Research on the investment performance evaluation of corporate venture capital with intuitionistic fuzzy information

Abstract

Corporate venture capital refers to the direct venture investment in private companies made by non-financial corporations with specific core business. Along with rapid growth of Chinese economy, domestic venture capital has gone through a fantastic development among which corporate venture capital is an outstanding one. The study on the different types of venture captital, especially corporate venture capital, is required by the establishment and improvement of the mutiple-level capital market in China. Only on the basis of through understanding of the traits, advantages and disadvantages of corporate venture capital evaluation, we can set a better platform for its further development, contributing to the promotion of the entire venture capital market. In this paper, we utilize Hamacher operations to develop some intuitionistic fuzzy aggregation operators: intuitionistic fuzzy Hamacher weighted geometric (IFHWG) operator, intuitionistic fuzzy Hamacher ordered weighted geometric (IFHOWG) operator and intuitionistic fuzzy Hamacher hybrid geometric (IFHHG) operator. The prominent characteristic of these proposed operators are studied. Then, we have utilized these operators to develop some approaches to solve the intuitionistic fuzzy multiple attribute decision making problems for investment performance evaluation of corporate venture capital. Finally, a practical example for investment performance evaluation of corporate venture capital is given to verify the developed approach and to demonstrate its practicality and effectiveness.

Keywords

Intuitionistic fuzzy numbers operational laws intuitionistic fuzzy Hamacher hybrid geometric (IFHHG) operator investment performance evaluation corporate venture capital

1 Introduction

In China, venture capital from the mid-1980 s, originally used to support the development of the scientific enterprise, and introduced and studied from abroad as imported goods. Venture Capital since its birth, has been a great concern of the Chinese government and the community, and also achieved fruitful results. Today, venture capital has developed into an important pillar of China’s high-tech industry. Specifically, the issuance of Interim Measures for the Administration of Venture Capital Enterprises in 2005, as well as the launch of the GEM market in 2009, pushes the development of the venture to the climax. Associated with the transformation of the mode of economic development as well as the development of strategic emerging industries, the role of venture capital investment will be. More prominent, and more broad space for development. As the microscopic carrier of the venture capital development, venture capital firms operating conditions is a direct manifestation of the development of the venture capital industry. The course of development of the venture capital firms is very tortuous and full of thorns, and there were prosperity as well as recession. The overseas listed blocked, the domestic capital market in the doldrums, the context of the primary market and the secondary market upside down, in such situation, the development of venture capital firms once again at a crossroads. Generally speaking, the problem of China venture capital company mainly excessive government intervention and lack of post-investment management, homogeneous competition severe, venture capital firms operating conditions of the non-public, again to cover up these problems, the venture capital industry in the blind in the front row. Clearly, the performance and efficiency of the venture capital firms covered up, not conducive to the healthy development of the industry. Based disease where, by building a comprehensive performance evaluation system of venture capital firms operating conditions form, to guide the development of the industry, it is an effective way to solve the current problems of China venture capital companies. The performance evaluation of the venture capital company is a complicated systematic project involving capital raising, investment decision-making, post-investment management, capital exit and other links.

The prosperous development of venture capital in our country had played a great role in promoting the financing environment and development for small and medium-sized technology-based enterprises, in the other hands, it also had Provided a more broad space for the development of building the multi-level capital market in China. Due to the imperfect domestic regulation and the influence of Profit-seeking behavior in the industry, the venture capital industry is facing unprecedented market crisis. In recent years, foreign scholars played universal attention to corporate venture capital. From the view of the development process of the abroad, the corporate venture capital could maintain stable profit and investment when the Industry as a whole under the malaise. Therefore, the research of corporate venture capital could help set up a new way to get rid of the current competition development for the venture capital industry in China. For the past few years, Scholars at home and abroad showed a lot of attention on corporate venture capital investment. But related research system is still in its early stage. In the context of the existing research, it mainly based on the related theory research. There is no related empirical analysis. So, carrying out the related investment performance evaluation about venture capital investment investments is very necessary.

