Multi-Q Fermatean Hesitant Fuzzy Soft Sets and Their Application in Decision-Making

Abstract

The concept of Multi Q-Fermatean hesitant fuzzy soft sets (MQFHFSS), derived from the integration of multi-Q fuzzy soft sets and Fermatean hesitant fuzzy sets, can be applied in practice to optimise the resolution of complex multi-criteria decision-making problems. The method exceeds traditional approaches such as Fermatean hesitant fuzzy sets, fuzzy soft sets, and Pythagorean fuzzy sets in enhancing the ability to capture higher levels of uncertainty, hesitation, and symmetry in multi-criteria evaluations, thereby supporting more balanced judgments in complex decision-making situations. In this study, we investigate the novel MQFHFSS concept along with the associated operations. The fundamental characteristics of aggregation operators derived from MQFHFSS have been examined to address some complex decision-making issues. Moreover, we discuss some key algebraic features and their different cases, emphasizing the role of symmetry under the influence of MQFHFSS. Finally, we illustrate some numerical examples and solve the real-world decision-making problem by using the proposed technique.

1. Introduction

Most contemporary issues [1] in health, engineering as a career, and social disciplines involve information that is inaccurate and unclear due to ambiguity. Techniques used in classical mathematics have become unproductive due to the involvement of various types of uncertainty. In 1995, Zadeh [2] established the concept of making linguistic terms computable, that is, the concept of the fuzzy set (FS), explaining that its basic component is the membership degree ($\mathcal{MD}$) belonging to the range [0, 1]. However, this does not clarify the confusion in many matters. Atanassov [3] expanded the idea of the fuzzy set to the intuitionistic fuzzy set ($\mathcal{IFS}$) [4], containing a degree of ($\mathcal{MD}$) and degree of non-membership function (ND). They belong to the range [0, 1], provided that the sum of the two grades does not exceed one. Yager proposed an eternalized form named the Pythagorean fuzzy set ($\mathcal{PFS}$) [5] under the concept that the sum of the squares of the degree of $\mathcal{MD}$ and $\mathcal{ND}$ is less than or equal to one. The Fermatean fuzzy set ($\mathcal{FFS}$) [6] was proposed by [7] as a superior approach to $\mathcal{IFS}$ and $\mathcal{PFS}$; here, the cubic sum of $\mathcal{MD}$ and $\mathcal{ND}$ is less than or equal to one. Since 2009, researchers have begun to investigate whether the $\mathcal{MD}$ and $\mathcal{ND}$ of a set is a subset of [0, 1]. Hence, lTora and Narukawa [8] established the initial approach of the $\mathcal{HFS}$, which we benefit from in cases of hesitation in determining the $\mathcal{MD}$ or $\mathcal{ND}$ of an element in a group. The concept of the $\mathcal{FFS}$ has been generalized to $\mathcal{HFS}$ [9,10] to study hesitant cases under the conditions of $\mathcal{FFS}$, which can more precisely represent individuals’ hesitance to articulate their preferences through item selection. The concept of the multi-Q $\mathcal{FS}$ integrated with the Fermatean hesitant set [7,11,12,13] provides a more accurate and flexible representation of uncertainties. However, each of these sets has its own difficulties; thus, Adam and Hassan [14] proposed the soft set [15] as a new difficulty-free concept for dealing with uncertainty, characterized by the absence of any condition to describe the objects as in the fuzzy set. This makes it easier for the researcher to choose the parameter needed to make the decision. Adam et al. introduced the novel notion of the Q-fuzzy soft set ($\mathcal{FSS}$), which combines the Q-FS [16] and the soft set. They then developed this into the multi-Q $\mathcal{SFS}$ [17,18,19], explored its features, and used it in decision-making [20]. Korucuk and Aytekin’s use of interval-valued Fermatean fuzzy sets within the SWARA framework enables a more precise and flexible evaluation of logistics criteria and enhances decision-making power [21]. Additionally, it allows for a thorough evaluation of anti-corruption strategies in the face of uncertainty. This method was successful in prioritising governance reforms that are vital to increasing transparency and accountability [22,23].

While some decisions are made intuitively, complex situations often require more rigorous approaches. Hence, decision-making processes arise when a person or institution must select the most appropriate option under given conditions. In this context, the present study introduces a combination of the multi-Q $\mathcal{FSS}$ and $\mathcal{FHFS}$, incorporating symmetry in their algebraic structures to enhance the effectiveness of decision-making.

