Model for evaluating the transformation and upgrading ability of small and medium enterpries with the hesitant fuzzy uncertain linguistic information

Abstract

In this paper, we investigate the multiple attribute decision making problems based on the power aggregation operators with hesitant fuzzy uncertain linguistic information. Then, we have developed some power aggregation operators for aggregating hesitant fuzzy uncertain linguistic information: hesitant fuzzy uncertain linguistic power weighted average (HFULPWA) operator and hesitant fuzzy uncertain linguistic power weighted geometric (HFULPWG) operator. Then, we have utilized these operators to develop some approaches to solve the hesitant fuzzy uncertain linguistic multiple attribute decision making problems. Finally, a practical example for evaluating the transformation and upgrading ability of small and medium enterpries is given to verify the developed approach and to demonstrate its practicality and effectiveness.

Keywords

Multiple attribute decision making hesitant fuzzy uncertain linguistic values hesitant fuzzy uncertain linguistic power weighted average (HFULPWA) operator hesitant fuzzy uncertain linguistic power weighted geometric (HFULPWG) operator transformation and upgrading ability of small and medium enterpries

1. Introduction

Multiple attribute decision making (MADM) problems are to find a desirable solution from a finite number of feasible alternatives assessed on multiple attributes, both quantitative and qualitative [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18]. In the recent years, MADM has received a great deal of attention from researchers in many disciplines [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33]. Lin et al. [34] proposed the concept of hesitant fuzzy linguistic set and extended the hesitant fuzzy linguistic set to hesitant fuzzy uncertain linguistic set and have developed some arithmetic aggregation operators for aggregating hesitant fuzzy uncertain linguistic information. Li et al. [35] defined some hesitant fuzzy uncertain linguistic geometric aggregation operators: hesitant fuzzy uncertain linguistic weighted geometric (HFULWG) operator, hesitant fuzzy uncertain linguistic ordered weighted geometric (HFULOWG) operator and hesitant fuzzy uncertain linguistic hybrid geometric (HFULHG) operator.

SMEs (small and medium-sized enterprise) are not only the basic micro entity in the market economy, but also the active factors and key forces in improving market economy system. At present, the steady development of economic society is facing severe challenges from the transformation of production mode and structural transformation of the economy. It is of important realistic significance regarding how to prompt the transformation and upgrading of SMEs, so as to further promote fair competition, effective allocation of resources, technological progress and upgrading of an industrial structure. However, the concentration of SMEs in the space, and it’s derivatives brought to the development of industrial cluster, has increasingly become the essential characteristics of regional economic development and it contains many potential possibilities for regional economical development. Against the backdrop of regional economic development, how to combine the advantages brought by the industrial cluster’s development, and how to bring the transition and upgrading of individual enterprises under cluster, into full play, is a good subject worth our study and research. In this paper, we investigate the multiple attribute decision making problems for evaluating the transformation and upgrading ability of small and medium enterpries with hesitant fuzzy uncertain linguistic information. Then, motivated by the ideal of traditional power aggregation operators [16], we have developed the hesitant fuzzy uncertain linguistic power weighted average (HFULPWA) operator and hesitant fuzzy uncertain linguistic power weighted geometric (HFULPWG) operator. Then, we have utilized these operators to develop some approaches to solve the hesitant fuzzy uncertain linguistic multiple attribute decision making problems. Finally, a practical example for evaluating the transformation and upgrading ability of small and medium enterpries is given to verify the developed approach and to demonstrate its practicality and effectiveness.

2. Preliminaries

In the section, Lin et al. [34] proposed some basic concepts related to hesitant fuzzy uncertain linguistic set.

Definition 1. Given a fixed set $X$ and uncertain linguistic set $\tilde{{S}}$ , then a hesitant fuzzy uncertain linguistic set (HFULS) on $X$ is in terms of a function that when applied to $X$ returns a sunset of $\left[{0,1}\right]$ . To be easily understood, the HFULS can be expressed by mathematical symbol as follows:

$\tilde{{A}}=\left({\left\langle{x,\tilde{{s}}_{\theta\left(x\right)},h_{\tilde% {{A}}}\left(x\right)}\right\rangle\left|{x\in X}\right.}\right)$ (1)

where $h_{A}\left(x\right)$ is a set of some values in $\left[{0,1}\right]$ , denoting the possible membership degree of the element $x\in X$ to the uncertain linguistic set $\tilde{{s}}_{\theta\left(x\right)}$ . For convenience, we called $\tilde{{a}}=\left\langle{\tilde{{s}}_{\theta\left(x\right)},h_{\tilde{{A}}}% \left(x\right)}\right\rangle=\left\langle{\left[{s_{\theta^{L}\left(x\right)},% s_{\theta^{R}\left(x\right)}}\right],h_{\tilde{{A}}}\left(x\right)}\right\rangle$ a hesitant fuzzy uncertain linguistic element (HFULE) and $\tilde{{A}}$ the set of all HFULEs.

3. Power aggregation operators with hesitant fuzzy uncertain linguistic information

Yager [36] developed a nonlinear weighted average aggregation operator called power average (PA) operator, which can be defined as follows:

$\textit{PA}\left({a_{1},a_{2},\ldots,a_{n}}\right)=\frac{\sum\limits_{i=1}^{n}% {\left({1+T\left({a_{i}}\right)}\right)a_{i}}}{\sum\limits_{i=1}^{n}{\left({1+% T\left({a_{i}}\right)}\right)}}$ (2)

where $T\left({a_{i}}\right)=\sum\nolimits_{j=1,j\neq i}^{n}{\textit{Sup}\left({a_{i}% ,a_{j}}\right)}$ , and $\textit{Sup}\left({a,b}\right)$ is the support for $a$ from $b$ , which satisfies the following three properties: (1) $\textit{Sup}\left({a,b}\right)\in\left[{0,1}\right]$ ; (2) $\textit{Sup}\left({a,b}\right)=\textit{Sup}\left({b,a}\right)$ ; (3) $\textit{Sup}\left({a,b}\right)\geqslant\textit{Sup}\left({x,y}\right)$ , if $\left|{a-b}\right|<\left|{x-y}\right|$ . Obvoiusly, the support (Sup) measure is essentially a similarity index. The more similar, the closer two values, and the more they support each other.