Atanassov [1, 2] introduced the concept of intuitionistic fuzzy set(IFS) which is a generalization of the concept of fuzzy set [3]. The intuitionistic fuzzy set has received more and more attention since its appearance [4 –14]. Xu [15] developed some geometric aggregation operators, such as the intuitionistic fuzzy weighted geometric (IFWG) operator, the intuitionistic fuzzy ordered weighted geometric (IFOWG) operator, and the intuitionistic fuzzy hybrid geometric (IFHG) operator and gave an application of the IFHG operator to multiple attribute group decision making with intuitionistic fuzzy information. Wei [16] proposed some geometric aggregation operators such as the dynamic intuitionistic fuzzy weighted geometric (DIFWG) operator and uncertain dynamic intuitionistic fuzzy weighted geometric (UDIFWG) operator to aggregate dynamic or uncertain dynamic intuitionistic fuzzy information. Wei [17] proposed two new aggregation operators: induced intuitionistic fuzzy ordered weighted geometric (I-IFOWG) operator and induced Intuitionistic fuzzy ordered weighted geometric (I-IIFOWG) operator are proposed, and some desirable properties of the I-IFOWG and I-IIFOWG operators are studied, such as commutativity, idempotency and monotonicity. Wei and Zhao [18] developed the induced intuitionistic fuzzy correlated geometric (I-IFCG) operator are developed and some desirable properties of the I-IFCG operators are studied, such as commutativity, idempotency and monotonicity. Yu et al. [19] proposed the Intuitionistic fuzzy prioritized weighted geometric (IFPWG) operator. Xu [20] developed a series of operators for aggregating intuitionistic fuzzy numbers, establish various properties of these power aggregation operators.

In this paper, we utilize Hamacher operations to develop some intuitionistic fuzzy aggregation operators: intuitionistic fuzzy Hamacher weighted geometric (IFHWG) operator, intuitionistic fuzzy Hamacher ordered weighted geometric (IFHOWG) operator and intuitionistic fuzzy Hamacher hybrid geometric (IFHHG) operator. The prominent characteristic of these proposed operators are studied. Then, we have utilized these operators to develop some approaches to solve the intuitionistic fuzzy multiple attribute decision making problems for investment performance evaluation of corporate venture capital. Finally, a practical example for investment performance evaluation of corporate venture capital is given to verify the developed approach and to demonstrate its practicality and effectiveness.

2 Preliminaries

In the following, we introduce some basic concepts of the intuitionistic fuzzy sets. Atanassov [1, 2] extended the fuzzy set to the IFS, shown as follows:

Definition 1. [1, 2] An IFS A in X is given by $A = {〈 x, μ_{A} (x), ν_{A} (x) 〉 | x \in X}$ (1)

Where μ_A : X → [0, 1] and ν_A : X → [0, 1], with the condition 0 ≤ μ_A (x) + ν_A (x) ≤ 1, ∀ x ∈ X . The numbers μ_A (x) and ν_A (x) represent, respectively, the membership degree and non- membership degree of the element x to the set A [1, 2].

Definition 2. [15] Let $\tilde{a} = (μ, ν)$ be an intuitionistic fuzzy number, a score function S of an intuitionistic fuzzy value can be represented as follows: $S (\tilde{a}) = μ - ν, S (\tilde{a}) \in [- 1, 1] .$ (2)

Definition 3. [15] Let $\tilde{a} = (μ, ν)$ be an intuitionistic fuzzy number, a accuracy function H of an intuitionistic fuzzy value can be represented as follows: $H (\tilde{a}) = μ + ν, H (\tilde{a}) \in [0, 1] .$ (3) to evaluate the degree of accuracy of the intuitionistic fuzzy value $\tilde{a} = (μ, ν)$ , where $H (\tilde{a}) \in [0, 1]$ . The larger the value of $H (\tilde{a})$ , the more the degree of accuracy of the intuitionistic fuzzy value $\tilde{a}$ .

Xu [15] give an order relation between two intuitionistic fuzzy values.

Definition 4. [15] Let ${\tilde{a}}_{1} = (μ_{1}, ν_{1})$ and ${\tilde{a}}_{2} = (μ_{2}, ν_{2})$ be two intuitionistic fuzzy values, $s ({\tilde{a}}_{1}) = μ_{1} - ν_{1}$ and $s ({\tilde{a}}_{2}) = μ_{2} - ν_{2}$ be the scores of $\tilde{a}$ and $\tilde{b}$ , respectively, and let $H ({\tilde{a}}_{1}) = μ_{1} + ν_{1}$ and $H ({\tilde{a}}_{2}) = μ_{2} + ν_{2}$ be the accuracy degrees of $\tilde{a}$ and $\tilde{b}$ , respectively, then if $S (\tilde{a}) < S (\tilde{b})$ , then $\tilde{a}$ is smaller than $\tilde{b}$ , denoted by $\tilde{a} < \tilde{b}$ ; if $S (\tilde{a}) = S (\tilde{b})$ , then

(1) if $H (\tilde{a}) = H (\tilde{b})$ , then $\tilde{a}$ and $\tilde{b}$ represent the same information, denoted by $\tilde{a} = \tilde{b}$ ; (2) if $H (\tilde{a}) < H (\tilde{b})$ , $\tilde{a}$ is smaller than $\tilde{b}$ , denoted by $\tilde{a} < \tilde{b}$ [15].

