Fermatean fuzzy TOPSIS method with Dombi aggregation operators and its application in multi-criteria decision making

Abstract

Algebraic operations are used effectively in decision-making problems. Especially, Dombi, Hamacher and Einstein algebraic operators are used frequently in the decision-making field. On the other hand, it is known that aggregation operators affect the decision-making process in decision-making problems. In this paper, we used Dombi operations to develop some Fermatean fuzzy aggregation operators. Arithmetic and geometric analysis of each aggregation method were performed. We defined the following operators: Fermatean fuzzy Dombi weighted average operator, Fermatean fuzzy Dombi weighted geometric operator, Fermatean fuzzy Dombi ordered weighted average operator, Fermatean fuzzy Dombi ordered weighted geometric operator, Fermatean fuzzy Dombi hybrid weighted average operator, Fermatean fuzzy Dombi hybrid weighted geometric operator. Also, an analysis was performed for the beta value of the Dombi parameter. Properties of proposed operators were presented, and operators were defined on Fermatean fuzzy sets. Finally, proposed operators were compared with the existing aggregation operators. To understand the impact of the proposed operators on the decision-making process, Fermatean fuzzy TOPSIS was established.

Keywords

Fermatean fuzzy sets multi-criteria decision making TOPSIS Dombi operations Fermatean fuzzy aggregation operators

1 Introduction

Unclearness or incomplete information in many areas poses major problems. Likewise, it can be said that there is incomplete and unclear information in decision-making problems. The solution of decision-making problems depends largely on the imprecision, uncertain and incomplete information. Real numbers are insufficient to solve uncertain situations. Therefore, fuzzy sets constitute the solution in an uncertain case. Fuzzy sets are introduced by Zadeh [48]. In particular, finding the degree of membership of a value within a fuzzy set helps to solve the uncertain situation. In this sense, fuzzy sets have made progress in many fields of engineering and technology. However, the elements of classical fuzzy sets are interpreted based on only the degree of membership. So, unclearness or incomplete information in a dataset only be represented by one membership function. The generalization of the concept of the classical fuzzy set is defined by Atanassov [2]. This structure is called an intuitionistic fuzzy set (IFS).

While classical fuzzy sets are represented by the only degree of membership, IFSs are represented by two values: the degree of membership and degree of non-membership. IFSs have been used, effectively in many areas such as pattern recognition, medical diagnosis, image segmentation, fuzzy time series forecasting [11 , 37]. The sum of the degree of membership and non-membership degree is defined in the unit interval. Likewise, each degree is in the [0,1] interval. When the sum of membership and non-membership values is greater than 1 then when we cannot define the IFS. For example, if both membership degrees are greater than 0.5, IFS is not provided. To generalize the IFS concept, the Pythagorean fuzzy set (PFS) was defined by Yager [46]. In PFS, the membership and non-membership degrees are in the [0,1] interval. However, the sum of squares of membership degrees varies in the unit closed interval. In this case, the PFS is larger fuzzy sets than the IFSs. Then, we can say that PFS offers more advantages than IFS in solving real-world problems. For example, pattern recognition, supplier selection, industrial accident early warning, recommender systems [14 , 49].

Although PFS provides a larger set, it appears to be expandable. This can be illustrated with an example. Suppose that membership degree is 0.8 and the non-membership degree is 0.7. For IFS, 0.8+0.7 < 1 condition is not provided. Similarly, for PFS, 0 . 8² + 0 .7² > 1. The sum of membership and non-membership degrees for IFS and PFS appears to be out of unit interval. So, it is considered that defining a more general fuzzy set will be effective in solving problems. Senapati and Yager proposed Fermatean fuzzy set (FFS) [33]. FFS provides a more general perspective for fuzzy sets. In FFS, the sum of cubes of membership and non-membership degrees is in the unit interval. For example, 0.8³+0.7³ > 1 condition is provided in FFS. Currently, it can be said that FFS is the most extended set among fuzzy sets. So far, basic issues for fuzzy sets have been addressed. In the following part of the introduction section, FFS and aggregation operators on FFS will be examined.

Aggregation methods are widely used in multiple criteria decision making (MCDM) and PFS. Wei and Lu used power aggregation operators in decision-making problems [43]. Yager introduced ordered weighted averaging aggregation operators for MCDM [45]. Khan et al. developed an aggregation operator for multi-attribute group decision making and it is emphasized that decision-makers and attributes are at different priority levels [21]. Merigo and Casanovas presented induced aggregation operators [29]. Depending on the decision-maker’s attitude, the reorder-based provides the parametrized family of distance aggregation operators. Merigó and Casanovas introduced fuzzy generalized hybrid aggregation operators and these operators are applied in the decision-making problem [30].

Algebraic operators are used for many aggregation structures. These are Dombi, Einstein, Hamacher operators. The Dombi triangular norm(t-norm) and triangular conorm(t-conorm) operator was defined in 1982 by Dombi [12]. The main advantage of Dombi norms is that it depends on the parameter value. As the parameter value changes, the resulting norm changes. Chen and Ye [4] proposed Dombi weighted aggregation operators for single-valued neutrosophic sets and applied on investment alternatives. Shi and Ye are used Dombi aggregation operators of neutrosophic cubic sets for travel decision-making [36]. Jana et al. presented bipolar fuzzy Dombi operators for the evaluation of emerging technology commercialization with multiple-attribute decision making [15]. With the widespread application of Dombi operators on Pythagorean fuzzy sets, Khan et al. proposed Pythagorean fuzzy Dombi aggregation operators on investment decision making [20]. Wei defined aggregation method for Hamacher and delivered a comparative analysis for decision making [41]. It is seen that the TOPSIS method is applied to various fields and gives effective results. Data Envelopment Analysis (DEA) was combined with the TOPSIS method [40]. The efficiency of the end-of-life vehicle reverse logistics industry was analyzed by DEA TOPSIS. In another study, the TOPSIS method is presented for probabilistic linguistic sets in multiple attribute group decision making using entropy weights. Also, probabilistic linguistic sets with TOPSIS was applied to Supplier Selection of New Agricultural Machinery Products [27]. TOPSIS extended with the Choquet integral based method [39]. Besides, the TOPSIS-Choquet integral method was used in the interval-valued hesitant Pythagorean fuzzy environment.

In 2016, the q-rung orthopair fuzzy set was introduced by Yager [47]. Intuitionistic fuzzy sets were expanded from the qth degree. On the other hand, aggregation methods have been described on q-rung fuzzy sets [26 , 44]. Also, in q-rung orthopair fuzzy sets were used Dombi operator [16]. However, this study, a comprehensive TOPSIS method for multiple decision making on Fermatean fuzzy sets is included. The distance method presented for IFSs was extended for FFSs and more accurate distance calculation was performed [19].

Senapati and Yager introduced new aggregated operators in [34]. The operators are presented with practical examples. These operators will also be included in our study. Senapati and Yager defined basic operators over the FFSs. On the other hand, division and subtraction operations on FFSs were defined in [35]. They implemented the weighted product model on MCDM. Then, their model was applied to bridge construction selection problems. In [25], it was focused on a distance method for distance measure for Fermatean fuzzy linguistic term sets. The proposed structure was tested on TODIM and TOPSIS. Fermatean fuzzy weighted average(FFWA), Fermatean fuzzy weighted geometric(FFWG), Fermatean fuzzy weighted power average(FFWPA), Fermatean fuzzy weighted power geometric(FFWPG) have already been applied [34]. Considering all the above studies, Dombi t-norm is very important in between aggregation methods with changing parameters. In this study, Dombi aggregation operators will be examined on FFSs. In this study, Fermatean fuzzy Dombi weighted average (FFDWA) operator, Fermatean fuzzy Dombi weighted geometric (FFDWG) operator, Fermatean fuzzy Dombi ordered weighted average (FFDOWA) operator, Fermatean fuzzy Dombi ordered weighted geometric (FFDOWG) operator, Fermatean fuzzy Dombi hybrid weighted average (FFDHWA) operator, Fermatean fuzzy Dombi hybrid weighted geometric (FFDHWG) operator are used on FFS and also tested on Fermatean-TOPSIS method. The beta(β) parameter of the Dombi is analyzed. Ranks were compared for each aggregation method. On the other hand, the distance method for IFSs proposed in [19] has been extended for FFSs in our study in cases where the Hamming and Euclidean distance are unsuccessful.

The rest of the work is designed as follows. In the second section, basic information about the extension of fuzzy sets is given. Besides, general definitions and theorems on FFS are mentioned. In the third section, as a new approach, the Dombi operator is defined on FFSs. Necessary proofs and properties are included. In the fourth section, the modified distance function is mentioned. In the fifth section, the application of the proposed method and the results are presented. The comparison analysis based on the Dombi parameter is performed. In the last section, the advantages of the proposed method are discussed. Also, this section contains recommendations for future studies.

2 Preliminaries

Theorems and definitions used in the study are given in this section. T-norm and T-conorm definitions, an extension of fuzzy set, fundamental properties of Fermatean fuzzy set, aggregations on Fermatean fuzzy set are given.

Definition 1. ([22]) T:[0,1] X [0,1]⟶[0,1] and For ∀ x, y, z, x’, y’. if T is a t-norm, the following properties are provided.

T(x,0)=0, T(x,1) = x boundary condition

T (x, y) = T (y, x) commutativity

(x≤ x’, y≤y’) ⟶T (x, y) ≤T (x’, y’)monotonicity

T (T (x, y), z) = T (x, T (y, z) associativity

Definition 2. ([22]) T:[0,1] X [0,1]⟶[0,1] T:,0,1.X,0,1.]⟶[0,1] and For ∀ x, y, z, x’, y’. if S is a t-conorm (s-norm), the following properties are provided.

S(x,0) = x, S(x,1) = 1 boundary condition

S (x, y) = S (y, x) commutativity

(x≤ x’, y≤ y’) ⟶S (x, y) ≤S (x’, y’) monotonicity

S (S (x, y), z) = S (x, S (y, z) associativity

Definition 3. ([1]) Let A be a universal set. Intuitionistic fuzzy set, I is defined on set A as follows. $I = {< x, μ_{I} (x), v_{I} (x) > | x \in A}$ (1) where μ:A⟶[0,1] is the membership degree 0≤μ_I (x)≤1 and v:A⟶[0,1] is the non-membership degree 0≤v_I(x)≤1. For each x∈A element in set I, 0≤μ_I(x)+v_I(x)≤1. Also, hesitant degree π_I = 1-μ_I (x)-v_I(x).