The power average [36] operators, however, have usually been used in situations where the input arguments are the exact values. In this Section, we shall investigate the PA operator under hesitant fuzzy uncertain linguistic environments. Then, we give the definition of the hesitant fuzzy uncertain linguistic power weighted average (HFULPWA) operator as follows:

Definition 2. Let $\tilde{{a}}_{j}=\left({\tilde{{s}}_{\theta\left({\tilde{{a}}_{j}}\right)},h% \left({\tilde{{a}}_{j}}\right)}\right)=([s_{\theta^{L}\left({\tilde{{a}}_{j}}% \right)},$ $s_{\theta^{R}\left({\tilde{{a}}_{j}}\right)}],h\left({\tilde{{a}}_{j}}\right))% \left({j=1,2,\ldots,n}\right)$ be a collection of HFULEs, $\omega=\left({\omega_{1},\omega_{2},\ldots,\omega_{n}}\right)^{T}$ be the weighting vector of HFULEs $\tilde{{a}}_{j}\left({j=1,2,\ldots,n}\right)$ and $\omega_{j}\in\left[{0,1}\right]$ , $\sum\nolimits_{j=1}^{n}{\omega_{j}}=1$ , then we define the hesitant fuzzy uncertain linguistic power weighted average (HFULPWA) operator as follows:

$\displaystyle\textit{HFULPWA}\left({\tilde{{a}}_{1},\tilde{{a}}_{2},\ldots,% \tilde{{a}}_{n}}\right)=$ $\displaystyle\frac{\begin{array}[]{l}\omega_{1}\left({1+T\left({\tilde{{a}}_{1% }}\right)}\right)\tilde{{a}}_{1}\oplus\omega_{2}\left({1+T\left({\tilde{{a}}_{% 2}}\right)}\right)\tilde{{a}}_{2}\oplus\ldots\\ \oplus∼{}\omega_{n}\left({1+T\left({\tilde{{a}}_{n}}\right)}\right)\tilde{{a}}% _{n}\end{array}}{\sum\limits_{j=1}^{n}{\omega_{j}\left({1+T\left({\tilde{{a}}_% {j}}\right)}\right)}}$ (3)

where

$T\left({\tilde{{a}}_{j}}\right)=\sum\limits_{\begin{subarray}{l}i=1\\ i\neq j\\ \end{subarray}}^{n}{\omega_{i}\textit{Sup}\left({\tilde{{a}}_{j},\tilde{{a}}_{% i}}\right)}$ (4)

and $\textit{Sup}\left({\tilde{{a}}_{j},\tilde{{a}}_{i}}\right)$ is the support for $a_{j}$ from $a_{i}$ , with the conditions:

$\textit{Sup}\left({\tilde{{a}}_{i},\tilde{{a}}_{j}}\right)\in\left[{0,1}\right];$

$\textit{Sup}\left({\tilde{{a}}_{i},\tilde{{a}}_{j}}\right)=\textit{Sup}\left({% \tilde{{a}}_{j},\tilde{{a}}_{i}}\right);$

$\textit{Sup}\left({\tilde{{a}}_{i},\tilde{{a}}_{j}}\right)\geqslant\textit{Sup% }\left({\tilde{{a}}_{s},\tilde{{a}}_{t}}\right),$ if $d\left({\tilde{{a}}_{i},\tilde{{a}}_{j}}\right)\geqslant d\left({\tilde{{a}}_{% s},\tilde{{a}}_{t}}\right)$ , where $d$ is a distance measure.

Especially, if $\omega=\left({\frac{1}{n},\frac{1}{n},\ldots,\frac{1}{n}}\right)^{T}$ , then theHFULPWA operator reduces to hesitant fuzzy uncertain linguistic power average (HFULPA) operator:

$\displaystyle\textit{HFULPA}\left({a_{1},a_{2},\ldots,a_{n}}\right)=$ $\displaystyle\frac{\begin{array}[]{l}\left({1+T\left({\tilde{{a}}_{1}}\right)}% \right)\tilde{{a}}_{1}\oplus\left({1+T\left({\tilde{{a}}_{2}}\right)}\right)% \tilde{{a}}_{2}\oplus\ldots\\ \oplus\left({1+T\left({\tilde{{a}}_{n}}\right)}\right)\tilde{{a}}_{n}\end{% array}}{\sum\limits_{j=1}^{n}{\left({1+T\left({\tilde{{a}}_{j}}\right)}\right)}}$ (5)

where

$T\left({a_{j}}\right)=\frac{1}{n}\sum\limits_{\begin{subarray}{l}i=1\\ i\neq j\\ \end{subarray}}^{n}{\textit{Sup}\left({\tilde{{a}}_{j},\tilde{{a}}_{i}}\right)}$ (6)

Based on operations of the hesitant fuzzy uncertain linguistic values, we can drive the Theorem 1.

Theorem 1. Let $\tilde{{a}}_{j}=\left({\tilde{{s}}_{\theta\left({\tilde{{a}}_{j}}\right)},h% \left({\tilde{{a}}_{j}}\right)}\right)=([s_{\theta^{L}\left({\tilde{{a}}_{j}}% \right)},$ $s_{\theta^{R}\left({\tilde{{a}}_{j}}\right)}],h\left({\tilde{{a}}_{j}}\right))% \left(j=1,2,\ldots,n\right)$ be a collection of HFULEs, then their aggregated value by using the HFULPWA operator is also a HFULE, and

