Models for economic benefit evaluation of investment project of electric power enterprise with triangular fuzzy information

Abstract

Einstein product is a t-norm and Einstein sum is a t-conorm. They are good alternatives to algebraic product and algebraic sum, respectively. Nevertheless, it seems that most of the existing triangular fuzzy aggregation operators are based on the algebraic operations. In this paper, we utilize Einstein operations to develop some triangular fuzzy aggregation operators: triangular fuzzy Einstein weighted average (TFEWA) operator, triangular fuzzy Einstein ordered weighted average (TFEOWA) operator and triangular fuzzy Einstein hybrid average (TFEHA) operator. Then, we have utilized these operators to develop some approaches to solve the triangular fuzzy comprehensive evaluation problems. Finally, a practical example for evaluating the economic benefit evaluation of investment project of electric power enterprise is given to verify the developed approach.

Keywords

Fuzzy comprehensive evaluation triangular fuzzy information Einstein aggregation operators triangular fuzzy Einstein hybrid average (TFEHA) operator electric power enterprise

1 Introduction

At present, China has clearly put forward the great strategic goal to become an innovative country by 2020. Since the enterprises are the basis and subject of the national innovation system, how to construct an innovative enterprise will be the kernel and one of the main contents of building innovative country [1]. In 2006, China has implemented “technical innovation guidance program”, aimed at promoting innovative enterprises’ building. How to evaluate the innovative enterprises scientifically, objectively and comprehensively has become an important issue of the innovative enterprise theory and practice in China [2, 3]. As the theoretical system of innovative enterprises is not comprehensive at present, many scholars’ studies are staying on the technological innovation, taking it as the entire contents for innovative enterprises research, failing to construct the theory from the entire enterprise innovation system. This state is incompatible with current innovative enterprises’ construction, and it is an urgent need to establish a new theory to guide the construction of innovative enterprises [4, 5].

A multiple attribute decision making problem is to find a desirable solution from a finite number of feasible alternatives assessed on multiple attributes, both quantitative and qualitative [7 –9]. Xu [10] 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 [11] proposed the fuzzy ordered weighted harmonic averaging (FOWHA) operator. Wei [38] developed the fuzzy induced ordered weighted harmonic mean (FIOWHM) operator and applied it to the group decision making.

In this paper, we utilize Einstein operations to develop some triangular fuzzy aggregation operators: triangular fuzzy Einstein weighted average (TFEWA) operator, triangular fuzzy Einstein ordered weighted average (TFEOWA) operator and triangular fuzzy Einstein hybrid average (TFEHA) operator. Then, we have utilized these operators to develop some approaches to solve the triangular fuzzy comprehensive evaluation problems. Finally, a practical example for evaluating the enterprise sustainable innovation capacity based on the community responsibilityis is given to verify the developed approach.

2 Preliminaries

In the following, we briefly describe some basic concepts and basic operational laws related to triangular fuzzy numbers.

Definition 1. [12] A triangular fuzzy numbers $\tilde{a}$ can be defined by a triplet (a^L, a^M, a^R). The membership function $μ_{\tilde{a}} (x)$ is defined as: $μ_{\tilde{a}} (x) = {\begin{matrix} 0, x < a^{L}, \\ \frac{x - a^{L}}{a^{M} - a^{L}}, a^{L} ⩽ x ⩽ a^{M}, \\ \frac{x - a^{R}}{a^{M} - a^{R}}, a^{M} ⩽ x ⩽ a^{R}, \\ 0, x ⩾ a^{R} . \end{matrix}$ (1) where 0 < a^L ⩽ a^M ⩽ a^R, a^L and a^R stand for the lower and upper values of the support of $\tilde{a}$ , respectively, and a^M for the modal value.

Definition 2. [13] If A is a triangular fuzzy variable $[a^{L}, a^{M}, a^{R}]$ , according to the Equation.(2), then the expected value of A is $E (A) = \frac{1}{3} (a^{L} + a^{M} + a^{R})$ (2)

