Some q-rung orthopair fuzzy hybrid aggregation operators and TOPSIS method for multi-attribute decision-making

Abstract

The q-rung orthopair fuzzy numbers (q-ROFNs) are used to deal with vague and uncertain information and they are superior to the intuitionistic fuzzy numbers (IFNs) and the Pythagorean fuzzy numbers (PFNs). In this paper, we introduce two operators namely q-rung orthopair fuzzy hybrid weighted arithmetic geometric aggregation (q-ROFHWAGA) operator and q-rung orthopair fuzzy hybrid ordered weighted arithmetic geometric aggregation (q-ROFHOWAGA) operator. The suggested operators q-ROFHWAGA and q-ROFHOWAGA are superior to the existing operators defined on q-ROFNs. We present an application of the proposed operator of q-ROFHWAGA to multiple-attribute decision-making (MADM) in computer numerical control (CNC) machine. Furthermore, we present TOPSIS method based on q-ROFNs for MADM in transport policy problem.

Keywords

q-rung orthopair fuzzy numbers q-rung orthopair fuzzy hybrid aggregation operators multi-attribute decision-making CNC machine

1 Introduction

For many years, the issue of vague and imperfect information has been at the forefront. Information aggregation is the key factor for the decision management in the areas of business, management, engineering, psychology, social sciences, medical sciences, and artificial intelligence. Traditionally, the information about an alternative has been believed to be a crisp number or linguistic number. Nevertheless, information can not be aggregated in a simple form due to its uncertainty. MADM is a critical framework for decision making science, the purpose of which is to identify the most exceptional goals among the most feasible ones. The person needs to assess the choices made by different types of assessment criteria, such as crisp numbers and intervals, in the actual decision-making process. However, in many cases, due to the presence of a number of data anomalies that may arise due to lack of knowledge or human error, it is difficult for a person to choose the correct choice. Consequently, in order to measure these inconsistencies and to analyze the mechanism, a large number of theories have been suggested. Zadeh [55] initiated the idea of fuzzy set with the help of membership function. Subsequently, the intuitionistic fuzzy set (IFS) was proposed by Atanassov [5] as an extension of the fuzzy set by means of the membership and the non-membership functions. Yager [47 –49] introduced Pythagorean fuzzy set (PFS) as an extension of Atanassov’s IFS. Fuzzy sets, extensions of fuzzy sets, rough sets and soft sets with real life applications have been studied by many researchers; Akram [1, 2], Ali [3, 4], Chi and Lui [6], Feng et al. [7 –9], Garg and Arora [10, 11], Hashmi et al. [13], Hashmi and Riaz [14], Jose and Kuriaskose [15], Kaur and Garg [16], Kumar and Garg [17], Karaaslan [18], Liu et al. [20], Naeem et al. [21 –23], Peng and Yang [24], Peng and Yuan [25], Peng et al. [26], Peng and Selvachandran [27], Peng and Liu [28],Yu et al. [54], Riaz et al. [29 –33], Riaz and Hashmi [34 –36], Riaz and Tehrim [37 –39], Shabir and Naz [40] and Tehrim and Riaz [42].

Xu [43 –45] initiated IF and hesitant fuzzy aggregation operators. Ye [51, 52] introduced some operators of hesitant fuzzy set. Zhan et al. [56, 57] presented the concepts of rough soft hemirings, soft rough covering. Zhang et al. [58 –61] established fuzzy soft β-covering based fuzzy rough sets and fuzzy soft coverings based fuzzy rough sets. Some hybrid weighted aggregation operators have been studied by Garg and Nancy [12], Shi and Yuan [41] and Ye [53].

In 2017, Yager introduced the idea of q-ROFS as the extension of PFS [50], in which the sum of membership degree $F_{A} (\bar{C})$ and non membership degree $T_{A} (\bar{C})$ satisfy the condition $0 \leq F_{A} (\bar{C})^{q} + T_{A} (\bar{C})^{q} \leq 1, (q \geq 1)$ and degree of indeterminacy is given by $π_{A} (\bar{C}) = (F_{A} (\bar{C})^{q} + T_{A} (\bar{C})^{q} - F_{A} (\bar{C})^{q} T_{A} (\bar{C})^{q})^{1 / q}$ . There is no condition on q other than q ⩾ 1. We can easily check that 99% area is covered when we put q = 10 of unit square [0, 1] × [0, 1].

The researchers have developed many techniques to solve complex MADM problems based on q-ROF information. How to aggregate q-ROF information is a very important research topic for the researchers. If we use maximum and minimum operations to aggregate q-ROF information then only one value participate in final answer. So, we use different aggregation operator to aggregate the information and get comprehensive answer depending upon all values.

Liu and Wang [19], purposed q-rung orthopair fuzzy weighted averaging operator (q-ROFWA) and q-rung orthopair fuzzy weighted geometric operator (q-ROFWG). The q-ROFWG and q-ROFWA operators are two regular aggregation operators in the field of data fusion and decision-making within q-ROFN aggregation. In some cases, however, q-ROFWA and q-ROFWG operators have some deficiencies. For example, the aggregated result of the q-ROFWA and q-ROFWG operators tends towards to the maximum value or maximum weight value. In order to address the down sides of q-ROFWA and q-ROFWG operators in the aggregation of q-ROFNs, this paper introduced q-ROFHWAGA and q-ROFHOWAGA for more effective aggregation.

The remaining article is structured in the following manner. In Section 2, basic concepts related to q-ROFS are given. In Section 3, novel concept of q-ROFHWAGA and q-ROFHOWAGA operators is presented. In Section 4, MADM approach relating to proposed operators is presented and numerical example is given as well. In section 5 TOPSIS method based on q-ROFNs for MADM in transport policy problem is presented. Section 6 summarizes the results of this analytical research.

2 Preliminaries

We present description relating to the q-ROFS set in this section.

Definition 2.1. [50] A q-ROFS in a finite universe of discourse $X$ is $\hat{A} = {< \bar{C}, F_{\hat{A}} (\bar{C}), T_{\hat{A}} (\bar{C}) > : \bar{C} \in X}$ where $F_{\hat{A}} (\bar{C}) : X \to [0, 1]$ shows the degree of membership and $T_{\hat{A}} (\bar{C}) : X \to [0, 1]$ shows the degree of non-membership of the element $\bar{C} \in X$ to the set $\hat{A}$ , respectively, with the condition that $0 \leq F_{\hat{A}} (\bar{C})^{q} + T_{\hat{A}} (\bar{C})^{q} \leq 1, (q \geq 1)$ . The degree of indeterminacy is given as $π_{\hat{A}} (\bar{C}) = \sqrt[q]{(F_{\hat{A}} (\bar{C})^{q} + T_{\hat{A}} (\bar{C})^{q} - F_{\hat{A}} (\bar{C})^{q} T_{\hat{A}} (\bar{C})^{q})}$

A basic element of the form $〈 F_{\hat{A}} (\bar{C}), T_{\hat{A}} (\bar{C}) 〉$ in a q-ROF set $\hat{A}$ is called q-ROFN. It is denoted by $T = 〈 F_{\hat{A}}, T_{\hat{A}} 〉$ .

2.1 Operational laws of q-ROFSs

Definition 2.2. [50] Let ${\hat{A}}_{1} = 〈 F_{{\hat{A}}_{1}} (\bar{C}), T_{{\hat{A}}_{1}} (\bar{C}) 〉$ and ${\hat{A}}_{2} = 〈 F_{{\hat{A}}_{2}} (\bar{C}), T_{{\hat{A}}_{2}} (\bar{C}) 〉$ be q-ROF sets on a $X$ . Then we have the following operations:

(1) $\bar{{\hat{A}}_{1}} = 〈 T_{{\hat{A}}_{1}} (\bar{C}), F_{{\hat{A}}_{1}} (\bar{C}) 〉$ .

(2) ${\hat{A}}_{1} \subseteq {\hat{A}}_{2}$ iff $F_{{\hat{A}}_{1}} (\bar{C}) ⩽ F_{{\hat{A}}_{2}} (\bar{C})$ and $T_{{\hat{A}}_{2}} (\bar{C}) ⩽ T_{{\hat{A}}_{1}} (\bar{C})$ .

(3) ${\hat{A}}_{1} = {\hat{A}}_{2}$ iff ${\hat{A}}_{1} \subseteq {\hat{A}}_{2}$ and ${\hat{A}}_{2} \subseteq {\hat{A}}_{1}$ .

(4) ${\hat{A}}_{1} \cup {\hat{A}}_{2} = {〈 \bar{C}, \max {F_{{\hat{A}}_{1}} (\bar{C}), F_{{\hat{A}}_{2}} (\bar{C})}, min {T_{{\hat{A}}_{1}} (\bar{C}), T_{{\hat{A}}_{2}} (\bar{C})} 〉 ∣ \bar{C} \in X}$ .

(5) ${\hat{A}}_{1} \cap {\hat{A}}_{2} = {〈 \bar{C}, min {F_{{\hat{A}}_{1}} (\bar{C}), F_{{\hat{A}}_{2}} (\bar{C})}, \max {T_{{\hat{A}}_{1}} (\bar{C}), T_{{\hat{A}}_{2}} (\bar{C})} 〉 ∣ \bar{C} \in X}$ .

(6) ${\hat{A}}_{1} + {\hat{A}}_{2} = {〈 \bar{C}, (F_{{\hat{A}}_{1}} (\bar{C})^{q} + F_{{\hat{A}}_{2}} (\bar{C})^{q} - F_{{\hat{A}}_{1}} (\bar{C})^{q} F_{{\hat{A}}_{2}} (\bar{C})^{q})^{1 / q}, T_{{\hat{A}}_{1}} (\bar{C}) T_{{\hat{A}}_{2}} (\bar{C}) 〉 ∣ \bar{C} \in X}$ .

