Fuzzy comprehensive evaluation model for evaluating the competitiveness of high technological parks with fuzzy information

Abstract

In this paper, we study on the multiple attribute decision making problems for evaluating the competitiveness of high technological parks with triangular fuzzy information. Inspired by the idea of Bonferroni mean (BM) and geometric Bonferroni mean (GBM), we develop the triangular fuzzy power Bonferroni mean (TFPBM) operator, triangular fuzzy weighted power Bonferroni mean (TFWPBM) operator, the triangular fuzzy power geometric Bonferroni mean (TFPGBM) operator, triangular fuzzy weighted power geometric Bonferroni mean (TFWPGBM) operator. Using the proposed operator, we propose the program for multiple attribute decision making with the triangular fuzzy environments. In the end, a practical example for evaluating the competitiveness of high technological parks with triangular fuzzy information is given to testify the performance of the given approach.

Keywords

multiple attribute decision making triangular fuzzy numbers power average (PA)Bonferroni mean triangular fuzzy weighted power Bonferroni mean (TFWPBM) operator triangular fuzzy weighted power geometric Bonferroni mean (TFWPGBM) operator high technological parks

1 Introduction

With the rapid development of social economy, and in the regional government to vigorously promote, nationwide, various types of parks mushroomed, rapid development, continue to expand the region, high-tech enterprises have be settled down. With the continuous expansion of the scale of the park, the number of enterprises settled in the park is increasing, the requirements of the construction of the park is becoming more and more high, the traditional way of the park management will be faced with a lot of challenges and problems.

The information aggregation operators are an interesting research topic, which is receiving increasing attention. The fundamental aspect of the OWA operator is a reordering step in which the input arguments are rearranged in descending order [1]. Since its appearance, it has been studied and applied in a wide range of problems [2 –11]. The ordered weighted geometric (OWG) operator is an aggregation operator that is based on the OWA operator and the geometric mean [12, 13]. In some situations, however, the input arguments take the form of fuzzy data rather than numerical ones because of time pressure, lack of knowledge, and the decision maker’s limited attention and information processing capabilities. Therefore, Xu [14] and Fan and Wang [15] developed the fuzzy ordered weighted averaging (FOWA) operator. Xu [16] introduced the fuzzy ordered weighted geometric (FOWG) operator. Xu and Wu [17] proposed the fuzzy induced ordered weighted averaging (FIOWA) operator. Xu and Da [18] developed the fuzzy induced ordered weighted geometric (FIOWG) operator. Xu [19] developed some fuzzy harmonic mean operators, such as fuzzy weighted harmonic mean (FWHM) operator, fuzzy ordered weighted harmonic mean (FOWHM) operator, fuzzy hybrid harmonic mean (FHHM) operator. Wei [20] proposed the fuzzy ordered weighted harmonic averaging(FOWHA) operator. Wei [21] developed the fuzzy induced ordered weighted harmonic mean (FIOWHM) operator and applied it to the group decision making. Wei [22] proposed the generalized triangular fuzzy correlated averaging operator and applied these operators to multiple attribute decision making. Merigo [23] presented the fuzzy probabilistic ordered weighted averaging (FPOWA) operator which is an aggregation operator that unifies the fuzzy probabilistic aggregation and the fuzzy OWA (FOWA) operator in the same formulation considering the degree of importance that each concept has in the analysis. Merigo and Casanovas [24] introduced several generalizations of the HA operator by using generalized and quasi-arithmetic means, fuzzy numbers and order inducing variables in the reordering step of the aggregation process. they presented the fuzzy generalized hybrid averaging (FGHA) operator, the fuzzy induced generalized hybrid averaging (FIGHA) operator, the Quasi-FHA operator and the Quasi-FIHA operator. The main advantage of these operators is that they generalize a wide range of fuzzy aggregation operators that can be used in a wide range of applications such as decision making problems. For example, they mentioned the fuzzy induced hybrid averaging (FIHA), the fuzzy weighted generalized mean (FWGM) and the fuzzy induced generalized OWA (FIGOWA). Merigo and Gil-Lafuente [25] proposed a wide range of fuzzy induced generalized aggregation operators such as the fuzzy induced generalized ordered weighted averaging (FIGOWA) and the fuzzy induced quasi-arithmetic OWA (Quasi-FIOWA) operator. They are aggregation operators that use the main characteristics of the fuzzy OWA (FOWA) operator, the induced OWA (IOWA) operator and the generalized (or quasi-arithmetic) OWA operator. Therefore, they use uncertain information represented in the form of fuzzy numbers, generalized (or quasi-arithmetic) means and order inducing variables. The main advantage of these operators is that they include a wide range of mean operators such as the FOWA, the IOWA, the induced Quasi-OWA, the fuzzy IOWA, the fuzzy generalized mean and the fuzzy weighted quasi-arithmetic average (Quasi-FWA). They further generalize this approach by using Choquet integrals, obtaining the fuzzy induced quasi-arithmetic Choquet integral aggregation (Quasi-FICIA) operator. Xu [26] considered situations with linguistic, interval or fuzzy preference information, and develop some fuzzy ordered distance measures, such as linguistic ordered weighted distance measure, uncertain ordered weighted distance measure, linguistic hybrid weighted distance measure, and uncertain hybrid weighted distance measure, etc. After that, based on hybrid weighted distance measures, they established a consensus reaching process of group decision making with linguistic, interval, triangular or trapezoidal fuzzy preference information

In this paper, we research on the multiple attribute decision making problems [27 –36] for evaluating the human resource value accounting measurement with triangular fuzzy information. Motivated by the idea of Bonferroni mean (BM) and geometric Bonferroni mean (GBM), we develop the triangular fuzzy power Bonferroni mean (TFPBM) operator, triangular fuzzy weighted power Bonferroni mean (TFWPBM) operator, the triangular fuzzy power geometric Bonferroni mean (TFPGBM) operator, triangular fuzzy weighted power geometric Bonferroni mean (TFWPGBM) operator. Using the proposed operator, we propose the program for multiple attribute decision making with the triangular fuzzy environments. In the end, a practical example for evaluating the human resource value accounting measurement with triangular fuzzy information is given to testify the performance of the given approach.

