TOPSIS Model for evaluating the corporate environmental performance under intuitionistic fuzzy environment

Abstract

With global resource waste and environmental pollution becoming increasingly serious, corporate environmental performance (CEP) has received much attention from researchers over the past decade. As an important part of economic development, enterprises also pay increasingly attention to environmental protection and pollution control. CEP is regarded as the result of corporate environmental management. Assessing CEP can not only make enterprises focus on the environmental protection and management, but also promote sustainable social development. And it is frequently viewed as a multi-attribute group decision-making (MAGDM) issue. Thus, a novel MAGDM method is needed to tackle it. Depending on the conventional TOPSIS (Technique for Order Preferenceby Similarity to Ideal Solution) method and intuitionistic fuzzy sets (IFSs), this essay design a novel intuitive distance based IF-TOPSIS method to assess CEP. First of all, a related literature review is conducted. What’s more, some necessary theories related to IFSs are briefly reviewed. In addition, since subjective randomness frequently exists in determining criteria weights, the weights of criteria is decided objectively by utilizing CRITIC method. Afterwards, relying on novel distance measures between IFNs, the conventional TOPSIS method is extended to the intuitionistic fuzzy environment to calculate assessment score of each enterprise. Eventually, an application about CEP evaluation and some comparative analysis have been given to demonstrate the superiority of the designed method. The results illustrate that the designed framework is useful for assessing CEP.

Keywords

MAGDM issues intuitionistic fuzzy sets (IFSs)TOPSIS method CRITIC method corporate environmental performance (CEP)

1. Introduction

Zadeh [1] initially presented the theory of fuzzy sets (FSs). Atanassov [2] introduced the concept of intuitionistic fuzzy sets (IFSs), which is a generalization of the concept of fuzzy set. In IFSs, there are three functions expressing the degrees of membership, non-membership and hesitancy. And it must satisfy the only condition that the sum of three degrees cannot exceed 1. Garg [3] presented a method related to group decision-making on the basis of intuitionistic fuzzy multiplicative preference relations and defined several geometric operators. Gou, Xu and Lei [4] pointed out a novel exponential operational law about IFNs and offered a method which was utilized to aggregate intuitionistic fuzzy information. Garg [5] developed some intuitionistic fuzzy averaging aggregation operators by taking the degrees of hesitation between the membership functions into consideration. He, He and Huang [6] integrated the power averaging operators with IFSs and defined several intuitionistic fuzzy power interaction aggregation operators. Liu, Liu and Chen [7] presented some novel intuitionistic fuzzy operators by extending the BM operator on the basis of the Dombi operations and designed some MAGDM methods. Zhang and He [8] defined the extensions of intuitionistic fuzzy geometric interaction aggregation operators by utilizing the t-norm and the corresponding t-conorm means. Gupta, Arora and Tiwari [9] extended the fuzzy entropy to IFSs with axiomatic justification and proposed importance of parameter alpha. Li and Wu [10] presented a comprehensive decision method relying on the intuitionistic fuzzy cross entropy distance and the grey correlation analysis.

In addition, Khan, Lohani and Ieee [11] put forward a novel similarity measure about IFNs depending on the distance measure of double sequence of bounded variation. Li, Liu, Liu, Su and Wu [12] developed a grey target decision making method in the form of IFNs on the basis of grey incidence analysis. Bao, Xie, Long and Wei [13] put forward a decision method depending on the prospect theory and the evidential reasoning method under intuitionistic fuzzy environment. Jin, Ni, Chen and Li [14] developed two group decision making (GDM) methods which can derive the normalized intuitionistic fuzzy priority weights from IFPRs on the basis of the order consistency and the multiplicative consistency. Chen, Cheng and Lan [15] developed a novel MCDM method on the basis of the TOPSIS method and similarity measures in the context of intuitionistic fuzzy. Gan and Luo [16] employed a hybrid method on the basis of DEMATEL and IFSs to examine the cause-effect relationships among factors. Gupta, Mehlawat, Grover and Chen [17] modified the superiority and inferiority ranking (SIR) method and combined it with intuitionistic fuzzy. Hao, Xu, Zhao and Zhang [18] presented a novel intuitionistic fuzzy decision making method depending on the theory of decision field. Krishankumar, Arvinda, Amrutha, Premaladha, Ravichandran and Ieee [19] integrated analytic hierarchy process (AHP) with IFSs to design a group decision making (GDM) method for effective cloud vendor selection. Krishankumar, Ravichandran and Saeid [20] developed IFSP (intuitionistic fuzzy set based PROMETHEE) which was a novel ranking method. Luo and Wang [21] combined IFSs with VIKOR method relying on a novel distance measure which taking the waver of intuitionistic fuzzy information into consideration. Rouyendegh [22] integrated the ELECTRE method with IFSs to tackle some MCDM issues. Cali and Balaman [23] extended ELECTRE I with VIKOR method in the context of intuitionistic fuzzy to reflect the decision makers’ preferences. Phochanikorn and Tan [24] incorporated DEMATEL with ANP to determine uncertainties and interdependencies among criteria and modified VIKOR to evaluate the sustainable supplier performance’s desired level under intuitionistic fuzzy context. Liang, He, Wang, Chen and Li [25] extended MABAC method to IFSs by utilizing the novel distance measures.

