# A Modified TODIM Based on Compromise Distance for MAGDM withq-Rung Orthopair Trapezoidal Fuzzy Numbers

Authors: Benting Wan; Juelin Huang; Xiaolu Zhang
Journal: Complexity
 (2021)

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## Abstract

Theq-rung orthopair fuzzy number (q-ROFN) has been recently developed by Yager and has been widely applied in handling real-life decision-making problems. To enhance its usefulness in dealing with complex practical issues, this paper first proposes the new concept of q-rung orthopair trapezoidal fuzzy numbers (q-ROTrFNs) which is a new and useful extension of q-ROFNs. Then, we investigate the operation of q-ROTrFNs and develop a new ranking method for q-ROTrFNs. We also propose a new q-rung orthopair trapezoidal fuzzy Hamming distance measure. More important, we develop a useful q-rung orthopair trapezoidal fuzzy modified TODIM group decision-making method. In this method, a new q-rung orthopair trapezoidal fuzzy weighted aggregating (q-ROTrFWA) operator is developed to integrate individual decision matrices into the collective decision matrix, and a q-rung orthopair trapezoidal fuzzy distance measure-based compromise approach is proposed to determine the relative dominance degree of alternatives. It is worth to mention that the modified TODIM method not only expands the freedom of decision makers but also allows decision makers to choose the appropriate risk preference parameter. Finally, a case study on health management of hypertensive patients is conducted to demonstrate the feasibility of the modified TODIM group decision-making method, and the developed method is further verified by comparison analysis with the existing methods and sensitive analysis of different parameters.


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## Body

## 1. Introduction

Due to the complexity and uncertainty in real-life decision-making processes, it is difficult for experts to timely and accurately provide the estimated results in form of exact real numbers. To this end, Zadeh [1] proposed fuzzy set theory and advised experts to use membership functions instead of exact real (nonfuzzy) numbers to describe the estimated results in practical multiattribute decision-making (MADM) problems. Atanassov [2] further proposed the concept of intuitionistic fuzzy sets (IFSs) which include the degree of membership, the degree of nonmembership degree, and the hesitation degree. One important constraint is that the membership degree u and nonmembership degree v in IFS satisfy u+v≤1. In order to make IFSs suitable for more complex decision-making environment, Yager [3] proposed the concept of Pythagorean fuzzy set (PFS) in which the square sum of the membership and nonmemberships is less than one. Zhang and Xu [4] originally provided the mathematical expression form of PFSs and built the basic theoretical framework of PFSs. Subsequently, Zhang [5] extended PFSs into interval-valued fuzzy environment and developed interval-valued PFSs. Despite the usefulness of Pythagorean fuzzy theory [6, 7], practical applications have shown that PFSs still have some limitations. Therefore, Yager [8] recently developed the concept of generalized orthopair fuzzy set (is also called q-rung orthopair fuzzy set and is known simply as q-ROFS) in which uq+vq≤1 (q ≥ 1). The proposed q-ROFSs well incorporate the advantages of IFSs and PFSs and describe a wider range of information within different values of the parameter q. The q-ROFSs have been a research hotspot in recent years, and many excellent research results have been achieved, for example, the q-rung orthopair fuzzy approximate reasoning method [9], integrals and differentials of q-rung orthopair fuzzy functions [10–12], various q-rung orthopair fuzzy aggregation operators [13–16], q-rung orthopair fuzzy information measures [17, 18], and different q-rung orthopair fuzzy decision-making methods [19, 20] have been developed.Furthermore, with the development of fuzzy sets, the application of trapezoidal fuzzy numbers has been popularized. The trapezoidal fuzzy function is increased and maintained for a period of time and then decreased. Its maximum membership degree can be maintained for a period of time so that it can fit more realistic scenes. If the uncertainty of the alternative presents such a trend and it is presented in the form of interval, the trapezoidal fuzzy number is the most suitable for describing it. However, due to the complexity of the real-life decision-making environment, in some situations, the sum of the membership degree and the nonmembership degree are sometimes greater than 1, and the trapezoidal fuzzy number is not enough to adapt to the realistic decision-making situations. Motivated by the ideas ofq-ROFS [8] and trapezoidal fuzzy numbers, one important work of this paper is to introduce the concept of q-rung orthopair trapezoidal fuzzy numbers (q-ROTrFNs). We will investigate the operation of q-ROTrFNs and propose a ranking method for comparing q-ROTrFNs. We also propose a new q-rung orthopair trapezoidal fuzzy Hamming distance measure. As a useful extension of q-ROFS, the q-ROTrFNs have also wide application fields, especially in decision-making fields. In order to well handle complex practical MADM or group MADM (MAGDM) problems under q-ROTrFNs environment, it is necessary to develop correspondingly effective decision-making methods. TODIM (interactive and multicriteria decision-making) is a multicriteria method proposed by Gomes and Lima [21] on the basis of prospect theory. The biggest advantage of this method is that it can take into account the psychological behavior of decision makers and can well solve decision-making problems with risk characteristics. Gomes et al. [22] developed a Choquet integral-based TODIM method to deal with attributes’ interactions. To deal with complex practical MADM issues, many new extensions of TODIM methods, for instance, the hybrid TODIM method [23], the intuitionistic fuzzy TODIM method [24], the hesitant fuzzy TODIM method [25], the interval type-2 fuzzy TODIM method [26], the Pythagorean fuzzy TODIM approach [27], the double hierarchy hesitant fuzzy linguistic entropy-based TODIM approach [28], unbalanced hesitant fuzzy linguistic TODIM approach [29], linguistic distribution TODIM method [30], and hesitant fuzzy linguistic TODIM method [31]have been developed. Tian et al. [32] conducted a detail state-of the-art survey for the development of TODIM with different types of fuzzy sets. Although these studies are able to effectively deal with MAGDM problems, it is indispensable to further extend the TODIM method. On the one hand, the classical TODIM method distinguished decision data by subtraction, which cannot solve the MAGDM problem in the q-rung orthopair trapezoidal fuzzy environment. On the other hand, the modified TODIM method in this paper uses the compromise distance to capture the difference between q-ROTrFNs based on the compromise theory. Not only can the distance parameter combined with risk parameter of TODIM reflect the attitude of decision makers facing risks but also the integrity of the information will be ensured to a large extent so that the result is more in line with the actual situation.Therefore, we attempt to develop a modified TODIM group decision-making method based onq-ROTrFNs data. We first develop a new q-rung orthopair trapezoidal fuzzy weighted average aggregating (q-ROTrFWA) operator to integrate the individual decision matrix into the collective decision matrix. On the basis of the positive-ideal Hamming distance and the negative-ideal Hamming distance, a new q-rung orthopair trapezoidal fuzzy compromise distance measure is developed for calculating the relative dominance degrees of alternatives. This paper is organized as follows. Section 2 briefly reviews the basic concepts of q-ROFNs and Pythagorean trapezoidal fuzzy numbers (PTrFNs). In Section 3, the concept of q-ROTrFNs is defined, the operation of q-ROTrFNs, the q-ROTrFWA operator, and the new the ranking method, and Hamming distance are developed for q-ROTrFNs. In Section 4, the modified TODIM method is proposed, and the group decision-making method based on q-ROTrFNs is also given. In Section 5, an example is given to demonstrate the feasibility of the modified TODIM group decision-making method, and Section 6 summarizes this article.


## 2. Preliminaries

This section describes some basic knowledge involved in this paper, including the definition and operation ofq-ROFS and the definition and operation of PTrFNs.Definition 1.
(see [8]). Let X be the universe of discourse; A=x,uAx,vAx|x∈X is q-ROFS on the universe X, the degree of membership uA:X⟶0,1, and the degree of nonmembership vA:X⟶0,1, and they are satisfied by the following equation.(1)0≤uAxq+vAxq≤1,where q≥1, and the degree of hesitation πAx can be expressed as(2)πAx=1−uAxq+vAxqq,q≥1.Definition 2.
(see [8]). Let a=ua,va and β=uβ,vβ be two q-ROFNs; γ>0 is a random number; then, the following equations hold:(3)a⊕β=uaq+uβq−uaquβq1/q,vavβ,a⊗β=uauβ,vaq+vβq−vaqvβq1/q,γa=1−1−uaqγq,vaγ,aγ=uaγ,1−1−vaqγq.Definition 3.
(see [33]). Let a^=a,b,c,d;ua^,va^ be a PTrFN, and its membership function and nonmembership function, respectively, satisfy equations (4) and (5):(4)ua^x=x−aua^b−a,ifa≤x<b,ua^,ifb≤x≤c,x−dua^d−c,ifc<x≤d,0,ifx<a,x>d,(5)va^x=b−x+va^x−ab−a,ifa≤x<b,va^,ifb≤x≤c,x−c+va˜d−xd−c,ifc<x≤d,1,ifx<a,x>d,where ua^ is the maximum degree of membership of a^ and va^ is the minimum degree of nonmembership of a^, which satisfy 0≤ua^≤1, 0≤va^≤1, and 0≤ua^2+va^2≤1.Definition 4.
(see [33]). Let a^=aa^,ba^,ca^,da^;ua^,va^ and b^=ab^,bb^,cb^,db^;ub^,vb^ be two PTrFNs; γ>0 is random number; then, the following equations hold:(6)a^⊕b^=aa^+ab^,ba^+bb^,ca^+cb^,da^+db^;ua^2+ub^2−ua^2ub^2,va^vb^,(7)a^⊗b^=aa^ab^,ba^bb^,ca^cb^,da^db^;ua^ub^,va^2+vb^2−va^2vb^2,(8)γa^=γaa^,γba^,γca^,γda^;1−1−ua^2γ,va^γ,(9)a^γ=da^γ,ca^γ,ba^γ,aa^γ;ua^γ,1−1−va^2γ.