$\displaystyle\textit{HFULPWA}\left({\tilde{{a}}_{1},\tilde{{a}}_{2},\ldots,% \tilde{{a}}_{n}}\right)$ $\displaystyle=\frac{\begin{array}[]{l}\omega_{1}\left({1+T\left({\tilde{{a}}_{% 1}}\right)}\right)\tilde{{a}}_{1}\oplus\omega_{2}\left({1+T\left({\tilde{{a}}_% {2}}\right)}\right)\tilde{{a}}_{2}\oplus\\ \ldots\oplus\omega_{n}\left({1+T\left({\tilde{{a}}_{n}}\right)}\right)\tilde{{% a}}_{n}\end{array}}{\sum\limits_{j=1}^{n}{\omega_{j}\left({1+T\left({\tilde{{a% }}_{j}}\right)}\right)}}$ $\displaystyle=\mathop{\oplus}\limits_{j=1}^{n}\left({\frac{\omega_{j}\left({1+% T\left({\tilde{{a}}_{j}}\right)}\right)\tilde{{a}}_{j}}{\sum\limits_{j=1}^{n}{% \omega_{j}\left({1+T\left({\tilde{{a}}_{j}}\right)}\right)}}}\right)$ $\displaystyle=\left\langle\left[\sum\limits_{j=1}^{n}\frac{\omega_{j}\left({1+% T\left({\tilde{{a}}_{j}}\right)}\right)s_{\theta^{L}\left({\tilde{{a}}_{j}}% \right)}}{\sum\limits_{j=1}^{n}{\omega_{j}\left({1+T\left({\tilde{{a}}_{j}}% \right)}\right)}},\right.\right.$ $\displaystyle\quad∼{}\left.\left.\sum\limits_{j=1}^{n}{\frac{\omega_{j}\left({% 1+T\left({\tilde{{a}}_{j}}\right)}\right)s_{\theta^{R}\left({\tilde{{a}}_{j}}% \right)}}{\sum\limits_{j=1}^{n}{\omega_{j}\left({1+T\left({\tilde{{a}}_{j}}% \right)}\right)}}}\right],\right.$ $\displaystyle\left.\Bigg{(}\cup_{\gamma\left({\tilde{{a}}_{1}}\right)\in h% \left({\tilde{{a}}_{1}}\right),\gamma\left({\tilde{{a}}_{2}}\right)\in h\left(% {\tilde{{a}}_{2}}\right),\ldots,\gamma\left({\tilde{{a}}_{n}}\right)\in h\left% ({\tilde{{a}}_{n}}\right)}\right.$ $\displaystyle\left.\left.\left\{1-\prod\limits_{j=1}^{n}{\left({1-\gamma\left(% {\tilde{{a}}_{j}}\right)}\right)}^{\frac{\omega_{j}\left({1+T\left({\tilde{{a}% }_{j}}\right)}\right)}{\sum\limits_{j=1}^{n}{\omega_{j}\left({1+T\left({\tilde% {{a}}_{j}}\right)}\right)}}}\right\}\right)\right\rangle$ (7)

where

$T\left({\tilde{{a}}_{j}}\right)=\sum\limits_{\begin{subarray}{l}i=1\\ i\neq j\\ \end{subarray}}^{n}{\omega_{i}\textit{Sup}\left({\tilde{{a}}_{j},\tilde{{a}}_{% i}}\right)}$ (8)

Based on the HFULPWA operator and the geometric mean [37, 38, 39, 40], here we define a hesitant fuzzy uncertain linguistic power weighted geometric (HFULPWG) operator:

Definition 3. Let $\tilde{{a}}_{j}=\left({\tilde{{s}}_{\theta\left({\tilde{{a}}_{j}}\right)},h% \left({\tilde{{a}}_{j}}\right)}\right)=([s_{\theta^{L}\left({\tilde{{a}}_{j}}% \right)},$ $s_{\theta^{R}\left({\tilde{{a}}_{j}}\right)}],h\left({\tilde{{a}}_{j}}\right))% \left({j=1,2,\ldots,n}\right)$ be a collection of HFULEs, $\omega=(\omega_{1},\omega_{2},\ldots,\omega_{n})^{T}$ be the weighting vector of HFULEs, $\tilde{{a}}_{j}(j=1,2,\ldots,n)$ and $\omega_{j}\in[0,1]$ , $\sum\nolimits_{j=1}^{n}{\omega_{j}}=1$ , then we define the hesitant fuzzy uncertain linguistic power weighted geometric (HFULPWG) operator as follows:

$\displaystyle\textit{HFULPWG}\left(\tilde{{a}}_{1},\tilde{{a}}_{2},\ldots,% \tilde{{a}}_{n}\right)$ $\displaystyle=\tilde{{a}}_{1}^{\frac{\omega_{1}\left({1+T\left({\tilde{{a}}_{1% }}\right)}\right)}{\sum\limits_{j=1}^{n}{\omega_{j}\left({1+T\left({\tilde{{a}% }_{j}}\right)}\right)}}}\otimes\tilde{{a}}_{2}^{\frac{\omega_{2}\left({1+T% \left({\tilde{{a}}_{2}}\right)}\right)}{\sum\limits_{j=1}^{n}{\omega_{j}\left(% {1+T\left({\tilde{{a}}_{j}}\right)}\right)}}}\otimes\ldots$ $\displaystyle\quad∼{}\otimes\tilde{{a}}_{n}^{\frac{\omega_{n}\left({1+T\left({% \tilde{{a}}_{n}}\right)}\right)}{\sum\limits_{j=1}^{n}{\omega_{j}\left({1+T% \left({\tilde{{a}}_{j}}\right)}\right)}}}$ (9)

where

$T\left({\tilde{{a}}_{j}}\right)=\sum\limits_{\begin{subarray}{l}i=1\\ i\neq j\\ \end{subarray}}^{n}\omega_{i}\textit{Sup}\left(\tilde{{a}}_{j},\tilde{{a}}_{i}\right)$ (10)

and $\textit{Sup}\left({\tilde{{a}}_{j},\tilde{{a}}_{i}}\right)$ is the support for $a_{j}$ from $a_{i}$ , with the conditions:

(1) (1)

$\textit{Sup}\left({\tilde{{a}}_{i},\tilde{{a}}_{j}}\right)\in\left[{0,1}\right];$

(2)

$\textit{Sup}\left({\tilde{{a}}_{i},\tilde{{a}}_{j}}\right)=\textit{Sup}\left({% \tilde{{a}}_{j},\tilde{{a}}_{i}}\right);$

(3)

$\textit{Sup}\left({\tilde{{a}}_{i},\tilde{{a}}_{j}}\right)\geqslant\textit{Sup% }\left({\tilde{{a}}_{s},\tilde{{a}}_{t}}\right),$ if $d\left({\tilde{{a}}_{i},\tilde{{a}}_{j}}\right)\geqslant d(\tilde{{a}}_{s},% \tilde{{a}}_{t})$ , where $d$ is a distance measure.