TOPSIS (Technique for Order Preferenceby Similarity to Ideal Solution) was initially developed by Hwang and Yoon [26] to tackle MAGDM issues. Compare with other MAGDM methods, TOPSIS method can consider the distances of each alternative from positive ideal solution and negative ideal solution. This method has been extended to various fuzzy environments [27, 28, 29, 30, 31, 32]. Garg and Arora [33] created a novel correlation measure and developed TOPSIS under intuitionistic fuzzy soft set Set pair analysis (SAP) is an effective extension of fuzzy set. Kumar and Garg [34] proposed the connection number (CN) under interval-valued intuitionistic fuzzy set (IVIFS) by using the concepts of SAP and CN, and established a new TOPSIS multi-attribute decision making method on this basis Kumar and Garg [35] investigated the intuitionistic fuzzy TOPSIS method based on CN. After in-depth understanding of SPA, CN and other knowledge, Garg and Kumar [36] developed a new distance formula which was used in the TOPSIS method so that an improved MADM method was born. Acuna-Soto, Liern and Perez-Gladish [37] believed that the ideal solution kept in line with any value between the minimum and maximum values of the criterion range rather than the optimal value of the decision criterion and further developed ideal similarity TOPSIS (IS-TOPSIS) method as a tool of MADM In order to help manufacturing enterprises choose a suitable flexible manufacturing system, Mathew, Chakrabortty and Ryan [38] combined analytical hierarchical process (AHP) to construct spherical fuzzy TOPSIS method. Akram, Garg and Zahid [39] reconstructed TOPSIS method and ELECTRE method, respectively, under complex Pythagorean fuzzy environment. Ali, Mahmood and Yang [40] discussed the model about complex spherical fuzzy TOPSIS method.

This paper’s goal is to extend the TOPSIS method to intuitionistic fuzzy environment and build a new decision making model for actual MADM problems. This new model inherits the advantages of the TOPSIS method in terms of computational simplicity, intelligibility, and so on. And it also ensures the integrity of information and accuracy of data processing to a greater extent In summary, the contribution of this paper is as follows: (1) a new multi-attribute decision making method is constructed; (2) a new distance formula is introduced into the TOPSIS method; (3) the TOPSIS method is combined with IFN and CRITIC method to extend the MADM model in the intuitionistic fuzzy environment; (4) The effectiveness of this method is proved successfully and applied to the evaluation of CEP The reminder of our essay proceeds as follows. Some necessary knowledge of IFSs is concisely reviewed in Section 2. The improved TOPSIS method is integrated with IFNs and the calculating procedures is simply depicted in Section 3. An empirical application about evaluating CEP is given to show the superiority of this approach and some comparative analysis are also offered to further prove the merits of this method in Section 4. At last, we make an overall conclusion of our work in Section 5.

2. Preliminaries

2.1 Intuitionistic fuzzy sets

Definition 1 [2]. An intuitionistic fuzzy set (IFS) on the universe $X$ is an object of the form

$\displaystyle I=\left\{{\left\langle{x,{\mu_{I}}\left(x\right),{\nu_{I}}\left(% x\right)}\right\rangle\left|{x\in X}\right.}\right\}$ (1)

where ${\mu_{I}}\left(x\right)\in\left[{0,1}\right]$ is called the “degree of membership of $I$ ” and ${\nu_{I}}\left(x\right)\in\left[{0,1}\right]$ is called the “degree of non-membership of $I$ ”, and ${\mu_{I}}\left(x\right)$ , ${\nu_{I}}\left(x\right)$ satisfy the following condition: $0\leqslant{\mu_{I}}\left(x\right)+{\nu_{I}}\left(x\right)\leqslant 1$ , $\forall\;x\in X$ . Then for $x\in X$ , ${\pi_{I}}\left(x\right)=1-\left({{\mu_{I}}\left(x\right)+{\nu_{I}}\left(x% \right)}\right)$ could be called the degree of hesitancy of $x$ in $I$ .

Definition 2 [41]. Let ${I_{1}}=\left({{\mu_{1}},{\nu_{1}}}\right)$ and ${I_{2}}=\left({{\mu_{2}},{\nu_{2}}}\right)$ be two intuitionistic fuzzy numbers (IFNs), the operation formula of them can be defined:

$\displaystyle{I_{1}}\oplus{I_{2}}=\left({{\mu_{1}}+{\mu_{2}}-{\mu_{1}}{\mu_{2}% },{\nu_{1}}{\nu_{2}}}\right)$ (2) $\displaystyle{I_{1}}\otimes{I_{2}}=\left({{\mu_{1}}{\mu_{2}},{\nu_{1}}+{\nu_{2% }}-{\nu_{1}}{\nu_{2}}}\right)$ (3) $\displaystyle\lambda{I_{1}}=\left({1-{{\left({1-{\mu_{1}}}\right)}^{\lambda}},% \nu_{1}^{\lambda}}\right),\lambda>0$ (4) $\displaystyle I_{1}^{\lambda}=\left({\mu_{1}^{\lambda},1-{{\left({1-{\nu_{1}}}% \right)}^{\lambda}}}\right),\lambda>0$ (5)

Definition 3 [42]. Let ${I_{1}}=\left({{\mu_{1}},{\nu_{1}}}\right)$ and ${I_{2}}=\left({{\mu_{2}},{\nu_{2}}}\right)$ be IFNs, the score and accuracy functions of ${I_{1}}$ and ${I_{2}}$ can be expressed:

$\displaystyle S\left({{I_{1}}}\right)={\mu_{1}}+{\mu_{1}}\left({1-{\mu_{1}}-{% \nu_{1}}}\right),$ (6) $\displaystyle S\left({{I_{2}}}\right)={\mu_{2}}+{\mu_{2}}\left({1-{\mu_{2}}-{% \nu_{2}}}\right)$ $\displaystyle H\left({{I_{1}}}\right)={\mu_{1}}+{\nu_{1}},H\left({{I_{2}}}% \right)={\mu_{2}}+{\nu_{2}}$ (7)