Hamacher operation [21] include the Hamacher product and Hamacher sum, which are examples of t-norms and t-conorms, respectively. They are defined as follows:

Hamacher product ⊗ is a t-norm and Hamacher sum ⊕ is a t-conorm, where $T (a, b) = a \otimes b = \frac{ab}{γ + (1 - γ) (a + b - ab)}$ (4) $T * (a, b) = a \oplus b = \frac{a + b - ab - (1 - γ) ab}{1 - (1 - γ) ab}$ (5)

Then, we shall introduce the Hamacher operations on intuitionistic fuzzy sets and analyze some desirable properties of these operations. Motivated by (4-5), let the t-norm T and t-conorm S be Hamacher product T” and Hamacher sum S” respectively, then the generalised intersection and union on two IFSs A and B become the Hamacher product (denoted by ${\tilde{a}}_{1} \otimes {\tilde{a}}_{2}$ ) and Hamacher sum (denoted by ${\tilde{a}}_{1} \oplus {\tilde{a}}_{2}$ ) on two IFSs ${\tilde{a}}_{1}$ and ${\tilde{a}}_{2}$ , respectively, as follows. $\begin{matrix} {\tilde{a}}_{1} \oplus {\tilde{a}}_{2} = (\frac{μ_{1} + μ_{2} - μ_{1} μ_{2} - (1 - γ) μ_{1} μ_{2}}{1 - (1 - γ) μ_{1} μ_{2}}, \\ \frac{ν_{1} ν_{2}}{γ + (1 - γ) (ν_{1} + ν_{2} - ν_{1} ν_{2})}) \end{matrix}$ $\begin{matrix} {\tilde{a}}_{1} \otimes {\tilde{a}}_{2} = (\frac{μ_{1} μ_{2}}{γ + (1 - γ) (μ_{1} + μ_{2} - μ_{1} μ_{2})}, \\ \frac{ν_{1} + ν_{2} - ν_{1} ν_{2} - (1 - γ) ν_{1} ν_{2}}{1 - (1 - γ) ν_{1} ν_{2}}) \end{matrix}$ $\begin{matrix} λ {\tilde{a}}_{1} = (\frac{{(1 + (γ - 1) μ_{1})}^{λ} - {(1 - μ_{1})}^{λ}}{{(1 + (γ - 1) μ_{1})}^{λ} + (γ - 1) {(1 - μ_{1})}^{λ}}, \\ \frac{γ {(ν_{1})}^{λ}}{{(1 + (γ - 1) (1 - ν_{1}))}^{λ} + (γ - 1) {(ν_{1})}^{λ}}) λ > 0 . \end{matrix}$ $\begin{matrix} {({\tilde{a}}_{1})}^{λ} = ((\frac{γ {(μ_{1})}^{λ}}{{(1 + (γ - 1) (1 - μ_{1}))}^{λ} + (γ - 1) {(μ_{1})}^{λ}}, \\ \frac{{(1 + (γ - 1) ν_{1})}^{λ} - {(1 - ν_{1})}^{λ}}{{(1 + (γ - 1) ν_{1})}^{λ} + (γ - 1) {(1 - ν_{1})}^{λ}}), λ > 0 \end{matrix}$

3 Intuitionistic fuzzy hamacher geometric aggregation operators

In the section, we shall propose the intuitionistic fuzzy geometric aggregation operators with the help of the Hamacher operations.

Definition 5. Let ${\tilde{a}}_{j} = (μ_{j}, ν_{j}) (j = 1, 2, \dots, n)$ (j = 1, 2, ⋯ , n) be a collection of intuitionistic fuzzy values, and let IFHWG: Qⁿ → Q, if ${IFHWG}_{ω} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) = \otimes_{j = 1}^{n} {({\tilde{a}}_{j})}^{ω_{j}}$ (6) where ω = (ω₁, ω₂, ⋯ , ω_n) ^T be the weight vector of ${\tilde{a}}_{j} (j = 1, 2, \dots, n)$ , and ω_j > 0, $\sum_{j = 1}^{n} ω_{j} = 1$ , then IFHWG is called the Intuitionistic fuzzy Hamacher weighted geometric (IFHWG) operator.

Based on Hamacher product operations of the Intuitionistic fuzzy numbers described, we can drive the Theorem 1.

