Novel fusion strategies for continuous interval-valued q-rung orthopair fuzzy information: a case study in quality assessment of SmartWatch appearance design

Abstract

The notion of Yager’s q-rung orthopair fuzzy set (QROFS) have gained considerable and continuously increasing attention as a useful tool for imprecision and uncertainty representation due to its capability to discard the constraints on the membership and nonmembership functions as generally required by its intuitionistic fuzzy counterpart. Among the generalizations and variants established in the past few years, the interval-valued QROFSs (IVQROFSs) have been diffusely considered to be a powerful generalization of the interval-valued fuzzy sets. The continuous ordered weighted averaging (COWA) operator has been extended successfully to some special cases of IVQROFSs, including interval-valued intuitionistic and Pythagorean fuzzy sets. Thus, to expand on previous studies, several continuous IVQROF (C-IVQROF) aggregation operators are proposed in this study. First, the dual C-GOWA operator is defined on the basis of the continuous generalized ordered weighted averaging (C-GOWA) operator and Yager class of fuzzy negation. Subsequently, the C-IVQROFOWA operator with two independent parameters is constructed, and the weighted C-IVQROFOWA operator is then proposed for aggregating a collection of IVQROFSs. The C-IVQROFOWA operator and its weighted version can model commendably the attitudinal characteristics of the decision-maker. Second, a parameter optimization model and its algorithm-solving strategy driven by consensus measures are built to develop a group decision-making method. Finally, a case study to evaluate the SmartWatch design alternatives is provided to demonstrate the proposed approach, and the results of a comparative analysis verify the rationality and efficiency of the proposed operators.

1 Introduction

Q-rung orthopair fuzzy sets [1,2,3] (QROFSs) are fuzzy sets [4] in which the membership grades of an element are the pairs of values in the unit interval, $( \mu ( x ),v( x ) )$, one of which indicates support for membership in the fuzzy set and the other represents support against membership. QROFS presents the following prominent features:

• Large space for membership the constraining relationship between the support for and against memberships is ${{( \mu ( x ) )}^{q}}+{{( v( x ) )}^{q}}\le 1$ with $q \ge 1$;

• Strong degeneracy of sets the Atanassov’s intuitionistic fuzzy sets [5] and what Yager called Pythagorean fuzzy sets [6] are special cases of QROFSs with $q=1$ and $q=2$, respectively;

• Flexibility in the application it allows the decision-maker the freedom to provide information in support or against the membership of an element in a set.

QROFSs have received considerable critical attention from researchers since its introduction. Operational laws and aggregation operators are proposed to aggregate q-rung orthopair fuzzy (QROF) numbers [7,8,9,10,11,12,13,14,15,16], including QROF Heronian mean, Hamy mean, Bonferroni mean, Maclaurin symmetric mean, and Power mean operators. Information measures, such as distance measure, similarity measure, entropy, and inclusion measure, are used to analyze the relationship between different QROFSs [17,18,19,20,21,22]; the intrinsic correlation of these measures have also been explored [23]. QROF integrals and differentials are constructed to aggregate QROF continuous information [24,25,26,27]. QROF preference relation has been proposed to deal with the QROF decision-making problems of the comparison matrix [28, 29]. QROF multiple attribute decision-making (MADM) method are developed to solve MADM problems with hierarchical interacting criteria under QROF environment [30]. Existing research results are used to consolidate the theoretical construction of QROFSs and promote the application of QROFSs in various decision-making problems.

With the continuous progress of society and the rapid development of the economy, decision-making problems have become increasingly complicated with their noticeable uncertainty and fuzzy human thinking [31,32,33]. Therefore, in the decision-making process, unlike the formal representation of precise values, the representation of input parameters in the form of interval values is more suitable in the current decision-making environment. For this reason, a generalization of QROFSs has been introduced by some researchers [34, 35] in terms of interval-valued fuzzy sets called interval-valued QROFSs (IVQROFSs). To date, existing studies include mainly the operations of union, intersection, addition, and multiplication (with real scalars) of IVQROFSs. On this basis, the interval-valued q-rung orthopair fuzzy weighted averaging (IVQROFWA) [35] and Maclaurin symmetric mean operators are developed [36] to aggregate different IVQROFSs. Moreover, a new notion of the IVQROF graph (IVQROFG) [37] is introduced to investigate the common problem of determining the shortest path in a traffic network. The interval-valued QROF (IVQROF) operators mentioned above are constructed by directly extending the QROF operators to the interval-valued fuzzy environment. The characteristics of its aggregation process are that the endpoints of interval-valued membership (or non-membership) are directly aggregated, and the information aggregation results are dominated by the interval endpoint values, and the contribution of other points in the interval is rarely considered.

The continuous ordered weighted averaging (COWA) operator was introduced by Yager in [38] and is a continuation of the OWA operator [39, 40] when the given argument is a continuous-valued interval rather than an exact argument. Since its introduction, the COWA operator has been used in many fields, including preference relations [41, 42], information measures [43,44,45,46], and aggregation operators [47,48,49,50]. The COWA operator is applied to propose the weighted continuous interval-valued intuitionistic fuzzy ordered weighted averaging (WC-IVIFOWA) operator [47,48,49]. Another interesting extension of the OWA operator is the continuous ordered weighted quadratic averaging (C-OWQA) operator [51], which is applied to propose the weighted continuous interval-valued Pythagorean fuzzy ordered weighted averaging (WC-IVPFOWA) operator [52]. As mentioned earlier, intuitionistic and Pythagorean fuzzy sets are two special cases of IVQROFSs and the COWA and C-OWQA operators are the core tools for constructing WC-IVIFOWA and WC-IVPFOWA operators, respectively. Correspondingly, the COWA and C-OWQA operators are two special cases of a generalization of ordered weighted generalized averaging (OWGA) operator [53] called the continuous OWGA (COWGA) operator [51]. The aggregation of group information and multi-attribute information is a necessary process to obtain comprehensive information of alternatives and an important stage to deal with multi-attribute group decision-making (MAGDM) problems. For the MAGDM approaches proposed in references [47,48,49, 52], as the core tools, the WC-IVPFOWA and WC-IVIFOWA operators can effectively aggregate weighted group evaluation information, reduce the computational complexity of information aggregation and improve the accuracy of decision-making.However, the COWA operator has not been extended to the IVQROF environment. Furthermore, for any given interval-valued intuitionistic fuzzy numbers (IVIFNs), decision makers can not obtain any IFNs contained in the IVIFNs by using the C-IVIFOWA operator. The C-IVPFOWA operator in interval-valued Pythagorean fuzzy environment exhibits the same shortcoming alike. From a mathematical point of view, the reason for the aforementioned shortcoming is that the values of attitudinal characteristics of interval-valued membership and interval-valued non-membership are the same, and there is no independent setting. The deficiency of the operators makes it easier for decision makers to ignore most of the information in the given interval-valued fuzzy numbers, and then it is difficult to accurately model the attitude preference of decision makers.

These facts motivate us to design a robust decision-making framework with the application of continuous ordered weighted averaging operator in solving the group decision-making (GDM) scenarios under IVQROF environment. Concretely, this study proposes continuous IVQROF aggregation operators by combining the COWGA operator with IVQROFSs. The dual COWGA (DCOWGA) is constructed using Yager’s class negation [54, 55] and the COWGA operator. With COWGA and DCOWGA operators as the aggregation tools, the continuous IVQROF ordered weighted averaging (C-IVQROFOWA) and weighted C-IVQRFOWA (WC-IVQROFOWA) operators are proposed to aggregate IVQROF numbers. Based on these operators, a GDM method with consensus-improving is developed with the help of a parameter optimization model. Several novel consensus measures are defined in the use of the proposed WC-IVQROFOWA operator. This method is used to solve problems in the evaluation of SmartWatch appearance design alternatives.

The remainder of this paper is organized as follows. In Sect. 2, we briefly review basic concepts, such as the IVQROFSs, GOWA operator, and DGOWA operator. In Sect. 3, the WC-IVQROFOWA operator is proposed. A decision-making approach based on the WC-IVQROFOWA operator and parameter optimization model is presented in Sect. 4, and an illustrative example is examined in Sect. 5. Comparison analysis is performed between the aggregation operators proposed in this study and the existing aggregation operators in Sect. 6. The main conclusions of the study are drawn in Sect. 7.

2 Preliminaries

In this section, we provide the necessary background for our subsequent developments.

2.1 Q-rung orthopair fuzzy set interval-valued q-rung orthopair fuzzy set

Yager [1] generalizes Atanassov’s intuitionistic fuzzy set theory [5] with the concept of QROFS as defined below.

Definition 1

[1] Let X be a universe of discourse. A QROFS P in X is expressed as

$$\begin{aligned} P=\left\{ \left\langle x,{{\mu }_{P}}\left( x \right) ,{{v}_{P}}\left( x \right) \right\rangle \left| x\in X \right. \right\} , \end{aligned}$$

(1)

where the function ${{\mu }_{P}}:X\rightarrow [ 0,1 ]$ defines the degree of membership and ${{v}_{P}}:X\rightarrow [ 0,1 ]$ defines the degree of non-membership of the element $x\in X$ to P, respectively. For every $x\in X$, it holds that ${{( {{\mu }_{P}}( x ) )}^{q}}+{{( {{v}_{P}}( x ) )}^{q}}\le 1$. The degree of indeterminacy is ${{\pi }_{P}}( x )={{( 1-{{( {{\mu }_{P}}( x ) )}^{q}}-{{( {{v}_{P}}( x ) )}^{q}} )}^{{1}/{q}\;}}$.

For simplicity, we define $( {{\mu }_{P}}( x ),{{v}_{P}}( x ) )$ as a q-rung orthopair fuzzy number (QROFN) as denoted by $P=( {{\mu }_{P}},{{v}_{P}} )$, where ${{\mu }_{P}},{{v}_{P}}\in [ 0,1 ]$,${{\pi }_{P}}={{( 1-{{\mu }^{q}}-{{v}^{q}} )}^{{1}/{q}\;}}$ and ${{\mu }^{q}}\text {+}{{v}^{q}}\le 1$. It can be conveniently observed that the intuitionistic fuzzy sets (IFSs) [5] are QROFS with $q=1$ and Yager’s Pythagorean fuzzy sets (PFSs) [6] are QROFS with $q=2$.

Theorem 1

[1] If $\alpha$ is a QROFN on X and if ${{q}_{1}}>q$, then $\alpha$ is also a QROFN on X.

Corollary 1

[1] (1) Any IFS is a QROFS for all $q\ge 2$;

(2) An IFS is a PFS;

(3) Any PFS is a QROFS for $q\ge 2$.

Definition 2

[17] Let $\alpha =( \mu ,v )$ be a QROFN solid at q, then $s( \alpha )\text {=}{{\mu }^{q}}-{{v}^{q}}$ is the score function of $\alpha$ and $h( \alpha )\text {=}{{\mu }^{q}}+{{v}^{q}}$ is the accuracy function of $\alpha$.

Based on the defined score function and the accuracy function, a ranking method was proposed in the following definition.

Definition 3

[17] Let ${{\alpha }_{i}}=( {{\mu }_{i}},{{v}_{i}} )$ be two QROFNs solid at q.

• 1. If $s( {{\alpha }_{1}} )<s( {{\alpha }_{2}} )$, then ${{\alpha }_{1}}\prec {{\alpha }_{2}}$;

• 2. If $s( {{\alpha }_{1}} )=s( {{\alpha }_{2}} )$, (i) $h( {{\alpha }_{1}} )<h( {{\alpha }_{2}} )$, ${{\alpha }_{1}}\prec {{\alpha }_{2}}$, and (ii) if $s( {{\alpha }_{1}} )=s( {{\alpha }_{2}} )$ and $h( {{\alpha }_{1}} )=h( {{\alpha }_{2}} )$, then ${{\alpha }_{1}}={{\alpha }_{2}}$.

Definition 4

[34, 35] Let X be a universe of discourse. An IVQROFS $\tilde{P}$ in X is expressed as

$$\begin{aligned} \tilde{P}=\left\{ \left\langle x,{{{\tilde{\mu }}}_{P}}\left( x \right) ,{{{\tilde{v}}}_{P}}\left( x \right) \right\rangle \left| x\in X \right. \right\} , \end{aligned}$$

(2)

where the function ${{\tilde{\mu }}_{P}}( x )=[ \mu _{P}^{-}( x ),\mu _{P}^{+}( x ) ]$ defines the degree of membership and ${{\tilde{v}}_{P}}( x )=\left[ v_{P}^{-}( x ),v_{P}^{+}( x ) \right]$ defines the degree of non-membership of the element $x\in X$ to P, respectively; and for every $x\in X$, it holds that ${{( \mu _{P}^{+}( x ) )}^{p}}+{{( v_{P}^{+}( x ) )}^{p}}\le 1$. The degree of indeterminacy is

$$\begin{aligned}&{{\tilde{\pi }}_{P}}( x )=[ \pi _{P}^{-}( x ),\pi _{P}^{+}( x ) ]\\&\quad =\left[ {{\left( 1-{{\left( \mu _{P}^{+}\left( x \right) \right) }^{p}}-{{\left( v_{P}^{+}\left( x \right) \right) }^{p}} \right) }^{{1}/{p}\;}},\right. \\&\quad \quad \left. {{\left( 1-{{\left( \mu _{P}^{-}\left( x \right) \right) }^{p}}-{{\left( v_{P}^{-}\left( x \right) \right) }^{p}} \right) }^{{1}/{p}\;}} \right] . \end{aligned}$$

For simplicity, we call $( {{{\tilde{\mu }}}_{P}}( x ),{{{\tilde{v}}}_{P}}( x ) )$ an interval-valued q-rung orthopair fuzzy number (IVQROFN) denoted by $P=( {{{\tilde{\mu }}}_{P}},{{{\tilde{v}}}_{P}} )=( [ \mu _{P}^{-},\mu _{P}^{+} ], [ v_{P}^{-},v_{P}^{+} ] )$, where ${{( \mu _{P}^{+} )}^{p}}+{{( v_{P}^{+} )}^{p}}\le 1$. Let $P=( [ {{\mu }^{-}},{{\mu }^{+}} ], [ {{v}^{-}},{{v}^{+}} ] )$ be an IVQROFN solid at q, then

$$\begin{aligned} S( \alpha )\text {=}\frac{1}{2}+\frac{{{\left( {{\mu }^{-}} \right) }^{q}}+{{\left( {{\mu }^{+}} \right) }^{q}}-{{\left( {{v}^{-}} \right) }^{q}}-{{\left( {{v}^{+}} \right) }^{q}}}{4} \end{aligned}$$

is the score function [35] of P and

$$\begin{aligned} H\left( \alpha \right) \text {=}\frac{{{\left( {{\mu }^{-}} \right) }^{q}}+{{\left( {{\mu }^{+}} \right) }^{q}}\text {+}{{\left( {{v}^{-}} \right) }^{q}}\text {+}{{\left( {{v}^{+}} \right) }^{q}}}{2} \end{aligned}$$

is the accuracy function [35].

A ranking method of IVQROFNs is proposed based on the above score and the accuracy functions [35]. Let ${{\tilde{\alpha }}_{i}}=( [ \mu _{i}^{-},\mu _{i}^{+} ],[ v_{i}^{-},v_{i}^{+}] )( i=1,2 )$ be two IVQROFN solids at q.