## 3.q-Rung Orthopair Trapezoidal Fuzzy Numbers

This section first puts forward the definition ofq-ROTrFN and investigates the operation low of q-ROTrFNs. Then, a new ranking method of q-ROTrFNs, a q-ROTrFWA operator, and q-rung orthopair trapezoidal fuzzy Hamming distance measure are developed, respectively.Definition 5.
Leta˜=a¯,a1,a2,a¯;ua˜max,va˜min be a q-ROTrFN; its membership function ua˜ is shown as(10)ua˜x=x−a¯ua˜maxa1−a¯,ifa¯≤x<a1,ua˜maxifa1≤x≤a2,a¯−xua˜maxa¯−a2,ifa2<x≤a¯,0,ifx<a¯,x>a¯.
And, the nonmembership functionva˜is shown as(11)va˜x=a1−x+va˜minx−a¯a1−a¯,ifa¯≤x<a1,va˜min,ifa1≤x≤a2,x−a2+va˜mina¯−xa¯−a2,ifa2<x≤a¯,1,ifx<a¯,x>a¯,where 0≤ua˜max≤1, 0≤va˜min≤1, and 0≤ua˜maxq+va˜minq≤1, q≥1.
As shown in Figures1 and 2, the q-ROTrFN a˜ is different from the PTrFN and the intuitionistic trapezoidal fuzzy number (ITrFN), and obviously, the q-ROTrFNs are more flexible because of 0≤ua˜maxq+va˜minq≤1.
πa˜x is the degree of indeterminacy of the q-ROTrFN a˜ and is shown as(12)πa˜x=1−ua˜qx−va˜qxq.
It is easy to see that trapezoidal fuzzy numbers, ITrFN and PTrFN, are the special cases of the developedq-ROTrFN. That is to say, the q-ROTrFN degenerates to an ITrFN when q = 1, namely, 0≤ua˜max+va˜min≤1, and the q-ROTrFN degenerates into the PTrFN when q = 2, namely, 0≤ua˜max2+va˜min2≤1.Figure 1
q-ROTrFN a˜=a¯,a1,a2,a¯;ua˜max,va˜min.Figure 2
Information space with different values ofq.Definition 6.
Letγ be a random number and γ>0 and a˜=a¯,a1,a2,a¯;ua˜max,va˜min and b˜=b¯,b1,b2,b¯;ub˜max,vb˜min be two q-ROTrFNs; then, the following equations (13)–(16) hold:(13)a˜⊕b˜=a¯+b¯,a1+b1,a2+b2,a¯+b¯;ua˜maxq+ub˜maxq−ua˜maxqub˜maxqq,va˜minvb˜min,(14)a˜⊗b˜=a¯b¯,a1b1,a2b2,a¯b¯;ua˜maxub˜max,va˜minq+vb˜minq−va˜minqvb˜minqq,(15)γa˜=γa¯,γa1,γa2,γa¯;1−1−ua˜maxqγq,va˜minγ,(16)a˜γ=a¯γ,a2γ,a1γ,a¯γ;ua˜maxγ,1−1−va˜minqγq.Theorem 1.
Given twoq-ROTrFNs a˜=a¯,a1,a2,a¯;ua˜max,va˜min and b˜=b¯,b1,b2,b¯;ub˜max,vb˜min, and γ>0,λ>0; then, we have(1)
a˜⊕b˜=b˜⊕a˜(2)
a˜⊗b˜=b˜⊗a˜(3)
γa˜⊕b˜=γa˜⊕γb˜(4)
γa˜⊕λa˜=γ+λa˜Proof.
(17)1a˜⊕b˜=a¯+b¯,a1+b1,a2+b2,a¯+b¯;ua˜maxq+ub˜maxq−ua˜maxqub˜maxqq,va˜minvb˜min=b¯+a¯,b1+a1,b2+a2,b¯+a¯;ub˜maxq+ua˜maxq−ub˜maxqua˜maxqq,vb˜minva˜min=b˜⊕a˜.(2)
The same as (1) can be proved,a˜⊗b˜=b˜⊗a˜.(18)3γa˜⊕b˜=γa¯+b¯,a1+b1,a2+b2,a¯+b¯;ua˜maxq+ub˜maxq−ua˜maxqub˜maxqq,va˜minvb˜min=γa¯+b¯,γa1+b1,γa2+b2,γa¯+b¯;1−1−ua˜maxq1−ub˜maxqγq,va˜minvb˜minγ=γa¯+b¯,γa1+b1,γa2+b2,γa¯+b¯;1−1−ua˜maxqγ1−ub˜maxqγq,va˜minvb˜minγ=γa¯+b¯,γa1+b1,γa2+b2,γa¯+b¯;1−1−ua˜maxqγ+1−1−ub˜maxqγ−1−1−ub˜maxqγ−1−ua˜maxqγ+1−ua˜maxqγ1−ub˜maxqγq,va˜minvb˜minγ=γa¯+b¯,γa1+b1,γa2+b2,γa¯+b¯;1−1−ua˜maxqγ+1−1−ub˜maxqγ−1−1−ub˜maxqγ1−1−ub˜maxqγq,va˜minvb˜minγ=γa˜⊕γb˜.(19)4γa˜=γa¯,γa1,γa2,γa¯;1−1−ua˜maxqγq,va˜minγλa˜=λa¯,λa1,λa2,λa¯;1−1−ua˜maxqλq,va˜minλγa˜⊕λa˜=γ+λa¯,γ+λa1,γ+λa2,γ+λa¯;1−1−ua˜maxqγ+1−1−ua˜maxqλ−1−1−ua˜maxqγ1−1−ua˜maxqλq,va˜minγva˜minλ=γ+λa¯,γ+λa1,γ+λa2,γ+λa¯;1−1−ua˜maxqγ1−ua˜maxqλq,va˜minγva˜minλ=γ+λa¯,γ+λa1,γ+λa2,γ+λa¯;1−1−ua˜maxqγ+λq,va˜minγ+λ=γ+λa˜.Example 1.
Leta˜=2,3,5,7;0.7,0.5 and b˜=1,4,6,8;0.8,0.3 be two q-ROTrFNs; γ=2,λ=3, and q = 3; then, we have(20)a˜⊕b˜=2+1,3+4,5+6,7+8;0.73+0.83−0.73×0.833,0.5×0.3=3,7,11,15;0.8791,0.15γa˜⊕b˜=2×3,2×7,2×11,2×15;1−1−0.73+0.83−0.73×0.8323,0.5×0.32=6,14,22,30;0.9645,0.0225γa˜⊕λa˜=2+3×2,2+3×3,2+3×5,2+3×7;1−1−0.732+1−1−0.733−1−1−0.732×1−1−0.7333,0.52×0.53=10,15,25,35;0.9574,0.03125.Definition 7.
Leta˜=a¯,a1,a2,a¯;ua˜max,va˜min be a q-ROTrFN, q≥1; then, the score function of a˜ is shown as(21)Sa˜=a¯+a1+a2+a¯4×ua˜maxq−va˜minq.Definition 8.
Leta˜=a¯,a1,a2,a¯;ua˜max,va˜min be a q-ROTrFN, q≥1; then, the exact function of a˜ is shown as(22)Ha˜=a¯+a1+a2+a¯4×ua˜maxq+va˜minq.
Obviously,Ha˜ reflects the degree of accuracy of a˜, and a˜ has become more accurate as Ha˜ increases. Then, the comparison method of q-ROTrFNs according to Definitions 7 and 8 can be determined.Definition 9.
Leta˜=a¯,a1,a2,a¯;ua˜max,va˜min and b˜=b¯,b1,b2,b¯;ub˜max,vb˜min be two q-ROTrFNs; q≥1, Sa˜, Sb˜, Ha˜, and Hb˜ are the score function and exact function of a˜ and b˜, respectively; then, the ranking method for q-ROTrFNs is introduced as follows:(1)
IfSa˜>Sb˜, then a˜>b˜(2)
IfSa˜<Sb˜, then a˜<b˜(3)
WhenSa˜=Sb˜, if Ha˜>Hb˜, then a˜>b˜IfHa˜<Hb˜, then a˜<b˜IfHa˜=Hb˜, then a˜=b˜Example 2.
Given twoq-ROTrFNs a˜=1,5,6,7;0.8,0.5 and b˜=4,7,8,9;0.3,0.9 and q = 2, then we use the developed ranking method to compare the magnitudes of these two q-ROTrFNs as below:(23)Sa˜=1+5+6+74×0.64−0.25=1.8525,Sb˜=4+7+8+94×0.09−0.81=−5.04.
Obviously,Sb˜<Sa˜, so it can be found that a˜>b˜.
However, ifa˜=4,5,5,6;0.4,0.3 and b˜=5,7,7,9;0.3,0.2, q = 2, it can be obtained that Sa˜=Sb˜=0.35. Then, the relationship between a˜ and b˜ has to be determined by an exact function:(24)Ha˜=4+5+5+64×0.16+0.09=1.25,Hb˜=5+7+7+94×0.09+0.04=0.91.
Obviously,Hb˜<Ha˜, which means a˜>b˜.
On the contrary, ifa˜=1,5,6,7;0.8,0.5 and b˜=4,7,8,9;0.3,0.9, q = 3, we have(25)Sa˜=1+5+6+74×0.512−0.125=1.83825,Sb˜=4+7+8+94×0.027−0.729=−4.914.
It is easy to finda˜>b˜. Similarly, if a˜=4,5,5,6;0.4,0.3 and b˜=5,7,7,9;0.3,0.2, q = 3, it can be obtained that Sa˜=0.185>Sb˜=0.133. Therefore, a˜>b˜.
Next, we propose a newq-ROTrFWA operator and investigate its properties.Definition 10.
Leta˜=a˜1,a˜2,…,a˜nT be a set of q-ROTrFNs and ω=ω1,ω2,…,ωnT be the weight vector of a˜ii=1,2,…,n,ωi∈0,1 and ∑i=1nωi=1; then, the definition of q-ROTrFWA is shown as(26)q−ROTrFWAa˜1,a˜2,…,a˜n=∑i=1nωia˜i=ω1a˜1⊕ω2a˜2⊕⋯⊕ωna˜n.Theorem 2.
The aggregating result obtained byq−ROTrFWAa˜1,a˜2,…,a˜n is still a q-ROTrFN, which is shown as(27)q−ROTrFWAa˜1,a˜2,…,a˜n=∑i=1nωia˜ia¯,∑i=1nωia˜ia1,∑i=1nωia˜ia2,∑i=1nωia˜ia¯;1−∏i=1n1−ua˜imaxqωiq,∏i=1nva˜iminωi.Proof.
Theorem2 can be proved by mathematical induction as follows:(1)
Obviously, whenn = 1, the equation holds.(2)
Assume that the result obtained is true whenn = m, and the result is as follows:(28)q−ROTrFWAa˜1,a˜2,…,a˜m=∑i=1mωia˜ia¯,∑i=1mωia˜ia1,∑i=1mωia˜ia2,∑i=1mωia˜ia¯;1−∏i=1m1−ua˜imaxqωiq,∏i=1mva˜iminωi.
Whenn = m + 1,(29)ωm+1a˜m+1=ωm+1a˜m+1a¯,ωm+1a˜m+1a1,ωm+1a˜m+1a2,ωm+1a˜m+1a¯;1−1−ua˜m+1maxqωm+1q,vm+1minωm+1.
According to equation (8), the result is as follows:(30)q−ROTrFWAa˜1,a˜2,…,a˜m⊕ωm+1a˜m+1=∑i=1mωia˜ia¯+ωm+1a˜m+1a¯,∑i=1mωia˜ia1+ωm+1a˜m+1a1,∑i=1mωia˜ia2+ωm+1a˜m+1a2,∑i=1mωia˜ia¯+ωm+1a˜m+1a¯,1−∏i=1m1−ua˜imaxqωi+1−1−ua˜m+1maxqωm+1−1−∏i=1m1−ua˜imaxqωi×1−1−ua˜m+1maxqωm+1q,∏i=1mva˜iminωi×vm+1minωm+1=∑i=1m+1ωia˜ia¯,∑i=1m+1ωia˜ia1,∑i=1m+1ωia˜ia2,∑i=1m+1ωia˜ia¯;1−∏i=1m+11−ua˜m+1maxqωiq,∏i=1m+1va˜iminωi=q−ROTrFWAa˜1,a˜2,…,a˜m+1.
It can be seen from the proof that, for any integern, the collective result obtained by the q-ROTrFWA operator is still a q-ROTrFN.Example 3.
Given fourq-ROTrFNs a˜=0.2,0.3,0.5,0.6;0.6,0.3, b˜=0.3,0.5,0.6,0.7;0.3,0.8, c˜=0.2,0.4,0.5,0.6;0.8,0.5, and d˜=0.1,0.5,0.7,0.8;0.7,0.5, and q=2 and the corresponding weights of four q-ROTrFNs ωa˜=0.15, ωb˜=0.35, ωc˜=0.2, and ωd˜=0.3, then the result obtained by the q-ROTrFWA is shown as follows:(31)q−ROTrFWAa˜,b˜,c˜,d˜=ωaa˜⊕ωbb˜⊕ωcc˜⊕ωdd˜=0.15×0.2,0.3,0.5,0.6;0.6,0.3⊕0.35×0.3,0.5,0.6,0.7;0.3,0.8⊕0.2×0.2,0.4,0.5,0.6;0.8,0.5⊕0.3×0.1,0.5,0.7,0.8;0.7,0.5=0.03,0.05,0.08,0.09;0.25,0.83⊕0.11,0.18,0.21,0.25;0.18,0.92⊕0.02,0.1,0.14,0.16;0.35,0.87⊕0.03,0.15,0.21,0.24;0.43,0.81=0.19,0.48,0.64,0.74;0.59,0.54.Definition 11.
Leta˜=a¯,a1,a2,a¯;ua˜max,va˜min and b˜=b¯,b1,b2,b¯;ub˜max,vb˜min be two q-ROTrFNs; the Hamming distance da˜,b˜ between a˜ and b˜ is show as(32)da˜,b˜=181+ua˜maxq−va˜minqqa¯−1+ub˜maxq−vb˜minqqb¯+1+ua˜maxq−va˜minqqa1−1+ub˜maxq−vb˜minqqb1+1+ua˜maxq−va˜minqqa2−1+ub˜maxq−vb˜minqqb2+1+ua˜maxq−va˜minqqa¯−1+ub˜maxq−vb˜minqqb¯.Example 4.
Given twoq-ROTrFNs a˜=0.4,0.7,0.8,1;0.6,0.5 and b˜=0.5,0.7,1,2;0.6,0.7 and q = 3, we get the Hamming distance between them as follows:(33)da˜,b˜=181+0.63−0.533×0.4−1+0.63−0.733×0.5+1+0.63−0.533×0.7−1+0.63−0.733×0.7+1+0.63−0.533×0.8−1+0.63−0.733×1+1+0.63−0.533×1−1+0.63−0.733×2=0.141.
As a classical distance measure, the developed Hamming distance provides a way to quantify the difference betweenq-ROTrFNs, which indicates the closeness of two q-ROTrFNs and takes into account the multifactor differences of q-ROTrFNs.


