A Novel Decision-Making Framework for Sustainable Supplier Selection Considering Interaction among Criteria with Heterogeneous Information

Abstract

Sustainable supplier selection has become a strategic activity to enhance the competitiveness of sustainable supply chain management. Research on sustainable supplier selection is considering increasingly more practical factors, such as the uncertainty of decision context and the fuzzy recognition of experts. Evaluation values on different criteria with different characteristic should be represented in their suitable information types to reflect the characteristic accurately and represent experts’ judgments entirely. Moreover, it is difficult, or costly, to build a decision criteria set in which all criteria are independent to each other because of the interaction of technical, economic, environmental and social factors. Therefore, the aim of this paper is to propose a novel decision-making framework for sustainable supplier selection which considers the interaction among criteria with heterogeneous decision information. The proposed framework can not only allow the experts to express their judgments completely, but also improve the efficiency of decision-making. First, a normalized dominance decision matrix based on normalized closeness is built with the heterogeneous decision matrix. Then, a defined discrete Choquet integral multi-criteria distance measure is used to compute the comprehensive associated closeness and rank the alternative sustainable suppliers. This framework provides a new way to handle the interaction among criteria for sustainable supplier selection from the perspective of multi-criteria distance measure, and a novel methodology to solve the problems that the evaluation values cannot be aggregated directly. Finally, an example is given to illustrate the proposed framework for sustainable supplier selection with a comparison analysis.

1. Introduction

Nowadays, with the development of technology, economy, environment and society, sustainable supply chain management (SSCM) has become a key issue for corporates to keep their sustainable competitive advantage [1]. In addition to technical and economic factors, decision makers have been paying more attention to environmental and social benefits, which leads to sustainable development. The term sustainability, which increasingly refers to an integration of economic, environmental and social benefits, has appeared often in the literature of business fields such as management and operations recently. Carter and Rogers introduced the concept of sustainability to the field of supply chain management and demonstrate the relationships among environmental, social, and economic performance within a supply chain management context [2]. An effective way to achieve sustainable business development is to build a sustainable supply chain that simultaneously considers technical, economic, social and environmental factors in the process of supply chain management. It combines broader environmental sustainability, economic sustainability and social sustainability. In other words, the sustainability comes from a balance among the goals of technical, economic, environmental and social benefits. As the complexity and uncertainty of SSCM issues have increased, researchers have been paying attention to the uncertainty of SSCM. Martino et al. focused on the particular field of the fashion retail industry and prioritized the list of identified risk factors by adopting the Analytic Network Process approach [3]. Fera et al. defined a system dynamics model to assess the competitiveness coming from the positioning of the order in different supply chain locations, and a Taguchi analysis was implemented to create a decision map for identifying possible strategic decisions under different scenarios [4]. Megahed and Goetschalckx developed a two-stage stochastic programming model for the comprehensive tactical planning of supply chains under demand and supply uncertainty [5]. The sustainability of main suppliers is a critical factor determining the competitiveness of sustainable supply chain. Organizational sustainability, at a broader level, consists of three components: the environment, society and economic performance. To choose the most suitable sustainable supplier means to choose the supplier that is able to simultaneously consider and balance economic, environmental and social goals. It can optimize the production costs, reduce the time to market and enhance the competitiveness of the product. To choose the most suitable sustainable supplier has become a strategic activity to enhance the competitiveness of SSCM. In most cases, decision makers must evaluate suppliers from different perspectives to meet the goals of decision makers and other stakeholders. Xiao et al. argued that sustainability and other business aims are not always compatible, particularly in an emerging market context [6]. Sustainable supplier selection is a particular and complex multi-criteria decision-making (MCDM) process. Recently, several influential researches on methods for sustainable supplier selection have been achieved [7,8,9,10,11,12,13,14,15,16,17,18]. Sometimes, due to the characteristics of the decision criteria and other uncertainties, evaluation values on some criteria of the particular supplier selection issues depend on the expert’s professional judgments based on certain knowledge background, and are given in the form of fuzzy information. This type of supplier selection problem is typical fuzzy MCDM problem. Büyüközkan and Çifçi proposed a novel fuzzy multi-criteria decision framework for sustainable supplier selection with incomplete information [19]. Büyüközkan presented an integrated fuzzy multi-criteria group decision making approach for green supplier evaluation [20]. Govindan et al. proposed a fuzzy multi criteria approach for measuring sustainability performance of a supplier based on triple bottom line approach [21]. Orji and Wei integrated the fuzzy-logic and systems dynamics innovatively in sustainable supplier selection [22]. A review of models supporting sustainable supplier selection, monitoring and development was proposed by Zimmer et al. [23]. Quan et al. proposed a hybrid MCDM approach for large group green supplier selection with uncertain linguistic information [24]. A comparison of fuzzy DEA(Data Envelopment Analysis) and fuzzy TOPSIS(Technique for Order Preference by Similarity to an Ideal Solution) in sustainable supplier selection was carried out by Rashidi and Cullinane [25]. Yu et al. put forward a group decision making sustainable supplier selection approach using extended TOPSIS under interval-valued Pythagorean fuzzy environment [26]. Liu et al. built an integrated MCDM model for sustainable supplier selection under interval-valued intuitionistic uncertain linguistic environment [27]. Dos Santos et al. proposed an entropy-TOPSIS-F method for performance evaluation of green suppliers [28]. Xu et al. put forward a sustainable supplier selection method based on AHPSort II in interval type-2 fuzzy environment [29]. Memari et al. proposed a multi-criteria intuitionistic fuzzy TOPSIS method for sustainable supplier selection [30].

As can be seen from the literature, there is a trend in the research on methods for sustainable supplier selection, that is, researchers are taking consideration into more decision-making practical factors, such as the uncertainty of decision-making context and the ambiguity of the evaluation values resulted from the experts’ fuzzy cognition. More and more integrated decision methods combining classical decision techniques, fuzzy set theories and other multi-objective optimization methods have been proposed to improve the decision quality for sustainable supplier selection.

Although sustainable supplier selection methods considering different types of fuzzy or uncertain information have been proposed, most of the methods are based on the settings for which the evaluation values in decision matrix are represented by only one specific type. This means that for one decision matrix, the evaluation values on each criteria are all given in the type of crisp number, or one type of fuzzy number. With the economic and social development, the complexity and uncertainty of the business environment are increasing. Therefore, it is very important to handle the uncertainty appropriately to keep sustainability. Bussee et al. examined the information processing problem in SSCM, and pointed out that the sustainability-related uncertainty originating from the supply chain must be managed appropriately by the firm [31]. For the practice of sustainable supplier selection, in order to cope with the high complexity and uncertainty in these issues, the evaluation values on different criteria with different decision characteristics should be represented in the types of information which are more suitable to reflect the decision features and more in line with the expert’s preference. This is more reasonable and flexible especially due to the limited time and decision information. Furthermore, it is consistent with the point of Carter and Rogers: firms need to adopt even longer-term and more flexible supply chain solutions to ensure their long-term viability [2]. This decision mechanism can not only allow the experts to express their judgments completely in the types of information they prefer according to the characteristics of decision criteria, but also shorten the decision time and improve the efficiency of the decision-making. In order to improve the flexibility and rationality of such decisions for sustainable supplier selection, the main research points in this paper consider the cases where the evaluation values on different criteria with different features are represented by heterogeneous information.

In addition, due to the characteristics of sustainable supply management, the impacts of technical, economic, environmental and social factors on the competitiveness of the sustainable supply chain are not independent to each other, but are of certain internal relationships. At the intersection of economic, environmental and social performance, organizations can carry out activities that not only affect the environment and society positively, but which also result in long-term economic benefits. For example, one company that emphasizes environmental benefit will be able to gain better social identity, thus leading to economic development. The same is true for the decision criteria of sustainable supplier selection. On one hand, it is difficult or costly for sustainable supplier selection to build decision criteria set of which all criteria are independent to each other, because there is a great possibility of interaction among the technical, economic, environmental and social decision factors. Suppliers undertaking environmental and social programs can reduce costs while also improving corporate reputation. On the other hand, assessment criteria to be built for sustainability are not necessarily independent to each other completely. Unlike traditional supplier selection, decision methods that assumed that all decision criteria are completely independent, decision makers should consider the interaction among criteria to develop a reasonable and flexible sustainable supplier selection framework to address the features of sustainable supplier selection more appropriately.

Therefore, this paper proposes a novel MCDM framework for sustainable supplier selection considering the interaction among criteria with heterogeneous information, which can not only allow the experts to express their judgments entirely, but also shorten the decision time and improve the efficiency of decision-making. This paper expands the concept of sustainable supplier selection, which is based on a more practical and flexible basis and facilitates the implementation of managerial decision-making practices. As for this decision framework for sustainable supplier selection, the key is how to properly handle the interaction among criteria in the decision matrix with heterogeneous decision information to improve the decision quality for sustainable supplier selection, and to achieve sustainable development.

In this paper, the proposal of the MCDM framework for sustainable supplier selection considering the interaction among criteria with heterogeneous decision information is motivated and initiated by the follows:

• There is a trend of the research on sustainable supplier selection, that is, researchers are considering more decision-making practical factors, such as the uncertainty of decision-making context and the fuzziness of the evaluation values on decision criteria resulting from the ambiguity of expert’s cognition.
• For sustainable supplier selection, in order to cope with the high complexity and uncertainty, the evaluation values on different criteria with different decision characteristics should be represented in the types of information which are more suitable for the decision features and more in line with the expert’s preference. It is more reasonable and flexible especially when decision-making time and decision information are limited.
• It is difficult or costly for sustainable supplier selection to build decision criteria set of which all criteria are independent to each other. Interaction among the technical, economic and social decision factors is of great possibility. For example, one company that emphasizes environmental benefit will be able to gain better social identity.

