A Fuzzy Multicriteria Group Decision-Making Method with New Entropy of Interval-Valued Intuitionistic Fuzzy Sets

Abstract

A new entropy measure of interval-valued intuitionistic fuzzy set (IVIFS) is proposed by using cotangent function, which overcomes several limitations in the existing methods for calculating entropy of IVIFS. The efficiency of the new entropy is demonstrated by comparing it with several classical entropies. Moreover, an entropy weight model is established to determine the entropy weights for fuzzy multicriteria group decision-making (FMCGDMs) problems, which depends on incomplete weight information of criteria in IVIFSs setting. Finally, an illustrative supplier selection problem is used to demonstrate the practicality and effectiveness of the proposed method. It is capable of the handling the FMCGDM problems with incomplete known weights for criteria.

1. Introduction

The theory of fuzzy sets (FSs) proposed by Zadeh [1] has achieved a great success in various fields. A lot of generalized forms of FSs have been proposed. The classical sets include interval-valued fuzzy sets (IVFSs) [2], intuitionistic fuzzy sets (IFSs) [3, 4], interval-valued intuitionistic fuzzy sets (IVIFSs) [5], R-fuzzy sets [6], and intuitionistic linguistic fuzzy sets (ILFSs) [7]. From [4, 8–10], it turns out that IVFS theory is equivalent to IFS theory, and IVIFS theory extends IFS theory.

As an important topic in the theory of fuzzy sets, entropy measures of IFSs have been investigated widely by many researchers from different views. Burillo and Bustince [11] introduced the notion that entropy of IVFSs and IFSs can be used to evaluate the degree of intuitionism of an IVFS or IFS. Szmidt and Kacprzyk [12] proposed a nonprobabilistic-type entropy measure with a geometric interpretation of IFSs. Hung and Yang [13] gave their axiomatic definitions of entropy of IFSs and IVFSs by using the concept of probability. Wei et al. [14] gave a new entropy measure for IVIFSs to overcome the disadvantages of those three entropy measures defined independently by Szmidt and Kacprzyk [12], Wang and Lei [15], and Huang and Liu [16]. Different entropy formulas for IFS [15, 17], IVFS [18, 19], and vague set [16, 20, 21] were also proposed by other researchers.

The entropy of IFSs has been applied widely in decision making [22, 23]. On the one hand, due to the increasing complexity of the social-economic environment and a lack of information about the problem domains, the decision information may be provided with IVIFSs, whose membership degree and nonmembership degree are intervals, instead of real numbers. Entropy is concerned as a measure of fuzziness. Therefore, it is highly necessary and significant to study the entropy of IVIFSs. And on the other hand, a proper assessment of attribute weights plays an essential role in the MADM process [24]. In terms of determining weights, the entropy method is one of the most representative approaches, which expresses the relative intensities of attribute importance to signify the average intrinsic information transmitted to the DM [22, 25, 26]. The following are some of the research findings.

Ye [27] proposed two entropy measures for IVIFSs and established an entropy weight model, which could be used to determine the criteria weights on alternatives. Zhang et al. [28] proposed a new information entropy measure of IVIFS by using membership interval and nonmembership interval of IVIFSs, which complied with the extended form of De Luca and Termini [29] axioms for fuzzy entropy. Wei et al. [14] also proposed an entropy measure for IVIFSs, and they applied the new entropy measure to solve problem on multicriteria fuzzy decision making.

However, the entropy is seldom applied in multiexpert and multicriteria decision making, which is one of the most important branches of decision-making method. Due to limited investigation on multiexpert and multicriteria decision-making issues, this study proposed a novel formula to calculate the entropy of an IVIFS on the basis of the argument on the relationship among the entropies of IFSs given in [27, 30]. For interval-valued intuitionistic fuzzy multicriteria group decision-making problem, in which the information on the weights of criteria is incomplete, a linear fuzzy programming model based on intuitionistic fuzzy entropy is constructed to obtain the criteria weights.

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. In Section 4, the method and procedure for solving FMCGDM problems with the new entropy measure of interval-valued intuitionistic fuzzy set are developed in detail. An illustrative supplier selection problem was employed to demonstrate how to apply the proposed approach in Section 5. Short conclusion is given in Section 6.

