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

Authors: Xiaohong Chen; Li Yang; Pei Wang; Wei Yue
Journal: Advances in Mathematical Physics
 (2013)

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## 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.


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## Body

## 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]).
LetD[0,1] be the set of all closed subintervals of the interval [0,1] and X an ordinary finite nonempty set. An interval-valued intuitionistic fuzzy set (IVIFS) in X is an expression given by
(1)A={〈x,μ~A(x),ν~A(x)〉∣x∈X},xin the set A, respectively, where
(2)μ~A:X⟶D[0,1],ν~A:X⟶D[0,1]
with the condition 0≤sup(μ~A(x))+sup(νA(x))≤1 for any x∈X.
The intervalsμ~A(x) and ν~A(x) denote the degree of membership and nonmembership of the element x in the set A, respectively.
For convenience, letμ~A(x)=[μAL(x),μAU(x)], ν~A(x)=[νAL(x),νAU(x)]; then
(3)A={〈x,[μAL(x),μAU(x)],[νAL(x),νAU(x)]〉∣x∈X},
where 0≤μAU(x)+νAU(x)≤1,  μAL(x)≥0, νAL(x)≥0.
For each elementx, we can compute the hesitancy degree of an intuitionistic fuzzy interval of x∈X in A defined as follows:
(4)π~A(x)=1-μ~A(x)-ν~A(x)=[1-μAU(x)-νAU(x),1-μAL(x)-νAL(x)].
For convenience, an IVIFS value is denoted byA=([a,b],[c,d]).Definition 2 (see [37]).
LetA,B∈IVIFS(X); then some operations can be defined as follows:
(5)A∪B={〈xi,[μAL(xi)∨μBL(xi),μAU(xi)∨μBU(xi)],[νAL(xi)∧νBL(xi),νAU(xi)∧νBU(xi)]〉},A∩B={〈xi,[μAL(xi)∧μBL(xi),μAU(xi)∧μBU(xi)],[νAL(xi)∨νBL(xi),νAU(xi)∨νBU(xi)]〉}.
The following expressions are defined in [37] for all A,B∈IVFSs(X):A⊆B
if and only ifμAL≤μBL, μAU≤μBU, νAL≥νBL and νAU≥νBU for all x∈X,A=B if and only if A⊆B and B⊆A for all x∈X,AC={〈x,[νAL(x),νAU(x)],[μAL(x),μAU(x)]〉∣x∈X}.
In the following, we introduce two weighed aggregation operators related to IVIFSs [52].Definition 3 (see [52]).
LetAj(j=1,2,…,n)∈IVIFS(X). The weighed geometric average operator (IVIF-WGA operator) is defined by
(6)Fw(A1,A2,…,An)=([∏j=1nμAjL(x)wj,∏j=1nμAjU(x)wj],[1-∏j=1n(1-νAjL(x))wj,1-∏j=1n(1-νAjU(x))wj]),
where wj is the weight of Aj(j=1,2,…,n), wj∈[0,1] and ∑j=1nwj=1. In particular, assuming that wj=1/n(j=1,2,…,n), then Fw is called an arithmetic average operator for IVIFSs.Definition 4 (see [53]).
LetA=([a,b],[c,d]) be an interval-valued intuitionistic fuzzy number; a score function S of an interval-valued intuitionistic fuzzy value can be represented as follows:
(7)S(A)=a-c+b-d2.Definition 5 (see [53]).
LetA1=([a1,b1],[c1,d1]) and A2=([a2,b2],[c2,d2]) be two interval-valued intuitionistic fuzzy values and S(A1)=(a1-c1+b1-d1)/2 and S(A2)=(a2-c2+b2-d2)/2 the scores of A1 and A2, respectively; then if S(A1)<S(A2), then A1 is smaller than A2, denoted by A1<A2.Definition 6 (see [43]).
A real-valued functionE:IVIFS(X)→[0,1] is called an entropy measure on IVIFS(X) if it satisfies the following axiomatic requirements:(P1)
E(A)=0, if and only if A is a crisp set,(P2)
E(A)=1, if and only if μ~A(xi)=ν~A(xi) for all xi∈X,(P3)
E(A)=E(AC) for all xi∈IVIFS(X),(P4)
E(A)≤E(B) if A is less fuzzy than B, that is, μAL(xi)≤νAL(xi),  μAU(xi)≤νAU(xi) and A⊆B for all xi∈X or μAL(xi)≥νAL(xi),μAU(xi)≥νAU(xi) and B⊆A for all xi∈X.


