# Interval-Valued q-Rung Orthopair Fuzzy QUALIFLEX Decision Analysis Method with Dombi Operators

Authors: Benting Wan; Xiaolu Zhang; Meifang Xiong; Zhenwu Wang
Journal: Discrete Dynamics in Nature and Society
 (2022)

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## Abstract

As a new and useful decision tool, interval-valued q-rung orthopair fuzzy set (IVq-ROFS), has strong ability to address uncertain or imprecision information in practical decision-making processes. This paper develops a new interval-valued q-rung orthopair fuzzy QUALIFLEX decision analysis method with Dombi operators for handling the group decision-making problems with IVq-ROFSs data. We also conduct a case study on the health management of hypertensive patients for demonstrating the feasibility and superiority of the developed method. The main contributions of this paper are as follows: (1) we first propose interval-valued q-rung orthopair fuzzy Dombi operational laws and investigate their desirable properties; (2) using the developed Dombi operational laws, we develop an interval-valued q-rung orthopair fuzzy Dombi weighted average (IVq-ROFDWA) operator, and we prove that the IVq-ROFDWA operator satisfies the properties of idempotency, commutativity, boundedness, and monotonicity; (3) we introduce the concept of interval-valued q-rung orthopair fuzzy ideal group opinions with the help of IVq-ROFDWA operator and propose a new weight-determining method for identifying the weights of experts; (4) we develop a new interval-valued q-rung orthopair fuzzy distance measure. With the help of the developed distance measure, we calculate interval-valued q-rung orthopair fuzzy concordance/discordance index under each permutation, and then a new interval-valued q-rung orthopair fuzzy QUALIFLEX decision analysis method is developed for addressing the complex multiattribute group decision-making problems with IVq-ROFSs.


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## Body

## 1. Introduction

With the increasing complexity and uncertainty in real-life decision-making processes, many useful extensions of fuzzy sets, such as the classical intuitionistic fuzzy sets (IFSs) [1], Pythagorean fuzzy set (PFS) [2, 3], interval-valued PFS [4], and the generalized orthopair fuzzy set [5], have been developed successively to describe the uncertain estimated results in complex practical multiattribute decision-making (MADM) problems [6]. The generalized orthopair fuzzy set is also called q-rung orthopair fuzzy set and is known simply as q-ROFS in the existing literature. Compared with the IFS and the PFS, the q-ROFS can contain more fuzzy information. In the practical decision-making process, IFS and PFS can be treated as two special forms of the q-ROFS for q = 1 and q = 2, respectively. The q-ROFSs have gradually drawn attention of scholars in recent years and many excellent research results have been achieved, for example, the q-rung orthopair fuzzy approximate reasoning [7], integrals and differentials of q-rung orthopair fuzzy functions [8–10], various q-rung orthopair aggregation operators [11–15], q-rung orthopair fuzzy information measures [16–18], and lots of different q-rung orthopair fuzzy decision-making methods [19, 20].To quantify the satisfying degrees and nonsatisfying degrees in terms of interval values instead of a single numeric value, IVq-ROFS [21, 22] has recently been developed. Jin [23] developed an interval-valued q-rung orthopair fuzzy-additive ratio assessment method. Yang et al. [24] investigated fusion strategies of continuous interval-valued q-rung orthopair fuzzy information, Yang and Chang [25] proposed the interval q-rung orthopair fuzzy power Muirhead mean operator. As an extension of interval-valued PFS from the perspective of the constraint condition, IVq-ROFSs have wide application fields, especially in decision-making fields. In order to well handle complex MADM or multiattribute group decision-making (MAGDM) problems under IVq-ROFSs environment, it is necessary to develop correspondingly effective decision-making methods. QUALIFLEX (qualitative flexible multiple criteria method) [26] is one of the well-known outranking methods to solve the MADM problems with exact real numbers. The most important characteristic of the QUALIFLEX method is the correct treatment of cardinal and ordinal information [4]. To adapt different fuzzy decision-making environments, several excellent studies have been devoted to fuzzy extensions of the QUALIFLEX approach in the literature. For example, Chen et al. [27, 28] proposed interval type-2 trapezoidal fuzzy QUALIFLEX method, Zhang and Xu [29] developed hesitant fuzzy QUALIFLEX method, and Chen [30] developed an interval-valued intuitionistic fuzzy QUALIFLEX method. More recently, some new extensions, Pythagorean fuzzy QUALIFLEX method [4], heterogeneous QUALIFLEX method [31], and SMAA-QUALIFLEX methodology with q-rung fuzzy sets [20], have been developed. Although these aforementioned methods are very effective for solving different issues, they fail to deal with IVq-ROFSs decision data in the complex decision process.Therefore, this paper attempts to develop a new interval-valued q-rung orthopair fuzzy QUALIFLEX decision analysis method with Dombi operators. Dombi operations developed by Domb [32] have an advantage of flexibility within the working behavior of parameters [33] and have been extended into various fuzzy environments [33–35]. Thus, this paper first defines the interval-valued q-rung orthopair fuzzy Dombi operations based on the t-module and t-comodule, including interval-valued q-rung orthopair fuzzy Dombi addition, multiplication, number multiplication, and power operations and investigates the relevant operation properties. Secondly, using the developed interval-valued q-rung orthopair fuzzy Dombi operations, the interval-valued q-rung orthopair fuzzy Dombi weighted average (IVq-ROFDWA) operator is proposed, and, using mathematical induction, we prove that the calculated results are still interval-valued q-rung orthopair fuzzy numbers (IVq-ROFNs), and at the same time we prove that the IVq-ROFDWA operator satisfies the properties of idempotency, commutativity, boundedness, and monotonicity. Thirdly, with the help of IVq-ROFDWA operator, we propose a new weight-determining method for identifying the weights of experts. Fourthly, the concordance/discordance index and the weighted concordance/discordance indices under IVq-ROFNs environment can be calculated and the ranking order of alternatives can be determined by comparing the comprehensive index. Finally, the algorithm of the developed interval-valued q-rung orthopair fuzzy QUALIFLEX decision analysis method is introduced.The remainder of this paper is constructed as follows: Section2 recalls the knowledge of IVq-ROFSs and proposes the new distance measure for IVq-ROFSs. Section 3 develops the interval-valued q-rung orthopair fuzzy Dombi operations and the IVq-ROFDWA operator. Section 4 proposes an interval-valued q-rung orthopair fuzzy QUALIFLEX decision analysis method. An illustrating example is presented in Section 5 to demonstrate the feasibility and superiority of the proposed method. Section 6 summarizes this study.


## 2. Basic Concepts and New Distance Measure

This section introduces the definition and operation of IVq-ROFS involved in this paper and proposes the new distance measure for IVq-ROFNs.Definition 1 (see [5]).
LetX be a fixed set; a q-rung orthopair fuzzy set (q-ROFS) A in X is an object of the form A=<x,uAx,vAx>|x∈X.μAx and νAx are the degree of membership and the degree of nonmembership of the element x∈X to A, respectively, and they satisfy(1)0≤uAxq+vAxq≤1.
The degree of hesitationπAx can be expressed as follows:(2)πAx=1−uAxq+vAxqq,q≥1.Definition 2 (see [21, 22]).
LetX be an ordinary finite nonempty set; the IVq-ROFS A in X is defined as(3)A=<x,uAx,vAx>|x∈X.The membership function is a mapping range of values to meet uAx=uA−x,uA+x⊆0,1 and the nonmembership function is a mapping range of values to meet vAx=vA−x,vA+x⊆0,1; 0≤uA+xq+vA+xq≤1,q≥1.
The degree of hesitation ofA is shown in the following formula:(4)πAx=πA−x,πA+x=1−uA+xq−vA+xqq,1−uA−xq−vA−xqq,q≥1.Definition 3 (see [21, 22]).
Leta=u−,u+,v−,v+, a1=ua1−,ua1+,va1−,va1+, and a2=ua2−,ua2+,va2−,va2+ be three IVq-ROFNs and λ>0; the basic operations on them are given as follows:(5)1a1⊕a2=ua1−q+ua2−q−ua1−qua2−qq,ua1+q+ua2+q−ua1+qua2+qq,va1−va2−,va1+va2+,2a1⊗a2=ua1−ua2−,ua1+ua2+,va1−q+va2−q−va1−qva2−qq,va1+q+va2+q−va1+qva2+qq,3λa=1−1−u−qλq,1−1−u+qλq,v−λ,v+λ,4aλ=u−λ,u+λ,1−1−v−qλq,1−1−v+qλq.Definition 4 (see [21, 22]).
Given an IVq-ROFNa=u−,u+,v−,v+, the scoring function of a is shown in the following formula:(6)Sa=12ua−q+ua+q−va−q−va+q,q≥1.Definition 5 (see [21, 22]).
Given an IVq-ROFNa=ua−,ua+,va−,va+, its accuracy value function is defined as follows:(7)Ha=12ua−q+ua+q+va−q+va+q,q≥1.Definition 6 (see [21, 22]).
Given two IVq-ROFNsa1=ua1−,ua1+,va1−,va1+ and a2=ua2−,ua2+,va2−,va2+, the comparison law of IVq-ROFNs is proposed as follows:(1)
IfSa1>Sa2, then a1>a2;(2)
IfSa1<Sa2, then a1<a2;(3)
IfSa1=Sa2 and(a)
ifHa1>Ha2, then a1>a2;(b)
ifHa1<Ha2, then a1<a2;(c)
ifHa1=Ha2, then a1=a2.Definition 7.
Given two IVq-ROFNsa1=ua1−,ua1+,va1−,va1+ and a2=ua2−,ua2+,va2−,va2+ and q≥1, the distance measure between them is developed as follows:(8)da1,a2=14ua1−q−ua2−q+ua1+q−ua2+q+va1−q−va2−q+va1+q−va2+q+πa1−q−πa2−q+πa1+q−πa2+q.Theorem 1.
Leta,b,c be three IVq-ROFNs; then the following results hold:(1)
0≤da,b≤1, da,b=0, if and only if a=b;(2)
da,b=db,a;(3)
da,b≤da,c+dc,b.Proof. It is easy to see that (2) holds according to the definition of distance measure; we next prove (1) and (3) as follows:(9)1da,b=14ua−q−ub−q+ua+q−ub+q+va−q−vb−q+va+q−vb+q+πa−q−πb−q+πa+q−πb+q≤14ua−q+va−q+πa−q+ub−q+vb−q+πb−q+ua+q+va+q+πa+q+ub+q+vb+q+πb+q≤141+1+1+1≤1.According to Definition7, it is easy to obtain 0≤da,b and da,b=0 if and only if a=b. Obviously, (1) holds.According to Definition7, it is easy to obtain(10)da,b=14ua−q−uc−q+uc−q−ub−q+ua+q−uc+q+uc+q−ub+q+va−q−vc−q+vc−q−vb−q+va+q−vc+q+vc+q−vb+q+πa−q−πc−q+πc−q−πb−q+πa+q−πc+q+πac+q−πb+q≤14ua−q−uc−q+uc−q−ub−q+ua+q−uc+q+uc+q−ub+q+va−q−vc−q+vc−q−vb−q+va+q−vc+q+vc+q−vb+q+πa−q−πc−q+πc−q−πb−q+πa+q−πc+q+πac+q−πb+q≤da,c+dc,b.Obviously, (3) holds.