2. Interval-Valued Intuitionistic Fuzzy Sets

Definition 1 (see [3]). Let 𝑋 be an ordinary finite nonempty set. An intuitionistic fuzzy set (IFS) in 𝑋 is an object of the form: 𝐴=𝑥,𝜇_𝐴(𝑥),𝜈_𝐴(𝑥)∣𝑥∈𝑋, (1) where 𝜇_𝐴∶𝑋⟶[0,1],𝜈_𝐴∶𝑋⟶[0,1] (2) with the condition 0≤𝜇_𝐴(𝑥)+𝜈_𝐴(𝑥)≤1, for all 𝑥∈𝑋.

The numbers 𝜇_𝐴(𝑥), 𝜈_𝐴(𝑥) denote the degree of membership and nonmembership of the element 𝑥 in the set 𝐴, respectively.

For each IFS, we call 𝜋_𝐴(𝑥)=1−𝜇_𝐴(𝑥)−𝜈_𝐴(𝑥) the intuitionistic index of the element 𝑥 in the set 𝐴. It also denotes the hesitancy degree of 𝑥 to 𝐴.

Definition 2 (see [5]). Let 𝐷[0,1] be the set of all closed subintervals of the interval [0,1], and let 𝑋 be an ordinary finite nonempty set. An interval-valued intuitionistic fuzzy set (IVIFS) in 𝑋 is an object of the form: 𝐴=𝑥,𝜇^∼_𝐴(𝑥),𝜈^∼_𝐴(𝑥)∣𝑥∈𝑋, (3)𝑥 in the set 𝐴, where 𝜇^∼_𝐴∶𝑋→𝐷[0,1], 𝜈^∼_𝐴∶𝑋→𝐷[0,1], with the condition 0≤sup(𝜇^∼_𝐴(𝑥))+sup(𝜈^∼_𝐴(𝑥))≤1, for all 𝑥∈𝑋.

The intervals 𝜇^∼_𝐴(𝑥), 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, (4) where 0≤𝜇_𝐴𝑈(𝑥)+𝜈_𝐴𝑈(𝑥)≤1, 𝜇_𝐴𝐿(𝑥)≥0, and 𝜈_𝐴𝐿(𝑥)≥0.

For each element 𝑥, we can compute the intuitionistic index 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). (5)

For convenience, an IVIFS value is denoted by 𝐴=([𝑎,𝑏],[𝑐,𝑑]).

Definition 3 (see [5]). Assume 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. (6)

The following expressions are defined in [5] 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 weighted aggregation operators related to IVIFSs.

Definition 4 (see [31]). Let 𝐴_𝑗  (𝑗=1,2,…,𝑛)∈IVIFS(𝑋). The weighted 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, (7) where 𝑤_𝑗 is the weight of 𝐴_𝑗(𝑗=1,2,…,𝑛),𝑤_𝑗∈[0,1] and ∑^𝑛_𝑗=1𝑤_𝑗=1. Particularly, if 𝑤_𝑗=1/𝑛(𝑗=1,2,…,𝑛), then 𝐹_𝑤 is called a geometric average operator for IVIFSs.

Definition 5 (see [31]). Let 𝐴_𝑗(𝑗=1,2,…,𝑛)∈IVIFS(𝑋). The hybrid averaging operator (IVIF-HA operator) is defined by 𝐻_𝑤𝐴₁,𝐴₂,…,𝐴_𝑛=^𝑛_𝑗=1𝜇_𝐵𝑗𝐿(𝑥)^𝑤𝑗,^𝑛_𝑗=1𝜇_𝐵𝑗𝑈(𝑥)^𝑤𝑗,1−^𝑛_𝑗=11−𝜈_𝐵𝑗𝐿(𝑥)^𝑤𝑗,1−^𝑛_𝑗=11−𝜈_𝐵𝑗𝑈(𝑥)^𝑤𝑗, (8) where 𝑤_𝑗 is the weight of 𝐵_𝑗(𝑗=1,2,…,𝑛), 𝑤_𝑗∈[0,1] and ∑^𝑛_𝑗=1𝑤_𝑗=1. 𝐵_𝑗 is the 𝑗th largest one of all values (𝑛𝜔₁𝐴₁,𝑛𝜔₂𝐴₂,…,𝑛𝜔_𝑛𝐴_𝑛), and 𝜔_𝑘 is the weight of 𝐴_𝑘(𝑘=1,2,…,𝑛), satisfying 𝜔_𝑘∈[0,1] and ∑^𝑛_𝑘=1𝜔_𝑘=1. 𝑛 is a balance factor. Here, 𝑊=(𝑤₁,𝑤₂,…,𝑤_𝑛)^𝑇 can be obtained by the following formula: 𝑤_𝑗𝑗=𝑄𝑙−𝑄𝑗−1𝑙,𝑗=1,2,…,𝑛, (9) where 𝑄=0,𝑟<𝑎,𝑟−𝑎𝑎−𝑏,𝑎≤𝑟≤𝑏,1,𝑟>𝑏. (10)

We utilize the principle of antonym pairs most, at least half, as many as possible, where the parameters (𝑎,𝑏) are equal to (0.3,0.8), (0,05), and (0.5,1), respectively.