The remainder of this paper is organized as follows. In Section 2, the normalized distance measures and priority comparison for several widely employed information sets are introduced, research on multi-criteria distance measure is reviewed, and a discrete Choquet integral multi-criteria distance measure is defined, which are used as basics for the proposed MCDM framework for sustainable supplier selection with heterogeneous information. A novel MCDM framework for sustainable supplier selection considering the interaction among criteria with heterogeneous information is designed in Section 3. In Section 4, an example is given to illustrate the proposed framework for sustainable supplier selection with comparison analysis. Finally, conclusions are drawn in Section 5.

2. Basics of the Methodology for Sustainable Supplier Selection with Particular Features

Considering the characteristics of heterogeneous information-based sustainable supplier selection, in order to improve the flexibility and decision quality of such decisions, we deal with such decision characteristics from the perspective of multi-criteria distance measurement. In this section, we briefly introduce the normalized distance measures and priority comparisons for several widely employed information sets; moreover, research on multi-criteria distance measure is introduced and discrete Choquet integral multi-criteria distance measure is defined, which are used as basics for the proposed MCDM framework for sustainable supplier selection with heterogeneous information and other decision features.

2.1. Normalized Distance Measures, Priority Comparisons for Several Information Sets

Here, we introduce and develop several normalized distance measures and superior comparison approaches for several types of information set used in the proposed MCDM framework for sustainable supplier selection with heterogeneous information.

(1) Fuzzy sets

For two normalized fuzzy sets A, B on $X=\{x_1,x_2,\dots ,x_n\}$, various distance measures have been proposed. The ones most widely employed in practice are:

The Hamming distance:

$$d_h(A,B)={\displaystyle {\sum }_{i=1}^n|u_A(x_i)-u_B(x_i)|}$$

(1)

The normalized Hamming distance:

$$d_{nh}(A,B)=\frac{1}{n}{\displaystyle {\sum }_{i=1}^n|u_A(x_i)-u_B(x_i)|}$$

(2)

The Euclidean distance:

$$d_e(A,B)={\left({\displaystyle {\sum }_{i=1}^n|u_A(x_i)-u_B(x_i){|}^2}\right)}^{1/2}$$

(3)

The normalized Euclidean distance:

$$d_{ne}(A,B)=\frac{1}{n}{\left({\displaystyle {\sum }_{i=1}^n|u_A(x_i)-u_B(x_i){|}^2}\right)}^{1/2}$$

(4)

where $u_A(x_i)$ and $u_B(x_i)$ are the membership functions of A and B, respectively, with the condition that $0\le u_A(x_i),u_B(x_i)\le 1$, for $x_i\in X,i=1,2,\dots ,n$.

(2) Linguistic terms

In the process of uncertain decision making, it is suitable and straightforward for experts to express their opinions by using linguistic information that is close to human’s cognitive processes sometimes. A common approach to model linguistic information is the fuzzy linguistic approach proposed by Zadeh [32] in 1975, which represents qualitative information as linguistic variables. Although it is less precise than a number, the linguistic variable, defined as “a variable whose values are not numbers but words or sentences in a natural or artificial language,” enhances the flexibility and reliability of decision models and provides good results in different fields.

Xu [33] introduced a subscript-symmetric linguistic evaluation scale, which can be defined as $S=\{S_{\alpha }|-\tau ,\dots ,-1,0,1,\dots ,\tau \}$, where the mid linguistic label s₀ represents an assessment of ‘‘indifference’’, and the rest of them are placed symmetrically around it.

S satisfies the following conditions:

(1)

If a > b, then s_a > s_b;

(2)

The negation operator is defined as: $neg(s_{\alpha })=s_{-\alpha }$.

Distance measure between two linguistic terms is given by Xu [34], which is as follows:

Let $s_{\alpha },s_{\beta }$ be two linguistic terms in $S=\{S_{\alpha }|-\tau ,\dots ,-1,0,1,\dots ,\tau \}$, then the distance measure between $s_{\alpha },s_{\beta }$ is defined as follows:

$$d(s_{\alpha },s_{\beta })=\frac{|\alpha -\beta |}{2\tau +1}$$

(5)

(3) Interval fuzzy numbers

Let X = {x₁, x₂,…,x_n} be the universal set, I[0, 1] be the set of all closed subintervals of the interval [0, 1]. An interval-valued fuzzy set (IVFS) in X, denoted by $A=\{\langle x,A(x)\rangle x\in X\}$, is defined as mapping:

A: X→I([0, 1])

$x\to A(x)=[\underset{\_}{A}(x_i),\overline{A}(x_i)]\in I([0,1])$.

Several formulas for distance measure between IVFSs have been proposed [35,36]. Here, two of them used in this paper are briefly described as follows:

The generalized distance:

$$D_g^{IVFS}={\left(\frac{1}{2n}{\displaystyle \sum _{i=1}^n[|\underset{\_}{A}(x_i)-\underset{\_}{B}(x_i){|}^p+|\overline{A}(x_i)-\overline{B}(x_i){|}^p]}\right)}^{1/p}$$

(6)

The normalized Hamming distance:

$$D_{nh}^{IVFS}=\frac{1}{2n}{\displaystyle \sum _{i=1}^n[|\underset{\_}{A}(x_i)-\underset{\_}{B}(x_i){|}^{}+|\overline{A}(x_i)-\overline{B}(x_i){|}^{}]}$$

(7)

Definition 1.

Let A = (a^L, a^U) and B = (b^L, b^U) be two interval fuzzy numbers, the preference degree of A over B (A > B) is defined as follows:

$$P(A>B)=\frac{\max (0,a^U-b^L)-\max (0,a^L-b^U)}{(a^U-a^L)+(b^U-b^L)}$$

(8)

It is obviously that, P(A > B) + P(B > A) = 1; and P(A > B) = P(B > A) = 0.5, if A = B.

(4) Intuitionistic fuzzy numbers (IFNs)

Definition 2.

[37]. An intuitionistic fuzzy sets A on $X=\{x_1,x_2,\dots ,x_n\}$ is defined as:

$$A=\{(x_i,u_A(x_i),u_A(x_i))|x_i\in X\}$$

where $u_A(x_i),u_A(x_i)$ denote the membership degree and non-membership degree of $x_i$ to A in X with conditions that $0\le u_A(x_i)\le 1,0\le v_A(x_i)\le 1,u_A(x_i)+v_A(x_i)\le 1$, and ${\pi }_A(x_i)=1-u_A(x_i)-v_A(x_i)$ is the hesitation degree of $x_i$ to A in X.

Suppose two intuitionistic fuzzy sets A, B on X, the normalized Hamming distance is as follows [38]:

$$d_{}(A,B)=\frac{1}{n}{\displaystyle \sum _{i=1}^n(|u_A(x_i)-u_B(x_i)}|+|v_A(x_i)-v_B(x_i)|+|{\pi }_A(x_i)-{\pi }_B(x_i)|)$$

Specifically, let ${\alpha }_1=(u_{{\alpha }_1},v_{{\alpha }_1})$, ${\alpha }_2=(u_{{\alpha }_2},v_{{\alpha }_2})$ be two intuitionistic fuzzy numbers (IFNs):

$$d_{}({\alpha }_1,{\alpha }_2)=|{\alpha }_1-{\alpha }_2|=\frac{1}{2}(|u_{{\alpha }_1}-u_{{\alpha }_2}|+|v_{{\alpha }_1}-v_{{\alpha }_2}|+|{\pi }_{{\alpha }_1}-{\pi }_{{\alpha }_2}|)$$

(9)

Definition 3.

[39]. Let ${\alpha }_1=(u_{{\alpha }_1},v_{{\alpha }_1})$, ${\alpha }_2=(u_{{\alpha }_2},v_{{\alpha }_2})$ be two IFNs, $s({\alpha }_i)=u_{{\alpha }_i}-v_{{\alpha }_i}$ be the score of ${\alpha }_i$, $h({\alpha }_i)=u_{{\alpha }_i}+v_{{\alpha }_i}$ be the accuracy degree of ${\alpha }_i$, I = 1, 2, then,

• If $s({\alpha }_1)$>$s({\alpha }_2)$, then ${\alpha }_1$ is superior to ${\alpha }_2$, denoted as ${\alpha }_1$<${\alpha }_2$;
• If $s({\alpha }_1)$=$s({\alpha }_2)$, and,

  ·

  if $h({\alpha }_1)$=$h({\alpha }_2)$, then $s({\alpha }_1)$ is indifferent to $s({\alpha }_2)$, denoted as ${\alpha }_1~{\alpha }_2$;

  ·

  if $h({\alpha }_1)$>$h({\alpha }_2)$, then $s({\alpha }_1)$ is superior to $s({\alpha }_2)$, denoted as ${\alpha }_1>{\alpha }_2$.