• Motivation and contribution:

  1.

  Yager [7] developed the FS concept. The primary stipulation in this set theory is that the aggregate of $\mathcal{MD}$ and $\mathcal{ND}$ must not surpass 1. Building on this idea, we delineate and investigate the various characteristics of $\mathcal{FFS}$.

  2.

  Torra [24] expanded the $\mathcal{FS}$ notion to incorporate the hesitant $\mathcal{FS}$ model. The challenge in constructing the $\mathcal{MD}$ can be addressed when it stems from indecision among certain values instead of from a margin of error or a particular allocation of chance of the probable values [25]. Adam provided the initial MQFSS concept and discussed the application of such sets in decision-making.

  3.

  Our proposed $\mathcal{MQFHFSS}$ concept is derived from the integration of $\mathcal{MQFSS}$ and $\mathcal{FHFS}$, and can be applied in practice to streamline the resolution of complicated multi-criteria $\mathcal{DM}$ challenges. The fundamental characteristics of aggregation operators derived from $\mathcal{MQFHFSS}$ have been examined, highlighting their inherent symmetry; in addition, numerical examples are illustrated to solve real-world decision-making problems using the proposed technique.

The thesis is arranged as follows. In Section 2, we establish the definition of the multi-$\mathcal{FHFS}$ and operations on it as well as its properties. In Section 3, we expand on $\mathcal{MQFHFSS}$, including the concept and many of its theories and propositions. Finally, in Section 4 we focus on applications of $\mathcal{MQFHFSS}$ in $\mathcal{DM}$ processes, explaining the results that can be achieved using this set.

2. Preliminaries

Definition 1 

([11]). The following set

$$F_H=\left\{(u,U_{F_H}\left(u\right),V_{F_H}\left(u\right))\mid u\in U\right\}$$

is known as $\mathcal{FHFS}$, where:

1. 

Every entity $u\in U$, $U_{F_H}\left(u\right)$, $V_{F_H}\left(u\right)$ is transformed from U to $[0, 1]$, illustrating a realistic $\mathcal{MD}$ and $\mathcal{ND}$ of entity $u\in U$ in $F_H$, respectively.

2. 

$\forall {\mu }_{F_H}\left(u\right)\in U_{F_H}\left(u\right)$, $\exists v_{F_H}\left(u\right)\in V_{F_H}\left(u\right)$, such that

$$0\le {\mu }_{F_H}^3\left(u\right)+v_{F_H}^3\left(u\right)\le 1$$

.

3. 

$\forall v_{F_H}\left(u\right)\in V_{F_H}\left(u\right)$, $\exists {\mu }_{F_H}\left(u\right)\in U_{F_H}\left(u\right)$; therefore,

$$0\le {\mu }_{F_H}^3\left(u\right)+v_{F_H}^3\left(u\right)\le 1$$

.

Definition 2 

([15]). A pair $(F,A)$ is known as a soft set across U, while F is a transformation. In other words, a soft-set over U is a function from attributes to $\mathcal{P}\left(U\right)$, which is the power set of U.

Definition 3.

Suppose that X is a fixed set and $Q\ne \varnothing$. Then, a Q-$\mathcal{FHFS}$  F in X can be described as follows:

$$F=\{⟨(\mathfrak{m},\varrho ),U(\mathfrak{m},\varrho ),V(\mathfrak{m},\varrho )⟩\mid \mathfrak{m}\in X,\varrho \in Q\}.$$

Therefore, $U(\mathfrak{m},\varrho )$ and $V(\mathfrak{m},\varrho )$ are functions from $X\times Q$ to $[0,1]$ and denote the set of alternatives $\mathcal{MD}$ and ND of X to F, respectively:

$$\forall \mu (\mathfrak{m},\varrho )\in U(\mathfrak{m},\varrho ),\exists v(\mathfrak{m},\varrho )\in V(\mathfrak{m},\varrho )\mathit{and}\forall v(\mathfrak{m},\varrho )\in V(\mathfrak{m},\varrho ),$$

$$\exists \mu (\mathfrak{m},\varrho )\in U(\mathfrak{m},\varrho )\mathrm{s}.\mathrm{t}.0\le {\mu }^3(\mathfrak{m},\varrho )+v^3(\mathfrak{m},\varrho )\le 1.$$

3. Multi-Q Fermatean Hesitant Fuzzy Set

Definition 4.