Theorem 1. Let ${\tilde{a}}_{j} = (μ_{j}, ν_{j}) (j = 1, 2, \dots, n)$ be a collection of Intuitionistic fuzzy numbers , then their aggregated value by using the IF H W G operator is also a n IF N , and $\begin{matrix} {IFHWG}_{ω} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) = \otimes_{j = 1}^{n} {({\tilde{a}}_{j})}^{ω_{j}} \\ = (\frac{γ \prod_{j = 1}^{n} μ_{j}^{ω_{j}}}{\prod_{j = 1}^{n} {(1 + (γ - 1) (1 - μ_{j}))}^{ω_{j}} + (γ - 1) \prod_{j = 1}^{n} μ_{j}^{ω_{j}}}, \\ \frac{\prod_{j = 1}^{n} {(1 + (γ - 1) ν_{j})}^{ω_{j}} - \prod_{j = 1}^{n} {(1 - ν_{j})}^{ω_{j}}}{\prod_{j = 1}^{n} {(1 + (γ - 1) ν_{j})}^{ω_{j}} + (γ - 1) \prod_{j = 1}^{n} {(1 - ν_{j})}^{ω_{j}}}) \end{matrix}$ (7) where ω = (ω₁, ω₂, ⋯ , ω_n) ^T be the weight vector of ${\tilde{a}}_{j} (j = 1, 2, \dots, n)$ , and ω_j > 0, $\sum_{j = 1}^{n} ω_{j} = 1$ .

Now, we can discuss some special cases of the IFHWG operator with respect to the parameter γ.

When γ = 1, IFHWG operator reduces to the intuitionistic fuzzy weighted geometric (IFWG) operator as follows:

$\begin{matrix} {IFWG}_{ω} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) = \otimes_{j = 1}^{n} {({\tilde{a}}_{j})}^{ω_{j}} \\ = (\prod_{j = 1}^{n} μ_{j}^{ω_{j}}, 1 - \prod_{j = 1}^{n} {(1 - ν_{j})}^{ω_{j}}) \end{matrix}$ (8)

When γ = 2, IFHWG operator reduces to the intuitionistic fuzzy Einstein weighted geometric (IFEWG) operator as follows:

\begin{matrix} {IFEWG}_{ω} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) = \otimes_{j = 1}^{n} {({\tilde{a}}_{j})}^{ω_{j}} \\ = (\frac{2 \prod_{j = 1}^{n} μ_{j}^{ω_{j}}}{\prod_{j = 1}^{n} {(2 - μ_{j})}^{ω_{j}} + \prod_{j = 1}^{n} μ_{j}^{ω_{j}}}, \\ \frac{\prod_{j = 1}^{n} {(1 + ν_{j})}^{ω_{j}} - \prod_{j = 1}^{n} {(1 - ν_{j})}^{ω_{j}}}{\prod_{j = 1}^{n} {(1 + ν_{j})}^{ω_{j}} + \prod_{j = 1}^{n} {(1 - ν_{j})}^{ω_{j}}}) \end{matrix}

(9)

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

Theorem 2. (Idempotency) If all ${\tilde{a}}_{j} (j = 1, 2, \dots, n)$ are equal, i.e. ${\tilde{a}}_{j} = \tilde{a}$ for all j, then ${IFHWG}_{ω} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) = \tilde{a}$ (10)

Theorem 3. (Boundedness) Let ${\tilde{a}}_{j} (j = 1, 2, \dots, n)$ be a collection of IF N , and let ${\tilde{a}}^{-} = min_{j} {\tilde{a}}_{j}, {\tilde{a}}^{+} = max_{j} {\tilde{a}}_{j}$ Then ${\tilde{a}}^{-} \leq {IFHWG}_{ω} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) \leq {\tilde{a}}^{+}$ (11)

Theorem 4. (Monotonicity) Let ${\tilde{a}}_{j} (j = 1, 2, \dots, n)$ and ${\tilde{a}}_{j}^{'} (j = 1, 2, \dots, n)$ be two set of IF N s, if ${\tilde{a}}_{j} \leq {\tilde{a}}_{j}^{'}$ , for all j, then $\begin{matrix} {IFHWG}_{ω} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) \\ \leq {IFHWG}_{ω} ({\tilde{a}}_{1}^{'}, {\tilde{a}}_{2}^{'}, \dots, {\tilde{a}}_{n}^{'}) \end{matrix}$ (12)

Furthermore, we shall develop the intuitionistic fuzzy Hamacher ordered weighted geometric (IFHOWG) operator.

Definition 6. Let ${\tilde{a}}_{j} = (μ_{j}, ν_{j}) (j = 1, 2, \dots, n)$ be a collection of Intuitionistic fuzzy numbers. An Intuitionistic fuzzy Hamacher ordered weighted geometric (IFHOWG) operator of dimension n is a mapping IFHOWG: Qⁿ → Q, that has an associated vector w = (w₁, w₂, ⋯ , w_n) ^T such that w_j > 0 and $\sum_{j = 1}^{n} w_{j} = 1$ . Furthermore, ${IFHOWG}_{w} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) = \otimes_{j = 1}^{n} {({\tilde{a}}_{σ (j)})}^{w_{j}}$ (13) where (σ (1) , σ (2) , ⋯ , σ (n)) is a permutation of (1, 2, ⋯ , n), such that ${\tilde{α}}_{σ (j - 1)} \geq {\tilde{α}}_{σ (j)}$ for all j = 2, ⋯ , n.