(5) Hesitant fuzzy linguistic terms set (HFLTS)

For some fuzzy decision-making problems, the experts usually feel more comfortable to express their opinions by linguistic variables (or linguistic terms) because this approach is more realistic and it is close to the human’s cognitive processes. Furthermore, sometimes decision makers may hesitate between several linguistic terms, or they need a complex linguistic term set instead of a single linguistic term to represent their judgments. Then, the HFLTS was proposed by Rodriguez et al. [40], and some operators including distance measure on HFLTS have been studied. In the work of Liao et al. [41], a definition for the distance of two HFLTSs directly based on the hesitant fuzzy element index is given.

Definition 4.

[40] Let $X=\{x_1,x_2,\dots ,x_N\}$ be a reference set, and $S=\{S_{\alpha }|-\tau ,\dots ,-1,0,1,\dots ,\tau \}$ be a linguistic term set. A hesitant fuzzy linguistic term set (HFLTS) on X is mathematically shown in terms of:

$$H_S=\{\langle x_i,h_S(x_i)\rangle |x_i\in X\}$$

(10)

here, $h_S(x_i)$ is a set of some possible values in the linguistic term set S and can be characterized by:

$$h_S(x_i)=\{s_{{\delta }_l}(x_i)|s_{{\delta }_l}(x_i)\in S,l=1,2,\dots ,L\}$$

(11)

where L denotes the number of linguistic terms in $h_S(x_i)$.

(a) Comparison approach for HFLTSs

Suppose that $h_S=\{s_{{\delta }_l}|s_{{\delta }_l}\in S,l=1,2,\dots ,L\}$ is a HFLE, $u(h_S)$ and $v(h_S)$ is defined to compare two HFLEs, which are as follows:

$$u(h_S)=\frac{1}{L}{\displaystyle \sum _{i=1}^L{\delta }_l};v(h_S)=\sqrt{\frac{1}{(L){}_2}{\displaystyle \sum _{l\ne k=1}^L{({\delta }_l-{\delta }_k)}^2}};\mathrm{where}(L){}_2=\frac{L!}{(L-2)!2!}$$

For two HFLEs $h_S^1$ and $h_S^2$, the comparison rules are as follows:

• if $u(h_S^1)>u(h_S^2)$, then $h_S^1\succ h_S^2$;
• if $u(h_S^1)=u(h_S^2)$, then

  ·

  if $v(h_S^1)=v(h_S^2)$, then $h_S^1\approx h_S^2$;

  ·

  if $v(h_S^1)>v(h_S^2)$, then $h_S^1\prec h_S^2$.

(2) Distance measure forHFLTS based on 2-tuple linguistic approach

The 2-tuple linguistic information is the evaluation result expressed in 2-tuples (s_k, a_k), where s_k is the k_th element in the predefined language evaluation set S, it means the closest language phrase in the linguistic assessment information given or gotten to original language evaluation set; a_k is called symbolic transfer value, which represents the deviation between s_k and the evaluation result, and satisfy a_k ∈ [−0.5, 0.5].

Suppose that a real number β ∈ [0, Z], where Z is the number of elements in set S, β is represented as 2-tuple information by the following function Δ:

$$\begin{gathered} \mathsf{\Delta }:[0,Z]\to S\times [-0.5,0.5) \\ \mathsf{\Delta }(\beta )=\{\begin{array}{l}{s}_k,k=round(\beta )\hfill \\ a_k=\beta -k,a_k\in [-0.5,0.5)\hfill \end{array} \end{gathered}$$

(12)

where round( ) is a rounding function.

Given a 2-tuple linguistic value (s_k, a_k), there is an inverse function Δ⁻¹, which can convert (s_k, a_k) into a numerical value β∈ [0, Z]:

$$\begin{gathered} {\mathsf{\Delta }}^{-1}:S\times [-0.5,0.5)\to [0,Z] \\ {\mathsf{\Delta }}^{-1}(s_k,a_k)=k+a_k=\beta \end{gathered}$$

(13)

Suppose that $H_1={\cup }_{(s_{{\delta }_l^1},a_{{\delta }_l^1})\in H_1}\{(s_{{\delta }_l^1},a_{{\delta }_l^1})|l=1,\dots ,\#H_1\}$ and $H_2={\cup }_{(s_{{\delta }_l^2},a_{{\delta }_l^2})\in H_2}\{(s_{{\delta }_l^2},a_{{\delta }_l^2})|l=1,\dots ,\#H_2\}$ are two 2-tuple HFLTSs on $X=\{x_1,x_2,\dots ,x_n\}$ defined on linguistic term set S, in which the elements of the 2-tuple HFLSs are represented as 2-tuple linguistic representation (s, a). When all the symbolic transfer values $a$ in a 2-tuple HFLTS are equal to 0，the 2-tuple HFLTS become a common HFLTS. $\#H_1$ and $\#H_2$ are the numbers of HFLTSs H₁ and H₂, respectively. In this paper, we use the symbol #( ) as a function returning the number of the elements in a set. Let L = max{$\#H_1$, $\#H_2$}. The process of new distance measure method for 2-tuple HFLTSs based on 2-tuple linguistic approach is as follows:

(1)

If $\#H_1\ne \#H_2$, the shorter one would be extended to have the same number of linguistic terms as the other one by adding $|\#H_1-\#H_2|$ the lowest (resp. greatest) term in the shorter one, if the decision maker in certain decision making situation is pessimistic (resp. optimistic), which depends on the decision maker’s risk preference, and the decision making situations.

(2)

Suppose that $H_1{}^{\prime }={\cup }_{(s_{{\delta }_l^1},a_{{\delta }_l^1})\in H_1{}^{\prime }}\{(s_{{\delta }_l^1},a_{{\delta }_l^1})|l=1,\dots ,L\}$ and $H_2{}^{\prime }={\cup }_{(s_{{\delta }_l^2},a_{{\delta }_l^2})\in H_2{}^{\prime }}\{(s_{{\delta }_l^2},a_{{\delta }_l^2})|l=1,\dots ,L\}$ are the extended 2-tuple HFLTSs with same numbers of elements.

Then Hamming distance and the Euclidean distance between 2-tuple HFLTSs are computed as follows:

$$d_{hd}(H_1,H_2)=\frac{1}{L}{\displaystyle \sum _{l=1}^L\frac{|{\mathsf{\Delta }}^{-1}((s_{{\delta }_l^1},a_{{\delta }_l^1}))-{\mathsf{\Delta }}^{-1}((s_{{\delta }_l^2},a_{{\delta }_l^2}))|}{2\tau }}$$

(14)

$$d_{ed}(H_1,H_2)={\left(\frac{1}{L}{{\displaystyle \sum _{l=1}^L\left(\frac{|{\mathsf{\Delta }}^{-1}((s_{{\delta }_l^1},a_{{\delta }_l^1}))-{\mathsf{\Delta }}^{-1}((s_{{\delta }_l^2},a_{{\delta }_l^2}))|}{2\tau }\right)}}^2\right)}^{1/2}$$

(15)

In this definition, $d(H_1,H_2)$ satisfy the following properties:

(1)

$0\le d(H_1,H_2)\le 1$,

(2)

$d(H_1,H_2)=0$, if $H_1=H_2$,

(3)

$d(H_1,H_2)=d(H_2,H_1)$.

2.2. Discrete Choquet Integral Multi-Criteria Distance Measure

Distance and similarity measures are very useful techniques widely used to measure the deviation and closeness degrees between two elements, or two sets. Varity of distance measure approaches have been introduced over the last decades, such as the well-known Hamming distance, Euclidean distance, Hausdorff distance, which have been applied to the MCDM problem to measure the distance between alternatives to reference solution to get the optimal alternative. Some MCDM methodologies, from the perspective of distance measure have also been proposed and the most classical and popular ones, are TOPSIS [42], VIKOR(VlseKriterijumska Optimizacija I Kompromisno Resenje in Serbian, meaning multicriteria optimization and compromise solution) [43] and their fuzzy extensions [43,44], which have been successfully applied in practice. Furthermore, in order to enrich and develop the decision-making methods based on distance measure, some new and powerful distance measures containing more information are put forward for MCDM. Motivated by the idea of the ordered weighted averaging (OWA) operator, Xu and Chen [45] put forward the ordered weighted averaging distance (OWAD) measure, which emphasizes the importance of ordered position of the individual distances. Merigo [46] proposed the probabilistic weighted averaging distance (PWAD) operator unifying the probability and the weighted averaging (WA). Moreover, distance measures combining with different types of fuzzy numbers for MCDM have been also developed. Zeng and Su [47] developed an intuitionistic fuzzy ordered weighted distance (IFOWD) operator. Peng et al. [48] presented a generalized hesitant fuzzy synergetic weighted distance (GHFSWD) measure, and utilized the proposed GHFSWD measure to develop a MCDM method with hesitant fuzzy information. However, it is noted that the premise of using these distance measures is to suppose that the criteria are independent to each other, or the criteria are built to be independent to each other.

