# International Journal of Computational Intelligence Systems

Volume 12, Issue 2, 2019, Pages 955 - 969

# A Novel Distance Measure for Pythagorean Fuzzy Sets and its Applications to the Technique for Order Preference by Similarity to Ideal Solutions

Authors
Fang Zhou1, Ting-Yu Chen2, 3, 4, *
1Graduate Institute of Business and Management, Chang Gung University, Department of Economic and Management, Suzhou Vocational Institute of Industrial Technology, No.1, Zhineng Avenue, Suzhou City, 215104, China
2Graduate Institute of Business and Management, Chang Gung University, No. 259, Wenhua 1st Rd., Guishan District, Taoyuan City 33302, Taiwan
3Department of Industrial and Business Management, Chang Gung University, No. 259, Wenhua 1st Rd., Guishan District, Taoyuan City 33302, Taiwan
4Department of Nursing, Linkou Chang Gung Memorial Hospital, No. 259, Wenhua 1st Rd., Guishan District, Taoyuan City 33302, Taiwan
*Corresponding author. Email: tychen@mail.cgu.edu.tw
Corresponding Author
Ting-Yu Chen
Received 25 May 2019, Accepted 19 August 2019, Available Online 9 September 2019.
DOI
10.2991/ijcis.d.190820.001How to use a DOI?
Keywords
Pythagorean fuzzy set; Multicriteria decision-making (MCDM); Distance measure; Technique for order preference by similarity to ideal solutions (TOPSIS)
Abstract

Ever since the introduction of Pythagorean fuzzy (PF) sets, many scholars have focused on solving multicriteria decision-making (MCDM) problems with PF information. The technique for order preference by similarity to ideal solutions (TOPSIS) is a well-known and effective method for MCDM problems. The objective of this study is to extend the TOPSIS to tackle MCDM problems under the PF environment. In this study, we develop a novel distance measure that considers the length, the angle, and the greater space, which reflect the properties of PF sets. Then, we apply the proposed distance measure in PF-TOPSIS to calculate the distances from the PF positive ideal solution and the PF negative ideal solution. Finally, we take the evaluation of emerging technology commercialization as an MCDM problem to illustrate the proposed approaches, and we then compare these approaches to demonstrate the scalar type PF-TOPSIS is the most feasible and effective approach in practice.

Open Access

## 1. INTRODUCTION

Multicriteria decision-making (MCDM) refers to the identification of the optimal alternative or the determination of the ranking order of all alternatives. Due to the inherent vagueness of decision makers' preferences in practice, Zadeh [1] introduced the concept of fuzzy sets for dealing with MCDM problems, which use the membership degree to denote the alternative satisfies the criterion. However, in practice, a decision maker usually points out the degree to which the alternative fails to satisfy the criterion. Atanassov [2] initially developed the intuitionistic fuzzy (IF) sets, which utilize the membership degree and the nonmembership degree and allow the sum of the two degrees to be equal to or less than one. However, we may encounter the problem that the sum of the two degrees is greater than one in a real decision process. To overcome this restriction, Yager [3,4] introduced the Pythagorean fuzzy (PF) sets, with the condition that the sum of the squares of the membership degree and the nonmembership degree is equal to or less than one. PF sets outperform IF sets and can deal with more ambiguous and uncertain information in MCDM practice.

Ever since the introduction of PF sets, many scholars have focused on solving MCDM problems under the PF environment. Zhang and Xu [5], Zeng et al. [6], Gary [7], Biswas and Sarkar [8], Yu et al. [9], and Akram et al. [10] extended the technique for order preference by similarity to ideal solutions (TOPSIS) method with PF information to solve MCDM problems. Ren et al. [11] and Biswas and Sarkar [12] proposed the PF-TODIM (acronym in Portuguese for interactive MCDM) approach for MCDM problems. Zhang [13] developed the Pythagorean fuzzy qualitative flexible multiple criteria method (PF-QUALIFLEX) for dealing with hierarchical MCDM problems. Chen [14], Gul et al. [15], and Liang et al. [16] proposed the PF-Vise Kriterijumska Optimizacija I Kompromisno Resenje (VIKOR) method for dealing with MCDA problems. Wang and Chen [17] developed an effective assignment-based method using correlation-based precedence indices for MCDM problems within the PF uncertain environment. Haktanir and Kahraman [18] proposed the interval-valued Pythagorean fuzzy quality function development (IVPF-QFD) method for handling solar photovoltaic technology problems. Liu et al. [19] integrated QFD and QUALIFLEX to solve the robot selection problem under the PF environment. Yang et al. [20] developed an IVPF Frank power weighted average operator-based technique to deal with multiple attribute group decision-making problems.

TOPSIS [21] is a well-known and widely accepted method for MCDM problems, which follows the strategy of selecting the solution that has the shortest distance from the positive ideal solution (PIS) and the farthest distance from the negative ideal solution (NIS). Because TOPSIS is a simple yet effective approach for MCDM problems, researchers have applied TOPSIS for MCDM problems within the PF environment. Zhang and Xu [5] extended TOPSIS with PF information based on score function and a score-based ranking. Zeng et al. [6] introduced the PF ordered weighted averaging weighted average distance into TOPSIS by integrating the subjective information and the attitudinal characteristics of decision makers' preferences. Gary [7] introduced an improved score function into TOPSIS with IVPF information. Wan et al. [22] developed a novel ranking method according to the arc-length-based relative closeness (RC) degree, which was inspired by the strategy of TOPSIS. Onar et al. [8] used PF-TOPSIS to evaluate cloud service providers. Yu et al. [9] extended the TOPSIS method via the integration of the distance and the similarity to evaluate suppliers' performance using IVPF information. Akram et al. [10] extended TOPSIS to address group decision-making problems in the PF scenario.

The distance measure is essential in PF-TOPSIS due to the fundamental TOPSIS strategy of considering the distances to both the Pythagorean fuzzy positive ideal solution (PFPIS) and the Pythagorean fuzzy negative ideal solution (PFNIS). Distance measure between fuzzy sets is an important means to quantify the degree of difference in fuzzy mathematics. A large number of scholars have presented various distance measurement formulas from different viewpoints [2327]. Li et al. [26,27] proposed the divergence measure based on dissimilarity function and fuzzy equivalence. Szmidt and Kacprzyk [28] considered three parameters that reflects the properties of IF sets and extended distance measures to IF environments. Grzegorzewsk [29] developed the distance measure for IF sets based on the Hausdorff metric. Montes et al. [30] defined the divergence to measure the distance between two IF sets. He et al. [31] defined the IF dissimilarity function based on quaternary functions.

Ever since the appearance of the PF sets, some researchers have extended the distance measure of IF sets to PF sets. Zhang and Xu [5] considered three parameters of PFSs, namely, the membership degree, the nonmembership degree, and the hesitation degree, while ignoring the direction of commitment, the strength of commitment, and the radian. Li and Zeng [32] considered four basic parameters (the membership degree, the nonmembership degree, the strength of commitment, and the direction of commitment) of PF sets in the distance measure equation. Zeng et al. [33] incorporated a parameter, namely, the hesitation degree, into Li and Zeng's method [32]; both approaches ignore the angle and the procedure is directly extended from the IF sets but does not consider the greater space of the PF sets. Wang et al. [34] introduced a distance measure that is based on the length distance and the angular distance in a bidirectional projection model under the PF environment. Yu et al. [35] proposed a new distance formula that employs induced ordered weighted averaging (IOWA) with PF information; however, this basic distance formula considers only three parameters, which are the same as the parameters that are considered in the method of Zhang and Xu [5]. Peng and Li [36] proposed a new distance measure for IVPF sets that has two parameters (the membership degree and the nonmembership degree) for resolving the counterintuitive situation.

Although scholars have focused on distance measures for PF sets, two main difficulties remain: failure to consider the unique parameters and maintenance of the greater space of PF sets. Zhang and Xu [5] considered the greater space of PF sets utilizing the sum of squared deviations of the membership degree, nonmembership degree, and hesitation degree, but they ignored the strength of commitment, the direction of commitment, and the radian. This limitation resulted in failure to satisfy the fundamental properties of PF sets, which obtained unreasonable results. Li and Zeng [32] and Zeng et al. [33] considered the direction of commitment and the strength of commitment, but the procedure of the distance formulas simply utilized the sum of deviations of the corresponding parameters. This difficulty resulted in failure to ensure the greater space because the maximum distance of PF sets couldn't arrive to one. Hence, the motivation of this paper is to propose a novel distance measure to overcome the difficulties of the existing distance measures for PF sets, ensure the greater space of the PF sets, consider all the related parameters that reflects the unique properties of PF sets, and satisfy the basic properties of distance measure.