## 4. A Modified TODIM Group Decision-Making Method forq-ROTrFNs

In this section, we develop a newq-rung orthopair trapezoidal fuzzy modified TODIM group decision-making method for solving MAGDM problems with q-ROTrFNs. We here consider a common MAGDM problem with a set of feasible alternatives A=A1,A2,…,Am; each alternative is evaluated based on n attributes C=C1,C2,…,Cn, where the weight of attribute Cj can be expressed as ωj, and fulfills 0≤ωj≤1 and ∑j=1nωj=1. Let P=P1,P2,…,Pt be the set of experts; the weight of the expert Pk can be expressed as λk and the weights of different experts satisfy ∑k=1tλk=1,λk≥0k=1,2,…,t. For each expert, the decision matrix can be obtained according to different alternatives and attributes. For the kth expert, the decision matrix is expressed as Ak=a˜ijkm×n,k=1,2,…,t. The element a˜ijk in the decision matrix means that the evaluated data of the alternative Ai on the attribute Cj is provided by the expert Pk, and each a˜ijk is a q-ROTrFN. On the basis of the developed q-ROTrFWA operator, we can integrate the individual decision matrices Ak=a˜ijkm×nk=1,2,…,t into the collective decision matrix R=r˜ijm×n as follows:(34)r˜ij=q−ROTrFWAa˜ij1,a˜ij2,…,a˜ijt,i=1,2,…,m,j=1,2,…,n.After obtaining the collective decision matrixR=r˜ijm×n, we next measure the relative dominance of each alternative over the others by constructing the prospect value function [25]. Firstly, the attribute with the highest weight can be regarded as the reference attribute according to the TODIM method [21, 25] and then the relative weight ωcr of the attribute Cc to the reference attribute Cr can be obtained as(35)ωcr=ωcωr,where ωr=maxωc|c=1,2,…,n is the reference weight.To calculate the relative dominance degree of each alternative, this section first proposes a newq-rung orthopair trapezoidal fuzzy distance measure-based compromise approach. The compromise method can take fully into account the importance of positive-ideal alternative and negative-ideal alternative in decision-making and well avoid the defect of Hamming distance that sometimes cannot reflect the distinction of q-ROTrFNs.Definition 12.
IfA=a˜ijm×n=a¯,a1,a2,a¯;ua˜max,va˜minijm×n is a q-ROTrFNs matrix given by the expert, then the positive-ideal set A+ and the negative-ideal set A− of the alternatives are shown as(36)A+=a˜1+,a˜2+,…,a˜m+,where(37)a˜i+=max1≤j≤na¯ij,max1≤j≤na1ij,max1≤j≤na2ij,max1≤j≤na¯ij;max1≤j≤nua˜ijmax,min1≤j≤nva˜ijmin,A−=a˜1−,a˜2−,…,a˜m−,where(38)a˜i−=min1≤j≤na¯ij,min1≤j≤na1ij,min1≤j≤na2ij,min1≤j≤na¯ij;min1≤j≤nua˜ijmax,max1≤j≤nva˜ijmin.Definition 13.
The distanceD+ between each element aij in the q-ROTrFNs matrix and the positive-ideal alternative A+=a˜1+,a˜2+,…,a˜m+, and the distance D− between each element aij and the negative-ideal alternative A−=a˜1−,a˜2−,…,a˜m− are defined as equations (39) and (40), respectively:(39)D+a˜ij=ωjda˜ij,a˜i+=18ωj1+ua˜ijmaxq−va˜ijminqqa¯ij−1+ua˜i+maxq−va˜i+minqqa¯i++1+ua˜ijmaxq−va˜ijminqqa1ij−1+ua˜i+maxq−va˜i+minqqa1i++1+ua˜ijmaxq−va˜ijminqqa2ij−1+ua˜i+maxq−va˜i+minqqa2i++1+ua˜ijmaxq−va˜ijminqqa¯ij−1+ua˜i+maxq−va˜i+minqqa¯i+,(40)D−a˜ij=ωjda˜ij,a˜i−=18ωj1+ua˜ijmaxq−va˜ijminqqa¯ij−1+ua˜i−maxq−va˜i−minqqa¯i−+1+ua˜ijmaxq−va˜ijminqqa1ij−1+ua˜i−maxq−va˜i−minqqa1i−+1+ua˜ijmaxq−va˜ijminqqa2ij−1+ua˜i−maxq−va˜i−minqqa2i−+1+ua˜ijmaxq−va˜ijminqqa¯ij−1+ua˜i−maxq−va˜i−minqqa¯i−,where ωj is the relative weight of the attribute, 0≤ωj≤1 and ∑j=1mωj=1. Obviously, D+ and D− are the further expansion of q-ROTrFNs based on Hamming distance, which can fully consider the impacts of benefit and cost on decision-making. However, how to deal with MADM via D+ and D− is still the focus of the follow-up research.
In order to find a compromise, a method to express the relative importance ofD+ and D− by the parameters α and 1−α is as follows, where 0≤α≤1. Based on the proposed distance measure function, the distance-based compromise with q-ROTrFNs is defined as follows.Definition 14.
LetA=a˜ijm×n=a¯,a1,a2,a¯;ua˜max,va˜minijm×n be a q-rung orthopair trapezoidal fuzzy evaluation matrix; the compromise value of the element aij is shown as(41)Da˜ij=αD−a˜ij−1−αD+a˜ij.
It is easy to find that0≤Da˜ij≤1. By setting an appropriate distance parameter α, the compromise value of a˜i can be obtained according to its positive-ideal distance D+ and negative-ideal distance D−. On this basis, a distance measure-based compromise approach is proposed to distinguish q-ROTrFNs.Definition 15.
Leta˜=a¯,a1,a2,a¯;ua˜max,va˜min and b˜=b¯,b1,b2,b¯;ub˜max,vb˜min be two q-ROTrFNs; then, the distance measure-based compromise value Cda˜,b˜ between a˜ and b˜ is shown as(42)Cda˜,b˜=Da˜−Db˜,
Compared with the Hamming distance proposed in Definition11, the compromise distance is more accurate and comprehensive when applied to decision-making methods. Based on the Hamming distance, the compromise distance can fully consider the importance of the two aspects of the ideal alternatives in the decision-making and is more feasible in actual decision-making. Next, we use Example 5 to show the advantage of the compromise distance.Example 5.
Consider aq-rung orthopair trapezoidal fuzzy matrix R as(43)R=0.52,0.40,0.40,0.34;0.71,0.46,0.45,0.42,0.55,0.47;0.57,0.53,0.38,0.33,0.43,0.44;0.71,0.46,0.44,0.4,0.42,0.40;0.48,0.660.49,0.34,0.39,0.43;0.70,0.46,0.35,0.34,0.40,0.37;0.41,0.68,0.40,0.53,0.64,0.50;0.51,0.58,0.38,0.44,0.3,0.29;0.57,0.570.42,0.38,0.41,0.44;0.62,0.51,0.39,0.41,0.36,0.48;0.49,0.58,0.41,0.45,0.28,0.32;0.29,0.67,0.55,0.74,0.74,0.43;0.50,0.700.57,0.67,0.46,0.22;0.67,0.58,0.52,0.34,0.30,0.38;0.37,0.75,0.37,0.42,0.5,0.46;0.61,0.6,,0.37,0.47,0.45,0.52;0.52,0.65.
According to equations (36) and (38), the positive-ideal set A+ and the negative-ideal set A− of the alternatives can be obtained as(44)A+=a˜1+,a˜2+,a˜3+,a˜4+=0.52,0.42,0.55,0.47;0.71,0.46,0.49,0.53,0.64,0.50;0.70,0.46,0.55,0.74,0.74,0.48;0.62,0.51,0.57,0.67,0.50,0.52;0.67,0.58,A−=a˜1−,a˜2−,a˜3−,a˜4−=0.38,0.33,0.40,0.34;0.48,0.66,0.35,0.34,0.30,0.29;0.41,0.68,0.39,0.38,0.28,0.32;0.29,0.70,0.37,0.34,0.30,0.22;0.37,0.71.
Letα=0.5 and q = 2; the Hamming distance and the compromise distance for R are shown in the Figure 3.
Obviously, the Hamming distance is greater than the compromise distance according to Figure3. The reason for the situation is that the compromise distance takes into account the difference between the positive- and negative-ideal distances of alternatives compared to the Hamming distance. In the light of this characteristic of the compromise distance, it will be more comprehensive when dealing with the deviation between q-ROTrFNs, and it can also be seen from the figure that the compromise distance is more stable. Therefore, this paper will further apply the compromise distance to modify the TODIM method for group decision-making.
By using the above developed compromise distance, we can calculate the relative dominance degree of the alternativeAi over the alternative Aj concerning the attribute Cc using the following expression:(45)ΦcAi,Aj=ωcr∑c=1nωcrCdr˜ic,r˜jc,ifr˜ic>r˜jc,0,ifr˜ic=r˜jc,−1θ∑c=1nωcrωcrCdr˜ic,r˜jc,ifr˜ic<r˜jc,where r˜ic represents a q-ROTrFN in the matrix and Cdr˜ic,r˜jc reflects the compromise distance between two q-ROTrFNs. θθ>0 is the attenuation factor, and the smaller the value of the parameter θ, the higher loss avoidance of the expert.
Obviously, there exist three cases in equation (45). (1) If r˜ic>r˜jc, then ΦcAi,Aj represents a gain. (2) If r˜ic=r˜jc, then ΦcAi,Aj represents a nil. (3) If r˜ic<r˜jc, then ΦcAi,Aj represents a loss.
By aggregatingϕcAi,Aj with each attribute Cc, the dominance of the alternative Ai over the alternative Aj can be obtained as follows:(46)ϑAi,Aj=∑c=1nϕcAi,Aj,i,j=1,2,…,m.
Finally, we calculate the overall dominance degree of the alternativeAi according to the following expression:(47)ξAi=∑k=1mϑAi,Ak−mini∑k=1mϑAi,Akmaxi∑k=1mϑAi,Ak−mini∑k=1mϑAi,Ak,i=1,2,…,m.
Obviously,0≤ξAi≤1, and the greater ξAi is, the better the alternative Ai will be. Therefore, the ranking order of all alternatives Aii=1,2,…,m can be determined according to the increasing order of the dominance degree of the alternative Aii=1,2,…,m. The algorithm of the q-rung orthopair trapezoidal fuzzy modified TODIM group decision-making method is introduced in Algorithm 1.
The process of the group decision-making method is shown in Figure4.Algorithm 1: Algorithm of the modified TODIM method.
Input: q-ROTrFNs matricesOutput: ranking result according to the value of ξiProcessStep 1: to eliminate the influence of different physical dimensions on decision-making, transform the evaluation matrix into a standard matrixAk,k=1,2,…,t.Step 2: decision maker chooses the value ofθ according to the risk preference and gets the value of q by the observation method or the traversal method. The value of q needs to hold 0≤ua˜ijkmaxq+va˜ijkminq≤1in matrix Ak,k=1,2,…,t.Step 3: aggregateAk by equation (34), a˜ijt represents q-ROTrFNs in the decision matrix given by the tth expert, and get the collective matrix R=r˜ijm×n.Step 4: get the relative weight of each attributeCc using equation (35).Step 5: obtain the positive-ideal alternativeA+=a˜1+,a˜2+,…,a˜m+ and the negative-ideal alternative A−=a˜1−,a˜2−,…,a˜m− of A=A1,A2,…,Am according to equations (36) and (38) and using equations (39) and (40) to get the distance between them and each element in R=r˜ijm×n.Step 6: the compromise distances of elements inr˜ij are obtained via equations (41) and (42).Step 7: get the dominance of alternativeAi relative to alternative Aj by equations (45) and (46).Step 8: calculate the overall dominanceξi by equation (47).Step 9: according to the value ofξi, the ranking result of the alternatives is the same as rank of ξi, and the alternative with the largest ξi is the optimal.Figure 3
Difference between Hamming distance and compromise distance.Figure 4
The process of the group method withq-ROTrFNs.


## 5. Case Study on Health Management of Hypertensive Patients

In this section, a case study on health management of hypertensive patients is conducted to demonstrate the feasibility of the modified TODIM group decision-making method, and the comparison analysis with the existing methods and sensitive analysis of different parameters are investigated in detail.