Motivated by the Definition of the Einstein operations, let a t-norm T be the Einstein product ⊗_ɛ and t-conormS be the Einstein sum⊕_ɛ, then the generalized intersection and union on two fuzzy numbers ${\tilde{a}}_{1} = [a_{1}^{L}, a_{1}^{M}, a_{1}^{R}]$ and ${\tilde{a}}_{2} = [a_{2}^{L}, a_{2}^{M}, a_{2}^{R}]$ become the Einstein product(denoted by ${\tilde{a}}_{1} \otimes_{ɛ} {\tilde{a}}_{2}$ ) and Einstein sum(denoted by ${\tilde{a}}_{1} \oplus_{ɛ} {\tilde{a}}_{2}$ ) of two fuzzy numbers ${\tilde{a}}_{1} = [a_{1}^{L}, a_{1}^{M}, a_{1}^{R}]$ and ${\tilde{a}}_{2} = [a_{2}^{L}, a_{2}^{M}, a_{2}^{R}]$ , respectively, as follows:

(1) ${\tilde{a}}_{1} \oplus_{ɛ} {\tilde{a}}_{2} = (\frac{a_{1}^{L} + a_{2}^{L}}{1 + a_{1}^{L} a_{2}^{L}}, \frac{a_{1}^{M} + a_{2}^{M}}{1 + a_{1}^{M} a_{2}^{M}}, \frac{a_{1}^{R} + a_{2}^{R}}{1 + a_{1}^{R} a_{2}^{R}});$

(2) ${\tilde{a}}_{1} \otimes_{ɛ} {\tilde{a}}_{2} = (\frac{a_{1}^{L} a_{2}^{L}}{1 + (1 - a_{1}^{L}) (1 - a_{2}^{L})}, \frac{a_{1}^{M} a_{2}^{M}}{1 + (1 - a_{1}^{M}) (1 - a_{2}^{M})}, \frac{a_{1}^{R} a_{2}^{R}}{1 + (1 - a_{1}^{R}) (1 - a_{2}^{R})});$

(3) $λ ._{ɛ} {\tilde{a}}_{1} = (\frac{{(1 + a_{1}^{L})}^{λ} - {(1 - a_{1}^{L})}^{λ}}{{(1 + a_{1}^{L})}^{λ} + {(1 - a_{1}^{L})}^{λ}}, \frac{{(1 + a_{1}^{M})}^{λ} - {(1 - a_{1}^{M})}^{λ}}{{(1 + a_{1}^{M})}^{λ} + {(1 - a_{1}^{M})}^{λ}}, \frac{{(1 + a_{1}^{R})}^{λ} - {(1 - a_{1}^{R})}^{λ}}{{(1 + a_{1}^{R})}^{λ} + {(1 - a_{1}^{R})}^{λ}}), λ > 0$ ;

(4) ${\tilde{a}}_{1}^{\land_{ɛ} λ} = (\frac{2 {(a_{1}^{L})}^{λ}}{(2 - a_{1}^{L})^{λ} + {(a_{1}^{L})}^{λ}}, \frac{2 {(a_{1}^{M})}^{λ}}{(2 - a_{1}^{M})^{λ} + {(a_{1}^{M})}^{λ}}, \frac{2 {(a_{1}^{R})}^{λ}}{(2 - a_{1}^{R})^{λ} + {(a_{1}^{R})}^{λ}}), λ > 0$ .

3 Triangular fuzzy Einstein aggregation operators

In the following, we shall develop some fuzzy Einstein arithmetic aggregation operator based on the operations of fuzzy numbers and Einstein sum.

Definition 3. Let ${\tilde{a}}_{j} = (a_{j}^{L}, a_{j}^{M}, a_{j}^{R}) (j = 1, 2, \dots, n)$ be a set of triangular fuzzy numbers, then we define the triangular fuzzy Einstein weighted average (TFEWA) operator as follows: ${TFEWA}_{ω} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) = {\oplus_{ɛ}}_{j = 1}^{n} ω_{j} {\tilde{a}}_{j}$ (3) where ω = (ω₁, ω₂, ⋯ , ω_n) ^T be the weight vector of ${\tilde{a}}_{j} = (a_{j}^{L}, a_{j}^{M}, a_{j}^{R}) (j = 1, 2, \dots, n)$ , and ω_j > 0, $\sum_{j = 1}^{n} ω_{j} = 1$ .

Based on Einstein sum operations of the triangular fuzzy numbers described, we can drive theTheorem 1.

