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

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 𝐷[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 𝐴=𝑥,𝜇_𝐴(𝑥),𝜈_𝐴(𝑥)∣𝑥∈𝑋, (1)𝑥in the set 𝐴, respectively, where 𝜇_𝐴[],𝜈∶𝑋⟶𝐷0,1_𝐴∶𝑋⟶𝐷[0,1] (2) with the condition 0≤sup(𝜇_𝐴(𝑥))+sup(𝜈_𝐴(𝑥))≤1 for any 𝑥∈𝑋.
The intervals 𝜇_𝐴(𝑥) and 𝜈_𝐴(𝑥) denote the degree of membership and nonmembership of the element 𝑥 in the set 𝐴, respectively.
For convenience, let 𝜇_𝐴(𝑥)=[𝜇_𝐴𝐿(𝑥),𝜇_𝐴𝑈(𝑥)], 𝜈_𝐴(𝑥)=[𝜈_𝐴𝐿(𝑥),𝜈_𝐴𝑈(𝑥)]; then 𝐴=𝑥,𝜇_𝐴𝐿(𝑥),𝜇_𝐴𝑈(𝑥),𝜈_𝐴𝐿(𝑥),𝜈_𝐴𝑈(𝑥)∣𝑥∈𝑋, (3) where 0≤𝜇_𝐴𝑈(𝑥)+𝜈_𝐴𝑈(𝑥)≤1, 𝜇_𝐴𝐿(𝑥)≥0, 𝜈_𝐴𝐿(𝑥)≥0.
For each element 𝑥, we can compute the hesitancy degree of an intuitionistic fuzzy interval of 𝑥∈𝑋 in 𝐴 defined as follows: 𝜋_𝐴(𝑥)=1−𝜇_𝐴(𝑥)−𝜈_𝐴(𝑥)=1−𝜇_𝐴𝑈(𝑥)−𝜈_𝐴𝑈(𝑥),1−𝜇_𝐴𝐿(𝑥)−𝜈_𝐴𝐿(𝑥). (4)
For convenience, an IVIFS value is denoted by 𝐴=([𝑎,𝑏],[𝑐,𝑑]).

Definition 2 (see [37]). Let 𝐴,𝐵∈IVIFS(𝑋); then some operations can be defined as follows: 𝐴∪𝐵=𝑥_𝑖,𝜇_𝐴𝐿𝑥_𝑖∨𝜇_𝐵𝐿𝑥_𝑖,𝜇_𝐴𝑈𝑥_𝑖∨𝜇_𝐵𝑈𝑥_𝑖,𝜈_𝐴𝐿𝑥_𝑖∧𝜈_𝐵𝐿𝑥_𝑖,𝜈_𝐴𝑈𝑥_𝑖∧𝜈_𝐵𝑈𝑥_𝑖,𝐴∩𝐵=𝑥_𝑖,𝜇_𝐴𝐿𝑥_𝑖∧𝜇_𝐵𝐿𝑥_𝑖,𝜇_𝐴𝑈𝑥_𝑖∧𝜇_𝐵𝑈𝑥_𝑖,𝜈_𝐴𝐿𝑥_𝑖∨𝜈_𝐵𝐿𝑥_𝑖,𝜈_𝐴𝑈𝑥_𝑖∨𝜈_𝐵𝑈𝑥_𝑖. (5)
The following expressions are defined in [37] for all 𝐴,𝐵∈IVFSs(𝑋):𝐴⊆𝐵 if and only if 𝜇_𝐴𝐿≤𝜇_𝐵𝐿, 𝜇_𝐴𝑈≤𝜇_𝐵𝑈, 𝜈_𝐴𝐿≥𝜈_𝐵𝐿 and 𝜈_𝐴𝑈≥𝜈_𝐵𝑈 for all 𝑥∈𝑋, 𝐴=𝐵 if and only if 𝐴⊆𝐵 and 𝐵⊆𝐴 for all 𝑥∈𝑋, 𝐴^𝐶={⟨𝑥,[𝜈_𝐴𝐿(𝑥),𝜈_𝐴𝑈(𝑥)],[𝜇_𝐴𝐿(𝑥),𝜇_𝐴𝑈(𝑥)]⟩∣𝑥∈𝑋}.
In the following, we introduce two weighed aggregation operators related to IVIFSs [52].

