# Preference Attitude-Based Method for Ranking Intuitionistic Fuzzy Numbers and Its Application in Renewable Energy Selection

Authors: Jian Lin; Fanyong Meng; Riqing Chen; Qiang Zhang
Journal: Complexity
 (2018)

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## Abstract

Many applications of intuitionistic fuzzy sets depend on ranking or comparing intuitionistic fuzzy numbers. This paper presents a novel ranking method for intuitionistic fuzzy numbers based on the preference attitudinal accuracy and score functions. The proposed ranking method considers not only the preference attitude of decision maker, but also all the possible values in feasible domain. Some desirable properties of preference attitudinal accuracy and score functions are verified in detail. A total order on the set of intuitionistic fuzzy numbers is established by using the proposed two functions. The proposed ranking method is also applied to select renewable energy. The advantage and validity of the proposed method are shown by comparing with some representative ranking methods.


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## Body

## 1. Introduction

Atanassov [1] introduced the concept of intuitionistic fuzzy sets characterized by a membership function, a nonmembership function, and a hesitancy function. Due to the increasing complexity of real life problems, intuitionistic fuzzy set is very suitable for representing fuzzy information under complicated and uncertain settings as an extension of traditional fuzzy set. Intuitionistic fuzzy set theory has been deeply discussed by many scholars since the notations appearance and applied in various fields, such as decision-making [2–7], supplier selection [8–10], pattern recognition [11–14], medical diagnosis [15, 16], and artificial intelligence [17, 18].Many applications of intuitionistic fuzzy sets depend on ranking or comparing intuitionistic fuzzy numbers. A number of researchers focus on the order relation of intuitionistic fuzzy numbers over the past two decades. Chen and Tan [19] proposed a score function to evaluate the score of intuitionistic fuzzy values. Through analysis on the limitation of Chen and Tan’s score function, Hong and Choi [20] improved their ranking method by adding an accuracy function. Xu [21] gave a kind of order relation to rank the intuitionistic fuzzy numbers by combining the score and accuracy function. Wang et al. [22] introduced a novel score function to measure the degree of suitability. Some desirable properties of the novel score function were discussed. Ye [23] developed an improved algorithm for score functions based on hesitancy degree. By using the intuitionistic fuzzy point operators, Liu and Wang [24] defined a series of new score functions for dealing with multicriteria decision-making problems. Jafarian and Rezvani [25] presented a method for mapping the intuitionistic fuzzy numbers into the crisp values and described the concept of spread value of intuitionistic fuzzy number. To analyze the fuzzy meaning of an intuitionistic fuzzy value, Yu et al. [26] formalized an intuitionistic fuzzy value as a fuzzy subset and determined the dominance relation between two intuitionistic fuzzy values. Guo [27] built a new ranking model based on the viewpoint of amount of information. A total order which extended the usual partial order was analyzed in deep. Zhang and Xu [28] used a special function to define the order of intuitionistic fuzzy numbers. Some good mathematical properties on algebraic intuitionistic fuzzy numbers were also given. Lakshmana et al. [29] derived a total order on the entire class of intuitionistic fuzzy number by applying upper lower dense sequence to the interval. Gupta et al. [30] utilized relative comparisons based on the advantage and disadvantage scores of intuitionistic fuzzy numbers. The relative comparison of intuitionistic fuzzy numbers took the membership, nonmembership, and hesitancy degree into account. Xing et al. [31] proposed an Euclidean distance-based approach to ranking intuitionistic fuzzy numbers. However, the above-mentioned methods for ranking intuitionistic fuzzy numbers do not consider the risk attitude of decision maker which is very flexible and useful in real-world applications. A valid ranking method should be established in accordance with the preference attitude of decision maker. To do this, Chen [32] conducted a comparative analysis of score functions for ranking intuitionistic fuzzy numbers. A parameterized score function was developed to represent a mixed result of positive and negative outcome expectations. Wang et al. [33] proposed a new score function based on relative entropy. The risk attitudes of decision makers were defined by risk preference indexes. Wan et al. [34] utilized the closeness degree to characterize the amount of information based on TOPSIS method. A novel risk attitudinal ranking measure is developed to rank the intuitionistic fuzzy numbers. Nevertheless, the above preference attitude-based ranking methods just focus on the extreme points of feasible domain; the other possible values in feasible domain are ignored. Therefore, these ranking methods may lose some valuable information, which can be useful in determining the order relation of intuitionistic fuzzy numbers. To overcome the shortages of existing ranking methods, this paper proposes a preference attitudinal method for ranking intuitionistic fuzzy numbers based on preference attitudinal accuracy and score functions. The proposed ranking method considers not only the preference attitude of decision maker, but also all the possible values in feasible domain. Some desirable properties of preference attitudinal accuracy and score functions are discussed. Moreover, the proposed method is applied to deal with renewable energy selection problem. The advantage and validity of the proposed method are shown in detail by comparing with some representative ranking methods.The rest of this paper is structured as follows. Section2 introduces some basic concepts on intuitionistic fuzzy set. In Section 3, the preference attitudinal accuracy and score functions of intuitionistic fuzzy numbers are proposed. Section 4 presents the order relation between intuitionistic fuzzy numbers. Numeral examples and comparison are shown in Section 5.


## 2. Preliminaries

In the following, some basic concepts on intuitionistic fuzzy set are introduced to facilitate future discussions.Definition 1 (see [1]).
LetX be a universe of discourse. An intuitionistic fuzzy set A over X is expressed as(1)A=x,μAx,vAx∣x∈X,where μA(x) and vA(x) are the membership degree and nonmembership degree of x to A, respectively, such that μA(x),vA(x)∈[0,1], μA(x)+vA(x)≤1. The hesitation degree of x to A is denoted by πA(x)=1-μA(x)-vA(x).Obviously, for eachx∈X, we have πA(x)∈[0,1]. For simplicity, α=(μ,v) is called an intuitionistic fuzzy number (IFN), such that μ,v∈[0,1], μ+v≤1, and π(α)=1-μ-v. The set of all IFNs is denoted by IFN.Letα1=(μ1,v1) and α2=(μ2,v2) be two IFNs. Clearly, α1=α2 if and only if μ1=μ2 and v1=v2. Based on the score function [19] and accuracy function [20], Xu [21] introduced the operational laws and ordering relation among intuitionistic fuzzy numbers as follows.Definition 2.
Letα1=(μ1,v1) and α2=(μ2,v2) be two IFNs; then (1)
α1⊕α2=(μ1+μ2-μ1μ2,v1v2),(2)
k⊗α1=(1-(1-μ1)k,v1k), k>0.Definition 3.
Letα=(μ,v) be an IFN. The accuracy and score function of α are, respectively, represented by(2)AXα=μ+v,(3)SXα=μ-v.It is clear that AX(α)∈[0,1] and SX(α)∈[-1,1]. From (2) and (3), we have AX(α)=SX(α)+2v. Thus, AX(α)≥SX(α) holds.Definition 4.
Letα1=(μ1,v1) and α2=(μ2,v2) be two IFNs. The ordering relation is established as follows:(1)
IfSX(α1)<SX(α2), then α1≺Xα2.(2)
IfSX(α1)=SX(α2), then
(a)
ifAX(α1)=AX(α2), then α1=α2,(b)
ifAX(α1)<AX(α2), then α1≺Xα2.


## 3. The Preference Attitudinal Accuracy and Score Functions of Intuitionistic Fuzzy Numbers

In this section, we will propose the definitions of preference attitudinal accuracy and score functions by adding an attitudinal parameter in double integral.