Theorem 1. Let ${\tilde{a}}_{j} = (a_{j}^{L}, a_{j}^{M}, a_{j}^{R}) (j = 1, 2, \dots, n)$ be a set of triangular fuzzy numbers, then their aggregated value by using the TFEWA operator is also a triangular fuzzy number, and $\begin{matrix} {TFEWA}_{ω} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) = {\oplus_{ɛ}}_{j = 1}^{n} ω_{j} {\tilde{a}}_{j} \\ = (\frac{\prod_{j = 1}^{n} {(1 + a_{j}^{L})}^{ω_{j}} - \prod_{j = 1}^{n} {(1 - a_{j}^{L})}^{ω_{j}}}{\prod_{j = 1}^{n} {(1 + a_{j}^{L})}^{ω_{j}} + \prod_{j = 1}^{n} {(1 - a_{j}^{L})}^{ω_{j}}}, \frac{\prod_{j = 1}^{n} {(1 + a_{j}^{M})}^{ω_{j}} - \prod_{j = 1}^{n} {(1 - a_{j}^{M})}^{ω_{j}}}{\prod_{j = 1}^{n} {(1 + a_{j}^{M})}^{ω_{j}} + \prod_{j = 1}^{n} {(1 - a_{j}^{M})}^{ω_{j}}}, \frac{\prod_{j = 1}^{n} {(1 + a_{j}^{R})}^{ω_{j}} - \prod_{j = 1}^{n} {(1 - a_{j}^{R})}^{ω_{j}}}{\prod_{j = 1}^{n} {(1 + a_{j}^{R})}^{ω_{j}} + \prod_{j = 1}^{n} {(1 - a_{j}^{R})}^{ω_{j}}}) \end{matrix}$ (4) where ω = (ω₁, ω₂, ⋯ , ω_n) ^T be the weight vector of ${\tilde{a}}_{j} = (a_{j}^{L}, a_{j}^{M}, a_{j}^{R}) (j = 1, 2, \dots, n)$ , and ω_j > 0, $\sum_{j = 1}^{n} ω_{j} = 1$ .

It can be easily proved that the TFEWA operator has the following properties.

Theorem 2. (Idempotency) If all ${\tilde{a}}_{j} = (a_{j}^{L}, a_{j}^{M}, a_{j}^{R}) (j = 1, 2, \dots, n)$ are equal, i.e. ${\tilde{a}}_{j} = \tilde{a}$ for all j, then ${TFEWA}_{ω} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) = \tilde{a}$ (5)

Theorem 3. (Boundedness) Let ${\tilde{a}}_{j} = (a_{j}^{L}, a_{j}^{M}, a_{j}^{R}) (j = 1, 2, \dots, n)$ be a set of triangular fuzzy numbers, and let $\begin{matrix} {\tilde{a}}^{-} & = & min_{j} {\tilde{a}}_{j} = min_{j} (a_{j}^{L}, a_{j}^{M}, a_{j}^{R}), \\ {\tilde{a}}^{+} & = & max_{j} {\tilde{a}}_{j} = max_{j} (a_{j}^{L}, a_{j}^{M}, a_{j}^{R}) \end{matrix}$

Then ${\tilde{a}}^{-} ⩽ {TFEWA}_{ω} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) ⩽ {\tilde{a}}^{+}$ (6)

Theorem 4. (Monotonicity) Let ${\tilde{a}}_{j} = (a_{j}^{L}, a_{j}^{M}, a_{j}^{R}) (j = 1, 2, \dots, n)$ and ${\tilde{a}}_{j}^{'} = (a_{j}^{' L}, a_{j}^{' M}, a_{j}^{' U}) (j = 1, 2, \dots, n)$ be two set of triangular fuzzy numbers, if ${\tilde{a}}_{j} ⩽ {\tilde{a}}_{j}^{'}$ , for all j, then $\begin{matrix} {TFEWA}_{ω} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) \\ ⩽ {TFEWA}_{ω} ({\tilde{a}}_{1}^{'}, {\tilde{a}}_{2}^{'}, \dots, {\tilde{a}}_{n}^{'}) \end{matrix}$ (7)

Definition 4. Let ${\tilde{a}}_{j} = (a_{j}^{L}, a_{j}^{M}, a_{j}^{R})$ (j = 1, 2, ⋯ , n) be a set of triangular fuzzy numbers, then we define the triangular fuzzy Einstein ordered weighted average (TFEOWA) operator as follows:

${TFEOWA}_{w} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) = {\oplus_{ɛ}}_{j = 1}^{n} w_{j} {\tilde{a}}_{σ (j)}$ (8) where(σ (1) , σ (2) , ⋯ , σ (n)) is a permutation of (1, 2, ⋯ , n), such that ${\tilde{a}}_{σ (j - 1)} ⩾ {\tilde{a}}_{σ (j)}$ for all j = 2, ⋯ , n, and w = (w₁, w₂, ⋯ , w_n) ^T is the aggregation-associated weight vector such that w_j ∈ [0, 1] and $\sum_{j = 1}^{n} w_{j} = 1$ .

