1 Introduction

Since the financial crisis of 2008, Western countries have wondered whether the past overemphasis on financial and light industry development has led to an imbalance in industrial diversification. After this event, these countries realized the importance of the role of manufacturing in the economic development of a nation [1]. Undoubtedly, the manufacturing sector is the main engine driving economic growth. For example, many countries in East Asia have increased their export rates as a result of industrialization, which in turn has increased average national incomes and government tax revenues [2]. However, industrialization has also exacerbated the damage to human life and the natural environment. There is an urgent need for enterprises and organizations to develop their economies in a way that is both environmentally friendly and socially responsible [3].

Global climate change, water and air pollution in Europe, child labor problems in Asia, labor rights issues in North America, political instability in South America, and human rights issues in Africa are some of the challenges that business executives have to face today. Moldavska and Welo [4] argued that manufacturing is an important participant in achieving sustainable development, and its role in boosting the economy, creating jobs, reducing environmental pollution, and improving social equity has received increasing attention in the field of sustainable development. Pope et al. [5] defined that the concept of sustainability as “the balance of resources, technology creation, and institutional change that an enterprise or organization makes to meet human needs, while maintaining environmental balance and social harmony, and delivering benefits that meet public expectations.” Since then, many studies have discussed sustainability across different industries based on the three dimensions of sustainability: economic, social, and environmental [6, 7].

To date, despite the emphasis on economic, environmental, and social sustainability in the literature, there has been little consideration of the institutional and risk control perspectives of sustainability. Chang et al. [8] suggest that the implementation of subsidized financial lending by central or local governments can help companies to transform themselves towards sustainability. Their findings echo the institutional sustainability advocated by Agrawal [9]. In addition, risk control is often overlooked in sustainability evaluation frameworks and is not even included as a criterion in the evaluation system [10]. For instance, although many enterprises already had the foundational conditions for sustainability in place, they still experienced operational crises in the face of the global COVID-19 outbreak in 2020. Furthermore, unforeseen natural and man-made disasters (such as droughts, floods, power outages, and terrorist attacks) can hurt business sustainability. To redress this imbalance, this paper proposes a more comprehensive framework for evaluating enterprise sustainability and investigates the mutually influential relationships of these sustainability criteria in the manufacturing context. Specifically, the paper aims to achieve the following objectives:

  1. (i)

    Identify the five dimensions of enterprise sustainability (economic, social, environmental, institutional, and risk control) in the proposes framework for sustainability evaluation in the manufacturing context;

  2. (ii)

    Explore the mutually influential relationships and influences among sustainability criteria based on the methodology of expert-assisted decision support;

  3. (iii)

    Present a management discussion and practical implications to support enterprises in developing more appropriate improvement strategies.

To achieve these goals, an extensive literature review was first conducted within the discipline of corporate sustainable development to identify potential sustainability criteria, alon with several rounds of reviews by managers and experts from multiple manufacturers prior to proposing a comprehensive framework for the evaluation of sustainability in manufacturing. The selected sustainability criteria are conflicting and constrain one another, which is considered to be characteristic of a classic Multiple-Criteria Decision-Making (MCDM) problem. The literature review shows that most researchers have assumed these criteria to be independent; however, in fact, the criteria have direct or indirect influences on one another. Common MCDM approaches to explore criterion dependence include the Criteria Importance Through Inter-criteria Correlation (CRITIC) [11], Interpretive Structural Modeling (ISM) [12], and Decision-making Trial and Evaluation Laboratory (DEMATEL) [13] methods. CRITIC is an objective weighting method that uses the correlation coefficient between criteria as a calculation parameter. When the sample size is too small to obey the normal distribution, it is difficult to accurately reflect the correlation between the variables. In contrast, ISM uses the variables 0 and 1 to identify whether the criteria have an influential relationship in evaluating influences. Although ISM is simple to perform, it can only tell whether there is an influential relationship instead of describing the magnitude of the influence. DEMATEL uses a five-scale semantic variable to optimize the ISM and can construct an influential network relation map (INRM) of the criteria [14]. In addition, there have been many studies that demonstrate the effectiveness of DEMATEL in exploring criterion dependency issues [15,16,17,18]. It is an important task to effectively identify the mutual influential relationships among the criteria in the uncertain environment and under the fuzzy information, and it is also vital work to collect the judgments of many experts.

This study proposes a novel group decision-making approach to explore the above issues, namely the Rough-Fermatean Fuzzy DEMATEL (RFF-DEMATEL) technique. The approach combines Fermatean fuzzy sets (FFs) [19] and rough set theory (RST) [20] with DEMATEL, with the FFs used to reflect the uncertainty of the experts’ evaluation and RST to aggregate the judgments of the multiple experts. FFs is a relatively novel variant of intuitionistic fuzzy theory, which was proposed by Senapati and Yager [19]. The FFs enhance the ability of Pythagorean fuzzy sets (PFs) to measure uncertainty, and to cover a wider range of information uncertainty, avoiding the loss of potential information in semantic transformations [21]. Many studies have applied the FFs-based approach to discuss MCDM issues in the real world, illustrating the effectiveness and reliability of FFs in reflecting expert’s uncertainty. [19, 21]. On the other hand, most previous studies have used the method of arithmetic averaging to integrate the evaluation data from multiple experts; however, this approach is also prone to potential information loss. RST is not only a reliable method of aggregating multiple data but can also generate an interval value to reflect the upper and lower bounds of the data [22]. The RFF-DEMATEL does not require a complex calculation as DANP (DEMATEL-based ANP) does, but only needs to determine the total influence of the criteria to generate the influence weights. The results show that the proposed approach significantly improves the shortcomings of the conventional DEMATEL and Fermatean Fuzzy DEMATEL (FF-DEMATEL), and thus it more accurately translates expert judgments into computable quantitative data. A survey of the electronics manufacturers in Taiwan is used as a demonstration case for this study. The analysis results have important implications for the concept of enterprise sustainability and will be helpful in developing effective corporate policies and improvement strategies to achieve sustainable development in the electronics manufacturing industry. The specific contributions and features of this study can be summarized as follows:

  1. (i)

    The proposed evaluation framework and methodology are novel and can generate new ideas for research in the field of sustainable development.

  2. (ii)

    RFF-DEMATEL not only reflects the uncertainty of the experts’ evaluation, but also encompasses a wider range of ambiguities. In addition, RST allows effective integration of the judgments of multiple experts and determines the level of group uncertainty.

  3. (iii)

    The electronics manufacturing industry in Taiwan is used as a demonstration case in this study, but the process can be followed to allow other industries to analyze sustainability performance.

  4. (iv)

    Decision-makers/researchers can use the influence of INRM and the criteria to develop appropriate strategies to support enterprises in moving toward sustainability.

The rest of this paper is structures as follows. In Sect. 2, a review of the literature on enterprise sustainability is presented, and the developed evaluation framework is introduced. In Sect. 3, the theoretical basis and implementation steps of the proposed RFF-DEMATEL approach are described in detail. In Sect. 4, the feasibility and effectiveness of the proposed approach are demonstrated using a real-life case, and the results of the analysis are discussed in Sect. 5. Finally, in Sect. 6 some conclusions and recommendations for future research are given.

2 Framework for Evaluation of Enterprise Sustainable Development

The study of enterprise sustainability has expanded from conceptual qualitative discussions to rigorous expert systems analysis. Many researchers have applied the MCDM model for this type of analysis [23]. In this section, the literature on enterprise sustainability is first reviewed and the gaps filled in by this study are illustrated. Then, the proposed evaluation framework and criteria are presented.

Enterprise sustainability has expanded from its traditional focus on just the financial dimension to three dimensions to incorporate environmental and social aspects into the evaluation of corporate performance [24]. Economic sustainability refers to the need to be economically profitable. This has two implications: first, only economically profitable development projects can be promoted and their sustainability maintained; second, economically loss-making projects must be subsidized by other profitable projects for the organization to break even and maintain normal operations [25]. However, social and environmental performance, including the obligations of organizations to society and the natural environment, have long been overlooked or devalued. Social sustainability refers to the responsibility of the business to the workforce, the surrounding community, and society as a whole throughout its operations. Sustainability does not require a return to a primitive society, even though the human species at that time would have done minimal damage to the environment [3]. Environmental performance considers the environmental impacts caused by the organizations, including environmental resource consumption, pollution emissions, carbon footprint, solid and water waste, recycling and reuse emissions. [26]. Although these definitions and principles are widely accepted, different companies and organizations often have different ideologies, perspectives, and value judgments about sustainability, which means they will interpret the concept of sustainability differently. Currently, most of the existing enterprise sustainability studies use these three dimensions as the basis for performance evaluation.

To make enterprise sustainability evaluations more comprehensive and rigorous, Chang et al. [8] proposed a novel model for assessing sustainable suppliers that adds institutional sustainability to illustrate how government policy support can significantly influence enterprise sustainability. In addition, sustainability-related issues are increasingly being enriched by the inclusion of institutional sustainability in industry sustainability evaluation questions [27, 28]. In order to ensure that a company’s operations are robust and are not affected by the external environment or internal operations, companies should pay attention to operational risk management. Recently, the outbreak of the COVID-19 pandemic has led many to become interested in the impact of the pandemic on business sustainability [29, 30]. The criteria for corporate preparedness measures are often subsumed under social sustainability, and even many risk control criteria are included in the social perspective, which can create an imbalance in the sustainability evaluation system. Furthermore, the assumption of criterion independence already violates the realities of natural and social science. Unfortunately, there have been only a few studies using the dependency-based MCDM approach to discuss the mutual influence relationships among the various criteria. To overcome these problems, this study develops a novel sustainability framework that incorporates economic, social, environmental, institutional, and risk control dimensions, as shown in Fig. 1. In addition, the proposed RFF-DEMATEL explores the mutually influential relationships among the criteria in the evaluation.

Fig. 1
figure 1

The sustainability framework developed in this study

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First, 14 experts from industry, government, and academia were invited to study the sustainability of the electronics manufacturing industry in Taiwan. They were invited to form an expert decision-making team for consultation for this study. Following an extensive literature review and five meetings with this team, we established 15 evaluation criteria organized under the five aspects, as shown in Table 1.