Definition 4. ([46]) Let A be a universal set. Pythagorean fuzzy set, P is defined on set A as follows. $P = {< x, μ_{P} (x), v_{P} (x) > | x \in A}$ (2) where μ:A⟶[0,1] is the membership degree 0≤μ_P (x)≤1 and v:A⟶[0,1] is the non-membership degree. For each x∈A element in set I, 0≤μ_I(x)+v_I(x)≤1. Also, hesitant degree π_I = 1-μ_I (x)-v_I(x)

where μ : A → [0, 1] is the membership degree 0 ⩽ μ_I (x) ⩽1 and v : A → [0, 1] is the non-membership degree 0≤v_P(x)≤1. For each x ∈ A element in set P, 0≤(μ_I(x))²+(v_I(x))²≤1. Also, hesitant degree of Pythagorean fuzzy set $π_{P} (x) = \sqrt{1 - {(μ_{P} (x))}^{2} - (v_{P} (x))^{2}}$ Definition 5. ([33]) Let A be a universal set. Fermatean fuzzy set, F is defined on set A as follows. $F = {〈 x, μ_{F} (x), v_{F} (x) 〉 | x ɛ A}$ (3)

Where μ : A → [0, 1] is the membership degree 0 ⩽ μ_F (x) ⩽ 1 and v : A → [0, 1] is the non-membership degree 0 ⩽ v_F (x) ⩽ 1. For each x ∈ A element in set F, 0 ⩽ (μ_F (x)) ³ + (v_F (x)) ³ ⩽ 1. Also, hesitant degree of Fermatean fuzzy set, $π_{F} (x) = \sqrt[3]{1 - {(μ_{F} (x))}^{3} - (v_{F} (x))^{3}}$ .

For clarity and simplicity, Instead of F = {〈x, μ_F (x) , v_F (x) 〉|xɛA}, F = (μ_F, v_F) is used. In the following, some operations for FFNs (Fermatean fuzzy numbers) will be mentioned.

Definition 6. ([33]) Let F = (μ_F, v_F), F₁ = (μ_{F
₁}, v_{F
₁}) and F₂ = (μ_{F
₂}, v_{F
₂}) be FFNs. The following operations can be defined for FFNs.

F₁ ∩ F₂ = (min { μ_F
₁, μ_F
₂ } , max { v_F
₁, v_F
₂ } ) (Intersection)

F₁ ∪ F₂ = (max { μ_F
₁, μ_F
₂ } , min { v_F
₁, v_F
₂ } ) ;

(Union)

F^c = (v_F, μ_F). (Complement)

F₁ ⩾ F₂ if and only if μ_F
₁ ⩾ μ_F
₂ and v_F
₁ ⩽ v_F
₂

Definition 7. ([33]) Let F = (μ_F, v_F), F₁ = (μ_{F
₁}, v_{F
₁}) and F₂ = (μ_{F
₂}, v_{F
₂}) be FFNs. For λ > 0 . The following operations can be defined for FFNs.

$F_{1} \cap F_{2} = (\sqrt[3]{μ_{F_{1}}^{3} + μ_{F_{2}}^{3} - μ_{F_{1}}^{3} μ_{F_{2}}^{3}}, v_{F_{1}} . v_{F_{2}})$ ; (Addition on FFNs)

$F_{1} ⊠ F_{2} = (μ_{F_{1}} . μ_{F_{2}}, \sqrt[3]{v_{F_{1}}^{3} + v_{F_{2}}^{3} - v_{F_{1}}^{3} v_{F_{2}}^{3}});$ (Multiplication on FFNs)

$λ . F = (\sqrt[3]{1 - (1 - μ_{F}^{3})}, v_{F}^{λ})$ ; (Multiplication by Scalar)

$F^{λ} = (μ_{F}^{λ}, \sqrt[3]{1 - (1 - v_{F}^{3})})$ .

(Power of FFNs)

Theorem 1. ([33]) For F = (μ_F, v_F), F₁ = (μ_{F
₁}, v_{F
₁}) and F₂ = (μ_{F
₂}, v_{F
₂}) FFNs and λ > 0, λ₁ > 0, λ₂ > 0.

F₁ ∩ F₂ = F₂ ∩ F₁;

F₁ ⊠ F₂ = F₂ ⊠ F₁;

λ (F₁ ∩ F₂) = λ F₁ ∩ λF₂;

(λ₁+ λ₂) F = λ₁F ∩ λ₂F ;

${(F_{1} ⊠ F_{2})}^{λ} = F_{1}^{λ} ⊠ F_{2}^{λ};$

$F_{1}^{λ} ⊠ F_{2}^{λ} = F^{(λ_{1} + λ_{2})}$ .

Definition 8. [33] Let F = (μ_F, v_F) be a FFNs. Score function and accuracy of F can be represented as

(1) Score(F) = $(μ_{F}^{3} - v_{F}^{3}) ɛ [- 1, 1]$ .

(2) Accuracy(F) = $(μ_{F}^{3} + v_{F}^{3}) ɛ [0, 1]$ .

Definition 9. ([33]) Let F₁ = (μ_{F
₁}, v_{F
₁}) and F₂ = (μ_{F
₂}, v_{F
₂}) be FFNs. Score(F₁), Score(F₂) are the score values. Accuracy(F₁), Accuracy(F₂) are accuracy values.

If Score(F₁) < Score(F₂) then, F₁ < F₂;

If Score(F₁) > Score(F₂) then, F₁ > F₂;

If Score(F₁) = Score(F₂) then,

If Accuracy(F₁) < Accuracy(F₂), then F₁ < F₂;

If Accuracy(F₁) > Accuracy(F₂), then F₁ > F₂;

If Accuracy(F₁) = Accuracy(F₂), then F₁ = F₂;

The aggregations on FFNs presented by Yager as follows

Definition 10. ([34]) Let F_i = (μ_{F
_i}, v_{F
_i}) be FFNs and w = (w₁, w₂, …, w_n) ^T be weight vector of F_i (i = 1, 2, … n) (i = 1, 2, … n) and $\sum_{i = 1}^{n} w_{i} = 1$ Fermatean fuzzy weighted average operator is a function FFWA: Fⁿ → F, where

$FFWA (F_{1}, F_{2}, \dots, F_{n}) = (\sum_{i = 1}^{n} w_{i} μ_{F_{i}}, \sum_{i = 1}^{n} w_{i} v_{F_{i}})$ (4)

Definition 11. ([34]). Let F_i = (μ_{F
_i}, v_{F
_i}) be FFNs and w = (w₁, w₂, …, w_n) ^T be weight vector of F_i (i = 1, 2, … n) and $\sum_{i = 1}^{n} w_{i} = 1 .$ Fermatean fuzzy weighted geometric operator is a function FFWG: Fⁿ → F, where $FFWPA (F_{1}, F_{2}, \dots, F_{n}) = (\prod_{i = 1}^{n} μ_{F_{i}}^{w_{i}}, \prod_{i = 1}^{n} v_{F_{i}}^{w_{i}})$ (5)

Definition 12. ([34]). Let F_i = (μ_{F
_i}, v_{F
_i}) be FFNs and w = (w₁, w₂, …, w_n) ^T be weight vector of F_i (i = 1, 2, … n) and $\sum_{i = 1}^{n} w_{i} = 1$ . Fermatean fuzzy weighted power average operator is a function FFWPA: Fⁿ → F, where

$\begin{matrix} FFWPA (F_{1}, F_{2}, \dots, F_{n}) \\ = ({(\sum_{i = 1}^{n} w_{i} μ_{F_{i}}^{3})}^{\frac{1}{3}}, {(\sum_{i = 1}^{n} w_{i} v_{F_{i}}^{3})}^{\frac{1}{3}}) \end{matrix}$ (6)

Definition 13. ([34]) Let F_i = (μ_{F
_i}, v_{F
_i}) be FFNs and w = (w₁, w₂, …, w_n) ^T be weight vector of F_i (i = 1, 2, … n) and $\sum_{i = 1}^{n} w_{i} = 1$ . Fermatean fuzzy weighted power geometric operator is a function FFWPG: Fⁿ → F, where

$\begin{matrix} FFWPG (F_{1}, F_{2}, \dots, F_{n}) \\ = ({(1 - \prod_{i = 1}^{n} {(1 - μ_{F_{i}}^{3})}^{w_{i}})}^{\frac{1}{3}}, {(1 - \prod_{i = 1}^{n} {(1 - v_{F_{i}}^{3})}^{w_{i}})}^{\frac{1}{3}}) \end{matrix}$ (7)

Definition 14. ([12]) Let (a,b) be element of (0,1)X(0.1). For β ⩾ 1, The following operators are the Dombi norm.

$T (a, b) = \frac{1}{1 + {{(\frac{1 - a}{a})}^{β} + {(\frac{1 - b}{b})}^{β}}^{\frac{1}{β}}}$ (8) $S (a, b) = 1 - \frac{1}{1 + {{(\frac{a}{1 - a})}^{β} + {(\frac{b}{1 - b})}^{β}}^{\frac{1}{β}}}$ (9)

3 Fermatean fuzzy dombi operators

In this section, using definition 7 the basic Dombi operations on FFSs are defined. Fermatean Fuzzy Dombi Operators is a special case of Dombi aggregation of q-rung orthopair fuzzy numbers [16]. For β = 3 Fermatean fuzzy Dombi is obtained.

Definition 15. Let F₁ = (μ_{F
₁}, v_{F
₁}) and Let F₂ = (μ_{F
₂}, v_{F
₂}) be two FFNs. For β ⩾ 1, λ ⩾ 0 the basic operations of FFNs based on Dombi operation are defined as

$F_{1} \cap F_{2} = (\sqrt[3]{S (μ_{F_{1}}^{3}, μ_{F_{2}}^{3})}, \sqrt[3]{T (v_{F_{1}}^{3}, v_{F_{2}}^{3})})$

$F_{1} ⊠ F_{2} = (\sqrt[3]{T (μ_{F_{1}}^{3}, μ_{F_{2}}^{3})}, \sqrt[3]{S (v_{F_{1}}^{3}, v_{F_{2}}^{3})})$

λ . F₁ = $(\sqrt[3]{S (λ . μ_{F_{1}}^{3})}$ , $\sqrt[3]{T (λ . v_{F_{1}}^{3})})$ $\begin{matrix} = & (\sqrt[3]{1 - \frac{1}{1 + {λ {(\frac{μ_{F_{1}}^{3}}{1 - μ_{F_{1}}^{3}})}^{β}}^{\frac{1}{β}}}}, \\ \frac{1}{\sqrt[3]{1 + {λ {(\frac{1 - v_{F_{1}}^{3}}{v_{F_{1}}^{3}})}^{β}}^{\frac{1}{β}}}}) \end{matrix}$

$F_{1}^{λ} = (\sqrt[3]{T (λ . μ_{F_{1}}^{3})}$ , $\sqrt[3]{S (λ . v_{F_{1}}^{3})})$ $\begin{matrix} = & (\frac{1}{\sqrt[3]{1 + {λ {(\frac{1 - μ_{F_{1}}^{3}}{μ_{F_{1}}^{3}})}^{β}}^{\frac{1}{β}}}}, \\ \sqrt[3]{1 - \frac{1}{1 + {λ {(\frac{v_{F_{1}}^{3}}{1 - v_{F_{1}}^{3}})}^{β}}^{\frac{1}{β}}}}) . \end{matrix}$

3.1 Fermatean fuzzy Dombi averaging aggregation operators

In this section, weighted averaging operators are defined on FFSs via Dombi operators. Basic properties and proofs are mentioned.