For two IFNs ${I_{1}}$ and ${I_{2}}$ , regarding on the Definition 3, then

if $s\left({{I}_{1}}\right)<s\left({{I}_{2}}\right),$ then ${{I}_{1}}<{{I}_{2}}$ ;

if $s\left({{I}_{1}}\right)>s\left({{I}_{2}}\right),$ then ${{I}_{1}}>{{I}_{2}}$ ;

if $s\left({{I}_{1}}\right)=s\left({{I}_{2}}\right),h\left({{I}_{1}}\right)<h\left% ({{I}_{2}}\right)$ , then ${{I}_{1}}<{{I}_{2}}$ ;

if $s\left({{I}_{1}}\right)=s\left({{I}_{2}}\right),h\left({{I}_{1}}\right)>h\left% ({{I}_{2}}\right)$ , then ${{I}_{1}}>{{I}_{2}}$ ;

if $s\left({{I}_{1}}\right)=s\left({{I}_{2}}\right),h\left({{I}_{1}}\right)=h\left% ({{I}_{2}}\right)$ , then ${{I}_{1}}={{I}_{2}}$ .

Definition 4 [21]. Let ${I_{1}}=\left({{\mu_{1}},{\nu_{1}}}\right)$ and ${I_{2}}=\left({{\mu_{2}},{\nu_{2}}}\right)$ be IFNs, the novel Euclidean distances between two IFNs can be defined as follows:

$\displaystyle\begin{aligned} \displaystyle\textit{IFED}\left({{I_{1}},{I_{2}}}% \right)=\sqrt{\frac{1}{{6}}\left({{{\left({{\ell_{1}}}\right)}^{2}}+{{\left({{% \ell_{2}}}\right)}^{2}}+{{\left({{\ell_{3}}}\right)}^{2}}}\right)}\end{aligned}$ (8)

where ${\ell_{1}}=\frac{{\left|{{\mu_{1}}-{\mu_{2}}}\right|+\left|{{\nu_{1}}-{\nu_{2}% }}\right|+\left|{\left({{\mu_{1}}+1-{\nu_{1}}}\right)-\left({{\mu_{2}}+1-{\nu_% {2}}}\right)}\right|}}{{2}}$

${\ell_{2}}=\frac{{{\pi_{1}}+{\pi_{2}}}}{2}$

${\ell_{3}}=\max\left({\left|{{\mu_{1}}-{\mu_{2}}}\right|,\left|{{\nu_{1}}-{\nu% _{2}}}\right|,\frac{{\left|{{\pi_{1}}-{\pi_{2}}}\right|}}{2}}\right)$

2.2 Intuitionistic fuzzy aggregation operators

Under the context of the intuitionistic fuzzy, some aggregation operators will be introduced in this chapter, including intuitionistic fuzzy weighted averaging (IFWA) operator and intuitionistic fuzzy weighted geometric (IFWG) operator.

Definition 5 [41]. Let ${I_{j}}=\left({{\mu_{{I_{j}}}},{\nu_{{I_{j}}}}}\right)(j=1,2,\cdots,$ $n)$ be a collection of IFNs, the intuitionistic fuzzy weighted averaging (IFWA) operator can be defined as:

$\displaystyle{\textit{IFWA}_{\omega}}({I_{1}},{I_{2}},\ldots,{I_{n}})=\mathop{% \oplus}\limits_{j=1}^{n}\left({{\omega_{j}}{I_{j}}}\right)$ (9)

where $\omega={\left({{\omega_{1}},{\omega_{2}},...,{\omega_{n}}}\right)^{T}}$ be the weight vector of ${I_{j}}\left({j=1,2,...,n}\right)$ and ${\omega_{j}}>0,\sum\limits_{j=1}^{n}{{\omega_{j}}}=1$ .

Derived from the Definition 5, the following result can be obtained:

Theorem 1. The aggregated value by using IFWA operator is also a IFN, where

$\displaystyle\textit{IFW}{A_{\omega}}({I_{1}},{I_{2}},\ldots,{I_{n}})=\mathop{% \oplus}\limits_{j=1}^{n}\left({{\omega_{j}}{I_{j}}}\right)$ (10) $\displaystyle\quad=\left({1-{{\prod\limits_{j=1}^{n}{\left({1-{\mu_{{I_{j}}}}}% \right)}}^{{\omega_{j}}}},\prod\limits_{j=1}^{n}{{{\left({{\nu_{{I_{j}}}}}% \right)}^{{\omega_{j}}}}}}\right)$

where $\omega={\left({{\omega_{1}},{\omega_{2}},...,{\omega_{n}}}\right)^{T}}$ be the weight vector of ${I_{j}}\left({j=1,2,...,n}\right)$ and ${\omega_{j}}>0,\sum\limits_{j=1}^{n}{{\omega_{j}}}=1$

Definition 6 [41]. Let ${I_{j}}\left({j=1,2,\cdots,n}\right)$ be a collection of IFNs, the intuitionistic fuzzy weighted geometric (IFWG) operator can be defined as:

$\displaystyle{\textit{IFWG}_{\omega}}({I_{1}},{I_{2}},\ldots,{I_{n}})=\mathop{% \otimes}\limits_{j=1}^{n}{\left({{I_{j}}}\right)^{{\omega_{j}}}}$ (11)

where $\omega={\left({{\omega_{1}},{\omega_{2}},...,{\omega_{n}}}\right)^{T}}$ be the weight vector of ${I_{j}}\left({j=1,2,...,n}\right)$ and ${\omega_{j}}>0,\sum\limits_{j=1}^{n}{{\omega_{j}}}=1$ .