The purpose of this paper is to propose a novel distance measurement method that considers the length distance, the angular distance, and the greater space, which is applied in PF-TOPSIS for MCDM problems. We first define a novel distance formula that resolves these limitations by considering the membership degree, the nonmembership degree, and the strength of commitment as the parameters for the length distance measure; by considering the direction of commitment and the radian as the parameters for the angular distance measure; and by using the square deviations of the membership degree, the nonmembership degree, and the strength of commitment, the deviation of the direction of commitment and the sin value of the angle to maintain the greater space of the PF sets. Next, we propose the PF-TOPSIS approach for MCDM problems, in which we utilize four approaches to identify the PFPIS and the PFNIS: the first is the classical type that is utilized by the union and intersection operators; the second is the fixed type that is utilized by the fixed extremum; the third is the scored type that is utilized by the score function; the fourth is the scalar type that is utilized by scalar function. Then, we use the proposed distance measurement method to calculate the distances of each alternative from the PFPIS and the PFNIS to fully reflect the properties of the PF sets and we employ both the relative closeness index and the revised closeness index to obtain the ranking order and to identify the optimal solution. Moreover, as a practical MCDM problem, this paper considers the evaluation of emerging technology commercialization to evaluate the proposed approaches and to conduct a comparative analysis among these approaches, the results of which demonstrate that the scalar type approach is the most feasible, effective, and credible approach for MCDM problems in practice. Finally, this paper proposes directions for future research.

The main contributions of this study relative to the existing distance measurement method and the existing PF-TOPSIS method are summarized as follows:

1. A novel distance measurement method for PF sets is proposed, the parameters of which reflect the length distance and angular distance based on the characteristics of PF sets and the procedure of which ensures that the greater space of PF sets is maintained. Therefore, the proposed distance measurement method measures the distance between two PF sets more accurately.

2. A scalar function is used to determine the PFPIS and the PFNIS in the PF-TOPSIS approach. Due to the unique properties of PF sets, the scalar function compares the magnitudes more accurately according to Zeng et al. [33]. However, the scalar function has yet to be applied in PF-TOPSIS. In this paper, we utilize the scalar function as an approach for comparing the magnitudes of PF sets for the determination of the PFPIS and the PFNIS, which may yield more accurate results.

3. The proposed distance measurement method is applied in PF-TOPSIS to calculate the distances from the PFPIS and the PFNIS. Due to the effectiveness of the proposed distance measurement method, which is based on the properties of the PF sets, it can calculate the distances from the PFPIS and the PFNIS more precisely.

4. Substantial improvement has been realized by the PF-TOPSIS methodology in solving MCDM problems. This paper demonstrates the scalar type approach to determine the PFPIS and the PFNIS and utilizes the novel distance measurement method to calculate the distances from the PFPIS and the PFNIS, both of which reflect the properties and characteristics of PF sets, and to make the scalar type approach more effective and practicable for solving MCDM problems compared to the classical type, fixed type, and the scored type approaches.

The remainder of this paper is organized as follows: In Section 2, we briefly recall the basic concepts of PF sets, PF numbers, operations, and magnitude comparison methods. In Section 3, we compare various existing distance measurement methods and propose a novel distance measurement method that considers five parameters of PF sets. In Section 4, we propose PF-TOPSIS for solving MCDM problems with PF information. In Section 5, we consider the “evaluation of emerging technology commercialization” as an illustrative example on which to demonstrate the proposed method and conduct a comparative analysis. In Section 6, we present the conclusions of this study and discuss future work.

## 2. PRELIMINARIES

In this section, we recall basic concepts, properties, operations, and magnitude comparison methods of PF sets and PF numbers.

### Definition 1.

[5] Let X be a set of finite universal sets. A PF set is an object that has the following form:

p=<x,Pμpx,νpx|xX>,
where the function μpx:X0,1 denotes the membership degree of element xX to set p and the function νpx:X0,1 denotes the nonmembership degree of element xX to set p. For any PF set p and xX,
0μpx2+νpx21.

The function hpx denotes the hesitation degree of element xX to set p. For any PF set p and xX,

hpx=1μpx2νpx2.

For convenience, Zhang and Xu [5] defined the Pythagorean fuzzy number (PFN) as follows: A PF set Pμpx,νpx can be expressed as a PFN Pμp,νp, where μp, νp0,1, hp=(1(μp)2(νp)2)1/2, and 0μp2+νp21.

Yager [3] proposed another PFN formulation, namely, p=Prp,dp, where rp defines the strength of commitment and rp0,1. The larger the value of rp, the stronger the commitment and the lower the uncertainty of the commitment. dp denotes the direction of commitment. rp and dp are associated with a pair of membership grades, namely, μp and νp, which correspond to the support for and against the membership; μp=rpcosθp and νp=rpsinθp, where θp denotes the radian, which is in the range of 0,π/2 and defined as θp=1dpπ/2. Alternatively, Prp, dp can be expressed in polar coordinates rp,θp. The relationship between dp and θp can be expressed as dp=12θp/π.

The parameters of PFNs include μp, νp, hp, rp, dp, and θp, which are based on the concepts and properties of PFNs. From μp, νp0,1, it follows that μp2μp, νp2νp; hence, a PFN has a larger membership grade range compared to an IF number.

### Example 1.

Let p=P0.8,0.3 be a PFN, then six parameters of the PFN p are calculated in accordance with aforementioned definition as below, and the graphical representation is presented in Fig. 1. Here, μp=0.8, νp=0.3, hp=10.820.321/2=0.5196, rp=0.82+0.321/2=0.8544, θp=arctan0.3/0.8=0.3588, and dp=12×0.3588/π=0.7716.

### Definition 2.

[3] Let p1=Pμp1,νp1, p2=Pμp2,νp2, and p=Pμp,νp be three PFNs. The basic operations (the union operator, the intersection operator, and the complement operator) on PFNs can be expressed as follows:

1. p1p2=Pmaxμp1,μp2,minνp1,νp2.

2. p1p2=Pminμp1,μp2,maxνp1,νp2.

3. pc=Pνp,μp.

Zeng et al. [33] further proposed the following operation with respect to dp between two PFNs:

4. |dp1dp2       c|=|dp1       cdp2|.

### Example 2.

Let p1=P0.8,0.3,p2=P0.7,0.2 be two PFNs. According to Definition 2, we have:

p1p2=Pmax0.8,0.7,min0.3,0.2=P0.8,0.2, p1p2=Pmin0.8,0.7,max0.3,0.2=P0.7,0.3, p1c=P0.3,0.8, p2c=P0.2,0.7. |dp1dp2c|= |0.77160.1772|=0.5944, and |dp1cdp2|=|0.22840.8228| =0.5944, then |dp1dp2c|=|dp1cdp2|.

After the inception of PF sets and PFNs, the magnitude comparison of PFNs attracted substantial attention from scholars.

### Definition 3.

[5] Let p=Pμp,νp be a PFN. The score function of p is defined as follows:

sp=μp2νp2,
where sp1,1.

According to Peng and Yang [37], the score function was not sufficiently effective if the same values of the score function occurred in different PFNs. Then, they developed the accuracy function for improving the effectiveness of the magnitude comparison for PFNs.

### Definition 4.

[37] Let p=Pμp,νp be a PFN. The accuracy function of p is defined as follows:

ap=μp2+νp2,
where ap0,1.

Zhang [13] utilized the closeness index to compare the magnitudes between two PFNs.

### Definition 5.

[13] Let p=Pμp,νp be a PFN. The closeness index of p is calculated as follows:

cp=1νp22μp2νp2,
where cp0,1.

Yager [3] utilized the scalar function to compare the magnitudes of PFNs, which considers parameters rp, dp, and θp.

### Definition 6.

[3] Let p=Prp,dp be a PFN. The scalar function of p is defined as follows:

Vp=12+rpdp12=12+rp122θpπ.

### Example 3.

Let p1=P0.8,0.3,p2=P0.7,0.2 be two PFNs. According to Definitions 36, we have the magnitude comparison results between p1 and p2, as shown in Table 1.

p1 p2 Comparison Results
sp 0.5500 0.4500 p1>p2
ap 0.8747 0.8670
cp 0.7165 0.6531 p1>p2
Vp 0.7321 0.7350 p1<p2

PFN, Pythagorean fuzzy number.

Table 1

Magnitude comparison results between two PFNs.

When we employ the score function in Eq. (5) and the closeness index in Eq. (7), we can obtain p1>p2. When we employ the scalar function in Eq. (7), we can obtain p1<p2. The result from the scalar function is inconsistent with the results from the score function and the closeness index. Since the scalar function takes the parameters of the direction of the commitment and the angular degree into consideration, it can reflect the unique characteristics of the PFNs, while the other three methods ignore these important parameters. Therefore, the comparison results by scalar functions are more reasonable than the results from score functions, accuracy functions, and closeness indices.

## 3. NOVEL DISTANCE MEASUREMENT METHOD FOR PFNs

In this section, we analyze several existing distance measurement methods and identify their limitations. Then, we propose a novel distance measurement method that reflects the properties of PFNs.

## 3.1. Limitations of the Existing Distance Measurement Methods for PFNs

Distance measurement is essential for fuzzy sets and has received extensive attention in the past decades. For PFNs, scholars have proposed various distance measurement methods, which are based on three main distances: the Hamming distance, the Euclidean distance, and the generalized distance.