Based on Hamacher product operations of the Intuitionistic fuzzy values described, we can drive the Theorem 5.

Theorem 5. Let ${\tilde{a}}_{j} = (μ_{j}, ν_{j}) (j = 1, 2, \dots, n)$ be a collection of Intuitionistic fuzzy numbers . An Intuitionistic fuzzy Hamacher ordered weighted geometric (IF H OW G) operator of dimension n is a mapping IF H OW G: Qⁿ → Q, that has an associated vector w = (w₁, w₂, ⋯ , w_n) ^T such that w_j > 0 and $\sum_{j = 1}^{n} w_{j} = 1$ . Furthermore, $\begin{matrix} {IFHOWG}_{w} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) = \otimes_{j = 1}^{n} {({\tilde{a}}_{σ (j)})}^{w_{j}} \\ = (\frac{γ \prod_{j = 1}^{n} μ_{σ (j)}^{w_{j}}}{\prod_{j = 1}^{n} {(1 + (γ - 1) (1 - μ_{σ (j)}))}^{w_{j}} + (γ - 1) \prod_{y} {j = 1}^{n} μ_{σ (j)}^{w_{j}}}, \\ \frac{\prod_{j = 1}^{n} {(1 + (γ - 1) ν_{σ (j)})}^{w_{j}} - \prod_{j = 1}^{n} {(1 - ν_{σ (j)})}^{w_{j}}}{\prod_{j = 1}^{n} {(1 + (γ - 1) ν_{σ (j)})}^{w_{j}} + (γ - 1) \prod_{j = 1}^{n} {(1 - ν_{σ (j)})}^{w_{j}}}) \end{matrix}$ (14) where (σ (1) , σ (2) , ⋯ , σ (n)) is a permutation of (1, 2, ⋯ , n), such that ${\tilde{α}}_{σ (j - 1)} \geq {\tilde{α}}_{σ (j)}$ for all j = 2, ⋯ , n.

Now, we can discuss some special cases of the IFHOWG operator with respect to the parameter γ.

When γ = 1, IFHOWG operator reduces to the intuitionistic fuzzy ordered weighted geometric (IFOWG) operator as follows:

$\begin{matrix} {IFOWG}_{w} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) = \otimes_{j = 1}^{n} {({\tilde{a}}_{σ (j)})}^{w_{j}} \\ = (\prod_{j = 1}^{n} μ_{σ (j)}^{w_{j}}, 1 - \prod_{j = 1}^{n} {(1 - ν_{σ (j)})}^{w_{j}}) \end{matrix}$ (15)

When γ = 2, IFHOWG operator reduces to the intuitionistic fuzzy Einstein ordered weighted geometric (HFEOWG) operator as follows:

\begin{matrix} {IFEOWG}_{w} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) = \otimes_{j = 1}^{n} {({\tilde{a}}_{σ (j)})}^{w_{j}} \\ = (\frac{2 \prod_{j = 1}^{n} μ_{σ (j)}^{w_{j}}}{\prod_{j = 1}^{n} {(2 - μ_{σ (j)})}^{w_{j}} + \prod_{j = 1}^{n} μ_{σ (j)}^{w_{j}}}, \\ \frac{\prod_{j = 1}^{n} {(1 + ν_{σ (j)})}^{w_{j}} - \prod_{j = 1}^{n} {(1 - ν_{σ (j)})}^{w_{j}}}{\prod_{j = 1}^{n} {(1 + ν_{σ (j)})}^{w_{j}} + \prod_{j = 1}^{n} {(1 - ν_{σ (j)})}^{w_{j}}}) \end{matrix}

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

Theorem 6. (Idempotency) If all ${\tilde{a}}_{j} (j = 1, 2, \dots, n)$ are equal, i.e. ${\tilde{a}}_{j} = \tilde{a}$ for all j, then ${IFHOWG}_{w} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) = \tilde{a}$ (17)

Theorem 7. (Boundedness) Let ${\tilde{a}}_{j} (j = 1, 2, \dots, n)$ be a collection of IF N s, and let ${\tilde{a}}^{-} = min_{j} {\tilde{a}}_{j}, {\tilde{a}}^{+} = max_{j} {\tilde{a}}_{j}$

Then ${\tilde{a}}^{-} \leq {IFHOWG}_{w} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) \leq {\tilde{a}}^{+}$ (18)