## 3. Interval-Valued q-Rung Orthopair Fuzzy Dombi Operator

### 3.1. Interval-Valued q-Rung Orthopair Fuzzy Dombi Operation

Definition 8 (see [32]).
For any two real numbersa, b∈[0, 1], the Dombi sum and Dombi product of a and b are defined as follows:(11)a⊕Db=11+a/1−aφ+b/1−bφ1/φ,a⊗Db=11+1−a/aφ+1−b/bφ1/φ,where φ≥1.
Then, the new interval-valued q-rung orthopair fuzzy Dombi operations are introduced in Definition9.Definition 9.
Leta=ua−,ua+,va−,va+, a1=ua1−,ua1+,va1−,va1+, and a2=ua2−,ua2+,va2−,va2+ be three IVq-ROFNs, and φ≥1; the interval-valued q-rung orthopair fuzzy Dombi operational laws are shown in the four following equations:(12)a1⊕Da2=1−11+ua1−q/1−ua1−qφ+ua2−q/1−ua2−qφ1/φq,1−11+ua1+q/1−ua1+qφ+ua2+q/1−ua2+qφ1/φq,11+1−va1−q/va1−qφ+1−va2−q/va2−qφ1/φq,11+1−va1+q/va1+qφ+1−va2+q/va2+qφ1/φq,(13)a1⊗Da2=11+ua1−q/1−ua1−qφ+ua2−q/1−ua2−qφ1/φq,11+ua1+q/1−ua1+qφ+ua2+q/1−ua2+qφ1/φq,1−11+1−va1−q/va1−qφ+1−va2−q/va2−qφ1/φq,1−11+1−va1+q/va1+qφ+1−va2+q/va2+qφ1/φq,(14)δa=1−11+δua−q/1−ua−qφ1/φq,1−11+δua+q/1−ua+qφ1/φq,11+δ1−va−q/va−qφ1/φq,11+δ1−va+q/ua+qφ1/φq,(15)aδ=11+δua−q/1−ua−qφ1/φq,11+δua+q/1−ua+qφ1/φq,1−11+δva−q/1−va−qφ1/φq,1−11+δ1−va+q/va+qφ1/φq.
It can be verified that interval-valued q-rung orthopair fuzzy Dombi addition operation and multiplication operation given in this paper satisfy the operation rules of t-conorms and t-norms. Number multiplication and power Dombi operations of IVq-ROFNs also satisfy the operation rules of t-norms and t-conorms.Theorem 2.
Leta=ua−,ua+,va−,va+, a1=ua1−,ua1+,va1−,va1+, and a2=ua2−,ua2+,va2−,va2+ be three IVq-ROFNs, and δ,δ1,δ2>0; then the six following properties are satisfied and are shown in the six following equations:(16)a1⊕Da2=a2⊕Da1,(17)a1⊗Da2=a2⊗Da1,(18)δa1⊕Da2=δa2⊕Dδa1,(19)a1⊗Da2δ=a2δ⊗Da1δ,(20)δ1a⊕Dδ2a=δ1+δ2a,(21)aδ1⊗Daδ2=aδ1+δ2.Theorem2 can be proved by the interval-valued q-rung orthopair fuzzy Dombi operation laws. We only prove (18) and the others can be proved similarly.Proof. According to (12) and (14), the value of δa1⊕Da2 can be obtained as(22)δa1⊕Da2=1−11+δua1−q/1−ua1−qφ+ua2−q/1−ua2−qφ1/φq,1−11+δua1+q/1−ua1+qφ+ua2+q/1−ua2+qφ1/φq,11+δ1−va1−q/va1−qφ+1−va2−q/va2−qφ1/φq,11+δ1−va1+q/va1+qφ+1−va2+q/va2+qφ1/φq,and, according to (14), δa1 and δa2 are(23)δa1=1−11+δua1−q/1−ua1−qφ1/φq,1−11+δua1+q/1−ua1+qφ1/φq,11+δ1−va1−q/va1−qφ1/φq,11+δ1−va1+q/va1+qφ1/φq,δa2=1−11+δua2−q/1−ua2−qφ1/φq,1−11+δua2+q/1−ua2+qφ1/φq,11+δ1−va2−q/va2−qφ1/φq,11=δ1−va2+q/va2+qφ1/φq.According to (12), δa2⊕Dδa1 can be obtained as(24)δa1⊕Dδa2=1−11+δua1−q/1−ua1−qφ+ua2−q/1−ua2−qφ1/φq,1−11+δua1+q/1−ua1+qφ+ua2+q/1−ua2+qφ1/φq,11+δ1−va1−q/va1−qφ+1−va2−q/va2−qφ1/φq,11+δ1−va1+q/va1+qφ+1−va2+q/va2+qφ1/φq.Therefore,δa1⊕Da2=δa2⊕Dδa1 in (18) is certified.


### 3.2. Interval-Valued q-Rung Orthopair Fuzzy Dombi Weighted Average Operator

Definition 10.
Letaj=uaj−,uaj+,vaj−,vaj+ (j = 1, 2, …, n) be a set of IVq-ROFNs; ω=ω1,ω2,…,ωnT for the weight vector and satisfies ∑j=1nωj=1,ωj∈0,1,j=1,2,…,n. Then, the IVq-ROFDWA operator is defined as follows:(25)IVq−ROFDWAa1,a2,…,an=⊕Di=1nωiai.Theorem 3.
Letaj=uaj−,uaj+,vaj−,vaj+ (j = 1, 2, 3, …, n) be a set of IVq-ROFNs; the aggregation result of IVq−ROFDWAa1,a2,a3,…an is shown as follows:(26)IVq−ROFDWAa1,a2,a3,…an=1−11+∑inωiuai−q/1−uai−qφ1/φq,1−11+∑inωiuai+q/1−uai+qφ1/φq,11+∑inωi1−vai−q/vai−qφ1/φq,11+∑inωi1−vai+q/vai+qφ1/φq.Proof.Whenn = 2,(27)ω1a1=1−11+ω1ua1−q/1−ua1−qφ1/φq,1−11+ω1ua1+q/1−ua1+qφ1/φq,11+ω11−va1−q/va1−qφ1/φq,11+ω11−va1+q/va1+qφ1/φq,ω2a2=1−11+ω2ua2−q/1−ua2−qφ1/φq,1−11+ω2ua2+q/1−ua2+qφ1/φq,11+ω21−va2−q/va2−qφ1/φq,11+ω21−va2+q/va2+qφ1/φq,and thenIVq−ROFDWAa1,a2 can be obtained as follows:(28)IVq−ROFDWAa1,a2=⊕Di=12ωiai=ω1a1⊕Dω2a2=1−11+ω1ua1−q/1−ua1−qφ+ω2ua2−q/1−ua2−qφ1/φq,1−11+ω1ua1+q/1−ua1+qφ+ω2ua2+q/1−ua2+qφ1/φq,11+ω11−va1−q/va1−qφ+ω21−va2−q/va2−qφ1/φq,1ω11−va1+q/va1+qφ+ω21−va2+q/va2+qφ1/φq=1−11+∑i2ωiuai−q/1−uai−qφ1/φq,1−11+∑i2ωiuai+q/1−uai+qφ1/φq,11+∑i2ωi1−vai−q/vai−qφ1/φq,11+∑i2ωi1−vai+q/vai+qφ1/φq.Therefore, this result holds forn = 2. Suppose that (26) holds for n = k; that is to say,(29)IVq−ROFDWAa1,a2,a3,…ak=1−11+∑ikωiuai−q/1−uai−qφ1/φq,1−11+∑ikωiuai+q/1−uai+qφ1/φq,11+∑ikωi1−vai−q/vai−qφ1/φq,11+∑ikωi1−vai+q/vai+qφ1/φq.Whenn = k + 1, we have(30)IVq−ROFDWAa1,a2,a3,…ak+1=IVq−ROFDWAa1,a2,a3,…ak⊕Dωk+1ak+1=1−11+∑ikωiuai−q/1−uai−qφ1/φq,1−11+∑ikωiuai+q/1−uai+qφ1/φq,11+∑ikωi1−vai−q/vai−qφ1/φq,11+∑ikωi1−vai+q/vai+qφ1/φq⊕D1−11+ωk+1uak+1−q/1−uak+1−qφ1/φq,1−11+ωk+1uak+1+q/1−uak+1+qφ1/φq,11+ωk+11−vak+1−q/vak+1−qφ1/φq,11+ωk+11−vak+1−q/vak+1−qφ1/φq,and, according to formulas (12) and (14), we get(31)IVq−ROFDWAa1,a2,a3,…ak+1=1−11+∑ik+1ωiuai−q/1−uai−qφ1/φq,1−11+∑ik+1ωiuai+q/1−uai+qφ1/φq,11+∑ik+1ωi1−vai−q/vai−qφ1/φq,11+∑ik+1ωi1−vai−q/vai−qφ1/φq.Therefore, (30) holds for n = k + 1, and, according to the mathematical induction, Theorem 3 can be proved.Theorem 4 (idempotency).
Letaj=uaj−,uaj+,vaj−,vaj+ (j = 1, 2, 3, …, n) be a set of IVq-ROFNs and aj=a; then we have(32)IVq−ROFDWAa1,a2,a3,…,an=a.Proof. For any j (j = 1, 2, 3, …, n), aj=a; then the following result can be obtained:(33)IVq−ROFDWAa1,a2,a3,…an=1−11+∑inωiua−q/1−ua−qφ1/φq,1−11+∑inωiua+q/1−ua+qφ1/φq,11+∑inωi1−va−q/va−qφ1/φq,11+∑inωi1−va+q/va+qφ1/φq=1−11+ua−q/1−ua−qq,1−11+ua+q/1−ua+qq,11+1−va−q/va−qq,11+1−va+q/va+qq=ua−,ua+,va−,va+=aTheorem 5 (boundedness).
Letaj=uaj−,uaj+,vaj−,vaj+ (j = 1, 2, 3, …, n) be a set of IVq-ROFNs; then it is easy to obtain(34)amin≤IVq−ROFDWAa1,a2,a3,…,an≤amax,amax=maxjaj,amin=minjaj.The proof of Theorem5 can be deduced by the interval-valued q-rung orthopair fuzzy Dombi operations, which is ignored here.Theorem 6 (monotonicity).
Letaj,aj∗ (j = 1, 2, …, n) be two sets of IVq-ROFNs, and aj≤aj∗ (j = 1, 2, …, n); then we have(35)IVq−ROFDWAa1,a2,a3,…,an≤IVq−ROFDWAa1∗,a2∗,a3∗,…,an∗.Proof. According to (18), we can see(36)IVq−ROFDWAa1,a2,a3,...,an=ω1a1⊕Dω2a2⊕Dω3a3⊕D...⊕Dωnan,(37)IVq−ROFDWAa1∗,a2∗,a3∗,…,an∗=ω1a1∗⊕Dω2a2∗⊕Dω3a3∗⊕D…⊕Dωnan∗.Comparing the right sides of (36) and (37), (35) holds.Theorem 7 (commutativity).
Letα1′,α2′,…,αn′ be a set of IVq-ROFNs, and let α1,α2,…,αn be any of the permutations of α1′,α2′,…,αn′; then, we have(38)IVq−ROFDWAα1,α2,…,αn=IVq−ROFDWAα1′,α2′,…,αn′.The proof of Theorem7 can be deduced by the interval-valued q-rung orthopair fuzzy Dombi operations, which is ignored here.


