Some improved pythagorean fuzzy Dombi power aggregation operators with application in multiple-attribute decision making

Abstract

Pythagorean fuzzy set (PyFS) is an extension of various fuzzy concepts, such as fuzzy set (FS), intuitionistic FS, and it is enhanced mathematical gizmo to pact with uncertain and vague information. In this article, some drawbacks in the Dombi operational rules for Pythagorean fuzzy numbers (PyFNs) are examined and some improved Dombi operational laws for PyFNs are developed. We also find out that the value aggregated using the existing Dombi aggregation operators (DAOs) is not a PyFN. Furthermore, we developed two new aggregations, improved existing aggregation operators (AOs) for aggregating Pythagorean fuzzy information (PyFI) and are applied to multiple-attribute decision making (MADM). To acquire full advantage of power average (PA) operators proposed by Yager, the Pythagorean fuzzy Dombi power average (PyFDPA) operator, the Pythagorean fuzzy Dombi weighted power average (PyFDWPA) operator, Pythagorean fuzzy Dombi power geometric (PyFDPG) operator, Pythagorean fuzzy Dombi weighted geometric (PyFDPWG) operator, improved the existing AOs and their desirable properties are discussed. The foremost qualities of these developed Dombi power aggregation operators is that they purge the cause of discomfited data and are more supple due to general parameter. Additionally, based on these Dombi power AOs, a novel MADM approach is instituted. Finally, a numerical example is given to show the realism and efficacy of the proposed approach and judgment with the existing approaches is also specified.

Keywords

Pythagorean fuzzy set PA operator Dombi t-norm and Dombi t-conorm MADM

1 Introduction

One of the complexities in practical MADM problems is how to indicate the attribute values in fuzzy and vague decision making environments. Fuzzy set (FS) initiated by Zadeh [1] emanated as a gizmo for depicting and imparting unpredictability and fuzziness. While its origination, FS has acquired an exceptional concentration from scholars all over the world. These scholars studied its real and theoretical features. Various expansions of FS have been proposed, such as interval valued FS (IVFS) [2], which explicated the membership degree (MD) and is a subset of the [0, 1], and intuitionistic fuzzy set (IFS) [3] developed by Atanassov, which explicated the MD and non-membership degree (NMD) and fulfill the clause that summation of MD and NMD must be less or equal to one. Consequently, IFS explicates fuzziness and unpredictability more extensively than FS. Though, the captivating situation emanated when the MD and NMD of an entity is permitted from the unit interval and the summation of MD and NMD exceeds from one. In such situation the traditional IFS fails to deal with such information. Therefore, an extensive mathematical model is needed to deal with such type of situations.

To deal with the above defined situation Yager [4 –6] developed the notion of PyFS and is an extension of IFSs. The only dissimilarity among PyFS and IFS is that, the summation of square of MD and NMD in PyFS is less or equal to 1, while the summation of MD and NMD in IFS is less or equal to 1. After the initiation of PyFS, various studies have been conducted by many researchers, such as distance measure [7 –10], correlation co-efficient [11]. Zhang and Xu [12] delivered the comprehensive mathematical look for PyFS and proposed the idea of PyFN, then they also proposed a MADM approach based on Pythagorean fuzzy TOPSIS to deal with PyFNs. Ren et al. [13] developed the TODIM approach to obtain the optimal alternative in decision making utilizing PyFNs.

In decision making, the AO plays ruling role. From the past few years, information AOs have put on a great perception from the scholars and they developed different AOs and their extensions, for example, the traditional AO presented by Xu and Xu and Yager [14, 15] can only amassed a set of real number in a single real number. Currently these traditional AOs be further enlarged by several scholars, for example, Garg [16, 17] proposed generalized AOs, based on Einstein t-norm and t-conorm and pertained these AOs to MADM. Wei [18] put forward some Pythagorean fuzzy interaction AOs and applied these to MADM. Obviously, different AOs have different characteristics. Some AOs considered the interrelationship among input arguments, such as Bonferroni mean (BM) [19], Heronian mean (HM) [20], Muirhead mean [21], Maclaurin symmetric mean operator [22]. These AOs were further extended to deal with various fuzzy environments [23 –26]. Some AOs can diminish the influence of bad data such as PA operator initially presented by Yager [27], and it was extended by many researchers from all over the world to deal with various environments, such as, Wei and Lu [28] put forward Pythagorean fuzzy power AOs and pertains these AOs to MADM, Wei et al. [29] developed entropy information, taxonomy method for PyFSs and applied these to select the optimal tourism destination. Zhou et al. [30] initiated a new divergence measure for PyFS and confer its application in medical diagnosis. Shahzadi et al. [31] introduced some Yager AOs for PyFNs. Li et al. [32] developed some interactive hybrid AOs for PyFNs. Tang et al. [33] proposed some HM operators fo PyF dual hesitant fuzzy sets.

For aggregating PyFNs, several AOs are proposed using different T-norm (TN) and T-conorm (TCN), such as algebraic, Einstein, Hamacher and frank norm. Generally, Archimedean TN and TCN are the extensions of Dombi TN, Dombi TCN [34] and of the above defined TN and TCN. Dombi TN and TCN consist of a general parameter and be able to compose aggregation process more flexible. In recent years, some scholars proposed Dombi operational laws (DOLs) for various fuzzy sets and developed various AOs on these DOLs [35 –42]. Recently, Khan et al. [43] proposed DOLs for PyFNs and presented some Pythagorean fuzzy Dombi weighted AOs and applied these to MADM. However, the proposed DOLs for PyFNs and the Dombi weighted AOs have several limitations, which will discussed in Section 3 and Section 4.

As a result, the core purposes of this article is to present improved DOLs, improve the existing Dombi AOs and develop some new AOs, such as PyFPWA operator and its weighted form, PyPGA operator and its weighted form, and discussed several basic properties of these newly proposed AOs and apply them to MADM. These newly developed aggregation operators have several advantages over the existing Dombi aggregation operators for PyFNs, 1) the developed aggregation operators can remove the effect of awkward data on the final ranking result, 2) these aggregation operators are based on improved Dombi operational laws, 3) the result obtained by utilizing these aggregation operators is a PyFN. While the result obtained by utilizing the existing aggregation operators may not be a PyFN, 4) the proposed aggregation operators are more flexible to be use in solving MADM problems.

To do so, the rest of article is organized as follows: In section 2, some vital definitions about PyFS, PA operator, Dombi TN and Dombi TCN are given. In section 3, some drawbacks in the Dombi operational laws for PyFNs are discussed and some improved DOLs are proposed to over these limitations. In section 4, drawbacks in existing Dombi AOs are discussed and some improved AOs are proposed. In section 5, two newly Pythagorean fuzzy Dombi power AOs are proposed and discussed their desirable properties. In section 6, based on these newly developed AOs, a novel approach to MADM is developed. In section 7, conclusion, future work and references are given.

2 Preliminaries

In this part, some essential definitions about PyFSs, PyFN, PA operator, Dombi t-norm (DTN) and Dombi t-conorm (DTCN) and their related properties are argued.

2.1 Pythagorean fuzzy sets and their operational laws

In this subpart, definition of TN and TCN adapted from [44, 45], PyFSs [4], PyFNs [12] are given.

Definition 1. [44, 45]. A function $\bar{\bar{TN}} : \tilde{ℑ} \times \tilde{ℑ} \to \tilde{ℑ}$ is alleged to be triangular-norm (TN) if, for every element, $\bar{\bar{TN}}$ assures the following axioms:

$\bar{\bar{TN}}$ is associative, monotonic and commutative;

$\bar{\bar{TN}} (\bar{\bar{tn}}, 1) = \bar{\bar{tn}}$ , for each $\bar{\bar{tn}} \in \bar{\bar{TN}}$ .

Where $\tilde{ℑ} = [0, 1]$ .

Definition 2. [44, 45]. A function $\bar{\bar{TCN}} : \tilde{ℑ} \times \tilde{ℑ} \to \tilde{ℑ}$ is alleged to be triangular-conorm (TCN) if, for every element, $\bar{\bar{TCN}}$ assures the following axioms:

$\bar{\bar{TCN}}$ is associative, monotonic and commutative;

$\bar{\bar{TCN}} (\bar{\bar{tcn}}, 0) = \bar{\bar{tcn}}$ , for each $\bar{\bar{tcn}} \in \bar{\bar{TCN}}$ .

Where $\tilde{ℑ} = [0, 1]$ .

Definition 3. [4]. Let the universe of discourse set is signified by $\bar{\bar{ℵ}}$ . A PyFS $\bar{\bar{PY}}$ in $\bar{\bar{ℵ}}$ is mathematically signified by $\bar{\bar{PY}} = {〈 \bar{\bar{pe}} (\bar{\bar{ς}}), \bar{\bar{ne}} (\bar{\bar{ς}}) 〉 | \bar{\bar{ς}} \in \bar{\bar{ℵ}}};$ (1)

where $\bar{\bar{pe}} (\bar{\bar{ς}}) : \bar{\bar{ℵ}} \to [0, 1], \bar{\bar{ne}} (\bar{\bar{ς}}) : \bar{\bar{ℵ}} \to [0, 1]$ are two functions respectively, representing MD and NMD of each element $\bar{\bar{ς}} \in \bar{\bar{ℵ}}$ . The above two functions must assure the clause that $0 ⩽ {(\bar{\bar{pe}} (\bar{\bar{ς}}))}^{2} + {(\bar{\bar{ne}} (\bar{\bar{ς}}))}^{2} ⩽ 1$ . The hesitancy degree of $\bar{\bar{PY}}$ is denoted by ${\bar{\bar{π}}}_{\bar{\bar{PY}}} (\bar{\bar{ς}}) = \sqrt{1 - {(\bar{\bar{pe}} (\bar{\bar{ς}}))}^{2} + {(\bar{\bar{ne}} (\bar{\bar{ς}}))}^{2}}$ . Zhang and Xu [12] called $〈 \bar{\bar{pe}} (\bar{\bar{ς}}), \bar{\bar{ne}} (\bar{\bar{ς}}) 〉$ as a PyFN, which can be denoted by $\bar{\bar{py}} = 〈 \bar{\bar{pe}}, \bar{\bar{ne}} 〉$ .

Definition 4. [12]. Let $\bar{\bar{py}} = 〈 \bar{\bar{pe}}, \bar{\bar{ne}} 〉, {\bar{\bar{py}}}_{1} = 〈 {\bar{\bar{pe}}}_{1}, {\bar{\bar{ne}}}_{1} 〉$ and ${\bar{\bar{py}}}_{2} = 〈 {\bar{\bar{pe}}}_{2}, {\bar{\bar{ne}}}_{2} 〉$ be any three PyFNs, and ζ > 0. Then, $\begin{matrix} (1) {\bar{\bar{py}}}_{1} \oplus {\bar{\bar{py}}}_{2} = \\ 〈 \sqrt{{({\bar{\bar{pe}}}_{1})}^{2} + {({\bar{\bar{pe}}}_{2})}^{2} - {({\bar{\bar{pe}}}_{1})}^{2} {({\bar{\bar{pe}}}_{2})}^{2}}, {\bar{\bar{ne}}}_{1} {\bar{\bar{ne}}}_{2} 〉; \end{matrix}$ (2) $\begin{matrix} (2) {\bar{\bar{py}}}_{1} \otimes {\bar{\bar{py}}}_{2} = \\ 〈 {\bar{\bar{pe}}}_{1} {\bar{\bar{pe}}}_{2}, \sqrt{{({\bar{\bar{ne}}}_{1})}^{2} + {({\bar{\bar{ne}}}_{2})}^{2} - {({\bar{\bar{ne}}}_{1})}^{2} {({\bar{\bar{ne}}}_{2})}^{2}} 〉; \end{matrix}$ (3) $(3) ζ \bar{\bar{py}} = 〈 \sqrt{1 - {(1 - {(\bar{\bar{pe}})}^{2})}^{ζ}}, {\bar{\bar{ne}}}^{ζ} 〉;$ (4) $(4) {\bar{\bar{py}}}^{ζ} = 〈 {\bar{\bar{pe}}}^{ζ}, \sqrt{1 - {(1 - {(\bar{\bar{ne}})}^{2})}^{ζ}} 〉 .$ (5)

In order to compare two PyFNs, Wei and Lu [31] proposed the following comparison laws.

Definition 5. [28]. Let $\bar{\bar{py}} = 〈 \bar{\bar{pe}}, \bar{\bar{ne}} 〉$ be a PyFN. Then, the score and accuracy functions for PyFN are delineated as follows: $\bar{\bar{SE}} = \frac{1}{2} (1 + {(\bar{\bar{pe}})}^{2} - {(\bar{\bar{ne}})}^{2}); \bar{\bar{SE}} \in [0, 1],$ (6) $\bar{\bar{AC}} = {(\bar{\bar{pe}})}^{2} + {(\bar{\bar{ne}})}^{2}, \bar{\bar{AC}} \in [0, 1] .$ (7)

Definition 6. [28]. Let ${\bar{\bar{py}}}_{1} = 〈 {\bar{\bar{pe}}}_{1}, {\bar{\bar{ne}}}_{1} 〉$ and ${\bar{\bar{py}}}_{2} = 〈 {\bar{\bar{pe}}}_{2}, {\bar{\bar{ne}}}_{2} 〉$ be any two PyFNs, then the judgment rules are delineated as go after:

If $\bar{\bar{SE}} ({\bar{\bar{py}}}_{1}) > \bar{\bar{SE}} ({\bar{\bar{py}}}_{2})$ , then ${\bar{\bar{py}}}_{1}$ is greater than ${\bar{\bar{py}}}_{2}$ and is denoted as ${\bar{\bar{py}}}_{1} > {\bar{\bar{py}}}_{2}$ ;

If $\bar{\bar{SE}} ({\bar{\bar{py}}}_{1}) = \bar{\bar{SE}} ({\bar{\bar{py}}}_{2})$ , and $\bar{\bar{AC}} ({\bar{\bar{py}}}_{1}) > \bar{\bar{AC}} ({\bar{\bar{py}}}_{2})$ , then ${\bar{\bar{py}}}_{1}$ is greater than ${\bar{\bar{py}}}_{2}$ and is denoted as ${\bar{\bar{py}}}_{1} > {\bar{\bar{py}}}_{2}$ ;

If $\bar{\bar{SE}} ({\bar{\bar{py}}}_{1}) = \bar{\bar{SE}} ({\bar{\bar{py}}}_{2})$ , and $\bar{\bar{AC}} ({\bar{\bar{py}}}_{1}) = \bar{\bar{AC}} ({\bar{\bar{py}}}_{2})$ , then ${\bar{\bar{py}}}_{1}$ is equal to ${\bar{\bar{py}}}_{2}$ and is denoted as ${\bar{\bar{py}}}_{1} = {\bar{\bar{py}}}_{2}$ .

Definition 7. [12]. Let ${\bar{\bar{py}}}_{1} = 〈 {\bar{\bar{pe}}}_{1}, {\bar{\bar{ne}}}_{1} 〉$ and ${\bar{\bar{py}}}_{2} = 〈 {\bar{\bar{pe}}}_{2}, {\bar{\bar{ne}}}_{2} 〉$ be any two PyFNs, then the Hamming distance among ${\bar{\bar{py}}}_{1}$ and ${\bar{\bar{py}}}_{2}$ is identified as follows: $\begin{matrix} \bar{\bar{D}} ({\bar{\bar{py}}}_{1}, {\bar{\bar{py}}}_{2}) = \frac{1}{2} (| {({\bar{\bar{pe}}}_{1})}^{2} - {({\bar{\bar{pe}}}_{2})}^{2} | + \\ | {({\bar{\bar{ne}}}_{1})}^{2} - {({\bar{\bar{ne}}}_{2})}^{2} | + | {({\bar{\bar{π}}}_{1})}^{2} - {({\bar{\bar{π}}}_{2})}^{2} |) \end{matrix}$ (8)

2.2 PA operator

PA operator initially developed by Yager [27] for crisp number. The core benefit of the PA operator is its capability to lessen the insufficient consequence of unduly high and low arguments on the final results.

Definition 8. [27]. Let ${\bar{\bar{a}}}_{l} (1, . . ., q)$ be a set of crisp numbers, then the PA operator is symbolized as follows: $PA ({\bar{\bar{a}}}_{1}, {\bar{\bar{a}}}_{2}, . . ., {\bar{\bar{a}}}_{q}) = \sum_{l = 1}^{q} (\frac{(1 + T ({\bar{\bar{a}}}_{l})) {\bar{\bar{a}}}_{l}}{\sum_{m = 1}^{q} (1 + T ({\bar{\bar{a}}}_{m}))}),$ (9) where $T ({\bar{\bar{a}}}_{m}) = \sum_{n = 1, m \neq n}^{q} Supt ({\bar{\bar{a}}}_{m}, {\bar{\bar{a}}}_{n})$ and $Supt ({\bar{\bar{a}}}_{m}, {\bar{\bar{a}}}_{n})$ is identified as the support degree (SupD) for ${\bar{\bar{a}}}_{m}$ and ${\bar{\bar{a}}}_{n}$ . The SupD must assure the following axioms:

$Supt ({\bar{\bar{a}}}_{m}, {\bar{\bar{a}}}_{n}) \in [0, 1];$

$Supt ({\bar{\bar{a}}}_{m}, {\bar{\bar{a}}}_{n}) = Supt ({\bar{\bar{a}}}_{n}, {\bar{\bar{a}}}_{m});$

If $\bar{\bar{D}} ({\bar{\bar{a}}}_{m}, {\bar{\bar{a}}}_{n}) < \bar{\bar{D}} ({\bar{\bar{a}}}_{u}, {\bar{\bar{a}}}_{v})$ then $Supt ({\bar{\bar{a}}}_{m}, {\bar{\bar{a}}}_{n}) > Supt ({\bar{\bar{a}}}_{u}, {\bar{\bar{a}}}_{v})$ , where $\bar{\bar{D}} ({\bar{\bar{a}}}_{m}, {\bar{\bar{a}}}_{n})$ articulate the distance measure among ${\bar{\bar{a}}}_{m}$ and ${\bar{\bar{a}}}_{n}$ .