Definition 3 (see [52]). Let 𝐴_𝑗(𝑗=1,2,…,𝑛)∈IVIFS(𝑋). The weighed geometric average operator (IVIF-WGA operator) is defined by 𝐹_𝑤𝐴₁,𝐴₂,…,𝐴_𝑛=^𝑛_𝑗=1𝜇_𝐴𝑗𝐿(𝑥)^𝑤𝑗,^𝑛_𝑗=1𝜇_𝐴𝑗𝑈(𝑥)^𝑤𝑗,1−^𝑛_𝑗=11−𝜈_𝐴𝑗𝐿(𝑥)^𝑤𝑗,1−^𝑛_𝑗=11−𝜈_𝐴𝑗𝑈(𝑥)^𝑤𝑗, (6) where 𝑤_𝑗 is the weight of 𝐴_𝑗(𝑗=1,2,…,𝑛), 𝑤_𝑗∈[0,1] and ∑^𝑛_𝑗=1𝑤_𝑗=1. In particular, assuming that 𝑤_𝑗=1/𝑛(𝑗=1,2,…,𝑛), then 𝐹_𝑤 is called an arithmetic average operator for IVIFSs.

Definition 4 (see [53]). Let 𝐴=([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: 𝑆(𝐴)=𝑎−𝑐+𝑏−𝑑2. (7)

Definition 5 (see [53]). Let 𝐴₁=([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 function 𝐸∶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(𝑥_𝑖)=𝜈_𝐴(𝑥_𝑖) for all 𝑥_𝑖∈𝑋,(P3)𝐸(𝐴)=𝐸(𝐴^𝐶) for all 𝑥_𝑖∈IVIFS(𝑋),(P4)𝐸(𝐴)≤𝐸(𝐵) if 𝐴 is less fuzzy than 𝐵, that is, 𝜇_𝐴𝐿(𝑥_𝑖)≤𝜈_𝐴𝐿(𝑥_𝑖), 𝜇_𝐴𝑈(𝑥_𝑖)≤𝜈_𝐴𝑈(𝑥_𝑖) and 𝐴⊆𝐵 for all 𝑥_𝑖∈𝑋 or 𝜇_𝐴𝐿(𝑥_𝑖)≥𝜈_𝐴𝐿(𝑥_𝑖),𝜇_𝐴𝑈(𝑥_𝑖)≥𝜈_𝐴𝑈(𝑥_𝑖) and 𝐵⊆𝐴 for all 𝑥_𝑖∈𝑋.

3. An Effective Interval-Valued Intuitionistic Fuzzy Entropy

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

Let us suppose that 𝐸(𝐴_𝑖) is the entropy of IVIFSs for convenience.

Vlachos’ entropy measure [50]:𝐸¹(𝐴)=1−12𝑛^𝑛_𝑖=1𝜇_𝐴𝐿𝑥_𝑖−𝜈_𝐴𝐿𝑥_𝑖²+𝜇_𝐴𝑈𝑥_𝑖−𝜈_𝐴𝑈𝑥_𝑖². (8)

Example 7. Let 𝐴=([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 that 𝐵 is more fuzzy than 𝐴. Now, we calculate the 𝐸¹(𝐴) and 𝐸¹(𝐵) by (8), and we can obtain 𝐸¹(𝐴)=1−12|0.4−0.3|²+|0.5−0.4|²𝐸=0.9,₁(𝐵)=1−12|0.2−0.1|²+|0.3−0.2|²=0.9, (9)
which indicate that 𝐸¹(𝐴)=𝐸¹(𝐵) is not consistent with our intuition.
Ye’s entropy measures [51]:𝐸²(𝐴)=sin𝜋×1+𝜇_𝐴𝐿𝑥_𝑖+𝑝𝑊_𝜇𝐴𝑥_𝑖−𝜈_𝐴𝐿𝑥_𝑖−𝑞𝑊_𝜈𝐴𝑥_𝑖4+sin𝜋×1−𝜇_𝐴𝐿𝑥_𝑖−𝑝𝑊_𝜇𝐴𝑥_𝑖+𝜈_𝐴𝐿𝑥_𝑖+𝑞𝑊_𝜈𝐴𝑥_𝑖4×1−1√,2−1 (10)
where 𝑝,𝑞∈[0,1] representes two fixed numbers, 𝑊_𝜇𝐴(𝑥)=𝜇_𝐴𝑈(𝑥)−𝜇_𝐴𝐿(𝑥) and 𝑊_𝜈𝐴(𝑥)=𝜈_𝐴𝑈(𝑥)−𝜈_𝐴𝐿(𝑥).