Derived from the Definition 6, the following result can be obtained:

Theorem 2. The aggregated value by using IFWG operator is also a IFN, where

$\displaystyle{\textit{IFWG}_{\omega}}({I_{1}},{I_{2}},\ldots,{I_{n}})=\mathop{% \otimes}\limits_{j=1}^{n}{\left({{I_{j}}}\right)^{{\omega_{j}}}}$ (12) $\displaystyle\quad=\left({{{\prod\limits_{j=1}^{n}{\left({{\mu_{{I_{j}}}}}% \right)}}^{{\omega_{j}}}},1-\prod\limits_{j=1}^{n}{{{\left({1-{\nu_{{I_{j}}}}}% \right)}^{{\omega_{j}}}}}}\right)$

where $\omega={\left({{\omega_{1}},{\omega_{2}},...,{\omega_{n}}}\right)^{T}}$ be the weight vector of ${I_{j}}\left({j=1,2,...,n}\right)$ and ${\omega_{j}}>0,\sum\limits_{j=1}^{n}{{\omega_{j}}}=1$

3. The TOPSIS method with intuitionistic fuzzy information

Integrating the TOPSIS method with intuitionistic fuzzy environment, we build the IF-TOPSIS method which the assessment values are given by IFNs. In order to be more accurate, we not only use intuitionistic fuzzy numbers to express varieties of information, but also choose a new distance formula in the improved TOPSIS proposed in this paper. In addition, objectivity is guaranteed because of the CRITIC method as a way of determining weights. The calculating procedures of the developed method can be described subsequently.

Let $R=\left\{{{R_{1}},{R_{2}},\ldots{R_{n}}}\right\}$ be the set of attributes, $r=\left\{{{r_{1}},{r_{2}},\ldots{r_{n}}}\right\}$ be the weight vector of attributes ${R_{j}}$ , where ${r_{j}}\in\left[{0,1}\right],j=1,2,\ldots,n,\sum\nolimits_{j=1}^{n}{{r_{j}}}=1$ . Assume $H=\left\{{{H_{1}},{H_{2}},\ldots{H_{l}}}\right\}$ be a set of decision makers that have significant degree of $h=\left\{{{h_{1}},{h_{2}},\ldots{h_{l}}}\right\}$ , where ${h_{k}}\in\left[{0,1}\right]$ , $k=1,2,\ldots,l.$ $\sum\nolimits_{k=1}^{l}{{h_{k}}}=1$ . Let $F=\left\{{{F_{1}},{F_{2}},\ldots{F_{m}}}\right\}$ be a discrete collection of alternatives. And $Q={\left({q_{ij}}\right)_{m\times n}}$ is the overall intuitionistic fuzzy decision matrix, $q_{ij}$ means the value of alternative ${F_{i}}$ regarding the attribute ${R_{j}}$ .Subsequently, the specific calculating procedures will be depicted.

Step 1. Set up each decision maker’s intuitionistic fuzzy decision matrix ${Q^{\left(k\right)}}={\left({q_{ij}^{k}}\right)_{m\times n}}$ and calculate the overall intuitionistic fuzzy decision matrix $Q={\left({{q_{ij}}}\right)_{m\times n}}$ .

$\displaystyle{Q^{\left(k\right)}}={\left[{q_{ij}^{k}}\right]_{m\times n}}=% \left[{\begin{array}[]{*{20}{c}}{q_{11}^{k}}&{q_{12}^{k}}&\ldots&{q_{1n}^{k}}% \\ {q_{21}^{k}}&{q_{22}^{k}}&\ldots&{q_{2n}^{k}}\\ \vdots&\vdots&\vdots&\vdots\\ {q_{m1}^{k}}&{q_{m2}^{k}}&\ldots&{q_{mn}^{k}}\end{array}}\right]$ (13) $\displaystyle Q={\left[{q{}_{ij}}\right]_{m\times n}}=\left[{\begin{array}[]{*% {20}{c}}{{q_{11}}}&{{q_{12}}}&\ldots&{{q_{1n}}}\\ {{q_{21}}}&{{q_{22}}}&\ldots&{{q_{2n}}}\\ \vdots&\vdots&\vdots&\vdots\\ {{q_{m1}}}&{{q_{m2}}}&\ldots&{{q_{mn}}}\end{array}}\right]$ (14)

$\displaystyle{q_{ij}}\!=\!\left({1-{{\prod\limits_{k=1}^{l}\!{\left({1-{\mu_{q% _{ij}^{k}}}}\right)}}^{{h_{k}}}},\prod\limits_{k=1}^{l}\!{{{\left({{\nu_{q_{ij% }^{k}}}}\right)}^{{h_{k}}}}}}\right)$ (15)

where $q_{ij}^{k}$ is the assessment value of the alternative ${F_{i}}\left({i=1,2,\ldots,m}\right)$ on the basis of the attribute ${R_{j}}\left({j=1,2,\ldots,n}\right)$ and the decision maker.

Step 2. Normalize the overall intuitionistic fuzzy decision matrix $Q={\left({{q_{ij}}}\right)_{m\times n}}$ to ${Q^{N}}={\left[{q_{ij}^{N}}\right]_{m\times n}}$ .