Suppose that I is an interval $[0, 1]$, k is a positive integer, X is a set of discourse, and Q is a non-empty set.

Then, an M^K $\mathcal{QFHFS}$  $F_Q$ can be described as follows:

$$F_Q=\{⟨(\mathfrak{m},\varrho ),U(\mathfrak{m},\varrho ),V(\mathfrak{m},\varrho )⟩\mid (\mathfrak{m}\in X,\varrho \in Q\}$$

where $U(\mathfrak{m},\varrho )$ and $V(\mathfrak{m},\varrho )\ne \varnothing$ are finite subsets on $[0, 1]$ and denote all existing $\mathcal{MD}$ and $\mathcal{ND}$, respectively, $\forall {\mu }_i(\mathfrak{m},\varrho )\in U(\mathfrak{m},\varrho )$, $\exists v_i(\mathfrak{m},\varrho )\in V(\mathfrak{m},\varrho )$ and $\forall v_i(\mathfrak{m},\varrho )\in V(\mathfrak{m},\varrho )$, $\exists {\mu }_i(\mathfrak{m},\varrho )\in U(\mathfrak{m},\varrho )s.t$.

Here, ${\mu }_i:X\times Q\to I^k$ are the $\mathcal{MD}$s of the multi-Q $\mathcal{FHFS}$ $F_Q$, while $v_i:X\times Q\to I^k$ are the $\mathcal{ND}$s of the multi-Q $\mathcal{FHFS}$ $F_Q$ satisfying $0\le {\mu }_i^3(\mathfrak{m},\varrho )+v_i^3(\mathfrak{m},\varrho )\le 1$ for $i=1,...,k$, where k has a dimension of $F_Q$.

Example 1.

Let $X=\{x_1,x_2,x_3,x_4\}$ be set of discourse, $Q=\{q,p\}\ne \varnothing$, and $k=2$ be a positive integer. If $F_Q$ is a mapping from

$$X\times Q\to I^2,\mathit{then}\mathit{the}\mathit{set}$$

$$F_Q=\left\{\begin{array}{c}\left((x_1,q),\left(\{0.31,0.51\}\right),\left(\{0.15,0.42\}\right)\right)\hfill \\ \left((x_3,p),\left(\{0.14,0.22\}\right),\left(\{0.12,0.14\}\right)\right)\hfill \end{array}\right\}$$

is a multi-Q $\mathcal{FHFS}$.

3.1. Operations on Multi-Q Fermatean Hesitant Fuzzy Sets

Let $\Theta$ and $\Phi$ be two $M^k$-$\mathcal{QFHFS}$ and let $\lambda >0$. Then:

(a)

$\Theta \cup \Phi =\left\{⟨(x,\varsigma ),max({\mu }_{i\Theta }(x,\varsigma ),{\mu }_{i\Phi }(x,\varsigma )),min(v_{i\Theta }(x,\varsigma ),v_{i\Phi }(x,\varsigma ))⟩\right\}$

(b)

$\Theta \cap \Phi =\left\{⟨(x,\varsigma ),min({\mu }_{i\Theta }(x,\varsigma ),{\mu }_{i\Phi }(x,\varsigma )),max(v_{i\Theta }(x,\varsigma ),v_{i\Phi }(x,\varsigma ))⟩\right\}$

(c)

$\lambda \Theta =⟨{\bigcup }_{{\mu }_i\in U_{\Theta }}\left\{\sqrt[3]{1-{(1-{\mu }_i^3)}^{\lambda }}\right\},{\bigcup }_{v_i\in V_{\Theta }}\left\{v_i^{\lambda }\right\}⟩$

(d)

${\Theta }^{\lambda }=⟨{\bigcup }_{{\mu }_i\in U_{\Theta }}\left\{{\mu }_i^{\lambda }\right\},{\bigcup }_{v_i\in V_{\Theta }}\left\{\sqrt[3]{1-{(1-v_i^2)}^{\lambda }}\right\}⟩$.

Theorem 1.