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

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

Let us suppose thatE(Ai) is the entropy of IVIFSs for convenience.Vlachos’ entropy measure [50]:(8)E1(A)=1-12n∑i=1n(|μAL(xi)-νAL(xi)|)2+(|μAU(xi)-νAU(xi)|)2.Example 7.
LetA=([0.4,0.5],[0.3,0.4]) and B=([0.2,0.3],[0.1,0.2]) be two IVIFSs in X.
Intuitively, we can see thatB is more fuzzy than A. Now, we calculate the E1(A) and E1(B) by (8), and we can obtain
(9)E1(A)=1-12(|0.4-0.3|2+|0.5-0.4|2)=0.9,E1(B)=1-12(|0.2-0.1|2+|0.3-0.2|2)=0.9,
which indicate thatE1(A)=E1(B) is not consistent with our intuition.
Ye’s entropy measures [51]:(10)E2(A)={sinπ×[1+μAL(xi)+pWμA(xi)-νAL(xi)-qWνA(xi)]4+sinπ×[1-μAL(xi)-pWμA(xi)+νAL(xi)+qWνA(xi)]4-1sinπ×[1+μAL(xi)+pWμA(xi)-νAL(xi)-qWνA(xi)]4}×12-1,
wherep,q∈[0,1] representes two fixed numbers, WμA(x)=μAU(x)-μAL(x) and WνA(x)=νAU(x)-νAL(x).Example 8.
LetA=([0.6,0.6],[0.2,0.2]) and B=([0.7,0.7],[0.3,0.3]) be two IVIFSs in X.
Intuitively,A is more fuzzy than B. Now the E2(A) and E2(B) can be gained by (10), and taking p=q=0.5, the results are
(11)E2(A)={sinπ×[1+0.6-0.2]4+sinπ×[1-0.6+0.2]4-1}×12-1=0.833,E2(B)={sinπ×[1+0.7-0.3]4+sinπ×[1-0.7+0.3]4-1}×12-1=0.833,
which indicates that E2(A)=E2(B) 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 thatA∈IVIFS(X), then an interval-valued intuitionistic fuzzy entropy measure can be defined as
(12)E(A)=1-12n∑i=1n((μAL(xi)-νAL(xi))×(1-πAL(xi)))2+((μAU(xi)-νAU(xi))×(1-πAU(xi)))2.Theorem 10.
The mappingE(A), 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)
LetA be a crisp set. Then, we have π~A(xi)=0,μ~A(xi)=1 and ν~A(xi)=0, or π~A(xi)=0, μ~A(xi)=0 and ν~A(xi)=1. So, E(A)=0.(P2)
Letμ~A(xi)=ν~B(xi); we can obtain E(A)=1.(P3)
It is clear thatAC={〈xi,[νAL(xi),νAU(xi)],[μAL(xi),μAU(xi)]〉∣xi∈X}. By applying (12), we have E(A)=E(AC).(P4)
In order to show that (12) fulfills the requirement of (P4), they suffice to prove the following function:
(13)((μAL(xi))2-(νAL(xi))2)2+((μAU(xi))2-(νAU(xi))2)2≥((μBL(xi))2-(νBL(xi))2)2+((μBU(xi))2-(νBU(xi))2)2,where function E is monotonically decreasing.
Suppose thatμBL(xi)≤νBL(xi), μBU(xi)≤νBU(xi) and A⊆B; in order to prove (13), namely, we prove (14):
(14)(νAL(xi))2-(μAL(xi))2≥(νBL(xi))2-(μBL(xi))2,(νAU(xi))2-(μAU(xi))2≥(νBU(xi))2-(μBU(xi))2.
If νAL(xi)≥νBL(xi) and  μAL(xi)≤μBL(xi), then it follows that (νAL(xi))2-(μAL(xi))2≥(νBL(xi))2-(μBL(xi))2. If νAU(xi)≥νBU(xi) and μAU(xi)≤μBU(xi), then we have (νAU(xi))2-(μAU(xi))2≥(νBU(xi))2-(μBU(xi))2. So, we can get that (14) holds. Therefore, E(A)≤E(B).
Similarly, whenμBL(xi)≥νBL(xi), μBU(xi)≥νBU(xi) and B⊆A, we can also prove that E(A)≤E(B).