Especially, if $\omega=\left(\frac{1}{n},\frac{1}{n},\ldots,\frac{1}{n}\right)^{T}$ , then theHFULPWG operator reduces to hesitant fuzzy uncertain linguistic power geometric (HFULPG) operator:

$\displaystyle{\rm HFULPG}\left({\tilde{{a}}_{1},\tilde{{a}}_{2},\ldots,\tilde{% {a}}_{n}}\right)$ $\displaystyle=\tilde{{a}}_{1}^{\frac{\left({1+T\left({\tilde{{a}}_{1}}\right)}% \right)}{\sum\limits_{j=1}^{n}{\left({1+T\left({\tilde{{a}}_{j}}\right)}\right% )}}}\otimes\tilde{{a}}_{2}^{\frac{\left({1+T\left({\tilde{{a}}_{2}}\right)}% \right)}{\sum\limits_{j=1}^{n}{\left({1+T\left({\tilde{{a}}_{j}}\right)}\right% )}}}\otimes\ldots$ $\displaystyle\quad∼{}\otimes\tilde{{a}}_{n}^{\frac{\left({1+T\left({\tilde{{a}% }_{n}}\right)}\right)}{\sum\limits_{j=1}^{n}{\left({1+T\left({\tilde{{a}}_{j}}% \right)}\right)}}}$ (11)

where

$T\left({a_{j}}\right)=\frac{1}{n}\sum\limits_{\begin{subarray}{l}i=1\\ i\neq j\\ \end{subarray}}^{n}{\textit{Sup}\left({\tilde{{a}}_{j},\tilde{{a}}_{i}}\right)}$ (12)

Based on operations of the hesitant fuzzy uncertain linguistic values, we can drive the Theorem 2.

Theorem 2. Let $\tilde{{a}}_{j}=\left({\tilde{{s}}_{\theta\left({\tilde{{a}}_{j}}\right)},h% \left({\tilde{{a}}_{j}}\right)}\right)=([s_{\theta^{L}\left({\tilde{{a}}_{j}}% \right)},$ $s_{\theta^{R}\left({\tilde{{a}}_{j}}\right)}],h\left({\tilde{{a}}_{j}}\right))% \left({j=1,2,\ldots,n}\right)$ be a collection of HFULEs, then their aggregated value by using the HFULPWG operator is also a HFULE, and

$\displaystyle\textit{HFULPWG}\left({\tilde{{a}}_{1},\tilde{{a}}_{2},\ldots,% \tilde{{a}}_{n}}\right)$ $\displaystyle=\tilde{{a}}_{1}^{\frac{\omega_{1}\left({1+T\left({\tilde{{a}}_{1% }}\right)}\right)}{\sum\limits_{j=1}^{n}{\omega_{j}\left({1+T\left({\tilde{{a}% }_{j}}\right)}\right)}}}\otimes\tilde{{a}}_{2}^{\frac{\omega_{2}\left({1+T% \left({\tilde{{a}}_{2}}\right)}\right)}{\sum\limits_{j=1}^{n}{\omega_{j}\left(% {1+T\left({\tilde{{a}}_{j}}\right)}\right)}}}\otimes\ldots$ $\displaystyle\quad∼{}\otimes\tilde{{a}}_{n}^{\frac{\omega_{n}\left({1+T\left({% \tilde{{a}}_{n}}\right)}\right)}{\sum\limits_{j=1}^{n}{\omega_{j}\left({1+T% \left({\tilde{{a}}_{j}}\right)}\right)}}}=\mathop{\otimes}\limits_{j=1}^{n}% \tilde{{a}}_{j}^{\frac{\omega_{j}\left({1+T\left({\tilde{{a}}_{j}}\right)}% \right)}{\sum\limits_{j=1}^{n}{\omega_{j}\left({1+T\left({\tilde{{a}}_{j}}% \right)}\right)}}}$ $\displaystyle=\left\langle\left[\prod\limits_{j=1}^{n}\left({s_{\theta^{L}% \left({\tilde{{a}}_{j}}\right)}}\right)^{{\omega_{j}\left({1+T\left({\tilde{{a% }}_{j}}\right)}\right)}/{\sum\limits_{j=1}^{n}{\omega_{j}\left({1+T\left({% \tilde{{a}}_{j}}\right)}\right)}}},\right.\right.$ $\displaystyle\quad∼{}\left.\left.\prod\limits_{j=1}^{n}{\left({s_{\theta^{R}% \left({\tilde{{a}}_{j}}\right)}}\right)^{{\omega_{j}\left({1+T\left({\tilde{{a% }}_{j}}\right)}\right)}/{\sum\limits_{j=1}^{n}{\omega_{j}\left({1+T\left({% \tilde{{a}}_{j}}\right)}\right)}}}}\right],\right.$ $\displaystyle\left(\cup_{\gamma\left({\tilde{{a}}_{1}}\right)\in h\left({% \tilde{{a}}_{1}}\right),\gamma\left({\tilde{{a}}_{2}}\right)\in h\left({\tilde% {{a}}_{2}}\right),\ldots,\gamma\left({\tilde{{a}}_{n}}\right)\in h\left({% \tilde{{a}}_{n}}\right)}\left\{\prod\limits_{j=1}^{n}\right.\right.$ $\displaystyle\left.\left.\left.\gamma\left({\tilde{{a}}_{j}}\right)^{{\omega_{% j}\left({1+T\left({\tilde{{a}}_{j}}\right)}\right)}/{\sum\limits_{j=1}^{n}{% \omega_{j}\left({1+T\left({\tilde{{a}}_{j}}\right)}\right)}}}\right\}\right)\right\rangle$ (13)

where

$T\left({\tilde{{a}}_{j}}\right)=\sum\limits_{\begin{subarray}{l}i=1\\ i\neq j\\ \end{subarray}}^{n}{\omega_{i}\textit{Sup}\left({\tilde{{a}}_{j},\tilde{{a}}_{% i}}\right)}$ (14)

4. An approach to multiple attribute decision making with hesitant fuzzy uncertain linguistic information

Let $A=\left\{{A_{1},A_{2},\ldots,A_{m}}\right\}$ be a discrete set of alternatives, and $G=\left\{{G_{1},G_{2},\ldots,G_{n}}\right\}$ be the state of nature. If the decision makers provide several values for the alternative $A_{i}$ under the state of nature $G_{j}$ with respect to $\tilde{{s}}_{\theta_{ij}}$ with anonymity, these values can be considered as a hesitant fuzzy uncertain linguistic element $\left\langle{\tilde{{s}}_{\theta_{ij}},h_{ij}}\right\rangle$ . Suppose that the decision matrix $\tilde{{H}}=(\tilde{{h}}_{ij})_{m\times n}=\left({\left\langle{\tilde{{s}}_{% \theta_{ij}},h_{ij}}\right\rangle}\right)_{m\times n}$ is the hesitant fuzzy uncertain linguistic decision matrix, where $\left\langle{\tilde{{s}}_{\theta_{ij}},h_{ij}}\right\rangle\left({i=1,2,\ldots% ,m,j=1,2,\ldots,n}\right)$ are in the form of HFULEs.

