1 Introduction
Fuzzy sets (FSs) (Zadeh, 1965) were initiated primarly due to the consideration of ambiguous human evaluations when dealing with realistic problems. Alternatively, the FSs doctrine can manage the reality beyond computational observation and comprehension, that is ambiguity, partial belongingness, inaccuracy and sharpless limits (Zhan et al., 2022). Later on, intuitionistic FSs (Atanassov, 1986) were introduced as a generalization of FSs to tackle problems that have deficient data. Since IFSs were proposed, many scholars have conducted in-depth studies (Chen and Chang, 2016; Chen et al., 2016; Garg, 2017b, 2019; Kumar and Chen, 2021; Mishra et al., 2019, 2020; Zeng et al., 2019; Zou et al., 2020). Yet, the relevance of intuitionistic FS is restricted because of given limitation, which is the sum of belongingness grade (BG) μ and non-belongingness grade (NBG) ν cannot surpass one, that is $\mu +\nu \leqslant 1$. However, it was later discovered that the aforesaid limitation is not satisfied based on expert preferences for complicated decision-making difficulties. For example, if an expert favours BG 0.7 and NBG 0.5 during the usage of IFSs, at that point, obviously, their sum surpasses 1. To overcome this sort of circumstance, Yager (2013a; 2013b) pioneered the notion of Pythagorean FSs with BG μ and NBG ν complying with the condition ${\mu ^{2}}+{\nu ^{2}}\leqslant 1$. As a result, Pythagorean FSs are preferred over intuitionistic FSs for expressing ambiguous data. Numerous studies have been conducted based on Pythagorean FSs. Yager and Abbasov (2013) developed a decision-making (DM) approach with Pythagorean fuzzy (PF) sets. An extension of TOPSIS tool under PF sets setting was realized by Zhang and Xu (2014) to resolve DM issues. Ma and Xu (2016) presented the symmetric PF weighted aggregation operators (AOs). Garg (2016a) defined some new generalized PF information AOs and applied them to DM problems. Garg (2016b) also presented a DM process based on correlation coefficients of PF sets. A confidence level-based methodology was presented by Garg (2017a) with PF information. Peng et al. (2017) built up a series of PF information measures and applied them in DM. Garg (2018) developed generalized geometric interactive AOs based on PF sets and Einstein operations. Mardani et al. (2018) extensively reviewed decision making methods based on fuzzy aggregation operators. Nguyen et al. (2019) introduced PF exponential similarity measures to tackle pattern recognition problems. Nie et al. (2019) provided a DM strategy with PFSs using Shapley fuzzy measures and the partitioned normalized weighted Bonferroni mean operator. Jana et al. (2019a) used PF Dombi operators to tackle MADM problems. Rani et al. (2019) presented a VIKOR approach with entropy and divergence measures of PFSs. To assess waste treatment technologies, Rani et al. (2020) again proposed a new DM framework. Ejegwa (2021) proposed a generalized tri-parametric correlation coefficient for PF sets.
The q-rung orthopair fuzzy (qROF) set, introduced by Yager (2017), reserves the constraint that the sum of qth power of the BG and the NBG must be the value in $[0,1]$, i.e. $0\leqslant {\mu ^{q}}+{\nu ^{q}}\leqslant 1$. Clearly, qROF sets are extended versions of intuitionistic FSs (for $q=1$) and PF sets (for $q=2$). For the last couple of years, information aggregation has been a popular topic due to its significance and close connection to the issues of multi-criteria group decision-making (MCGDM) under the qROF setting. The qROF weighted averaging and geometric (qROFWA and qROFWG, respectively) were announced by Liu and Wang (2018). Based on the ideas of certainty and possibility, Yager and Alajlan (2017) recommended approximate reasoning on qROFSs. Peng et al. (2018) proposed exponential operators and acquired satisfactory outputs after using them in the assessment of the teaching management system. In a qROFSs context, Yager et al. (2018) managed strong coordination between probability, certainty, believability, and faith. Liu and Liu (2018) proposed qROF weighted BM operators and used them for MCGDM problems. Wei et al. (2018) introduced qROF weighted Heronian mean AOs in DM issues. Jana et al. (2019b) proposed qROF Dombi weighted averaging and geometric operators for aggregating criteria values. Mi et al. (2019) settled a multi-criteria DM (MCDM) issue utilizing a qROFS-VIKOR strategy. Xing et al. (2019a) proposed a new group of weighted AOs to amassed qROF data which takes part in the rearrangement of BG and NBG in qROFNs as per different principles. Qin et al. (2019a) presented Archimedean Muirhead mean (MM) AOs of qROFNs and furthermore indicated its conceivable application in settling MCGDM problem. In view of association operations and dual Hamy mean (HM) operation, Xing et al. (2019b) introduced qROF interaction dual HM AO to solve a MCGDM problem. Qin et al. (2019b) built up the Archimedean power partitioned MM of qROFNs to tackle the MCGDM strategy. Zhong et al. (2019) introduced qROF Dombi power partitioned weighted Heronian mean (HEM) AO to decrease the negative impact of some criteria degrees during the aggregation process. Darko and Liang (2020) built up some qROF Hamacher AOs to extend EDAS technique for solving MCDM concern. Yang and Pang (2020) developed qROF Bonferroni mean (BM) Dombi operators for a site selection problem. Yang et al. (2020) developed an online shopping structure for utilizing the qROF interaction weighted HEM operator. Joshi and Gegov (2020) conveyed the commonality level of DEs with considered elements for starting appraisals on qROF setting and suggested some AOs to combine the required information. Liu and Wang (2020) introduced qROF generalized MSM operator (qROFGMSM) and qROF generalized geometric MSM operator (qROFGGMSM), which might access BGs and NBGs in the range $[0,1]$, respectively, and admit different criteria. Using qROF-MULTIMOORA methodology and qROF Dombi-Prioritized weighted AOs, Aydemir and Gunduz (2020) solved a MCDM problem. Garg and Chen (2020) presented qROF weighted neutrality operators by using the notion of proportional distribution procedures of the BGs and NBGs. Liu et al. (2022a) developed group decision-making tool using linguistic qROF generalized point weighted AOs.
1.1 Research Motivation
The interrelationship between multiple criteria can be seen in different realistic situations. Many of the existing studies (Liu and Wang, 2018; Jana et al., 2019b; Liu and Wang, 2019; Garg and Chen, 2020) cannot tackle this situation. Although few operators (Liu and Liu, 2018; Yang and Pang, 2020) have been developed earlier, none of them is capable of handling this situation as they consider dependency between two criteria only. Although the Archimedean Muir-head mean operator (Qin et al., 2019a) and generalized MSM operator (Liu and Wang, 2018) can meet this requirement, they fail to eliminate the impact of extreme evaluating criteria values from some biased experts with diverse preference attitudes. To address such circumstances, Liu et al. (2020) proposed qROF power MSM operator. The method of Liu et al. (2020) has the constraint that it fails to distinguish the priority orders of alternatives in certain cases, specifically when among the qROF numbers considered one qROF number has a non-belongingness grade that equals to 0 (or a belongingness grade that equals to 1). Thus, it is essential to develop a novel MCGDM approach to overcome the limitation of the existing method (Liu et al., 2020) and the existing power weighted MSM operator (Liu et al., 2020).
1.2 Contributions
To overcome the shortcomings of Liu et al.’s (2020) method, in this paper, the followings have been incorporated:
The rest of the paper is arranged as given below:
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1. Some new operational laws are presented in order to fair treatment of belongingness and non-belongingness grades.
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2. Four new operators, namely qROF improved power weighted averaging and geometric (qROFIPWA and qROFIPWG, resp.) operators, qROF improved power weighted averaging and geometric MSM (qROFIPWAMSM and qROFIPWGMSM, resp.) operators are developed.
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3. A novel DM approach is developed based on the proposed operators. This proposed approach can resolve the limitations of Liu et al. (2020).
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4. To show the efficiency of the proposed methodology, a personnel selection problem is considered under qROF setting.
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5. A detailed comparative investigation is demonstrated to validate the superiority of the proposed model.
Some essential concepts related to qROF sets are briefly discussed in Section 2. Section 3 presents some new operations between qROF numbers. This section also puts forward the qROFPWA operator, qROFIPWG operator, qROFIPWAMSM operator and qROFIPWGMSM operator along with their characteristics. In Section 4, a decision-making methodology using the developed operators is provided. A case study of personnel selection problem is demonstrated in Section 5 to show the applicability of the developed approach. The solution of the case study, effect of the parameter and comparative study are also demonstrated in this section. Section 6 concludes the paper along with future research directions.
2 Preliminaries
2.1 q-Rung Orthopair Fuzzy Sets (qROFSs)
Some important concepts on qROFNs, basic operations between qROFNs and qROF weighted neutral AOs are highlighted as follows:
Definition 1 (Yager, 2017).
Let U be the discourse set. Then a $\text{qROFS}$ Θ on U is given by
where $\Delta (t)$ and $\nabla (t)$ represent the BG and NBG, respectively, of $t\in U$ with the constraint $0\leqslant \Delta (t)$, $\nabla (t)\leqslant 1$ and $0\leqslant {(\Delta (t))^{q}}+{(\nabla (t))^{q}}\leqslant 1$, $(q\geqslant 1)$.
Next, the hesitancy grade of $t\in U$ in Θ is given by $\pi (t)=\sqrt[q]{1-{(\Delta (t))^{q}}-{(\nabla (t))^{q}}}$. Obviously, $0\leqslant \pi (t)\leqslant 1$. Also, Yager (2017) called the pair $\langle \Delta (t),\nabla (t)\rangle $ a $\text{qROFN}$. For easiness, the symbol $\Theta =\langle \Delta ,\nabla \rangle $ is used to signify a $\text{qROFN}$. Suppose ${\Sigma ^{U}}$ denotes the collection of all qROFNs over U.
Definition 2 (Liu and Wang, 2018).
Clearly, $-1\leqslant {S_{c}}(\Theta )\leqslant 1$. It should be mentioned that the score value cannot be effectively utilized to separate numerous qROFNs for the situation when score values become identical. As a result, when comparing qROFNs, it is not recommended to rely solely on their score values. To manage such an issue, Liu and Wang (2018) proposed the idea of accuracy value of a qROFN.
Definition 3 (Liu and Wang, 2018).
According to the score function and accuracy function, a comparison scheme of qROFNs is given as follows:
Definition 4 (Liu and Wang, 2018).
Let ${\Theta _{1}}=\langle {\Delta _{1}},{\nabla _{1}}\rangle $ and ${\Theta _{2}}=\langle {\Delta _{2}},{\nabla _{2}}\rangle $ be two qROFNs. Then
Definition 5 (Liu and Wang, 2018).
Let ${\Theta _{1}}=\langle {\Delta _{1}},{\nabla _{1}}\rangle $ and ${\Theta _{2}}=\langle {\Delta _{2}},{\nabla _{2}}\rangle $ be two qROFNs and $\lambda >0$. Then the basic operations are defined by
(3)
\[\begin{aligned}{}(\mathrm{i})& \hspace{2.5pt}{\Theta _{1}}\otimes {\Theta _{2}}=\big\langle \sqrt[q]{1-\big(1-{\Delta _{1}^{q}}\big)\big(1-{\Delta _{2}^{q}}\big)},{\nabla _{1}}{\nabla _{2}}\big\rangle ,\end{aligned}\](4)
\[\begin{aligned}{}(\mathrm{ii})& \hspace{2.5pt}{\Theta _{1}}\otimes {\Theta _{2}}=\big\langle {\Delta _{1}}{\Delta _{2}},\sqrt[q]{\big(1-\big(1-{\nabla _{1}^{q}}\big)\big(1-{\nabla _{2}^{q}}\big)\big)}\big\rangle ,\end{aligned}\]Definition 6.
(Liu et al., 2020). Let ${\Theta _{1}}=\langle {\Delta _{1}},{\nabla _{1}}\rangle $ and ${\Theta _{2}}=\langle {\Delta _{2}},{\nabla _{2}}\rangle $ be two qROFNs. Then the normalized Hamming distance between them is expressed as:
2.2 Power Averaging Operator (PAO)
The PAO, discovered by Yager (2001), can relegate weights to the aggregated elements’ values by means of processing the degree of support among the elements. The conventional definition of PAO is given by:
Definition 7 (Yager, 2001).
Let ${b_{1}},{b_{2}},\dots ,{b_{n}}$ be a collection of crisp values. Then the power averaging operator (PAO) of these numbers is defined as follows:
where $\psi ({b_{i}})={\textstyle\sum _{j=1,j\ne i}^{n}}\textit{Supp}({b_{i}},{b_{j}})$.
(8)
\[ PA({b_{1}},{b_{2}},\dots ,{b_{n}})=\frac{{\textstyle\textstyle\sum _{i=1}^{n}}(1+\psi ({b_{i}})){b_{i}}}{{\textstyle\textstyle\sum _{i=1}^{n}}(1+\psi ({b_{i}}))},\]Here, $\textit{Supp}({b_{i}},{b_{j}})$ denotes the support of ${b_{i}}$ from ${b_{j}}$ and has the three axioms as
3 qROF Improved Power Weighted Operators
3.1 New Operations Between qROFNs
A few new operations are introduced between qROFNs and the basic laws are investigated.
Definition 8.