Definition 16. Let F_i = (μ_{F
_i}, v_{F
_i}) be a number of FFNs. Fermatean fuzzy Dombi weighted averaging is a function FFNⁿ → FFN, so that $FFDWA (F_{1}, F_{2}, \dots, F_{n}) = \cap_{i = 1}^{n} λ_{i} F_{i}$ (10) where λ_i ⩾ 0 . The sum of λ_i is 1 (i = 1, 2, … n). Also, λ_i are weight vectors of F_i.

Theorem 2. Let F_i = (μ_{F
_i}, v_{F
_i}) be several FFNs. Fermatean fuzzy Dombi weighted averaging operator is defined using Dombi operator. For β ⩾ 1, such that, $\begin{matrix} FFDWA (F_{1}, F_{2}, \dots, F_{n}) \\ = (\sqrt[3]{1 - \frac{1}{1 + {\sum_{i = 1}^{n} λ_{i} {(\frac{μ_{F_{i}}^{3}}{1 - μ_{F_{i}}^{3}})}^{β}}^{\frac{1}{β}}}}, \\ \frac{1}{\sqrt[3]{1 + {\sum_{i = 1}^{n} λ_{i} {(\frac{1 - v_{F_{i}}^{3}}{v_{F_{i}}^{3}})}^{β}}^{\frac{1}{β}}}}) \end{matrix}$ (11) where $\sum_{i = 1}^{n} λ_{i} = 1$ , λ_i > 0. λ = (λ₁, λ₂, … λ_n) is weight vector of F_i.

Proof. Theorem 2 can be proved by the method of mathematical induction. Step-by-step operations are as follows:

(i) For n = 2, from Definition 15, using (I) and (III);

λ₁F₁ and λ₂F₂ are obtained by using (III) (Definition 15) $λ_{1} F_{1} = (\sqrt[3]{S (λ_{1} . μ_{F_{1}}^{3})}, \sqrt[3]{T (λ_{1} . v_{F_{1}}^{3})})$ $λ_{2} F_{2} = (\sqrt[3]{S (λ_{2} . μ_{F_{2}}^{3})}, \sqrt[3]{T (λ_{2} . v_{F_{2}}^{3})})$

λ₁F₁ ∩ λ₂F₂ is obtained by using (I) (Definition 15) $\begin{matrix} λ_{1} F_{1} \cap λ_{2} F_{2} \\ = (\sqrt[3]{S (λ_{1} . μ_{F_{1}}^{3}, λ_{2} μ_{F_{2}}^{3})}, \sqrt[3]{T (λ_{1} . v_{F_{1}}^{3}, λ_{2} v_{F_{2}}^{3})}) \end{matrix}$ $= (\sqrt[3]{\sum_{i = 1}^{2} S (λ_{i} . μ_{F_{i}}^{3})}, \sqrt[3]{\sum_{i = 1}^{2} T (λ_{i} . μ_{F_{i}}^{3})})$

(ii) Assume that (11) true for n = k; $\begin{matrix} FFDWA (F_{1}, F_{2}, \dots, F_{k}) \\ = (\sqrt[3]{\sum_{i = 1}^{k} S (λ_{i} . μ_{F_{i}}^{3})}, \sqrt[3]{\sum_{i = 1}^{k} T (λ_{i} . μ_{F_{i}}^{3})}) \end{matrix}$

(iii) We have to show that (11) true in n = k+1.

$FFDWA (F_{1}, F_{2}, \dots, F_{k}, F_{k + 1}) = \cap_{i = 1}^{k} λ_{i} F_{i} \cap λ_{k + 1} F_{k + 1}$

$\begin{matrix} FFDWA (F_{1}, F_{2}, \dots, F_{k}, F_{k + 1}) = \\ (\sqrt[3]{\sum_{i = 1}^{k} S (λ_{i} . μ_{F_{i}}^{3})}, \sqrt[3]{\sum_{i = 1}^{k} T (λ_{i} . μ_{F_{i}}^{3})}) \\ \cap (\sqrt[3]{S (λ_{k + 1} . μ_{F_{k + 1}}^{3})}, \sqrt[3]{T (λ_{k + 1} . μ_{F_{k + 1}}^{3})}) \\ = (\sqrt[3]{\sum_{i = 1}^{k + 1} S (λ_{i} . μ_{F_{i}}^{3})}, \sqrt[3]{\sum_{i = 1}^{k + 1} T (λ_{i} . μ_{F_{i}}^{3})}) \end{matrix}$

Thus, (11) is true for n = k+1, it true for all N. So, as a result of (i), (ii) and (iii) is (11) proved.

Definition 17. Let F_i = (μ_{F
_i}, v_{F
_i}) be a number of FFNs. Fermatean fuzzy Dombi order weighted averaging is a function FFNⁿ → FFN, So that, $FFDOWA (F_{1}, F_{2}, \dots, F_{n}) = \cap_{i = 1}^{n} λ_{i} F_{σ (i)}$ (12) where λ_i ⩾ 0 . The sum of λ_i is 1 (i = 1, 2, … n). F_σ(i) is ith biggest of weighted value. So, total order: F_σ(1) ⩾ F_σ(2) ⩾ … ⩾ F_σ(n) . (σ (1) , σ (2) , …, σ (n)) is the permutation of σ (i) .

Theorem 3. Let F_i = (μ_{F
_i}, v_{F
_i}) be a number of FFNs. Fermatean fuzzy Dombi order weighted averaging operator is defined using Dombi operator. For β ⩾ 1, Such that,

$\begin{matrix} FFDOWA (F_{1}, F_{2}, \dots, F_{n}) \\ (\sqrt[3]{1 - \frac{1}{1 + {\sum_{i = 1}^{n} λ_{i} {(\frac{μ_{F_{σ}^{(i)}}^{3}}{1 - μ_{F_{σ}^{(i)}}^{3}})}^{β}}^{\frac{1}{β}}}}, \\ \frac{1}{\sqrt[3]{1 + {\sum_{i = 1}^{k} λ_{i} {(\frac{1 - v_{F_{σ}^{(i)}}^{3}}{v_{F_{σ}^{(i)}}^{3}} - 1)}^{β}}^{\frac{1}{β}}}}) \end{matrix}$ (13) where $\sum_{i = 1}^{n} λ_{i} = 1$ , λ_i > 0.λ = (λ₁, λ₂ … λ_n) is the weight vector of F_i. F_σ(i) is ith biggest of weighted value. So, total order: F_σ(1) ⩾ F_σ(2) ⩾ ⋯ ⩾ F_σ(n). It shall be noted that the score-based comparison is performed.

Proof. The proof is similar to Theorem 2.

Definition 18. Let $F_{i}^{*} = (μ_{Fi}, v_{Fi})$ be a number of FFNs. Fermatean fuzzy Dombi hybrid weighted averaging is a function FFNⁿ → FFN, so that $FFDWG (F_{1}^{*}, F_{2}^{*}, \dots, F_{n}^{*}) = ⊠_{i = 1}^{n} λ_{i} F_{σ (i)}^{*_{i}}$ (14) where λ_i ⩾ 0_. Sum of λ_i is 1 (i = 1, 2, … n). $F_{σ (i)}^{*}$ , ith biggest weighted value and $F_{i}^{*} = n λ_{i} F_{i}$ . So, total order: $F_{σ (1)}^{*} ⩾ F_{σ (2)}^{*} ⩾ \dots ⩾ F_{σ (n)}^{*}$ where n is the balancing coefficient. If $w = (\frac{1}{n}, \frac{1}{n}, \dots, \frac{1}{n})$ , then FFDWA operator is a special case FFDHWA operator. If $λ = (\frac{1}{n}, \frac{1}{n}, \dots, \frac{1}{n})$ , then FFDOWA is a special case of the operator FFDHWA.

Theorem 4. Let $F_{i}^{*} = (μ_{F_{i}}, v_{F_{i}})$ be a number of FFNs. Fermatean fuzzy Dombi hybrid weighted averaging operator is defined using Dombi operator. For β ⩾ 1, such that, $\begin{matrix} FDHWA (F_{1}^{*}, F_{2}^{*}, \dots, F_{n}^{*}) \\ = (\sqrt[3]{1 - \frac{1}{1 + {\sum_{i = 1}^{n} λ_{i} {(\frac{μ_{F_{σ (i)}^{*}}^{3}}{1 - μ_{F_{σ (i)}^{*}}^{3}})}^{β}}^{\frac{1}{β}}}}, \\ \frac{1}{\sqrt[3]{1 + {\sum_{i = 1}^{n} λ_{i} {(\frac{1 - v_{F_{σ (i)}^{*}}^{3}}{v_{F_{σ (i)}^{*}}^{3}})}^{β}}^{\frac{1}{β}}}}) \end{matrix}$ (15) where $\sum_{i = 1}^{n} λ_{i} = 1$ , λ_i > 0. λ = (λ₁, λ₂, … λ_n) is the weight vector of F_i. $F_{σ (i)}^{*}$ , ith biggest weighted value and $F_{i}^{*} = n λ_{i} F_{i}$ . So, total order: $F_{σ (1)}^{*} ⩾ F_{σ (2)}^{*} ⩾ \dots ⩾ F_{σ (n)}^{*} .$ (σ (1) , σ (2) , …, σ (n)) is the permutation of σ (i) .

Proof. The proof is similar to Theorem 2.

3.2 Fermatean Fuzzy Dombi Geometric Aggregation Operators

Definition 19. Let F_i = (μ_{F
_i}, v_{F
_i}) be a number of FFNs. Fermatean fuzzy Dombi weighted geometric is a function FFNⁿ → FFN, so that $FFDWG (F_{1}, F_{2}, \dots, F_{n}) = ⊠_{i = 1}^{n} F_{i}^{λ_{i}}$ (16) where λ_i ⩾ 0 . The sum of λ_i is 1 (i = 1, 2, … n). Also, λ_i are weight vectors of F_i.