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

Example 11.
LetA=([0.4,0.5],[0.3,0.4]) and B=([0.2,0.3],[0.1,0.2]) be two IVIFSs in X.
Intuitively, we can see thatB is more fuzzy than A. Now we calculate the E(A) and E(B) by (12); we can obtain(15)E(A)=1-12((|0.4-0.3|×(1-(1-0.4-0.3)))2+(|0.5-0.4|×(1-(1-0.5-0.4)))2)=0.919,E(B)=1-12((|0.2-0.1|×(1-(1-0.2-0.1)))2+(|0.3-0.2|×(1-(1-0.3-0.2)))2)=0.959,
which indicate thatE(A)<E(B)  is consistent with our intuition. This result is better than the result in Example 7.Example 12.
LetA=([0.6,0.6],[0.2,0.2]) and B=([0.7,0.7],[0.3,0.3]) be two IVIFSs in X.
Intuitively, we can see thatA is more fuzzy than B. Now we calculate the E(A) and E(B) by (12); we can obtain(16)E(A)=1-12((|0.6-0.2|×(1-(1-0.6-0.2)))2+(|0.6-0.2|×(1-(1-0.6-0.2)))2)=0.680,E(B)=1-12((|0.7-0.3|×(1-(1-0.7-0.3)))2+(|0.3-0.2|×(1-(1-0.7-0.3)))2)=0.600,
which indicate thatE(A)>E(B) is consistent with our intuition. This result is better than the result in Example 8.


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

Let us suppose thatE(Ai) is the entropy of IVIFSs for convenience.Vlachos’ entropy measure [50]:(8)E1(A)=1-12n∑i=1n(|μAL(xi)-νAL(xi)|)2+(|μAU(xi)-νAU(xi)|)2.Example 7.
LetA=([0.4,0.5],[0.3,0.4]) and B=([0.2,0.3],[0.1,0.2]) be two IVIFSs in X.
Intuitively, we can see thatB is more fuzzy than A. Now, we calculate the E1(A) and E1(B) by (8), and we can obtain
(9)E1(A)=1-12(|0.4-0.3|2+|0.5-0.4|2)=0.9,E1(B)=1-12(|0.2-0.1|2+|0.3-0.2|2)=0.9,
which indicate thatE1(A)=E1(B) is not consistent with our intuition.
Ye’s entropy measures [51]:(10)E2(A)={sinπ×[1+μAL(xi)+pWμA(xi)-νAL(xi)-qWνA(xi)]4+sinπ×[1-μAL(xi)-pWμA(xi)+νAL(xi)+qWνA(xi)]4-1sinπ×[1+μAL(xi)+pWμA(xi)-νAL(xi)-qWνA(xi)]4}×12-1,
wherep,q∈[0,1] representes two fixed numbers, WμA(x)=μAU(x)-μAL(x) and WνA(x)=νAU(x)-νAL(x).Example 8.
LetA=([0.6,0.6],[0.2,0.2]) and B=([0.7,0.7],[0.3,0.3]) be two IVIFSs in X.
Intuitively,A is more fuzzy than B. Now the E2(A) and E2(B) can be gained by (10), and taking p=q=0.5, the results are
(11)E2(A)={sinπ×[1+0.6-0.2]4+sinπ×[1-0.6+0.2]4-1}×12-1=0.833,E2(B)={sinπ×[1+0.7-0.3]4+sinπ×[1-0.7+0.3]4-1}×12-1=0.833,
which indicates that E2(A)=E2(B) 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 thatA∈IVIFS(X), then an interval-valued intuitionistic fuzzy entropy measure can be defined as
(12)E(A)=1-12n∑i=1n((μAL(xi)-νAL(xi))×(1-πAL(xi)))2+((μAU(xi)-νAU(xi))×(1-πAU(xi)))2.Theorem 10.
The mappingE(A), 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)
LetA be a crisp set. Then, we have π~A(xi)=0,μ~A(xi)=1 and ν~A(xi)=0, or π~A(xi)=0, μ~A(xi)=0 and ν~A(xi)=1. So, E(A)=0.(P2)
Letμ~A(xi)=ν~B(xi); we can obtain E(A)=1.(P3)
It is clear thatAC={〈xi,[νAL(xi),νAU(xi)],[μAL(xi),μAU(xi)]〉∣xi∈X}. By applying (12), we have E(A)=E(AC).(P4)
In order to show that (12) fulfills the requirement of (P4), they suffice to prove the following function:
(13)((μAL(xi))2-(νAL(xi))2)2+((μAU(xi))2-(νAU(xi))2)2≥((μBL(xi))2-(νBL(xi))2)2+((μBU(xi))2-(νBU(xi))2)2,where function E is monotonically decreasing.
Suppose thatμBL(xi)≤νBL(xi), μBU(xi)≤νBU(xi) and A⊆B; in order to prove (13), namely, we prove (14):
(14)(νAL(xi))2-(μAL(xi))2≥(νBL(xi))2-(μBL(xi))2,(νAU(xi))2-(μAU(xi))2≥(νBU(xi))2-(μBU(xi))2.
If νAL(xi)≥νBL(xi) and  μAL(xi)≤μBL(xi), then it follows that (νAL(xi))2-(μAL(xi))2≥(νBL(xi))2-(μBL(xi))2. If νAU(xi)≥νBU(xi) and μAU(xi)≤μBU(xi), then we have (νAU(xi))2-(μAU(xi))2≥(νBU(xi))2-(μBU(xi))2. So, we can get that (14) holds. Therefore, E(A)≤E(B).
Similarly, whenμBL(xi)≥νBL(xi), μBU(xi)≥νBU(xi) and B⊆A, we can also prove that E(A)≤E(B).