• 1. If $S( {{{\tilde{\alpha }}}_{1}} )<S( {{{\tilde{\alpha }}}_{2}} )$, then ${{\tilde{\alpha }}_{1}}\prec {{\tilde{\alpha }}_{2}}$;

• 2. If $S( {{{\tilde{\alpha }}}_{1}} )=S( {{{\tilde{\alpha }}}_{2}} )$, (i) $H( {{{\tilde{\alpha }}}_{1}} )<H( {{{\tilde{\alpha }}}_{2}} )$, ${{\tilde{\alpha }}_{1}}\prec {{\tilde{\alpha }}_{2}}$, and (ii) if $S( {{{\tilde{\alpha }}}_{1}} )=S( {{{\tilde{\alpha }}}_{2}} )$ and $H( {{{\tilde{\alpha }}}_{1}} )=H( {{{\tilde{\alpha }}}_{2}})$, then ${{\tilde{\alpha }}_{1}}\sim {{\tilde{\alpha }}_{2}}$.

2.2 GOWA and DGOWA operators

Definition 5

[56] Fuzzy negation is a mapping denoted by ${\mathcal {N}}:[ 0,1 ]\rightarrow [ 0,1 ]$, which satisfies the following properties:

• 1. Boundary conditions: ${\mathcal {N}}( 0 )=1$ and $N( 1 )=0$;

• 2. Monotonicity: for all $a,b\in [ 0,1 ]$, if $a\le b$, then ${\mathcal {N}}( a )\ge {\mathcal {N}}( b )$;

• 3. Continuity;

• 4. Involution: ${\mathcal {N}}( {\mathcal {N}}( a ) )=a$ for all $a\in [ 0,1 ]$.

The Yager class of fuzzy negation [54, 55] is defined by ${{\mathcal {N}}_{q}}( a )={{( 1-{{a}^{q}} )}^{{1}/{q}\;}}$, where $q\in ( 0,\infty )$. When $q=1$, this function becomes the classical fuzzy negation ${{\mathcal {N}}_{1}}( a )=1-a$, whereas $q=2$ will make this function become the Pythagorean negation [6] ${{\mathcal {N}}_{2}}( a )=\sqrt{1-{{a}^{2}}}$.

Definition 6

[56] An aggregation function is a function of $n>1$ arguments that maps the (n-dimensional) unit cube onto the unit interval $\mathsf{{F}}:{{[ 0,1 ]}^{n}}\rightarrow [ 0,1 ]$ with the following properties:

• 1. $\mathsf{{F}}( 0,0,\ldots ,0 )=0$ and $\mathsf{{F}}( 1,1,\ldots ,1 )=1$;

• 2. ${{a}_{j}}\ge {{b}_{j}}$implies $\mathsf{{F}}( {{a}_{1}},{{a}_{2}},\ldots ,{{a}_{n}} )\ge \mathsf{{F}}( {{b}_{1}},{{b}_{2}},\ldots ,{{b}_{n}} )$ for all j.

The dual of aggregation function with respect to fuzzy negation is given as follows:

Definition 7

[56] Let $\mathsf{{F}}:{{[ 0,1 ]}^{n}}\rightarrow [ 0,1 ]$ be an aggregation function. Then, the aggregation function ${\mathsf{{F}}_{d}}$ is given by

$$\begin{aligned} {\mathsf{{F}}_{d}}\left( {{x}_{1}},{{x}_{2}},\ldots ,{{x}_{n}} \right) ={\mathcal {N}}\left( \mathsf{{F}}\left( {\mathcal {N}}\left( {{x}_{1}} \right) , {\mathcal {N}}\left( {{x}_{2}} \right) ,\ldots , {\mathcal {N}}\left( {{x}_{n}} \right) \right) \right) , \end{aligned}$$

(3)

which is the dual of $\mathsf{{F}}$ with respect to ${\mathcal {N}}$, where ${\mathcal {N}}$ is a negation function.

Definition 8

[53] A mapping $\mathsf{{GOWA}}:{{R}^{n}}\rightarrow R$ is called a generalized ordered weighted averaging (GOWA) operator of dimension n if

$$\begin{aligned} \mathsf{{GOWA}}\left( {{x}_{1}},{{x}_{2}},\ldots ,{{x}_{n}} \right) ={{\left( \sum \limits _{i=1}^{n}{{{w}_{j}}x_{\sigma \left( j \right) }^{p}} \right) }^{{1}/{p}\;}}, \end{aligned}$$

(4)

where $\sigma$ is an index function to ensure that ${{x}_{\sigma ( j )}}$ is the $j\text {-th}$ largest of the ${{x}_{i}}$ and the W with components ${{w}_{j}}$ for $j=1$, such that ${{w}_{j}}\in [ 0,1 ]$ and $\sum \nolimits _{j=1}^{n}{{{w}_{j}}=1}$. p is a parameter with $p \in [ -\infty ,+\infty ]$ and $p \ne 0$.

We can obtain the OWA weights by using

$$\begin{aligned} {{w}_{j}}=Q( \frac{j}{n} )-Q( \frac{j-1}{n} ),\quad j=1,2,\ldots ,n, \end{aligned}$$

(5)

where basic unit-interval monotonic (BUM) function [40] $Q:[ 0,1 ]\rightarrow [ 0,1 ]$ is monotonic with the following properties: (1) $Q( 0 )=0$; (2) $Q( 1 )=1$; and (3) $Q( x )\ge Q( y )$ if $x>y$. Hence, these weights satisfy the conditions ${{w}_{j}}\in [ 0,1 ]( j=1,2,\ldots ,n )$ and $\sum \nolimits _{j=1}^{n}{{{w}_{j}}=1}$. When $p =1$, the GOWA operator becomes the OWA operator as follows [39]:

$$\begin{aligned} \mathsf{{OWA}}\left( {{x}_{1}},{{x}_{2}},\ldots ,{{x}_{n}} \right) =\sum \limits _{i=1}^{n}{{{w}_{j}}{{x}_{\sigma \left( j \right) }}}. \end{aligned}$$

(6)

Definition 9

[1] Let $\mathsf{{GOWA}}:{{R}^{n}}\rightarrow R$ be a generalized ordered weighted averaging aggregation operator of dimension n and its dual generalized ordered weighted averaging aggregation (DGOWA) operator with respect to a q-rung negation ${{\mathcal {N}}_{q}}$ is defined as follows:

$$\begin{aligned}&\mathsf{{DGOWA}}\left( {{x}_{1}},{{x}_{2}},\ldots ,{{x}_{n}} \right) \nonumber \\&\quad ={{\mathcal {N}}_{q}}\left( \mathsf{{GOWA}}\left( {{\mathcal {N}}_{q}}\left( {{x}_{1}} \right) ,\ldots ,{{\mathcal {N}}_{q}}\left( {{x}_{n}} \right) \right) \right) . \end{aligned}$$

(7)

According to the above definitions, we can obtain

$$\begin{aligned}&\mathsf{{DGOWA}}\left( {{x}_{1}},{{x}_{2}},\ldots ,{{x}_{n}} \right) \nonumber \\&\quad ={{\mathcal {N}}_{q}}\left( {{\left( \sum \limits _{i=1}^{n}{{{w}_{j}}{{\left( {{\mathcal {N}}_{q}}\left( {{x}_{\delta \left( j \right) }} \right) \right) }^{p}}} \right) }^{{1}/{p}\;}} \right) , \end{aligned}$$

(8)

where $\delta$ is an index function to ensure that ${{\mathcal {N}}_{q}}( {{x}_{\delta ( j )}} )$ is the $j\text {-th}$ largest of ${{\mathcal {N}}_{q}}( {{x}_{i}} )$.

3 WC-IVQROFWA operator

This section defines the weighted continuous interval-valued q-rung orthopais fuzzy ordered weighted averaging (C-IVQROFOWA) and studies its properties. Based on this operator, a new score and accuracy functions are introduced.

3.1 Dual continuous generalized ordered weighted averaging (DC-GOWA) operator

Definition 10

[51] The continuous generalized ordered weighted averaging (C-GOWA) operator is defined as follows:

$$\begin{aligned} {{\phi }_{Q}}\left( \left[ a,b \right] \right) ={{\left( \int _{0}^{1}{\frac{dQ\left( y \right) }{dy}\left( {{b}^{p}}-y\left( {{b}^{p}}-{{a}^{p}} \right) \right) dy} \right) }^{{1}/{p}\;}}. \end{aligned}$$

(9)

When $p=1$ the C-GOWA operator becomes the C-OWA operator, it can be defined as [38]

$$\begin{aligned}&\mathsf{{C-OWA}}\left( {{x}_{1}},{{x}_{2}},\ldots ,{{x}_{n}} \right) \nonumber \\&\quad =\int _{0}^{1}{\frac{dQ\left( y \right) }{dy} \left( b-y\left( b-a \right) \right) dy}. \end{aligned}$$

(10)

In the following, we present the establishment of the C-GOWA operator.

Theorem 2

[51] (Monotonicity) Let ${{\phi }_{Q}}$ be the C-GOWA operator. If ${{a}_{1}}\ge {{a}_{2}},{{b}_{1}}\ge {{b}_{2}}$ for all Q, then ${{\phi }_{Q}}([ {{a}_{1}},{{b}_{1}} ] )\ge {{\phi }_{Q}}( [ {{a}_{2}},{{b}_{2}} ] )$.

Theorem 3

[51] (Boundness) Let ${{\phi }_{Q}}$ be the C-GOWA operator. For all Q, we have $a\le {{\phi }_{Q}}([ a,b ] )\le b$.

Theorem 4

[51] If ${{Q}_{1}}\ge {{Q}_{2}}$, then ${{\phi }_{{{Q}_{1}}}}( [ {{a}_{1}},{{b}_{1}} ])\ge {{\phi }_{{{Q}_{2}}}}( [ {{a}_{2}},{{b}_{2}} ] )$.

Definition 11

The dual continuous generalized ordered weighted averaging (DC-GOWA) operator is defined as follows:

$$\begin{aligned}&\phi _{Q}^{d}\left( \left[ a,b \right] \right) ={{\mathcal {N}}_{q}}\left( \left( \int _{0}^{1}\frac{dQ\left( y \right) }{dy}\left( {{\left( {{\mathcal {N}}_{q}}\left( a \right) \right) }^{p}}\right. \right. \right. \nonumber \\&\quad \left. \left. \left. -y\left( {{\left( {{\mathcal {N}}_{q}}\left( a \right) \right) }^{p}}-{{\left( {{\mathcal {N}}_{q}}\left( b \right) \right) }^{p}} \right) \right) dy \right) ^{{1}/{p}\;} \right) , \end{aligned}$$

(15)

where ${{\mathcal {N}}_{q}}$ is the q-rung negation function.

The derivative of the DC-GOWA operator is analyzed based on the establishment of the C-GOWA operator.

Theorem 5

(Monotonicity) If ${{a}_{1}}\ge {{a}_{2}},{{b}_{1}}\ge {{b}_{2}}$ for all Q, then $\phi _{Q}^{d}( [ {{a}_{1}},{{b}_{1}} ] )\ge \phi _{Q}^{d}( [ {{a}_{2}},{{b}_{2}} ] )$.

Proof

According to Definitions 10 and 11, we have

$$\begin{aligned} \phi _{Q}^{d}\left( \left[ a,b \right] \right) ={{\mathcal {N}}_{q}}\left( {{\phi }_{Q}}\left[ {{\mathcal {N}}_{q}}\left( b \right) ,{{\mathcal {N}}_{q}}\left( a \right) \right] \right) . \end{aligned}$$

From ***Theorem 2, if ${{a}_{1}}\ge {{a}_{2}},{{b}_{1}}\ge {{b}_{2}}$ for all , then

$$\begin{aligned} {{\phi }_{Q}}\left( \left[ {{a}_{1}},{{b}_{1}} \right] \right) \ge {{\phi }_{Q}}\left( \left[ {{a}_{2}},{{b}_{2}} \right] \right) . \end{aligned}$$

Therefore, we have

$$\begin{aligned} {{\phi }_{Q}}\left( \left[ {{\mathcal {N}}_{q}}\left( {{b}_{1}} \right) ,{{\mathcal {N}}_{q}}\left( {{a}_{1}} \right) \right] \right) \le {{\phi }_{Q}}\left( \left[ {{\mathcal {N}}_{q}}\left( {{b}_{2}} \right) ,{{\mathcal {N}}_{q}}\left( {{a}_{2}} \right) \right] \right) \end{aligned}$$

and

$$\begin{aligned}&{{\mathcal {N}}_{q}}\left( {{\phi }_{Q}}\left( \left[ {{\mathcal {N}}_{q}}\left( {{b}_{1}} \right) ,{{\mathcal {N}}_{q}}\left( {{a}_{1}} \right) \right] \right) \right) \\&\quad \ge {{\mathcal {N}}_{q}}\left( {{\phi }_{Q}}\left( \left[ {{\mathcal {N}}_{q}}\left( {{b}_{2}} \right) ,{{\mathcal {N}}_{q}}\left( {{a}_{2}} \right) \right] \right) \right) . \end{aligned}$$

From this and ${{\mathcal {N}}_{q}}\left( {{x}_{1}} \right) \le {{\mathcal {N}}_{q}}\left( {{x}_{2}} \right)$ for ${{x}_{1}}\ge {{x}_{2}}$ and ${{x}_{1}},{{x}_{2}}\in [ 0,1 ]$. Thus,

$$\begin{aligned} \phi _{Q}^{d}\left( \left[ {{a}_{1}},{{b}_{1}} \right] \right) \ge \phi _{Q}^{d}\left( \left[ {{a}_{2}},{{b}_{2}} \right] \right) . \end{aligned}$$

The proof is completed. $\square$

Theorem 6

(Boundness) For all Q, we have $a\le \phi _{Q}^{d}([ a,b ] )\le b$.

Proof

According to Definitions 10 and 11, we have

$$\begin{aligned} \phi _{Q}^{d}\left( \left[ a,b \right] \right) ={{\mathcal {N}}_{q}}\left( {{\phi }_{Q}}\left[ {{\mathcal {N}}_{q}}\left( b \right) ,{{\mathcal {N}}_{q}}\left( a \right) \right] \right) . \end{aligned}$$

From Theorem 3, For all Q, we have

$$\begin{aligned} {{\mathcal {N}}_{q}}\left( b \right) \le {{\phi }_{Q}}\left[ {{\mathcal {N}}_{q}}\left( b \right) ,{{\mathcal {N}}_{q}}\left( a \right) \right] \le {{\mathcal {N}}_{q}}\left( a \right) . \end{aligned}$$

Therefore, we have

$$\begin{aligned}&a={{\mathcal {N}}_{q}}\left( {{\mathcal {N}}_{q}}\left( a \right) \right) \le {{\mathcal {N}}_{q}}\left( {{\phi }_{Q}}\left[ {{\mathcal {N}}_{q}}\left( b \right) ,{{\mathcal {N}}_{q}}\left( a \right) \right] \right) \\&\quad \le {{\mathcal {N}}_{q}}\left( {{\mathcal {N}}_{q}}\left( b \right) \right) =b. \end{aligned}$$

From this and (i) ${{\mathcal {N}}_{q}}( {{x}_{1}} )\le {{\mathcal {N}}_{q}}( {{x}_{2}} )$ for ${{x}_{1}}\ge {{x}_{2}}$ and ${{x}_{1}},{{x}_{2}}\in [ 0,1 ]$; (ii) ${{\mathcal {N}}_{q}} ( {{\mathcal {N}}_{q}}( x ) )=x$ for all . Thus,

$$\begin{aligned} a\le \phi _{Q}^{d}\left( \left[ a,b \right] \right) \le b. \end{aligned}$$

The proof is completed. $\square$

Definition 12

[38] If ${{Q}_{1}}$ and ${{Q}_{2}}$ are such that ${{Q}_{1}}( x )\ge {{Q}_{2}}( x )$ for all $x\in [ 0,1 ]$, we denote this as ${{Q}_{1}}\ge {{Q}_{2}}$.