Based on Einstein sum operations of the triangular fuzzy numbers described, we can drive the Theorem 5.

Theorem 5. Let ${\tilde{a}}_{j} = (a_{j}^{L}, a_{j}^{M}, a_{j}^{R}) (j = 1, 2, \dots, n)$ be a set of triangular fuzzy numbers, then their aggregated value by using the TFEOWA operator is also a triangular fuzzy numbers, and $\begin{matrix} {TFEOWA}_{w} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) = {\oplus_{ɛ}}_{j = 1}^{n} w_{j} {\tilde{a}}_{σ (j)} \\ = (\frac{\prod_{j = 1}^{n} {(1 + a_{σ (j)}^{L})}^{w_{j}} - \prod_{j = 1}^{n} {(1 - a_{σ (j)}^{L})}^{w_{j}}}{\prod_{j = 1}^{n} {(1 + a_{σ (j)}^{L})}^{w_{j}} + \prod_{j = 1}^{n} {(1 - a_{σ (j)}^{L})}^{w_{j}}}, \\ \frac{\prod_{j = 1}^{n} {(1 + a_{σ (j)}^{M})}^{w_{j}} - \prod_{j = 1}^{n} {(1 - a_{σ (j)}^{M})}^{w_{j}}}{\prod_{j = 1}^{n} {(1 + a_{σ (j)}^{M})}^{w_{j}} + \prod_{j = 1}^{n} {(1 - a_{σ (j)}^{M})}^{w_{j}}}, \\ \frac{\prod_{j = 1}^{n} {(1 + a_{σ (j)}^{R})}^{w_{j}} - \prod_{j = 1}^{n} {(1 - a_{σ (j)}^{R})}^{w_{j}}}{\prod_{j = 1}^{n} {(1 + a_{σ (j)}^{R})}^{w_{j}} + \prod_{j = 1}^{n} {(1 - a_{σ (j)}^{R})}^{w_{j}}}) \end{matrix}$ (9) where (σ (1) , σ (2) , ⋯ , σ (n)) is a permutation of (1, 2, ⋯ , n), such that ${\tilde{a}}_{σ (j - 1)} ⩾ {\tilde{a}}_{σ (j)}$ for all j = 2, ⋯ , n, and w = (w₁, w₂, ⋯ , w_n) ^T is the aggregation-associated weight vector such that w_j ∈ [0, 1] and $\sum_{j = 1}^{n} w_{j} = 1$ .

It can be easily proved that the TFEOWA operator has the following properties.

Theorem 6. (Idempotency) If all ${\tilde{a}}_{j} = (a_{j}^{L}, a_{j}^{M}, a_{j}^{R}) (j = 1, 2, \dots, n)$ are equal, i.e. ${\tilde{a}}_{j} = \tilde{a}$ for allj, then ${TFEOWA}_{w} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) = \tilde{a}$ (10)

Theorem 7. (Boundedness) Let ${\tilde{a}}_{j} = (a_{j}^{L}, a_{j}^{M}, a_{j}^{R}) (j = 1, 2, \dots, n)$ be a set of triangular fuzzy numbers, and let $\begin{matrix} {\tilde{a}}^{-} & = & min_{j} {\tilde{a}}_{j} = min_{j} (a_{j}^{L}, a_{j}^{M}, a_{j}^{R}), \\ {\tilde{a}}^{+} & = & max_{j} {\tilde{a}}_{j} = max_{j} (a_{j}^{L}, a_{j}^{M}, a_{j}^{R}) \end{matrix}$

Then ${\tilde{a}}^{-} ⩽ {TFEOWA}_{w} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) ⩽ {\tilde{a}}^{+}$ (11)