Zhang and Xu [5] presented the Hamming distance measurement method for PFNs, which considers μp, νp, and hp.

### Definition 7.

[5] Let pi=Pμpi,νpii=1,2 be two PFNs. The normalized Hamming distance measure between p1 and p2 can be defined as follows:

DZH(p1,p2)=12((μp1)2(μp2)2+(vp1)2(vp2)2+(hp1)2(hp2)2).

The parameters of Zhang and Xu's distance measurement formula are directly extended from the normalized Hamming distance measure of IF numbers [36]. Only three parameters that reflect the properties of IF numbers are considered; the unique characteristics of PFNs, such as the direction of commitment, the strength of commitment and the radian, are ignored.

Li and Zeng [32] proposed a new distance measurement equation that includes four parameters, namely, μp, νp, rp, dp, of PFNs and overcomes the main limitation of Zhang and Xu's method [5].

### Definition 8.

[32] Let pi=Pμpi,νpii=1,2 be two PFNs. The normalized Hamming distance measure, the normalized Euclidean distance measure, and the normalized generalized distance measure between p1 and p2 can be defined, respectively, as

DLHp1,p2=14|μp1μp2|+|νp1νp2|14+|rp1rp2|+|dp1dp2|),
DLE(p1,p2)=[14((μp1μp2)2+(νp1νp2)2+(rp1rp2)2+(dp1dp2)2)]1/2,
DLG(p1,p2)=[14(|μp1μp2|λ+|νp1νp2|λ+|rp1rp2|λ+|dp1dp2|λ)14]1/λ,
where λ1.

Zeng et al. [33] incorporated the hesitation degree into the distance measurement equation that is based on Li and Zeng's method [32].

### Definition 9.

[33] Let pi=Pμpi,νpii=1,2 be two PFNs. The normalized Hamming distance measure, the normalized Euclidean distance measure, and the normalized generalized distance measure between p1 and p2 are defined respectively as follows:

DZHp1,p2=15|μp1μp2|+|νp1νp2|+|hp1hp2|+|rp1rp2|+|dp1dp2|,
DZH(p1,p2)=[15((μp1μp2)2+(νp1νp2)2+(hp1hp2)2+(rp1rp2)2+(dp1dp2)2)]1/2,
DZH(p1,p2)=[15(|μp1μp2|λ+|νp1νp2|λ+|hp1hp2|λ+|rp1rp2|λ+|dp1dp2|λ)15]1/λ.
where λ1.

### Example 4.

Let p1=P0.8,0.3 and p2=P0.7,0.2 be two PFNs. According to Definitions 79, the computed distances between p1 and p2 are shown as follows: DZHp1,p2=0.2, DLHp1,p2=0.1043, DLEp1,p2=0.1120, DLGp1,p2=0.0364 λ=3, DZHp1,p2=0.1087, DZEp1,p2=0.1150, and DZGp1,p2=0.1204 λ=3. For instance,

DZH(p1,p2)=0.5×(|0.820.72|+|0.320.22|+|0.5220.692|)=0.2,
DLE(p1,p2)=(0.25×((0.80.7)2+(0.30.2)2+(0.850.73)2+(0.770.82)2))1/2=0.1120,
and
DZH(p1,p2)=(0.2×(|0.80.7|3+|0.30.2|3+|0.520.69|3+|0.850.73|3+|0.770.82|3))1/3=01204.

We identify several limitations in Li and Zeng' method [32] and Zeng et al.'s method [33]: First, both distance measurement methods ignore the radian. According to the properties of PFNs that were analyzed by Chen [38], the angle θp in degrees plays an important role in the magnitude comparison, which determines the radians and, thus, influences the distance between two PFNs.

Second, the procedures of both distance measurement methods were simply extended from the distance measure of IF numbers [39], which employed the absolute deviations of the parameters. In line with the properties of the PFNs, the space of PFNs is larger than the space of IF numbers with the condition that μp2+νp21. To ensure the specified scope, we should utilize the squared values of the μp,νp,hp, and rp to specify their relationships. Hence, aforementioned formulas ignored the greater space of PFNs, which couldn't precisely reflect the characteristics of the PFNs.

Third, the distance measure formulas from Li and Zeng's [32] and Zeng et al. [33] didn't satisfy the basic properties of distance measure. An effective distance measurement formula of PFNs should satisfy the properties that the distance is in the range of [0,1]; however, the maximum distance values of Li and Zeng's [32] and Zeng et al.'s [33] formulas don't arrive to one. These limitations will be demonstrated in details in the subsequent comparative analysis of the distance measures for PFNs using specific cases.

## 3.2. Novel Distance Measurement Method for PFNs

To overcome the restrictions of existing distance measurement methods for PFNs, we present a novel distance measurement method that considers five parameters that represent the characteristics of PFNs and the procedure reflects the properties of PFNs.

In the novel distance measurement method, we take the parameters μp, νp,  rp, dp, θp into the equation, but eliminate the parameter hp.

From hp2=1μp2νp2, rp2=μp2+ νp2, it follows that |hp12hp22| =|(1(μp1)2(νp1)2)(1(μp2)2(vp2)2)| =|μp12+νp12μp22+νp22|, and |rp12rp22|= |μp12+νp12 μp22+νp22|. Thus, |hp12hp22| =|rp12rp22|. Therefore, |hp12hp22| plays the same role as |rp12rp22| in the distance measure of PFNs. We only incorporate one of them into the distance measurement formula under the PF environment. In this paper, we select parameter rp in the proposed distance measurement method for PFNs.

We use sinθp to represent the influence of the angle on the distance of PFNs. According to the properties of PFNs, the distance between two PFNs is influenced by the angle θp. Since θp0,π/2, sinθp and cosθp are strictly monotone, where sinθp,cosθp0,1. They have the same properties in the distance measure.

The proposed distance measurement method for PFNs is presented below.

### Definition 10.

Let pi=Pμpi,νpii=1,2 be two PFNs. The parameters of the PFNs are μp,νp,rp, dp,θp. The normalized Hamming distance measure between p1 and p2 complies with the following rule:

D(p1,p2)=14(|(μp1)2(μp2)2|+|(νp1)2(νp2)2|+|(rp1)2(rp2)2|+|dp1dp2|12+|sin(θp1)sin(θp2)|).12

### Example 5.

Let p1=P(0.8,0.3), p2=P(0.7,0.2)  be two PFNs. According to Definition 10, the normalized Hamming distance between p1 and p2 is calculated as below:

Dp1,p2=0.25×|0.820.72|+|0.320.22|+|0.8520.732|+|0.770.82|+|sin(0.36)sin(0.28)|=0.1319.

### Theorem 1.

Let pi=Pμpi,νpii=1,2 be two PFNs. Then, 0Dp1,p21.

### Proof:

It follows from μp1,μp2,νp1,νp20,1, θp1,θp20,π/2, 0μp12+νp121, 0μp22+νp221, rp12=μp12+νp12, rp22=μp22+νp22, |dp1dp2|=π/2×|θp1θp2|, |dp1dp2|0,1, and |sinθp1sinθp2|0,1 that

D(p1,p2)=14(|(μp1)2(μp2)2|+|(νp1)2(νp2)2|+|(rp1)2(rp2)2|+|dp1dp2|+sin(θp1)sin(θp2)|(vp1)2(vp2)2|)=14(|(μp1)2(μp2)2|+|(νp1)2(νp2)2|+|(μp1)2+(νp1)2||(μp2)2+(νp2)2|+(2/π)×|θp1θp2|+|sin(θp1)sin(θp2)||(μp1)2(μp2)2|)14((μp1)2+(νp1)2+(μp1)2+(νp1)2+(2/π)×|θp1θp2|+|sin(θp1)sin(θp2)|(μp1)2+(vp1)2)14(1+1+1+1)=1.

Moreover, all the absolute deviations in Eq. (15) are equal to or greater than zero, which completes the proof of Theorem 1.

### Theorem 2.

Let pi=Pμpi,νpii=1,2 be two PFNs. Then, Dp1,p2=0 if and only if p1=p2.

### Proof:

Because all the absolute deviations in Eq. (15) are equal to or greater than zero, if Dp1,p2=0, then each absolute deviation is equal to zero.

If |μp12μp22|=0, |νp12νp22|=0, |rp12rp22|=0, |sinθp1sinθp2|=0, |dp1dp2|=0, μp1, μp2, νp1, νp20,1, and θp1,θp20,π/2, then μp1=μp2, νp1=νp2, rp1=rp2, θp1=θp2, and dp1=dp2.

Via any magnitude comparison method, we can obtain the result p1=p2, which completes the proof of Theorem 2.

### Theorem 3.

Let pi=Pμpi,νpii=1,2 be two PFNs. Then, Dp1,p2=Dp2,p1.

### Proof:

It follows from |μp12μp22|= |μp22μp12|, |νp12νp22|= |νp22νp12|, |rp12rp22|= |rp22rp12|, |dp1dp2|= |dp2dp1|, and |sinθp1sinθp2|= |sinθp2sinθp1|, that Dp1,p2= Dp2,p1, which completes the proof of Theorem 3.