Theorem 7

If ${{Q}_{1}}\ge {{Q}_{2}}$, then $\phi _{{{Q}_{1}}}^{d}( [ a,b ] )\le \phi _{{{Q}_{2}}}^{d}([ a,b ] )$.

Proof

According to Definitions 10 and 11, we have

$$\begin{aligned} \phi _{Q}^{d}\left( \left[ a,b \right] \right) ={{\mathcal {N}}_{q}}\left( {{\phi }_{Q}}\left[ {{\mathcal {N}}_{q}}\left( b \right) ,{{\mathcal {N}}_{q}}\left( a \right) \right] \right) . \end{aligned}$$

From Theorem 4, if ${{Q}_{1}}\ge {{Q}_{2}}$, then

$$\begin{aligned} {{\phi }_{{{Q}_{1}}}}\left( \left[ a,b \right] \right) \ge {{\phi }_{{{Q}_{2}}}}\left( \left[ a,b \right] \right) . \end{aligned}$$

Therefore, we have

$$\begin{aligned} {{\phi }_{{{Q}_{1}}}}\left[ {{\mathcal {N}}_{q}}\left( b \right) ,{{\mathcal {N}}_{q}}\left( a \right) \right] \ge {{\phi }_{{{Q}_{2}}}}\left[ {{\mathcal {N}}_{q}}\left( b \right) ,{{\mathcal {N}}_{q}}\left( a \right) \right] \end{aligned}$$

and

$$\begin{aligned} {{\mathcal {N}}_{q}}\left( {{\phi }_{{{Q}_{1}}}}\left[ {{\mathcal {N}}_{q}}\left( b \right) ,{{\mathcal {N}}_{q}}\left( a \right) \right] \right) \le {{\mathcal {N}}_{q}}\left( {{\phi }_{{{Q}_{2}}}}\left[ {{\mathcal {N}}_{q}}\left( b \right) ,{{\mathcal {N}}_{q}}\left( a \right) \right] \right) . \end{aligned}$$

From this and ${{\mathcal {N}}_{q}}( {{x}_{1}} )\le {{\mathcal {N}}_{q}} ( {{x}_{2}} )$ for ${{x}_{1}}\ge {{x}_{2}}$ and ${{x}_{1}},{{x}_{2}}\in [ 0,1 ]$. Thus,

$$\begin{aligned} \phi _{{{Q}_{1}}}^{d}\left( \left[ a,b \right] \right) \le \phi _{{{Q}_{2}}}^{d}\left( \left[ a,b \right] \right) . \end{aligned}$$

$\square$

Yager [38] pointed out that $\lambda =\int _{0}^{1}{Q( y )dy}$ is the attitudinal character of Q and $\lambda \in [ 0,1 ]$. Some properties of C-GOWA and DC-GOWA operators with respect to the attitudinal character $\lambda$ are discussed.

Theorem 8

[51] If $\lambda$ is the attitudinal character of Q, then

$$\begin{aligned} {{\phi }_{Q}}\left( \left[ a,b \right] \right) ={{\left( \lambda {{b}^{p}}+\left( 1-\lambda \right) {{a}^{p}} \right) }^{{1}/{p}\;}}. \end{aligned}$$

(20)

Theorem 9

If $\lambda$ is the attitudinal character of Q, then

$$\begin{aligned} \phi _{Q}^{d}\left( \left[ a,b \right] \right) ={{\mathcal {N}}_{q}}{{\left( \lambda {{\left( {{\mathcal {N}}_{q}}\left( a \right) \right) }^{p}}+\left( 1-\lambda \right) {{\left( {{\mathcal {N}}_{q}}\left( b \right) \right) }^{p}} \right) }^{{1}/{p}\;}}. \end{aligned}$$

(21)

Proof

According to Definitions 10 and 11, we have

$$\begin{aligned} \phi _{Q}^{d}\left( \left[ a,b \right] \right) ={{\mathcal {N}}_{q}}\left( {{\phi }_{Q}}\left[ {{\mathcal {N}}_{q}}\left( b \right) ,{{\mathcal {N}}_{q}}\left( a \right) \right] \right) . \end{aligned}$$

From Theorem 8, we have

$$\begin{aligned} \phi _{Q}^{d}\left( \left[ a,b \right] \right) ={{\mathcal {N}}_{q}}{{\left( \lambda {{\left( {{\mathcal {N}}_{q}}\left( a \right) \right) }^{p}}+\left( 1-\lambda \right) {{\left( {{\mathcal {N}}_{q}}\left( b \right) \right) }^{p}} \right) }^{{1}/{p}\;}}. \end{aligned}$$

$\square$

Remark 1

For convenience, we denote ${{\phi }_{Q}}$ and $\phi _{Q}^{d}$ as ${{\phi }_{\lambda }}$ and $\phi _{\lambda }^{d}$, respectively.

3.2 C-IVQROFOWA operator

Consider the situation where $\tilde{\alpha }=( [ {{\mu }^{-}},{{\mu }^{+}} ], [ {{v}^{-}},{{v}^{+}} ] )$ is a QROFN. Use ${{\phi }_{Q}}$ and $\phi _{Q}^{d}$ to aggregate the continuous interval-valued $[ {{\mu }^{-}},{{\mu }^{+}} ]$ and $[ {{v}^{-}},{{v}^{+}} ]$, respectively. Further, assume the power of $p=q$ in this case, then

$$\begin{aligned} {{\phi }_{Q}}( [ a,b ] )={{( \int _{0}^{1}{\frac{dQ( y )}{dy} ( {{b}^{q}}-y ( {{b}^{q}}-{{a}^{q}} ) )dy} )}^{{1}/{q}\;}} \end{aligned}$$

(22)

and

$$\begin{aligned} \phi _{Q}^{d}\left( \left[ a,b \right] \right) \text {=}{{\left( 1-\int _{0}^{1}{\frac{dQ\left( y \right) }{dy}\left( 1-{{a}^{q}}-y\left( {{b}^{q}}-{{a}^{q}} \right) \right) dy} \right) }^{{1}/{q}\;}} \end{aligned}$$

(23)

If $\lambda$ is the attitudinal character of Q, then

$$\begin{aligned}&{{\phi }_{\lambda }}\left( \left[ a,b \right] \right) ={{\left( \lambda {{b}^{q}}+\left( 1-\lambda \right) {{a}^{q}} \right) }^{{1}/{q}\;}},\phi _{\lambda }^{d}\left( \left[ a,b \right] \right) \nonumber \\&\quad ={{\left( \lambda {{a}^{q}}+\left( 1-\lambda \right) {{b}^{q}} \right) }^{{1}/{q}\;}}. \end{aligned}$$

(24)

Definition 13

Let $\tilde{\alpha }=( [ {{\mu }^{-}},{{\mu }^{+}} ], [ {{v}^{-}},{{v}^{+}} ] )$ be an IVQROFN solid at q and the continuous interval-valued q-rung orthopair fuzzy ordered weighted averaging (C-IVQROFOWA) operator is defined as follows:

$$\begin{aligned} {{G}_{{{\lambda }_{1}},{{\lambda }_{2}}}}\left( {\tilde{\alpha }} \right) =\left( {{\phi }_{{{\lambda }_{1}}}}\left( \left[ {{\mu }^{-}},{{\mu }^{+}} \right] \right) ,\phi _{{{\lambda }_{2}}}^{d}\left( \left[ {{v}^{-}},{{v}^{+}} \right] \right) \right) , \end{aligned}$$

(25)

where ${{\lambda }_{1}}$ and ${{\lambda }_{2}}$ are the attitudinal character of BUM function ${{Q}_{1}}$ and ${{Q}_{2}}$, respectively.

Example 1

Let $\tilde{\alpha }=( [ 0.5,0.6 ], [ 0.7,0.8 ] )$ be an IVQROFN with $q=2$. Let ${{Q}_{\text {1}}}( x )={{x}^{2}}$ and ${{Q}_{\text {2}}}( x )=x$ be the related BUM functions. The attitudinal character of ${{Q}_{i}}( i=1,2 )$ and the C-IVQROFOWA operator of $\tilde{\alpha }$ can be computed on the basis of Definition 13 as follows:

$$\begin{aligned}\left\{ \begin{aligned}&{{\lambda }_{1}}=\int _{0}^{1}{{{Q}_{\text {1}}}\left( x \right) {{d}_{x}}=\frac{1}{3},{{\lambda }_{2}}=\int _{0}^{1}{{{Q}_{\text {2}}}\left( x \right) {{d}_{x}}=\frac{1}{2}}}, \\&{{G}_{{{\lambda }_{1}},{{\lambda }_{2}}}}\left( \alpha \right) =\left( {{\left( \frac{1}{3}{{0.6}^{2}}+\frac{2}{3}{{0.5}^{2}} \right) }^{{1}/{2}\;}},{{\left( \frac{1}{2}{{0.7}^{2}}+\frac{1}{2}{{0.8}^{2}} \right) }^{{1}/{2}\;}} \right) =\left( 0.5354,0.7517 \right) . \end{aligned} \right. \end{aligned}$$

Theorem 10

Let $\tilde{\alpha }=( [ {{\mu }^{-}},{{\mu }^{+}} ], [ {{v}^{-}},{{v}^{+}} ] )$ be an IVQROFN solid at q, then ${{G}_{{{\lambda }_{1}},{{\lambda }_{2}}}}( {\tilde{\alpha }} )$ is also an IVQROFN solid at q.

Proof

According to Definition 11, we have

$$\begin{aligned} {{\phi }_{{{\lambda }_{1}}}}\left( \left[ {{\mu }^{-}},{{\mu }^{+}} \right] \right) \in \left[ {{\mu }^{-}},{{\mu }^{+}} \right] ,\phi _{{{\lambda }_{2}}}^{d}\left( \left[ {{v}^{-}},{{v}^{+}} \right] \right) \in \left[ {{v}^{-}},{{v}^{+}} \right] , \end{aligned}$$

where ${{G}_{{{\lambda }_{1}},{{\lambda }_{2}}}}( {\tilde{\alpha }} )$ is also an IVQROFN solid at q. $\square$

Definition 14

Let $\tilde{\alpha }=( [ {{\mu }^{-}},{{\mu }^{+}} ], [ {{v}^{-}},{{v}^{+}} ] )$ be an IVQROFN solid at q. For any QROFN $\alpha$, if $\mu \in [ {{\mu }^{-}},{{\mu }^{+}} ]$ and $v\in [ {{v}^{-}},{{v}^{+}} ]$, then $\alpha$ belongs to $\tilde{\alpha }$. We write $\alpha \in \tilde{\alpha }$ and read “$\alpha$ is a q-rung orthopair fuzzy element of $\tilde{\alpha }$”.

3.3 New score and accuracy functions based on the C-IVQROFOWA operator

For any IVPFN $\tilde{\alpha }$, its related C-IVQROFOWA operator ${{G}_{{{\lambda }_{1}},{{\lambda }_{2}}}}( {\tilde{\alpha }} )$ is QROFN. Based on the score and accuracy functions of QROF, the new score and accuracy functions of IVQROF are defined as follows.

Definition 15

Let $\tilde{\alpha }=( \tilde{\mu },\tilde{v} )=( [ {{\mu }^{-}},{{\mu }^{+}} ], [ {{v}^{-}},{{v}^{+}} ] )$ be an IVQROFN solid at q and the new score function $Sc{{o}_{{{\lambda }_{1}},{{\lambda }_{2}}}}( {\tilde{\alpha }} )$ and accuracy function $Ac{{c}_{{{\lambda }_{1}},{{\lambda }_{2}}}}( {\tilde{\alpha }} )$ are defined as follows:

$$\begin{aligned}&Sc{{o}_{{{\lambda }_{1}},{{\lambda }_{2}}}}\left( {\tilde{\alpha }} \right) ={{\left( {{\phi }_{{{\lambda }_{1}}}}\left( {\tilde{\mu }} \right) \right) }^{q}}-{{\left( \phi _{{{\lambda }_{2}}}^{d}\left( {\tilde{v}} \right) \right) }^{q}}, \nonumber \\&Ac{{c}_{{{\lambda }_{1}},{{\lambda }_{2}}}}\left( {\tilde{\alpha }} \right) ={{\left( {{\phi }_{{{\lambda }_{1}}}}\left( {\tilde{\mu }} \right) \right) }^{q}}+{{\left( \phi _{{{\lambda }_{2}}}^{d}\left( {\tilde{v}} \right) \right) }^{q}}, \end{aligned}$$

(26)

where ${{G}_{{{\lambda }_{1}},{{\lambda }_{2}}}}( {\tilde{\alpha }} )=( {{\phi }_{{{\lambda }_{1}}}}( {\tilde{\mu }} ),\phi _{{{\lambda }_{2}}}^{d}( {\tilde{v}} ) )$ is the C-IVQROFOWA operator of $\tilde{\alpha }$.

The ranking method is proposed based on the score function and accuracy function. Assume that ${{\tilde{\alpha }}_{i}}=( [ \mu _{i}^{-},\mu _{i}^{+} ], [ v_{i}^{-},v_{i}^{+} ] ) ( i=1,2 )$ are two IVPFNs solid at q.

(1) If $Sc{{o}_{{{\lambda }_{1}},{{\lambda }_{2}}}}( {{{\tilde{\alpha }}}_{1}} )<Sc{{o}_{{{\lambda }_{1}},{{\lambda }_{2}}}} ( {{{\tilde{\alpha }}}_{2}} )$, then ${{\tilde{\alpha }}_{1}}{{\prec }_{\lambda }}{{\tilde{\alpha }}_{2}}$.

(2) If $Sc{{o}_{{{\lambda }_{1}},{{\lambda }_{2}}}}( {{{\tilde{\alpha }}}_{1}} )=Sc{{o}_{{{\lambda }_{1}},{{\lambda }_{2}}}}$and $Ac{{c}_{{{\lambda }_{1}},{{\lambda }_{2}}}}( {{{\tilde{\alpha }}}_{1}} )<Ac{{c}_{{{\lambda }_{1}},{{\lambda }_{2}}}}( {{{\tilde{\alpha }}}_{2}} )$, then ${{\tilde{\alpha }}_{1}}{{\prec }_{\lambda }}{{\tilde{\alpha }}_{2}}$.