### 3.1. The Preference Attitudinal Accuracy Function

Definition 5.
Letα=(μ,v) be an IFN. The feasible domain of α is represented by(4)Dα=x,y∣μ≤x≤μ+π,v≤y≤v+π,x+y≤1,where π=1-u-v. By considering the hesitation degree of α, Dα denotes the set of feasible intuitionistic fuzzy numbers with respect to α. The feasible domain Dα is illustrated in Figure 1.Figure 1
The feasible domain with respect toα.Definition 6.
Letα=(μ,v) be an IFN. The preference attitudinal accuracy function of α is defined by(5)Aλα=1areaDα∬Dαλx+1-λydxdy,where Dα is the feasible domain of intuitionistic fuzzy number α and area(Dα) is the area of feasible domain Dα. The parameter λ can be regarded as the level of decision maker or assessor’s preference attitude towards positive information, λ∈[0,1]. In particular, decision maker or assessor is said to be neutral to the positive and negative information when λ=0.5.From (5), Aλ(α) represents the average value of accuracy function over feasible domain Dα. ∬Dα(λx+(1-λ)y)dxdy is equal to the volume of the oblique triangular prism P1 shown in Figure 2. It means that Aλ(α) can be regarded as the average height of oblique triangular prism P1. By Definition 5, we have (6)areaDα=12π2=121-μ-v2.Since(7)∬Dαλx+1-λydxdy=∫μ1-v∫v1-xλx+1-λydydx=161-μ-v21-μ+2v+λ21-μ-v2μ-v,we get the following expression of preference attitudinal accuracy function(8)Aλα=1-μ+2v3+λμ-vwhich can be equivalently written as(9)Aλα=λ2μ-v+13+1-λ2v-μ+13.The preference attitudinal accuracy function satisfies the following desirable properties.Figure 2
The graph of oblique triangular prismP1.Property 7.
Letα=(μ,v) be an IFN. For each parameter λ∈[0,1], we have (10)0≤min⁡2μ-v+13,2v-μ+13≤Aλα≤max⁡2μ-v+13,2v-μ+13≤1.Proof.
Property7 can be proved in two cases, shown as follows:
(1) Ifμ≥v, then we have 0≤2v-μ+1≤2μ-v+1≤3. For all λ∈[0,1], it holds that (11)2v-μ+13≤λ2μ-v+13+1-λ2v-μ+13≤2μ-v+13.Since 2v-μ+1/3≥0 and (2μ-v+1)/3≤1, by (8), we have(12)0≤2v-μ+13≤Aλα≤2μ-v+13≤1.
(2) Ifμ<v, then we have 0≤2μ-v+1<2v-μ+1≤3. For all λ∈[0,1], it holds that (13)2μ-v+13≤λ2μ-v+13+1-λ2v-μ+13≤2v-μ+13.Since (2v-μ+1)/3≤1 and 2μ-v+1/3≥0, by (8), we have(14)0≤2μ-v+13≤Aλα≤2v-μ+13≤1.In sum, Property 7 is proved.Property 8.
Letα1=(μ1,v1) and α2=(μ2,v2) be two IFNs. If v1-v2≤μ1-μ2≤2(v1-v2), then Aλ(α1)≥Aλ(α2) holds for all λ∈[0,1].Proof.
From (8), we have(15)Aλα1=1-μ1+2v13+λμ1-v1,Aλα2=1-μ2+2v23+λμ2-v2.It follows that (16)Aλα1-Aλα2=13μ2-μ1+2v1-v2+λμ1-μ2+v2-v1.Since v1-v2≤μ1-μ2≤2(v1-v2) and 0≤λ≤1, we have Aλ(α1)-Aλ(α2)≥0.
This completes the proof of Property8.Property 9.
For any two IFNsα1=(μ1,v1) and α2=(μ2,v2), A0.5(α1)>A0.5(α2) if and only if AX(α1)>AX(α2).Proof.
Whenλ=0.5, we have A0.5(αi)=(μi+vi+2)/6, i=1,2. Based on (8), we get A0.5(αi)=(1/6)AX(αi)+1/3, i=1,2. Thus, AX(α1)>AX(α2) can be equivalently expressed as (17)A0.5α1=16AXα1+13>16AXα2+13=A0.5α2.This completes the proof of Property 9.Property 10.
Letα=(μ,v) be an IFN. α-=(v,μ) is the complementary intuitionistic fuzzy number of α. For each attitudinal parameter λ∈[0,1], we have Aλ(α)=A1-λ(α-).Proof.
From (9), it follows that(18)A1-λα-=1-λ2v-μ+13+λ2μ-v+13=Aλα.Thus, Property 10 is proved.Property 11 (sensitivity analysis).
Given two IFNsα1=(μ1,v1) and α2=(μ2,v2). Let λ be the attitudinal parameter under which it has satisfied the fact that Aλ(α1)≤Aλ(α2). Let Δλ be a perturbation of the attitudinal parameter λ with λ+Δλ∈[0,1]. Then, Aλ+Δλ(α1)≤Aλ+Δλ(α2) holds if and only if(19)-λ≤Δλ≤min⁡Aλα2-Aλα1μ1-μ2+v2-v1,1-λ,μ1-μ2>v1-v2;max⁡Aλα2-Aλα1μ1-μ2+v2-v1,-λ≤Δλ≤1-λ,μ1-μ2<v1-v2;-λ≤Δλ≤1-λ,μ1-μ2=v1-v2.Proof.
Since0≤λ+Δλ≤1, we have -λ≤Δλ≤1-λ. Considering of (20)Aλ+Δλα1=Aλα1+Δλμ1-v1,Aλ+Δλα2=Aλα2+Δλμ2-v2,the following inequalities are equivalent:(21)Aλ+Δλα1≤Aλ+Δλα2⟺Aλα2-Aλα1≥Δλμ1+v2-μ2-v1.We can consider (21) in three cases, shown as follows:
(a) Ifμ1-μ2=v1-v2, then inequalities Aλ+Δλ(α1)≤Aλ+Δλ(α2) and Aλ(α2)-Aλ(α1)≥0 are equivalent.
It means that (21) is true for all Δλ with -λ≤Δλ≤1-λ.
(b) Ifμ1-μ2>v1-v2, then(22)Aλ+Δλα1≤Aλ+Δλα2⟺Δλ≤Aλα2-Aλα1μ1-μ2+v2-v1.
Notice thatΔλ is subject to the constraint -λ≤Δλ≤1-λ; inequality Aλ+Δλ(α1)≤Aλ+Δλ(α2) is equivalent to (23)-λ≤Δλ≤min⁡Aλα2-Aλα1μ1-μ2+v2-v1,1-λ.
(c) Ifμ1-μ2<v1-v2, then(24)Aλ+Δλα1≤Aλ+Δλα2⟺Δλ≥Aλα2-Aλα1μ1-μ2+v2-v1.
Therefore, inequalityAλ+Δλ(α1)≤Aλ+Δλ(α2) is equivalent to(25)max⁡Aλα2-Aλα1μ1-μ2+v2-v1,-λ≤Δλ≤1-λ.In sum, Property 11 is proved.Remark 12.
As everyone knows, it is very difficult to determine the exact value of parameterλ. To deal with this problem, Table 1 is presented for estimating the appropriate value of parameter λ.Table 1
Preference attitude towards information and the corresponding value ofλ.
The assessor’s preference attitude towards information The value ofλ The negative information is absolutely preferred (NA) 0 The negative information is extremely preferred (NE) 0.1 The negative information is strongly preferred (NST) 0.2 The negative information is moderately preferred (NM) 0.3 The negative information is slightly preferred (NSL) 0.4 The assessor is neutral to the positive and negative information (NPN) 0.5 The positive information is slightly preferred (PSL) 0.6 The positive information is moderately preferred (PM) 0.7 The positive information is strongly preferred (PST) 0.8 The positive information is extremely preferred (PE) 0.9 The positive information is absolutely preferred (PA) 1It is clear that Table1 proposed some discrete values of parameter λ corresponding to the linear preference of assessor. If the preference of assessor is nonlinear and the value of λ is continuous, some nonlinear functions can be used here to measure the value of λ: that is, f:Ω→[0,1] such that f(NA)=0, f(NPN)=0.5 and f(PA)=1. Ω is the set of assessor’s preference attitudes towards information.


