Comprehensive evaluation of regional economy development level in Jiangsu Province with 2-tuple linguistic information

Abstract

Since China’s adoption of reform and open-up policy in 1978, Jiangsu Province has achieved a fast economic development, with the average annual increasement rate of GDP per capita at 11.3% (counted with comparative price). In this process, the role of infrastructure in pushing the economic growth receives great attention. Study on the pushing effect of infrastructure on the economic growth will help the stipulation of proper infrastructure development strategy, so as to push forward economy and narrow the regional gap. In this paper, we investigates the comprehensive evaluation problems of regional economy development level in Jiangsu Province with 2-tuple linguistic information. Then we propose dependent 2-tuple linguistic Maclaurin symmetric mean (D2TLMSM) operator and then apply D2TLMSM operator for evaluating the regional economy development level in Jiangsu Province with 2-tuple linguistic information. At last, a practical example is proposed to test the effectiveness of the given method.

Keywords

Comprehensive evaluation linguistic information Maclaurin symmetric mean (MSM) operator dependent 2-tuple linguistic Maclaurin symmetric mean (D2TLMSM) operator regional economy development level

1 Introduction

In recent years, China’s economy develops with a high speed. Between the year 1978 to 2009, the average annual GDP growth was 9%, GDP grown more than 8% per capita average annual, and per capita income raised substantially, people’s living standards improved gradually, creating a remarkable “China miracle”. At the same time, China’s regional economy stepped into the overall high growth period, and a rapid increase in total economic output and per capita amount in each region had been seen, playing an important role in China’s sustained, rapid and healthy development. However, since China has a vast area with many ethnic groups, society, customs, cultural differences, historical traditions, and regional endowments, plus the impact of a number of factors such as geographic location and preferential policies, China’s regional economic development is very uneven among regions and there is still large gap between regions which gradually becomes an important problem constraining the healthy and sustainable development of China’s economic harmony. Therefore, how to correctly face the changes of regional disparities, reduce regional economic development gap and promote the harmony, sustainable development of our country and the region’s economy becomes hot issues that China’s economic circles, business community as well as the Government concern about.

However, in many situations, the input arguments take the form of 2-tuples linguistic variables due to the time pressure, and lack of knowledge [1 –15]. The comprehensive evaluation problems for evaluating the financial management performance with 2-tuple linguistic information is the multiple attribute decision making (MADM) problems [16 –28]. In this paper, we investigates the comprehensive evaluation problems of regional economy development level in Jiangsu Province with 2-tuple linguistic information. Then we propose dependent 2-tuple linguistic Maclaurin symmetric mean (D2TLMSM) operator and then apply D2TLMSM operator for evaluating the regional economy development level in Jiangsu Province with 2-tuple linguistic information. Finally, a practical example is proposed to test the effectiveness of the proposed method.

2 Preliminaries

The Maclaurin symmetric mean (MSM) was firstly discussed by Maclaurin in [29], that is a powerful technology characterized by the capability to obtain the relationship between the multi-input arguments. Afterwards, we propose the Definition of MSM:

Definition 1. [29] Assume that a_j (j = 1, 2, ⋯ , n) is a set of nonnegative numbers, and k = 1, 2, ⋯ , n. If

$\begin{matrix} {MSM}^{(k)} (a_{1}, a_{2}, \dots, a_{n}) \\ = {(\frac{\sum_{1 \leq i_{1} \leq \dots \leq i_{k} \leq n} \prod_{j = 1}^{k} a_{ij}}{C_{n}^{k}})}^{1 / k} \end{matrix}$ (1) then MSM^(k) is called the Maclaurin symmetric mean (MSM), where (i₁, i₂, ⋯ , i_k) traversal all the k-tuple combination of (1, 2, ⋯ , n), $C_{n}^{k}$ is the binomial coefficient.