Remark 2

According to the ranking method of IVQROFNs in [35], we have

$$\begin{aligned} {{\tilde{\alpha }}_{1}}\prec {{\tilde{\alpha }}_{2}}\Leftrightarrow {{\tilde{\alpha }}_{1}}{{\prec }_{\lambda }}{{\tilde{\alpha }}_{2}}\left( {{\lambda }_{1}}\text {=}{{\lambda }_{2}}\text {=}0.5 \right) \end{aligned}$$

(27)

with the following

$$\begin{aligned}&S\left( {{{\tilde{\alpha }}}_{1}} \right)<S\left( {{{\tilde{\alpha }}}_{2}} \right) \Leftrightarrow Sc{{o}_{{{\lambda }_{1}},{{\lambda }_{2}}}}\left( {{{\tilde{\alpha }}}_{1}} \right)<Sc{{o}_{{{\lambda }_{1}},{{\lambda }_{2}}}}\left( {{{\tilde{\alpha }}}_{2}} \right) ,\\&H\left( {{{\tilde{\alpha }}}_{1}} \right)<H\left( {{{\tilde{\alpha }}}_{2}} \right) \Leftrightarrow Ac{{c}_{{{\lambda }_{1}},{{\lambda }_{2}}}}\left( {{{\tilde{\alpha }}}_{1}} \right) <Ac{{c}_{{{\lambda }_{1}},{{\lambda }_{2}}}}\left( {{{\tilde{\alpha }}}_{2}} \right) . \end{aligned}$$

Theorem 11

Let $\tilde{\alpha }=([ {{\mu }^{-}},{{\mu }^{+}} ],[ {{v}^{-}},{{v}^{+}} ] )$ be an IVQROFN solid at q, then ${{G}_{{{\lambda }_{1}},{{\lambda }_{2}}}}( {\tilde{\alpha }} )$ increases with respect to ${{\lambda }_{1}}$ and ${{\lambda }_{2}}$, and ${{G}_{0,0}}( {\tilde{\alpha }} )\le {{G}_{{{\lambda }_{1}},{{\lambda }_{2}}}}( {\tilde{\alpha }} )\le {{G}_{1,1}}( {\tilde{\alpha }} )$.

Proof

Suppose that ${{\lambda }_{1}}<{{\bar{\lambda }}_{1}}$ and ${{\lambda }_{2}}<{{\bar{\lambda }}_{2}}$. According to Definition 15, we have

$$\begin{aligned}&Sc{{o}_{{{\lambda }_{1}},{{\lambda }_{2}}}}\left( {\tilde{\alpha }} \right) -Sc{{o}_{{{{\bar{\lambda }}}_{1}},{{{\bar{\lambda }}}_{2}}}}\left( {\tilde{\alpha }} \right) \\&\quad ={{\phi }_{{{\lambda }_{1}}}}\left( \left[ {{\mu }^{-}},{{\mu }^{+}} \right] \right) -{{\phi }_{{{{\bar{\lambda }}}_{1}}}}\left( \left[ {{\mu }^{-}},{{\mu }^{+}} \right] \right) +\phi _{{{{\bar{\lambda }}}_{2}}}^{d}\left( \left[ {{v}^{-}},{{v}^{+}} \right] \right) -\phi _{{{\lambda }_{2}}}^{d}\left( \left[ {{v}^{-}},{{v}^{+}} \right] \right) . \end{aligned}$$

Given that ${{\phi }_{\lambda }}$ increases with respect to $\lambda$ and $\phi _{\lambda }^{d}$ decreases with respect to $\lambda$, then

$$\begin{aligned}&{{\phi }_{{{\lambda }_{1}}}}\left( \left[ {{\mu }^{-}},{{\mu }^{+}} \right] \right)<{{\phi }_{{{{\bar{\lambda }}}_{1}}}}\left( \left[ {{\mu }^{-}},{{\mu }^{+}} \right] \right) ,\\&\phi _{{{{\bar{\lambda }}}_{2}}}^{d}\left( \left[ {{v}^{-}},{{v}^{+}} \right] \right) <\phi _{{{\lambda }_{2}}}^{d}\left( \left[ {{v}^{-}},{{v}^{+}} \right] \right) . \end{aligned}$$

Therefore, $Sc{{o}_{{{\lambda }_{1}},{{\lambda }_{2}}}}( {\tilde{\alpha }} )-Sc{{o}_{{{{\bar{\lambda }}}_{1}},{{{\bar{\lambda }}}_{2}}}} ( {\tilde{\alpha }} )<0$, then ${{G}_{{{\lambda }_{1}},{{\lambda }_{2}}}}( {\tilde{\alpha }} )\prec {{G}_{{{{\bar{\lambda }}}_{1}},{{{\bar{\lambda }}}_{2}}}} ( {\tilde{\alpha }} )$. Thus, ${{G}_{{{\lambda }_{1}},{{\lambda }_{2}}}} ( {\tilde{\alpha }} )$ increases with respect to ${{\lambda }_{1}}$ and ${{\lambda }_{2}}$.

The proof is completed. $\square$

3.4 Weighted continuous interval-valued q-rung orthopair fuzzy aggregation operators

Definition 16

[1] Let ${{\alpha }_{i}}\text {=}( {{\mu }_{{{\alpha }_{i}}}},{{v}_{{{\alpha }_{i}}}} ) ( i=1,2,\ldots ,n )$ be a collection of QROFNs, then

$$\begin{aligned}&\mathsf{{Agg}}\left( {{\alpha }_{1}},{{\alpha }_{2}},\ldots ,{{\alpha }_{n}} \right) \nonumber \\&\quad =\left( {\mathbb {E}}\left( {{\mu }_{{{\alpha }_{1}}}},{{\mu }_{{{\alpha }_{2}}}},\ldots ,{{\mu }_{{{\alpha }_{n}}}} \right) ,{{\mathbb {E}}_{d}}\left( {{v}_{{{\alpha }_{1}}}},{{v}_{{{\alpha }_{2}}}},\ldots ,{{v}_{{{\alpha }_{n}}}} \right) \right) , \end{aligned}$$

(28)

is their aggregation operator, where

$$\begin{aligned}&{{\mathbb {E}}_{d}}\left( {{v}_{{{\alpha }_{1}}}},{{v}_{{{\alpha }_{2}}}},\ldots ,{{v}_{{{\alpha }_{n}}}} \right) \nonumber \\&\quad ={{\mathcal {N}}_{q}}\left( {\mathbb {E}}\left( {{\mathcal {N}}_{q}}\left( {{v}_{{{\alpha }_{1}}}} \right) ,{{\mathcal {N}}_{q}}\left( {{v}_{{{\alpha }_{2}}}} \right) ,\ldots ,{{\mathcal {N}}_{q}}\left( {{v}_{{{\alpha }_{n}}}} \right) \right) \right) \end{aligned}$$

(29)

with a negation function ${{\mathcal {N}}_{q}}$.

Theorem 12

[1] If each ${{\alpha }_{1}}$ is an orthopair fuzzy set solid at rung q, which is a QROFN and the negation used to define E is taken with respect to q, which is the q-th rung negation function ${{N}_{q}}$, then Agg is an orthopair fuzzy set solid at rung q, which is a QROFN.

By using the general form in Definition 16, Yager [1] introduced some important mean-type aggregation operators.

Definition 17

[1] Let ${{\alpha }_{i}}\text {=}( {{\mu }_{{{\alpha }_{i}}}},{{v}_{{{\alpha }_{i}}}} ) ( i=1,2,\ldots ,n )$ be a collection of QROFNs and $w={{( {{w}_{1}},{{w}_{2}},\ldots ,{{w}_{n}} )}^{T}}$ be the weighting vector to satisfy $\sum \nolimits _{i=1}^{n}{{{w}_{i}}=1}$ and ${{w}_{i}}\ge 0 ( i=1,2,\ldots ,n )$. The q-rung orthopair fuzzy weighted averaging (Q-ROFWA) operator is presented as follows:

$$\begin{aligned}&\mathsf{{QROFWA}}\left( {{\alpha }_{1}},{{\alpha }_{2}},\ldots ,{{\alpha }_{n}} \right) \nonumber \\&\quad =\left( {{\left( \sum \limits _{i=1}^{n}{{{w}_{i}}\mu _{i}^{q}} \right) }^{{1}/{q}\;}},{{\left( \sum \limits _{i=1}^{n}{{{w}_{i}}v_{i}^{q}} \right) }^{{1}/{q}\;}} \right) . \end{aligned}$$

(30)

Some limiting cases are also provided.

• If $q=1$ , then $\mathsf{{QROFWA}}( {{\alpha }_{1}},{{\alpha }_{2}},\ldots ,{{\alpha }_{n}} )= ( \sum \nolimits _{i=1}^{n}{{{w}_{i}}{{\mu }_{i}}},\sum \nolimits _{i=1}^{n}{{{w}_{i}}{{v}_{i}}} )$.

• If $q=2$ , then $\mathsf{{QROFWA}}( {{\alpha }_{1}},{{\alpha }_{2}},\ldots ,{{\alpha }_{n}} )= ( {{ ( \sum \nolimits _{i=1}^{n}{{{w}_{i}}\mu _{i}^{2}} )}^{{1}/{2}\;}},{{( \sum \nolimits _{i=1}^{n}{{{w}_{i}}v_{i}^{2}} )}^{{1}/{2}\;}} )$.

Definition 18

Let ${{\tilde{\alpha }}_{i}}=( {{{\tilde{\mu }}}_{i}},{{{\tilde{v}}}_{i}})=( [ \mu _{i}^{-},\mu _{i}^{+} ], [ v_{i}^{-},v_{i}^{+} ] ) ( i=1,\ldots ,n )$ be a collection of IVQROFNs solid at q, and the weighted C-IVQROFOWA (WC-IVQROFOWA) operator is defined as follows:

$$\begin{aligned}&\mathsf{{WC-IVQROFOWA}}\left( {{{\tilde{\alpha }}}_{1}},\ldots ,{{{\tilde{\alpha }}}_{n}} \right) \nonumber \\&\quad {=}\left( {{\left( \sum \limits _{i=1}^{n}{{{w}_{i}}{{\left( {{\phi }_{{{\lambda }_{1}}}}\left( {{{\tilde{\mu }}}_{i}} \right) \right) }^{q}}} \right) }^{{1}/{q}\;}},{{\left( \sum \limits _{i=1}^{n}{{{w}_{i}}{{\left( \phi _{{{\lambda }_{2}}}^{d}\left( {{{\tilde{v}}}_{i}} \right) \right) }^{q}}} \right) }^{{1}/{q}\;}} \right) . \end{aligned}$$

(31)

Example 2

Let ${{\tilde{\alpha }}_{1}}=([ 0.5,0.6 ], [ 0.7,0.8 ]),{{\tilde{\alpha }}_{2}}= ( [ 0.2,0.4 ], [ 0.8,0.9 ] ),{{\tilde{\alpha }}_{3}}= ( [ 0.7,0.8], [ 0.4,0.6 ] )$ be three IVQROFNs solid at $q=2$,and $w={{( 0.2,0.3,0.5 )}^{T}}$ be the weighting vector. The WC-IVQRFOWA operator is conputed as foolows with ${{Q}_{\text {1}}}( x )={{x}^{2}}$ and ${{Q}_{\text {2}}}( x )=x$:

$$\begin{aligned}&\mathsf{{WC-IVQRFOWA}}\left( {{\alpha }_{1}},{{\alpha }_{2}},{{\alpha }_{3}} \right) \\&\quad =\left( \begin{aligned}&{{\left( 0.2\left( \frac{1}{3}{{0.6}^{2}}+\frac{2}{3}{{0.5}^{2}} \right) \text {+}0.3\left( \frac{1}{3}{{0.4}^{2}}+\frac{2}{3}{{0.2}^{2}} \right) \text {+}0.5\left( \frac{1}{3}{{0.8}^{2}}+\frac{2}{3}{{0.7}^{2}} \right) \right) }^{{1}/{2}\;}} \\&{{\left( 0.2\left( \frac{1}{2}{{0.7}^{2}}+\frac{1}{2}{{0.8}^{2}} \right) \text {+}0.3\left( \frac{1}{2}{{0.8}^{2}}+\frac{1}{2}{{0.9}^{2}} \right) \text {+}0.5\left( \frac{1}{2}{{0.4}^{2}}+\frac{1}{2}{{0.6}^{2}} \right) \right) }^{{1}/{2}\;}} \\ \end{aligned} \right) \\&\quad =\left( 0.3513,0.4605 \right) . \\ \end{aligned}$$

Theorem 13

Let ${{\tilde{\alpha }}_{i}}=\left( {{{\tilde{\mu }}}_{i}},{{{\tilde{v}}}_{i}} \right) =\left( \left[ \mu _{i}^{-},\mu _{i}^{+} \right] ,\left[ v_{i}^{-},v_{i}^{+} \right] \right) \left( i=1,\ldots ,n \right)$ be a collection of IVQROFNs solid at q, then $\mathsf{{WC-IVQROFOWA}}\left( {{{\tilde{\alpha }}}_{1}},\ldots ,{{{\tilde{\alpha }}}_{n}} \right)$ is a QROFN.

Theorem 14

Let ${{\tilde{\alpha }}_{i}}=\left( {{{\tilde{\mu }}}_{i}},{{{\tilde{v}}}_{i}} \right) =\left( \left[ \mu _{i}^{-},\mu _{i}^{+} \right] ,\left[ v_{i}^{-},v_{i}^{+} \right] \right) \left( i=1,\ldots ,n \right)$ be a collection of IVQROFNs solid at q, then we have

$$\begin{aligned}&s\left( \mathsf{{WC-IVQROFOWA}}\left( {{{\tilde{\alpha }}}_{1}},\ldots ,{{{\tilde{\alpha }}}_{n}} \right) \right) =\sum \limits _{i=1}^{n}{{{w}_{i}}Sc{{o}_{{{\lambda }_{1}},{{\lambda }_{2}}}}\left( {{{\tilde{\alpha }}}_{i}} \right) }\nonumber \\&h\left( \mathsf{{WC-IVQROFOWA}}\left( {{{\tilde{\alpha }}}_{1}},\ldots ,{{{\tilde{\alpha }}}_{n}} \right) \right) =\sum \limits _{i=1}^{n}{{{w}_{i}}Ac{{c}_{{{\lambda }_{1}},{{\lambda }_{2}}}}\left( {{{\tilde{\alpha }}}_{i}} \right) } \end{aligned}$$

(32)

Proof

According to the definition of the score and accuracy functions, we have

$$\begin{aligned}&s\left( \mathsf{{WC-IVQROFOWA}}\left( {{{\tilde{\alpha }}}_{1}},\ldots ,{{{\tilde{\alpha }}}_{n}} \right) \right) \\&\quad ={{\left( {{\left( \sum \limits _{i=1}^{n}{{{w}_{i}}{{\left( {{\phi }_{{{\lambda }_{1}}}}\left( {{{\tilde{\mu }}}_{i}} \right) \right) }^{q}}} \right) }^{{1}/{q}\;}} \right) }^{q}}\\&\quad \quad -{{\left( {{\left( \sum \limits _{i=1}^{n}{{{w}_{i}}{{\left( \phi _{{{\lambda }_{2}}}^{d}\left( {{{\tilde{v}}}_{i}} \right) \right) }^{q}}} \right) }^{{1}/{q}\;}} \right) }^{q}} \\&\quad =\sum \limits _{i=1}^{n}{{{w}_{i}}\left( {{\left( {{\phi }_{{{\lambda }_{1}}}}\left( {{{\tilde{\mu }}}_{{{{\tilde{\alpha }}}_{i}}}} \right) \right) }^{q}}-{{\left( \phi _{{{\lambda }_{2}}}^{d}\left( {{{\tilde{v}}}_{{{{\tilde{\alpha }}}_{i}}}} \right) \right) }^{q}} \right) }\\&\quad =\sum \limits _{i=1}^{n}{{{w}_{i}}Sc{{o}_{{{\lambda }_{1}},{{\lambda }_{2}}}}\left( {{{\tilde{\alpha }}}_{i}} \right) } \end{aligned}$$