### 5.1. Decision Results Obtained by the Developed Modified TODIM Method

At present, there are about 245 million hypertensive patients in China, and related diseases caused by hypertension have brought a heavy burden to residents and society. In order to improve the health of residents, the health management of hypertensive patients has been highly valued by the public and the government. In order to simplify the management process, a community hospital plans to develop hypertension health management software to facilitate the treatment and management of patients by community doctors, which means that doctors need to evaluate the risk level based on the residents’ inspection. The main observation data include blood pressure measurement, cardiovascular damage, related diseases (mainly related to cardiovascular and cerebrovascular diseases), and related risk factors (age, smoking, drinking and sleeping, eating more sodium and less potassium, greasy, obesity, work and life pressure, etc.), and the weights of them areω=0.4,0.3,0.2,0.1. In order to better provide prompts to doctors, the risk level of residents is indicated by different colors: red indicates high risk, orange indicates moderate risk, blue indicates mild danger, and green indicates normal, which are expressed as A=A1,A2,A3,A4.Considering that these four indicators are abnormal to the normal process, the process is ascending, adjusting, and gradually recovering. The fuzzy decision model is used to construct the system for processing. In order to facilitate the flexible evaluation of experts, theq-ROTrFNs are used for evaluation. Now, a 65-year-old resident has a systolic blood pressure of 140 mmHg and a diastolic blood pressure of 99 mmHg. He suffers from hyperglycemia, and parents suffer from high blood pressure. Besides, he smokes and drinks frequently, and his taste tends to be heavy. Four experts are invited to evaluate the risk level of this resident, and the weights of experts are λ=0.3,0.3,0.2,0.2. The evaluation data are obtained in Tables 1–4.Table 1
Evaluation value matrix by expert1A1.
C1C2C3C4A1(0.3, 0.5, 0.6, 0.8; 0.9, 0.3)(0.1, 0.2, 0.3, 0.4; 0.6, 0.3)(0.5, 0.6, 0.8, 0.9; 0.8, 0.1)(0.4, 0.5, 0.6, 0.7; 0.7, 0.4)A2(0.6, 0.7, 0.8, 0.9; 0.8, 0.5)(0.5, 0.6, 0.7, 0.8; 0.7, 0.3)(0.4, 0.5, 0.7, 0.8; 0.9, 0.4)(0.1, 0.3, 0.5, 0.6; 0.8, 0.3)A3(0.1, 0.2, 0.4, 0.5; 0.7, 0.4)(0.2, 0.3, 0.5, 0.6; 0.5, 0.4)(0.5, 0.6, 0.7, 0.8; 0.6, 0.2)(0.3, 0.4, 0.5, 0.7; 0.6, 0.5)A4(0.3, 0.4, 0.5, 0.6; 0.2, 0.7)(0.1, 0.2, 0.4, 0.5; 0.4, 0.5)(0.2, 0.3, 0.5, 0.7; 0.5, 0.5)(0.1, 0.3, 0.4, 0.5; 0.4, 0.7)Table 2
Evaluation value matrix by expert2A2.
C1C2C3C4A1(0.2, 0.3, 0.5, 0.7; 0.8, 0.3)(0.1, 0.2, 0.3, 0.4; 0.6, 0.2)(0.5, 0.6, 0.8, 0.9; 0.9, 0.1)(0.2, 0.3, 0.5, 0.7; 0.7, 0.4)A2(0.4, 0.5, 0.6, 0.9; 0.6, 0.2)(0.5, 0.6, 0.7, 0.8; 0.7, 0.4)(0.4, 0.5, 0.6, 0.8; 0.8, 0.3)(0.1, 0.2, 0.4, 0.5; 0.6, 0.3)A3(0.1, 0.3, 0.5, 0.7; 0.5, 0.4)(0.2, 0.3, 0.4, 0.6; 0.4, 0.8)(0.3, 0.6, 0.7, 0.8; 0.7, 0.2)(0.3, 0.4, 0.5, 0.8; 0.6, 0.2)A4(0.2, 0.3, 0.5, 0.6; 0.4, 0.7)(0.2, 0.4, 0.5, 0.6; 0.5, 0.7)(0.2, 0.3, 0.5, 0.7; 0.5, 0.5)(0.4, 0.6, 0.7, 0.9; 0.3, 0.2)Table 3
Evaluation value matrix by expert3A3.
C1C2C3C4A1(0.3, 0.4, 0.6, 0.7; 0.8, 0.4)(0.1, 0.2, 0.3, 0.5; 0.6, 0.2)(0.4, 0.6, 0.8, 0.9; 0.8, 0.1)(0.3, 0.5, 0.6, 0.8; 0.7, 0.4)A2(0.2, 0.3, 0.4, 0.5; 0.7, 0.1)(0.4, 0.5, 0.7, 0.8; 0.5, 0.3)(0.4, 0.5, 0.6, 0.7; 0.6, 0.1)(0.2, 0.3, 0.5, 0.7; 0.6, 0.3)A3(0.2, 0.3, 0.5, 0.6; 0.5, 0.3)(0.3, 0.4, 0.5, 0.6; 0.4, 0.2)(0.5, 0.7, 0.8, 0.9; 0.5, 0.2)(0.3, 0.4, 0.5, 0.7; 0.6, 0.2)A4(0.3, 0.4, 0.5, 0.6; 0.2, 0.6)(0.1, 0.2, 0.4, 0.5; 0.5, 0.4)(0.3, 0.4, 0.5, 0.7; 0.4, 0.5)(0.1, 0.2, 0.4, 0.5; 0.4, 0.2)Table 4
Evaluation value matrix by expert4A4.
C1C2C3C4A1(0.6, 0.7, 0.8, 0.9; 0.8, 0.5)(0.4, 0.6, 0.8, 0.9; 0.6, 0.3)(0.2, 0.5, 0.8, 0.9; 0.8, 0.3)(0.5, 0.6, 0.8, 0.9; 0.9, 0.2)A2(0.3, 0.4, 0.5, 0.7; 0.7, 0.2)(0.2, 0.4, 0.5, 0.6; 0.5, 0.5)(0.3, 0.4, 0.5, 0.7; 0.7, 0.4)(0.2, 0.4, 0.5, 0.6; 0.9, 0.2)A3(0.4, 0.5, 0.7, 0.8; 0.5, 0.4)(0.2, 0.4, 0.5, 0.6; 0.4, 0.5)(0.1, 0.5, 0.7, 0.8; 0.4, 0.5)(0.2, 0.3, 0.4, 0.6; 0.6, 0.4)A4(0.2, 0.4, 0.5, 0.7; 0.1, 0.7)(0.1, 0.3, 0.4, 0.5; 0.3, 0.3)(0.2, 0.3, 0.4, 0.7; 0.4, 0.6)(0.1, 0.2, 0.3, 0.5; 0.3, 0.8)Now, we employ the developed modifiedq-rung orthopair trapezoidal fuzzy TODIM group decision-making method to help the doctor identify the risk level of hypertensive patients. In this case, we consider q = 2, θ=1, and α=0.5; then, we aggregate the evaluation matrix Akk=1,2,3,4 given by these four experts to obtain the collective matrix R=r˜ijm×n by equation (34), as shown in Table 5.Table 5
Aggregating matrixR=r˜ijm×n.
C1C2C3C4A1(0.33, 0.46, 0.61, 0.77; 0.84, 0.35)(0.16, 0.28, 0.40, 0.52; 0.60, 0.24)(0.42, 0.58, 0.80, 0.90; 0.84, 0.12)(0.34, 0.46, 0.61, 0.76; 0.76, 0.35)A2(0.40, 0.50, 0.60, 0.78; 0.71, 0.23)(0.42, 0.54, 0.66, 0.76; 0.64, 0.36)(0.38, 0.48, 0.61, 0.76; 0.80, 0.28)(0.14, 0.29, 0.47, 0.59; 0.76, 0.28)A3(0.18, 0.31, 0.51, 0.64; 0.58, 0.38)(0.22, 0.34, 0.47, 0.60; 0.43, 0.45)(0.36, 0.60, 0.72, 0.82; 0.59, 0.24)(0.28, 0.38, 0.48, 0.71; 0.60, 0.30)A4(0.25, 0.37, 0.50, 0.62; 0.27, 0.68)(0.13, 0.28, 0.43, 0.53; 0.44, 0.48)(0.22, 0.32, 0.48, 0.70; 0.46, 0.52)(0.19, 0.35, 0.47, 0.62; 0.35, 0.38)According to the definition of compromise distance, we can calculate the compromise distance between two risk levels under each attribute as below:(48)Cdr˜11r˜21=8.69×10−3,Cdr˜11r˜31=3.14×10−2,Cdr˜11r˜41=1.67×10−2,Cdr˜21r˜31=4.01×10−2,Cdr˜21r˜41=2.54×10−2,Cdr˜31r˜41=1.47×10−2,Cdr˜12r˜22=4.87×10−2,Cdr˜12r˜32=1.41×10−2,Cdr˜12r˜42=3.11×10−2,Cdr˜22r˜32=3.46×10−2,Cdr˜22r˜42=1.76×10−2,Cdr˜32r˜42=1.70×10−2,Cdr˜13r˜23=1.42×10−2,Cdr˜13r˜33=8.97×10−3,Cdr˜13r˜43=2.01×10−2,Cdr˜23v33=5.21×10−3,Cdr˜23r˜43=5.97×10−3,Cdr˜33r˜43=1.12×10−2,Cdr˜14r˜24=7.95×10−3,Cdr˜14r˜34=2.75×10−3,Cdr˜14r˜44=3.90×10−4,Cdr˜24r˜34=5.19×10−3,Cdr˜24r˜44=8.34×10−3,Cdr˜34r˜44=3.14×10−3.Then, the