A total distance ranking approach to fuzzy AHP-based MCDM method for selecting sustainable manufacturing facility location

Abstract

Selecting a sustainable facility location is a crucial strategy for manufacturing companies to achieve long-term success in today’s competitive environment. Various quantitative and qualitative criteria with different importance in a multiple level structure must be considered and aggregated to assist the company in decision-making. How to determine these criteria weights and select the sustainable manufacturing facility locations have become research questions. To resolve this problem, this paper proposes a total distance ranking approach to fuzzy analytic hierarchical process (AHP)-based multiple criteria decision-making (MCDM) method. Currently, the membership functions of fuzzy weighted ratings in the MCDM with a multilevel structure cannot be developed. A ranking method is needed to defuzzify those fuzzy numbers used for fuzzy AHP and qualitative criteria in the MCDM for better executing decision-making procedure. The total distance ranking method related to the centroid on x-axis, centroid on y-axis and the distances of centroids point to the two tangent lines of a fuzzy number are considered in the proposed ranking approach. Formulas of the proposed approach will be presented, and some properties will be investigated to derive formulas for trapezoidal and triangular fuzzy numbers. A comparison with relevant ranking methods will be made to show the advantages of the proposed ranking approach. The proposed ranking approach is then applied to defuzzify the fuzzy numbers used in fuzzy AHP and linguistic values under qualitative criteria to obtain the criteria weights under multi-level structure and crisp values under qualitative criteria, respectively. The final scores of alternatives can be obtained by aggregating crisp criteria values and their corresponding weights by simple additive weighting method to obtain the ranking result. A numerical example will be conducted to show the effectiveness of the proposed model. Finally, a comparison with Best-Worst method (BWM) will be presented to show the persuasiveness of the proposed method.

Keywords

Total distance tangent line fuzzy AHP MCDM sustainable manufacturing facility location

1 Introduction

The facility location issue involves the facilities’ geographical sitting for an organizational unit, for example a firm [1]. Facility location decisions are seen as an important part in long-term planning for service and manufacturing organizations, and an incorrect location selection may lead to inadequately skilled manpower, operating expenditures increase, inadequate means of transportation, primary materials shortage, or even adverse consequences on the firms as a result of the intervention of political and social [2]. Therefore, selecting facility location is a crucial component of a manufacturing company’s strategy for achieving long-term success in the free market economy.

The process of selecting a facility location involves making strategic-level decisions, considering the significant time scale and costs associated with constructing these facilities [3]. This decision cannot be solely approached from an economic standpoint, as establishing facilities and their interconnections have substantial environmental and social impacts. Depending on the facility’s location, waste treatment facilities and the management of emissions (solid, liquid, or gas) may exhibit considerable variations [3]. Furthermore, when expanding a manufacturing network into multiple countries, it is essential to consider corporate social responsibility (CSR) aspects, including cultural differences and ethical values. In some countries, underage labor might be viewed as a necessary means for survival, but it is not ethically acceptable in many developed countries [1]. The global firms are broadening the viewpoints of the manufacturing footprint from only economic aspect to multi-aspect, which include environmental and social concerns; thus, sustainability is a critical factor in selecting the manufacturing facilities location [1]. Therefore, a manufacturing company must consider all the sustainability elements in the facility sitting decisions.

Recently, environmental impacts and social issues have been considered while selecting a location toward a sustainable development of companies [4 –6]. Terouhid et al. [7] indicated that sustainable facility location is not only needed for noxious industries but it also should be applied for all kinds of industries. They further indicated that the need to take a comprehensive approach including three sustainable facets (i.e. economic, social and environmental facets) to achieve the shifting paradigm towards sustainable facility location is higher and more justified than ever before. Thus, comprehensively considering the three areas in making decisions related to facility location selection plays a key role for manufacturing firms seeking competitive advantage.

In the evaluation of sustainable manufacturing facility locations, many factors under the three above mentioned areas must be considered. For example, Chen et al. [1] proposed a three-level structure to describe the sustainability factors when selecting the manufacturing facility location. Evidently, evaluation of sustainable manufacturing facility locations is seen as an issue to solve by multiple criteria decision-making. Some of these criteria are quantitative and have numerical values, such as potential demand and labor cost, whereas some are qualitative and usually subjectively evaluated, such as competitiveness and political stability. Quantitative factors can be sorted into benefit and cost factors. The high score of benefit factors and low score of cost factors will be preferred. In addition, different criteria in the multilevel structure may have different importance levels. Aggregation of numerous criteria and their corresponding weights in the multilevel structure forms a critical component of decision-making regarding the selection of a sustainable manufacturing location. Thus, this paper suggests a fuzzy AHP-based MCDM method under an uncertain environment to overcome the above problems.

Among MCDM methods, AHP, introduced by Saaty [8], has found widespread application in evaluating complex multi-criteria alternatives across various domains [9, 10]. It excels in terms of ease of use, systematic problem structuring, and the ability to calculate both criteria weights and alternative priorities efficiently [11]. Fuzzy sets, initially introduced by Zadeh [12] as a prominent approach for managing imprecision, have been integrated with AHP to create the fuzzy AHP (FAHP) method. This combination retains the benefits of AHP while incorporating the ability to handle imprecise data. Consequently, FAHP has gained extensive application in various fields [13]. Fuzzy set theory presented the mathematical method to represent the linguistic terms by fuzzy numbers which is a useful tool for evaluating the qualitative criteria. In the proposed method, ratings of alternative locations versus qualitative factors are obtained by using linguistic values represented by triangular fuzzy numbers, and fuzzy AHP is employed to produce weights to display difference importance of the criteria. In this study, fuzzy AHP is utilized to produce the criteria weights because of the ambiguous nature of human subjective cognition for linguistic evaluation in a pairwise comparison (i.e., fuzzy numbers are employed in the pairwise comparison matrices instead of crip numbers). However, the procedure for calculating the fuzzy numbers in these pairwise comparison matrices is complicated. Furthermore, the procedure for aggregating fuzzy weights and the linguistic values described by fuzzy numbers of locations with respect to qualitative criteria in the MCDM is also complicated. Currently, the exact membership functions of fuzzy weighted ratings in the MCDM with a multilevel structure cannot be developed. Therefore, a ranking method is needed to defuzzify those fuzzy numbers that are used for fuzzy AHP and qualitative criteria in the MCDM for better executing the decision-making procedure. In 2000, Yao and Wu [14] proposed the signed distance based on the α–cut to order the fuzzy number; despite the benefits, this method cannot provide the distinguishable result for ranking some fuzzy numbers because of using the average operator. To solve this limitation, this study proposes a total distance ranking approach to conduct the defuzzification by using the addition of the centroid on x-axis, centroid on y-axis and the distances of centroids point to the tangent lines of a fuzzy number, in which both the centroids and shape of a fuzzy number are considered. Formulas of the proposed ranking approached will be clearly presented and some properties will be investigated to derive formulas for trapezoidal and triangular fuzzy numbers. Moreover, a comparison with the other relevant techniques, including Yao and Wu [14], Wang et al. [15], Deng et al. [16], Asady & Zendehnam [17], will be made to show the worthiness of the proposed approach. The proposed ranking approach is then applied to defuzzify the fuzzy numbers used in fuzzy AHP and linguistic values under qualitative criteria to obtain the criteria weights under multi-level structure and crisp values under qualitative criteria, respectively. The final scores can be produced by aggregating crisp criteria values and their corresponding weights by simple additive weighting method to obtain the ranking result. The contributions of this study are highlighted as follows.

A fuzzy AHP-based MCDM method is proposed to evaluate sustainable manufacturing locations;

A novel ranking approach considering the centroid on x-axis, centroid on y-axis and the distances of centroids point to the tangent lines of a fuzzy number is proposed;

Formulas of the proposed ranking method are presented, some properties are investigated;

A comparison with relevant methods is made to show the merits of the proposed approach;

A numerical example of evaluating sustainable manufacturing locations is conducted to illustrate the practicality of the proposed model;

A comparison with BWM is presented to indicate effectiveness of the proposed method.

The remaining parts of this paper are organized as follows. The following section presents the literature review on sustainable facility location, AHP, and fuzzy number ranking. The third section shows the fundamental notion of fuzzy set theory. Section 4 discusses about the proposed ranking approach, property study and comparison. Section 5 introduces the proposed model. In Section 6, a numerical experiment is shown to present feasibility of the model and a comparison with the BWM is performed in section 6.1 to exhibit the effectiveness of the proposed model. Finally, the conclusions and future works are presented in Section 7.

2 Literature review

Facility location selection is crucial for all types of firms both service and industrial organizations. Thus, location selection for the expansion of an existing company or new company is the key factor affecting the firm’s prosperity or failure. The decision makers have to choose a location which not only enhances the business performance but also is pliable enough to adapt to the required changes in the future [2].

Facility siting decisions have been investigated for a long time. Since the pioneering work of Alfred Weber [18], many researchers from various academic disciplines have investigated the model facility location decisions [19]. Terouhid et al. [7], Li et al. [20], Arabani and Farahani [21], Gülpınar et al. [22], Trivedi and Singh [23], Schöbel [24] conducted a review and comparison of these articles. Facility location selection has been traditionally concentrated on economic aspects; however, with the growing concern for global sustainability, the environmental and social dimensions are being included in the development of location selection models [7].

Studies on industrial sitting facility and sustainability began to be published in 1997 [1], when Eagan and Joeres [25] developed a facility-based environmental performance indicator involving economic sustainability in 1977. Anvari and Turkay [3] integrated three dimensions of sustainability into the facility location decision framework. Chen et al. [1] indicated that in many industrial enterprises, sustainable issue has been arisen as a strategic objective; however, it is rarely built into the decision support framework. Ghasempour et al. [26] presented the MCDM to select the solar plants sites and solar plants technologies. Wang et al. [27] used three types of belief degree-constrained mathematical techniques to address a sustainable facility location-allocation issue under the Hurwicz criterion, optimistic and pessimistic value criterion. Khanjarpanah and Jabbarzadeh [28] introduced a novel method using data envelopment analysis combined with fuzzy-cross-efficiency and cross-efficiency methods to determine the optimal locations for establishing the wind plants. Some additional recent studies related to this topic include those of Tajbakhsh and Shamsi [6], Farahbakhsh and Forghani [29] and Hrabec et al. [30]. Budak et al. [31] introduced the combination of DEMATEL and ANP under interval-valued intuitionistic fuzzy environment to select the most appropriate real-time location systems. To select the dry pot locations in the Western Balkans region,Tadić et al. [32] integrated MCDM models including Delphi, AHP (Analytical Hierarchy Process), and CODAS under grey environment. Ulutaş et al. [33] introduced Grey MCDM model to select the warehouse location.

During the sustainable manufacturing facility locations evaluation process, various criteria must be determined. Chen et al. [1] classified these criteria into the three areas of environmental, social, and economic aspects. They developed a three-level structure, including sub-factors and sub-sub-factors, to depict the relationships among these three factors. Some of these factors are quantitative, such as potential demand and labor cost, whereas some are qualitative, such as competitiveness and political stability. In this study, a four-level structure of criteria is developed based on the three-level structure of Chen et al. [1] and United Nations publication [34]. Therefore, evaluating sustainable manufacturing facility locations is a multicriteria decision-making problem. For effective decision-making, MCDM methods can be employed, which assist companies to pick up the optimal alternative under many criteria and are usually employed for addressing the ranking and evaluating issues which involve in various conflicting criteria [35]. A center of area method was suggested by Chu and Varma [36] to defuzzify triangular fuzzy numbers and applied it to a multiple levels MCDM modelfor evaluate suppliers under fuzzy environment. Numerous MCDM techniques and applications in location selection have been explored. Meta analyses of MCDM methods have been presented by Boutkhoum et al. [4] and Terouhid et al. [7]. MCDM methods can be applied in various areas which can be seen in Tan et al. [37], Hussain et al. [38], Liao et al. [39]. In MCDM problems, the qualitative criteria do not have numerical values and are usually evaluated subjectively.

The performance rating of the sustainable location for the retail companies under different qualitative factors and their level of importance are frequently presented in linguistic values and the MCDM methods integrated with fuzzy methods can be used to address the data uncertainties [40]. Fuzzy set theory [12] is a useful tool to evaluate the qualitative criteria employing linguistic values represented by fuzzy numbers. Several other recently published fuzzy sets have also been developed, such as intuitionistic fuzzy sets, hesitant fuzzy sets, Z-number, and utility numbers. Kahraman et al. [41] reviewed and compared these fuzzy sets. Gupta et al. [42] extended the VIKOR method using trapezoidal intuitionistic fuzzy numbers to address the plant location selection issue. Saravi et al. [43] proposed a novel approach based on the Z-number DEA model to determine the optimum locations for biomass power plants. Ji et al. [44] combined the ELECTRE method and multi hesitant fuzzy linguistic term sets to select a hotel’s location.