Table 1 Description of the framework and criteria for evaluating the performance of sustainable development for the electronics manufacturing industry
Full size table

3 Proposed RFF-DEMATEL Approach

In the 1970s, the science of expert-assisted decision-making was in its infancy, and more often than not the inputs for decision-making methods were obtained through expert surveys. DEMATEL is a technique for identifying mutually influential relationships among various criteria. DEMATEL requires experts to perform pairwise comparisons of criteria to determine the indirect and direct influence between them. The total and net influence of these criteria are then used as parameters to build the influence relationship of the system [15]. This technique has been widely used in various decision-making problems, such as the discussion of medical trends [15], blockchain-based life cycle evaluation [16], customer relationship management evaluation [17], and supply chain risk resilience evaluation [18].

Due to the ambiguity of the evaluation environment and information sources, scholars have combined various fuzzy theories in DEMATEL to reflect the uncertainty of the experts in their judgment. Ataei and Norouzi Masir [56] applied the triangular fuzzy numbers (TFN) in the DEMATEL to assess sustainability of bauxite mining. Similar to Ataei and Norouzi Masir [56]’s fuzzy DEMATEL, Abdullah and Rahim [57] used the method to measure the urban sustainable development performance. Giri et al. [58] proposed Pythagorean DEMATEL to select the most appropriate supplier in sustainable supply chain management. Gonzales et al. [59] developed Fermatean DEMATEL and maximum mean de-entropy algorithm to analyze the barriers of implementing education 4.0. In addition, there are several hybrid models. For example, Braga et al. [60] integrated cognitive mapping technique and DEMATEL to develop a MCDM model that assist decision-makers in exploring smart city determinants in a collaborative manner. Singh and Sarkar [61] combined fuzzy Delphi and DEMATEL to measure performance of sustainable product development.

In this study, FFs are integrated into DEMATEL to reflect the uncertainty of the experts’ evaluations, and RST is introduced to aggregate the judgments of the multiple experts, and to obtain more reliable criteria-influence results. FFs is an extended sets of q-rung orthopair fuzzy sets, which is FFs when q = 3. FFs enhance the ability to measure uncertainty compared to intuitionistic fuzzy and Pythagorean fuzzy. The scope of information uncertainty covered by FFs is broader, which is instrumental in avoiding potential information missing in qualitative conversion to quantification. FFs are mainly composed of membership (μ) and non-membership (v) parameters. In addition, several studies have indicated that FFs are effective and reliable in dealing with expert’s uncertainty [62,63,64,65]. For detailed definitions and operations of FFs, please refer to Senapati and Yager [19]. Furthermore, in practice, MCDM issues are usually not discussed by only one expert/decision-maker. Deciding an appropriate technique to integrate/aggregate individual judgments and observations from multiple experts/decision-makers is a critical task for group decision-making. Based on concept of rough set theory [20], Zhai et al. [66] proposed a rough number technique to construct lower and upper approximations of decision groups. This technique has been extensively incorporated into many MCDM methodologies (e.g. rough-fuzzy DANP [67], fuzzy-rough AHP-VIKOR [68], rough-fuzzy TOPSIS [69], etc.). And it has been shown that the technique can effectively address the shortcomings of the arithmetic mean method.

The detailed RFF-DEMATEL concept and calculation procedure are described below.

Step 1 Establish an expert decision-making team.

Experts with professional expertise and experience in the topic of interest are invited to form a decision-making team. Expert k can be written as Ek, where k = 1, 2,…, p.

Step 2 Confirm the criteria for the evaluation system.

The criteria used in the evaluation system are established after discussion by the expert team. Criterion j can be written as Cj, where j = 1, 2,…, n

Step 3 Build the FFs direct relationship matrix.

There are n criteria for which we need to evaluate their influence on one another in pairwise comparisons. Expert k determines the direct influence of criterion i on criterion j based on the linguistic term for the Fermatean fuzzy set (Table 2) to form the FFs direct relationship matrix for expert k, as shown in Eq. 1.

$$ X^{\left( k \right)} = \left[ {\begin{array}{*{20}c} {\left( {\mu_{\text{F}} \left( {x_{11}^{\left( k \right)}} \right),v_{\text{F}} \left( {x_{11}^{\left( k \right)}} \right)} \right)} & {\left( {\mu_{\text{F}} \left( {x_{12}^{\left( k \right)}} \right),v_{\text{F}} \left( {x_{12}^{\left( k \right)}} \right)} \right)} & \cdots & {\left( {\mu_{\text{F}} \left( {x_{1n}^{\left( k \right)}} \right),v_{\text{F}} \left( {x_{1n}^{\left( k \right)}} \right)} \right)} \\ {\left( {\mu_{\text{F}} \left( {x_{21}^{\left( k \right)}} \right),v_{\text{F}} \left( {x_{21}^{\left( k \right)}} \right)} \right)} & {\left( {\mu_{\text{F}} \left( {x_{22}^{\left( k \right)}} \right),v_{\text{F}} \left( {x_{22}^{\left( k \right)}} \right)} \right)} & \cdots & {\left( {\mu_{\text{F}} \left( {x_{2n}^{\left( k \right)}} \right),v_{\text{F}} \left( {x_{2n}^{\left( k \right)}} \right)} \right)} \\ \vdots & \vdots & \ddots & \vdots \\ {\left( {\mu_{\text{F}} \left( {x_{n1}^{\left( k \right)}} \right),v_{\text{F}} \left( {x_{n1}^{\left( k \right)}} \right)} \right)} & {\left( {\mu_{\text{F}} \left( {x_{n2}^{\left( k \right)}} \right),v_{\text{F}} \left( {x_{n2}^{\left( k \right)}} \right)} \right)} & \cdots & {\left( {\mu_{\text{F}} \left( {x_{nn}^{\left( k \right)}} \right),v_{\text{F}} \left( {x_{nn}^{\left( k \right)}} \right)} \right)} \\ \end{array}} \right],\;i = j = \, 1, \, 2,...,n, $$
(1)
Table 2 Linguistic terms for the FFs [21]
Full size table