2.3 Dombi t-norm and t-conorm

Definition 9. [39]. Assume that $(\bar{\bar{tn}}, \bar{\bar{tcn}}) \in (0, 1) \times (0, 1)$ are any real numbers with γ ⩾ 1. Then, DTN and DTCN are depicted as follows: $\bar{\bar{TN}} (\bar{\bar{tn}}, \bar{\bar{tcn}}) = \frac{1}{1 + {{(\frac{1 - \bar{\bar{tn}}}{\bar{\bar{tn}}})}^{γ} + (\frac{1 - \bar{\bar{tcn}}}{\bar{\bar{tcn}}})}^{\frac{1}{γ}}};$ (10) $\bar{\bar{TCN}} (\bar{\bar{tn}}, \bar{\bar{tcn}}) = \frac{1}{1 + {{(\frac{\bar{\bar{tn}}}{1 - \bar{\bar{tn}}})}^{γ} + (\frac{\bar{\bar{tcn}}}{1 - \bar{\bar{tcn}}})}^{\frac{1}{γ}}} .$ (11)

3 Dombi operational laws for PyFNs

3.1 Existing Dombi operational laws for PyFNs

In this subpart, we discuss drawbacks in the existing DOLs anticipated by Khan el al. [43].

Definition 10. Let $\bar{\bar{py}} = 〈 \bar{\bar{pe}}, \bar{\bar{ne}} 〉, {\bar{\bar{py}}}_{1} = 〈 {\bar{\bar{pe}}}_{1}, {\bar{\bar{ne}}}_{1} 〉$ and ${\bar{\bar{py}}}_{2} = 〈 {\bar{\bar{pe}}}_{2}, {\bar{\bar{ne}}}_{2} 〉$ be any three PyFNs. Then, Khan et al. [43] identified the DOLs for PyFNs as follow:

$\begin{matrix} (1) & {\bar{\bar{py}}}_{1} \oplus {\bar{\bar{py}}}_{2} = \\ 〈 \sqrt{1 - \frac{1}{1 + {{(\frac{1}{{\bar{\bar{pe}}}_{1}^{2}} - 1)}^{2 γ} + {(\frac{1}{{\bar{\bar{pe}}}_{2}^{2}} - 1)}^{2 γ}}^{\frac{1}{γ}}}}, \\ \frac{1}{\sqrt{1 + {{(\frac{1}{{\bar{\bar{ne}}}_{1}^{2}} - 1)}^{2 γ} + {(\frac{1}{{\bar{\bar{ne}}}_{2}^{2}} - 1)}^{2 γ}}^{\frac{1}{γ}}}} 〉; \end{matrix}$

$(2) {\bar{\bar{py}}}_{1} \otimes {\bar{\bar{py}}}_{2} = 〈 \frac{1}{\sqrt{1 + {{(\frac{1}{{\bar{\bar{pe}}}_{1}^{2}} - 1)}^{2 γ} + {(\frac{1}{{\bar{\bar{pe}}}_{2}^{2}} - 1)}^{2 γ}}^{\frac{1}{γ}}}}, \sqrt{1 - \frac{1}{1 + {{(\frac{1}{{\bar{\bar{ne}}}_{1}^{2}} - 1)}^{2 γ} + {(\frac{1}{{\bar{\bar{ne}}}_{2}^{2}} - 1)}^{2 γ}}^{\frac{1}{γ}}}} 〉;$ (12)

$(3) β \bar{\bar{py}} = 〈 \sqrt{1 - \frac{1}{1 + {β {(\frac{1}{{\bar{\bar{pe}}}^{2}} - 1)}^{2 γ}}^{\frac{1}{γ}}}}, \frac{1}{\sqrt{1 + {β {(\frac{1}{{\bar{\bar{ne}}}^{2}} - 1)}^{2 γ}}^{\frac{1}{γ}}}} 〉; β > 0 .$ (13)

$(4) {\bar{\bar{py}}}^{β} = 〈 \frac{1}{\sqrt{1 + {β {(\frac{1}{{\bar{\bar{pe}}}^{2}} - 1)}^{2 γ}}^{\frac{1}{γ}}}}, \sqrt{1 - \frac{1}{1 + {β {(\frac{1}{{\bar{\bar{ne}}}^{2}} - 1)}^{2 γ}}^{\frac{1}{γ}}}} 〉, β > 0 .$ (14)

In the above DOLs, there exist some drawbacks in the sum and multiplication operations which are argued below:

Example 1. Let ${\bar{\bar{py}}}_{1} = 〈 0.5, 0.8 〉$ and ${\bar{\bar{py}}}_{2} = 〈 0.6, 0.7 〉$ be two PyFNs, then by utilizing (11), we have ${\bar{\bar{py}}}_{1} \oplus {\bar{\bar{py}}}_{2} = 〈 0.9514, 0.6854 〉 (γ = 2) .$

When we add the squares of these two MD and NMD, it becomes 1.3749, which is not a PyFN. Similarly, let ${\bar{\bar{py}}}_{1} = 〈 0.7, 0.4 〉$ and ${\bar{\bar{py}}}_{2} = 〈 0.8, 0.4 〉$ be two PyFNs, then by utilizing Equation (12), we have ${\bar{\bar{py}}}_{1} \otimes {\bar{\bar{py}}}_{2} = 〈 0.6854, 0.9874 〉 (γ = 2) .$

When we add the squares of these two MD and NMD, it becomes 1.4448, which is not a PyFN.

Hence, there is a need to improve the above DOLs for PyFNs. So, in the next subpart, we develop improved DOLs for PyFNs.

3.2 Improved Dombi operational laws for PyFNs

In this subpart, we suggest a few new improved DOLs for PyFNs.

Definition 11. Let $\bar{\bar{py}} = 〈 \bar{\bar{pe}}, \bar{\bar{ne}} 〉, {\bar{\bar{py}}}_{1} = 〈 {\bar{\bar{pe}}}_{1}, {\bar{\bar{ne}}}_{1} 〉$ and ${\bar{\bar{py}}}_{2} = 〈 {\bar{\bar{pe}}}_{2}, {\bar{\bar{ne}}}_{2} 〉$ be any PyFNs, then the improved DOLs for PyFNs based on DTN and DTCN are defined as follows:

$(1) {\bar{\bar{py}}}_{1} \oplus {\bar{\bar{py}}}_{2} = 〈 \sqrt{1 - \frac{1}{1 + {{(\frac{{\bar{\bar{pe}}}_{1}^{2}}{1 - {\bar{\bar{pe}}}_{1}^{2}})}^{γ} + {(\frac{{\bar{\bar{pe}}}_{2}^{2}}{1 - {\bar{\bar{pe}}}_{2}^{2}})}^{γ}}^{\frac{1}{γ}}}}, \frac{1}{1 + {{(\frac{1 - {\bar{\bar{ne}}}_{1}^{2}}{{\bar{\bar{ne}}}_{1}^{2}})}^{γ} + {(\frac{1 - {\bar{\bar{ne}}}_{2}^{2}}{{\bar{\bar{ne}}}_{2}^{2}})}^{γ}}^{\frac{1}{γ}}} 〉;$ (15)

$(2) {\bar{\bar{py}}}_{1} \otimes {\bar{\bar{py}}}_{2} = 〈 \frac{1}{1 + {{(\frac{1 - {\bar{\bar{pe}}}_{1}^{2}}{{\bar{\bar{pe}}}_{1}^{2}})}^{γ} + {(\frac{1 - {\bar{\bar{pe}}}_{2}^{2}}{{\bar{\bar{pe}}}_{2}^{2}})}^{γ}}^{\frac{1}{γ}}}, \sqrt{1 - \frac{1}{1 + {{(\frac{{\bar{\bar{ne}}}_{1}^{2}}{1 - {\bar{\bar{ne}}}_{1}^{2}})}^{γ} + {(\frac{{\bar{\bar{ne}}}_{2}^{2}}{1 - {\bar{\bar{ne}}}_{2}^{2}})}^{γ}}^{\frac{1}{γ}}}} 〉;$ (16)

$\bar{\bar{py}} = 〈 \sqrt{1 - \frac{1}{1 + {ζ {(\frac{\bar{\bar{pe}}}{1 - \bar{\bar{pe}}})}^{γ}}^{\frac{1}{γ}}}}, \frac{1}{1 + {ζ {(\frac{1 - \bar{\bar{ne}}}{\bar{\bar{ne}}})}^{γ}}^{\frac{1}{γ}}} 〉, ζ > 0;$ (17)

${\bar{\bar{py}}}^{ζ} = 〈 \frac{1}{1 + {ζ {(\frac{1 - \bar{\bar{pe}}}{\bar{\bar{pe}}})}^{γ}}^{\frac{1}{γ}}}, \sqrt{1 - \frac{1}{1 + {ζ {(\frac{\bar{\bar{ne}}}{1 - \bar{\bar{ne}}})}^{γ}}^{\frac{1}{γ}}}} 〉, ζ > 0 .$ (18)

Now, re-consider Example 1, and by utilizing Equation (15), we have ${\bar{\bar{py}}}_{1} \oplus {\bar{\bar{py}}}_{2} = 〈 0.6288, 0.4581 〉 (γ = 2) .$

When we add the square of these two MD and NMD, it becomes 0.6052, which is a PyFN. Similarly, let ${\bar{\bar{py}}}_{1} = 〈 0.7, 0.4 〉$ and ${\bar{\bar{py}}}_{2} = 〈 0.8, 0.4 〉$ be two PyFNs, then by utilizing Equation (16), we have ${\bar{\bar{py}}}_{1} \otimes {\bar{\bar{py}}}_{2} = 〈 0.4581, 0.4607 〉 (γ = 2) .$

When we add the square of these two MD and NMD, it becomes 0.4220, which is a PyFN.

4 Dombi aggregation operators

In this part, firstly, we review the existing Dombi AOs (DAOs) based on existing DOLs and discuss its drawbacks. Secondly, we develop improved DAOs based on improved DOLs.

4.1 Existing aggregation operators

In this subpart, we give some existing DAOs developed by Khan et al. [43] and examined drawbacks of these proposed AOs.

Definition 12. [43]. Let ${\bar{\bar{py}}}_{h} = 〈 {\bar{\bar{pe}}}_{h}, {\bar{\bar{ne}}}_{h} 〉 (h = 1, . . ., g)$ be a set of PyFNs, then the PyFDWA operator is described as go after:

$PyFDWA ({\bar{\bar{py}}}_{1}, {\bar{\bar{py}}}_{2}, . . ., {\bar{\bar{py}}}_{o}) = \sum_{h = 1}^{o} ς_{h} {\bar{\bar{py}}}_{h},$ (19) where ς = (ς₁, ς₂, . . . , ς_g) ^T is importance degree (ID) with ς_h ∈ [0, 1] and $\sum_{h = 1}^{o} ς_{h} = 1$ .

Theorem 1. [43]. Let ${\bar{\bar{py}}}_{h} = 〈 {\bar{\bar{pe}}}_{h}, {\bar{\bar{ne}}}_{h} 〉 (h = 1, . . ., o)$ be a set of PyFNs, then the structure of PyFWA operator is described utilizing Dombi operations with γ > 0;

$PyFDWA ({\bar{\bar{py}}}_{1}, {\bar{\bar{py}}}_{2}, . . ., {\bar{\bar{py}}}_{o}) = 〈 \sqrt{1 - \frac{1}{1 + {\sum_{h = 1}^{o} ς_{h} {(\frac{1}{{({\bar{\bar{pe}}}_{h})}^{2}} - 1)}^{2 γ}}^{\frac{1}{γ}}}}, \frac{1}{\sqrt{1 + {\sum_{h = 1}^{o} ς_{h} {(\frac{1}{{({\bar{\bar{ne}}}_{h})}^{2}} - 1)}^{2 γ}}^{\frac{1}{γ}}}} 〉;$ (20)

In some cases, when we aggregate PyFNs by (19), the obtained aggregated value is not a PyFN.

Example 2. Let ${\bar{\bar{py}}}_{1} = 〈 0.3, 0.7 〉, {\bar{\bar{py}}}_{2} = 〈 0.4, 0.6 〉, {\bar{\bar{py}}}_{3} = 〈 0.5, 0.8 〉$ and ${\bar{\bar{py}}}_{4} = 〈 0.6, 0.7 〉$ be four PyFNs and w = (0.3, 0.2, 0.4, 0.1) ^Tbe the ID of these four PyFNs. Then, by utilizing Equation (19), we have (γ = 2). $PyFDWA ({\bar{\bar{py}}}_{1}, {\bar{\bar{py}}}_{2}, {\bar{\bar{py}}}_{3}, {\bar{\bar{py}}}_{4}) = 〈 0.9914, 0.6222 〉 .$

Now, if the squares of MD and NMD of the aggregated value is added, then its value is 1.3700. So it violates the necessary clause that summation of squares of MD and NMD is less or equal to 1. Therefore, the aggregated value is not a PyFN. Similar limitations exists in other structures such as Pythagorean fuzzy Dombi ordered weighted averaging (PyFDOWA) operator and Pythagorean fuzzy Dombi hybrid weighted averaging (PyFDHWA) operator.

Definition 13. [43]. Let ${\bar{\bar{py}}}_{h} = 〈 {\bar{\bar{pe}}}_{h}, {\bar{\bar{ne}}}_{h} 〉 (h = 1, . . ., o)$ be a set of PyFNs, then the PyFDWG operator is described as follows: $PyFDWA ({\bar{\bar{py}}}_{1}, {\bar{\bar{py}}}_{2}, . . ., {\bar{\bar{py}}}_{o}) = \prod_{h = 1}^{o} {({\bar{\bar{py}}}_{h})}^{ς_{h}},$ (21)

where ς = (ς₁, ς₂, . . . , ς_g) ^T is ID with ς_h ∈ [0, 1] and $\sum_{h = 1}^{o} ς_{h} = 1$ .

Theorem 2. [43]. Let ${\bar{\bar{py}}}_{h} = 〈 {\bar{\bar{pe}}}_{h}, {\bar{\bar{ne}}}_{h} 〉 (h = 1, . . ., g)$ be a group of PyFNs, then the structure of PyFWG operator is described by utilizing Dombi operations with γ > 0;

$PyFDWG ({\bar{\bar{py}}}_{1}, {\bar{\bar{py}}}_{2}, . . ., {\bar{\bar{py}}}_{o}) = 〈 \frac{1}{\sqrt{1 + {\sum_{h = 1}^{o} ς_{h} {(\frac{1}{{({\bar{\bar{pe}}}_{h})}^{2}} - 1)}^{2 γ}}^{\frac{1}{γ}}}}, \sqrt{1 - \frac{1}{1 + {\sum_{h = 1}^{o} ς_{h} {(\frac{1}{{({\bar{\bar{ne}}}_{h})}^{2}} - 1)}^{2 γ}}^{\frac{1}{γ}}}} 〉;$ (22)

In some cases, when we aggregate PyFNs utilizing Equation (21), the obtained aggregated value is not a PyFN.

Example 3. Let ${\bar{\bar{py}}}_{1} = 〈 0.7, 0.5 〉, {\bar{\bar{py}}}_{2} = 〈 0.8, 0.3 〉, {\bar{\bar{py}}}_{3} = 〈 0.8, 0.2 〉$ and ${\bar{\bar{py}}}_{4} = 〈 0.9, 0.1 〉$ be any four PyFNs and ς = (0.3, 0.2, 0.4, 0.1) ^T be the ID of these four PyFNs. Then, by utilizing Equation (21), we have (γ = 2). $PyFDWG ({\bar{\bar{py}}}_{1}, {\bar{\bar{py}}}_{2}, {\bar{\bar{py}}}_{3}, {\bar{\bar{py}}}_{4}) = 〈 0.7803, 0.9998 〉 .$

Now, if the square of the summation of MD and NMD of the aggregated value is added, then its value is 1.6086. So it violates the condition that summation of square of MD and NMD is less or equal to 1. Therefore, the aggregated value is not a PyFN. Similar limitations exists in other structures such as Pythagorean fuzzy Dombi ordered weighted geometric (PyFDOWG) operator and Pythagorean fuzzy Dombi hybrid weighted geometric (PyFDHWG) operator.

4.2 Improved Dombi aggregation operators

The definitions of the Dombi weighted AOs remain the same as those defined by Khan et al. [43]. We just improved some theorems.