$\displaystyle q_{ij}^{N}=\left\{{\begin{array}[]{*{20}{l}}{\left({\mu_{ij},\nu% _{ij}}\right),{R_{j}}∼{}\textit{is a benefit criterion}}\\ {\left({\nu_{ij},\mu_{ij}}\right),{R_{j}}∼{}\textit{is a cost criterion}}\end{% array}}\right.$ (16)

Step 3. Utilize CRITIC method to determine the weighting matrix of attributes.

CRiteria Importance Through Intercriteria Correlation (CRITIC) method will be proposed in this part which is utilized to decide attributes’ weights. This method was initially put forward by Diakoulaki, Mavrotas and Papayannakis [43] which took the correlations between attributes into consideration. Subsequently, the calculating procedures of this method will be presented.

(1) Depending on the normalized overall intuitionistic fuzzy decision matrix ${Q^{N}}={\left({q_{ij}^{N}}\right)_{m\times n}}$ , the correlation coefficient between attributes can be calculated.

$\displaystyle{\textit{IFCC}_{jt}}=,j,t=1,2,\ldots,n$ (17)

where $S\left(q_{j}^{N}\right)=\frac{1}{m}\sum\limits_{i=1}^{m}{S\left({q_{ij}^{N}}% \right)}$ and $S\left({q_{t}^{N}}\right)=\frac{1}{m}$ $\sum\limits_{i=1}^{m}{S\left({q_{it}^{N}}\right)}$ .

(2) Calculate attributes’ standard deviation.

$\displaystyle{\textit{IFSD}_{j}}=\sqrt{\frac{1}{{m-1}}\sum\limits_{i=1}^{m}{{{% \left({S\left({q_{ij}^{N}}\right)-S\left({q_{j}^{N}}\right)}\right)}^{2}}}},j=% 1,2,\ldots,n$ (18)

where $S\left({q_{j}^{N}}\right)=\frac{1}{m}\sum\limits_{i=1}^{m}{S\left({q_{ij}^{N}}% \right)}$ .

(3) Calculate the attributes’ weights.

$\displaystyle{r_{j}}=\frac{{{\textit{IFSD}_{j}}\sum\limits_{t=1}^{n}{\left({1-% {\textit{IFCC}_{jt}}}\right)}}}{{\sum\limits_{j=1}^{n}{\left({{\textit{IFSD}_{% j}}\sum\limits_{t=1}^{n}{\left({1-{\textit{IFCC}_{jt}}}\right)}}\right)}}},j=1% ,2\ldots,n$ (19)

where $r_{j}\in\left[{0,1}\right]$ and $\sum\limits_{j=1}^{n}{{r_{j}}}=1$ .

Step 4. Define the intuitionistic fuzzy positive ideal solution (IF-PIS) $A_{j}^{+}$ and the intuitionistic fuzzy negative ideal solution (IF-NIS) $A_{j}^{-}$ as:

$\displaystyle A_{j}^{+}=\mathop{\max}\limits_{i}\left({q_{ij}^{N}}\right)=% \left({\mathop{\max}\limits_{i}\left({\mu_{ij}^{N}}\right),\mathop{\min}% \limits_{i}\left({\nu_{ij}^{N}}\right)}\!\right)$ (20) $\displaystyle A_{j}^{-}=\mathop{\min}\limits_{i}\left({q_{ij}^{N}}\right)=% \left({\mathop{\min}\limits_{i}\left({\mu_{ij}^{N}}\right),\mathop{\max}% \limits_{i}\left({\nu_{ij}^{N}}\right)}\!\right)$ (21)

where $q_{ij}^{N}$ denotes the normalized intuitionistic fuzzy nummbers.

Step 5. Compute the positive distances $d_{i}^{+}$ between each aternative and IF-PIS and the negative distances between each aternative and IF-PIS as:

$\displaystyle d_{i}^{+}\!=\sum\limits_{j=1}^{n}r_{j}\textit{IFED}\left(q_{ij}^% {N},A_{j}^{+}\right),i=1,2,\ldots,m$ (22) $\displaystyle d_{i}^{-}\!=\sum\limits_{j=1}^{n}{{r_{j}}\textit{IFED}\left({q_{% ij}^{N},A_{j}^{-}}\right)},i=1,2,\ldots,m$ (23)

where $\textit{IFED}\left({q_{ij}^{N},A_{j}^{+}}\right)$ and $\textit{IFED}\left({q_{ij}^{N},A_{j}^{-}}\right)$ denote the intuitionistic fuzzy Euclidean distances given in Definition 4, and ${r_{j}}$ is the weighting vector of attributes.

Step 6. Compute each alternative’s correlation coefficient as:

$\displaystyle{C_{i}}=\frac{{d_{i}^{-}}}{{d_{i}^{-}+d_{i}^{+}}},i=1,2,\ldots,m$ (24)

Step 7. According to the computing values of the ${C_{i}}({i=1,2,\ldots,m})$ , the highest value of ${C_{i}}(i=1,2,$ $\ldots,m)$ is, the optimal alternative is designed.

4. The empirical example and comparative analysis

4.1 An empirical example

With the prosperity of economy, human beings have been blindly requesting and abusing environmental resources while pursuing their own benefits, which has contributed to increasingly prominent environmental issues. As the major responsibility body of environmental pollution, enterprises must play an essential role in protecting environment to achieve sustainable and stable development of economy. In order to effectively alleviate the conflicts between enterprises and environment, it is necessary to establish environmental performance evaluation system. For enterprises, a scientific environmental performance evaluation system can not only prompt themselves to conduct periodically evaluation about environmental behaviors, but also help them discover their existing environmental deficiencies and guide them to correct their deficiencies. In this chapter, an empirical application of evaluating CEP will be provided by making use of IF-TOPSIS method.