Let ${\Theta _{1}}=\langle {\Delta _{1}},{\nabla _{1}}\rangle $ and ${\Theta _{2}}=\langle {\Delta _{2}},{\nabla _{2}}\rangle $ be two qROFNs and $\lambda >0$. Then we define:
(9)
\[\begin{aligned}{}(\mathrm{i})& \hspace{2.5pt}{\Theta _{1}}\tilde{\oplus }{\Theta _{2}}=\left\langle \sqrt[q]{1-{\prod \limits_{r=1}^{2}}\big(1-{\Delta _{r}^{q}}\big)},\sqrt[q]{{\prod \limits_{r=1}^{2}}\big(1-{\Delta _{r}^{q}}\big)-{\prod \limits_{r=1}^{2}}\big(1-{\Delta _{r}^{q}}-{\nabla _{r}^{q}}\big)}\right\rangle ;\end{aligned}\](10)
\[\begin{aligned}{}(\mathrm{ii})& \hspace{2.5pt}{\Theta _{1}}\tilde{\otimes }{\Theta _{2}}=\left\langle \sqrt[q]{{\prod \limits_{r=1}^{2}}\big(1-{\nabla _{r}^{q}}\big)-{\prod \limits_{r=1}^{2}}\big(1-{\Delta _{r}^{q}}-{\nabla _{r}^{q}}\big)},\sqrt[q]{1-{\prod \limits_{r=1}^{2}}\big(1-{\nabla _{r}^{q}}\big)}\right\rangle ;\end{aligned}\]To understand the superiority of the developed operations, four examples are considered as follows:
Example 1.
Let us consider two qROFNs ${\Theta _{1}}=\langle 0.4,0.7\rangle $ and ${\Theta _{2}}=\langle 0.8,0\rangle $. Then using the basic operational laws (Liu and Wang, 2018) of qROFNs, we have ${\Theta _{1}}\oplus {\Theta _{2}}=\langle 0.8352,0\rangle $, which means that the non-zero non-belongingness grade has no impact on the output. This makes the operation ‘⊕’ unreasonable. But based on the proposed operations ‘$\tilde{\oplus }$’ and ‘$\tilde{\otimes }$’, we have ${\Theta _{1}}\tilde{\oplus }{\Theta _{2}}=\langle 0.8352,0.42\rangle $ and ${\Theta _{1}}\tilde{\otimes }{\Theta _{2}}=\langle 0.6196,0.7\rangle $.
Example 2.
Let us consider two qROFNs ${\Theta _{1}}=\langle 1,0\rangle $ and ${\Theta _{2}}=\langle 0.8,0.5\rangle $. Then using the basic operational laws (Liu and Wang, 2018) of qROFNs, we have ${\Theta _{1}}\oplus {\Theta _{2}}=\langle 1,0\rangle $, which means that the belongingness grade which is not equals to ‘1’ has no impact on the output. This again makes the operation ‘⊕’ unreasonable. But based on the proposed operation ‘$\tilde{\otimes }$’, we get ${\Theta _{1}}\tilde{\otimes }{\Theta _{2}}=\langle 0.8660,0.5\rangle $.
Example 3.
Let us consider two qROFNs ${\Theta _{1}}=\langle 0,0.7\rangle $ and ${\Theta _{2}}=\langle 0.8,0.3\rangle $. Then using the basic operational laws (Liu and Wang, 2018) of qROFNs, we have,${\Theta _{1}}\otimes {\Theta _{2}}=\langle 0,0.7320\rangle $, which means that the non-zero belongingness grade has no impact on the output. This makes the operation ‘⊗’ unreasonable. But based on the proposed operations ‘$\tilde{\oplus }$’ and ‘$\tilde{\otimes }$’, we have ${\Theta _{1}}\tilde{\oplus }{\Theta _{2}}=\langle 0.8,0.4714\rangle $ and ${\Theta _{1}}\tilde{\otimes }{\Theta _{2}}=\langle 0.5713,0.7320\rangle $.
Example 4.
Let us consider two qROFNs ${\Theta _{1}}=\langle 0,1\rangle $ and ${\Theta _{2}}=\langle 0.6,0.6\rangle $. Then using the basic operational laws (Liu and Wang, 2018) of qROFNs, we have ${\Theta _{1}}\otimes {\Theta _{2}}=\langle 0,1\rangle $, which means that the non-belongingness grade which is not equals to ‘1’ has no impact on the output. This again makes the operation ‘⊗’ unreasonable. But based on the proposed operation ‘$\tilde{\oplus }$’, we have ${\Theta _{1}}\tilde{\oplus }{\Theta _{2}}=\langle 0.6,0.8\rangle $.
From the above four examples, it is clear that our proposed operations are more sensible.
Theorem 1.
Let ${\Theta _{1}}=\langle {\Delta _{1}},{\nabla _{1}}\rangle $ and ${\Theta _{2}}=\langle {\Delta _{2}},{\nabla _{2}}\rangle $ be two qROFNs and $\lambda ,{\lambda _{1}},{\lambda _{2}}>0$. Then:
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(i) ${\Theta _{1}}\tilde{\oplus }\hspace{0.1667em}{\Theta _{2}}={\Theta _{2}}\tilde{\oplus }{\Theta _{1}}$;
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(ii) ${\Theta _{1}}\tilde{\otimes }\hspace{0.1667em}{\Theta _{2}}={\Theta _{2}}\tilde{\otimes }\hspace{0.1667em}{\Theta _{1}}$;
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(iii) $\lambda ({\Theta _{1}}\tilde{\oplus }{\Theta _{2}})=\lambda {\Theta _{1}}\tilde{\oplus }\hspace{0.1667em}\lambda {\Theta _{2}}$;
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(iv) ${({\Theta _{1}}\tilde{\otimes }\hspace{0.1667em}{\Theta _{2}})^{\lambda }}={\Theta _{1}^{\lambda }}\tilde{\otimes }\hspace{0.1667em}{\Theta _{2}^{\lambda }}$;
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(v) $({\lambda _{1}}+{\lambda _{2}}){\Theta _{1}}={\lambda _{1}}{\Theta _{1}}\tilde{\oplus }\hspace{0.1667em}{\lambda _{2}}{\Theta _{1}}$;
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(vi) ${\Theta _{1}^{{\lambda _{1}}+{\lambda _{2}}}}={\Theta _{1}^{{\lambda _{1}}}}\tilde{\otimes }\hspace{0.1667em}{\Theta _{1}^{{\lambda _{2}}}}$.
3.2 qROF Improved Power Weighted Averaging Operators
In this paper, qROF improved power weighted averaging (qROFIPWA) and qROF improved power weighted averaging MSM (qROFIPWAMSM) operators are developed as follows.
Definition 9.
Let ${\Theta _{r}}=\langle {\Delta _{r}},{\nabla _{r}}\rangle (r=1(1)n)\in {\Sigma ^{U}}$. Then the qROFIPWA operator is defined by:
In Eq. (13), $\frac{{\varpi _{r}}(1+\psi ({\Theta _{r}}))}{{\textstyle\sum _{r=1}^{n}}{\varpi _{r}}(1+\psi ({\Theta _{r}}))}$ is called the power weight of ${\Theta _{r}}$, where ${\varpi _{r}}$ is the weight of ${\Theta _{r}}$ satisfying ${\varpi _{r}}\geqslant 0$ and ${\textstyle\sum _{r=1}^{n}}{\varpi _{r}}=1$. To keep things simple, ${\Omega _{r}}$ is used denote the power weight of ${\Theta _{r}}$. Then Eq. (13) can be re-written as:
Theorem 2.
Let ${\Theta _{r}}=\langle {\Delta _{r}},{\nabla _{r}}\rangle (r=1(1)n)\in {\Sigma ^{U}}$. Then the aggregated value qROFIPWA$({\Theta _{1}},{\Theta _{2}},\dots ,{\Theta _{n}})$ is also a qROFN and
(15)
\[\begin{aligned}{}& \text{qROFIPWA}({\Theta _{1}},{\Theta _{2}},\dots ,{\Theta _{n}})\\ {} & \hspace{1em}=\left\langle {\Bigg(1-{\prod \limits_{r=1}^{n}}{\big(1-{\Delta _{r}^{q}}\big)^{{\Omega _{r}}}}\Bigg)^{\frac{1}{q}}},{\Bigg({\prod \limits_{r=1}^{n}}{\big(1-{\Delta _{r}^{q}}\big)^{{\Omega _{r}}}}-{\prod \limits_{r=1}^{n}}{\big(1-{\Delta _{r}^{q}}-{\nabla _{r}^{q}}\big)^{{\Omega _{r}}}}\Bigg)^{\frac{1}{q}}}\right\rangle .\end{aligned}\]The following Theorems readily follow from Theorem 2.
Theorem 3 (Idempotency).
Let ${\Theta _{r}}=\langle {\Delta _{r}},{\nabla _{r}}\rangle (r=1(1)n)\in {\Sigma ^{U}}$ with ${\Theta _{r}}={\Theta _{0}}(r=1(1)n)$. Then, qROFIPW$({\Theta _{1}},{\Theta _{2}},\dots ,{\Theta _{n}})={\Theta _{0}}$.
Theorem 4 (Boundedness).
Let ${\Theta _{r}}=\langle {\Delta _{r}},{\nabla _{r}}\rangle (r=1(1)n)\in {\Sigma ^{U}}$. If ${\Theta ^{-}}=\langle {\min _{r}}{\Delta _{r}},\sqrt[q]{{\max _{r}}({\Delta _{r}^{q}}+{\nabla _{r}^{q}})-{\min _{r}}{\Delta _{r}^{q}}}\rangle $ and ${\Theta ^{+}}=\langle {\max _{r}}{\Delta _{r}},\hslash \rangle $, then we have ${\Theta ^{-}}\prec \text{qROFIPWA}({\Theta _{1}},{\Theta _{2}},\dots ,{\Theta _{n}})\prec {\Theta ^{+}}$, where
\[ \hslash =\left\{\begin{array}{l@{\hskip4.0pt}l}0,\hspace{1em}& \textit{if}\hspace{2.5pt}{\min _{r}}({\Delta _{r}^{q}}+{\nabla _{r}^{q}})\leqslant {\max _{r}}{\Delta _{r}^{q}},\\ {} \sqrt[q]{{\min _{r}}({\Delta _{r}^{q}}+{\nabla _{r}^{q}})-{\max _{r}}{\Delta _{r}^{q}}},\hspace{1em}& \textit{if}\hspace{2.5pt}{\min _{r}}({\Delta _{r}^{q}}+{\nabla _{r}^{q}})\geqslant {\max _{r}}{\Delta _{r}^{q}}.\end{array}\right.\]
Theorem 5 (Monotonicity).
Let ${\Theta _{r}}=\langle {\Delta _{r}},{\nabla _{r}}\rangle $ and ${\Theta ^{\prime }_{r}}=\langle {\Delta ^{\prime }_{r}},{\nabla ^{\prime }_{r}}\rangle (r=1(1)n)\in {\Sigma ^{U}}$ such that ${\Delta _{r}}\leqslant {\Delta ^{\prime }_{r}}$, ${\nabla _{r}}\geqslant {\nabla ^{\prime }_{r}}$. Then,
Definition 10.
Let ${\Theta _{r}}=\langle {\Delta _{r}},{\nabla _{r}}\rangle (r=1(1)n)\in {\Sigma ^{U}}$. Then the qROFIPWAMSM operator is defined by:
where ${t_{1}},{t_{2}},\dots ,{t_{p}}\geqslant 0$, p is a parameter, ${^{n}}{c_{p}}$ stands for binomial coefficient, (${t_{1}},{t_{2}},\dots ,{t_{p}}$) denotes a p-tuple combination of ($1,2,\dots ,n$).
(16)
\[ {\text{qROFIPWAMSM}^{(p)}}({\Theta _{1}},{\Theta _{2}},\dots ,{\Theta _{n}})={\Bigg(\frac{1}{{^{n}}{c_{p}}}\hspace{-0.1667em}\hspace{-0.1667em}\underset{1\leqslant {t_{1}}<{t_{2}}<\cdots <{t_{p}}\leqslant n}{\bigoplus }\Bigg(\hspace{-0.1667em}{\underset{j=1}{\overset{p}{\bigotimes }}}(n{\Omega _{{t_{j}}}}{\Theta _{{t_{j}}}})\Bigg)\Bigg)^{\frac{1}{p}}},\]In Eq. (16), ${\Omega _{{t_{j}}}}=\frac{{\varpi _{{t_{j}}}}(1+\psi ({\Theta _{{t_{j}}}}))}{{\textstyle\sum _{t=1}^{n}}{\varpi _{{t_{j}}}}(1+\psi ({\Theta _{{t_{j}}}}))}$ is called the power weight of ${\Theta _{{t_{j}}}}$, where ${\varpi _{k}}$ is the weight of ${\Theta _{k}}$ satisfying ${\varpi _{k}}\geqslant 0$ and ${\textstyle\sum _{k=1}^{n}}{\varpi _{k}}=1$.
Theorem 6.
