An Effective Interval-Valued Intuitionistic Fuzzy Entropy to Evaluate Entrepreneurship Orientation of Online P 2 P Lending Platforms

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.

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][31][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 -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 nonprobabilistictype 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][48][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.

Preliminaries
Here, we give a brief review of some preliminaries.
The intervals μ () and ] () denote the degree of membership and nonmembership of the element  in the set , respectively.
For convenience, let 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  = ([, ], [, ]).

An Effective Interval-Valued Intuitionistic Fuzzy Entropy
which indicate that  1 () =  1 () is not consistent with our intuition.

New Interval-Valued Intuitionistic Fuzzy Entropy.
A new interval-valued intuitionistic fuzzy entropy measure is introduced in what follows.
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.
(P4) In order to show that (12) fulfills the requirement of (P4), they suffice to prove the following function: where function  is monotonically decreasing.

Numeric Examples for the New Interval-Valued
Intuitionistic Fuzzy Entropy 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. Intuitively, we can see that  is more fuzzy than .Now we calculate the () and () by (12); we can obtain Advances in Mathematical Physics 5 which indicate that () > () is consistent with our intuition.This result is better than the result in Example 8.

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.
If the information about weight   of the criterion   ( = 1, 2, . . ., ) 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 ( 1 ,  2 , . . .,   )  .
In summary, the evaluation procedure proposed is listed below.
Step 1. Collect data for three given attributes.
Step 3. Utilize the decision information given in matrix  and the IVIF-WGA operator to derive the collective overall values   ( = 1, 2, . . ., ) of the alternative   .

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,  = { 1 , Moreover, the lending expert can only provide his incomplete information on the weights as follows: Step 2. Calculate the weight vector  = ( 1 ,  2 ,  3 )  by solving model ( 20

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.