2 Preliminaries

2.1 Triangular fuzzy sets

In this section, we will simply introduce some basic concepts and basic operational rules, which are corresponding to triangular fuzzy numbers.

Definition 1. [37] A triangular fuzzy numbers $\tilde{a}$ can be defined via a triplet (a^L, a^M, a^U).The membership function $μ_{\tilde{a}} (x)$ is defined as: $μ_{\tilde{a}} (x) = {\begin{matrix} 0, x < a^{L}, \\ \frac{x - a^{L}}{a^{M} - a^{L}}, a^{L} \leq x \leq a^{M}, \\ \frac{x - a^{U}}{a^{M} - a^{U}}, a^{M} \leq x \leq a^{U}, \\ 0, x \geq a^{U} . \end{matrix}$ (1) where 0 < a^L ≤ a^M ≤ a^U, a^L and a^U represent the lower and upper values of the support of $\tilde{a}$ , respectively, and a^M is used to represent the modal value.

Definition 2 [38 –41]. Let $\tilde{b} = [b^{L}, b^{M}, b^{U}]$ and $\tilde{a} = [a^{L}, a^{M}, a^{U}]$ be two triangular fuzzy numbers, then the degree of possibility of a ≥ b is defined as $\begin{matrix} p (a \geq b) = λ max {1 - max [\frac{b^{M} - a^{L}}{a^{M} - a^{L} + b^{M} - b^{L}}, 0], 0} + \\ (1 - λ) max {1 - max [\frac{b^{U} - a^{M}}{a^{U} - a^{M} + b^{U} - b^{M}}, 0], 0} \end{matrix}$ (2) where the value λ refers to an index of rating attitude, which can reflect the decision maker’s risk-bearing attitude. If λ is larger than 0.5, the decision maker is risk lover. If λ is equal to 0.5, the decision maker may be neutral to risk. If the value of parameter λ is smaller than 0.5, the decision maker is risk averters.

2.2 Power aggregation operator

Yager [42] developed a power average (PA) operator. $PA (x_{1}, x_{2}, \dots, x_{n}) = \frac{\sum_{i = 1}^{n} (1 + T (x_{i})) x_{i}}{\sum_{i = 1}^{n} (1 + T (x_{i}))}$ (3) where $T (x_{i}) = \sum_{j = 1, j \neq i}^{n} Sup (x_{i}, x_{j})$ , and Sup(x_i, x_j) is the support for x_i from x_j, which satisfies the following three properties: (1)Sup (x_i, x_j) ∈ [0, 1]; (2)Sup (x_i, x_j) = Sup (x_j, x_i); (3) Sup (c, d) ≥ Sup (x, y), if |c - d| < |x - y|. Obviously, the support (Sup) measure is essentially a similarity index. The more similar, the closer two values, and the more they support each other.

2.3 Bonferroni mean operator

Bonferroni [43] originally introduced Bonferroni mean, which can provide for aggregation lying between the max, min operators and the logical “or” and “and” operators.

Definition 3. [43] Let p, q ≥ 0 and a_i (i = 1, 2, ⋯ , n) be a collection of non-negative real numbers. Then the aggregation functions: ${BM}^{p, q} (a_{1}, a_{2}, \dots, a_{n}) = {(\frac{1}{n (n - 1)} \sum_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} a_{i}^{p} a_{j}^{q})}^{\frac{1}{p + q}}$ (4) is called the Bonferroni mean (BM) operator.

3 TFPBM and TFWPBM operators

3.1 TFPBM operator

In the following, we shall develop the triangular fuzzy power Bonferroni mean (TFPBM) aggregation operator based on the operations of TFNs and power average [42].

Definition 4. Let ${\tilde{a}}_{j} = [a_{j}^{L}, a_{j}^{M}, a_{j}^{U}]$ (j = 1, 2, … , n) be a collection of TFNs, p, q ≥ 0, then we define the triangular fuzzy power Bonferroni mean (TFPBM) operator as follows: $\begin{matrix} TFPBM ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) \\ = {(\sum_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} (\frac{1 + T ({\tilde{a}}_{i})}{\sum_{t = 1}^{n} (1 + T ({\tilde{a}}_{t}))} \frac{1 + T ({\tilde{a}}_{j})}{\sum_{t = 1}^{n} (1 + T ({\tilde{a}}_{t}))} {({\tilde{a}}_{i})}^{p} \times {({\tilde{a}}_{j})}^{q}))}^{\frac{1}{p + q}} \end{matrix}$ (5)

where $T ({\tilde{a}}_{j}) = \sum_{\begin{matrix} i = 1 \\ i \neq j \end{matrix}}^{n} Sup ({\tilde{a}}_{j}, {\tilde{a}}_{i})$ (6) and $Sup ({\tilde{a}}_{j}, {\tilde{a}}_{i})$ is the support for ${\tilde{a}}_{j}$ from ${\tilde{a}}_{i}$ , with the conditions:

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

$Sup ({\tilde{a}}_{j}, {\tilde{a}}_{i}) = Sup ({\tilde{a}}_{i}, {\tilde{a}}_{j})$ ;

$Sup ({\tilde{a}}_{j}, {\tilde{a}}_{i}) \geq Sup ({\tilde{a}}_{s}, {\tilde{a}}_{t})$ , if $d ({\tilde{a}}_{j}, {\tilde{a}}_{i}) < d ({\tilde{a}}_{s}, {\tilde{a}}_{t})$ , where d is a distance measure.