Theorem 3. Let ${\bar{\bar{py}}}_{h} = 〈 {\bar{\bar{pe}}}_{h}, {\bar{\bar{ne}}}_{h} 〉 (h = 1, . . ., g)$ be a set of PyFNs, then the structure of improved PyFWA (IPyFWA) operator is described as follows by utilizing improved DOLs with γ > 0;

$IPyFDWA ({\bar{\bar{py}}}_{1}, {\bar{\bar{py}}}_{2}, . . ., {\bar{\bar{py}}}_{o}) = 〈 \sqrt{1 - \frac{1}{1 + {\sum_{h = 1}^{o} ς_{h} {(\frac{{\bar{\bar{pe}}}_{h}^{2}}{1 - {\bar{\bar{pe}}}_{h}^{2}})}^{γ}}^{\frac{1}{γ}}}}, \frac{1}{1 + {\sum_{h = 1}^{o} ς_{h} {(\frac{1 - {\bar{\bar{ne}}}_{h}}{{\bar{\bar{ne}}}_{h}})}^{γ}}^{\frac{1}{γ}}} 〉 .$ (23) Now consider Example 2, and aggregate those values by utilizing Equation (22), we have (γ = 2). $IPyFDWA ({\bar{\bar{py}}}_{1}, {\bar{\bar{py}}}_{2}, {\bar{\bar{py}}}_{3}, {\bar{\bar{py}}}_{4}) = 〈 0.4765, 0.6979 〉;$

Now, if the square of MD and NMD of the aggregated value is added, then its value is 0.7141. Hence, the obtained aggregated value is a PyFN.

Similarly, the improved structure of other Dombi AOs such as improved PyFDWOA (IPyFDWOA) operator and improved PyFDHWA (IPyFDHWA) operator are explained as follows.

$IPyFDOWA ({\bar{\bar{py}}}_{1}, {\bar{\bar{py}}}_{2}, . . ., {\bar{\bar{py}}}_{o}) = 〈 \sqrt{1 - \frac{1}{1 + {\sum_{h = 1}^{o} ς_{h} {(\frac{{\bar{\bar{pe}}}_{σ (h)}^{2}}{1 - {\bar{\bar{pe}}}_{σ (h)}^{2}})}^{γ}}^{\frac{1}{γ}}}}, \frac{1}{1 + {\sum_{h = 1}^{o} ς_{h} {(\frac{1 - {\bar{\bar{ne}}}_{σ (h)}}{{\bar{\bar{ne}}}_{σ (h)}})}^{γ}}^{\frac{1}{γ}}} 〉 .$ (24)

$IPyFDHWA ({\bar{\bar{py}}}_{1}, {\bar{\bar{py}}}_{2}, . . ., {\bar{\bar{py}}}_{o}) = 〈 \sqrt{1 - \frac{1}{1 + {\sum_{h = 1}^{o} ς_{h} {(\frac{{({\bar{\bar{pe}}}_{σ (h)}^{*})}^{2}}{1 - {({\bar{\bar{pe}}}_{σ (h)}^{*})}^{2}})}^{γ}}^{\frac{1}{γ}}}}, \frac{1}{1 + {\sum_{h = 1}^{o} ς_{h} {(\frac{1 - {\bar{\bar{ne}}}_{σ (h)}^{*}}{{\bar{\bar{ne}}}_{σ (h)}^{*}})}^{γ}}^{\frac{1}{γ}}} 〉 .$ (25)

Theorem 4. Let ${\bar{\bar{py}}}_{h} = 〈 {\bar{\bar{pe}}}_{h}, {\bar{\bar{ne}}}_{h} 〉 (h = 1, . . ., g)$ be a group of PyFNs, then the structure of improved PyFWG (IPyFWG) operator is described by utilizing improved DOLs with γ > 0 as

$IPyFDWG ({\bar{\bar{py}}}_{1}, {\bar{\bar{py}}}_{2}, . . ., {\bar{\bar{py}}}_{o}) = 〈 \frac{1}{1 + {\sum_{h = 1}^{o} ς_{h} {(\frac{1 - {\bar{\bar{pe}}}_{h}}{{\bar{\bar{pe}}}_{h}})}^{γ}}^{\frac{1}{γ}}}, \sqrt{1 - \frac{1}{1 + {\sum_{h = 1}^{o} ς_{h} {(\frac{{\bar{\bar{ne}}}_{h}^{2}}{1 - {\bar{\bar{ne}}}_{h}^{2}})}^{γ}}^{\frac{1}{γ}}}} 〉 .$ (26)

Now consider Example 3, and aggregate those values by utilizing Equation (25), we have (γ = 2). $IPyFDWG ({\bar{\bar{py}}}_{1}, {\bar{\bar{py}}}_{2}, {\bar{\bar{py}}}_{3}, {\bar{\bar{py}}}_{4}) = 〈 0.7655, 0.3993 〉 .$

Now, if the square of MD and NMD of the aggregated value is added, then its value is 0.7455. Hence, the obtained aggregated value is a PyFN.

Similarly, the improved structure of other Dombi aggregation operators such as improved PyFDWOG (IPyFDWOG) operator and improved PyFDHWG (IPyFDHWG) operator are described as follows.

$IPyFDOWG ({\bar{\bar{py}}}_{1}, {\bar{\bar{py}}}_{2}, . . ., {\bar{\bar{py}}}_{o}) = 〈 \frac{1}{1 + {\sum_{h = 1}^{o} ς_{h} {(\frac{1 - {\bar{\bar{pe}}}_{σ (h)}}{{\bar{\bar{pe}}}_{σ (h)}})}^{γ}}^{\frac{1}{γ}}}, \sqrt{1 - \frac{1}{1 + {\sum_{h = 1}^{o} ς_{h} {(\frac{{\bar{\bar{ne}}}_{σ (h)}^{2}}{1 - {\bar{\bar{ne}}}_{σ (h)}^{2}})}^{γ}}^{\frac{1}{γ}}}} 〉 .$ (27)

$IPyFDHWG ({\bar{\bar{py}}}_{1}, {\bar{\bar{py}}}_{2}, . . ., {\bar{\bar{py}}}_{o}) = 〈 \frac{1}{1 + {\sum_{h = 1}^{o} ς_{h} {(\frac{1 - {\bar{\bar{pe}}}_{σ (h)}^{*}}{{\bar{\bar{pe}}}_{σ (h)}^{*}})}^{γ}}^{\frac{1}{γ}}}, \sqrt{1 - \frac{1}{1 + {\sum_{h = 1}^{o} ς_{h} {(\frac{{({\bar{\bar{ne}}}_{σ (h)}^{*})}^{2}}{1 - {({\bar{\bar{ne}}}_{σ (h)}^{*})}^{2}})}^{γ}}^{\frac{1}{γ}}}} 〉 .$ (28)

5 Pythagorean fuzzy Dombi power AOs

In this part, we develop two new Dombi AOs to deal with Pythagorean fuzzy information.

5.1 Pythagorean Fuzzy Dombi Power Arithmetic AOs

Definition 14. Let ${\bar{\bar{py}}}_{h} 〈 {\bar{\bar{pe}}}_{h}, {\bar{\bar{ne}}}_{h} 〉 (h = 1, . . ., o)$ be a group of PyFNs, then we describe the PyFDPA operator as follows:

$PyFDPA ({\bar{\bar{py}}}_{1}, {\bar{\bar{py}}}_{2}, . . ., {\bar{\bar{py}}}_{o}) = \frac{\oplus_{h = 1}^{o} ((1 + T ({\bar{\bar{py}}}_{h})) {\bar{\bar{py}}}_{h})}{\sum_{h = 1}^{o} (1 + T ({\bar{\bar{py}}}_{h}))}$ (29) Where $T ({\bar{\bar{py}}}_{h}) = \sum_{k = 1 k \neq h}^{o} Supt ({\bar{\bar{py}}}_{h}, {\bar{\bar{py}}}_{k})$ (30) and $Supt ({\bar{\bar{py}}}_{h}, {\bar{\bar{py}}}_{k})$ is the SupD for ${\bar{\bar{py}}}_{h}$ from ${\bar{\bar{py}}}_{k}$ , which must assure the following clause:

$Supt ({\bar{\bar{py}}}_{h}, {\bar{\bar{py}}}_{k}) \in [0, 1];$

$Supt ({\bar{\bar{py}}}_{h}, {\bar{\bar{py}}}_{k}) = Supt ({\bar{\bar{py}}}_{k}, {\bar{\bar{py}}}_{h});$

$Supt ({\bar{\bar{py}}}_{h}, {\bar{\bar{py}}}_{k}) ⩾ Supt ({\bar{\bar{py}}}_{m}, {\bar{\bar{py}}}_{n})$ , if $\bar{\bar{D}} ({\bar{\bar{py}}}_{h}, {\bar{\bar{py}}}_{k}) < \bar{\bar{D}} ({\bar{\bar{py}}}_{m}, {\bar{\bar{py}}}_{n})$ , where $\bar{\bar{D}}$ stand for the distance measure provided in Definition (7).

In order to write Equation (28) in uncomplicated form, we presume that ${\tilde{ξ}}_{h} = \frac{(1 + T ({\bar{\bar{py}}}_{h}))}{\sum_{h = 1}^{o} (1 + T ({\bar{\bar{py}}}_{h}))}$ (31)

Hence, Equation (28) becomes $PyFDPA ({\bar{\bar{py}}}_{1}, {\bar{\bar{py}}}_{2}, . . ., {\bar{\bar{py}}}_{o}) = \oplus_{h = 1}^{o} {\tilde{ξ}}_{h} {\bar{\bar{py}}}_{h}$ (32)

Based on Definition 14, we include the subsequent result.

Theorem 5. Let ${\bar{\bar{py}}}_{h} 〈 {\bar{\bar{pe}}}_{h}, {\bar{\bar{ne}}}_{h} 〉 (h = 1, . . ., o)$ be a set of PyFNs, then the value acquired by utilizing Equation (28) is still PyFN, and

$PyFDPA ({\bar{\bar{py}}}_{1}, {\bar{\bar{py}}}_{2}, . . ., {\bar{\bar{py}}}_{o}) = 〈 \sqrt{1 - \frac{1}{1 + {\sum_{h = 1}^{o} {\tilde{ξ}}_{h} {(\frac{{\bar{\bar{pe}}}_{h}^{2}}{1 - {\bar{\bar{pe}}}_{h}^{2}})}^{γ}}^{\frac{1}{γ}}}},$

$\frac{1}{1 + {\sum_{h = 1}^{o} {\tilde{ξ}}_{h} {(\frac{1 - {\bar{\bar{ne}}}_{h}}{{\bar{\bar{ne}}}_{h}})}^{γ}}^{\frac{1}{γ}}} 〉 .$ (33)

Proof.

Based on DOLs, we have

${\tilde{ξ}}_{h} {\bar{\bar{py}}}_{h} = 〈 \sqrt{1 - \frac{1}{1 + {{\tilde{ξ}}_{h} {(\frac{{\bar{\bar{pe}}}_{h}^{2}}{1 - {\bar{\bar{pe}}}_{h}^{2}})}^{γ}}^{\frac{1}{γ}}}}, \frac{1}{1 + {{\tilde{ξ}}_{h} {(\frac{1 - {\bar{\bar{ne}}}_{h}}{{\bar{\bar{ne}}}_{h}})}^{γ}}^{\frac{1}{γ}}} 〉 .$ (34)

We prove Equation (32) by utilizing mathematical induction.

If g = 2, then from Equation (33), we enclose $\begin{matrix} {\tilde{ξ}}_{1} {\bar{\bar{py}}}_{1} = 〈 \sqrt{1 - \frac{1}{1 + {{\tilde{ξ}}_{1} {(\frac{{\bar{\bar{pe}}}_{1}^{2}}{1 - {\bar{\bar{pe}}}_{1}^{2}})}^{γ}}^{\frac{1}{γ}}}}, \\ \frac{1}{1 + {{\tilde{ξ}}_{1} {(\frac{1 - {\bar{\bar{ne}}}_{1}}{{\bar{\bar{ne}}}_{1}})}^{γ}}^{\frac{1}{γ}}} 〉; \end{matrix}$ and $\begin{matrix} {\tilde{ξ}}_{2} {\bar{\bar{py}}}_{2} = 〈 \sqrt{1 - \frac{1}{1 + {{\tilde{ξ}}_{2} {(\frac{{\bar{\bar{pe}}}_{2}^{2}}{1 - {\bar{\bar{pe}}}_{2}^{2}})}^{γ}}^{\frac{1}{γ}}}}, \\ \frac{1}{1 + {{\tilde{ξ}}_{2} {(\frac{1 - {\bar{\bar{ne}}}_{2}}{{\bar{\bar{ne}}}_{2}})}^{γ}}^{\frac{1}{γ}}} 〉, \end{matrix}$

So,

$\begin{matrix} PyFDPA ({\bar{\bar{py}}}_{1}, {\bar{\bar{py}}}_{2}) = ξ_{1} {\bar{\bar{py}}}_{1} \oplus ξ_{2} {\bar{\bar{py}}}_{2}; \\ = 〈 \sqrt{1 - \frac{1}{1 + {{\tilde{ξ}}_{1} {(\frac{{\bar{\bar{pe}}}_{1}^{2}}{1 - {\bar{\bar{pe}}}_{1}^{2}})}^{γ}}^{\frac{1}{γ}}}}, \frac{1}{1 + {{\tilde{ξ}}_{1} {(\frac{1 - {\bar{\bar{ne}}}_{1}}{{\bar{\bar{ne}}}_{1}})}^{γ}}^{\frac{1}{γ}}} 〉 \oplus 〈 \sqrt{1 - \frac{1}{1 + {{\tilde{ξ}}_{2} {(\frac{{\bar{\bar{pe}}}_{2}^{2}}{1 - {\bar{\bar{pe}}}_{2}^{2}})}^{γ}}^{\frac{1}{γ}}}}, \frac{1}{1 + {{\tilde{ξ}}_{2} {(\frac{1 - {\bar{\bar{ne}}}_{2}}{{\bar{\bar{ne}}}_{2}})}^{γ}}^{\frac{1}{γ}}} 〉; \\ = 〈 \sqrt{1 - \frac{1}{1 + {\sum_{h = 1}^{2} {\tilde{ξ}}_{h} {(\frac{{\bar{\bar{pe}}}_{h}^{2}}{1 - {\bar{\bar{pe}}}_{h}^{2}})}^{γ}}^{\frac{1}{γ}}}}, \frac{1}{1 + {\sum_{h = 1}^{2} {\tilde{ξ}}_{h} {(\frac{1 - {\bar{\bar{ne}}}_{h}}{{\bar{\bar{ne}}}_{h}})}^{γ}}^{\frac{1}{γ}}} 〉; \end{matrix}$

Consequently, $PyFDPA ({\bar{\bar{py}}}_{1}, {\bar{\bar{py}}}_{2}) = 〈 \sqrt{1 - \frac{1}{1 + {\sum_{h = 1}^{2} {\tilde{ξ}}_{h} {(\frac{{\bar{\bar{pe}}}_{h}^{2}}{1 - {\bar{\bar{pe}}}_{h}^{2}})}^{γ}}^{\frac{1}{γ}}}}, \frac{1}{1 + {\sum_{h = 1}^{2} {\tilde{ξ}}_{h} {(\frac{1 - {\bar{\bar{ne}}}_{h}}{{\bar{\bar{ne}}}_{h}})}^{γ}}^{\frac{1}{γ}}} 〉 .$ (35)

Hence Equation (32), is true for g = 2.

If o = x, by Equation (32), we acquire the following equation: $PyFDPA ({\bar{\bar{py}}}_{1}, {\bar{\bar{py}}}_{2}, . . ., {\bar{\bar{py}}}_{x}) = 〈 \sqrt{1 - \frac{1}{1 + {\sum_{h = 1}^{x} {\tilde{ξ}}_{h} {(\frac{{\bar{\bar{pe}}}_{h}^{2}}{1 - {\bar{\bar{pe}}}_{h}^{2}})}^{γ}}^{\frac{1}{γ}}}}, \frac{1}{1 + {\sum_{h = 1}^{x} {\tilde{ξ}}_{h} {(\frac{1 - {\bar{\bar{ne}}}_{h}}{{\bar{\bar{ne}}}_{h}})}^{γ}}^{\frac{1}{γ}}} 〉 .$ (36)

If o = x + 1, then based on Equations (34) and (35), we include the subsequent formula: $\begin{matrix} \begin{matrix} PyFDPA ({\bar{\bar{py}}}_{1}, {\bar{\bar{py}}}_{2}, . . ., {\bar{\bar{py}}}_{x}, {\bar{\bar{py}}}_{x + 1}) = \oplus_{h = 1}^{x} {\tilde{ξ}}_{h} {\bar{\bar{py}}}_{h} \oplus {\tilde{ξ}}_{x + 1} {\bar{\bar{py}}}_{x + 1}; \\ = 〈 \sqrt{1 - \frac{1}{1 + {\sum_{h = 1}^{x} {\tilde{ξ}}_{h} {(\frac{{\bar{\bar{pe}}}_{h}^{2}}{1 - {\bar{\bar{pe}}}_{h}^{2}})}^{γ}}^{\frac{1}{γ}}}}, \frac{1}{1 + {\sum_{h = 1}^{x} {\tilde{ξ}}_{h} {(\frac{1 - {\bar{\bar{ne}}}_{h}}{{\bar{\bar{ne}}}_{h}})}^{γ}}^{\frac{1}{γ}}} 〉 \oplus \\ 〈 \sqrt{1 - \frac{1}{1 + {{\tilde{ξ}}_{x + 1} {(\frac{{\bar{\bar{pe}}}_{x + 1}^{2}}{1 - {\bar{\bar{pe}}}_{x + 1}^{2}})}^{γ}}^{\frac{1}{γ}}}}, \frac{1}{1 + {{\tilde{ξ}}_{x + 1} {(\frac{1 - {\bar{\bar{ne}}}_{x + 1}}{{\bar{\bar{ne}}}_{x + 1}})}^{γ}}^{\frac{1}{γ}}} 〉; \\ = 〈 \sqrt{1 - \frac{1}{1 + {\sum_{h = 1}^{x + 1} {\tilde{ξ}}_{h} {(\frac{{\bar{\bar{pe}}}_{h}^{2}}{1 - {\bar{\bar{pe}}}_{h}^{2}})}^{γ}}^{\frac{1}{γ}}}}, \frac{1}{1 + {\sum_{h = 1}^{x + 1} {\tilde{ξ}}_{h} {(\frac{1 - {\bar{\bar{ne}}}_{h}}{{\bar{\bar{ne}}}_{h}})}^{γ}}^{\frac{1}{γ}}} 〉 . \end{matrix} \end{matrix}$

Thus Equation (32) is true for all g . Hence Theorem 5 is true. The proof is completed.