Theorem 8. (Monotonicity) Let ${\tilde{a}}_{j} = (a_{j}^{L}, a_{j}^{M}, a_{j}^{R}) (j = 1, 2, \dots, n)$ and ${\tilde{a}}_{j}^{'} = (a_{j}^{' L}, a_{j}^{' M}, a_{j}^{' R}) (j = 1, 2, \dots, n)$ be two set of triangular fuzzy numbers, if ${\tilde{a}}_{j} ⩽ {\tilde{a}}_{j}^{'}$ , for all j, then $\begin{matrix} {TFEOWA}_{w} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) \\ ⩽ {TFEOWA}_{w} ({\tilde{a}}_{1}^{'}, {\tilde{a}}_{2}^{'}, \dots, {\tilde{a}}_{n}^{'}) \end{matrix}$ (12)

Theorem 9. (Commutativity) Let ${\tilde{a}}_{j} = (a_{j}^{L}, a_{j}^{M}, a_{j}^{R}) (j = 1, 2, \dots, n)$ and ${\tilde{a}}_{j}^{'} = (a_{j}^{' L}, a_{j}^{' M}, a_{j}^{' R}) (j = 1, 2, \dots, n)$ be two set of triangular fuzzy numbers, then $\begin{matrix} {TFEOWA}_{w} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) \\ = {TFEOWA}_{w} ({\tilde{a}}_{1}^{'}, {\tilde{a}}_{2}^{'}, \dots, {\tilde{a}}_{n}^{'}) \end{matrix}$ (13) where ${\tilde{a}}_{j}^{'} (j = 1, 2, \dots, n)$ is any permutation of ${\tilde{a}}_{j} (j = 1, 2, \dots, n)$ .

In the following we shall propose the triangular fuzzy Einstein hybrid average (TFEHA) operator.

Definition 5. The triangular fuzzy Einstein hybrid average (TFEHA) operator is defined as follows: ${TFEHA}_{w, ω} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) = {\oplus_{ε}}_{j = 1}^{n} w_{\dot{\tilde{j}}} a_{σ (j)}$ (14) where w = (w₁, w₂, ⋯ , w_n) is the associated weighting vector, with w_j ∈ [0, 1], $\sum_{j = 1}^{n} w_{j} = 1$ , and ${\dot{\tilde{a}}}_{σ (j)}$ is the j-th largest element of the triangular fuzzy arguments ${\dot{\tilde{a}}}_{σ (j)} ({\dot{\tilde{a}}}_{σ} (j) = n ω_{j} ._{ɛ} {\tilde{a}}_{j}, j = 1, 2, \dots, n)$ , ω = (ω₁, ω₂, ⋯ , ω_n) is the weighting vector of triangular fuzzy arguments ${\tilde{a}}_{j} (j = 1, 2, \dots, n)$ , with ω_i ∈ [0, 1], $\sum_{i = 1}^{n} ω_{i} = 1$ , and n is the balancing coefficient. Especially, if w = (1/ - n, 1/ - n, ⋯ , 1/ - n) ^T, then TFEHA is reduced to the triangular fuzzy Einstein weighted average (TFEWA) operator; if ω = (1/ - n, 1/ - n, ⋯ , 1/ - n), then TFEHA is reduced to the triangular fuzzy Einstein ordered weighted average (TFEOWA) operator.

Based on Einstein sum operations of the triangular fuzzy numbers described, we can drive the Theorem 10.