Let ${\Theta _{r}}=\langle {\Delta _{r}},{\nabla _{r}}\rangle (r=1(1)n)\in {\Sigma ^{U}}$. Then the aggregated value ${\text{qROFIPWAMSM}^{(p)}}({\Theta _{1}},{\Theta _{2}},\dots ,{\Theta _{n}})$ is also a qROFN and
(17)
\[\begin{aligned}{}& {\text{qROFIPWAMSM}^{(p)}}({\Theta _{1}},{\Theta _{2}},\dots ,{\Theta _{n}})\\ {} & \hspace{1em}=\Bigg\langle \Bigg(\Bigg(1-\Bigg(\prod \limits_{1\leqslant {t_{1}}<{t_{2}}<\cdots <{t_{p}}\leqslant n}\Bigg(1-{\prod \limits_{j=1}^{p}}\big(1-{\big(1-{({\Delta _{{t_{j}}}})^{q}}\big)^{n{\Omega _{{t_{j}}}}}}\\ {} & \hspace{2em}+{\big(1-{({\Delta _{{t_{j}}}})^{q}}-{({\nabla _{{t_{j}}}})^{q}}\big)^{n{\Omega _{{t_{j}}}}}}\big)+{\prod \limits_{j=1}^{p}}{\big(1-{({\Delta _{{t_{j}}}})^{q}}-{({\nabla _{{t_{j}}}})^{q}}\big)^{n{\Omega _{{t_{j}}}}}}\Bigg)\Bigg){^{\frac{1}{{^{n}}{c_{p}}}}}\\ {} & \hspace{2em}+{\Bigg(\prod \limits_{1\leqslant {t_{1}}<{t_{2}}<\cdots <{t_{p}}\leqslant n}\Bigg({\prod \limits_{j=1}^{p}}{\big(1-{({\Delta _{{t_{j}}}})^{q}}-{({\nabla _{{t_{j}}}})^{q}}\big)^{n{\Omega _{{t_{j}}}}}}\Bigg)\Bigg)^{\frac{1}{{^{n}}{c_{p}}}}}\Bigg){^{\frac{1}{r}}}\\ {} & \hspace{2em}-{\Bigg({\Bigg(\prod \limits_{1\leqslant {t_{1}}<{t_{2}}<\cdots <{t_{p}}\leqslant n}\Bigg({\prod \limits_{j=1}^{p}}{\big(1-{({\Delta _{{t_{j}}}})^{q}}-{({\nabla _{{t_{j}}}})^{q}}\big)^{n{\Omega _{{t_{j}}}}}}\Bigg)\Bigg)^{\frac{1}{{^{n}}{c_{p}}}}}\Bigg)^{\frac{1}{p}}}\Bigg){^{\frac{1}{q}}},\\ {} & \hspace{2em}\Bigg(1-\Bigg(1-\Bigg(\prod \limits_{1\leqslant {t_{1}}<{t_{2}}<\cdots <{t_{p}}\leqslant n}\Bigg(1-{\prod \limits_{j=1}^{p}}\big(1-{\big(1-{({\Delta _{{t_{j}}}})^{q}}\big)^{n{\Omega _{{t_{j}}}}}}\\ {} & \hspace{2em}+{\big(1-{({\Delta _{{t_{j}}}})^{q}}-{({\nabla _{{t_{j}}}})^{q}}\big)^{n{\Omega _{{t_{j}}}}}}\big)+{\prod \limits_{j=1}^{p}}{\big(1-{({\Delta _{{t_{j}}}})^{q}}-{({\nabla _{{t_{j}}}})^{q}}\big)^{n{\Omega _{{t_{j}}}}}}\Bigg)\Bigg){^{\frac{1}{{^{n}}{c_{p}}}}}\\ {} & \hspace{2em}+{\Bigg(\prod \limits_{1\leqslant {t_{1}}<{t_{2}}<\cdots <{t_{p}}\leqslant n}\Bigg({\prod \limits_{j=1}^{p}}{\big(1-{({\Delta _{{t_{j}}}})^{q}}-{({\nabla _{{t_{j}}}})^{q}}\big)^{n{\Omega _{{t_{j}}}}}}\Bigg)\Bigg)^{\frac{1}{{^{n}}{c_{p}}}}}\Bigg){^{\frac{1}{p}}}\Bigg){^{\frac{1}{q}}}\Bigg\rangle .\end{aligned}\]The following Theorems readily follow from Theorem 6.
Theorem 7 (Idempotency).
For ${\Theta _{r}}=\langle {\Delta _{r}},{\nabla _{r}}\rangle (r=1(1)n)\in {\Sigma ^{U}}$ with ${\Theta _{r}}={\Theta _{0}}(r=1(1)n)$, ${\text{qROFIPWAMSM}^{(p)}}({\Theta _{1}},{\Theta _{2}},\dots ,{\Theta _{n}})={\Theta _{0}}$.
Theorem 8 (Boundedness).
Let ${\Theta _{r}}=\langle {\Delta _{r}},{\nabla _{r}}\rangle (r=1(1)n)\in {\Sigma ^{U}}$. If ${\Theta ^{-}}=\langle {\min _{r}}{\Delta _{r}},\sqrt[q]{{\max _{r}}({\Delta _{r}^{q}}+{\nabla _{r}^{q}})-{\min _{r}}{\Delta _{r}^{q}}}\rangle $ and ${\Theta ^{+}}=\langle {\max _{r}}{\Delta _{r}},\hslash \rangle $, then we have, ${\Theta ^{-}}\prec \text{qROFIPWAMSM}({\Theta _{1}},{\Theta _{2}},\dots ,{\Theta _{n}})\prec {\Theta ^{+}}$, where
\[ \hslash =\left\{\begin{array}{l@{\hskip4.0pt}l}0,\hspace{1em}& \textit{if}\hspace{2.5pt}{\min _{r}}({\Delta _{r}^{q}}+{\nabla _{r}^{q}})\leqslant {\max _{r}}{\Delta _{r}^{q}},\\ {} \sqrt[q]{{\min _{r}}({\Delta _{r}^{q}}+{\nabla _{r}^{q}})-{\max _{r}}{\Delta _{r}^{q}}},\hspace{1em}& \textit{if}\hspace{2.5pt}{\min _{r}}({\Delta _{r}^{q}}+{\nabla _{r}^{q}})\geqslant {\max _{r}}{\Delta _{r}^{q}}.\end{array}\right.\]
Theorem 9 (Monotonicity).
Let ${\Theta _{r}}=\langle {\Delta _{r}},{\nabla _{r}}\rangle $ and ${\Theta ^{\prime }_{r}}=\langle {\Delta ^{\prime }_{r}},{\nabla ^{\prime }_{r}}\rangle (r=1(1)n)\in {\Sigma ^{U}}$ such that ${\Delta _{r}}\leqslant {\Delta ^{\prime }_{r}}$, ${\nabla _{r}}\geqslant {\nabla ^{\prime }_{r}}$. Then, ${\text{qROFIPAWMSM}^{(p)}}({\Theta _{1}},{\Theta _{2}},\dots ,{\Theta _{n}})\hspace{2.5pt}\prec {\text{qROFIPWAMSM}^{(p)}}({\Theta ^{\prime }_{1}},{\Theta ^{\prime }_{2}},\dots ,{\Theta ^{\prime }_{n}})$.
3.3 qROF Improved Power Weighted Geometric Operators
This paper develops qROF improved power weighted geometric (qROFIPWG) operator and qROF improved power weighted geometric MSM (qROFIPWGMSM) operator as follows:
In Eq. (18), ${\Omega _{r}}=\frac{{\varpi _{r}}(1+\psi ({\Theta _{r}}))}{{\textstyle\sum _{r=1}^{n}}{\varpi _{r}}(1+\psi ({\Theta _{r}}))}$ is called the power weight of ${\Theta _{r}}$, where ${\varpi _{r}}$ is the weight of ${\Theta _{r}}$ satisfying ${\varpi _{r}}\geqslant 0$ and ${\textstyle\sum _{r=1}^{n}}{\varpi _{r}}=1$.
Theorem 10.
Let ${\Theta _{r}}=\langle {\Delta _{r}},{\nabla _{r}}\rangle (r=1(1)n)\in {\Sigma ^{U}}$. Then the aggregated value $\text{qROFIPWG}({\Theta _{1}},{\Theta _{2}},\dots ,{\Theta _{n}})$ is also a qROFN and
(19)
\[\begin{aligned}{}& \text{qROFIPWG}({\Theta _{1}},{\Theta _{2}},\dots ,{\Theta _{n}})\\ {} & \hspace{1em}=\left\langle {\Bigg({\prod \limits_{r=1}^{n}}{\big(1-{\nabla _{r}^{q}}\big)^{{\Omega _{r}}}}-{\prod \limits_{r=1}^{n}}{\big(1-{\Delta _{r}^{q}}-{\nabla _{r}^{q}}\big)^{{\Omega _{r}}}}\Bigg)^{\frac{1}{q}}},\right.\left.{\Bigg(1-{\prod \limits_{r=1}^{n}}{\big(1-{\nabla _{r}^{q}}\big)^{{\Omega _{r}}}}\Bigg)^{\frac{1}{q}}}\right\rangle .\end{aligned}\]The following Theorems readily follow from Theorem 10.
Theorem 11.
(Idempotency) Let ${\Theta _{r}}=\langle {\Delta _{r}},{\nabla _{r}}\rangle (r=1(1)n)\in {\Sigma ^{U}}$ such that ${\Theta _{r}}={\Theta _{0}}(r=1(1)n)$, then we have $\text{qROFIPWG}({\Theta _{1}},{\Theta _{2}},\dots ,{\Theta _{n}})={\Theta _{0}}$.
Theorem 12.
(Boundedness) Let ${\Theta _{r}}=\langle {\Delta _{r}},{\nabla _{r}}\rangle (r=1(1)n)\in {\Sigma ^{U}}$. If ${\Theta ^{-}}=\langle {\min _{r}}{\nabla _{r}},\sqrt[q]{{\max _{r}}({\Delta _{r}^{q}}+{\nabla _{r}^{q}})-{\min _{r}}{\nabla _{r}^{q}}}\rangle $ an ${\Theta ^{+}}=\langle {\max _{r}}{\nabla _{r}},\hslash \rangle $, then we have ${\Theta ^{-}}\prec \text{qROFIPWG}({\Theta _{1}},{\Theta _{2}},\dots ,{\Theta _{n}})\prec {\Theta ^{+}}$, where
\[ \hslash =\left\{\begin{array}{l@{\hskip4.0pt}l}0,\hspace{1em}& \textit{if}\hspace{2.5pt}{\min _{r}}({\Delta _{r}^{q}}+{\nabla _{r}^{q}})\leqslant {\max _{r}}{\nabla _{r}^{q}},\\ {} \sqrt[q]{{\min _{r}}({\Delta _{r}^{q}}+{\nabla _{r}^{q}})-{\max _{r}}{\nabla _{r}^{q}}},\hspace{1em}& \textit{if}\hspace{2.5pt}{\min _{r}}({\Delta _{r}^{q}}+{\nabla _{r}^{q}})\geqslant {\max _{r}}{\nabla _{r}^{q}}.\end{array}\right.\]
Theorem 13.
(Monotonicity) Let ${\Theta _{r}}=\langle {\Delta _{r}},{\nabla _{r}}\rangle $ and ${\Theta ^{\prime }_{r}}=\hspace{2.5pt}\langle {\Delta ^{\prime }_{r}},{\nabla ^{\prime }_{r}}\rangle (r=1(1)n)\in {\Sigma ^{U}}$ such that ${\Delta _{r}}\leqslant {\Delta ^{\prime }_{r}}$, ${\nabla _{r}}\geqslant {\nabla ^{\prime }_{r}}$. Then, $\text{qROFIPWG}({\Theta _{1}},{\Theta _{2}},\dots ,{\Theta _{n}})\hspace{2.5pt}\prec \text{qROFIPWG}({\Theta ^{\prime }_{1}},{\Theta ^{\prime }_{2}},\dots ,{\Theta ^{\prime }_{n}})$.
Definition 12.
Let ${\Theta _{r}}=\langle {\Delta _{r}},{\nabla _{r}}\rangle (r=1(1)n)\in {\Sigma ^{U}}$. Then the qROFIPWGMSM operator is defined by:
where ${t_{1}},{t_{2}},\dots ,{t_{p}}\geqslant 0$, p is a parameter, ${^{n}}{c_{p}}$ stands for binomial coefficient, (${t_{1}},{t_{2}},\dots ,{t_{p}}$) denotes a p-tuple combination of ($1,2,\dots ,n$).
(20)
\[ {\text{qROFIPWGMSM}^{(p)}}({\Theta _{1}},{\Theta _{2}},\dots ,{\Theta _{n}})=\frac{1}{p}{\Big(\underset{1\leqslant {t_{1}}<{t_{2}}<\cdots <{t_{p}}\leqslant n}{\widetilde{\bigotimes }}\Big({\underset{j=1}{\overset{p}{\widetilde{\bigoplus }}}}{\Theta _{{t_{j}}}^{{\Omega _{{t_{j}}}}}}\Big)\Big)^{\frac{1}{{^{n}}{c_{p}}}}},\]In Eq. (20), ${\Omega _{{t_{j}}}}=\frac{{\varpi _{{t_{j}}}}(1+\psi ({\Theta _{{t_{j}}}}))}{{\textstyle\sum _{t=1}^{n}}{\varpi _{{t_{j}}}}(1+\psi ({\Theta _{{t_{j}}}}))}$ is called the power weight of ${\Theta _{{t_{j}}}}$, where ${\varpi _{k}}$ is the weight of ${\Theta _{k}}$ satisfying ${\varpi _{k}}\geqslant 0$ and ${\textstyle\sum _{k=1}^{n}}{\varpi _{k}}=1$.
Theorem 14.