Example 8. Let 𝐴=([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 𝐸²(𝐴)=[]sin𝜋×1+0.6−0.24[]+sin𝜋×1−0.6+0.24×1−1√𝐸2−1=0.833,₂(𝐵)=[]sin𝜋×1+0.7−0.34[]+sin𝜋×1−0.7+0.34×1−1√2−1=0.833, (11) 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 that 𝐴∈IVIFS(𝑋), then an interval-valued intuitionistic fuzzy entropy measure can be defined as 𝐸(𝐴)=1−12𝑛^𝑛_𝑖∑₌₁𝜇_𝐴𝐿𝑥_𝑖−𝜈_𝐴𝐿𝑥_𝑖×1−𝜋_𝐴𝐿𝑥_𝑖²+𝜇_𝐴𝑈𝑥_𝑖−𝜈_𝐴𝑈𝑥_𝑖×1−𝜋_𝐴𝑈𝑥_𝑖². (12)

Theorem 10. The mapping 𝐸(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)Let 𝐴 be a crisp set. Then, we have 𝜋_𝐴(𝑥_𝑖)=0,𝜇_𝐴(𝑥_𝑖)=1 and 𝜈_𝐴(𝑥_𝑖)=0, or 𝜋_𝐴(𝑥_𝑖)=0, 𝜇_𝐴(𝑥_𝑖)=0 and 𝜈_𝐴(𝑥_𝑖)=1. So, 𝐸(𝐴)=0.(P2)Let 𝜇_𝐴(𝑥_𝑖)=𝜈_𝐵(𝑥_𝑖); we can obtain 𝐸(𝐴)=1.(P3)It is clear that 𝐴^𝐶={⟨𝑥_𝑖,[𝜈_𝐴𝐿(𝑥_𝑖),𝜈_𝐴𝑈(𝑥_𝑖)],[𝜇_𝐴𝐿(𝑥_𝑖),𝜇_𝐴𝑈(𝑥_𝑖)]⟩∣𝑥_𝑖∈𝑋}. By applying (12), we have 𝐸(𝐴)=𝐸(𝐴^𝐶).(P4)In order to show that (12) fulfills the requirement of (P4), they suffice to prove the following function: 𝜇_𝐴𝐿(𝑥_𝑖)²−𝜈_𝐴𝐿(𝑥_𝑖)²²+𝜇_𝐴𝑈(𝑥_𝑖)²−𝜈_𝐴𝑈(𝑥_𝑖)²²≥𝜇_𝐵𝐿𝑥_𝑖²−𝜈_𝐵𝐿𝑥_𝑖²²+𝜇_𝐵𝑈𝑥_𝑖²−𝜈_𝐵𝑈𝑥_𝑖²², (13)where function 𝐸 is monotonically decreasing.
Suppose that 𝜇_𝐵𝐿(𝑥_𝑖)≤𝜈_𝐵𝐿(𝑥_𝑖), 𝜇_𝐵𝑈(𝑥_𝑖)≤𝜈_𝐵𝑈(𝑥_𝑖) and 𝐴⊆𝐵; in order to prove (13), namely, we prove (14): 𝜈_𝐴𝐿(𝑥_𝑖)²−𝜇_𝐴𝐿𝑥_𝑖²≥𝜈_𝐵𝐿𝑥_𝑖²−𝜇_𝐵𝐿𝑥_𝑖²,𝜈_𝐴𝑈(𝑥_𝑖)²−𝜇_𝐴𝑈𝑥_𝑖²≥𝜈_𝐵𝑈𝑥_𝑖²−𝜇_𝐵𝑈𝑥_𝑖². (14) If 𝜈_𝐴𝐿(𝑥_𝑖)≥𝜈_𝐵𝐿(𝑥_𝑖) and 𝜇_𝐴𝐿(𝑥_𝑖)≤𝜇_𝐵𝐿(𝑥_𝑖), then it follows that (𝜈_𝐴𝐿(𝑥_𝑖))²−(𝜇_𝐴𝐿(𝑥_𝑖))²≥(𝜈_𝐵𝐿(𝑥_𝑖))²−(𝜇_𝐵𝐿(𝑥_𝑖))². If 𝜈_𝐴𝑈(𝑥_𝑖)≥𝜈_𝐵𝑈(𝑥_𝑖) and 𝜇_𝐴𝑈(𝑥_𝑖)≤𝜇_𝐵𝑈(𝑥_𝑖), then we have (𝜈_𝐴𝑈(𝑥_𝑖))²−(𝜇_𝐴𝑈(𝑥_𝑖))²≥(𝜈_𝐵𝑈(𝑥_𝑖))²−(𝜇_𝐵𝑈(𝑥_𝑖))². So, we can get that (14) holds. Therefore, 𝐸(𝐴)≤𝐸(𝐵).
Similarly, when 𝜇_𝐵𝐿(𝑥_𝑖)≥𝜈_𝐵𝐿(𝑥_𝑖), 𝜇_𝐵𝑈(𝑥_𝑖)≥𝜈_𝐵𝑈(𝑥_𝑖) 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. Let 𝐴=([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 that 𝐵 is more fuzzy than 𝐴. Now we calculate the 𝐸(𝐴) and 𝐸(𝐵) by (12); we can obtain𝐸(𝐴)=1−12(|0.4−0.3|×(1−(1−0.4−0.3)))²+(|0.5−0.4|×(1−(1−0.5−0.4)))²𝐸(𝐵)=0.919,=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, (15)
which indicate that 𝐸(𝐴)<𝐸(𝐵) is consistent with our intuition. This result is better than the result in Example 7.