### 3.2. The Preference Attitudinal Score Function

Similar to the preference attitudinal accuracy function, the preference attitudinal score function can be defined as follows.Definition 13.
Letα=(μ,v) be an IFN. The preference attitudinal score function of α is defined by(26)Sλα=1areaDα∬Dαλx-1-λydxdy,where Dα is the feasible domain of intuitionistic fuzzy number α and area(Dα) is the area of feasible domain Dα.From (26), Sλ(α) represents the average value of score function over feasible domain Dα. Denote the volume of P2 by Vup and the volume of P3 by Vbelow, respectively. Then, we have ∬Dαλx-1-λydxdy=Vup-Vbelow. The graphs of cubic P2 and P3 are illustrated in Figure 3. Accordingly, Sλ(α) can be regarded as the average values of variable z over feasible domain Dα. Since(27)∬Dαλx-1-λydxdy=∫μ1-v∫v1-xλx-1-λydydx=161-μ-v2μ-2v-1+λ61-μ-v2μ+v+2,based on (6) and (26), the following expression of preference attitudinal score function is derived:(28)Sλα=μ-2v-13+λ3μ+v+2which can be equivalently written as(29)Sλα=λ2μ-v+13+1-λμ-2v-13.The preference attitudinal score function satisfies the following desirable properties.Figure 3
The graphs of cubicP2 and P3.Property 14.
Letα1=(μ1,v1) and α2=(μ2,v2) be two IFNs. If μ1≥μ2 and v1≤v2, then Sλ(α1)≥Sλ(α2) holds for all λ∈[0,1].Proof.
From (28), we have(30)Sλα1=μ1-2v1-13+λ3μ1+v1+2,Sλα2=μ2-2v2-13+λ3μ2+v2+2.It follows that (31)Sλα1-Sλα2=λ32μ1-μ2+v2-v1+1-λ3μ1-μ2+2v2-v1.Since μ1≥μ2, v1≤v2, and 0≤λ≤1, we have Sλ(α1)-Sλ(α2)≥0.
Thus, Property14 is proved.Property 15.
Letα=(μ,v) be an IFN. The preference attitudinal score function Sλ(α) is monotonic with respect to the parameter λ; that is, if λ1≥λ2, then Sλ1(α)≥Sλ2(α).Proof.
From (28), we have(32)Sλ1α=μ-2v-13+λ13μ+v+2,Sλ2α=μ-2v-13+λ23μ+v+2.Thus, we have Sλ1(α)-Sλ2(α)=(1/3)(λ1-λ2)(μ+v+2)≥0.
This completes the proof of Property15.Property 16.
Letα=(μ,v) be an IFN. For each parameter λ∈[0,1], we have -1≤μ-2v-1/3≤Sλ(α)≤(2μ-v+1)/3≤1.Proof.
From (29), we get S1(α)=(2μ-v+1)/3 and S0(α)=(μ-2v-1)/3. Since 0≤λ≤1, by Property 15, we have S0(α)≤Sλ(α)≤S1(α). Notice that μ-2v-1≥-3 and 2μ-v+1≤3; we have(33)Sλα≥S0α=μ-2v-13≥-1,Sλα≤S1α=2μ-v+13≤1.Therefore, Property 16 is proved.Because the addition operation of intuitionistic fuzzy number is nonlinear, the following inequalities of preference attitudinal score function are accordingly derived.Property 17.
Letα1=(μ1,v1) and α2=(μ2,v2) be two IFNs. Denoting λ0=(1+2v1+2v2-μ1μ2-2v1v2)/(μ1μ2-v1v2+v1+v2+2), we have λ0∈[0,1], and it follows that(1)
ifλ=λ0, then Sλ(α1⊕α2)=Sλ(α1)+Sλ(α2);(2)
if0≤λ<λ0, then Sλ(α1⊕α2)>Sλ(α1)+Sλ(α2);(3)
ifλ0<λ≤1, then Sλ(α1⊕α2)<Sλ(α1)+Sλ(α2).Proof.
Since0≤μ1μ2≤1 and -1≤(v2-1)(1-v1)≤0, we have(34)μ1μ2-v1v2+v1+v2+2=μ1μ2+v2-11-v1+3>0,1+2v1+2v2-μ1μ2-2v1v2=2v2-11-v1+3-μ1μ2≥0.Thus, we have λ0≥0. Considering 2μ1μ2-(v2-1)(1-v1)≥0 and(35)μ1μ2-v1v2+v1+v2+2=2v2-11-v1+3-μ1μ2+2μ1μ2-v2-11-v1,we have λ0≤1.
In sum, we haveλ0∈[0,1].
Denotingf(λ)=Sλ(α1)+Sλ(α2)-Sλ(α1⊕α2), from (28) and Definition 2, we have(36)fλ=13μ1μ2+2v1v2-2v1-2v2-1+λ3μ1μ2-v1v2+v1+v2+2.If λ=λ0=1+2v1+2v2-μ1μ2-2v1v2/μ1μ2-v1v2+v1+v2+2, then we have f(λ0)=0.
Therefore,Sλ0(α1⊕α2)=Sλ0(α1)+Sλ0(α2) holds.
Sinceμ1μ2-v1v2+v1+v2+2>0, function f(λ) is monotonically increasing with respect to the parameter λ. Thus, if 0≤λ<λ0, then f(λ)<f(λ0)=0. Based on the monotonicity of function f(λ), we have Sλ(α1⊕α2)>Sλ(α1)+Sλ(α2).
Similarly, ifλ0<λ≤1, then f(λ)>f(λ0)=0; we have Sλ(α1⊕α2)<Sλ(α1)+Sλ(α2).
This completes the proof of Property17.Property 18.
Letα=(μ,v) be an IFN. α-=(v,μ) is the complementary intuitionistic fuzzy number of α. For each attitudinal parameter λ∈[0,1], we have Sλ(α)=-S1-λ(α-).Proof.
From (29), it follows that(37)S1-λα-=1-λ2v-μ+13+λv-2μ-13=-λ2μ-v+13-1-λμ-2v-13=-Sλα.Thus, Property 18 is proved.Property 19 (sensitivity analysis).
Given two IFNsα1=(μ1,v1) and α2=(μ2,v2). Let λ be the attitudinal parameter under which it has satisfied the fact that Sλ(α1)≤Sλ(α2). Let Δλ be a perturbation of the attitudinal parameter λ with λ+Δλ∈[0,1]. Then, Sλ+Δλ(α1)≤Sλ+Δλ(α2) holds if and only if(38)-λ≤Δλ≤min⁡3Sλα2-3Sλα1μ1+v1-μ2-v2,1-λ,μ1+v1>μ2+v2;max⁡3Sλα2-3Sλα1μ1+v1-μ2-v2,-λ≤Δλ≤1-λ,μ1+v1<μ2+v2;-λ≤Δλ≤1-λ,μ1+v1=μ2+v2.Proof.
Since0≤λ+Δλ≤1, we have -λ≤Δλ≤1-λ. Considering(39)Sλ+Δλα1=Sλα1+Δλ3μ1+v1+2,Sλ+Δλα2=Sλα2+Δλ3μ2+v2+2,the following inequalities are equivalent:(40)Sλ+Δλα1≤Sλ+Δλα2⟺Sλα2-Sλα1≥Δλ3μ1+v1-μ2-v2.We can consider (40) in three cases, shown as follows:
(a) Ifμ1+v1=μ2+v2, then inequalities Sλ+Δλ(α1)≤Sλ+Δλ(α2) and Sλ(α2)-Sλ(α1)≥0 are equivalent.
It means that (40) is true for all Δλ with -λ≤Δλ≤1-λ.
(b) Ifμ1+v1>μ2+v2, then(41)Sλ+Δλα1≤Sλ+Δλα2⟺Δλ≤3Sλα2-3Sλα1μ1+v1-μ2-v2.Notice that Δλ is subject to the constraint -λ≤Δλ≤1-λ, inequality Sλ+Δλ(α1)≤Sλ+Δλ(α2) is equivalent to (42)-λ≤Δλ≤min⁡3Sλα2-3Sλα1μ1+v1-μ2-v2,1-λ.
(c) Ifμ1+v1<μ2+v2, then(43)Sλ+Δλα1≤Sλ+Δλα2⟺Δλ≥3Sλα2-3Sλα1μ1+v1-μ2-v2.Therefore, inequality Sλ+Δλ(α1)≤Sλ+Δλ(α2) is equivalent to(44)max⁡3Sλα2-3Sλα1μ1+v1-μ2-v2,-λ≤Δλ≤1-λ.In sum, Property 19 is proved.


### 3.3. The Relations among Score Functions and Accuracy Functions

In the following, the relations among score functions and accuracy functions are studied in detail.Property 20.
Letα=(μ,v) be an IFN. The score function SX(α), accuracy function AX(α), and preference attitudinal score function Sλ(α) are related as(45)Sλα=12SXα+2λ-16AXα+2λ-13.In particular, S0.5(α)=1/2SX(α) holds.Proof.
The proof is straightforward, so omitted.Property 21.
Letα=(μ,v) be an IFN. The score function SX(α), accuracy function AX(α), and preference attitudinal accuracy function Aλ(α) are related as(46)Aλα=16AXα+2λ-12SXα+13.In particular, A0.5(α)=(1/6)AX(α)+1/3 holds.Proof.
The proof is straightforward, so omitted.Property 22.
Letα=(μ,v) be an IFN. The preference attitudinal score function Sλ(α) and accuracy function Aλ(α) are related as(47)μ+v+2Aλα+3v-3μSλα=232μ-v+12v-μ+1.Proof.
From (8) and (28), we have (48)μ+v+2Aλα+3v-3μSXα=μ+v+21-μ+2v3+λμ-v+3v-3μμ-2v-13+λ3μ+v+2=13μ+v+21-μ+2v+133v-3μμ-2v-1=232μ-v+12v-μ+1.Thus, Property 22 is proved.Property 23.
Letα=(μ,v) be an IFN. Given two attitudinal parameters λ1 and λ2, such that λ1<λ2. The score function SX(α) and preference attitudinal accuracy function Aλ(α) are related as follows:(1)
IfSX(α)>0, then Aλ1(α)<Aλ2(α).(2)
IfSX(α)<0, then Aλ1(α)>Aλ2(α).(3)
IfSX(α)=0, then Aλ1(α)=Aλ2(α).Proof.
From (8), we have(49)Aλ1α=1-μ+2v3+λ1μ-v,Aλ2α=1-μ+2v3+λ2μ-v.Thus, we have Aλ1(α)-Aλ2(α)=(λ1-λ2)(μ-v). Since λ1<λ2 and SX(α)=μ-v, items (1), (2), and (3) are obviously hold.
Therefore, Property23 is proved.Property 24.
Letα=(μ,v) be an IFN. For all λ∈[0,1], Aλ(α)≥Sλ(α) holds. In particular, we have A1(α)=S1(α), A0(α)=-S0(α), Aλ((1,0))=Sλ((1,0)), and Aλ((0,1))=-Sλ((0,1)).Proof.
From (8) and (28), we have(50)Aλα-Sλα=132+4v-2μ1-λ.Since 2+4v-2μ≥0 and λ∈[0,1], we have Aλ(α)-Sλ(α)≥0.
Obviously,A1(α)=S1(α)=(2μ-v+1)/3, A0(α)=-S0(α)=(2v-μ+1)/3.
Whenα=(1,0), we have Aλ(α)=Sλ(α)=λ. When α=(0,1), we have Aλ(α)=-Sλ(α)=1-λ.
This completes the proof of Property24.