Theorem 1. The aggregated value by using TFBPM operator is also a TFN, where $\begin{matrix} TFPBM ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) \\ = {(\sum_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} (\frac{1 + T ({\tilde{a}}_{i})}{\sum_{t = 1}^{n} (1 + T ({\tilde{a}}_{t}))} \frac{1 + T ({\tilde{a}}_{j})}{\sum_{t = 1}^{n} (1 + T ({\tilde{a}}_{t}))} {({\tilde{a}}_{i})}^{p} \times {({\tilde{a}}_{j})}^{q}))}^{\frac{1}{p + q}} \\ = {\begin{matrix} {(\sum_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} (\frac{1 + T ({\tilde{a}}_{i})}{\sum_{t = 1}^{n} (1 + T ({\tilde{a}}_{t}))} \frac{1 + T ({\tilde{a}}_{j})}{\sum_{t = 1}^{n} (1 + T ({\tilde{a}}_{t}))} {(a_{i}^{L})}^{p} \times {(a_{j}^{L})}^{q}))}^{\frac{1}{p + q}}, \\ {(\sum_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} (\frac{1 + T ({\tilde{a}}_{i})}{\sum_{t = 1}^{n} (1 + T ({\tilde{a}}_{t}))} \frac{1 + T ({\tilde{a}}_{j})}{\sum_{t = 1}^{n} (1 + T ({\tilde{a}}_{t}))} {(a_{i}^{M})}^{p} \times {(a_{j}^{M})}^{q}))}^{\frac{1}{p + q}}, \\ {(\sum_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} (\frac{1 + T ({\tilde{a}}_{i})}{\sum_{t = 1}^{n} (1 + T ({\tilde{a}}_{t}))} \frac{1 + T ({\tilde{a}}_{j})}{\sum_{t = 1}^{n} (1 + T ({\tilde{a}}_{t}))} {(a_{i}^{U})}^{p} \times {(a_{j}^{U})}^{q}))}^{\frac{1}{p + q}} \end{matrix}} \end{matrix}$ (7) where $T ({\tilde{a}}_{j}) = \sum_{\begin{matrix} i = 1 \\ i \neq j \end{matrix}}^{n} Sup ({\tilde{a}}_{j}, {\tilde{a}}_{i})$ (8)

The TFPBM operator has the following properties.

Property 1. (Idempotency) Let ${\tilde{a}}_{i} = [a_{i}^{L},$ $a_{i}^{M}, a_{i}^{U}] (i = 1, 2, \dots, n)$ be a set of triangular fuzzy numbers. If all ${\tilde{a}}_{j} ({\tilde{a}}_{j} = [a_{j}^{L}, a_{j}^{M}, a_{j}^{U}])$ are equal, i.e. ${\tilde{a}}_{j} ({\tilde{a}}_{j} = [a_{j}^{L}, a_{j}^{M}, a_{j}^{U}]) = \tilde{a} (\tilde{a} = [a^{L}, a^{M}, a^{U}])$ for all j, then $TFPBM ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) = \tilde{a}$ (9)

Property 2. (Boundedness) Let ${\tilde{a}}_{i} = [a_{i}^{L},$ $a_{i}^{M}, a_{i}^{U}] (i = 1, 2, \dots, n)$ be a set of triangular fuzzy numbers, and let ${\tilde{a}}^{-} \underset{j}{= min} {\tilde{a}}_{j}, {\tilde{a}}^{+} \underset{j}{= min} {\tilde{a}}_{j}$

Then ${\tilde{a}}^{-} \leq TFPBM ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) \leq {\tilde{a}}^{+}$ (10)

Property 3. (Monotonicity) Let ${\tilde{a}}_{i} = [a_{i}^{L}, a_{i}^{M}, a_{i}^{U}] (i = 1, 2, \dots, n)$ and ${\tilde{a}}_{i}^{'} = [a_{i}^{L'},$ $a_{i}^{M}, a_{i}^{U'}] (i = 1, 2, \dots, n)$ be two sets of triangular fuzzy numbers, if ${\tilde{a}}_{j} \leq {\tilde{a}}_{j}^{'}$ , for all j, then $TFPBM ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) \leq TFPBM ({\tilde{a}}_{1}^{'}, {\tilde{a}}_{2}^{'}, \dots, {\tilde{a}}_{n}^{'})$ (11)

Property 4. (Commutativity) Let ${\tilde{a}}_{i} = [a_{i}^{L},$ $a_{i}^{M}, a_{i}^{U}] (i = 1, 2, \dots, n)$ and ${\tilde{a}}_{i}^{'} = [a_{i}^{L'},$ $a_{i}^{M'}, a_{i}^{U'}] (i = 1, 2, \dots, n)$ be two sets of triangular fuzzy numbers, where ${\tilde{a}}_{i}^{'} = [a_{i}^{L'}, a_{i}^{M'}, a_{i}^{U'}] (i = 1, 2, \dots, n)$ is any permutation of ${\tilde{a}}_{i} = [a_{i}^{L}, a_{i}^{M}, a_{i}^{U}] (i = 1, 2, \dots, n)$ , then $TFPBM ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) = TFPBM ({\tilde{a}}_{1}^{'}, {\tilde{a}}_{2}^{'}, \dots, {\tilde{a}}_{n}^{'})$ (12)

3.2 TFWPBM operator

Because the input arguments have different importance, we present the definition of the triangular fuzzy weighted power Bonferroni mean (TFWPBM) operator.

Definition 5. Let ${\tilde{a}}_{i} = [a_{i}^{L}, a_{i}^{M}, a_{i}^{U}] (i = 1, 2,$ ⋯, n) be a set of triangular fuzzy numbers and p, q > 0, w = (w₁, w₂, ⋯ , w_n) ^Tis the weight vector of ${\tilde{a}}_{i} = [a_{i}^{L}, a_{i}^{M}, a_{i}^{U}] (i = 1, 2, \dots, n)$ , where w_i indicates the importance degree of ${\tilde{a}}_{i}$ , satisfying w_i > 0 (i = 1, 2, ⋯ , n), and $\sum_{i = 1}^{n} w_{i} = 1$ . If $\begin{matrix} TFWPBM ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) \\ = {(\sum_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} (\frac{w_{i} (1 + T ({\tilde{a}}_{i}))}{\sum_{t = 1}^{n} w_{t} (1 + T ({\tilde{a}}_{t}))} \frac{w_{j} (1 + T ({\tilde{a}}_{j}))}{\sum_{t = 1}^{n} w_{t} (1 + T ({\tilde{a}}_{t}))} {({\tilde{a}}_{i})}^{p} \times {({\tilde{a}}_{j})}^{q}))}^{\frac{1}{p + q}} \\ = {\begin{matrix} {(\sum_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} (\frac{w_{i} (1 + T ({\tilde{a}}_{i}))}{\sum_{t = 1}^{n} w_{t} (1 + T ({\tilde{a}}_{t}))} \frac{w_{j} (1 + T ({\tilde{a}}_{j}))}{\sum_{t = 1}^{n} w_{t} (1 + T ({\tilde{a}}_{t}))} {(a_{i}^{L})}^{p} \times {(a_{j}^{L})}^{q}))}^{\frac{1}{p + q}}, \\ {(\sum_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} (\frac{w_{i} (1 + T ({\tilde{a}}_{i}))}{\sum_{t = 1}^{n} w_{t} (1 + T ({\tilde{a}}_{t}))} \frac{w_{j} (1 + T ({\tilde{a}}_{j}))}{\sum_{t = 1}^{n} w_{t} (1 + T ({\tilde{a}}_{t}))} {(a_{i}^{M})}^{p} \times {(a_{j}^{M})}^{q}))}^{\frac{1}{p + q}}, \\ {(\sum_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} (\frac{w_{i} (1 + T ({\tilde{a}}_{i}))}{\sum_{t = 1}^{n} w_{t} (1 + T ({\tilde{a}}_{t}))} \frac{w_{j} (1 + T ({\tilde{a}}_{j}))}{\sum_{t = 1}^{n} w_{t} (1 + T ({\tilde{a}}_{t}))} {(a_{i}^{U})}^{p} \times {(a_{j}^{U})}^{q}))}^{\frac{1}{p + q}} \end{matrix}} \end{matrix}$ (13) where $T ({\tilde{a}}_{j}) = \sum_{\begin{matrix} i = 1 \\ i \neq j \end{matrix}}^{n} w_{i} Sup ({\tilde{a}}_{j}, {\tilde{a}}_{i})$ (14)