This paper proposes a MCDM model under an uncertain environment to assess and choose sustainable manufacturing facility locations, where fuzzy AHP is applied to find out the weights of the criteria. AHP is considered as a flexible and powerful methodology for addressing complicated decision problems and has been extensively studied and applied [4]. Nghiem and Chu [45] proposed an AHP-based ELECTRE I method for sustainable conceptual design evaluation under a two-level criteria framework. An overview of the AHP applications has been presented by Emrouznejad and Marra [10], Russo and Camanho [46]. AHP is an extensively used MCDM method; however, it has been generally criticized due to accept a nine-point scale, which cannot deal with the uncertainty and ambiguity involved in determining the weights of different criteria [47]. It is very hard for the decision maker to evaluate the subjective and qualitative criteria which requires using exact numerical values to express the preferences and providing exact pairwise comparison judgments [47]. By contrast, fuzzy AHP is the effective technique to manage with both quantitative and qualitative criteria in MCDM problems [47]. Due to the ambiguous nature of human subjective cognition for linguistic evaluation in a pairwise comparison, fuzzy numbers are used. Therefore, the fuzzification of 1–9 scale is sensible and rational.

Various fuzzy AHP applications have been investigated so far. Du and Wang [48] proposed the combination of GIS and fuzzy AHP approach to select suitable monitoring locations for the Tomur World Natural Heritage in China. Calabrese et al. [49] conducted a literature review for fuzzy AHP. Under fuzzy environment, Chou et al. [50] integrated AHP and TOPSIS to evaluate HRST performance of Southeast Asian nations. Wang et al. [51] presented the SWOT based fuzzy AHP approach to assess the renewable resources of Pakistan. Guler and Yomralioglu [52] used Fuzzy AHP and TOPSIS under GIS environment to select the electric vehicle fast charging station. Alosta et al. [53] combined AHP and RAFSI model to resolve location selection problem. Ayyildiz et al. [54] provided the real case study of refugee camp location selection in Turkey using the combination of AHP and WASPAS. Garg et al. [55] introduced the hybrid trapezoidal fuzzy FUCOM-AHP model to identify the reasons that cause the spread of the monkeypox virus. Other recent studies in this field include those of Khan et al. [56] and Boral et al. [57].

For most of fuzzy AHP studies, the reciprocal value of a linear fuzzy number is seen as a linear fuzzy number; however, this reciprocal value does not produce a linear membership function. In addition, the weights obtained from fuzzy AHP are more complicated than those obtained from crisp AHP, because fuzzy numbers are employed in pairwise comparison matrices instead of crisp numbers. Moreover, the membership functions of the aggregation of fuzzy numbers evaluating qualitative criteria in a multilevel structure cannot be precisely obtained. To resolve these limitations, a defuzzification or ranking method is necessary. This paper uses a fuzzy AHP method in which the values on the 1–9 scale are defuzzified before being used in the pairwise comparison matrices. In addition, the fuzzy numbers used in the alternatives’ evaluation with respect to qualitative factors in the MCDM model are defuzzified before being aggregated. In this study, triangular fuzzy numbers are utilized to display the uncertain and subjective nature of the linguistic evaluation on the AHP scale and the evaluation of alternatives under qualitative criteria.

Ranking fuzzy numbers methods have been extensively researched. Chen [58] suggested a new raking fuzzy numbers method for solving tool steel materials evaluation; Wang and Kerre [59] proposed a review of the ranking fuzzy numbers methods; some other recent studies can be found in Deng [60]. Cheng [61] proposed the centroid-based distance method to rank fuzzy number by using the Euclidean distances from the origin to the centroid point. The error of centroid formula provided by Cheng [61] was found by Wang et al. [15]; thus, they proposed the correct centroid formulae for fuzzy numbers from analytical geometry perspective. Based on the standard deviations and centroid points, Chen and Chen [62] introduced an approach to rank the generalized trapezoidal fuzzy numbers. Wang et al. [63] proposed a ranking method based on left and right deviation degrees derived from maximal and minimal reference sets. Asady [64] provided the revise of distance minimization method ranking fuzzy numbers by using the epsilon –neighborhood of fuzzy numbers. Nejad and Mashinchi [65] developed a technique based on the left and right areas to resolves the shortcomings the method of Wang et al. [63].

In 2000, Yao and Wu [14] introduced the signed distance to rank both positive and negative fuzzy numbers by using the average of the closed interval α–cut. Despite the merits, Yao and Wu [14] method cannot provide discernible result in ranking some fuzzy numbers such as symmetric ones due to the used average operator. Other than the centroids on the x-axis and y-axis, the distance of the centroids point to both left and right membership function of a fuzzy number should be investigated. To the best of our knowledge, there is no solution for calculating the distance of the centroids point to both left and right membership function of a non-linear fuzzy number. Bivens [66] found that tangent line can be considered as the best linear approximation of the curve. Therefore, this study proposes a novel approach by using the centroids on x-axis and y-axis and the distances of centroids point to the tangent lines of a fuzzy number to defuzzify fuzzy numbers. The proposed ranking approach is used to defuzzify the fuzzy numbers used in fuzzy AHP and the linguistic values of qualitative criteria under MCDM structure. The final evaluated values of alternative locations then can be received through additive weighting to obtain the ranking result.

3 Fuzzy set theory

3.1 Fuzzy sets

A fuzzy set $\tilde{A}$ can denoted by $\tilde{A} = {(x, f_{\tilde{A}} (x)) | x \in U}$ , where U is the universe of discourse, x is an element in U, $\tilde{A}$ is a fuzzy set in U, $f_{\tilde{A}} (x)$ is the membership function of A at x [67]. The larger $f_{\tilde{A}} (x)$ , the stronger the grade of membership for x in $\tilde{A}$ .

3.2 Fuzzy numbers

A real fuzzy number $\tilde{A}$ is described as any fuzzy subset of the real line R with membership function f $\tilde{A}$ which possesses the following properties [68]:

f $\tilde{A}$ is a continuous mapping from R to [0, 1];

$f_{\tilde{A}} (x) = 0, \forall x \in (- \infty, a]$ ;

$f_{\tilde{A}}$ is strictly increasing on [a, b];

$f_{\tilde{A}} (x) = w, x \in [b, c]$ ;

$f_{\tilde{A}}$ is strictly decreasing on [c, d];

f_A (x) = 0, ∀ x ∈ [d, ∞);

The membership function of $f_{\tilde{A}}$ can be denoted as: $f_{\tilde{A}} (x) = {\begin{matrix} f_{\tilde{A}}^{L} (x), a ⩽ x ⩽ b \\ w, b ⩽ x ⩽ c \\ f_{\tilde{A}}^{R} (x), c ⩽ x ⩽ d \\ 0, othervise \end{matrix}$ (1) where a,b,c and d are real numbers. We may let a = -∞, or a = b, or b = c, or c = d, or d =+ ∞. Unless elsewhere specified, it is assumed that $\tilde{A}$ is convex, normal and bounded, i.e. - ∞ < a, d< ∞. For convenience, fuzzy number $\tilde{A}$ can be denoted by $\tilde{A} = [a, b, c, d; w]$ . Since $f_{\tilde{A}}^{L} : [a, b] \to [0, w]$ and $f_{\tilde{A}}^{R} : [c, d] \to [0, w]$ are continuous and strictly monotonic, the inverse function of $f_{\tilde{A}}^{L}$ and $f_{\tilde{A}}^{R}$ are also exist, denoted as $g_{\tilde{A}}^{L}$ and $g_{\tilde{A}}^{R}$ . Since $f_{\tilde{A}}^{L}$ and $f_{\tilde{A}}^{R}$ are continuous and strictly increasing, $g_{\tilde{A}}^{L} : [0, w] \to [a, b]$ and $g_{\tilde{A}}^{R} : [0, w] \to [c, d]$ are also continuous and strictly increasing.

Fuzzy number $\tilde{B}$ is a trapezoidal fuzzy number, denoted by (a, b, c, d ; 1). The membership function $f_{\tilde{B}}$ is given by [69]: $f_{\tilde{B}} (x) = {\begin{matrix} (x - a) / (b - a), a ⩽ x ⩽ b, \\ 1, b ⩽ x ⩽ c, \\ (x - d) / (c - d), c ⩽ x ⩽ d, \\ 0, otherwise . \end{matrix}$ (2)

Fuzzy number $\tilde{C}$ is a triangular fuzzy number, denoted by (a, b, d ; 1), when b = c in Equation (2) and its membership function $f_{\tilde{C}}$ is given by [69]: $f_{\tilde{C}} (x) = {\begin{matrix} (x - a) / (b - a), a ⩽ x ⩽ b, \\ (x - d) / (b - d), b ⩽ x ⩽ d, \\ 0, otherwise . \end{matrix}$ (3)

3.3 Linguistic values

Linguistic terms are common words in the daily life which involve uncertainty or imprecision. If decision-makers employ linguistic terms with values that have imprecise characteristics during decision-making procedures, the decision-making results would be more accurate. To solve the uncertainty or vagueness of the linguistic variables, Zadeh [12] introduced fuzzy set theory in 1965 to quantify these linguistic terms. The linguistic terms are used in this study to express the importance of the criteria, for example EI (equal important), SI (Slightly important), FI (Fairly important), EI (Essentially important), I (Important), MI (Moderate important), SIM (Strong important), VSI (Very strong important), EIM (Extremely important) and the linguistic terms for ranking the alternatives are VP (Very Poor), P (Poor), MP (Medium Poor), SP (Slightly Poor), F (Fair), SG (Slightly Good), MG (Medium Good), G (Good) and VG (Very Good). The triangular fuzzy numbers are employed to represent these linguistic terms such as EI = VP = (0, 0, 1.5), SI = P = (0.5, 1.5, 2), FI = MP = (1.5, 2, 3), EI = SP = (2, 3, 4), I = F = (3, 4.5, 6), MI = SG = (5, 6, 7), SIM = MG = (6, 7, 7.5), VSI = G = (7, 7.5, 8.5), EIM = VG = (7.5, 9, 9). Assume that the experts understand completely these linguistics’ meaning with the equivalent fuzzy numbers before making the decision.

4 A new ranking approach by total distance

In this section, a new approach [70] based on the centroid points on x-axis and y-axis and the distances between the centroid points and the tangent lines of a fuzzy number, $\tilde{A} = [a, b, c, d; w]$ in Equation (1), is proposed to rank fuzzy numbers as can be seen in Figure 1.

Step 1. Determine the tangent line

Fig. 1

Total distance ranking process.

–Calculate tangent points

The tangent at the point (x_T, f (x_T)) to the graph of y = f (x) is the line through that point with slope [71]. Goetschel and Voxman [72] pointed out that the tangent line segments yield the best local ‘linear’ approximation of a fuzzy function. The tangent line can be used as the approximation to the curve at one point [73]. If the tangent point is the most representative point, the tangent line at this point of the curve also can be the best approximation of the curve. Therefore, this study considers the middle points of the left and right membership functions of a fuzzy number as the most representative points which can be used to determine the tangent lines of the left and right membership functions, respectively. The tangent points of a fuzzy number can be produced as follows. Assume T₁ and T₂ are the tangent points of the left and right membership functions of fuzzy number A, respectively, which can be defined as: $T_{1} = (x_{T_{1}}, f (x_{T_{1}})) \in f_{\tilde{A}}^{L}, where f (x_{T_{1}}) = \frac{w}{2}$ (4) $T_{2} = (x_{T_{2}}, f (x_{T_{2}})) \in f_{\tilde{A}}^{R}, where f (x_{T_{2}}) = \frac{w}{2}$ (5) when $f (x_{T_{1}}) = f (x_{T_{2}}) = \frac{w}{2}$ , x_{T
₁} and x_{T
₂} can be determined based on the membership function of $f_{\tilde{A}}^{L} (x)$ and $f_{\tilde{A}}^{R} (x)$ because $T_{1} \in f_{\tilde{A}}^{L} (x)$ and $T_{2} \in f_{\tilde{A}}^{R} (x)$ .

–Produce the tangent line

Horwitz [74] defined the tangent line of a function at a point, and is used as follows.

