection of green building material suppliers, adding attributes related to green and sustainable ability without losing sight of assessing supplier quality and cost. The aim is to identify suppliers who can maximize green efficiency while ensuring stable corporate profits. The subsystem analysis results show that among the six suppliers evaluated in the case study, none significantly outperformed others in green and sustainable ability. The top-ranked suppliers were mostly those with fewer disadvantages in these attributes. This implies that most suppliers have room for improvement in green and sustainable ability, and every increment in these attributes could help them advance in the next evaluation, confirming the importance of assessing these attributes in green supplier selection and providing insights into the future development direction of green building material suppliers.

Lastly, the evaluation process considers the interaction of attributes, which has rarely been addressed in previous solutions to building material supplier evaluation problems [92,93,94]. However, attribute interaction is a common issue in the evaluation process, even without a green context. For example, cost can affect product quality, and under a green backdrop, the recyclability of materials can impact both cost and pollution emissions. This paper uses the Choquet integral and Shapley value to calculate weights that account for attribute interaction, leading to more accurate and realistic evaluation outcomes compared to previous methods.

This refined approach not only enhances the evaluation process but also aligns with the growing importance of sustainability in the construction industry, providing a robust framework for selecting suppliers that meet both economic and environmental goals. Specifically, several key elements merit attention for applying this framework within real business environments for selecting green building material suppliers.

Data availability: In this paper, evaluation data are mainly categorized into real numbers and IFNs, where the IFNs type of data are derived from expert evaluations and can be obtained by questionnaires or group meetings of experts (online or offline). The real number type of data are usually derived from surveys of alternative supplier firms. Collaboration with industry stakeholders such as green building material suppliers and construction companies can facilitate access to relevant data sources. In addition, initiatives to promote transparency and data sharing within the green building industry can improve the availability of essential information for supplier assessment.

Computational complexity: The computational procedure of the proposed framework can be mainly divided into four parts: expert weight calculation, attribute weight calculation, expert opinion assembly, and supplier ranking. Its computational process has been broken down into nine steps, as detailed in the overall framework in Section 4.2. In large-scale applications, the computational complexity of the proposed framework may need to be carefully considered. To meet this challenge, the use of efficient algorithms and computational techniques can optimize the decision-making process.

User acceptance: Whether the framework achieves successful implementation in practical situations is influenced by user acceptance. Engaging stakeholders in the decision-making process, including architects, engineers, project managers, and procurement professionals, and providing user training and support resources can also promote understanding and acceptance of framework methods and results.

6. Conclusions

6.1. Managerial and Practical Implications

This paper introduces a novel MAGDM method for green building material supplier selection, offering significant managerial and practical advantages. The innovative aspects of this research, particularly the integration of IFNs and the consideration of attribute interactions, provide a robust framework for decision makers to enhance the efficacy of green supplier selection processes.

Managerially, the application of IFNs in the proposed group decision-making method allows for a more granular and flexible assessment of supplier performance. This approach acknowledges the subjectivity and uncertainty inherent in expert evaluations, leading to more nuanced and reliable decision making. For strategic supplier management, this paper provides managers with a tool to identify suppliers that can contribute to the triple bottom line of profit, environment, and corporate performance by integrating a comprehensive attribute system that includes traditional and sustainable development-related standards, ensuring that the chosen suppliers not only meet operational requirements but also align with the company’s sustainability goals.

Practically, the methodology presented in this study adeptly consolidates expert opinions and accounts for attribute interactions, streamlining the supplier selection process and reducing the time and resources required for evaluation and negotiation. Consequently, this approach yields assessment outcomes that are more aligned with practical realities. Additionally, the green and sustainable ability attributes developed within this research serve as benchmarks for suppliers to evaluate their environmental performance. This can incentivize suppliers to enhance their eco-friendly practices, thereby fostering a competitive market that prioritizes sustainability. Furthermore, the insights from this paper can inform the development of policies and guidelines for sustainable procurement in the construction industry. By highlighting the significance of green supplier selection, this research has the potential to influence industry standards and regulatory frameworks.

6.2. Main Contributions and Innovations

Green buildings play a crucial role in addressing environmental pollution and climate change. The selection of green building material suppliers serves as a fundamental guarantee for achieving the desired outcomes of green buildings. A well-chosen supplier not only directly impacts the green performance of buildings but also contributes to the sustainability of the supply chain. This study expands the traditional supplier selection issue to the selection of green building material suppliers. It proposes a group decision-making framework to evaluate their performance and select the best supplier, thus promoting the sustainable development of the green building industry. The main contributions and innovations of this study can be summarized as follows:

(1)

This study constructs a group decision-making research framework for selecting green building material suppliers, which includes a more comprehensive evaluation attribute system and more suitable decision methods.

(2)

Distinguishing it from general studies on the evaluation of building material suppliers, this study incorporates green and sustainable capabilities into the evaluation attribute system of green building material suppliers, aiming to improve evaluation performance and meet evaluation requirements more effectively.

(3)

In terms of evaluation methods, this study extends the ExpTODIM method to a group decision-making environment, proposing the Group-IF-ExpTODIM method and utilizing Choquet integrals and the Shapley value to calculate attribute weights considering interactions between attributes. The proposed method not only ensures the scientificity and accuracy of the evaluation results but also promotes the development of the field of MAGDM.

(4)

The proposed framework is validated through a case study of the selection of timber suppliers for a building project in Wuhan, China. Subsystem analysis, comparative analysis, and robustness analysis verify the effectiveness and stability of the attribute system and the method utilized.

6.3. Limitations and Further Research

While the present study offers a robust framework for evaluating green building material suppliers, it is not without limitations. The model’s static nature may not be sufficiently flexible to accommodate the industry’s rapidly changing landscape, including shifting market dynamics and emerging environmental regulations. Furthermore, the study does not fully utilize recent technological advancements, such as artificial intelligence and blockchain for supply chain transparency. It would be beneficial for future research to address these gaps by incorporating dynamic, technology-driven decision-making frameworks that can adapt to real-time changes, enhancing the model’s applicability and effectiveness in the sustainable construction industry.