We can see that the MSM have the following properties: $\begin{matrix} {MSM}^{(k)} (0, 0, \dots, 0) = 0; \\ {MSM}^{(k)} (a, a, \dots, a) = a; \end{matrix}$ $\begin{matrix} {MSM}^{(k)} (a_{1}, a_{2}, \dots, a_{n}) \\ \leq {MSM}^{(k)} (b_{1}, b_{2}, \dots, b_{n}), \end{matrix}$ if a_i ≤ b_i for all i; $min_{i} {a_{i}} \leq {MSM}^{(k)} (a_{1}, a_{2}, \dots, a_{n}) \leq max_{i} {a_{i}} .$

Assume that S ={ s_i |i = 1, 2, ⋯ , t } is a linguistic term set with odd cardinality. Any label, s_i represents a possible value for a linguistic variable, and it should satisfy the following characteristics [5, 6]:

(1) The set is ranked: s_i > s_j, if i > j; (2) Max operator: max(s_i, s_j) = s_i, if s_i ≥ s_j; (3) Min operator: min(s_i, s_j) = s_i, if s_i ≤ s_j. Particularly, S can be defined as follows. $\begin{matrix} S & = & {s_{1} = extremely poor (EP), \\ s_{2} = very poor (VP), \\ s_{3} = poor (P), s_{4} = medium (M), \\ s_{5} = good (G), s_{6} = very good (VG), \\ s_{7} = extremely good (EG)} \end{matrix}$

Definition 2. [5, 6] Suppose that β is the result of an aggregation of the indices of a set of labels assessed in a linguistic term set S, β ∈ [1, t], being t the cardinality of S. Let i = round (β) and α = β - i be two values, such that, i ∈ [1, t] and α ∈ [- 0.5, 0.5) then α is named as a symbolic translation.

Definition 3. [5, 6] Let S ={ s₁, s₂, ⋯ , s_t } be a linguistic term set and β ∈ [1, t] is a number value which denotes the aggregation result of linguistic symbolic. Then the function Δ is able to achieve the 2-tuple linguistic information related to β is given as: $Δ : [1, t] \to S \times [- 0.5, 0.5)$ (2) $Δ (β) = {\begin{matrix} s_{i}, i = round (β) \\ α = β - i, α \in [- 0.5, 0.5) \end{matrix}$ (3) where the function round(.) means the usual round operation, s_i has the closest index label to β and α refers to value of the symbolic translation.

Definition 4. [5, 6] Suppose that S ={ s₁, s₂, ⋯ , s_t } is a linguistic term set and (s_i, α_i) be a 2-tuple. From a 2-tuple (s_i, α_i) it return the related numerical value β ∈ [1, t] ⊂ R, which is. $Δ^{- 1} : S \times [- 0.5, 0.5) \to [1, t]$ (4) $Δ^{- 1} (s_{i}, α) = i + α = β$ (5)

3 Dependent 2-tuple linguistic Maclaurin symmetric mean operators

Utilizing the Definition 1, Zhang et al. [30] gave the definition of the 2-tuple linguistic Maclaurin symmetric mean (2TLMSM) operator as follows:

Definition 5. [30] Assume that x = {(r₁, a₁) , (r₂, a₂) , …, (r_n, a_n)} is a set of 2-tuple, and let p, q > 0. If

$\begin{matrix} {2 TLMSM}^{(k)} ((r_{1}, a_{1}), (r_{2}, a_{2}), \dots, (r_{n}, a_{n})) \\ = Δ {(\frac{\sum_{1 \leq i_{1} \leq \dots \leq i_{k} \leq n} \prod_{j = 1}^{k} Δ^{- 1} (r_{ij}, a_{ij})}{C_{n}^{k}})}^{1 / k} \end{matrix}$ (6) where (i₁, i₂, ⋯ , i_k) traversal all the k-tuple combination of (1, 2, ⋯ , n), $C_{n}^{k}$ is the binomial coefficient, then 2TLMSM^(k) is called the 2-tuple linguistic Maclaurin symmetric mean (2TLMSM) operator.

Considering that the input arguments may have different importance, furthermore, Zhang et al. [30] defined the 2-tuple linguistic weighted Maclaurin symmetric mean (2TLWMSM) operator.