relative weight of the attributeCcc=1,2,3,4 can be determined by equation (35), and the result is as follows:(49)ωcr=1,0.75,0.5,0.25.Next, the relative dominance matrix of the risk levelAi relative to the risk level Aj can be determined by equations (45) and (46), respectively, and the results are shown as below:(50)ϑ=0−0.190.280.40−0.6000.270.36−1.10−0.8900.33−1.58−1.42−1.350.Finally, the overall dominanceξii=1,2,3,4 can be determined by equation (47), which is shown as below:(51)ξ1=1.16,ξ2=0.66,ξ3=−0.39,ξ4=−1.75.According to the overall dominances,ξ1>ξ2>ξ3>ξ4, the four risk levels are ranked as A1>A2>A3>A4, so blood pressure risk of the patient is high risk, and this hypertensive patient needs to control the blood pressure as soon as possible.


### 5.2. Sensitivity Analysis of Parameters

This section conducts the sensitive analysis for the ranking orders of risk levels by modifying the values of different parameters. Considering the actual decision-making situation, the values of the parameterq are, respectively, as 2, 3, and 4 by using the traversal method. According to the prospect theory, the values of the parameter θ in the experiments are, respectively, as 1, 1.5, and 2.25. In addition, the influence of the change of the distance parameter α on the ranking results is also not negligible. According to the different values of q, α, and θ, the comprehensive results are shown in Tables 6–8, respectively. The corresponding figures are shown in Figures 5–7.Table 6
The overall dominancevalues of risk levels with different values of α and θ in case of q = 2.
ξiα=0.1α=0.2α=0.3α=0.4α=0.5α=0.6α=0.7α=0.8α=0.9θ=1[1.35, 0.58, −0.35, −1.79][1.30, 0.59, −0.35, −1.78][1.26, 0.60, −0.33, −1.76][1.21, 0.63, −0.36, −1.76][1.16, 0.66, −0.39, −1.75][1.11, 0.69, −0.41, −1.74][1.03, 0.75, −0.44, −1.74][1.03, 0.72, −0.47, −1.74][1.02, 0.70, −0.50, −1.74]θ=1.5[1.66, 0.78, −0.21, −1.79][1.60, 0.79, −0.21, −1.78][1.54, 0.80, −0.19, −1.76][1.48, 0.84, −0.22, −1.76][1.42, 0.87, −0.24, −1.75][1.35, 0.91, −0.26, −1.74][1.25, 0.98, −0.29, −1.74][1.27, 0.94, −0.31, −1.74][1.26, 0.92, −0.34, −1.74]θ=2.25[1.89, 1.01, −0.05, −1.79][1.84, 1.02, −0.04, −1.78][1.79, 1.04, −0.03, −1.76][1.74, 1.07, −0.06, −1.76][1.67, 1.10, −0.07, −1.75][1.58, 1.15, −0.09, −1.74][1.46, 1.22, −0.11, −1.74][1.49, 1.18, −0.13, −1.74][1.50, 1.15, −0.15, −1.74]Table 7
The overall dominancevaluesof risk levels with different values of α and θ in case of q = 3.
ξiα=0.1α=0.2α=0.3α=0.4α=0.5α=0.6α=0.7α=0.8α=0.9θ=1[1.22, 0.58, −0.45, −1.82][1.19, 0.60, −0.46, −1.8][1.15, 0.62, −0.47, −1.79][1.11, 0.64, −0.48, −1.79][1.06, 0.67, −0.50, −1.78][1.01, 0.70, −0.50, −1.78][0.95, 0.76, −0.48, −1.78][0.95, 0.71, −0.43, −1.78][0.99, 0.69, −0.48, −1.78]θ=1.5[1.55, 0.79, −0.29, −1.82][1.49, 0.81, −0.30, −1.8][1.44, 0.83, −0.31, −1.79][1.38, 0.86, −0.32, −1.79][1.32, 0.89, −0.34, −1.78][1.25, 0.93, −0.34, −1.78][1.17, 0.99, −0.32, −1.78][1.18, 0.93, −0.29, −1.78][1.24, 0.91, −0.33, −1.78]θ=2.25[1.83, 1.04, −0.12, −1.82][1.78, 1.06, −0.12, −1.8][1.73, 1.08, −0.13, −1.79][1.66, 1.11, −0.14, −1.79][1.58, 1.14, −0.15, −1.78][1.49, 1.18, −0.16, −1.78][1.38, 1.24, −0.14, −1.78][1.42, 1.18, −0.12, −1.78][1.50, 1.15, −0.16, −1.78]Table 8
The overall dominancevalues of risk levels with different values of α and θ in case of q = 4.
ξiα=0.1α=0.2α=0.3α=0.4α=0.5α=0.6α=0.7α=0.8α=0.9θ=1[1.16, 0.57, −0.50, −1.82][1.13, 0.59, −0.50, −1.82][1.09, 0.62, −0.48, −1.82][1.05, 0.64, −0.47, −1.82][1.00, 0.67, −0.45, −1.82][0.95, 0.71, −0.43, −1.82][0.90, 0.74, −0.39, −1.82][0.94, 0.71, −0.41, −1.83][0.96, 0.69, −0.43, −1.83]θ=1.5[1.48, 0.79, −0.34, −1.82][1.43, 0.81, −0.33, −1.82][1.38, 0.84, −0.33, −1.82][1.32, 0.86, −0.32, −1.82][1.26, 0.90, −0.30, −1.82][1.19, 0.95, −0.29, −1.82][1.13, 0.97, −0.25, −1.82][1.19, 0.93, −0.28, −1.83][1.23, 0.91, −0.30, −1.83]θ=2.25[1.79, 1.05, −0.15, −1.82][1.75, 1.07, −0.15, −1.82][1.68, 1.09, −0.14, −1.82][1.61, 1.12, −0.14, −1.82][1.53, 1.16, −0.13, −1.82][1.43, 1.21, −0.12, −1.82][1.36, 1.23, −0.09, −1.82][1.45, 1.19, −0.12, −1.83][1.52, 1.17, −0.14, −1.83]Figure 5
The pictorial expressions of the dominance values with different values ofα and θ in case of q = 2.Figure 6
The pictorial expressions of the dominance values with different values ofα and θ in case of q = 3.Figure 7
The pictorial expressions of the dominance values with different values ofα and θ in case of q = 4.It can be seen from the above figures that the ranking result does not change with different values ofq, α, and θ, which are always A1>A2>A3>A4. In addition, as the risk parameter θ increases, the advantages of the risk level A1 are not changed in this case. It is worth mentioning that, in this case, the change of θ does not change the overall dominance of A1. From the curve in the figures, it can be seen that the change of q has little effect on the value of the overall dominance of risk levels. Besides, it is proved in practical that the increase of q and α will not change the ranking, although they affect the value of overall dominance. Furthermore, it can be known that the ranking results do not be changed when the values of θ are 1, 1.5, and 2.25, respectively. From the above analysis, the overall dominance of each risk level will not be changed although they are different trends with the change of q, α, and θ; the final results will not be affected in the group decision-making method proposed in this paper. The feasibility and effectiveness of this method are verified.