Let ${\Theta _{r}}=\langle {\Delta _{r}},{\nabla _{r}}\rangle (r=1(1)n)\in {\Sigma ^{U}}$. Then the aggregated value ${\text{qROFIPWGMSM}^{(p)}}({\Theta _{1}},{\Theta _{2}},\dots ,{\Theta _{n}})$ is also a qROFN and
(21)
\[\begin{aligned}{}& {\text{qROFIPWGMSM}^{(p)}}({\Theta _{1}},{\Theta _{2}},\dots ,{\Theta _{n}})\\ {} & \hspace{1em}=\Bigg\langle \Bigg(1-\Bigg(1-\Bigg(\prod \limits_{1\leqslant {t_{1}}<{t_{2}}<\cdots <{t_{p}}\leqslant n}\Bigg(1-{\prod \limits_{j=1}^{p}}\big(1-{\big(1-{({\nabla _{{t_{j}}}})^{q}}\big)^{n{\Omega _{{t_{j}}}}}}\\ {} & \hspace{2em}+{\big(1-{({\Delta _{{t_{j}}}})^{q}}-{({\nabla _{{t_{j}}}})^{q}}\big)^{n{\Omega _{{t_{j}}}}}}\big)+{\prod \limits_{j=1}^{p}}{\big(1-{({\Delta _{{t_{j}}}})^{q}}-{({\nabla _{{t_{j}}}})^{q}}\big)^{n{\Omega _{{t_{j}}}}}}\Bigg)\Bigg){^{\frac{1}{{^{n}}{c_{p}}}}}\\ {} & \hspace{2em}+{\Bigg(\prod \limits_{1\leqslant {t_{1}}<{t_{2}}<\cdots <{t_{p}}\leqslant n}\Bigg({\prod \limits_{j=1}^{p}}{\big(1-{({\Delta _{{t_{j}}}})^{q}}-{({\nabla _{{t_{j}}}})^{q}}\big)^{n{\Omega _{{t_{j}}}}}}\Bigg)\Bigg)^{\frac{1}{{^{n}}{c_{p}}}}}\Bigg){^{\frac{1}{p}}}\Bigg){^{\frac{1}{q}}},\\ {} & \hspace{2em}\Bigg(\Bigg(1-\Bigg(\prod \limits_{1\leqslant {t_{1}}<{t_{2}}<\cdots <{t_{p}}\leqslant n}\Bigg(1-{\prod \limits_{j=1}^{p}}\big(1-{\big(1-{({\nabla _{{t_{j}}}})^{q}}\big)^{n{\Omega _{{t_{j}}}}}}\\ {} & \hspace{2em}+{\big(1-{({\Delta _{{t_{j}}}})^{q}}-{({\nabla _{{t_{j}}}})^{q}}\big)^{n{\Omega _{{t_{j}}}}}}\big)+{\prod \limits_{j=1}^{p}}{\big(1-{({\Delta _{{t_{j}}}})^{q}}-{({\nabla _{{t_{j}}}})^{q}}\big)^{n{\Omega _{{t_{j}}}}}}\Bigg)\Bigg){^{\frac{1}{{^{n}}{c_{p}}}}}\\ {} & \hspace{2em}+{\Bigg(\prod \limits_{1\leqslant {t_{1}}<{t_{2}}<\cdots <{t_{p}}\leqslant n}\Bigg({\prod \limits_{j=1}^{p}}{\big(1-{({\Delta _{{t_{j}}}})^{q}}-{({\nabla _{{t_{j}}}})^{q}}\big)^{n{\Omega _{{t_{j}}}}}}\Bigg)\Bigg)^{\frac{1}{{^{n}}{c_{p}}}}}\Bigg){^{\frac{1}{p}}}\\ {} & \hspace{2em}-{\Bigg({\Bigg(\prod \limits_{1\leqslant {t_{1}}<{t_{2}}<\cdots <{t_{p}}\leqslant n}\Bigg({\prod \limits_{j=1}^{p}}{\big(1-{({\Delta _{{t_{j}}}})^{q}}-{({\nabla _{{t_{j}}}})^{q}}\big)^{n{\Omega _{{t_{j}}}}}}\Bigg)\Bigg)^{\frac{1}{{^{n}}{c_{p}}}}}\Bigg)^{\frac{1}{p}}}\Bigg){^{\frac{1}{q}}}\Bigg\rangle .\end{aligned}\]The following Theorems readily follow from Theorem 14.
Theorem 15.
(Idempotency) Let ${\Theta _{r}}=\langle {\Delta _{r}},{\nabla _{r}}\rangle (r=1(1)n)\in {\Sigma ^{U}}$ such that ${\Theta _{r}}={\Theta _{0}}(r=1(1)n)$, then we have ${\text{qROFIPWGMSM}^{(p)}}({\Theta _{1}},{\Theta _{2}},\dots ,{\Theta _{n}})={\Theta _{0}}$.
Theorem 16.
(Boundedness) Let ${\Theta _{r}}=\langle {\Delta _{r}},{\nabla _{r}}\rangle (r=1(1)n)\in {\Sigma ^{U}}$. If ${\Theta ^{-}}=\langle {\min _{r}}{\nabla _{r}},\sqrt[q]{{\max _{r}}({\Delta _{r}^{q}}+{\nabla _{r}^{q}})-{\min _{r}}{\nabla _{r}^{q}}}\rangle $ and ${\Theta ^{+}}=\langle {\max _{r}}{\nabla _{r}},\hslash \rangle $, then we have ${\Theta ^{-}}\prec {\text{qROFIPWGMSM}^{(p)}}({\Theta _{1}},{\Theta _{2}},\dots ,{\Theta _{n}})\prec {\Theta ^{+}}$, where
\[ \hslash =\left\{\begin{array}{l@{\hskip4.0pt}l}0,\hspace{1em}& \textit{if}\hspace{2.5pt}{\min _{r}}({\Delta _{r}^{q}}+{\nabla _{r}^{q}})\leqslant {\max _{r}}{\nabla _{r}^{q}},\\ {} \sqrt[q]{{\min _{r}}({\Delta _{r}^{q}}+{\nabla _{r}^{q}})-{\max _{r}}{\nabla _{r}^{q}}},\hspace{1em}& \textit{if}\hspace{2.5pt}{\min _{r}}({\Delta _{r}^{q}}+{\nabla _{r}^{q}})\geqslant {\max _{r}}{\nabla _{r}^{q}}.\end{array}\right.\]
Theorem 17 (Monotonicity).
Let ${\Theta _{r}}=\langle {\Delta _{r}},{\nabla _{r}}\rangle $ and ${\Theta ^{\prime }_{r}}=\langle {\Delta ^{\prime }_{r}},{\nabla ^{\prime }_{r}}\rangle (r=1(1)n)\in {\Sigma ^{U}}$ such that ${\Delta _{r}}\leqslant {\Delta ^{\prime }_{r}}$, ${\nabla _{r}}\geqslant {\nabla ^{\prime }_{r}}$. Then ${\text{qROFIPWGMSM}^{(p)}}({\Theta _{1}},{\Theta _{2}},\dots ,{\Theta _{n}})\hspace{2.5pt}\prec {\text{qROFIPWGMSM}^{(p)}}({\Theta ^{\prime }_{1}},{\Theta ^{\prime }_{2}},\dots ,{\Theta ^{\prime }_{n}})$.
4 Group Decision Making Methodology
Suppose m different alternatives ${X_{i}}(i=1(1)m)$ need to be assessed over n distinct attributes ${L_{j}}(j=1(1)n)$. Assume a set of l experts ${D_{d}}(d=1(1)l)$ with weights ${\eta _{d}}(d=1(1)l)$ with ${\eta _{d}}\geqslant 0$ and ${\textstyle\sum _{d=1}^{l}}{\eta _{d}}=1$ for the assessment of considered alternatives. The initial assessment result of the expert ${D_{d}}(d=1(1)l)$ is specified in terms of qROFNs ${\Theta _{ij}^{(d)}}=\langle {\Delta _{ij}^{(d)}},{\nabla _{ij}^{(d)}}\rangle $ subject to $0\leqslant {\Delta _{ij}^{(d)}}$, ${\nabla _{ij}^{(d)}}\leqslant 1$ and $0\leqslant {({\Delta _{ij}^{(d)}})^{q}}+{({\nabla _{ij}^{(d)}})^{q}}\leqslant 1$.
To find the best-suited alternative(s), the introduced operators are applied to propose a MCGDM methodology relating to the qROF data with the steps acquired as follows:
Step 1: The initial assessment results of experts are: ${\mathrm{\Im }_{d}}={[{\Theta _{ij}^{(d)}}]_{m\times n}}={[\langle {\Delta _{ij}^{(d)}},{\nabla _{ij}^{(d)}}\rangle ]_{m\times n}}(d=1(1)l)$.
Step 2: Normalize the decision matrices ${\mathrm{\Im }_{d}}={[{\Theta _{ij}^{(d)}}]_{m\times n}}(d=1(1)l)$.
The Normalized decision matrix is: ${\tilde{\mathrm{\Im }}_{d}}={[{\tilde{\Theta }_{ij}^{(d)}}]_{m\times n}}={[\langle {\tilde{\Delta }_{ij}^{(d)}},{\tilde{\nabla }_{ij}^{(d)}}\rangle ]_{m\times n}}(d=1(1)l)$ where:
Step 3: Find $\textit{Supp}({\tilde{\Theta }_{ij}^{(d)}},{\tilde{\Theta }_{ij}^{(s)}})(d,s=1(1)l;d\ne s)$ based on the following formula:
where $\textit{Dist}({\tilde{\Theta }_{ij}^{(d)}},{\tilde{\Theta }_{ij}^{(s)}})$ is the Hamming distance between the qROFNs [20].
(22)
\[ {\tilde{\Theta }_{ij}^{(d)}}=\left\{\begin{array}{l@{\hskip4.0pt}l}\langle {\Delta _{ij}^{(d)}},{\nabla _{ij}^{(d)}}\rangle \hspace{1em}& \text{if}\hspace{2.5pt}{C_{j}}\hspace{2.5pt}\text{is of benefit-type},\\ {} \langle {\nabla _{ij}^{(d)}},{\Delta _{ij}^{(d)}}\rangle \hspace{1em}& \text{if}\hspace{2.5pt}{C_{j}}\hspace{2.5pt}\text{is of cost-type}.\end{array}\right.\](23)
\[ \textit{Supp}\big({\tilde{\Theta }_{ij}^{(d)}},{\tilde{\Theta }_{ij}^{(s)}}\big)=1-\textit{Dist}\big({\tilde{\Theta }_{ij}^{(d)}},{\tilde{\Theta }_{ij}^{(s)}}\big)\big(d,s=1(1)l;d\ne s\big),\]
Step 4: Compute $\psi ({\tilde{\Theta }_{ij}^{(d)}})$ by
Step 5: Calculate the power weights ${\Omega _{ij}^{(d)}}(i=1(1)m;j=1(1)n;d=1(1)l)$ associated with the qROFNs ${\tilde{\Theta }_{ij}^{(d)}}$ by utilizing the weights ${\eta _{d}}$ of DEs ${D_{d}}(d=1(1)l)$, where
Step 6: Construct aggregated normalized qROF decision matrix $\mathrm{\Im }\ast ={[{\Theta _{ij}}]_{m\times n}}={[\langle {\Delta _{ij}},{\nabla _{ij}}\rangle ]_{m\times n}}$.
(24)
\[ \psi \big({\tilde{\Theta }_{ij}^{(d)}}\big)={\sum \limits_{s=1,s\ne d}^{l}}\textit{Supp}\big({\tilde{\Theta }_{ij}^{(d)}},{\tilde{\Theta }_{ij}^{(s)}}\big)\big(i=1(1)m;j=1(1)n;d=1(1)l\big).\](25)
\[ {\Omega _{ij}^{(d)}}=\frac{{\eta _{d}}(1+\psi ({\tilde{\Theta }_{ij}^{(d)}}))}{{\textstyle\textstyle\sum _{d=1}^{l}}{\eta _{d}}(1+\psi ({\tilde{\Theta }_{ij}^{(d)}}))}.\]The operator $\text{qROFIPWA}$ or $\text{qROFIPWG}$ can be applied for aggregating normalized qROFNs.
Step 7: Calculate the supports $\textit{Supp}({\Theta _{ij}},{\Theta _{iy}})(j,y=1(1)n;j\ne y)$ based on the following formula:
where $\textit{Dist}({\Theta _{ij}},{\Theta _{iy}})$ is the Hamming distance between the qROFNs (Liu et al., 2020).
(26)
\[\begin{aligned}{}& \text{qROFIPWA}\big({\tilde{\Theta }_{ij}^{(1)}},{\tilde{\Theta }_{ij}^{(2)}},\dots ,{\tilde{\Theta }_{ij}^{(l)}}\big)\\ {} & \hspace{1em}=\Bigg\langle {\Bigg(1-{\prod \limits_{d=1}^{l}}{\big(1-{\big({\tilde{\Delta }_{ij}^{(d)}}\big)^{q}}\big)^{{\Omega _{ij}^{(d)}}}}\Bigg)^{\frac{1}{q}}},\\ {} & \hspace{2em}{\Bigg({\prod \limits_{d=1}^{l}}{\big(1-{\big({\tilde{\Delta }_{ij}^{(d)}}\big)^{q}}\big)^{{\Omega _{ij}^{(d)}}}}-{\prod \limits_{d=1}^{l}}{\big(1-{\big({\tilde{\Delta }_{ij}^{(d)}}\big)^{q}}-{\big({\tilde{\nabla }_{ij}^{(d)}}\big)^{q}}\big)^{{\Omega _{ij}^{(d)}}}}\Bigg)^{\frac{1}{q}}}\Bigg\rangle ,\end{aligned}\](27)
\[\begin{aligned}{}& \text{qROFIPWG}\big({\tilde{\Theta }_{ij}^{(1)}},{\tilde{\Theta }_{ij}^{(2)}},\dots ,{\tilde{\Theta }_{ij}^{(l)}}\big)\\ {} & \hspace{1em}=\Bigg\langle {\Bigg({\prod \limits_{d=1}^{l}}{\big(1-{\big({\tilde{\nabla }_{ij}^{(d)}}\big)^{q}}\big)^{{\Omega _{ij}^{(d)}}}}-{\prod \limits_{d=1}^{l}}{\big(1-{\big({\tilde{\Delta }_{ij}^{(d)}}\big)^{q}},-{\big({\tilde{\nabla }_{ij}^{(d)}}\big)^{q}}\big)^{{\Omega _{ij}^{(d)}}}}\Bigg)^{\frac{1}{q}}},\\ {} & \hspace{2em}{\Bigg(1-{\prod \limits_{d=1}^{l}}{\big(1-{\big({\tilde{\nabla }_{ij}^{(d)}}\big)^{q}}\big)^{{\Omega _{ij}^{(d)}}}}\Bigg)^{\frac{1}{q}}}\Bigg\rangle .\end{aligned}\](28)
\[ \textit{Supp}({\Theta _{ij}},{\Theta _{iy}})=1-\textit{Dist}({\Theta _{ij}},{\Theta _{iy}})\big(j,y=1(1)n;j\ne y\big),\]
Step 8: Compute the values $\psi ({\Theta _{ij}})$ using the formula given by:
Step 9: Calculate the power weights ${\Omega _{ij}}(i=1(1)m;j=1(1)n)$ associated with the qROFNs ${\Theta _{ij}}$ by utilizing the weights ${\eta _{d}}$ of DEs ${D_{d}}(d=1(1)l)$, where
Step 10: Construct the final aggregated qROF decision matrix $\hat{\mathrm{\Im }}={[{\Theta _{i}}]_{m\times 1}}={[\langle {\Delta _{i}},{\nabla _{i}}\rangle ]_{m\times 1}}$.