Definition 6. Suppose that x = {(r₁, a₁) , (r₂, a₂) , …, (r_n, a_n)} is a set of 2-tuple and k > 0, w = (w₁, w₂, ⋯ , w_n) ^T is the weight vector of (r_i, a_i) (i = 1, 2, ⋯ , n), where w_i refers to the importance degree of (r_i, a_i), and the condition w_i > 0 (i = 1, 2, ⋯ , n), and $\sum_{i = 1}^{n} w_{i} = 1$ is satisfied. If

$\begin{matrix} {2 TLWMSM}_{w}^{(k)} ((r_{1}, a_{1}), (r_{2}, a_{2}), \dots, (r_{n}, a_{n})) \\ = Δ {(\frac{\sum_{1 \leq i_{1} \leq \dots \leq i_{k} \leq n} \prod_{j = 1}^{k} (w_{j} Δ^{- 1} (r_{ij}, a_{ij}))}{C_{n}^{k}})}^{1 / k} \end{matrix}$ (7) where (i₁, i₂, ⋯ , i_k) traversal all the k-tuple combination of (1, 2, ⋯ , n), $C_{n}^{k}$ denotes the binomial coefficient, then $2 {TLWMSM}_{w}^{(k)}$ is called the 2-tuple linguistic weighted Maclaurin symmetric mean (2TLWMSM) operator.

Definition 7. [31] Assume that x = {(r₁, a₁) , (r₂, a₂) , …, (r_n, a_n)} is a set of 2-tuples, and the 2-tuple arithmetic mean is calculated as follows.

$\begin{matrix} (\bar{r}, \bar{a}) = Δ (\frac{1}{n} \sum_{j = 1}^{n} Δ^{- 1} (r_{j}, a_{j})), \\ \bar{r} \in S, \bar{a} \in [- 0.5, 0.5) \end{matrix}$ (8)

Definition 8. [31] Assume that (r_i, a_i) and (r_j, a_j) are two 2-tuples, thus, we name

$\begin{matrix} d ((r_{i}, a_{i}), (r_{j}, a_{j})) \\ = \frac{| Δ^{- 1} (r_{i}, a_{i}) - Δ^{- 1} (r_{j}, a_{j}) |}{t} \end{matrix}$ (9) as the distance between (r_i, a_i) and (r_j, a_j).

Definition 9. [31] Assume that x = {(r₁, a₁) , (r₂, a₂) , …, (r_n, a_n)} is a set of 2-tuples, and let $(\bar{r}, \bar{a})$ the arithmetic mean of these 2-tuples. Based on the above definition, the following equation is satisfied.

$\begin{matrix} sim ((r_{j}, a_{j}), (\bar{r}, \bar{a})) \\ = 1 - \frac{d ((r_{j}, a_{j}), (\bar{r}, \bar{a}))}{\sum_{j = 1}^{n} d ((r_{j}, a_{j}), (\bar{r}, \bar{a}))}, j = 1, 2, \dots, n . \end{matrix}$ (10) the similarity between the given 2-tuples linguistic variables (r_j, a_j) and the mean $(\bar{r}, \bar{a})$ .

Utilizing Equation 7, the 2TLMSM weights can be defined as follows. $w_{j} = \frac{sim ((r_{j}, a_{j}), (\bar{r}, \bar{a}))}{\sum_{j = 1}^{n} sim ((r_{j}, a_{j}), (\bar{r}, \bar{a}))}, j = 1, 2, \dots, n$ (11)

We can see that w_j ≥ 0, j = 1, 2, ⋯ , n and $\sum_{j = 1}^{n} w_{j} = 1$ .

By (7), we have $\begin{matrix} {2 TLMSM}_{w}^{(k)} ((r_{1}, a_{1}), (r_{2}, a_{2}), \dots, (r_{n}, a_{n})) \\ = Δ {(\frac{\sum_{1 \leq i_{1} \leq \dots \leq i_{k} \leq n} \prod_{j = 1}^{k} (\frac{sim ((r_{j}, a_{j}), (\bar{r}, \bar{a})) Δ^{- 1} (r_{j}, a_{j})}{\sum_{j = 1}^{n} sim ((r_{j}, a_{j}), (\bar{r}, \bar{a}))})}{C_{n}^{k}})}^{1 / k} \end{matrix}$ (12)

We call (12) a dependent 2-tuple linguistic Maclaurin symmetric mean (D2TLMSM) operator.