The TFWPBM operator has the following properties.

Property 5. (Idempotency) Let ${\tilde{a}}_{i} = [a_{i}^{L}, a_{i}^{M}, a_{i}^{U}] (i = 1, 2, \dots, n)$ be a set of triangular fuzzy numbers. If all ${\tilde{a}}_{j} ({\tilde{a}}_{j} = [a_{j}^{L}, a_{j}^{M}, a_{j}^{U}])$ are equal, i.e. ${\tilde{a}}_{j} ({\tilde{a}}_{j} = [a_{j}^{L}, a_{j}^{M}, a_{j}^{U}]) = \tilde{a} (\tilde{a} = [a^{L}, a^{M}, a^{U}])$ for all j, then $TFWPBM ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) = \tilde{a}$ (15)

Property 6. (Boundedness) Let ${\tilde{a}}_{i} = [a_{i}^{L}, a_{i}^{M}, a_{i}^{U}] (i = 1, 2, \dots, n)$ be a set of triangular fuzzy numbers, and let ${\tilde{a}}^{-} \underset{j}{= min} {\tilde{a}}_{j}, {\tilde{a}}^{+} \underset{j}{= min} {\tilde{a}}_{j}$

Then ${\tilde{a}}^{-} \leq TFWPBM ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) \leq {\tilde{a}}^{+}$ (16)

Property 7. (Monotonicity) Let ${\tilde{a}}_{i} = [a_{i}^{L}, a_{i}^{M}, a_{i}^{U}] (i = 1, 2, \dots, n)$ and ${\tilde{a}}_{i}^{'} = [a_{i}^{L'}, a_{i}^{M'}, a_{i}^{U'}] (i = 1, 2, \dots, n)$ be two sets of triangular fuzzy numbers, if ${\tilde{a}}_{j} \leq {\tilde{a}}_{j}^{'}$ , for all j, then $\begin{matrix} TFWPBM ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) \\ \leq TFWPBM ({\tilde{a}}_{1}^{'}, {\tilde{a}}_{2}^{'}, \dots, {\tilde{a}}_{n}^{'}) \end{matrix}$ (17)

Property 8. (Commutativity) Let ${\tilde{a}}_{i} = [a_{i}^{L}, a_{i}^{M}, a_{i}^{U}] (i = 1, 2, \dots, n)$ and ${\tilde{a}}_{i}^{'} = [a_{i}^{L'}, a_{i}^{M'}, a_{i}^{U'}] (i = 1, 2, \dots, n)$ be two sets of triangular fuzzy numbers, where ${\tilde{a}}_{i}^{'} = [a_{i}^{L'}, a_{i}^{M'}, a_{i}^{U'}] (i = 1, 2, \dots, n)$ is any permutation of ${\tilde{a}}_{i} = [a_{i}^{L}, a_{i}^{M}, a_{i}^{U}] (i = 1, 2, \dots, n)$ , then $\begin{matrix} TFWPBM ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) \\ = TFWPBM ({\tilde{a}}_{1}^{'}, {\tilde{a}}_{2}^{'}, \dots, {\tilde{a}}_{n}^{'}) \end{matrix}$ (18)

4 TFPGBM and TFWPGBM operators

4.1 TFPGBM operator

Based on the PA operator [42] and geometric mean [44 –54], Xu and Yager [55] further defined a power geometric (PG) operator: $PG (a_{1}, a_{2}, \dots, a_{n}) = \prod_{i = 1}^{n} a_{i}^{\frac{1 + T (a_{i})}{\sum_{i = 1}^{n} (1 + T (a_{i}))}}$ (19)

Obviously, the PA and PG operators are two nonlinear weighted aggregation tools, whose weighting vectors depend upon the input values and allow values being aggregated to support and reinforce each other, that’s to say, the closer a_i and a_j, the more similar they are, and the more they support each other.

Definition 6 [5 5]. Let p, q ≥ 0 and a_i (i = 1, 2, ⋯ , n) be a collection of non-negative real numbers. Then the aggregation functions: $\begin{matrix} {GBM}^{p, q} (a_{1}, a_{2}, \dots, a_{n}) \\ = \frac{1}{p + q} {(\prod_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} ({pa}_{i} + {qa}_{j}))}^{\frac{1}{n (n - 1)}} \end{matrix}$ (20)

is called the geometric Bonferroni mean (GBM) operator.