(1) The classical Hamming distance, Euclidean distance and Hausdorff distance

For two evaluations sets $A=\{a_1,a_2,\dots ,a_m\}$ and $B=\{b_1,b_2,\dots ,b_m\}$ with positive crisp real numbers with respect to criteria set X of n elements, the associated criteria weight vector is $W=(w_1,w_2,\dots ,w_n)$, $w_i\ge 0$, i = 1, 2, … , n, ${\displaystyle {\sum }_{i=1}^n{w}_i}=1$. The distance measures most widely employed in practice are:

• The normalized weighed Hamming distance:

  $$d_h(A,B)=\frac{1}{n}{\displaystyle {\sum }_{i=1}^n{w}_i|a_i-b_i|}$$

  (16)
• The normalized weighed Euclidean distance:

  $$d_e(A,B)=\frac{1}{n}{\left({\displaystyle {\sum }_{i=1}^n\left(w_i|a_i-b_i{|}^2\right)}\right)}^{1/2}$$

  (17)
• The Hausdorff distance:

  $$d_{\mathrm{hd}}(A,B)=\underset{i}{\max }\{w_i|a_i-b_i|\}$$

  (18)

(2) The Hamming OWAD

The Hamming OWAD [45,49] is an extension of the normalized weighted Hamming distance containing more information for distance measure of two sets. It can be seen as the combination of normal Hamming distance and the OWA operator. By using this distance measure, the importance of ordered position of the individual distances is considered, which means the larger (or smaller) individual deviations are assigned with higher (or lower) weights representing their influence degree on the global distance measure.

Definition 5.

The Hamming OWAD of dimension n is a mapping OWAD: $R^n\times R^n\to R$, with its associated weight vector $W=(w_1,w_2,\dots ,w_n)$, $w_i\ge 0$, i = 1, 2, … , n, ${\displaystyle {\sum }_{i=1}^n{w}_i}=1$. Based on the evaluations sets A = {a₁, a₂,…, a_n} and B = {b₁, b₂,…, b_n} defined above, the OWAD measure is defined as follows:

$$OWAD(A,B)=OWAD(\langle a_1,b_1\rangle ,\langle a_2,b_2\rangle ,\dots ,\langle a_n,b_n\rangle )={\displaystyle \sum _{i=1}^n{w}_i}{d}_i$$

(19)

where $d_i$ represents the ith largest of the individual distance $|a_i-b_i|$.

The OWAD operator is commutative, monotonic, bounded and idempotent, and it provides a parameterized family of distance measures ranging from the minimum to the maximum individual distance.

The Choquet integral is a powerful tool to handle the interaction among the elements in a set. In order to deal with the interaction among criteria for MCDM, some aggregation operators using the Choquet integral are proposed. As for the situations where criteria are not independent, Xu [50] put forward several intuitionistic fuzzy aggregation correlated operators and interval-valued intuitionistic fuzzy correlated averaging operators; Yang and Chen [51] proposed several 2-tuple linguistic correlated aggregation operators when the evaluations are linguistic arguments; Ju et al. [52] developed some hesitant fuzzy aggregation operators based on Choquet integral for hesitant fuzzy information.

Although these aggregation operators considering the interaction among criteria were proposed, most of the distance measures for MCDM as another main part for MCDM research do not take into the interactive factors among criteria. It should be synchronously developed with the aggregation operators. Among the various distance measures for MCDM, to our knowledge, only one work has mentioned, but not focused on: the ranking approach based on distance measure considering the interaction among criteria [53]. Very little attention has been paid systematically or specifically to the study on distance measure considering the interaction among criteria.

Thus, inspired by the idea of Choquet integral aggregation operators, Choquet integral distance measure combining the distance measure methodologies with the popular Choquet integral is defined. It is consistent with the main idea dealing with interaction among criteria of the Choquet integral aggregation operation. For example, for the synergy (positive interaction) between two criteria in a MCDM problem, the aggregation result should be greater than the simple sum of the individual evaluations, which means a smaller distance to ideal solution than that of independent criteria. As for the redundancy (negative interaction) among criteria, a distance measure greater than that of independent criteria should be acquired.

2.2.1. Fuzzy Measure and Discrete Choquet Integral

Definition 6.

A fuzzy measure on a measure space set C is a set function μ: $𝒫$(C) → [0,1], which satisfies the following axioms:

(i) 

$\mu (\varphi )=0,\mu (C)=1$; (Boundary conditions)

(ii) 

$B\subseteq Q\subseteq C$ implies $\mu （B）\le \mu （Q）$.

For all disjoint subsets $B\subseteq Q\subseteq C$,

• if $\mu (B\cup \mathrm{Q})\le \mu (B)+\mu (Q)$, the fuzzy measure is said to be subadditive, which means there is a negative interaction between B and Q (or we say they are redundant or substitutive);
• if $\mu (B\cup Q)\ge \mu (B)+\mu (Q)$, the fuzzy measure is said to be superadditive, which means there is a positive synergetic interaction between B and Q (or we say they are complementary);
• Particularly, if for all $B\subseteq Q\subseteq C$, $\mu (B\cup Q)=\mu (B)+\mu (Q)$, we say the fuzzy measure is additive, which means there is no interaction between B and Q and all the elements in C are independent, we have:

  $$\mu (B)={\displaystyle \sum _{c_i\in B}\mu (c_i)},\mathrm{for}\mathrm{all}B\subseteq C$$

For a MCDM framework with fuzzy measure $\mu (B)$, the component $\mu (B)$ can be interpreted as the importance of subset $B\subseteq C$. A fuzzy measure for subsets of C is monotonous, which means that when new criteria are added to a subset of C, the importance of the expanded subset does not decrease. Moreover, due to the main feature of a fuzzy measure: non-additive, it is able to be used to represent various kinds of interactions among the elements of a set (the decision criteria), the interactions may range from redundancy (negative interactions) to synergy (positive interaction).

In practice, fuzzy measure plays a role similar to the one of weights in the weighted mean operators. They are used to represent the importance or relevance of a set. Thus, with the separate weights of criteria, weights of any combination of criteria can also be defined. Additionally, the ability of the Choquet integral to deal with interaction among criteria is due to the fact that a weight of importance is attributed to every subset of criteria.

When a fuzzy measure is employed to represent the importance degree of the subsets of criteria set, a suitable aggregation function is called the Choquet integral.

Definition 7.

Let $\mu$ be a fuzzy measure on C, f be a positive real-valued function on C, the discrete Choquet integral on f with respective to $\mu$ is defined as follows:

$$C_{\mu }(f)={\displaystyle \sum _{i=1}^{n}f(c_{\sigma (i)})[\mu (C_{\sigma (i)})-\mu (C_{\sigma (i+1)})]}$$

(20)

where $(\sigma (1),\sigma (2),\dots ,\sigma (n))$ is a permutation of $(1,2,\dots ,n)$, so that $0\le f_{\sigma (1)}\le f_{\sigma (2)}\le ,\dots ,f_{\sigma (n)}$, $C_{\sigma (i)}=\{c_{\sigma (i)},c_{\sigma (i+1)},\dots ,c_{\sigma (n)}\}$, $C_{\sigma (n+1)}=\varphi$.

With the definition of fuzzy measure and Choquet integral, an intuitionistic fuzzy Choquet integral aggregation was defined [50].

Definition 8.

let $\mu$ be a fuzzy measure on C，let evaluations set $\tilde{A}=({\tilde{a}}_1,{\tilde{a}}_2,\dots ,{\tilde{a}}_n)$ be an intuitionistic fuzzy set whose elements ${\tilde{a}}_i=(u_{{\tilde{a}}_i},v_{{\tilde{a}}_i})$, i = 1，2,…, n, are intuitionistic fuzzy numbers (IFNs), then the intuitionistic fuzzy discrete Choquet integral aggregation (IFCA) operator is defined as follows:

$$IFCA({\tilde{a}}_1,{\tilde{a}}_2,\dots ,{\tilde{a}}_n)=\underset{i=1}{\overset{n}{\oplus }}{\tilde{a}}_{\sigma (i)}[\mu (C_{\sigma (i)})-\mu (C_{\sigma (i+1)})]$$

(21)

where $(\sigma (1),\sigma (2),\dots ,\sigma (n))$ is a permutation of $(1,2,\dots ,n)$, so that ${\tilde{a}}_{\sigma (1)}\le {\tilde{a}}_{\sigma (2)}\le ,\dots ,\le {\tilde{a}}_{\sigma (n)}$, $C_{\sigma (i)}=\{c_{\sigma (i)},c_{\sigma (i+1)},\dots ,c_{\sigma (n)}\}$, $C_{\sigma (n+1)}=\varphi$.

With the operation rules of IFNs, the IFCA operator can be transformed into the following form:

$$IFCA({\tilde{a}}_1,{\tilde{a}}_2,\dots ,{\tilde{a}}_n)=\left(1-{\displaystyle \prod _{i=1}^n{(1-u_{{\tilde{a}}_{\sigma (i)}})}^{\mu (C_{\sigma (i)})-\mu (C_{\sigma (i+1)})}}，{\displaystyle \prod _{i=1}^n{v}_{{\tilde{a}}_{\sigma (i)}}{}^{\mu (C_{\sigma (i)})-\mu (C_{\sigma (i+1)})}}\right)$$

(22)

In a special case, if the fuzzy measure is an additive measure, the intuitionistic fuzzy discrete Choquet integral aggregation (IFCA) operator reduces to the intuitionistic fuzzy weighted averaging operator.