### Theorem 4.

Let pi=Pμpi,νpii=1,2,3 be three PFNs. If p1p2p3, then Dp1,p2Dp1,p3 and Dp2,p3Dp1,p3.

### Proof:

According to the quasi-ordering on the space of Pythagorean membership grades from Yager [3], if p1p2p3 we can obtain μp1μp2μp3 and νp1νp2νp3. Then,

|μp12μp22||μp12μp32|,|νp12νp22||νp12νp32|,|rp12rp22|=|μp12+νp12μp22+νp22|=|μp12μp22+νp12νp22||μp12μp32+νp12νp32|=|μp12+νp12μp32+νp32|=|rp12rp32|.

From θp=arctanνp/μp, tanθp=νp/μp, θp 0,π/2, tanθp1tanθp2tanθp3, and the function tanθp being strictly monotone, it follows that θp1θp2θp3.

|dp1dp2|=2/π|θp1θp2|2/π|θp1θp3|=|dp1dp3|,
and |sinθp1sinθp2||sinθp1sinθp3|. Thus,
D(p1,p2)=14(|(μp1)2(μp2)2|+|(νp1)2(νp2)2|+|(rp1)2(rp2)2|+|dp1dp2|+|sin(θp1)sin(θp2)||(μp1)2(μp2)2|)14(|(μp1)2(μp3)2|+|(νp1)2(νp3)2|+|(rp1)2(rp3)2|+|dp1dp3|+|sin(θp1)sin(θp3)||(μp1)2(μp3)2|)=D(p1,p3).

Similarly, we can also prove Dp2,p3Dp1,p3, which completes the proof of Theorem 4.

In this analysis, we have proved that the proposed normalized Hamming distance of PFNs is nonnegative, symmetric, and transitivity; moreover, the distance is in the range of [0,1].

## 3.3. Comparative Analysis of the Distance Measures for PFNs

In order to illustrate the advantages of the proposed distance measure for PFNs, we conduct a comparison between the proposed distance measure and the existing distance measures from Zhang and Xu [5], Li and Zeng [32], and Zeng et al. [33]. We explore a series of cases expressed as PFNs based on the fundamental properties of the distance measure and the deviations of the magnitudes of the PFNs by scalar functions. The comparison results from different distance measures are summarized in Table 2. It is clear that the proposed distance measure can overcome the difficulties of obtaining the illogical results of the existing methods. We will investigate the following four major limitations of the existing distance measures in details.

 Case 1 Case 2 Case 3 p1 P(1.00, 0.00) P(1.00, 0.00) P(0.50, 0.71) p2 P(0.00, 1.00) P(0.00, 0.00) P(0.00, 1.00) V(p1) 1 1 0.4063 V(p2) 0 0.5 0 DZH [5] 1 1 0.5 DLH [32] 0.75 0.5 0.4212 DLE [32] 0.75 0.5 0.1849 DLG [32] 0.75 0.5 0.0838 DZH′ [33] 0.6 0.6 0.3627 DZE′ [33] 0.6 0.6 0.1515 DZG′ [33] 0.6 0.6 0.0675 D (proposed) 1 0.5 0.3938 Case 4 Case 5 Case 6 p1 P(0.40, 0.20) P(0.40, 0.20) P(0.71, 0.50) p2 P(0.50, 0.30) P(0.14, 0.30) P(0.00, 1.00) V(p1) 0.5916 0.5916 0.5937 V(p2) 0.5909 0.4265 0 DZH [5] 0.14 0.1375 0.75 DLH [32] 0.0827 0.2018 0.5788 DLE [32] 0.0073 0.0606 0.3425 DLG [32] 0.0007 0.0214 0.2071 DZH′ [33] 0.0934 0.1838 0.4899 DZE′ [33] 0.0095 0.0510 0.2776 DZG′ [33] 0.0010 0.0174 0.1662 D (proposed) 0.0990 0.2830 0.6327

PFN, Pythagorean fuzzy number.

Results in orange shaded cells represent the unreasonable distance values.

Table 2

Comparison results of six cases from different distance measures for PFNs.

First, the property of the distance measure in Theorem 1 is not satisfied by use of DLH,DLE,DLG,DZH,DZE and DZG in Case 1. Because p1=P1.00,0.00 and p2=P0.00,1.00 are the maximum PFN and the minimum PFN by scalar functions, namely, VP1.00,0.00=1 and VP0.00,1.00=0, it is expected that the distance between p1=P1.00,0.00 and p2=P0.00,1.00 is the maximum distance, arriving to one. However, DLH= DLE= DLG= 0.75 and DZH=DZE=DZG=0.6, none of which are up to one. The key problem is that the procedure of distance measures from Li and Zeng [32] and Zeng et al. [33] directly extended from the IF numbers, not considered the squared deviations of the parameters. Therefore, the results from Li and Zeng [32] and Zeng et al. [33] cannot ensure the greater space of the PFNs and do not satisfy the property of the distance measure in Theorem 1.

Second, the distance value DZH= 1 in Cases 1 and 2 is unreasonable. Although they have the same p1 in the two cases, p2=P0.00,1.00 (in Case 1) and p2=P1.00,0.00 (in Case 2) are obviously different. The same obtained distance values from the different PFNs are illogical and unreasonable. According to the deviations of magnitudes by the scalar function, the deviation in Case 2 is 0.5. It is respected that the distance value is equal to or close to 0.5, so DZH= 1 is unreasonable in Case 2. The key problem is that Zhang and Xu [5] took the parameters μp, νp, and hp into the distance measure, but ignored the rp, dp, and θp of the PFNs. This limitation will result in the same distance values from the different PFNs. The similar results occur in DZH,DZE, and DZG (in Cases 1 and 2). Although Zeng et al. [33] took account of the parameters μp, νp, hp, rp, and dp of PFNs, they neglected θp of the PFNs. This limitation will also result in the same distance values for different PFNs. Hence, the distance measures from Zhang and Xu [5] and Zeng et al. [33] are demonstrated to distinguish the differences of the PFNs ineffectively in Cases 1 and 2.

Third, it is easily seen that the property of the distance measure in Theorem 4 is not satisfied by use of DZH in Cases 4 and 5. Based on the magnitudes of the PFNs by scalar function, VP0.40,0.20> VP0.40,0.20>VP0.50,0.30>VP0.15,0.30, as shown in Table 2. We obtain VP0.40,0.20VP0.50,0.30 <VP0.40,0.20VP0.15,0.30. It is expected that the distance value between p1=P0.40,0.20 and p2=P0.50,0.30 (in Case 4) is greater than the distance value between p1=P0.40,0.20 and p2=P0.14,0.30 (in Case 5). However, the DZH value in Case 4 is greater than that in Case 5, which leads to a counterintuitive result. Therefore, the distance measure from Zhang and Xu [5] is infeasible and unreasonable in Cases 4 and 5.

Fourth, it is clearly seen that the distance values by use of DLE,DLG,DZE, and DZG in Cases 3 and 6 are unreasonable. Since P(0.50, 0.71) and P(0.71, 0.50) are compliment, they have the same rp, dp, and θp. Meanwhile, in line with the magnitudes by scalar functions, VP0.50,0.71VP0.00,1.00=0.4063 and VP0.71,0.50VP0.00,1.00=0.5937. It is logically respected that the distance values between p1 and p2 are close to 0.4063 in Case 3 and 0.5937 in Case 6. However, we find out that the obtained results by use of DLE,DLG,DZE, and DZG have great disparities with the expected deviations. The similar results occur in the DLE,DLG,DZE, and DZG values in Cases 4 and 5, which are obviously less than the deviations of the magnitudes. Therefore, the Euclidean distance measures and the generalized distance measures from Li and Zeng [32] and Zeng et al. [33], respectively, are not effective to measure the distance for PFNs in these cases.

Based on the comparative discussions, we find out that the distance measure from Zhang and Xu [5] couldn't precisely calculate the distance between two PFNs owing to the ignorance of some important parameters representing the characteristics of PFNs. The distance measures from Li and Zeng [31] and Zeng et al. [32] are not satisfied with the maximum value of the distance between two PFNs, because the relevant procedures do not utilize the squared deviations of corresponding parameters to ensure the greater space of the PFNs. The distance values of the Euclidean distance measure and the generalized distance measure have great disparities with the deviations of the magnitudes between the PFNs. In contrast, the proposed distance measure for PFNs can effectively conquer the difficulties of illogical results from the existing distance measures and satisfy the properties of the distance measure in Theorems 14. Moreover, the obtained results are closer to the deviations of the magnitudes between PFNs.

## 4. PROPOSED PF-TOPSIS APPROACH FOR MCDM PROBLEMS

In this section, we introduce the MCDM problems under the PF environment and apply the novel distance measure in the PF-TOPSIS approach to deal with the MCDM problems.