Theorem 5. Let F_i = (μ_{F
_i}, v_{F
_i}) be a number of FFNs. Fermatean fuzzy Dombi weighted geometric operator is defined using Dombi operator. For β ⩾ 1, such that, $\begin{matrix} FFDWG (F_{1}, F_{2}, \dots, F_{n}) \\ = (\frac{1}{\sqrt[3]{1 + {\sum_{i = 1}^{n} λ_{i} {(\frac{1 - μ_{F_{i}}^{3}}{μ_{F_{i}}^{3}})}^{β}}^{\frac{1}{β}}}}, \\ \sqrt[3]{1 - \frac{1}{1 + {\sum_{i = 1}^{n} λ_{i} {(\frac{v_{F_{i}}^{3}}{1 - v_{F_{i}}^{3}})}^{β}}^{\frac{1}{β}}}}) \end{matrix}$ (17) where $\sum_{i = 1}^{n} λ_{i} = 1$ , λ_i > 0. λ = (λ₁, λ₂, … λ_n) is weight vector of F_i.

Proof. The proof is similar to Theorem 2 (Using (I) and (IV)).

Definition 20. Let F_i = (μ_{F
_i}, v_{F
_i}) be a number of FFNs. Fermatean fuzzy Dombi order weighted geometric is a function FFNⁿ → FFN, so that $FFDOWG (F_{1}, F_{2}, \dots, F_{n}) = ⊠_{i = 1}^{n} F_{σ (i)}^{λ_{i}}$ (18) where λ_i ⩾ 0. The sum of λ_i is 1 (i = 1, 2, … n). F_σ(i) is ith biggest of weighted value. So, total order: F_σ(1) ⩾ F_σ(2) ⩾ … ⩾ F_σ(n). (σ (1) , σ (2) , …, σ (n)) is the permutation of σ (i).

Theorem 6. Let F_i = (μ_{F
_i}, v_{F
_i}) be a number of FFNs. Fermatean fuzzy Dombi order weighted geometric operator is defined using Dombi operator. For β ⩾ 1, such that, $\begin{matrix} FFDOWG (F_{1}, F_{2}, \dots, F_{n}) \\ = (\frac{1}{\sqrt[3]{1 + {\sum_{i = 1}^{n} λ_{i} {(\frac{1 - μ_{F_{σ (i)}}^{3}}{μ_{F_{σ (i)}}^{3}})}^{β}}^{\frac{1}{β}}}}, \\ \sqrt[3]{1 - \frac{1}{1 + {\sum_{i = 1}^{n} λ_{i} {(\frac{v_{F_{σ (i)}}^{3}}{1 - v_{F_{σ (i)}}^{3}})}^{β}}^{\frac{1}{β}}}}) \end{matrix}$ (19) where $\sum_{i = 1}^{n} λ_{i} = 1$ , λ_i > 0. λ = (λ₁, λ₂, … λ_n) is weight vector of F_i.F_σ(i) is ith biggest of weighted value. So, total order: F_σ(1) ⩾ F_σ(2) ⩾ … ⩾ F_σ(n). The proof is similarly Theorem 2.

Definition 21. Let $F_{i}^{*} = (μ_{F_{i}}, v_{F_{i}})$ be a number of FFNs. Fermatean fuzzy Dombi hybrid weighted geometric is a function FFNⁿ → FFN, so that $FFDHWG (F_{1}^{*}, F_{2}^{*}, \dots, F_{n}^{*}) = ⊠_{i = 1}^{n} {(F_{σ (i)}^{*})}^{λ_{i}}$ (20) where λ_i ⩾ 0 . The sum of λ_i is 1 (i = 1, 2, … n). $F_{σ (i)}^{*}$ , ith biggest weighted value and $F_{i}^{*} = n λ_{i} F_{i}$ . So, total order: $F_{σ (1)}^{*} ⩾ F_{σ (2)}^{*} ⩾ \dots ⩾ F_{σ (n)}^{*}$ , where n is the balancing coefficient. If $w = (\frac{1}{n}, \frac{1}{n}, \dots, \frac{1}{n})$ , then FFDWA operator is a special case FFDHWA operator. If $λ = (\frac{1}{n}, \frac{1}{n}, \dots, \frac{1}{n})$ , then FFDOWA is a special case of the operator FFDHWA.

Theorem 7. Let F_i = (μ_{F
_i}, v_{F
_i}) be a number of FFNs. Fermatean fuzzy Dombi hybrid weighted geometric operator is defined using Dombi operator. For β ⩾ 1, Such that, $\begin{matrix} FFDHWG (F_{1}^{*}, F_{2}^{*}, \dots, F_{n}^{*}) \\ = (\frac{1}{\sqrt[3]{1 + {\sum_{i = 1}^{n} λ_{i} {(\frac{1 - μ_{F_{σ (i)}^{*}}^{3}}{μ_{F_{σ (i)}^{*}}^{3}})}^{β}}^{\frac{1}{β}}}}, \\ \sqrt[3]{1 - \frac{1}{1 + {\sum_{i = 1}^{n} λ_{i} {(\frac{v_{F_{σ (i)}^{*}}^{3}}{1 - v_{F_{σ (i)}^{*}}^{3}})}^{β}}^{\frac{1}{β}}}}) \end{matrix}$ (21) where $\sum_{i = 1}^{n} λ_{i} = 1$ , λ_i > 0. λ = (λ₁, λ₂, … λ_n) is weight vector of F_i. $F_{σ (i)}^{*}$ , ith biggest weighted value and $F_{i}^{*} = n λ_{i} F_{i}$ . So, total order: $F_{σ (1)}^{*} ⩾ F_{σ (2)}^{*} ⩾ \dots ⩾ F_{σ (n)}^{*}$ . The proof is similarly Theorem 2. (σ (1) , σ (2) , …, σ (n)) is the permutation of σ (i).

3.3 Some properties of operators

Now, we show that operators provide some properties. Such as, boundedness, idempotency and monotonicity.

Theorem 8 (Boundedness) Let F_i = (μ_{F
_i}, v_{F
_i}) be a number of FFNs. $\sum_{i = 1}^{n} λ_{i} = 1 . λ = (λ_{1}, λ_{2}, \dots λ_{n})$ is weight vector of F_i. Also, let = 1, 2, …, n, $μ_{F}^{-} = \min {μ_{F_{i}}}$ , $μ_{F}^{+} = max {μ_{F_{i}}}$ , $v_{F}^{-} = \min {v_{F_{i}}}$ , $v_{F}^{+} = max {v_{F_{i}}}$ ,

$(μ_{F}^{-}, v_{F}^{+}) ⩽ FFDWA (F_{1}, F_{2}, \dots, F_{n}) ⩽ (μ_{F}^{+}, v_{F}^{-})$ ;

$(μ_{F}^{-}, v_{F}^{+}) ⩽ FFDOWA (F_{1}, F_{2}, \dots, F_{n}) ⩽ (μ_{F}^{+}, v_{F}^{-});$

$(μ_{F}^{-}, v_{F}^{+}) ⩽ FFDHWA (F_{1}, F_{2}, \dots, F_{n}) ⩽ (μ_{F}^{+}, v_{F}^{-});$

$(μ_{F}^{-}, v_{F}^{+}) ⩽ FFDWG (F_{1}, F_{2}, \dots, F_{n}) ⩽ (μ_{F}^{+}, v_{F}^{-});$

$(μ_{F}^{-}, v_{F}^{+}) ⩽ FFDOWG (F_{1}, F_{2}, \dots, F_{n}) ⩽ (μ_{F}^{+}, v_{F}^{-});$

$(μ_{F}^{-}, v_{F}^{+}) ⩽ FFDHWG (F_{1}, F_{2}, \dots, F_{n}) ⩽ (μ_{F}^{+}, v_{F}^{-}) .$

Proof. Proofs for (i), (ii) and (iii) will be shown. Other aggregations are shown similar manner.

(i) F_i = (μ_{F
_i}, v_{F
_i}), $μ_{F}^{-} ⩽ μ_{F_{i}} ⩽ μ_{F}^{+}$ and $v_{F}^{-} ⩽ v_{F_{i}} ⩽ v_{F}^{+}$ . Suppose that $F_{\min} = (μ_{F}^{-}, v_{F}^{+})$ and $F_{\max} = (μ_{F}^{+}, v_{F}^{-})$ .

$μ_{F}^{-} = \sqrt[3]{\sum_{i = 1}^{n} S (λ_{i} . (μ_{F}^{-})^{3})}, μ_{F}^{+} = \sqrt[3]{\sum_{i = 1}^{n} S (λ_{i} . (μ_{F}^{+})^{3})}$ $v_{F}^{-} = \sqrt[3]{\sum_{i = 1}^{n} T (λ_{i} . (v_{F}^{-})^{3})}, v_{F}^{+} = \sqrt[3]{\sum_{i = 1}^{n} T (λ_{i} . (v_{F}^{+})^{3})}$

So, $\begin{matrix} \sqrt[3]{\sum_{i = 1}^{n} S (λ_{i} . (μ_{F}^{-})^{3})} ⩽ \sqrt[3]{\sum_{i = 1}^{n} S (λ_{i} . μ_{F_{i}}^{3})} \\ ⩽ \sqrt[3]{\sum_{i = 1}^{n} S (λ_{i} . (μ_{F}^{+})^{3})} \end{matrix}$

Other hand, $\begin{matrix} \sqrt[3]{\sum_{i = 1}^{n} S (λ_{i} . (v_{F}^{-})^{3})} ⩽ \sqrt[3]{\sum_{i = 1}^{n} S (λ_{i} . v_{F_{i}}^{3})} \\ ⩽ \sqrt[3]{\sum_{i = 1}^{n} S (λ_{i} . (v_{F}^{+})^{3})} \end{matrix}$ $score (F_{\min}) = {(μ_{F}^{-})}^{3} - {(v_{F}^{+})}^{3}$

$= {(\sqrt[3]{\sum_{i = 1}^{n} S (λ_{i} . (μ_{F}^{-})^{3})})}^{3} - {(\sqrt[3]{\sum_{i = 1}^{n} T (λ_{i} . (v_{F}^{+})^{3})})}^{3}$

score $(F_{\max}) = {(μ_{F}^{+})}^{3} - {(v_{F}^{-})}^{3}$

$= {(\sqrt[3]{\sum_{i = 1}^{n} S (λ_{i} . (μ_{F}^{+})^{3})})}^{3} - {(\sqrt[3]{\sum_{i = 1}^{n} T (λ_{i} . (v_{F}^{-})^{3})})}^{3}$

$score (FFDWA (F_{1}, F_{2}, \dots, F_{n})) =$ ${(\sqrt[3]{\sum_{i = 1}^{n} S (λ_{i} . μ_{F_{i}}^{3})})}^{3} - {(\sqrt[3]{\sum_{i = 1}^{n} S (λ_{i} . v_{F_{i}}^{3})})}^{3}$

Finally,

score(F_min) ⩽ score (FFDWA (F₁, F₂, …, F_n)) ⩽ score (F_max).