Definition 7. Let ${\tilde{a}}_{i} = [a_{i}^{L}, a_{i}^{M}, a_{i}^{U}] (i = 1, 2,$ ⋯, n) be a collection of TFNs, p, q ≥ 0, then we define the triangular fuzzy power geometric Bonferroni mean (TFPGBM) operator as follows: $\begin{matrix} TFPGBM ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) \\ = \frac{1}{p + q} \prod_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} {(p {\tilde{a}}_{i} + q {\tilde{a}}_{j})}^{\frac{1 + T ({\tilde{a}}_{i})}{\sum_{t = 1}^{n} (1 + T ({\tilde{a}}_{t}))} \frac{1 + T ({\tilde{a}}_{j})}{\sum_{t = 1}^{n} (1 + T ({\tilde{a}}_{t}))}} \end{matrix}$ (21)

where $T ({\tilde{a}}_{j}) = \sum_{\begin{matrix} i = 1 \\ i \neq j \end{matrix}}^{n} Sup ({\tilde{a}}_{j}, {\tilde{a}}_{i})$ (22) and $Sup ({\tilde{a}}_{j}, {\tilde{a}}_{i})$ is the support for ${\tilde{a}}_{j}$ from ${\tilde{a}}_{i}$ , with the conditions:

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

$Sup ({\tilde{a}}_{j}, {\tilde{a}}_{i}) = Sup ({\tilde{a}}_{i}, {\tilde{a}}_{j})$ ;

Theorem 2. The aggregated value by using TFPGBM operator is also a TFN, where $\begin{matrix} TFPGBM = \frac{1}{p + q} \prod_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} {(p {\tilde{a}}_{i} + q {\tilde{a}}_{j})}^{\frac{1 + T ({\tilde{a}}_{i})}{\sum_{t = 1}^{n} (1 + T ({\tilde{a}}_{t}))} \frac{1 + T ({\tilde{a}}_{j})}{\sum_{t = 1}^{n} (1 + T ({\tilde{a}}_{t}))}} \\ = {\begin{matrix} \frac{1}{p + q} \prod_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} {({pa}_{i}^{L} + {qa}_{j}^{L})}^{\frac{1 + T ({\tilde{a}}_{i})}{\sum_{t = 1}^{n} (1 + T ({\tilde{a}}_{t}))} \frac{1 + T ({\tilde{a}}_{j})}{\sum_{t = 1}^{n} (1 + T ({\tilde{a}}_{t}))}}, \\ \frac{1}{p + q} \prod_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} {({pa}_{i}^{M} + q {\tilde{a}}_{j}^{M})}^{\frac{1 + T ({\tilde{a}}_{i})}{\sum_{t = 1}^{n} (1 + T ({\tilde{a}}_{t}))} \frac{1 + T ({\tilde{a}}_{j})}{\sum_{t = 1}^{n} (1 + T ({\tilde{a}}_{t}))}}, \\ \frac{1}{p + q} \prod_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} {({pa}_{i}^{U} + q {\tilde{a}}_{j}^{U})}^{\frac{1 + T ({\tilde{a}}_{i})}{\sum_{t = 1}^{n} (1 + T ({\tilde{a}}_{t}))} \frac{1 + T ({\tilde{a}}_{j})}{\sum_{t = 1}^{n} (1 + T ({\tilde{a}}_{t}))}} \end{matrix}} \end{matrix}$ (23)

where $T ({\tilde{a}}_{j}) = \sum_{\begin{matrix} i = 1 \\ i \neq j \end{matrix}}^{n} Sup ({\tilde{a}}_{j}, {\tilde{a}}_{i})$ (24)

The TFPGBM operator has the following properties.

Property 9. (Idempotency) Let ${\tilde{a}}_{i} = [a_{i}^{L}, a_{i}^{M}, a_{i}^{U}] (i = 1, 2, \dots, n)$ be a set of triangular fuzzy numbers. If all ${\tilde{a}}_{j} ({\tilde{a}}_{j} = [a_{j}^{L}, a_{j}^{M}, a_{j}^{U}])$ are equal, i.e. ${\tilde{a}}_{j} ({\tilde{a}}_{j} = [a_{j}^{L}, a_{j}^{M}, a_{j}^{U}]) = \tilde{a} (\tilde{a} = [a^{L}, a^{M}, a^{U}])$ for all j, then $TFPGBM ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) = \tilde{a}$ (25)

Property 10. (Boundedness) Let ${\tilde{a}}_{i} = [a_{i}^{L}, a_{i}^{M}, a_{i}^{U}] (i = 1, 2, \dots, n)$ be a set of triangular fuzzy numbers, and let ${\tilde{a}}^{-} \underset{j}{= min} {\tilde{a}}_{j}, {\tilde{a}}^{+} \underset{j}{= min} {\tilde{a}}_{j}$

Then ${\tilde{a}}^{-} \leq TFPGBM ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) \leq {\tilde{a}}^{+}$ (26)

Property 11. (Monotonicity) Let ${\tilde{a}}_{i} = [a_{i}^{L}, a_{i}^{M}, a_{i}^{U}] (i = 1, 2, \dots, n)$ and ${\tilde{a}}_{i}^{'} = [a_{i}^{L'}, a_{i}^{M'}, a_{i}^{U'}] (i = 1, 2, \dots, n)$ be two sets of triangular fuzzy numbers, if ${\tilde{a}}_{j} \leq {\tilde{a}}_{j}^{'}$ , for all j, then $TFPGBM ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) \leq TFPGBM ({\tilde{a}}_{1}^{'}, {\tilde{a}}_{2}^{'}, \dots, {\tilde{a}}_{n}^{'})$ (27)

Property 12. (Commutativity) Let ${\tilde{a}}_{i} = [a_{i}^{L}, a_{i}^{M}, a_{i}^{U}] (i = 1, 2, \dots, n)$ and ${\tilde{a}}_{i}^{'} = [a_{i}^{L'}, a_{i}^{M'}, a_{i}^{U'}] (i = 1, 2, \dots, n)$ be two sets of triangular fuzzy numbers, where ${\tilde{a}}_{i}^{'} = [a_{i}^{L'}, a_{i}^{M'}, a_{i}^{U'}] (i = 1, 2, \dots, n)$ is any permutation of ${\tilde{a}}_{i} = [a_{i}^{L}, a_{i}^{M}, a_{i}^{U}] (i = 1, 2, \dots, n)$ , then $TFPGBM ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) = TFPGBM ({\tilde{a}}_{1}^{'}, {\tilde{a}}_{2}^{'}, \dots, {\tilde{a}}_{n}^{'})$ (28)

4.2 TFWPGBM operator

Because the input arguments have different importance, we present the definition of the triangular fuzzy weighted power geometric Bonferroni mean (TFWPGBM) operator.