2.2.2. Discrete Choquet Integral Distance Measure

For two evaluation values sets A = {a₁, a₂,…, a_n} and B = {b₁, b₂,…, b_n} with respect to criteria set C of n elements, let $\mu$ be a fuzzy measure on C，let d (a_i, b_i) (i = 1, 2, … , n), be normalized positive real-valued individual distance measure functions with respect to the type of the values, a Choquet integral distance measure $CH{D}_{\mu }(A,B)=\underset{i=1}{\overset{n}{\oplus }}{d}_{\sigma (i)}(a_i,b_i)[\mu (C_{\sigma (i)})-\mu (C_{\sigma (i+1)})]$ is a definition that satisfies the following axioms:

(i)

$0\le CH{D}_{\mu }\left(A,B\right)\le 1$;

(ii)

$CH{D}_{\mu }(A,B)=0$, if A=B;

(iii)

$CH{D}_{\mu }\left(A,B)=CH{D}_{\mu }\right(B,A)$.

For the sets of crisp real numbers, some Choquet integral distance measures are defined as follows:

Definition 9.

A normalized Choquet integral Hamming distance (CHD) of dimension n is a mapping ${[0,1]}^n\times {[0,1]}^n\to [0,1]$, which is defined as follows:

$$CH{D}_{\mu }(A,B)=\frac{1}{n}{\displaystyle \sum _{i=1}^n\left(|a_{\sigma (i)}-b_{\sigma (i)}|\cdot [\mu (C_{\sigma (i)})-\mu (C_{\sigma (i+1)})]\right)}$$

(23)

where $(\sigma (1),\sigma (2),\dots ,\sigma (n))$ is a permutation of $(1,2,\dots ,n)$, so that $|a_{\sigma (1)}-b_{\sigma (1)}|\le |a_{\sigma (2)}-b_{\sigma (2)}|\le ,\dots ,\le |a_{\sigma (n)}-b_{\sigma (n)}|$, $C_{\sigma (i)}=\{c_{\sigma (i)},c_{\sigma (i+1)},\dots ,c_{\sigma (n)}\}$, $C_{\sigma (n+1)}=\varphi$.

The definition satisfies the Choquet integral distance measure properties (і) (іі) (ііі).

Proof. 

(i)

Because $\mu (C_{\sigma (i)})$ and $\mu (C_{\sigma (i+1)})$ are two fuzzy measures whose values are in the interval $[0,1]$, for $\mu (C_{\sigma (i)})\supseteq \mu (C_{\sigma (i+1)})$, because $\mu (C_{\sigma (i)})\ge \mu (C_{\sigma (i+1)})$, so $0\le \mu (C_{\sigma (i)})-\mu (C_{\sigma (i+1)})\le 1$; $|a_{\sigma (i)}-b_{\sigma (i)}|$, (i = 1, 2, …, n) are individual Hamming distance measures, $0\le |a_{\sigma (i)}-b_{\sigma (i)}|\le 1$; particularly, when i = n, $\mu (C_{\sigma (i)})-\mu (C_{\sigma (i+1)})$ = 1, if $|a_{\sigma (n)}-b_{\sigma (n)}|=1$, then $CH{D}_{\mu }\left(A,B\right)$= 1, thus, we can get that $0\le \frac{1}{n}{\displaystyle \sum _{i=1}^n\left(|a_{\sigma (i)}-b_{\sigma (i)}|\cdot [\mu (C_{\sigma (i)})-\mu (C_{\sigma (i+1)})]\right)}\le 1$.

(ii)

If A = B, $a_i=b_i$ (i = 1, 2, …, n), then $|a_{\sigma (i)}-b_{\sigma (i)}|=0$ (i = 1, 2, …, n),

$$\frac{1}{n}{\displaystyle \sum _{i=1}^n\left(|a_{\sigma (i)}-b_{\sigma (i)}|\cdot [\mu (C_{\sigma (i)})-\mu (C_{\sigma (i+1)})]\right)}=0$$

(iii)

For $|a_{\sigma (i)}-b_{\sigma (i)}|=|b_{\sigma (i)}-a_{\sigma (i)}|$ (i = 1, 2, …, n), then

$$\begin{gathered} \frac{1}{n}{\displaystyle \sum _{i=1}^n\left(|a_{\sigma (i)}-b_{\sigma (i)}|\cdot [\mu (C_{\sigma (i)})-\mu (C_{\sigma (i+1)})]\right)}=\frac{1}{n}{\displaystyle \sum _{i=1}^n\left(|b_{\sigma (i)}-a_{\sigma (i)}|\cdot [\mu (C_{\sigma (i)})-\mu (C_{\sigma (i+1)})]\right)} \\ CH{D}_{\mu }\left(A,B)=CH{D}_{\mu }\right(B,A)\square \end{gathered}$$

Definition 10.

A normalized Choquet integral Euclidean distance (CED) of dimension n is a mapping ${[0,1]}^n\times {[0,1]}^n\to [0,1]$, which is defined as follows:

$$CE{D}_{\mu }(A,B)=\frac{1}{n}{\left({\displaystyle \sum _{i=1}^n|a_{\sigma (i)}-b_{\sigma (i)}{|}^2\cdot [\mu (C_{\sigma (i)})-\mu (C_{\sigma (i+1)})]}\right)}^{1/2}$$

(24)

where $(\sigma (1),\sigma (2),\dots ,\sigma (n))$ is a permutation of $(1,2,\dots ,\mathrm{n})$, so that:

$|a_{\sigma (i)}-b_{\sigma (i)}{|}^2\le |a_{\sigma (2)}-b_{\sigma (2)}{|}^2\le ,\dots ,\le |a_{\sigma (n)}-b_{\sigma (n)}{|}^2$, $C_{\sigma (i)}=\{c_{\sigma (i)},c_{\sigma (i+1)},\dots ,c_{\sigma (n)}\}$, $C_{\sigma (n+1)}=\varphi$.

The definition satisfies the Choquet integral distance measure properties (і) (іі) (ііі), and the proof is similar with that of definition of Choquet integral Hamming distance.

Definition 11.

A Choquet integral Hausdorff distance (CHFD) of dimension n is a mapping ${[0,1]}^n\times {[0,1]}^n\to [0,1]$, which is defined as:

$$CHF{D}_{\mu }(A,B)=\underset{i}{\max }\left(|a_{\sigma (i)}-b_{\sigma (i)}|\cdot [\mu (C_{\sigma (i)})-\mu (C_{\sigma (i+1)})]\right)$$

(25)

where $(\sigma (1),\sigma (2),\dots ,\sigma (n))$ is a permutation of $(1,2,\dots ,n)$, so that:

$$|a_{\sigma (1)}-b_{\sigma (1)}|\le |a_{\sigma (2)}-b_{\sigma (2)}|\le ,\dots ,\le |a_{\sigma (n)}-b_{\sigma (n)}|,C_{\sigma (i)}=\{c_{\sigma (i)},c_{\sigma (i+1)},\dots ,c_{\sigma (n)}\},C_{\sigma (n+1)}=\varphi$$

For $0\le |a_{\sigma (i)}-b_{\sigma (i)}|\le 1$, $0\le \mu (C_{\sigma (i)})-\mu (C_{\sigma (i+1)})\le 1$, so $0\le CHF{D}_{\mu }\left(A,B\right)\le 1$.

Compared to the normal distance measures, which suppose the elements in the sets are independent, the Choquet integral distance measures can take into consideration the interaction among criteria. It can be used to deal with the non-additive distance measure for sets. This defined discrete Choquet integral multi-criteria distance measure will be used to support the sustainable supplier selection with particular decision features in the following selection.

3. A Novel MCDM Framework for Sustainable Supplier Selection

In this section, a description of a particular sustainable supplier selection problem with particular decision features is introduced, and a corresponding novel MCDM framework for sustainable supplier selection issues considering the interaction among criteria with heterogeneous decision information is built.

3.1. Description of Sustainable Supplier Selection Issue Considering the Interaction among Criteria with Heterogeneous Information

An effective way to achieve sustainable business development is to conduct a sustainable supply chain, which takes into account the technical, economic, social and environmental factors during the process of supply chain management. It is to pursue an integration of broader dimensions of environmental sustainability, economic sustainability and social sustainability. In SSCM, the procurement of raw materials or components is a key activity determining the cost and quality of the production. It is important to choose the most suitable sustainable supplier to keep the competitiveness of the sustainable supply chain.

As declared in the research of Carter and Rogers [2], environmental performance and social performance can influence economic performance for SSCM, and firms need to adopt longer-term and more flexible supply chain solutions to ensure their long-term viability. The interaction among different dimensions and management flexibility should be also considered during the sustainable supplier selection. Thus, we expand the concept of sustainable supplier selection based on their conceptualization of sustainable supply chain management. The sustainable supplier selection in this paper refers to the issue of supplier selection considering the role of performance in each dimension and the flexibility of decision-making practice. It is based on a more practical and flexible basis and facilitates the implementation of managerial decision-making practices.

Due to the features of sustainable supply management, the impacts of technical, economic, social and environmental factors on the competitiveness of sustainable supply chain is not independent, and is of certain internal correlation. For example, one company that emphasizes environmental benefits will be able to get better social identity. The same as sustainable supplier selection, there is no reason to suppose that the assessment criteria for sustainability are independent to each other, as that in traditional supplier selection methods. Instead, a reasonable and flexible sustainable supplier selection framework considering interaction among criteria to handle the problem more appropriately should be developed for sustainable supplier selection.

Moreover, with economic and social development, the complexity and uncertainty of the business environment are increasing. For sustainable supplier selection, in order to cope with the high complexity and uncertainty in these issues, evaluation values on different criteria with different decision characteristics should be represented in the types of information which are more suitable to the features and in line with the experts’ preference. It is more reasonable and flexible due to the limited time and decision information. This decision mechanism not only can allow the experts to express their judgments completely according to the type of information they prefer based on criteria and decision characteristics, but also shorten the decision time and improve the efficiency of decision-making. In order to improve the flexibility and rationality of such decisions, one of the main points is to consider the case where the evaluation values of the sustainable supplier selection are represented by heterogeneous information.