Suppose that $F_Q$ is an $M^k$-$\mathcal{QFHFS}$; then, the complement of $F_Q$ is represented by $F_Q^c$ and defined as follows:

$$\begin{array}{cc}\hfill F_Q^c& =\left\{(\mathfrak{m},\mathfrak{k}),U^c(\mathfrak{m},\mathfrak{k}),V^c(\mathfrak{m},\mathfrak{k})\right\}\hfill \\ & =\left\{(\mathfrak{m},\mathfrak{k}),\bigcup _{v_i(\mathfrak{m},\mathfrak{k})\in V(\mathfrak{m},\mathfrak{k})}\left\{v_i(\mathfrak{m},\mathfrak{k})\right\},\bigcup _{{\mu }_i(\mathfrak{m},\mathfrak{k})\in U(\mathfrak{m},\mathfrak{k})}\left\{{\mu }_i(\mathfrak{m},\mathfrak{k})\right\}\right\}\hfill \end{array}$$

such that

$$0\le {\left({\mu }_i{(\mathfrak{m},\mathfrak{k})}^3,v_i(\mathfrak{m},\mathfrak{k})\right)}^3\le 1.$$

Proof. 

$$\begin{array}{cc}\hfill F_Q& =\left\{⟨(\mathfrak{m},\mathfrak{k}),U(\mathfrak{m},\mathfrak{k}),V(\mathfrak{m},\mathfrak{k})⟩\right\}\hfill \\ & ={\left\{⟨(\mathfrak{m},\mathfrak{k}),\bigcup _{{\mu }_i(\mathfrak{m},\mathfrak{k})\in U(\mathfrak{m},\mathfrak{k})}\left\{{\mu }_i(\mathfrak{m},\mathfrak{k})\right\},\bigcup _{v_i(\mathfrak{m},\mathfrak{k})\in V(\mathfrak{m},\mathfrak{k})}\left\{v_i(\mathfrak{m},\mathfrak{k})\right\}⟩\right\}}^c\hfill \\ & =\left\{⟨(\mathfrak{m},\mathfrak{k}),\bigcup _{v_i(\mathfrak{m},\mathfrak{k})\in V(\mathfrak{m},\mathfrak{k})}\left\{v_i(\mathfrak{m},\mathfrak{k})\right\},\bigcup _{{\mu }_i(\mathfrak{m},\mathfrak{k})\in U(\mathfrak{m},\mathfrak{k})}\left\{{\mu }_i(\mathfrak{m},\mathfrak{k})\right\}⟩\right\}\hfill \\ & =F_Q^c\hfill \end{array}$$

□

Definition 5.

Let Θ and Φ be two multi-Q $\mathcal{FHFS}$; then:

(1) 

$\Theta \oplus \Phi$ is defined as

$$\begin{array}{cc}\hfill \Theta \oplus \Phi & =⟨\bigcup _{{\mu }_{i_{\Theta }}(\mathfrak{m},\mathfrak{k})\in U_{\Theta }(\mathfrak{m},\mathfrak{k}),{\mu }_{i_{\Phi }}(\mathfrak{m},\mathfrak{k})\in U_{\Phi }(\mathfrak{m},\mathfrak{k})}\left\{\sqrt[3]{{\mu }_{i_{\Theta }}^3+{\mu }_{i_{\Phi }}^3-{\mu }_{i_{\Theta }}^3{\mu }_{i_{\Phi }}^3}\right\},\hfill \\ & \bigcup _{v_{i_{\Theta }}(\mathfrak{m},\mathfrak{k})\in V_{\Theta }(\mathfrak{m},\mathfrak{k}),v_{i_{\Phi }}(\mathfrak{m},\mathfrak{k})\in V_{\Phi }(\mathfrak{m},\mathfrak{k})}\left\{v_{i_{\Theta }}{v}_{i_{\Phi }}\right\}⟩.\hfill \end{array}$$

(2) 

$\Theta \otimes \Phi$ is defined as

$$\begin{array}{cc}\hfill \Theta \otimes \Phi & =⟨\bigcup _{{\mu }_{i_{\Theta }}(\mathfrak{m},\mathfrak{k})\in U_{\Theta }(\mathfrak{m},\mathfrak{k}),{\mu }_{i_{\Phi }}(\mathfrak{m},\mathfrak{k})\in U_{\Phi }(\mathfrak{m},\mathfrak{k})}\left\{{\mu }_{i_{\Theta }}{\mu }_{i_{\Phi }}\right\},\hfill \\ & \bigcup _{v_{i_{\Theta }}(\mathfrak{m},\mathfrak{k})\in V_{\Theta }(\mathfrak{m},\mathfrak{k}),v_{i_{\Phi }}(\mathfrak{m},\mathfrak{k})\in V_{\Phi }(\mathfrak{m},\mathfrak{k})}\left\{\sqrt[3]{v_{i_{\Theta }}^3+v_{i_{\Phi }}^3-v_{i_{\Theta }}^3{v}_{i_{\Phi }}^3}\right\}⟩.\hfill \end{array}$$

3.2. Properties of Multi-Q Fermatean Hesitant Fuzzy Sets

Proposition 1.