Theorem 8. (Monotonicity) Let ${\tilde{a}}_{j} (j = 1, 2, \dots, n)$ and ${\tilde{a}}_{j}^{'} (j = 1, 2, \dots, n)$ be two set of IF N s, if ${\tilde{a}}_{j} \leq {\tilde{a}}_{j}^{'}$ , for all j, then $\begin{matrix} {IFHOWG}_{w} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) \\ \leq {IFHOWG}_{w} ({\tilde{a}}_{1}^{'}, {\tilde{a}}_{2}^{'}, \dots, {\tilde{a}}_{n}^{'}) \end{matrix}$ (19)

Theorem 9. (Commutativity) Let ${\tilde{a}}_{j} (j = 1, 2, \dots, n)$ and ${\tilde{a}}_{j}^{'} (j = 1, 2, \dots, n)$ be two set of IF N s, then $\begin{matrix} {IFHOWG}_{w} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) \\ = {IFHOWG}_{w} ({\tilde{a}}_{1}^{'}, {\tilde{a}}_{2}^{'}, \dots, {\tilde{a}}_{n}^{'}) \end{matrix}$ (20) where ${\tilde{a}}_{j}^{'} (j = 1, 2, \dots, n)$ is any permutation of ${\tilde{a}}_{j} (j = 1, 2, \dots, n)$ .

From Definitions 5 and 6, we know that the IFHWG operator weights only the intuitionistic fuzzy values, while the IFHOWG operator weights only the ordered positions of the intuitionistic fuzzy values instead of weighting the intuitionistic fuzzy values themselves. Therefore, weights represent different aspects in both the IFHWG and IFHOWG operators. However, both the operators consider only one of them. To solve this drawback, in the following we shall propose an intuitionistic fuzzy Hamacher hybrid geometric (IFHHG).

Definition 7. An intuitionistic fuzzy Hamacher hybrid geometric (IFHHG) operator of dimension n is a mapping IFHHG: Qⁿ → Q, that has an associated vector w = (w₁, w₂, ⋯ , w_n) ^T such that w_j > 0 and $\sum_{j = 1}^{n} w_{j} = 1$ . Furthermore, $\begin{array}{l} {IFHHG}_{ω, w} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) = \otimes_{j = 1}^{n} ({\overset{\dot{\sim}}{a}}_{σ (j)})^{w_{j}} \\ = (\frac{γ \prod_{j = 1}^{n} {\dot{μ}}_{σ {(j)}^{w_{j}}}}{\prod_{j = 1}^{n} {(1 + (γ - 1) (1 - {\dot{μ}}_{σ (j)}))}^{w_{j}} + (γ - 1) \prod_{j = 1}^{n} {\dot{μ}}_{σ {(j)}^{w_{j}}}}, \\ \frac{\prod_{j = 1}^{n} {(1 + (γ - 1) {\dot{ν}}_{σ (j)})}^{w_{j}} - \prod_{j = 1}^{n} {(1 - {\dot{ν}}_{σ (j)})}^{w_{j}}}{\prod_{j = 1}^{n} {(1 + (γ - 1) {\dot{ν}}_{σ (j)})}^{w_{j}} + (γ - 1) \prod_{j = 1}^{n} {(1 - {\dot{ν}}_{σ (j)})}^{w_{j}}}) \end{array}$ (21) where $\overset{\dot{\sim}}{a_{σ}} (j)$ is the jth largest of the weighted intuitionistic fuzzy values $\overset{\dot{\sim}}{a_{σ}} ({\overset{\dot{\sim}}{a}}_{j} = {({\tilde{a}}_{j})}^{n ω j}, j = 1, 2,, n),$ ω = (ω₁,ω₂,. . …,ω_n)^T be the weight vector of ~ a_j (j = 1,2, …,n), and $ω_{j} > 0, \sum_{j = 1}^{n} ω_{j} = 1$ , and n is the balancing coefficient. Let w = (1 /. n, …,1 /. n), then the IFHWG operator is a special case of the IFHHG operator; let ω = (1 /. n,1 /. n, …,1 /. n), then the IFHOWG operator is a special case of the IFHHG operator. So we know that the IFHHG operator generalizes both the IFHWG and IFHOWG operators, and reflects the importance degrees of both the given arguments and their ordered positions.

Now, we can discuss some special cases of the IFHHG operator with respect to the parameter γ.