## 3.1. Interval-Valued q-Rung Orthopair Fuzzy Dombi Operation

Definition 8 (see [32]).
For any two real numbersa, b∈[0, 1], the Dombi sum and Dombi product of a and b are defined as follows:(11)a⊕Db=11+a/1−aφ+b/1−bφ1/φ,a⊗Db=11+1−a/aφ+1−b/bφ1/φ,where φ≥1.
Then, the new interval-valued q-rung orthopair fuzzy Dombi operations are introduced in Definition9.Definition 9.
Leta=ua−,ua+,va−,va+, a1=ua1−,ua1+,va1−,va1+, and a2=ua2−,ua2+,va2−,va2+ be three IVq-ROFNs, and φ≥1; the interval-valued q-rung orthopair fuzzy Dombi operational laws are shown in the four following equations:(12)a1⊕Da2=1−11+ua1−q/1−ua1−qφ+ua2−q/1−ua2−qφ1/φq,1−11+ua1+q/1−ua1+qφ+ua2+q/1−ua2+qφ1/φq,11+1−va1−q/va1−qφ+1−va2−q/va2−qφ1/φq,11+1−va1+q/va1+qφ+1−va2+q/va2+qφ1/φq,(13)a1⊗Da2=11+ua1−q/1−ua1−qφ+ua2−q/1−ua2−qφ1/φq,11+ua1+q/1−ua1+qφ+ua2+q/1−ua2+qφ1/φq,1−11+1−va1−q/va1−qφ+1−va2−q/va2−qφ1/φq,1−11+1−va1+q/va1+qφ+1−va2+q/va2+qφ1/φq,(14)δa=1−11+δua−q/1−ua−qφ1/φq,1−11+δua+q/1−ua+qφ1/φq,11+δ1−va−q/va−qφ1/φq,11+δ1−va+q/ua+qφ1/φq,(15)aδ=11+δua−q/1−ua−qφ1/φq,11+δua+q/1−ua+qφ1/φq,1−11+δva−q/1−va−qφ1/φq,1−11+δ1−va+q/va+qφ1/φq.
It can be verified that interval-valued q-rung orthopair fuzzy Dombi addition operation and multiplication operation given in this paper satisfy the operation rules of t-conorms and t-norms. Number multiplication and power Dombi operations of IVq-ROFNs also satisfy the operation rules of t-norms and t-conorms.Theorem 2.
Leta=ua−,ua+,va−,va+, a1=ua1−,ua1+,va1−,va1+, and a2=ua2−,ua2+,va2−,va2+ be three IVq-ROFNs, and δ,δ1,δ2>0; then the six following properties are satisfied and are shown in the six following equations:(16)a1⊕Da2=a2⊕Da1,(17)a1⊗Da2=a2⊗Da1,(18)δa1⊕Da2=δa2⊕Dδa1,(19)a1⊗Da2δ=a2δ⊗Da1δ,(20)δ1a⊕Dδ2a=δ1+δ2a,(21)aδ1⊗Daδ2=aδ1+δ2.Theorem2 can be proved by the interval-valued q-rung orthopair fuzzy Dombi operation laws. We only prove (18) and the others can be proved similarly.Proof. According to (12) and (14), the value of δa1⊕Da2 can be obtained as(22)δa1⊕Da2=1−11+δua1−q/1−ua1−qφ+ua2−q/1−ua2−qφ1/φq,1−11+δua1+q/1−ua1+qφ+ua2+q/1−ua2+qφ1/φq,11+δ1−va1−q/va1−qφ+1−va2−q/va2−qφ1/φq,11+δ1−va1+q/va1+qφ+1−va2+q/va2+qφ1/φq,and, according to (14), δa1 and δa2 are(23)δa1=1−11+δua1−q/1−ua1−qφ1/φq,1−11+δua1+q/1−ua1+qφ1/φq,11+δ1−va1−q/va1−qφ1/φq,11+δ1−va1+q/va1+qφ1/φq,δa2=1−11+δua2−q/1−ua2−qφ1/φq,1−11+δua2+q/1−ua2+qφ1/φq,11+δ1−va2−q/va2−qφ1/φq,11=δ1−va2+q/va2+qφ1/φq.According to (12), δa2⊕Dδa1 can be obtained as(24)δa1⊕Dδa2=1−11+δua1−q/1−ua1−qφ+ua2−q/1−ua2−qφ1/φq,1−11+δua1+q/1−ua1+qφ+ua2+q/1−ua2+qφ1/φq,11+δ1−va1−q/va1−qφ+1−va2−q/va2−qφ1/φq,11+δ1−va1+q/va1+qφ+1−va2+q/va2+qφ1/φq.Therefore,δa1⊕Da2=δa2⊕Dδa1 in (18) is certified.


## 3.2. Interval-Valued q-Rung Orthopair Fuzzy Dombi Weighted Average Operator

Definition 10.
Letaj=uaj−,uaj+,vaj−,vaj+ (j = 1, 2, …, n) be a set of IVq-ROFNs; ω=ω1,ω2,…,ωnT for the weight vector and satisfies ∑j=1nωj=1,ωj∈0,1,j=1,2,…,n. Then, the IVq-ROFDWA operator is defined as follows:(25)IVq−ROFDWAa1,a2,…,an=⊕Di=1nωiai.Theorem 3.
Letaj=uaj−,uaj+,vaj−,vaj+ (j = 1, 2, 3, …, n) be a set of IVq-ROFNs; the aggregation result of IVq−ROFDWAa1,a2,a3,…an is shown as follows:(26)IVq−ROFDWAa1,a2,a3,…an=1−11+∑inωiuai−q/1−uai−qφ1/φq,1−11+∑inωiuai+q/1−uai+qφ1/φq,11+∑inωi1−vai−q/vai−qφ1/φq,11+∑inωi1−vai+q/vai+qφ1/φq.Proof.Whenn = 2,(27)ω1a1=1−11+ω1ua1−q/1−ua1−qφ1/φq,1−11+ω1ua1+q/1−ua1+qφ1/φq,11+ω11−va1−q/va1−qφ1/φq,11+ω11−va1+q/va1+qφ1/φq,ω2a2=1−11+ω2ua2−q/1−ua2−qφ1/φq,1−11+ω2ua2+q/1−ua2+qφ1/φq,11+ω21−va2−q/va2−qφ1/φq,11+ω21−va2+q/va2+qφ1/φq,and thenIVq−ROFDWAa1,a2 can be obtained as follows:(28)IVq−ROFDWAa1,a2=⊕Di=12ωiai=ω1a1⊕Dω2a2=1−11+ω1ua1−q/1−ua1−qφ+ω2ua2−q/1−ua2−qφ1/φq,1−11+ω1ua1+q/1−ua1+qφ+ω2ua2+q/1−ua2+qφ1/φq,11+ω11−va1−q/va1−qφ+ω21−va2−q/va2−qφ1/φq,1ω11−va1+q/va1+qφ+ω21−va2+q/va2+qφ1/φq=1−11+∑i2ωiuai−q/1−uai−qφ1/φq,1−11+∑i2ωiuai+q/1−uai+qφ1/φq,11+∑i2ωi1−vai−q/vai−qφ1/φq,11+∑i2ωi1−vai+q/vai+qφ1/φq.Therefore, this result holds forn = 2. Suppose that (26) holds for n = k; that is to say,(29)IVq−ROFDWAa1,a2,a3,…ak=1−11+∑ikωiuai−q/1−uai−qφ1/φq,1−11+∑ikωiuai+q/1−uai+qφ1/φq,11+∑ikωi1−vai−q/vai−qφ1/φq,11+∑ikωi1−vai+q/vai+qφ1/φq.Whenn = k + 1, we have(30)IVq−ROFDWAa1,a2,a3,…ak+1=IVq−ROFDWAa1,a2,a3,…ak⊕Dωk+1ak+1=1−11+∑ikωiuai−q/1−uai−qφ1/φq,1−11+∑ikωiuai+q/1−uai+qφ1/φq,11+∑ikωi1−vai−q/vai−qφ1/φq,11+∑ikωi1−vai+q/vai+qφ1/φq⊕D1−11+ωk+1uak+1−q/1−uak+1−qφ1/φq,1−11+ωk+1uak+1+q/1−uak+1+qφ1/φq,11+ωk+11−vak+1−q/vak+1−qφ1/φq,11+ωk+11−vak+1−q/vak+1−qφ1/φq,and, according to formulas (12) and (14), we get(31)IVq−ROFDWAa1,a2,a3,…ak+1=1−11+∑ik+1ωiuai−q/1−uai−qφ1/φq,1−11+∑ik+1ωiuai+q/1−uai+qφ1/φq,11+∑ik+1ωi1−vai−q/vai−qφ1/φq,11+∑ik+1ωi1−vai−q/vai−qφ1/φq.Therefore, (30) holds for n = k + 1, and, according to the mathematical induction, Theorem 3 can be proved.Theorem 4 (idempotency).
Letaj=uaj−,uaj+,vaj−,vaj+ (j = 1, 2, 3, …, n) be a set of IVq-ROFNs and aj=a; then we have(32)IVq−ROFDWAa1,a2,a3,…,an=a.Proof. For any j (j = 1, 2, 3, …, n), aj=a; then the following result can be obtained:(33)IVq−ROFDWAa1,a2,a3,…an=1−11+∑inωiua−q/1−ua−qφ1/φq,1−11+∑inωiua+q/1−ua+qφ1/φq,11+∑inωi1−va−q/va−qφ1/φq,11+∑inωi1−va+q/va+qφ1/φq=1−11+ua−q/1−ua−qq,1−11+ua+q/1−ua+qq,11+1−va−q/va−qq,11+1−va+q/va+qq=ua−,ua+,va−,va+=aTheorem 5 (boundedness).
Letaj=uaj−,uaj+,vaj−,vaj+ (j = 1, 2, 3, …, n) be a set of IVq-ROFNs; then it is easy to obtain(34)amin≤IVq−ROFDWAa1,a2,a3,…,an≤amax,amax=maxjaj,amin=minjaj.The proof of Theorem5 can be deduced by the interval-valued q-rung orthopair fuzzy Dombi operations, which is ignored here.Theorem 6 (monotonicity).
Letaj,aj∗ (j = 1, 2, …, n) be two sets of IVq-ROFNs, and aj≤aj∗ (j = 1, 2, …, n); then we have(35)IVq−ROFDWAa1,a2,a3,…,an≤IVq−ROFDWAa1∗,a2∗,a3∗,…,an∗.Proof. According to (18), we can see(36)IVq−ROFDWAa1,a2,a3,...,an=ω1a1⊕Dω2a2⊕Dω3a3⊕D...⊕Dωnan,(37)IVq−ROFDWAa1∗,a2∗,a3∗,…,an∗=ω1a1∗⊕Dω2a2∗⊕Dω3a3∗⊕D…⊕Dωnan∗.Comparing the right sides of (36) and (37), (35) holds.Theorem 7 (commutativity).
Letα1′,α2′,…,αn′ be a set of IVq-ROFNs, and let α1,α2,…,αn be any of the permutations of α1′,α2′,…,αn′; then, we have(38)IVq−ROFDWAα1,α2,…,αn=IVq−ROFDWAα1′,α2′,…,αn′.The proof of Theorem7 can be deduced by the interval-valued q-rung orthopair fuzzy Dombi operations, which is ignored here.