### 5.3. Comparison Analysis

To well conduct a comparison analysis with the method proposed in [33], in this case, we set the value of q as 2 and the q-ROTrFNs in the above example are transformed into the PTrFNs. Then, the PTFWA operator developed in [33] is introduced as follows:(52)z^ik=PTFWAz^i1k,z^i2k,…,z^ink=∑j=1nωja¯z^ijk,∑j=1nωja1z^ijk,∑j=1nωja2z^ijk,∑j=1nωja¯z^ijk,1−∏j=1n1−ua^maxz^ijk2ωj,∏j=1nva^minz^ijk2ωj.By equation (52), we can get the individual overall preference Pythagorean trapezoidal fuzzy values as follows:(53)z^11=0.29,0.43,0.55,0.69;0.81,0.25,z^12=0.48,0.59,0.72,0.82;0.81,0.39,z^13=0.23,0.33,0.50,0.61;0.62,0.36,z^14=0.20,0.31,0.46,0.58;0.37,0.59,z^21=0.23,0.33,0.50,0.65;0.78,0.22,z^22=0.40,0.50,0.61,0.81;0.68,0.28,z^23=0.19,0.37,0.51,0.70;0.54,0.40,z^24=0.22,0.36,0.52,0.65;0.45,0.58,z^31=0.26,0.39,0.55,0.69;0.75,0.25,z^32=0.30,0.40,0.54,0.65;0.62,0.16,z^33=0.30,0.42,0.56,0.67;0.49,0.24,z^34=0.22,0.32,0.46,0.58;0.38,0.46,z^41=0.45,0.62,0.80,0.90;0.77,0.35,z^42=0.26,0.40,0.50,0.66;0.69,0.30,z^43=0.26,0.45,0.61,0.72;0.47,0.45,z^44=0.16,0.33,0.43,0.62;0.27,0.53.Next, the following PTFHA operator developed in [33] is used to obtain the collective overall preference Pythagorean trapezoidal fuzzy values:(54)z^i=PTFHAz^1k,z^2k,…,z^ik=∑k=1tλka¯σz^ik,∑k=1tλka1σz^ik,∑k=1tλka2σz^ik,∑k=1tλka¯σz^ik,1−∏k=1t1−ua^maxσz^ik2λk,∏k=1tva^minσz^ik2λk.The following results are obtained:(55)z^1=0.29,0.42,0.58,0.72;0.77,0.25,z^2=0.35,0.46,0.58,0.73;0.69,0.25,z^3=0.24,0.40,0.54,0.68;0.53,0.34,z^4=0.21,0.33,0.48,0.61;0.39,0.53.And, the Pythagorean trapezoidal fuzzy distance between each risk level and the positive-ideal solutionz^+=1,1,1,1;1,0 is as below:(56)d^z^1,z^+=0.27,d^z^2,z^+=0.28,d^z^3,z^+=0.35,d^z^4,z^+=0.42.It is easy to see that(57)d^z^1,z^+<d^z^2,z^+<d^z^3,z^+<d^z^4,z^+.According to the idea of the developed method in [33], the closer distance to the positive-ideal solution z^+ is, the better the solution is. Thus, the ranking order of risk levels is A1>A2>A3>A4.Furthermore, the comparison analysis results obtained by the developed modified TODIM method and the method proposed in [33] are shown in Table 9 and Figure 8. It can be easily found that the best alternative according to the method proposed in this paper and the method developed in [33] is the same, i.e., A1. From the calculation results of our developed method, it can be found that A1 has obvious advantages over other risk levels, which means that the method proposed in this paper can highlight the optimal solution among all risk levels in group decision-making. Comparing with the method introduced in [33], the biggest advantage of the developed method is that it can take into account the psychological behavior of decision makers and can well solve decision-making problems with risk characteristics. Meanwhile, the compromise distance proposed in this paper takes both positive-ideal solution and negative-ideal solution into consideration, which is more objective and reasonable comparing with the method introduced in [33] that only considers the influence of the positive-ideal solution z^+=1,1,1,1;1,0.Table 9
The comparison analysis results with different methods.
MethodsCalculation resultsRanking orders of risk levelsThe modified TODIM proposed in this paperThe overall dominances [1.16, 0.66, −0.39, −1.75]A1>A2>A3>A4Method proposed in [33] based on PTFHADistances between collective value and PIS [0.27, 0.28, 0.35, 0.42]A1>A2>A3>A4Figure 8
The comparison analysis results with different methods.