Sustainable supplier selection is a typical MCDM problem with particular decision features. Generally, MCDM problem involves a process of selecting the best alternative(s) from a set of feasible alternatives with respect to multiple criteria, being either of qualitative or quantitative character. Then, sustainable supplier selection problem considering the interaction among criteria with heterogeneous information is a particular MCDM problem. For this particular MCDM issue, the decision criteria are not independent to each other and the evaluation values on different criteria with different decision characteristics are represented in different types of information which are more suitable to reflect the decision features and more in line with the expert’s preference.

For specific sustainable supplier selection problems, due to the high complexity, uncertainty and the fuzzy recognition of experts, the most appropriate type of evaluation value on different criteria can be: crisp number, fuzzy number, intuitionistic fuzzy number, hesitant fuzzy value in the hesitant context or linguistic variable in the linguistic context, and so on that can accurately characterize the features of the decision making problems or to fully reflect the judgments of the decision makers.

3.2. A Novel MCDM Framework for Sustainable Supplier Selection Considering the Interaction among Criteria with Heterogeneous Information

As for the sustainable supplier selection problem considering the interaction among criteria with heterogeneous information, the key is to build a decision framework to handle the interaction factors among criteria in the decision matrix with heterogeneous information properly.

The core idea or methodology of multi-criteria decision-making methods for sustainable supplier selection based on aggregation operator is that: first, for each alternative, aggregate its evaluations on the various criteria by using a specific aggregation operator; then, based on the appropriate comparison rules, compare the aggregated or collective evaluations of each alternative, and sort the alternatives; then the most suitable sustainable supplier is gotten. As comparison, the main idea or methodology of TOPSIS, VIKOR decision-making methods and so on from the perspective of multi-criteria distance measure is that: first, determine the reference values on each criterion for the alternatives, i.e., determine the ideal solutions; then, calculate the distance of each solution to the reference solution, and sort it based on certain rules, which are different for TOPSIS and VIKOR.

Some MCDM methods based on aggregation operator considering the interaction among criteria have been proposed. However, it is not able to handle the heterogeneous information. Thus, in this section, a novel MCDM method considering the interaction among criteria with heterogeneous information for sustainable supplier selection based on the Choquet integral multi-criteria distance measure is built from the perspective of distance measure.

Consider a general sustainable supplier selection problem taking into account the interaction among criteria in the decision context of heterogeneous decision information with several information types, of which the alternatives set is $A=\{A_1,A_2,\dots ,A_m\}$, and the decision criteria set is $C=\{C_1,C_2,\dots ,C_n\}$, the evaluation value of the ith alternative on the jth criterion given by expert or obtained from other information sources in information type t to accurately characterize decision features or to fully reflect the judgments of decision makers is denoted as $x_{ij(t)}^{}$. Suppose T is the number of information types. Then the decision matrix with heterogeneous information with T types of values can be denoted as follows:

$$X=\left[\begin{array}{ccccc}{x}_{11(\mathrm{type}1)}^{}& \cdots & x_{1j(\mathrm{type}t)}^{}& \cdots & x_{1n(\mathrm{type}T)}^{}\\ ⋮& & ⋮& & ⋮\\ x_{i1(\mathrm{type}1)}^{}& \ddots & x_{ij(\mathrm{type}t)}^{}& \ddots & x_{in(\mathrm{type}T)}^{}\\ ⋮& & ⋮& & ⋮\\ x_{m1(\mathrm{type}1)}^{}& \cdots & x_{mj(\mathrm{type}t)}^{}& \cdots & x_{mn(\mathrm{type}T)}^{}\end{array}\right]$$

In order to reflect the main points of this paper, suppose that the criteria have the same weights, i.e., the associated weight vector with criteria is $W_{}=(1/n,1/n,\dots ,1/n)$. Suppose the criteria in decision criteria set are not independent to each other, and the fuzzy measure on criteria set C and the subsets of C is μ: $𝒫$(C) → [0,1], which can be determined by the existing approaches.

A novel sustainable supplier selection decision-making framework based on the Choquet integral multi-criteria distance measure with heterogeneous information is visualized in Figure 1.

Step 1: Determine the heterogeneous ideal solutions.

The most popular used ideal solutions are the positive-ideal solution ${\eta }^{+}=({\eta }_1^{+},{\eta }_2^{+},\dots ,{\eta }_n^{+})$ and the negative-ideal solution ${\eta }^{-}=({\eta }_1^{-},{\eta }_2^{-},\dots {\eta }_n^{-})$, j = 1, 2, ..., n. Let C₁ be a collection of benefit criteria and C₂ be a collection of cost criteria. Sometimes, the evaluations may be represented as fuzzy numbers. For fuzzy evaluations, researchers have proposed some methods to compare different types of fuzzy numbers, such as the comparison methods for intuitionistic fuzzy numbers, hesitant fuzzy numbers and linguistic terms. In order to highlight the generality of the method proposed in this paper, the methods to determine the ideal solution for each criterion of different types of values are represented as two generalized function, denoted as g-max( ) for the positive-ideal solution and g-min( ) for the negative-ideal solution, respectively. For example, if the evaluations are expressed as intuitionistic fuzzy numbers, then the g-max( ), or g-min( ) is used to return the biggest or smallest evaluation by the score function proposed by Xu [54] operated on the intuitionistic fuzzy numbers.

$${\eta }^{+}=({\eta }_1^{+},{\eta }_2^{+},\dots ,{\eta }_j^{+},\dots ,{\eta }_n^{+}),j=1,2,\dots ,n$$

(26)

where ${\eta }_j^{+}=\underset{i}{g-\max }(p_{ij})\mathrm{if}j\in C_1$, ${\eta }_j^{+}=\underset{i}{g-\min }(p_{ij})$$\mathrm{if}j\in C_2,j=1,2,\dots ,n$.

$${\eta }^{-}=({\eta }_1^{-},{\eta }_2^{-},\dots ,{\eta }_j^{-},\dots ,{\eta }_n^{-}),j=1,2,\dots ,n$$

(27)

where ${\eta }_j^{-}=\underset{i}{g-\min }(p_{ij})\mathrm{if}j\in C_1,{\eta }_j^{-}=\underset{i}{g-\max }(p_{ij})$$\mathrm{if}j\in C_2,j=1,2,\dots ,n$.

Step 2: Build the normalized dominances decision matrix.

First, compute the normalized closeness between each evaluation value and the ideal value in the data column it located. Here, we choose the most popular used positive-ideal solution and negative-ideal solution as the ideal values set. The normalized closeness decision matrix P^NC is as follows:

$$P^{NC}={[p_{ij}^{NC}]}_{m\times n}={[\frac{d_t^N(x_{ij(t)}^{},{\eta }_{j(t)}^{+})}{d_t^N(x_{ij(t)}^{},{\eta }_{j(t)}^{+})+d_t^N(x_{ij(t)}^{},{\eta }_{j(t)}^{-})}]}_{m\times n}$$

(28)

where:

$$\begin{gathered} d_t^N(x_{ij(t)}^{},{\eta }_{j(t)}^{+})=d^t(x_{ij(t)}^{},{\eta }_{oj(t)}^{+})/d^t(\max (x_{ij(t)}^{})，\min (x_{ij(t)}^{})) \\ {d}_t^N(x_{ij(t)}^{},{\eta }_{j(t)}^{-})=d^t(x_{ij(t)}^{},{\eta }_{oj(t)}^{-})/d^t(\max (x_{ij(t)}^{})，\min (x_{ij(t)}^{})) \end{gathered}$$

Then, the normalized dominances decision matrix ($P^{ND}$) is gotten as follows:

$$P^{ND}={[1-p_{ij}^{NC}]}_{m\times n}$$

Step 3: Confirm the fuzzy measure on criteria set C and the subsets of C, which is denoted as μ: $𝒫$(C) → [0,1].

Several methods for determining the fuzzy measure on one particular set have been proposed, such as, the heuristic-based methods [55], the fuzzy identification method based on the semantics [56], genetic algorithms [57], the novel λ-fuzzy measure method [58], and so on.

Step 4: Get the comprehensive associated closeness (CAC) by Choquet integral multi-criteria distance measure based on the normalized dominances decision matrix (P^ND).