where \(\mu_{\text{F}} \left( {x_{ij}^{\left( k \right)}} \right)\) and \(v_{\text{F}} \left( {x_{ij}^{\left( k \right)}} \right)\) are the membership (μ) and non-membership (v) evaluations for the x event, in which \(0 \le \mu_{\text{F}} \left( {x_{ij}^{\left( k \right)}} \right) \le 1\) and \(0 \le v_{\text{F}} \left( {x_{ij}^{\left( k \right)}} \right) \le 1\). The conventional DEMATEL requires that the diagonal element in the matrix must be 0, i.e., \(\left( {\mu_{\text{F}} \left( {x_{ii}^{\left( k \right)}} \right),v_{\text{F}} \left( {x_{ii}^{\left( k \right)}} \right)} \right)\) = 0. On the other hand, it is certain that the sum of the cube of the membership and non-membership value falls between 0 and 1, i.e., \(0 \le \left( {\mu_{\text{F}} \left( {x_{ij}^{\left( k \right)}} \right)} \right)^{3} + \left( {v_{\text{F}} \left( {x_{ij}^{\left( k \right)}} \right)} \right)^{3} \le 1\). According to the definition of the FFs [19], the degree of uncertainty of expert k can be calculated by

$$ \pi_{\text{F}} \left( {x_{ij}^{\left( k \right)}} \right) = \sqrt[3]{{1 - \left( {\mu_{\text{F}} \left( {x_{ij}^{\left( k \right)}} \right)} \right)^{3} - \left( {v_{\text{F}} \left( {x_{ij}^{\left( k \right)}} \right)} \right)^{3}}},\;0 \le \pi_{\text{F}} \left( {x_{ij}^{\left( k \right)}} \right) \le 1 $$
(2)

Step 4 Use the rough set theory to aggregate the judgments of multiple experts to construct the rough FFs direct relationship matrix.

The rough FFs direct relationship matrix is obtained by calculating the lower and upper approximations of the elements in the matrix according to the RST, as in Eq. 3. For more details about the rough number calculation procedure the interested reader can consult Lo et al. [22]’s. Previously, the integration of expert judgments has been obtained by averaging them, but this results in the loss of potential information. The use of RST as the basis for group decision-making has been widely promoted [70].

$$ \Theta X = \left[ {\begin{array}{*{20}c} {\Theta \left( {\mu _{{\text{F}}} \left( {x_{{11}} } \right),v_{{\text{F}}} \left( {x_{{11}} } \right)} \right)} & {\Theta \left( {\mu _{{\text{F}}} \left( {x_{{12}} } \right),v_{{\text{F}}} \left( {x_{{12}} } \right)} \right)} & \cdots & {\Theta \left( {\mu _{{\text{F}}} \left( {x_{{1n}} } \right),v_{{\text{F}}} \left( {x_{{1n}} } \right)} \right)} \\ {\Theta \left( {\mu _{{\text{F}}} \left( {x_{{21}} } \right),v_{{\text{F}}} \left( {x_{{21}} } \right)} \right)} & {\Theta \left( {\mu _{{\text{F}}} \left( {x_{{22}} } \right),v_{{\text{F}}} \left( {x_{{22}} } \right)} \right)} & \cdots & {\Theta \left( {\mu _{{\text{F}}} \left( {x_{{2n}} } \right),v_{{\text{F}}} \left( {x_{{2n}} } \right)} \right)} \\ \vdots & \vdots & \ddots & \vdots \\ {\Theta \left( {\mu _{{\text{F}}} \left( {x_{{n1}} } \right),v_{{\text{F}}} \left( {x_{{n1}} } \right)} \right)} & {\Theta \left( {\mu _{{\text{F}}} \left( {x_{{n2}} } \right),v_{{\text{F}}} \left( {x_{{n2}} } \right)} \right)} & \cdots & {\Theta \left( {\mu _{{\text{F}}} \left( {x_{{nn}} } \right),v_{{\text{F}}} \left( {x_{{nn}} } \right)} \right)} \\ \end{array} } \right],\quad i = j = 1,2,...,n, $$
(3)

where \(\Theta \left( {\mu_{\text{F}} \left( {x_{ij}} \right),v_{\text{F}} \left( {x_{ij}} \right)} \right) = \left\{ {\left[ {\Theta \underline {\mu }_{\text{F}} \left( {x_{ij}} \right),\Theta \overline{\mu }_{\text{F}} \left( {x_{ij}} \right)} \right],\left[ {\Theta \underline {v}_{\text{F}} \left( {x_{ij}} \right),\Theta \overline{v}_{\text{F}} \left( {x_{ij}} \right)} \right]} \right\}\). Here, “\(\Theta\)” is the code for the rough number, and \(\Theta \underline {\mu }_{\text{F}} \left( {x_{ij}} \right)\) and \(\Theta \overline{\mu }_{\text{F}} \left( {x_{ij}} \right)\) are the lower and upper approximations of FFs membership, respectively. Similarly, \(\Theta \underline {v}_{\text{F}} \left( {x_{ij}} \right)\) and \(\Theta \overline{v}_{\text{F}} \left( {x_{ij}} \right)\) are the lower and upper approximations of FFs non-membership, respectively.