(29)
\[ \psi ({\Theta _{ij}})={\sum \limits_{y=1,y\ne j}^{n}}\textit{Supp}({\Theta _{ij}},{\Theta _{iy}})\big(i=1(1)m;j=1(1)n\big).\](30)
\[ {\Omega _{ij}}=\frac{{\varpi _{j}}(1+\psi ({\Theta _{ij}}))}{{\textstyle\textstyle\sum _{j=1}^{n}}{\varpi _{j}}(1+\psi ({\Theta _{ij}}))}\big(i=1(1)m;j=1(1)n\big).\]The final aggregated qROF decision matrix is constructed based on the $\text{qROFIPWAMSM}$ or $\text{qROFIPWGMSM}$ operator.
Step 11: Estimate the score values of ${\Theta _{i}}(i=(1)m)$ by utilizing Definition 2.
(31)
\[\begin{aligned}{}{\Theta _{i}}& ={\text{qROFIPWAMSM}^{(p)}}({\Theta _{i1}},{\Theta _{i2}},\dots ,{\Theta _{in}})\\ {} & ={\Bigg(\frac{1}{{^{n}}{c_{p}}}\underset{1\leqslant {t_{1}}<{t_{2}}<\cdots <{t_{p}}\leqslant n}{\bigoplus }\Bigg({\underset{j=1}{\overset{p}{\bigotimes }}}(n{\Omega _{i{t_{j}}}}{\Theta _{i{t_{j}}}})\Bigg)\Bigg)^{\frac{1}{p}}},\end{aligned}\](32)
\[\begin{aligned}{}{\Theta _{i}}& ={\text{qROFIPWGMSM}^{(p)}}({\Theta _{i1}},{\Theta _{i2}},\dots ,{\Theta _{in}})\\ {} & =\frac{1}{p}{\Big(\underset{1\leqslant {t_{1}}<{t_{2}}<\cdots <{t_{p}}\leqslant n}{\widetilde{\bigotimes }}\Big({\underset{j=1}{\overset{\hspace{0.1667em}p}{\widetilde{\bigoplus }}}}{\Theta _{i{t_{j}}}^{n{\Omega _{i{t_{j}}}}}}\Big)\Big)^{\frac{1}{{^{n}}{c_{p}}}}}.\end{aligned}\]If two score values ${S_{c}}({\Theta _{i}})$ and ${S_{c}}({\Theta _{u}})$ are same, then accuracy values (Definition 3) should be computed.
Step 12: Obtain the priority order of alternatives ${A_{i}}(i=1(1)m)$ indicated by the Definition 4 and subsequently choose the optimal one.
5 Application of the Proposed Methodology
5.1 Problem Description
Personnel selection plays a significant role for tracking down the adequate information quality for an organization/industry. Personnel selection is the most common way of picking the people who match the capabilities needed to play out a characterized work in the most ideal manner. A personnel selection problem can be viewed as a MCGDM problem due to the fact that a group of experts and many attributes are considered in the selection process of suitable personnel. qROFS theory can be considered as an essential tool to provide an efficient decision framework to tackle personnel selection problems. Now, let’s think about an Engineering Institute (Under Graduate level), which desires to appoint a Placement officer for ‘Training and Placement Cell’. Suppose five candidates ${X_{i}}$ $(i=1(1)5)$ are shortlisted for personal interview based on their scores of written tests. A team of three experts (Principal, Director and HR manager) is formed to assess the five candidates on the grounds of industry experience (${L_{1}}$), communication skill (${L_{2}}$), networking skill (${L_{3}}$), and academic qualifications (${L_{4}}$).
5.2 Problem Solution
Step 1: Present the initial assessments of each expert as: ${\mathrm{\Im }_{d}}={[{\Theta _{ij}^{(d)}}]_{5\times 4}}={[\langle {\Delta _{ij}^{(d)}},{\nabla _{ij}^{(d)}}\rangle ]_{5\times 4}}(d=1(1)3)$ (Table 1).
Table 1
Initial assessment results of the experts.
| Expert | Alternative | ${L_{1}}$ | ${L_{2}}$ | ${L_{3}}$ | ${L_{4}}$ |
| ${D_{1}}$ | ${X_{1}}$ | $\langle 0.2,0.6\rangle $ | $\langle 0.4,0.6\rangle $ | $\langle 0.4,0.3\rangle $ | $\langle 0.5,0.6\rangle $ |
| ${X_{2}}$ | $\langle 0.6,0.5\rangle $ | $\langle 0.6,0.5\rangle $ | $\langle 0.5,0.4\rangle $ | $\langle 0.5,0.2\rangle $ | |
| ${X_{3}}$ | $\langle 0.8,0.2\rangle $ | $\langle 0.5,0.2\rangle $ | $\langle 0.5,0.3\rangle $ | $\langle 0.4,0.3\rangle $ | |
| ${X_{4}}$ | $\langle 0.5,0.6\rangle $ | $\langle 0.3,0.5\rangle $ | $\langle 0.5,0.2\rangle $ | $\langle 0.5,0.2\rangle $ | |
| ${X_{5}}$ | $\langle 0.5,0.3\rangle $ | $\langle 0.4,0.7\rangle $ | $\langle 0.6,0.4\rangle $ | $\langle 0.6,0.6\rangle $ | |
| ${X_{1}}$ | $\langle 0.4,0.6\rangle $ | $\langle 0.2,0.2\rangle $ | $\langle 0.5,0.4\rangle $ | $\langle 0.4,0.6\rangle $ | |
| ${D_{2}}$ | ${X_{2}}$ | $\langle 0.5,0.1\rangle $ | $\langle 0.6,0.4\rangle $ | $\langle 0.5,0.5\rangle $ | $\langle 0.4,0.3\rangle $ |
| ${X_{3}}$ | $\langle 0.7,0.3\rangle $ | $\langle 0.4,0.2\rangle $ | $\langle 0.4,0.1\rangle $ | $\langle 0.5,0.4\rangle $ | |
| ${X_{4}}$ | $\langle 0.5,0.4\rangle $ | $\langle 0.5,0.7\rangle $ | $\langle 0.5,0.6\rangle $ | $\langle 0.3,0.8\rangle $ | |
| ${X_{5}}$ | $\langle 0.6,0.4\rangle $ | $\langle 0.3,0.3\rangle $ | $\langle 0.6,0.3\rangle $ | $\langle 0.4,0.2\rangle $ | |
| ${X_{1}}$ | $\langle 0.7,0.7\rangle $ | $\langle 0.5,0.4\rangle $ | $\langle 0.2,0.4\rangle $ | $\langle 0.4,0.6\rangle $ | |
| ${D_{3}}$ | ${X_{2}}$ | $\langle 0.4,0.2\rangle $ | $\langle 0.5,0.4\rangle $ | $\langle 0.6,0.3\rangle $ | $\langle 0.5,0.1\rangle $ |
| ${X_{3}}$ | $\langle 0.5,0.3\rangle $ | $\langle 0.4,0.2\rangle $ | $\langle 0.4,0.3\rangle $ | $\langle 0.6,0.4\rangle $ | |
| ${X_{4}}$ | $\langle 0.3,0.5\rangle $ | $\langle 0.5,0.4\rangle $ | $\langle 0.5,0.2\rangle $ | $\langle 0.8,0.2\rangle $ | |
| ${X_{5}}$ | $\langle 0.4,0.6\rangle $ | $\langle 0.6,0.3\rangle $ | $\langle 0.4,0.4\rangle $ | $\langle 0.6,0.1\rangle $ |
For each of the remaining steps, $q=2$ is taken since the least value of q that satisfies ${({\Delta _{ij}^{(d)}})^{q}}+{({\nabla _{ij}^{(d)}})^{q}}\leqslant 1$ is ‘2’.
Step 2: Since all the criteria are of benefit type, normalization is not required. Hence, ${\mathrm{\Im }_{d}}={[{\Theta _{ij}^{(d)}}]_{5\times 4}}={[\langle {\Delta _{ij}^{(d)}},{\nabla _{ij}^{(d)}}\rangle ]_{5\times 4}}={[\langle {\tilde{\Delta }_{ij}^{(d)}},{\tilde{\nabla }_{ij}^{(d)}}\rangle ]_{5\times 4}}={[{\tilde{\Theta }_{ij}^{(d)}}]_{5\times 4}}={\tilde{\mathrm{\Im }}_{d}}$ ($d=1(1)l$).
Step 3: The supports are calculated as $\textit{Supp}({\tilde{\Theta }_{ij}^{(d)}},{\tilde{\Theta }_{ij}^{(s)}})$ ($d,s=1(1)l;d\ne s$) using Eq. (23). For sake of simplicity, ${S_{ds}}(d\ne s;d,s=1(1)3)$ is used here to represent $\textit{Supp}({\tilde{\Theta }_{ij}^{(d)}},{\tilde{\Theta }_{ij}^{(s)}})$ ($d,s=1(1)l;d\ne s$) and, consequently, the following matrices are obtained:
\[\begin{aligned}{}& {[{S_{12}}]_{5\times 4}}={[{S_{21}}]_{5\times 4}}=\left[\begin{array}{c@{\hskip4.0pt}c@{\hskip4.0pt}c@{\hskip4.0pt}c}0.88\hspace{1em}& 0.56\hspace{1em}& 0.84\hspace{1em}& 0.91\\ {} 0.65\hspace{1em}& 0.91\hspace{1em}& 0.91\hspace{1em}& 0.91\\ {} 0.85\hspace{1em}& 0.91\hspace{1em}& 0.83\hspace{1em}& 0.84\\ {} 0.8\hspace{1em}& 0.6\hspace{1em}& 0.68\hspace{1em}& 0.4\\ {} 0.82\hspace{1em}& 0.53\hspace{1em}& 0.93\hspace{1em}& 0.48\end{array}\right],\\ {} & {[{S_{13}}]_{5\times 4}}={[{S_{31}}]_{5\times 4}}=\left[\begin{array}{c@{\hskip4.0pt}c@{\hskip4.0pt}c@{\hskip4.0pt}c}0.42\hspace{1em}& 0.8\hspace{1em}& 0.88\hspace{1em}& 0.91\\ {} 0.59\hspace{1em}& 0.8\hspace{1em}& 0.89\hspace{1em}& 0.97\\ {} 0.61\hspace{1em}& 0.91\hspace{1em}& 0.91\hspace{1em}& 0.73\\ {} 0.73\hspace{1em}& 0.84\hspace{1em}& 1\hspace{1em}& 0.61\\ {} 0.73\hspace{1em}& 0.6\hspace{1em}& 0.8\hspace{1em}& 0.65\end{array}\right],\\ {} & {[{S_{23}}]_{5\times 4}}={[{S_{32}}]_{5\times 4}}=\left[\begin{array}{c@{\hskip4.0pt}c@{\hskip4.0pt}c@{\hskip4.0pt}c}0.54\hspace{1em}& 0.67\hspace{1em}& 0.79\hspace{1em}& 1\\ {} 0.91\hspace{1em}& 0.89\hspace{1em}& 0.84\hspace{1em}& 0.91\\ {} 0.76\hspace{1em}& 1\hspace{1em}& 0.92\hspace{1em}& 0.89\\ {} 0.84\hspace{1em}& 0.67\hspace{1em}& 0.68\hspace{1em}& 0.4\\ {} 0.8\hspace{1em}& 0.73\hspace{1em}& 0.8\hspace{1em}& 0.8\end{array}\right].\end{aligned}\]
Step 4: According to Eq. (24), the values $\psi ({\tilde{\Theta }_{ij}^{(d)}})$ ($i=1(1)5$; $j=1(1)4$; $d=1(1)3$) are calculated as given by the following matrices:
\[\begin{aligned}{}& {\big[\psi \big({\tilde{\Theta }_{ij}^{(1)}}\big)\big]_{5\times 4}}=\left[\begin{array}{c@{\hskip4.0pt}c@{\hskip4.0pt}c@{\hskip4.0pt}c}1.3\hspace{1em}& 1.36\hspace{1em}& 1.72\hspace{1em}& 1.82\\ {} 1.24\hspace{1em}& 1.71\hspace{1em}& 1.8\hspace{1em}& 1.88\\ {} 1.46\hspace{1em}& 1.82\hspace{1em}& 1.74\hspace{1em}& 1.57\\ {} 1.53\hspace{1em}& 1.44\hspace{1em}& 1.68\hspace{1em}& 1.01\\ {} 1.55\hspace{1em}& 1.13\hspace{1em}& 1.73\hspace{1em}& 1.13\end{array}\right],\\ {} & {\big[\psi \big({\tilde{\Theta }_{ij}^{(2)}}\big)\big]_{5\times 4}}=\left[\begin{array}{c@{\hskip4.0pt}c@{\hskip4.0pt}c@{\hskip4.0pt}c}1.42\hspace{1em}& 1.23\hspace{1em}& 1.63\hspace{1em}& 1.91\\ {} 1.56\hspace{1em}& 1.8\hspace{1em}& 1.75\hspace{1em}& 1.82\\ {} 1.61\hspace{1em}& 1.91\hspace{1em}& 1.75\hspace{1em}& 1.73\\ {} 1.64\hspace{1em}& 1.27\hspace{1em}& 1.36\hspace{1em}& 0.8\\ {} 1.62\hspace{1em}& 1.26\hspace{1em}& 1.73\hspace{1em}& 1.28\end{array}\right],\\ {} & {\big[\psi \big({\tilde{\Theta }_{ij}^{(3)}}\big)\big]_{5\times 4}}=\left[\begin{array}{c@{\hskip4.0pt}c@{\hskip4.0pt}c@{\hskip4.0pt}c}0.96\hspace{1em}& 1.47\hspace{1em}& 1.67\hspace{1em}& 1.91\\ {} 1.5\hspace{1em}& 1.69\hspace{1em}& 1.73\hspace{1em}& 1.88\\ {} 1.37\hspace{1em}& 1.91\hspace{1em}& 1.83\hspace{1em}& 1.62\\ {} 1.57\hspace{1em}& 1.51\hspace{1em}& 1.68\hspace{1em}& 1.01\\ {} 1.53\hspace{1em}& 1.33\hspace{1em}& 1.6\hspace{1em}& 1.45\end{array}\right].\end{aligned}\]
Step 5: According to Eq. (25), the weights ${\eta _{d}}(d=1(1)3)$ of DEs are utilized to calculate the power weights ${\Omega _{ij}^{(d)}}$ ($i=1(1)5$; $j=1(1)4$; $d=1(1)3$) associated with the qROFNs and the following matrices are obtained:
\[\begin{aligned}{}& {\big[{\Omega _{ij}^{(1)}}\big]_{5\times 4}}=\left[\begin{array}{c@{\hskip4.0pt}c@{\hskip4.0pt}c@{\hskip4.0pt}c}0.355722\hspace{1em}& 0.353671\hspace{1em}& 0.356354& 0.342887\\ {} 0.322236\hspace{1em}& 0.346041\hspace{1em}& 0.354751& 0.352941\\ {} 0.344746\hspace{1em}& 0.342887\hspace{1em}& 0.346647& 0.339883\\ {} 0.342685\hspace{1em}& 0.357397\hspace{1em}& 0.367555& 0.365265\\ {} 0.346871\hspace{1em}& 0.334005\hspace{1em}& 0.354217& 0.328414\end{array}\right],\\ {} & {\big[{\Omega _{ij}^{(2)}}\big]_{5\times 4}}=\left[\begin{array}{c@{\hskip4.0pt}c@{\hskip4.0pt}c@{\hskip4.0pt}c}0.427751\hspace{1em}& 0.381931\hspace{1em}& 0.393786\hspace{1em}& 0.404378\\ {} 0.420879\hspace{1em}& 0.408609\hspace{1em}& 0.398190\hspace{1em}& 0.394958\\ {} 0.418018\hspace{1em}& 0.404378\hspace{1em}& 0.397614\hspace{1em}& 0.412620\\ {} 0.408669\hspace{1em}& 0.379996\hspace{1em}& 0.369906\hspace{1em}& 0.373832\\ {} 0.407307\hspace{1em}& 0.405018\hspace{1em}& 0.404819\hspace{1em}& 0.401762\end{array}\right],\\ {} & {\big[{\Omega _{ij}^{(3)}}\big]_{5\times 4}}=\left[\begin{array}{c@{\hskip4.0pt}c@{\hskip4.0pt}c@{\hskip4.0pt}c}0.216527\hspace{1em}& 0.264397\hspace{1em}& 0.249859\hspace{1em}& 0.252736\\ {} 0.256884\hspace{1em}& 0.245348\hspace{1em}& 0.247059\hspace{1em}& 0.252101\\ {} 0.237237\hspace{1em}& 0.252736\hspace{1em}& 0.255738\hspace{1em}& 0.247497\\ {} 0.248645\hspace{1em}& 0.262607\hspace{1em}& 0.262539\hspace{1em}& 0.260903\\ {} 0.245822\hspace{1em}& 0.260977\hspace{1em}& 0.240964\hspace{1em}& 0.269824\end{array}\right].\end{aligned}\]
Step 6: According to the qROFIPWA operator expressed by Eq. (26), the matrices ${\tilde{\mathrm{\Im }}_{d}}={[{\tilde{\Theta }_{ij}^{(d)}}]_{5\times 4}}(d=1(1)3)$ are aggregated (taking $q=2$) to form an integrated decision matrix $\mathrm{\Im }\ast ={[{\Theta _{ij}}]_{5\times 4}}$, as shown in the following Table 2.