The tangent line of the left membership function at the point T₁ can be obtained as: $\begin{matrix} d_{T_{1}} : y = f (x_{T_{1}}) + f^{'} (x_{T_{1}}) (x - x_{T_{1}}) or \\ f^{'} (x_{T_{1}}) \cdot x - y + f (x_{T_{1}}) - f^{'} (x_{T_{1}}) \cdot x_{T_{1}} = 0 \end{matrix}$ (6) where is the derivative of $f_{A}^{L} (x)$ at x = x_{T
₁}, $f (x_{T_{1}}) = \frac{w}{2}$

The tangent line of the right membership function at the point T₂ is defined as: $\begin{matrix} d_{T_{2}} : y = f (x_{T_{2}}) + f^{'} (x_{T_{2}}) (x - x_{T_{2}}) or \\ f^{'} (x_{T_{2}}) \cdot x - y + f (x_{T_{2}}) - f^{'} (x_{T_{2}}) \cdot x_{T_{2}} = 0 \end{matrix}$ (7) where is the derivative of $f_{A}^{R} (x)$ at x = x_{T
₂}, $f (x_{T_{2}}) = \frac{w}{2}$

Step 2. Determine the centroids point

The centroid is also known as the “center of gravity” or the “center of mass” [75]. Møller [76] defined center of gravity (centroid) as a center of mass corresponding to a kind of average position of the mass in the system. Cheng [61] pointed out that the ranking method with centroid point relates to find the geometric center of a fuzzy number. Following Wang et al. [15], the centroids point $\bar{A} = (x_{\bar{A}}, y_{\bar{A}})$ of a fuzzy number can be calculated as shown in Equations (8, 9). $x_{\bar{A}} = \frac{\int_{a}^{b} x . f_{\tilde{A}}^{L} (x) dx + \int_{b}^{c} (x . w) dx + \int_{c}^{d} x . f_{\tilde{A}}^{R} (x) dx}{\int_{a}^{b} f_{\tilde{A}}^{L} (x) dx + \int_{b}^{c} (w) dx + \int_{c}^{d} f_{\tilde{A}}^{R} (x) dx}$ (8) $y_{\bar{A}} = \frac{\int_{0}^{w} y . (g_{\tilde{A}}^{R} (y) - g_{\tilde{A}}^{L} (y)) dy}{\int_{0}^{w} (g_{\tilde{A}}^{R} (y) - g_{\tilde{A}}^{L} (y)) dy}$ (9)

In case of the fuzzy number $\tilde{A} = [a, a, a, a; w]$ , which means $0 ⩽ f_{\tilde{A}}^{L} = f_{\tilde{A}}^{R} ⩽ w, x = a$ and $g_{\tilde{A}}^{L} = g_{\tilde{A}}^{R} = a$ ; therefore, the Equations (8, 9) cannot be used. In such case, the centroid point on x-axis of $\tilde{A}$ can be represented by a and the centroid point on y-axis of $\tilde{A}$ can be represented by $\frac{w}{2}$ which denotes the average height of this fuzzy number, as shown in Equations (10, 11). $x_{\tilde{A}} = a$ (10) $y_{\tilde{A}} = \frac{w}{2}$ (11)

Step 3. Calculate the distance between centroids point and the tangent line

The distance of a point from a line was defined by Anton [77] as $d ((x_{0}, y_{0}), a_{1} x + b_{1} y + c_{1} = 0) = \frac{| a_{1} x_{0} + b_{1} y_{0} + c_{1} |}{\sqrt{a_{1}^{2} + b_{1}^{2}}}$ . This concept is applied in this study to calculate the distances between the centroids point and the tangent lines of the left and right membership functions, respectively.

–The distance of centroids point to the tangent line of the left membership function at the point T₁ can be obtained by Anton [77] concept and Equation (6) as shown in Equation (12).

$\begin{array}{l} d (\bar{A}, f_{\tilde{A}}^{L}) = \\ \frac{| f^{'} (x_{T_{1}}) \cdot x_{\bar{A}} - y_{\bar{A}} + f (x_{T_{1}}) - f^{'} (x_{T_{1}}) \cdot x_{T_{1}} |}{\sqrt{f^{'} {(x_{T_{1}})}^{2} + {(- 1)}^{2}}} \end{array}$ (12)

–The distance of centroids pointto the tangent line of the right membership function at the point T₂ can be obtained by Anton [77] concept and Equation (7) as shown in Equation (13). $\begin{array}{l} d (\bar{A}, f_{\tilde{A}}^{R}) = \\ \frac{| f^{'} (x_{T_{2}}) \cdot x_{\bar{A}} - y_{\bar{A}} + f (x_{T_{2}}) - f^{'} (x_{T_{2}}) \cdot x_{T_{2}} |}{\sqrt{f^{'} {(x_{T_{2}})}^{2} + {(- 1)}^{2}}} \end{array}$ (13)

Step 4. Calculate the total distance

The larger distance of centroids point to the right membership function of the fuzzy number leads to the larger distance of the right membership function to the y-axis. Whereas the smaller distance of centroids point to the left membership function leads to the larger distance of the left membership function to the y-axis. A fuzzy number with the larger distance of right and/or smaller distance of left membership function(s) to the y-axis generates larger ranking value. Herein, the relative closeness coefficient of a fuzzy number is defined as shown in Equation (14) to assist in ranking fuzzy numbers. $R_{\bar{A}} = \frac{d (\bar{A}, f_{\tilde{A}}^{R})}{1 + d (\bar{A}, f_{\tilde{A}}^{L})}$ (14)

Due to the reason that either the distance of $x_{\bar{A}}$ , $y_{\bar{A}}$ or $R_{\bar{A}}$ is larger, the fuzzy number $\tilde{A}$ is larger, three of them must be simultaneously considered. This study proposes the total distance by using three of them to rank fuzzy numbers as shown Equation (15). If the value of $d (\tilde{A})$ is larger, the ranking of $\tilde{A}$ is higher. $d (\tilde{A}) = x_{\bar{A}} + y_{\bar{A}} + R_{\bar{A}}$ (15)

Proposition 1. In case of trapezoidal fuzzy number $\tilde{B} (a, b, c, d; 1)$ in Equation (2), the tangent line of the left membership function of the fuzzy number at T₁ in Equation (6) and the tangent line of the right membership function of the fuzzy number at T₂ in Equation (7) are the left and right membership function of $\tilde{B} (a, b, c, d; 1)$ , respectively.

Proof.

From Equations (2) and (4),

T₁ (x_{T
₁}, f (x_{T
₁})), where $f (x_{T_{1}}) = \frac{1}{2}$ $\frac{x_{T_{1}} - a}{b - a} = \frac{1}{2}$ $\Rightarrow x_{T_{1}} = \frac{b - a}{2} + a \Rightarrow T_{1} (\frac{b - a}{2} + a, \frac{1}{2})$

From Equations (2) and (5),

T₂ (x_{T
₂}, f (x_{T
₂})), where $f (x_{T_{2}}) = \frac{1}{2}$ $\frac{x_{T_{2}} - d}{c - d} = \frac{1}{2}$ $\Rightarrow x_{T_{2}} = \frac{c - d}{2} + d \Rightarrow T_{2} (\frac{c - d}{2} + d, \frac{1}{2})$

From Equation (6), the tangent line at T₁ of the left function of $\tilde{B}$ are defined as: $\frac{1}{b - a} \cdot x - y + \frac{1}{2} - (\frac{1}{b - a} \cdot (\frac{b - a}{2} + a)) = 0$ $\Rightarrow \frac{x}{b - a} - \frac{a}{b - a} - y = 0$

$\Rightarrow \frac{x - a}{b - a} = y$ which is the left membership function of $\tilde{B}$ in Equation (2).

From Equation (7), the tangent line at T₂ of the right function of $\tilde{B}$ is defined as: $\frac{1}{c - d} \cdot x - y + \frac{1}{2} - (\frac{1}{c - d} \cdot (\frac{c - d}{2} + d)) = 0$ $\Rightarrow \frac{x}{c - d} - \frac{d}{c - d} - y = 0$

$\Rightarrow \frac{x - d}{c - d} = y$ which is the right membership function of $\tilde{B}$ in Equation (2). □

Remark 1. The total distance of the trapezoidal fuzzy number $\tilde{B} (a, b, c, d; 1)$ can be obtained as follows: $By Eq . (12), d (\bar{B}, f_{\tilde{B}}^{L}) = \frac{| \frac{x_{\bar{B}} - a}{b - a} - y_{\bar{B}} |}{\sqrt{{(\frac{1}{b - a})}^{2} + {(- 1)}^{2}}}$ (16) $By Eq . (13), d (\bar{B}, f_{\tilde{B}}^{R}) = \frac{| \frac{x_{\bar{B}} - d}{c - d} - y_{\bar{B}} |}{\sqrt{{(\frac{1}{c - d})}^{2} + {(- 1)}^{2}}}$ (17) $By Eq . (14), R_{\bar{B}} = \frac{\frac{| \frac{x_{\bar{B}} - d}{c - d} - y_{\bar{B}} |}{\sqrt{{(\frac{1}{c - d})}^{2} + {(- 1)}^{2}}}}{1 + \frac{| \frac{x_{\bar{B}} - a}{b - a} - y_{\bar{B}} |}{\sqrt{{(\frac{1}{b - a})}^{2} + {(- 1)}^{2}}}}$ (18)

By Equation (15), the total distance of a trapezoidal fuzzy number can be produced by summation of Equations (8, 9) and (18).

Proposition 2. In case of triangular fuzzy number $\tilde{C} (a, b, d; 1)$ in Equation (3), the tangent line of the left membership function of the fuzzy number at T₁ in Equation (6) and the tangent line of the right membership function of the fuzzy number at T₂ in Equation (7) are the left and right membership function of $\tilde{C} (a, b, d; 1)$ , respectively.

Proof.

From Equations (3) and (4),

From Equations (3) and (5),

T₂ (x_{T
₂}, f (x_{T
₂})), where $f (x_{T_{2}}) = \frac{1}{2}$ $\frac{x_{T_{2}} - d}{b - d} = \frac{1}{2}$ $\Rightarrow x_{T_{2}} = \frac{b - d}{2} + d \Rightarrow T_{2} (\frac{b - d}{2} + d, \frac{1}{2})$

From Equation (6), the tangent line at T₁ of the left function of $\tilde{C}$ are defined as: $\frac{1}{b - a} \cdot x - y + \frac{1}{2} - (\frac{1}{b - a} \cdot (\frac{b - a}{2} + a)) = 0$ $\Rightarrow \frac{x}{b - a} - \frac{a}{b - a} - y = 0$

$\Rightarrow \frac{x - a}{b - a} = y$ = Equation (3) left membership function of $\tilde{C}$

From Equation (7), the tangent line at T₂ of the right function of $\tilde{C}$ is defined as: $\frac{1}{b - d} \cdot x - y + \frac{1}{2} - (\frac{1}{b - d} \cdot (\frac{b - d}{2} + d)) = 0$ $\Rightarrow \frac{x}{b - d} - \frac{d}{b - d} - y = 0$

$\Rightarrow \frac{x - d}{b - d} = y = Equation . (3)$ = right membership function of $\tilde{C}$ . □

Remark 2. The total distance of the triangular fuzzy number $\tilde{C} (a, b, d; 1)$ can be obtained as follows. $By Eq . (12), d (\bar{C}, f_{\tilde{C}}^{L}) = \frac{| \frac{x_{\bar{C}} - a}{b - a} - y_{\bar{C}} |}{\sqrt{{(\frac{1}{b - a})}^{2} + {(- 1)}^{2}}}$ (19) $By Eq . (13), d (\bar{C}, f_{\tilde{C}}^{R}) = \frac{| \frac{x_{\bar{C}} - d}{b - d} - y_{\bar{C}} |}{\sqrt{{(\frac{1}{b - d})}^{2} + {(- 1)}^{2}}}$ (20) $By Eq . (14), R_{\bar{C}} = \frac{\frac{\frac{x_{\bar{C}} - d}{b - d} - y_{\bar{C}}}{\sqrt{{(\frac{1}{b - d})}^{2} + {(- 1)}^{2}}}}{1 + \frac{| \frac{x_{\bar{C}} - a}{b - a} - y_{\bar{C}} |}{\sqrt{{(\frac{1}{b - a})}^{2} + {(- 1)}^{2}}}}$ (21)

By Equation (15), the total distance of the triangular fuzzy number can be produced by the summation of Equations (8, 9) and (21).

