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Advances in Mathematical Physics

Volume 2013 (2013), Article ID 467215, 8 pages

http://dx.doi.org/10.1155/2013/467215

## An Effective Interval-Valued Intuitionistic Fuzzy Entropy to Evaluate Entrepreneurship Orientation of Online P2P Lending Platforms

Business School, Central South University, Changsha 410083, China

Received 25 January 2013; Accepted 2 August 2013

Academic Editor: Christian Maes

Copyright © 2013 Xiaohong Chen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper describes an approach to measure the entrepreneurship orientation of online P2P lending platforms. The limitations of existing methods for calculating entropy of interval-valued intuitionistic fuzzy sets (IVIFSs) are significantly improved by a new entropy measure of IVIFS considered in this paper, and then the essential properties of the proposed entropy are introduced. Moreover, an evaluation procedure is proposed to measure entrepreneurship orientation of online P2P lending platforms. Finally, a case is used to demonstrate the effectiveness of this method.

#### 1. Introduction

With the enormous popularity of online communities, a new way of loan origination has entered the credit market: online peer-to-peer (P2P) lending [1]. Online P2P lending platforms are financial institutions operating without the participation of traditional financial intermediaries. Although online P2P lending is a relatively young field of research, an increasing amount of scientific contributions has been published in recent years. The main research is focused on stakeholders (Herzenstein et al., 2008 [2]; Klafft, 2008 [3]; Galloway, 2009 [4]), funding success factors (Kumar, 2007 [5]; Lin, 2009 [6]; Larrimore et al., 2009 [7]; Duarte et al., 2012 [8]; Burtch et al., 2013 [9]), determinants of interest rates factors (Pope and Sydnor, 2011 [10]; Iyer et al., 2009 [11]; Collier and Hampshire, 2010 [12]; Brandes et al., 2011 [13]), and lenders’ behavior (Shen et al., 2010 [14]; Yum et al., 2012 [15]; Dezs and Loewenstein, 2012 [16]; Zhang and Liu [17]). The online P2P lending platforms are firms, and they need to acquire entrepreneurial competences to survive. The processes of strategy-making and the styles of firms engaging in entrepreneurial activities are together referred to as “entrepreneurship orientation” (EO) [18]. Several studies have found a positive relationship between EO and firm performance (e.g., [19, 20]). Consequently, measuring EO of online P2P lending platforms is of special importance for firms and for organizations, such as venture capitalists, business angels, and governments.

Variations in the weights often influence the rankings of the alternatives [21]. The weights can be classified into subjective weights and objective weights depending on the information source. The most well-known method of generating objective weights is the entropy method [22]. Entropy has been the main tool for measuring uncertain information since information theory was conceived in the work of Shannon [23] more than sixty years ago. Fuzziness, a feature of imperfect information, results from the lack of crisp distinction between the elements belonging and not belonging to a set; that is, the boundaries of the set under consideration are not sharply defined [24]. Entropy has also been concerned as a measure of fuzziness since fuzzy entropy was first mentioned in 1965 by Zadeh [25]. de Luca and Termini [26] proposed a fuzzy entropy based on Shannon’s function. Kaufmann [27] proposed to measure the degree of fuzziness of any fuzzy set by a metric distance between its membership function and the membership function (characteristic function) of its nearest crisp set. Another way given by Yager [28] was to view the degree of fuzziness in terms of a lack of distinction between the fuzzy set and its complement. Higashi and Klir [29] showed the entropy measure as the difference between two fuzzy sets. Kosko [30–32] investigated the fuzzy entropy in relation to a measure of subsethood. Parkash et al. [33] have developed two new measures of weighed fuzzy entropy, the findings of which have been applied to study the principle of maximum weighed fuzzy entropy. Lotfi and Fallahnejad [34] extended Shannon’s entropy method for fuzzy data based on * α*-level sets.

The theory of fuzzy sets (FSs) proposed by Zadeh [25] has achieved a great success in various fields. Later, a lot of generalized forms of FSs have been proposed. The classical sets include interval-valued fuzzy sets (IVFSs) [35], intuitionistic fuzzy sets (IFSs) [36], interval-valued intuitionistic fuzzy sets (IVIFSs) [37], and vague sets [38]. Burillo and Bustince [39] introduced the notion that entropy of IVFSs and IFSs can be used to evaluate the degree of intuitionism of an IVFS or IFS. Szimidt and Kacprzyk [24] proposed a nonprobabilistic-type entropy measure with a geometric interpretation of IFSs. Hung and Yang [40] gave their axiomatic definitions of entropy of IFSs and IVFSs by exploiting the concept of probability. Zhang and Jiang [41] proposed nonprobabilistic entropy of a vague set by means of the intersection and union of the membership degree and nonmembership degree of the vague set. Ye [42] proposed two effective entropy measures for IVFSs. Wei et al. [43] gave an entropy measure for IVIFSs based on three entropy measures defined independently by Szmidt and Kacprzyk [24], Wang and Lei [44], and Huang and Liu [45]. Zhang et al. [46] proposed a new information entropy measure of interval-valued intuitionistic fuzzy set by using membership interval and nonmembership interval of IVIFSs, which complied with the extended form of de Luca and Termini [26] axioms for fuzzy entropy.