(7) ${\hat{A}}_{1} . {\hat{A}}_{2}$ = ${〈 \bar{C}, (F_{{\hat{A}}_{1}} (\bar{C}) F_{{\hat{A}}_{2}} (\bar{C}), T_{{\hat{A}}_{1}} (\bar{C})^{q} + T_{{\hat{A}}_{2}} (\bar{C})^{q} - T_{{\hat{A}}_{1}} (\bar{C})^{q} T_{{\hat{A}}_{2}} (\bar{C})^{q})^{1 / q} 〉 ∣ \bar{C} \in X}$ .

(8) $σ {\hat{A}}_{1}$ = ${〈 \bar{C}, (1 - (1 - F_{{\hat{A}}_{1}} (\bar{C})^{q})^{σ}))^{1 / q}, T_{{\hat{A}}_{1}} (\bar{C})^{σ} 〉}$ .

(9) ${\hat{A}}_{1}^{σ} = {〈 \bar{C}, F_{{\hat{A}}_{1}} (\bar{C})^{σ}, (1 - (1 - T_{{\hat{A}}_{1}} (\bar{C})^{q})^{σ}))^{1 / q}, 〉} .$

2.2 Operational laws of q-ROFNs

Definition 2.3. [19] Suppose $T_{1} = 〈 F_{1}, T_{1} 〉$ and $T_{2} = 〈 F_{2}, T_{2} 〉$ are the two q-ROFNs. Then

(1) $\bar{T_{1}} = 〈 T_{1}, F_{1} 〉$

(2) $T_{1} \lor T_{2} = 〈 \max {F_{1}, T_{1}}, \min {F_{2}, T_{2}} 〉$

(3) $T_{1} \land T_{2} = 〈 \min {F_{1}, T_{1}}, \max {F_{2}, T_{2}} 〉$

(4) $T_{1} \oplus T_{2} = 〈 \sqrt[q]{F_{1}^{q} + F_{2}^{q} - F_{1}^{q} F_{2}^{q}}, T_{1} T_{2} 〉$

(5) $T_{1} \otimes T_{2} = 〈 F_{1} F_{2}, \sqrt[q]{T_{1}^{q} + T_{2}^{q} - T_{1}^{q} T_{2}^{q}} 〉$

(6) $σ T_{1} = 〈 (1 - (1 - F_{1}^{q})^{σ})^{1 / q}, T_{1}^{σ} 〉$

(7) $T_{1}^{T_{1}} = 〈 F_{1}^{σ}, (1 - (1 - T_{1}^{q})^{σ})^{1 / q} 〉$

Theorem 2.4. [19] Suppose $T_{1} = 〈 F_{1}, T_{1} 〉$ and $T_{2} = 〈 F_{2}, T_{2} 〉$ are any q-ROFNs on a $X$ , and n₁, n₂ > 0, then

(1) $T_{1} \oplus T_{2} = T_{2} \oplus T_{1}$

(2) $T_{1} \otimes T_{2} = T_{2} \otimes T_{1}$

(3) $n (T_{1} \oplus T_{2}) = n T_{1} \oplus n T_{2}$

(4) $n_{1} T_{1} \oplus n_{2} T_{2} = (n_{1} + n_{2}) T_{1}$

(5) $T_{1}^{n_{1}} \otimes T_{1}^{n_{2}} = T_{1}^{n_{1} + n_{2}}$

(6) $T_{1}^{n} \otimes T_{2}^{n} = (T_{1} \otimes T_{2})^{n}$

Definition 2.5. [50] Suppose that $\hat{A} = 〈 F, T 〉$ is a q-ROFN, then a score function $E$ of $\hat{A}$ is defined as $E (\hat{A}) = F^{q} - T^{q}$ $E (\hat{A}) \in [- 1, 1]$ . If score is large then the q-ROFN is greater. However score function cannot be useful in many cases of q - ROFN. For example: Let $T_{1} = 〈 0.6138, 0.2534 〉$ and $T_{2} = 〈 0.7147, 0.4453 〉$ be two q-ROFNs. If we take q = 2. Then score function of $T_{1}$ is $E (T_{1}) = 0.3125$ and score function of $T_{2}$ is $E (T_{2}) = 0.3125$ . This shows the score function isn’t enough to compare the q-ROFNs. We use another function named as accuracy function to solve this problem. Definition 2.6. [50] Let $\hat{A} = 〈 F, T 〉$ be the q-ROFN. Then the accuracy function $R$ of $\hat{A}$ is defined as $R (\hat{A}) = F^{q} + T^{q}$ where $R (\hat{A}) \in [0, 1]$ . The larger accuracy degree $R (\hat{A})$ represent greater value of $\hat{A}$ .

For $T_{1} = 〈 0.6138, 0.2534 〉$ and $T_{2} = 〈 0.7147, 0.4453 〉$ . The accuracy functions are $R (T_{1}) = 0.4410$ and $R (T_{2}) = 0.7091$ . Thus we can write $T_{1} < T_{2}$ .

Definition 2.7. Let $\tilde{Q} = 〈 F_{\tilde{Q}}, T_{\tilde{Q}} 〉$ and $\tilde{P} = 〈 F_{\tilde{P}}, T_{\tilde{P}} 〉$ be any two q-ROFNs. Let $E (\tilde{Q}), E (\tilde{P})$ be the score functions of $\tilde{Q}$ and $\tilde{P}$ and $R (\tilde{Q}), R (\tilde{P})$ be the accuracy functions of $\tilde{Q}$ and $\tilde{P}$ , respectively. Then,

(1) If $E (\tilde{Q}) > E (\tilde{P})$ , then $\tilde{Q} > \tilde{P}$

(2) If $E (\tilde{Q}) = E (\tilde{P})$ , then

If $R (\tilde{Q}) > R (\tilde{P})$ then $\tilde{Q} > \tilde{P}$ ,

If $R (\tilde{Q}) = R (\tilde{P})$ , then $\tilde{Q} = \tilde{P}$ .

2.3 Some basic aggregation operators on q-ROFNs

Definition 2.8. (Liu and Wang [19]) Let ${\overset{˘}{T}}_{k} = 〈 F_{k}, T_{k} 〉$ be the assemblage of q-ROFNs. Define (q - ROFWA) : Tⁿ → T given by

$\begin{matrix} (q - ROFWA) ({\overset{˘}{T}}_{1}, \dots, {\overset{˘}{T}}_{n}) & = \\ = \sum_{k = 1}^{n} {\overset{˘}{ħ}}_{k} {\overset{˘}{T}}_{k} \\ = {\overset{˘}{ħ}}_{1} {\overset{˘}{T}}_{1} \oplus {\overset{˘}{ħ}}_{2} {\overset{˘}{T}}_{2} \oplus \dots, {\overset{˘}{ħ}}_{n} {\overset{˘}{T}}_{n} \end{matrix}$

where Tⁿ is the set of all q-ROFNs, and $\overset{˘}{ħ} = ({\overset{˘}{ħ}}_{1}, {\overset{˘}{ħ}}_{2}, \dots, {\overset{˘}{ħ}}_{n})^{T}$ is weight vector (WV) of $({\overset{˘}{T}}_{1}, {\overset{˘}{T}}_{2}, \dots {\overset{˘}{T}}_{n})$ , such that $0 ⩽ {\overset{˘}{ħ}}_{k} ⩽ 1$ and $\sum_{k = 1}^{n} {\overset{˘}{ħ}}_{k} = 1$ . Then, the q-ROFWA is called the q-rung orthopair fuzzy weighted averaging operator.

On the basis of operational laws of q-ROFNs we can calculate q-ROFWA as given by the following theorem.

Theorem 2.9. Let ${\overset{˘}{T}}_{k} = 〈 F_{k}, T_{k} 〉 (k = 1, 2, \dots, n)$ be the assemblage of q-ROFNs,we can find q-ROFWA by $(md q - ROFWA) ({\overset{˘}{T}}_{1}, {\overset{˘}{T}}_{2}, \dots, {\overset{˘}{T}}_{n})$

$= 〈 \sqrt[q]{(1 - \prod_{k = 1}^{n} (1 - F_{k}^{q})^{{\overset{˘}{ħ}}_{k}})}, \prod_{k = 1}^{n} T_{k}^{{\overset{˘}{ħ}}_{k}} 〉$

(2.1) In the next example we explain the above idea.

Example 2.10. Let ${\overset{˘}{T}}_{1} = (0.7, 0.5), {\overset{˘}{T}}_{2} = (0.3, 0.5),$ and ${\overset{˘}{T}}_{3} = (0.6, 0.7)$ be the three q-ROFNs, $\overset{˘}{ħ} = (0.3, 0.3, 0.4)$ be the WV of $({\overset{˘}{T}}_{1}, {\overset{˘}{T}}_{2}, {\overset{˘}{T}}_{3})$ , and if we take q = 3. We use q-ROFWA operator to aggregate the three q-ROFNs by using (2.1).

$(md q - ROFWA) ({\overset{˘}{T}}_{1}, {\overset{˘}{T}}_{2}, {\overset{˘}{T}}_{3})$ $= 〈 \sqrt[q]{(1 - \prod_{k = 1}^{3} (1 - F_{k}^{q})^{{\overset{˘}{ħ}}_{k}})}, \prod_{k = 1}^{3} T_{k}^{{\overset{˘}{ħ}}_{k}} 〉$ = (0.591, 0.572).