Proposition 3. In the case of fuzzy number $\tilde{A} = [a, a, c, d; w]$ when a = b in Equation (22), the tangent line of the left membership function of the fuzzy number at T₁ in Equation (6) cannot be defined because . In this case, the line x = a or x - a = 0 in Figure 2 is the most representative of the left membership function of $\tilde{A} = [a, a, c, d; w]$ which can be considered as the tangent line of the left membership function at the point T₁. The distance of centroids point to the tangent line of the left membership function at the point T₁ in Equation (12) can be defined as in Equation (23). $f_{\tilde{A}} (x) = {\begin{matrix} f_{\tilde{A}}^{L} (x) = 0, x < a \\ w, a ⩽ x ⩽ c \\ f_{\tilde{A}}^{R} (x), c ⩽ x ⩽ d \\ 0, othervise \end{matrix}$ (22)

Fig. 2

Fuzzy number $\tilde{A} = [a, a, c, d; w]$ .

Following Anton [77], the distance of centroids point to the line x - a = 0 can be defined as: $d (\bar{A}, f_{\tilde{A}}^{L}) = \frac{| x_{\bar{A}} - a |}{\sqrt{1^{2} + 0}} = | x_{\bar{A}} - a |$ (23)

Proposition 4. Similarly, in the case of fuzzy number $\tilde{A} = [a, b, d, d; w]$ when c = d in Equation (24), the tangent line of the left membership function of the fuzzy number at T₂ in Equation (7) cannot be defined because . In this case, the line x = d or x - d = 0 in Figure 3 is the most representative of the left membership function of $\tilde{A} = [a, b, d, d; w]$ which can be considered as the tangent line of the right membership function at the point T₂ in Equation (7). The distance of centroids point to the tangent line of the right membership function at the point T₂ in Equation (13) can be defined as in Equation (25). $f_{\tilde{A}} (x) = {\begin{matrix} f_{\tilde{A}}^{L} (x), a ⩽ x ⩽ b \\ w, a ⩽ x ⩽ d \\ f_{\tilde{A}}^{R} (x) = 0, x > d \\ 0, othervise \end{matrix}$ (24)

Fig. 3

Fuzzy number $\tilde{A} = [a, b, d, d; w]$ .

Following Anton [77], the distance of centroids point to the line x - d = 0 can be defined as: $d (\bar{A}, f_{\tilde{A}}^{R}) = \frac{| x_{\bar{A}} - d |}{\sqrt{1^{2} + 0}} = | x_{\bar{A}} - d |$ (25)

Proposition 5. In case of fuzzy number $\tilde{A} = [a, a, a, a; w]$ , $d (\bar{A}, f_{\tilde{A}}^{L}) = d (\bar{A}, f_{\tilde{A}}^{R}) = 0$ because the centroids point lies in the left and right membership function. Therefore, the distance of the centroids point to the left and right membership functions is zero.

4.1 Comparison

Consider two fuzzy numbers $\tilde{A} = (0.1, 0.4, 0.8)$ and $\tilde{B} = (0, 0.5, 0.7)$ which can be seen in Fig.4.

Fig. 4

Fuzzy number $\tilde{A} = (0.1, 0.4, 0.8) and \tilde{B} = (0, 0.5, 0.7)$ .

By Yao and Wu [14] method, the signed distance of two fuzzy numbers $\tilde{A}, \tilde{B}$ can be calculated as shown in Equation (26). $\begin{matrix} d (\tilde{A}, \tilde{B}) = d (\tilde{A}, 0) - d (\tilde{B}, 0) \\ = \frac{1}{2} \int_{0}^{1} ({\tilde{A}}^{L} + {\tilde{A}}^{R} - {\tilde{B}}^{L} - {\tilde{B}}^{R}) d α \end{matrix}$ (26) where $\tilde{A} > \tilde{B}$ if and only if $d (\tilde{A}, \tilde{B}) > 0$ ; $\tilde{A} < \tilde{B}$ if and only if $d (\tilde{A}, \tilde{B}) < 0$ and $\tilde{A} = \tilde{B}$ if and only if $d (\tilde{A}, \tilde{B}) = 0$ .

Table 1

Comparison with other methods

	Fuzzy number	Set 1	Set 2	Set 3	Set 4
Yao and Wu [14]	$\tilde{C}$	0.6	0.575	0.5	0.475
	$\tilde{D}$	0.7	0.65	0.625	0.525
	$\tilde{E}$	0.8	0.7	0.55	0.525
Result		$\tilde{C} < \tilde{D} < \tilde{E}$	$\tilde{C} < \tilde{D} < \tilde{E}$	$\tilde{C} < \tilde{E} < \tilde{D}$	$\tilde{C}$ lt; $\tilde{D}$ ∼ $\tilde{E}$
Wang et al. [15]	$\tilde{C}$	0.21	0.26	0.17	0.2
	$\tilde{D}$	0.23	0.21	0.27	0.18
	$\tilde{E}$	0.25	0.25	0.19	0.17
Result		$\tilde{C} < \tilde{D} < \tilde{E}$	$\tilde{D} < \tilde{E} < \tilde{C}$	$\tilde{C} < \tilde{E} < \tilde{D}$	$\tilde{E} < \tilde{D} < \tilde{C}$
Deng at al. [16]	$\tilde{C}$	0.264	0.313	0.207	0.252
	$\tilde{D}$	0.29	0.264	0.336	0.226
	$\tilde{E}$	0.318	0.288	0.237	0.213
Result		$\tilde{C} < \tilde{D} < \tilde{E}$	$\tilde{D} < \tilde{E} < \tilde{C}$	$\tilde{C} < \tilde{E} < \tilde{D}$	$\tilde{E} < \tilde{D} < \tilde{C}$
Asady & Zendehnam[17]	$\tilde{C}$	0.60	0.58	0.50	0.48
	$\tilde{D}$	0.70	0.65	0.63	0.53
	$\tilde{E}$	0.80	0.70	0.55	0.53
Result		$\tilde{C} < \tilde{D} < \tilde{E}$	$\tilde{C} < \tilde{D} < \tilde{E}$	$\tilde{C} < \tilde{E} < \tilde{D}$	$\tilde{C}$ lt; $\tilde{D}$ ∼ $\tilde{E}$
Proposed method	$\tilde{C}$	1.11	1.12	0.96	1.12
	$\tilde{D}$	1.19	1.13	1.25	1.04
	$\tilde{E}$	1.26	1.17	1.05	1.02
Result		$\tilde{C} < \tilde{D} < \tilde{E}$	$\tilde{C} < \tilde{D} < \tilde{E}$	$\tilde{C} < \tilde{E} < \tilde{D}$	$\tilde{E} < \tilde{D} < \tilde{C}$

However, by the Equation (26), the sign distance of two fuzzy numbers $\tilde{A}$ and $\tilde{B}$ is $d (\tilde{A}, \tilde{B}) = 0$ , producing the ranking result $\tilde{A} = \tilde{B}$ , which seems unreasonable. Following the method of Yao and Wu [14], if $\int_{0}^{1} \frac{{\tilde{A}}^{L} + {\tilde{A}}^{R}}{2} d α = \int_{0}^{1} \frac{{\tilde{B}}^{L} + {\tilde{B}}^{R}}{2} d α$ , the fuzzy number $\tilde{A}$ and $\tilde{B}$ will receive same ranking result due to the use of average operator. By the proposed total distance method. i.e., Equation (15), the distances of two fuzzy numbers $\tilde{A}$ and $\tilde{B}$ are $d (\tilde{A}) = 0.98$ and $d (\tilde{B}) = 0.93$ , respectively. Therefore, the final ranking result is $\tilde{A} > \tilde{B}$ because 0.98 > 0.93. The proposed method can provide more distinguishable result than that of the Yao and Wu [14] method. Moreover, the method of Yao and Wu [14] cannot deal with the fuzzy numbers with different grades. For example, the signed distance of two fuzzy numbers $\tilde{A} = (9, 9, 9; 0.8)$ and $\tilde{B} = (9, 9, 9; 0.5)$ cannot be obtained by Equation (26) because the grades of two fuzzy numbers are 0.8 and 0.5, respectively. However, the proposed method provides the final result $\tilde{A} > \tilde{B}$ because $d (\tilde{A}) = 9.4 > d (\tilde{B}) = 9.25$ through Equation (18).

In addition, a comparison is made by using the four sets of fuzzy numbers in Yao and Wu [14] as shown in Table 1.

Set 1: $\tilde{C} = (0.4, 0.5, 1), \tilde{D} = (0.4, 0.7, 1),$ $\tilde{E} = (0.4, 0.9, 1) .$

Set 2: $\tilde{C} = (0.3, 0.4, 0.7, 0.9), \tilde{D} = (0.3, 0.7 .$ $0.9), \tilde{E} = (0.5, 0.7, 0.9) .$

Set 3: $\tilde{C} = (0.3, 0.5, 0.7), \tilde{D} = (0.3, 0.5, 0.8,$ $0.9), \tilde{E} = (0.3, 0.5, 0.9) .$

Set 4: $\tilde{C} = (0, 0.4, 0.7, 0.8), \tilde{D} = (0.2, 0.5,$ $0.9), \tilde{E} = (0.1, 0.6, 0.8)$

From Table 1, the proposed method has the same ranking result as that of the other four papers in set 1 and set 3. In set 2, the proposal method has the same result as that of Yao & Wu [14] and Asady & Zendehnam [17], which is $\tilde{C} < \tilde{D} < \tilde{E}$ . However, Wang et al. [15] and Deng at al. [16] provide the ranking result as $\tilde{D} < \tilde{E} < \tilde{C}$ which seems unreasonable because intuitively $\tilde{C}$ should be smaller than $\tilde{D}$ and $\tilde{E}$ as shown in Figure 5. In set 4 shown in Figure 6, the ranking results from Yao & Wu [14] and Asady & Zendehnam [17] are $\tilde{C}$ < $\tilde{D}$ ∼ $\tilde{E}$ . Clearly, their methods cannot provide distinguishable result between $\tilde{D}$ and $\tilde{E}$ ; whereas, the proposed method can distinguish $\tilde{D}$ and $\tilde{E}$ by the ranking order $\tilde{E} < \tilde{D} < \tilde{C}$ , which is the same result by Wang et al. [15] and Deng et al. [16].

Fig. 5

Fuzzy numbers in set 2.

Fig. 6

Fuzzy numbers in set 4.

5 Model establishment

The proposed AHP based MCDM method under uncertain environment is introduced as the following steps [70].

Step 1. Establish multi-level structure for criteria weights $\begin{matrix} c_{x_{1} x_{2} \cdot \cdot \cdot x_{i} \cdot \cdot \cdot x_{m}}, m \in N, \\ c_{m} = 1 \sim n_{m}, n_{m} ⩾ 0, n_{m} \in N . \end{matrix}$ (27)

The criteria weights in a multi-level structure can be obtained by the concepts from Chu and Varma [36] and Nghiem and Chu [45] as shown in Equation (27). The weights of major criteria in the structure can be denoted by w_{x
₁}, x₁ = 1 ∼ n₁, n₁ ⩾ 1, n₁ ∈ N, $\sum_{x_{1} = 1}^{n_{1}} w_{x_{1}} = 1$ ; the weights of criteria in the second level can be denoted by w_{x
₁
x
₂}, x₂ = 1 ∼ n₂, n₂ ⩾ 0, n₂ ∈ N, $\sum_{x_{2} = 1}^{n_{2}} w_{x_{1} x_{2}} = 1$ ; and the weights of criteria in the ith level can be denoted by w_{x₁x₂···x_i}, x_i = 1 ∼ n_i, n_i ⩾ 0, n_i ∈ N, $\sum_{x_{i} = 1}^{n_{i}} w_{x_{1} x_{2} \cdot \cdot \cdot x_{i}} = 1$ . Finally, the weights of criteria in the bottom level can be denoted by w_{x₁x₂···x_i···x_m}, m ∈ N, x_m = 1 ∼ n_m, n_m ⩾ 0, n_m ∈ N, $\sum_{x_{m} = 1}^{n_{m}} w_{x_{1} x_{2} \cdot \cdot \cdot x_{i} \cdot \cdot \cdot x_{m}} = 1$ .