When γ = 1, IFHHG operator reduces to the intuitionistic fuzzy hybrid geometric (HFHG) operator as follows:

\begin{array}{l} {IFHG}_{ω, w} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) = \otimes_{j = 1}^{n} ({\overset{\dot{\sim}}{a}}_{σ (j)})^{w_{j}} \\ = (\prod_{j = 1}^{n} {\dot{μ}}_{σ (j)}^{w_{j}}, 1 - \prod_{j = 1}^{n} {(1 - {\dot{ν}}_{σ (j)})}^{w_{j}}) \end{array}

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When γ = 2, IFHHG operator reduces to the intuitionistic fuzzy Einstein hybrid geometric (IFEHG) operator as follows:

\begin{array}{l} {IFEHG}_{w, ω} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) = \otimes_{j = 1}^{n} ({\overset{\dot{\sim}}{a}}_{σ (j)})^{w_{j}} \\ = (\frac{2 \prod_{j = 1}^{n} {\dot{μ}}_{σ {(j)}^{w_{j}}}}{\prod_{j = 1}^{n} {(2 - {\dot{μ}}_{σ (j)})}^{w_{j}} + \prod_{j = 1}^{n} {\dot{μ}}_{σ {(j)}^{w_{j}}}}, \\ \frac{\prod_{j = 1}^{n} {(1 + {\dot{ν}}_{σ (j)})}^{w_{j}} - \prod_{j = 1}^{n} {(1 - {\dot{ν}}_{σ (j)})}^{w_{j}}}{\prod_{j = 1}^{n} {(1 + {\dot{ν}}_{σ (j)})}^{w_{j}} + \prod_{j = 1}^{n} {(1 - {\dot{ν}}_{σ (j)})}^{w_{j}}}) \end{array}

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4 Model for multiple attribute decision making with intuitionistic fuzzy information

In this section, we shall investigate the multiple attribute decision making (MADM) problems based on the IFHHG operator with intuitionistic fuzzy numbers. Let A = { A₁,A₂, …,A_m }be a discrete set of alternatives, and G = { G₁,G₂, …,G_n }be the set of attributes. The information about attribute weights is completely known. Let w = (w₁,w₂, …,w_n) ∈ H be the weight vector of attributes, where w_j ≥ 0, j = 1,2, …,n, ∑_j = 1ⁿ w_j = 1. Suppose that ~ R = (~ r_ij)_m × n = (μ_ij,ν_ij)_m × n is the intuitionistic fuzzy decision matrix, where μ_ij indicates the degree that the alternative A_i satisfies the attribute G_j given by the decision maker, ν_ij indicates the degree that the alternative A_i doesn’t satisfy the attribute G_j given by the decision maker, μ_ij ⊂ [0,1], ν_ij ⊂ [0,1], μ_ij + ν_ij ≤ 1, i = 1,2, …,m, j = 1,2, …,n.

In the following, we apply the IFHHG operator to multiple attribute decision making based on intuitionistic fuzzy information. The method involves the following steps:

Step 1. Utilize the decision information given in matrix ~ R, and the IFHHA operator

$\begin{matrix} {\tilde{r}}_{i} & = (μ_{i}, ν_{i}) = {IFHHG}_{ω, w} ({\tilde{r}}_{i 1}, {\tilde{r}}_{i 2}, \dots, {\tilde{r}}_{in}), \\ i & = 1, 2, \dots, m . \end{matrix}$ (24) to derive the collective overall preference values ~ r_i (i = 1,2, …,m)of the alternative A_i , where ω = (ω₁,ω₂, …,ω_n)^T is the weighting vector of the attributes and w = (w₁,w₂, …,w_n)^T is the associated vector of the IFHHA operator, such that w_j >0and ∑_j = 1ⁿ w_j = 1.

Step 2. calculate the scores S (~ r_i) (i = 1,2, …,m) of the collective overall values ~ r_i (i = 1,2, …,m)to rank all the alternatives A_i (i = 1,2, …,m) and then to select the best one(s)

Step 3. Rank all the alternatives A_i (i = 1,2, …,m) and select the best one(s) in accordance with S (~ r_i) and H (~ r_i) (i = 1,2, …,m).

Step 4. End.