Step 5 Calculate the rough FFs score function and the degree of group uncertainty.

The rough FFs score function is calculated according to the basic rules [19], as shown in Eq. 4.

$$ {\text{score }}f\left( {\Theta \left( {\mu_{\text{F}} \left( {x_{ij}} \right),v_{\text{F}} \left( {x_{ij}} \right)} \right)} \right) = \frac{{\left( {\Theta \underline {\mu }_{\text{F}} \left( {x_{ij}} \right)} \right)^{3} + \left( {\Theta \overline{\mu }_{\text{F}} \left( {x_{ij}} \right)} \right)^{3} - \left( {\Theta \underline {v}_{\text{F}} \left( {x_{ij}} \right)} \right)^{3} - \left( {\Theta \overline{v}_{\text{F}} \left( {x_{ij}} \right)} \right)^{3}}}{2},\quad - 1 \le {\text{score }}f\left( {\Theta \left( {\mu_{\text{F}} \left( {x_{ij}} \right),v_{\text{F}} \left( {x_{ij}} \right)} \right)} \right) \le 1 $$
(4)

However, the score function has a value between 0 and 1. Equation 5 is used to make the range converge to between 0 and 1.

$$ \varphi_{ij} = \frac{{1{ + } {\text{score}} \, f\left( {\Theta \left( {\mu_{\text{F}} \left( {x_{ij}} \right),v_{\text{F}} \left( {x_{ij}} \right)} \right)} \right)}}{2},\quad 0 \le \varphi_{ij} \le 1 $$
(5)

Through Eqs. 4 and 5, the rough FFs direct relationship matrix can be converted into the group relationship matrix, as shown in Eq. 6.

$$ \varphi = \left[ {\begin{array}{*{20}c} {\varphi _{{11}} } & {\varphi _{{12}} } & \cdots & {\varphi _{{1n}} } \\ {\varphi _{{21}} } & {\varphi _{{22}} } & \cdots & {\varphi _{{2n}} } \\ \vdots & \vdots & \ddots & \vdots \\ {\varphi _{{n1}} } & {\varphi _{{n2}} } & \cdots & {\varphi _{{nn}} } \\ \end{array} } \right],\quad i = j = 1,{\text{ }}2,...,n. $$
(6)

In addition, Eq. 7 can be used to determine the uncertainty of the group.

$$ \partial \pi_{\text{F}} \left( {x_{ij}} \right) = \sqrt[3]{{1 - \Theta \overline{\mu }_{\text{F}} \left( {x_{ij}} \right)^{{3}} - \Theta \underline {v}_{\text{F}} \left( {x_{ij}} \right)^{{3}}}},\quad {0} \le \partial \pi_{\text{F}} \left( {x_{ij}} \right) \le {1} $$
(7)

Step 6 Obtain the normalized group relationship matrix

Next, the influence relationships among the criteria can be identified simply by following the regular DEMATEL operation. In this step, a normalization procedure is performed to obtain the normalized group relationship matrix, as shown in Eq. 8.

$$ \gamma = \left[ {\begin{array}{*{20}c} {\varepsilon \cdot \varphi _{{11}} } & {\varepsilon \cdot \varphi _{{12}} } & \cdots & {\varepsilon \cdot \varphi _{{1n}} } \\ {\varepsilon \cdot \varphi _{{21}} } & {\varepsilon \cdot \varphi _{{22}} } & \cdots & {\varepsilon \cdot \varphi _{{2n}} } \\ \vdots & \vdots & \ddots & \vdots \\ {\varepsilon \cdot \varphi _{{n1}} } & {\varepsilon \cdot \varphi _{{n2}} } & \cdots & {\varepsilon \cdot \varphi _{{nn}} } \\ \end{array} } \right],\quad i = j = 1,{\text{ }}2,...,n. $$
(8)

where \(\varepsilon = \frac{1}{{\max \left\{ {\sum\limits_{j = 1}^{n} {\varphi_{ij} ,\sum\limits_{i = 1}^{n} {\varphi_{ij}} }} \right\}}}\).

Step 7 Generate the total group influence matrix.

The mutual influential relationship among the criteria may be direct or indirect. To ensure that all mutual potentially influential relationships are taken into account, the matrix is sub-multiplied and summed up as in Eq. 9. The total group influence matrix is presented as in Eq. 10.

$$ T = \gamma + \gamma ^{2} + \gamma ^{3} + \cdots \gamma ^{\infty } = \gamma \left( {I + \gamma + \gamma ^{2} + \gamma ^{3} + \cdots + \gamma ^{{\infty - 1}} } \right) = \gamma \left( {I - \gamma ^{\infty } } \right)\left( {I - \gamma } \right)^{{ - 1}} = \gamma \left( {I - \gamma } \right)^{{ - 1}} $$
(9)

where \(\gamma^{\infty } = [0]\)\(_{n \times n}\), and \({I}\) is the identity matrix.

$$ T = \left[ {\begin{array}{*{20}c} {t_{{11}} } & {t_{{12}} } & \cdots & {t_{{1n}} } \\ {t_{{21}} } & {t_{{22}} } & \cdots & {t_{{2n}} } \\ \vdots & \vdots & \ddots & \vdots \\ {t_{{n1}} } & {t_{{n2}} } & \cdots & {t_{{nn}} } \\ \end{array} } \right],\quad i = j = 1,{\text{ }}2,...,n. $$
(10)

Step 8 Obtain the influence weights of the criteria and construct the INRM.