Step 2. Build up fuzzy AHP

AHP is introduced by Saaty [8]. In this work, the weights in each level are produced by fuzzy AHP (FAHP). The fuzzy AHP starts creating a pairwise comparison matrix $\tilde{A}$ . Further suppose each pairwise comparison matrix $\tilde{A}$ is conducted by a committee ofdecision makers, D_t, t = 1 ∼ q. The matrix $\tilde{A}$ is a n×n real matrix, where n is the number of evaluation criteria considered. Due to human subjective cognition with ambiguous nature for linguistic evaluation in the pairwise comparison, the fuzzy number ${\tilde{a}}_{ij}$ is used. Each entry ${\tilde{a}}_{ij}$ of the matrix $\tilde{A}$ represents the importance of the ith criterion relative to the jth criterion. If ${\tilde{a}}_{ij} > \tilde{1}$ , then the ith criterion is more important than the jth criterion, while if ${\tilde{a}}_{ij} < \tilde{1}$ , then the ith criterion is less important than the jth criterion. If two criteria have the same importance, then the entry ${\tilde{a}}_{ij}$ is $\tilde{1}$ . The relative importance between two criteria is measured according to a fuzzy numerical scale from $\tilde{1}$ to $\tilde{9}$ as shown in Table 2. Herein the triangular fuzzy numbers such as $\tilde{1}$ = (0, 0, 1.5), $\tilde{2}$ = (0.5, 1.5, 2), $\tilde{3}$ = (1.5, 2, 3), $\tilde{4}$ = (2, 3, 4), $\tilde{5}$ = (3, 4.5, 6), $\tilde{6}$ = (5, 6, 7), $\tilde{7}$ = (6, 7, 7.5), $\tilde{8}$ = (7, 7.5, 8.5), $\tilde{9}$ = (7.5, 9, 9), are used and are displayed in Figure 7.

Fig. 7

Triangular fuzzy numbers of $\tilde{1}$ to $\tilde{9}$ .

Step 3. Establish a defuzzified $\tilde{A}$

Because the calculation of fuzzy numbers in the pairwise comparison matrix $\tilde{A}$ is complicated, the defuzzification of those fuzzy numbers in $\tilde{A}$ is needed. By Equations (4)–(15), fuzzy numbers ${\tilde{a}}_{ij}$ in Table 1 can be defuzzified to be crisp values a_ij as also shown in Table 2. When the crisp pairwise comparison matrix A is obtained, the entries a_ij and a_ji satisfy the constraint a_ij × a_ji = 1. Obviously, a_ii = 1 for all i.

Table 2

Linguistic evaluation, fuzzy numbers and defuzzified Values

Linguistic evaluation	${\tilde{a}}_{ij}$ value	a_ij value
i and j are equally important	$\tilde{1}$	1.02
i is fairly more important than j	$\tilde{3}$	2.74
i is more important than j	$\tilde{5}$	5.19
i is strongly more important than j	$\tilde{7}$	7.49
i is extremely more important than j	$\tilde{9}$	9.47
For compromise between the above values	$\tilde{2}$ , $\tilde{4}$ , $\tilde{6}$ , $\tilde{8}$	1.99,3.65,6.65,8.24

Step 4. Produce weights

Once the crisp matrix A is built, the normalized pairwise comparison matrix of A is derived by making the sum of the entries on each column equal to one. The criterion weight can be obtained by averaging the entries on each row of A. The weight can be obtained by applying Equation (28). $w_{i} = \frac{1}{m} \sum_{j = 1}^{m} (a_{ij} / - \sum_{i = 1}^{m} a_{ij}), i, j = 1 \sim m,$ (28)

Step 5. Test consistency

The consistency in each pairwise comparison matrix is conducted by the following procedure (Satty, 1980). The is produced by Equation (29). The estimated eigenvalue λ_max can be obtained as Equation (30). ${w^{'}}_{i} = \sum_{j = 1}^{n} (a_{i j} \times w_{j}), w_{j} = w_{i}$ (29) $λ_{\max} = (\frac{1}{n}) \times \sum_{i = 1}^{n} (\frac{{w^{'}}_{i}}{w_{i}})$ (30)

Table 3

Values of the random index [8]

m	1	2	3	4	5	6	7	8	9	10
RI	0	0	0.58	0.9	1.12	1.24	1.32	1.41	1.45	1.49

Hence, A is completely consistent when λmax = n, while it becomes increasingly inconsistent as the eigenvalue grows. And CI is the consistency index as shown in Equation (31) and RI is the random consistency index, i.e. the consistency index when the entries of A are completely random. The values of RI for small problems (m≤10) are shown in Table 3. The consistency ratio (CR) is presented as Equation (32). A matrix is consistent when CR < 0.1. $CI = \frac{λ_{max} - m}{m - 1}$ (31) $CR = \frac{CI}{RI}$ (32)

Step 6. Obtain weights of criteria

The aggregated weight for each final criterion in the structure can be produced by the following equation. The summation of weights of the end criteria equals one. $\begin{matrix} c (w_{x_{1} x_{2} \cdot \cdot \cdot x_{i}}) = w_{x_{1}} \times w_{x_{1} x_{2}} \times \cdot \cdot \cdot \times w_{x_{1} x_{2} \cdot \cdot \cdot x_{i}}, \\ where x_{i} = 1 \sim n_{i}, n_{i} ⩾ 0, n_{i} \in N . \end{matrix}$ (33)

Step 7. Establish MCDM matrix

Suppose the final criteria in Equation (27), denoted by c_{f
_i}, i = 1 ∼ n, and their corresponding final weights in Equation (33), denoted by w_{f
_i}, i = 1 ∼ n, are identified. Criteria c_{f
_i} include qualitative and quantitative ones. Quantitative criteria can be classified to benefit (B) and cost (C) ones. Qualitative criteria are benefit. The decision-making matrix can be shown as follows. $\begin{matrix} \begin{matrix} c_{f_{1}} \dots c_{f_{j}} \dots c_{f_{n}} \end{matrix} \\ \begin{matrix} A_{1} \\ ⋮ \\ A_{p} \\ ⋮ \\ A_{m} \end{matrix} [\begin{matrix} e_{1 f_{1}} \dots e_{1 f_{i}} \dots e_{1 f_{n}} \\ ⋮ ⋮ ⋮ \\ e_{p f_{1}} \dots e_{p f_{i}} \dots e_{p f_{n}} \\ ⋮ ⋮ ⋮ \\ e_{m f_{1}} \dots e_{m f_{i}} \dots e_{m f_{n}} \end{matrix}] \end{matrix}$ (34) where e_{pf
_i} represents the performance value of alternative facility location A_p, p = 1 ∼ m, versus criterion c_{f
_i}, i = 1 ∼ n.

Step 8. Average values of alternatives versus each qualitative criterion

Assume that values of alternatives versus qualitative criteria are evaluated by decision makers using linguistic values and fuzzy numbers in section 3.3, denoted by ${\tilde{e}}_{p f_{j}}$ . Those fuzzy numbers are further defuzzified to e_{pf
_j} by Equations (4)–(15). The average value of each alternative versus each subjective criterion in the criteria structure Equation (34) is produced by the committee of decision makers, D_t, t = 1 ∼ q, as follows. $e_{p f_{j}} = \frac{1}{q} \sum_{t = 1}^{q} e_{p f_{j} t}$ (35) where e_{pf
_i} represents the average value of each alternative A_p versus each qualitative criterion c_{f
_i} from q decision makers.

Step 9. Normalize alternative values versus criteria

Values of alternatives versus criteria can be classified into benefit (B) and cost (C) ones and must be normalized because they have different units. Benefit criteria include qualitative criteria and benefit quantitative criteria. Benefit criteria have the characteristics of larger better, while cost criteria have the characteristics of smaller better. The following two formulas are applied. $r_{p f_{i}} = \frac{e_{p f_{i}}}{max_{p} e_{p f_{i}}}, c_{p f_{i}} \in B$ (36) $r_{p f_{i}} = \frac{min_{p} e_{p f_{i}}}{e_{p f_{i}}}, c_{p f_{i}} \in C$ (37) where r_{pf
_i} denotes the normalized value of e_{pf
_i}.

Step 10. Obtain final evaluation

Finally, the additive weighted evaluation of each alternative can be obtained by using multiplication and addition to aggregate the normalized values and their corresponding weights as follows. Alternative A_p has higher priority if its additive weighted value is larger. $A_{p} = r_{p f_{i}} \times w_{f_{i}}, p = 1 \sim m, i = 1 \sim n .$ (38) where r_{pf
_i} denotes the normalized value of e_{pf
_i} and w_{f
_i} denoted the weight of the final criterion, that is c (w_{x₁x₂···x_i}) in Equation (33).

6 Numerical example

Assume that a manufacturing company must select a sustainable facility location to build a new plant [70]. After preliminary screening, five locations A_i, i = 1∼5, are chosen for further evaluation. A committee of three decision-makers D_t, t = 1∼3, is formed to conduct the evaluation and selection of the five locations by the proposed method as follows. The decision-makers may include the CEO, the heads of departments, or experts who possess a comprehensive understanding of both the company and the locations in this area. When choosing the manufacturing location, there are numerous options to be considered. To streamline the process, decision-makers must screen the alternatives to narrow down their choices. Supposed five alternatives were chosen by the decision makers after conducting an initial screening. These alternatives may be situated either within the same area or across different locations. After preliminary screening, five locations A_i, i = 1∼5, are chosen for further evaluation.

Step 1. Establish multiple levels structure for criteria weights

Chen et al. [1] provided a three-level structure to summarize the sustainability factors when choosing manufacturing facility location. Assume that a four-level structure based on Chen et al.’s [1] three-level structure is established to depict the relationship among criteria as displayed in Figure 7. In Figure 7, the first level includes C₁: Economic, C₂: Environmental, C₃: Social; the second level includes C₁₁: Cost, C₁₂: Market, C₁₃: Economic stability, C₁₄: Suppliers, C₁₅: Growth; C₂₁: Ecosystem vitality, C₂₂: Environmental health, C₂₃: Environmental factors within production; C₃₁: Governance, C₃₂: Education, C_33 :Individual, C₃₄: Community; the third level includes C₁₁₁: Labor cost, C₁₁₂: Energy cost, C₁₁₃: Material cost, C₁₁₄: Facility cost, C₁₁₅: Logistic cost, C₁₂₁: International markets, C₁₂₂: Potential demand, C₁₃₁: Favorable tax, C₁₃₂: Exchange rate fluctuations, C₁₄₁: Proximity to key suppliers, C₁₄₂: Potential high quality suppliers, C₁₅₁: Industry growth, C₁₅₂: Competitiveness in the host region, C₂₁₁: Air pollution related to the ecosystem, C₂₁₂: Water quality related to the ecosystem, C₂₁₃: Biodiversity protection, C₂₁₄: Climate change performance, C₂₂₁: Environmental burden of disease, C₂₂₂: Air pollution related to humans, C₂₂₃: Water quality related to humans, C₂₃₁: Material use, C₂₃₂: Energy consumption, C₂₃₃: Renewable resources, C₂₃₄: Waste generation and management, C₂₃₅: Recycling of material, energy, and waste, C₃₁₁: Corruption, C₃₁₂: Political stability, C₃₁₃: Trade and tariff barriers, C₃₂₁: Literacy, C₃₂₂: General education level, C₃₃₁: Civil liberties, C₃₃₂: Human rights, C₃₄₁: Equity, C₃₄₂: Safety, C₃₄₃: Cohesion, C₃₄₄: Local technology. In addition, according to a report from United Nations publication [34], some sub-sub-criteria can be further classified to sub-sub-sub criteria in the fourth level, such as Energy consumption (C₂₃₂) can be further classified to Annual energy consumption, total and by main user category (C₂₃₂₁) and Intensity of energy use, total and by economic activity (C₂₃₂₂); Waste generation and management (C₂₃₄) can be further classified to Generation of hazardous waste (C₂₃₄₁) and Waste treatment and disposal (C₂₃₄₂); and General education level (C₃₂₂) can be further classified to Gross intake ratio to last grade of primary education (C₃₂₂₁), Adult secondary (tertiary) schooling attainment level (C₃₂₂₂) and Net enrolment rate in primary education (C₃₂₂₃), as displayed in Figure 8.

Herein, criteria can be classified to benefit (B) and cost (C) ones such as C₁₁₁ (C, US$), C₁₁₂ (C, US$), C₁₁₃ (C, million US$), C₁₁₄ (C, US$), C₁₁₅ (C, US$), C₁₃₁ (B, %), C₁₃₂ (C, %), C₁₅₁ (B, %), C₂₁₄ (C, index), C₂₂₁ (C, %), C₂₂₂ (C, air quality index), C₂₂₃ (C, %), C₂₃₁ (C, ton), C₂₃₃ (B, %), C₂₃₅ (B, ton), C₃₁₁ (C, index), C₃₂₁ (B, index), C₃₄₄ (B, number), C₂₃₂₁ (C, ton), C₂₃₂₂ (B, index), C₂₃₄₁ (C, ton), C₂₃₄₂ (B, %), C₃₂₂₁ (B, %), C₃₂₂₂ (B, %) and C₃₂₂₃ (B, %). And C₁₂₁, C₁₂₂, C₁₄₁, C₁₄₂, C₁₅₂, C₂₁₁, C₂₁₂, C₂₁₃, C₃₁₂, C₃₁₃, C₃₃₁, C₃₃₂, C₃₄₁, C₃₄₂, C₃₄₃ are treated as qualitative criteria, and all qualitative criteria are benefit criteria.