Example 12. Let 𝐴=([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 that 𝐴 is more fuzzy than 𝐵. Now we calculate the 𝐸(𝐴) and 𝐸(𝐵) by (12); we can obtain𝐸(𝐴)=1−12(|0.6−0.2|×(1−(1−0.6−0.2)))²+(|0.6−0.2|×(1−(1−0.6−0.2)))²𝐸(𝐵)=0.680,=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, (16)
which indicate that 𝐸(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 are 𝑛 alternatives 𝑋={𝑥₁,𝑥₂,…,𝑥_𝑛} and 𝑚 evaluation criteria 𝐶={𝑐₁,𝑐₂,…,𝑐_𝑚} with weight vector 𝑊=(𝑤₁,𝑤₂,…,𝑤_𝑚)^𝑇 associated with 𝐶, where 𝑤_𝑗∈[0,1] and ∑^𝑚_𝑗=1𝑤_𝑗=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 𝐴_𝑖 is represented by the following IVIFS: 𝐴_𝑖=𝐶_𝑗,𝜇_𝐴𝑖𝐿𝐶_𝑗,𝜇_𝐴𝑖𝑈𝐶_𝑗,𝜈_𝐴𝑖𝐿𝐶_𝑗,𝜈_𝐴𝑖𝑈𝐶_𝑗∣𝐶_𝑗∈𝐶, (17) where 0≤𝜇_𝐴𝑖𝑈(𝐶_𝑗)+𝜈_𝐴𝑖𝑈(𝐶_𝑗)≤1, 𝜇_𝐴𝑖𝐿(𝐶_𝑗)≥0, 𝜇_𝐴𝑖𝐿(𝐶_𝑗)≥0, 𝑖=1,2,…,𝑛, and 𝑗=1,2,…,𝑚. The IVIFS value, that is, the pair of intervals 𝜇_𝐴𝑖(𝐶_𝑗)=[𝑎_𝑖𝑗,𝑏_𝑖𝑗], 𝜈_𝐴𝑖(𝐶_𝑗)=[𝑐_𝑖𝑗,𝑑_𝑖𝑗] for 𝐶_𝑗∈𝐶, is denoted by 𝑟_𝑖𝑗=([𝑎_𝑖𝑗,𝑏_𝑖𝑗],[𝑐_𝑖𝑗,𝑑_𝑖𝑗]) for convenience. Here, we can elicit an interval-valued intuitionistic fuzzy evaluation matrix 𝐷=(𝑟_𝑖𝑗)_𝑛×𝑚.

If the information about weight 𝑤_𝑗 of the criterion 𝐶_𝑗(𝑗=1,2,…,𝑚) is incomplete, for determining the criterion weight from the evaluation matrix 𝐷=(𝑟_𝑖𝑗)_𝑛×𝑚, we can establish a model of interval-valued intuitionistic fuzzy entropy weights.