Definition 2.11. (Liu and Wang [19]) Assume that ${\overset{˘}{T}}_{k} = 〈 F_{k}, T_{k} 〉$ is a assemblage of q-ROFN, and (mdq - ROFWG) : Tⁿ → T, if

$\begin{matrix} (md q - ROFWG) ({\overset{˘}{T}}_{1}, {\overset{˘}{T}}_{2}, \dots {\overset{˘}{T}}_{n}) & = \sum_{k = 1}^{n} {\overset{˘}{T}}_{k}^{{\overset{˘}{ħ}}_{k}} \\ = {\overset{˘}{T}}_{1}^{{\overset{˘}{ħ}}_{1}} \otimes {\overset{˘}{T}}_{2}^{{\overset{˘}{ħ}}_{2}} \otimes \dots, {\overset{˘}{T}}_{n}^{{\overset{˘}{ħ}}_{n}} \end{matrix}$

where Tⁿ is the set of all q-ROFNs, and $\overset{˘}{ħ} = ({\overset{˘}{ħ}}_{1}, {\overset{˘}{ħ}}_{2}, \dots, {\overset{˘}{ħ}}_{n})^{T}$ is WV of $({\overset{˘}{T}}_{1}, {\overset{˘}{T}}_{2}, \dots, {\overset{˘}{T}}_{n})$ , such that $0 ⩽ {\overset{˘}{ħ}}_{k} ⩽ 1$ and $\sum_{k = 1}^{n} {\overset{˘}{ħ}}_{k} = 1$ . Then, the q-ROFWG is called the q-rung orthopair fuzzy weighted geometric operator. On the basis of operational laws of q-ROFNs we can also find q-ROFWG by the following theorem.

Theorem 2.12. Let ${\overset{˘}{T}}_{k} = 〈 F_{k}, T_{k} 〉 (k = 1, 2, \dots, n)$ be the assemblage of q-ROFNs,we can find q-ROFWG by $(md q - ROFWG) ({\overset{˘}{T}}_{1}, {\overset{˘}{T}}_{2}, \dots {\overset{˘}{T}}_{n})$ $= 〈 \prod_{k = 1}^{n} F_{k}^{{\overset{˘}{ħ}}_{k}}, (1 - \prod_{k = 1}^{n} (1 - T_{k}^{q})^{{\overset{˘}{ħ}}_{k}})^{1 / q} 〉$ .

Example 2.13. Let ${\overset{˘}{T}}_{1} = (0.7, 0.5), {\overset{˘}{T}}_{2} = (0.3, 0.5),$ and ${\overset{˘}{T}}_{3} = (0.6, 0.7)$ be the three q-ROFNs, $\overset{˘}{ħ} = (0.3, 0.3, 0.4)$ be the WV of $({\overset{˘}{T}}_{1}, {\overset{˘}{T}}_{2}, {\overset{˘}{T}}_{3})$ , and if we take q = 3. We use q-ROFWG operator to aggregate the three q-ROFNs. $(md q - ROFWG) ({\overset{˘}{T}}_{1}, {\overset{˘}{T}}_{2}, {\overset{˘}{T}}_{3})$ $= 〈 \prod_{k = 1}^{3} F_{k}^{{\overset{˘}{ħ}}_{k}}, (1 - \prod_{k = 1}^{3} (1 - T_{k}^{q})^{{\overset{˘}{ħ}}_{k}})^{1 / q} 〉$ = (0.510, 0.603).

2.4 Some drawbacks of q-ROFWA and q-ROFWG operators

As we know, in many MADM issues, q-ROFWA and q-ROFWG operators are being used to consolidate the data. However, their consolidated values can suggest certain irrational outcomes when those values converge toward the maximum arguments or weights. Here, we consider two cases and take q = 3 and σ = 0.5.

bfCase 1: Take two q-ROFNs such that ${\overset{˘}{T}}_{1} = (0.001, 0)$ , ${\overset{˘}{T}}_{2} = (1, 0)$ with weights ${\overset{˘}{ħ}}_{1} = 0.9$ and ${\overset{˘}{ħ}}_{2} = 0.1$ . By (1) and (2) we get $md q - ROFWA ({\overset{˘}{T}}_{1}, {\overset{˘}{T}}_{2}) = (1, 0)$ and $md q - ROFWG ({\overset{˘}{T}}_{1}, {\overset{˘}{T}}_{2}) = (0.002, 0)$ bfCase 2: Take two q-ROFNs such that ${\overset{˘}{T}}_{1} = (0.001, 0)$ , ${\overset{˘}{T}}_{2} = (1, 0)$ with weights ${\overset{˘}{ħ}}_{1} = 0.1$ and ${\overset{˘}{ħ}}_{2} = 0.9$ . By (1) and (2) we get $md q - ROFWA ({\overset{˘}{T}}_{1}, {\overset{˘}{T}}_{2}) = (1, 0)$ and $md q - ROFWG ({\overset{˘}{T}}_{1}, {\overset{˘}{T}}_{2}) = (0.501, 0)$ From these results, we note that in these two cases, q-ROFWA and q-ROFWG operators are failed to give the reasonable results. Therefore, we must improve the operators of aggregation to overcome these shortcomings or weaknesses.

3 Some hybrid aggregation operators of q-ROFNs

In this section,we propose a new hybrid aggregation operator to overcome the shortcoming of the q-ROFWA and q-ROFGA operators.

3.1 q-ROFHWAGA operator

Assume that ${\overset{˘}{T}}_{k} = 〈 F_{k}, T_{k} 〉$ is a assemblage of q-ROFN, and md (q - ROFHWAGA) : Tⁿ → T, if $md (q - ROFHWAGA) ({\overset{˘}{T}}_{1}, {\overset{˘}{T}}_{2}, \dots, {\overset{˘}{T}}_{n})$ $= (\sum_{k = 1}^{n} {\overset{˘}{ħ}}_{k} {\overset{˘}{T}}_{k})^{σ} (\sum_{k = 1}^{n} {\overset{˘}{T}}_{k}^{{\overset{˘}{ħ}}_{k}})^{1 - σ}$ where Tⁿ is the set of all q-ROFNs, σ is any real number in the interval [0, 1] and $\overset{˘}{ħ} = ({\overset{˘}{ħ}}_{1}, {\overset{˘}{ħ}}_{2}, \dots, {\overset{˘}{ħ}}_{n})^{T}$ is WV of $({\overset{˘}{T}}_{1}, {\overset{˘}{T}}_{2}, \dots, {\overset{˘}{T}}_{n})$ , such that $0 ⩽ {\overset{˘}{ħ}}_{k} ⩽ 1$ and $\sum_{k = 1}^{n} {\overset{˘}{ħ}}_{k} = 1$ . Then, the q-ROFHWAGA is called the q-ROF Hybrid weighted arithmetic and geometric aggregation operator. On the basis of operational laws of q-ROFNs we can also find q-ROFHWAGA by the following theorem.

Theorem 3.1. Let ${\overset{˘}{T}}_{k} = 〈 F_{k}, T_{k} 〉 (k = 1, 2, \dots, n)$ is a assemblage of q-ROFN,we can find q-ROFHWAGA by $md (q - ROFHWAGA) ({\overset{˘}{T}}_{1}, {\overset{˘}{T}}_{2}, \dots, {\overset{˘}{T}}_{n})$ $= (\sum_{k = 1}^{n} {\overset{˘}{ħ}}_{k} {\overset{˘}{T}}_{k})^{σ} (\sum_{k = 1}^{n} {\overset{˘}{T}}_{k}^{{\overset{˘}{ħ}}_{k}})^{1 - σ}$ $= 〈 (1 - \prod_{k = 1}^{n} (1 - (F_{k})^{q})^{w_{k}})^{\frac{σ}{q}} (\prod_{k = 1}^{n} F_{k}^{w_{k}})^{1 - σ}$ , $\sqrt[q]{1 - (1 - (\prod_{k = 1}^{n} (T_{k})^{w_{k}})^{q})^{σ} (\prod_{k = 1}^{n} (1 - (T_{k})^{q})^{w_{k}})^{1 - σ}} 〉$ (3) where σ is any real number in the interval [0, 1]. $\overset{˘}{ħ} = ({\overset{˘}{ħ}}_{1}, {\overset{˘}{ħ}}_{2}, \dots, {\overset{˘}{ħ}}_{n})^{T}$ is WV of $({\overset{˘}{T}}_{1}, {\overset{˘}{T}}_{2}, \dots, {\overset{˘}{T}}_{n})$ , such that $0 ⩽ {\overset{˘}{ħ}}_{k} ⩽ 1$ and $\sum_{k = 1}^{n} {\overset{˘}{ħ}}_{k} = 1$ .