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

Example 11.
LetA=([0.4,0.5],[0.3,0.4]) and B=([0.2,0.3],[0.1,0.2]) be two IVIFSs in X.
Intuitively, we can see thatB is more fuzzy than A. Now we calculate the E(A) and E(B) by (12); we can obtain(15)E(A)=1-12((|0.4-0.3|×(1-(1-0.4-0.3)))2+(|0.5-0.4|×(1-(1-0.5-0.4)))2)=0.919,E(B)=1-12((|0.2-0.1|×(1-(1-0.2-0.1)))2+(|0.3-0.2|×(1-(1-0.3-0.2)))2)=0.959,
which indicate thatE(A)<E(B)  is consistent with our intuition. This result is better than the result in Example 7.Example 12.
LetA=([0.6,0.6],[0.2,0.2]) and B=([0.7,0.7],[0.3,0.3]) be two IVIFSs in X.
Intuitively, we can see thatA is more fuzzy than B. Now we calculate the E(A) and E(B) by (12); we can obtain(16)E(A)=1-12((|0.6-0.2|×(1-(1-0.6-0.2)))2+(|0.6-0.2|×(1-(1-0.6-0.2)))2)=0.680,E(B)=1-12((|0.7-0.3|×(1-(1-0.7-0.3)))2+(|0.3-0.2|×(1-(1-0.7-0.3)))2)=0.600,
which indicate thatE(A)>E(B) 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 aren alternatives X={x1,x2,…,xn} and m evaluation criteria C={c1,c2,…,cm} with weight vector W=(w1,w2,…,wm)T associated with C, where wj∈[0,1] and ∑j=1mwj=1. Three evaluated criteria are considered, including innovativeness (c1), risk-taking (c2), and proactiveness (c3) [54, 55]. In this case, the characteristic of the alternative Ai is represented by the following IVIFS:
(17)Ai={〈Cj,[μAiL(Cj),μAiU(Cj)],[νAiL(Cj),νAiU(Cj)]〉∣Cj∈C},
where 0≤μAiU(Cj)+νAiU(Cj)≤1, μAiL(Cj)≥0, μAiL(Cj)≥0, i=1,2,…,n, and j=1,2,…,m. The IVIFS value, that is, the pair of intervals μAi(Cj)=[aij,bij], νAi(Cj)=[cij,dij] for Cj∈C, is denoted by rij=([aij,bij],[cij,dij]) for convenience. Here, we can elicit an interval-valued intuitionistic fuzzy evaluation matrix D=(rij)n×m.If the information about weightwj of the criterion Cj(j=1,2,…,m) is incomplete, for determining the criterion weight from the evaluation matrix D=(rij)n×m, we can establish a model of interval-valued intuitionistic fuzzy entropy weights.For the criteriaCj, the entropy of the alternative Ai can be given as(18)E(rij)=1-12((μAL(rij)-νAL(rij))×(1-πAL(rij)))2+((μAU(rij)-νAU(rij))×(1-πAU(rij)))2.And the overall entropy for the alternativeAi is given as
(19)E(Ai)=∑j=1m(1-12((μAL(rij)-νAL(rij))×(1-πAL(rij)))2+((μAU(rij)-νAU(rij))×(1-πAU(rij)))2).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.LetH be the set of incomplete information about criteria weights; to get the optimal weight vector, the following model can be constructed:(20)MinE(Ai)=∑i=1n∑j=1m(1-12((μAL(rij)-νAL(rij))×(1-πAL(rij)))2+((μAU(rij)-νAU(rij))×(1-πAU(rij)))2)s.t.w∈H,∑j=1mwi=1,wj≥0,j=1,2,…,m.By solving model (20) with Excel software, we get the optimal solution (w1,w2,…,wm)T.In summary, the evaluation procedure proposed is listed below.Step 1.
Collect data for three given attributes.Step 2.
Calculate the weight vectorW=(w1,w2,…,wm)T  by solving model (20).Step 3.
Utilize the decision information given in matrixD and the IVIF-WGA operator to derive the collective overall values zi(i=1,2,…,n) of the alternative Ai. 									(21)zi=Fw(ri1,ri2,…,rim),i=1,2,…,n.Step 4.
Calculate the scoresS(zi)(i=1,2,…,n) of the collective overall values to rank all the alternatives Ai(i=1,2,…,n).