Fig. 8

Four-Level Structure of Criteria.

Step 2. Build up fuzzy AHP

Assume that the pairwise comparisons are conducted by the three decision makers by using the triangular fuzzy numbers in Table 2 as shown in Tables 4–22. The shapes of those trapezoidal fuzzy numbers are displayed in Figure 6.

Step 3. Establish a defuzzified $\tilde{A}$

By Equations (4)–(15), triangular fuzzy numbers can be defuzzified to be crisp values as also presented in Tables 4–22.

Step 4. Produce weights

The weights of criteria for each matrix in the structure can be obtained by Equation (28) as shown in Table 23.

Step 5. Test consistency

By Equations (29)–(32), consistency ratio (CR) can be produced for each matrix as 0.06, 0.08, 0.02, 0.09, 0.03, 0.07, 0.05,0.08, 0, 0, 0.04, 0.06, 0, 0, 0, 0, 0, 0, 0.05 for Tables 4–22. All ratios are less than 0.1, which shows consistency.

Step 6. Obtain weights of criteria

The aggregated weight for each final criterion in the structure can be produced by Equation (33) as shown in Table 24.

Step 7. Establish MCDM matrix

The MCDM matrix can be obtained as displayed in Equation (34) by using the final criteria from Equation (27) and the final weights from Equation (33).

Step 8. Average values of alternatives versus each qualitative criterion

Assume that alternatives versus qualitative criteria can be evaluated by decision makers using linguistic values in section 3.3 as shown in Table 25. Those linguistic values can be defuzzified by Equations (4)–(15) and averaged by Equation (35) as also shown in Table 25.

Step 9. Normalize alternatives values versus criteria

Assume that numerical values of alternatives versus quantitative criteria are shown in Table 26. By Equations (36)-(37), normalized values of alternatives versus all criteria can be obtained as shown in Table 27.

Step 10. Obtain final evaluation

Finally, the additive weighted evaluation of each alternative can be obtained by Equation (38) as shown in Table 28. From Table 28, the ranking order of the five locations is A₄ > A₁ > A₅ > A₂ > A₃. The location A₄ is selected with priority because it has the highest evaluation with 0.92034.

6.1 Comparison with best-worst method

6.1.1 Best-worst method

The key elements in MCDM involve two main aspects, including the evaluation of criteria importance and the ranking of alternative options [78]. When assessing criteria importance, numerous subjective MCDM techniques are available such as the Analytic Hierarchy Process (AHP), Analytic Network Process (ANP), and Best-Worst Method (BWM) [78]. Among these, AHP and BWM stand out as prominent and robust techniques for deriving criteria weights through pairwise comparisons. Despite their widespread applications, these methods are not without limitations. For instance, the AHP method imposes a workload due to the exhaustive nature of pairwise comparisons [79]. Addressing this limitation, Rezaei [80] introduced the BWM in 2015 and further refined it in 2016 [81]. BWM presents some advantages compared to alternative MCDM methods because it requires less pairwise comparison data than a full pairwise comparison matrix and it tends to yield more consistent results compared to other MCDM methods employing a full pairwise comparison matrix [82]. However, a drawback of BWM is the absence of a consistency threshold to validate result reliability, as highlighted by Liang et al. [83]. Consequently, decision-makers face the challenge of determining when to revise their judgments and a comparative analysis of the two methods becomes needed. Therefore, in this work, the proposed method is compared with the best-worst method.

The above numerical example is used to compare with the Best–Worst method [80] to show the effectiveness of proposed method, in which BWM is used to derive the criteria weights. Decision makers can determine a pairwise comparison vector for the best (e.g., most desirable and most important) criterion in the structure by using Equation (39), as shown in Table 29, and that for the worst (e.g., least desirable and least important) criterion by using Equation (40), as shown in Table 30. $E_{B} = (e_{B 1}, \dots, e_{Bi}, . . ., e_{Bn})$ (39) $E_{W} = (e_{1 W}, \dots, e_{iW}, . . ., e_{nW})$ (40) where e_Bi indicates the preference of the best criterion B over criterion i, e_Wi indicates the preference of the criterion i over the worst criterion W.

The BWM model is shown as Equation (41) [80]: $\begin{matrix} | \frac{w_{B}}{w_{i}} - e_{Bi} | ⩽ ξ, for all i \\ | \frac{w_{i}}{w_{W}} - e_{iW} | ⩽ ξ, for all j \\ \sum_{i} w_{i} = 1, w_{i} ⩾ 0 for all j \end{matrix}$ (41)

And the consistency ration of the matrix is obtained by the Equation (42). $CR = \frac{ξ^{*}}{CI}$ (42) where ξ^* is the optimal solution of Equation (41). Consistency index (CI) can be seen as in Table 31.

Next, the weights can be obtained using Equations (41) and (42), as shown in Table 32. The final weights can be obtained using Equation (33), as also shown in Table 32. The final ranking can be obtained using Equations (34)–(38), as shown in Table 28.

6.1.2 Comparison results

Both BWM and AHP rely on a decision maker’s evaluations of the relative priorities of the decision-making elements, which are represented in a comprehensive pairwise comparison matrix [83], however, BWM requires fewer comparisons compared to AHP [84]. BWM and AHP stand out as the most popular methods for determining criteria weights. Therefore, conducting a comparison between these two methods becomes essential to validate the results.

The results of the proposed method can be validated using BWM, because both methods produce the same ranking order for the five alternatives. Although BWM uses less comparison data than the other MCDM methods, it lacks consistency threshold for verifying the reliability of the results [83]. With no consistency threshold, the decision makers are left with addressing the major problem of determining when their judgements should be revised, along with the consideration of the number of criteria and evaluation scale, making the situation more complicated [83]. In case of inconsistency, a matrix review process must be followed, which requires decision makers to provide new comparison judgments [49]. Furthermore, the original BWM utilizes a nonlinear model that can lead to multiple optimal weights, implying that the weight of each criterion is presented as an interval [85]. Whereas AHP provides the consistency threshold (i.e. 10%), which means that the matrix is consistent if the consistency ratio is smaller than this threshold. Moreover, AHP method provides a single optimal result for the criteria weights, making it straightforward for calculation in ranking alternatives. With all the reasons above, this study utilized AHP method to produce the criteria weight.

7 Theoretical and practical implications

7.1 Theoretical implications

This study provideda Fuzzy AHP-Based MCDM method for selecting sustainable manufacturing facility location. In this study, a novel total distance ranking approach is developed to defuzzify the fuzzy numbers. A new topic of sustainable manufacturing facilitylocation selection is put forward. This paper firstcontributed to establishing the comprehensive list of criteria for sustainable manufacturing location selection which considers fourth-level structure of criteria. Numerous quantitative and qualitative criteria with different weights in a multilevel structure must be comprehensively considered and aggregated to assist the company in decision-making. These criteria are classified as benefit and cost criteria. Besides that,this study proposed a total distance approach to fuzzy AHP-based MCDM method to resolve this problem. Formulas for the proposed total distance ranking approach have been clearly developed for ranking fuzzy numbers. Some properties have been investigated to derive formulas for trapezoidal and triangular fuzzy numbers. A comparison with some relevant ranking methods including Yao and Wu [14], Wang et al. [15], Deng at al. [16], Asady & Zendehnam [17] has been made to reveal the advantage of the proposed approach. The proposed ranking approach was employed to defuzzify the fuzzy numbers used in fuzzy AHP and linguistic values under qualitative criteria to obtain the criteria weights under multi-level structure and crisp values under qualitative criteria, respectively. The final criteria and their corresponding weights were further used to establish an MCDM model to come up with the final evaluation values. The ranking of alternative locations was then obtained by the final evaluation values through simple additive weighting method. A numerical example with a four-level structure for evaluating and selecting sustainable manufacturing facility locations was utilized to illustrate the proposed method feasibility. Finally, the proposed method was compared with BWM to validate its effectiveness in yielding the same ranking order of alternatives.

7.2 Practical implications

In addition to its theoretical implications, this study also offers valuable contributions to the practical aspects of sustainable manufacturing facility selection process. Firstly, the identified evaluation criteria provide decision-makers with the opportunity to target appropriate locations in terms of sustainability dimensions, encompassing environment, society, and economic aspects. This enables firms to focus on improving the most crucial aspects from a sustainable standpoint. Secondly, the fuzzy AHP-based MCDM method can be conveniently employed using tools like Excel or other applications, facilitating more accurate and precise selection results. This approach can prove beneficial for smart PSS developers and other professionals involved in sustainable decision-making. Thirdly, the model developed in this study holds potential for application in various other fuzzy MCDM problems in future endeavors, such as green supplier selection, sustainable facility layout selection, green marketing tools selection, and more. This versatility enhances its value in different areas of decision-making and sustainable practices. This study offers a comprehensive set of criteria for the selection process of sustainable manufacturing facilities. However, policymakers have the flexibility to adjust the number of criteria to align with the firm’s size or the company’s goals. Furthermore, this proposed method can be adapted for other MCDM issues by modifying the list of criteria and alternatives.

8 Conclusion and discussion

Selecting a facility location with a focus on sustainability, encompassing economic, environmental, and social aspects, is a crucial aspect of a manufacturing company’s strategy for long-term success in today’s competitive environment of sustainable development. This complex decision-making process involves considering numerous quantitative and qualitative criteria, each carrying different weights within a multilevel structure. To address this complex problem, this study introduces a total distance approach to the fuzzy AHP-based MCDM method. The proposed total distance ranking approach offers a clear formulation for ranking fuzzy numbers, with specific investigation into properties concerning trapezoidal and triangular fuzzy numbers. A comparison with relevant ranking methods highlights the advantages of the proposed approach. By employing this ranking approach, fuzzy numbers used in fuzzy AHP and linguistic values under qualitative criteria are defuzzified to obtain criteria weights within the multi-level structure and crisp values under qualitative criteria, respectively.

These final criteria and their corresponding weights are then incorporated into an MCDM model to derive the final evaluation values. Subsequently, the alternative facility locations are ranked based on the final evaluation values using the simple additive weighting method. To demonstrate the feasibility of the proposed method, a numerical example featuring a four-level structure for evaluating and selecting sustainable manufacturing facility locations is presented.Furthermore, the effectiveness of the proposed method is validated by comparing it with the Best Worst Method (BWM) to ensure both methods yield the same ranking order of alternatives. Overall, this study showcases a comprehensive and effective approach for sustainable facility location selection, taking into account economic, environmental, and social aspects, and contributes to the advancement of decision-making practices in sustainable manufacturing.

The proposed method exhibits several strengths. Firstly,this study offers a comprehensive set of criteria for the selection process of sustainable manufacturing facilities. To address this complex problem, this study introduces a total distance approach to the fuzzy AHP-based MCDM methodwhich yields more accurate ranking results compared to other ranking methods such as Yao and Wu [14], Wang et al. [15], Deng et al. [16], and Asady & Zendehnam [17]. Additionally, it offers the convenience of easy implementation through tools like Excel or other applications. Moreover, its versatility allows it to be applied to address various other fuzzy MCDM problems in future applications. However, some problems require closer consideration. First, different ranking approach may lead to various ranking result. Second, weights may be different if a method other than fuzzy AHP is used. Third, the number of criteriacan be adjusted as per the requirement, and the change in ranking result may happen if different criteria and levels are used. Finally, the ranking result may change if, for example, different fuzzy numbers are used, different aggregation method is used, or different normalization formulae are used, etc.