Proof. Based on q-ROFWA and q-ROFGA operators and the operational laws of q-ROF sets. $md (q - ROFHWAGA) ({\overset{˘}{T}}_{1}, {\overset{˘}{T}}_{2}, \dots {\overset{˘}{T}}_{n})$ $= (\sum_{k = 1}^{n} {\overset{˘}{ħ}}_{k} {\overset{˘}{T}}_{k})^{σ} (\sum_{k = 1}^{n} {\overset{˘}{T}}_{k}^{{\overset{˘}{ħ}}_{k}})^{1 - σ}$ $= 〈 (1 - \prod_{k = 1}^{n} (1 - F_{k}^{q})^{{\overset{˘}{ħ}}_{k}})^{1 / q}, \prod_{k = 1}^{n} T_{k}^{{\overset{˘}{ħ}}_{k}}) 〉^{σ}$ $〈 (\prod_{k = 1}^{n} F_{k}^{{\overset{˘}{ħ}}_{k}}, (1 - \prod_{k = 1}^{n} (1 - T_{k}^{q})^{{\overset{˘}{ħ}}_{k}})^{1 / q}) 〉^{1 - σ}$ $= 〈 ((1 - \prod_{k = 1}^{n} (1 - F_{k}^{q})^{{\overset{˘}{ħ}}_{k}})^{σ / q}$ , $(1 - (1 - (\prod_{k = 1}^{n} T_{k}^{{\overset{˘}{ħ}}_{k}})^{q})^{σ})^{\frac{1}{q}})$ $((\prod_{k = 1}^{n} F_{k}^{{\overset{˘}{ħ}}_{k}})^{1 - σ}, (1 - (\prod_{k = 1}^{n} (1 - T_{k}^{q})^{{\overset{˘}{ħ}}_{k}})^{1 - σ})^{\frac{1}{q}}) 〉$ $= 〈 (1 - \prod_{k = 1}^{n} (1 - F_{k}^{q})^{{\overset{˘}{ħ}}_{k}})^{σ / q} (\prod_{k = 1}^{n} F_{k}^{{\overset{˘}{ħ}}_{k}})^{1 - σ}$ , $((1 - (1 - (\prod_{k = 1}^{n} T_{k}^{{\overset{˘}{ħ}}_{k}})^{q})^{σ})^{\frac{q}{q}} + (1 - (\prod_{k = 1}^{n} (1 - T_{k}^{q})^{{\overset{˘}{ħ}}_{k}})^{1 - σ})^{\frac{q}{q}} - (1 - (1 - (\prod_{k = 1}^{n} T_{k}^{{\overset{˘}{ħ}}_{k}})^{q})^{σ}) (1 - (\prod_{k = 1}^{n} (1 - T_{k}^{q})^{{\overset{˘}{ħ}}_{k}})^{1 - σ}))^{1 / q} 〉$ $= 〈 (1 - \prod_{k = 1}^{n} (1 - (F_{k})^{q})^{{\overset{˘}{ħ}}_{k}})^{\frac{σ}{q}} (\prod_{k = 1}^{n} F_{k}^{{\overset{˘}{ħ}}_{k}})^{1 - σ},$ $((1 - (1 - (\prod_{k = 1}^{n} T_{k}^{{\overset{˘}{ħ}}_{k}})^{q})^{σ})$ $+ (1 - (\prod_{k = 1}^{n} (1 - T_{k}^{q})^{{\overset{˘}{ħ}}_{k}})^{1 - σ})^{\frac{q}{q}}$ $- 1 + (\prod_{k = 1}^{n} (1 - T_{k}^{q})^{{\overset{˘}{ħ}}_{k}})^{1 - σ})$ $+ (1 - (\prod_{k = 1}^{n} T_{k}^{{\overset{˘}{ħ}}_{k}})^{q})^{σ})$ $- (1 - (\prod_{k = 1}^{n} (1 - T_{k}^{q})^{{\overset{˘}{ħ}}_{k}})^{1 - σ}) (1 - (\prod_{k = 1}^{n} T_{k}^{{\overset{˘}{ħ}}_{k}})^{q})^{σ}))^{1 / q} 〉$ $= 〈 (1 - \prod_{k = 1}^{n} (1 - (F_{k})^{q})^{{\overset{˘}{ħ}}_{k}})^{\frac{σ}{q}} (\prod_{k = 1}^{n} F_{k}^{{\overset{˘}{ħ}}_{k}}),$ $(1 - (1 - (\prod_{k = 1}^{n} (T_{k})^{{\overset{˘}{ħ}}_{k}})^{q})^{σ} (\prod_{k = 1}^{n} (1 - (T_{k})^{q})^{{\overset{˘}{ħ}}_{k}})^{1 - σ})^{1 / q} 〉$ Therefore, this complete the proof of (3) .□

Remark. For different values of σ ∈ [0, 1], it is possible to investigate the various families of the q-ROFHWAGA operator individually. If we consider some special case like σ = 1 the q-ROFHWAGA operator is reduced to q-ROFWA operator. If σ = 0, the q-ROFHWAGA operator is reduced to q-ROFWG operator. If σ = 0.5, the q-ROFHWAGA operator is mean of the q-ROFWA and q-ROFWG operators.

Example 3.2. Let ${\overset{˘}{T}}_{1} = (0.71, 0.52), {\overset{˘}{T}}_{2} = (0.34, 0.56),$ and ${\overset{˘}{T}}_{3} = (0.57, 0.68)$ be the three q-ROFNs, $\overset{˘}{ħ} = (0.5, 0.3, 0.2)$ be the WV of $({\overset{˘}{T}}_{1}, {\overset{˘}{T}}_{2}, {\overset{˘}{T}}_{3}$ ), σ = 0.5 and if we take q = 3. We use q-ROFHWAGA operator to aggregate the three q-ROFNs by using (3). $md (q - ROFHWAGA) ({\overset{˘}{T}}_{1}, {\overset{˘}{T}}_{2}, {\overset{˘}{T}}_{3})$

$= 〈 (1 - \prod_{k = 1}^{3} (1 - (F_{k})^{3})^{{\overset{˘}{ħ}}_{k}})^{\frac{0.5}{3}} (\prod_{k = 1}^{3} F_{k}^{{\overset{˘}{ħ}}_{k}})^{1 - 0.5}$ , $\sqrt[3]{1 - (1 - (\prod_{k = 1}^{3} (T_{k})^{{\overset{˘}{ħ}}_{k}})^{3})^{0.5} (\prod_{k = 1}^{3} (1 - (T_{k})^{3})^{{\overset{˘}{ħ}}_{k}})^{1 - 0.5}} 〉$ = (0.582, 0.567).

According to properties of the q-ROFWA and q-ROFWG operators, it is clear that the q-ROFHWAGA operator also have properties of idempotency, boundedness and monotonicity.

Theorem 3.3. Let ${\overset{˘}{T}}_{k} = 〈 F_{k}, T_{k} 〉$ is a assemblage of q-ROFNs. Then, 1. (Idempotency) if ${\overset{˘}{T}}_{k} = \overset{˘}{T} = 〈 F, T 〉$ for all k, then $md q - ROFHWAGA ({\overset{˘}{T}}_{1}, {\overset{˘}{T}}_{2}, \dots, {\overset{˘}{T}}_{n}) = \overset{˘}{T}$ .2.(Boundedness) if ${\overset{˘}{T}}^{-} = (\min (F_{k}), \max (T_{k}))$ and ${\overset{˘}{T}}^{+} = (\max (F_{k}), \min (T_{k}))$ , then we have ${\overset{˘}{T}}^{-} \leq md q - ROFHWAGA ({\overset{˘}{T}}_{1}, {\overset{˘}{T}}_{2}, \dots, {\overset{˘}{T}}_{n}) \leq {\overset{˘}{T}}^{+}$ 3. (Monotonicity) If ${\overset{˘}{T}}_{k} = 〈 F_{k}, T_{k} 〉$ and ${\overset{˘}{T}}_{k}^{*} = 〈 F_{k}^{*}, T_{k}^{*} 〉$ (k = 1, 2, …, n) are two sets of q-ROFNs. If $F_{k} \geq F_{k}^{*}, T_{k} \leq T_{k}^{*}$ for all k then $md q - ROFHWAGA ({\overset{˘}{T}}_{1}, {\overset{˘}{T}}_{2}, \dots, {\overset{˘}{T}}_{n})$ $\geq md q - ROFHWAGA ({\overset{˘}{T}}_{1}^{*}, {\overset{˘}{T}}_{2}^{*}, \dots, {\overset{˘}{T}}_{n}^{*})$

3.2 q-ROFHOWAGA operator

Assume that ${\overset{˘}{T}}_{k} = 〈 F_{k}, T_{k} 〉$ is a assemblage of q-ROFNs, and md (q - ROFHOWAGA) : Tⁿ → T, if $md (q - ROFHOWAGA) ({\overset{˘}{T}}_{1}, {\overset{˘}{T}}_{2}, \dots {\overset{˘}{T}}_{n})$ $= (\sum_{k = 1}^{n} {\overset{˘}{ħ}}_{k} {\overset{˘}{T}}_{σ (k)})^{σ} (\sum_{k = 1}^{n} {\overset{˘}{T}}_{σ (k)}^{{\overset{˘}{ħ}}_{k}})^{1 - σ}$ where Tⁿ is the set of all q-ROFNs,(σ (1) , σ (2) , …, σ (k)) is a permutation of (1, 2, …, n) such that ${\overset{˘}{T}}_{σ (j - 1)} \geq {\overset{˘}{T}}_{σ (j)}$ for any k, σ is any real number in the interval [0, 1] and $\overset{˘}{ħ} = ({\overset{˘}{ħ}}_{1}, {\overset{˘}{ħ}}_{2}, \dots, {\overset{˘}{ħ}}_{n})^{T}$ is WV of $({\overset{˘}{T}}_{1}, {\overset{˘}{T}}_{2}, \dots {\overset{˘}{T}}_{n})$ , such that $0 ⩽ {\overset{˘}{ħ}}_{k} ⩽ 1$ and $\sum_{k = 1}^{n} {\overset{˘}{ħ}}_{k} = 1$ . Then, the q-ROFHOWAGA is called the q-ROF Hybrid ordered weighted arithmetic and geometric aggregation operator. On the basis of operational laws of q-ROFNs we can also find q-ROFHOWAGA by the following theorem.

Theorem 3.4. Let ${\overset{˘}{T}}_{k} = 〈 F_{k}, T_{k} 〉$ is a assemblage of q-ROFNs,we can find q-ROFHOWAGA by $md (q - ROFHWAGA) ({\overset{˘}{T}}_{1}, {\overset{˘}{T}}_{2}, \dots, {\overset{˘}{T}}_{n})$ $= (\sum_{k = 1}^{n} {\overset{˘}{ħ}}_{k} {\overset{˘}{T}}_{σ (k)})^{σ} (\sum_{k = 1}^{n} {\overset{˘}{T}}_{σ (k)}^{{\overset{˘}{ħ}}_{k}})^{1 - σ}$ $= 〈 (1 - \prod_{k = 1}^{n} (1 - (F_{σ (k)})^{q})^{{\overset{˘}{ħ}}_{k}})^{\frac{σ}{q}} (\prod_{k = 1}^{n} F_{σ (k)}^{{\overset{˘}{ħ}}_{k}})^{1 - σ},$ $(1 - (1 - (\prod_{k = 1}^{n} (T_{σ (k)})^{{\overset{˘}{ħ}}_{k}})^{q})^{σ}$ $(\prod_{k = 1}^{n} (1 - (T_{σ (k)})^{q})^{{\overset{˘}{ħ}}_{k}})^{1 - σ})^{1 / q} 〉$ (4)where σ is any real number in the interval [0, 1]. (σ (1) , σ (2) , …, σ (k)) is a permutation of (1, 2, …, n) such that ${\overset{˘}{T}}_{σ (k - 1)} \geq {\overset{˘}{T}}_{σ (k)}$ for any k, $\overset{˘}{ħ} = ({\overset{˘}{ħ}}_{1}, {\overset{˘}{ħ}}_{2}, \dots, {\overset{˘}{ħ}}_{n})^{T}$ is WV of $({\overset{˘}{T}}_{1}, {\overset{˘}{T}}_{2}, \dots {\overset{˘}{T}}_{n})$ , such that $0 ⩽ {\overset{˘}{ħ}}_{k} ⩽ 1$ and $\sum_{k = 1}^{n} {\overset{˘}{ħ}}_{k} = 1$ .