In the following, we apply the HFULPWA (or HFULPWG) operator to the MADM problems with hesitant fuzzy uncertain linguistic information.

Step 3. Step 1.
We utilize the decision information given in matrix $H$ , and the HFULPWA operator

$\displaystyle\tilde{{h}}_{i}=\left({\left\langle{\tilde{{s}}_{\theta_{i}},h_{i% }}\right\rangle}\right)={\rm HFULPWA}(\tilde{{h}}_{11},$ $\displaystyle\tilde{{h}}_{12},\tilde{{h}}_{13},\tilde{{h}}_{14}),i=1,2,\ldots,m.$ (15)

Or the hesitant fuzzy uncertain linguistic power weighted geometric (HFULPWG) operator:

$\displaystyle\tilde{{h}}_{i}=\left({\left\langle{\tilde{{s}}_{\theta_{i}},h_{i% }}\right\rangle}\right)={\rm HFULPWG}$ $\displaystyle(\tilde{{h}}_{11},\tilde{{h}}_{12},\tilde{{h}}_{13},\tilde{{h}}_{% 14})$ (16)

to derive the overall preference values $\tilde{{h}}_{i}(i$ $=1,2,\ldots,m)$ of the alternative $A_{i}$ .
Step 2.
Calculate the scores $S(\tilde{{h}}_{i})(i=1,2,\ldots,m)$ of the overall hesitant fuzzy uncertain linguistic preference values $\tilde{{h}}_{i}(i=1,2,\ldots,$ $m)$ .
Step 3.
Rank all the alternatives $A_{i}(i=1,2,\ldots,$ $m)$ and select the best one(s) in accordance with $S(\tilde{{h}}_{i})\;\left({i=1,2,\ldots,m}\right)$ .
Step 4.
End.

Table 1
Hesitant fuzzy uncertain linguistic decision matrix

G ${}_{1}$ G ${}_{2}$ G ${}_{3}$ G ${}_{4}$

A ${}_{1}$ $<$ [s ${}_{2}$ , s ${}_{3}$ ], (0.5,0.7) $>$ $<$ [s ${}_{1}$ , s ${}_{2}$ ], 0.5) $>$ $<$ [s ${}_{4}$ , s ${}_{5}$ ], (0.7,0.9) $>$ $<$ [s ${}_{6}$ , s ${}_{7}$ ], (0.3,0.4) $>$

A ${}_{2}$ $<$ [s ${}_{3}$ , s ${}_{4}$ ], (0.7,0.8) $>$ $<$ [s ${}_{1}$ , s ${}_{2}$ ], (0.7,. 9) $>$ $<$ [s ${}_{3}$ , s ${}_{4}$ ], (0.2,0.3) $>$ $<$ [s ${}_{1}$ , s ${}_{2}$ ], (0.6,0.7) $>$

A ${}_{3}$ $<$ [s ${}_{1}$ , s ${}_{2}$ ], (0.4,0.5) $>$ $<$ [s ${}_{3}$ , s ${}_{4}$ ], (0.8,0.9) $>$ $<$ [s ${}_{4}$ , s ${}_{5}$ ], (0.3,0.5) $>$ $<$ [s ${}_{2}$ , s ${}_{3}$ ], (0.7,0.8) $>$

A ${}_{4}$ $<$ [s ${}_{2}$ , s ${}_{3}$ ], (0.6,0.7) $>$ $<$ [s ${}_{4}$ , s ${}_{5}$ ], 0.4,0.5) $>$ $<$ [s ${}_{2}$ , s ${}_{3}$ ], (0.3,0.4,0.5) $>$ $<$ [s ${}_{4}$ , s ${}_{5}$ ], (0.6,0.9) $>$

A ${}_{5}$ $<$ [s ${}_{4}$ , s ${}_{5}$ ], (0.3, 0.7) $>$ $<$ [s ${}_{2}$ , s ${}_{3}$ ], 0.6,0.7) $>$ $<$ [s ${}_{5}$ , s ${}_{6}$ ], (0.4,0.6) $>$ $<$ [s ${}_{1}$ , s ${}_{2}$ ], (0.7,0.8) $>$

Table 2
The score values for the small and medium enterpries

HFULPWA HFULPWG

A ${}_{1}$ $\left[{{\rm s}_{{\rm 1.22}},{\rm s}_{{\rm 1.76}}}\right]$ $\left[{{\rm s}_{{\rm 1.54}},{\rm s}_{{\rm 2.45}}}\right]$

A ${}_{2}$ $\left[{{\rm s}_{{\rm 1.31}},{\rm s}_{{\rm 1.87}}}\right]$ $\left[{{\rm s}_{{\rm 1.67}},{\rm s}_{{\rm 2.21}}}\right]$

A ${}_{3}$ $\left[{{\rm s}_{{\rm 1.61}},{\rm s}_{{\rm 2.45}}}\right]$ $\left[{{\rm s}_{{\rm 2.14}},{\rm s}_{{\rm 2.53}}}\right]$

A ${}_{4}$ $\left[{{\rm s}_{{\rm 1.78}},{\rm s}_{{\rm 2.89}}}\right]$ $\left[{{\rm s}_{{\rm 2.42}},{\rm s}_{{\rm 2.65}}}\right]$

A ${}_{5}$ $\left[{{\rm s}_{{\rm 1.07}},{\rm s}_{{\rm 1.54}}}\right]$ $\left[{{\rm s}_{{\rm 1.34}},{\rm s}_{{\rm 1.76}}}\right]$

Table 3
Ordering of the small and medium enterpries by utilizing the HFULPWA and HFULPWG operators