Similarly, we can obtain

$$\begin{aligned}&Ac{{c}_{{{\lambda }_{1}},{{\lambda }_{2}}}}\left( \mathsf{{WC-IVQROFOWA}}\left( {{{\tilde{\alpha }}}_{1}},\ldots ,{{{\tilde{\alpha }}}_{n}} \right) \right) \\&\quad =\sum \limits _{i=1}^{n}{{{w}_{i}}Ac{{c}_{{{\lambda }_{1}},{{\lambda }_{2}}}}\left( {{{\tilde{\alpha }}}_{i}} \right) }. \end{aligned}$$

The proof is completed. $\square$

Theorem 15

(Monotonicity) Let ${{\tilde{\alpha }}_{i}}=\left( {{{\tilde{\mu }}}_{{{{\tilde{\alpha }}}_{i}}}},{{{\tilde{v}}}_{{{{\tilde{\alpha }}}_{i}}}} \right)$ and ${{\tilde{\beta }}_{i}}=\left( {{{\tilde{\mu }}}_{{{{\tilde{\beta }}}_{i}}}},{{{\tilde{v}}}_{{{{\tilde{\beta }}}_{i}}}} \right) \left( i=1,\ldots ,n \right)$ be two collections of IVPFNs solid at q. If ${{\tilde{\alpha }}_{i}}{{\prec }_{\lambda }}{{\tilde{\beta }}_{i}}\left( i=1,\ldots ,n \right)$, then

$$\begin{aligned}&\mathsf{{WC-IVQROFOWA}}\left( {{{\tilde{\alpha }}}_{1}},\ldots ,{{{\tilde{\alpha }}}_{n}} \right) \nonumber \\&\quad \prec \mathsf{{WC-IVQROFOWA}}\left( {{{\tilde{\beta }}}_{1}},\ldots ,{{{\tilde{\beta }}}_{n}} \right) . \end{aligned}$$

(33)

Proof

If ${{\tilde{\alpha }}_{i}}{{\prec }_{\lambda }}{{\tilde{\beta }}_{i}}\left( i=1,\ldots ,n \right)$,

Case1. $Sc{{o}_{{{\lambda }_{1}},{{\lambda }_{2}}}}\left( {{{\tilde{\alpha }}}_{i}} \right) <Sc{{o}_{{{\lambda }_{1}},{{\lambda }_{2}}}}\left( {{{\tilde{\beta }}}_{i}} \right) \left( i=1,\ldots ,n \right)$, by Theorem 14,

$$\begin{aligned}&s\left( \mathsf{{WC-IVQROFOWA}}\left( {{{\tilde{\alpha }}}_{1}},\ldots ,{{{\tilde{\alpha }}}_{n}} \right) \right) \\&\quad \quad -s\left( \mathsf{{WC-IVQROFOWA}}\left( {{{\tilde{\beta }}}_{1}},\ldots ,{{{\tilde{\beta }}}_{n}} \right) \right) \\&\quad =\sum \limits _{i=1}^{n}{{{w}_{i}}\left( Sc{{o}_{{{\lambda }_{1}},{{\lambda }_{2}}}}\left( {{{\tilde{\alpha }}}_{i}} \right) -Sc{{o}_{{{\lambda }_{1}},{{\lambda }_{2}}}}\left( {{{\tilde{\beta }}}_{i}} \right) \right) }<0 \end{aligned}$$

then $\mathsf{{WC-IVQROFOWA}}\left( {{{\tilde{\alpha }}}_{1}},\ldots ,{{{\tilde{\alpha }}}_{n}} \right) \prec \mathsf{{WC-IVQROFOWA}}\left( {{{\tilde{\beta }}}_{1}},\ldots ,{{{\tilde{\beta }}}_{n}} \right) .$

Case2. $Sc{{o}_{{{\lambda }_{1}},{{\lambda }_{2}}}}\left( {{{\tilde{\alpha }}}_{i}} \right) =Sc{{o}_{{{\lambda }_{1}},{{\lambda }_{2}}}}\left( {{{\tilde{\beta }}}_{i}} \right)$ and $Ac{{c}_{{{\lambda }_{1}},{{\lambda }_{2}}}}\left( {{{\tilde{\alpha }}}_{i}} \right) <Ac{{c}_{{{\lambda }_{1}},{{\lambda }_{2}}}}\left( {{{\tilde{\beta }}}_{i}} \right)$, by Theorem 14,

$$\begin{aligned}&h\left( \mathsf{{WC-IVQROFOWA}}\left( {{{\tilde{\alpha }}}_{1}},\ldots ,{{{\tilde{\alpha }}}_{n}} \right) \right) \\&\quad \quad -h\left( \mathsf{{WC-IVQROFOWA}}\left( {{{\tilde{\beta }}}_{1}},\ldots ,{{{\tilde{\beta }}}_{n}} \right) \right) \\&\quad =\sum \limits _{i=1}^{n}{{{w}_{i}}\left( Ac{{c}_{{{\lambda }_{1}},{{\lambda }_{2}}}}\left( {{{\tilde{\alpha }}}_{i}} \right) -Ac{{c}_{{{\lambda }_{1}},{{\lambda }_{2}}}}\left( {{{\tilde{\beta }}}_{i}} \right) \right) }<0 \end{aligned}$$

then $\mathsf{{WC-IVQROFOWA}}\left( {{{\tilde{\alpha }}}_{1}},\ldots ,{{{\tilde{\alpha }}}_{n}} \right) \prec \mathsf{{WC-IVQROFOWA}}\left( {{{\tilde{\beta }}}_{1}},\ldots ,{{{\tilde{\beta }}}_{n}} \right) .$

The proof is completed. $\square$

Theorem 16

(Idempotency) Let ${{\tilde{\alpha }}_{i}}=\left( {{{\tilde{\mu }}}_{i}},{{{\tilde{v}}}_{i}} \right) =\left( \left[ \mu _{i}^{-},\mu _{i}^{+} \right] ,\left[ v_{i}^{-},v_{i}^{+} \right] \right) \left( i=1,\ldots ,n \right)$ be a collection of IVPFNs. If ${{\tilde{\alpha }}_{i}}=\tilde{\alpha }=\left( \left[ {{\mu }^{-}},{{\mu }^{+}} \right] ,\left[ {{v}^{-}},{{v}^{+}} \right] \right)$ for all i, then

$$\begin{aligned} \mathsf{{WC-IVQROFOWA}}\left( {{{\tilde{\alpha }}}_{1}},\ldots ,{{{\tilde{\alpha }}}_{n}} \right) =\tilde{\alpha }. \end{aligned}$$

(34)

Theorem 17

(Boundness) Let ${{\tilde{\alpha }}_{i}}=\left( \left[ \mu _{i}^{-},\mu _{i}^{+} \right] ,\left[ v_{i}^{-},v_{i}^{+} \right] \right) \left( i=1,\ldots ,n \right)$ be a collection of IVPFNs,

$$\begin{aligned} {{\tilde{\alpha }}_{\min }}\le \mathsf{{WC-IVQROFOWA}}\left( {{{\tilde{\alpha }}}_{1}},\ldots ,{{{\tilde{\alpha }}}_{n}} \right) \le {{\tilde{\alpha }}_{\max }}, \end{aligned}$$

(35)

where ${{\tilde{\alpha }}_{\min }}=\left( {{\min }_{i}}\left\{ \mu _{i}^{-} \right\} ,ma{{x}_{i}}\left\{ v_{i}^{+} \right\} \right)$ and ${{\tilde{\alpha }}_{\max }}=\left( {{\max }_{i}}\left\{ \mu _{i}^{-} \right\} ,{{\min }_{i}}\left\{ v_{i}^{+} \right\} \right)$.

Proof

According to Definition 18 and Theorem 17, we have

$$\begin{aligned}&\mathsf{{WC-IVQROFOWA}}\left( {{{\tilde{\alpha }}}_{1}},\ldots ,{{{\tilde{\alpha }}}_{n}} \right) \\&\quad =\mathsf{{QROFWA}}\left( {{G}_{{{\lambda }_{1}},{{\lambda }_{2}}}}\left( {{{\tilde{\alpha }}}_{1}} \right) ,\ldots ,{{G}_{{{\lambda }_{1}},{{\lambda }_{2}}}}\left( {{{\tilde{\alpha }}}_{n}} \right) \right) \end{aligned}$$

and

$$\begin{aligned} {{\tilde{\alpha }}_{\min }}&\le \left( \mu _{i}^{-},v_{i}^{+} \right) ={{G}_{0,0}}\left( {{{\tilde{\alpha }}}_{i}} \right) \\&\le {{G}_{{{\lambda }_{1}},{{\lambda }_{2}}}}\left( {{{\tilde{\alpha }}}_{i}} \right) \le {{G}_{1,1}}\left( {{{\tilde{\alpha }}}_{i}} \right) =\left( \mu _{i}^{+},v_{i}^{-} \right) \le {{\tilde{\alpha }}_{\max }}. \end{aligned}$$

Therefore,

$$\begin{aligned} {{\tilde{\alpha }}_{\min }}\le \mathsf{{WC-IVQROFOWA}}\left( {{{\tilde{\alpha }}}_{1}},\ldots ,{{{\tilde{\alpha }}}_{n}} \right) \le {{\tilde{\alpha }}_{\max }}. \end{aligned}$$

The proof is completed. $\square$

The WC-IVQROFOWA operator proposed in this section is used mainly to aggregate interval-valued q-rung orthopair fuzzy numbers and support the research and construction of the decision-making method in the next section.

4 Decision-making approach based on the WC-IVQROFOWA operator

To select the best alternative for a given application, we put forward a novel framework based on the WC-IVQROFOWA operator for solving fuzzy multiple attributes decision-making problems. Hence, the proposed approach consists of the following main stages: (1) determining the parameter value of aggregation operators and (2) obtaining the collective matrix, computing the collective evaluation values of alternatives, and obtaining the ranking orders of alternatives.

In the first stage, the performance rating of alternatives of each attribute provided by decision-makers is linguistic terms expressed in IVQROFNs. Given that the WC-IVQROFOWA operator is used to aggregate the individual decision matrix, the parameter value of the aggregation operator is calculated by using an optimization model and then the QROF collective matrix can be obtained. In the second stage, the q-rung orthopair fuzzy weighted averaging (QROFWA) operator is used to compute the collective evaluation values of alternatives, and the ranking method used to determine the ranking orders of alternatives.

Fig. 1

Flowchart of the proposed decision-making approach

Figure 1 delineates the flowchart of the proposed IVQROF decision-making approach and its detailed explanations are given in the following subsections.

4.1 Determining the parameter value and obtaining the collective matrix

Considering the decision-making problems of general fuzzy multiple attributes with a set of alternatives $\left\{ {{x}_{1}},\ldots ,{{x}_{m}} \right\}$, which are evaluated by t decision-makers ${{e}_{k}}\left( k=1,\ldots ,t \right)$ on a set of attributes $\left\{ {{c}_{1}},\ldots ,{{c}_{n}} \right\}$. Suppose that ${{\tilde{R}}_{k}}={{\left( \tilde{\alpha }_{ij}^{k} \right) }_{m\times n}}$ is a decision matrix, where $\tilde{\alpha }_{ij}^{k}$ is the performance or rating of alternative ${{x}_{i}}$ with respect to attribute ${{c}_{j}}$ and given by the decision-maker ${{e}_{k}}$ and $\tilde{\alpha }_{ij}^{k}=\left( \tilde{\mu }_{ij}^{k},\tilde{v}_{ij}^{k} \right) =\left( \left[ \mu _{ij}^{k+},\mu _{ij}^{k+} \right] ,\left[ v_{ij}^{k+},v_{ij}^{k+} \right] \right)$ is an IVQROFN indicating the range of degrees to which the alternative ${{x}_{i}}$ satisfies and dissatisfies attribute ${{c}_{j}}$, respectively. Given that decision-makers may come from different departments and have different backgrounds and expertise, each decision-maker is given a weight ${{w}_{k}}$, $k=1,\ldots ,t$, (where $\sum \nolimits _{k=1}^{t}{{{w}_{k}}=1}$ and ${{w}_{k}}>0$) to reflect his/her influence on the overall decision results. Accordingly, the IVQROFS theory is adopted to handle the uncertain assessments of alternatives provided by the decision-makers.

(1) Consensus degree (CD) with interval-valued q-rung orthopair fuzzy sets

Before using the WC-IVQROFOWA operator to aggregate t decision matrices ${{\tilde{R}}_{k}}$,$k=1,\ldots ,t$, the parameters ${{\lambda }_{1}}$ and ${{\lambda }_{2}}$ need to be determined.

Consensus measure plays an important role in the MAGDM process [57,58,59,60]. In the process of GDM, different experts having different opinions on certain attributes of evaluation alternatives are unavoidable because of the differences in their professional fields [61]. Therefore, defining the element, attribute, and group consensus measures is necessary to identify the decision-making results with low group consensus effectively and lay the foundation for the follow-up model.

This study constructs the consensus evaluation index and defines the CD of each attribute, ${{c}_{j}}\left( j=1,\ldots ,n \right)$ based on the proximity index between opinions.

• The CD on the evaluation element $\tilde{\alpha }_{ij}^{k}$.