Next, we will analyze several special cases of the D2TLMSM operator by taking various values of the parameter k.

Case 1. If k = 1, the D2TLMSM operator can be defined as follows.

$\begin{matrix} {D 2 TLMSM}^{(k)} ((r_{1}, a_{1}), (r_{2}, a_{2}), \dots, (r_{n}, a_{n})) \\ = Δ {(\frac{\sum_{1 \leq i_{1} \leq \dots \leq i_{k} \leq n} \prod_{j = 1}^{k} (\frac{sim ((r_{j}, a_{j}), (\bar{r}, \bar{a})) Δ^{- 1} (r_{j}, a_{j})}{\sum_{j = 1}^{n} sim ((r_{j}, a_{j}), (\bar{r}, \bar{a}))})}{C_{n}^{k}})}^{1 / k} \\ = {D 2 TLMSM}^{(1)} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) \\ = Δ {(\frac{\underset{1 \leq i \leq n}{\oplus} (\otimes_{j = 1}^{1} \sum_{1 \leq i_{1} \leq \dots \leq i_{k} \leq n} \prod_{j = 1}^{k} (\frac{sim ((r_{j}, a_{j}), (\bar{r}, \bar{a})) Δ^{- 1} (r_{j}, a_{j})}{\sum_{j = 1}^{n} sim ((r_{j}, a_{j}), (\bar{r}, \bar{a}))}))}{C_{n}^{1}})}^{1 / 1} \\ = Δ (\frac{\underset{1 \leq i \leq n}{\oplus} (\otimes_{j = 1}^{1} \sum_{1 \leq i_{1} \leq \dots \leq i_{k} \leq n} \prod_{j = 1}^{k} (\frac{sim ((r_{j}, a_{j}), (\bar{r}, \bar{a})) Δ^{- 1} (r_{j}, a_{j})}{\sum_{j = 1}^{n} sim ((r_{j}, a_{j}), (\bar{r}, \bar{a}))}))}{C_{n}^{1}}) \end{matrix}$ (13)

Case 2. If k = 2, utilizing the D2TLMSM operator, we can obtain the following equation.

4 Comprehensive evaluation of regional economy development level in Jiangsu Province with 2-tuple linguistic information

We use the D2TLMSM to comprehensive evaluation for evaluating the regional economy development level in Jiangsu Province with 2-tuple linguistic information. Assume that A ={ A₁, A₂, ⋯ , A_m } is a discrete set of alternatives, and G ={ G₁, G₂, ⋯ , G_n } is the set of attributes, whose weight vector is ω = (ω₁, ω₂, ⋯ , ω_n), with ω_j ≥ 0, j = 1, 2, ⋯ , n, $\sum_{j = 1}^{n} ω_{j} = 1$ . Assume that that R = (r_ij) _m×n is the comprehensive evaluation matrix, where r_ij ∈ S is the attribute values, that is used to generate the linguistic variable utilizing the alternative A_i ∈ A corresponding to the attribute G_j ∈ G.

Afterwards, we use the D2TLMSM operator to design a method to comprehensive evaluation problems for evaluating the regional economy development level in Jiangsu Province with 2-tuple linguistic information, which is able to be illustrated as follows.

Step 1. Convert the linguistic decision matrix $R = {(r_{ij})}_{m \times n}$ to 2-tuple linguistic decision matrix $\tilde{R} = {(r_{ij}, 0)}_{m \times n}$ .