Since the government wants to select one enterprise with the best environmental behaviors and awareness to reward, there are five potential enterprises ${F_{i}}\left({i=1,2,3,4,5}\right)$ preparing to evaluate their CEP. In order to assess these enterprises fairly, five experts $H=\left\{{{H_{1}},{H_{2}},{H_{3}}}\right\}$ (expert’s weight $h=\left(0.30,0.50,0.20\right)$ ) are invited. All experts give their assessment information depending on the four subsequently attributes: (1) ${R_{1}}$ is environmental training and awareness; (2) ${R_{2}}$ is waste disposal cost; (3) ${R_{3}}$ is the ability to implement environment concerned design; (4) ${R_{4}}$ is environmental protection equipment investment. Evidently, ${R_{2}}$ is the cost attribute, while ${R_{1}}$ ${R_{3}}$ and ${R_{4}}$ are the benefit attributes, and the decision making matrix are given in Tables 1–3.

Table 1
Decision making information given by ${H_{1}}$

	$R_{1}$	$R_{2}$	$R_{3}$	$R_{4}$
F ${}_{1}$	(0.6,0.3)	(0.5,0.2)	(0.3,0.6)	(0.4,0.2)
F ${}_{2}$	(0.5,0.4)	(0.2,0.4)	(0.1,0.3)	(0.7,0.1)
F ${}_{3}$	(0.8,0.1)	(0.4,0.5)	(0.5,0.4)	(0.4,0.3)
F ${}_{4}$	(0.5,0.4)	(0.3,0.3)	(0.1,0.3)	(0.5,0.2)
F ${}_{5}$	(0.4,0.5)	(0.2,0.4)	(0.2,0.5)	(0.4,0.1)

Table 2

Decision making information given by ${H_{2}}$

	$R_{1}$	$R_{2}$	$R_{3}$	$R_{4}$
F ${}_{1}$	(0.4,0.2)	(0.3,0.4)	(0.5,0.1)	(0.2,0.7)
F ${}_{2}$	(0.3,0.5)	(0.5,0.1)	(0.4,0.2)	(0.3,0.3)
F ${}_{3}$	(0.6,0.2)	(0.3,0.6)	(0.5,0.1)	(0.6,0.2)
F ${}_{4}$	(0.1,0.1)	(0.4,0.5)	(0.2,0.5)	(0.4,0.3)
F ${}_{5}$	(0.3,0.2)	(0.5,0.2)	(0.1,0.3)	(0.4,0.5)

Table 3

Decision making information given by ${H_{3}}$

	$R_{1}$	$R_{2}$	$R_{3}$	$R_{4}$
F ${}_{1}$	(0.7,0.2)	(0.8,0.1)	(0.4,0.2)	(0.3,0.5)
F ${}_{2}$	(0.4,0.1)	(0.3,0.6)	(0.5,0.2)	(0.4,0.3)
F ${}_{3}$	(0.6,0.2)	(0.2,0.4)	(0.3,0.1)	(0.5,0.4)
F ${}_{4}$	(0.4,0.5)	(0.2,0.2)	(0.4,0.1)	(0.3,0.4)
F ${}_{5}$	(0.5,0.2)	(0.1,0.2)	(0.3,0.4)	(0.5,0.3)

Table 4

Overall intuitionistic fuzzy evaluation information

	$R_{1}$	$R_{2}$	$R_{3}$	$R_{4}$
F ${}_{1}$	(0.5858,0.2305)	(0.5885,0.1986)	(0.4026,0.2353)	(0.3078,0.4041)
F ${}_{2}$	(0.4086,0.2719)	(0.3414,0.2934)	(0.3489,0.2305)	(0.5054,0.2042)
F ${}_{3}$	(0.6862,0.1569)	(0.3069,0.4924)	(0.4413,0.1625)	(0.5038,0.2897)
F ${}_{4}$	(0.3169,0.2763)	(0.3037,0.3090)	(0.2418,0.2458)	(0.4077,0.2862)
F ${}_{5}$	(0.4065,0.2756)	(0.2844,0.2549)	(0.2051,0.3945)	(0.4350,0.2405)

Table 5

The normalized intuitionistic fuzzy evaluation information

	$R_{1}$	$R_{2}$	$R_{3}$	$R_{4}$
F ${}_{1}$	(0.5858,0.2305)	(0.1986, 0.5885)	(0.4026,0.2353)	(0.3078,0.4041)
F ${}_{2}$	(0.4086,0.2719)	(0.2934,0.3414)	(0.3489,0.2305)	(0.5054,0.2042)
F ${}_{3}$	(0.6862,0.1569)	(0.4924, 0.3069)	(0.4413,0.1625)	(0.5038,0.2897)
F ${}_{4}$	(0.3169,0.2763)	(0.3090, 0.3037)	(0.2418,0.2458)	(0.4077,0.2862)
F ${}_{5}$	(0.4065,0.2756)	(0.2549, 0.2844)	(0.2051,0.3945)	(0.4350,0.2405)

Step 1. Based on the decision making information ${Q^{\left(k\right)}}={\left({q_{ij}^{k}}\right)_{m\times n}}\left({i=1,2,% \ldots,m,j=1,2,\ldots,n}\right)$ given in Tables 1–3 and the expert’s weights $h=\left({0.30,0.50,0.20}\right)$ , we can derive the overall intuitionistic fuzzy evaluation information $Q={\left({q_{ij}}\right)_{m\times n}}(i=1,$ $2,\ldots,m,j=1,2,\ldots,n)$ according to Eq. (16), the computing results are list in Table 4.