## 3.1. The Preference Attitudinal Accuracy Function

Definition 5.
Letα=(μ,v) be an IFN. The feasible domain of α is represented by(4)Dα=x,y∣μ≤x≤μ+π,v≤y≤v+π,x+y≤1,where π=1-u-v. By considering the hesitation degree of α, Dα denotes the set of feasible intuitionistic fuzzy numbers with respect to α. The feasible domain Dα is illustrated in Figure 1.Figure 1
The feasible domain with respect toα.Definition 6.
Letα=(μ,v) be an IFN. The preference attitudinal accuracy function of α is defined by(5)Aλα=1areaDα∬Dαλx+1-λydxdy,where Dα is the feasible domain of intuitionistic fuzzy number α and area(Dα) is the area of feasible domain Dα. The parameter λ can be regarded as the level of decision maker or assessor’s preference attitude towards positive information, λ∈[0,1]. In particular, decision maker or assessor is said to be neutral to the positive and negative information when λ=0.5.From (5), Aλ(α) represents the average value of accuracy function over feasible domain Dα. ∬Dα(λx+(1-λ)y)dxdy is equal to the volume of the oblique triangular prism P1 shown in Figure 2. It means that Aλ(α) can be regarded as the average height of oblique triangular prism P1. By Definition 5, we have (6)areaDα=12π2=121-μ-v2.Since(7)∬Dαλx+1-λydxdy=∫μ1-v∫v1-xλx+1-λydydx=161-μ-v21-μ+2v+λ21-μ-v2μ-v,we get the following expression of preference attitudinal accuracy function(8)Aλα=1-μ+2v3+λμ-vwhich can be equivalently written as(9)Aλα=λ2μ-v+13+1-λ2v-μ+13.The preference attitudinal accuracy function satisfies the following desirable properties.Figure 2
The graph of oblique triangular prismP1.Property 7.
Letα=(μ,v) be an IFN. For each parameter λ∈[0,1], we have (10)0≤min⁡2μ-v+13,2v-μ+13≤Aλα≤max⁡2μ-v+13,2v-μ+13≤1.Proof.
Property7 can be proved in two cases, shown as follows:
(1) Ifμ≥v, then we have 0≤2v-μ+1≤2μ-v+1≤3. For all λ∈[0,1], it holds that (11)2v-μ+13≤λ2μ-v+13+1-λ2v-μ+13≤2μ-v+13.Since 2v-μ+1/3≥0 and (2μ-v+1)/3≤1, by (8), we have(12)0≤2v-μ+13≤Aλα≤2μ-v+13≤1.
(2) Ifμ<v, then we have 0≤2μ-v+1<2v-μ+1≤3. For all λ∈[0,1], it holds that (13)2μ-v+13≤λ2μ-v+13+1-λ2v-μ+13≤2v-μ+13.Since (2v-μ+1)/3≤1 and 2μ-v+1/3≥0, by (8), we have(14)0≤2μ-v+13≤Aλα≤2v-μ+13≤1.In sum, Property 7 is proved.Property 8.
Letα1=(μ1,v1) and α2=(μ2,v2) be two IFNs. If v1-v2≤μ1-μ2≤2(v1-v2), then Aλ(α1)≥Aλ(α2) holds for all λ∈[0,1].Proof.
From (8), we have(15)Aλα1=1-μ1+2v13+λμ1-v1,Aλα2=1-μ2+2v23+λμ2-v2.It follows that (16)Aλα1-Aλα2=13μ2-μ1+2v1-v2+λμ1-μ2+v2-v1.Since v1-v2≤μ1-μ2≤2(v1-v2) and 0≤λ≤1, we have Aλ(α1)-Aλ(α2)≥0.
This completes the proof of Property8.Property 9.
For any two IFNsα1=(μ1,v1) and α2=(μ2,v2), A0.5(α1)>A0.5(α2) if and only if AX(α1)>AX(α2).Proof.
Whenλ=0.5, we have A0.5(αi)=(μi+vi+2)/6, i=1,2. Based on (8), we get A0.5(αi)=(1/6)AX(αi)+1/3, i=1,2. Thus, AX(α1)>AX(α2) can be equivalently expressed as (17)A0.5α1=16AXα1+13>16AXα2+13=A0.5α2.This completes the proof of Property 9.Property 10.
Letα=(μ,v) be an IFN. α-=(v,μ) is the complementary intuitionistic fuzzy number of α. For each attitudinal parameter λ∈[0,1], we have Aλ(α)=A1-λ(α-).Proof.
From (9), it follows that(18)A1-λα-=1-λ2v-μ+13+λ2μ-v+13=Aλα.Thus, Property 10 is proved.Property 11 (sensitivity analysis).
Given two IFNsα1=(μ1,v1) and α2=(μ2,v2). Let λ be the attitudinal parameter under which it has satisfied the fact that Aλ(α1)≤Aλ(α2). Let Δλ be a perturbation of the attitudinal parameter λ with λ+Δλ∈[0,1]. Then, Aλ+Δλ(α1)≤Aλ+Δλ(α2) holds if and only if(19)-λ≤Δλ≤min⁡Aλα2-Aλα1μ1-μ2+v2-v1,1-λ,μ1-μ2>v1-v2;max⁡Aλα2-Aλα1μ1-μ2+v2-v1,-λ≤Δλ≤1-λ,μ1-μ2<v1-v2;-λ≤Δλ≤1-λ,μ1-μ2=v1-v2.Proof.
Since0≤λ+Δλ≤1, we have -λ≤Δλ≤1-λ. Considering of (20)Aλ+Δλα1=Aλα1+Δλμ1-v1,Aλ+Δλα2=Aλα2+Δλμ2-v2,the following inequalities are equivalent:(21)Aλ+Δλα1≤Aλ+Δλα2⟺Aλα2-Aλα1≥Δλμ1+v2-μ2-v1.We can consider (21) in three cases, shown as follows:
(a) Ifμ1-μ2=v1-v2, then inequalities Aλ+Δλ(α1)≤Aλ+Δλ(α2) and Aλ(α2)-Aλ(α1)≥0 are equivalent.
It means that (21) is true for all Δλ with -λ≤Δλ≤1-λ.
(b) Ifμ1-μ2>v1-v2, then(22)Aλ+Δλα1≤Aλ+Δλα2⟺Δλ≤Aλα2-Aλα1μ1-μ2+v2-v1.
Notice thatΔλ is subject to the constraint -λ≤Δλ≤1-λ; inequality Aλ+Δλ(α1)≤Aλ+Δλ(α2) is equivalent to (23)-λ≤Δλ≤min⁡Aλα2-Aλα1μ1-μ2+v2-v1,1-λ.
(c) Ifμ1-μ2<v1-v2, then(24)Aλ+Δλα1≤Aλ+Δλα2⟺Δλ≥Aλα2-Aλα1μ1-μ2+v2-v1.
Therefore, inequalityAλ+Δλ(α1)≤Aλ+Δλ(α2) is equivalent to(25)max⁡Aλα2-Aλα1μ1-μ2+v2-v1,-λ≤Δλ≤1-λ.In sum, Property 11 is proved.Remark 12.
As everyone knows, it is very difficult to determine the exact value of parameterλ. To deal with this problem, Table 1 is presented for estimating the appropriate value of parameter λ.Table 1
Preference attitude towards information and the corresponding value ofλ.
The assessor’s preference attitude towards information The value ofλ The negative information is absolutely preferred (NA) 0 The negative information is extremely preferred (NE) 0.1 The negative information is strongly preferred (NST) 0.2 The negative information is moderately preferred (NM) 0.3 The negative information is slightly preferred (NSL) 0.4 The assessor is neutral to the positive and negative information (NPN) 0.5 The positive information is slightly preferred (PSL) 0.6 The positive information is moderately preferred (PM) 0.7 The positive information is strongly preferred (PST) 0.8 The positive information is extremely preferred (PE) 0.9 The positive information is absolutely preferred (PA) 1It is clear that Table1 proposed some discrete values of parameter λ corresponding to the linear preference of assessor. If the preference of assessor is nonlinear and the value of λ is continuous, some nonlinear functions can be used here to measure the value of λ: that is, f:Ω→[0,1] such that f(NA)=0, f(NPN)=0.5 and f(PA)=1. Ω is the set of assessor’s preference attitudes towards information.