## 4. Interval-Valued q-Rung Orthopair Fuzzy QUALIFLEX Decision Analysis Approach

### 4.1. Description of MAGDM Problem with IVq-ROFNs

LetX=x1,x2,…,xm be a set of alternatives, let C=c1,c2,…,cn be the attributes set, and let D=d1,d2,…,dt be the experts set. We denote the attribute weighting vector by W=w1,w2,…,wnT, where wj is the relative weight of attribute cj and ∑j=1nwj=1,wj∈0,1j=1,2,…,n. Let ϖk be the weight of expert dkk=1,2,…,t and the expert weights are unknown in advance in this study. In real-world MAGDM process, each expert provides an evaluated value for an alternative under one attribute, and all evaluated information provided by the expert constructs a decision matrix. For the k-th expert, the decision matrix is expressed as Ak=aijkm×n, and in this study the evaluated values provided by the experts are represented by IVq-ROFNs and thus the element aijk=uaijk−,uaijk+,vaijk−,vaijk+ in the decision matrix Ak takes the form of IVq-ROFN satisfying uaijk+q+vaijk+q≤1. The value of parameter q is determined by DMs according to the actual decision-making situation.The dimensions and measurements of attribute values are often different because the types of evaluated attributes are different. To eliminate the effect of different physical dimensions and measurements on the final decision results, we transform the attribute values of the cost type into the attribute values of the benefit type by using the following equation:(39)rijk=aijk,aijk∈Ω1,aijkc,aijk∈Ω2,,i=1,2,…,m;j=1,2,…,n,where Ω1 indicates benefit type and Ω2 indicates cost type, and(40)aijkc=vaij−,vaij+,uaij−,uaij+.In what follows, we develop an interval-valued q-rung orthopair fuzzy QUALIFLEX analysis approach to solve the above MAGDM problems with incomplete weight information in which the evaluation values are expressed in the format of IVq-ROFNs. One important task of this method is to determine the weights of experts.


### 4.2. Identification of Experts’ Weights

In practical group decision-making process, the group opinion should be consistent with each expert’s opinion to the greatest extent [36, 37]. Motivated by the study in [38], the group opinions in this paper are divided into four parts, that is, the positive ideal group opinion, the negative ideal group opinion, the negative ideal left group opinion, and the negative ideal right group opinion. That is to say, the weight of one expert should be assigned a greater value when the deviation value between the positive group opinion and this expert is much smaller. Similarly, the weight of one expert should be assigned a smaller value when the deviation value between the negative group opinion and this expert is much smaller.Thus, we first denote the positive ideal group opinion byCri+∗=Cri1+∗,Cri2+∗,…,Crin+∗, and the element Crij+∗ can be determined by the following formula:(41)Crij+∗=uCrij+∗−,uCrij+∗+,vCrij+∗−,vCrij+∗+=IVq−ROFDWArij1,rij2,…,rijt.Then, the negative ideal group opinion is denoted byCri−∗=Cri1−∗,Cri2−∗,…,Crin−∗, and the element Crij−∗=uCrij−∗−,uCrij−∗+,vCrij−∗−,vCrij−∗+ can be determined according to (40) as follows:(42)uCrij−∗−=vCrij+∗−,uCrij−∗+=vCrij+∗+,vCrij−∗−=uCrij+∗−,vCrij−∗+=uCrij+∗+.Meanwhile, the negative ideal right opinion is denoted byIri+∗=Iri1+∗,Iri2+∗,⋯,Irin+∗ and the element Irij+∗=uIrij+∗−,uIrij+∗+,vIrij+∗−,vIrij+∗+ can be determined by the following formula:(43)uIrij+∗−=maxk=1turijk−,uIrij+∗+=maxk=1turijk+,vIrij+∗−=maxk=1tvrijk−,vIrij+∗+=maxk=1tvrijk+.Similarity, the negative ideal left group opinion is denoted byIri−∗=Iri1−∗,Iri2−∗,…,Irin−∗ and Irij−∗=uIrij−∗−,uIrij−∗+,vIrij−∗−,vIrij−∗+ which can be determined by the following formula:(44)uIrij−∗−=mink=1turijk−,uIrij−∗+=mink=1turijk+,vIrij−∗−=mink=1tvrijk−,vIrij−∗+=mink=1tvrijk+.Then, the closeness index for expertdk is determined by the following formula:(45)CIk=∑j=1n∑i=1mdrijk,Cri−∗+∑j=1n∑i=1mdrijk,Iri+∗+∑j=1n∑i=1mdrijk,Iri−∗∑j=1n∑i=1mdrijk,Cri+∗+∑j=1n∑i=1mdrijk,Cri−∗+∑j=1n∑i=1mdrijk,Iri+∗+∑j=1n∑i=1mdrijk,Iri−∗.Thus, the weights of expert can be determined by the following formula:(46)ϖk=CIk∑k=1tCIk.


### 4.3. Interval-Valued q-Rung Orthopair Fuzzy QUALIFLEX Method

Another important task is to sort the alternatives and identify the best optimal alternative. On the basis of the obtained weights of experts, we next integrate the individual decision matricesRk=rijkm×nk=1,2,…,t into the collective decision matrix R=rijm×n by the developed IVq-ROFDWA operator as follows:(47)rij=IVq−ROFDWArij1,rij2,…,rijt,i=1,2,…,m;j=1,2,…,n.Based on the collective decision matrix, the interval-valued q-rung orthopair fuzzy QUALIFLEX method is developed to obtain the ranking order of alternatives, which starts with the computation of the concordance/discordance index based on the successive permutations of all possible rankings of alternatives.For the set of alternativesX=x1,x2,…,xm, there are m! permutations of the ranking of all alternatives, and let Pρρ=1,2,…,m! denote the ρth permutation as(48)Pρ=…,xξ,…,xζ,….xξ,xζ∈X; the alternative xξ is ranked higher than or equal to xζ.Considering that the decision information takes the form of IVq-ROFNs, this study utilizes the ranking approach introduced in Definition6 to compare the magnitudes of evaluated values and compute the corresponding concordance/discordance index by the distance measure. Then, the concordance/discordance index ϕjρxξ,xζ for each pair of alternatives xξ,xζ, xξ,xζ∈X, at the level of preorder according to the jth attribute and the ranking corresponding to the ρth permutation, can be defined as follows:(49)ϕjρxξ,xζ=drξj,rζj,ifrξj>rζj,0,ifrξj=rζj,−drξj,rζj,ifrξj<rζj.Thus, the weighted concordance/discordance indexϕjρxξ,xζ for each pair of alternatives xξ,xζ, at the level of preorder according to n attributes and the ranking corresponding to the ρth permutation, can be defined as follows:(50)ϕρxξ,xζ=∑j=1nwjϕjρxξ,xζ.Finally, the comprehensive concordance/discordance indexϕρ for the ρth permutation can be defined as follows:(51)ϕρ=∑xξ,xζ∈Xϕρxξ,xζ=∑xξ,xζ∈X∑j=1nwjϕjρxξ,xζ.It is found that the more the comprehensive concordance/discordance index value is, the better the ranking order of alternatives is. Therefore, we can determine the optimal ranking order of alternatives by comparing the valuesϕρ of each permutation Pρ.Based on the above analysis, the decision processes of the proposed interval-valued q-rung orthopair fuzzy QUALIFLEX method can be summarized as follows.Step 1.
Formulate interval-valued q-rung orthopair fuzzy MAGDM problem and identify the interval-valued q-rung orthopair fuzzy decision matrix, and transform the decision matrixAk into a standard matrix Rkk=1,2,…,t by using (39) and (40).Step 2.
Determine the weights of experts by using (46).Step 3.
Integrate the individual decision matricesRk=rijkm×nk=1,2,…,t into to the collective decision matrix R=rijm×n by (47).Step 4.
List all of the possiblem! permutations that should be tested in the next steps. Let Pρ denote the ρth permutation using (48).Step 5.
Calculate the concordance/discordance indexϕjρxξ,xζ using (49).Step 6.
Calculate the weighted concordance/discordance indicesϕρxξ,xζ using (50).Step 7.
Calculate the comprehensive concordance/discordance indexϕρ for permutation Pρ using (51).Step 8.
Determine the optimal ranking order of the alternatives. On the basis of the results instep 7, the permutation with the maximal comprehensive concordance/discordance index ϕρ is the optimal ranking order of the alternatives; namely,(52)P∗=maxρ=1m!ϕρ.
The decision processes of the proposed interval-valued q-rung orthopair fuzzy QUALIFLEX method are described in Figure1.Figure 1
Frame diagram of interval-valued q-rung orthopair fuzzy QUALIFLEX method.


## 4.1. Description of MAGDM Problem with IVq-ROFNs

LetX=x1,x2,…,xm be a set of alternatives, let C=c1,c2,…,cn be the attributes set, and let D=d1,d2,…,dt be the experts set. We denote the attribute weighting vector by W=w1,w2,…,wnT, where wj is the relative weight of attribute cj and ∑j=1nwj=1,wj∈0,1j=1,2,…,n. Let ϖk be the weight of expert dkk=1,2,…,t and the expert weights are unknown in advance in this study. In real-world MAGDM process, each expert provides an evaluated value for an alternative under one attribute, and all evaluated information provided by the expert constructs a decision matrix. For the k-th expert, the decision matrix is expressed as Ak=aijkm×n, and in this study the evaluated values provided by the experts are represented by IVq-ROFNs and thus the element aijk=uaijk−,uaijk+,vaijk−,vaijk+ in the decision matrix Ak takes the form of IVq-ROFN satisfying uaijk+q+vaijk+q≤1. The value of parameter q is determined by DMs according to the actual decision-making situation.The dimensions and measurements of attribute values are often different because the types of evaluated attributes are different. To eliminate the effect of different physical dimensions and measurements on the final decision results, we transform the attribute values of the cost type into the attribute values of the benefit type by using the following equation:(39)rijk=aijk,aijk∈Ω1,aijkc,aijk∈Ω2,,i=1,2,…,m;j=1,2,…,n,where Ω1 indicates benefit type and Ω2 indicates cost type, and(40)aijkc=vaij−,vaij+,uaij−,uaij+.In what follows, we develop an interval-valued q-rung orthopair fuzzy QUALIFLEX analysis approach to solve the above MAGDM problems with incomplete weight information in which the evaluation values are expressed in the format of IVq-ROFNs. One important task of this method is to determine the weights of experts.