Definition 6 (see [32]). 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. (11)

Definition 7 (see [32]). Let 𝐴₁=([𝑎₁,𝑏₁],[𝑐₁,𝑑₁]) and 𝐴₂=([𝑎₂,𝑏₂],[𝑐₂,𝑑₂]) be two interval-valued intuitionistic fuzzy values, and let 𝑆(𝐴₁)=(𝑎₁−𝑐₁+𝑏₁−𝑑₁)/2 and 𝑆(𝐴₂)=(𝑎₂−𝑐₂+𝑏₂−𝑑₂)/2 be the scores of 𝐴₁ and 𝐴₂, respectively, then if 𝑆(𝐴₁)<𝑆(𝐴₂), 𝐴₁ is smaller than 𝐴₂, denoted by 𝐴₁<𝐴₂.

3. Interval-Valued Intuitionistic Fuzzy Entropy

Definition 8 (see [23]). 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.

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

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

Vlachos’ entropy measure [30] is as follows 𝐸¹(𝐴)=1−12𝑛^𝑛_𝑖=1𝜇_𝐴𝐿𝑥_𝑖−𝜈_𝐴𝐿𝑥_𝑖²+𝜇_𝐴𝑈𝑥_𝑖−𝜈_𝐴𝑈𝑥_𝑖². (12)

Example 9. Let 𝐴=([0.4,0.5],[0.3,0.4]) and 𝐵=([0.1,0.2],[0,0.1]) be two IVIFSs in 𝑋.
Intuitively, we can see that 𝐵 is more fuzzy than 𝐴. If we calculate the 𝐸¹(𝐴) and 𝐸¹(𝐵) by (12), then we can obtain 𝐸¹(𝐴)=1−12|0.4−0.3|²+|0.5−0.4|²𝐸=0.9,₁(𝐵)=1−12|0.1−0|²+|0.2−0.1|²=0.9, (13) which indicate that 𝐸¹(𝐴)=𝐸¹(𝐵) and is not consistent with our intuition.
Ye’s entropy measures [27] are as follows 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 (14) where 𝑝,𝑞∈[0,1] are two fixed numbers, 𝑊_𝜇𝐴(𝑥)=𝜇_𝐴𝑈(𝑥)−𝜇_𝐴𝐿(𝑥), and 𝑊_𝜈𝐴(𝑥)=𝜈_𝐴𝑈(𝑥)−𝜈_𝐴𝐿(𝑥).

Example 10. Let 𝐴=([0.5,0.5],[0.1,0.1]) and 𝐵=([0.6,0.6],[0.2,0.2]) be two IVIFSs in 𝑋.
Intuitively, 𝐴 is more fuzzy than 𝐵. Now the 𝐸²(𝐴) and 𝐸²(𝐵) can be gained by (14) and take 𝑝=𝑞=0.5, and the results are 𝐸²(𝐴)=[]sin𝜋×1+0.5−0.14[]+sin𝜋×1−0.5+0.14×1−1√𝐸2−1=0.833,₂(𝐵)=[]sin𝜋×1+0.6−0.24[]+sin𝜋×1−0.6+0.24×1−1√2−1=0.833, (15) which indicate that 𝐸²(𝐴)=𝐸²(𝐵) and are not consistent with our intuition.