## 4.1. Description of the MCDM Problems within the PF Environment

An MCDM problem is expressed as a decision matrix, the elements of which are the assessment values of all alternatives for every criterion. Given an MCDM problem with PF information, let X=x1,x2,,xm m2 be a set of alternatives, C=C1,C2,,Cn n2 be a set of criteria, ω=ω1,ω2,,ωnT 0ωj1j=1,2,,n, and j=1nωj=1 be the weight vector for each criterion. Pμpij,νpij denotes the assessment value of the ith alternative for the jth criterion, namely, Cjxi=Pμpij,νpij, and R=Cjxim×n denotes the PF decision matrix, which is concisely expressed as follows:

R=Pμp11,νp11Pμp12,νp12Pμp1n,νp1nPμp21,νp21Pμp22,νp22Pμp2n,νp2n          Pμpm1,νpm1Pμpm2,νm2Pμpmn,νpmn.

## 4.2. Process of the Proposed Approach

To solve these MCDM problems within the PF environment, we propose the PF-TOPSIS approach, which is based on the principle that the optimal solution has the shortest distance from the PFPIS and the farthest distance from the PFNIS.

First, we determine the PFPIS x+ and the PFNIS x. The classical TOPSIS introduced the PIS and NIS, which were determined by the union operator and the intersection operator of fuzzy sets according to the properties of the criteria. The benefit criteria belong to J1 and the cost criteria belong to J2. Akram et al. [10] extended the PFPIS and the PFNIS from the classical TOPSIS method. Let xc+ and xc denote the classical type PFPIS and PFNIS, respectively, using the following formulas:

xc+=Cj,maxμpjxi,minμpjxi|CjJ1,Cj,minμpjxi,maxμpjxi|CjJ2=C1,Pμp1+,νp1+,C2,Pμp2+,νp2+,,Cn,Pμpn+,νpn+,j=1,2,,n,
xc=Cj,minμpjxi,maxμpjxi|CjJ1,Cj,maxμpjxi,minμpjxi|CjJ2=C1,Pμp1,νp1,C2,Pμp2+,νp2,,Cn,Pμpn,νpn,j=1,2,,n.

The traditional TOPSIS also introduced the PIS and the NIS determined the fixed number 1 or 0. Wu et al. [40] employed the PFN P(1, 0) or P(0, 1) as the fixed type PFPIS xf+ and the fixed type PFNIS xf, respectively, as follows:

xf+=Cj,P(1,0)|CjJ1,Cj,P(0,1)|CjJ2,
xf=Cj,P(0,1)|CjJ1,Cj,P(1,0)|CjJ2.

Zhang and Xu [5] developed the score function in Eq. (4) for identifying the PFPIS and the PFNIS, which narrowed the distances from the PFPIS and the PFNIS compared to the classical TOPSIS. Let xs+ and xs denote the scored type PFPIS and PFNIS, respectively, using the following formulas:

xs+=Cj,maxsCj(xi)|j=1,2,,n=C1,Pμp1+,νp1+,C2,Pμp2+,νp2+,,Cn,Pμpn+,νpn+,
xs=Cj,minsCj(xi)|j=1,2,,n=C1,Pμp1,νp1,C2,Pμp2,νp2,,Cn,Pμpn,νpn.

Yager [3] developed the scalar function to identify the magnitudes of the PFNs. We employ the scalar function in Eq. (7) to determine the PFPIS and the PFNIS. Let xv+ and xv denote the scalar type PFPIS and PFNIS, respectively, using the following formulas:

xv+=Cj,maxVCj(xi)|CjJ1,Cj,minVCj(xi)|CjJ2=C1,Pμp1+,νp1+, C2,Pμp2+,νp2+,,Cn,Pμpn+,νpn+, j=1,2,,n,
xv=Cj,minVCj(xi)|CjJ1,Cj,maxVCj(xi)|CjJ2=C1,Pμp1,νp1, C2,Pμp2,νp2,,Cn,Pμpn,νpn, j=1,2,,n.

Next, we calculate the distances from each alternative to the PFPIS Dxi,x+ and the PFNIS Dxi,x in MCDM practice because we may not find the PFPIS x+ or the PFNIS x if it is outside the feasible region, namely, if x+ or xX. Moreover, in practice, decision makers typically express the importance of each element relative to the others based on various preferences. We consider ϖj with respect to the importance of the criteria and propose the weighted distance measure for PFNs.

### Definition 11.

Let pi=Pμpij,νpij|i=1,2;j=1,2,,n be two PFNs on C=C1,C2,,Cn. ωj is the importance of criterion j; the vector of importances for all criteria is expressed as ω=ω1,ω2,,ωnT. The weighted distance measure equation between p1 and p2 is expressed as follows:

DW(p1,p2)=14j=1nωj(|(μp1j)2(μp2j)2|+|(νp1j)2(νp2j)2|+|(rp1j)2(rp2j)2|+|dp1jdp2j|++|sin(θp1j)sin(θp2j)||(μp1j)2(μp2j)2|),
where 0ωj1j=1,2,,n and j=1nωj=1.

### Example 6.

Let p1=P0.8,0.3,P0.7,0.3,P0.6,0.3 and p2=P0.7,0.2,P0.6,0.2,P0.8,0.2 be two sets of PFNs with respect to the set of criteria C=C1,C2,C3. The weight vector of the criteria is ω=0.2,0.3,0.5T. According to Definition 11, the weighted Hamming distance between p1 and p2 is calculated as follows:

DWp1,p2=0.25×0.2×|0.820.72|+|0.320.22|+|0.8520.732|+|0.770.82|+|sin(0.36)sin(0.28)|+0.3×|0.720.62|+|0.320.22|+|0.7620.632|+|0.740.80|+|sin(0.40)sin(0.32)|+0.5×|0.620.82|+|0.320.22|+|0.6720.822|+|0.700.84|+|sin(0.46)sin(0.24)|=0.2439.

### Theorem 5.

Let pi=Pμpij,νpij|i=1,2; j=1,2,,n be two PFNs on C=C1,C2,,Cn. The vector of importance for all criteria is expressed as ω=ω1,ω2,,ωnT. Then, 0DWp1,p21.

### Proof:

Since 0ωj1j=1,2,,n and j=1nωj=1, if DWHp1,p2=1, we should set the distance of each criterion between two PFNs to Dp1,p2=1; if DWHp1,p2=0, we should also set it to Dp1,p2=0. Since all the normalized Hamming distances of PFNs satisfy the properties that Dp1,p20,1, we obtain DWp1,p20,1, which completes the proof of Theorem 5.

### Theorem 6.

Let pi=Pμpij,νpij|i=1,2;j=1,2,,n be two PFNs on C=C1,C2,,Cn. The vector of importance for all criteria is expressed as ω=ω1,ω2,,ωnT. Then, DWp1,p2=0 if and only if p1=p2.

### Proof:

Since all the absolute deviations in Eq. (25) are equal to or greater than zero, if DWp1,p2=0, each absolute deviation is equal to zero. In addition, μp1j,μp2j,νp1j,νp2j,rp1j,rp2j0,1, θp1j, and θp2j0,π/2. Therefore, μp1j=μp2j, νp1j=νp2j, rp1j=rp2j, θp1j=θp2j, and dp1j=dp2j. Employing any magnitude comparison method, we can obtain p1=p2 under the same criterion, which completes the proof of Theorem 6.

### Theorem 7.

Let pi=Pμpij,νpij|i=1,2;j=1,2,,n be two PFNs on C=C1,C2,,Cn. The vector of importance for all criteria is expressed as ω=ω1,ω2,,ωnT. Then, DWp1,p2=DWp2,p1.

### Theorem 8.

Let pi=Pμpij,νpij|i=1,2,3;j=1,2,,n be three PFNs on C=C1,C2,,Cn. The vector of importance for all criteria is expressed as ω=ω1,ω2,,ωnT. Then, DWp1,p2DWp1,p3 and DWp2,p3DWp1,p3.

### Proof:

In line with Theorem 4, if p1p2p3 for each criterion, we can obtain μp1μp2μp3 and νp1νp2 νp3. Then, Dp1,p2Dp1,p3 and Dp2,p3 Dp1,p3. For two PFNs on C=C1,C2,,Cn, DWp1,p2 is equal to the weighted sum of each Dp1,p2. Since the weight of the jth criterion is the same, DWp1,p2DWp1,p3 remains satisfied.

Similarly, we can also prove DWp2,p3 DWp1,p3 by the same way, which completes the proof of Theorem 8.

We obtain the distance of alternative xi from PFPIS x+ based on Eq. (25), which can be defined as follows:

Dxi,x+=j=1nωjDCjxi,Cjx+=14j=1nωj|μpij2μpj+2|+|νpij2νpj+2|+|rpij2rpj+2|+|dpijdpj+|+|sinθpijsinθpj+|,
where i=1,2,,m and x+ represents xc+, xf+ or xv+.

According to the principle of TOPSIS, the smaller Dxi,x+ is, the better the alternative xi is. Let:

Dminxi,x+=mini=1mDxi,x+.

The distance between alternative xi and the PFNIS x can be defined as follows:

Dxi,x=j=1nωjDCjxi,Cjx=14j=1nωj|μpij2μpj2|+|νpij2νpj2|+|rpij2rpj2|+|dpijdpj|+|sinθpijsinθpj|,
where i=1,2,,m and x represents xc, xf or xv.