Step 7: The supports are calculated as $\textit{Supp}({\Theta _{ij}},{\Theta _{iy}})(j,y=1(1)4$; ($j\ne y$) using Eq. (28). For sake of simplicity, the symbol ${S^{jy}}(j\ne y;j,y=1(1)4)$ is used to represent $\textit{Supp}({\Theta _{ij}},{\Theta _{iy}})$ ($j,y=1(1)4$; $j\ne y$) and the following values are obtained:
\[\begin{aligned}{}& {S^{12}}={S^{21}}=(0.612624,0.784901,0.535690,0.842471,0.909825),\\ {} & {S^{13}}={S^{31}}=(0.639889,0.840685,0.556273,0.931132,0.815082),\\ {} & {S^{14}}={S^{41}}=(0.662497,0.784869,0.565558,0.926220,0.896442),\\ {} & {S^{23}}={S^{32}}=(0.934186,0.843704,0.847702,0.842471,0.815082),\\ {} & {S^{24}}={S^{42}}=(0.781172,0.716582,0.789510,0.842471,0.896442),\\ {} & {S^{34}}={S^{43}}=(0.808437,0.772366,0.810093,0.846851,0.815082).\end{aligned}\]
Step 8: According to Eq. (29) the values $\psi ({\Theta _{ij}})$ ($i=1(1)5$; $j=1(1)4$) are calculated, as presented in the following matrix:
\[ \psi =\left[\begin{array}{c@{\hskip4.0pt}c@{\hskip4.0pt}c@{\hskip4.0pt}c}1.915011\hspace{1em}& 2.327982\hspace{1em}& 2.382513\hspace{1em}& 2.252106\\ {} 2.410456\hspace{1em}& 2.345187\hspace{1em}& 2.456756\hspace{1em}& 2.273818\\ {} 1.657522\hspace{1em}& 2.172903\hspace{1em}& 2.214069\hspace{1em}& 2.165161\\ {} 2.699824\hspace{1em}& 2.527413\hspace{1em}& 2.620454\hspace{1em}& 2.615542\\ {} 2.621349\hspace{1em}& 2.621349\hspace{1em}& 2.445245\hspace{1em}& 2.607966\end{array}\right].\]
Step 9: The power weights ${\Omega _{ij}}$ ($i=1(1)5$; $j=1(1)4$) are calculated using Eq. (30). These values are presented in the following matrix:
\[ \Omega =\left[\begin{array}{c@{\hskip4.0pt}c@{\hskip4.0pt}c@{\hskip4.0pt}c}0.1804\hspace{1em}& 0.1030\hspace{1em}& 0.3140\hspace{1em}& 0.4026\\ {} 0.2028\hspace{1em}& 0.0995\hspace{1em}& 0.3083\hspace{1em}& 0.3894\\ {} 0.1726\hspace{1em}& 0.1030\hspace{1em}& 0.3131\hspace{1em}& 0.4112\\ {} 0.2041\hspace{1em}& 0.0973\hspace{1em}& 0.2996\hspace{1em}& 0.3990\\ {} 0.2033\hspace{1em}& 0.1016\hspace{1em}& 0.2901\hspace{1em}& 0.4050\end{array}\right].\]
Step 10: Based on the qROFIPWAMSM operator expressed by Eq. (31), the final aggregated qROFNs are derived (taking $q=2$, $r=2$), as given by:
\[\begin{aligned}{}& {\Theta _{1}}=\langle 0.44483774,0.564039033\rangle ,\\ {} & {\Theta _{2}}=\langle 0.509610528,0.351422694\rangle ,\\ {} & {\Theta _{3}}=\langle 0.532907336,0.302081533\rangle ,\\ {} & {\Theta _{4}}=\langle 0.531043417,0.491528853\rangle ,\\ {} & {\Theta _{5}}=\langle 0.536489152,0.414240749\rangle .\end{aligned}\]
Step 11: The scores ${V_{i}}=V({\Theta _{i}})(i=1(1)5)$ are calculated by utilizing Eq. (1), as follows:
\[ {V_{1}}=-0.1202,\hspace{1em}{V_{2}}=0.1362,\hspace{1em}{V_{3}}=0.1927,\hspace{1em}{V_{4}}=0.0404,\hspace{1em}{V_{5}}=0.1162.\]
Step 12: Since ${V_{3}}>{V_{2}}>{V_{5}}>{V_{4}}>{V_{1}}$, the priority order is ${X_{3}}\succ {X_{2}}>{X_{5}}>{X_{4}}>{X_{2}}$, hence, the most suitable alternative is ${X_{3}}$.Table 2
Aggregated normalized decision matrix.
| Alternative | ${L_{1}}$ | ${L_{2}}$ | ${L_{3}}$ | ${L_{4}}$ |
| ${X_{1}}$ | $\langle 0.457546,0.727680\rangle $ | $\langle 0.377018,0.456022\rangle $ | $\langle 0.411592,0.370298\rangle $ | $\langle 0.438196,0.600809\rangle $ |
| ${X_{2}}$ | $\langle 0.516059,0.345696\rangle $ | $\langle 0.578450,0.442128\rangle $ | $\langle 0.528034,0.422266\rangle $ | $\langle 0.464399,0.226244\rangle $ |
| ${X_{3}}$ | $\langle 0.710191,0.264441\rangle $ | $\langle 0.438196,0.200154\rangle $ | $\langle 0.438588,0.246261\rangle $ | $\langle 0.500551,0.375744\rangle $ |
| ${X_{4}}$ | $\langle 0.461582,0.510394\rangle $ | $\langle 0.443096,0.594864\rangle $ | $\langle 0.5,0.425597\rangle $ | $\langle 0.578171,0.513749\rangle $ |
| ${X_{5}}$ | $\langle 0.526362,0.432299\rangle $ | $\langle 0.437910,0.502261\rangle $ | $\langle 0.562724,0.362961\rangle $ | $\langle 0.534898,0.427268\rangle $ |
If the proposed qROFIPWG operator is applied in Step 6 and the proposed qROFIPWGMSM operator is applied in Step 10, then the following values are obtained:
Since ${V_{3}}>{V_{2}}>{V_{5}}>{V_{4}}>{V_{1}}$, the priority order of the alternatives is ${X_{3}}\succ {X_{2}}>{X_{5}}>{X_{4}}>{X_{2}}$, hence, the optimal choice is ${X_{3}}$.
5.3 Effects of the Parameter ‘p’ on Ranking Orders
Table 3
Effects ofthe parameter p when the operators qROFIPWA and qROFIPWAMSM are used.
| Parameter | Score value | Ranking order |
| $p=1$ | ${V_{1}}=-0.1497$, ${V_{2}}=0.1332$, ${V_{3}}=0.1849$, ${V_{4}}=0.0222$, ${V_{5}}=0.1092$ | ${X_{3}}\succ {X_{2}}>{X_{5}}>{X_{4}}>{X_{2}}$ |
| $p=2$ | ${V_{1}}=-0.1202$, ${V_{2}}=0.1362$, ${V_{3}}=0.1927$, ${V_{4}}=0.0404$, ${V_{5}}=0.1162$ | ${X_{3}}\succ {X_{2}}>{X_{5}}>{X_{4}}>{X_{2}}$ |
| $p=3$ | ${V_{1}}=-0.1040$, ${V_{2}}=0.1388$, ${V_{3}}=0.2002$, ${V_{4}}=0.0579$, ${V_{5}}=0.1242$ | ${X_{3}}\succ {X_{2}}>{X_{5}}>{X_{4}}>{X_{2}}$ |
| $p=4$ | ${V_{1}}=-0.0641$, ${V_{2}}=0.0307$, ${V_{3}}=0.0484$, ${V_{4}}=-0.0109$, ${V_{5}}=0.0199$ | ${X_{3}}\succ {X_{2}}>{X_{5}}>{X_{4}}>{X_{2}}$ |
Here, all possible values of p are considered in the proposed MCGDM technique to get the solution of the case study, as discussed in Section 5.1 (taking $q=2$). To illustrate the impact of ‘p’ upon priority order, qROFIPWA operator is used in Step 6 and the proposed qROFIPWAMSM operator is used in Step 10. The related score values of alternatives and their priority position for various values of ‘p’ (taking $q=2$) are presented in Table 3. To illustrate the effect of ‘p’ upon priority order, qROFIPWG operator is utilized in Step 6 and the proposed qROFIPWGMSM operator is used in Step 10. The related score values of alternatives and their priority position for various values of ‘p’ (taking $q=2$) are presented in Table 4. With the increasing value of p increases, the priority order of alternatives changes in couple of cases due to the fact that the developed methodology considers interrelationships among criteria, but the best alternative (A3) remains unaltered for any value of p when $q=2$. For the case study presented in Section 5.1, four criteria are considered. So, maximum possible integral value of p is 4. When $p=1$, all the criteria are independent. For $p=2$, pairs of criteria are dependent, and for $p=3$, any of the three criteria will be interrelated. But for $p=4$, all the four criteria will be dependent. Depending on the given number of dependent criteria, expert/decision-maker will choose appropriate value of the parameter p.
Table 4
Effects ofthe parameter p when the operators qROFIPWG and qROFIPWGMSM are used.
| Parameter | Score value | Ranking order |
| $p=1$ | ${V_{1}}=0.0173$, ${V_{2}}=0.0043$, ${V_{3}}=0.0014$, ${V_{4}}=0.0020$, ${V_{5}}=-0.0017$ | ${X_{1}}\succ {X_{2}}>{X_{4}}>{X_{3}}>{X_{5}}$ |
| $p=2$ | ${V_{1}}=-0.0788$, ${V_{2}}=0.1435$, ${V_{3}}=0.1823$, ${V_{4}}=-0.0452$, ${V_{5}}=0.1155$ | ${X_{3}}\succ {X_{2}}>{X_{5}}>{X_{4}}>{X_{2}}$ |
| $p=3$ | ${V_{1}}=-0.1772$, ${V_{2}}=0.0349$, ${V_{3}}=0.1132$, ${V_{4}}=-0.1631$, ${V_{5}}=0.0176$ | ${X_{3}}\succ {X_{2}}>{X_{5}}>{X_{4}}>{X_{2}}$ |
| $p=4$ | ${V_{1}}=-0.1280$, ${V_{2}}=0.0609$, ${V_{3}}=0.0883$, ${V_{4}}=-0.0983$, ${V_{5}}=0.0538$ | ${X_{3}}\succ {X_{2}}>{X_{5}}>{X_{4}}>{X_{2}}$ |
5.4 Comparative Analysis with Existing Methods
To verify the effectiveness of our developed methodology based on the developed operators, an investigation has been conducted for the purpose of comparison between the existing methods of Jana et al. (2019b) and qROF Dombi weighted averaging (qROFDWA) operator; Wei et al. (2018) and qROF generalized weighted Heronian mean (qROFGWHM) operator and qROF generalized weighted geometric Heronian mean (qROFGWGHM) operator; Liu and Liu (2018) and qROF weighted Bonferroni mean (qROFWBM) operator; Yang and Pang (2020) and qROF weighted Bonferroni Dombi averaging (qROFWBMDA) operator; Liu and Wang (2018) and qROF weighted averaging (qROFWA) operator; Garg and Chen (2020) and qROF weighted neutrality (qROFWN) operator, and Liu et al. (2020) with qROF power weighted MSM (qROFPWMSM) operator. These methods are applied to the same case study presented at the beginning of Section 5. In Table 5, priority values of the considered alternatives are presented along with their ranking order. From Table 5, it is found that the ranking order obtained by our proposed method is exactly the same as obtained by other existing methods (Liu and Liu, 2018; Liu and Wang, 2018; Wei et al., 2018; Jana et al., 2019b; Yang and Pang, 2020; Garg and Chen, 2020; Liu et al., 2020). Hence, the developed methodology based on the proposed operators is effective and feasible.