## 3.2. The Preference Attitudinal Score Function

Similar to the preference attitudinal accuracy function, the preference attitudinal score function can be defined as follows.Definition 13.
Letα=(μ,v) be an IFN. The preference attitudinal score function of α is defined by(26)Sλα=1areaDα∬Dαλx-1-λydxdy,where Dα is the feasible domain of intuitionistic fuzzy number α and area(Dα) is the area of feasible domain Dα.From (26), Sλ(α) represents the average value of score function over feasible domain Dα. Denote the volume of P2 by Vup and the volume of P3 by Vbelow, respectively. Then, we have ∬Dαλx-1-λydxdy=Vup-Vbelow. The graphs of cubic P2 and P3 are illustrated in Figure 3. Accordingly, Sλ(α) can be regarded as the average values of variable z over feasible domain Dα. Since(27)∬Dαλx-1-λydxdy=∫μ1-v∫v1-xλx-1-λydydx=161-μ-v2μ-2v-1+λ61-μ-v2μ+v+2,based on (6) and (26), the following expression of preference attitudinal score function is derived:(28)Sλα=μ-2v-13+λ3μ+v+2which can be equivalently written as(29)Sλα=λ2μ-v+13+1-λμ-2v-13.The preference attitudinal score function satisfies the following desirable properties.Figure 3
The graphs of cubicP2 and P3.Property 14.
Letα1=(μ1,v1) and α2=(μ2,v2) be two IFNs. If μ1≥μ2 and v1≤v2, then Sλ(α1)≥Sλ(α2) holds for all λ∈[0,1].Proof.
From (28), we have(30)Sλα1=μ1-2v1-13+λ3μ1+v1+2,Sλα2=μ2-2v2-13+λ3μ2+v2+2.It follows that (31)Sλα1-Sλα2=λ32μ1-μ2+v2-v1+1-λ3μ1-μ2+2v2-v1.Since μ1≥μ2, v1≤v2, and 0≤λ≤1, we have Sλ(α1)-Sλ(α2)≥0.
Thus, Property14 is proved.Property 15.
Letα=(μ,v) be an IFN. The preference attitudinal score function Sλ(α) is monotonic with respect to the parameter λ; that is, if λ1≥λ2, then Sλ1(α)≥Sλ2(α).Proof.
From (28), we have(32)Sλ1α=μ-2v-13+λ13μ+v+2,Sλ2α=μ-2v-13+λ23μ+v+2.Thus, we have Sλ1(α)-Sλ2(α)=(1/3)(λ1-λ2)(μ+v+2)≥0.
This completes the proof of Property15.Property 16.
Letα=(μ,v) be an IFN. For each parameter λ∈[0,1], we have -1≤μ-2v-1/3≤Sλ(α)≤(2μ-v+1)/3≤1.Proof.
From (29), we get S1(α)=(2μ-v+1)/3 and S0(α)=(μ-2v-1)/3. Since 0≤λ≤1, by Property 15, we have S0(α)≤Sλ(α)≤S1(α). Notice that μ-2v-1≥-3 and 2μ-v+1≤3; we have(33)Sλα≥S0α=μ-2v-13≥-1,Sλα≤S1α=2μ-v+13≤1.Therefore, Property 16 is proved.Because the addition operation of intuitionistic fuzzy number is nonlinear, the following inequalities of preference attitudinal score function are accordingly derived.Property 17.
Letα1=(μ1,v1) and α2=(μ2,v2) be two IFNs. Denoting λ0=(1+2v1+2v2-μ1μ2-2v1v2)/(μ1μ2-v1v2+v1+v2+2), we have λ0∈[0,1], and it follows that(1)
ifλ=λ0, then Sλ(α1⊕α2)=Sλ(α1)+Sλ(α2);(2)
if0≤λ<λ0, then Sλ(α1⊕α2)>Sλ(α1)+Sλ(α2);(3)
ifλ0<λ≤1, then Sλ(α1⊕α2)<Sλ(α1)+Sλ(α2).Proof.
Since0≤μ1μ2≤1 and -1≤(v2-1)(1-v1)≤0, we have(34)μ1μ2-v1v2+v1+v2+2=μ1μ2+v2-11-v1+3>0,1+2v1+2v2-μ1μ2-2v1v2=2v2-11-v1+3-μ1μ2≥0.Thus, we have λ0≥0. Considering 2μ1μ2-(v2-1)(1-v1)≥0 and(35)μ1μ2-v1v2+v1+v2+2=2v2-11-v1+3-μ1μ2+2μ1μ2-v2-11-v1,we have λ0≤1.
In sum, we haveλ0∈[0,1].
Denotingf(λ)=Sλ(α1)+Sλ(α2)-Sλ(α1⊕α2), from (28) and Definition 2, we have(36)fλ=13μ1μ2+2v1v2-2v1-2v2-1+λ3μ1μ2-v1v2+v1+v2+2.If λ=λ0=1+2v1+2v2-μ1μ2-2v1v2/μ1μ2-v1v2+v1+v2+2, then we have f(λ0)=0.
Therefore,Sλ0(α1⊕α2)=Sλ0(α1)+Sλ0(α2) holds.
Sinceμ1μ2-v1v2+v1+v2+2>0, function f(λ) is monotonically increasing with respect to the parameter λ. Thus, if 0≤λ<λ0, then f(λ)<f(λ0)=0. Based on the monotonicity of function f(λ), we have Sλ(α1⊕α2)>Sλ(α1)+Sλ(α2).
Similarly, ifλ0<λ≤1, then f(λ)>f(λ0)=0; we have Sλ(α1⊕α2)<Sλ(α1)+Sλ(α2).
This completes the proof of Property17.Property 18.
Letα=(μ,v) be an IFN. α-=(v,μ) is the complementary intuitionistic fuzzy number of α. For each attitudinal parameter λ∈[0,1], we have Sλ(α)=-S1-λ(α-).Proof.
From (29), it follows that(37)S1-λα-=1-λ2v-μ+13+λv-2μ-13=-λ2μ-v+13-1-λμ-2v-13=-Sλα.Thus, Property 18 is proved.Property 19 (sensitivity analysis).
Given two IFNsα1=(μ1,v1) and α2=(μ2,v2). Let λ be the attitudinal parameter under which it has satisfied the fact that Sλ(α1)≤Sλ(α2). Let Δλ be a perturbation of the attitudinal parameter λ with λ+Δλ∈[0,1]. Then, Sλ+Δλ(α1)≤Sλ+Δλ(α2) holds if and only if(38)-λ≤Δλ≤min⁡3Sλα2-3Sλα1μ1+v1-μ2-v2,1-λ,μ1+v1>μ2+v2;max⁡3Sλα2-3Sλα1μ1+v1-μ2-v2,-λ≤Δλ≤1-λ,μ1+v1<μ2+v2;-λ≤Δλ≤1-λ,μ1+v1=μ2+v2.Proof.
Since0≤λ+Δλ≤1, we have -λ≤Δλ≤1-λ. Considering(39)Sλ+Δλα1=Sλα1+Δλ3μ1+v1+2,Sλ+Δλα2=Sλα2+Δλ3μ2+v2+2,the following inequalities are equivalent:(40)Sλ+Δλα1≤Sλ+Δλα2⟺Sλα2-Sλα1≥Δλ3μ1+v1-μ2-v2.We can consider (40) in three cases, shown as follows:
(a) Ifμ1+v1=μ2+v2, then inequalities Sλ+Δλ(α1)≤Sλ+Δλ(α2) and Sλ(α2)-Sλ(α1)≥0 are equivalent.
It means that (40) is true for all Δλ with -λ≤Δλ≤1-λ.
(b) Ifμ1+v1>μ2+v2, then(41)Sλ+Δλα1≤Sλ+Δλα2⟺Δλ≤3Sλα2-3Sλα1μ1+v1-μ2-v2.Notice that Δλ is subject to the constraint -λ≤Δλ≤1-λ, inequality Sλ+Δλ(α1)≤Sλ+Δλ(α2) is equivalent to (42)-λ≤Δλ≤min⁡3Sλα2-3Sλα1μ1+v1-μ2-v2,1-λ.
(c) Ifμ1+v1<μ2+v2, then(43)Sλ+Δλα1≤Sλ+Δλα2⟺Δλ≥3Sλα2-3Sλα1μ1+v1-μ2-v2.Therefore, inequality Sλ+Δλ(α1)≤Sλ+Δλ(α2) is equivalent to(44)max⁡3Sλα2-3Sλα1μ1+v1-μ2-v2,-λ≤Δλ≤1-λ.In sum, Property 19 is proved.


## 3.3. The Relations among Score Functions and Accuracy Functions

In the following, the relations among score functions and accuracy functions are studied in detail.Property 20.
Letα=(μ,v) be an IFN. The score function SX(α), accuracy function AX(α), and preference attitudinal score function Sλ(α) are related as(45)Sλα=12SXα+2λ-16AXα+2λ-13.In particular, S0.5(α)=1/2SX(α) holds.Proof.
The proof is straightforward, so omitted.Property 21.
Letα=(μ,v) be an IFN. The score function SX(α), accuracy function AX(α), and preference attitudinal accuracy function Aλ(α) are related as(46)Aλα=16AXα+2λ-12SXα+13.In particular, A0.5(α)=(1/6)AX(α)+1/3 holds.Proof.
The proof is straightforward, so omitted.Property 22.
Letα=(μ,v) be an IFN. The preference attitudinal score function Sλ(α) and accuracy function Aλ(α) are related as(47)μ+v+2Aλα+3v-3μSλα=232μ-v+12v-μ+1.Proof.
From (8) and (28), we have (48)μ+v+2Aλα+3v-3μSXα=μ+v+21-μ+2v3+λμ-v+3v-3μμ-2v-13+λ3μ+v+2=13μ+v+21-μ+2v+133v-3μμ-2v-1=232μ-v+12v-μ+1.Thus, Property 22 is proved.Property 23.
Letα=(μ,v) be an IFN. Given two attitudinal parameters λ1 and λ2, such that λ1<λ2. The score function SX(α) and preference attitudinal accuracy function Aλ(α) are related as follows:(1)
IfSX(α)>0, then Aλ1(α)<Aλ2(α).(2)
IfSX(α)<0, then Aλ1(α)>Aλ2(α).(3)
IfSX(α)=0, then Aλ1(α)=Aλ2(α).Proof.
From (8), we have(49)Aλ1α=1-μ+2v3+λ1μ-v,Aλ2α=1-μ+2v3+λ2μ-v.Thus, we have Aλ1(α)-Aλ2(α)=(λ1-λ2)(μ-v). Since λ1<λ2 and SX(α)=μ-v, items (1), (2), and (3) are obviously hold.
Therefore, Property23 is proved.Property 24.
Letα=(μ,v) be an IFN. For all λ∈[0,1], Aλ(α)≥Sλ(α) holds. In particular, we have A1(α)=S1(α), A0(α)=-S0(α), Aλ((1,0))=Sλ((1,0)), and Aλ((0,1))=-Sλ((0,1)).Proof.
From (8) and (28), we have(50)Aλα-Sλα=132+4v-2μ1-λ.Since 2+4v-2μ≥0 and λ∈[0,1], we have Aλ(α)-Sλ(α)≥0.
Obviously,A1(α)=S1(α)=(2μ-v+1)/3, A0(α)=-S0(α)=(2v-μ+1)/3.
Whenα=(1,0), we have Aλ(α)=Sλ(α)=λ. When α=(0,1), we have Aλ(α)=-Sλ(α)=1-λ.
This completes the proof of Property24.