Definition 15. Let ${\bar{\bar{py}}}_{h} 〈 {\bar{\bar{pe}}}_{h}, {\bar{\bar{ne}}}_{h} 〉 (h = 1, . . ., o)$ be a group of PyFNs, Θ = (Θ₁, Θ₂, . . . , Θ_o) ^T be the ID of ${\bar{\bar{py}}}_{h} (h = 1, 2, . . ., o)$ , such that Θ_h ∈ [0, 1] and $\sum_{h = 1}^{o} Θ_{h} = 1$ . Then, the PyFDPWA operator is described as follows: $PyFDPWA ({\bar{\bar{py}}}_{1}, {\bar{\bar{py}}}_{2}, . . ., {\bar{\bar{py}}}_{o}) = \frac{\oplus_{h = 1}^{o} (Θ_{h} (1 + T ({\bar{\bar{py}}}_{h})) {\bar{\bar{py}}}_{h})}{\sum_{h = 1}^{o} Θ_{h} (1 + T ({\bar{\bar{py}}}_{h}))}$ (37)

Where $T ({\bar{\bar{py}}}_{h}) = \sum_{k = 1 k \neq h}^{o} Supt ({\bar{\bar{py}}}_{h}, {\bar{\bar{py}}}_{k})$ (38) and $Supt ({\bar{\bar{py}}}_{h}, {\bar{\bar{py}}}_{k})$ is the SupD for ${\bar{\bar{py}}}_{h}$ from ${\bar{\bar{py}}}_{k}$ , which should oblige the subsequent axioms:

$Supt ({\bar{\bar{py}}}_{h}, {\bar{\bar{py}}}_{k}) \in [0, 1];$

$Supt ({\bar{\bar{py}}}_{h}, {\bar{\bar{py}}}_{k}) = Supt ({\bar{\bar{py}}}_{k}, {\bar{\bar{py}}}_{h});$

$Supt ({\bar{\bar{py}}}_{h}, {\bar{\bar{py}}}_{k}) ⩾ Supt ({\bar{\bar{py}}}_{m}, {\bar{\bar{py}}}_{n})$ , if $\bar{\bar{D}} ({\bar{\bar{py}}}_{h}, {\bar{\bar{py}}}_{k}) < \bar{\bar{D}} ({\bar{\bar{py}}}_{m}, {\bar{\bar{py}}}_{n})$ , where $\bar{\bar{D}}$ signify the distance measure provided in Definition (7).

In order to write Equation (36) in uncomplicated form, we presume that

Ψ_{h} = \frac{Θ_{h} (1 + T ({\bar{\bar{py}}}_{h}))}{\sum_{h = 1}^{o} Θ_{h} (1 + T ({\bar{\bar{py}}}_{h}))}

(39)

Hence, Equation (36) becomes $PyFDPWA ({\bar{\bar{py}}}_{1}, {\bar{\bar{py}}}_{2}, . . ., {\bar{\bar{py}}}_{o}) = \oplus_{h = 1}^{o} Ψ_{h} {\bar{\bar{py}}}_{h}$ (40)

Based on Definition 15, we include the subsequent result.

Theorem 6. Let ${\bar{\bar{py}}}_{h} 〈 {\bar{\bar{pe}}}_{h}, {\bar{\bar{ne}}}_{h} 〉 (h = 1, . . ., g)$ be a set of PyFNs, then the value acquired by utilizing Equation (36) is still PyFN, and $PyFDPWA ({\bar{\bar{py}}}_{1}, {\bar{\bar{py}}}_{2}, . . ., {\bar{\bar{py}}}_{o}) = 〈 \sqrt{1 - \frac{1}{1 + {\sum_{h = 1}^{o} Ψ_{h} {(\frac{{\bar{\bar{pe}}}_{h}^{2}}{1 - {\bar{\bar{pe}}}_{h}^{2}})}^{γ}}^{\frac{1}{γ}}}}, \frac{1}{1 + {\sum_{h = 1}^{o} Ψ_{h} {(\frac{1 - {\bar{\bar{ne}}}_{h}}{{\bar{\bar{ne}}}_{h}})}^{γ}}^{\frac{1}{γ}}} 〉 .$ (41)

Proof. Proof of Theorem 6 is same as Theorem 5, omitted here.

It can be straightforwardly proved that the PyFDPWA operator has the subsequent properties:

Property 1. (Idempotency) If all ${\bar{\bar{py}}}_{h} 〈 {\bar{\bar{pe}}}_{h}, {\bar{\bar{ne}}}_{h} 〉 (h = 1, . . ., o)$ are equal, that is ${\bar{\bar{py}}}_{h} = \bar{\bar{py}}$ for all h. Then $PyFDPWA ({\bar{\bar{py}}}_{1}, {\bar{\bar{py}}}_{2}, . . ., {\bar{\bar{py}}}_{o}) = \bar{\bar{py}} .$ (42)

Property 2. (Boundedness) Let ${\bar{\bar{py}}}_{h} 〈 {\bar{\bar{pe}}}_{h}, {\bar{\bar{ne}}}_{h} 〉 (h = 1, . . ., o)$ be a group of PyFNs, and let ${\bar{\bar{py}}}^{-} = min_{h} {\bar{\bar{py}}}_{h}, {\bar{\bar{py}}}^{+} = max_{h} {\bar{\bar{py}}}_{h}$ . Then ${\bar{\bar{py}}}^{-} ⩽ PyFDPWA ({\bar{\bar{py}}}_{1}, {\bar{\bar{py}}}_{2}, . . ., {\bar{\bar{py}}}_{o}) ⩽ {\bar{\bar{py}}}^{+} .$ (43)

Property 3. (Monotonicity) Let ${\bar{\bar{py}}}_{h} 〈 {\bar{\bar{pe}}}_{h}, {\bar{\bar{ne}}}_{h} 〉 (h = 1, . . ., o)$ and ${\bar{\bar{py}}}_{h}^{\sim} 〈 {\bar{\bar{pe}}}_{h}^{\sim}, {\bar{\bar{ne}}}_{h}^{\sim} 〉 (h = 1, . . ., o)$ be two groups of PyFNs, if ${\bar{\bar{py}}}_{h} ⩽ {\bar{\bar{py}}}_{h}^{\sim}$ for all h. Then $PyFDPWA ({\bar{\bar{py}}}_{1}, {\bar{\bar{py}}}_{2}, . . ., {\bar{\bar{py}}}_{o}) ⩽ PyFDPWA ({\bar{\bar{py}}}_{1}^{\sim}, {\bar{\bar{py}}}_{2}^{\sim}, . . ., {\bar{\bar{py}}}_{o}^{\sim}) .$ (44)

5.2 Pythagorean fuzzy Dombi power geometric aggregation (PyFDPGA) operators

Definition 16. Let ${\bar{\bar{py}}}_{h} 〈 {\bar{\bar{pe}}}_{h}, {\bar{\bar{ne}}}_{h} 〉 (h = 1, . . ., o)$ be a group of PyFNs, then we describe the PyFDPG operator as follows: $PyFDPG ({\bar{\bar{py}}}_{1}, {\bar{\bar{py}}}_{2}, . . ., {\bar{\bar{py}}}_{o}) = \otimes_{h = 1}^{o} {({\bar{\bar{py}}}_{h})}^{\frac{(1 + T ({\bar{\bar{py}}}_{h}))}{\sum_{h = 1}^{o} (1 + T ({\bar{\bar{py}}}_{h}))}}$ (45)

Where $T ({\bar{\bar{py}}}_{h}) = \sum_{k = 1 k \neq h}^{o} Supt ({\bar{\bar{py}}}_{h}, {\bar{\bar{py}}}_{k})$ (46) and $Supt ({\bar{\bar{py}}}_{h}, {\bar{\bar{py}}}_{k})$ is the SupD for ${\bar{\bar{py}}}_{h}$ from ${\bar{\bar{py}}}_{k}$ , which must assure the subsequent conditions:

$Supt ({\bar{\bar{py}}}_{h}, {\bar{\bar{py}}}_{k}) \in [0, 1];$

$Supt ({\bar{\bar{py}}}_{h}, {\bar{\bar{py}}}_{k}) = Supt ({\bar{\bar{py}}}_{k}, {\bar{\bar{py}}}_{h});$

$Supt ({\bar{\bar{py}}}_{h}, {\bar{\bar{py}}}_{k}) ⩾ Supt ({\bar{\bar{py}}}_{m}, {\bar{\bar{py}}}_{n})$ , if $\bar{\bar{D}} ({\bar{\bar{py}}}_{h}, {\bar{\bar{py}}}_{k}) < \bar{\bar{D}} ({\bar{\bar{py}}}_{m}, {\bar{\bar{py}}}_{n})$ , where $\bar{\bar{D}}$ symbolized the distance measure provided in Definition (7).

In order to write Equation (44) in uncomplicated form, we presume that ${\tilde{ξ}}_{h} = \frac{(1 + T ({\bar{\bar{py}}}_{h}))}{\sum_{h = 1}^{o} (1 + T ({\bar{\bar{py}}}_{h}))}$ (47)

Hence, Equation (44) becomes $PyFDPG ({\bar{\bar{py}}}_{1}, {\bar{\bar{py}}}_{2}, . . ., {\bar{\bar{py}}}_{o}) = \otimes_{h = 1}^{o} {({\bar{\bar{py}}}_{h})}^{{\tilde{ξ}}_{h}}$ (48)

Based on Definition 16, we include the subsequent result.

Theorem 7. Let ${\bar{\bar{py}}}_{h} 〈 {\bar{\bar{pe}}}_{h}, {\bar{\bar{ne}}}_{h} 〉 (h = 1, . . ., o)$ be a set of PyFNs, then the value acquired by utilizing Equation (44) is still PyFN, and $PyFDPG ({\bar{\bar{py}}}_{1}, {\bar{\bar{py}}}_{2}, . . ., {\bar{\bar{py}}}_{o}) = 〈 \frac{1}{1 + {\sum_{h = 1}^{o} {\tilde{ξ}}_{h} {(\frac{1 - {\bar{\bar{pe}}}_{h}}{{\bar{\bar{pe}}}_{h}})}^{γ}}^{\frac{1}{γ}}}, \sqrt{1 - \frac{1}{1 + {\sum_{h = 1}^{o} {\tilde{ξ}}_{h} {(\frac{{\bar{\bar{ne}}}_{h}^{2}}{1 - {\bar{\bar{ne}}}_{h}^{2}})}^{γ}}^{\frac{1}{γ}}}} 〉 .$ (49)

Proof. Based on DOLs, we have ${({\bar{\bar{py}}}_{h})}^{{\tilde{Ω}}_{h}} = 〈 \frac{1}{1 + {{\tilde{ξ}}_{h} {(\frac{1 - {\bar{\bar{pe}}}_{h}}{{\bar{\bar{pe}}}_{h}})}^{γ}}^{\frac{1}{γ}}}, \sqrt{1 - \frac{1}{1 + {{\tilde{ξ}}_{h} {(\frac{{\bar{\bar{ne}}}_{h}^{2}}{1 - {\bar{\bar{ne}}}_{h}^{2}})}^{γ}}^{\frac{1}{γ}}}} 〉 .$ (50) We prove Equation (48) by utilizing mathematical induction. If o = 2, then from Equation (49), we have ${({\bar{\bar{py}}}_{1})}^{{\tilde{ξ}}_{1}} = 〈 \frac{1}{1 + {{\tilde{ξ}}_{1} {(\frac{1 - {\bar{\bar{pe}}}_{1}}{{\bar{\bar{pe}}}_{1}})}^{γ}}^{\frac{1}{γ}}}, \sqrt{1 - \frac{1}{1 + {{\tilde{ξ}}_{1} {(\frac{{\bar{\bar{ne}}}_{1}^{2}}{1 - {\bar{\bar{ne}}}_{1}^{2}})}^{γ}}^{\frac{1}{γ}}}} 〉;$ and ${({\bar{\bar{py}}}_{2})}^{{\tilde{ξ}}_{2}} = 〈 \frac{1}{1 + {{\tilde{ξ}}_{2} {(\frac{1 - {\bar{\bar{pe}}}_{2}}{{\bar{\bar{pe}}}_{2}})}^{γ}}^{\frac{1}{γ}}}, \sqrt{1 - \frac{1}{1 + {{\tilde{ξ}}_{2} {(\frac{{\bar{\bar{ne}}}_{2}^{2}}{1 - {\bar{\bar{ne}}}_{2}^{2}})}^{γ}}^{\frac{1}{γ}}}} 〉,$ So, $\begin{matrix} PyFDPG ({\bar{\bar{py}}}_{1}, {\bar{\bar{py}}}_{2}) = {({\bar{\bar{py}}}_{1})}^{{\tilde{ξ}}_{1}} \otimes {({\bar{\bar{py}}}_{2})}^{{\tilde{ξ}}_{2}}; \\ = 〈 \frac{1}{1 + {{\tilde{ξ}}_{1} {(\frac{1 - {\bar{\bar{pe}}}_{1}}{{\bar{\bar{pe}}}_{1}})}^{γ}}^{\frac{1}{γ}}}, \sqrt{1 - \frac{1}{1 + {{\tilde{ξ}}_{1} {(\frac{{\bar{\bar{ne}}}_{1}^{2}}{1 - {\bar{\bar{ne}}}_{1}^{2}})}^{γ}}^{\frac{1}{γ}}}} 〉 \otimes 〈 \frac{1}{1 + {{\tilde{ξ}}_{2} {(\frac{1 - {\bar{\bar{pe}}}_{2}}{{\bar{\bar{pe}}}_{2}})}^{γ}}^{\frac{1}{γ}}}, \\ \sqrt{1 - \frac{1}{1 + {{\tilde{ξ}}_{2} {(\frac{{\bar{\bar{ne}}}_{2}^{2}}{1 - {\bar{\bar{ne}}}_{2}^{2}})}^{γ}}^{\frac{1}{γ}}}} 〉; \\ = 〈 \frac{1}{1 + {\sum_{h = 1}^{2} {\tilde{ξ}}_{h} {(\frac{1 - {\bar{\bar{pe}}}_{h}}{{\bar{\bar{pe}}}_{h}})}^{γ}}^{\frac{1}{γ}}}, \sqrt{1 - \frac{1}{1 + {\sum_{h = 1}^{2} {\tilde{ξ}}_{h} {(\frac{{\bar{\bar{ne}}}_{h}^{2}}{1 - {\bar{\bar{ne}}}_{h}^{2}})}^{γ}}^{\frac{1}{γ}}}} 〉; \end{matrix}$ Therefore, $PyFDPG ({\bar{\bar{py}}}_{1}, {\bar{\bar{py}}}_{2}) = 〈 \frac{1}{1 + {\sum_{h = 1}^{2} {\tilde{ξ}}_{h} {(\frac{1 - {\bar{\bar{pe}}}_{h}}{{\bar{\bar{pe}}}_{h}})}^{γ}}^{\frac{1}{γ}}}, \sqrt{1 - \frac{1}{1 + {\sum_{h = 1}^{2} {\tilde{ξ}}_{h} {(\frac{{\bar{\bar{ne}}}_{h}^{2}}{1 - {\bar{\bar{ne}}}_{h}^{2}})}^{γ}}^{\frac{1}{γ}}}} 〉 .$ (51)

Hence Equation (48), is true for o = 2.