Let Θ and Φ be two M-$\mathcal{QFHFS}$; then:

(1) 

$\Theta \oplus \Phi =\Phi \oplus \Theta$

(2) 

$\Theta \otimes \Phi =\Phi \otimes \Theta$

(3) 

$\lambda (\Theta \oplus \Phi )=\lambda \Theta \oplus \lambda \Phi$, where $\lambda >0$

(4) 

$({\lambda }_1+{\lambda }_2)\Theta ={\lambda }_1\Theta \oplus {\lambda }_2\Theta$, where ${\lambda }_1, {\lambda }_2>0$

(5) 

${(\Theta \otimes \Phi )}^{\lambda }={\Theta }^{\lambda }\otimes {\Phi }^{\lambda }$, where $\lambda >0$

(6) 

${\Theta }^{({\lambda }_1+{\lambda }_2)}={\Theta }^{{\lambda }_1}\otimes {\Theta }^{{\lambda }_2}$, where ${\lambda }_1, {\lambda }_2>0$.

Proof.  

(1)

$$\begin{array}{cc}\hfill \Theta \oplus \Phi & =\left(\bigcup _{{\mu }_{i_{\Theta }}\in U_{\Theta },{\mu }_{i_{\Phi }}\in U_{\Phi }}\left\{\sqrt[3]{{\mu }_{i_{\Theta }}^3+{\mu }_{i_{\Phi }}^3-{\mu }_{i_{\Theta }}^3{\mu }_{i_{\Phi }}^3}\right\},\bigcup _{v_{i_{\Theta }}\in V_{\Theta },v_{i_{\Phi }}\in V_{\Phi }}\left\{v_{i_{\Theta }}{v}_{i_{\Phi }}\right\}\right)\hfill \\ & =\left(\bigcup _{{\mu }_{i_{\Phi }}\in U_{\Phi },{\mu }_{i_{\Theta }}\in U_{\Theta }}\left\{\sqrt[3]{{\mu }_{i_{\Phi }}^3+{\mu }_{i_{\Theta }}^3-{\mu }_{i_{\Phi }}^3{\mu }_{i_{\Theta }}^3}\right\},\bigcup _{v_{i_{\Phi }}\in V_{\Phi },v_{i_{\Theta }}\in V_{\Theta }}\left\{v_{i_{\Phi }}{v}_{i_{\Theta }}\right\}\right)=\Phi \oplus \Theta \hfill \end{array}$$

(2)

$$\begin{array}{cc}\hfill \Theta \otimes \Phi & =\left(\bigcup _{{\mu }_{i_{\Theta }}\in U_{\Theta },{\mu }_{i_{\Phi }}\in U_{\Phi }}\left\{{\mu }_{i_{\Theta }}{\mu }_{i_{\Phi }}\right\},\bigcup _{v_{i_{\Theta }}\in V_{\Theta },v_{i_{\Phi }}\in V_{\Phi }}\left\{\sqrt[3]{v_{i_{\Theta }}^3+v_{i_{\Phi }}^3-v_{i_{\Theta }}^3{v}_{i_{\Phi }}^3}\right\}\right)\hfill \\ & =\left(\bigcup _{{\mu }_{i_{\Phi }}\in U_{\Phi },{\mu }_{i_{\Theta }}\in U_{\Theta }}\left\{{\mu }_{i_{\Phi }}{\mu }_{i_{\Theta }}\right\},\bigcup _{v_{i_{\Phi }}\in V_{\Phi },v_{i_{\Theta }}\in V_{\Theta }}\left\{\sqrt[3]{v_{i_{\Phi }}^3+v_{i_{\Theta }}^3-v_{i_{\Phi }}^3{v}_{i_{\Theta }}^3}\right\}\right)=\Phi \otimes \Theta \hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill \lambda (\Theta \oplus \Phi )& =\lambda \left(\bigcup _{{\mu }_{i_{\Theta }}\in U_{\Theta },{\mu }_{i_{\Phi }}\in U_{\Phi }}\left\{\sqrt[3]{{\mu }_{i_{\Theta }}^3+{\mu }_{i_{\Phi }}^3-{\mu }_{i_{\Theta }}^3{\mu }_{i_{\Phi }}^3}\right\},\bigcup _{v_{i_{\Theta }}\in V_{\Theta },v_{i_{\Phi }}\in V_{\Phi }}\left\{v_{i_{\Theta }}{v}_{i_{\Phi }}\right\}\right)\hfill \\ & =\left(\bigcup _{{\mu }_{i_{\Theta }}\in U_{\Theta },{\mu }_{i_{\Phi }}\in U_{\Phi }}\left\{\sqrt[3]{1-{(1-{\mu }_{i_{\Theta }}^3-{\mu }_{i_{\Phi }}^3+{\mu }_{i_{\Theta }}^3{\mu }_{i_{\Phi }}^3)}^{\lambda }}\right\},\bigcup _{v_{i_{\Theta }}\in V_{\Theta },v_{i_{\Phi }}\in V_{\Phi }}{\left\{v_{i_{\Theta }}{v}_{i_{\Phi }}\right\}}^{\lambda }\right)\hfill \end{array}$$