According to the principle of TOPSIS, the larger Dxi,x is, the better the alternative xi is. Let:

Dmaxxi,x=maxi=1mDxi,x.

Traditionally, we calculate the RC of the alternative xi with respect to the PFPIS x+ and the PFNIS x in line with the basic principle of classical TOPSIS. The formula for RCxi is expressed as follows:

RC(xi)=Dxi,xDxi,x++Dxi,x.

In view of Hadi-Vencheh and Mirjaberi's formula [41], according to which the optimal solution has the shortest distance from the PIS and the farthest distance from the NIS, concurrently, Zhang and Xu [5] and Akram et al. [10] utilized a revised index, which is denoted as ζxi, to identify the ranking order, which is expressed as follows:

ζxi=Dxi,xDmaxxi,xDxi,x+Dminxi,x+.

RCxi is the classical index for determining the ranking order of alternatives, which has been extensively used and can reflect the performance of TOPSIS. The index ζxi considers the optimal solution to be close to PFPIS and far from PFNIS simultaneously. To examine the feasibility of the proposed approach, we adopt both indices in this paper. According to RCxi or ζxi, we obtain the ranking order of alternatives xi, which we use to determine the optimal solution according to the maximum value of RCxi or ζxi.

x*=xi:i=RCxi=maxi=1mRCxi,
x*=xi:i=ζxi=maxi=1mζxi.

## 4.3. Algorithm of the Proposed Approach

In line with the above analysis, the proposed PF-TOPSIS approach is separated into four paths for identifying the PFPIS and the PFNIS: the classical type applies the classical TOPSIS union operator or intersection operator; the fixed type applies the fixed PFNs of P(1, 0) or P(0, 1); the scored type applies the score function; the scalar type applies the scalar function. We describe the algorithm of the proposed PF-TOPSIS method in the following seven steps. We present a flowchart that illustrates the process of the proposed approach in Fig. 2.

Step 1. Construct a PF decision matrix R=Cjxim×n for an MCDM problem under the PF environment, where each element Cjxi is the assessment value of the ith alternative with respect to the jth criterion.

Step 2. Identify the PFPIS. Utilize Eqs. (17), (19), (21), and (23) to identify the PFPIS x+=C1x+,C2x+,,Cnx+ (x+ represents xc+, xf+, xs+ or xv+) for the classical type, the fixed type, the scored type, and the scalar type, respectively.

Step 3. Identify the PFNIS. Utilize Eqs. (18), (20), (22), and (24) to identify the PFNIS x=C1x,C2x,,Cnx (x represents xc, xf, xs or xv) for the classical type, the fixed type, the scored type, and the scalar type, respectively.

Step 4. Employ Eq. (26) to calculate the distance of alternative xi from the PFPIS and use Eq. (27) to determine the minimum distance from the PFPIS.

Step 5. Employ Eq. (28) to calculate the distance of alternative xi from the PFNIS and use Eq. (29) to determine the maximum distance from the PFNIS.

Step 6. Utilize Eqs. (30) and (31) to compute the RC index RCxi and the revised closeness index ζxi, respectively, of alternative xi.

Step 7. Employ Eqs. (32) and (33) to obtain the ranking order of alternatives xi and identify the optimal solution, which corresponds to the maximum value of RCxi or ζxi.

## 5. ILLUSTRATIVE EXAMPLE

In this section, we consider an example evaluation for emerging technology commercialization that was adapted from Wei and Lu [42] as an MCDM problem on which to evaluate the feasibility of the proposed approaches and conduct a comparison analysis among four types of approaches under the PF environment.

## 5.1. Description of the Example

The example from Wei and Lu [42] was used to evaluate the commercialization of emerging technology companies. Five potential emerging technology companies are expressed as a set of alternatives X=x1,x2,x3,x4,x5. The experts selected four major criteria (C1: the technical advancement; C2: the potential market risk; C3: the industrialization infrastructure, human resources, and financial conditions; and C4: the employment criterion and the development of science and technology), which are expressed as a set of criteria C=C1,C2,C3,C4. All the criteria are benefit attributes. The weight vector of the criteria that is provided by the decision makers is expressed as ω=0.2,0.1,0.3,0.4T.

The assessment values of the five companies with respect to the four criteria that were specified by the decision makers are expressed as PFNs, as listed in Table 3. For example, the element C1x1=P0.5,0.8 specifies that the degree to which alternative x1 satisfies criterion C1 is 0.5 and the degree to which alternative x1 dissatisfies criterion C1 is 0.8.

C1 C2 C3 C4
x1 P(0.5,0.8) P(0.6,0.3) P(0.3,0.6) P(0.5,0.7)
x2 P(0.7,0.5) P(0.7,0.2) P(0.9,0.2) P(0.8,0.5)
x3 P(0.6,0.2) P(0.5,0.2) P(0.5,0.3) P(0.6,0.3)
x4 P(0.4,0.2) P(0.6,0.3) P(0.3,0.4) P(0.5,0.4)
x5 P(0.6,0.4) P(0.4,0.8) P(0.7,0.6) P(0.5,0.8)

PF, Pythagorean fuzzy.

Table 3

PF decision matrix that was specified by the decision maker.

## 5.2. Decision Process of the Proposed Approach

According to the algorithm of the proposed PF-TOPSIS approach, we apply the proposed PF-TOPSIS approach to deal with the MCDM problem that is discussed in Subsection 5.1.

Step 1. We utilize Eqs. (17) and (18) to calculate the classical type PFPIS xc+ and the classical type PFNIS (xc), respectively. Then we obtain the results of xc+ and xc as follows:

xc+=P0.7,0.2,P0.7,0.2,P0.9,0.2,P0.8,0.3,xc=P0.4,0.8,P0.4,0.8,P0.3,0.6,P0.5,0.8.

We utilize Eqs. (19) and (20) to calculate the fixed type PFPIS xf+ and the fixed type PFNIS (xf), respectively. We obtain the results of xf+ and xf as follows:

xf+=P1.0,0.0,P1.0,0.0,P1.0,0.0,P1.0,0.0,xf=P0.0,1.0,P0.0,1.0,P0.0,1.0,P0.0,1.0.

We utilize Eqs. (21) and (22) to calculate the scored type PFPIS xs+ and the scored type PFNIS (xs), respectively. We obtain the results of xs+ and xs as follows:

xs+=P0.6,0.2,P0.6,0.3,P0.9,0.2,P0.8,0.5,xs=P0.5,0.8,P0.4,0.8,P0.3,0.6,P0.5,0.8.

We utilize Eqs. (23) and (24) to calculate the scalar type PFPIS xv+ and the scalar type PFNIS (xv), respectively. Then, we obtain the results of xv+ and xv as follows

xv+=P0.6,0.2,P0.5,0.2,P0.9,0.2,P0.6,0.3,xv=P0.5,0.8,P0.4,0.8,P0.3,0.6,P0.5,0.8.

Step 2. We utilize Eqs. (26) and (27) and Eqs. (28) and (29) to calculate the distances of each alternative xi from the PFPIS as well as the PFNIS, respectively. The results of Dxi,xc+ and Dxi,xc, Dxi,xf+ and Dxi,xf, Dxi,xs+ and Dxi,xs, as well as Dxi,xv+ and Dxi,xv are listed in Tables 47, respectively. Alternative x2 has the shortest distance from PFPIS for the classical type, the fixed type, and the scored type approaches, whereas x3 does for the scalar type approach. Meanwhile, alternative x3 has the farthest distance from the PFNIS for all types of approaches.

Step 3. We employ Eqs. (30) and (31) to compute the RCxi and the ζxi for each alternative xi from four types of approaches, which are also listed in Tables 47.

Dxi,xc+ Dxi,xc RCxi(Rank) ζxi(Rank)
x1 0.5116 0.1740 0.2538(5) −2.6807(5)
x2 0.1688 0.4978 0.7467(1) 0.0000(1)
x3 0.2505 0.5316 0.6797(2) −0.4160(2)
x4 0.3930 0.4106 0.5110(3) −1.5026(3)
x5 0.4888 0.2827 0.3665(4) −2.3271(4)

PF-TOPSIS, Pythagorean fuzzy technique for order preference by similarity to ideal solutions; RC, relative closeness.

Table 4

Results that were obtained from the classical type PF-TOPSIS approach.

Dxi,xf+ Dxi,xf RCxi(Rank) ζxi(Rank)
x1 0.8243 0.6900 0.4556(5) −1.0036(5)
x2 0.4403 0.7945 0.6434(1) 0.0000(1)
x3 0.6013 0.8636 0.5895(2) −0.2780(2)
x4 0.7437 0.8436 0.5315(3) −0.6270(3)
x5 0.7738 0.6797 0.4676(4) −0.9017(4)

PF-TOPSIS, Pythagorean fuzzy technique for order preference by similarity to ideal solutions; RC, relative closeness.

Table 5

Results that were obtained from the fixed type PF-TOPSIS approach.