Appendix

Table 4

Pairwise comparison matrix –Level 1

	Economic	Environment	Social
Economic	EI (1.02)	SIM (7.49)	I (5.19)
Environment	LSIM (0.13)	EI (1.02)	LFI (0.36)
Social	LI (0.19)	FI (2.74)	EI (1.02)

Table 5

Pairwise comparison matrix under economic –Level 2

	Cost	Market	Economic stability	Suppliers	Growth
Cost	EI (1.02)	FI (2.74)	ESI (3.65)	SI (1.99)	I (5.19)
Market	LFI (0.36)	EI (1.02)	FI (2.74)	LESI (0.27)	MI (6.65)
Economic stability	LESI (0.27)	LFI (0.36)	EI (1.02)	LMI (0.15)	FI (2.74)
Suppliers	LSI (0.50)	ESI (3.65)	MI (6.65)	EI (1.02)	SIM (7.49)
Growth	LI (0.19)	LMI (0.15)	LFI (0.36)	LSIM (0.13)	EI (1.02)

Table 6

Pairwise comparison matrix under environment –Level 2

	Ecosystem vitality	Environmental health	Environmental factors within production
Ecosystem vitality	EI (1.02)	ESI (3.65)	SIM (7.49)
Environmental health	LESI (0.27)	EI (1.02)	SI (1.99)
Environmental factors within production	LSIM (0.13)	LSI (0.50)	EI (1.02)

Table 7

Pairwise comparison matrix under Social –Level 2

	Governance	Education	Individual	Community
Governance	EI (1.02)	FI (2.74)	ESI (3.65)	SI (1.99)
Education	LFI (0.36)	EI (1.02)	I (5.19)	LSI (0.50)
Individual	LESI (0.27)	LI (0.19)	EI (1.02)	LI (0.19)
Community	LSI (0.50)	SI (1.99)	I (5.19)	EI (1.02)

Table 8

Pairwise comparison matrix under cost –Level 3

	Labor cost	Energy cost	Material cost	Facility cost	Logistic cost
Labor cost	EI (1.02)	FI (2.74)	LSI (0.50)	ESI (3.65)	FI (2.74)
Energy cost	LFI (0.36)	EI (1.02)	LI (0.19)	LSI (0.50)	LFI (0.36)
Material cost	SI (1.99)	I (5.19)	EI (1.02)	I (5.19)	MI (6.65)
Facility cost	LESI (0.27)	SI (1.99)	LI (0.19)	EI (1.02)	SI (1.99)
Logistic cost	LFI (0.36)	FI (2.74)	LMI (0.15)	LSI (0.50)	EI (1.02)

Table 9

Pairwise comparison matrix under market –Level 3

	International markets	Potential demand
International markets	EI (1.02)	LFI (0.36)
Potential demand	FI (2.74)	EI (1.02)

Table 10

Pairwise comparison matrix under economic stability –Level 3

	Favorable tax	Exchange rate fluctuations
Favorable tax	EI (1.02)	ESI (3.65)
Exchange rate fluctuations	LESI (0.27)	EI (1.02)

Table 11

Pairwise comparison matrix under suppliers –Level 3

	Proximity to key suppliers	Potential high quality suppliers
Proximity to key suppliers	EI (1.02)	ESI (3.65)
Potential high quality suppliers	LESI (0.27)	EI (1.02)

Table 12

Pairwise comparison matrix under growth –Level 3

	Industry growth	Competitiveness in the host region
Industry growth	EI (1.02)	LFI (0.36)
Competitiveness in the host region	FI (2.74)	EI (1.02)

Table 13

Pairwise comparison matrix under ecosystem vitality –Level 3

	Air pollution related	Water quality related	Biodiversity	Climate change
	to the ecosystem	to the ecosystem	protection	performance
Air pollution related to the ecosystem	EI (1.02)	FI (2.74)	SIM (7.49)	I (5.19)
Water quality related to the ecosystem	LFI (0.36)	EI (1.02)	I (5.19)	ESI (3.65)
Biodiversity protection	LSIM (0.13)	LI (0.19)	EI (1.02)	LSI (0.50)
Climate change performance	LI (0.19)	LESI (0.27)	SI (1.99)	EI (1.02)

Table 14

Pairwise comparison matrix under environmental health- Level 3

	Environmental burden	Air pollution	Water quality
	of disease	related to humans	related to humans
Environmental burden of disease	EI (1.02)	LMI (0.15)	LI (0.19)
Air pollution related to humans	MI (6.65)	EI (1.02)	FI (2.74)
Water quality related to humans	I (5.19)	LFI (0.36)	EI (1.02)

Table 15

Pairwise comparison matrix under environmental factors within production- Level 3

	Material	Energy	Renewable	Waste generation	Recycling of material,
	use	consumption	resources	and management	energy, and waste
Material use	EI (1.02)	ESI (3.65)	I (5.19)	MI (6.65)	SIM (7.49)
Energy consumption	LESI (0.27)	EI (1.02)	FI (2.74)	SI (1.99)	ESI (3.65)
Renewable resources	LI (0.19)	LFI (0.36)	EI (1.02)	LFI (0.36)	SI (1.99)
Waste generation and management	LMI (0.15)	LSI (0.50)	FI (2.74)	EI (1.02)	SI (1.99)
Recycling of material, energy, and waste	LSIM (0.13)	LESI (0.27)	LSI (0.50)	LSI (0.50)	EI (1.02)

Table 16

Pairwise comparison matrix under governance –Level 3

	Corruption	Political stability	Trade and tariff barriers
Corruption	EI (1.02)	LSIM (0.13)	LVSI (0.12)
Political stability	SIM (7.49)	EI (1.02)	SI (1.99)
Trade and tariff barriers	VSI (8.24)	LSI (0.50)	EI (1.02)

Table 17

Pairwise comparison matrix under education- Level 3

	Literacy	General education level
Literacy	EI (1.02)	LFI (0.36)
General education level	FI (2.74)	EI (1.02)

Table 18

Pairwise comparison matrix under individual- Level 3

	Civil liberties	Human rights
Civil liberties	EI (1.02)	LFI (0.36)
Human rights	FI (2.74)	EI (1.02)

Table 19

Pairwise comparison matrix under community- Level 3

	Equity	Safety	Cohesion	Local technology
Equity	EI (1.02)	LESI (0.27)	FI (2.74)	LESI (0.27)
Safety	ESI (3.65)	EI (1.02)	MI (6.65)	SI (1.99)
Cohesion	LFI (0.36)	LMI (0.15)	EI (1.02)	LSIM (0.13)
Local technology	ESI (3.65)	LSI (0.50)	SIM (7.49)	EI (1.02)

Table 20

Pairwise comparison matrix under Energy consumption –Level 4

	Annual energy consumption,	Intensity of energy use,
	total and by main user category	total and by economic activity
Annual energy consumption, total and by main user category	EI (1.02)	LSI (0.50)
Intensity of energy use, total and by economic activity	SI (1.99)	EI (1.02)

Table 21

Pairwise comparison matrix under Waste generation and management –Level 4

	Generation of hazardous waste	Waste treatment and disposal
Generation of hazardous waste	EI (1.02)	LFI (0.36)
Waste treatment and disposal	FI (2.74)	EI (1.02)

Table 22

Pairwise comparison matrix under education level –Level 4

	Gross intake ratio to last	Adult secondary (tertiary)	Net enrolment rate in
	grade of primary education	schooling attainment level	primary education
Gross intake ratio to last grade of primary education	EI (1.02)	LI (0.19)	LFI (0.36)
Adult secondary (tertiary) schooling attainment level	I (5.19)	EI (1.02)	ESI (3.65)
Net enrolment rate in primary education	FI (2.74)	LESI (0.27)	EI (1.02)

Table 23

Weights of criteria in the structure

First level	Second level	Third level		Fourth level
w₁ = 0.738	w₁₁ = 0.268	w₁₁₁ = 0.067	w₂₂₂ = 0.010	w₂₃₂₁ = 0.00051
w₂ = 0.082	w₁₂ = 0.124	w₁₁₂ = 0.018	w₂₂₃ = 0.005	w₂₃₂₂ = 0.00101
w₃ = 0.181	w₁₃ = 0.058	w₁₁₃ = 0.127	w₂₃₁ = 0.004	w₂₃₄₁ = 0.00026
	w₁₄ = 0.257	w₁₁₄ = 0.030	w₂₃₂ = 0.002	w₂₃₄₂ = 0.00071
	w₁₅ = 0.031	w₁₁₅ = 0.026	w₂₃₃ = 0.001	w₃₂₂₁ = 0.00286
	w₂₁ = 0.058	w₁₂₁ = 0.033	w₂₃₄ = 0.001	w₃₂₂₂ = 0.01802
	w₂₂ = 0.016	w₁₂₂ = 0.091	w₂₃₅ = 0.0004	w₃₂₂₃ = 0.00627
	w₂₃ = 0.008	w₁₃₁ = 0.046	w₃₁₁ = 0.005
	w₃₁ = 0.077	w₁₃₂ = 0.012	w₃₁₂ = 0.044
	w₃₂ = 0.037	w₁₄₁ = 0.202	w₃₁₃ = 0.029
	w₃₃ = 0.013	w₁₄₂ = 0.055	w₃₂₁ = 0.010
	w₃₄ = 0.054	w₁₅₁ = 0.008	w₃₂₂ = 0.027
		w₁₅₂ = 0.022	w₃₃₁ = 0.003
		w₂₁₁ = 0.032	w₃₃₂ = 0.009
		w₂₁₂ = 0.017	w₃₄₁ = 0.007
		w₂₁₃ = 0.003	w₃₄₂ = 0.025
		w₂₁₄ = 0.006	w₃₄₃ = 0.003
		w₂₂₁ = 0.001	w₃₄₄ = 0.019

Table 24

Average weights of criteria

Criteria	Weight	Criteria	Weight
C_f₁	0.067	C_f₂₁	0.004
C_f₂	0.018	C_f₂₂	0.001
C_f₃	0.127	C_f₂₃	0.000
C_f₄	0.030	C_f₂₄	0.005
C_f₅	0.026	C_f₂₅	0.044
C_f₆	0.033	C_f₂₆	0.029
C_f₇	0.091	C_f₂₇	0.010
C_f₈	0.046	C_f₂₈	0.003
C_f₉	0.012	C_f₂₉	0.009
C_f₁₀	0.202	C_f₃₀	0.007
C_f₁₁	0.055	C_f₃₁	0.025
C_f₁₂	0.008	C_f₃₂	0.003
C_f₁₃	0.022	C_f₃₃	0.019
C_f₁₄	0.032	C_f₃₄	0.001
C_f₁₅	0.017	C_f₃₅	0.001
C_f₁₆	0.003	C_f₃₆	0.000
C_f₁₇	0.006	C_f₃₇	0.001
C_f₁₈	0.001	C_f₃₈	0.003
C_f₁₉	0.010	C_f₃₉	0.018
C_f₂₀	0.005	C_f₄₀	0.006

Table 25

Linguistic values of alternatives versus qualitative criteria and averaged values

A1				A2				A3				A4				A5
	D ₁	D ₂	D ₃	Ave.	D ₁	D ₂	D ₃	Ave.	D ₁	D ₂	D ₃	Ave.	D ₁	D ₂	D ₃	Ave.	D ₁	D ₂	D ₃	Ave.
C_f6	F	VG	G	7.63	P	MP	P	2.24	P	SP	P	2.54	VG	VG	G	9.06	P	MP	SP	2.79
C_f7	SP	MG	MP	4.63	F	VG	G	7.63	MP	F	MP	3.56	G	G	VG	8.65	G	VG	F	7.63
C_f10	VG	G	F	7.63	MP	SP	F	3.86	MP	SG	P	3.79	F	VG	G	7.63	G	VG	F	7.63
C_f11	VG	G	F	7.63	G	VG	F	7.63	SP	MG	MP	4.63	F	VG	G	7.63	SP	MG	MP	4.63
C_f13	G	F	VG	7.63	SG	F	MP	4.86	G	VG	F	7.63	MG	G	F	6.97	F	VG	F	6.62
C_f14	P	MP	SP	2.79	F	SG	SG	6.16	G	MG	VG	8.40	G	G	MG	7.99	G	G	MG	7.99
C_f15	MP	F	SP	3.86	G	F	G	7.22	F	SG	MG	6.44	G	MG	F	6.97	MP	F	MP	3.56
C_f16	F	VG	SG	7.10	VP	P	P	1.67	F	MG	VG	7.38	SG	F	VG	7.10	MP	SG	P	3.79
C_f25	G	VG	F	7.63	G	MG	G	7.99	G	F	VG	7.63	G	G	VG	8.65	MP	SG	P	3.79
C_f26	G	VG	MG	8.40	SG	MG	G	7.46	F	MG	SG	6.44	G	SG	MG	7.46	F	VG	G	7.63
C_f28	P	SP	P	2.54	VG	G	F	7.63	G	F	VG	7.63	F	G	VG	7.63	MP	SG	P	3.79
C_f29	MP	F	MP	3.56	VG	MG	SG	7.87	VG	G	F	7.63	F	VG	F	6.62	SP	MG	MP	4.63
C_f30	F	SG	F	5.68	F	VG	VG	8.04	VG	F	G	7.63	VG	VG	G	9.06	MP	SG	P	3.79
C_f31	MP	SG	P	3.79	VG	G	MG	8.40	F	VG	VG	8.04	VG	F	VG	8.04	SP	MG	MP	4.63
C_f32	SP	MG	MP	4.63	F	G	F	6.21	VG	VG	G	9.06	F	VG	G	7.63	F	VG	F	6.62