Theorem 10. Let ${\tilde{a}}_{j} = (a_{j}^{L}, a_{j}^{M}, a_{j}^{R}) (j = 1, 2, \dots, n)$ be a set of triangular fuzzy numbers, then their aggregated value by using the TFEHA operator is also a triangular fuzzy number, and $\begin{matrix} {TFEHA}_{w, ω} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) = {\oplus_{ɛ}}_{j = 1}^{n} w_{\dot{\tilde{j}}} a_{σ ˜} (j) \\ = (\frac{\prod_{j = 1}^{n} {(1 + {\dot{a}}_{σ (j)}^{L})}^{w_{j}} - \prod_{j = 1}^{n} {(1 - {\dot{a}}_{σ (j)}^{L})}^{w_{j}}}{\prod_{j = 1}^{n} {(1 + {\dot{a}}_{σ (j)}^{L})}^{w_{j}} + \prod_{j = 1}^{n} {(1 - {\dot{a}}_{σ (j)}^{L})}^{w_{j}}}, \\ \frac{\prod_{j = 1}^{n} {(1 + {\dot{a}}_{σ (j)}^{M})}^{w_{j}} - \prod_{j = 1}^{n} {(1 - {\dot{a}}_{σ (j)}^{M})}^{w_{j}}}{\prod_{j = 1}^{n} {(1 + {\dot{a}}_{σ (j)}^{M})}^{w_{j}} + \prod_{j = 1}^{n} {(1 - {\dot{a}}_{σ (j)}^{M})}^{w_{j}}}, \\ \frac{\prod_{j = 1}^{n} {(1 + {\dot{a}}_{σ (j)}^{R})}^{w_{j}} - \prod_{j = 1}^{n} {(1 - {\dot{a}}_{σ (j)}^{R})}^{w_{j}}}{\prod_{j = 1}^{n} {(1 + {\dot{a}}_{σ (j)}^{R})}^{w_{j}} + \prod_{j = 1}^{n} {(1 - {\dot{a}}_{σ (j)}^{R})}^{w_{j}}}) \end{matrix}$ (15) where w = (w₁, w₂, ⋯ , w_n) is the associated weighting vector, with w_j ∈ [0, 1], $\sum_{j = 1}^{n} w_{j} = 1$ , and ${\dot{\tilde{a}}}_{σ (j)}$ is the j-th largest element of the triangular fuzzy arguments ${\dot{\tilde{a}}}_{σ (j)} ({\dot{\tilde{a}}}_{σ (j)} = n ω_{j} ._{ɛ} {\tilde{a}}_{j}, j = 1, 2, \dots, n)$ , ω = (ω₁, ω₂, ⋯ , ω_n) is the weighting vector of triangular fuzzy arguments ${\tilde{a}}_{j} (j = 1, 2, \dots, n)$ , with ω_i ∈ [0, 1], $\sum_{i = 1}^{n} ω_{i} = 1$ , and n is the balancing coefficient.

4 Numerical example

Thus, in this section we shall present a numerical example for evaluating the economic benefit evaluation of investment project of electric power enterprise with triangular fuzzy information in order to illustrate the method proposed in this paper. Suppose an organization plans to evaluate the economic benefit evaluation of investment project of electric power enterprises. Project term choose five potential electric power enterprise A_i (i = 1, 2, ⋯ , 5) as candidates. The Project team selects four attributes to evaluate the electric power enterprise: (1) sustainable technology innovation ability - G₁, (2) organize and coordinate sustainable innovation ability-G₂, (3) entrepreneurs sustainable innovation ability-G₃ ; (4) enterprise sustainable marketing innovation ability-G₄. The five possible enterprises A_i (i = 1, 2, ⋯ , 5) are to be evaluated using the triangular fuzzy numbers under the above four attributes, and construct the following matrix $A = {({\tilde{a}}_{ij})}_{5 \times 4}$ is shown inTable 1.

Table 1
Decision matrix A

$\begin{matrix} G_{1} \end{matrix}$ G ₂ G ₃ G ₄

A ₁ (0.73,0.74,0.75) (0.73,0.75,0.76) (0.56,0.59,0.69) (0.72,0.73,0.76)

A ₂ (0.57,0.59,0.62) (0.69,0.73,0.76) (0.56,0.58,0.64) (0.53,0.56,0.57)

A ₃ (0.65,0.67,0.70) (0.38,0.45,0.49) (0.63,0.68,0.73) (0.48,0.56,0.59)

A ₄ (0.35,0.42,0.46) (0.43,0.47,0.53) (0.63,0.66,0.68) (0.43,0.55,0.59)

A ₅ (0.51,0.55,0.57) (0.69,0.75,0.79) (0.68,0.72,0.74) (0.67,0.68,0.78)

	$\begin{matrix} G_{1} \end{matrix}$	G ₂	G ₃	G ₄
A ₁	(0.73,0.74,0.75)	(0.73,0.75,0.76)	(0.56,0.59,0.69)	(0.72,0.73,0.76)
A ₂	(0.57,0.59,0.62)	(0.69,0.73,0.76)	(0.56,0.58,0.64)	(0.53,0.56,0.57)
A ₃	(0.65,0.67,0.70)	(0.38,0.45,0.49)	(0.63,0.68,0.73)	(0.48,0.56,0.59)
A ₄	(0.35,0.42,0.46)	(0.43,0.47,0.53)	(0.63,0.66,0.68)	(0.43,0.55,0.59)
A ₅	(0.51,0.55,0.57)	(0.69,0.75,0.79)	(0.68,0.72,0.74)	(0.67,0.68,0.78)

The information about the attribute weights is known as follows: ω = (0.3, 0.2, 0.4, 0.1).

Then, we utilize the approach developed to get the most desirable electric power enterprise.