If o = a, by Equation (48), we acquire the subsequent equation: $PyFDPG ({\bar{\bar{py}}}_{1}, {\bar{\bar{py}}}_{2}, . . ., {\bar{\bar{py}}}_{a}) = 〈 \frac{1}{1 + {\sum_{h = 1}^{a} {\tilde{ξ}}_{h} {(\frac{1 - {\bar{\bar{pe}}}_{h}}{{\bar{\bar{pe}}}_{h}})}^{γ}}^{\frac{1}{γ}}}, \sqrt{1 - \frac{1}{1 + {\sum_{h = 1}^{a} {\tilde{ξ}}_{h} {(\frac{{\bar{\bar{ne}}}_{h}^{2}}{1 - {\bar{\bar{ne}}}_{h}^{2}})}^{γ}}^{\frac{1}{γ}}}} 〉 .$ (52)

If o = a + 1, then based on Equations (50) and (51), we include the subsequent formula: $\begin{matrix} PyFDPG ({\bar{\bar{py}}}_{1}, {\bar{\bar{py}}}_{2}, . . ., {\bar{\bar{py}}}_{a}, {\bar{\bar{py}}}_{a + 1}) = \otimes_{h = 1}^{a} {({\bar{\bar{py}}}_{h})}^{{\tilde{ξ}}_{h}} \otimes {({\bar{\bar{py}}}_{a + 1})}^{{\tilde{ξ}}_{a + 1}}; \\ = 〈 \frac{1}{1 + {\sum_{h = 1}^{a} {\tilde{ξ}}_{h} {(\frac{1 - {\bar{\bar{pe}}}_{h}}{{\bar{\bar{pe}}}_{h}})}^{γ}}^{\frac{1}{γ}}}, \sqrt{1 - \frac{1}{1 + {\sum_{h = 1}^{a} {\tilde{ξ}}_{h} {(\frac{{\bar{\bar{ne}}}_{h}^{2}}{1 - {\bar{\bar{ne}}}_{h}^{2}})}^{γ}}^{\frac{1}{γ}}}} 〉 \otimes 〈 \frac{1}{1 + {{\tilde{ξ}}_{a + 1} {(\frac{1 - {\bar{\bar{pe}}}_{a + 1}}{{\bar{\bar{pe}}}_{a + 1}})}^{γ}}^{\frac{1}{γ}}}, \\ \sqrt{1 - \frac{1}{1 + {{\tilde{ξ}}_{a + 1} {(\frac{{\bar{\bar{ne}}}_{a + 1}^{2}}{1 - {\bar{\bar{ne}}}_{a + 1}^{2}})}^{γ}}^{\frac{1}{γ}}}} 〉; \\ = 〈 \frac{1}{1 + {\sum_{h = 1}^{a + 1} {\tilde{ξ}}_{h} {(\frac{1 - {\bar{\bar{pe}}}_{h}}{{\bar{\bar{pe}}}_{h}})}^{γ}}^{\frac{1}{γ}}}, \sqrt{1 - \frac{1}{1 + {\sum_{h = 1}^{a + 1} {\tilde{ξ}}_{h} {(\frac{{\bar{\bar{ne}}}_{h}^{2}}{1 - {\bar{\bar{ne}}}_{h}^{2}})}^{γ}}^{\frac{1}{γ}}}} 〉 . \end{matrix}$

Thus Equation (48) is true for o. Hence Theorem 7 is true. The proof is completed.

Definition 17. Let ${\bar{\bar{py}}}_{h} 〈 {\bar{\bar{pe}}}_{h}, {\bar{\bar{ne}}}_{h} 〉 (h = 1, . . ., o)$ be a group of PyFNs, Θ = (Θ₁, Θ₂, . . . , Θ_o) ^T be the ID of ${\bar{\bar{py}}}_{h} (h = 1, . . ., o)$ , such that Θ_h ∈ [0, 1] and $\sum_{h = 1}^{o} Θ_{h} = 1$ . Then the PyFDPWG operator is described as follows: $PyFDPWG ({\bar{\bar{py}}}_{1}, {\bar{\bar{py}}}_{2}, . . ., {\bar{\bar{py}}}_{o}) = \otimes_{h = 1}^{o} {({\bar{\bar{py}}}_{h})}^{\frac{Θ_{h} (1 + T ({\bar{\bar{py}}}_{h}))}{\sum_{h = 1}^{o} Θ_{h} (1 + T ({\bar{\bar{py}}}_{h}))}}$ (53)

Where $T ({\bar{\bar{py}}}_{h}) = \sum_{k = 1 k \neq h}^{o} Supt ({\bar{\bar{py}}}_{h}, {\bar{\bar{py}}}_{k})$ (54) and $Supt ({\bar{\bar{py}}}_{h}, {\bar{\bar{py}}}_{k})$ is the SupD for ${\bar{\bar{py}}}_{h}$ from ${\bar{\bar{py}}}_{k}$ , which must assure the subsequent clause:

$Supt ({\bar{\bar{py}}}_{h}, {\bar{\bar{py}}}_{k}) \in [0, 1];$

$Supt ({\bar{\bar{py}}}_{h}, {\bar{\bar{py}}}_{k}) = Supt ({\bar{\bar{py}}}_{k}, {\bar{\bar{py}}}_{h});$

$Supt ({\bar{\bar{py}}}_{h}, {\bar{\bar{py}}}_{k}) ⩾ Supt ({\bar{\bar{py}}}_{m}, {\bar{\bar{py}}}_{n})$ , if $\bar{\bar{D}} ({\bar{\bar{py}}}_{h}, {\bar{\bar{py}}}_{k}) < \bar{\bar{D}} ({\bar{\bar{py}}}_{m}, {\bar{\bar{py}}}_{n})$ , where $\bar{\bar{D}}$ symbolized the distance measure provided in Definition (7).

In order to write Equation (52) in uncomplicated form, we presume that $Ψ_{h} = \frac{Θ_{h} (1 + T ({\bar{\bar{py}}}_{h}))}{\sum_{h = 1}^{o} Θ_{h} (1 + T ({\bar{\bar{py}}}_{h}))}$ (55)

Hence, Equation (52) becomes $PyFDPWG ({\bar{\bar{py}}}_{1}, {\bar{\bar{py}}}_{2}, . . ., {\bar{\bar{py}}}_{o}) = \otimes_{h = 1}^{o} {({\bar{\bar{py}}}_{h})}^{Ψ_{h}}$ (56)

Based on Definition (17), we include the subsequent result.

Theorem 8. Let ${\bar{\bar{py}}}_{h} 〈 {\bar{\bar{pe}}}_{h}, {\bar{\bar{ne}}}_{h} 〉 (h = 1, . . ., o)$ be a group of PyFNs, then the value acquired by utilizing Equation (52) is still PyFN, and $PyFDPWG ({\bar{\bar{py}}}_{1}, {\bar{\bar{py}}}_{2}, . . ., {\bar{\bar{py}}}_{o}) = 〈 \frac{1}{1 + {\sum_{h = 1}^{o} Ψ_{h} {(\frac{1 - {\bar{\bar{pe}}}_{h}}{{\bar{\bar{pe}}}_{h}})}^{γ}}^{\frac{1}{γ}}}, \sqrt{1 - \frac{1}{1 + {\sum_{h = 1}^{o} Ψ_{h} {(\frac{{\bar{\bar{ne}}}_{h}^{2}}{1 - {\bar{\bar{ne}}}_{h}^{2}})}^{γ}}^{\frac{1}{γ}}}} 〉 .$ (57)

Proof. Proof of Theorem 8 is same as Theorem 7, omitted here.

It can be straightforwardly proved that the PyFDPWG operator has the subsequent properties.

Property 1. (Idempotency) If all ${\bar{\bar{py}}}_{h} 〈 {\bar{\bar{pe}}}_{h}, {\bar{\bar{ne}}}_{h} 〉 (h = 1, . . ., o)$ are equal, that is ${\bar{\bar{py}}}_{h} = \bar{\bar{py}}$ for all h, then $PyFDPWG ({\bar{\bar{py}}}_{1}, {\bar{\bar{py}}}_{2}, . . ., {\bar{\bar{py}}}_{o}) = \bar{\bar{py}} .$ (58)

Property 2. (Boundedness) Let ${\bar{\bar{py}}}_{h} 〈 {\bar{\bar{pe}}}_{h}, {\bar{\bar{ne}}}_{h} 〉 (h = 1, . . ., o)$ be a set of PyFNs, and let ${\bar{\bar{py}}}^{-} = min_{h} {\bar{\bar{py}}}_{h}, {\bar{\bar{py}}}^{+} = max_{h} {\bar{\bar{py}}}_{h}$ , then ${\bar{\bar{py}}}^{-} ⩽ PyFDPWG ({\bar{\bar{py}}}_{1}, {\bar{\bar{py}}}_{2}, . . ., {\bar{\bar{py}}}_{o}) ⩽ {\bar{\bar{py}}}^{+} .$ (59)

Property 3. Monotonicity) Let ${\bar{\bar{py}}}_{h} 〈 {\bar{\bar{pe}}}_{h}, {\bar{\bar{ne}}}_{h} 〉 (h = 1, . . ., o)$ and ${\bar{\bar{py}}}_{h}^{\sim} 〈 {\bar{\bar{pe}}}_{h}^{\sim}, {\bar{\bar{ne}}}_{h}^{\sim} 〉 (h = 1, . . ., o)$ be two groups of PyFNs, if ${\bar{\bar{py}}}_{h} ⩽ {\bar{\bar{py}}}_{h}^{\sim}$ for all h, then $PyFDPWG ({\bar{\bar{py}}}_{1}, {\bar{\bar{py}}}_{2}, . . ., {\bar{\bar{py}}}_{o}) ⩽ PyFDPWG ({\bar{\bar{py}}}_{1}^{\sim}, {\bar{\bar{py}}}_{2}^{\sim}, . . ., {\bar{\bar{py}}}_{o}^{\sim}) .$ (60)

6 Models for MADM with pythagorean fuzzy information

In this part, we shall employ the PyFDPWA operator and PyFDPWG operator to MADM with Pythagorean fuzzy information (PyFI). The following notions or assumptions are utilized to express the MADM problems for potential evaluation of emerging technology commercialization (PEETC) with PyFI.

Let the set of distinct alternative is expressed by $\bar{\bar{ATr}} = {{\bar{\bar{ATr}}}_{1}, {\bar{\bar{ATr}}}_{2}, . . ., {\bar{\bar{ATr}}}_{g}}$ and the set of attributes is represented by $\bar{\bar{AB}} = {{\bar{\bar{AB}}}_{1}, {\bar{\bar{AB}}}_{2}, . . ., {\bar{\bar{AB}}}_{q}}$ . Let the importance degree of the attributes is designated by $\tilde{Θ} = {({\tilde{Θ}}_{1}, {\tilde{Θ}}_{2}, . . ., {\tilde{Θ}}_{q})}^{T}$ such that ${\tilde{Θ}}_{p} \in [0, 1]$ and $\sum_{p = 1}^{q} {\tilde{Θ}}_{p} = 1 .$ Assume that $\bar{\bar{Y}} = {({\bar{\bar{pe}}}_{hp}, {\bar{\bar{ne}}}_{hp})}_{g \times q}$ is the PyF decision matrix, where ${\bar{\bar{pe}}}_{hp}$ identified the degree that the alternative ${\bar{\bar{ATr}}}_{h}$ suits the attribute ${\bar{\bar{AB}}}_{p}$ provided by decision maker (DM), ${\bar{\bar{ne}}}_{hp}$ identified the degree that the alternative ${\bar{\bar{ATr}}}_{h}$ does not suit the attribute ${\bar{\bar{AB}}}_{p}$ provided by DM, ${\bar{\bar{pe}}}_{hp} \in [0, 1], {\bar{\bar{ne}}}_{hp} \in [0, 1]$ , ${({\bar{\bar{pe}}}_{hp})}^{2} + {({\bar{\bar{ne}}}_{hp})}^{2} ⩽ 1, h = 1, 2, . . ., g; p = 1, 2, . . ., q$ .

Subsequently, we employ the PyFDPWA operator and PyFDPWG operator to MADM problems for PEETC with PyFI. The steps are as go after:

Step 1. Find out the SupDs by utilizing the formula: $Supt ({\bar{\bar{py}}}_{hp}, {\bar{\bar{py}}}_{hl}) = 1 - \bar{\bar{D}} ({\bar{\bar{py}}}_{hp}, {\bar{\bar{py}}}_{hl});$ (61)

where $\bar{\bar{D}} ({\bar{\bar{py}}}_{hp}, {\bar{\bar{py}}}_{hl})$ symbolized as the distance measure among two PyFNs ${\bar{\bar{py}}}_{hp}$ and ${\bar{\bar{py}}}_{hl}$ provided in Definition 7.

Step 2. Find out the weighted Sup $T ({\bar{\bar{py}}}_{hp})$ of the PyFN ${\bar{\bar{py}}}_{hp}$ by the other PyFNs ${\bar{\bar{py}}}_{hl} (p, l = 1, 2, . . ., q)$ by the formula: $T ({\bar{\bar{py}}}_{hp}) = \sum_{l = 1 l \neq p}^{q} Supt ({\bar{\bar{py}}}_{hp}, {\bar{\bar{py}}}_{hl}); (p, l = 1, 2, . . ., q) .$ (62)

Step 3. Find out the weight Ψ_hp associated with PyFN ${\bar{\bar{py}}}_{hp}$ , by utilizing the formula: $Ψ_{hp} = \frac{{\tilde{Θ}}_{p} (1 + T ({\bar{\bar{py}}}_{p}))}{\sum_{p = 1}^{q} (1 + T ({\bar{\bar{py}}}_{hp}))}, h = 1, 2, . . ., g; p = 1, 2, . . ., q .$ (63)

Step 4. Find out the overall assessment value of every alternative ${\bar{\bar{py}}}_{h}$ by utilizing the formula:

$PyFDPWA ({\bar{\bar{py}}}_{h 1}, {\bar{\bar{py}}}_{h 2}, . . ., {\bar{\bar{py}}}_{hq}) = 〈 \sqrt{1 - \frac{1}{1 + {\sum_{p = 1}^{q} Ψ_{hp} {(\frac{{\bar{\bar{pe}}}_{hp}^{2}}{1 - {\bar{\bar{pe}}}_{hp}^{2}})}^{γ}}^{\frac{1}{γ}}}}, \frac{1}{1 + {\sum_{p = 1}^{q} Ψ_{hp} {(\frac{1 - {\bar{\bar{ne}}}_{hp}}{{\bar{\bar{ne}}}_{hp}})}^{γ}}^{\frac{1}{γ}}} 〉;$ (64)

$PyFDPWG ({\bar{\bar{py}}}_{h 1}, {\bar{\bar{py}}}_{h 2}, . . ., {\bar{\bar{py}}}_{hq}) = 〈 \frac{1}{1 + {\sum_{p = 1}^{q} Ψ_{hp} {(\frac{1 - {\bar{\bar{pe}}}_{hp}}{{\bar{\bar{pe}}}_{hp}})}^{γ}}^{\frac{1}{γ}}}, \sqrt{1 - \frac{1}{1 + {\sum_{p = 1}^{q} Ψ_{hp} {(\frac{{\bar{\bar{ne}}}_{hp}^{2}}{1 - {\bar{\bar{ne}}}_{hp}^{2}})}^{γ}}^{\frac{1}{γ}}}} 〉 .$ (65)

Step 5. Find out the score values $\bar{\bar{SE}} ({\bar{\bar{py}}}_{h})$ of the collective PyFNs ${\bar{\bar{py}}}_{h} (h = 1, 2, . . ., g)$ to rank the alternatives ${\bar{\bar{AT}}}_{h} (h = 1, 2, . . ., g)$ and then pick the finest one(s). If the score values of PyFNs are same then we find out the accuracy values and rank the alternatives according to their accuracy values, and pick the finest one(s).

Step 6. End.

6.1 Numerical examples

In this subpart, we shall provide two numerical examples, one adapted from [28] and the other is real life example to exemplify the efficacy and practicality of the extended MADM model under PyFI.