## 4.2. Identification of Experts’ Weights

In practical group decision-making process, the group opinion should be consistent with each expert’s opinion to the greatest extent [36, 37]. Motivated by the study in [38], the group opinions in this paper are divided into four parts, that is, the positive ideal group opinion, the negative ideal group opinion, the negative ideal left group opinion, and the negative ideal right group opinion. That is to say, the weight of one expert should be assigned a greater value when the deviation value between the positive group opinion and this expert is much smaller. Similarly, the weight of one expert should be assigned a smaller value when the deviation value between the negative group opinion and this expert is much smaller.Thus, we first denote the positive ideal group opinion byCri+∗=Cri1+∗,Cri2+∗,…,Crin+∗, and the element Crij+∗ can be determined by the following formula:(41)Crij+∗=uCrij+∗−,uCrij+∗+,vCrij+∗−,vCrij+∗+=IVq−ROFDWArij1,rij2,…,rijt.Then, the negative ideal group opinion is denoted byCri−∗=Cri1−∗,Cri2−∗,…,Crin−∗, and the element Crij−∗=uCrij−∗−,uCrij−∗+,vCrij−∗−,vCrij−∗+ can be determined according to (40) as follows:(42)uCrij−∗−=vCrij+∗−,uCrij−∗+=vCrij+∗+,vCrij−∗−=uCrij+∗−,vCrij−∗+=uCrij+∗+.Meanwhile, the negative ideal right opinion is denoted byIri+∗=Iri1+∗,Iri2+∗,⋯,Irin+∗ and the element Irij+∗=uIrij+∗−,uIrij+∗+,vIrij+∗−,vIrij+∗+ can be determined by the following formula:(43)uIrij+∗−=maxk=1turijk−,uIrij+∗+=maxk=1turijk+,vIrij+∗−=maxk=1tvrijk−,vIrij+∗+=maxk=1tvrijk+.Similarity, the negative ideal left group opinion is denoted byIri−∗=Iri1−∗,Iri2−∗,…,Irin−∗ and Irij−∗=uIrij−∗−,uIrij−∗+,vIrij−∗−,vIrij−∗+ which can be determined by the following formula:(44)uIrij−∗−=mink=1turijk−,uIrij−∗+=mink=1turijk+,vIrij−∗−=mink=1tvrijk−,vIrij−∗+=mink=1tvrijk+.Then, the closeness index for expertdk is determined by the following formula:(45)CIk=∑j=1n∑i=1mdrijk,Cri−∗+∑j=1n∑i=1mdrijk,Iri+∗+∑j=1n∑i=1mdrijk,Iri−∗∑j=1n∑i=1mdrijk,Cri+∗+∑j=1n∑i=1mdrijk,Cri−∗+∑j=1n∑i=1mdrijk,Iri+∗+∑j=1n∑i=1mdrijk,Iri−∗.Thus, the weights of expert can be determined by the following formula:(46)ϖk=CIk∑k=1tCIk.


## 4.3. Interval-Valued q-Rung Orthopair Fuzzy QUALIFLEX Method

Another important task is to sort the alternatives and identify the best optimal alternative. On the basis of the obtained weights of experts, we next integrate the individual decision matricesRk=rijkm×nk=1,2,…,t into the collective decision matrix R=rijm×n by the developed IVq-ROFDWA operator as follows:(47)rij=IVq−ROFDWArij1,rij2,…,rijt,i=1,2,…,m;j=1,2,…,n.Based on the collective decision matrix, the interval-valued q-rung orthopair fuzzy QUALIFLEX method is developed to obtain the ranking order of alternatives, which starts with the computation of the concordance/discordance index based on the successive permutations of all possible rankings of alternatives.For the set of alternativesX=x1,x2,…,xm, there are m! permutations of the ranking of all alternatives, and let Pρρ=1,2,…,m! denote the ρth permutation as(48)Pρ=…,xξ,…,xζ,….xξ,xζ∈X; the alternative xξ is ranked higher than or equal to xζ.Considering that the decision information takes the form of IVq-ROFNs, this study utilizes the ranking approach introduced in Definition6 to compare the magnitudes of evaluated values and compute the corresponding concordance/discordance index by the distance measure. Then, the concordance/discordance index ϕjρxξ,xζ for each pair of alternatives xξ,xζ, xξ,xζ∈X, at the level of preorder according to the jth attribute and the ranking corresponding to the ρth permutation, can be defined as follows:(49)ϕjρxξ,xζ=drξj,rζj,ifrξj>rζj,0,ifrξj=rζj,−drξj,rζj,ifrξj<rζj.Thus, the weighted concordance/discordance indexϕjρxξ,xζ for each pair of alternatives xξ,xζ, at the level of preorder according to n attributes and the ranking corresponding to the ρth permutation, can be defined as follows:(50)ϕρxξ,xζ=∑j=1nwjϕjρxξ,xζ.Finally, the comprehensive concordance/discordance indexϕρ for the ρth permutation can be defined as follows:(51)ϕρ=∑xξ,xζ∈Xϕρxξ,xζ=∑xξ,xζ∈X∑j=1nwjϕjρxξ,xζ.It is found that the more the comprehensive concordance/discordance index value is, the better the ranking order of alternatives is. Therefore, we can determine the optimal ranking order of alternatives by comparing the valuesϕρ of each permutation Pρ.Based on the above analysis, the decision processes of the proposed interval-valued q-rung orthopair fuzzy QUALIFLEX method can be summarized as follows.Step 1.
Formulate interval-valued q-rung orthopair fuzzy MAGDM problem and identify the interval-valued q-rung orthopair fuzzy decision matrix, and transform the decision matrixAk into a standard matrix Rkk=1,2,…,t by using (39) and (40).Step 2.
Determine the weights of experts by using (46).Step 3.
Integrate the individual decision matricesRk=rijkm×nk=1,2,…,t into to the collective decision matrix R=rijm×n by (47).Step 4.
List all of the possiblem! permutations that should be tested in the next steps. Let Pρ denote the ρth permutation using (48).Step 5.
Calculate the concordance/discordance indexϕjρxξ,xζ using (49).Step 6.
Calculate the weighted concordance/discordance indicesϕρxξ,xζ using (50).Step 7.
Calculate the comprehensive concordance/discordance indexϕρ for permutation Pρ using (51).Step 8.
Determine the optimal ranking order of the alternatives. On the basis of the results instep 7, the permutation with the maximal comprehensive concordance/discordance index ϕρ is the optimal ranking order of the alternatives; namely,(52)P∗=maxρ=1m!ϕρ.
The decision processes of the proposed interval-valued q-rung orthopair fuzzy QUALIFLEX method are described in Figure1.Figure 1
Frame diagram of interval-valued q-rung orthopair fuzzy QUALIFLEX method.


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

In order to improve the management efficiency of rural doctors, it is planned to develop the Department of Hypertension Daily Follow-up Early Warning Management. After research, it is determined that the warning colors are 5 types,xi (i = 1, 2, 3, 4, 5): x1 means urgent handling and notifying residents to seek doctor and treatment immediately, shown in red; x2 means urgent treatment is required and informing residents to seek doctor in time, shown in orange; x3 means timely follow-up is needed to promote management, shown in yellow; x4 means short-term need to promote services, shown in blue; x5 means temporary no need to promote management services, shown in green. c1 (measurement value of blood pressure), c2 (cardiovascular disease), c3 (family history of hypertension), and c4 (obesity) are used to express their attributes, and the weights of them are w = (0.7, 0.1, 0.1, 0.1).In the absence of medication, for a 60-year-old male resident, a certain measurement value and related factors of collection are as follows: systolic blood pressure is 153 mmHg, dilation pressure is 98 mmHg, there are no related disease records, obesity, or family history of hypertension, the doctor has no follow-up records, the evaluation value given by four experts is expressed by q-ROFNs, because the cost type and benefit type do not require complicated calculations, and the amount of data is small, and the converted evaluation matrix is directly given in this paper. The evaluation values are given in Tables1–4.Table 1
Decision matrix given by Expert 1R1.
c1c2c3c4x1([0.7, 0.8], [0.2, 0.3])([0.5, 0.6], [0.6, 0.7])([0.8, 0.9], [0.1, 0.2])([0.8, 0.95], [0.1, 0.2])x2([0.8, 0.9], [0.1, 0.2])([0.5, 0.55], [0.4, 0.5])([0.8, 0.9], [0.1, 0.15])([0.8, 0.95], [0.1, 0.2])x3([0.9, 0.95], [0.1, 0.2])([0.5, 0.6], [0.3, 0.4])([0.8, 0.85], [0.2, 0.3])([0.8, 0.9], [0.1, 0.2])x4([0.6, 0.7], [0.4, 0.5])([0.4, 0.5], [0.3, 0.4])([0.75, 0.8], [0.2, 0.3])([0.7, 0.8], [0.2, 0.3])x5([0.1, 0.2], [0.8, 0.9])([0.2, 0.3], [0.5, 0.6])([0.7, 0.85], [0.3, 0.4])([0.6, 0.7], [0.2, 0.3])Table 2
Decision matrix given by Expert 2R2.
c1c2c3c4x1([0.6, 0.7], [0.2, 0.3])([0.6, 0.65], [0.4, 0.5])([0.7, 0.8], [0.1, 0.2])([0.6, 0.65], [0.2, 0.3])x2([0.85, 0.9], [0.1, 0.15])([0.5, 0.55], [0.4, 0.5])([0.6, 0.7], [0.1, 0.15])([0.7, 0.75], [0.1, 0.2])x3([0.9, 0.95], [0.1, 0.15])([0.5, 0.6], [0.3, 0.4])([0.5, 0.6], [0.2, 0.3])([0.45, 0.5], [0.2, 0.3])x4([0.2, 0.3], [0.7, 0.8])([0.4, 0.5], [0.3, 0.4])([0.70, 0.80], [0.2, 0.3])([0.50, 0.6], [0.1, 0.2])x5([0.1, 0.2], [0.8, 0.95])([0.2, 0.3], [0.5, 0.6])([0.7, 0.80], [0.3, 0.4])([0.5, 0.6], [0.1, 0.2])Table 3
Decision matrix given by Expert 3R3.
c1c2c3c4x1([0.7, 0.9], [0.2, 0.3])([0.5, 0.6], [0.4, 0.5])([0.8, 0.9], [0.1, 0.2])([0.7, 0.8], [0.1, 0.2])x2([0.7, 0.9], [0.1, 0.2])([0.5, 0.55], [0.4, 0.5])([0.8, 0.9], [0.1, 0.3])([0.7, 0.90], [0.1, 0.2])x3([0.8, 0.95], [0.1, 0.2])([0.6, 0.7], [0.3, 0.4])([0.8, 0.9], [0.1, 0.3])([0.8, 0.9], [0.1, 0.2])x4([0.5, 0.6], [0.4, 0.5])([0.5, 0.6], [0.3, 0.4])([0.75, 0.85], [0.1, 0.3])([0.7, 0.8], [0.3, 0.4])x5([0.1, 0.2], [0.8, 0.9])([0.4, 0.5], [0.7, 0.8])([0.75, 0.8], [0.2, 0.4])([0.75, 0.85], [0.1, 0.2])Table 4
Decision matrix given by Expert 4R4.
c1c2c3c4x1([0.7, 0.8], [0.2, 0.3])([0.55, 0.65], [0.3, 0.4])([0.7, 0.9], [0.15, 0.2])([0.70, 0.80], [0.1, 0.2])x2([0.8, 0.9], [0.15, 0.2])([0.55, 0.6], [0.4, 0.5])([0.7, 0.85], [0.1, 0.15])([0.70, 0.80], [0.2, 0.3])x3([0.8, 0.9], [0.15, 0.2])([0.5, 0.6], [0.3, 0.4])([0.7, 0.9], [0.1, 0.3])([0.8, 0.9], [0.1, 0.2])x4([0.6, 0.7], [0.6, 0.7])([0.4, 0.6], [0.6, 0.7])([0.6, 0.8], [0.2, 0.3])([0.7, 0.8], [0.1, 0.3])x5([0.1, 0.2], [0.85, 0.9])([0.4, 0.5], [0.75, 0.8])([0.7, 0.8], [0.3, 0.4])([0.7, 0.8], [0.1, 0.2])