For the criteria 𝐶_𝑗, the entropy of the alternative 𝐴_𝑖 can be given as𝐸𝑟_𝑖𝑗=1−12𝜇_𝐴𝐿𝑟_𝑖𝑗−𝜈_𝐴𝐿𝑟_𝑖𝑗×1−𝜋_𝐴𝐿𝑟_𝑖𝑗²+𝜇_𝐴𝑈𝑟_𝑖𝑗−𝜈_𝐴𝑈𝑟_𝑖𝑗×1−𝜋_𝐴𝑈𝑟_𝑖𝑗². (18)

And the overall entropy for the alternative 𝐴_𝑖 is given as 𝐸𝐴_𝑖=∑^𝑚_𝑗=11−12𝜇_𝐴𝐿𝑟_𝑖𝑗−𝜈_𝐴𝐿𝑟_𝑖𝑗×1−𝜋_𝐴𝐿𝑟_𝑖𝑗²+𝜇_𝐴𝑈𝑟_𝑖𝑗−𝜈_𝐴𝑈𝑟_𝑖𝑗×1−𝜋_𝐴𝑈𝑟_𝑖𝑗². (19)

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 𝐻 be the set of incomplete information about criteria weights; to get the optimal weight vector, the following model can be constructed:Min𝐸𝐴_𝑖=∑^𝑛𝑚_𝑖=1∑_𝑗=11−12𝜇_𝐴𝐿(𝑟_𝑖𝑗)−𝜈_𝐴𝐿(𝑟_𝑖𝑗)×1−𝜋_𝐴𝐿(𝑟_𝑖𝑗)²+𝜇_𝐴𝑈(𝑟_𝑖𝑗)−𝜈_𝐴𝑈(𝑟_𝑖𝑗)×1−𝜋_𝐴𝑈(𝑟_𝑖𝑗)²s.t.𝑤∈𝐻,∑^𝑚_𝑗=1𝑤_𝑖=1,𝑤_𝑗≥0,𝑗=1,2,…,𝑚. (20)

By solving model (20) with Excel software, we get the optimal solution (𝑤₁,𝑤₂,…,𝑤_𝑚)^𝑇.

In summary, the evaluation procedure proposed is listed below.

Step 1. Collect data for three given attributes.

Step 2. Calculate the weight vector 𝑊=(𝑤₁,𝑤₂,…,𝑤_𝑚)^𝑇 by solving model (20).

Step 3. Utilize the decision information given in matrix 𝐷 and the IVIF-WGA operator to derive the collective overall values 𝑧_𝑖(𝑖=1,2,…,𝑛) of the alternative 𝐴_𝑖. 𝑧_𝑖=𝐹_𝑤𝑟_𝑖1,𝑟_𝑖2,…,𝑟_𝑖𝑚,𝑖=1,2,…,𝑛. (21)

Step 4. Calculate the scores 𝑆(𝑧_𝑖)(𝑖=1,2,…,𝑛) of the collective overall values to rank all the alternatives 𝐴_𝑖(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],[],[],[],[],[],[],[],[],[],[],[],[.𝐷=([0.45,0.50.45,0.5])([0.5,0.70.15,0.25])([0.35,0.450.2,0.55])([0.4,0.50.45,0.5])([0.6,0.80.1,0.2])([0.25,0.550.3,0.45])([0.35,0.50.45,0.5])([0.2,0.30.2,0.4])([0.7,0.80.1,0.15])([0.25,0.40.45,0.5])([0.6,0.650.2,0.3])([0.5,0.60.2,0.3]) (22)
Moreover, the lending expert can only provide his incomplete information on the weights as follows: 𝐻=0.3≤𝑤₁≤0.45,0.25≤𝑤₂≤0.35,0.35≤𝑤₃≤0.4. (23)

Step 2. Calculate the weight vector 𝑊=(𝑤₁,𝑤₂,𝑤₃)^𝑇 by solving model (20): 𝑊=(0.3,0.35,0.35)^𝑇. (24)

Step 3. Utilize the evaluation information given in matrix 𝐷 and the IVIF-WGA operator to derive the collective overall values 𝑧_𝑖(𝑖=1,2,3,4) of the alternative 𝐴_𝑖(𝑖=1,2,3,4): 𝑧₁=],[,𝑧([0.436,0.5680.231,0.406])₂=],[,𝑧([0.437,0.6500.231,0.350])₃=],[,𝑧([0.467,0.5920.200,0.303])₄=([0.478,0.569],[0.255,0.350]). (25)

Step 4. Calculate the scores 𝑆(𝑧_𝑖)(𝑖=1,2,3,4) of the collective overall values 𝑧_𝑖(𝑖=1,2,3,4): 𝑆𝑧₁𝑆𝑧=0.367,₂𝑆𝑧=0.507,₃𝑆𝑧=0.555,₄=0.442. (26)
Rank all the alternatives 𝐴_𝑖(𝑖=1,2,3,4) in accordance with the score 𝑆(𝑧_𝑖) of the collective overall interval-valued intuitionistic fuzzy values 𝑧_𝑖(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.

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).