Example 6.1.1. Assume that there is a panel with five available emerging technologies enterprises (ETC) (alternatives) ${\bar{\bar{AT}}}_{h} (h = 1, . . ., 5)$ to be selected. Four attributes are considered by experts to evaluate the five available ETC including: (1) ${\bar{\bar{AB}}}_{1}$ is the technical improvement; (2) ${\bar{\bar{AB}}}_{2}$ is the capability of market and market risk; (3) ${\bar{\bar{AB}}}_{3}$ is the human resources, financial framework and industrialization framework; (4) ${\bar{\bar{AB}}}_{4}$ is the progress of science and technology and employment creation. The five available alternatives are assessed by the experts with respect to the above four attributes, whose weight vector is (0.2, 0.1, 0.3, 0.4) ^T, and assessment information is endow with in the form of PyFNs, and listed in the following decision matrix:

$\bar{\bar{Y}} = [\begin{matrix} 〈 0.5, 0.8 〉 & 〈 0.6, 0.3 〉 & 〈 0.3, 0.6 〉 & 〈 0.5, 0.7 〉 \\ 〈 0.7, 0.5 〉 & 〈 0.7, 0.2 〉 & 〈 0.9, 0.2 〉 & 〈 0.5, 0.4 〉 \\ 〈 0.6, 0.2 〉 & 〈 0.5, 0.2 〉 & 〈 0.5, 0.3 〉 & 〈 0.6, 0.5 〉 \\ \begin{matrix} 〈 0.4, 0.2 〉 \\ 〈 0.6, 0.4 〉 \end{matrix} & \begin{matrix} 〈 0.6, 0.3 〉 \\ 〈 0.4, 0.8 〉 \end{matrix} & \begin{matrix} 〈 0.3, 0.4 〉 \\ 〈 0.7, 0.6 〉 \end{matrix} & \begin{matrix} 〈 0.5, 0.4 〉 \\ 〈 0.5, 0.8 〉 \end{matrix} \end{matrix}]$

To pick the most excellent ETC, the following steps are involved:

Step 1. Find out the SupDs $Supt ({\bar{\bar{py}}}_{hp}, {\bar{\bar{py}}}_{hl})$ by employing the formula (60). For computational ease we shall signify the SupD by S_hp,hl instead of $Supt ({\bar{\bar{py}}}_{hp}, {\bar{\bar{py}}}_{lp}) (h = 1, . . ., 5, p, l = 1, . . ., 4)$ . $\begin{matrix} S_{11, 12} = S_{12, 11} = 0.45, S_{11, 13} = S_{13, 11} = 0.56, S_{11, 14} = S_{14, 11} = 0.85, S_{12, 13} = S_{13, 12} = 0.73, S_{12, 14} = S_{14, 12} = 0.6, S_{13, 14} = S_{14, 13} = 0.71; \\ S_{21, 22} = S_{22, 21} = 0.79, S_{21, 23} = S_{23, 21} = 0.68, S_{21, 24} = S_{24, 21} = 0.67, S_{22, 23} = S_{23, 22} = 0.68, S_{22, 24} = S_{24, 22} = 0.76, S_{23, 24} = S_{23, 24} = 0.44, \\ S_{31, 32} = S_{32, 31} = 0.89, S_{31, 33} = S_{33, 31} = 0.89, S_{31, 34} = S_{34, 31} = 0.79, S_{32, 33} = S_{33, 32} = 0.95, S_{32, 34} = S_{34, 32} = 0.68, S_{33, 34} = S_{34, 33} = 0.73; \\ S_{41, 42} = S_{42, 41} = 0.75, S_{41, 43} = S_{43, 41} = 0.88, S_{41, 44} = S_{44, 41} = 0.79, S_{42, 43} = S_{43, 42} = 0.73, S_{42, 44} = S_{44, 42} = 0.89, S_{43, 44} = S_{44, 43} = 0.84; \\ S_{51, 52} = S_{52, 51} = 0.52, S_{51, 53} = S_{53, 51} = 0.67, S_{51, 54} = S_{54, 51} = 0.52, S_{52, 53} = S_{53, 52} = 0.67, S_{52, 54} = S_{54, 52} = 0.91, S_{53, 54} = S_{54, 53} = 0.72 . \end{matrix}$

Step 2. Find out the weighted SupD $T ({\bar{\bar{py}}}_{hp})$ by exploiting formula (61). For ease, we shall signify $T ({\bar{\bar{py}}}_{hp})$ by T_hp (h = 1, . . . , 5 ; p = 1, . . . , 4). $\begin{matrix} T_{11} = 1.86, T_{12} = 1.78, T_{13} = 2, T_{14} = 2.16, T_{21} = 2.14, T_{22} = 2.23, T_{23} = 1.8, T_{24} = 1.87; \\ T_{31} = 2.57, T_{32} = 2.52, T_{33} = 2.57, T_{34} = 2.2, T_{41} = 2.42, T_{42} = 2.37, T_{43} = 2.45, T_{44} = 2.52; \\ T_{51} = 1.71, T_{52} = 2.1, T_{53} = 2.06, T_{54} = 2.15 . \end{matrix}$

Step 3. Find out the weight Ψ_hp associated with PyFN ${\bar{\bar{py}}}_{hp} (h = 1, . . ., 5; p = 1, . . ., 4)$ , by exploiting the formula (62). $\begin{matrix} Ψ_{11} = 0.1898, Ψ_{12} = 0.0922, Ψ_{13} = 0.2986, Ψ_{14} = 0.4194, Ψ_{21} = 0.2137, Ψ_{22} = 0.1099, Ψ_{23} = 0.2858, Ψ_{24} = 0.3906; \\ Ψ_{31} = 0.2090, Ψ_{32} = 0.1030, Ψ_{33} = 0.3134, Ψ_{34} = 0.3746, Ψ_{41} = 0.1975, Ψ_{42} = 0.0973, Ψ_{43} = 0.2988, Ψ_{44} = 0.4065; \\ Ψ_{51} = 0.1789, Ψ_{52} = 0.1023, Ψ_{53} = 0.3030, Ψ_{54} = 0.4158 . \end{matrix}$

Step 4. Find out the overall assessment value of each alternative ${\bar{\bar{py}}}_{h} (h = 1, . . ., 5)$ by exploiting the formula (63) (assume γ = 1). $\begin{matrix} {\bar{\bar{py}}}_{1} = 〈 0.470603, 0.6092 〉, {\bar{\bar{py}}}_{2} = 〈 0.7899, 0.2956 〉, {\bar{\bar{py}}}_{3} = 〈 0.5642, 0.2968 〉, \\ {\bar{\bar{py}}}_{4} = 〈 0.4524, 0.3252 〉, {\bar{\bar{py}}}_{5} = 〈 0.5956, 0.6251 〉 . \end{matrix}$

Similarly, if we exploit formula (64), we have $\begin{matrix} {\bar{\bar{py}}}_{1} = 〈 0.4224, 0.6917 〉, {\bar{\bar{py}}}_{2} = 〈 0.6406, 0.3735 〉, {\bar{\bar{py}}}_{3} = 〈 0.5539, 0.3801 〉, \\ {\bar{\bar{py}}}_{4} = 〈 0.4057, 0.3634 〉, {\bar{\bar{py}}}_{5} = 〈 0.5499, 0.7277 〉 . \end{matrix}$

Step 5. Find out the score values $\bar{\bar{SE}} ({\bar{\bar{py}}}_{h})$ by exploiting Definition 5, we have $\bar{\bar{SE}} ({\bar{\bar{py}}}_{1}) = 0.4252, \bar{\bar{SE}} ({\bar{\bar{py}}}_{2}) = 0.7683, \bar{\bar{SE}} ({\bar{\bar{py}}}_{3}) = 0.6152, \bar{\bar{SE}} ({\bar{\bar{py}}}_{4}) = 0.5495, \bar{\bar{SE}} ({\bar{\bar{py}}}_{5}) = 0.4820;$

or $\bar{\bar{SE}} ({\bar{\bar{py}}}_{1}) = 0.3500, \bar{\bar{SE}} ({\bar{\bar{py}}}_{2}) = 0.6354, \bar{\bar{SE}} ({\bar{\bar{py}}}_{3}) = 0.5812, \bar{\bar{SE}} ({\bar{\bar{py}}}_{4}) = 0.5163, \bar{\bar{SE}} ({\bar{\bar{py}}}_{5}) = 0.3864 .$

Thus, according to their score values the ranking order of the alternatives ${\bar{\bar{ATr}}}_{h} (h = 1, 2, 3, 4, 5)$ is ${\bar{\bar{ATr}}}_{2} > {\bar{\bar{ATr}}}_{3} > {\bar{\bar{ATr}}}_{4} > {\bar{\bar{ATr}}}_{5} > {\bar{\bar{ATr}}}_{1}$ and ${\bar{\bar{ATr}}}_{2} > {\bar{\bar{ATr}}}_{3} > {\bar{\bar{ATr}}}_{4} > {\bar{\bar{ATr}}}_{5} > {\bar{\bar{ATr}}}_{1}$ .

Hence, by exploiting both the AOs, we get the same ranking orders, So, according to the ranking order, the best ETC is ${\bar{\bar{ATr}}}_{2}$ and the worst one is ${\bar{\bar{ATr}}}_{5}$ .

Example 6.1.2

Assume that there have been fIVE transport agencies ${\bar{\bar{{ATr}_{1}}}, \bar{\bar{{ATr}_{2}}}, \bar{\bar{{ATr}_{3}}}, \bar{\bar{{ATr}_{4}}}, \bar{\bar{{ATr}_{5}}}}$ to be preferred for the appraisal. After thorough discussion, the DMs examine the four possible alternatives in conformity with the four attributes ${\bar{\bar{{AB}_{1}}}, \bar{\bar{{AB}_{2}}}, \bar{\bar{{AB}_{3}}}, \bar{\bar{{AB}_{4}}}}$ . $\bar{\bar{{AB}_{1}}}$ stands for accessibility; $\bar{\bar{{AB}_{2}}}$ stands for protection; $\bar{\bar{{AB}_{3}}}$ stands for consistency; $\bar{\bar{{AB}_{4}}}$ stands for resilience. PyF scales are described to express linguistic terms. The evaluation scales for alternatives and criteria are given in Tables 1 and 2. The corresponding PyF decision matrix and weight of the criteria are given in Tables 3 and 4.

Table 1
Scale for evaluation of criteria

Meaning $PFNs 〈 \bar{\bar{pe}}, \bar{\bar{ne}} 〉$

Very Insignificant (VI) 〈0.10, 0.90〉

Insignificant (I) 〈0.35, 0.60〉

Average (A) 〈0.50, 0.45〉

Imperative (IM) 〈0.75, 0.40〉

Very Significant (VS) 〈0.90, 0.10〉

Meaning	$PFNs 〈 \bar{\bar{pe}}, \bar{\bar{ne}} 〉$
Very Insignificant (VI)	〈0.10, 0.90〉
Insignificant (I)	〈0.35, 0.60〉
Average (A)	〈0.50, 0.45〉
Imperative (IM)	〈0.75, 0.40〉
Very Significant (VS)	〈0.90, 0.10〉

Table 2

Scale for evaluation of alternatives

Meaning	$PFNs 〈 \bar{\bar{pe}}, \bar{\bar{ne}} 〉$
Extremely Low (EL)	〈0.10, 0.99〉
Very Low (VL)	〈0.10, 0.97〉
Medium Low (ML)	〈0.25, 0.92〉
Low (L)	〈0.40, 0.87〉
Medium (M)	〈0.50, 0.80〉
High (H)	〈0.6, 0.71〉
Medium High (MH)	〈0.70, 0.60〉
Very High (VH)	〈0.80, 0.44〉
Extremely High (EH)	〈1, 0.0〉

Table 3

The assessments of the alternatives

Alternative/Criteria	${\bar{\bar{AB}}}_{1}$	${\bar{\bar{AB}}}_{2}$	${\bar{\bar{AB}}}_{3}$	${\bar{\bar{AB}}}_{4}$
${\bar{\bar{ATr}}}_{1}$	M	VH	L	ML
${\bar{\bar{ATr}}}_{2}$	H	MH	VH	L
${\bar{\bar{ATr}}}_{3}$	ML	H	VH	H
${\bar{\bar{ATr}}}_{4}$	VH	VH	ML	H
$\bar{\bar{{ATr}_{5}}}$	H	VH	L	VH

Table 4

Corresponding PyF decision matrix

Alternative/Criteria	${\bar{\bar{AB}}}_{1}$	${\bar{\bar{AB}}}_{2}$	${\bar{\bar{AB}}}_{3}$	${\bar{\bar{AB}}}_{4}$
${\bar{\bar{ATr}}}_{1}$	〈0.5, 0.80〉	〈0.80, 0.44〉	〈0.40, 0.87〉	〈0.25, 0.92〉
${\bar{\bar{ATr}}}_{2}$	〈0.6, 0.71〉	〈0.7, 0.6〉	〈0.80, 0.44〉	〈0.40, 0.87〉
${\bar{\bar{ATr}}}_{3}$	〈0.25, 0.92〉	〈0.6, 0.71〉	〈0.8, 0.44〉	〈0.6, 0.71〉
${\bar{\bar{ATr}}}_{4}$	〈0.80, 0.44〉	〈0.8, 0.44〉	〈0.25, 0.92〉	〈0.6, 0.71〉
$\bar{\bar{{ATr}_{5}}}$	〈0.6, 0.71〉	〈0.8, 0.44〉	〈0.40, 0.87〉	〈0.8, 0.44〉

Step 1. Since all the attributes are of benefit types, so there is no need to normalize it.

Step 2. Find out the SupDs $Supt ({\bar{\bar{py}}}_{hp}, {\bar{\bar{py}}}_{hl})$ by employing the formula (60). For computational ease we shall signify the SupD by S_hp,hl instead of $Supt ({\bar{\bar{py}}}_{hp}, {\bar{\bar{py}}}_{lp}) (h = 1, . . ., 5, p, l = 1, . . ., 4)$ . $\begin{matrix} S_{11, 12} = S_{12, 11} = 0.5536, S_{11, 13} = S_{13, 11} = 0.8831, S_{11, 14} = S_{14, 11} = 0.7936, S_{12, 13} = S_{13, 12} = 0.4367, S_{12, 14} = S_{14, 12} = 0.3472, S_{13, 14} = S_{14, 13} = 0.9025; \\ S_{21, 22} = S_{22, 21} = 0.8559, S_{21, 23} = S_{23, 21} = 0.6895, S_{21, 24} = S_{24, 21} = 0.7472, S_{22, 23} = S_{23, 22} = 0.8336, S_{22, 24} = S_{24, 22} = 0.6031, S_{23, 24} = S_{23, 24} = 0.4367, \\ S_{31, 32} = S_{32, 31} = 0.6577, S_{31, 33} = S_{33, 31} = 0.3472, S_{31, 34} = S_{34, 31} = 0.6577, S_{32, 33} = S_{33, 32} = 0.6895, S_{32, 34} = S_{34, 32} = 1.00, S_{33, 34} = S_{34, 33} = 0.6895; \\ S_{41, 42} = S_{42, 41} = 1.00, S_{41, 43} = S_{43, 41} = 0.3472, S_{41, 44} = S_{44, 41} = 0.6895, S_{42, 43} = S_{43, 42} = 0.3472, S_{42, 44} = S_{44, 42} = 0.6895, S_{43, 44} = S_{44, 43} = 0.6577; \\ S_{51, 52} = S_{52, 51} = 0.6895, S_{51, 53} = S_{53, 51} = 0.7472, S_{51, 54} = S_{54, 51} = 0.6895, S_{52, 53} = S_{53, 52} = 0.4367, S_{52, 54} = S_{54, 52} = 1.00, S_{53, 54} = S_{54, 53} = 0.4367 . \end{matrix}$

Step 3. Find out the weighted SupD $T ({\bar{\bar{py}}}_{hp})$ by exploiting formula (61). For ease, we shall signify $T ({\bar{\bar{py}}}_{hp})$ by T_hp (h = 1, . . . , 5 ; p = 1, . . . , 4). $\begin{matrix} T_{11} = 2.2303, T_{12} = 1.3375, T_{13} = 2.2223, T_{14} = 2.0433, T_{21} = 2.2926, T_{22} = 2.2926, T_{23} = 1.9598, T_{24} = 1.787; \\ T_{31} = 1.6626, T_{32} = 2.3472, T_{33} = 1.7262, T_{34} = 2.3472, T_{41} = 2.0367, T_{42} = 2.0367, T_{43} = 1.3521, T_{44} = 2.0367; \\ T_{51} = 2.1262, T_{52} = 2.1262, T_{53} = 1.6206, T_{54} = 2.1262 . \end{matrix}$

Step 4. Calculate the weight vector $\tilde{Θ}$ by the utilizing the following formula (65), and is given in Table 5.