### 5.1. The Calculated Results by the Developed Method

We next use the developed method to identify the optimal alternative. Owing to the fact that the weights of experts are unknown in advance in this case, we first need to objectively determine the weights of experts. Using (41), the positive ideal group opinion with the form of interval-valued q-rung orthopair fuzzy matrix CR+ is determined in Table 5.Table 5
The positive ideal group opinion matrixCR+.
c1c2c3c4x1([0.68, 0.85], [0.20, 0.30])([0.55, 0.63], [0.50, 0.60])([0.77, 0.89], [0.12, 0.20])([0.73, 0.91], [0.16, 0.25])x2([0.81, 0.90], [0.12, 0.19])([0.52, 0.57], [0.40, 0.50])([0.76, 0.88], [0.10, 0.24])([0.74, 0.92], [0.16, 0.25])x3([0.88, 0.94], [0.12, 0.19])([0.54, 0.64], [0.30, 0.40])([0.76, 0.88], [0.18, 0.30])([0.78, 0.89], [0.16, 0.25])x4([0.51, 0.61], [0.66, 0.76])([0.44, 0.56], [0.50, 0.60])([0.72, 0.82], [0.19, 0.30])([0.68, 0.78], [0.24, 0.34])x5([0.10, 0.20], [0.82, 0.93])([0.36, 0.45], [0.68, 0.76])([0.72, 0.82], [0.29, 0.40])([0.69, 0.79], [0.16, 0.25])The negative ideal group opinion is obtained by (42) and the calculated results are listed in Table 6.Table 6
The negative ideal group opinion matrixCR−.
c1c2c3c4x1([0.20, 0.30], [0.68, 0.85])([0.50, 0.60], [0.55, 0.63])([0.12, 0.20], [0.77, 0.89])([0.16, 0.25], [0.73, 0.91])x2([0.12, 0.19], [0.81, 0.90])([0.40, 0.50], [0.52, 0.57])([0.10, 0.24], [0.76, 0.88])([0.16, 0.25], [0.74, 0.92])x3([0.12, 0.19], [0.88, 0.94])([0.30, 0.40], [0.54, 0.64])([0.18, 0.30], [0.76, 0.88])([0.16, 0.25], [0.78, 0.89])x4([0.66, 0.76], [0.51, 0.61])([0.50, 0.60], [0.44, 0.56])([0.19, 0.30], [0.72, 0.82])([0.24, 0.34],[0.68, 0.78])x5([0.82, 0.93], [0.10, 0.20])([0.68, 0.76], [0.36, 0.45])([0.29, 0.40], [0.72, 0.82])([0.16, 0.25], [0.69, 0.79])Similarly, using (43) and (44), the negative ideal right group opinion and the negative ideal left group opinion are obtained, and the calculated results are listed in Tables 7 and 8.Table 7
The negative ideal right group opinion matrixIr+.
c1c2c3c4x1([0.70, 0.90], [0.20, 0.30])([0.60, 0.65], [0.60, 0.70])([0.80, 0.90], [0.15, 0.20])([0.80, 0.95], [0.20, 0.30])x2([0.85, 0.90], [0.15, 0.20])([0.55, 0.60], [0.40, 0.50])([0.80, 0.90], [0.10, 0.30])([0.80, 0.95], [0.20, 0.30])x3([0.90, 0.95], [0.15, 0.20])([0.60, 0.70], [0.30, 0.40])([0.80, 0.90], [0.20, 0.30])([0.80, 0.90], [0.20, 0.30])x4([0.60, 0.70], [0.70, 0.80])([0.50, 0.60], [0.60, 0.70])([0.75, 0.85], [0.20, 0.30])([0.70, 0.80], [0.30, 0.40])x5([0.10, 0.20], [0.85, 0.95])([0.40, 0.50], [0.75, 0.80])([0.75, 0.85], [0.30, 0.40])([0.75, 0.85], [0.20, 0.30])Table 8
The negative ideal left group opinion matrixIr−.
c1c2c3c4x1([0.60, 0.70], [0.20, 0.30])([0.50, 0.60], [0.30, 0.40])([0.70, 0.80], [0.10, 0.20])([0.60, 0.65], [0.10, 0.20])x2([0.70, 0.90], [0.10, 0.15])([0.50, 0.55], [0.40, 0.50])([0.60, 0.70], [0.10, 0.15])([0.70, 0.75], [0.10, 0.20])x3([0.80, 0.90], [0.10, 0.15])([0.50, 0.60], [0.30, 0.40])([0.50, 0.60], [0.10, 0.30])([0.45, 0.50], [0.10, 0.20])x4([0.20, 0.30], [0.40, 0.50])([0.40, 0.50], [0.30, 0.40])([0.60, 0.80], [0.10, 0.30])([0.50, 0.60], [0.10, 0.20])x5([0.10, 0.20], [0.80, 0.90])([0.20, 0.30], [0.50, 0.60])([0.70, 0.80], [0.20, 0.40])([0.50, 0.60], [0.10, 0.20])Then, using (45), the closeness indices of experts are obtained as follows:(53)CIk=0.9037,0.8084,0.8986,0.9014.On the basis of the closeness indices of experts, the expert weight vector is obtained by (46) as(54)ϖk=0.2573,0.2302,0.2559,0.2566.Furthermore, we use theIVq−ROFDWA operator (47) to integrate the individual decision matrices into to the collective decision matrix and the calculated results are listed in Table 9.Table 9
The collective decision matrixR.
c1c2c3c4x1([0.69, 0.85], [0.20, 0.30])([0.55, 0.63], [0.51, 0.61])([0.77, 0.89], [0.12, 0.20])([0.74, 0.91], [0.16, 0.25])x2([0.81, 0.90], [0.12, 0.19])([0.52, 0.57], [0.40, 0.50])([0.76, 0.88], [0.10, 0.24])([0.74, 0.92], [0.16, 0.25])x3([0.87, 0.94], [0.12, 0.19])([0.54, 0.64], [0.30, 0.40])([0.76, 0.88], [0.18, 0.30])([0.78, 0.89], [0.16, 0.25])x4([0.51, 0.61], [0.66, 0.76])([0.44, 0.57], [0.50, 0.60])([0.72, 0.82], [0.19, 0.30])([0.68, 0.78], [0.24, 0.34])x5([0.10, 0.20], [0.82, 0.92])([0.36, 0.46], [0.68, 0.76])([0.72, 0.82], [0.29, 0.40])([0.69, 0.79], [0.16, 0.25])Using (49), we can calculate the concordance/discordance index ϕjρxξ,xζρ=1,2,…,120;j=1,2,3,4;ξ,ζ=1,2,…,5, and to save the space we only provide the calculated results of the concordance/discordance index ϕjρxξ,xζ under ρ=1. These calculated results are listed in Table 10.Table 10
The results of the concordance/discordance index underP1.
P1ϕj1x1,x2ϕj1x1,x3ϕj1x1,x4ϕj1x1,x5ϕj2x2,x3c1−0.16−0.270.350.65−0.11c2−0.13−0.140.080.2−0.06c30.020.020.120.120.01c4−0.01−0.060.180.170.07P1ϕj2x2,x4ϕj2x2,x5ϕj2x3,x4ϕj2x3,x5ϕj2x4,x5c10.450.660.560.740.30c20.080.280.120.330.21c30.100.100.100.100.03c40.200.180.200.18−0.02Then, the weighted concordance/discordance indicesϕρxξ,xζ can be obtained by using (50). The calculated results are listed in Table 11.Table 11
The results of the weighted concordance/discordance index underP1.
P1ϕj1x1,x2ϕj1x1,x3ϕj1x1,x4ϕj1x1,x5ϕj2x2,x3c1−0.1112−0.19130.24200.4533−0.0800c2−0.0131−0.01390.00820.0197−0.0057c30.00210.00210.01180.01180.0009c4−0.0013−0.00590.01850.01690.0072P1ϕj2x2,x4ϕj2x2,x5ϕj2x3,x4ϕj2x3,x5ϕj2x4,x5c10.31530.46250.39530.51800.2113c20.00760.02820.01250.03310.0206c30.00980.00980.00980.00980.0027c40.01970.01810.01950.0179−0.0019Finally, we can calculate the comprehensive concordance/discordance indexϕρ for permutation Pρ using (51). The calculated results are listed in Table 12.Table 12
The results of the concordance/discordance index.
P1P2P3P4P5P6P7P8P9P102.492.031.620.460.87−0.012.652.181.940.91P11P12P13P14P15P16P17P18P19P201.140.440.91−0.241.070.03−1.28−1.13−0.17−1.04P21P22P23P24P25P26P27P28P29P30−0.01−0.72−1.75−1.592.742.271.860.711.120.24P31P32P33P34P35P36P37P38P39P403.162.692.601.591.691.131.300.151.720.72P41P42P43P44P45P46P47P48P49P50−0.86−0.440.11−0.760.53−0.03−1.32−0.913.072.60P51P52P53P54P55P56P57P58P59P602.361.321.560.863.312.852.751.751.841.28P61P62P63P64P65P66P67P68P69P701.800.762.051.04−0.240.010.56−0.150.810.24P71P72P73P74P75P76P77P78P79P80−0.71−0.460.35−0.810.51−0.53−1.84−1.690.60−0.56P81P82P83P84P85P86P87P88P89P901.020.01−1.56−1.140.92−0.111.170.17−1.12−0.87P91P92P93P94P95P96P97P98P99P100−2.85−2.69−2.60−2.18−2.27−2.03−1.17−2.05−1.02−1.72P101P102P103P104P105P106P107P108P109P110−2.75−2.60−0.92−1.80−0.51−1.07−2.36−1.94−0.60−1.30P111P112P113P114P115P116P117P118P119P120−0.35−0.91−1.86−1.62−3.31−3.16−3.07−2.65−2.74−2.49Using (52), the permutation with the maximal value of the comprehensive concordance/discordance indices is identified as ϕρ=3.31, and the corresponding permutation of alternatives is x3>x2>x1>x4>x5. Thus, x3 is the optimal alternative. This plan is a yellow warning, which means that timely follow-up with the doctor is needed and the patient should seek medical attention immediately. This decision result is basically consistent with medical experts.