Ordering

HFULPWA A ${}_{4}$ $>$ A ${}_{3}$ $>$ A ${}_{1}$ $>$ A ${}_{2}$ $>$ A ${}_{5}$

HFULPWG A ${}_{4}$ $>$ A ${}_{3}$ $>$ A ${}_{1}$ $>$ A ${}_{2}$ $>$ A ${}_{5}$

5. Numerical example

	G ${}_{1}$	G ${}_{2}$	G ${}_{3}$	G ${}_{4}$
A ${}_{1}$	$<$ [s ${}_{2}$ , s ${}_{3}$ ], (0.5,0.7) $>$	$<$ [s ${}_{1}$ , s ${}_{2}$ ], 0.5) $>$	$<$ [s ${}_{4}$ , s ${}_{5}$ ], (0.7,0.9) $>$	$<$ [s ${}_{6}$ , s ${}_{7}$ ], (0.3,0.4) $>$
A ${}_{2}$	$<$ [s ${}_{3}$ , s ${}_{4}$ ], (0.7,0.8) $>$	$<$ [s ${}_{1}$ , s ${}_{2}$ ], (0.7,. 9) $>$	$<$ [s ${}_{3}$ , s ${}_{4}$ ], (0.2,0.3) $>$	$<$ [s ${}_{1}$ , s ${}_{2}$ ], (0.6,0.7) $>$
A ${}_{3}$	$<$ [s ${}_{1}$ , s ${}_{2}$ ], (0.4,0.5) $>$	$<$ [s ${}_{3}$ , s ${}_{4}$ ], (0.8,0.9) $>$	$<$ [s ${}_{4}$ , s ${}_{5}$ ], (0.3,0.5) $>$	$<$ [s ${}_{2}$ , s ${}_{3}$ ], (0.7,0.8) $>$
A ${}_{4}$	$<$ [s ${}_{2}$ , s ${}_{3}$ ], (0.6,0.7) $>$	$<$ [s ${}_{4}$ , s ${}_{5}$ ], 0.4,0.5) $>$	$<$ [s ${}_{2}$ , s ${}_{3}$ ], (0.3,0.4,0.5) $>$	$<$ [s ${}_{4}$ , s ${}_{5}$ ], (0.6,0.9) $>$
A ${}_{5}$	$<$ [s ${}_{4}$ , s ${}_{5}$ ], (0.3, 0.7) $>$	$<$ [s ${}_{2}$ , s ${}_{3}$ ], 0.6,0.7) $>$	$<$ [s ${}_{5}$ , s ${}_{6}$ ], (0.4,0.6) $>$	$<$ [s ${}_{1}$ , s ${}_{2}$ ], (0.7,0.8) $>$

	HFULPWA	HFULPWG
A ${}_{1}$	$\left[{{\rm s}_{{\rm 1.22}},{\rm s}_{{\rm 1.76}}}\right]$	$\left[{{\rm s}_{{\rm 1.54}},{\rm s}_{{\rm 2.45}}}\right]$
A ${}_{2}$	$\left[{{\rm s}_{{\rm 1.31}},{\rm s}_{{\rm 1.87}}}\right]$	$\left[{{\rm s}_{{\rm 1.67}},{\rm s}_{{\rm 2.21}}}\right]$
A ${}_{3}$	$\left[{{\rm s}_{{\rm 1.61}},{\rm s}_{{\rm 2.45}}}\right]$	$\left[{{\rm s}_{{\rm 2.14}},{\rm s}_{{\rm 2.53}}}\right]$
A ${}_{4}$	$\left[{{\rm s}_{{\rm 1.78}},{\rm s}_{{\rm 2.89}}}\right]$	$\left[{{\rm s}_{{\rm 2.42}},{\rm s}_{{\rm 2.65}}}\right]$
A ${}_{5}$	$\left[{{\rm s}_{{\rm 1.07}},{\rm s}_{{\rm 1.54}}}\right]$	$\left[{{\rm s}_{{\rm 1.34}},{\rm s}_{{\rm 1.76}}}\right]$

	Ordering
HFULPWA	A ${}_{4}$ $>$ A ${}_{3}$ $>$ A ${}_{1}$ $>$ A ${}_{2}$ $>$ A ${}_{5}$
HFULPWG	A ${}_{4}$ $>$ A ${}_{3}$ $>$ A ${}_{1}$ $>$ A ${}_{2}$ $>$ A ${}_{5}$

The economic development of China is in an important period of strategic opportunities, it presents the new norm then our country should adapt to it and change the pattern of economic development, with the purpose of making the growth of economic driven by innovation not the investment any more, finally, completing the transformation and upgrading. Whether the enterprise can successfully carries on the transformation and upgrading is the key to the sustainable development of economy due to the enterprise is the small units of the economic development. In our country’s enterprises, SMEs are accounted for nearly 99% in the whole enterprises of our China. Whether it can keep up with the steps and implement the transformation and upgrading plays an important role for the sustainable development of economy. At present, there is a blindly following phenomenon in the transformation and upgrading of SMEs. Some companies didn’t set out from their ability before making strategic and not to scan at the ability of resource, caused the fruitless of their transformation and upgrading, some of it had to exit from the market. In this section, we present an empirical case study of evaluating the transformation and upgrading ability of small and medium enterpries. The project’s aim is to evaluate the best small and medium enterpries from the different small and medium enterprise. The transformation and upgrading ability of five possible small and medium enterpries $A_{i}\left({i=1,2,3,4,5}\right)$ is evaluated. After preliminary screening, five possible small and medium enterpries $A_{i}\left({i=1,2,\ldots,5}\right)$ have remained in the candidate list. Three decision makers (experts) form a committee to act as decision makers. The expert group must take a decision according to the following four attributes: (1) G ${}_{1}$ is the economic effectiveness; (2) G ${}_{2}$ is the technological innovation; (3) G ${}_{3}$ is the high-quality brand relationship; (4) G ${}_{4}$ is the structure optimization. In order to avoid influence each other, the decision makers are required to evaluate the five possible small and medium enterpries $A_{i}\left({i=1,2,3,4,5}\right)$ under the above four attributes in anonymity and the decision matrix $H=(\tilde{{h}}_{ij})_{5\times 4}=\left(\left\langle{\tilde{{s}}_{\theta_{ij}},% h_{ij}}\right\rangle\right)_{5\times 4}$ is presented in Table 1.

Then, we utilize the approach developed to get the most desirable small and medium enterprie.

Step 1. Step 1.
We utilize the decision information given in matrix $H$ , and the HFULPWA operator (or HFULPWG) to obtain the overall preference values $\tilde{{h}}_{i}$ of the small and medium enterpries $A_{i}\left({i=1,2,3,4,5}\right)$ and calculate the scores $S(\tilde{{h}}_{i})\left({i=1,2,3,4,5}\right)$ of the overall hesitant fuzzy uncertain linguistic preference values $h_{i}\left({i=1,2,3,4,5}\right)$ . The results are shown in Table 2.
Step 2.
Rank all the five possible small and medium enterpries Ai(i=1,2,3,4,5) in accordance with the scores $S(\tilde{{h}}_{i})\left({i=1,2,3,4,5}\right)$ . The ordering of small and medium enterpries is shown in Table 3. Note that $>$ means “preferred to”.