Step 2. Exploit the decision information shown in matrix $\tilde{R}$ , and the D2TLMSM operator $\begin{matrix} {\tilde{r}}_{ij} & = & (r_{ij}, a_{ij}) \\ = & {D 2 TLMSM}_{w}^{(k)} ((r_{i 1}, 0), (r_{i 2}, 0), \dots, (r_{in}, 0)) \\ = & Δ {(\frac{\sum_{1 \leq i_{1} \leq \dots \leq i_{k} \leq n} \prod_{j = 1}^{k} (\frac{sim ((r_{ij}, 0), ({\bar{r}}_{i}, {\bar{a}}_{i})) Δ^{- 1} (r_{ij}, 0)}{\sum_{j = 1}^{n} sim ((r_{ij}, 0), ({\bar{r}}_{i}, {\bar{a}}_{i}))})}{C_{n}^{k}})}^{1 / k} \\ i = 1, 2, \dots, m, j = 1, 2, \dots, n . \end{matrix}$ (15)

Step 3. Sort all the alternatives A_i (i = 1, 2, ⋯ , m) in accordance with the overall 2-tuple linguistic preference values ${\tilde{r}}_{i} (i = 1, 2, \dots, m)$ of the alternative A_i. The larger the overall preference values ${\tilde{r}}_{i}$ , the better the alternative A_i will be.

5 Numerical example

The world’s innovation of technology has shown deep influence in the world’s economy and become an important impetus to the world after the world war II. The government pay more attentions ton integrate resources of all fields and promoting technical innovation. Especially since the communication revolution growing rapidly, government is very important to encourage the regional economy development investment of both the society and enterprises, and leading the direction of the regional economy development input. Furthermore, it has become a vital factor of enhancing the competitiveness. Thus the new function of every country’s government is to enhance the competitiveness and promote economic development. Nowadays, to enhance innovation competence independently is one of Jiangsu’s development strategies. Although regional economy development has become the leading department in Jiangsu, governmental public regional economy development will still play an important role in a long time because of the limitation of corporations and other reasons. Therefore, it is significative to discover the functionary mechanism between public economic development, so as to exert an influence on Jiangsu’s economic development, and enhance regional innovation competence independently. This section presents a numerical example for evaluating the regional economy development level in Jiangsu Province with linguistic information to describe the method proposed in this paper. The expert group must determine in terms of four attributes: G₁ is the per capita social consumable total volume of retail sales; G₂ is the per capita income of public financial budget; G₃ is the per capita disposable income of urban households; G₄ is the per capita net income of rural residents. The five possible cities A_i (i = 1, 2, ⋯ , 5) can be estimated utilizing the linguistic term set $\begin{matrix} S & = & {s_{1} = extremely poor (EP), \\ s_{2} = very poor (VP), \\ s_{3} = poor (P), s_{4} = medium (M), \\ s_{5} = good (G), s_{6} = very good (VG), \\ s_{7} = extremely good (EG)} \end{matrix}$ by the decision makers under the above four attributes (whose weighting vector is ω = (0.2, 0.4, 0.3, 0.1)), and construct the decision matrix R = (r_ij) _5×4. The linguistic decision matrix is shown in Table 1.

Next, to choose the most suitable city, we exploit the 2TLWMSM operator to develop models for evaluating the regional economy development level in Jiangsu Province with linguistic information, that is able to described as follows.

Firstly, we transform the linguistic decision information given in Table 1 into 2-tuple linguistic decision matrix which are given in Table 2.

Then, aggregate all 2-tuple linguistic preference value ${\tilde{r}}_{ij} (j = 1, 2, \dots, n)$ by using D2TLMSM to derive the overall linguistic preference values ${\tilde{r}}_{i} (i = 1, 2, 3, 4, 5)$ of the enterprises A_i. The aggregating results are illustrated as follows.

In the end, in terms of the aggregating results given in Table 3, the rank of the cities are illustrated in Table 4. Note that the symbol > denotes “preferred to”. The ordering of the alternatives is seen in Table 4 and the best city is A₃.

6 Conclusion

Regional economy strength between different area exists object during the economy development. Economic strength distance between cities result from several reasons and goes arise or down in the dynamic development. But how to analyze this distance objective and justice is a key problem for the government to make a policy or notice the problem during the economic development. In this paper, we investigates the comprehensive evaluation problems of regional economy development level in Jiangsu Province with 2-tuple linguistic information. Then we propose dependent 2-tuple linguistic Maclaurin symmetric mean (D2TLMSM) operator and then apply D2TLMSM operator for evaluating the regional economy development level in Jiangsu Province with 2-tuple linguistic information. In the end, a practical example is proposed to testify the developed method. Next, we will investigate the comprehensive evaluation problems of regional economy development level in Jiangsu Province with other fuzzy evaluating models [32 –46].