3.2. New Interval-Valued Intuitionistic Fuzzy Entropy Based on Cotangent Function

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

Definition 11. Assuming that 𝐴∈IVIFS(𝑋), then an interval-valued intuitionistic fuzzy entropy measure can be defined as 𝐸(𝐴)=1𝑛^𝑛_𝑖=11cot41𝜋+2𝜇_𝐴𝐿𝑥_𝑖−𝜈_𝐴𝐿𝑥_𝑖+𝜇_𝐴𝑈𝑥_𝑖−𝜈_𝐴𝑈𝑥_𝑖×41+𝜋_𝐴𝑥_𝑖⁻¹𝜋=1𝑛^𝑛_𝑖=11cot4𝜋+𝜇_𝐴𝐿𝑥_𝑖−𝜈_𝐴𝐿𝑥_𝑖+𝜇_𝐴𝑈𝑥_𝑖−𝜈_𝐴𝑈𝑥_𝑖×44−𝜇_𝐴𝐿𝑥_𝑖−𝜈_𝐴𝐿𝑥_𝑖−𝜇_𝐴𝑈𝑥_𝑖−𝜈_𝐴𝑈𝑥_𝑖⁻¹𝜋. (16)

Theorem 12. The mapping 𝐸(𝐴), defined by (16), is an entropy measure for IVIFSs.

Proof. In order for (16) to be qualified as a sensible measure of interval-valued intuitionistic fuzzy entropy, it must satisfy the conditions (P1)–(P4) in Definition 8.
Let 0≤𝜇^∼_𝐴(𝑥_𝑖), 𝜈^∼_𝐴(𝑥_𝑖), and 𝜋^∼_𝐴(𝑥_𝑖)≤1; we have 0≤|𝜇_𝐴𝐿(𝑥_𝑖)−𝜈_𝐴𝐿(𝑥_𝑖)+𝜇_𝐴𝑈(𝑥_𝑖)−𝜈_𝐴𝑈(𝑥_𝑖)|≤2. It follows that (0≤|𝜇_𝐴𝐿(𝑥_𝑖)−𝜈_𝐴𝐿(𝑥_𝑖)+𝜇_𝐴𝑈(𝑥_𝑖)−𝜈_𝐴𝑈(𝑥_𝑖)|/8(1+𝜋_𝐴(𝑥_𝑖)))𝜋≤(1/4)𝜋. Thus 0≤𝐸(𝐴)≤1.(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), then 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 (11), we have 𝐸(𝐴)=𝐸(𝐴^𝐶).(P4) In order to show that (11) fulfill the requirement of (P4), it is suffice to to prove the following function: 𝜇_𝐴𝐿𝑥_𝑖−𝜈_𝐴𝐿𝑥_𝑖+𝜇_𝐴𝑈𝑥_𝑖−𝜈_𝐴𝑈𝑥_𝑖44−𝜇_𝐴𝐿𝑥_𝑖−𝜈_𝐴𝐿𝑥_𝑖−𝜇_𝐴𝑈𝑥_𝑖−𝜈_𝐴𝑈𝑥_𝑖≥𝜇_𝐵𝐿𝑥_𝑖−𝜈_𝐵𝐿𝑥_𝑖+𝜇_𝐵𝑈𝑥_𝑖−𝜈_𝐵𝑈𝑥_𝑖44−𝜇_𝐵𝐿𝑥_𝑖−𝜈_𝐵𝐿𝑥_𝑖−𝜇_𝐵𝑈𝑥_𝑖−𝜈_𝐵𝑈𝑥_𝑖, (17) where function 𝐸 is monotonic 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 (17); namely, we prove that 2−𝜇_𝐴𝐿𝑥_𝑖−𝜇_𝐴𝑈𝑥_𝑖2−𝜈_𝐵𝐿𝑥_𝑖−𝜈_𝐵𝑈𝑥_𝑖≥2−𝜇_𝐵𝐿𝑥_𝑖−𝜇_𝐵𝑈𝑥_𝑖2−𝜈_𝐴𝐿𝑥_𝑖−𝜈_𝐴𝑈𝑥_𝑖. (18) If 𝜇_𝐴𝐿(𝑥_𝑖)≤𝜇_𝐵𝐿(𝑥_𝑖) and 𝜇_𝐴𝑈(𝑥_𝑖)≤𝜇_𝐵𝑈(𝑥_𝑖), then 2−𝜇_𝐴𝐿(𝑥_𝑖)−𝜇_𝐴𝑈(𝑥_𝑖)≥2−𝜇_𝐵𝐿(𝑥_𝑖)−𝜇_𝐵𝑈(𝑥_𝑖)≥0. If 𝜈_𝐴𝐿(𝑥_𝑖)≥𝜈_𝐵𝐿(𝑥_𝑖) and 𝜇_𝐴𝑈(𝑥_𝑖)≥𝜇_𝐵𝑈(𝑥_𝑖), then we have 2−𝜈_𝐵𝐿(𝑥_𝑖)−𝜈_𝐵𝑈(𝑥_𝑖)≥2−𝜈_𝐴𝐿(𝑥_𝑖)−𝜈_𝐴𝑈(𝑥_𝑖)≥0. So we can get that (18) 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).