It is provided. ( $μ_{F}^{-}, v_{F}^{+}) ⩽ FFDWA (F_{1}, F_{2}, \dots, F_{n}) ⩽ (μ_{F}^{+}, v_{F}^{-})$ ;

F_i = (μ_{F
_i}, v_{F
_i}), $μ_{F}^{-} ⩽ μ_{F_{i}} ⩽ μ_{F}^{+}$ and $v_{F}^{-} ⩽ v_{F_{i}} ⩽ v_{F}^{+}$ . Suppose that $F_{\min} = (μ_{F}^{-}, v_{F}^{+})$ and $F_{\max} = (μ_{F}^{+}, v_{F}^{-})$ . Based on (i) for simplicity, it can be said that the following result is obtained directly. $F_{\min} \subseteq FFDOWA (F_{1}, F_{2}, \dots, F_{n}) \subseteq F_{\max}$

F_i = (μ_{F
_i}, v_{F
_i}), $μ_{F}^{-} ⩽ μ_{F_{i}} ⩽ μ_{F}^{+}$ and $v_{F}^{-} ⩽ v_{F_{i}} ⩽ v_{F}^{+}$ . Suppose that $F_{\min} = (μ_{F}^{-}, v_{F}^{+})$ and $F_{\max} = (μ_{F}^{+}, v_{F}^{-})$ . Based on (i) for simplicity, it can be said that the following result is obtained directly. $F_{\min} \subseteq FFDHWA (F_{1}, F_{2}, \dots, F_{n}) \subseteq F_{\max}$

Theorem 9 (Idempotency) Let F_i = (μ_{F
_i}, v_{F
_i}) be a number of FFNs. $\sum_{i = 1}^{n} λ_{i} = 1 . λ = (λ_{1}, λ_{2}, \dots λ_{n})$ is weight vector of F_i. Also, i = 1, 2, …, n, if F_i are identical FFNs F_i = F = (μ_{F
_i}, v_{F
_i}) then,

FFDWA (F₁, F₂, …, F_n) = F;

FFDOWA (F₁, F₂, …, F_n) = F;

FFDHWA (F₁, F₂, …, F_n) = F;

FFDWG (F₁, F₂, …, F_n) = F;

FFDOWG (F₁, F₂, …, F_n) = F;

FFDHWG (F₁, F₂, …, F_n) = F.

Proof. Proofs for (i), (ii) and (iii) will be shown. Other aggregations are shown similar manner. For clarity, aggregation operators are not included ((ii) and (iii)).

FFDWA (F₁, F₂, …, F_n) = FFDWA (F, F, …, F) = $\begin{matrix} (\sqrt[3]{1 - \frac{1}{1 + {\sum_{i = 1}^{n} λ_{i} {(\frac{μ_{F}^{3}}{1 - μ_{F}^{3}})}^{β}}^{\frac{1}{β}}}}, \\ \frac{1}{\sqrt[3]{1 + {\sum_{i = 1}^{n} λ_{i} {(\frac{1 - v_{F}^{3}}{v_{F}^{3}})}^{β}}^{\frac{1}{β}}}}) \\ = (μ_{F}, v_{F}) = F . \end{matrix}$

FFDOWA (F₁, F₂, …, F_n) = FFDOWA (F, F, …, F) = (μ_F, v_F) = F.

FFDHWA (F₁, F₂, …, F_n) = FFDHWA (F, F, …, F) = (μ_F, v_F) = F.

Theorem 10 (Monotonicity) Let F_i = (μ_{F
_i}, v_{F
_i}) and G_i = (μ_{G
_i}, v_{G
_i}) be two numbers of FFNs.λ = (λ₁, λ₂, … λ_n) is weight vector of F_i, if μ_{F
_i} ⩽ μ_{G
_i}, v_{F
_i} ⩾ v_{G
_i}, then

FFDWA (F₁, F₂, …, F_n) ⩽ FFDWA (G₁, G₂, …, G_n);

FFDOWA (F₁, F₂, …, F_n) ⩽ FFDOWA (G₁, G, …, G_n);

FFDHWA (F₁, F₂, …, F_n) ⩽ FFDHWA (G₁, G₂, …, G_n);

FFDWG (F₁, F₂, …, F_n) ⩽ FFDWG (G₁, G₂, …, G_n);

FFDOWG (F₁, F₂, …, F_n) ⩽ FFDOWG (G₁, G₂, …, G_n);

FFDHWG (F₁, F₂, …, F_n) ⩽ FFDHWG (G₁, G₂, …, G_n)

4 Modified distance method for Fermatean fuzzy sets

In this section, a distance method that provides more accurate results on FFSs is mentioned. Ke and Song et. al. proposed a new distance method on intuitionistic fuzzy sets [19]. Taking into account some examples, the presented method is more accurate than the existing Euclidean distance, Hamming distance, normalized Euclidean distance and normalized Hamming distance. Suppose that A and B are two FFSs defined in X ={ x₁, x₂, …, x_n }. A = {〈x, μ_A (x) , v_A (x) 〉|xɛX}, and B = {〈x, μ_B (x) , v_B (x) 〉|xɛX}, respectively, The distance between A and B is calculated by the above expression (22).

$\begin{matrix} D^{F} (A, B) = \frac{1}{n} \sum_{i = 1}^{n} \\ \sqrt{{(\frac{Sc (A)}{2} - \frac{Sc (B)}{2})}^{2} + \frac{1}{3} {(\frac{Acc (A)}{2} - \frac{Acc (B)}{2})}^{2}} \end{matrix}$ (22)

Theorem 11. For A, B, and C FFSs defined in X ={ x₁, x₂, …, x_n }.

D^F (A, B) = 0 if and only if A = B.

D^F (A, B) = D^F (B, A)

If A ⊆ B ⊆ C, then D^F (A, B) ⩽ D^F (A, C) and D^F (B, C) ⩽ D^F (A, C)

0 ⩽ D^F (A, B) ⩽ 1

Proof (I)

⇒ Suppose that D^F (A, B) =0. In that case $\frac{μ_{A}^{3} (x_{i}) - v_{A}^{3} (x_{i})}{2} - \frac{μ_{B}^{3} (x_{i}) - v_{B}^{3} (x_{i})}{2} = 0$ and $\frac{μ_{A}^{3} (x_{i}) + v_{A}^{3} (x_{i})}{2} - \frac{μ_{B}^{3} (x_{i}) + v_{B}^{3} (x_{i})}{2} = 0$ For ∀i = 1, 2, …, n. Therefore, the following equations can be written. $μ_{A}^{3} (x_{i}) - v_{A}^{3} (x_{i}) = μ_{B}^{3} (x_{i}) - v_{B}^{3} (x_{i})$ and $μ_{A}^{3} (x_{i}) +$ $v_{A}^{3} (x_{i}) = μ_{B}^{3} (x_{i}) + v_{B}^{3} (x_{i})$ left and right sides between each other added and subtracted $μ_{A}^{3} (x_{i}) = μ_{B}^{3} (x_{i})$ and $v_{A}^{3} (x_{i}) = v_{B}^{3} (x_{i})$ . Hence, ∀iɛ { 1, 2, …, n } , μ_A (x) = μ_B (x) and v_A (x) = v_B (x) are obtained. A = B can be concluded in the same way.

⇐ Suppose that A = B ∀iɛ{ 1, 2, …, n }, μ_A (x_i) = μ_B (x_i), v_A (x_i) = v_B (x_i), hence D^F (A, B) = 0.

Proof (II)

In terms of simplicity, suppose that $Sc (A) = μ_{A}^{3} (x_{i}) - v_{A}^{3} (x_{i})$ and $Acc (A) = μ_{A}^{3} (x_{i}) + v_{A}^{3} (x_{i})$ score value and accuracy value of FFSs, respectively.

$\begin{matrix} D^{F} (B, A) = \frac{1}{n} \sum_{i = 1}^{n} \\ \sqrt{{(\frac{Sc (B)}{2} - \frac{Sc (A)}{2})}^{2} + \frac{1}{3} {(\frac{Acc (B)}{2} - \frac{Acc (A)}{2})}^{2}} \end{matrix}$ $\begin{matrix} = \frac{1}{n} \sum_{i = 1}^{n} \\ \sqrt{{(\frac{Sc (A)}{2} - \frac{Sc (B)}{2})}^{2} + \frac{1}{3} {(\frac{Acc (A)}{2} - \frac{Acc (B)}{2})}^{2}} \\ = D^{F} (A, B) \end{matrix}$

Proof (III) For the proof of this proposition, the monotony status should be investigated with the help of basic derivative information. Let A = {〈x, μ_A (x) , v_A (x) 〉|xɛX}, B = {〈x, μ_B (x) , v_B (x) 〉|xɛX} and C = {〈x, μ_C (x) , v_C (x) 〉|xɛX} be three FFSs. μ_A (x_i) ⩽ μ_B (x_i) ⩽ μ_C (x_i) and v_A (x_i) ⩾ v_B (x_i) ⩾ v_C (x_i) is valid (from definition 6, IV) when A ⊆ B ⊆ C is satisfied.

Distance between A and B

$\begin{matrix} D^{F} (A, B) = \frac{1}{n} \sum_{i = 1}^{n} \\ \sqrt{{(\frac{Sc (A)}{2} - \frac{Sc (B)}{2})}^{2} + \frac{1}{3} {(\frac{Acc (A)}{2} - \frac{Acc (B)}{2})}^{2}} . \end{matrix}$

Distance between A and C

$\begin{matrix} D^{F} (A, C) = \frac{1}{n} \sum_{i = 1}^{n} \\ \sqrt{{(\frac{Sc (A)}{2} - \frac{Sc (C)}{2})}^{2} + \frac{1}{3} {(\frac{Acc (A)}{2} - \frac{Acc (C)}{2})}^{2}} \end{matrix}$

Distance between B and C

$\begin{matrix} D^{F} (B, C) = \frac{1}{n} \sum_{i = 1}^{n} \\ \sqrt{{(\frac{Sc (B)}{2} - \frac{Sc (C)}{2})}^{2} + \frac{1}{3} {(\frac{Acc (B)}{2} - \frac{Acc (C)}{2})}^{2}} \end{matrix}$

Let’s construct a f (x, y) function with two variables at this stage.

$f (x, y) = {((x^{3} - y^{3}) - (a^{3} - b^{3}))}^{2} + \frac{1}{3} {((x^{3} + y^{3}) - (a^{3} + b^{3}))}^{2}$

Where 0 ⩽ x ⩽ 1, 0 ⩽ y ⩽ 1, 0 ⩽ a ⩽ 1, 0 ⩽ b ⩽ 1. Let’s examine partial derivatives according to x and y variables for monotonicity.