Choose an appropriate Choquet integral distance measure to compute the distance between alternative and the ideal normalized dominances for each alternative. Take the positive-ideal normalized dominance and negative-ideal normalized dominance as reference solutions, then the CAC is gotten as follows:

$$\begin{array}{cc}\hfill CA{C}_i& =\frac{CH{D}_{\mu }\left(p_{ij}^{ND},p_{ij}^{ND}{}^{+}\right)}{CH{D}_{\mu }\left(p_{ij}^{ND},p_{ij}^{ND}{}^{+}\right)+CH{D}_{\mu }\left(p_{ij}^{ND},p_{ij}^{ND}{}^{-}\right)}\hfill \\ & =\frac{\underset{i=1}{\overset{n}{\oplus }}{d}_{\sigma (j)}(p_{ij}^{ND},p_{ij}^{ND}{}^{+})[\mu (C_{\sigma (j)})-\mu (C_{\sigma (j+1)})]}{\underset{i=1}{\overset{n}{\oplus }}{d}_{\sigma (j)}(p_{ij}^{ND},p_{ij}^{ND}{}^{+})[\mu (C_{\sigma (j)})-\mu (C_{\sigma (j+1)})]+\underset{i=1}{\overset{n}{\oplus }}{d}_{{\sigma }^{\prime }(j)}(p_{ij}^{ND},p_{ij}^{ND}{}^{-})[\mu (C_{{\sigma }^{\prime }(j)})-\mu (C_{{\sigma }^{\prime }(j+1)})]}\hfill \end{array}$$

(29)

where $p_{ij}^{NC}{}^{+}=\underset{i}{g-\max }(p_{ij}^{NC})，p_{ij}^{NC}{}^{-}=\underset{i}{g-\min }(p_{ij}^{NC})\mathrm{if}j\in C_1，$, $p_{ij}^{NC}{}^{+}=\underset{i}{g-\min }(p_{ij}^{NC})，$$p_{ij}^{NC}{}^{-}=\underset{i}{g-\max }(p_{ij}^{NC}),$$\mathrm{if}j\in C_2,j=1,2,\dots ,n$, $i=1,2,\dots ,m$; $(\sigma (1),\sigma (2),\dots ,\sigma (n))$ is a permutation of $(1,2,\dots ,n)$, so that $d_{\sigma (1)}(p_{ij}^{NC},p_{ij}^{NC}{}^{+})\le d_{\sigma (2)}(p_{ij}^{NC},p_{ij}^{NC}{}^{+})\le \dots \le d_{\sigma (n)}(p_{ij}^{NC},p_{ij}^{NC}{}^{+})$, $C_{\sigma (j)}=\{c_{\sigma (j)},c_{\sigma (j+1)},\dots ,c_{\sigma (n)}\}$, $C_{\sigma (n+1)}=\varphi$; $({\sigma }^{\prime }(1),{\sigma }^{\prime }(2),\dots ,{\sigma }^{\prime }(n))$ is a permutation of $(1,2,\dots ,n)$, so that $d_{{\sigma }^{\prime }(1)}(p_{ij}^{NC},p_{ij}^{NC}{}^{-})\le d_{{\sigma }^{\prime }(2)}(p_{ij}^{NC},p_{ij}^{NC}{}^{-})\le \dots \le d_{{\sigma }^{\prime }(n)}(p_{ij}^{NC},p_{ij}^{NC}{}^{-})$, $C_{{\sigma }^{\prime }(j)}=\{c_{{\sigma }^{\prime }(j)},c_{{\sigma }^{\prime }(j+1)},\dots ,c_{{\sigma }^{\prime }(n)}\}$, $C_{{\sigma }^{\prime }(n+1)}=\varphi$.

Step 5: Rank the alternatives based on the results of CAC by Choquet integral multi-criteria distance measure to get the most suitable sustainable supplier, which takes into consideration the interaction among criteria with heterogeneous information.

4. Case Study and Comparison Analysis

In this section, we illustrate the proposed MCDM framework for sustainable supplier selection which considers interaction among criteria in decision setting of heterogeneous information to an example and utilize the decision framework. A comparison to TOPSIS method with heterogeneous information is carried out to further validate the rationality of the proposed framework.

The numerical case is described as follows:

In the context of SSCM, one manufacturing company is going to select the most competent supplier from six alternative sustainable suppliers taking consideration into several influencing factors, which is a particular MCDM problem. The candidate suppliers are denoted as a set $X=\{x_1,x_2,x_3,x_4,x_5,x_6\}$. As for the decision influencing factors, i.e., decision criteria, decision makers have a reference in the choice of decision criteria for sustainable supplier selection to that in the literature of Zimmer et al. [23]. The decision criteria for sustainable supplier selection are reviewed and summarized very well in the literature and the focus of our research is not the determination of decision criteria. This is an independent example considering certain practical factors. Several critical criteria are chosen by the decision makers to compose the decision criteria set, which is denoted as $C=\{c_1,c_2,c_3,c_4,c_5,c_6\}$. The criteria are represented as follows: c₁: Donations for sustainable projects, c₂: Technical capability, c₃: Control ability of ecological impacts, c₄: Energy consumption, c₅: Staff training, c₆: Social management commitment.

Considering the complexity of the problem itself and the uncertainty of the decision context, it is important to accurately and flexibly evaluate the candidate suppliers. Thus, based on the decision characteristics of each criteria, the evaluation values on different criteria for the candidate sustainable suppliers are respectively represented by their appropriate information types, corresponding to crisp numbers, interval fuzzy numbers, linguistic terms, fuzzy numbers, intuitionistic fuzzy numbers and hesitant fuzzy linguistic terms set. This decision mechanism not only can allow the experts to express their judgments more fully in the type of information they prefer according to criteria and decision characteristics, but also shorten the decision time and improve the efficiency of the decision-making. For example, as for the evaluation values on the criteria “ c₅: Staff training”, because decision makers cannot get accurate judgments, instead, they have an intuitionistic judgments based on the decision materials and their professional knowledge background, so they are expressed in the type of intuitionistic fuzzy number, denoted as $\left(u(x_{ij})，v(x_{ij})\right)$, which reveals that the expert’s satisfaction and dissatisfaction with the alternative a_i satisfying the attribute c_j. For another example, as for the evaluation values on the criteria “c₁: Donations for sustainable projects “, because it can be accurately quantified based on precise mathematical calculation and historical data, thus, it is expressed in the form of crisp numbers.

Suppose the linguistic terms set defined for the evaluation values in the types of linguistic term and HFLTS is S = {s_-g,…, s₀,…,s_g}, g = 3,which is shown in Figure 2.

The evaluation values on the criteria of “Donations for sustainable projects” are in numerical range of {1,…, 10}, which are quantified during this levels. The evaluation values on criteria of “Social management commitment” are in form of HFLTS based on linguistic terms set S, for the reason that experts have high ambiguity about their cognition on it and are difficult to give specific judgments and hesitate between several evaluation linguistic terms.

Decision matrix with heterogeneous information shown in

Table 1. Heterogeneous decision matrix.

	C1	C2	C3	C4	C5	C6
A1	6	[0.2, 0.6]	s5	0.7	(0.7,0.2)	{s4, s5, s6}
A2	8	[0.5, 0.7]	s4	0.6	(0.6,0.4)	{s3, s4, s5}
A3	7	[0.4, 0.6]	s6	0.9	(0.4,0.6)	{s4, s5, s6}
A4	9	[0.8, 0.9]	s3	0.5	(0.6,0.3)	{s4, s5}
A5	5	[0.6, 0.7]	s5	0.8	(0.8,0.1)	{s3, s4}
A6	8	[0.7, 0.8]	s4	0.6	(0.4,0.5)	{s5, s6}

is as follows:

The decision process of the sustainable supplier selection with heterogeneous information considering the interaction among criteria is as follows:

Step 1: Determine the heterogeneous ideal solutions with heterogeneous information.

The positive-ideal solution ${\eta }^{+}=({\eta }_1^{+},{\eta }_2^{+},\dots ,{\eta }_n^{+})=(9，\left[0.8,0.9\right],s_6，0.5，\left(0.8,0.1\right)，\{s_5,s_6\})$;

The negative-ideal solution ${\eta }^{-}=({\eta }_1^{-},{\eta }_2^{-},\dots ,{\eta }_n^{-})=(5，\left[02,0.6\right],s_3，0.9，\left(0.4,0.5\right)，\{s_3,s_4\})$.

Step 2: Build the normalized dominances decision matrix.

First, build the normalized closeness matrix, which is as follows:

$$P^{NC}=\left[\begin{array}{cccc}0.25& 1& 0.67& 0.5\\ 0.75& 0.54& 0.33& 0.25\\ 0.5& 0.69& 0& 1\\ 0& 0& 1& 0\\ 1& 0.37& 0.67& 0.75\\ 0.75& 0.21& 0.33& 0.25\end{array}\begin{array}{c}0.33\\ 0.57\\ 0.73\\ 0.48\\ 0\\ 0.89\end{array}\begin{array}{c}0.24\\ 0.45\\ 0.24\\ 0.37\\ 0.55\\ 0\end{array}\right]$$

Then, we get the normalized dominances decision matrix, which is as follows:

$$P^{ND}=\left[\begin{array}{cccc}0.75& 0& 0.33& 0.5\\ 0.25& 0.46& 0.67& 0.75\\ 0.5& 0.31& 1& 0\\ 1& 1& 0& 1\\ 0& 0.63& 0.33& 0.25\\ 0.25& 0.79& 0.67& 0.75\end{array}\begin{array}{c}0.67\\ 0.43\\ 0.27\\ 0.52\\ 1\\ 0.11\end{array}\begin{array}{c}0.76\\ 0.55\\ 0.76\\ 0.63\\ 0.45\\ 1\end{array}\right]$$

(30)

Step 3: Confirm the fuzzy measures on criteria set C and the subsets of C by using the fuzzy measure methods mentioned in the previous section.