Example 13. Let 𝐴=([0.4,0.5],[0.3,0.4]) and 𝐵=([0.1,0.2],[0,0.1]) 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 (16), and we can obtain that 𝐸(𝐴)1=cot4||𝜋+0.4−0.3+0.5−0.44()𝜋𝐸(𝐵)14−0.4−0.3−0.5−0.4=0.877,=cot4||𝜋+0.1−0+0.2−0.14()𝜋4−0.1−0−0.2−0.1=0.916, (19) which indicate that 𝐸(𝐴)<𝐸(𝐵) and are consistent with our intuition.

Example 14. Let 𝐴=([0.5,0.5],[0.1,0.1]) and 𝐵=([0.6,0.6],[0.2,0.2]) 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 (16), and we can obtain that 𝐸(𝐴)1=cot4||𝜋+0.5−0.1+0.5−0.14()𝜋𝐸(𝐵)14−0.5−0.1−0.5−0.1=0.628,=cot4||𝜋+0.6−0.2+0.6−0.24()𝜋4−0.6−0.2−0.6−0.2=0.577, (20) which indicate that 𝐸(𝐴)>𝐸(𝐵) and are consistent with our intuition.

4. Fuzzy Multicriteria Group Decision-Making Method Based on the New Interval-Valued Intuitionistic Fuzzy Entropy

In this section, we propose a method for fuzzy group decision-making problems based on the interval-valued intuitionistic fuzzy entropy.

Consider an interval-valued intuitionistic fuzzy multicriteria group decision-making problem. Assume that there are 𝑛 alternatives 𝑋={𝑥₁,𝑥₂,…,𝑥_𝑛} and 𝑚 decision 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. Assume that there are 𝑡 decision makers 𝐷={𝑑₁,𝑑₂,…,𝑑_𝑡} whose corresponding weight vector is 𝜆=(𝜆₁,𝜆₂,…,𝜆_𝑡)^𝑇, where 𝜆_𝑘∈[0,1] and ∑^𝑡_𝑘=1𝜆_𝑘=1. In this case, the characteristic of the alternative 𝐴_𝑖 of the 𝑘th decision maker 𝑑_𝑘 is represented by the following IVIFS: 𝐴^𝑘_𝑖=𝐶^𝑘_𝑗,𝜇_{𝐴𝑘𝑖𝐿}𝐶^𝑘_𝑗,𝜇_{𝐴𝑘𝑖𝑈}𝐶^𝑘_𝑗,𝜈_𝐴^𝑘_𝑖𝐿𝐶^𝑘_𝑗,𝜈_𝐴^𝑘_𝑖𝑈𝐶^𝑘_𝑗∣𝐶^𝑘_𝑗∈𝐶, (21) where 0≤𝜇_{𝐴𝑘𝑖𝑈}(𝐶^𝑘_𝑗)+𝜈_𝐴^𝑘_𝑖𝑈(𝐶^𝑘_𝑗)≤1, 𝜇_{𝐴𝑘𝑖𝐿}(𝐶^𝑘_𝑗)≥0, 𝜇_{𝐴𝑘𝑖𝐿}(𝐶^𝑘_𝑗)≥0, 𝑖=1,2,…,𝑛, 𝑗=1,2,…,𝑚, and 𝑘=1,2,…,𝑡. The IVIFS value that is the pair of intervals 𝜇_𝐴𝑘𝑖(𝐶^𝑘_𝑗)=[𝑎^𝑘_𝑖𝑗,𝑏^𝑘_𝑖𝑗], 𝜈_𝐴^𝑘_𝑖(𝐶^𝑘_𝑗)=[𝑐^𝑘_𝑖𝑗,𝑑^𝑘_𝑖𝑗] for 𝐶^𝑘_𝑗∈𝐶 is denoted by 𝑟^𝑘_𝑖𝑗=([𝑎^𝑘_𝑖𝑗,𝑏^𝑘_𝑖𝑗],[𝑐^𝑘_𝑖𝑗,𝑑^𝑘_𝑖𝑗]). Here, we can elicit the interval-valued intuitionistic fuzzy decision matrix 𝑅^𝑘=(𝑟^𝑘_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 decision matrix we can establish a model of interval-valued intuitionistic fuzzy entropy weights.