Definition 19

Let ${{\tilde{R}}_{k}}={{\left( \tilde{\alpha }_{ij}^{k} \right) }_{m\times n}}\left( k\in \left\{ 1,\ldots ,t \right\} \right)$ be an individual evaluation matrix and ${{\tilde{R}}_{c}}={{\left( {{{\tilde{\alpha }}}_{ij}} \right) }_{m\times n}}$ is the collective matrix aggregated by the WC-IVQROFOWA operator, then their CD on the alternative ${{x}_{i}}$ for the attribute ${{c}_{j}}$ is

$$\begin{aligned}&C{{E}_{ij}}\left( {{{\tilde{R}}}_{k}},{{{\tilde{R}}}_{c}} \right) =1-\frac{1}{2}\left( \left| {{\left( {{\phi }_{{{\lambda }_{1}}}}\left( \tilde{\mu }_{ij}^{k} \right) \right) }^{q}}\right. \right. \nonumber \\&\quad \left. \left. -{{\left( {{\mu }_{ij}} \right) }^{q}} \right| +\left| {{\left( \phi _{{{\lambda }_{2}}}^{d}\left( \tilde{v}_{ij}^{k} \right) \right) }^{q}}-{{\left( {{v}_{ij}} \right) }^{q}} \right| \right) , \end{aligned}$$

(36)

where ${{\alpha }_{ij}}=\left( {{\mu }_{ij}},{{v}_{ij}} \right)$, which satisfies

$$\begin{aligned} {{\alpha }_{ij}}&=\mathsf{{WC-IVQROFOWA}}\left( \tilde{\alpha }_{ij}^{1},\ldots ,\tilde{\alpha }_{ij}^{t} \right) \nonumber \\&=\left( {{\left( \sum \limits _{l=1}^{t}{{{w}_{l}}{{\left( {{\phi }_{{{\lambda }_{1}}}}\left( \tilde{\mu }_{ij}^{l} \right) \right) }^{q}}} \right) }^{{1}/{q}\;}},\right. \nonumber \\&\quad \left. {{\left( \sum \limits _{l=1}^{t}{{{w}_{l}}{{\left( \phi _{{{\lambda }_{2}}}^{d}\left( \tilde{v}_{ij}^{l} \right) \right) }^{q}}} \right) }^{{1}/{q}\;}} \right) \end{aligned}$$

(37)

and ${{w}_{l}}$ is the weight of expert ${{e}_{l}}$, and ${{G}_{{{\lambda }_{1}},{{\lambda }_{2}}}}\left( \tilde{\alpha }_{ij}^{k} \right) =\left( {{\phi }_{{{\lambda }_{1}}}}\left( \tilde{\mu }_{ij}^{k} \right) ,\phi _{{{\lambda }_{2}}}^{d}\left( \tilde{v}_{ij}^{k} \right) \right)$ satisfies

$$\begin{aligned}&{{\phi }_{{{\lambda }_{1}}}}\left( \tilde{\mu }_{ij}^{k} \right) ={{\left( {{\lambda }_{1}}{{\left( \mu _{ij}^{k+} \right) }^{q}}+\left( 1-{{\lambda }_{1}} \right) {{\left( \mu _{ij}^{k-} \right) }^{q}} \right) }^{{1}/{q}\;}}, \end{aligned}$$

(38)

$$\begin{aligned}&\phi _{{{\lambda }_{2}}}^{d}\left( \tilde{v}_{ij}^{k} \right) ={{\left( {{\lambda }_{2}}{{\left( v_{ij}^{k-} \right) }^{q}}+\left( 1-{{\lambda }_{2}} \right) {{\left( v_{ij}^{k+} \right) }^{q}} \right) }^{{1}/{q}\;}}. \end{aligned}$$

(39)

Therefore, we have

$$\begin{aligned}&C{{E}_{ij}}\left( {{{\tilde{R}}}_{k}},{{{\tilde{R}}}_{c}} \right) \nonumber \\&\quad =1-\frac{1}{2}\left( \left| \sum \limits _{l=1}^{t}{{w}_{l}}\left( {{\lambda }_{1}}\left( {{\left( \mu _{ij}^{k+} \right) }^{q}}-{{\left( \mu _{ij}^{l+} \right) }^{q}} \right) \right. \right. \right. \nonumber \\&\quad \quad \left. \left. \left. +\left( 1-{{\lambda }_{1}} \right) \left( {{\left( \mu _{ij}^{k-} \right) }^{q}}-{{\left( \mu _{ij}^{l-} \right) }^{q}} \right) \right) \right| \right) \nonumber \\&\quad \quad -\frac{1}{2}\left( \left| \sum \limits _{l=1}^{t}{{w}_{l}}\left( {{\lambda }_{2}}\left( {{\left( v_{ij}^{k-} \right) }^{q}}-{{\left( v_{ij}^{l-} \right) }^{q}} \right) \right. \right. \right. \nonumber \\&\quad \quad \left. \left. \left. +\left( 1-{{\lambda }_{2}} \right) \left( {{\left( v_{ij}^{k+} \right) }^{q}}-{{\left( v_{ij}^{l+} \right) }^{q}} \right) \right) \right| \right) \end{aligned}$$

(40)

• The CD on the attribute ${{c}_{j}}$.

Definition 20

Let ${{\tilde{R}}_{k}}\left( k\in \left\{ 1,\ldots ,t \right\} \right)$ be an individual evaluation matrix and ${{\tilde{R}}_{c}}$ be the collective matrix, then their CD on attribute ${{c}_{j}}$ is

$$\begin{aligned} C{{C}_{j}}\left( {{{\tilde{R}}}_{k}},{{{\tilde{R}}}_{c}} \right) =\frac{1}{m}\sum \limits _{i=1}^{m}{C{{E}_{ij}}\left( {{{\tilde{R}}}_{k}},{{{\tilde{R}}}_{c}} \right) }, \end{aligned}$$

(41)

which is the CD on attribute ${{c}_{j}}$ with the group.

Definition 21

Let ${{\tilde{R}}_{k}}\left( k=1,\ldots ,t \right)$ be a collection of individual evaluation matrices, then the CD on attribute ${{c}_{j}}$ with the group is defined as

$$\begin{aligned} G\text {-}C{{C}_{j}}=\frac{1}{t}\sum \limits _{k=1}^{t}{C{{C}_{j}}\left( {{{\tilde{R}}}_{k}},{{{\tilde{R}}}_{c}} \right) }. \end{aligned}$$

(42)

Moreover, the CD on all attributes ${{c}_{j}}\left( j=1,\ldots ,n \right)$ with the group is

$$\begin{aligned} G\text {-}CD=\frac{1}{n}\sum \limits _{j=1}^{n}{G\text {-}C{{C}_{j}}}. \end{aligned}$$

(43)

Notice that

$$\begin{aligned} G\text {-}CD=\frac{1}{mtn}\sum \limits _{j=1}^{n}{\sum \limits _{k=1}^{t}{\sum \limits _{i=1}^{m}{C{{E}_{ij}}\left( {{{\tilde{R}}}_{k}},{{{\tilde{R}}}_{c}} \right) }}}. \end{aligned}$$

(44)

(2) Initial parameter optimization model driven by the group consensus measure

Based on the consensus measure defined above, a parameter optimization model is constructed to maximize the consensus measure $G\text {-}CD$.

In [40], Yager proposed the following parameterized family of regular increasing monotone (RIM) quantifiers $Q\left( x \right) ={{x}^{\alpha }}$ with $\alpha \in \left[ 0,+\infty \right)$ as an attitudinal parameter that reflects the attitude of the expert. In this study, RIM quantifiers $Q\left( x \right) ={{x}^{\alpha }}$ are used to construct the following parameter optimization model:

Model (M1)

$$\begin{aligned}&\max G\text {-}CD={\sum \limits _{j=1}^{n}{\sum \limits _{k=1}^{t}{\sum \limits _{i=1}^{m}{C{{E}_{ij}}\left( {{{\tilde{R}}}_{k}},{{{\tilde{R}}}_{c}} \right) }}}}/{mtn}\nonumber \\&s.t.\left\{ \begin{aligned}&0\le \kappa _{\mu }^{*}\le 1,0\le \kappa _{v}^{*}\le 1, \\&\lambda _{\mu }^{*}=\int _{0}^{1}{{{x}^{\kappa _{\mu }^{*}}}dx}=\frac{1}{1+\kappa _{\mu }^{*}},\lambda _{v}^{*}=\int _{0}^{1}{{{x}^{\kappa _{v}^{*}}}dx}=\frac{1}{1+\kappa _{v}^{*}} \\&C{{E}_{ij}}\left( {{{\tilde{R}}}_{k}},{{{\tilde{R}}}_{c}} \right) =1-\frac{1}{2}\left( \left| {{\left( {{\phi }_{\lambda _{\mu }^{*}}}\left( \tilde{\mu }_{ij}^{k} \right) \right) }^{q}}-{{\left( {{\mu }_{ij}} \right) }^{q}} \right| \right. \\&\quad \left. +\left| {{\left( \phi _{\lambda _{v}^{*}}^{d}\left( \tilde{v}_{ij}^{k} \right) \right) }^{q}}-{{\left( {{v}_{ij}} \right) }^{q}} \right| \right) \\&i=1,\ldots ,m;j=1,\ldots ,n;k=1,\ldots ,t. \\ \end{aligned} \right. \nonumber \\ \end{aligned}$$

(45)

The attributes or criteria are the basis of evaluation alternatives. Different experts may have different opinions on the determined attributes or criteria in the evaluation process. In the original model M1, we set the uniform parameters ($\lambda _{\mu }^{*}$ and $\lambda _{v}^{*}$), and assume that $0.5\le \lambda _{\mu }^{*},\lambda _{v}^{*} \le 1$ . We wish to obtain the maximum consensus by adjusting the parameters flexibly to improve the credibility of the decision results based on this model.

(3) Determining the parameter value and obtaining the collective matrix

Using the proposed original optimization model, the following Algorithm 1 is constructed to determine the parameter values and obtain the collective matrix ${{\tilde{R}}_{c}}$.

4.2 Computing the comprehensive evaluation values and ranking orders of alternatives

From Sect. 4.2, the collective matrix ${{\tilde{R}}_{c}}={{\left( {{{\tilde{\alpha }}}_{ij}} \right) }_{m\times n}}$ is obtained, where ${{\tilde{\alpha }}_{ij}}=\left( {{\mu }_{ij}},{{v}_{ij}} \right)$. Assume that the weights of attributes ${{c}_{j}}$ are ${{\omega }_{j}}\left( j=1,\ldots ,n \right)$, where $\sum \nolimits _{j=1}^{n}{{{\omega }_{j}}=1}$ and ${{\omega }_{j}}>0$ for all $j=1,\ldots ,n$. Use the QROFWA operator to compute the comprehensive evaluation values ${{\alpha }_{i}}$ of alternatives ${{X}_{i}}\left( i=1,\ldots ,m \right)$, where

$$\begin{aligned}&{{\tilde{\alpha }}_{i}}=\mathsf{{QROFWA}}\left( {{{\tilde{\alpha }}}_{i1}},{{{\tilde{\alpha }}}_{i2}},\ldots ,{{{\tilde{\alpha }}}_{in}} \right) =\left( {{\mu }_{i}},{{v}_{i}} \right) \nonumber \\&\quad =\left( {{\left( \sum \limits _{j=1}^{n}{{{\omega }_{j}}\mu _{ij}^{q}} \right) }^{{1}/{q}\;}},{{\left( \sum \limits _{j=1}^{n}{{{\omega }_{j}}v_{ij}^{q}} \right) }^{{1}/{q}\;}} \right) . \end{aligned}$$

(46)

Compute the score $s\left( {{\alpha }_{i}} \right) =\mu _{i}^{q}-v_{i}^{q}$ and accuracy $h\left( {{\alpha }_{i}} \right) =\mu _{i}^{q}+v_{i}^{q}$ values as follows:

• (i) If $s\left( {{\alpha }_{{{i}_{1}}}} \right) <s\left( {{\alpha }_{{{i}_{2}}}} \right)$, then ${{\alpha }_{{{i}_{1}}}}\prec {{\alpha }_{{{i}_{2}}}}$;

• (ii) If $s\left( {{\alpha }_{{{i}_{1}}}} \right) =s\left( {{\alpha }_{{{i}_{2}}}} \right)$ and $h\left( {{\alpha }_{{{i}_{1}}}} \right) <h\left( {{\alpha }_{{{i}_{2}}}} \right)$, then ${{\alpha }_{{{i}_{1}}}}\prec {{\alpha }_{{{i}_{2}}}}$;

• (iii) If $s\left( {{\alpha }_{{{i}_{1}}}} \right) =s\left( {{\alpha }_{{{i}_{2}}}} \right)$ and $h\left( {{\alpha }_{{{i}_{1}}}} \right) =h\left( {{\alpha }_{{{i}_{2}}}} \right)$, then ${{\alpha }_{{{i}_{1}}}}={{\alpha }_{{{i}_{2}}}}$.

According to the above method, rank the comprehensive evaluation values ${{\alpha }_{\sigma \left( 1 \right) }}\prec \cdots \prec {{\alpha }_{\sigma \left( n \right) }}$. Then, all the alternatives are ranked based on the increasing order of their related comprehensive evaluation values (Fig. 2).

Fig. 2

Flowchart of Algorithm 1

5 Numerical examples

In this section, the background in [62] is utilized to demonstrate the implementation process and effectiveness of the proposed decision approach in the previous section, which is an evaluation problem of SmartWatch design.

Fig. 3

Semantic difference method

Fig. 4

SmartWatch appearance design alternatives

A team of six experts and decision-makers is gathered to conduct the performance assessment and determine the most suitable alternative. These decision-makers include a product manager, a customer manager, a product designer, a senior user, an R&D manager, and a senior expert. The authors also assume the relative weights of experts are $w={{\left( 0.1,0.2,0.1,0.4,0.1,0.1 \right) }^{T}}$. The semantic difference method is used in the evaluation stage, as shown in Fig. 3. These decision-makers expressed their preferences on the rating of candidate design alternatives of SmartWatch appearance (Fig. 4) with respect to the four essential attributes (“Fashionability (${{C}_{1}}$) ”, “Science and Technology (${{C}_{2}}$)”, “Friendliness (${{C}_{3}}$),” and “Comfort (${{C}_{4}}$)”) by using linguistic variables according to their domain knowledge. Table 1 shows the linguistic variables used by the decision-makers and their corresponding IVQROFNs (solid at $q=2$). The obtained linguistic assessments of the 10 alternatives provided by the decision-makers are listed in Tables 2, 3 and 4.

Table 1 Linguistic terms and their corresponding IVQROFNs [63]

Table 2 Ratings of alternatives with respect to evaluation attribute

Table 3 Ratings of alternatives with respect to evaluation attribute

Table 4 Ratings of alternatives with respect to evaluation attribute

We then apply the developed method in this study to derive the order relation of ${{A}_{i}}\left( i=1,2,\ldots ,10 \right)$. The necessary steps of the method are provided as follows.

Stage 1. Obtain the q-rung orthopair fuzzy collective matrix.

• Stage 1.1. According to the linguistic variables and their corresponding IVQROFNs shown in Table 1 and the ratings of design alternatives with respect to attribute by the decision-makers shown in Table 2, 3 and 4, the IVQROFNs of the decision matrix assigned by the decision-makers are obtained through ${{\tilde{R}}_{k}}={{\left( \tilde{\alpha }_{ij}^{k} \right) }_{10\times 4}}\left( k\in \left\{ 1,\ldots ,6 \right\} \right)$. These fuzzy matrices are readily available in Tables 1, 2, 3 and 4. Hence, no specific data are provided in this study.

• Stage 1.2. Obtain a collective matrix ${{\tilde{R}}_{c}}$ (Table 5) based on Algorithm 2.

Table 5 Pythagorean fuzzy collective matrix ${{\tilde{R}}_{c}}$

Stage 2. Obtain the collective evaluation values and ranking orders of alternatives.

• Stage 2.1. Use the QROFWA operator to compute comprehensive values ${{\alpha }_{i}}\left( i=1,\ldots ,10 \right)$.

  $$\begin{aligned}&{{\alpha }_{1}}=\left( 0.7173,0.3597 \right) ,{{\alpha }_{2}}=\left( 0.5828,0.5531 \right) ,\nonumber \\&{{\alpha }_{3}}=\left( 0.6131,0.4918 \right) ,{{\alpha }_{4}}=\left( 0.6558,0.4216 \right) ,\nonumber \\&{{\alpha }_{5}}=\left( 0.6278,0.4704 \right) ,{{\alpha }_{6}}=\left( 0.6252,0.4817 \right) ,\nonumber \\&{{\alpha }_{7}}=\left( 0.5640,0.5145 \right) ,{{\alpha }_{8}}=\left( 0.5811,0.4896 \right) ,\nonumber \\&{{\alpha }_{9}}=\left( 0.6897,0.4099 \right) ,{{\alpha }_{10}}=\left( 0.5424,0.5627 \right) \end{aligned}$$

  (47)

• Stage 2.2. Rank the comprehensive evaluation values ${{\alpha }_{i}}\left( i=1,\ldots ,10 \right)$ as follows:

  $$\begin{aligned}&s\left( {{\alpha }_{1}} \right) =0.3850,s\left( {{{\tilde{\alpha }}}_{2}} \right) =0.0337,s\left( {{{\tilde{\alpha }}}_{3}} \right) =0.1340,\nonumber \\&s\left( {{{\tilde{\alpha }}}_{4}} \right) =0.2523,s\left( {{{\tilde{\alpha }}}_{5}} \right) =0.1729 \nonumber \\&s\left( {{{\tilde{\alpha }}}_{6}} \right) =0.1588,s\left( {{\alpha }_{7}} \right) =0.0534,s\left( {{{\tilde{\alpha }}}_{8}} \right) =0.0980,\nonumber \\&s\left( {{{\tilde{\alpha }}}_{9}} \right) =0.3077,s\left( {{{\tilde{\alpha }}}_{10}} \right) =0.2523 \end{aligned}$$

  (48)

  Obtain the rank of ${{A}_{i}}\left( i=1,\ldots ,10 \right)$:

  $$\begin{aligned} {{A}_{2}}\prec {{A}_{7}}\prec {{A}_{8}}\prec {{A}_{3}}\prec {{A}_{6}}\prec {{A}_{5}}\prec {{A}_{10}}\sim {{A}_{4}}\prec {{A}_{9}}\prec {{A}_{1}}. \end{aligned}$$

  (49)

Thus, according to the proposed method, the suggested design alternative is ${A}_{1}$.