In order to explain the problem more clearly and concisely, here we suppose that there are interaction among criteria $(c_1,c_2,c_3,c_4,c_5,c_6)$, and we suppose that the fuzzy measures are as follows:

$$\begin{gathered} \mu (c_1)=0.20,\mu (c_2)=0.15,\mu (c_3)=0.17,\mu (c_4)=0.30,\mu (c_5)=0.12,\mu (c_6)=0.20, \\ \mu (c_1,c_2)=0.76,\mu (c_1,c_3)=0.65,\mu (c_1,c_4)=0.50，\mu (c_1,c_5)=0.49,\mu (c_1,c_6)=0.30, \\ \mu (c_2,c_3)=0.29,\mu (c_2,c_4)=0.53,\mu (c_2,c_5)=0.31,\mu (c_2,c_6)=0.58, \\ \mu (c_1,c_2,c_3)=0.45,\mu (c_1,c_2,c_4)=0.59,\mu (c_1,c_3,c_4)=0.73,\mu (c_2,c_3,c_4)=0.61, \\ \mu (c_1,c_2,c_5)=0.25,\mu (c_1,c_2,c_6)=0.19,\mu (c_1,c_3,c_5)=0.23,\mu (c_1,c_3,c_6)=0.55, \\ \mu (c_2,c_3,c_5)=0.39,\mu (c_2,c_3,c_6)=0.57,\mu (c_4,c_5,c_6)=0.45,\mu (c_3,c_4,c_5)=0.65, \\ \mu (c_2,c_4,c_6)=0.25,\mu (c_3,c_4,c_6)=0.19,\mu (c_1,c_4,c_5)=0.23,\mu (c_1,c_4,c_6)=0.55, \\ \mu (c_1,c_5,c_6)=0.45,\mu (c_2,c_5,c_6)=0.38,\mu (c_3,c_5,c_6)=0.29,\mu (c_4,c_5,c_6)=0.56, \\ \mu (c_1,c_2,c_3,c_4)=0.49,\mu (c_1,c_2,c_3,c_5)=0.53,\mu (c_1,c_2,c_3,c_6)=0.31, \\ \mu (c_2,c_3,c_4,c_5)=0.61,\mu (c_2,c_3,c_4,c_6)=0.59,\mu (c_1,c_3,c_4,c_5)=0.46,\mu (c_1,c_3,c_4,c_6)=0.37, \\ \mu (c_1,c_2,c_5,c_6)=0.36,\mu (c_1,c_4,c_5,c_6)=0.35,\mu (c_3,c_4,c_5,c_6)=0.51,\mu (c_2,c_4,c_5,c_6)=0.71, \\ \mu (c_1,c_3,c_4,c_5,c_6)=0.39,\mu (c_2,c_3,c_4,c_5,c_6)=0.46,\mu (c_1,c_2,c_4,c_5,c_6)=0.50, \\ \mu (c_1,c_2,c_3,c_5,c_6)=0.61,\mu (c_1,c_2,c_3,c_4,c_5,c_6)=1.0. \end{gathered}$$

Step 4: Get the CAC by Choquet integral multi-criteria distance measure based on the normalized dominances decision matrix ($P^{ND}$).

Choose an appropriate Choquet integral multi-criteria distance measure to compute the distance between alternative suppliers and the ideal normalized dominances for each alternative. From the perspective of normalized dominance, the positive-ideal normalized dominance is ${\eta }_{ND}^{+}=(1,1,1,1,1,1)$ and the negative-ideal normalized dominance is ${\eta }_{ND}^{-}=(0,0,0,0,0,0)$. Then, the CAC is gotten as follows:

$$CAC(A_1)=0.47,CAC(A_2)=0.39,CAC(A_3)=0.63,CAC(A_4)=0.16,CAC(A_5)=0.51,CAC(A_6)=0.21.$$

Step 5: Rank the alternatives based on the results of the CAC by Choquet integral multi-criteria distance measure to get the most suitable sustainable supplier, which takes into consideration the interaction among criteria with heterogeneous decision information.

Because $CAC(A_4)<CAC(A_6)<CAC(A_2)<CAC(A_1)<CAC(A_5)<CAC(A_3)$, the candidate A₄ has the nearest distance from the positive-ideal normalized dominance, i.e., positive-ideal solution; hence, the supplier A₄ is the most suitable sustainable supplier.

Comparison analysis:

By using the ranking part of the heterogeneous TOPSIS decision method proposed by Lourenzutti and Krohling [59], we get the distance measure results $TOP(A_i)$ and a ranking list for alternative sustainable suppliers, which is as follows:

$$TOP(A_1)=0.36,TOP(A_2)=0.45,TOP(A_3)=0.57,TOP(A_4)=0.19,TOP(A_5)=0.69,TOP(A_6)=0.13.$$

For that, $TOP(A_6)<TOP(A_4)<TOP(A_1)<TOP(A_2)<TOP(A_3)<TOP(A_5)$, so we get,

$A_6\succ A_4\succ A_1\succ A_2\succ A_3\succ A_5$.

Comparison of the results of $CAC(A_i)$ and $TOP(A_i)$ is visualized as shown in Figure 3.

There are differences between the ranking results of the candidate sustainable suppliers by using the proposed method and TOPSIS with heterogeneous information. We can see these differences from the comparison of the results of $CAC(A_i)$ and $TOP(A_i)$ as shown in Figure 3. The reason is that the heterogeneous TOPSIS for sustainable supplier selection is suitable for the decision situations where the decision criteria are independent to each other, while the proposed method here for sustainable supplier selection is able to take into consideration the association factor when interaction existing among the criteria. Both of them deal with the problem from the perspective of distance measurement, but the two are essentially different. The further from the positive-ideal point, the worse the quality of the solution. Sometimes, MCDM based on distance measure is necessary and more appropriate when a suitable aggregation operator cannot be gotten or acquired. Moreover, distance from the ideal solution can be more intuitive to reflect the merits of the alternatives. The method proposed in this paper considers more practical factors to support decision makers or managers to make high-quality decision for sustainable supplier selection.

Take A₃ and A₅ as examples, $CAC(A_3)=$0.63, $CAC(A_5)=$0.51, while, $TOP(A_3)=$0.57, $TOP(A_5)=$0.69, the reason resulting the difference is that: synergy or redundancy among criteria which cannot be handled by heterogeneous TOPSIS leads to differences in distance calculations. This can also be explained from a practical perspective, for example, the criteria c₂ and c₆ is of synergy, which means one company that emphasizes technical capability and environmental benefits will be able to get a better social identity. In addition, we divide the sources of decision information into objective data that can be accurately measured and evaluated, as well as evaluation values that need to be judged by expert judgment which correspond to decision criteria with different characteristics for the sustainable supplier selection problem. In the processing of heterogeneous information, heterogeneous decision matrix for sustainable supplier selection is normalized into dominances decision matrix by the proposed decision framework, which is more adequate and intuitive in data processing. The research goal of sustainable supplier selection should be to help decision makers or supply chain managers make decisions more scientifically, flexibly and efficiently, and to consider more practical factors. In the proposed decision-making framework for sustainable supplier selection, evaluation values on different criteria with different decision characteristics can be represented in the types of information which are more suitable for the decision features and more in line with the expert’s preference.

5. Conclusions

In this paper, we expand the concept of sustainable supplier selection based on the conceptualization of SSCM. In the conceptualization of SSCM, environmental performance and social performance affect economic performance for SSCM and firms need to adopt longer-term and more flexible supply chain solutions to ensure their long-term viability. Thus, the interaction among different dimensions and management flexibility should be considered during the sustainable supplier selection. It is based on a more practical and flexible basis and facilitates the implementation of managerial decision-making practices. In order to improve the decision efficient and quality for sustainable supplier selection under uncertainty to achieve sustainable development, we propose a decision framework for sustainable supplier selection that considers the interaction among criteria under the context of heterogeneous decision information. More practical factors for sustainable supplier selection under uncertainty are taken consideration into the research. The decision-making mechanism established not only can allow the experts to express their judgments more fully in types of information they prefer according to the criteria and decision characteristics, but also improve the flexibility and efficiency of the decision-making. As for the proposed decision framework, the key is how to handle the interaction among criteria in the decision matrix with heterogeneous information properly. We deal with this from the perspective of normalized distance measure for information set and multi-criteria distance measure. As for the decision process for sustainable supplier selection under the setting of heterogeneous information: first, heterogeneous ideal solutions are determined; second, a normalized dominance decision matrix is built based on the normalized closeness by computing the normalized closeness distance between each evaluation value and the ideal value in the data column it located; third, a discrete Choquet integral multi-criteria distance measure is used to compute the CAC and rank the alternative sustainable suppliers. The rationality of the proposed method for sustainable supplier selection is further verified through a comparison with the heterogeneous TOPSIS.

Furthermore, the proposed decision framework can improve the decision-making quality for evaluating sustainable suppliers. The reasons are as follows: first, more uncertain and fuzzy factors can be modeled in the decision procedure by different types of information types; second, the interactive factors of high probability among decision criteria resulting from the technical, economic, environmental and social aspects for sustainable supplier selection are taken into consideration; more importantly, the established decision-making mechanism can not only allow the experts to express their judgments more fully in types of information they prefer according to the criteria and decision characteristics, but also improve the flexibility and efficiency of the decision-making for sustainable supplier selection.

For further research, with the development of economy, environment, society and the continuous advancement of decision-making technology, decision-making for sustainable supplier selection can model more uncertain and fuzzy practical factors, thus improving the quality of such decisions for sustainable supplier selection to achieve sustainable development. In addition, the characteristics of decision criteria and decision makers should also be more fully taken consideration into the decision process of SSCM.