## 5.1. Decision Results Obtained by the Developed Modified TODIM Method

At present, there are about 245 million hypertensive patients in China, and related diseases caused by hypertension have brought a heavy burden to residents and society. In order to improve the health of residents, the health management of hypertensive patients has been highly valued by the public and the government. In order to simplify the management process, a community hospital plans to develop hypertension health management software to facilitate the treatment and management of patients by community doctors, which means that doctors need to evaluate the risk level based on the residents’ inspection. The main observation data include blood pressure measurement, cardiovascular damage, related diseases (mainly related to cardiovascular and cerebrovascular diseases), and related risk factors (age, smoking, drinking and sleeping, eating more sodium and less potassium, greasy, obesity, work and life pressure, etc.), and the weights of them areω=0.4,0.3,0.2,0.1. In order to better provide prompts to doctors, the risk level of residents is indicated by different colors: red indicates high risk, orange indicates moderate risk, blue indicates mild danger, and green indicates normal, which are expressed as A=A1,A2,A3,A4.Considering that these four indicators are abnormal to the normal process, the process is ascending, adjusting, and gradually recovering. The fuzzy decision model is used to construct the system for processing. In order to facilitate the flexible evaluation of experts, theq-ROTrFNs are used for evaluation. Now, a 65-year-old resident has a systolic blood pressure of 140 mmHg and a diastolic blood pressure of 99 mmHg. He suffers from hyperglycemia, and parents suffer from high blood pressure. Besides, he smokes and drinks frequently, and his taste tends to be heavy. Four experts are invited to evaluate the risk level of this resident, and the weights of experts are λ=0.3,0.3,0.2,0.2. The evaluation data are obtained in Tables 1–4.Table 1
Evaluation value matrix by expert1A1.
C1C2C3C4A1(0.3, 0.5, 0.6, 0.8; 0.9, 0.3)(0.1, 0.2, 0.3, 0.4; 0.6, 0.3)(0.5, 0.6, 0.8, 0.9; 0.8, 0.1)(0.4, 0.5, 0.6, 0.7; 0.7, 0.4)A2(0.6, 0.7, 0.8, 0.9; 0.8, 0.5)(0.5, 0.6, 0.7, 0.8; 0.7, 0.3)(0.4, 0.5, 0.7, 0.8; 0.9, 0.4)(0.1, 0.3, 0.5, 0.6; 0.8, 0.3)A3(0.1, 0.2, 0.4, 0.5; 0.7, 0.4)(0.2, 0.3, 0.5, 0.6; 0.5, 0.4)(0.5, 0.6, 0.7, 0.8; 0.6, 0.2)(0.3, 0.4, 0.5, 0.7; 0.6, 0.5)A4(0.3, 0.4, 0.5, 0.6; 0.2, 0.7)(0.1, 0.2, 0.4, 0.5; 0.4, 0.5)(0.2, 0.3, 0.5, 0.7; 0.5, 0.5)(0.1, 0.3, 0.4, 0.5; 0.4, 0.7)Table 2
Evaluation value matrix by expert2A2.
C1C2C3C4A1(0.2, 0.3, 0.5, 0.7; 0.8, 0.3)(0.1, 0.2, 0.3, 0.4; 0.6, 0.2)(0.5, 0.6, 0.8, 0.9; 0.9, 0.1)(0.2, 0.3, 0.5, 0.7; 0.7, 0.4)A2(0.4, 0.5, 0.6, 0.9; 0.6, 0.2)(0.5, 0.6, 0.7, 0.8; 0.7, 0.4)(0.4, 0.5, 0.6, 0.8; 0.8, 0.3)(0.1, 0.2, 0.4, 0.5; 0.6, 0.3)A3(0.1, 0.3, 0.5, 0.7; 0.5, 0.4)(0.2, 0.3, 0.4, 0.6; 0.4, 0.8)(0.3, 0.6, 0.7, 0.8; 0.7, 0.2)(0.3, 0.4, 0.5, 0.8; 0.6, 0.2)A4(0.2, 0.3, 0.5, 0.6; 0.4, 0.7)(0.2, 0.4, 0.5, 0.6; 0.5, 0.7)(0.2, 0.3, 0.5, 0.7; 0.5, 0.5)(0.4, 0.6, 0.7, 0.9; 0.3, 0.2)Table 3
Evaluation value matrix by expert3A3.
C1C2C3C4A1(0.3, 0.4, 0.6, 0.7; 0.8, 0.4)(0.1, 0.2, 0.3, 0.5; 0.6, 0.2)(0.4, 0.6, 0.8, 0.9; 0.8, 0.1)(0.3, 0.5, 0.6, 0.8; 0.7, 0.4)A2(0.2, 0.3, 0.4, 0.5; 0.7, 0.1)(0.4, 0.5, 0.7, 0.8; 0.5, 0.3)(0.4, 0.5, 0.6, 0.7; 0.6, 0.1)(0.2, 0.3, 0.5, 0.7; 0.6, 0.3)A3(0.2, 0.3, 0.5, 0.6; 0.5, 0.3)(0.3, 0.4, 0.5, 0.6; 0.4, 0.2)(0.5, 0.7, 0.8, 0.9; 0.5, 0.2)(0.3, 0.4, 0.5, 0.7; 0.6, 0.2)A4(0.3, 0.4, 0.5, 0.6; 0.2, 0.6)(0.1, 0.2, 0.4, 0.5; 0.5, 0.4)(0.3, 0.4, 0.5, 0.7; 0.4, 0.5)(0.1, 0.2, 0.4, 0.5; 0.4, 0.2)Table 4
Evaluation value matrix by expert4A4.
C1C2C3C4A1(0.6, 0.7, 0.8, 0.9; 0.8, 0.5)(0.4, 0.6, 0.8, 0.9; 0.6, 0.3)(0.2, 0.5, 0.8, 0.9; 0.8, 0.3)(0.5, 0.6, 0.8, 0.9; 0.9, 0.2)A2(0.3, 0.4, 0.5, 0.7; 0.7, 0.2)(0.2, 0.4, 0.5, 0.6; 0.5, 0.5)(0.3, 0.4, 0.5, 0.7; 0.7, 0.4)(0.2, 0.4, 0.5, 0.6; 0.9, 0.2)A3(0.4, 0.5, 0.7, 0.8; 0.5, 0.4)(0.2, 0.4, 0.5, 0.6; 0.4, 0.5)(0.1, 0.5, 0.7, 0.8; 0.4, 0.5)(0.2, 0.3, 0.4, 0.6; 0.6, 0.4)A4(0.2, 0.4, 0.5, 0.7; 0.1, 0.7)(0.1, 0.3, 0.4, 0.5; 0.3, 0.3)(0.2, 0.3, 0.4, 0.7; 0.4, 0.6)(0.1, 0.2, 0.3, 0.5; 0.3, 0.8)Now, we employ the developed modifiedq-rung orthopair trapezoidal fuzzy TODIM group decision-making method to help the doctor identify the risk level of hypertensive patients. In this case, we consider q = 2, θ=1, and α=0.5; then, we aggregate the evaluation matrix Akk=1,2,3,4 given by these four experts to obtain the collective matrix R=r˜ijm×n by equation (34), as shown in Table 5.Table 5
Aggregating matrixR=r˜ijm×n.
C1C2C3C4A1(0.33, 0.46, 0.61, 0.77; 0.84, 0.35)(0.16, 0.28, 0.40, 0.52; 0.60, 0.24)(0.42, 0.58, 0.80, 0.90; 0.84, 0.12)(0.34, 0.46, 0.61, 0.76; 0.76, 0.35)A2(0.40, 0.50, 0.60, 0.78; 0.71, 0.23)(0.42, 0.54, 0.66, 0.76; 0.64, 0.36)(0.38, 0.48, 0.61, 0.76; 0.80, 0.28)(0.14, 0.29, 0.47, 0.59; 0.76, 0.28)A3(0.18, 0.31, 0.51, 0.64; 0.58, 0.38)(0.22, 0.34, 0.47, 0.60; 0.43, 0.45)(0.36, 0.60, 0.72, 0.82; 0.59, 0.24)(0.28, 0.38, 0.48, 0.71; 0.60, 0.30)A4(0.25, 0.37, 0.50, 0.62; 0.27, 0.68)(0.13, 0.28, 0.43, 0.53; 0.44, 0.48)(0.22, 0.32, 0.48, 0.70; 0.46, 0.52)(0.19, 0.35, 0.47, 0.62; 0.35, 0.38)According to the definition of compromise distance, we can calculate the compromise distance between two risk levels under each attribute as below:(48)Cdr˜11r˜21=8.69×10−3,Cdr˜11r˜31=3.14×10−2,Cdr˜11r˜41=1.67×10−2,Cdr˜21r˜31=4.01×10−2,Cdr˜21r˜41=2.54×10−2,Cdr˜31r˜41=1.47×10−2,Cdr˜12r˜22=4.87×10−2,Cdr˜12r˜32=1.41×10−2,Cdr˜12r˜42=3.11×10−2,Cdr˜22r˜32=3.46×10−2,Cdr˜22r˜42=1.76×10−2,Cdr˜32r˜42=1.70×10−2,Cdr˜13r˜23=1.42×10−2,Cdr˜13r˜33=8.97×10−3,Cdr˜13r˜43=2.01×10−2,Cdr˜23v33=5.21×10−3,Cdr˜23r˜43=5.97×10−3,Cdr˜33r˜43=1.12×10−2,Cdr˜14r˜24=7.95×10−3,Cdr˜14r˜34=2.75×10−3,Cdr˜14r˜44=3.90×10−4,Cdr˜24r˜34=5.19×10−3,Cdr˜24r˜44=8.34×10−3,Cdr˜34r˜44=3.14×10−3.Then, the relative weight of the attributeCcc=1,2,3,4 can be determined by equation (35), and the result is as follows:(49)ωcr=1,0.75,0.5,0.25.Next, the relative dominance matrix of the risk levelAi relative to the risk level Aj can be determined by equations (45) and (46), respectively, and the results are shown as below:(50)ϑ=0−0.190.280.40−0.6000.270.36−1.10−0.8900.33−1.58−1.42−1.350.Finally, the overall dominanceξii=1,2,3,4 can be determined by equation (47), which is shown as below:(51)ξ1=1.16,ξ2=0.66,ξ3=−0.39,ξ4=−1.75.According to the overall dominances,ξ1>ξ2>ξ3>ξ4, the four risk levels are ranked as A1>A2>A3>A4, so blood pressure risk of the patient is high risk, and this hypertensive patient needs to control the blood pressure as soon as possible.