6. Conclusion

In this paper, we have developed some power aggregation operators for aggregating hesitant fuzzy uncertain linguistic information: hesitant fuzzy uncertain linguistic power weighted average (HFULPWA) operator and hesitant fuzzy uncertain linguistic power weighted geometric (HFULPWG) operator. Then, we have utilized these operators to develop some approaches to solve the hesitant fuzzy uncertain linguistic multiple attribute decision making problems for evaluating the transformation and upgrading ability of small and medium enterpries. Finally, a practical example for evaluating the transformation and upgrading ability of small and medium enterpries is given to verify the developed approach and to demonstrate its practicality and effectiveness. In the future, we shall extend the proposed methods to other domains [41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60].

References

Dyer

J.S.

Fishburn

P.C.

and Steuer

R.E.

, Multiple criteria decision making, multiattribute utility theory: The next ten years, Management Science 38(5) (1992), 645–654.

Abo-Sinna

M.A.

and Amer

A.H.

, Extensions of TOPSIS for multi-objective large-scale nonlinear programming problems, Applied Mathematics and Computation 162 (2005), 243–256.

Wei

G.W.

Wang

J.M.

and Chen

, Potential optimality and robust optimality in multiattribute decision analysis with incomplete information: A comparative study, Decision Support Systems 55(3) (2013), 679–684.

Hwang

C.L.

and Yoon

, Multiple Attribute Decision Making, Springer, Verlag, Berlin, 1981.

Torra

, Hesitant fuzzy sets, International Journal of Intelligent Systems 25 (2010), 529–539.

Xia

and Xu

Z.S.

, Hesitant fuzzy information aggregation in decision making, International Journal of Approximate Reasoning 52(3) (2011), 395–407.

Z.S.

Xia

and Chen

, Some hesitant fuzzy aggregation operators with their application in group decision making, Group Decision and Negotiation 22(2) (2013), 259–279.

Zhang

Z.M.

, Hesitant fuzzy power aggregation operators and their application to multiple attribute group decision making, Information Sciences 234 (2013), 150–181.

Wei

G.W.

, Hesitant fuzzy prioritized operators and their application to multiple attribute group decision making, Knowledge-Based Systems 31 (2012), 176–182.

10.

Zhu

Z.S.

and Xia

M.M.

, Hesitant fuzzy geometric Bonferroni means, Information Sciences 205(1) (2012), 72–85.

11.

Wei

G.W.

, FIOWHM operator and its application to multiple attribute group decision making, Expert Systems with Applications 38(4) (2011), 2984–2989.

12.

Z.S.

and Xia

M.M.

, Hesitant fuzzy entropy and cross-entropy and their use in multiattribute decision-making, International Journal of Intelligent Systems 27(9) (2012), 799–822.

13.

Wei

G.W.

and Wang

J.M.

, A comparative study of robust efficiency analysis and data envelopment analysis with imprecise data, Expert Systems with Applications 81 (2017), 28–38.

14.

Wei

G.W.

, Approaches to interval intuitionistic trapezoidal fuzzy multiple attribute decision making with incomplete weight information, International Journal of Fuzzy Systems 17(3) (2015), 484–489.

15.

Chen

T.Y.

, The inclusion-based TOPSIS method with interval-valued intuitionistic fuzzy sets for multiple criteria group decision making, Appl Soft Comput 26 (2015), 57–73.

16.

D.F.

, Mathematical-programming approach to matrix games with payoffs represented by Atanassov’s interval-valued intuitionistic fuzzy sets, IEEE Transactions on Fuzzy Systems 18 (2010), 1112–1128.

17.

D.F.

, Closeness coefficient based nonlinear programming method for interval-valued intuitionistic fuzzy multiattribute decision making with incomplete preference information, Applied Soft Computing 11 (2011), 3402–3418.

18.

Jiang

X.P.

and Wei

G.W.

, Some Bonferroni mean operators with 2-tuple linguistic information and their application to multiple attribute decision making, Journal of Intelligent and Fuzzy Systems 27 (2014), 2153–2162.

19.

Wang

H.J.

Zhao

X.F.

and Wei

G.W.

, Dual hesitant fuzzy aggregation operators in multiple attribute decision making, Journal of Intelligent and Fuzzy Systems 26(5) (2014), 2281–2290.

20.

Chen

T.Y.

, The inclusion-based TOPSIS method with interval-valued intuitionistic fuzzy sets for multiple criteria group decision making, Applied Soft Computing 26 (2015), 57–73.

21.

Wei

G.W.

and Zhao

X.F.

, Induced hesitant interval-valued fuzzy Einstein aggregation operators and their application to multiple attribute decision making, Journal of Intelligent and Fuzzy Systems 24 (2013), 789–803.

22.

Zhou

L.Y.

Lin

Zhao

X.F.

and Wei

G.W.

, Uncertain linguistic prioritized aggregation operators and their application to multiple attribute group decision making, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 21(4) (2013), 603–627.

23.

Chen

T.Y.

, An interval-valued intuitionistic fuzzy permutation method with likelihood-based preference functions and its application to multiple criteria decision analysis, Applied Soft Computing 42 (2016), 390–409.

24.

Zhao

X.F.

Lin

and Wei

G.W.

, Fuzzy prioritized operators and their application to multiple attribute group decision making, Applied Mathematical Modelling 37 (2013), 4759–4770.

25.

Zhao

X.F.

and Wei

G.W.

, Some intuitionistic fuzzy Einstein hybrid aggregation operators and their application to multiple attribute decision making, Knowledge-Based Systems 37 (2013), 472–479.

26.

Wei

G.W.

, Gray relational analysis method for intuitionistic fuzzy multiple attribute decision making, Expert Systems with Applications 38(9) (2011), 11671–11677.

27.

, Cosine similarity measures for intuitionistic fuzzy sets and their applications, Mathematical and Computer Modelling 53(1–2) (2011), 91–97.

28.

Wei

G.W.

, Grey relational analysis method for 2-tuple linguistic multiple attribute group decision making with incomplete weight information, Expert Systems with Applications 38(5) (2011), 4824–4828.

29.

Wei

G.W.

, FIOWHM operator and its application to multiple attribute group decision making, Expert Systems with Applications 38(4) (2011), 2984–2989.

30.

Wei

G.W.