Proof. The proof can be made by similar way to proof of Theorem 3.1, so we omit the proof.□

Example 3.5. Let $T_{1} = (0.71, 0.52), T_{2} = (0.34, 0.56),$ and $T_{3} = (0.57, 0.68)$ be the three q-ROFNs, $\overset{˘}{ħ} = (0.5, 0.3, 0.2)$ be the WV of $(T_{1}, T_{2}, T_{3}$ ), σ = 0.5 and if we take q = 4.

By score function we rank these q-ROFNs, $E (T_{1}) = 0.181$ $E (T_{2}) = - 0.085$ $E (T_{3}) = - 0.108$ now ${\overset{˘}{T}}_{1} = \ddot{a_{1}}, {\overset{˘}{T}}_{2} = T_{2}, {\overset{˘}{T}}_{3} = T_{3} .$ We use q-ROFHOWAGA operator to aggregate by using (4). $md (q - ROFHOWAGA) ({\overset{˘}{T}}_{1}, {\overset{˘}{T}}_{2}, {\overset{˘}{T}}_{3})$

$= 〈 (1 - \prod_{k = 1}^{3} (1 - (F_{σ (k)})^{4})^{{\overset{˘}{ħ}}_{k}})^{\frac{0.5}{4}}$ $(\prod_{k = 1}^{3} F_{σ (k)}^{{\overset{˘}{ħ}}_{k}})^{1 - 0.5}$ , $(1 - (1 - (\prod_{k = 1}^{3} (T_{σ (k)})^{{\overset{˘}{ħ}}_{k}})^{4})^{0.5}$ $(\prod_{k = 1}^{3} (1 - (T_{σ (k)})^{4})^{{\overset{˘}{ħ}}_{k}})^{1 - 0.5})^{1 / 4} 〉$ = (0.582, 0.567).

Theorem 3.6. Let ${\overset{˘}{T}}_{k} = 〈 F_{k}, T_{k} 〉$ is a assemblage of q-ROFNs. Then, 1. (Idempotency) If ${\overset{˘}{T}}_{k} = \overset{˘}{T} = 〈 F, T 〉$ for all k, then $q - ROFHOWAGA ({\overset{˘}{T}}_{1}, {\overset{˘}{T}}_{2}, \dots, {\overset{˘}{T}}_{n}) = \overset{˘}{T}$ .2.(Boundedness) If ${\overset{˘}{T}}^{-} = (\min (F_{k}), \max (v_{k}))$ and ${\overset{˘}{T}}^{+} = (\max (F_{k}), \min (T_{k}))$ , then we have ${\overset{˘}{T}}^{-} \leq md q - ROFHOWAGA ({\overset{˘}{T}}_{1}, {\overset{˘}{T}}_{2}, \dots, {\overset{˘}{T}}_{n}) \leq {\overset{˘}{T}}^{+}$ 3.(Monotonicity)If ${\overset{˘}{T}}_{k} = 〈 F_{k}, T_{k} 〉$ and ${\overset{˘}{T}}_{k}^{*} = 〈 F_{k}^{*}, T_{k}^{*} 〉$ (k = 1, 2, …, n) are two sets of q-ROFNs. If $F_{k} \geq F_{k}^{*}, T_{k} \leq T_{k}^{*}$ for all k then $md q - ROFHOWAGA ({\overset{˘}{T}}_{1}, {\overset{˘}{T}}_{2}, \dots, {\overset{˘}{T}}_{n})$ $\geq md q - ROFHOWAGA ({\overset{˘}{T}}_{1}^{*}, {\overset{˘}{T}}_{2}^{*}, \dots, {\overset{˘}{T}}_{n}^{*})$

3.3 Numerical example

To show the appropriateness of the aggregated values of q-ROFHWAGA and q-ROFHOWAGA operators, we take first case of section 2.4. If we take σ = 0.5 and q = 3, we use the q-ROFHWAGA and q-ROFHOWAGA operators for the case.

For case 1, by (3), there is $md q - ROFHWAGA ({\overset{˘}{T}}_{1}, {\overset{˘}{T}}_{2}) = (0.045, 0)$ which is between $md q - ROFWA ({\overset{˘}{T}}_{1}, {\overset{˘}{T}}_{2}) = (1, 0)$ and $q - ROFWG ({\overset{˘}{T}}_{1}, {\overset{˘}{T}}_{2}) = (0.002, 0) .$

In the above case, newly developed operators indicate the moderate values. it is obvious that these operators can overcome the shortcoming of q-ROFWA and q-ROFWG operators. Therefore the q-ROFHWAGA and q-ROFHOWAGA operators are more effective and reasonable in the information aggregation.

4 MADM with q-ROFHWAGA andq-ROFHOWAGA operators

Suppose that $\overset{˘}{T} = {({\overset{˘}{T}}_{1}, {\overset{˘}{T}}_{2}, \dots, {\overset{˘}{T}}_{p})}$ and $C = {C_{1}, C_{2}, \dots, C_{q}}$ is the set of alternatives and criterions, respectively. Let $\overset{˘}{ħ}$ be the WV of attributes, such that ${\overset{˘}{ħ}}_{j} \in [0, 1]$ and $\sum_{j = 1}^{n} {\overset{˘}{ħ}}_{j} = 1$ , (j = 1, 2, …, n) and ${\overset{˘}{ħ}}_{j}$ represent the weight of $C_{j}$ . An alternatives on criterions is evaluated by the decision maker and the evaluation values must be in q-ROFNs. Assume that $(H_{ij}^{'})_{p \times q} = 〈 F_{ij}, T_{ij} 〉$ is the decision matrix provided by decision maker. $(H_{ij}^{'})$ represent a q-ROFNs for alternative ${\overset{˘}{T}}_{i}$ associated with the criterions $C_{j}$ . Here we have some conditions such that, 1. $F_{ij}$ and $T_{ij} \in [0, 1]$ 2. $0 \leq (F_{ij})^{q} + (T_{ij})^{q} \leq 1, (q \geq 1)$ . Now, we develop an Algorithm 1 to solve given problem.

bfAlgorithm 1

Step 1. Evaluate the decision matrix for input. $(H_{ij}^{'})_{p \times q} = 〈 F_{ij}, T_{ij} 〉$ Step 2. Normalize the decision matrix. If we have different types of criterions or attributes like cost and benefit, then we normalize the decision matrix by taking complement of criteria like cost.

Step 3. Evaluate $H_{i}^{'} = md q - ROFHWAGA (H_{i 1}^{'}, H_{i 2}^{'}, \dots H_{in}^{'})$ or $H_{i}^{'} = md q - ROFHOWAGA (H_{i 1}^{'}, H_{i 2}^{'}, \dots H_{in}^{'})$ for each i = 1, 2, …, q.

Step 4. Determine the score functions for all $H_{i}^{'}$ for the collective overall q-ROFNs.

Step 5. Rank all the $H_{i}^{'}$ (i = 1, 2, …, p) according to the score values.

4.1 Case study

CNC stands for “Computer Numeric Control”. It is the result of the older term “NC”, which means “Numerical Control”. This refers to the idea of controlling machine tools through a computer. CNC machines are the same robot. With the term “NC” parents, computers need not be involved.. CNC machines are computer-controlled machine tools. Before CNC, machine tools were manually operated by machine experts. With CNC, computers control servos that serve machines. Essentially, they process robots. NC, and later CNC, allow for an incredible increase in capacity of machine tools, as they can be operated automatically without the constant attention of their operators.

CNC processing is a manufacturing process in which a computer runs a program that controls how the machine produces parts. CNC programs using these CNC software can control everything from machine movement to pivot speed, by starting or stopping the coolant and more. The computer language used for CNC programming machines is called G-Code. G-Code is a common term for the computer language industry that most CNC machines use to control their movements and the way they produce parts. CNC machines only understand the instructions provided by the G-code command. The list of CNC machines has been expanded. Here some other possibilities not mentioned:

(1) CNC Milling (2) CNC Lathes(3) CNC Routers(4) CNC Plasma Cutters, Lasers, and Water jets(5) CNC Electronic Discharge Machines (“EDM’s”)(6) CNC Foam Cutters(7) CNC Wire Bending and Tubing Bending (8) CNC Punches(9) CNC Embroidery (10) CNC Pattern Cutters (11) CNC Food Makers (12) CNC Pottery Printers Cutting tools are nothing new. From the first stone shaft to the modern final factory, mankind has invented and perfected tools since the earliest times. Most common materials you can use in CNC machine are:Carbon steelHigh-speed steel (HSS)Solid carbideCeramicsCubic boron nitrideDiamond Ceramics-alumina Cast cobalt alloys.

4.2 Application to selection of cutting tool ofCNC machine

Example 4.1. Let consider a very simple decision-making problem of selection of cutting tool of computer numerical control (CNC) machine. There are four types of material are available. The set of alternatives $D = {H_{i}^{'}, i = 1, 2, 3, 4}$ where $\begin{matrix} H_{1}^{'} & = & md carbonsteel, \\ H_{2}^{'} & = & md highspeedsteel, \\ H_{3}^{'} & = & md solidcarbide, \\ H_{4}^{'} & = & md diamond \end{matrix}$ To evaluate the alternatives, we consider a set of attributes $P = {C_{i}, i = 1, 2, 3, 4}$ given as $\begin{matrix} C_{1} & = & md shockresistance, \\ C_{2} & = & md chemicalstabilityorinertness, \\ C_{3} & = & md lowfriction, \\ C_{4} & = & md hothardness \end{matrix}$ In this problem we use q-ROFNs to evaluate the four possible alternatives $H_{i}^{'}$ (i = 1, 2, 3, 4) under the four above attributes. Let q = 3, a WV $\overset{˘}{ħ}$ is (0.2, 0.3, 0.1, 0.4) ^T and the controlling index is σ = 0.5.