## 4. The Order Relation between Intuitionistic Fuzzy Numbers

In this section, a total order is established based on the proposed score and accuracy functions.Property 25.
Letα1=(μ1,v1) and α2=(μ2,v2) be two IFNs. For each λ∈[0,1], we can get the equivalence as follows:(51)α1=α2⟺Aλα1=Aλα2,Sλα1=Sλα2,πα1=πα2.Proof.
“⇒” If α1=α2, we have μ1=μ2 and v1=v2. Obviously, Aλ(α1)=Aλ(α2), Sλ(α1)=Sλ(α2), and π(α1)=π(α2) holds.
“⇐” In the following, the proof of sufficiency is considered in three cases.
(1) Whenλ∈(0,1), since Aλ(α1)=Aλ(α2) and Sλ(α1)=Sλ(α2), we have(52)1-μ1+2v13+λμ1-v1=1-μ2+2v23+λμ2-v2,μ1-2v1-13+λ3μ1+v1+2=μ2-2v2-13+λ3μ2+v2+2. Equation (52) can be further written as(53)3λ-1μ1-μ2=3λ-2v1-v2,(54)λ+1μ1-μ2=2-λv1-v2.From (54), we have(55)μ1-μ2=2-λλ+1v1-v2.Applying (55) to (53), we have(56)2-λ3λ-1λ+1v1-v2=3λ-2v1-v2which can be equivalently expressed by λ(1-λ)(v1-v2)=0.
Consideringλ∈(0,1), we get v1=v2. From (54) we have μ1=μ2.
(2) Whenλ=0, since A0(α1)=A0(α2), S0(α1)=S0(α2), and π(α1)=π(α2), we have μ1-μ2=2(v1-v2) and μ1+v1=μ2+v2. Thus, we get μ1=μ2 and v1=v2.
(3) Whenλ=1, since A1(α1)=A1(α2), S1(α1)=S1(α2), and π(α1)=π(α2), we have v1-v2=2(μ1-μ2) and μ1+v1=μ2+v2. Therefore, we also get μ1=μ2 and v1=v2.
In sum, for eachλ∈[0,1], we have μ1=μ2 and v1=v2. Namely, α1=α2 holds.The order relation on the set of intuitionistic fuzzy numbers can be defined as follows.Definition 26.
Letα1=(μ1,v1) and α2=(μ2,v2) be two IFNs. ∀λ∈[0,1],  α1 is said to be inferior to α2, denoted by α1≺λα2, if and only if one of the following conditions is true:(1)
Sλ(α1)<Sλ(α2).(2)
Sλ(α1)=Sλ(α2) and Aλ(α1)<Aλ(α2).(3)
Sλ(α1)=Sλ(α2), Aλ(α1)=Aλ(α2), and π(α1)>π(α2).Theorem 27.
The relation≺λ on the set of IFNs is a strict order, that is, the relation ≺λ satisfying the following:(1)
Irreflexivity:∀α∈INF, α≺λα does not hold.(2)
Asymmetry:∀α1,α2∈INF, if α1≺λα2, then α2≺λα1 does not hold.(3)
Transitivity:∀α1,α2,α3∈INF, if α1≺λα2 and α2≺λα3, then α1≺λα3.Proof.
According to Definition26, Properties (1) and (2) obviously hold here. In the following, we focus on the proof of Property (3).
Sinceα1≺λα2, from Definition 26, the proof of transitivity property is considered in three cases.(a)
Sλ(α1)<Sλ(α2). Because α2≺λα3, we have Sλ(α2)≤Sλ(α3). Thus, Sλ(α1)<Sλ(α3) holds. Namely, from Definition 26, we have α1≺λα3.(b)
Sλ(α1)=Sλ(α2) and Aλ(α1)<Aλ(α2). Because α2≺λα3, one has the following:
(b1)
IfSλ(α2)<Sλ(α3), then Sλ(α1)=Sλ(α2)<Sλ(α3). We conclude that α1≺λα3.(b2)
IfSλ(α2)=Sλ(α3) and Aλ(α2)<Aλ(α3), then Sλ(α1)=Sλ(α3) and Aλ(α1)<Aλ(α2)<Aλ(α3). We have α1≺λα3.(b3)
IfSλ(α2)=Sλ(α3), Aλ(α2)=Aλ(α3), and π(α2)>π(α3), then Sλ(α1)=Sλ(α3), Aλ(α1)<Aλ(α2)=Aλ(α3). We get α1≺λα3.(c)
Sλ(α1)=Sλ(α2), Aλ(α1)=Aλ(α2) and π(α1)>π(α2). Because α2≺λα3, then
(c1)
ifSλ(α2)<Sλ(α3), then Sλ(α1)=Sλ(α2)<Sλ(α3). We conclude that α1≺λα3.(c2)
ifSλ(α2)=Sλ(α3) and Aλ(α2)<Aλ(α3), then Sλ(α1)=Sλ(α3) and Aλ(α1)=Aλ(α2)<Aλ(α3). We have α1≺λα3.(c3)
ifSλ(α2)=Sλ(α3), Aλ(α2)=Aλ(α3), and π(α2)>π(α3), then Sλ(α1)=Sλ(α3), Aλ(α1)=Aλ(α3), and π(α1)>π(α2)>π(α3). We get α1≺λα3.
In sum, ifα1≺λα2 and α2≺λα3, then α1≺λα3 always hold here. Therefore, Theorem 27 is proved.Property 28.
Letα1=(μ1,v1) and α2=(μ2,v2) be two IFNs. ∀λ∈[0,1], if SX(α1)=SX(α2) and AX(α1)<AX(α2) (which imply that α1≺Xα2), then one has the following:(1)
Ifλ>1/2, then Sλ(α1)<Sλ(α2); we get α1≺λα2.(2)
Ifλ=1/2, then Sλ(α2)=Sλ(α1) and Aλ(α1)<Aλ(α2); we get α1≺λα2.(3)
Ifλ<1/2, then Sλ(α2)<Sλ(α1); we get α2≺λα1.Proof.
SinceSX(α1)=SX(α2) and AX(α1)<AX(α2), we have μ1-v1=μ2-v2 and μ1+v1<μ2+v2. Based on (28), we have(57)Sλα1-Sλα2=13μ1-2v1-μ2+2v2+λ3μ1+v1-μ2-v2.
(1) Ifλ>1/2, by considering μ1+v1-μ2-v2<0, we have(58)Sλα1-Sλα2<12μ1-μ2+v2-v1=0.Namely, Sλ(α1)<Sλ(α2) holds. Thus, we have α1≺λα2.
(2) Ifλ=1/2, then S0.5(α1)-S0.5(α2)=(1/2)(μ1-μ2+v2-v1)=0.
Namely,S0.5(α1)=S0.5(α2) holds.
Moreover, from (8), we get A0.5(α1)-A0.5(α2)=(1/6)(μ1+v1-μ2-v2)<0.
Therefore, we haveα1≺λα2.
(3) Ifλ<1/2, by considering μ1+v1-μ2-v2<0, we have(59)Sλα1-Sλα2>12μ1-μ2+v2-v1=0.Namely, Sλ(α2)<Sλ(α1) holds. Thus, we have α2≺λα1.
In sum, Property28 is proved.Remark 29.
Whenλ=0.5, from Properties 20 and 21, we have S0.5(α)=1/2SX(α) and A0.5(α)=1/6AX(α)+1/3. In this case, the order relation proposed in Definition 26 is equivalent to Xu’s method in Definition 4. That is to say, Xu’s method is a special case of the proposed method.


## 5. Numerical Examples and Comparison

In this section, the proposed ranking method is applied to select renewable energy. Moreover, some comparison examples are employed to show the advantage and validity of the proposed method.


### 5.1. Application of the Preference Attitudinal Score and Accuracy Functions in Renewable Energy Selection

Due to the increasing consumption of fossil fuels, the shortage of energy resources and the pollution caused by burning energy become more and more serious. It is necessary to relieve energy shortage and environmental pollution by developing renewable energy sources in the future. To find efficient alternative energy sources, many governments around the world focus on the renewable energy selection problems in the past several decades (adapted from [36]). Suppose there is a panel with four renewable energy resources D1 (wind energy), D2 (hydropower), D3 (geothermal), and D4 (solar power) to be chose. Government wants to select a renewable energy in the best option according to the following five attributes: g1 (environmental), g2 (sociopolitical), g3 (quality of the energy source), g4 (technological), and g5 (economic). The weighting vector of attributes is W=(w1,w2,w3,w4,w5)=(0.2,0.1,0.2,0.4,0.1). Since intuitionistic fuzzy set is practical and flexible in dealing with uncertainty and ambiguity, the four possible renewable energy resources D1, D2, D3, and D4 are evaluated using intuitionistic fuzzy numbers under five attributes. Then, the intuitionistic fuzzy decision matrix R=(rij)4×5 is constructed as follows: (60)R=g1g2g3g4g5D1D2D3D40.2,0.40.2,0.50.3,0.40.3,0.20.2,0.30.2,0.50.3,0.60.1,0.40.1,0.20.3,0.30.1,0.30.4,0.20.2,0.40.4,0.50.3,0.60.3,0.40.1,0.30.3,0.60.2,0.60.1,0.5.Based on (28), the preference attitudinal score matrix Sλ(R)=(Sλ(rij))4×5 corresponding to attitudinal parameter λ is calculated as(61)SλR=13·-1.6+2.6λ-1.4+2.5λ-1.3+2.3λ-1.6+2.6λ-1.5+2.4λ-1.8+2.7λ-1.8+2.7λ-1.3+2.6λ-1.6+2.9λ-1.9+2.9λ-1.5+2.7λ-1.9+2.9λ-1.5+2.4λ-1.9+2.9λ-2.0+2.8λ-1.1+2.5λ-1.7+2.5λ-1.0+2.6λ-1.5+2.7λ-1.9+2.6λ.In the following, we discuss the order relation of alternatives in three cases.(1) The collective preference attitudinal score values of alternativesD1, D2, D3, and D4 are derived as follows:(62)SλD1=∑j=15wjSλr1j=13-1.51+2.51λ;SλD2=∑j=15wjSλr2j=13-1.63+2.78λ;SλD3=∑j=15wjSλr3j=13-1.75+2.75λ;SλD4=∑j=15wjSλr4j=13-1.38+2.61λ.It follows that(63)SλD4-SλD1=130.13+0.10λ>0;SλD4-SλD2=130.25-0.17λ>0;SλD4-SλD3=130.37-0.14λ>0;SλD3-SλD2=13-0.12-0.03λ<0.Accordingly, for all λ∈[0,1], we have Sλ(D4)>Sλ(D2)>Sλ(D3) and Sλ(D4)>Sλ(D1). From Definition 26, we have D4>D2>D3 and D4>D1.(2) Moreover, we haveSλ(D1)-Sλ(D2)=(1/3)(0.12-0.27λ).Ifλ∈[0,4/9), then we have Sλ(D1)>Sλ(D2). Namely, D1>D2.Ifλ∈(4/9,1], then we have Sλ(D2)>Sλ(D1). Namely, D2>D1.Ifλ=4/9, then we have Sλ(D2)=Sλ(D1). In this case, we need to consider the preference attitudinal accuracy values as follows:(64)A4/9r11=0.44,A4/9r12=0.42,A4/9r13=0.39,A4/9r14=0.44,A4/9r15=0.41;A4/9r21=0.47,A4/9r22=0.47,A4/9r23=0.43,A4/9r24=0.49,A4/9r25=0.50.Therefore, the collective preference attitudinal accuracy values of alternatives D1 and D2 are derived as(65)A4/9D1=∑j=15wjA4/9r1j=0.425,A4/9D2=∑j=15wjA4/9r2j=0.473.It follows that A4/9(D2)>A4/9(D1). From Definition 26, we have D2>D1.(3) In addition,Sλ(D1)-Sλ(D3)=(1/3)(0.24-0.24λ). Obviously, ∀λ∈[0,1) we have Sλ(D1)>Sλ(D3): that is, D1>D3.We observe thatSλ(D1)=Sλ(D3) if and only if λ=1. In this case, we cannot decide the best one between D1 and D3 based on collective preference attitudinal score values. To select the best one, we consider the preference attitudinal accuracy values as follows:(66)A1r11=13,A1r12=1130,A1r13=13,A1r14=13,A1r15=310;A1r31=25,A1r32=13,A1r33=310,A1r34=13,A1r35=415.Thus, the collective preference attitudinal accuracy values of alternatives D1 and D3 are derived as A1D1=∑j=15wjA1r1j=1/3 and A1(D3)=∑j=15wjA1(r3j)=1/3.From Definition26, we need further to consider the hesitation degree as follows:(67)πr11=0.4,πr12=0.5,πr13=0.7,πr14=0.4,πr15=0.6;πr31=0.3,πr32=0.1,πr33=0.6,πr34=0.1,πr35=0.2.Accordingly, the collective hesitation degree of alternatives D1 and D3 is calculated as (68)πD1=∑j=15wjπr1j=0.49,πD3=∑j=15wjπr3j=0.25.It follows that π(D1)>π(D3). From Definition 26, we have D3>D1.In sum, we have the following order relations of alternatives:(a)
Ifλ∈[0,4/9), then D4>D1>D2>D3.(b)
Ifλ∈[4/9,1), then D4>D2>D1>D3.(c)
Ifλ=1, then D4>D2>D3>D1.