References

Managing the consensus in group decision making in an unbalanced fuzzy linguistic context with incomplete information, Knowledge-Based Systems 23(2) (2010), 169–181.

Herrera

, Martínez

and Sánchez

P.J.

, Managing non-homogeneous information in group decision making, European Journal of Operational Research 166(1) (2005), 115–132.

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.

Herrera

, Herrera-Viedma

and Martínez

, A fuzzy linguistic methodology to deal with unbalanced linguistic term sets, IEEE Transactions on Fuzzy Systems 16(2) (2008), 354–370.

Herrera-Viedma

and Lopez-Herrera

A.G.

, A review on information accessing systems based on fuzzy linguistic modelling, International Journal of Computational Intelligence Systems 3 (2010), 420–437.

Herrera

and Martinez

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

Herrera

and Martinez

, A model based on linguistic 2-tuples for dealing with multigranularity hierarchical linguistic contexts in multiexpert decision-making, IEEE Transactions on Systems, Man and Cybernetics-Part B: Cybernetics 31(2) (2001), 227–234.

Wei

G.W.

, Extension of TOPSIS method for 2-tuple linguistic multiple attribute group decision making with incomplete weight information, Knowledge and Information Systems 25 (2010), 623–634.

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 (2011), 4824–4828.

10.

Wei

G.W.

, Lin

, Zhao

X.F.

and Wang

H.J.

, Models for multiple attribute group decision making with 2-tuple linguistic assessment information, International Journal of Computational Intelligence Systems 3 (2010), 315–324.

11.

Wei

G.W.

, Some harmonic averaging operators with 2-tuple linguistic assessment information and their application to multiple attribute group decision making, International Journal of Uncertainty, Fuzziness and Knowledge- Based Systems 19(6) (2011), 977–998.

12.

Merigo

J.M.

, Gil-Lafuente

A.M.

, Zhou

L.G.

and Chen

H.Y.

, Generalization of the linguistic aggregation operator and its application in decision making, Journal of Systems Engineering and Electronics 22 (2011), 593–603.

13.

Merigó

J.M.

, Casanovas

and Martínez

, Linguistic aggregation operators for linguistic decision making based on the Dempster-Shafer theory of Evidence, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 18(3) (2010), 287–304.

14.

Wei

, Some generalized aggregating operators with linguistic information and their application to multiple attribute group decision making, Computers & Industrial Engineering 61(1) (2011), 32–38.

15.

Wei

, Some linguistic power aggregating operators and their application to multiple attribute group decision making, Journal of Intelligent and Fuzzy Systems 25 (2013), 695–707.

16.

Merigo

J.M.

and Casanovas

, Decision making with distance measures and linguistic aggregation operators, International Journal of Fuzzy Systems 12 (2010), 190–198.

17.

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.

18.

Liao

X.W.

, Li

and Lu

, A model for selecting an ERP system based on linguistic information processing, Information Systems 32 (2007), 1005–1017.

19.

Zhou

S.H.

, Chang

W.B.

and Xiong

Z.J.

, Risk assessment model with 2-tuple temporal linguistic variable. Shanghai, China: Trans Tech Publications, 2011, pp. 2540–2545.

20.

D.F.

, Multiattribute group decision making method using extended linguistic variables, International Journal of Uncertainty Fuzziness and Knowledge-Based Systems 17 (2009), 793–806.

21.

Y.J.

and Wang

H.M.

, Approaches based on 2-tuple linguistic power aggregation operators for multiple attribute group decision making under linguistic environment, Applied Soft Computing 11(5) (2011), 3988–3997.

22.

Zhang

and Guo

C.H.

, A method for multi-granularity uncertain linguistic group decision making with incomplete weight information, Knowledge-based Systems 26 (2012), 111–119.

23.

Liu

P.D.

, A novel method for hybrid multiple attribute decision making, Knowledge-based Systems 22(5) (2009), 388–391.

24.

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.

25.

Moreno

J.M.