For the criteria 𝐶_𝑗, the entropy of the alternative 𝐴_𝑖 of the 𝑘th decision maker can be given as 𝐸𝑟^𝑘_𝑖𝑗1=cot4𝜋+𝜇_𝐴𝐿𝑟^𝑘_𝑖𝑗−𝜈_𝐴𝐿𝑟^𝑘_𝑖𝑗+𝜇_𝐴𝑈𝑟^𝑘_𝑖𝑗−𝜈_𝐴𝑈𝑟^𝑘_𝑖𝑗×44−𝜇_𝐴𝐿𝑟^𝑘_𝑖𝑗−𝜈_𝐴𝐿𝑟^𝑘_𝑖𝑗−𝜇_𝐴𝑈𝑟^𝑘_𝑖𝑗−𝜈_𝐴𝑈𝑟^𝑘_𝑖𝑗⁻¹𝜋. (22)

And the entropy for the alternative 𝐴^𝑘_𝑖 of the 𝑘th decision maker is given as 𝐸𝐴^𝑘_𝑖=^𝑚_𝑗=11cot4𝜋+𝜇_𝐴𝐿𝑟^𝑘_𝑖𝑗−𝜈_𝐴𝐿𝑟^𝑘_𝑖𝑗+𝜇_𝐴𝑈𝑟^𝑘_𝑖𝑗−𝜈_𝐴𝑈𝑟^𝑘_𝑖𝑗×44−𝜇_𝐴𝐿𝑟^𝑘_𝑖𝑗−𝜈_𝐴𝐿𝑟^𝑘_𝑖𝑗−𝜇_𝐴𝑈𝑟^𝑘_𝑖𝑗−𝜈_𝐴𝑈𝑟^𝑘_𝑖𝑗⁻¹𝜋. (23)

As each alternative is made in a fair competitive environment, and the fuzzy entropy of each alternative is from a same criteria weight coefficient, the alternatives should be combined. The overall entropy for the alternative 𝐴_𝑖 is given as 𝐸𝐴_𝑖=^𝑡𝑚_𝑘=1𝑗=11cot4𝜋+𝜇_𝐴𝐿𝑟^𝑘_𝑖𝑗−𝜈_𝐴𝐿𝑟^𝑘_𝑖𝑗+𝜇_𝐴𝑈𝑟^𝑘_𝑖𝑗−𝜈_𝐴𝑈𝑟^𝑘_𝑖𝑗×44−𝜇_𝐴𝐿𝑟^𝑘_𝑖𝑗−𝜈_𝐴𝐿𝑟^𝑘_𝑖𝑗−𝜇_𝐴𝑈𝑟^𝑘_𝑖𝑗−𝜈_𝐴𝑈𝑟^𝑘_𝑖𝑗⁻¹𝜋. (24)

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 to a bigger weight value. Then the smaller the value of (24) is, the better the 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: (𝐴)=min𝐸_{𝑛𝑡𝑖=1𝑚𝑘=1𝑗=1}𝑤_𝑗1×cot4𝜋+𝜇_𝐴𝐿𝑟^𝑘_𝑖𝑗−𝜈_𝐴𝐿𝑟^𝑘_𝑖𝑗+𝜇_𝐴𝑈𝑟^𝑘_𝑖𝑗−𝜈_𝐴𝑈𝑟^𝑘_𝑖𝑗×44−𝜇_𝐴𝐿𝑟^𝑘_𝑖𝑗−𝜈_𝐴𝐿𝑟^𝑘_𝑖𝑗−𝜇_𝐴𝑈𝑟^𝑘_𝑖𝑗−𝜈_𝐴𝑈𝑟^𝑘_𝑖𝑗⁻¹𝜋,s.t.𝑤∈𝐻,^𝑚_𝑗=1𝑤_𝑗=1,𝑤_𝑗≥0,𝑗=1,2,…,𝑚. (25)

By solving model (25) with Lingo software, we get the optimal solution (𝑤₁,𝑤₂,…,𝑤_𝑚)^𝑇.

In summary, the main procedure of the decision method proposed is listed in the following.

Step 1. Calculate the weight vector 𝑊=(𝑤₁,𝑤₂,…,𝑤_𝑚)^𝑇 by solving model (25).