Definition 8. ${\tilde{a}}_{i} = [a_{i}^{L}, a_{i}^{M}, a_{i}^{U}] (i = 1, 2, \dots, n)$ be a set of triangular fuzzy numbers and p, q > 0, w = (w₁, w₂, ⋯ , w_n) ^T is the weight vector of ${\tilde{a}}_{i} = [a_{i}^{L}, a_{i}^{M}, a_{i}^{U}] (i = 1, 2, \dots, n)$ , wherew_i indicates the importance degree of ${\tilde{a}}_{i}$ , satisfying w_i > 0 (i = 1, 2, ⋯ , n), and $\sum_{i = 1}^{n} w_{i} = 1$ . If $\begin{matrix} TFWPGBM = \\ \frac{1}{p + q} \prod_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} {(p {\tilde{a}}_{i} + q {\tilde{a}}_{j})}^{\frac{ω_{i} (1 + T (A_{i}))}{\sum_{t = 1}^{n} ω_{t} (1 + T (A_{t}))} \frac{ω_{j} (1 + T (A_{j}))}{\sum_{t = 1}^{n} ω_{t} (1 + T (A_{t}))}} \\ = {\begin{matrix} \frac{1}{p + q} \prod_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} {({pa}_{i}^{L} + {qa}_{j}^{L})}^{\frac{ω_{i} (1 + T (A_{i}))}{\sum_{t = 1}^{n} ω_{t} (1 + T (A_{t}))} \frac{ω_{j} (1 + T (A_{j}))}{\sum_{t = 1}^{n} ω_{t} (1 + T (A_{t}))}}, \\ \frac{1}{p + q} \prod_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} {({pa}_{i}^{M} + q {\tilde{a}}_{j}^{M})}^{\frac{ω_{i} (1 + T (A_{i}))}{\sum_{t = 1}^{n} ω_{t} (1 + T (A_{t}))} \frac{ω_{j} (1 + T (A_{j}))}{\sum_{t = 1}^{n} ω_{t} (1 + T (A_{t}))}}, \\ \frac{1}{p + q} \prod_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} {({pa}_{i}^{U} + q {\tilde{a}}_{j}^{U})}^{\frac{ω_{i} (1 + T (A_{i}))}{\sum_{t = 1}^{n} ω_{t} (1 + T (A_{t}))} \frac{ω_{j} (1 + T (A_{j}))}{\sum_{t = 1}^{n} ω_{t} (1 + T (A_{t}))}} \end{matrix}} \end{matrix}$ (29)

where $T ({\tilde{a}}_{j}) = \sum_{\begin{matrix} i = 1 \\ i \neq j \end{matrix}}^{n} w_{i} Sup ({\tilde{a}}_{j}, {\tilde{a}}_{i})$ (30)

The TFWPGBM operator has the following properties.

Property 13. (Idempotency) Let ${\tilde{a}}_{i} = [a_{i}^{L}, a_{i}^{M}, a_{i}^{U}] (i = 1, 2, \dots, n)$ be a set of triangular fuzzy numbers. If all ${\tilde{a}}_{j} ({\tilde{a}}_{j} = [a_{j}^{L}, a_{j}^{M}, a_{j}^{U}])$ are equal, i.e. ${\tilde{a}}_{j} ({\tilde{a}}_{j} = [a_{j}^{L}, a_{j}^{M}, a_{j}^{U}]) = \tilde{a} (\tilde{a} = [a^{L}, a^{M}, a^{U}])$ for all j, then $TFWPGBM ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) = \tilde{a}$ (31)

Property 14. (Boundedness) Let ${\tilde{a}}_{i} = [a_{i}^{L}, a_{i}^{M}, a_{i}^{U}] (i = 1, 2, \dots, n)$ be a set of triangular fuzzy numbers, and let ${\tilde{a}}^{-} \underset{j}{= min} {\tilde{a}}_{j}, {\tilde{a}}^{+} \underset{j}{= min} {\tilde{a}}_{j}$

Then ${\tilde{a}}^{-} \leq TFWPGBM ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) \leq {\tilde{a}}^{+}$ (32)

Property 15. (Monotonicity) Let ${\tilde{a}}_{i} = [a_{i}^{L}, a_{i}^{M}, a_{i}^{U}] (i = 1, 2, \dots, n)$ and ${\tilde{a}}_{i}^{'} = [a_{i}^{L'}, a_{i}^{M'}, a_{i}^{U'}] (i = 1, 2, \dots, n)$ be two sets of triangular fuzzy numbers, if ${\tilde{a}}_{j} \leq {\tilde{a}}_{j}^{'}$ , for all j, then $\begin{matrix} TFWPGBM ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) \\ \leq TFWPGBM ({\tilde{a}}_{1}^{'}, {\tilde{a}}_{2}^{'}, \dots, {\tilde{a}}_{n}^{'}) \end{matrix}$ (33)

Property 16. (Commutativity) Let ${\tilde{a}}_{i} = [a_{i}^{L}, a_{i}^{M}, a_{i}^{U}] (i = 1, 2, \dots, n)$ and ${\tilde{a}}_{i}^{'} = [a_{i}^{L'}, a_{i}^{M'}, a_{i}^{U'}] (i = 1, 2, \dots, n)$ be two sets of triangular fuzzy numbers, where ${\tilde{a}}_{i}^{'} = [a_{i}^{L'}, a_{i}^{M'}, a_{i}^{U'}] (i = 1, 2, \dots, n)$ is any permutation of ${\tilde{a}}_{i} = [a_{i}^{L}, a_{i}^{M}, a_{i}^{U}] (i = 1, 2, \dots, n)$ , then $\begin{matrix} TFWPGBM ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) \\ = TFWPGBM ({\tilde{a}}_{1}^{'}, {\tilde{a}}_{2}^{'}, \dots, {\tilde{a}}_{n}^{'}) \end{matrix}$ (34)

6 The proposed multiple attribute decision making method with triangular fuzzy information

Based on the above analysis, in this part, we study on the multiple attribute decision making problems with triangular fuzzy information, let A ={ A₁, A₂, ⋯ , A_m } be a discrete set of alternatives, G ={ G₁, G₂, ⋯ , G_n } be the set of attributes. Suppose that $A = {({\tilde{a}}_{ij})}_{m \times n} = {[a_{ij}^{L}, a_{ij}^{M}, a_{ij}^{U}]}_{m \times n}$ is the decision making matrix. Then, we exploit the triangular fuzzy weighted geometric Bonferroni mean (TFWGBM) operator to develop an approach for multiple attribute decision making problems:

Step 1. Normalize the value ${\tilde{a}}_{ij}$ into corresponding values ${\tilde{r}}_{ij}$ by using the following formulas:

$\begin{matrix} {\begin{matrix} r_{ij}^{L} = a_{ij}^{L} / - \sum_{i = 1}^{m} a_{ij}^{U} \\ r_{ij}^{M} = a_{ij}^{M} / - \sum_{i = 1}^{m} a_{ij}^{M} \\ r_{ij}^{U} = a_{ij}^{U} / - \sum_{i = 1}^{m} a_{ij}^{L} \end{matrix}, \\ for benefit attribute G_{j}, \\ i = 1, 2, \dots, m, j = 1, 2, \dots, n \end{matrix}$ (35)

$\begin{matrix} {\begin{matrix} r_{ij}^{L} = (1 / - a_{ij}^{U}) / - \sum_{i = 1}^{m} (1 / - a_{ij}^{L}) \\ r_{ij}^{M} = (1 / - a_{ij}^{M}) / - \sum_{i = 1}^{m} (1 / - a_{ij}^{M}) \\ r_{ij}^{U} = ((1 / - a_{ij}^{L})) / - \sum_{i = 1}^{m} (1 / - a_{ij}^{U}) \end{matrix}, \\ for cost attribute G_{j}, \\ i = 1, 2, \dots, m, j = 1, 2, \dots, n \end{matrix}$ (36)

Step 2. Utilize the matrix $\tilde{R}$ by the TFWPBM operator (in general, we let p = q = 1) $\begin{matrix} \begin{matrix} {\tilde{r}}_{i} = (r_{i}^{L}, r_{i}^{M}, r_{i}^{U}) = TFWPBM ({\tilde{r}}_{i 1}, {\tilde{r}}_{i 2}, \dots, {\tilde{r}}_{in}) \\ = {(\sum_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} (\frac{w_{i} (1 + T ({\tilde{a}}_{i}))}{\sum_{t = 1}^{n} w_{t} (1 + T ({\tilde{a}}_{t}))} \frac{w_{j} (1 + T ({\tilde{a}}_{j}))}{\sum_{t = 1}^{n} w_{t} (1 + T ({\tilde{a}}_{t}))} {({\tilde{a}}_{i})}^{p} \times {({\tilde{a}}_{j})}^{q}))}^{\frac{1}{p + q}} \end{matrix} \end{matrix}$

$\begin{matrix} \begin{matrix} = {\begin{matrix} {(\sum_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} (\frac{w_{i} (1 + T ({\tilde{a}}_{i}))}{\sum_{t = 1}^{n} w_{t} (1 + T ({\tilde{a}}_{t}))} \frac{w_{j} (1 + T ({\tilde{a}}_{j}))}{\sum_{t = 1}^{n} w_{t} (1 + T ({\tilde{a}}_{t}))} {(a_{i}^{L})}^{p} \times {(a_{j}^{L})}^{q}))}^{\frac{1}{p + q}}, \\ {(\sum_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} (\frac{w_{i} (1 + T ({\tilde{a}}_{i}))}{\sum_{t = 1}^{n} w_{t} (1 + T ({\tilde{a}}_{t}))} \frac{w_{j} (1 + T ({\tilde{a}}_{j}))}{\sum_{t = 1}^{n} w_{t} (1 + T ({\tilde{a}}_{t}))} {(a_{i}^{M})}^{p} \times {(a_{j}^{M})}^{q}))}^{\frac{1}{p + q}}, \\ {(\sum_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} (\frac{w_{i} (1 + T ({\tilde{a}}_{i}))}{\sum_{t = 1}^{n} w_{t} (1 + T ({\tilde{a}}_{t}))} \frac{w_{j} (1 + T ({\tilde{a}}_{j}))}{\sum_{t = 1}^{n} w_{t} (1 + T ({\tilde{a}}_{t}))} {(a_{i}^{U})}^{p} \times {(a_{j}^{U})}^{q}))}^{\frac{1}{p + q}} \end{matrix}} \end{matrix} \\ i = 1, 2, \dots, m, j = 1, 2, \dots, n \end{matrix}$ 37

Or the TFWPGBM operator (in general, we let p = q = 1) $\begin{matrix} {\tilde{r}}_{i} = (r_{i}^{L}, r_{i}^{M}, r_{i}^{U}) = TFWPGBM ({\tilde{r}}_{i 1}, {\tilde{r}}_{i 2}, \dots, {\tilde{r}}_{in}) \\ = \frac{1}{p + q} \prod_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} {(p {\tilde{a}}_{i} + q {\tilde{a}}_{j})}^{\frac{ω_{i} (1 + T (A_{i}))}{\sum_{t = 1}^{n} ω_{t} (1 + T (A_{t}))} \frac{ω_{j} (1 + T (A_{j}))}{\sum_{t = 1}^{n} ω_{t} (1 + T (A_{t}))}} \\ = {\begin{matrix} \frac{1}{p + q} \prod_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} {({pa}_{i}^{L} + {qa}_{j}^{L})}^{\frac{ω_{i} (1 + T (A_{i}))}{\sum_{t = 1}^{n} ω_{t} (1 + T (A_{t}))} \frac{ω_{j} (1 + T (A_{j}))}{\sum_{t = 1}^{n} ω_{t} (1 + T (A_{t}))}}, \\ \frac{1}{p + q} \prod_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} {({pa}_{i}^{M} + q {\tilde{a}}_{j}^{M})}^{\frac{ω_{i} (1 + T (A_{i}))}{\sum_{t = 1}^{n} ω_{t} (1 + T (A_{t}))} \frac{ω_{j} (1 + T (A_{j}))}{\sum_{t = 1}^{n} ω_{t} (1 + T (A_{t}))}}, \\ \frac{1}{p + q} \prod_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} {({pa}_{i}^{U} + q {\tilde{a}}_{j}^{U})}^{\frac{ω_{i} (1 + T (A_{i}))}{\sum_{t = 1}^{n} ω_{t} (1 + T (A_{t}))} \frac{ω_{j} (1 + T (A_{j}))}{\sum_{t = 1}^{n} ω_{t} (1 + T (A_{t}))}} \end{matrix}} \end{matrix} i = 1, 2, \dots, m, j = 1, 2, \dots, n$ 38

Step 3. To obtain the ranking score of these collective overall preference values ${\tilde{r}}_{i} (i = 1, 2, \dots, m)$ and adding all the elements which is located in each row of the matrix P, we have $p_{i} = \sum_{j = 1}^{m} p_{ij}, i = 1, 2, \dots, m$ (39)

Step 4. Ranking all the alternatives A_i (i = 1, 2, ⋯ , m) and then choose the best element according to the overall preference valuesp_i (i = 1, 2, ⋯ , m).