Step 2. Normalize the intuitionistic fuzzy evaluation matrix $Q={\left[{q_{ij}}\right]_{m\times n}}$ to ${Q^{N}}={\left[{q_{ij}^{N}}\right]_{m\times n}}$ .

Step 3. Decide the attribute weights ${r_{j}}(j=1,2,\ldots,$ $n)$ by making use of the CRITIC method as recorded in Table 6.

Table 6

The attributes weights ${r_{j}}$

	$R_{1}$	$R_{2}$	$R_{3}$	$R_{4}$
${r_{j}}$	0.2799	0.2171	0.2606	0.2423

Table 7

Evaluation results of dissimilar methods

Methods	Ranking order	The best alternative	The worst alternative
IFWA operator	${F_{3}}>{F_{2}}>{F_{1}}>{F_{4}}>{F_{5}}$	${F_{3}}$	${F_{5}}$
IFWG operator	${F_{3}}>{F_{2}}>{F_{4}}>{F_{5}}>{F_{1}}$	${F_{3}}$	${F_{1}}$
The modified VIKOR	${F_{3}}>{F_{2}}>{F_{4}}>{F_{1}}>{F_{5}}$	${F_{3}}$	${F_{5}}$
The GRA method	${F_{3}}>{F_{2}}>{F_{4}}>{F_{5}}>{F_{1}}$	${F_{3}}$	${F_{1}}$
The CODAS method	${F_{3}}>{F_{2}}>{F_{4}}>{F_{5}}>{F_{1}}$	${F_{3}}$	${F_{1}}$
SPA-TOPSIS	${F_{3}}>{F_{2}}>{F_{4}}>{F_{5}}>{F_{1}}$	${F_{3}}$	${F_{1}}$
Distance measures for connection number sets based on set pair analysis	${F_{3}}>{F_{2}}>{F_{4}}>{F_{5}}>{F_{1}}$	${F_{3}}$	${F_{1}}$
The developed method	${F_{3}}>{F_{2}}>{F_{5}}>{F_{4}}>{F_{1}}$	${F_{3}}$	${F_{1}}$

Step 4. Calculate the intuitionistic fuzzy positive ideal solution (IF-PIS) $A_{j}^{+}$ and the intuitionistic fuzzy negative ideal solution (IF-NIS) $A_{j}^{-}$ according to equation (20) and (21);

$\displaystyle A_{j}^{+}=\left\{\begin{array}[]{l}\left(0.6862,0.1569\right),% \left(0.4924,0.2844\right),\\ \left(0.4413,0.1625\right),\left(0.5054,0.2042\right)\end{array}\right\}$ $\displaystyle A_{j}^{-}=\left\{\begin{array}[]{l}\left(0.3169,0.2763\right),% \left(0.1986,0.5885\right),\\ \left(0.2051,0.3945\right),\left(0.3078,0.4041\right)\end{array}\right\}$

Step 5. Compute the distances $d_{i}^{+}$ and $d_{i}^{-}$ , the results are list as follows.

$\displaystyle d_{1}^{+}=0.1879,d_{2}^{+}=0.1765,d_{2}^{+}=0.1065,d_{2}^{+}=0.2% 099,d_{2}^{+}=0.2074;$ $\displaystyle d_{1}^{-}=0.1102,d_{2}^{-}=0.1618,d_{2}^{-}=0.2424,d_{2}^{-}=0.1% 300,d_{2}^{-}=0.1331.$

Step 6. Compute each alternative’s correlation coefficient ${C_{i}}$ by equation (24), the results are list as:

$\displaystyle{C_{1}}=0.3697,{C_{2}}=0.4781,{C_{3}}=0.6948,$ $\displaystyle{C_{4}}=0.3824,{C_{5}}=0.3908$

Step 9. Relying on the calculating results of ${C_{i}}$ , all the alternatives can be ranked, the higher value of ${C_{i}}$ is, the optimal alternative selected will be. Evidently, the rank of all alternatives is ${F_{3}}>{F_{2}}>{F_{5}}>{F_{4}}>{F_{1}}$ and ${F_{3}}$ is the optimal enterprise.

4.2 Compare analysis

In this part, our developed method is made comparison with some other methods to illustrate its superiority.

First of all, our presented method is compared with IFWA and IFWG operators [41]. For the IFWA operator, the calculating result is: $S\left(F_{1}\right)={0.0790},S\left(F_{2}\right)={0.1411},S\left(F_{3}\right)=% {0.3351},$ $S\left(F_{4}\right)={0.0443},S\left(F_{5}\right)={\text{0.0403.}}$ Thus, the ranking order is ${F_{3}}>{F_{2}}>{F_{1}}>{F_{4}}>{F_{5}}$ . For the IFWG operator, the calculating result is $S\left(F_{1}\right)=-0.0102,S\left(F_{2}\right)=0.1227,S\left(F_{3}\right)=0.3% 018,$ $S\left(F_{4}\right)=0.0352,S\left(F_{5}\right)=0.0099.$ So the ranking order is ${F_{3}}>{F_{2}}>{F_{4}}>{F_{5}}>{F_{1}}$ .