Step 2. Aggregate the values of each decision maker, and utilize the IVIF-WGA operator to derive the values 𝑧^𝑘_𝑖 (𝑖=1,2,…,𝑛) of each decision maker as follows: 𝑧^𝑘_𝑖=𝐹_𝑤𝑟^𝑘_𝑖1,𝑟^𝑘_𝑖2,…,𝑟^𝑘_𝑖𝑚,𝑖=1,2,…,𝑛,𝑘=1,2,…,𝑡. (26)

Step 3. Aggregate the values of all decision makers, and utilize the IVIF-HA operator to derive the collective values 𝑧_𝑖 of the alternative 𝑥_𝑖 as follows 𝑧_𝑖=𝐻_𝜆𝑧¹_𝑖,𝑧²_𝑖,…,𝑧^𝑡_𝑖,𝑖=1,2,…,𝑛, (27) where the associated weighting vector 𝜆=(𝜆₁,𝜆₂,…,𝜆_𝑡) of the IVIF-HA operator.

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) and then to select the best one(s).

5. An Illustration for Solving the Supplier Selection Problem

This section adopts a supplier selection problem in [33, 34] to demonstrate how to apply the proposed approach. With continual business development, globalized markets become increasingly competitive. Establishing effective supply chain management (SCM) becomes a critical activity because a sound SCM system can reduce supply chain risk, maximize revenue, optimize business processes, and allow a company to maintain a dominant position in the market [34, 35]. On the other hand, it is a hard problem since supplier selection is typically a multicriteria group decision-making problem involving several conflicting criteria on which decision maker’s knowledge is usually vague and imprecise [34]. Previous research concerning supplier selection often used exact numbers to measure criterion weights. In this study, considering that the decision maker may have difficulty in eliciting precise criterion weights, the proposed approach is proposed to select appropriate supplier in group decision-making environment. It should be noted that, as suggested and illustrated by Merigo and Gil-Lafuente [36], the proposed approach can be easily applied to a host of practical decision problems that involve choosing an optimal alternative from a list of alternatives when multiple attributes must be considered.

Suppose that a high-tech company which manufactures electronic products intends to evaluate and select a supplier of USB connectors. There are four suppliers 𝑥₁, 𝑥₂, 𝑥₃, and 𝑥₄ which are chosen as candidates. A committee of three decision makers 𝑑₁, 𝑑₂, and 𝑑₃ is established, which are an engineering expert, financial expert, and quality control expert, respectively. Four evaluated criteria are considered, including finance (𝑐₁), performance (𝑐₂), technique (𝑐₃), and organizational culture (𝑐₄). The expert weight vector is given by 𝜆=(0.35,0.35,0.3)^𝑇. The interval-valued intuitionistic fuzzy decision matrices of criterion values are constructed as follows: 𝑅¹=],[],[],[],[],[],[],[],[],[],[],[],[],[],[],[],[,𝑅([0.3,0.40.4,0.6])([0.6,0.70.1,0.2])([0.5,0.70.2,0.3])([0.7,0.80.0,0.1])([0.7,0.80.1,0.2])([0.6,0.70.2,0.3])([0.2,0.30.4,0.6])([0.5,0.60.1,0.3])([0.5,0.80.1,0.2])([0.7,0.80.0,0.1])([0.5,0.50.4,0.5])([0.2,0.30.2,0.4])([0.2,0.30.4,0.5])([0.5,0.70.1,0.3])([0.6,0.70.1,0.2])([0.4,0.50.1,0.3])₂=],[],[],[],[],[],[],[],[],[],[],[],[],[],[],[],[,𝑅([0.4,0.50.3,0.4])([0.5,0.60.2,0.2])([0.6,0.70.2,0.3])([0.7,0.80.1,0.2])([0.6,0.80.1,0.2])([0.5,0.60.3,0.4])([0.4,0.50.3,0.4])([0.4,0.60.3,0.4])([0.5,0.60.3,0.4])([0.5,0.70.1,0.2])([0.5,0.60.3,0.4])([0.3,0.40.2,0.5])([0.5,0.60.3,0.4])([0.7,0.80.0,0.1])([0.4,0.50.2,0.4])([0.5,0.70.1,0.2])₃=],[],[],[],[],[],[],[],[],[],[],[],[],[],[],[],[.([0.4,0.60.3,0.4])([0.5,0.70.1,0.2])([0.5,0.60.2,0.4])([0.6,0.80.1,0.2])([0.5,0.80.1,0.2])([0.3,0.50.2,0.3])([0.3,0.60.2,0.4])([0.4,0.50.2,0.4])([0.5,0.60.0,0.1])([0.5,0.80.1,0.2])([0.4,0.70.2,0.3])([0.2,0.40.2,0.3])([0.5,0.70.1,0.3])([0.4,0.60.0,0.1])([0.3,0.50.2,0.4])([0.7,0.90.0,0.1]) (28)