From [47–49], it turns out that IVFS theory is equivalent to IFS theory, which is equivalent to vague set theory, and IVIFS theory extends IFS theory. At the same time, some existing entropy measures for IVIFSs are not always effective in some cases. The focus of this study is on apparent weaknesses of these entropy measures. In this paper, we propose a novel formula to calculate the entropy of an IVIFS on the basis of the argument on the relationship among the entropies given in [50, 51] and use it to evaluate entrepreneurship orientation of online P2P lending platforms. The rest of this paper is organized as follows. In Section 2, we introduce some basic notions of IFS and IVIFSs. In Section 3, we propose a new entropy measure of interval-valued intuitionistic fuzzy set by using cotangent function. Two numeral examples are given to demonstrate the effectiveness by the comparison of the proposed entropy and existing entropy [50, 51]. In Section 4, we use the new entropy to evaluate entrepreneurship orientation of online P2P lending platforms. Concluding remarks are made in Section 5.

#### 2. Preliminaries

Here, we give a brief review of some preliminaries.

*Definition 1 (see [37]). * Let be the set of all closed subintervals of the interval and an ordinary finite nonempty set. An interval-valued intuitionistic fuzzy set (IVIFS) in is an expression given by
in the set , respectively, where
with the condition for any .

The intervals and denote the degree of membership and nonmembership of the element in the set , respectively.

For convenience, let , ; then
where , , .

For each element , we can compute the hesitancy degree of an intuitionistic fuzzy interval of in defined as follows:

For convenience, an IVIFS value is denoted by .

*Definition 2 (see [37]). *Let ; then some operations can be defined as follows:

The following expressions are defined in [37] for all : if and only if , , and for all , if and only if and for all , .

In the following, we introduce two weighed aggregation operators related to IVIFSs [52].

*Definition 3 (see [52]). * Let . The weighed geometric average operator (IVIF-WGA operator) is defined by
where is the weight of , and . In particular, assuming that , then is called an arithmetic average operator for IVIFSs.

*Definition 4 (see [53]). *Let be an interval-valued intuitionistic fuzzy number; a score function of an interval-valued intuitionistic fuzzy value can be represented as follows:

*Definition 5 (see [53]). *Let and be two interval-valued intuitionistic fuzzy values and and the scores of and , respectively; then if , then is smaller than , denoted by .

*Definition 6 (see [43]). * A real-valued function is called an entropy measure on if it satisfies the following axiomatic requirements:(P1), if and only if is a crisp set,(P2), if and only if for all ,(P3) for all ,(P4) if is less fuzzy than , that is, , and for all or and for all .

#### 3. An Effective Interval-Valued Intuitionistic Fuzzy Entropy

##### 3.1. The Limitations of the Existing Interval-Valued Intuitionistic Fuzzy Entropy

Let us suppose that is the entropy of IVIFSs for convenience.

Vlachos’ entropy measure [50]:

*Example 7. * Let and be two IVIFSs in .

Intuitively, we can see that is more fuzzy than . Now, we calculate the and by (8), and we can obtain

which indicate that is not consistent with our intuition.

Ye’s entropy measures [51]:

where representes two fixed numbers, and .

*Example 8. *Let and be two IVIFSs in .

Intuitively, is more fuzzy than . Now the and can be gained by (10), and taking , the results are
which indicates that is not consistent with our intuition.

##### 3.2. New Interval-Valued Intuitionistic Fuzzy Entropy

A new interval-valued intuitionistic fuzzy entropy measure is introduced in what follows.

*Definition 9. *Assuming that , then an interval-valued intuitionistic fuzzy entropy measure can be defined as

Theorem 10. *The mapping , defined by (12), is an entropy measure for IVIFSs.*

*Proof. *In order for (12) to be qualified as a sensible measure of interval-valued intuitionistic fuzzy entropy, it must satisfy the conditions (P1)–(P4) in Definition 6.(P1)Let be a crisp set. Then, we have and , or , and . So, .(P2)Let ; we can obtain .(P3)It is clear that . By applying (12), we have .(P4)In order to show that (12) fulfills the requirement of (P4), they suffice to prove the following function:
where function is monotonically decreasing.