Table 5

Weights of the attributes

Attributes	Linguistic term	PFNs $〈 \bar{\bar{pe}}, \bar{\bar{ne}} 〉$	Weights
${\bar{\bar{AB}}}_{1}$	A	〈0.50, 0.45〉	0.1956
${\bar{\bar{AB}}}_{2}$	VS	〈0.90, 0.10〉	0.2819
${\bar{\bar{AB}}}_{3}$	IM	〈0.75, 0.4〉	0.2405
${\bar{\bar{AB}}}_{4}$	VS	〈0.90, 0.10〉	0.2819

$Θ_{p} = \frac{{\bar{\bar{pe}}}_{p} + {\bar{\bar{π}}}_{p} ({\bar{\bar{pe}}}_{p} / ({\bar{\bar{pe}}}_{p} + {\bar{\bar{ne}}}_{p}))}{\sum_{p = 1}^{m} {\bar{\bar{pe}}}_{p} + {\bar{\bar{π}}}_{p} ({\bar{\bar{pe}}}_{p} / ({\bar{\bar{pe}}}_{p} + {\bar{\bar{ne}}}_{p}))}$ (66)

Step 5. Find out the weight Ψ_hp associated with PyFN ${\bar{\bar{py}}}_{hp} (h = 1, . . ., 5; p = 1, . . ., 4)$ , by exploiting the formula (62). $\begin{matrix} Ψ_{11} = 0.2161, Ψ_{12} = 0.2254, Ψ_{13} = 0.2651, Ψ_{14} = 0.2934, Ψ_{21} = 0.2098, Ψ_{22} = 0.3024, Ψ_{23} = 0.2319, Ψ_{24} = 0.2559; \\ Ψ_{31} = 0.1700, Ψ_{32} = 0.3080, Ψ_{33} = 0.2140, Ψ_{34} = 0.3080, Ψ_{41} = 0.2068, Ψ_{42} = 0.2981, Ψ_{43} = 0.1970, Ψ_{44} = 0.2981; \\ Ψ_{51} = 0.2035, Ψ_{52} = 0.2933, Ψ_{53} = 0.2098, Ψ_{54} = 0.2933 . \end{matrix}$ (67)

Step 6. Find out the overall assessment value of each alternative ${\bar{\bar{py}}}_{h} (h = 1, . . ., 5)$ by exploiting the formula (63) (assume γ = 1). $\begin{matrix} {\bar{\bar{py}}}_{1} = 〈 0.5931, 0.7112 〉, {\bar{\bar{py}}}_{2} = 〈 0.6820, 0.6170 〉, {\bar{\bar{py}}}_{3} = 〈 0.6517, 0.6590 〉, \\ {\bar{\bar{py}}}_{4} = 〈 0.7203, 0.5613 〉, {\bar{\bar{py}}}_{5} = 〈 0.7382, 0.5373 〉 . \end{matrix}$ (68)

Similarly, if we exploit formula (64), we have $\begin{matrix} {\bar{\bar{py}}}_{1} = 〈 0.3921, 0.8616 〉, {\bar{\bar{py}}}_{2} = 〈 0.5843, 0.7435 〉, {\bar{\bar{py}}}_{3} = 〈 0.5065, 0.7858 〉, \\ {\bar{\bar{py}}}_{4} = 〈 0.5219, 0.7756 〉, {\bar{\bar{py}}}_{5} = 〈 0.6262, 0.7073 〉 . \end{matrix}$ (69)

Step 7. Find out the score values $\bar{\bar{SE}} ({\bar{\bar{py}}}_{h})$ by exploiting Definition 5, we have $\bar{\bar{SE}} ({\bar{\bar{py}}}_{1}) = 0.4230, \bar{\bar{SE}} ({\bar{\bar{py}}}_{2}) = 0.5422, \bar{\bar{SE}} ({\bar{\bar{py}}}_{3}) = 0.4952, \bar{\bar{SE}} ({\bar{\bar{py}}}_{4}) = 0.6019, \bar{\bar{SE}} ({\bar{\bar{py}}}_{5}) = 0.6281;$ (70)

or $\bar{\bar{SE}} ({\bar{\bar{py}}}_{1}) = 0.2057, \bar{\bar{SE}} ({\bar{\bar{py}}}_{2}) = 0.3943, \bar{\bar{SE}} ({\bar{\bar{py}}}_{3}) = 0.3195, \bar{\bar{SE}} ({\bar{\bar{py}}}_{4}) = 0.3354, \bar{\bar{SE}} ({\bar{\bar{py}}}_{5}) = 0.4459 .$ (71)

Thus, according to their score values the ranking order of the alternatives ${\bar{\bar{ATr}}}_{h} (h = 1, 2, 3, 4, 5)$ is ${\bar{\bar{ATr}}}_{5} > {\bar{\bar{ATr}}}_{4} > {\bar{\bar{ATr}}}_{2} > {\bar{\bar{ATr}}}_{3} > {\bar{\bar{ATr}}}_{1}$ and ${\bar{\bar{ATr}}}_{5} > {\bar{\bar{ATr}}}_{2} > {\bar{\bar{ATr}}}_{4} > {\bar{\bar{ATr}}}_{3} > {\bar{\bar{ATr}}}_{1}$ .

Hence, by exploiting both the AOs, we get the same ranking orders, So, according to the ranking order, the best alternative is ${\bar{\bar{ATr}}}_{5}$ and the worst one is ${\bar{\bar{ATr}}}_{1}$ .

6.2 Effect of the parameter γ on decision results

In this subpart, we will further argue the outcome of the parameter γ on the final ranking result of this example, then we take up the different values of the parameter γ to rank the alternatives. The results are shown in Table 1. From Table 1, we can observe that for different values of the parameter γ the ranking orders are different. From Table 1, we preserve that when the value of the parameter γ raise while utilizing PyFDPWA operator, the score values also raises, and when utilizing the PyFDPWG operator the score values decreases.

6.3 Comparison with other approaches

In this subpart, we judge our presented approach with those proposed by Khan et al. [43], Wei and Lu [28] and confer the compensation of the presented approach, and results are shown in Table 7.

From Table 7, we can observe that the ranking order obtained from the suggested approach and that of Khan et al. [43] is totally different. The core motive behind these dissimilar ranking orders is that the suggested AOs are based on improved DOLs can prevail over the existing inadequacy in [43].

From Table 6, we can also observe that the ranking order acquire by exploiting Wei and Lu [28] approach based on PyFPWA and PyFPWG operator is the same as that of the suggested approach based on the suggested AO. This shows the validity of the suggested approach based on the proposed AOs. This result is also explained because when γ = 1, the proposed AOs will reduce to PyFPWA and PyFPWG operator in [28], so it is reasonable. The advantages of the proposed method and Wei and Lu [28], method are that they can remove the bad effect of awkward data on final decision result, however, the Wei and Lu [28], method is defined by using algebraic t-norm and t-conorm while the suggested method is based on DTN and DTCN, and it is more flexible by general parameter. So the decision maker may choose the value of general parameter according to the actual need of the situation.

Table 6
Effect of the Parameter γ on Decision results

Parameter Value Score Values utilizing PyFDPWA operator Score Values utilizing PyFDPWG operator Ranking order

γ = 2 $\begin{matrix} \bar{\bar{SE}} ({\bar{\bar{py}}}_{1}) = 0.4741, \bar{\bar{SE}} ({\bar{\bar{py}}}_{2}) = 0.8148, \\ \bar{\bar{SE}} ({\bar{\bar{py}}}_{3}) = 0.6250, \bar{\bar{SE}} ({\bar{\bar{py}}}_{4}) = 0.5674, \\ \bar{\bar{SE}} ({\bar{\bar{py}}}_{5}) = 0.5292 . \end{matrix}$ $\begin{matrix} \bar{\bar{SE}} ({\bar{\bar{py}}}_{1}) = 0.3245, \bar{\bar{SE}} ({\bar{\bar{py}}}_{2}) = 0.5964, \\ \bar{\bar{SE}} ({\bar{\bar{py}}}_{3}) = 0.5629, \bar{\bar{SE}} ({\bar{\bar{py}}}_{4}) = 0.5042, \\ \bar{\bar{SE}} ({\bar{\bar{py}}}_{5}) = 0.3565 . \end{matrix}$ ${\bar{\bar{ATr}}}_{2} > {\bar{\bar{ATr}}}_{3} > {\bar{\bar{ATr}}}_{4} > {\bar{\bar{ATr}}}_{5} > {\bar{\bar{ATr}}}_{1}$

${\bar{\bar{ATr}}}_{2} > {\bar{\bar{ATr}}}_{3} > {\bar{\bar{ATr}}}_{4} > {\bar{\bar{ATr}}}_{5} > {\bar{\bar{ATr}}}_{1}$

γ = 5 $\begin{matrix} \bar{\bar{SE}} ({\bar{\bar{py}}}_{1}) = 0.5539, \bar{\bar{SE}} ({\bar{\bar{py}}}_{2}) = 0.8575, \\ \bar{\bar{SE}} ({\bar{\bar{py}}}_{3}) = 0.6407, \bar{\bar{SE}} ({\bar{\bar{py}}}_{4}) = 0.6033, \\ \bar{\bar{SE}} ({\bar{\bar{py}}}_{5}) = 0.5998 . \end{matrix}$ $\begin{matrix} \bar{\bar{SE}} ({\bar{\bar{py}}}_{1}) = 0.2791, \bar{\bar{SE}} ({\bar{\bar{py}}}_{2}) = 0.5492, \\ \bar{\bar{SE}} ({\bar{\bar{py}}}_{3}) = 0.5357, \bar{\bar{SE}} ({\bar{\bar{py}}}_{4}) = 0.4854, \\ \bar{\bar{SE}} ({\bar{\bar{py}}}_{5}) = 0.3154 . \end{matrix}$ ${\bar{\bar{ATr}}}_{2} > {\bar{\bar{ATr}}}_{3} > {\bar{\bar{ATr}}}_{4} > {\bar{\bar{ATr}}}_{5} > {\bar{\bar{ATr}}}_{1}$

${\bar{\bar{ATr}}}_{2} > {\bar{\bar{ATr}}}_{3} > {\bar{\bar{ATr}}}_{4} > {\bar{\bar{ATr}}}_{5} > {\bar{\bar{ATr}}}_{1}$

γ = 10 $\begin{matrix} \bar{\bar{SE}} ({\bar{\bar{py}}}_{1}) = 0.5919, \bar{\bar{SE}} ({\bar{\bar{py}}}_{2}) = 0.8718, \\ \bar{\bar{SE}} ({\bar{\bar{py}}}_{3}) = 0.6499, \bar{\bar{SE}} ({\bar{\bar{py}}}_{4}) = 0.6285, \\ \bar{\bar{SE}} ({\bar{\bar{py}}}_{5}) = 0.6325 . \end{matrix}$ $\begin{matrix} \bar{\bar{SE}} ({\bar{\bar{py}}}_{1}) = 0.2526, \bar{\bar{SE}} ({\bar{\bar{py}}}_{2}) = 0.5259, \\ \bar{\bar{SE}} ({\bar{\bar{py}}}_{3}) = 0.5198, \bar{\bar{SE}} ({\bar{\bar{py}}}_{4}) = 0.4754, \\ \bar{\bar{SE}} ({\bar{\bar{py}}}_{5}) = 0.2907 . \end{matrix}$ ${\bar{\bar{ATr}}}_{2} > {\bar{\bar{ATr}}}_{3} > {\bar{\bar{ATr}}}_{5} > {\bar{\bar{ATr}}}_{4} > {\bar{\bar{ATr}}}_{1}$

${\bar{\bar{ATr}}}_{2} > {\bar{\bar{ATr}}}_{3} > {\bar{\bar{ATr}}}_{4} > {\bar{\bar{ATr}}}_{5} > {\bar{\bar{ATr}}}_{1}$

γ = 17 $\begin{matrix} \bar{\bar{SE}} ({\bar{\bar{py}}}_{1}) = 0.6097, \bar{\bar{SE}} ({\bar{\bar{py}}}_{2}) = 0.8774, \\ \bar{\bar{SE}} ({\bar{\bar{py}}}_{3}) = 0.6541, \bar{\bar{SE}} ({\bar{\bar{py}}}_{4}) = 0.6413, \\ \bar{\bar{SE}} ({\bar{\bar{py}}}_{5}) = 0.6461 . \end{matrix}$ $\begin{matrix} \bar{\bar{SE}} ({\bar{\bar{py}}}_{1}) = 0.2411, \bar{\bar{SE}} ({\bar{\bar{py}}}_{2}) = 0.5153, \\ \bar{\bar{SE}} ({\bar{\bar{py}}}_{3}) = 0.5118, \bar{\bar{SE}} ({\bar{\bar{py}}}_{4}) = 0.4710, \\ \bar{\bar{SE}} ({\bar{\bar{py}}}_{5}) = 0.2780 . \end{matrix}$ ${\bar{\bar{ATr}}}_{2} > {\bar{\bar{ATr}}}_{3} > {\bar{\bar{ATr}}}_{5} > {\bar{\bar{ATr}}}_{4} > {\bar{\bar{ATr}}}_{1}$

${\bar{\bar{ATr}}}_{2} > {\bar{\bar{ATr}}}_{3} > \bar{\bar{AT}} r_{4} > {\bar{\bar{ATr}}}_{5} > {\bar{\bar{ATr}}}_{1}$

γ = 25 $\begin{matrix} \bar{\bar{SE}} ({\bar{\bar{py}}}_{1}) = 0.6178, \bar{\bar{SE}} ({\bar{\bar{py}}}_{2}) = 0.8799, \\ \bar{\bar{SE}} ({\bar{\bar{py}}}_{3}) = 0.6560, \bar{\bar{SE}} ({\bar{\bar{py}}}_{4}) = 0.6472, \\ \bar{\bar{SE}} ({\bar{\bar{py}}}_{5}) = 0.6522 . \end{matrix}$ $\begin{matrix} \bar{\bar{SE}} ({\bar{\bar{py}}}_{1}) = 0.2359, \bar{\bar{SE}} ({\bar{\bar{py}}}_{2}) = 0.5104, \\ \bar{\bar{SE}} ({\bar{\bar{py}}}_{3}) = 0.5081, \bar{\bar{SE}} ({\bar{\bar{py}}}_{4}) = 0.4691, \\ \bar{\bar{SE}} ({\bar{\bar{py}}}_{5}) = 0.2721 . \end{matrix}$ ${\bar{\bar{ATr}}}_{2} > {\bar{\bar{ATr}}}_{3} > {\bar{\bar{ATr}}}_{5} > {\bar{\bar{ATr}}}_{4} > {\bar{\bar{ATr}}}_{1}$

${\bar{\bar{ATr}}}_{2} > {\bar{\bar{ATr}}}_{3} > {\bar{\bar{ATr}}}_{4} > {\bar{\bar{ATr}}}_{5} > {\bar{\bar{ATr}}}_{1}$

γ = 50 $\begin{matrix} \bar{\bar{SE}} ({\bar{\bar{py}}}_{1}) = 0.6265, \bar{\bar{SE}} ({\bar{\bar{py}}}_{2}) = 0.8825, \\ \bar{\bar{SE}} ({\bar{\bar{py}}}_{3}) = 0.6580, \bar{\bar{SE}} ({\bar{\bar{py}}}_{4}) = 0.6536, \\ \bar{\bar{SE}} ({\bar{\bar{py}}}_{5}) = 0.6587 . \end{matrix}$ $\begin{matrix} \bar{\bar{SE}} ({\bar{\bar{py}}}_{1}) = 0.2304, \bar{\bar{SE}} ({\bar{\bar{py}}}_{2}) = 0.5052, \\ \bar{\bar{SE}} ({\bar{\bar{py}}}_{3}) = 0.5040, \bar{\bar{SE}} ({\bar{\bar{py}}}_{4}) = 0.4670, \\ \bar{\bar{SE}} ({\bar{\bar{py}}}_{5}) = 0.2660 . \end{matrix}$ ${\bar{\bar{ATr}}}_{2} > {\bar{\bar{ATr}}}_{5} > {\bar{\bar{ATr}}}_{3} > {\bar{\bar{ATr}}}_{4} > {\bar{\bar{ATr}}}_{1}$

${\bar{\bar{ATr}}}_{2} > {\bar{\bar{ATr}}}_{3} > {\bar{\bar{ATr}}}_{4} > {\bar{\bar{ATr}}}_{5} > {\bar{\bar{ATr}}}_{1}$

γ = 100 $\begin{matrix} \bar{\bar{SE}} ({\bar{\bar{py}}}_{1}) = 0.6307, \bar{\bar{SE}} ({\bar{\bar{py}}}_{2}) = 0.8837, \\ \bar{\bar{SE}} ({\bar{\bar{py}}}_{3}) = 0.6590, \bar{\bar{SE}} ({\bar{\bar{py}}}_{4}) = 0.6568, \\ \bar{\bar{SE}} ({\bar{\bar{py}}}_{5}) = 0.6618 . \end{matrix}$ $\begin{matrix} \bar{\bar{SE}} ({\bar{\bar{py}}}_{1}) = 0.2277, \bar{\bar{SE}} ({\bar{\bar{py}}}_{2}) = 0.5026, \\ \bar{\bar{SE}} ({\bar{\bar{py}}}_{3}) = 0.5020, \bar{\bar{SE}} ({\bar{\bar{py}}}_{4}) = 0.4660, \\ \bar{\bar{SE}} ({\bar{\bar{py}}}_{5}) = 0.2630 . \end{matrix}$ ${\bar{\bar{ATr}}}_{2} > {\bar{\bar{ATr}}}_{5} > {\bar{\bar{ATr}}}_{3} > {\bar{\bar{ATr}}}_{4} > {\bar{\bar{ATr}}}_{1}$

${\bar{\bar{ATr}}}_{2} > {\bar{\bar{ATr}}}_{3} > {\bar{\bar{ATr}}}_{4} > {\bar{\bar{ATr}}}_{5} > {\bar{\bar{ATr}}}_{1}$