Table 5
Comparison: existing vs. proposed (taking $q=2$).
| Method | Score value | Ranking order |
| Jana et al. (2019b) with qROFDWA operator | ${V_{1}}=-0.0212$, ${V_{2}}=0.1885$, ${V_{3}}=0.2743$, ${V_{4}}=0.0522$, ${V_{5}}=0.1233$ | ${X_{3}}\succ {X_{2}}\succ {X_{5}}\succ {X_{4}}\succ {X_{1}}$ |
| Wei et al. (2018) with qROFGWHM operator | ${V_{1}}=-0.2942$, ${V_{2}}=-0.0703$, ${V_{3}}=-0.0272$, ${V_{4}}=-0.1913$, ${V_{5}}=-0.1324$ | ${X_{3}}\succ {X_{2}}\succ {X_{5}}\succ {X_{4}}\succ {X_{1}}$ |
| Wei et al. (2018) with qROFGWGHM operator | ${V_{1}}=0.0884$, ${V_{2}}=0.3162$, ${V_{3}}=0.3317$, ${V_{4}}=0.2660$, ${V_{5}}=0.3294$ | ${X_{3}}\succ {X_{2}}\succ {X_{5}}\succ {X_{4}}\succ {X_{1}}$ |
| Liu and Liu (2018) with qROFWBM operator | ${V_{1}}=-0.7144$, ${V_{2}}=-0.5922$, ${V_{3}}=-0.5780$, ${V_{4}}=-0.6416$, ${V_{5}}=-6163$ | ${X_{3}}\succ {X_{2}}\succ {X_{5}}\succ {X_{4}}\succ {X_{1}}$ |
| Yang and Pang (2020) with qROFWBMDA operator | ${V_{1}}=-0.4993$, ${V_{2}}=-0.2204$, ${V_{3}}=-0.0846$, ${V_{4}}=-0.4227$, ${V_{5}}=-0.2986$ | ${X_{3}}\succ {X_{2}}\succ {X_{5}}\succ {X_{4}}\succ {X_{1}}$ |
| Liu and Wang (2018) with qROFWA operator | ${V_{1}}=-0.0919$, ${V_{2}}=0.1568$, ${V_{3}}=0.2051$, ${V_{4}}=0.0309$, ${V_{5}}=0.1130$ | ${X_{3}}\succ {X_{2}}\succ {X_{5}}\succ {X_{4}}\succ {X_{1}}$ |
| Garg and Chen (2020) with qROFWNA operator | ${V_{1}}=0.5931$, ${V_{2}}=0.7337$, ${V_{3}}=0.7525$, ${V_{4}}=0.6783$, ${V_{5}}=0.7411$ | ${X_{3}}\succ {X_{2}}\succ {X_{5}}\succ {X_{4}}\succ {X_{1}}$ |
| Liu et al. (2020) with qROFPWMSM operator | ${V_{1}}=-0.1154$, ${V_{2}}=0.1143$, ${V_{3}}=0.1656$, ${V_{4}}=0.0348$, ${V_{5}}=0.0934$ | ${X_{3}}\succ {X_{2}}\succ {X_{5}}\succ {X_{4}}\succ {X_{1}}$ |
| Proposed method with qROFIPWA operator and qROFIPWAMSM operator | ${V_{1}}=-0.1202$, ${V_{2}}=0.1362$, ${V_{3}}=0.1927$, ${V_{4}}=0.0404$, ${V_{5}}=0.1162$ | ${X_{3}}\succ {X_{2}}\succ {X_{5}}\succ {X_{4}}\succ {X_{1}}$ |
| Proposed method with qROFIPWG operator and qROFIPWGMSM operator | ${V_{1}}=-0.0788$, ${V_{2}}=0.1435$, ${V_{3}}=0.1823$, ${V_{4}}=-0.0452$, ${V_{5}}=0.1156$ | ${X_{3}}\succ {X_{2}}\succ {X_{5}}\succ {X_{4}}\succ {X_{1}}$ |
5.5 Comparative Analysis Based on Biasness of Experts
When evaluating alternatives in a realistic decision-making environment, experts may attempt to manipulate some initial data due to an inclination or biasness toward a particular alternative. As a result, the ranking order of alternatives may change. To reflect the actual situation, the case study presented in Section 5.1 must be modified in order to demonstrate the biased nature of experts. Assume that expert ${D_{2}}$ prefers alternative ${X_{2}}$ and has some reservations about alternative ${X_{3}}$ to an extent that the criteria value ${\tilde{\Theta }_{31}^{(2)}}$ changes to $\langle 0.1,0.1\rangle $ from $\langle 0.7,0.3\rangle $ and the criteria value ${\tilde{\Theta }_{24}^{(2)}}$ changes to $\langle 0.4,0.2\rangle $ from $\langle 0.4,0.3\rangle $ due to biased nature of the expert D2. The remaining assessment values remain the same as shown in Tables 1. The outcomes from various existing methods are recorded in Table 6 (for $q=2$).
Table 6
Comparison: existing vs. proposed (taking $q=2$).
| Method | Score value | Ranking order |
| Jana et al. (2019b) with qROFDWA operator | ${V_{1}}=-0.0343$, ${V_{2}}=0.2170$, ${V_{3}}=0.2042$, ${V_{4}}=0.0133$, ${V_{5}}=0.1445$ | ${X_{2}}\succ {X_{3}}\succ {X_{5}}\succ {X_{1}}\succ {X_{4}}$ |
| Wei et al. (2018) with qROFGWHM operator | ${V_{1}}=-0.2538$, ${V_{2}}=-0.0407$, ${V_{3}}=-0.0642$, ${V_{4}}=-0.2449$, ${V_{5}}=-0.1066$ | ${X_{2}}\succ {X_{3}}\succ {X_{5}}\succ {X_{4}}\succ {X_{1}}$ |
| Wei et al. (2018) with qROFGWGHM operator | ${V_{1}}=0.1324$, ${V_{2}}=0.3172$, ${V_{3}}=0.3312$, ${V_{4}}=0.2084$, ${V_{5}}=0.3526$ | ${X_{5}}\succ {X_{3}}\succ {X_{2}}\succ {X_{4}}\succ {X_{1}}$ |
| Liu and Liu (2018) with qROFWBM operator | ${V_{1}}=-0.7144$, ${V_{2}}=-0.5845$, ${V_{3}}=-0.5640$, ${V_{4}}=-0.6416$, ${V_{5}}=-6163$ | ${X_{3}}\succ {X_{2}}\succ {X_{5}}\succ {X_{4}}\succ {X_{1}}$ |
| Yang and Pang (2020) with qROFWBMDA operator | ${V_{1}}=-0.4570$, ${V_{2}}=-0.1966$, ${V_{3}}=-0.0703$, ${V_{4}}=-0.44442$, ${V_{5}}=-0.2833$ | ${X_{3}}\succ {X_{2}}\succ {X_{5}}\succ {X_{4}}\succ {X_{1}}$ |
| Liu and Wang (2018) with qROFWA operator | ${V_{1}}=-0.0405$, ${V_{2}}=0.1789$, ${V_{3}}=0.1774$, ${V_{4}}=-0.0122$, ${V_{5}}=0.1350$ | ${X_{2}}\succ {X_{3}}\succ {X_{5}}\succ {X_{4}}\succ {X_{1}}$ |
| Garg and Chen (2020) with qROFWNA operator | ${V_{1}}=0.5930$, ${V_{2}}=0.7395$, ${V_{3}}=0.7113$, ${V_{4}}=0.6782$, ${V_{5}}=0.7410$ | ${X_{5}}\succ {X_{3}}\succ {X_{2}}\succ {X_{4}}\succ {X_{1}}$ |
| Liu et al. (2020) with qROFPWMSM operator | ${V_{1}}=-0.1154$, ${V_{2}}=0.1161$, ${V_{3}}=0.1556$, ${V_{4}}=0.0348$, ${V_{5}}=0.0934$ | ${X_{3}}\succ {X_{2}}\succ {X_{5}}\succ {X_{4}}\succ {X_{1}}$ |
| Proposed method with qROFIPWA operator and qROFIPWAMSM operator | ${V_{1}}=-0.1202$, ${V_{2}}=0.1422$, ${V_{3}}=0.1645$, ${V_{4}}=0.0404$, ${V_{5}}=0.1162$ | ${X_{3}}\succ {X_{2}}\succ {X_{5}}\succ {X_{4}}\succ {X_{1}}$ |
| Proposed method with qROFIPWG operator and qROFIPWGMSM operator | ${V_{1}}=-0.0788$, ${V_{2}}=0.1523$, ${V_{3}}=0.1562$, ${V_{4}}=-0.0452$, ${V_{5}}=0.1155$ | ${X_{3}}\succ {X_{2}}\succ {X_{5}}\succ {X_{4}}\succ {X_{1}}$ |
Table 7
Initial assessment results of the experts.
| Expert | Alternative | ${L_{1}}$ | ${L_{2}}$ | ${L_{3}}$ | ${L_{4}}$ |
| ${D_{1}}$ | X1 | $\langle 0.5,0\rangle $ | $\langle 0.4,0.4\rangle $ | $\langle 0.4,0.3\rangle $ | $\langle 0.5,0\rangle $ |
| ${X_{2}}$ | $\langle 0.7,0.2\rangle $ | $\langle 0.51,0\rangle $ | $\langle 0.5,0.1\rangle $ | $\langle 0.6,0.2\rangle $ | |
| ${X_{3}}$ | $\langle 0.3,0.2\rangle $ | $\langle 0.38,0.4\rangle $ | $\langle 0.6,0\rangle $ | $\langle 0.4,0.3\rangle $ | |
| ${X_{4}}$ | $\langle 0.5,0\rangle $ | $\langle 0.502,0.4\rangle $ | $\langle 0.5,0.3\rangle $ | $\langle 0.5,0\rangle $ | |
| ${X_{5}}$ | $\langle 0.6,0.1\rangle $ | $\langle 0.3,0\rangle $ | $\langle 0.6,0.2\rangle $ | $\langle 0.6,0.3\rangle $ | |
| ${X_{1}}$ | $\langle 0.5,0.3\rangle $ | $\langle 0.5,0\rangle $ | $\langle 0.6,0.3\rangle $ | $\langle 0.6,0.3\rangle $ | |
| ${D_{2}}$ | ${X_{2}}$ | $\langle 0.6,0\rangle $ | $\langle 0.6,0.3\rangle $ | $\langle 0.206,0.2\rangle $ | $\langle 0.4,0\rangle $ |
| ${X_{3}}$ | $\langle 0.4,0.3\rangle $ | $\langle 0.7,0\rangle $ | $\langle 0.35,0.2\rangle $ | $\langle 0.6,0.4\rangle $ | |
| ${X_{4}}$ | $\langle 0.45,0.4\rangle $ | $\langle 0.5,0.2\rangle $ | $\langle 0.6,0\rangle $ | $\langle 0.5,0.2\rangle $ | |
| ${X_{5}}$ | $\langle 0.4,0\rangle $ | $\langle 0.6,0.3\rangle $ | $\langle 0.4,0.4\rangle $ | $\langle 0.6,0\rangle $ | |
| ${X_{1}}$ | $\langle 0.5,0.3\rangle $ | $\langle 0.6,0.3\rangle $ | $\langle 0.5,0\rangle $ | $\langle 0.6,0.3\rangle $ | |
| ${D_{3}}$ | ${X_{2}}$ | $\langle 0.6,0.1\rangle $ | $\langle 0.49,0.2\rangle $ | $\langle 0.7,0\rangle $ | $\langle 0.2,0.2\rangle $ |
| ${X_{3}}$ | $\langle 0.7,0\rangle $ | $\langle 0.6,0.2\rangle $ | $\langle 0.71,0.4\rangle $ | $\langle 0.4,0\rangle $ | |
| ${X_{4}}$ | $\langle 0.5,0.3\rangle $ | $\langle 0.7,0\rangle $ | $\langle 0.5,0.2\rangle $ | $\langle 0.6,0.1\rangle $ | |
| ${X_{5}}$ | $\langle 0.62,0.3\rangle $ | $\langle 0.6,0.3\rangle $ | $\langle 0.3,0\rangle $ | $\langle 0.4,0.2\rangle $ |
Table 6 shows that changing the criteria values has a significant effect on the ranking order of alternatives for the related existing methods (Liu and Wang, 2018; Wei et al., 2018; Jana et al., 2019b; Garg and Chen, 2020). The priority order of alternatives acquired by Jana et al. (2019b) with qROFDWA operator is changed from ${X_{3}}\succ {X_{2}}>{X_{5}}>{X_{4}}>{X_{2}}$ to ${X_{2}}\succ {X_{3}}>{X_{5}}>{X_{1}}>{X_{4}}$ such that best alternative is transformed from the alternative X3 to the alternative X2. The best alternative is changed from the alternative X3 to the alternative X2 and the ranking order is changed from ${X_{3}}\succ {X_{2}}>{X_{5}}>{X_{4}}>{X_{2}}$ to ${X_{2}}\succ {X_{3}}>{X_{5}}>{X_{4}}>{X_{1}}$ if the approach of Wei et al. (2018) is used with qROFGWHM operator. In addition, the priority order generated by Wei et al. (2018) with qROFGWGHM operator is changed from ${X_{3}}\succ {X_{2}}>{X_{5}}>{X_{4}}>{X_{2}}$ to ${X_{5}}\succ {X_{3}}>{X_{2}}>{X_{4}}>{X_{1}}$ and the best alternative is transformed from the alternative X3 to the alternative X5. The best alternative changes from X3 to X2 and the related ranking output changes from ${X_{3}}\succ {X_{2}}>{X_{5}}>{X_{4}}>{X_{2}}$ to ${X_{2}}\succ {X_{3}}>{X_{5}}>{X_{4}}>{X_{1}}$ when the method of Liu and Wang (2018) is utilized with qROFWA operator. Moreover, the ranking order generated by Garg and Chen (2020) with qROFNWA operator changes from ${X_{3}}\succ {X_{2}}>{X_{5}}>{X_{4}}>{X_{2}}$ to ${X_{5}}\succ {X_{3}}>{X_{2}}>{X_{4}}>{X_{1}}$, and the best alternative changes from X3 to X5. Thus the existing methods (Wei et al., 2018; Liu and Wang, 2018; Jana et al., 2019b; Garg and Chen, 2020) are unreasonable for the reason that the best alternative changes due to the biased nature of the expert ${D_{2}}$. However, the methods of Liu and Liu (2018) with qROFWBM operator, Yang and Pang (2020) with qROFWBMDA operator and Liu et al. (2020) with qROFPWMSM) operator and the developed approach still have rational and unaltered ranking of the alternatives. The methods of Liu and Liu (2018), Yang and Pang (2020), Liu et al. (2020) and the developed approach can diminish the impact of unreasonable assessment criteria values from a biased expert.