The element tij of the total group influence matrix can be interpreted as the sum of the direct and indirect influence of criterion i on criterion j. Therefore, the influence of criterion i (ri) can be obtained through Eq. 11. Similarly, the degree of influence (si) of criterion i can be obtained as shown in Eq. 12.

$$ r_{i} = t_{11} + t_{12} + \cdots t_{1n} = \sum\limits_{j = 1}^{n} {t_{ij}} , $$
(11)
$$ s_{i} = t_{11} + t_{21} + \cdots t_{n1} = \left[ {\sum\limits_{i = 1}^{n} {t_{ij}} } \right]^{{{\text{Transpose}}}} , $$
(12)

According to Eqs. 11 and 12, the total influence of the criterion i, which includes ri and si, is defined as ri + si. The influence weight of criterion i in the evaluation system is obtained through

$$ w_{i} = \frac{{r_{i} + s_{i}}}{{\sum\nolimits_{i = 1}^{n} {\left( {r_{i} + s_{i}} \right)}}}. $$
(13)

On the other hand, ri—si reflects the net influence of criterion i. When ri − si is greater than 0, it means that criterion i significantly influences the other criteria. Conversely, when ri − si is less than 0, it means that criterion i is significantly influenced by the other criteria. By taking ri + si and ri − si as the horizontal and vertical axes of INRM, the relative position of each criterion can be delineated. In this paper, the direction of the arrows is used to describe the mutual influential relationships among the criteria.

4 Case Study: Electronics Manufacturing in Taiwan

In Taiwan, the electronics manufacturing sector accounts for the highest percentage of all industrial manufacturing. The government has implemented several policies to encourage enterprises to engage in various innovative activities to enhance manufacturing technology capabilities [47]. In fact, Taiwan’s electronics manufacturing sector invests more money in R&D than other industries. In the past two decades, Taiwan has become one of the world’s largest suppliers of computers, circuit boards, power supplies, monitors, hardware accessories, and consumer electronics [71]. As a consequence of the policy of promoting the standardization of computer assembly and peripheral components, and encouraging multinational semiconductor enterprises to set up factories in Taiwan, enterprises have started to develop their own technologies and gradually build up a complete electronic manufacturing eco-chain from upstream to downstream.

However, the boom in electronics manufacturing has also had many negative impacts on the environment and society. A comprehensive sustainability evaluation framework is needed to guide enterprises toward improved sustainability and the achievement of each of these sustainability criteria. As described in Sect. 2, a team consisting of 14 experienced experts worked together to develop the evaluation framework and provided professional judgment and advice to finalize the computational inputs required for the RFF-DEMATEL. Each of the 14 experts was asked to individually complete the RFF-DEMATEL questionnaire and to carefully review its content the day after completing it, to ensure that the evaluation information was consistent with their own ideas. For example, let us look at the information obtained from Expert 1 after completing the questionnaire. According to the linguistic term (Table 2), starting with criterion E1, the expert was asked to carefully consider its influence on the other criteria in pairwise comparison, expressed in linguistic terms (see Table 2), until all criteria had been compared. For example, Expert 1 considered the influence of E1 on E2 to be “High Influence,” and therefore filled in the corresponding cell with the code “H”. The remaining items were also judged in this way to obtain Table 5 in Appendix.

Next, the RFF-DEMATEL calculation procedure and the corresponding formulas are explained step by step below. Using the terms in Table 2, the qualitative semantic judgment of the expert is converted into quantitative values to obtain the FFs direct relationship matrix (Eq. 1) for Expert 1, as shown in Table 6 in Appendix. The elements of the FFs direct relationship matrix and Eq. 2 can be used to calculate the uncertainty of the expert’s evaluation. For example, the uncertainty of Expert 1 is 0.36 on average, as shown in Table 7 in Appendix. The rough set concept is used to integrate the FFs direct relationship matrices of the 14 experts, by compiling the 14 datasets, and the rough FFs direct relationship matrix (Eq. 3), as shown in Table 8 in Appendix. The elements in this matrix represent four pieces of information, i.e., the lower bound of affiliation, the upper bound of affiliation, the lower bound of non-affiliation, and the upper bound of non-affiliation. The influence affiliation of E1 with respect to E2 is [0.522, 0.659], and the non-affiliation is [0.838, 0.902]. The FFs-based score function formula (Eq. 4) converts the elements of the rough FFs direct relationship matrix into crisp values. The score functions of its matrix elements are shown in Table 9 in Appendix. However, the value of the score function is bound between -1 and 1, which is not convenient for the subsequent evolution of the DEMATEL program. Therefore, the group relationship matrix (Table 10 in Appendix) can be formed using Eq. 5 to convert the score function to a value between 0 and 1. Similarly, the elements of the rough FFs direct relationship matrix can be determined by applying Eq. 7 to determine the degree of uncertainty of the group, as illustrated in Table 11 in Appendix.