### 5.2. Sensitive Analysis of Parameters

In order to reflect the influence of parameters on the final sorting results, we next conduct a sensitivity analysis by modifying the values of parametersq and φ. We first calculate scores of alternatives by changing the values of parameter q from 3 to 7 for verifying the influence of the change of q in the q-rung orthogonal fuzzy environment on the group decision-making method. The calculation results are shown in Figure 2. It can be verified that the results of group decision-making remain unchanged when the value of parameter q is changed from 3 to 7.Figure 2
Sensitivity analysis of parameterq.Then, we modify simultaneously the values ofq and φ; namely, the value of q is changed from 3 to 6, and the value of φ is changed from 3 to 6; the calculated results are shown in Figure 3. It can be seen from Figure 3 that the ranking results of alternatives do not change with the change of the values of parameters q and φ. From the above analysis, the ranking results of alternatives will not be changed although they are different trends with the change of q and φ; 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.Figure 3
Sensitivity analysis of parametersq and φ.


### 5.3. Comparative Analysis

To show the advantage of the developed method, on the one hand, we next conduct a comparison analysis with various aggregating operators-based decision-making methods proposed in [39, 40]. Therefore, we next use the IVPFWA, IVPFOWA, and IVPFWG operators proposed in [39, 40] to obtain the rankings of alternatives. It is worthy to point out that these operators proposed in [39, 40] can only be used to calculate the IVPFNs but failed to deal with the IVq-ROFNs. Thus, in this comparison analysis, we set the value of the parameter as 2 and all IVq-ROFNs data in the above health management of hypertensive patients are transformed IVPFNs data. Then, the calculated results are obtained and shown in Table 13.Table 13
Comparison analysis of the calculation results obtained by different methods.
MethodsRanking scores of alternativesRankings of alternativesx1x2x3x4x5IVPFWA0.440.610.69−0.04−0.44x3 > x2 > x1 > x4 > x5IVPFOWA0.370.450.490.08−0.19x3 > x2 > x1 > x4 > x5IVPFWG0.230.320.37−0.24−0.63x3 > x2 > x1 > x4 > x5IVq-ROFDWA0.971.271.48−0.08−0.70x3 > x2 > x1 > x4 > x5IVq-ROF QUALIFLEX—————x3 > x2 > x1 > x4 > x5It is noticed that the obtained ranking orders of alternatives are the same. Nevertheless, the information loss always happens in the process of information aggregation [41] when these aggregating operators-based decision-making methods [39, 40] are used. Meanwhile our developed method not only is capable of dealing with the situations where the weight information of the experts is unknown but also can reduce the information loss.On the other hand, we conduct an analysis to compare between our method and the IVq-ROF MULTIMOORA method [42]. According to the IVq-ROF MULTIMOORA method, we first determine the ranking of alternatives based on the ratio system by using the following equation:(55)y˜∗ri=q−RIVOFWAri1,ri2,…,rin=1−∏j=1n1−urij−qwiq,1−∏j=1n1−urij+qwiq,∏j=1nvrij−wi,∏j=1nvrij+wi.Then, the scores of alternatives are obtained asx1 = 0.53, x2 = 0.66, x3 = 0.73, x4 = 0.04, and x5 = -0.26, and the ranking of alternatives is x3 > x2 > x1 > x4 > x5.Next, we determine the ranking of alternatives based on the reference point approach by using the following equation:(56)dri,r∗=∑j=1n14urij−q−1+urij+q−1+vrij−q−0+vrij+q−0+πrij−q−0+πrij+q−0.The ideal distances of alternatives from ([1, 1], [0, 0]) are determined asx1 = 0.63, x2 = 0.49, x3 = 0.34, x4 = 0.91, and x5 = 0.93, and the ranking of alternatives is x3 > x2 > x1 > x4 > x5.Finally, we determine the ranking of alternatives based on the full multiplicative form by using the following equation:(57)U˜∗ri=q−RIVOFWGri1,ri2,…,rin=∏j=1nurij−wi,∏j=1nurij+wi,1−∏j=1n1−vrij−qwiq,1−∏j=1n1−vrij+qwiq.The scores of alternatives are got as x1 = 0.39, x2 = 0.46, x3 = 0.53, x4 = −0.29, and x5 = −0.74, and, the ranking of alternatives is x3 > x2 > x1 > x4 > x5. According to the dominance theory, the ranking of alternatives is x3 > x2 > x1 > x4 > x5. It is noticed that the ranking orders of alternatives obtained by our developed method and by the IVq-ROF MULTIMOORA method [42] are the same, which further confirms the effectiveness of our developed method.


## 5.1. The Calculated Results by the Developed Method

We next use the developed method to identify the optimal alternative. Owing to the fact that the weights of experts are unknown in advance in this case, we first need to objectively determine the weights of experts. Using (41), the positive ideal group opinion with the form of interval-valued q-rung orthopair fuzzy matrix CR+ is determined in Table 5.Table 5
The positive ideal group opinion matrixCR+.
c1c2c3c4x1([0.68, 0.85], [0.20, 0.30])([0.55, 0.63], [0.50, 0.60])([0.77, 0.89], [0.12, 0.20])([0.73, 0.91], [0.16, 0.25])x2([0.81, 0.90], [0.12, 0.19])([0.52, 0.57], [0.40, 0.50])([0.76, 0.88], [0.10, 0.24])([0.74, 0.92], [0.16, 0.25])x3([0.88, 0.94], [0.12, 0.19])([0.54, 0.64], [0.30, 0.40])([0.76, 0.88], [0.18, 0.30])([0.78, 0.89], [0.16, 0.25])x4([0.51, 0.61], [0.66, 0.76])([0.44, 0.56], [0.50, 0.60])([0.72, 0.82], [0.19, 0.30])([0.68, 0.78], [0.24, 0.34])x5([0.10, 0.20], [0.82, 0.93])([0.36, 0.45], [0.68, 0.76])([0.72, 0.82], [0.29, 0.40])([0.69, 0.79], [0.16, 0.25])The negative ideal group opinion is obtained by (42) and the calculated results are listed in Table 6.Table 6
The negative ideal group opinion matrixCR−.
c1c2c3c4x1([0.20, 0.30], [0.68, 0.85])([0.50, 0.60], [0.55, 0.63])([0.12, 0.20], [0.77, 0.89])([0.16, 0.25], [0.73, 0.91])x2([0.12, 0.19], [0.81, 0.90])([0.40, 0.50], [0.52, 0.57])([0.10, 0.24], [0.76, 0.88])([0.16, 0.25], [0.74, 0.92])x3([0.12, 0.19], [0.88, 0.94])([0.30, 0.40], [0.54, 0.64])([0.18, 0.30], [0.76, 0.88])([0.16, 0.25], [0.78, 0.89])x4([0.66, 0.76], [0.51, 0.61])([0.50, 0.60], [0.44, 0.56])([0.19, 0.30], [0.72, 0.82])([0.24, 0.34],[0.68, 0.78])x5([0.82, 0.93], [0.10, 0.20])([0.68, 0.76], [0.36, 0.45])([0.29, 0.40], [0.72, 0.82])([0.16, 0.25], [0.69, 0.79])Similarly, using (43) and (44), the negative ideal right group opinion and the negative ideal left group opinion are obtained, and the calculated results are listed in Tables 7 and 8.Table 7
The negative ideal right group opinion matrixIr+.
c1c2c3c4x1([0.70, 0.90], [0.20, 0.30])([0.60, 0.65], [0.60, 0.70])([0.80, 0.90], [0.15, 0.20])([0.80, 0.95], [0.20, 0.30])x2([0.85, 0.90], [0.15, 0.20])([0.55, 0.60], [0.40, 0.50])([0.80, 0.90], [0.10, 0.30])([0.80, 0.95], [0.20, 0.30])x3([0.90, 0.95], [0.15, 0.20])([0.60, 0.70], [0.30, 0.40])([0.80, 0.90], [0.20, 0.30])([0.80, 0.90], [0.20, 0.30])x4([0.60, 0.70], [0.70, 0.80])([0.50, 0.60], [0.60, 0.70])([0.75, 0.85], [0.20, 0.30])([0.70, 0.80], [0.30, 0.40])x5([0.10, 0.20], [0.85, 0.95])([0.40, 0.50], [0.75, 0.80])([0.75, 0.85], [0.30, 0.40])([0.75, 0.85], [0.20, 0.30])Table 8
The negative ideal left group opinion matrixIr−.
c1c2c3c4x1([0.60, 0.70], [0.20, 0.30])([0.50, 0.60], [0.30, 0.40])([0.70, 0.80], [0.10, 0.20])([0.60, 0.65], [0.10, 0.20])x2([0.70, 0.90], [0.10, 0.15])([0.50, 0.55], [0.40, 0.50])([0.60, 0.70], [0.10, 0.15])([0.70, 0.75], [0.10, 0.20])x3([0.80, 0.90], [0.10, 0.15])([0.50, 0.60], [0.30, 0.40])([0.50, 0.60], [0.10, 0.30])([0.45, 0.50], [0.10, 0.20])x4([0.20, 0.30], [0.40, 0.50])([0.40, 0.50], [0.30, 0.40])([0.60, 0.80], [0.10, 0.30])([0.50, 0.60], [0.10, 0.20])x5([0.10, 0.20], [0.80, 0.90])([0.20, 0.30], [0.50, 0.60])([0.70, 0.80], [0.20, 0.40])([0.50, 0.60], [0.10, 0.20])Then, using (45), the closeness indices of experts are obtained as follows:(53)CIk=0.9037,0.8084,0.8986,0.9014.On the basis of the closeness indices of experts, the expert weight vector is obtained by (46) as(54)ϖk=0.2573,0.2302,0.2559,0.2566.Furthermore, we use theIVq−ROFDWA operator (47) to integrate the individual decision matrices into to the collective decision matrix and the calculated results are listed in Table 9.Table 9
The collective decision matrixR.
c1c2c3c4x1([0.69, 0.85], [0.20, 0.30])([0.55, 0.63], [0.51, 0.61])([0.77, 0.89], [0.12, 0.20])([0.74, 0.91], [0.16, 0.25])x2([0.81, 0.90], [0.12, 0.19])([0.52, 0.57], [0.40, 0.50])([0.76, 0.88], [0.10, 0.24])([0.74, 0.92], [0.16, 0.25])x3([0.87, 0.94], [0.12, 0.19])([0.54, 0.64], [0.30, 0.40])([0.76, 0.88], [0.18, 0.30])([0.78, 0.89], [0.16, 0.25])x4([0.51, 0.61], [0.66, 0.76])([0.44, 0.57], [0.50, 0.60])([0.72, 0.82], [0.19, 0.30])([0.68, 0.78], [0.24, 0.34])x5([0.10, 0.20], [0.82, 0.92])([0.36, 0.46], [0.68, 0.76])([0.72, 0.82], [0.29, 0.40])([0.69, 0.79], [0.16, 0.25])Using (49), we can calculate the concordance/discordance index ϕjρxξ,xζρ=1,2,…,120;j=1,2,3,4;ξ,ζ=1,2,…,5, and to save the space we only provide the calculated results of the concordance/discordance index ϕjρxξ,xζ under ρ=1. These calculated results are listed in Table 10.Table 10
The results of the concordance/discordance index underP1.
P1ϕj1x1,x2ϕj1x1,x3ϕj1x1,x4ϕj1x1,x5ϕj2x2,x3c1−0.16−0.270.350.65−0.11c2−0.13−0.140.080.2−0.06c30.020.020.120.120.01c4−0.01−0.060.180.170.07P1ϕj2x2,x4ϕj2x2,x5ϕj2x3,x4ϕj2x3,x5ϕj2x4,x5c10.450.660.560.740.30c20.080.280.120.330.21c30.100.100.100.100.03c40.200.180.200.18−0.02Then, the weighted concordance/discordance indicesϕρxξ,xζ can be obtained by using (50). The calculated results are listed in Table 11.Table 11
The results of the weighted concordance/discordance index underP1.
P1ϕj1x1,x2ϕj1x1,x3ϕj1x1,x4ϕj1x1,x5ϕj2x2,x3c1−0.1112−0.19130.24200.4533−0.0800c2−0.0131−0.01390.00820.0197−0.0057c30.00210.00210.01180.01180.0009c4−0.0013−0.00590.01850.01690.0072P1ϕj2x2,x4ϕj2x2,x5ϕj2x3,x4ϕj2x3,x5ϕj2x4,x5c10.31530.46250.39530.51800.2113c20.00760.02820.01250.03310.0206c30.00980.00980.00980.00980.0027c40.01970.01810.01950.0179−0.0019Finally, we can calculate the comprehensive concordance/discordance indexϕρ for permutation Pρ using (51). The calculated results are listed in Table 12.Table 12
The results of the concordance/discordance index.
P1P2P3P4P5P6P7P8P9P102.492.031.620.460.87−0.012.652.181.940.91P11P12P13P14P15P16P17P18P19P201.140.440.91−0.241.070.03−1.28−1.13−0.17−1.04P21P22P23P24P25P26P27P28P29P30−0.01−0.72−1.75−1.592.742.271.860.711.120.24P31P32P33P34P35P36P37P38P39P403.162.692.601.591.691.131.300.151.720.72P41P42P43P44P45P46P47P48P49P50−0.86−0.440.11−0.760.53−0.03−1.32−0.913.072.60P51P52P53P54P55P56P57P58P59P602.361.321.560.863.312.852.751.751.841.28P61P62P63P64P65P66P67P68P69P701.800.762.051.04−0.240.010.56−0.150.810.24P71P72P73P74P75P76P77P78P79P80−0.71−0.460.35−0.810.51−0.53−1.84−1.690.60−0.56P81P82P83P84P85P86P87P88P89P901.020.01−1.56−1.140.92−0.111.170.17−1.12−0.87P91P92P93P94P95P96P97P98P99P100−2.85−2.69−2.60−2.18−2.27−2.03−1.17−2.05−1.02−1.72P101P102P103P104P105P106P107P108P109P110−2.75−2.60−0.92−1.80−0.51−1.07−2.36−1.94−0.60−1.30P111P112P113P114P115P116P117P118P119P120−0.35−0.91−1.86−1.62−3.31−3.16−3.07−2.65−2.74−2.49Using (52), the permutation with the maximal value of the comprehensive concordance/discordance indices is identified as ϕρ=3.31, and the corresponding permutation of alternatives is x3>x2>x1>x4>x5. Thus, x3 is the optimal alternative. This plan is a yellow warning, which means that timely follow-up with the doctor is needed and the patient should seek medical attention immediately. This decision result is basically consistent with medical experts.


