Proof of Theorem 6, which we provided you in the main manuscript..

This paper proposes a new multi-criteria group decision-making (MCGDM) method utilizing q-rung orthopair fuzzy (qROF) sets, improved power weighted operators and improved power weighted Maclaurin symmetric mean (MSM) operators. The power weighted averaging operator and power weighted Maclaurin symmetric mean (MSM) operator used in the existing MCGDM methods have the drawback of being unable to distinguish the priority order of alternatives in some scenarios, especially when one of the qROF numbers being considered has a non-belongingness grade of 0 or a belongingness grade of 1. To address this limitation of existing MCGDM methods, four operators, namely qROF improved power weighted averaging (qROFIPWA), qROF improved power weighted geometric (qROFIPWG), qROF improved power weighted averaging MSM (qROFIPWAMSM) and qROF improved power weighted geometric MSM (qROFIPWGMSM), are proposed in this paper. These operators mitigate the effects of erroneous assessment of information from some biased decision-makers, making the decision-making process more reliable. Following that, a group decision-making methodology is developed that is capable of generating a reasonable ranking order of alternatives when one of the qROF numbers considered has a non-belongingness grade of 0 or a belongingness grade of 1. To investigate the applicability of the proposed approach, a case study is also presented and a comparison-based investigation is used to demonstrate the superiority of the approach.

Fuzzy sets (FSs) (Zadeh,

The

The interrelationship between multiple criteria can be seen in different realistic situations. Many of the existing studies (Liu and Wang,

To overcome the shortcomings of Liu

Some new operational laws are presented in order to fair treatment of belongingness and non-belongingness grades.

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.

A novel DM approach is developed based on the proposed operators. This proposed approach can resolve the limitations of Liu

To show the efficiency of the proposed methodology, a personnel selection problem is considered under qROF setting.

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

Some important concepts on qROFNs, basic operations between qROFNs and qROF weighted neutral AOs are highlighted as follows:

Let

Next, the hesitancy grade of

Let

Clearly,

Let

According to the score function and accuracy function, a comparison scheme of qROFNs is given as follows:

Let

If

If

if

if

Let

(Liu

The PAO, discovered by Yager (

Let

Here,

A few new operations are introduced between qROFNs and the basic laws are investigated.

Let

To understand the superiority of the developed operations, four examples are considered as follows:

Let us consider two qROFNs

Let us consider two qROFNs

Let us consider two qROFNs

Let us consider two qROFNs

From the above four examples, it is clear that our proposed operations are more sensible.

Follows from Definition

In this paper, qROF improved power weighted averaging (qROFIPWA) and qROF improved power weighted averaging MSM (qROFIPWAMSM) operators are developed as follows.

Let

In Eq. (

Straightforward. □

The following Theorems readily follow from Theorem

Let

In Eq. (

Added in the Supplementary material. □

The following Theorems readily follow from Theorem

This paper develops qROF improved power weighted geometric (qROFIPWG) operator and qROF improved power weighted geometric MSM (qROFIPWGMSM) operator as follows:

Let

In Eq. (

Straightforward. □

The following Theorems readily follow from Theorem

(Idempotency)

(Boundedness)

(Monotonicity)

Let

In Eq. (

Similar to Theorem

The following Theorems readily follow from Theorem

(Idempotency)

(Boundedness)

Suppose

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:

The Normalized decision matrix is:

The operator

The final aggregated qROF decision matrix is constructed based on the

If two score values

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

Initial assessment results of the experts.

Expert | Alternative | ||||

For each of the remaining steps,

Aggregated normalized decision matrix.

Alternative | ||||

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

Effects ofthe parameter

Parameter | Score value | Ranking order |

Here, all possible values of _{3}) remains unaltered for any value of

Effects ofthe parameter

Parameter | Score value | Ranking order |

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

Comparison: existing vs. proposed (taking

Method | Score value | Ranking order |

Jana |
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Wei |
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Wei |
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Liu and Liu ( |
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Yang and Pang ( |
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Liu and Wang ( |
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Garg and Chen ( |
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Liu |
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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 _{2}. The remaining assessment values remain the same as shown in Tables

Comparison: existing vs. proposed (taking

Method | Score value | Ranking order |

Jana |
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Wei |
||

Wei |
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Liu and Liu ( |
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Yang and Pang ( |
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Liu and Wang ( |
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Garg and Chen ( |
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Liu |
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Initial assessment results of the experts.

Expert | Alternative | ||||

X_{1} |
|||||

Table _{3} to the alternative X_{2}. The best alternative is changed from the alternative X_{3} to the alternative X_{2} and the ranking order is changed from _{3} to the alternative X_{5}. The best alternative changes from X_{3} to X_{2} and the related ranking output changes from _{3} to X_{5}. Thus the existing methods (Wei

To show the disadvantages of the methods of Liu and Liu (

Comparison: proposed vs. existing methods (Liu and Liu,

Method | Score value | Ranking order |

Liu and Liu ( |
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Yang and Pang ( |
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Liu |
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Initial assessment results of the experts.

Expert | Alternative | ||||

X_{1} |
|||||

The same case study was examined again using the initial assessment matrix, as shown in Table

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

Comparison: proposed vs. existing methods (Liu

Method | Score value | Ranking order |

Yang and Pang ( |
||

Liu |
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Initial assessment results.

Alternative | ||||

Comparison: proposed vs. Yang and Peng’s method (Yang and Pang,

Method | Score value | Ranking order |

Yang and Pang ( |
Cannot be determined due to division by zero | Cannot be generated |

In real decision-making problems, the interrelationship between criteria can be seen. Methods of Wei

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.