The normalized group relationship matrix and the total group influence matrix are obtained using Eqs. 8–10 and are shown in Tables 12 and 13 in Appendix. Table 3 presents the results of the RRF-DEMATEL analysis. Undoubtedly, criterion R3 has the highest weight (influence) value of 0.112. The total influence (rR3 + sR3 = 4.016) and net influence (rR3—sR3 = 0.314) indicate that R3 influences the other criteria more than it is influenced by the other criteria. In addition, criteria G1, E1, S2, and G3 are the second to fifth highest ranked criteria, respectively. Further discussion and derivation of the management implications are presented in Sect. 5.

Table 3 The weights and ranking of the criteria (Eqs. 11–13)
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5 Discussion and Management Implications

The study of enterprise sustainability has become a popular MCDM issue. However, few have yet to explore the mutually influential relationships among the sustainability criteria, especially in manufacturing industries. This paper uses a conceptual approach based on an expert support system to address sustainability issues. Expert support systems are suitable for analyzing difficult-to-measure metrics and inaccurate or incomplete information. In general, expert systems perform knowledge-intensive solving tasks that can significantly reduce the survey expenses and training costs of traditional statistical algorithms. Therefore, 14 experts with recognized experience and extensive knowledge about the subject were invited to discuss how manufacturing industries can implement sustainable development.

The RFF-DEMATEL method elaborated here overcomes many of the drawbacks of the conventional DEMATEL technique. It not only uses a wider range of uncertainty information, but also effectively integrates the judgment of multiple experts. The INRM generated by RFF-DEMATEL shows the position and influence of the criteria in the evaluation system. In Fig. 2, the arrows shown are those that are more significant. Generally speaking, the criteria in the upper-right area of the INRM affect the criteria in the lower-left area. Several management implications are summarized and discussed in this paper.

  1. (i)

    “Goodwill and corporate image (R3),” “zero emission of pollution and waste discharge (G1),” “corporate governance (E1),” “use of clean energy (G2),” “innovative service (E3),” “social feedback and contributions (S1),” “product carbon footprint (G3),” and “stakeholder’ rights protection (S2)” have the closest mutual influential relationships.

Fig. 2
figure 2

INRM results generated by RFF-DEMATEL

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The eight criteria have close mutually influential relationships, which echoes the findings in many sustainability studies which interlock economic, social, and environmental perspectives into a circle-like relationship. Often, the enterprise profitability can lead to more socially beneficial activities, such as caring for disadvantaged groups, donating materials, and building local infrastructure. In addition, such enterprises could also implement better green designs and manufacturing policies, and even work towards the creation of a green supply chain [5, 8]. It is clear from the INRM that these eight criteria produce a complex set of arrows that reflect the need to focus on multiple indicators for a enterprise to move toward sustainability.

  1. (ii)

    “Goodwill and corporate image (R3),” “zero emission of pollution and waste discharge (G1),” and “corporate governance (E1)” play a crucial role in the sustainability of the manufacturing industry. Their influence weights are 0.112, 0.092, and 0.088, respectively.

    In the past, corporate development was mostly evaluated based on financial indicators. However, in today’s highly environmentally conscious climate, reflection on economic development and capitalism has prompted corporate operators to pay more attention to environmental and societal issues and incorporate related changes into their operations. For example, the G1 indicator is highly valued by experts for minimizing environmental impact at the beginning of investment and subsequently in daily operations [26]. On the other hand, the overall cumulative effect of practicing the various factors related to corporate governance is positively reflected in the R3 indicator. This is the main reason why investors in recent years have placed such a high value on environmental, social and corporate governance (ESG) indicators when judging the stability of companies and selecting investment targets [54, 55]. Moreover, the establishment of E1 has received a lot of attention from management and the board of directors, as having a significant impact on the enterprise’s long-term development plans and strategies, as well as corporate values. It even helps management to develop business development plan principles. Studies have shown that a better corporate governance rating leads to improved corporate social responsibility and is also a factor contributing to success at the corporate level [31, 32].

  2. (iii)

    “Corporate governance (E1)” and the “use of clean energy (G2)” are also mediating factors in the overall evaluation system (other criteria indirectly influence others through them).

    E1 and G2 also play a mediating role in enterprise sustainability. Companies proposing continuous investment in energy conservation or green energy-related environmental sustainability plans, which result in tangible improvements in environmental or social benefits, have good corporate governance E1 performance. In terms of enterprise sustainability, the most immediate benefit of actively promoting G2 development is the reduction of environmental damage while promoting technological innovation, an opportunity to build a more livable environment, a more equitable and better world [42, 43]. Clearly, it is logical that E1 and G2 would play mediating roles in sustainable business development.

  3. (iv)

    “Government’s national development fund (I1),” “state of cluster industrial development (I2),” “intellectual property rights (I3),” and “product quality and reliability (E2)” are the criteria most influenced by others.

Both I1 and I2 are systems promoted by governments and related organizations to encourage the sustainable development and upgrading of industry, to promote enterprise sustainability and their contribution to national economic and social development. However, these systems require a variety of economic and social supports to implement [8]. In addition, both I3 and E2 are significantly influenced by E1 and R3. When corporate governance and corporate image are positive, the performance of I3 and E2 can be effectively enhanced because business operators are willing to spend more resources to protect the enterprise’s intellectual property and product quality to stabilize competitiveness.