To show the disadvantages of the methods of Liu and Liu (2018), Yang and Pang (2020), Liu et al. (2020), the same case study is considered with the initial assessment matrix, as given in Table 7. The ranking outcomes are presented in Table 8. From this table, it follows that the priority order of alternatives, as given by Liu and Liu (2018), are unreasonable due to the fact that they fail to distinguish the priority of alternatives. The method of Yang and Pang (2020) and the proposed approach continue to have a reasonable ranking, and the best alternative remains the same for these two approaches. This implies that both approaches are capable of mitigating the effects of unreasonable assessment criteria values provided by a biased expert.
Table 8
| Method | Score value | Ranking order |
| Liu and Liu (2018) with qROFWBM operator | ${V_{1}}=0.019$, ${V_{2}}=0.019$, ${V_{3}}=0.019$, ${V_{4}}=0.019$, ${V_{5}}=0.019$ | ${X_{1}}={X_{2}}={X_{3}}={X_{4}}={X_{5}}$ |
| Yang and Pang (2020) with qROFWBMDA operator | ${V_{1}}=-0.0749$, ${V_{2}}=0.0459$, ${V_{3}}=-0.0633$, ${V_{4}}=-0.0262$, ${V_{5}}=-0.0352$ | ${X_{2}}\succ {X_{4}}\succ {X_{5}}\succ {X_{3}}\succ {X_{1}}$ |
| Liu et al. (2020) with qROFPWMSM operator | ${V_{1}}=0.264$, ${V_{2}}=0.264$, ${V_{3}}=0.264$, ${V_{4}}=0.264$, ${V_{5}}=0.264$ | ${X_{1}}={X_{2}}={X_{3}}={X_{4}}={X_{5}}$ |
| Proposed method with qROFIPWA operator and qROFIPWAMSM operator | ${V_{1}}=0.2204$, ${V_{2}}=0.2458$, ${V_{3}}=0.2010$, ${V_{4}}=0.2327$, ${V_{5}}=0.2278$ | ${X_{2}}\succ {X_{4}}\succ {X_{5}}\succ {X_{1}}\succ {X_{3}}$ |
| Proposed method with qROFIPWG operator and qROFIPWGMSM operator | ${V_{1}}=0.2165$, ${V_{2}}=0.2463$, ${V_{3}}=0.2031$, ${V_{4}}=0.2297$, ${V_{5}}=0.2257$ | ${X_{2}}\succ {X_{4}}\succ {X_{5}}\succ {X_{1}}\succ {X_{3}}$ |
Table 9
Initial assessment results of the experts.
| Expert | Alternative | ${L_{1}}$ | ${L_{2}}$ | ${L_{3}}$ | ${L_{4}}$ |
| ${D_{1}}$ | X1 | $\langle 1,0\rangle $ | $\langle 0.4,0.4\rangle $ | $\langle 0.4,0.3\rangle $ | $\langle 1,0\rangle $ |
| ${X_{2}}$ | $\langle 0.7,0.2\rangle $ | $\langle 0.1,0\rangle $ | $\langle 0.5,0.1\rangle $ | $\langle 0.6,0.2\rangle $ | |
| ${X_{3}}$ | $\langle 1,0\rangle $ | $\langle 0.5,0.4\rangle $ | $\langle 1,0\rangle $ | $\langle 0.4,0.3\rangle $ | |
| ${X_{4}}$ | $\langle 0.5,0.3\rangle $ | $\langle 0.5,0.4\rangle $ | $\langle 0.5,0.3\rangle $ | $\langle 1,0\rangle $ | |
| ${X_{5}}$ | $\langle 0.6,0.1\rangle $ | $\langle 1,0\rangle $ | $\langle 0.6,0.2\rangle $ | $\langle 0.6,0.3\rangle $ | |
| ${X_{1}}$ | $\langle 0.5,0.3\rangle $ | $\langle 1,0\rangle $ | $\langle 0.6,0.3\rangle $ | $\langle 0.6,0.3\rangle $ | |
| ${D_{2}}$ | ${X_{2}}$ | $\langle 1,0\rangle $ | $\langle 0.6,0.3\rangle $ | $\langle 1,0\rangle $ | $\langle 0.4,0.5\rangle $ |
| ${X_{3}}$ | $\langle 0.4,0.3\rangle $ | $\langle 0.7,0.2\rangle $ | $\langle 0.4,0.2\rangle $ | $\langle 1,0\rangle $ | |
| ${X_{4}}$ | $\langle 0.5,0.4\rangle $ | $\langle 1,0\rangle $ | $\langle 0.6,0.3\rangle $ | $\langle 0.5,0.4\rangle $ | |
| ${X_{5}}$ | $\langle 1,0\rangle $ | $\langle 0.6,0.3\rangle $ | $\langle 1,0\rangle $ | $\langle 0.6,0.4\rangle $ | |
| ${X_{1}}$ | $\langle 0.5,0.3\rangle $ | $\langle 0.6,0.3\rangle $ | $\langle 1,0\rangle $ | $\langle 0.6,0.3\rangle $ | |
| ${D_{3}}$ | ${X_{2}}$ | $\langle 1,0\rangle $ | $\langle 0.5,0.2\rangle $ | $\langle 0.7,0.2\rangle $ | $\langle 1,0\rangle $ |
| ${X_{3}}$ | $\langle 0.7,0.2\rangle $ | $\langle 1,0\rangle $ | $\langle 0.8,0.2\rangle $ | $\langle 0.4,0.5\rangle $ | |
| ${X_{4}}$ | $\langle 1,0\rangle $ | $\langle 0.7,0.1\rangle $ | $\langle 1,0\rangle $ | $\langle 0.6,0.1\rangle $ | |
| ${X_{5}}$ | $\langle 0.6,0.3\rangle $ | $\langle 0.6,0.3\rangle $ | $\langle 0.3,0.5\rangle $ | $\langle 1,0\rangle $ |
The same case study was examined again using the initial assessment matrix, as shown in Table 9. Table 10 exhibits the ranking results. According to Table 10, the ranking results of the alternatives obtained by Liu et al. (2020) are unreasonable because they fail to distinguish the priority of alternatives. The results obtained by Yang and Pang (2020) and using the proposed approach still have a reasonable ranking and the best alternative does not change for these two approaches. This suggests that both the approaches can mitigate the effects of unreasonable assessment criteria values from a biased expert.
It is known that when all experts give the same assessment values and if all the experts have the same importance, then a MCGDM problem reduces to a MCDM problem. Suppose the initial assessment matrix for the case study is given in Table 11. Table 12 shows the scores and priorities of the alternatives. The results revealed that the method of Yang and Pang (2020) failed to generate score values and alternative preference order, as well as to solve certain decision-making problems, rendering it inefficient. However, the developed methodology is capable of producing accurate ranking of alternatives.
Table 10
| Method | Score value | Ranking order |
| Yang and Pang (2020) with qROFWBMDA operator | ${V_{1}}=0.6133$, ${V_{2}}=0.8372$, ${V_{3}}=0.6602$, ${V_{4}}=0.5890$, ${V_{5}}=0.6265$ | ${X_{2}}\succ {X_{3}}\succ {X_{5}}\succ {X_{1}}\succ {X_{4}}$ |
| Liu et al. (2020) with qROFPWMSM operator | ${V_{1}}={V_{2}}={V_{3}}={V_{4}}={V_{5}}=1$. | ${X_{1}}={X_{2}}={X_{3}}={X_{4}}={X_{5}}$ |
| Proposed method with qROFIPWG operator and qROFIPWGMSM operator | ${V_{1}}=0.5623$, ${V_{2}}=0.7078$, ${V_{3}}=0.6156$, ${V_{4}}=0.5533$, ${V_{5}}=0.5863$ | ${X_{2}}\succ {X_{3}}\succ {X_{5}}\succ {X_{1}}\succ {X_{4}}$ |
Table 11
Initial assessment results.
| Alternative | ${L_{1}}$ | ${L_{2}}$ | ${L_{3}}$ | ${L_{4}}$ |
| ${X_{1}}$ | $\langle 0.5,0\rangle $ | $\langle 0.4,0.4\rangle $ | $\langle 0.4,0.3\rangle $ | $\langle 0.5,0.6\rangle $ |
| ${X_{2}}$ | $\langle 0.7,0.2\rangle $ | $\langle 0.5,0\rangle $ | $\langle 0.5,0.1\rangle $ | $\langle 0.6,0.4\rangle $ |
| ${X_{3}}$ | $\langle 0.3,0.6\rangle $ | $\langle 0.3,0.4\rangle $ | $\langle 0.6,0\rangle $ | $\langle 0.4,0.3\rangle $ |
| ${X_{4}}$ | $\langle 0.5,0\rangle $ | $\langle 0.7,0.5\rangle $ | $\langle 0.5,0.3\rangle $ | $\langle 0.5,0.8\rangle $ |
| ${X_{5}}$ | $\langle 0.6,0.5\rangle $ | $\langle 0.3,0\rangle $ | $\langle 0.6,0.4\rangle $ | $\langle 0.6,0.3\rangle $ |
Table 12
Comparison: proposed vs. Yang and Peng’s method (Yang and Pang, 2020) (taking $q=2$).
| Method | Score value | Ranking order |
| Yang and Pang (2020) with qROFWBMDA operator | Cannot be determined due to division by zero | Cannot be generated |
| Proposed method with qROFIPWA operator and qROFIPWAMSM operator | ${V_{1}}=0.0443$, ${V_{2}}=0.2818$, ${V_{3}}=0.0709$, ${V_{4}}=0.0453$, ${V_{5}}=0.2666$ | ${X_{2}}\succ {X_{5}}\succ {X_{3}}\succ {X_{4}}\succ {X_{1}}$ |
| Proposed method with qROFIPWG operator and qROFIPWGMSM operator | ${V_{1}}=-0.0014$, ${V_{2}}=0.2545$, ${V_{3}}=0.0589$, ${V_{4}}=-0.0926$, ${V_{5}}=0.1826$ | ${X_{2}}\succ {X_{5}}\succ {X_{3}}\succ {X_{1}}\succ {X_{4}}$ |
In real decision-making problems, the interrelationship between criteria can be seen. Methods of Wei et al. (2018), Liu and Liu (2018) and Yang and Pang (2020) can consider the dependency of two criteria, but they do not consider interrelationships between multiple criteria. There may be a situation in which all of the considered criteria are independent, and these methods are not appropriate for dealing with this type of decision-making problem, and may generate irrational preference of alternatives. Although the method of Garg and Chen (2020) is capable of mitigating the impact of some unreasonable assessing criteria values from some biased decision-makers and taking into account the dependency among multiple criteria, it fails to distinguish the priority orders of alternatives in some circumstances, as shown in Tables 7 and 8, respectively.
6 Conclusions
This paper presents a qROFS-based decision-making model to resolve the drawbacks of the existing methods. To develop the model, four operators, namely qROFIPWA, qROFIPWG, qROFIPWAMSM and qROFIPWGMSM, are proposed in this paper. The main advantages of the last two operators are: (1) they reduce the effects of outrageous assessing information from some biased experts, (ii) they consider the interrelationship among multiple number of criteria. A group decision-making methodology is developed based on these operators. The developed method can generate sensible ranking order of alternatives when among the qROF numbers considered, one qROF number has a (i) non-belongingness grade that equals to 0, or a (ii) belongingness grade that equals to 1. For the verification of feasibility of the proposed MCGDM method, one case study regarding personnel selection is considered. The superiority of the developed MCGDM approach is shown by comparison with existing approaches. The proposed method has two limitations: (i) it does not address the process of reaching consensus for large-scale decision-making, and (ii) it does not address the hesitancy of choosing membership and non-membership values. To address these issues, hesitant q-ROF based large scale decision-making with consensus reaching process can be developed in the future by extending the proposed operators. The proposed methodology can also be used to solve other decision-making problems and can be further extended by incorporating hesitant, probabilistic hesitant, linguistic, and probabilistic linguistic concepts.