The relationship between the parameters and consensus measure is analyzed according to Algorithm 2.

• According to Step 1, the consensus measure $G\text {-}C{{D}^{*}}=0.7849$ and parameter values $\left( \lambda _{\mu }^{*},\lambda _{v}^{*} \right) =\left( 1.0,0.5 \right)$ under the initial iteration state ($z=1$) can be obtained. Attribute consensus measure $G\text {-}CC_{j}^{\left( 1 \right) }$ and group CD $G\text {-}C{{D}^{\left( 1 \right) }}=0.7849$ obtained are lower than the threshold value $\theta =0.8$. The relevant results are presented in Table 6.

• Given that the consensus measure of the attribute ${{C}_{4}}$ is the lowest ($G\text {-}CC_{\text {4}}^{\left( \text {1} \right) }=0.7732$), the parameters of ${{C}_{4}}$ are first released with constraints $\lambda _{\mu \left( 1 \right) }^{i\text {4}k},\lambda _{v\left( 1 \right) }^{i\text {4}k}\in \left[ 0,1 \right]$. By solving the model M1, the new group consensus measure ($G\text {-}C{{D}^{\left( 2 \right) }}=0.7948$) and the corresponding consensus measure ($G\text {-}CC_{\text {4}}^{\left( \text {2} \right) }=0.\text {8130}$) of the attribute ${{C}_{4}}$ are obtained. The relevant results can be found in Table 6. The parameter values $\lambda _{\mu \left( 2 \right) }^{i\text {2}k}$ and $\lambda _{v\left( 2 \right) }^{i\text {2}k}$ are shown in Table 7.

• After the second iteration, the group consensus measure $G\text {-}C{{D}^{\left( 2 \right) }}=0.7948$ is lower than the threshold and the consensus measure of the attribute ${{C}_{\text {2}}}$ is the lowest. Hence, the parameter constraints of attribute ${{C}_{\text {2}}}$ are released ($\lambda _{\mu \left( \text {2} \right) }^{i\text {2}k},\lambda _{v\left( \text {2} \right) }^{i\text {2}k}\in \left[ 0,1 \right]$). By solving the model M1, the new group consensus measure ($G\text {-}C{{D}^{\left( \text {3} \right) }}=0.\text {8042}$) and the corresponding consensus measure ($G\text {-}CC_{\text {2}}^{\left( \text {3} \right) }=0.\text {8117}$) of the attribute ${{C}_{\text {2}}}$ are obtained. The relevant results can be seen in Table 6. The parameter values $\lambda _{\mu \left( \text {3} \right) }^{i\text {2}k}$ and $\lambda _{v\left( \text {3} \right) }^{i\text {2}k}$ are shown in Table 8, respectively. The new group consensus $G\text {-}C{{D}^{\left( \text {3} \right) }}=0.\text {8042}$ is greater than the threshold, and thus, the iteration stops, and the calculation results are considered the output.

Table 6 Consensus measure with the different iteration processes

Table 7 Membership/non-membership element parameters $\left( \lambda _{\mu }^{ijk},\lambda _{v}^{ijk}\right)$ with respect to attribute ${{C}_{4}}$

Table 8 Membership/non-membership element parameters $\left( \lambda _{\mu }^{ijk},\lambda _{v}^{ijk}\right)$ with respect to attribute ${{C}_{2}}$

6 Comparison analysis

6.1 Existing continuous interval-valued fuzzy aggregation operators

• C-IVIFOWA and C-IVPFOWQA operators

Definition 22

[48, 49] Let $\tilde{\alpha }=\left( \tilde{\mu },\tilde{v} \right) =\left( \left[ {{\mu }^{-}},{{\mu }^{+}} \right] ,\left[ {{v}^{-}},{{v}^{+}} \right] \right)$ be an IVIFN and the continuous interval-valued intuitionistic fuzzy ordered weighted averaging (C-IVIFOWA) operator $A\text {-}{{F}_{Q}}$ is defined as follows:

$$\begin{aligned} A\text {-}{{F}_{Q}}\left( {\tilde{\alpha }} \right) =\left( {{f}_{\lambda }}\left( {\tilde{\mu }} \right) ,{{f}_{\lambda }}\left( {\tilde{v}} \right) \right) =\left( \left( 1-\lambda \right) {{\mu }^{-}}+\lambda {{\mu }^{+}},\left( 1-\lambda \right) {{v}^{-}}+\lambda {{v}^{+}} \right) . \end{aligned}$$

(50)

However, some examples show the C-IVIFOWA operator fails in boundary accessibility and monotonicity with respect to the BUM function, and the relevant details, which are not elaborated in this study, can be found in references [48, 49]. The improved C-IVIFOWA operator $B\text {-}{{F}_{Q}}$ is proposed in [48, 49].

Definition 23

[48, 49] Let $\tilde{\alpha }=\left( \tilde{\mu },\tilde{v} \right) =\left( \left[ {{\mu }^{-}},{{\mu }^{+}} \right] ,\left[ {{v}^{-}},{{v}^{+}} \right] \right)$ be an IVIFN and the improved C-IVIFOWA operator $B\text {-}{{F}_{Q}}$ is defined as follows:

$$\begin{aligned} B\text {-}{{F}_{Q}}\left( {\tilde{\alpha }} \right) =\left( {{f}_{\lambda }}\left( {\tilde{\mu }} \right) ,{{f}_{1-\lambda }}\left( {\tilde{v}} \right) \right) =\left( \left( 1-\lambda \right) {{\mu }^{-}}+\lambda {{\mu }^{+}},\lambda {{v}^{-}}+\left( 1-\lambda \right) {{v}^{+}} \right) . \end{aligned}$$

(51)

For convenience, we let $B\text {-}{{F}_{Q}}\left( {\tilde{\alpha }} \right) =\left( \mu ,v \right)$. According to Definition 23, the relationship between membership degree $\mu$ and non-membership degree v can be obtained as follows:

$$\begin{aligned}&\left\{ \begin{aligned}&\mu \text {=}\left( 1-\lambda \right) {{\mu }^{-}}+\lambda {{\mu }^{+}} \\&v=\lambda {{v}^{-}}+\left( 1-\lambda \right) {{v}^{+}} \\ \end{aligned} \right. \nonumber \\&\quad \Rightarrow {{l}_{1}}:\left\{ \begin{aligned}&v=-\frac{{{v}^{+}}-{{v}^{-}}}{{{\mu }^{+}}-{{\mu }^{-}}}\times \mu +\frac{{{\mu }^{+}}{{v}^{+}}-{{\mu }^{-}}{{v}^{-}}}{{{\mu }^{+}}-{{\mu }^{-}}} \\&\mu \in \left[ {{\mu }^{-}},{{\mu }^{+}} \right] ,v\in \left[ {{v}^{-}},{{v}^{+}} \right] \\ \end{aligned} \right. \end{aligned}$$

(52)

The above results show all IFNs obtained by the operator $B\text {-}{{F}_{Q}}\left( {\tilde{\alpha }} \right)$ that form the line segment ${{l}_{1}}$ when all parameters in the unit intervals are taken $0\le \lambda \le 1$. The details are illustrated in Fig. 5.

Fig. 5

IVIFN $\tilde{\alpha }$ and its related $B\text {-}{{F}_{Q}}\left( {\tilde{\alpha }} \right)$

Fig. 6

IVPFN $\tilde{\beta }$ and its related ${{E}_{Q}}\left( {\tilde{\beta }} \right)$

Definition 24

[52] Let $\tilde{\beta }=\left( \tilde{\rho },\tilde{\sigma } \right) =\left( \left[ {{\rho }^{-}},{{\rho }^{+}} \right] ,\left[ {{\sigma }^{-}},{{\sigma }^{+}} \right] \right)$ be an IVPFN and the continuous interval-valued Pythagorean fuzzy ordered weighted averaging quadratic (C-IVIFOWQA) operator is defined as follows:

$$\begin{aligned} {{E}_{Q}}\left( {\tilde{\beta }} \right)&=\left( {{g}_{\lambda }}\left( {\tilde{\rho }} \right) ,{{g}_{1-\lambda }}\left( {\tilde{\sigma }} \right) \right) \nonumber \\&=\left( \sqrt{\left( 1-\lambda \right) {{\left( {{\rho }^{-}} \right) }^{2}}+\lambda {{\left( {{\rho }^{+}} \right) }^{2}}},\sqrt{\lambda {{\left( {{\sigma }^{-}} \right) }^{2}}+\left( 1-\lambda \right) {{\left( {{\sigma }^{+}} \right) }^{2}}} \right) . \end{aligned}$$

(53)

For convenience, we let ${{E}_{Q}}\left( {\tilde{\beta }} \right) =\left( \rho ,\sigma \right)$. According to Definition 24, the relationship between membership degree $\rho$ and non-membership degree $\sigma$ can be obtained as follows:

$$\begin{aligned}&\left\{ \begin{aligned}&\rho =\sqrt{\left( 1-\lambda \right) {{\left( {{\rho }^{-}} \right) }^{2}}+\lambda {{\left( {{\rho }^{+}} \right) }^{2}}} \\&\sigma =\sqrt{\lambda {{\left( {{\sigma }^{-}} \right) }^{2}}+\left( 1-\lambda \right) {{\left( {{\sigma }^{+}} \right) }^{2}}} \\ \end{aligned} \right. \nonumber \\&\quad \Rightarrow {{l}_{2}}:\left\{ \begin{aligned}&y=\sqrt{-\frac{{{\left( {{\sigma }^{+}} \right) }^{2}}-{{\left( {{\sigma }^{-}} \right) }^{2}}}{{{\left( {{\rho }^{+}} \right) }^{2}}-{{\left( {{\rho }^{-}} \right) }^{2}}}{{\rho }^{2}}+\frac{{{\left( {{\rho }^{+}} \right) }^{2}}{{\left( {{\sigma }^{+}} \right) }^{2}}-{{\left( {{\rho }^{-}} \right) }^{2}}{{\left( {{\sigma }^{-}} \right) }^{2}}}{{{\left( {{\rho }^{+}} \right) }^{2}}-{{\left( {{\rho }^{-}} \right) }^{2}}}} \\&\rho \in \left[ {{\rho }^{-}},{{\rho }^{+}} \right] ,\sigma \in \left[ {{\sigma }^{-}},{{\sigma }^{+}} \right] \\ \end{aligned} \right. \end{aligned}$$

(54)

The above results show that all PFNs obtained by the operator ${{E}_{Q}}\left( {\tilde{\beta }} \right)$ form the curve ${{l}_{2}}$ when all the parameters in the unit intervals are taken $0\le \lambda \le 1$. The details are demonstrated in Fig. 6.

According to Figs. 5 and 6, we can summarize the shortcomings of the C-IVIFOWA and C-IVPFOWQA operators as follows:

• From Fig. 5, for all $\alpha \in {{l}_{1}}$, a parameter value $\lambda \in \left[ 0,1 \right]$ always satisfy $B\text {-}{{F}_{Q}}\left( {\tilde{\alpha }} \right) =\alpha$; for all ${{\alpha }^{*}}\notin {{l}_{1}}$ and ${{\alpha }^{*}}\in \tilde{\alpha }$, no parameter value $\lambda \in \left[ 0,1 \right]$ can satisfy $B\text {-}{{F}_{Q}}\left( {\tilde{\alpha }} \right) ={{\alpha }^{*}}$. However, all IFNs on all line ${{l}_{1}}$ ($\alpha \in {{l}_{1}}$) account for only a very small part of the region of IVIFN $\tilde{\alpha }$ (Rectangular ABCD).

• From Fig. 6, for all $\beta \in {{l}_{2}}$, a parameter value $\lambda \in \left[ 0,1 \right]$ always satisfy ${{E}_{Q}}\left( {\tilde{\beta }} \right) =\beta$; for all ${{\beta }^{*}}\notin {{l}_{2}}$ and ${{\beta }^{*}}\in \tilde{\beta }$, no parameter value $\lambda \in \left[ 0,1 \right]$ can satisfy ${{E}_{Q}}\left( {\tilde{\beta }} \right) ={{\beta }^{*}}$. However, all PFNs on all line ${{l}_{2}}$ ($\beta \in {{l}_{2}}$) account for only a very small part of the region of IVPFN $\tilde{\beta }$ (Rectangular ABCD).

Fig. 7

IVQROFN $\tilde{\alpha }$ and its related ${{G}_{{{\lambda }_{1}},{{\lambda }_{2}}}}\left( {\tilde{\alpha }} \right)$

Then, the continuous interval-valued q-rung orthopair fuzzy ordered weighted averaging (C-IVQROFOWA) operator constructed in this study is analyzed further. Let $\tilde{\alpha }=\left( \left[ {{\mu }^{-}},{{\mu }^{+}} \right] ,\left[ {{v}^{-}},{{v}^{+}} \right] \right)$ be an IVQROFN solid at q. By using Definition 15, we have

$$\begin{aligned} {{G}_{{{\lambda }_{1}},{{\lambda }_{2}}}}\left( {\tilde{\alpha }} \right)&=\left( {{\phi }_{{{\lambda }_{1}}}}\left( \left[ {{\mu }^{-}},{{\mu }^{+}} \right] \right) ,\phi _{{{\lambda }_{2}}}^{d}\left( \left[ {{v}^{-}},{{v}^{+}} \right] \right) \right) \nonumber \\&=\left( {{\left( \left( 1-{{\lambda }_{1}} \right) {{\left( {{\mu }^{-}} \right) }^{q}}+{{\lambda }_{1}}{{\left( {{\mu }^{+}} \right) }^{q}} \right) }^{{1}/{q}}},\right. \nonumber \\&\quad \left. {{\left( {{\lambda }_{2}}{{\left( {{v}^{-}} \right) }^{q}}+\left( 1-{{\lambda }_{2}} \right) {{\left( {{v}^{+}} \right) }^{q}} \right) }^{{1}/{q}\;}} \right) . \end{aligned}$$

(55)

First, the relationship between parameters (${{\lambda }_{1}},{{\lambda }_{2}}\in \left[ 0,1 \right]$) and the C-IVQROFOWA operator is discussed in Theorem 18. Figure 7 illustrates the results of Theorem 18.