To illustrate how the proposed RFF-DEMATEL differs from previous DEMATEL techniques, several model comparisons are performed here. We use the data from case study (Sect. 4) to perform five different DEMATEL operations, including original DEMATEL [18], triangular fuzzy DEMATEL [56], Z-DEMATEL [15], FF-DEMATEL, and RFF-DEMATEL (this study proposed). The criteria weights and rankings obtained by the five DEMATEL techniques are presented in Table 4. No matter in which technique, criterion R3 is the most important and most influential factor. However, the rankings of criteria E1, E3, S1, S2, G1, G2, G3, I2, and I3 all have significant changes. The criteria weights change of the five DEMATEL techniques can be observed through Fig. 3. Undoubtedly, the proposed RFF-DEMATEL can not only characterize the importance of the criteria more clearly, but also can cover the uncertainty evaluation information of the multiple experts. The main advantage of proposed RFF-DEMATEL is that it reflects the broader potential information of individual expert judgments through the Fermatean fuzy theory, and integrates the Fermatean fuzy numbers obtained by the multiple experts through the rough set theory.

Table 4 Criteria weights and rankings of five DEMATEL techniques
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Fig. 3
figure 3

Scatter diagram of criteria weights of five DEMATEL techniques

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6 Conclusions and Further Work

The challenge of sustainability for businesses today has shifted from “whether” to include consideration of the economic, social, and environmental impact of business in regular management decisions to “how” to do so. Companies now need to think about how they can be more sustainable and how they can more effectively engage all of their stakeholders on this issue. Therefore, it is important to be able to identify sustainability criteria and consider their influence to make appropriate sustainability management decisions. This study develops a novel framework for evaluating enterprise sustainability, the RFF-DEMATEL methodology, which can be used to explore the mutually influential relationships among various criteria. This study demonstrates how RFF-DEMATEL can be used in real-world decision-making processes. The Fermatean fuzzy set method is utilized to take into account the uncertainty in the expert evaluation process, to more accurately convert expert judgments into numerical values. In addition, the rough set technique is used to aggregate the judgments of multiple experts. This integration method prevents potential information from being missed. These results can be used by the decision-makers to develop management strategies to improve corporate sustainability performance. The research concepts in this paper can be extended to apply to decision-making issues in other industries, especially where the interdependence of the various factors and their ranking needs to be explored.

Several management implications are proposed which would have an impact on corporate decisions made related to sustainability. The mutually influential relationships between all criteria are reflected in the INRM, which helps to identify the key influences in the sustainability process, thus reducing the occurrence of failures and mistakes. The results of the study confirm that the mutual influential relationships among R3 (goodwill and corporate image), G1 (zero pollution and waste), E1 (corporate governance), G2 (clean energy use), E3 (innovative services), S1 (social feedback and contribution), G3 (product carbon footprint), and S2 (stakeholder benefits) are the strongest of all. The eight criteria are closely related to each other, which implies that they not only have a high degree of mutual influence but also are indispensable. Therefore, the eight criteria serve as the basis for basic sustainable development, requiring cross-departmental collaboration and stakeholder participation, which will facilitate the goal of sustainable business operations. Furthermore, R3 (goodwill and corporate image maintenance), G1 (zero pollution and waste emissions), and E1 (corporate governance) play a crucial role in corporate sustainable development. This study adds risk control as the fifth sustainability dimension, while R3 is one of the criteria in this dimension, and it has the highest influence weight. The global market’s evaluation of corporate brands is no longer limited to the products and services themselves, yet is more concerned with the performance of corporate ESG implementation and risk management. Due to the continuous natural and man-made disasters in recent years, the threats and challenges posed to the enterprises have intensified. It is more difficult for the enterprises to solve the risk dilemma they face depending on the past experiences. The establishment of a more comprehensive risk control and disaster prevention mechanism is an urgent task for sustainable enterprises. On the other hand, I1 (National Development Fund), I2 (Policy Promotion), I3 (Intellectual and Intellectual Property Protection), and E2 (Product Quality and Reliability) are the criteria most influenced by the other criteria.

It is worth mentioning that this paper proposes the risk control perspective on the issue of sustainable supplier assessment for the first time. Traditionally, sustainability consists of three dimensions, economic, social, and environmental. It was not until recent years that the institutional aspect was added. However, the diversity of sustainability criteria can allow enterprises or governments to have a broader basis for assessment. In summary, this paper contributes to the issue of evaluating corporate sustainability performance, both in terms of storytelling and methodology. In addition to manufacturing, industries from all walks of life can use this conception as a research basis to develop a performance evaluation system that fits their business or institution.

However, there are still some limitations to this work which have yet to be overcome. Since the number of criteria that can be used to evaluate corporate sustainability is quite large, the expert decision-making team decided to combine some of the criteria to simplify the complexity of the analysis, so that only 15 criteria were ultimately included in the evaluation system. In the future, a more scientific methodology can be used to select and narrow down the criteria. In addition, the studies interested in corporate sustainability can add potential key indicators to enhance the comprehensiveness and completeness of the evaluation system.