### 5.2. Comparison with Other Works

Suppose that we need to rank four intuitionistic fuzzy numbers as follows:(69)α1=0.4,0.2,α2=0.5,0.3,α3=0.3,0.4,α4=0.4,0.5.By (28), the preference attitudinal score values of α1, α2, α3, and α4 are calculated as(70)Sλα1=13-1+2.6λ,Sλα2=13-1.1+2.8λ,Sλα3=13-1.5+2.7λ,Sλα4=13-1.6+2.9λ.Since λ∈[0,1], it follows that(71)Sλα3-Sλα1=-0.5+0.1λ<0,Sλα4-Sλα1=-0.6+0.3λ<0,Sλα3-Sλα2=-0.4-0.1λ<0,Sλα4-Sλα2=-0.5+0.1λ<0.Clearly, we have Sλ(α3)<Sλ(α1), Sλ(α4)<Sλ(α1), Sλ(α3)<Sλ(α2), and Sλ(α4)<Sλ(α2). Based on Definition 26, we have α3≺λα1, α4≺λα1, α3≺λα2, and α4≺λα2.Moreover, we haveSλ(α2)-Sλ(α1)=-0.1+0.2λ and Sλ(α4)-Sλ(α3)=-0.1+0.2λ. If λ∈[0,0.5), then Sλ(α2)<Sλ(α1) and Sλ(α4)<Sλ(α3); we have α2≺λα1 and α4≺λα3. If λ∈(0.5,1], then Sλ(α2)>Sλ(α1) and Sλ(α4)>Sλ(α3); we have α1≺λα2 and α3≺λα4. We observe that S0.5(α1)=S0.5(α2) and S0.5(α3)=S0.5(α4). For ranking these intuitionistic fuzzy numbers, we consider the preference attitudinal accuracy values of α1, α2, α3, and α4 as follows:(72)A0.5α1=0.43,A0.5α2=0.47,A0.5α3=0.45,A0.5α4=0.48.From Definition26, we have α1≺0.5α2 and α3≺0.5α4.In sum, we have the following ranking order in two cases:(a)
Ifλ∈[0,0.5), then α4≺α3≺α2≺α1.(b)
Ifλ∈[0.5,1], then α3≺α4≺α1≺α2.For comparison, we use Xu’s method [21], Liu and Wang’s method [24], Wang et al.’s method [22], Lin et al.’s method [35], and Ye’s methods [9] to rank the four intuitionistic fuzzy numbers. The comparative results are listed in Table 2.Table 2
Comparative results of different methods.
Method Attitudinal parameter Ranking Xu [21] Null α 3 ≺ α 4 ≺ α 1 ≺ α 2 Liu and Wang [24] Null α 3 ≺ α 4 ≺ α 1 ≺ α 2 Wang et al. [22] Null α3≺α4≺α1≺α2 Lin et al. [35] Null α1≺α3≺α2≺α4 Ye [23] Null α4≺α3≺α2≺α1 Ye [9] Null α3≺α1≺α4≺α2 Proposed λ ∈ [ 0,0.5 ) α4≺α3≺α2≺α1 Proposed λ ∈ [ 0.5,1 ] α3≺α4≺α1≺α2From Table2, we observe that our result is similar to those of Ye’s method [23] and differs from those of other methods when λ∈[0,0.5). On the other hand, when λ∈[0.5,1], our result is similar to those of Xu’s method [21], Liu and Wang’s method [24], and Wang et al.’s method [22] and differs from those of other methods. We need to notice that the membership degree of α1 is equal to α4, and the nonmembership degree of α1 is smaller than α4. Clearly, the ranking result of α1 and α4 should be α4≺α1. However, the result is α1≺α4 in Lin et al. [35] and Ye [9], which is illogical and unreasonable. A valid method should be chosen in accordance with the preference attitude of decision maker. The proposed method considers the influence of preference attitude on ranking results comprehensively and makes the ranking results more flexible than other methods.For ranking the intuitionistic fuzzy numbers in a proper way, the decision maker’s preference has been considered in score function. Chen [32] proposed a novel score function by using an attitudinal parameter λ for ranking intuitionistic fuzzy numbers. Given an intuitionistic fuzzy number α=(μ,v), Chen’s score function SλC(α) is shown as follows:(73)SλCα=λμ+1-λ1-v,λ∈0,1.0.5<λ≤1 and 0≤λ<0.5 denote the decision maker’s attitude of optimism and pessimism, respectively. When λ=0.5, the preference attitude of decision maker is indifferent.By (73), the score values of α1, α2, α3, and α4 are calculated as(74)SλCα1=0.8-0.4λ,SλCα2=0.7-0.2λ,SλCα3=0.6-0.3λ,SλCα4=0.5-0.1λ.Since λ∈[0,1], we have SλC(α3)<SλC(α1), SλC(α3)<SλC(α2), and SλC(α4)<SλC(α2). However, we notice that when λ=1, the equality SλC(α1)=SλC(α4) holds. That is to say, the order relation between intuitionistic fuzzy numbers α1 and α4 cannot be determined by Chen’s score function. The dynamic score values of Chen’s method are shown in Figure 4. Chen’s method just considers the extreme points of feasible domain Dα, but the shape of Dα and the other possible values in Dα are ignored. Thus, it may lose some valuable information, which can be useful in determining the order relation of intuitionistic fuzzy numbers. Therefore, Chen’s method for ranking intuitionistic fuzzy numbers is illogical and invalid in some special cases. In the previous calculations, by using the proposed score function, we obtain the ranking result α4≺λα1. The proposed method satisfies some desirable properties and consider all the possible values in Dα; the ranking of intuitionistic fuzzy numbers can be obtained effectively.Figure 4
The dynamic score values of Chen’s method.


## 5.1. Application of the Preference Attitudinal Score and Accuracy Functions in Renewable Energy Selection

Due to the increasing consumption of fossil fuels, the shortage of energy resources and the pollution caused by burning energy become more and more serious. It is necessary to relieve energy shortage and environmental pollution by developing renewable energy sources in the future. To find efficient alternative energy sources, many governments around the world focus on the renewable energy selection problems in the past several decades (adapted from [36]). Suppose there is a panel with four renewable energy resources D1 (wind energy), D2 (hydropower), D3 (geothermal), and D4 (solar power) to be chose. Government wants to select a renewable energy in the best option according to the following five attributes: g1 (environmental), g2 (sociopolitical), g3 (quality of the energy source), g4 (technological), and g5 (economic). The weighting vector of attributes is W=(w1,w2,w3,w4,w5)=(0.2,0.1,0.2,0.4,0.1). Since intuitionistic fuzzy set is practical and flexible in dealing with uncertainty and ambiguity, the four possible renewable energy resources D1, D2, D3, and D4 are evaluated using intuitionistic fuzzy numbers under five attributes. Then, the intuitionistic fuzzy decision matrix R=(rij)4×5 is constructed as follows: (60)R=g1g2g3g4g5D1D2D3D40.2,0.40.2,0.50.3,0.40.3,0.20.2,0.30.2,0.50.3,0.60.1,0.40.1,0.20.3,0.30.1,0.30.4,0.20.2,0.40.4,0.50.3,0.60.3,0.40.1,0.30.3,0.60.2,0.60.1,0.5.Based on (28), the preference attitudinal score matrix Sλ(R)=(Sλ(rij))4×5 corresponding to attitudinal parameter λ is calculated as(61)SλR=13·-1.6+2.6λ-1.4+2.5λ-1.3+2.3λ-1.6+2.6λ-1.5+2.4λ-1.8+2.7λ-1.8+2.7λ-1.3+2.6λ-1.6+2.9λ-1.9+2.9λ-1.5+2.7λ-1.9+2.9λ-1.5+2.4λ-1.9+2.9λ-2.0+2.8λ-1.1+2.5λ-1.7+2.5λ-1.0+2.6λ-1.5+2.7λ-1.9+2.6λ.In the following, we discuss the order relation of alternatives in three cases.(1) The collective preference attitudinal score values of alternativesD1, D2, D3, and D4 are derived as follows:(62)SλD1=∑j=15wjSλr1j=13-1.51+2.51λ;SλD2=∑j=15wjSλr2j=13-1.63+2.78λ;SλD3=∑j=15wjSλr3j=13-1.75+2.75λ;SλD4=∑j=15wjSλr4j=13-1.38+2.61λ.It follows that(63)SλD4-SλD1=130.13+0.10λ>0;SλD4-SλD2=130.25-0.17λ>0;SλD4-SλD3=130.37-0.14λ>0;SλD3-SλD2=13-0.12-0.03λ<0.Accordingly, for all λ∈[0,1], we have Sλ(D4)>Sλ(D2)>Sλ(D3) and Sλ(D4)>Sλ(D1). From Definition 26, we have D4>D2>D3 and D4>D1.(2) Moreover, we haveSλ(D1)-Sλ(D2)=(1/3)(0.12-0.27λ).Ifλ∈[0,4/9), then we have Sλ(D1)>Sλ(D2). Namely, D1>D2.Ifλ∈(4/9,1], then we have Sλ(D2)>Sλ(D1). Namely, D2>D1.Ifλ=4/9, then we have Sλ(D2)=Sλ(D1). In this case, we need to consider the preference attitudinal accuracy values as follows:(64)A4/9r11=0.44,A4/9r12=0.42,A4/9r13=0.39,A4/9r14=0.44,A4/9r15=0.41;A4/9r21=0.47,A4/9r22=0.47,A4/9r23=0.43,A4/9r24=0.49,A4/9r25=0.50.Therefore, the collective preference attitudinal accuracy values of alternatives D1 and D2 are derived as(65)A4/9D1=∑j=15wjA4/9r1j=0.425,A4/9D2=∑j=15wjA4/9r2j=0.473.It follows that A4/9(D2)>A4/9(D1). From Definition 26, we have D2>D1.(3) In addition,Sλ(D1)-Sλ(D3)=(1/3)(0.24-0.24λ). Obviously, ∀λ∈[0,1) we have Sλ(D1)>Sλ(D3): that is, D1>D3.We observe thatSλ(D1)=Sλ(D3) if and only if λ=1. In this case, we cannot decide the best one between D1 and D3 based on collective preference attitudinal score values. To select the best one, we consider the preference attitudinal accuracy values as follows:(66)A1r11=13,A1r12=1130,A1r13=13,A1r14=13,A1r15=310;A1r31=25,A1r32=13,A1r33=310,A1r34=13,A1r35=415.Thus, the collective preference attitudinal accuracy values of alternatives D1 and D3 are derived as A1D1=∑j=15wjA1r1j=1/3 and A1(D3)=∑j=15wjA1(r3j)=1/3.From Definition26, we need further to consider the hesitation degree as follows:(67)πr11=0.4,πr12=0.5,πr13=0.7,πr14=0.4,πr15=0.6;πr31=0.3,πr32=0.1,πr33=0.6,πr34=0.1,πr35=0.2.Accordingly, the collective hesitation degree of alternatives D1 and D3 is calculated as (68)πD1=∑j=15wjπr1j=0.49,πD3=∑j=15wjπr3j=0.25.It follows that π(D1)>π(D3). From Definition 26, we have D3>D1.In sum, we have the following order relations of alternatives:(a)
Ifλ∈[0,4/9), then D4>D1>D2>D3.(b)
Ifλ∈[4/9,1), then D4>D2>D1>D3.(c)
Ifλ=1, then D4>D2>D3>D1.