Dxi,xs+ Dxi,xs RCxi(Rank) ζxi(Rank)
x1 0.4411 0.1597 0.2657(5) −3.7959(5)
x2 0.1076 0.4924 0.8207(1) −0.0642(1)
x3 0.2521 0.5262 0.6761(2) −1.3423(2)
x4 0.3295 0.4123 0.5570(3) −2.2747(3)
x5 0.3973 0.2215 0.3579(4) −3.2713(4)

PF-TOPSIS, Pythagorean fuzzy technique for order preference by similarity to ideal solutions; RC, relative closeness.

Table 6

Results that were obtained from the scored type PF-TOPSIS approach.

Step 4. We obtain the ranking order of the five alternatives in accordance of the results from RCxi and ζxi, which are listed in Tables 47.

The ranking results that are based on RCxi and ζxi in this example are completely consistent from classical type, fixed type, and scored type approaches, namely, x2x3x4x5x1, among which x2 is the best alternative. However, the ranking results from the scalar type approach is x3x2x4x5x1, among which x3 is the best alternative.

Dxi,xv+ Dxi,xv RCxi(Rank) ζxi(Rank)
x1 0.4855 0.1597 0.2475(5) −1.7367(5)
x2 0.2380 0.4924 0.6742(2) −0.0642(2)
x3 0.1217 0.5262 0.8122(1) 0.4887(1)
x4 0.2979 0.4143 0.5817(3) −0.4644(3)
x5 0.4579 0.2774 0.3772(4) −1.3971(4)

PF-TOPSIS, Pythagorean fuzzy technique for order preference by similarity to ideal solutions; RC, relative closeness.

Table 7

Results that were obtained from the scalar type PF-TOPSIS approach.

## 5.3. Comparative Analysis

The main differences among the approaches are the identification of the PFPIS, the PFNIS, and the distance measure of the alternative from the PFPIS and the PFNIS according to the previously described analysis. Therefore, we compare the scalar type PF-TOPSIS with the classical type PF-TOPSIS and the fixed type PF-TOPSIS to analyze how the PFPIS and the PFNIS influence the ranking order results for the same distance measure. Meanwhile, we compare the scalar type approach with the scored type approach that employs the distance measure from Zhang and Xu [5] to analyze the influence of the PFPIS, the PFNIS, and the distance measure on the ranking order results.

First, we calculate the PFPIS and the PFNIS via the above four approaches. The PFPIS x+ and PFNIS x results are listed in Table 8.

x+ C1 C2 C3 C4
I P(0.7,0.2) P(0.7,0.2) P(0.9,0.2) P(0.8,0.3)
II P(1.0,0.0) P(1.0,0.0) P(1.0,0.0) P(1.0,0.0)
III P(0.6,0.2) P(0.6,0.3) P(0.9,0.2) P(0.8,0.5)
IV P(0.6,0.2) P(0.5,0.2) P(0.9,0.2) P(0.6,0.3)

x C1 C2 C3 C4

I P(0.4,0.8) P(0.4,0.8) P(0.3,0.6) P(0.5,0.8)
II P(0.0,1.0) P(0.0,1.0) P(0.0,1.0) P(0.0,1.0)
III P(0.5,0.8) P(0.4,0.8) P(0.3,0.6) P(0.5,0.8)
IV P(0.5,0.8) P(0.4,0.8) P(0.3,0.6) P(0.5,0.8)

PFNIS, Pythagorean fuzzy negative ideal solution; PFPIS, Pythagorean fuzzy positive ideal solution; PF-TOPSIS, Pythagorean fuzzy technique for order preference by similarity to ideal solutions.

I: The classical type PF-TOPSIS;

II: The fixed type PF-TOPSIS;

III: The scored type PF-TOPSIS;

IV: The scalar type PF-TOPSIS.

Table 8

PFPIS and PFNIS results that are based on four PF-TOPSIS approaches.

The differences between the results of x+ and x that were obtained via the methods of the scored type and the scalar type are smaller compared to the methods of the classical type and the fixed type, as listed in Table 8. Therefore, the results of the scored type and the scalar type narrow the range of the ideal solution, which directly influence the distance from x+ and x. In addition, the main difference between the methods of the scored type and the scalar type depends mainly on C2 and C4 with respect to x+; we should further compare the magnitudes of three PFNs: P0.5,0.2, P0.6,0.3, and P0.8,0.5.

According to Li and Zeng [32] and Zeng et al. [33], the magnitude comparison methods from score functions are extended directly from IF numbers. Considering the properties of PFNs, especially the direction of commitment and the angle in degrees, they compared the effectiveness of the magnitude measure between the score function and the scalar function that was based on the same PFN. They concluded that the scalar function was more accurate. Chen [39] further analyzed the desirable and important properties of scalar function Vp: Vp0,1; Vp=0.5 if rp,θp=0,π/4; Vp=1 if rp,θp=1,0; Vp decreases as θp increases if rp is fixed; Vp increases as rp increases if θp is fixed and θp < π/4; Vp decreases as rp increases if θp is fixed and θp>π/4. Considering the unique properties of the PFNs, the scalar function is more suitable for magnitude comparison of PFNs than the score function. We obtain V(P(0.5,0.2)=0.6388, V(P(0.6,0.3)=0.6374, and V(P(0.8,0.5)=0.6362; the magnitude comparison results, namely, P0.5,0.2 >P0.6,0.3 >P0.8,0.5, are reasonable, which precisely reflect the PFPIS. Additionally, the results of x that are obtained via the scored type approach are the same as those of the scalar type approach.

Second, we calculate the distances Dxi,x+ between alternative xi and the PFPIS and the distances Dxi,x between alternative xi and the PFNIS, which are reported in Tables 9 and 10, respectively.

I (Rank) II (Rank) III (Rank) IV (Rank)
x1 0.5116(5) 0.8243(5) 0.4920(5) 0.4855(5)
x2 0.1688(1) 0.4403(1) 0.1043(1) 0.2380(2)
x3 0.2505(2) 0.6013(2) 0.3640(2) 0.1217(1)
x4 0.3930(3) 0.7437(3) 0.4480(4) 0.2979(3)
x5 0.4888(4) 0.7738(4) 0.3698(3) 0.4579(4)
Table 9

Dxi,x+ results based on four approaches.

I (Rank) II (Rank) III (Rank) IV (Rank)
x1 0.1740(5) 0.6900(4) 0.1538(5) 0.1597(5)
x2 0.4978(2) 0.7945(3) 0.5400(1) 0.4924(2)
x3 0.5316(1) 0.8636(1) 0.5073(2) 0.5262(1)
x4 0.4106(3) 0.8436(2) 0.4838(3) 0.4143(3)
x5 0.2827(4) 0.6797(5) 0.3120(4) 0.2774(4)
Table 10

Dxi,x results based on four approaches.

To facilitate understanding, we present the Dxi,x+ and Dxi,x comparison results graphically in Figs. 3 and 4, respectively. The distance values of the fixed type approach are larger than those of the other approaches due to the PFPIS and PFNIS. The methods of the classical type, the fixed type, and the scalar type employed the proposed distance measurement method; however, the results are partially inconsistent (x2 has the minimum distance from x+ with the classical type approach and the fixed type approach, whereas x3 has the minimum distance with the scalar type approach) due to the differences in their PFPISs and PFNISs. Moreover, the difference between the scored type approach and the scalar type approach mainly depends on x2 and x3 with respect to both Dxi,x+ and Dxi,x, namely, x2 has the minimum distance from the PFPIS and the maximum distance from the PFNIS with the scored type approach, whereas x3 does with the scalar type approach. In addition, Zhang and Xu's distance measure [5] with the scored type approach considers three parameters (μp, νp, hp) of PFNs in the distance measure equation, whereas the proposed distance measurement method with the scalar type approach considers five parameters (μp, νp, rp, dp, θp) in the distance measure equation, which fully represents the properties of PFNs and is more effective and feasible.

Third, we obtain the results regarding the RCxi and the ζxi, which are utilized to identify the ranking order results via four approaches, which are listed in Table 11.

RC I (Rank) II (Rank) III (Rank) IV Rrank)
x1 0.2538(5) 0.4556(5) 0.2381(5) 0.2475(5)
x2 0.7467(1) 0.6434(1) 0.8382(1) 0.6742(2)
x3 0.6797(2) 0.5895(2) 0.5822(2) 0.8122(1)
x4 0.5110(3) 0.5315(3) 0.5192(3) 0.5817(3)
x5 0.3665(4) 0.4676(4) 0.4576(4) 0.3772(4)

ζ I (Rank) II (Rank) III (Rank) IV (Rank)

x1 −2.6807(5) −1.0036(5) −4.4163(5) −1.7367(5)
x2 0.0000(1) 0.0000(1) 0.0646(1) −0.0642(2)
x3 −0.4160(2) −0.278(2) −2.4916(2) 0.4887(1)
x4 −1.5026(3) −0.627(3) −3.3437(4) −0.4644(3)
x5 −2.3271(4) −0.9017(4) −2.9317(3) −1.3971(4)

RC, relative closeness.