7 Numerical example

In this section, we exploit a practical multiple attribute decision making model for evaluating the competitiveness of high technological parks. The competitiveness of high technological parks is to be evaluated according to four attributes: (1) G₁: factors of individual; (2) G₂: factors of organization; (3) G₃: factors of society; (4) G₄: factors of development. The five possible technology enterprises A_i (i = 1, 2, ⋯ , 5) are to be evaluated adopting the triangular fuzzy numbers through the decision makers with the given four attributes (weighting vector of which is ω = (0.3, 0.2, 0.4, 0.1)), and make up the following matrix $A = {({\tilde{a}}_{ij})}_{5 \times 4}$ is shown in Table 1.

Table 1
Evaluation matrix A

G ₁ G ₂ G ₃ G ₄

A ₁ (0.68,0.69,0.71) (0.64,0.67,0.69) (0.50,0.52,0.55) (0.66,0.68,0.75)

A ₂ (0.70,0.74,0.80) (0.67,0.70,0.74) (0.64,0.66,0.69) (0.82,0.84,0.88)

A ₃ (0.69,0.76,0.82) (0.73,0.76,0.79) (0.33,0.40,0.43) (0.86,0.90,0.92)

A ₄ (0.54,0.56,0.60) (0.68,0.74,0.78) (0.71,0.72,0.73) (0.74,0.76,0.79)

A ₅ (0.50,0.52,0.56) (0.55,0.57,0.59) (0.56,0.58,0.61) (0.69,0.72,0.76)

	G ₁	G ₂	G ₃	G ₄
A ₁	(0.68,0.69,0.71)	(0.64,0.67,0.69)	(0.50,0.52,0.55)	(0.66,0.68,0.75)
A ₂	(0.70,0.74,0.80)	(0.67,0.70,0.74)	(0.64,0.66,0.69)	(0.82,0.84,0.88)
A ₃	(0.69,0.76,0.82)	(0.73,0.76,0.79)	(0.33,0.40,0.43)	(0.86,0.90,0.92)
A ₄	(0.54,0.56,0.60)	(0.68,0.74,0.78)	(0.71,0.72,0.73)	(0.74,0.76,0.79)
A ₅	(0.50,0.52,0.56)	(0.55,0.57,0.59)	(0.56,0.58,0.61)	(0.69,0.72,0.76)

In the following, to choose the most suitable cities, the TFWGBM operator is utilized to design a method to multiple attribute decision making problems evaluating the competitiveness of high technological parks with triangular fuzzy information, which can be described as following:

Step 1. Computing the normalized decision matrix $\tilde{R}$ . The results are illustrated in Table 2.

Table 2

Decision matrix $\tilde{R}$

	₁	G ₂	G ₃	G ₄
A ₁	(0.178,0.194,0.216)	(0.235,0.248,0.265)	(0.151,0.158,0.166)	(0.242,0.246,0.276)
A ₂	(0.166,0.175,0.190)	(0.176,0.181,0.217)	(0.168,0.190,0.208)	(0.241,0.248,0.257)
A ₃	(0.178,0.190,0.213)	(0.176,0.181,0.191)	(0.178,0.187,0.200)	(0.169,0.178,0.192)
A ₄	(0.157,0.168,0.197)	(0.187,0.195,0.213)	(0.164,0.180,0.198)	(0.179,0.187,0.219)
A ₅	(0.229,0.233,0.252)	(0.207,0.229,0.258)	(0.195,0.212,0.236)	(0.225,0.238,0.246)

Step 2. Aggregate all triangular fuzzy preference value ${\tilde{r}}_{ij} (j = 1, 2, \dots, n)$ by using the TFWGBM to derive the overall triangular fuzzy preference values ${\tilde{r}}_{i} (i = 1, 2, 3, 4, 5)$ of the high technological parks A_i. $\begin{matrix} {\tilde{r}}_{1} = (0 . 172, 0 . 191, 0 . 212), {\tilde{r}}_{2} = (0 . 185, 0 . 202, 0 . 220) \\ {\tilde{r}}_{3} = (0 . 212, 0 . 225, 0 . 237), {\tilde{r}}_{4} = (0 . 187, 0 . 201, 0 . 217) \\ {\tilde{r}}_{5} = (0 . 170, 0 . 179, 0 . 192) \end{matrix}$ (40)

Step 3. Utilizing the aggregating results and the equation of degree of possibility (2), ranking all the high technological parks A_i (i = 1, 2, 3, 4, 5)according to scores p_i (i = 1, 2, ⋯ , 5): A₃ ≻ A₅ ≻ A₂ ≻ A₄ ≻ A₁, and then the most suitable high technological parks is A₃.

8 Conclusion

In this work, we focus on the multiple attribute decision making problems for evaluating the competitiveness of high technological parks with triangular fuzzy information. Inspired by the idea of Bonferroni mean (BM) and geometric Bonferroni mean (GBM), we develop the triangular fuzzy power Bonferroni mean (TFPBM) operator, triangular fuzzy weighted power Bonferroni mean (TFWPBM) operator, the triangular fuzzy power geometric Bonferroni mean (TFPGBM) operator, triangular fuzzy weighted power geometric Bonferroni mean (TFWPGBM) operator. Using the proposed operator, we propose the program for multiple attribute decision making with the triangular fuzzy environments. In the end, a practical example for evaluating the competitiveness of high technological parks with triangular fuzzy information is given to testify the performance of the given approach.

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