Suppose that , and ; in order to prove (13), namely, we prove (14):
If and , then it follows that . If and , then we have . So, we can get that (14) holds. Therefore, .

Similarly, when , and , we can also prove that .

##### 3.3. Numeric Examples for the New Interval-Valued Intuitionistic Fuzzy Entropy

*Example 11. *Let and be two IVIFSs in .

Intuitively, we can see that is more fuzzy than . Now we calculate the and by (12); we can obtain

which indicate that is consistent with our intuition. This result is better than the result in Example 7.

*Example 12. * Let and be two IVIFSs in .

Intuitively, we can see that is more fuzzy than . Now we calculate the and by (12); we can obtain

which indicate that is consistent with our intuition. This result is better than the result in Example 8.

#### 4. Evaluating Entrepreneurship Orientation of Online P2P Lending Platforms

In this section, we apply the proposed entropy to evaluate entrepreneurship orientation of online P2P lending platforms.

##### 4.1. The Evaluation Procedure

Evaluate entrepreneurship orientation of online P2P lending platforms. Assume that there are alternatives and evaluation criteria with weight vector associated with , where and . Three evaluated criteria are considered, including innovativeness (), risk-taking (), and proactiveness () [54, 55]. In this case, the characteristic of the alternative is represented by the following IVIFS: where , , , , and . The IVIFS value, that is, the pair of intervals , for , is denoted by for convenience. Here, we can elicit an interval-valued intuitionistic fuzzy evaluation matrix .

If the information about weight of the criterion is incomplete, for determining the criterion weight from the evaluation matrix , we can establish a model of interval-valued intuitionistic fuzzy entropy weights.

For the criteria , the entropy of the alternative can be given as

And the overall entropy for the alternative is given as

According to the entropy theory, if the entropy value for an alternative is smaller across alternatives, it can provide decision makers with the useful information. Therefore, the criteria should be assigned a bigger weight. Then the smaller the value of (19) is, the better weight we should assign to the criteria.

Let be the set of incomplete information about criteria weights; to get the optimal weight vector, the following model can be constructed:

By solving model (20) with Excel software, we get the optimal solution .

In summary, the evaluation procedure proposed is listed below.

*Step 1. * Collect data for three given attributes.

*Step 2. *Calculate the weight vector by solving model (20).

*Step 3. *Utilize the decision information given in matrix and the IVIF-WGA operator to derive the collective overall values of the alternative .

*Step 4. * Calculate the scores of the collective overall values to rank all the alternatives .

##### 4.2. Implementation in Case

In this section, we apply the proposed methodology to evaluate the EO of four online P2P lending platforms and rank them based on their final EO scores.

*Step 1. *Suppose that a lending expert in a financial management firm is assessing the EO of four online P2P lending platforms, . In addition, the lending expert is only comfortable with providing his assessment of each alternative on each attribute as an IVIFS and the evaluation matrix is

Moreover, the lending expert can only provide his incomplete information on the weights as follows:

*Step 2. * Calculate the weight vector by solving model (20):

*Step 3. *Utilize the evaluation information given in matrix and the IVIF-WGA operator to derive the collective overall values of the alternative :

*Step 4. * Calculate the scores of the collective overall values :

Rank all the alternatives in accordance with the score of the collective overall interval-valued intuitionistic fuzzy values : .

#### 5. Conclusion

The ability to assess the level of entrepreneurship of online P2P lending platforms is an important management tool for the platforms themselves and for other organizations. This paper describes an approach to measure the entrepreneurship orientation of online P2P lending platforms. Though many information measures have been developed, still there is scope that better measures can be developed, which will find applications in a variety of fields. In this paper, firstly, some existing measures of entropy are reviewed. Then some examples are applied to show that some existing entropy measures are not always effective in some cases. At the same time, a new entropy measure of interval-valued intuitionistic fuzzy sets is proposed by using cotangent function to overcome limitations of the existing methods. Two examples are made to show that the proposed entropy measure is more reasonable than some existing entropy measures. Therefore, the proposed entropy measure can provide a useful way for measuring the fuzziness of IVIFSs more effectively. Moreover, an evaluation procedure is proposed to measure entrepreneurship orientation of online P2P lending platforms. Finally, a case is used to demonstrate the effectiveness of this method.

#### Conflict of Interests

The authors have declared that no conflict of interests exists.

#### Acknowledgment

This paper was supported by the National Natural Science Foundation of China (no. 71221061 and no. 71210003).

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