Now we use Algorithm 1 to solve MADM problem of CNC machine. The procedural steps are elaborated as given below:

Step 1. Evaluate the decision matrix given by the person consist on q-ROF information.

	$C_{1}$	$C_{2}$	$C_{3}$	$C_{4}$
$H_{1}^{'}$	(0.32, 0.61)	(0.41, 0.45)	(0.33, 0.56)	(0.66, 0.53)
$H_{2}^{'}$	(0.61, 0.52)	(0.34, 0.56)	(0.57, 0.68)	(0.61, 0.49)
$H_{3}^{'}$	(0.72, 0.32)	(0.36, 0.49)	(0.60, 0.42)	(0.70, 0.21)
$H_{4}^{'}$	(0.76, 0.12)	(0.82, 0.32)	(0.91, 0.12)	(0.60, 0.13)

Step 2. The decision matrix is already in normalized form.

Step 3. Compute $H_{i}^{'} = md q - ROFHWAGA (H_{i 1}^{'}, H_{i 2}^{'}, \dots, H_{in}^{'})$ for each i. Thus we find aggregated q-ROFNs by using Equation (3). $H_{1}^{'} = (0.498, 0.527)$ , $H_{2}^{'} = (0.531, 0.539)$ , $H_{3}^{'} = (0.602, 0.351)$ , $H_{4}^{'} = (0.742, 0.199)$ .

Step 4. Evaluate the score functions for all $H_{i}^{'}$ for the collective overall q-ROFNs. $E (H_{1}^{'}) = - 0.023,$ $E (H_{2}^{'}) = - 0.007,$ $E (H_{3}^{'}) = 0.175,$ $E (H_{4}^{'}) = 0.401$ .

Step 5. Rank all the $H_{i}^{'}$ (i = 1, 2, 3, 4) according to the score values.

$H_{4}^{'} ≻ H_{3}^{'} ≻ H_{2}^{'} ≻ H_{1}^{'}$ and thus $H_{4}^{'}$ ia the most desirable alternative.

4.3 Comparison analysis

The proposed operator’s result is compared with two q-ROFWA and q-ROFWG operators, as indicated in the Table 2 as given below listing the results of the comparison in the final ranking of 4 alternatives. It can be observed in the comparison Table 2, the best selection made by the proposed operator is comparable with the already established operators which is expressive in itself and approves the reliability and validity of the proposed operator. The calculation of decision results based on various aggregation operator are given in Table 1 and pictorial view is in Figure 1.

Table 1
Calculation of decision results based on various aggregation operator

Aggregation Operators Aggregation Result Score Value

q-ROFHWAGA (Proposed) $H_{1}^{'} = < 0.498, 0.528 >$ $E (H_{1}^{'}) = - 0.023$

$H_{2}^{'} = < 0.531, 0.539 >$ $E (H_{2}^{'}) = - 0.007$

$H_{3}^{'} = < 0.602, 0.351 >$ $E (H_{3}^{'}) = 0.175$

$H_{4}^{'} = < 0.742, 0.199 >$ $E (H_{4}^{'}) = 0.401$

q-ROFWA (Liu and Wang [19]) $H_{1}^{'} = < 0.536, 0.522 >$ $E (H_{1}^{'}) = 0.012$

$H_{2}^{'} = < 0.555, 0.533 >$ $E (H_{2}^{'}) = 0.020$

$H_{3}^{'} = < 0.638, 0.316 >$ $E (H_{3}^{'}) = 0.228$

$H_{4}^{'} = < 0.765, 0.166 >$ $E (H_{4}^{'}) = 0.443$

q-ROFWG (Liu and Wang [19]) $H_{1}^{'} = < 0.462, 0.533 >$ $E (H_{1}^{'}) = - 0.053$

$H_{2}^{'} = < 0.508, 0.544 >$ $E (H_{2}^{'}) = - 0.030$

$H_{3}^{'} = < 0.567, 0.378 >$ $E (H_{3}^{'}) = 0.128$

$H_{4}^{'} = < 0.720, 0.225 >$ $E (H_{4}^{'}) = 0.362$

Aggregation Operators	Aggregation Result	Score Value
q-ROFHWAGA (Proposed)	$H_{1}^{'} = < 0.498, 0.528 >$	$E (H_{1}^{'}) = - 0.023$
	$H_{2}^{'} = < 0.531, 0.539 >$	$E (H_{2}^{'}) = - 0.007$
	$H_{3}^{'} = < 0.602, 0.351 >$	$E (H_{3}^{'}) = 0.175$
	$H_{4}^{'} = < 0.742, 0.199 >$	$E (H_{4}^{'}) = 0.401$
q-ROFWA (Liu and Wang [19])	$H_{1}^{'} = < 0.536, 0.522 >$	$E (H_{1}^{'}) = 0.012$
	$H_{2}^{'} = < 0.555, 0.533 >$	$E (H_{2}^{'}) = 0.020$
	$H_{3}^{'} = < 0.638, 0.316 >$	$E (H_{3}^{'}) = 0.228$
	$H_{4}^{'} = < 0.765, 0.166 >$	$E (H_{4}^{'}) = 0.443$
q-ROFWG (Liu and Wang [19])	$H_{1}^{'} = < 0.462, 0.533 >$	$E (H_{1}^{'}) = - 0.053$
	$H_{2}^{'} = < 0.508, 0.544 >$	$E (H_{2}^{'}) = - 0.030$
	$H_{3}^{'} = < 0.567, 0.378 >$	$E (H_{3}^{'}) = 0.128$
	$H_{4}^{'} = < 0.720, 0.225 >$	$E (H_{4}^{'}) = 0.362$

Table 2

Comparison analysis of final ranking

Operators	Ranking of alternatives	The optimal alternative
q-ROFHWAGA	$H_{4}^{'} ≻ H_{3}^{'} ≻ H_{2}^{'} ≻ H_{1}^{'}$	$H_{4}^{'}$
q-ROFWA	$H_{4}^{'} ≻ H_{3}^{'} ≻ H_{2}^{'} ≻ H_{1}^{'}$	$H_{4}^{'}$
q-ROFWG	$H_{4}^{'} ≻ H_{3}^{'} ≻ H_{2}^{'} ≻ H_{1}^{'}$	$H_{4}^{'}$

Fig. 1

Graphical representation of calculation of decision results.

5 MADM with q-ROFS TOPSIS method

In this section, we explore how q-ROFS can be used in MADM using TOPSIS. First, we will extend TOPSIS to the q-ROFS and then consider an example of a method that has been proposed. Of the immense approaches found in the literature, TOPSIS has a central role to play in solving these problems. Depending on the nature of the problem, every technique has its own benefits and drawbacks.

Now we present q-ROFS TOPSIS method for MADM as given in Algorithm 2 below.

bfAlgorithm 2

q-ROFS TOPSIS method

Step 1: Distinguish the problem: Let $D = {M_{i} : i \in N}$ be a assemblage of decision makers/experts, $A = {{\ddot{z}}_{i} : i \in N}$ is the finite assemblage of alternatives and C = {C_i : i ∈ mathdsN} is a finite family of parameters/criterion.

Step 2: Choosing the linguistic terms from the Table 3, constructing weighted parameter matrix $Q$ as $Q = [y_{ij}]_{n \times m} = (\begin{matrix} y_{11} & y_{12} & \dots & y_{1 m} \\ y_{21} & y_{22} & \dots & y_{2 m} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ y_{i 1} & y_{i 2} & \dots & y_{im} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ y_{n 1} & y_{n 2} & \dots & y_{nm} \end{matrix})$

where y_ij is the weight assigned by the expert $M_{i}$ to the alternative ${\ddot{z}}_{i}$ by assuming linguistic variables as given (for example) in Table 3. bfstep 3: Construct normalized weighted matrix $\hat{N} = [{\hat{n}}_{ij}]_{n \times m} = (\begin{matrix} {\hat{n}}_{11} & {\hat{n}}_{12} & \dots & {\hat{n}}_{1 m} \\ {\hat{n}}_{21} & {\hat{n}}_{22} & \dots & {\hat{n}}_{2 m} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ {\hat{n}}_{i 1} & {\hat{n}}_{i 2} & \dots & {\hat{n}}_{im} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ {\hat{n}}_{n 1} & {\hat{n}}_{n 2} & \dots & {\hat{n}}_{nm} \end{matrix})$

where ${\hat{n}}_{ij} = \frac{y_{ij}}{\sqrt{Σ_{i = 1}^{n} y_{ij}^{2}}}$ and obtaining the weighted vector $W = (y_{1}, y_{2}, \dots, y_{m})$ , where $y_{i} = \frac{y_{i}}{Σ_{α = 1}^{n} y_{α i}}$ and $y_{j} = \frac{Σ_{i = 1}^{n} {\hat{n}}_{ij}}{n}$ . Notice that $Σ_{α = 1}^{n} y_{α i}$ represents sum of weights of i^th column in the matrix $Q$ in step 2 above.