, Grey relational analysis model for dynamic hybrid multiple attribute decision making, Knowledge-Based Systems 24(5) (2011), 672–679.

31.

X.Y.

and Wei

G.W.

, GRA method for multiple criteria group decision making with incomplete weight information under hesitant fuzzy setting, Journal of Intelligent and Fuzzy Systems 27 (2014), 1095–1105.

32.

Tang

Wen

L.L.

and Wei

G.W.

, Approaches to multiple attribute group decision making based on the generalized Dice similarity measures with intuitionistic fuzzy information, International Journal of Knowledge-based and Intelligent Engineering Systems 21(2) (2017), 85–95.

33.

Z.S.

, Fuzzy harmonic mean operators, International Journal of Intelligent Systems 24 (2009), 152–172.

34.

Lin

Zhao

X.F.

and Wei

G.W.

, Models for selecting an ERP system with hesitant fuzzy linguistic information, Journal of Intelligent and Fuzzy Systems 26(5) (2014), 2155–2165.

35.

Q.X.

Zhao

X.F.

and Wei

G.W.

, Model for transformation and upgrading ability of small and medium enterpries evaluation with hesitant fuzzy uncertain linguistic information, Journal of Intelligent and Fuzzy Systems 26(6) (2014), 2639–2647.

36.

Yager

R.R.

, The power average operator, IEEE Transactions on Systems, Man, and Cybernetics-Part A 31(6) (2001), 724–731.

37.

Z.S.

and Da

Q.L.

, An overview of operators for aggregating information, International Journal of Intelligent System 18 (2003), 953–969.

38.

Ran

L.G.

and Wei

G.W.

, Uncertain prioritized operators and their application to multiple attribute group decision making, Technological and Economic Development of Economy 21(1) (2015), 118–139.

39.

Lin

Zhao

X.F.

Wang

H.J.

and Wei

G.W.

, Hesitant fuzzy linguistic aggregation operators and their application to multiple attribute decision making, Journal of Intelligent and Fuzzy Systems 27 (2014), 49–63.

40.

Lin

Zhao

X.F.

and Wei

G.W.

, Fuzzy number intuitionistic fuzzy prioritized operators and their application to multiple attribute decision making, Journal of Intelligent and Fuzzy Systems 24 (2013), 879–888.

41.

Wei

G.W.

, Picture fuzzy cross-entropy for multiple attribute decision making problems, Journal of Business Economics and Management 17(4) (2016), 491–502.

42.

Wei

G.W.

Alsaadi

F.E.

Hayat

and Alsaedi

, Hesitant fuzzy linguistic arithmetic aggregation operators in multiple attribute decision making, Iranian Journal of Fuzzy Systems 13(4) (2016), 1–16.

43.

, Vector similarity measures of simplified neutrosophic sets and their application in multicriteria decision making, International Journal of Fuzzy Systems 16(2) (2014), 204–211.

44.

Wei

G.W.

, Interval valued hesitant fuzzy uncertain linguistic aggregation operators in multiple attribute decision making, International Journal of Machine Learning and Cybernetics 7(6) (2016), 1093–1114.

45.

Zhang

Wang

Tian

and Li

, Induced generalized hesitant fuzzy operators and their application to multiple attribute group decision making, Computers & Industrial Engineering 67 (2014), 116–138.

46.

Wei

G.W.

Alsaadi

F.E.

Hayat

and Alsaedi

, Hesitant fuzzy linguistic arithmetic aggregation operators in multiple attribute decision making, Iranian Journal of Fuzzy Systems 13(4) (2016), 1–16.

47.

, Vector similarity measures of simplified neutrosophic sets and their application in multicriteria decision making, International Journal of Fuzzy Systems 16(2) (2014), 204–211.

48.

Hung

K.C.

, Applications of medical information: Using an enhanced likelihood measured approach based on intuitionistic fuzzy sets, IIE Transactions on Healthcare Systems Engineering 2(3) (2012), 224–231.

49.

Wei

G.W.

Alsaadi

F.E.

Hayat

and Alsaedi

, A linear assignment method for multiple criteria decision analysis with hesitant fuzzy sets based on fuzzy measure, International Journal of Fuzzy Systems 19(3) (2017), 607–614.

50.

Hung

W.L.

and Yang

M.S.

, Similarity measures of intuitionistic fuzzy sets based on Hausdorff distance, Pattern Recognition Letters 25 (2004), 1603–1611.

51.

Liu

H.W.

, New similarity measures between intuitionistic fuzzy sets and between elements, Mathematical and Computer Modelling 42 (2005), 61–70.

52.

Hung

K.C.

, Applications of medical information: Using an enhanced likelihood measured approach based on intuitionistic fuzzy sets, IIE Transactions on Healthcare Systems Engineering 2(3) (2012), 224–231.

53.

, Multiple attribute group decision-making methods with unknown weights in intuitionistic fuzzy setting and interval-valued intuitionistic fuzzy setting, International Journal of General Systems 42(5) (2013), 489–502.

54.

Herrera-Viedma

Martinez

Mata

and Chiclana

, A consensus support system model for group decision-making problems with multigranular linguistic preference relations, IEEE Transactions on Fuzzy Systems 13 (2005), 644–658.

55.

Merigo

J.M.

and Casanovas

, Induced aggregation operators in decision making with the Dempster-Shafer belief structure, International Journal of Intelligent Systems 24 (2009), 934–954.

56.

Herrera

and Herrera-Viedma

, Linguistic decision analysis: Steps for solving decision problems under linguistic information, Fuzzy Sets and Systems 115 (2000), 67–82.

57.

Herrera

and Martínez

, A 2-tuple fuzzy linguistic representation model for computing with words, IEEE Transactions on Fuzzy Systems 8 (2000), 746–752.

58.

Wan

S.P.

and Li

D.F.

, Fuzzy mathematical programming approach to heterogeneous multiattribute decision-making with interval-valued intuitionistic fuzzy truth degrees, Information Science 325 (2015), 484–503.

59.

Wei

G.W.

and Zhao

X.F.

, Some induced correlated aggregating operators with intuitionistic fuzzy information and their application to multiple attribute group decision making, Expert Systems with Applications 39(2) (2012), 2026–2034.

60.

Park

D.G.

Kwun

Y.C.

Park

J.H.

and Park

I.Y.

, Correlation coefficient of interval-valued intuitionistic fuzzy sets and its application to multiple attribute group decision-making problems, Mathematical and Computer Modelling 50 (2009), 1279–293.