## 5.2. Sensitive Analysis of Parameters

In order to reflect the influence of parameters on the final sorting results, we next conduct a sensitivity analysis by modifying the values of parametersq and φ. We first calculate scores of alternatives by changing the values of parameter q from 3 to 7 for verifying the influence of the change of q in the q-rung orthogonal fuzzy environment on the group decision-making method. The calculation results are shown in Figure 2. It can be verified that the results of group decision-making remain unchanged when the value of parameter q is changed from 3 to 7.Figure 2
Sensitivity analysis of parameterq.Then, we modify simultaneously the values ofq and φ; namely, the value of q is changed from 3 to 6, and the value of φ is changed from 3 to 6; the calculated results are shown in Figure 3. It can be seen from Figure 3 that the ranking results of alternatives do not change with the change of the values of parameters q and φ. From the above analysis, the ranking results of alternatives will not be changed although they are different trends with the change of q and φ; 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.Figure 3
Sensitivity analysis of parametersq and φ.


## 5.3. Comparative Analysis

To show the advantage of the developed method, on the one hand, we next conduct a comparison analysis with various aggregating operators-based decision-making methods proposed in [39, 40]. Therefore, we next use the IVPFWA, IVPFOWA, and IVPFWG operators proposed in [39, 40] to obtain the rankings of alternatives. It is worthy to point out that these operators proposed in [39, 40] can only be used to calculate the IVPFNs but failed to deal with the IVq-ROFNs. Thus, in this comparison analysis, we set the value of the parameter as 2 and all IVq-ROFNs data in the above health management of hypertensive patients are transformed IVPFNs data. Then, the calculated results are obtained and shown in Table 13.Table 13
Comparison analysis of the calculation results obtained by different methods.
MethodsRanking scores of alternativesRankings of alternativesx1x2x3x4x5IVPFWA0.440.610.69−0.04−0.44x3 > x2 > x1 > x4 > x5IVPFOWA0.370.450.490.08−0.19x3 > x2 > x1 > x4 > x5IVPFWG0.230.320.37−0.24−0.63x3 > x2 > x1 > x4 > x5IVq-ROFDWA0.971.271.48−0.08−0.70x3 > x2 > x1 > x4 > x5IVq-ROF QUALIFLEX—————x3 > x2 > x1 > x4 > x5It is noticed that the obtained ranking orders of alternatives are the same. Nevertheless, the information loss always happens in the process of information aggregation [41] when these aggregating operators-based decision-making methods [39, 40] are used. Meanwhile our developed method not only is capable of dealing with the situations where the weight information of the experts is unknown but also can reduce the information loss.On the other hand, we conduct an analysis to compare between our method and the IVq-ROF MULTIMOORA method [42]. According to the IVq-ROF MULTIMOORA method, we first determine the ranking of alternatives based on the ratio system by using the following equation:(55)y˜∗ri=q−RIVOFWAri1,ri2,…,rin=1−∏j=1n1−urij−qwiq,1−∏j=1n1−urij+qwiq,∏j=1nvrij−wi,∏j=1nvrij+wi.Then, the scores of alternatives are obtained asx1 = 0.53, x2 = 0.66, x3 = 0.73, x4 = 0.04, and x5 = -0.26, and the ranking of alternatives is x3 > x2 > x1 > x4 > x5.Next, we determine the ranking of alternatives based on the reference point approach by using the following equation:(56)dri,r∗=∑j=1n14urij−q−1+urij+q−1+vrij−q−0+vrij+q−0+πrij−q−0+πrij+q−0.The ideal distances of alternatives from ([1, 1], [0, 0]) are determined asx1 = 0.63, x2 = 0.49, x3 = 0.34, x4 = 0.91, and x5 = 0.93, and the ranking of alternatives is x3 > x2 > x1 > x4 > x5.Finally, we determine the ranking of alternatives based on the full multiplicative form by using the following equation:(57)U˜∗ri=q−RIVOFWGri1,ri2,…,rin=∏j=1nurij−wi,∏j=1nurij+wi,1−∏j=1n1−vrij−qwiq,1−∏j=1n1−vrij+qwiq.The scores of alternatives are got as x1 = 0.39, x2 = 0.46, x3 = 0.53, x4 = −0.29, and x5 = −0.74, and, the ranking of alternatives is x3 > x2 > x1 > x4 > x5. According to the dominance theory, the ranking of alternatives is x3 > x2 > x1 > x4 > x5. It is noticed that the ranking orders of alternatives obtained by our developed method and by the IVq-ROF MULTIMOORA method [42] are the same, which further confirms the effectiveness of our developed method.


## 6. Conclusion

In this paper, we have developed an interval-valued q-rung orthopair fuzzy QUALIFLEX decision analysis method with Dombi operators. We have first defined interval-valued q-rung orthopair fuzzy Dombi operational laws according to t-conorms and t-norms algorithms of the Dombi operations. Second, based on the developed interval-valued q-rung orthopair fuzzy Dombi operational laws, we have proposed the IVq-ROFDWA operator and studied its proprieties. Third, with the help of IVq-ROFDWA operator, we have determined the ideal group opinions and constructed a new weight-determining method for identifying the weights of experts. The biggest advantage of the weight-determining method is that it can make sure that each expert’s opinion is consistent with the group opinion to the greatest extent. Furthermore, we have proposed a new interval-valued q-rung orthopair fuzzy distance measure. With the help of the developed distance measure, we have defined the concept of the interval-valued q-rung orthopair fuzzy concordance/discordance index. By calculating the comprehensive concordance/discordance index under each permutation, the final ranking orders of alternatives have been identified. The algorithm of interval-valued q-rung orthopair fuzzy QUALIFLEX decision analysis method has been introduced for addressing the complex MAGDM problems with IVq-ROFSs. Apparently, our developed method does not need to provide expert weights and attribute weights in advance, so it can improve the decision-making efficiency and save decision cost. Nevertheless, our developed method is only applicable to the case of a small number of experts and a small number of schemes, since it needs to traverse the ranking of all schemes in the decision-making process. With the increasing complexity of the system and the rapid growth of uncertainty, it is necessary to combine big data and artificial intelligence to deal with complex problems in the decision-making process. For this reason, we will study the group of complex systems of big data decision problems on the basis of the developed method. On the other hand, in real-life decision processes, decision-makers often give preference information between schemes and provide a judgment matrix instead of a decision matrix according to their own experience and knowledge. Therefore, group decision-making methods based on interval-valued q-rung orthopair fuzzy preference relations are our research focus.



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*Source: 4898098-2022-02-22.xml*