Table 3
Linguistic terms for judging alternatives

Linguistic Terms Fuzzy Weights

Very High Importance (VHI) 0.95

High Importance (HI) 0.80

Average Importance (AI) 0.50

Low Importance (LI) 0.35

Linguistic Terms	Fuzzy Weights
Very High Importance (VHI)	0.95
High Importance (HI)	0.80
Average Importance (AI)	0.50
Low Importance (LI)	0.35

Step 4: Construct q-ROFS decision matrix $Z_{i} = [z_{jk}^{i}]_{l \times m} = (\begin{matrix} z_{11}^{i} & z_{12}^{i} & \dots & z_{1 m}^{i} \\ z_{21}^{i} & z_{22}^{i} & \dots & z_{2 m}^{i} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ z_{j 1}^{i} & z_{j 2}^{i} & \dots & z_{jm}^{i} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ z_{l 1}^{i} & z_{l 2}^{i} & \dots & z_{lm}^{i} \end{matrix})$

where $z_{jk}^{i}$ is a q-ROFS-element, for i^th decision maker. Then acquire the aggregating matrix $A = \frac{Z_{1} + Z_{2} + \dots + Z_{n}}{n} = [{\dot{z}}_{jk}]_{l \times m}$

Step 5: Acquire the weighted q-ROFS decision matrix $J = [{\ddot{z}}_{jk}]_{l \times m} = (\begin{matrix} {\ddot{z}}_{11} & {\ddot{z}}_{12} & \dots & {\ddot{z}}_{1 m} \\ {\ddot{z}}_{21} & {\ddot{z}}_{22} & \dots & {\ddot{z}}_{2 m} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ {\ddot{z}}_{j 1} & {\ddot{z}}_{j 2} & \dots & {\ddot{z}}_{jm} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ {\ddot{z}}_{l 1} & {\ddot{z}}_{l 2} & \dots & {\ddot{z}}_{lm} \end{matrix})$

where ${\ddot{z}}_{jk} = y_{k} \times {\dot{z}}_{jk}$ .

Step 6: Acquire q-ROFS-valued positive ideal solution (q-ROFSV-PIS) and q-ROFS-valued negative ideal solution (q-ROFSV-NIS), employing in order

$\begin{matrix} md q - ROFSV - PIS & = {{\ddot{z}}_{1}^{+}, {\ddot{z}}_{2}^{+}, \dots, {\ddot{z}}_{l}^{+}} \\ = {(\lor_{k} {\ddot{z}}_{jk}, \land_{k} {\ddot{z}}_{jk}) : k = 1, 2, \dots, m} \end{matrix}$

and

$\begin{matrix} md q - ROFSV - NIS & = {{\ddot{z}}_{1}^{-}, {\ddot{z}}_{2}^{-}, \dots, {\ddot{z}}_{l}^{-}} \\ = {(\land_{k} {\ddot{z}}_{jk}, \lor_{k} {\ddot{z}}_{jk}) : k = 1, 2, \dots, m} \end{matrix}$

where ∨ stands for q-ROFS union and ∧ represents q-ROFS intersection.

Step 7: Compute q-ROFS-Euclidean distances of each alternative from q-ROFSV-PIS and q-ROFSV-NIS, respectively, utilizing $(t_{j}^{+})^{2} = Σ_{k = 1}^{m} {\frac{1}{2} (F_{jk}^{+} -^{+} F_{k}^{+})^{2} + \frac{1}{2} (T_{jk}^{+} -^{+} T_{k}^{+})^{2} + \frac{1}{2} (F_{jk}^{-} -^{+} F_{k}^{-})^{2} + \frac{1}{2} (T_{jk}^{-} -^{+} T_{k}^{-})^{2}} (6)$ and $(t_{j}^{-})^{2} = Σ_{k = 1}^{m} {\frac{1}{2} (F_{jk}^{+} -^{-} F_{k}^{+})^{2} + \frac{1}{2} (T_{jk}^{+} -^{-} T_{k}^{+})^{2} + \frac{1}{2} (F_{jk}^{-} -^{-} F_{k}^{-})^{2} + \frac{1}{2} (T_{jk}^{-} -^{-} T_{k}^{-})^{2}} (7)$

Step 8: The closeness coefficient of each alternative with ideal solution by making use of $R^{*} ({\ddot{v}}_{j}) = \frac{t_{j}^{-}}{t_{j}^{+} + t_{j}^{-}} \in [0, 1]$ Step 9: Prosperous the alternatives in decreasing (or increasing) order.

Flow chart of Algorithm 2 given in Figure 2.

Fig. 2

Flow chart of q-ROF TOPSIS

5.1 Case study

Transport is essential need for daily routine life in this modern era. The modes of transport is buses, cars, train, bikes and lot of others. When public transport used, we have a lot of benefits like, reduces air pollution, increased fuel efficiency, increases mobility, saves money and it is safer. Here, we have a problem of a city transportation group which want to implement a transport system in the city. The available transport system are Car-share, Ride-share and Park and ride.

Car-share is the most popular in America. It is different from the traditional car rental. It is for those people who do not own the car and have used the car for several hours. People reserve their vehicle over the internet or call and get the vehicle at the required time and at the nearby destination. Another gain of car-share is that it allows you to access a car at any time, not just during business hours. Ride-sharing is a service which is used for one-way transportation on urgent basis. In ride sharing client booked their ride via mobile application or call at the end of ride client pay rent by bank account or in the form of cash. It is very common to go for college, universities and workplaces. One can see a ride near by on mobile application and booked this ride.

In park and ride car owner parked car near the bus station or railway station outside the city and connect with public transport. By park and ride people saved their time and money. The car is left in parking during the day and people get their cars when they returns.

Decision makers have different criterions to make a good decision, there are two types of criterions one is cost type and other one is benefit. Cost type criteria is suitable when its value is lower but benefit type criteria is suitable when its value is higher value.

5.2 Application of q-ROF TOPSIS

Example 5.1. Let a city transportation group implement a transport system which is most suitable for city. There are three systems available Car-share, Ride-share and Park and ride.

Step 1: Identifying the problem: Assume that $D = {M_{i} : i = 1, 2, 3, 4}$ is the set decision makers, $A = {{\ddot{z}}_{i} : i = 1, 2, 3}$ is the assemblage of alternatives under consideration and C = {C_i : i = 1, 2, ⋯ , 13} is a finite family of parameters/criterion. In following Table 4 we have listed some criterions related to transport system problem.

Table 4
Criterions for transport system problem

Criteria Definition Type

C ₁ Rates Operative cost for running the transport Cost

C ₂ Protection Safety obtained by the transport Benefit

C ₃ Reliability Capability to accomplish the service Benefit

C ₄ Air pollution Air contaminants from the transport Cost

C ₅ Noise Noise created by the transport Cost

C ₆ Fuels Usage of fuels petrol, CNG and diesel Cost

C ₇ Travel costs Costs for travel among stations Cost

C ₈ Energy Energy consumption by transport Cost

C ₉ Economy Like labor employment etc. Benefit

C ₁₀ Competency Modern technology Benefit

C ₁₁ Mobility Ability to provision transport area Benefit

C ₁₂ Productivity Ability to attain targets Benefit

C ₁₃ Excellence Excellence of service Benefit

	Criteria	Definition	Type
C ₁	Rates	Operative cost for running the transport	Cost
C ₂	Protection	Safety obtained by the transport	Benefit
C ₃	Reliability	Capability to accomplish the service	Benefit
C ₄	Air pollution	Air contaminants from the transport	Cost
C ₅	Noise	Noise created by the transport	Cost
C ₆	Fuels	Usage of fuels petrol, CNG and diesel	Cost
C ₇	Travel costs	Costs for travel among stations	Cost
C ₈	Energy	Energy consumption by transport	Cost
C ₉	Economy	Like labor employment etc.	Benefit
C ₁₀	Competency	Modern technology	Benefit
C ₁₁	Mobility	Ability to provision transport area	Benefit
C ₁₂	Productivity	Ability to attain targets	Benefit
C ₁₃	Excellence	Excellence of service	Benefit

Step 2: In this step we construct weighted parameter matrix, in which each decision maker assign linguistic terms from Table 3 to each parameter. Following is the Table 5 of importance which is given by decision makers to each cataria.

Table 5

linguistic terms to each parameter by decision maker

Criteria	Decision Maker
	D ₁	D ₂	D ₃	D ₄
C ₁	VHI	HI	HI	VHI
C ₂	HI	HI	VHI	AI
C ₃	AI	VHI	LI	HI
C ₄	VHI	VHI	HI	VHI
C ₅	VHI	HI	HI	HI
C ₆	HI	LI	AI	AI
C ₇	HI	AI	AI	LI
C ₈	LI	LI	AI	AI
C ₉	HI	HI	AI	HI
C ₁₀	VHI	AI	VHI	LI
C ₁₁	AI	VHI	LI	VHI
C ₁₂	LI	HI	AI	LI
C ₁₃	VHI	LI	AI	VHI

From Table 5 we construct weighted parameter matrix $Q$

$\begin{matrix} Q & = [y_{ij}]_{4 \times 13} \\ = (\begin{matrix} VHI & HI & AI & VHI & VHI & HI & HI & LI & HI & VHI & AI & LI & VHI \\ HI & HI & VHI & VHI & HI & LI & AI & LI & HI & AI & VHI & HI & LI \\ HI & VHI & LI & HI & HI & AI & AI & AI & AI & VHI & LI & AI & AI \\ VHI & AI & HI & VHI & HI & AI & LI & AI & HI & LI & VHI & LI & VHI \end{matrix}) \\ = (\begin{matrix} 0.95 & 0.80 & 0.50 & 0.95 & 0.95 & 0.80 & 0.80 & 0.80 & 0.80 & 0.80 & 0.80 & 0.80 & 0.80 \\ 0.50 & 0.35 & 0.95 & 0.50 & 0.35 & 0.35 & 0.80 & 0.80 & 0.80 & 0.80 & 0.80 & 0.80 & 0.80 \\ 0.35 & 0.35 & 0.50 & 0.35 & 0.95 & 0.80 & 0.80 & 0.80 & 0.80 & 0.80 & 0.80 & 0.80 & 0.80 \\ 0.50 & 0.80 & 0.95 & 0.35 & 0.35 & 0.50 & 0.80 & 0.80 & 0.80 & 0.80 & 0.80 & 0.80 & 0.80 \end{matrix}) \end{matrix}$

where w_ij is the weight provided by the decision maker D_i (row-wise) to each parameter C_j (column-wise).