Parameter Value	Score Values utilizing PyFDPWA operator	Score Values utilizing PyFDPWG operator	Ranking order
γ = 2	$\begin{matrix} \bar{\bar{SE}} ({\bar{\bar{py}}}_{1}) = 0.4741, \bar{\bar{SE}} ({\bar{\bar{py}}}_{2}) = 0.8148, \\ \bar{\bar{SE}} ({\bar{\bar{py}}}_{3}) = 0.6250, \bar{\bar{SE}} ({\bar{\bar{py}}}_{4}) = 0.5674, \\ \bar{\bar{SE}} ({\bar{\bar{py}}}_{5}) = 0.5292 . \end{matrix}$	$\begin{matrix} \bar{\bar{SE}} ({\bar{\bar{py}}}_{1}) = 0.3245, \bar{\bar{SE}} ({\bar{\bar{py}}}_{2}) = 0.5964, \\ \bar{\bar{SE}} ({\bar{\bar{py}}}_{3}) = 0.5629, \bar{\bar{SE}} ({\bar{\bar{py}}}_{4}) = 0.5042, \\ \bar{\bar{SE}} ({\bar{\bar{py}}}_{5}) = 0.3565 . \end{matrix}$	${\bar{\bar{ATr}}}_{2} > {\bar{\bar{ATr}}}_{3} > {\bar{\bar{ATr}}}_{4} > {\bar{\bar{ATr}}}_{5} > {\bar{\bar{ATr}}}_{1}$
			${\bar{\bar{ATr}}}_{2} > {\bar{\bar{ATr}}}_{3} > {\bar{\bar{ATr}}}_{4} > {\bar{\bar{ATr}}}_{5} > {\bar{\bar{ATr}}}_{1}$
γ = 5	$\begin{matrix} \bar{\bar{SE}} ({\bar{\bar{py}}}_{1}) = 0.5539, \bar{\bar{SE}} ({\bar{\bar{py}}}_{2}) = 0.8575, \\ \bar{\bar{SE}} ({\bar{\bar{py}}}_{3}) = 0.6407, \bar{\bar{SE}} ({\bar{\bar{py}}}_{4}) = 0.6033, \\ \bar{\bar{SE}} ({\bar{\bar{py}}}_{5}) = 0.5998 . \end{matrix}$	$\begin{matrix} \bar{\bar{SE}} ({\bar{\bar{py}}}_{1}) = 0.2791, \bar{\bar{SE}} ({\bar{\bar{py}}}_{2}) = 0.5492, \\ \bar{\bar{SE}} ({\bar{\bar{py}}}_{3}) = 0.5357, \bar{\bar{SE}} ({\bar{\bar{py}}}_{4}) = 0.4854, \\ \bar{\bar{SE}} ({\bar{\bar{py}}}_{5}) = 0.3154 . \end{matrix}$	${\bar{\bar{ATr}}}_{2} > {\bar{\bar{ATr}}}_{3} > {\bar{\bar{ATr}}}_{4} > {\bar{\bar{ATr}}}_{5} > {\bar{\bar{ATr}}}_{1}$
			${\bar{\bar{ATr}}}_{2} > {\bar{\bar{ATr}}}_{3} > {\bar{\bar{ATr}}}_{4} > {\bar{\bar{ATr}}}_{5} > {\bar{\bar{ATr}}}_{1}$
γ = 10	$\begin{matrix} \bar{\bar{SE}} ({\bar{\bar{py}}}_{1}) = 0.5919, \bar{\bar{SE}} ({\bar{\bar{py}}}_{2}) = 0.8718, \\ \bar{\bar{SE}} ({\bar{\bar{py}}}_{3}) = 0.6499, \bar{\bar{SE}} ({\bar{\bar{py}}}_{4}) = 0.6285, \\ \bar{\bar{SE}} ({\bar{\bar{py}}}_{5}) = 0.6325 . \end{matrix}$	$\begin{matrix} \bar{\bar{SE}} ({\bar{\bar{py}}}_{1}) = 0.2526, \bar{\bar{SE}} ({\bar{\bar{py}}}_{2}) = 0.5259, \\ \bar{\bar{SE}} ({\bar{\bar{py}}}_{3}) = 0.5198, \bar{\bar{SE}} ({\bar{\bar{py}}}_{4}) = 0.4754, \\ \bar{\bar{SE}} ({\bar{\bar{py}}}_{5}) = 0.2907 . \end{matrix}$	${\bar{\bar{ATr}}}_{2} > {\bar{\bar{ATr}}}_{3} > {\bar{\bar{ATr}}}_{5} > {\bar{\bar{ATr}}}_{4} > {\bar{\bar{ATr}}}_{1}$
			${\bar{\bar{ATr}}}_{2} > {\bar{\bar{ATr}}}_{3} > {\bar{\bar{ATr}}}_{4} > {\bar{\bar{ATr}}}_{5} > {\bar{\bar{ATr}}}_{1}$
γ = 17	$\begin{matrix} \bar{\bar{SE}} ({\bar{\bar{py}}}_{1}) = 0.6097, \bar{\bar{SE}} ({\bar{\bar{py}}}_{2}) = 0.8774, \\ \bar{\bar{SE}} ({\bar{\bar{py}}}_{3}) = 0.6541, \bar{\bar{SE}} ({\bar{\bar{py}}}_{4}) = 0.6413, \\ \bar{\bar{SE}} ({\bar{\bar{py}}}_{5}) = 0.6461 . \end{matrix}$	$\begin{matrix} \bar{\bar{SE}} ({\bar{\bar{py}}}_{1}) = 0.2411, \bar{\bar{SE}} ({\bar{\bar{py}}}_{2}) = 0.5153, \\ \bar{\bar{SE}} ({\bar{\bar{py}}}_{3}) = 0.5118, \bar{\bar{SE}} ({\bar{\bar{py}}}_{4}) = 0.4710, \\ \bar{\bar{SE}} ({\bar{\bar{py}}}_{5}) = 0.2780 . \end{matrix}$	${\bar{\bar{ATr}}}_{2} > {\bar{\bar{ATr}}}_{3} > {\bar{\bar{ATr}}}_{5} > {\bar{\bar{ATr}}}_{4} > {\bar{\bar{ATr}}}_{1}$
			${\bar{\bar{ATr}}}_{2} > {\bar{\bar{ATr}}}_{3} > \bar{\bar{AT}} r_{4} > {\bar{\bar{ATr}}}_{5} > {\bar{\bar{ATr}}}_{1}$
γ = 25	$\begin{matrix} \bar{\bar{SE}} ({\bar{\bar{py}}}_{1}) = 0.6178, \bar{\bar{SE}} ({\bar{\bar{py}}}_{2}) = 0.8799, \\ \bar{\bar{SE}} ({\bar{\bar{py}}}_{3}) = 0.6560, \bar{\bar{SE}} ({\bar{\bar{py}}}_{4}) = 0.6472, \\ \bar{\bar{SE}} ({\bar{\bar{py}}}_{5}) = 0.6522 . \end{matrix}$	$\begin{matrix} \bar{\bar{SE}} ({\bar{\bar{py}}}_{1}) = 0.2359, \bar{\bar{SE}} ({\bar{\bar{py}}}_{2}) = 0.5104, \\ \bar{\bar{SE}} ({\bar{\bar{py}}}_{3}) = 0.5081, \bar{\bar{SE}} ({\bar{\bar{py}}}_{4}) = 0.4691, \\ \bar{\bar{SE}} ({\bar{\bar{py}}}_{5}) = 0.2721 . \end{matrix}$	${\bar{\bar{ATr}}}_{2} > {\bar{\bar{ATr}}}_{3} > {\bar{\bar{ATr}}}_{5} > {\bar{\bar{ATr}}}_{4} > {\bar{\bar{ATr}}}_{1}$
			${\bar{\bar{ATr}}}_{2} > {\bar{\bar{ATr}}}_{3} > {\bar{\bar{ATr}}}_{4} > {\bar{\bar{ATr}}}_{5} > {\bar{\bar{ATr}}}_{1}$
γ = 50	$\begin{matrix} \bar{\bar{SE}} ({\bar{\bar{py}}}_{1}) = 0.6265, \bar{\bar{SE}} ({\bar{\bar{py}}}_{2}) = 0.8825, \\ \bar{\bar{SE}} ({\bar{\bar{py}}}_{3}) = 0.6580, \bar{\bar{SE}} ({\bar{\bar{py}}}_{4}) = 0.6536, \\ \bar{\bar{SE}} ({\bar{\bar{py}}}_{5}) = 0.6587 . \end{matrix}$	$\begin{matrix} \bar{\bar{SE}} ({\bar{\bar{py}}}_{1}) = 0.2304, \bar{\bar{SE}} ({\bar{\bar{py}}}_{2}) = 0.5052, \\ \bar{\bar{SE}} ({\bar{\bar{py}}}_{3}) = 0.5040, \bar{\bar{SE}} ({\bar{\bar{py}}}_{4}) = 0.4670, \\ \bar{\bar{SE}} ({\bar{\bar{py}}}_{5}) = 0.2660 . \end{matrix}$	${\bar{\bar{ATr}}}_{2} > {\bar{\bar{ATr}}}_{5} > {\bar{\bar{ATr}}}_{3} > {\bar{\bar{ATr}}}_{4} > {\bar{\bar{ATr}}}_{1}$
			${\bar{\bar{ATr}}}_{2} > {\bar{\bar{ATr}}}_{3} > {\bar{\bar{ATr}}}_{4} > {\bar{\bar{ATr}}}_{5} > {\bar{\bar{ATr}}}_{1}$
γ = 100	$\begin{matrix} \bar{\bar{SE}} ({\bar{\bar{py}}}_{1}) = 0.6307, \bar{\bar{SE}} ({\bar{\bar{py}}}_{2}) = 0.8837, \\ \bar{\bar{SE}} ({\bar{\bar{py}}}_{3}) = 0.6590, \bar{\bar{SE}} ({\bar{\bar{py}}}_{4}) = 0.6568, \\ \bar{\bar{SE}} ({\bar{\bar{py}}}_{5}) = 0.6618 . \end{matrix}$	$\begin{matrix} \bar{\bar{SE}} ({\bar{\bar{py}}}_{1}) = 0.2277, \bar{\bar{SE}} ({\bar{\bar{py}}}_{2}) = 0.5026, \\ \bar{\bar{SE}} ({\bar{\bar{py}}}_{3}) = 0.5020, \bar{\bar{SE}} ({\bar{\bar{py}}}_{4}) = 0.4660, \\ \bar{\bar{SE}} ({\bar{\bar{py}}}_{5}) = 0.2630 . \end{matrix}$	${\bar{\bar{ATr}}}_{2} > {\bar{\bar{ATr}}}_{5} > {\bar{\bar{ATr}}}_{3} > {\bar{\bar{ATr}}}_{4} > {\bar{\bar{ATr}}}_{1}$
			${\bar{\bar{ATr}}}_{2} > {\bar{\bar{ATr}}}_{3} > {\bar{\bar{ATr}}}_{4} > {\bar{\bar{ATr}}}_{5} > {\bar{\bar{ATr}}}_{1}$

Table 7

Comparison with other approaches

Approach	Score values	Ranking order
PyFDWA operator [43](γ = 1)	$\begin{matrix} \bar{\bar{SE}} ({\bar{\bar{py}}}_{1}) = 0.9472, \bar{\bar{SE}} ({\bar{\bar{py}}}_{2}) = 0.8968, \\ \bar{\bar{SE}} ({\bar{\bar{py}}}_{3}) = 0.9206, \bar{\bar{SE}} ({\bar{\bar{py}}}_{4}) = 0.9844, \\ \bar{\bar{SE}} ({\bar{\bar{py}}}_{5}) = 0.8742 . \end{matrix}$	${\bar{\bar{ATr}}}_{4} > {\bar{\bar{ATr}}}_{1} > {\bar{\bar{ATr}}}_{3} > {\bar{\bar{ATr}}}_{2} > {\bar{\bar{ATr}}}_{5}$
PyFDWG operator [43](γ = 1)	$\begin{matrix} \bar{\bar{SE}} ({\bar{\bar{py}}}_{1}) = 0.0528, \bar{\bar{SE}} ({\bar{\bar{py}}}_{2}) = 0.1032, \\ \bar{\bar{SE}} ({\bar{\bar{py}}}_{3}) = 0.0794, \bar{\bar{SE}} ({\bar{\bar{py}}}_{4}) = 0.0156, \\ \bar{\bar{SE}} ({\bar{\bar{py}}}_{5}) = 0.1258 . \end{matrix}$	${\bar{\bar{ATr}}}_{5} > {\bar{\bar{ATr}}}_{2} > {\bar{\bar{ATr}}}_{3} > {\bar{\bar{ATr}}}_{1} > {\bar{\bar{ATr}}}_{4}$
PyFPWA operator [28]	$\begin{matrix} \bar{\bar{SE}} ({\bar{\bar{py}}}_{1}) = 0.4252, \bar{\bar{SE}} ({\bar{\bar{py}}}_{2}) = 0.7683, \\ \bar{\bar{SE}} ({\bar{\bar{py}}}_{3}) = 0.6152, \bar{\bar{SE}} ({\bar{\bar{py}}}_{4}) = 0.5495, \\ \bar{\bar{SE}} ({\bar{\bar{py}}}_{5}) = 0.4820; \end{matrix}$	${\bar{\bar{ATr}}}_{2} > {\bar{\bar{ATr}}}_{5} > {\bar{\bar{ATr}}}_{3} > {\bar{\bar{ATr}}}_{4} > {\bar{\bar{ATr}}}_{1}$
PyFPWG operator [28]	$\begin{matrix} \bar{\bar{SE}} ({\bar{\bar{py}}}_{1}) = 0.3647, \bar{\bar{SE}} ({\bar{\bar{py}}}_{2}) = 0.6497, \\ \bar{\bar{SE}} ({\bar{\bar{py}}}_{3}) = 0.5850, \bar{\bar{SE}} ({\bar{\bar{py}}}_{4}) = 0.5220, \\ \bar{\bar{SE}} ({\bar{\bar{py}}}_{5}) = 0.4056 . \end{matrix}$	${\bar{\bar{ATr}}}_{2} > {\bar{\bar{ATr}}}_{5} > {\bar{\bar{ATr}}}_{3} > {\bar{\bar{ATr}}}_{4} > {\bar{\bar{ATr}}}_{1}$
Proposed PyFDPWA (γ = 1)	$\begin{matrix} \bar{\bar{SE}} ({\bar{\bar{py}}}_{1}) = 0.6307, \bar{\bar{SE}} ({\bar{\bar{py}}}_{2}) = 0.8837, \\ \bar{\bar{SE}} ({\bar{\bar{py}}}_{3}) = 0.6590, \bar{\bar{SE}} ({\bar{\bar{py}}}_{4}) = 0.6568, \\ \bar{\bar{SE}} ({\bar{\bar{py}}}_{5}) = 0.6618 . \end{matrix}$	${\bar{\bar{ATr}}}_{2} > {\bar{\bar{ATr}}}_{5} > {\bar{\bar{ATr}}}_{3} > {\bar{\bar{ATr}}}_{4} > {\bar{\bar{ATr}}}_{1}$
Proposed PyFDPWG (γ = 1)	$\begin{matrix} \bar{\bar{SE}} ({\bar{\bar{py}}}_{1}) = 0.3500, \bar{\bar{SE}} ({\bar{\bar{py}}}_{2}) = 0.6354, \\ \bar{\bar{SE}} ({\bar{\bar{py}}}_{3}) = 0.5812, \bar{\bar{SE}} ({\bar{\bar{py}}}_{4}) = 0.5163, \\ \bar{\bar{SE}} ({\bar{\bar{py}}}_{5}) = 0.3864 . \end{matrix}$	${\bar{\bar{ATr}}}_{2} > {\bar{\bar{ATr}}}_{5} > {\bar{\bar{ATr}}}_{3} > {\bar{\bar{ATr}}}_{4} > {\bar{\bar{ATr}}}_{1}$

7 Conclusions

In realistic decision making, the available information is often incomplete and inconsistent, and the PyFS is better gizmo to represent such type of information. In this article, firstly, some drawbacks in the existing DOLs for PyFNs are argued and some improved DOLs for PyFNs are initiated. Secondly, we show that the value aggregated using the existing Dombi AOs is not a PyFN, and we improved the existing AO. Thirdly, we proposed two new AOs, PyFDPWA operator, PyFDPWG operator, and their weighted forms, and discussed their desirable properties. The leading qualities of these developed Dombi power AOs is that they eliminate the effect of awkward data and are more flexible due to general parameter. Moreover, based on these Dombi power AOs a novel MADM approach is instituted. Lastly, a numerical example is given to show the practicality and effectiveness of the proposed approach and comparison with the existing approaches is also given. The developed aggregation operators are easy and flexible to be used in solving MADM problems. However, the limitation of the proposed aggregation operators is that it can not deal with the situation where the interrelation among the input arguments is considered.

In the future research, we will apply the proposed approach to some new applications, such as supplier evaluation [45 –47], evaluation of human resources, and so on, or extend the proposed method to q-Rung Orthopair Fuzzy sets [48 –50] which are more general fuzzy sets.

Footnotes

Acknowledgments

This work is supported by the National Natural Science Foundation of China (No. 71771140), the Special Funds of Taishan Scholars Project of Shandong Province (No. ts201511045).

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Some improved pythagorean fuzzy Dombi power aggregation operators with application in multiple-attribute decision making

Abstract

Keywords

1 Introduction

2 Preliminaries

2.1 Pythagorean fuzzy sets and their operational laws

3.1 Existing Dombi operational laws for PyFNs

4.1 Existing aggregation operators

5.1 Pythagorean Fuzzy Dombi Power Arithmetic AOs

Table 1 Scale for evaluation of criteria Meaning PFNs 〈 pe ¯ ¯ , ne ¯ ¯ 〉 Very Insignificant (VI) 〈0.10, 0.90〉 Insignificant (I) 〈0.35, 0.60〉 Average (A) 〈0.50, 0.45〉 Imperative (IM) 〈0.75, 0.40〉 Very Significant (VS) 〈0.90, 0.10〉

6.3 Comparison with other approaches

Footnotes

Acknowledgments

References

Table 1
Scale for evaluation of criteria

Meaning $PFNs 〈 \bar{\bar{pe}}, \bar{\bar{ne}} 〉$

Very Insignificant (VI) 〈0.10, 0.90〉

Insignificant (I) 〈0.35, 0.60〉

Average (A) 〈0.50, 0.45〉

Imperative (IM) 〈0.75, 0.40〉

Very Significant (VS) 〈0.90, 0.10〉