## 5.2. Comparison with Other Works

Suppose that we need to rank four intuitionistic fuzzy numbers as follows:(69)α1=0.4,0.2,α2=0.5,0.3,α3=0.3,0.4,α4=0.4,0.5.By (28), the preference attitudinal score values of α1, α2, α3, and α4 are calculated as(70)Sλα1=13-1+2.6λ,Sλα2=13-1.1+2.8λ,Sλα3=13-1.5+2.7λ,Sλα4=13-1.6+2.9λ.Since λ∈[0,1], it follows that(71)Sλα3-Sλα1=-0.5+0.1λ<0,Sλα4-Sλα1=-0.6+0.3λ<0,Sλα3-Sλα2=-0.4-0.1λ<0,Sλα4-Sλα2=-0.5+0.1λ<0.Clearly, we have Sλ(α3)<Sλ(α1), Sλ(α4)<Sλ(α1), Sλ(α3)<Sλ(α2), and Sλ(α4)<Sλ(α2). Based on Definition 26, we have α3≺λα1, α4≺λα1, α3≺λα2, and α4≺λα2.Moreover, we haveSλ(α2)-Sλ(α1)=-0.1+0.2λ and Sλ(α4)-Sλ(α3)=-0.1+0.2λ. If λ∈[0,0.5), then Sλ(α2)<Sλ(α1) and Sλ(α4)<Sλ(α3); we have α2≺λα1 and α4≺λα3. If λ∈(0.5,1], then Sλ(α2)>Sλ(α1) and Sλ(α4)>Sλ(α3); we have α1≺λα2 and α3≺λα4. We observe that S0.5(α1)=S0.5(α2) and S0.5(α3)=S0.5(α4). For ranking these intuitionistic fuzzy numbers, we consider the preference attitudinal accuracy values of α1, α2, α3, and α4 as follows:(72)A0.5α1=0.43,A0.5α2=0.47,A0.5α3=0.45,A0.5α4=0.48.From Definition26, we have α1≺0.5α2 and α3≺0.5α4.In sum, we have the following ranking order in two cases:(a)
Ifλ∈[0,0.5), then α4≺α3≺α2≺α1.(b)
Ifλ∈[0.5,1], then α3≺α4≺α1≺α2.For comparison, we use Xu’s method [21], Liu and Wang’s method [24], Wang et al.’s method [22], Lin et al.’s method [35], and Ye’s methods [9] to rank the four intuitionistic fuzzy numbers. The comparative results are listed in Table 2.Table 2
Comparative results of different methods.
Method Attitudinal parameter Ranking Xu [21] Null α 3 ≺ α 4 ≺ α 1 ≺ α 2 Liu and Wang [24] Null α 3 ≺ α 4 ≺ α 1 ≺ α 2 Wang et al. [22] Null α3≺α4≺α1≺α2 Lin et al. [35] Null α1≺α3≺α2≺α4 Ye [23] Null α4≺α3≺α2≺α1 Ye [9] Null α3≺α1≺α4≺α2 Proposed λ ∈ [ 0,0.5 ) α4≺α3≺α2≺α1 Proposed λ ∈ [ 0.5,1 ] α3≺α4≺α1≺α2From Table2, we observe that our result is similar to those of Ye’s method [23] and differs from those of other methods when λ∈[0,0.5). On the other hand, when λ∈[0.5,1], our result is similar to those of Xu’s method [21], Liu and Wang’s method [24], and Wang et al.’s method [22] and differs from those of other methods. We need to notice that the membership degree of α1 is equal to α4, and the nonmembership degree of α1 is smaller than α4. Clearly, the ranking result of α1 and α4 should be α4≺α1. However, the result is α1≺α4 in Lin et al. [35] and Ye [9], which is illogical and unreasonable. A valid method should be chosen in accordance with the preference attitude of decision maker. The proposed method considers the influence of preference attitude on ranking results comprehensively and makes the ranking results more flexible than other methods.For ranking the intuitionistic fuzzy numbers in a proper way, the decision maker’s preference has been considered in score function. Chen [32] proposed a novel score function by using an attitudinal parameter λ for ranking intuitionistic fuzzy numbers. Given an intuitionistic fuzzy number α=(μ,v), Chen’s score function SλC(α) is shown as follows:(73)SλCα=λμ+1-λ1-v,λ∈0,1.0.5<λ≤1 and 0≤λ<0.5 denote the decision maker’s attitude of optimism and pessimism, respectively. When λ=0.5, the preference attitude of decision maker is indifferent.By (73), the score values of α1, α2, α3, and α4 are calculated as(74)SλCα1=0.8-0.4λ,SλCα2=0.7-0.2λ,SλCα3=0.6-0.3λ,SλCα4=0.5-0.1λ.Since λ∈[0,1], we have SλC(α3)<SλC(α1), SλC(α3)<SλC(α2), and SλC(α4)<SλC(α2). However, we notice that when λ=1, the equality SλC(α1)=SλC(α4) holds. That is to say, the order relation between intuitionistic fuzzy numbers α1 and α4 cannot be determined by Chen’s score function. The dynamic score values of Chen’s method are shown in Figure 4. Chen’s method just considers the extreme points of feasible domain Dα, but the shape of Dα and the other possible values in Dα are ignored. Thus, it may lose some valuable information, which can be useful in determining the order relation of intuitionistic fuzzy numbers. Therefore, Chen’s method for ranking intuitionistic fuzzy numbers is illogical and invalid in some special cases. In the previous calculations, by using the proposed score function, we obtain the ranking result α4≺λα1. The proposed method satisfies some desirable properties and consider all the possible values in Dα; the ranking of intuitionistic fuzzy numbers can be obtained effectively.Figure 4
The dynamic score values of Chen’s method.


## 6. Conclusion and Further Study

Ranking the intuitionistic fuzzy numbers plays an important role in practical applications under intuitionistic fuzzy settings. This paper proposes a novel ranking method for intuitionistic fuzzy numbers based on the preference attitudinal accuracy and score functions. Meanwhile, some desirable properties of the proposed accuracy and score functions are studied. According to the preference attitude of decision maker, a total order on the set of intuitionistic fuzzy numbers is established. The main advantages of this paper are shown as follows:(1)
The proposed accuracy and score functions take the preference attitude of decision maker into account and make the ranking results more flexible than other methods.(2)
The preference attitudinal accuracy and score functions consider the shape ofDα and all possible values in Dα, hence avoiding the loss of information in ranking process.(3)
The proposed order relation satisfies irreflexivity, asymmetry, and transitivity. It means that a total order on the set of intuitionistic fuzzy numbers is established.(4)
Whenλ=0.5, the proposed order relation is equivalent to Xu’s method in Definition 4. That is to say, Xu’s method is a special case of the proposed method.Moreover, we apply the preference attitudinal accuracy and score functions to deal with the renewable energy selection problem under intuitionistic fuzzy environment. In the following research, we intend to make further extensions of the proposed method for ranking other types of fuzzy information. We will consider several applications of the proposed ranking method in actual fields such as pattern recognition, medical diagnosis, and economic forecast.



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*Source: 6251384-2018-02-22.xml*