$\begin{matrix} \frac{\partial f}{\partial x} = 6 x^{2} ((x^{3} - y^{3}) - (a^{3} - b^{3})) + x^{2} ((x^{3} + y^{3}) \\ - (a^{3} + b^{3})) = 6 x^{2} (x^{3} - a^{3}) + 6 x^{2} (b^{3} - y^{3}) \\ + x^{2} (x^{3} - a^{3}) - x^{2} (b^{3} - y^{3}) = 7 x^{2} (x^{3} - a^{3}) \\ + 5 x^{2} (b^{3} - y^{3}) \end{matrix}$ $\begin{matrix} \frac{\partial f}{\partial y} = - 6 y^{2} ((x^{3} - y^{3}) - (a^{3} - b^{3})) + y^{2} ((x^{3} + y^{3}) \\ - (a^{3} + b^{3})) = - 6 y^{2} (x^{3} - a^{3}) - 6 y^{2} (b^{3} - y^{3}) \\ + y^{2} (x^{3} - a^{3}) - y^{2} (b^{3} - y^{3}) = - 5 y^{2} (x^{3} - a^{3}) \\ - 7 y^{2} (b^{3} - y^{3}) = 5 y^{2} (a^{3} - x^{3}) + 7 y^{2} (y^{3} - b^{3}) \end{matrix}$

There are 2 different situations under the given conditions.

0 ⩽ a ⩽ x ⩽ 1and 0 ⩽ y ⩽ b ⩽ 1. We can obtain that $\frac{\partial f}{\partial x} ⩾ 0$ and $\frac{\partial f}{\partial y} ⩽ 0$

In light of this information, f is an increasing function of variable x and a decreasing function of variable y. Suppose that a = μ_A (x_i), b = v_A (x_i), then a = μ_A (x_i) ⩽ μ_B (x_i) ⩽ μ_C (x_i) and b = v_A (x_i) ⩾ v_B (x_i) ⩾ v_C (x_i), ∀iɛ{ 1, 2, …, n }. Considering the monotonicity of f (x, y), f (μ_B (x_i) , v_B (x_i))⩽ f (μ_C (x_i) , v_C (x_i)) is obtained. So,

$D^{F} (A, C) = \frac{1}{2 n} \sum_{i = 1}^{n} \sqrt{f (μ_{C} (x_{i}), v_{C} (x_{i}))}$ , $D^{F} (A, B) = \frac{1}{2 n} \sum_{i = 1}^{n} \sqrt{f (μ_{B} (x_{i}), v_{B} (x_{i}))}$

So, D^F (A, B) ⩽ D^F (A, C) is obtain.

0 ⩽ x ⩽ a ⩽ 1 and 0 ⩽ b ⩽ y ⩽ 1. We can obtain that $\frac{\partial f}{\partial x} ⩽ 0$ and $\frac{\partial f}{\partial y} ⩾ 0$

f (x, y) is a decreasing function of variable x and increasing function of variable y. Suppose that a = μ_C (x_i), b = v_C (x_i), then μ_A (x_i) ⩽ μ_B (x_i) ⩽ μ_C (x_i) = a and v_A (x_i) ⩾ v_B (x_i) ⩾ v_C (x_i) = b, ∀iɛ{ 1, 2, …, n }. Considering the monotonicity of f (x, y), f (μ_B (x_i) , v_B (x_i))⩽ f (μ_A (x_i) , v_A (x_i)) is obtained. So, $\begin{matrix} D^{F} (A, C) = \frac{1}{2 n} \sum_{i = 1}^{n} \sqrt{f (μ_{A} (x_{i}), v_{A} (x_{i}))}, \\ D^{F} (B, C) = \frac{1}{2 n} \sum_{i = 1}^{n} \sqrt{f (μ_{B} (x_{i}), v_{B} (x_{i}))}, \end{matrix}$

So, D^F (B, C) ⩽ D^F (A, C) is obtain.

Finally, if (i) and (ii) are taken into account, we can conclude that D^F (A, B) ⩽ D^F (A, C) and D^F (B, C) ⩽ D^F (A, C).

Proof (IV) Because of description D^F (A, B) ⩾ 0. Let A ={< x, μ_A (x) , v_A (x) > |xɛX} and B ={< x, μ_B (x) , v_B (x) > |xɛX} be two FFSs. Other hand F_* ={< x, 0, 1 > |xɛX} and F^* ={< x, 1, 0 > |xɛX}. From definition 6, IV F_* ⊆ A ⊆ B ⊆ F^* implies that D^F (A, B) ⩽ D^F (F_*, F^*). Substituting F_* and F^* in equation (22), D^F (F_*, F^*) =1. D^F (A, B) ⩽ D^F (F_*, F^*) =1

Hence, D^F (A, B) ⩽ 1. Finally, D^F (A, B) ⩾ 0 and D^F (A, B) ⩽ 1. We can conclude that 0 ⩽ D^F (A, B) ⩽ 1.

5 Fermatean fuzzy TOPSIS method to MCDM problem for proposed aggregation methods

In this section, the proposed aggregation methods are applied to Fermatean Fuzzy TOPSIS. This section also includes the steps of the algorithm, the problem description and a practical MCDM example.

5.1 Algorithms of the proposed method

Firstly, MCDM is represented with FFNs. There are alternatives and criteria in the structure of the problem. The votes/rating given to the criteria of each alternative constitutes the MCDM structure. Suppose that there are m alternatives S_i (i = 1, 2, … m) and n criteria C_j (j = 1, 2, … n). Another hand, $\sum_{j = 1}^{n} λ_{i} = 1$ , λ_j > 0. λ = (λ₁, λ₂, … λ_n) is the weight vector of attributes. Let’s address the notations of evaluating alternative S_i according to the criterion C_j. a_ij = (u_ij, v_ij) and R = (a_ij) _mXn. R is a Fermatean fuzzy decision matrix. The proposed operators were used considering the stages in the study [33]. The steps of MCDM problem by TOPSIS method with the help of Dombi operators are as follows:

Step I. Fermatean fuzzy decision matrix(FFDM) is created. S_i are alternatives. C_j are criteria. Let’s normalize the FFDM matrix. If the criterion is benefit-type, no change is necessary. If the criterion is cost-type, the complement operator must be used (Definition 6 and (III)). In this case, since the cost type is only the last column, the complement applies only to the last column (Table 2). As below R matrix

$\begin{matrix} C_{1} C_{n} \\ S_{1} \\ {(\begin{matrix} (u_{11}, v_{11}) & \dots & (u_{1 n}, v_{1 n}) \\ ⋮ & ⋱ & ⋮ \\ (u_{m 1}, v_{m 1}) & \dots & (u_{mn}, v_{mn}) \end{matrix})}_{mXn} \\ S_{m} \end{matrix}$

Step II. In this step, proposed aggregation operators on FFSs are used. Geometric and average methods were tested separately. Aggregation methods: Fermatean fuzzy Dombi weighted average, Fermatean fuzzy Dombi weighted geometric, Fermatean fuzzy Dombi order weighted average, Fermatean fuzzy Dombi order weighted geometric, Fermatean fuzzy Dombi hybrid weighted average, Fermatean fuzzy Dombi hybrid weighted geometric. After applying aggregated $R = C_{j} (S_{i}) = C_{j} (S_{i}^{'})$

Step III. Determine Fermatean fuzzy positive and negative ideal solutions.

In this step, S⁺ positive and S^- negative ideal solutions are determined. $S^{+} = {\max_{i} < score (C_{j} (S_{i}^{'})) | j = 1, 2, ..., n >}$ $S^{-} = {\min_{i} < score (C_{j} (S_{i}^{'})) | j = 1, 2, ..., n >}$

Step IV. Distance between S_i and positive (S⁺), negative (S^-) ideal solution is determined. Euclidean distance is used.

Distance between S_i and S⁺, i = 1, 2, … m, j = 1, 2, …, n $D (S_{i}, S^{+}) = \frac{1}{n} \sum_{i = 1}^{n} λ_{j} \sqrt{{(\frac{μ_{ij}^{3} - v_{ij}^{3}}{2} - \frac{(μ_{j}^{+})^{3} - (v_{j}^{+})^{3}}{2})}^{2} + \frac{1}{3} {(\frac{μ_{ij}^{3} + v_{ij}^{3}}{2} - \frac{(μ_{j}^{+})^{3} + (v_{j}^{+})^{3}}{2})}^{2}}$

Distance between S_i and S^-, i = 1, 2, … m, j = 1, 2, …, n $D (S_{i}, S^{-}) = \frac{1}{n} \sum_{i = 1}^{n} λ_{j} \sqrt{{(\frac{μ_{ij}^{3} - v_{ij}^{3}}{2} - \frac{(μ_{j}^{-})^{3} - (v_{j}^{-})^{3}}{2})}^{2} + \frac{1}{3} {(\frac{μ_{ij}^{3} + v_{ij}^{3}}{2} - \frac{(μ_{j}^{-})^{3} + (v_{j}^{-})^{3}}{2})}^{2}}$

Step V. In this step, it is computed the revised closeness of the alternatives. Vencheh and Mirjaberi explain that S⁺ and S^- may not always be the feasible solution [13]. The fact that S_i is close to positive ideal solution does not mean that it is far from negative ideal solution. So, the smaller D (S_i, S⁺) the better S_i and the bigger D (S_i, S^-) better S_i. Therefore, alternative satisfies, $\begin{matrix} D_{\min} (S_{i}, S^{+}) = & min_{1 ⩽ i ⩽ m} D (S_{i}, S^{+}), D_{\max} (S_{i}, S^{-}) \\ = max_{1 ⩽ i ⩽ m} D (S_{i}, S^{-}) \end{matrix}$

The compromise solution, $ξ (S_{i}) = \frac{D (S_{i}, S^{-})}{D_{\max} (S_{i}, S^{-})} - \frac{D (S_{i}, S^{+})}{D_{\min} (S_{i}, S^{+})}$

ξ (S_i) alternative S_i closes to positive ideal solution does mean that it is far from negative ideal solution.

Step VI. Values of ξ (S_i) are sorted in decreasing order and the final result is obtained.

5.2 Problem definition I

The purpose of the first problem consists of planning a settlement with the most suitable location. There are several papers about this [24 , 34]. In this study, the same problem is included. A group of professors from Vidyasagar University wants to build their homes in a central location. A team is brought together to find the appropriate place. Namely, Senapati Construction Limited (SCL). SCL visited four different places: Ashoke Nagar (S₁), Judge Court (S₂), Patna Bazar (S₃) and Kshudiram Nagar (S₄) which are located within 10 kilometers of Vidyasagar University. These locations should be evaluated according to certain criteria. Criteria: lifestyle and neighbors (C₁), soil type (C₂), size, shape, orientation and slope of the block of land (C₃), existing roads and access to essential services (C₄), cost (C₅). The weights of the criteria were determined as w = (0.2, 0.2, 0.1, 0.3, 0.2) ^T. The Fermatean fuzzy decision matrix created by the SCL is as in Table 1.