### 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,A={A1,A2,A3,A4}. 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(22)D=[([0.45,0.5],[0.45,0.5])([0.5,0.7],[0.15,0.25])([0.35,0.45],[0.2,0.55])([0.4,0.5],[0.45,0.5])([0.6,0.8],[0.1,0.2])([0.25,0.55],[0.3,0.45])([0.35,0.5],[0.45,0.5])([0.2,0.3],[0.2,0.4])([0.7,0.8],[0.1,0.15])([0.25,0.4],[0.45,0.5])([0.6,0.65],[0.2,0.3])([0.5,0.6],[0.2,0.3])].
Moreover, the lending expert can only provide his incomplete information on the weights as follows:(23)H={0.3≤w1≤0.45,0.25≤w2≤0.35,0.35≤w3≤0.4}.Step 2.
Calculate the weight vectorW=(w1,w2,w3)T by solving model (20):
(24)W=(0.3,0.35,0.35)T.Step 3.
Utilize the evaluation information given in matrixD and the IVIF-WGA operator to derive the collective overall values zi(i=1,2,3,4) of the alternative Ai(i=1,2,3,4):
(25)z1=([0.436,0.568],[0.231,0.406]),z2=([0.437,0.650],[0.231,0.350]),z3=([0.467,0.592],[0.200,0.303]),z4=([0.478,0.569],[0.255,0.350]).Step 4.
Calculate the scoresS(zi)(i=1,2,3,4) of the collective overall values zi(i=1,2,3,4):
(26)S(z1)=0.367,S(z2)=0.507,S(z3)=0.555,S(z4)=0.442.
Rank all the alternativesAi(i=1,2,3,4) in accordance with the score S(zi) of the collective overall interval-valued intuitionistic fuzzy values zi(i=1,2,3,4): A3≻A2≻A4≻A1.