What’s more, our presented method is compared with the modified VIKOR method with IFSs [44]. Then we can obtain the calculating result. The closest ideal score values are determined as: ${\textit{CI}^{*}}\left(F_{1}\right)=0.9045,$ ${\textit{CI}^{*}}\left(F_{2}\right)=0.6782,$ ${\textit{CI}^{*}}\left(F_{3}\right)=0.0000,$ ${\textit{CI}^{*}}\left(F_{4}\right)=0.9832,$ ${\textit{CI}^{*}}\left(F_{5}\right)=0.9576.$ And the farthest worst score values are determined as: ${\textit{CI}^{-}}\left(F_{1}\right)=0.0128,$ ${\textit{CI}^{-}}\left(F_{2}\right)=0.3459,$ ${\textit{CI}^{-}}\left(F_{3}\right)=1.0000,$ ${\textit{CI}^{-}}\left(F_{4}\right)=0.0168,$ Then each alternatives’ relative closeness are calculated as: ${\textit{DRC}_{1}}=0.9860$ , ${\textit{DRC}_{2}}=0.6622$ , ${\textit{DRC}_{3}}=0.0000$ , ${\textit{DRC}_{4}}=0.9832$ , ${\textit{DRC}_{5}}=1.0000$ . Hence, the ranking order of alternatives is ${F_{3}}>{F_{2}}>{F_{4}}>{F_{1}}>{F_{5}}$ .

Besides, our presented method is compared with GRA-based intuitionistic fuzzy [45]. Then we can obtain the calculating result. The grey relational grades of each alternative are calculated as: ${\gamma_{1}}=0.8099$ , ${\gamma_{2}}=0.8775$ , ${\gamma_{3}}=0.9827$ , ${\gamma_{4}}=0.8240$ , ${\gamma_{5}}=0.8172$ . Therefore, the ranking order of alternatives is ${F_{3}}>{F_{2}}>{F_{4}}>{F_{5}}>{F_{1}}$ .

Then, intuitionistic fuzzy MCDM-based CODAS[46]. Then we can obtain the calculating result. The total assessment score (AS) of each alternative is calculated as: ${\textit{AS}_{1}}=-0.8019$ , ${\textit{AS}_{2}}=0.1404$ , ${\textit{AS}_{3}}={1.4818}$ , ${\textit{AS}_{4}}=-{0.3860}$ , ${\textit{AS}_{5}}=-{0.4343}$ . Therefore, the ranking order of alternatives is ${F_{3}}>{F_{2}}>{F_{4}}>{F_{5}}>{F_{1}}$ .

In the end, our presented method is also compared with SPA-TOPSIS [35] and distance measures for connection number sets based on set pair analysis [47], we could get the same ranking order of alternatives is ${F_{3}}>{F_{2}}>{F_{4}}>{F_{5}}>{F_{1}}$ .

Eventually, the results of dissimilar methods are recorded in Table 7.

Derived from the Table 7, it is evidently that the optimal enterprise is ${F_{3}}$ in the mentioned methods, while the worst choice is ${F_{1}}$ in most situations. In other words, these methods’ ranking results are slightly different. Different methods can effectively tackle MAGDM issues from different angles. IFWA and IFWG operator emphasize to aggregate evaluation information. The modified VIKOR method with IFSs emphasizes the closest to the ideal solution and the farthest to the worst solution. The GRA-based intuitionistic fuzzy emphasizes the degree of similarity or difference between two sequences on the basis of the relation. However, compared with the above methods, our designed method is more accurate, since it considers the distances of each alternative from positive ideal solution and negative ideal solution. What’s more, compared with the intuitionistic fuzzy MCDM-based CODAS approach, our designed method utilize novel distance measures and CRITIC method. The novel distance measures can not only reflect intuitionistic fuzzy information more comprehensive but also take waver in IFSs into consideration and do not generate counterintuitive situations. The CRITIC method can minimize subjective randomness while the criteria weights are determined.

5. Conclusion

The CEP is of great significance in the process of enterprise production, management and competition. Thus, it is urgent to for enterprises to adopt an effective CEP evaluation system. This essay offers an effective solution for this issue, since it designs a novel intuitive distance based IF-TOPSIS method to establish the evaluation system of CEP. And then a numerical example has been given to confirm that this novel method is reasonable. What’s more, to verify the validity and feasibility of the developed method, some comparative analysis is also conducted. By comparing with the existing methods, it is obvious about the advantages of the proposed method. First of all, TOPSIS emphasizes the calculation of the distance between each scheme and the positive and negative ideal points, which better ensures the accuracy. Secondly, the TOPSIS method proposed in this paper is different from the previous ones. And the new distance formula which can not only reflect intuitionistic fuzzy information more comprehensive but also take waver in IFSs into consideration and do not generate counterintuitive situations, is introduced into the calculation of TOPSIS method. Last but not least, the CRITIC method can minimize subjective randomness while the criteria weights are determined. However, the main drawback of this essay is that the number of DMs and attributes are small and interdependency of criteria is not taken into consideration, which may limit the application scope of the developed method to some extent.

Future research can tackle the interdependency of criteria by utilizing some methods including analytic network process (ANP). Furthermore, the developed method can be utilized to tackle many other MAGDM issues like risk evaluation, project selection and site selection. And it can also be applied to many other uncertain and ambiguous environments [48, 49].

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TOPSIS Model for evaluating the corporate environmental performance under intuitionistic fuzzy environment

Abstract

Keywords

1. Introduction

2. Preliminaries

2.1 Intuitionistic fuzzy sets

4.1 An empirical example

Table 1 Decision making information given by H 1

5. Conclusion

References

Table 1
Decision making information given by ${H_{1}}$