Step 1. We utilize the decision information given in matrix $\tilde{R} = {(r_{ij}, a_{ij})}_{5 \times 4}$ , and the TFEWA operator to derive the overall preference values ${\tilde{r}}_{i} = (r_{i}^{L}, r_{i}^{M}, r_{i}^{U}) (i = 1, 2, 3, 4, 5)$ of the enterprises A_i. The results are shown in Table 2.

Table 2

The overall preference values of the enterprises

	TFEWA
A ₁	(0.152,0.174,0.196)
$\begin{matrix} A_{2} \end{matrix}$	(0.191,0.232,0.257)
A ₃	(0.204,0.226,0.248)
A ₄	(0.163,0.189,0.236)
A ₅	(0.185,0.193,0.214)

Step 3. According to the aggregating results shown in Table 2 and the expected values (3), the ordering of the alternatives are shown in Table 3. Note that >means “preferred to”.

Table 3

Ordering of the enterprises by utilizing the TFEWA operator

	Ordering
TFEWA	A₂ > A₃ > A₄ > A₅ > A₁

5 Conclusion

In this paper, we utilize Einstein operations to develop some triangular fuzzy aggregation operators: triangular fuzzy Einstein weighted average (TFEWA) operator, triangular fuzzy Einstein ordered weighted average (TFEOWA) operator and triangular fuzzy Einstein hybrid average (TFEHA) operator. Then, we have utilized these operators to develop some approaches to solve the triangular fuzzy comprehensive evaluation problems. Finally, a practical example for evaluating the economic benefit evaluation of investment project of electric power enterprise is given to verify the developed approach. In the future, we shall extend the proposed methods to other domains [14 –19].

Footnotes

Acknowledgments

The work was supported by the 985 engineering project of the Jilin University.

References

Altuzarra

, Moreno-Jiménez

J.M.

and Salvador

, Consensus Building in AHP-Group Decision Making: A Bayesian Approach, Operations Research 58 (2010), 1755–1773.

and Lai

K.K.

, A distance-based group decision-making methodology for multi-person multi-criteria emergency decision support, Decision Support Systems 51(2) (2011), 307–315.

Wang

J.M.

, Robust optimization analysis for multiple attribute decision making problems with imprecise information, Annals of Operations Research 197(197), 109–122.

Ben-Arieh

and Chen

Z.F.

, Linguistic-labels aggregation and consensus measure for autocratic decision making using group recommendations, IEEE Transactions on Systems, Man and Cybernetics. Part A 36 (2006), 558–568.

Yue

Z.L.

, A method for group decision-making based on determining weights of decision makers using TOPSIS, Applied Mathematical Modelling 35 (2011), 1926–1936.

Herrera

and Herrera-Viedma

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

Herrera

and Herrera-Viedma

, Choice functions and mechanisms for linguistic preference relations, European Journal of Operational Research 120 (2000), 144–161.

X.H.

and Xu

Z.S.

, Graph-based multi-agent decision making, International Journal of Approximate Reasoning 53(4) (2012), 502–512.

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.

10.

Z.S.

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

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.

Van Laarhoven

P.J.M.

and Pedrycz

, A fuzzy extension of Saaty’s priority theory, Fuzzy Sets and Systems 11 (1983), 229–241.

13.

Zétényi

, Fuzzy Sets in Psychology, North-Holland, Amsterdam, (1988).

14.

Wang

W.P.

, A fuzzy linguistic computing approach to supplier evaluation, Applied Mathematical Modelling 34 (2010), 3130–3141.

15.

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), 1279–1293.

16.

Park

J.H.

, Park

, Young

C.K.

and Xue

, Extension of the TOPSIS method for decision making problems under interval-valued intuitionistic fuzzy environment, Applied Mathematical Modeling 35 (2011), 2544–2556.

17.

Chen

T.Y.

, Hsiao

P.W.

and Yen

Y.L.

, A multicriteria group decision-making approach based on interval-valued intuitionistic fuzzy sets: a comparative perspective, Expert Systems with Applications 38 (2011), 7647–7658.

18.

Chiclana

, Herrera

and Herrera-Viedma

, The ordered weighted geometric operator: Properties and application, In: Proc of 8th Int Conf on Information Processing and Management of Uncertainty in Knowledge-based Systems, Madrid, (2000), 985–991.

19.

Liao

X.W.

, Li

and Lu

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