## 4.1. The Evaluation Procedure

Evaluate entrepreneurship orientation of online P2P lending platforms. Assume that there aren alternatives X={x1,x2,…,xn} and m evaluation criteria C={c1,c2,…,cm} with weight vector W=(w1,w2,…,wm)T associated with C, where wj∈[0,1] and ∑j=1mwj=1. Three evaluated criteria are considered, including innovativeness (c1), risk-taking (c2), and proactiveness (c3) [54, 55]. In this case, the characteristic of the alternative Ai is represented by the following IVIFS:
(17)Ai={〈Cj,[μAiL(Cj),μAiU(Cj)],[νAiL(Cj),νAiU(Cj)]〉∣Cj∈C},
where 0≤μAiU(Cj)+νAiU(Cj)≤1, μAiL(Cj)≥0, μAiL(Cj)≥0, i=1,2,…,n, and j=1,2,…,m. The IVIFS value, that is, the pair of intervals μAi(Cj)=[aij,bij], νAi(Cj)=[cij,dij] for Cj∈C, is denoted by rij=([aij,bij],[cij,dij]) for convenience. Here, we can elicit an interval-valued intuitionistic fuzzy evaluation matrix D=(rij)n×m.If the information about weightwj of the criterion Cj(j=1,2,…,m) is incomplete, for determining the criterion weight from the evaluation matrix D=(rij)n×m, we can establish a model of interval-valued intuitionistic fuzzy entropy weights.For the criteriaCj, the entropy of the alternative Ai can be given as(18)E(rij)=1-12((μAL(rij)-νAL(rij))×(1-πAL(rij)))2+((μAU(rij)-νAU(rij))×(1-πAU(rij)))2.And the overall entropy for the alternativeAi is given as
(19)E(Ai)=∑j=1m(1-12((μAL(rij)-νAL(rij))×(1-πAL(rij)))2+((μAU(rij)-νAU(rij))×(1-πAU(rij)))2).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.LetH be the set of incomplete information about criteria weights; to get the optimal weight vector, the following model can be constructed:(20)MinE(Ai)=∑i=1n∑j=1m(1-12((μAL(rij)-νAL(rij))×(1-πAL(rij)))2+((μAU(rij)-νAU(rij))×(1-πAU(rij)))2)s.t.w∈H,∑j=1mwi=1,wj≥0,j=1,2,…,m.By solving model (20) with Excel software, we get the optimal solution (w1,w2,…,wm)T.In summary, the evaluation procedure proposed is listed below.Step 1.
Collect data for three given attributes.Step 2.
Calculate the weight vectorW=(w1,w2,…,wm)T  by solving model (20).Step 3.
Utilize the decision information given in matrixD and the IVIF-WGA operator to derive the collective overall values zi(i=1,2,…,n) of the alternative Ai. 									(21)zi=Fw(ri1,ri2,…,rim),i=1,2,…,n.Step 4.
Calculate the scoresS(zi)(i=1,2,…,n) of the collective overall values to rank all the alternatives Ai(i=1,2,…,n).


## 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,A={A1,A2,A3,A4}. 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(22)D=[([0.45,0.5],[0.45,0.5])([0.5,0.7],[0.15,0.25])([0.35,0.45],[0.2,0.55])([0.4,0.5],[0.45,0.5])([0.6,0.8],[0.1,0.2])([0.25,0.55],[0.3,0.45])([0.35,0.5],[0.45,0.5])([0.2,0.3],[0.2,0.4])([0.7,0.8],[0.1,0.15])([0.25,0.4],[0.45,0.5])([0.6,0.65],[0.2,0.3])([0.5,0.6],[0.2,0.3])].
Moreover, the lending expert can only provide his incomplete information on the weights as follows:(23)H={0.3≤w1≤0.45,0.25≤w2≤0.35,0.35≤w3≤0.4}.Step 2.
Calculate the weight vectorW=(w1,w2,w3)T by solving model (20):
(24)W=(0.3,0.35,0.35)T.Step 3.
Utilize the evaluation information given in matrixD and the IVIF-WGA operator to derive the collective overall values zi(i=1,2,3,4) of the alternative Ai(i=1,2,3,4):
(25)z1=([0.436,0.568],[0.231,0.406]),z2=([0.437,0.650],[0.231,0.350]),z3=([0.467,0.592],[0.200,0.303]),z4=([0.478,0.569],[0.255,0.350]).Step 4.
Calculate the scoresS(zi)(i=1,2,3,4) of the collective overall values zi(i=1,2,3,4):
(26)S(z1)=0.367,S(z2)=0.507,S(z3)=0.555,S(z4)=0.442.
Rank all the alternativesAi(i=1,2,3,4) in accordance with the score S(zi) of the collective overall interval-valued intuitionistic fuzzy values zi(i=1,2,3,4): A3≻A2≻A4≻A1.


## 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.



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*Source: 467215-2013-10-01.xml*