Table 1
Fermatean fuzzy decision matrix

C ₁ C ₂ C ₃ C ₄ C ₅

S ₁ (0.7, 0.3) (0.4, 0.6) (0.5, 0.5) (0.8, 0.2) (0.8, 0.4)

S ₂ (0.5, 0.8) (0.8, 0.6) (0.4, 0.5) (0.7, 0.4) (0.6, 0.5)

S ₃ (0.9, 0.6) (0.8, 0.1) (0.6, 0.4) (0.7, 0.5) (0.9, 0.3)

S ₄ (0.6, 0.7) (0.8, 0.3) (0.7, 0.2) (0.5, 0.3) (0.7, 0.3)

	C ₁	C ₂	C ₃	C ₄	C ₅
S ₁	(0.7, 0.3)	(0.4, 0.6)	(0.5, 0.5)	(0.8, 0.2)	(0.8, 0.4)
S ₂	(0.5, 0.8)	(0.8, 0.6)	(0.4, 0.5)	(0.7, 0.4)	(0.6, 0.5)
S ₃	(0.9, 0.6)	(0.8, 0.1)	(0.6, 0.4)	(0.7, 0.5)	(0.9, 0.3)
S ₄	(0.6, 0.7)	(0.8, 0.3)	(0.7, 0.2)	(0.5, 0.3)	(0.7, 0.3)

Table 2

Normalized Fermatean fuzzy decision matrix

	C ₁	C ₂	C ₃	C ₄	C ₅
S ₁	(0.7, 0.3)	(0.4, 0.6)	(0.5, 0.5)	(0.8, 0.2)	(0.4, 0.8)
S ₂	(0.5, 0.8)	(0.8, 0.6)	(0.4, 0.5)	(0.7, 0.4)	(0.5, 0.6)
S ₃	(0.9, 0.6)	(0.8, 0.1)	(0.6, 0.4)	(0.7, 0.5)	(0.3, 0.9)
S ₄	(0.6, 0.7)	(0.8, 0.3)	(0.7, 0.2)	(0.5, 0.3)	(0.3, 0.7)

5.2.1 Experiment results with comparison analysis

In this section, results are given for different Dombi parameters. As the Dombi parameter changes, it is observed how it affects the decision-making process. As a result of changing Dombi parameters, it is realized that it is stable after a certain value. In this study, the values of β up to 50 were investigated. But it seems to be stable before the value 50. This parameter value is only chosen for the convenience of appearance in the figures. Each result is also compared using an effective relative closeness method. Also, the order obtained in is as [25, 34] general ranking S₃ > S₁ > S₄ > S₂. Considering that revised closeness [13] is more accurate for comparison. On the other hand, as mentioned in the studies of Jana and Senapati et. al. [17] Dombi operator has been found to give appropriate results. The results obtained for the Dombi operators were represented by figures.

The results obtained from Figs. 1., 2. and 3 are shown in Table 3. In terms of simplicity: S₁ = 1, S₂ = 2, S₃ = 3, S₄ = 4.

Fig. 1

FFDWA (a) and FFDWG (b) aggregations results with different beta parameter.

Fig. 2

FFDOWA (a) and FFDOWG (b) aggregations results with different beta parameter.

Fig. 3

FFDHWA (a) and FFDHWG (b) aggregations results with different beta parameter.

Considering Table 3, we can observe the effect of the Dombi (beta) parameter for β = 50. After beta greater than 20, revised closeness values are observed to be stable. Because of the large variability in small values of beta, large values of beta are also taken into consideration. The optimal ranking order of the four places is S₃ > S₁ > S₄ > S₂ and the best alternative is S₃. Considering all the existing aggregation functions on FFSs, the Dombi produces consistent and effective results.

Table 3

Dombi and Existing Aggregation Methods

	ξ (S₁)	ξ (S₂)	ξ (S₃)	ξ (S₄)	Rank
FFWA	–0.469	–0.375	–0.347	–0.440	3≻2≻4≻1
FFWG	–0.415	–0.378	–0.350	–0.371	3≻4≻2≻1
FFWPA	–0.472	–0.364	–0.348	–0.476	3≻2≻1≻4
FFWPG	–0.444	–0.356	–0.317	–0.468	3≻2≻1≻4
FFDWA	–0.131	–0.167	–0.061	–0.148	3≻1≻4≻2
FFDWG	–0.232	–0.145	–0.609	–0.056	4≻2≻1≻3
FFDOWA	–0.165	–0.194	–0.095	–0.200	3≻1≻2≻4
FFDOWG	–0.192	–0.142	–0.671	–0.122	4≻2≻1≻3
FFDHWA	–0.124	–0.172	–0.052	–0.146	3≻1≻4≻2
FFDHWG	–0.411	–0.531	0	–0.502	3≻1≻4≻2

Table 4 shows a comparison of different methods. Among studies, it is seen that the Dombi operator gives consistent results on FFSs.

Table 4

Comparison of different methods

Approach	Ranking Result
Senapati and Yager[33]	3≻2≻1≻4
Senapati and Yager[34]	3≻1≻4≻2
Liu et. al.[24]	3≻1≻4≻2
Dombi Aggregation
FFDWA, FFDHWG, FFDHWA	3≻1≻4≻2
FFDWG, FFDOWG	4≻2≻1≻3
FFDOWA	3≻1≻2≻4

5.3 Problem definition II

The subject of the second problem consists of the selection of bridge construction. Bridge data information explored the construction of the concrete-based bridge superstructure for the Suhua Highway Alternative Road Project in Taiwan. Chen/Wang and Chen focus on this problem in many studies [5–7 , 38]. On the other hand, with FFSs for selection of bridge construction solution searched [35]. There are four commonly used methods for bridge construction. Methods: the advanced shoring method (S₁), the incremental launching method (S₂), the balanced cantilever method (S₃) and the precast segmental method (S₄). Eight criteria were determined to evaluate the methods of bridge construction. Consisting of durability (C₁), damage cost (C₂), construction cost (C₃), traffic effect (C₄), site condition (C₅), climatic condition (C₆), landscape (C₇) and environmental impact (C₈). Durability and site conditions are benefit types, remaining criteria are cost type. Table 5 contains the data.

Table 5
Fermatean fuzzy decision matrix

S ₁ S ₂ S ₃ S ₄

C ₁ (0.38, 0.52) (0.73, 0.28) (0.78, 0.23) (0.57, 0.44)

C ₂ (0.62, 0.22) (0.56, 0.87) (0.08, 0.81) (0.35, 0.65)

C ₃ (0.24, 0.80) (0.25, 0.47) (0.21, 0.65) (0.42, 0.57)

C ₄ (0.51, 0.47) (0.25, 0.71) (0.46, 0.48) (0.40, 0.51)

C ₅ (0.68, 0.35) (0.39, 0.33) (0.65, 0.34) (0.29, 0.63)

C ₆ (0.16, 0.75) (0.23, 0.79) (0.38, 0.45) (0.86, 0.61)

C ₇ (0.25, 0.42) (0.68, 0.29) (0.21, 0.75) (0.17, 0.87)

C ₈ (0.11, 0.90) (0.40, 0.31) (0.38, 0.63) (0.23, 0.76)

	S ₁	S ₂	S ₃	S ₄
C ₁	(0.38, 0.52)	(0.73, 0.28)	(0.78, 0.23)	(0.57, 0.44)
C ₂	(0.62, 0.22)	(0.56, 0.87)	(0.08, 0.81)	(0.35, 0.65)
C ₃	(0.24, 0.80)	(0.25, 0.47)	(0.21, 0.65)	(0.42, 0.57)
C ₄	(0.51, 0.47)	(0.25, 0.71)	(0.46, 0.48)	(0.40, 0.51)
C ₅	(0.68, 0.35)	(0.39, 0.33)	(0.65, 0.34)	(0.29, 0.63)
C ₆	(0.16, 0.75)	(0.23, 0.79)	(0.38, 0.45)	(0.86, 0.61)
C ₇	(0.25, 0.42)	(0.68, 0.29)	(0.21, 0.75)	(0.17, 0.87)
C ₈	(0.11, 0.90)	(0.40, 0.31)	(0.38, 0.63)	(0.23, 0.76)

The matrix in Table 5 is taken from [35]. Furthermore, as in step 1, the normalized version is in Table 6. C₁ and C₅ benefit type, C₂, C₃, C₄, C₆, C₇ and C₈ cost type.

Table 6

Normalized Fermatean fuzzy decision matrix

	S ₁	S ₂	S ₃	S ₄
C ₁	(0.38, 0.52)	(0.73, 0.28)	(0.78, 0.23)	(0.57, 0.44)
C ₂	(0.22, 0.62)	(0.87, 0.56)	(0.81, 0.08)	(0.65, 0.35)
C ₃	(0.80, 0.24)	(0.47, 0.25)	(0.65, 0.21)	(0.57, 0.42)
C ₄	(0.47, 0.51)	(0.71, 0.25)	(0.48, 0.46)	(0.51, 0.40)
C ₅	(0.68, 0.35)	(0.39, 0.33)	(0.65, 0.34)	(0.29, 0.63)
C ₆	(0.75, 0.16)	(0.79, 0.23)	(0.45, 0.38)	(0.61, 0.86)
C ₇	(0.42, 0.25)	(0.29, 0.68)	(0.75, 0.21)	(0.87, 0.17)
C ₈	(0.90, 0.11)	(0.31, 0.40)	(0.63, 0.38)	(0.76, 0.23)

Let C = {C₁, C₂, C₃, C₄, C₅, C₆, C₇, C₈} be the set of criteria and let S ={ S₁, S₂, S₃, S₄ } be the set of alternatives. The weights of the criteria are given as w = (0.142875, 0.059524, 0.214251, 0.142875, 0.119048, 0.059524, 0.166665) ^T.

5.3.1 Experiment results with comparison analysis

The Dombi parameter has different effects on arithmetic and geometric aggregations. FFSs, with order and hybrid of aggregation methods are expressed more accurately. Furthermore, in the FFDWA and FFDWG operators, weights are used according to the fuzzy values given. In the FFDOWA and FFDOWG operators, weights are used in the order of the Fermatean fuzzy values. But in both cases, they use their own weight. On the other hand, FFDHWA and FFDHWG reflect the order positions and degree of given Fermatean fuzzy values (Score -based).