## 5.2. Sensitivity Analysis of Parameters

This section conducts the sensitive analysis for the ranking orders of risk levels by modifying the values of different parameters. Considering the actual decision-making situation, the values of the parameterq are, respectively, as 2, 3, and 4 by using the traversal method. According to the prospect theory, the values of the parameter θ in the experiments are, respectively, as 1, 1.5, and 2.25. In addition, the influence of the change of the distance parameter α on the ranking results is also not negligible. According to the different values of q, α, and θ, the comprehensive results are shown in Tables 6–8, respectively. The corresponding figures are shown in Figures 5–7.Table 6
The overall dominancevalues of risk levels with different values of α and θ in case of q = 2.
ξiα=0.1α=0.2α=0.3α=0.4α=0.5α=0.6α=0.7α=0.8α=0.9θ=1[1.35, 0.58, −0.35, −1.79][1.30, 0.59, −0.35, −1.78][1.26, 0.60, −0.33, −1.76][1.21, 0.63, −0.36, −1.76][1.16, 0.66, −0.39, −1.75][1.11, 0.69, −0.41, −1.74][1.03, 0.75, −0.44, −1.74][1.03, 0.72, −0.47, −1.74][1.02, 0.70, −0.50, −1.74]θ=1.5[1.66, 0.78, −0.21, −1.79][1.60, 0.79, −0.21, −1.78][1.54, 0.80, −0.19, −1.76][1.48, 0.84, −0.22, −1.76][1.42, 0.87, −0.24, −1.75][1.35, 0.91, −0.26, −1.74][1.25, 0.98, −0.29, −1.74][1.27, 0.94, −0.31, −1.74][1.26, 0.92, −0.34, −1.74]θ=2.25[1.89, 1.01, −0.05, −1.79][1.84, 1.02, −0.04, −1.78][1.79, 1.04, −0.03, −1.76][1.74, 1.07, −0.06, −1.76][1.67, 1.10, −0.07, −1.75][1.58, 1.15, −0.09, −1.74][1.46, 1.22, −0.11, −1.74][1.49, 1.18, −0.13, −1.74][1.50, 1.15, −0.15, −1.74]Table 7
The overall dominancevaluesof risk levels with different values of α and θ in case of q = 3.
ξiα=0.1α=0.2α=0.3α=0.4α=0.5α=0.6α=0.7α=0.8α=0.9θ=1[1.22, 0.58, −0.45, −1.82][1.19, 0.60, −0.46, −1.8][1.15, 0.62, −0.47, −1.79][1.11, 0.64, −0.48, −1.79][1.06, 0.67, −0.50, −1.78][1.01, 0.70, −0.50, −1.78][0.95, 0.76, −0.48, −1.78][0.95, 0.71, −0.43, −1.78][0.99, 0.69, −0.48, −1.78]θ=1.5[1.55, 0.79, −0.29, −1.82][1.49, 0.81, −0.30, −1.8][1.44, 0.83, −0.31, −1.79][1.38, 0.86, −0.32, −1.79][1.32, 0.89, −0.34, −1.78][1.25, 0.93, −0.34, −1.78][1.17, 0.99, −0.32, −1.78][1.18, 0.93, −0.29, −1.78][1.24, 0.91, −0.33, −1.78]θ=2.25[1.83, 1.04, −0.12, −1.82][1.78, 1.06, −0.12, −1.8][1.73, 1.08, −0.13, −1.79][1.66, 1.11, −0.14, −1.79][1.58, 1.14, −0.15, −1.78][1.49, 1.18, −0.16, −1.78][1.38, 1.24, −0.14, −1.78][1.42, 1.18, −0.12, −1.78][1.50, 1.15, −0.16, −1.78]Table 8
The overall dominancevalues of risk levels with different values of α and θ in case of q = 4.
ξiα=0.1α=0.2α=0.3α=0.4α=0.5α=0.6α=0.7α=0.8α=0.9θ=1[1.16, 0.57, −0.50, −1.82][1.13, 0.59, −0.50, −1.82][1.09, 0.62, −0.48, −1.82][1.05, 0.64, −0.47, −1.82][1.00, 0.67, −0.45, −1.82][0.95, 0.71, −0.43, −1.82][0.90, 0.74, −0.39, −1.82][0.94, 0.71, −0.41, −1.83][0.96, 0.69, −0.43, −1.83]θ=1.5[1.48, 0.79, −0.34, −1.82][1.43, 0.81, −0.33, −1.82][1.38, 0.84, −0.33, −1.82][1.32, 0.86, −0.32, −1.82][1.26, 0.90, −0.30, −1.82][1.19, 0.95, −0.29, −1.82][1.13, 0.97, −0.25, −1.82][1.19, 0.93, −0.28, −1.83][1.23, 0.91, −0.30, −1.83]θ=2.25[1.79, 1.05, −0.15, −1.82][1.75, 1.07, −0.15, −1.82][1.68, 1.09, −0.14, −1.82][1.61, 1.12, −0.14, −1.82][1.53, 1.16, −0.13, −1.82][1.43, 1.21, −0.12, −1.82][1.36, 1.23, −0.09, −1.82][1.45, 1.19, −0.12, −1.83][1.52, 1.17, −0.14, −1.83]Figure 5
The pictorial expressions of the dominance values with different values ofα and θ in case of q = 2.Figure 6
The pictorial expressions of the dominance values with different values ofα and θ in case of q = 3.Figure 7
The pictorial expressions of the dominance values with different values ofα and θ in case of q = 4.It can be seen from the above figures that the ranking result does not change with different values ofq, α, and θ, which are always A1>A2>A3>A4. In addition, as the risk parameter θ increases, the advantages of the risk level A1 are not changed in this case. It is worth mentioning that, in this case, the change of θ does not change the overall dominance of A1. From the curve in the figures, it can be seen that the change of q has little effect on the value of the overall dominance of risk levels. Besides, it is proved in practical that the increase of q and α will not change the ranking, although they affect the value of overall dominance. Furthermore, it can be known that the ranking results do not be changed when the values of θ are 1, 1.5, and 2.25, respectively. From the above analysis, the overall dominance of each risk level will not be changed although they are different trends with the change of q, α, and θ; the final results will not be affected in the group decision-making method proposed in this paper. The feasibility and effectiveness of this method are verified.


## 5.3. Comparison Analysis

To well conduct a comparison analysis with the method proposed in [33], in this case, we set the value of q as 2 and the q-ROTrFNs in the above example are transformed into the PTrFNs. Then, the PTFWA operator developed in [33] is introduced as follows:(52)z^ik=PTFWAz^i1k,z^i2k,…,z^ink=∑j=1nωja¯z^ijk,∑j=1nωja1z^ijk,∑j=1nωja2z^ijk,∑j=1nωja¯z^ijk,1−∏j=1n1−ua^maxz^ijk2ωj,∏j=1nva^minz^ijk2ωj.By equation (52), we can get the individual overall preference Pythagorean trapezoidal fuzzy values as follows:(53)z^11=0.29,0.43,0.55,0.69;0.81,0.25,z^12=0.48,0.59,0.72,0.82;0.81,0.39,z^13=0.23,0.33,0.50,0.61;0.62,0.36,z^14=0.20,0.31,0.46,0.58;0.37,0.59,z^21=0.23,0.33,0.50,0.65;0.78,0.22,z^22=0.40,0.50,0.61,0.81;0.68,0.28,z^23=0.19,0.37,0.51,0.70;0.54,0.40,z^24=0.22,0.36,0.52,0.65;0.45,0.58,z^31=0.26,0.39,0.55,0.69;0.75,0.25,z^32=0.30,0.40,0.54,0.65;0.62,0.16,z^33=0.30,0.42,0.56,0.67;0.49,0.24,z^34=0.22,0.32,0.46,0.58;0.38,0.46,z^41=0.45,0.62,0.80,0.90;0.77,0.35,z^42=0.26,0.40,0.50,0.66;0.69,0.30,z^43=0.26,0.45,0.61,0.72;0.47,0.45,z^44=0.16,0.33,0.43,0.62;0.27,0.53.Next, the following PTFHA operator developed in [33] is used to obtain the collective overall preference Pythagorean trapezoidal fuzzy values:(54)z^i=PTFHAz^1k,z^2k,…,z^ik=∑k=1tλka¯σz^ik,∑k=1tλka1σz^ik,∑k=1tλka2σz^ik,∑k=1tλka¯σz^ik,1−∏k=1t1−ua^maxσz^ik2λk,∏k=1tva^minσz^ik2λk.The following results are obtained:(55)z^1=0.29,0.42,0.58,0.72;0.77,0.25,z^2=0.35,0.46,0.58,0.73;0.69,0.25,z^3=0.24,0.40,0.54,0.68;0.53,0.34,z^4=0.21,0.33,0.48,0.61;0.39,0.53.And, the Pythagorean trapezoidal fuzzy distance between each risk level and the positive-ideal solutionz^+=1,1,1,1;1,0 is as below:(56)d^z^1,z^+=0.27,d^z^2,z^+=0.28,d^z^3,z^+=0.35,d^z^4,z^+=0.42.It is easy to see that(57)d^z^1,z^+<d^z^2,z^+<d^z^3,z^+<d^z^4,z^+.According to the idea of the developed method in [33], the closer distance to the positive-ideal solution z^+ is, the better the solution is. Thus, the ranking order of risk levels is A1>A2>A3>A4.Furthermore, the comparison analysis results obtained by the developed modified TODIM method and the method proposed in [33] are shown in Table 9 and Figure 8. It can be easily found that the best alternative according to the method proposed in this paper and the method developed in [33] is the same, i.e., A1. From the calculation results of our developed method, it can be found that A1 has obvious advantages over other risk levels, which means that the method proposed in this paper can highlight the optimal solution among all risk levels in group decision-making. Comparing with the method introduced in [33], the biggest advantage of the developed method is that it can take into account the psychological behavior of decision makers and can well solve decision-making problems with risk characteristics. Meanwhile, the compromise distance proposed in this paper takes both positive-ideal solution and negative-ideal solution into consideration, which is more objective and reasonable comparing with the method introduced in [33] that only considers the influence of the positive-ideal solution z^+=1,1,1,1;1,0.Table 9
The comparison analysis results with different methods.
MethodsCalculation resultsRanking orders of risk levelsThe modified TODIM proposed in this paperThe overall dominances [1.16, 0.66, −0.39, −1.75]A1>A2>A3>A4Method proposed in [33] based on PTFHADistances between collective value and PIS [0.27, 0.28, 0.35, 0.42]A1>A2>A3>A4Figure 8
The comparison analysis results with different methods.


## 6. Conclusion

In this paper, we have introduced the concept ofq-ROTrFNs and developed the basic operation of q-ROTrFNs. On the basis of the developed score function and exact function of q-ROTrFNs, we have proposed a new ranking method for q-ROTrFNs. To integrate the individual decision matrices into the collective decision matrix, we have proposed a new q-ROTrFWA operator and investigated its properties. Next, a new distance measure-based compromise approach has been proposed to determine the relative dominance degree of alternatives. Then, we have developed a new q-rung orthopair trapezoidal fuzzy modified TODIM group decision-making method. The psychological state of decision makers in actual decision-making process can be taken full into account in the modified TODIM method. This method can handle decision-making problems with risk characteristics well and make the decision-making results more reasonable and authentic. Finally, the modified TODIM method is verified and analyzed by the illustrating example, and the obtained optimal solution is consistent with the results obtained by [33]. At the same time, it has been verified that the different values of q, α, and θ do not affect the ranking results. This analysis also proves the rationality and effectiveness of the group method proposed in this paper.However, there are still some limitations in the study. First, the proposed GDM method is restricted to theq-rung orthopair trapezoidal fuzzy environment and relies on the experience and knowledge of the decision maker to obtain ranking of alternatives. When the group of decision makers is too large or the external environment influences strongly, the GDM method may not be able to adapt to the changes in the subjective attitude of the decision makers in time, which leads to deviations in results. Second, consistency and consensus are paid more and more attention in MAGDM [34]. However, the consensus among decision makers has not been studied in depth in this study; individual judgment is the main basis for obtaining results. As a result, it makes sense to further build the consensus model and integrate it into the proposed method. It is interesting to build the consistency and consensus model for MAGDM problems with q-ROTrFNs considering decision makers’ risk expectation in the future, such as research on consistency and consensus model of q-rung orthopair trapezoidal fuzzy preference relations based on prospect theory. Meanwhile, with the development of big data and artificial intelligence, it needs to fully consider the psychological behavior of the decision maker in order to obtain accurate decision-making results in the actual decision-making process. The fusion of the developed decision-making method in this paper, machine learning, and big data are among our research directions in the future, and the developed group decision-making method in a heterogeneous environment is also the focus of our future research.



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