Table 26

Ratings of alternatives versus quantitative criteria

A.	C_f₁	C_f₂	C_f₃	C_f₄	C_f₅	C_f₈	C_f₉	C_f₁₂	C_f₁₇	C_f₁₈	C_f₁₉	C_f₂₀	C_f₂₁
A ₁	200	0.09	10	4	3	10	2	6	0.5	17	85	4	7.1
A ₂	275	0.1	8	5	4	10	3	4	0.6	20	70	3	4.5
A ₃	225	0.2	7	5	3	8	2	5	0.3	18	60	3	5
A ₄	300	0.15	8	6	2	9	4	6	0.7	19	75	3.5	6.3
A ₅	250	0.08	9	4.5	4	9	4	4	0.8	21	75	2.7	8
A.	C_f₂₂	C_f₂₃	C_f₂₄	C_f₂₇	C_f₃₃	C_f₃₄	C_f₃₅	C_f₃₆	C_f₃₇	C_f₃₈	C_f₃₉	C_f₄₀
A ₁	5	4	30	90.8	63	6	50	3	50	80	70	100
A ₂	2	5	33	98.5	74	5	30	4	45	85	75	90
A ₃	3	6	29	80.4	80	6	35	5	55	83	70	98
A ₄	4	4	27	100	75	5	45	4	50	87	78	99
A ₅	3.5	4.5	28	95.2	81	5.5	37	4.5	40	90	75	94

Table 27

Normalized values of location versus quantitative and qualitative criteria

A.	r _pf₁	r _pf₂	r _pf₃	r _pf₄	r _pf₅	r _pf₆	r _pf₇	r _pf₈	r _pf₉	r _pf₁₀	r _pf₁₁	r _pf₁₂	r _pf₁₃	r _pf₁₄
A ₁	1.00	0.89	0.70	1.00	0.67	0.84	0.53	1.00	1.00	1.00	1.00	1.00	1.00	0.33
A ₂	0.73	0.80	0.88	0.80	0.50	0.25	0.88	1.00	0.67	0.51	1.00	0.67	0.64	0.73
A ₃	0.89	0.40	1.00	0.80	0.67	0.28	0.41	0.80	1.00	0.50	0.61	0.83	1.00	1.00
A ₄	0.67	0.53	0.88	0.67	1.00	1.00	1.00	0.90	0.50	1.00	1.00	1.00	0.91	0.95
A ₅	0.80	1.00	0.78	0.89	0.50	0.31	0.88	0.90	0.50	1.00	0.61	0.67	0.87	0.95
A.	r_pf 15	r_pf 16	r_pf 17	r_pf 18	r_pf 19	r_pf 20	r_pf 21	r_pf 22	r_pf 23	r_pf 24	r_pf 25	r_pf 26	r_pf 27	r_pf 28
A ₁	0.53	0.96	0.60	1.00	0.71	0.68	0.63	1.00	0.67	0.90	0.88	1.00	0.91	0.33
A ₂	1.00	0.23	0.50	0.85	0.86	0.90	1.00	0.40	0.83	0.82	0.92	0.89	0.99	1.00
A ₃	0.89	1.00	1.00	0.94	1.00	0.90	0.90	0.60	1.00	0.93	0.88	0.77	0.80	1.00
A ₄	0.97	0.96	0.43	0.89	0.80	0.77	0.71	0.80	0.67	1.00	1.00	0.89	1.00	1.00
A ₅	0.49	0.51	0.38	0.81	0.80	1.00	0.56	0.70	0.75	0.96	0.44	0.91	0.95	0.50
A.	r_pf 29	r_pf 30	r_pf 31	r_pf 32	r_pf 33	r_pf 34	r_pf 35	r_pf 36	r_pf 37	r_pf 38	r_pf 39	r_pf 40
A ₁	0.45	0.63	0.45	0.51	0.78	0.83	1.00	1.00	0.91	0.89	0.90	1.00
A ₂	1.00	0.89	1.00	0.69	0.91	1.00	0.60	0.75	0.82	0.94	0.96	0.90
A ₃	0.97	0.84	0.96	1.00	0.99	0.83	0.70	0.60	1.00	0.92	0.90	0.98
A ₄	0.84	1.00	0.96	0.84	0.93	1.00	0.90	0.75	0.91	0.97	1.00	0.99
A ₅	0.59	0.42	0.55	0.73	1.00	0.91	0.74	0.67	0.73	1.00	0.96	0.94

Table 28

The evaluation values from proposed method

Location	Proposed method	BWM
	Final score	Final score
A ₁	0.84197	0.82538
A ₂	0.75764	0.77379
A ₃	0.73831	0.74494
A ₄	0.92034	0.88152
A ₅	0.82235	0.79267

Table 29

Pairwise comparison vector for the best criterion in the structure

Level 1	Economic	Environment	Social
Economic (B)	1.00	7.25	5
Level 2: Economic	Cost	Market	Economic stability	Suppliers	Growth
Cost (B)	1	2.75	3.88	2	5
Level 2: Environment	Ecosystem vitality	Environmental health	Environmental factors within production
Environmental health (B)	3.88	1	2
Level 2: Social	Governance	Education	Individual	Community
Governance (B)	1	2.75	3.88	2
Level 3: Ecosystem vitality	Air pollution related to the ecosystem	Water quality related to the ecosystem	Biodiversity protection	Climate change performance
Air pollution related to the ecosystem (B)	1	2.75	7.25	5
Level 3: Environmental health	Environmental burden of disease	Air pollution related to humans	Water quality related to humans
Air pollution related to humans (B)	6.13	1	2.75
Level 3: Environmental factors	Material use	Energy consumption	Renewable resources	Waste generation and management	Recycling of material, energy, and waste
Material use (B)	1	3.875	5	6.13	7.25
Level 3: Governance	Corruption	Political stability	Trade and tariff barriers
Political stability (B)	3.88	1	2
Level 3: Education	Literacy	General education level	Level 3: Individual	Civil liberties	Human rights
General education level (B)	2.75	1	Human rights	2.75	1
Level 3: Community	Equity	Safety	Cohesion	Local technology
Local technology	3.875	0.5	7.25	1
Level 3: Cost	Labor cost	Energy cost	Material cost	Facility cost	Logistic cost
Material cost (B)	2	5	1	5	6.13
Level 3: Market	International markets	Potential demand
International markets (B)	2.75	1
Level 3: Economic stability	Favorable tax	Exchange rate fluctuations
Favorable tax (B)	1.00	3.88
Level 3: Suppliers	Proximity to key suppliers	Potential high quality suppliers
Proximity to key suppliers (B)	1.00	3.88
Level 3: Growth	Industry growth	Competitiveness in the host region
Competitiveness in the host region (B)	2.75	1
Level 4: Energy consumption	Annual energy consumption, total and by main user category	Intensity of energy use, total and by economic activity
Intensity of energy use, total and by economic activity (B)	2.00	1
Level 4: Wase Generation	Generation of hazardous waste	Waste treatment and disposal
Waste treatment and disposal (B)	2.75	1
Level 4: Education level	Gross intake ratio to last grade of primary education	Adult secondary schooling attainment level	Net enrolment rate in primary education
Adult secondary (tertiary) schooling attainment level (B)	5	1	3.88

Table 30

Pairwise comparison matrix for the worst criterion

Level 1	Environment (W)
Economic	7.25
Environment	1.00
Social	2.75
Level 2: Economic	Growth (W)
Cost	5.00
Market	6.13
Economic stability	2.75
Suppliers	7.25
Growth	1
Level 2: Social	Individual (W)
Governance	3.875
Education	5
Individual	1
Level 2: Environment	Ecosystem vitality (W)
Ecosystem vitality	1
Environmental health	3.88
Environmental factors within production	5
Level 3: Environmental health	Water quality related to humans (W)
Environmental burden of disease	0.2
Air pollution related to humans	2.75
Water quality related to humans	1
Level 3: Environmental factors	Recycling of material, energy, and waste (W)
Material use	7.25
Energy consumption	3.88
Renewable resources	2
Waste generation and management	2
Recycling of material, energy, and waste	1
Level 3: Cost	Energy cost (W)
Labor cost	2.75
Energy cost	1.00
Material cost	5
Facility cost	2
Logistic cost	2.75
Level 3: Governance	Corruption (W)
Corruption	1
Political stability	3.88
Trade and tariff barriers	5
Level 3: Education	Literacy (W)
Literacy	1
General education level	2.75
Level 3: Individual	Civil liberties (W)
Civil liberties	1
Human rights	2.75
Level 3: Community	Cohesion (W)
Equity	2.75
Safety	6.13
Cohesion	1.00
Local technology	7.25
Level 3: Market	Potential demand (W)
International markets	1
Potential demand	2.75
Level 3: Economic stability	Exchange rate fluctuations (W)
Favorable tax	3.875
Exchange rate fluctuations	1
Level 3: Suppliers	Potential high quality suppliers (W)
Proximity to key suppliers	3.875
Potential high quality suppliers	1
Level 3: Growth	Industry growth (W)
Industry growth	1
Competitiveness in the host region	2.75
Level 4: Wase Generation	Generation of hazardous waste (W)
Generation of hazardous waste	1
Waste treatment and disposal	2.75
Level 4: Energy consumption	Annual energy consumption, total and by main user category (W)
Annual energy consumption, total and by main user category	1
Intensity of energy use, total and by economic activity	2.00
Level 4: Education level	Gross intake ratio to last grade of primary education (W)
Gross intake ratio to last grade of primary education	1
Adult secondary (tertiary) schooling attainment level	5
Net enrolment rate in primary education	2.75

Table 31

Consistency index (CI)

a _ BW	1	2	3	4	5	6	7	8	9
CI	0.00	0.44	1.00	1.63	2.30	3.00	3.73	4.47	5.23

Table 32

Final weights by Best-Worst Method

Level 1	Local Weight	Level 2	Local Weight	Level 3	Local Weight	Level 4	Local Weight	Final weight
W ₁	0.744	w ₁₁	0.380	w ₁₁₁	0.277			0.078
				w ₁₁₂	0.070			0.020
				w ₁₁₃	0.452			0.128
				w ₁₁₄	0.111			0.031
				w ₁₁₅	0.090			0.026
		w ₁₂	0.184	w ₁₂₁	0.267			0.037
				w ₁₂₂	0.733			0.101
		w ₁₃	0.131	w ₁₃₁	0.795			0.077
				w ₁₃₂	0.205			0.020
		w ₁₄	0.254	w ₁₄₁	0.795			0.150
				w ₁₄₂	0.205			0.039
		w ₁₅	0.051	w ₁₅₁	0.267			0.010
				w ₁₅₂	0.733			0.028
W ₂	0.091	w ₂₁	0.101	w ₂₁₁	0.565			0.005
				w ₂₁₂	0.238			0.002
				w ₂₁₃	0.066			0.001
				w ₂₁₄	0.131			0.001
		w ₂₂	0.547	w ₂₂₁	0.098			0.005
				w ₂₂₂	0.649			0.032
				w ₂₂₃	0.253			0.013
		w ₂₃	0.351	w ₂₃₁	0.545			0.017
				w ₂₃₂	0.162	w ₂₃₂₁	0.333	0.002
						w ₂₃₂₂	0.667	0.003
				w ₂₃₃	0.126			0.004
				w ₂₃₄	0.103	w ₂₃₄₁	0.267	0.001
						w ₂₃₄₂	0.733	0.002
				w ₂₃₅	0.064			0.002
W ₃	0.166	w ₃₁	0.429	w ₃₁₁	0.101			0.007
				w ₃₁₂	0.547			0.039
		w ₃₁₃	0.351	0.025
w ₃₂	0.210	w ₃₂₁	0.267	0.009
				w ₃₂₂	0.733	w ₃₂₂₁	0.114	0.003
						w ₃₂₂₂	0.681	0.017
						w ₃₂₂₃	0.204	0.005
		w ₃₃	0.072	w ₃₃₁	0.267			0.003
				w ₃₃₂	0.733			0.009
		w ₃₄	0.289	w ₃₄₁	0.115			0.006
				w ₃₄₂	0.480			0.023
				w ₃₄₃	0.062			0.003
				w ₃₄₄	0.343			0.016

Footnotes

Acknowledgments

The authors would like to sincerely thank the three anonymous reviewers, Editor and Editor-in-Chief for providing helpful comments and suggestions. Their insights and comments led to a better presentation of the ideas expressed in this paper. This paper was supported in part by the National Science and Technology Council, Taiwan, under Grant NSTC 112-2410-H-218-005-.

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