FFSs, interval IFSs, interval PFSs are applied to the bridge construction selection problem [5–10 , 38]. The results were also compared in this study. The applicability of Dombi aggregation methods in decision-making mechanisms is presented with comparisons. In the following figures, the revised closeness values for each Dombi aggregation according to the parameters are given

Although rare fluctuations appear in arithmetic operators, geometric operators appear to be stable. This situation depends on the Dombi parameter. While arithmetic operators depend on Dombi parameter, it can be said that geometric operators are more independent on parameter Dombi parameter.

Table 7 shows the results for β = 50. In all methods, a single ranking was included since the ranking resulting from the closeness average of β values and the ranking for β = 50 are the same but except for FFDWG. The closeness average value for FFDWG is given. Considering Fig. 4a, when β ∈ [1, 2], then ranking order S₁ > S₃ > S₂ > S₄. After β ⩾ 3, it is seen to be stable and S₁ > S₄ > S₂ > S₃ ranking is obtained. For FFDWA the best alternative is S₁. In Fig. 4b, when β ∈ [1, 27], S₃ > S₂ > S₄ > S₁, for remaining β values S₂ > S₃ > S₄ > S₁. But considering all β values, the average result is ranking S₃ > S₂ > S₄ > S₁. So, for FFDWG total ranking order S₃ > S₂ > S₄ > S₁ obtained. So, for FFDWG the best alternative is S₃. According to Fig. 5a, when β ∈ [1, 2], then ranking order S₃ > S₁ > S₂ > S₄. For β ∈ [5, 7], ranking order S₄ > S₁ > S₂ > S₃. But when β ∈ [3, 4] and β ∈ [8, 50] ranking order S₁ > S₄ > S₂ > S₃ obtained. So, For FFDOWA the best alternative is S₁. According to Fig. 5b, For β ∈ [1, 4], ranking order S₃ > S₂ > S₄ > S₁. When β ∈ [5, 50] ranking order S₃ > S₂ > S₁ > S₄ obtained. Hence, for FFDOWG the best alternative is S₃. Considering Fig. 6a, ordering varies at lower values of β. Therefore, the best alternative ranges are given. When β ∈ [1, 2], the best alternative is S₃, for β ∈ [3, 50], the best alternative is S₁. So, for FFDHWA total ranking order S₁ > S₄ > S₂ > S₃ and the best alternative is S₁. Finally, Considering Fig. 6b, When β ∈ [1, 2], ranking is S₃ > S₁ > S₂ > S₄ and for β = 3 ranking is S₁ > S₄ > S₂ > S₃. When β ∈ [4, 50], the total ranking order is S₁ > S₃ > S₄ > S₂. Hence, for FFDHWG the best alternative S₁ obtained. As a result, the best alternatives are S₁ or S₃. Considering all Dombi operators, the best alternative is S₁. The proposed method is compared with other studies in Table 8. Ranking results are consistent with other existing methods. On the other hand, it can be said that arithmetic operators are more stable in itself. However, geometric operators depend much less on the Dombi parameter.

Table 7
Dombi and existing aggregation methods

ξ (s₁) ξ (s₂) ξ (s₃) ξ (s₄) Rank

FFWA –1.305 –1.175 –0.429 –1.477 3 ≻ 2 ≻1 ≻ 4

FFWG –1.271 –1.176 –0.406 –1.351 3 ≻ 2 ≻1 ≻ 4

FFWPA –0.799 –0.879 –0.376 –1.237 3 ≻ 1 ≻2 ≻ 4

FFWPG –0.533 –0.705 –0.338 –1.080 3 ≻ 1 ≻2 ≻ 4

FFDWA –0.322 –0.469 –0.489 –0.434 1 ≻ 4 ≻2 ≻ 3

FFDWG –0.568 –0.136 –0.103 –0.383 3 ≻ 2 ≻4 ≻ 1

FFDOWA –0.277 –0.362 –0.424 –0.297 1 ≻ 4 ≻2 ≻ 3

FFDOWG –0.174 –0.158 0 –0.269 3 ≻ 2 ≻1 ≻ 4

FFDHWA –0.321 –0.474 –0.497 –0.433 1 ≻ 4 ≻2 ≻ 3

FFDHWG –0.207 –0.346 –0.273 –0.288 1 ≻ 3 ≻4 ≻ 2

	ξ (s₁)	ξ (s₂)	ξ (s₃)	ξ (s₄)	Rank
FFWA	–1.305	–1.175	–0.429	–1.477	3 ≻ 2 ≻1 ≻ 4
FFWG	–1.271	–1.176	–0.406	–1.351	3 ≻ 2 ≻1 ≻ 4
FFWPA	–0.799	–0.879	–0.376	–1.237	3 ≻ 1 ≻2 ≻ 4
FFWPG	–0.533	–0.705	–0.338	–1.080	3 ≻ 1 ≻2 ≻ 4
FFDWA	–0.322	–0.469	–0.489	–0.434	1 ≻ 4 ≻2 ≻ 3
FFDWG	–0.568	–0.136	–0.103	–0.383	3 ≻ 2 ≻4 ≻ 1
FFDOWA	–0.277	–0.362	–0.424	–0.297	1 ≻ 4 ≻2 ≻ 3
FFDOWG	–0.174	–0.158	0	–0.269	3 ≻ 2 ≻1 ≻ 4
FFDHWA	–0.321	–0.474	–0.497	–0.433	1 ≻ 4 ≻2 ≻ 3
FFDHWG	–0.207	–0.346	–0.273	–0.288	1 ≻ 3 ≻4 ≻ 2

Fig. 4

FFDWA (a) and FFDWG (b) aggregations results with different beta parameter.

Fig. 5

FFDOWA (a) and FFDOWG (b) aggregations results with different beta parameter.

Fig. 6

FFDHWA (a) and FFDHWG (b) aggregations results with different beta parameter.

Table 8

Comparison of different methods

Source	Methods	Evaluative rating	Criterion weight	Ranking result	Model category
Chen[5]	Nonlinear assignment-based method	IVIF values	Incomplete information	1 ≻ 3 ≻4 ≻ 2	Outranking model
	IVIF QUALIFLEX method	IVIF values	Incomplete information	4 ≻ 3 ≻1 ≻ 2	Outranking model
Chen[7]		IVIF values	Incomplete and inconsistent information	2 ≻ 3 ≻1 ≻ 4	Outranking model
Chen[6]	Extended linear assignment method	IVIF values	IVIF values	1 ≻ 3 ≻4 ≻ 2	Outranking model
Wang and Chen [38]	Likelihood-based assignment method with Algorithm A	IVIF values	IVIF values	1 ≻ 3 ≻2 ≻ 4	Outranking model
	Likelihood-based assignment method with Algorithm B	IVIF values	Incomplete information	1 ≻ 3 ≻2 ≻ 4	Outranking model
Chen[10]	IVPF outranking method	IVPF values	IVIF values	1 ≻ 3 ≻4 ≻ 2	Outranking model
	IVPF TOPSIS method with respect to displaced ideals	IVPF values	Ordinary (crisp and nonfuzzy) values	3 ≻ 2 ≻1 ≻ 4	Compromise model
	IVPF TOPSIS method with respect to fixed ideals	IVPF values	Ordinary (crisp and nonfuzzy) values	3 ≻ 1 ≻2 ≻ 4	Compromise model
Chen[8]	IVPF compromise approach with Algorithm I	IVPF values	Ordinary (crisp and nonfuzzy) values	3 ≻ 1 ≻2 ≻ 4	Compromise model
	IVPF compromise approach with Algorithm II	IVPF values	Ordinary (crisp and nonfuzzy) values	3 ≻ 1 ≻2 ≻ 4	Compromise model
Senapati and Yager [35]	Weighted product model	FF values	Ordinary (crisp and nonfuzzy) values	1 ≻ 3 ≻2 ≻ 4	Compromise model
The current paper	Dombi aggregation methods (FFDWA, FFDOWA, FFDHWA)	FF values	Ordinary (crisp and nonfuzzy) values	1 ≻ 4 ≻2 ≻ 3	Compromise model
	FFDWG	FF values	Ordinary (crisp and nonfuzzy) values	3 ≻ 2 ≻4 ≻ 1	Compromise model
	FFDOWG	FF values	Ordinary (crisp and nonfuzzy) values	3 ≻ 2 ≻1 ≻ 4	Compromise model
	FFDHWG	FF values	Ordinary (crisp and nonfuzzy) values	1 ≻ 3 ≻4 ≻ 2	Compromise model

6 Conclusions and future work

In this study, it is aimed to construct a more sensitive structure by using Dombi operators on FFSs. These operators are FFDWA, FFDWG, FFDOWA, FFDOWG, FFDHWA, FFDHWG operators. The changing beta parameter of Dombi operator provides flexibility. After beta greater than 20, beta values are observed to be stable. In addition, the idempotency, boundedness and monotonicity properties of Dombi operators on FFSs are mentioned. Finally, the TOPSIS method was extended with FFSs. Since FFSs are the extended version of intuitionistic fuzzy sets and Pythagorean fuzzy sets, FFSs can be said to give more flexible and more consistent results. In future studies, FFSs can be hybridized with different MCDM methods. Such as, Multimoora method [3], Vikor method [32]. On the other hand, FFSs with Dombi operator can be applied in several areas such as supplier selection, airline company and recommender system. Furthermore, the linguistic sets of FFSs sets are mentioned in the study [24]. As a next study, taking into account the criteria-based scores given to airline companies, linguistic variables with Dombi aggregation FFSs can be created and the best airline company can be selected according to the criteria.

Conflicts of interest

The authors declare that there is no conflict of interest regarding the publication of this paper.

Footnotes

Acknowledgments

This work is supported by scientific research project of Eskisehir Technical University under Grant 20DRP041.

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Fermatean fuzzy TOPSIS method with Dombi aggregation operators and its application in multi-criteria decision making

Abstract

Keywords

1 Introduction

2 Preliminaries

3.1 Fermatean fuzzy Dombi averaging aggregation operators

4 Modified distance method for Fermatean fuzzy sets

5.1 Algorithms of the proposed method

5.2 Problem definition I

Table 1 Fermatean fuzzy decision matrix C 1 C 2 C 3 C 4 C 5 S 1 (0.7, 0.3) (0.4, 0.6) (0.5, 0.5) (0.8, 0.2) (0.8, 0.4) S 2 (0.5, 0.8) (0.8, 0.6) (0.4, 0.5) (0.7, 0.4) (0.6, 0.5) S 3 (0.9, 0.6) (0.8, 0.1) (0.6, 0.4) (0.7, 0.5) (0.9, 0.3) S 4 (0.6, 0.7) (0.8, 0.3) (0.7, 0.2) (0.5, 0.3) (0.7, 0.3)

Conflicts of interest

Footnotes

Acknowledgments

References