Theorem 18

Let $\tilde{\alpha }=\left( \left[ {{\mu }^{-}},{{\mu }^{+}} \right] ,\left[ {{v}^{-}},{{v}^{+}} \right] \right)$ be an IVQROFN solid at q, ${{\lambda }_{1}},{{\lambda }_{2}}\in \left[ 0,1 \right]$, then

(i) $\forall \alpha \text {=}\left( \mu ,v \right) \in \tilde{\alpha },\exists |{{\lambda }_{1}},{{\lambda }_{2}}\in \left[ 0,1 \right]$, $s.t.\left\{ \begin{aligned}&\mu ={{\left( \left( 1-{{\lambda }_{1}} \right) {{\left( {{\mu }^{-}} \right) }^{q}}+{{\lambda }_{1}}{{\left( {{\mu }^{+}} \right) }^{q}} \right) }^{{1}/{q}\;}}, \\&v={{\left( {{\lambda }_{2}}{{\left( {{v}^{-}} \right) }^{q}}+\left( 1-{{\lambda }_{2}} \right) {{\left( {{v}^{+}} \right) }^{q}} \right) }^{{1}/{q}\;}} \\ \end{aligned} \right.$

(ii) $\forall {{\lambda }_{1}},{{\lambda }_{2}}\in \left[ 0,1 \right] ,\exists |\alpha \text {=}\left( \mu ,v \right) \in \tilde{\alpha }$, $s.t.\left\{ \begin{aligned}&{{\lambda }_{1}}={\left( {{\mu }^{q}}-{{\left( {{\mu }^{-}} \right) }^{q}} \right) }/{\left( {{\left( {{\mu }^{+}} \right) }^{q}}-{{\left( {{\mu }^{-}} \right) }^{q}} \right) }\; \\&{{\lambda }_{2}}={\left( {{\left( {{v}^{+}} \right) }^{q}}-{{v}^{q}} \right) }/{\left( {{\left( {{v}^{+}} \right) }^{q}}-{{\left( {{v}^{-}} \right) }^{q}} \right) }\; \\ \end{aligned} \right.$.

Proof

According to Definition 11, ${{G}_{{{\lambda }_{1}},{{\lambda }_{2}}}}\left( {\tilde{\alpha }} \right) =\left( {{\phi }_{{{\lambda }_{1}}}}\left( \left[ {{\mu }^{-}},{{\mu }^{+}} \right] \right) ,\phi _{{{\lambda }_{2}}}^{d}\left( \left[ {{v}^{-}},{{v}^{+}} \right] \right) \right)$, where

$$\begin{aligned}&{{\mu }^{-}}\le {{\phi }_{{{\lambda }_{1}}}}\left( \left[ {{\mu }^{-}},{{\mu }^{+}} \right] \right) =\left( {{\lambda }_{1}}{{\left( {{\mu }^{+}} \right) }^{q}}\right. \\&\quad \left. +\left( 1-{{\lambda }_{1}} \right) {{\left( {{\mu }^{-}} \right) }^{q}} \right) ^{{1}/{q}\;}\le {{\mu }^{+}} \end{aligned}$$

and

$$\begin{aligned}&{{v}^{-}}\le \phi _{{{\lambda }_{2}}}^{d}\left( \left[ {{v}^{-}},{{v}^{+}} \right] \right) \\&\quad ={{\left( \left( 1-{{\lambda }_{2}} \right) {{\left( {{v}^{+}} \right) }^{q}}+{{\lambda }_{2}}{{\left( {{v}^{-}} \right) }^{q}} \right) }^{{1}/{q}\;}}\le {{v}^{+}}. \end{aligned}$$

(i) For any $\alpha \text {=}\left( \mu ,v \right) \in \tilde{\alpha }$, given that $\mu \in \left[ {{\mu }^{-}},{{\mu }^{+}} \right]$ and $v\in \left[ {{v}^{-}},{{v}^{+}} \right]$, we can find a two-tuple $\left( {{\lambda }_{1}},{{\lambda }_{2}} \right)$that satisfies

$$\begin{aligned} \mu ={{\phi }_{{{\lambda }_{1}}}}\left( \left[ {{\mu }^{-}},{{\mu }^{+}} \right] \right) ,v=\phi _{{{\lambda }_{2}}}^{d}\left( \left[ {{v}^{-}},{{v}^{+}} \right] \right) . \end{aligned}$$

Assume that another two-tuple $\left( {{{\bar{\lambda }}}_{1}},{{{\bar{\lambda }}}_{2}} \right)$ exists to satisfy

$$\begin{aligned} \mu ={{\phi }_{{{{\bar{\lambda }}}_{1}}}}\left( \left[ {{\mu }^{-}},{{\mu }^{+}} \right] \right) ,v=\phi _{{{{\bar{\lambda }}}_{2}}}^{d}\left( \left[ {{v}^{-}},{{v}^{+}} \right] \right) . \end{aligned}$$

Then,

$$\begin{aligned} \left\{ \begin{aligned}&{{\lambda }_{1}}{{\left( {{\mu }^{+}} \right) }^{q}}+\left( 1-{{\lambda }_{1}} \right) {{\left( {{\mu }^{-}} \right) }^{q}}={{{\bar{\lambda }}}_{1}}{{\left( {{\mu }^{+}} \right) }^{q}}+\left( 1-{{{\bar{\lambda }}}_{1}} \right) {{\left( {{\mu }^{-}} \right) }^{q}}, \\&\left( 1-{{\lambda }_{2}} \right) {{\left( {{v}^{+}} \right) }^{q}}+{{\lambda }_{2}}{{\left( {{v}^{-}} \right) }^{q}}=\left( 1-{{{\bar{\lambda }}}_{2}} \right) {{\left( {{v}^{+}} \right) }^{q}}+{{{\bar{\lambda }}}_{2}}{{\left( {{v}^{-}} \right) }^{q}}. \\ \end{aligned} \right. \end{aligned}$$

Therefore, we have ${{\lambda }_{1}}={{\bar{\lambda }}_{1}},{{\lambda }_{2}}=\bar{\lambda _2 }$. $\square$

The above theorem shows that for any QROFN $\alpha$ in the IVQROFN $\tilde{\alpha }$ ($\alpha \in \tilde{\alpha }$), it can be obtained by selecting the appropriate parameter values (${{\lambda }_{1}},{{\lambda }_{2}}\in \left[ 0,1 \right]$) and using the operator ${{G}_{{{\lambda }_{1}},{{\lambda }_{2}}}}\left( {\tilde{\alpha }} \right)$. Moreover, a one-to-one correspondence exists between this QROFN $\alpha$ and parameters ${{\lambda }_{1}}$ and ${{\lambda }_{2}}$.

The above analysis shows the proposed C-IVQROFOWA operator can overcome the shortcomings of the existing operators C-IVIFOWA and C-IVPFOWQA and the decision-makers can acquire the corresponding fuzzy numbers by combining the C-IVQROFOWA operator with its own attitude characteristics. The operator has strong flexibility.

• WC-IVPFOWQA and WC-IVPFOWQA operators

Definition 25

[48, 49] Let ${{\tilde{\alpha }}_{i}}=\left( {{{\tilde{\mu }}}_{i}},{{{\tilde{v}}}_{i}} \right) =\left( \left[ \mu _{i}^{-},\mu _{i}^{+} \right] ,\left[ v_{i}^{-},v_{i}^{+} \right] \right) \left( i=1,2,\ldots ,n \right)$ be a collection of IVIFNs and the weighted C-IVIFOWA (WC-IVIFOWA) operator is defined as follows:

$$\begin{aligned}&\mathsf{{WC-IVIFOWA}}\left( {{{\tilde{\alpha }}}_{1}},\ldots ,{{{\tilde{\alpha }}}_{n}} \right) \nonumber \\&\quad =\left( 1-\prod \limits _{i=1}^{n}{{{\left( 1-{{f}_{\lambda }}\left( {{{\tilde{\mu }}}_{i}} \right) \right) }^{{{w}_{i}}}},\prod \limits _{i=1}^{n}{\left( {{f}_{1-\lambda }}{{\left( {{{\tilde{v}}}_{i}} \right) }^{{{w}_{i}}}} \right) }} \right) . \end{aligned}$$

(56)

Definition 26

[52] Let ${{\tilde{\beta }}_{i}}=\left( {{{\tilde{\rho }}}_{i}},{{{\tilde{\sigma }}}_{i}} \right) =\left( \left[ \rho _{i}^{-},\rho _{i}^{+} \right] ,\left[ \sigma _{i}^{-},\sigma _{i}^{+} \right] \right) \left( i=1,2,\ldots ,n \right)$ be a collection of IVPFNs and the weighted C-IVPFOWQA (WC-IVPFOWQA) operator is defined as follows:

$$\begin{aligned}&\mathsf{{WC-IVPFOWQA}}\left( {{{\tilde{\beta }}}_{1}},\ldots ,{{{\tilde{\beta }}}_{n}} \right) \nonumber \\&\quad =\left( \sqrt{1-\prod \limits _{i=1}^{n}{{{\left( 1-{{\left( {{g}_{\lambda }}\left( {{{\tilde{\rho }}}_{i}} \right) \right) }^{2}} \right) }^{{{w}_{i}}}}}},\prod \limits _{i=1}^{n}{\left( {{g}_{1-\lambda }}{{\left( {{{\tilde{\sigma }}}_{i}} \right) }^{{{w}_{i}}}} \right) } \right) . \end{aligned}$$

(57)

The comparative analysis of the above two types of operators (WC-IVIFOWA and WC-IVPFOWQA) and the WC-IVPFOWQA operator is carried out using a simple example. Three types of operators are used to aggregate a collection of IVIFNs and a collection of IVPFNs with the assumption that the parameters and weights have been given. The details are shown in Table 9.

Table 9 Comparative analysis of the three aggregation operators with given parameters

According to Table 9, we have

$$\begin{aligned}&\mathsf{{WC-IVIFOWA}}\left( {{{\tilde{\alpha }}}_{1}},\ldots ,{{{\tilde{\alpha }}}_{n}} \right) \\&\quad =\left( 1.0,0.0 \right) ,WC\text {-}IVPFOWQA\left( {{{\tilde{\alpha }}}_{1}},\ldots ,{{{\tilde{\alpha }}}_{n}} \right) =\left( 1.0,0.0 \right) . \end{aligned}$$

Although ${{w}_{2}}+{{w}_{3}}+{{w}_{4}}=0.09$ and ${{w}_{1}}=0.01$, the aggregation results of operators WC-IVIFOWA and WC-IVPFOWQA are both $\left( 1.0,0.0 \right)$. These results indicate IVIFNs (${{\tilde{\alpha }}_{i}}\left( i=2,3,4 \right)$) are ignored completely in the aggregation process. This phenomenon is clearly unreasonable. Similarly, we have

$$\begin{aligned} \mathsf{{WC-IVPFOWQA}}\left( {{{\tilde{\beta }}}_{1}},\ldots ,{{{\tilde{\beta }}}_{n}} \right) =\left( 1.0,0.0 \right) . \end{aligned}$$

This aggregation result is also unreasonable.

However, the results obtained via the WC-IVQROFOWA are reasonable.

$$\begin{aligned}&\mathsf{{WC-IVQROFOWA}}\left( {{{\tilde{\alpha }}}_{1}},\ldots ,{{{\tilde{\alpha }}}_{n}} \right) =\left( 0.1480,0.7535 \right) \\&\mathsf{{WC-IVQROFOWA}}\left( {{{\tilde{\beta }}}_{1}},\ldots ,{{{\tilde{\beta }}}_{n}} \right) =\left( 0.3782,0.7634 \right) \end{aligned}$$

The above analysis shows the proposed WC-IVQROFOWA operator can overcome the shortcomings of WC-IVIFOWA and WC-IVPFOWQA operators, but the WC-IVQROFOWA operator can obtain more reasonable aggregation results in solving GDM problems. The construction ideas of the WC-IVIFOWA [47,48,49], WC-IVPFOWQA [52], and WC-IVQROFOWA operators are summarized and the details presented in Fig. 8.

Fig. 8

Construction ideas of the continuous fuzzy operators

The advantages and advancement of the developed approach compared with existing work are summarized as below:

• The proposed fuzzy MAGDM method develops the methodology in the use of the IVROFS and the aggregation operators. This advances its degenerate used in [30] in which the ROFS has been implemented as the ROFS is a particular case of IVROFS.

• The C-IVIFOWA and C-IVPFOWQA operators proposed in the previous studies [47,48,49, 52] lay a solid foundation for this study. The developed C-IVQROFOWA operator not only covers the existing two types of operators, but also realizes the independence of attitude parameters(${{\lambda }_{\mu }}$ and ${{\lambda }_{v}}$), which overcomes the shortcomings of the two types of operators as shown in Figs. 5 and 6 because of the same parameter values ${{\lambda }_{\mu }}= {{\lambda }_{v}}$.

• The comparative analysis in Table 9 shows that the proposed WC-IVQROFOWA operator can overcome the shortcomings of the WC-IVIFOWA [48, 49] and WC-IVPFOWQA [52] operators, and these operators are two special cases of the WC-IVQRFOWA operator with $q=1$ and $q=2$, respectively. Moreover, the IVQROFWA and Maclaurin symmetric mean operators developed by [35] and [36] aggregate directly the endpoints of the interval, which makes decision makers unable to obtain most of the information in the interval according to their attitude preferences. The proposed WC-IVQROFOWA operator and its counterpart can avoid successfully this deficiency as the information contained in the interval will be adequately taken into account.

• Another advantage of the proposed method upon comparison with the existing methods is that it derives the consensus measure to develop the optimization model of operator parameters and, based on which, its iteration algorithm are constructed. The consensus measures can guide the change direction of the parameter constraints to obtain the parameter values that satisfy the threshold conditions.

7 Conclusions

In order to adapt to the rapid change of the complex MAGDM environment, the traditional fuzzy MAGDM methods have been expanded to interval-valued intuitionistic and Pythagorean fuzzy MAGDM methods. The emergence of IVQROFS greatly promotes the formation of generalized MAGDM paradigm and provides decision-makers with more flexible and broad application ideas.

This study proposes novel fusion strategies for continuous IVQROF group information and multiple attribute information, and it integrates the attitudinal characteristics of decision makers and considers the degree of consensus among individuals in the group in a bid to enhance the reliability of decision-making results. The work of this paper is mainly reflected in the following aspects. Firstly, this study revealed the shortcomings of existing operators in dealing with decision-making problems by clarifying the advantages and construction ideas of existing operators, and it proposed the WC-IVQROFOWA operator with a wider range of information processing and more flexible attitude preference, so as to consolidate the decision-making framework under the IVQROF environment. Secondly, by integrating the consensus measure between different individuals in the context of GDM, the dynamic adjustment mechanism of operator parameters is constructed, which enhances the adjustability of GDM process and improves the credibility of decision-making results; last but not least, the GDM method proposed in this study has been applied in quality assessment of SmartWatch appearance design with linguistic inputs, which shows the independence of the parameters in the WC-IVQROFOWA operator and the positive effect of the group consensus measurement on the decision-making process.

In the case study, the original evaluation information of the alternatives is obtained by semantic difference method from experts, which constitutes the linguistic evaluation matrix, and is quantified by IVQROFNs solid at $q=2$. As regards to our future research, the applications of the developed models in this study can be used to other types of fuzzy models, for instance, the interval linguistic labels [64] and the basic uncertain information soft sets [65].