5 Numerical example

In such a dynamic and complicated environment of increasingly volatile competition in both domestic and overseas markets, rapid technological change and continually new business model development at the present, it is of great importance for the established firms to know how to give a fast response to environmental change through innovation and transformation so as to gain and maintain sustainable competitive advantages. Open innovation beyond organizational boundary and corporate venturing featuring innovation and transformation become a critical path and also a key strategic choice for the companies to deal with the dynamic and complicated environment. Corporate venture capital, an important mode of open innovation and corporate venturing, has been quickly developed worldwide. As a rising star of corporate venture capital, China encounters such problems as inefficient use of capital, listed companies’ insufficient participation in corporate venture capital investments and so forth, which hedge the improvement of corporate venture capital investments performance as well as the innovation and transformation of the established companies. Corporate investors should pay close attention to and answer the following questions urgently: how is corporate venture capital investments performance (strategic performance: technological innovation; comprehensive performance: value creation)? What are the factors influencing corporate venture capital performance of technological innovation and value creation? How should the company organize corporate venture capital activities effectively? In this section, we present a numerical example for investment performance evaluation of corporate venture capital. The investment performance of five possible corporate venture capital enterprises A_i (i = 1,2,3,4,5)is evaluated. The investment company must take a decision according to the following five attributes: G1 is the debt paying ability; G2 is the operation capability; G3 is the earning capacity; G4 is the development capability. The five possible corporate venture capital enterprises A_i (i = 1,2,3,4,5)are to be evaluated using the intuitionistic fuzzy information by the decision maker under the above four attributes (whose weighting vector ω = (0.3,0.2,0.4,0.1)^T), as listed in the following matrix. $\tilde{R} = [\begin{matrix} (0.2, 0.5) & (0.2, 0.7) & (0.2, 0.6) & (0.1, 0.6) \\ (0.3, 0.7) & (0.6, 0.3) & (0.5, 0.3) & (0.3, 0.4) \\ (0.1, 0.6) & (0.4, 0.4) & (0.3, 0.4) & (0.5, 0.2) \\ (0.3, 0.8) & (0.2, 0.5) & (0.4, 0.5) & (0.1, 0.5) \\ (0.5, 0.1) & (0.3, 0.2) & (0.6, 0.2) & (0.4, 0.2) \end{matrix}]$

In the following, we apply the IFHHG operator to multiple attribute decision making for investment performance evaluation of corporate venture capital with intuitionistic fuzzy information. The method involves the following steps:

Step 1. Utilize the decision information given in matrix ~ R, and the IFHHG operator which has associated weighting vector w = (0.20,0.30,0.30,0.2)^T , we obtain the overall values ~ r_i of the alternatives A_i (i = 1,2, …,5). $\begin{matrix} {\tilde{r}}_{1} = (0.3525, 0.5445), {\tilde{r}}_{2} = (0.5725, 0.3494) \\ {\tilde{r}}_{3} = (0.4384, 0.4547), {\tilde{r}}_{4} = (0.4530, 0.3868) \\ {\tilde{r}}_{5} = (0.4666, 0.3015) \end{matrix}$

Step 2. Calculate the scores S (~ r_i) (i = 1,2, …,5)of the overall Intuitionistic fuzzy values ~ r_i (i = 1,2,..…,5) $\begin{matrix} S ({\tilde{r}}_{1}) = - 0.1920, S ({\tilde{r}}_{2}) = 0.2230 \\ S ({\tilde{r}}_{3}) = - 0.0163, S ({\tilde{r}}_{4}) = 0.0662 \\ S ({\tilde{r}}_{5}) = 0.1651 \end{matrix}$

Step 3. Rank all the corporate venture capital enterprises A_i (i = 1,2,3,4,5)in accordance with the scores S (~ r_i) (i = 1,2, …,5) of the overall values ~ r_i (i = 1,2, …,5): A₂ ≻ A₅^≻ A₃ ≻ A₄ ≻ A₁, and thus the most desirable corporate venture capital enterprise is A₂.

6 Conclusion

Corporate venture capital refers to the direct venture investment in private companies made by non-financial corporations with specific core business. Along with rapid growth of Chinese economy, domestic venture capital has gone through a fantastic development among which corporate venture capital is an outstanding one. The study on the different types of venture captital, especially corporate venture capital, is required by the establishment and improvement of the mutiple-level capital market in China. Only on the basis of throughly understanding of the traits, advantages and disadvantages of corporate venture capital evaluation, we can set a better platform for its further development, contributing to the promotion of the entire venture capital market. In this paper, we utilize Hamacher operations to develop some intuitionistic fuzzy aggregation operators: intuitionistic fuzzy Hamacher weighted geometric (IFHWG) operator, intuitionistic fuzzy Hamacher ordered weighted geometric (IFHOWG) operator and intuitionistic fuzzy Hamacher hybrid geometric (IFHHG) operator. The prominent characteristic of these proposed operators are studied. Then, we have utilized these operators to develop some approaches to solve the intuitionistic fuzzy multiple attribute decision making problems for investment performance evaluation of corporate venture capital. Finally, a practical example for investment performance evaluation of corporate venture capital is given to verify the developed approach and to demonstrate its practicality and effectiveness.

Footnotes

Acknowledgments

The paper is supported by the Supported by the Fundamental Research Funds for the Central Universities (JUSRP51330B), the National Social Science Foundation of China (14BGL019), the National Social Science Fund, “Research on Risk Warning and Emergency Management Mechanism after the International Financial Crisis”(11BGL047).

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