The incomplete information about the criterion weights are as follows (in this problem, the criterion weights are incomplete information. The specific weight calculation method can be found in [34]):𝐻=0.228≤𝑤₁≤0.8758,0.2285≤𝑤₂≤0.8789,0.1642≤𝑤₃≤0.7979,0.1419≤𝑤₄≤0.7824. (29)

Step 1. Calculate the weight vector 𝑊=(𝑤₁,𝑤₂,𝑤₃,𝑤₄)^𝑇 by solving model (25) as follows: 𝑊=(0.228,0.4659,0.1642,0.1419)^𝑇. (30)

Step 2. Aggregate the values of each decision maker, andutilize the IVIF-WGA operator to derive the values 𝑧^𝑘_𝑖(𝑖=1,2,…,𝑛) of each decision maker.
The integrated values for alternatives 𝑥₁, 𝑥₂, 𝑥₃ and 𝑥₄ of decision maker 𝑑₁ are, respectively, 𝑧¹₁=],[,𝑧([0.508,0.6280.183,0.320])₁₂=],[,𝑧([0.506,0.6140.203,0.342])₁₃=],[,𝑧([0.514,0.6440.130,0.249])₁₄=([0.405,0.550],[0.179,0.337]). (31)
The integrated values for alternatives 𝑥₁, 𝑥₂, 𝑥₃ and 𝑥₄ of decision maker 𝑑₂ are, respectively, 𝑧²₁=],[,𝑧([0.514,0.6150.211,0.267])₂₂=],[,𝑧([0.487,0.6220.259,0.359])₂₃=],[,𝑧([0.439,0.5840.198,0.331])₂₄=([0.546,0.649],[0.124,0.259]). (32)
The integrated values for alternatives 𝑥₁, 𝑥₂, 𝑥₃ and 𝑥₄ of decision maker 𝑑₃ are, respectively, 𝑧³₁=],[,𝑧([0.488,0.6720.166,0.285])₃₂=],[,𝑧([0.351,0.5730.178,0.312])₃₃=],[,𝑧([0.423,0.6640.111,0.211])₃₄=([0.435,0.639],[0.06,0.205]). (33)

Step 3. Aggregate the values of all decision makers, and utilize the IVIF-HA operator to derive the collective values 𝑧_𝑖 of the alternative 𝑥_𝑖.
The weighting vector of the IVIL-HA operator is obtained by the principle of antonym pairs many and (𝑎,𝑏) equal to (0.3, 0.8). So we can obtain 𝜔=(0.243,0.514,0.243) and the integrated values as follows: 𝑧₁=],[,𝑧([0.489,0.6170.202,0.297])₂=],[,𝑧([0.436,0.5930.236,0.357])₃=],[,𝑧([0.430,0.6250.144,0.262])₄=([0.463,0.606],[0.129,0.278]). (34)

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) as follows: 𝑆𝑧₁=0.607,𝑆𝑧₂𝑆𝑧=0.436,₃=0.649,𝑆𝑧₄=0.663. (35)

Step 5. 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): 𝐴₄≻𝐴₃≻𝐴₁≻𝐴₂, and thus the best alternative is 𝐴₄.

6. Conclusion

In this paper, a new entropy measure of IVIFS is proposed by using cotangent function, which can overcome limitations of some existing methods. And we provide several numerical examples to illustrate its validity. For interval-valued intuitionistic fuzzy multicriteria group decision-making problem with incomplete information on the weights of criteria, an entropy weight model is established to determine the entropy weights. In addition, the method and procedure are developed to solve FMCGDM problems. Finally, the supplier selection problem is used as an example to demonstrate how to apply the proposed multicriteria group decision-making approach.

Acknowledgments

The authors would like to thank Yufeng Xu, Fuqiang Wang, Mengjun Liao, Siying Wang, and the anonymous reviewers for their valuable comments on this paper. This paper was supported by the National Natural Science Foundation of China (no. 70921001).