Step 3: The normalized weighted matrix is $\begin{matrix} \hat{N} & = & [{\hat{n}}_{ij}]_{4 \times 13} \\ = & (\begin{matrix} 0.5409 & 0.5129 & 0.3613 & 0.5129 & 0.5655 & 0.7119 & 0.7119 & 0.4055 & 0.5774 & 0.6438 & 0.3388 & 0.3285 & 0.6438 \\ 0.4555 & 0.5129 & 0.6865 & 0.5129 & 0.4762 & 0.3115 & 0.4449 & 0.4055 & 0.5774 & 0.3388 & 0.6438 & 0.7509 & 0.2372 \\ 0.4555 & 0.5771 & 0.2529 & 0.4372 & 0.4762 & 0.4449 & 0.4449 & 0.5793 & 0.3609 & 0.6438 & 0.2372 & 0.4693 & 0.3388 \\ 0.5409 & 0.3206 & 0.5781 & 0.5129 & 0.4762 & 0.4449 & 0.3115 & 0.5793 & 0.5774 & 0.2372 & 0.6438 & 0.3285 & 0.6438 \end{matrix}) \end{matrix}$ and hence the weighted vector is $W = (0.08, 0.08, 0.07, 0.08, 0.08, 0.08, 0.08, 0.08, 0.08, 0.07, 0.07, 0.08, 0.07)$ .

Step 4: Keeping in view the track record of the companies under consideration, the q-ROFS decision matrix Z_i of each specialist is provided in which alternatives are represented row-wise whereas parameters are represented column-wise. Hence, the aggregated decision matrix comes out to be

$\begin{matrix} A & = & \frac{Z_{1} + Z_{2} + Z_{3} + Z_{4}}{4} \\ = & (\begin{matrix} (0.83, 0.53) & (0.84, 0.21) & (0.32, 0.16) & (0.93, 0.14) & (0.81, 0.23) & (0.73, 0.21) & (0.36, 0.79) & (0.26, 0.16) & (0.43, 0.71) & (0.16, 0.63) & (0.36, 0.71) & (0.96, 0.12) & (0.86, 0.32) \\ (0.72, 0.63) & (0.56, 0.72) & (0.32, 0.63) & (0.51, 0.71) & (0.26, 0.71) & (0.36, 0.36) & (0.24, 0.84) & (0.69, 0.34) & (0.49, 0.63) & (0.26, 0.61) & (0.61, 0.36) & (0.71, 0.36) & (0.26, 0.37) \\ (0.39, 0.79) & (0.61, 0.72) & (0.23, 0.26) & (0.56, 0.36) & (0.26, 0.71) & (0.31, 0.76) & (0.63, 0.49) & (0.39, 0.71) & (0.82, 0.26) & (0.71, 0.30) & (0.71, 0.21) & (0.38, 0.81) & (0.18, 0.47) \end{matrix}) \\ = & [{\dot{z}}_{jk}]_{3 \times 13} \end{matrix}$

Normalize the matrix A

$\begin{matrix} = & \frac{Z_{1} + Z_{2} + Z_{3} + Z_{4}}{4} \\ = & (\begin{matrix} (0.53, 0.83) & (0.84, 0.21) & (0.32, 0.16) & (0.14, 0.93) & (0.23, 0.81) & (0.21, 0.73) & (0.79, 0.36) & (0.16, 0.26) & (0.43, 0.71) & (0.16, 0.63) & (0.36, 0.71) & (0.96, 0.12) & (0.86, 0.32) \\ (0.63, 0.72) & (0.56, 0.72) & (0.32, 0.63) & (0.71, 0.51) & (0.71, 0.26) & (0.36, 0.36) & (0.84, 0.24) & (0.34, 0.69) & (0.49, 0.63) & (0.26, 0.61) & (0.61, 0.36) & (0.71, 0.36) & (0.26, 0.37) \\ (0.79, 0.39) & (0.61, 0.72) & (0.23, 0.26) & (0.36, 0.56) & (0.71, 0.26) & (0.76, 0.31) & (0.63, 0.49) & (0.71, 0.39) & (0.82, 0.26) & (0.71, 0.30) & (0.71, 0.21) & (0.38, 0.81) & (0.18, 0.47) \end{matrix}) \\ = & [{\dot{z}}_{jk}]_{3 \times 13} \end{matrix}$

Step 5: The weighted q-ROFS decision matrix

$\begin{matrix} J & = & [{\ddot{z}}_{jk}]_{3 \times 13} \\ = & (\begin{matrix} (0.04, 0.07) & (0.07, 0.02) & (0.01, 0.02) & (0.01, 0.07) & (0.02, 0.06) & (0.02, 0.06) & (0.06, 0.03) & (0.01, 0.02) & (0.03, 0.06) & (0.01, 0.04) & (0.03, 0.05) & (0.08, 0.01) & (0.07, 0.03) \\ (0.05, 0.02) & (0.04, 0.06) & (0.04, 0.02) & (0.06, 0.04) & (0.06, 0.02) & (0.03, 0.03) & (0.07, 0.02) & (0.03, 0.06) & (0.04, 0.05) & (0.01, 0.04) & (0.03, 0.05) & (0.08, 0.01) & (0.07, 0.03) \\ (0.06, 0.03) & (0.05, 0.06) & (0.02, 0.02) & (0.03, 0.04) & (0.06, 0.02) & (0.06, 0.02) & (0.05, 0.04) & (0.06, 0.03) & (0.07, 0.02) & (0.05, 0.02) & (0.04, 0.01) & (0.03, 0.06) & (0.01, 0.04) \end{matrix}) \end{matrix}$

where ${\ddot{z}}_{jk} = y_{k} \times {\dot{z}}_{jk}$ .

Step 6: Now, we locate q-ROFS valued positive ideal solution (q-ROFSV-PIS) and q-ROFS-valued negative ideal solution (q-ROFSV-NIS) and are listed, respectively, as $md q - ROFSV - PIS = {{\ddot{z}}_{1}^{+}, {\ddot{z}}_{2}^{+}, {\ddot{z}}_{3}^{+}} = {(0.08, 0.01), (0.07, 0.02), (0.07, 0.01)}$ and $md q - ROFSV - NIS = {{\ddot{z}}_{1}^{-}, {\ddot{z}}_{2}^{-}, {\ddot{z}}_{3}^{-}} = {(0.01, 0.07), (0.02, 0.06), (0.01, 0.06)}$

Step 7, 8: The q-ROFS-Euclidean distances of each alternative from q-ROFSV-PIS and q-ROFSV-NIS along with closeness coefficients are given in Table 6.

Table 6

Distance measures & closeness coefficient of each alternative

Alternative $({\ddot{z}}_{i})$	$t_{i}^{+} = t^{+} ({\ddot{z}}_{i}, md q - ROFSV - PIS)$	$t_{i}^{-} = t^{+} ({\ddot{z}}_{i}, md q - ROFSV - NIS)$	$R_{i}^{*}$
${\ddot{z}}_{1}$	0.16171	0.12728	0.44043
${\ddot{z}}_{2}$	0.13248	0.10247	0.43614
${\ddot{z}}_{3}$	0.14491	0.11467	0.44175

Step 9: The preference order of the alternatives, therefore, is ${\ddot{z}}_{3} ≻ {\ddot{z}}_{1} ≻ {\ddot{z}}_{2}$ So, ${\ddot{z}}_{3}$ alternative is the best and it is to be selected.

6 Conclusion

Aggregation operators based on q-ROFNs namely q-ROFWA operators and q-ROFWG operators are important mathematical tool for aggregating q-rung orthopair fuzzy information. In order to overcome some drawbacks implied by the existing q-ROFWA operators and q-ROFWG operators in some real world problems, we developed two operators namely q-rung orthopair fuzzy hybrid weighted arithmetic geometric aggregation (q-ROFHWAGA) operator and q-rung orthopair fuzzy hybrid ordered weighted arithmetic geometric aggregation (q-ROFHOWAGA) operator. We established certain properties of q-ROFHWAGA and q-ROFHOWAGA operators. The suggested operators are superior to the existing operators defined on q-ROFNs and efficient to deal with vague and uncertain information. We elaborated the proposed operators with the help of illustrations. We presented MADM method for selecting the best material for cutting tool of CNC machine based on q-ROFHWAGA and q-ROFHOWAGA operators. Furthermore, we established q-ROF TOPSIS method to solve the transport policy problem. In the future, we will extend our work to solve other real world MADM problems by using different aggregation operators, TOPSIS, AHP, VIKOR, ELECTRE family and PROMETHEE family based on different hybrid structures of fuzzy sets like Pythagorean m-polar fuzzy sets, linear Diophantine fuzzy sets, cubic bipolar fuzzy sets and cubic m-polar fuzzy sets.

Acknowledgment

The authors would like to thank the Editor-in-Chief and the referees very much for their valuable input and Recommendations for making our paper more accurate.

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Some q-rung orthopair fuzzy hybrid aggregation operators and TOPSIS method for multi-attribute decision-making

Abstract

Keywords

1 Introduction

2 Preliminaries

2.1 Operational laws of q-ROFSs

2.2 Operational laws of q-ROFNs

2.3 Some basic aggregation operators on q-ROFNs

2.4 Some drawbacks of q-ROFWA and q-ROFWG operators

3 Some hybrid aggregation operators of q-ROFNs

3.1 q-ROFHWAGA operator

3.2 q-ROFHOWAGA operator

3.3 Numerical example

4 MADM with q-ROFHWAGA andq-ROFHOWAGA operators

4.1 Case study

4.2 Application to selection of cutting tool ofCNC machine

4.3 Comparison analysis

Table 3 Linguistic terms for judging alternatives Linguistic Terms Fuzzy Weights Very High Importance (VHI) 0.95 High Importance (HI) 0.80 Average Importance (AI) 0.50 Low Importance (LI) 0.35

5.2 Application of q-ROF TOPSIS

Acknowledgment

References

Table 3
Linguistic terms for judging alternatives

Linguistic Terms Fuzzy Weights

Very High Importance (VHI) 0.95

High Importance (HI) 0.80

Average Importance (AI) 0.50

Low Importance (LI) 0.35