, del Castillo

J.M.M.

, Porcel

and Herrera-Viedma

, A quality evaluation methodology for health-related websites based on a 2-tuple fuzzy linguistic approach, Soft Computing 14 (2010), 887–897.

26.

Zhou

L.G.

and Chen

H.Y.

, A generalization of the power aggregation operators for linguistic environment and its application in group decision making, Knowledge-based Systems 26 (2012), 216–224.

27.

Ran

L.-G.

and Wei

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

28.

Wei

G.W.

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

29.

Maclaurin

, A second letter toMartin Folkes, Esq.; concerning the roots of equations, with demonstration of other rules of algebra, Philos Trans Roy Soc London Ser A 36 (1729), 59–96.

30.

Zhang

, Wei

and Zhou

, Approaches to comprehensive evaluation with 2-tuple linguistic information, Journal of Intelligent and Fuzzy Systems 28(1) (2015), 469–475.

31.

Wei

and Zhao

, Some dependent aggregation operators with 2-tuple linguistic information and their application to multiple attribute group decision making, Expert Systems with Applications 39 (2012), 5881–5886.

32.

Benyounes

, Hafaifa

, Guemana

and Turbine

, Modeling based on fuzzy clustering algorithm using experimental data, Applied Artificial Intelligence 30(1) (2016), 29–51.

33.

Zhou

, Lin

, Zhao

and Wei

, 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.

34.

, Wang

, Tan

and Liu

X.-H.

, A state recognition approach for complex equipment based on a fuzzy probabilistic neural network, Algorithms 9(2) (2016), 34–44.

35.

Lin

, Zhao

and Wan

, Takagi-sugeno fuzzy model identification using coevolution particle swarm optimization with multi-strategy, Appl Intell 45(1) (2016), 187–197.

36.

Wang

, Shen

and Zhu

, Dual hesitant fuzzy power aggregation operators based on Archimedean t-conorm and t-norm and their application to multiple attribute group decision making, Appl Soft Comput 38 (2016), 23–50.

37.

Zhao

, Li

and Wei

, Some prioritized aggregating operators with linguistic information and their application to multiple attribute group decision making, Journal of Intelligent and Fuzzy Systems 26(4) (2014), 1619–1630.

38.

Rui

, Wei

, Wang

and Zhao

, Choquet integrals of weighted triangular fuzzy linguistic information and their applications to multiple attribute decision making, Journal of Business Economics and Management 15(5) (2014), 795–809.

39.

Heidarzade

, Mahdavi

and Mahdavi-Amiri

, Supplier selection using a clustering method based on a new distance for interval type-2 fuzzy sets: A case study, Appl Soft Comput 38 (2016), 213–231.

40.

Wei

, Lin

, Zhao

and Wang

, An Approach to multiple attribute decision making based on the induced Choquet integral with fuzzy number intuitionistic fuzzy information, Journal of Business Economics and Management 15(2) (2014), 277–298.

41.

Olvera-García

M.Á.

, Hernández

J.J.C.

, Fernández

L.P.S.

and Hernández-Bautista

, Air quality assessment using a weighted fuzzy inference system, Ecological Informatics 33 (2016), 57–74.

42.

Gupta

and Mohanty

B.K.

, An algorithmic approach to group decision making problems under fuzzy and dynamic environment, Expert Syst Appl 55 (2016), 118–132.

43.

Zhao

, Xu

and Ren

, On typical hesitant fuzzy prioritized “or” operator in multi-attribute decision making, Int J Intell Syst 31(1) (2016), 73–100.

44.

Wang

and Xu

, Admissible orders of typical hesitant fuzzy elements and their application in ordered information fusion in multi-criteria decision making, Information Fusion 29 (2016), 98–104.

45.

Gou

, Xu

and Lei

, New operational laws and aggregation method of intuitionistic fuzzy information, Journal of Intelligent and Fuzzy Systems 30(1) (2016), 129–141.

46.

Liu

, Xu

and Liao

, The multiplicative consistency index of hesitant fuzzy preference relation, IEEE Trans Fuzzy Systems 24(1) (2016), 82–93.