Table 11

RC(xi),ζ(xi) and ranking results from four approaches.

To compare the differences of the RCxi and ζxi values that were obtained via the four approaches, we plot the results in Fig. 5. The RCxi values from the PF-TOPSIS of the classical type, the fixed type, and the scored type are very close among alternatives and may not differ significantly, whereas the disparity of RCxi from the scalar type approach is more significant among the alternatives. For ζxi, the results that were obtained with the classical type and the fixed type are closer than those that were obtained with the scored type and the scalar type; hence, the PF-TOPSIS with the scored type and the scalar type can distinguish the alternatives more clearly.

To provide a clearer view of the comparison of the ranking order results that are based on RCxi and ζxi from the four approaches, we present Radar charts of the results in Fig. 6. Moreover, we add the ranking order results from Wei and Lu's method [42] (denoted as V) to Fig. 6.

The main differences are those of x2 and x3 between the scalar type approach and each of the other approaches. The alternative x3 is the best alternative that is based on RCxi and ζxi from the scalar type approach, which is assigned the second ranking order from the other approaches. The scalar type approach utilizes the scalar function to identify the PFPIS and PFNIS, and the proposed distance measure to calculate the distance from PFPIS and PFNIS, both of which reflects the unique properties of the PFNs, and x3 is the best alternative for both Dxi,x+ and Dxi,x. Therefore, the results from the scalar type PF-TOPSIS are reasonable. In addition, the result from Wei and Lu' method [42] is the same as that of the scored type approach that is based on RCxi because Wei and Lu's method [42] employs the score function to determine the magnitudes of the PFNs and the distance measure, while ignoring key properties of PFNs, such as the direction of commitment, the strength of commitment, and the radian.

Moreover, we consider the weight that corresponds to the importance of the criterion, which is determined by the decision maker's preference and may influence the ranking result. To evaluate the stability, we assume that all criterion weights are equal. Then, the weight vector of the four criteria in the example in Subsection 5.1 is expressed as ω=0.25,0.25,0.25,0.25T. The RCxi and ζxi under the same weights that are obtained via the four approaches are listed in Table 12.

RC I (Rank) II (Rank) III (Rank) IV (Rank)
x1 0.2468(5) 0.4580(5) 0.2349(5) 0.2558(5)
x2 0.7292(1) 0.6355(1) 0.7932(1) 0.6643(2)
x3 0.6931(2) 0.5921(2) 0.6204(2) 0.8219(1)
x4 0.5382(3) 0.5363(3) 0.5548(3) 0.5992(3)
x5 0.3935(4) 0.4776(4) 0.4990(4) 0.4094(4)

ζ I (Rank) II (Rank) III (Rank) IV (Rank)

x1 −2.4847(5) −0.9238(5) −3.2283(5) −1.6707(5)
x2 0.0000(1) 0.0000(1) 0.0538(1) −0.0959(2)
x3 −0.2127(2) −0.2193(2) −1.2268(2) 0.5257(1)
x4 −1.1797(3) −0.5427(3) −1.8979(4) −0.3777(3)
x5 −1.9569(4) −0.7963(4) −1.8047(3) −1.2125(4)

RC, relative closeness.

Table 12

RC(xi),ζ(xi) and their corresponding ranking results from four approaches under the same weights.

Among the alternatives, RCxi and ζxi are larger under the same weights. To facilitate comparison, we collect all the ranking order results from the four approaches in Table 13.

Index Ranking Order
RCxi from I x2x3x4x5x1
RCxi from II x2x3x4x5x1
RCxi from III x2x3x4x5x1
RCxi from IV x3x2x4x5x1
ζxi from I x2x3x4x5x1
ζxi from II x2x3x4x5x1
ζxi from III x2x3x5x4x1
ζxi from IV x3x2x4x5x1

RC, relative closeness.

Table 13

Ranking order results from four approaches based on the same weights.

According to the results in Tables 12 and 13, the ranking order results that are based on both RCxi and ζxi from the scalar type approach are consistent and satisfy the previous weight consideration condition, which demonstrates the stability of the this approach. Comparing the results with the classical type and fixed type approaches, the differences are similar to the previous weight consideration condition. All the approaches are stable except the scored type approach; hence, the PFPIS, the PFNIS, and the distance measure formulas influence RCxi and ζxi as well as their corresponding ranking orders of the alternatives.

In line with the properties of the PFNs, the scalar type PF-TOPSIS approach utilizes the scalar function to identify the PFPIS and PFNIS. Moreover, the proposed distance measure can obtain the distances from PFPIS and PFNIS, which were demonstrated to be relatively stable and accurate for MCDM problems compared to the classical type, the fixed type, and the scored type PF-TOPSIS approaches.

## 6. CONCLUSIONS AND FUTURE RESEARCH

Distance measurement is an essential method for distinguishing the objects in fuzzy environment. Considering the properties of PFNs, we proposed a novel distance measure that includes five parameters: the membership degree, the nonmembership degree, the strength of commitment, the direction of commitment, and the radian. This measure calculates the distance between two PFNs more accurately. Then, we applied the novel distance measurement method in TOPSIS with PF information to solve MCDM problems. On an example, we demonstrate that the scalar type PF-TOPSIS approach is effective and accurate.

The main contributions of this paper are summarized as follows: (1) This paper proposed a novel distance measurement method that is based on the properties of the PF sets, considering the parameters of μp, νp, rp, dp, and θp, employing the absolute squared deviations of μp, νp, and rp, and using the absolute deviations of dp and sinθp. The proposed distance measure reflects both the length distance and the angular distance, which ensures that the space of the PF sets is larger than that of the IF sets. The proposed distance measure satisfies the useful properties in the proven theorems. Moreover, it is consistent with the deviations of magnitude values. The proposed distance measure effectively overcame the maximum distance problem using the measures by Li and Zeng [32] and Zeng et al. [33], the transitivity problem using the measures by Zhang and Xu [5], and the great disparities problem using the measures by Li and Zeng [32] and Zeng et al. [33] (2) This paper improved the PF-TOPSIS methodology substantial. For the PFPIS and the PFNIS, we compared the existing magnitude comparison methods and selected the scalar function that fully reflects the characteristics of the PFNs. Then, we applied the novel distance measurement method to calculate the distances from the PFPIS and the PFNIS, which has been demonstrated to be more effective and accurate. (3) This paper proposed the more practical and precise PF-TOPSIS approach for addressing MCDM problems. Due to the full consideration of the properties and characteristics of PF sets, the scalar type PF-TOPSIS approach could handle the MCDM problems more effectively and precisely under the PF environment compared to the classical type, the fixed type, and the scored type PF-TOPSIS approaches.

This study utilized sinθp to represent the radian of commitment; however, we haven't determined whether cosθp serves the same function as sinθp in distance measure for PFNs. Moreover, we haven't applied the Euclidean distance measure or the generalized distance measure to PF-TOPSIS for MCDM problems. In the future, we can draw on the experience of the distance measure of IF sets by quaternary functions [30] to explore new properties of PFNs. We should compare the differences between cosθp and sinθp in distance measures for PFNs, then apply the other two distance measurement methods in PF-TOPSIS to evaluate their effectiveness and feasibility for MCDM problems.

## DATA AVAILABILITY STATEMENT

The datasets that were generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.

## CONFLICT OF INTEREST

The authors declare that there is no conflict of interest regarding the publication of this paper.

## ETHICAL APPROVAL

This article does not contain any studies with human participants or animals that were performed by the authors.

## Funding Statement

The corresponding author Ting-Yu Chen is grateful for grant funding support from the Taiwan Ministry of Science and Technology (MOST 108-2410-H-182-014-MY2) and Chang Gung Memorial Hospital (BMRP 574 and CMRPD2F0203) during the completion of this study.

## ACKNOWLEDGMENTS

The authors acknowledge the assistance of the respected editor and the anonymous referees via their insightful and constructive comments, which helped improve the overall quality of the paper. The corresponding author Ting-Yu Chen is grateful for grant funding support from the Taiwan Ministry of Science and Technology (MOST 108-2410-H-182-014-MY2) and Chang Gung Memorial Hospital (BMRP 574 and CMRPD2F0203) during the completion of this study.

Journal
International Journal of Computational Intelligence Systems
Volume-Issue
12 - 2
Pages
955 - 969
Publication Date
2019/09/09
ISSN (Online)
1875-6883
ISSN (Print)
1875-6891
DOI
10.2991/ijcis.d.190820.001How to use a DOI?
Open Access

TY  - JOUR
AU  - Fang Zhou
AU  - Ting-Yu Chen
PY  - 2019
DA  - 2019/09/09
TI  - A Novel Distance Measure for Pythagorean Fuzzy Sets and its Applications to the Technique for Order Preference by Similarity to Ideal Solutions
JO  - International Journal of Computational Intelligence Systems
SP  - 955
EP  - 969
VL  - 12
IS  - 2
SN  - 1875-6883
UR  - https://doi.org/10.2991/ijcis.d.190820.001
DO  - 10.2991/ijcis.d.190820.001
ID  - Zhou2019
ER  -