$$\begin{array}{cc}\hfill \lambda \Theta \oplus \lambda \Phi & =\left(\bigcup _{{\mu }_{i_{\Theta }}\in U_{\Theta }}\left\{\sqrt[3]{1-{(1-{\mu }_{i_{\Theta }}^3)}^{\lambda }}\right\},\bigcup _{v_{i_{\Theta }}\in V_{\Theta }}{\left\{v_{i_{\Theta }}\right\}}^{\lambda }\right)\oplus \hfill \\ & \left(\bigcup _{{\mu }_{i_{\Phi }}\in U_{\Phi }}\left\{\sqrt[3]{1-{(1-{\mu }_{i_{\Phi }}^3)}^{\lambda }}\right\},\bigcup _{v_{i_{\Phi }}\in V_{\Phi }}{\left\{v_{i_{\Phi }}\right\}}^{\lambda }\right)\hfill \\ & =\left(\bigcup _{{\mu }_{i_{\Theta }}\in U_{\Theta },{\mu }_{i_{\Phi }}\in U_{\Phi }}\left\{\sqrt[3]{1-{(1-{\mu }_{i_{\Theta }}^3)}^{\lambda }{(1-{\mu }_{i_{\Phi }}^3)}^{\lambda }}\right\},\bigcup _{v_{i_{\Theta }}\in V_{\Theta },v_{i_{\Phi }}\in V_{\Phi }}{\left\{v_{i_{\Theta }}{v}_{i_{\Phi }}\right\}}^{\lambda }\right)\hfill \\ & =\lambda (\Theta \oplus \Phi )\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill ({\lambda }_1+{\lambda }_2)\Theta & =\left(\bigcup _{{\mu }_{i_{\Theta }}\in U_{\Theta }}\left\{\sqrt[3]{1-{(1-{\mu }_{i_{\Theta }}^3)}^{{\lambda }_1+{\lambda }_2}}\right\},\bigcup _{v_{i_{\Theta }}\in V_{\Theta }}{\left\{v_{i_{\Theta }}\right\}}^{{\lambda }_1+{\lambda }_2}\right)\hfill \\ & =\left(\bigcup _{{\mu }_{i_{\Theta }}\in U_{\Theta }}\left\{\sqrt[3]{1-{(1-{\mu }_{i_{\Theta }}^3)}^{{\lambda }_1}{(1-{\mu }_{i_{\Theta }}^3)}^{{\lambda }_2}}\right\},\bigcup _{v_{i_{\Theta }}\in V_{\Theta }}{\left\{v_{i_{\Theta }}\right\}}^{{\lambda }_1+{\lambda }_2}\right)\hfill \\ & =\left(\bigcup _{{\mu }_{i_{\Theta }}\in U_{\Theta }}\left\{\sqrt[3]{1-{(1-{\mu }_{i_{\Theta }}^3)}^{{\lambda }_1}}\right\},\bigcup _{v_{i_{\Theta }}\in V_{\Theta }}{\left\{v_{i_{\Theta }}\right\}}^{{\lambda }_1}\right)\oplus \hfill \\ & \left(\bigcup _{{\mu }_{i_{\Theta }}\in U_{\Theta }}\left\{\sqrt[3]{1-{(1-{\mu }_{i_{\Theta }}^3)}^{{\lambda }_2}}\right\},\bigcup _{v_{i_{\Theta }}\in V_{\Theta }}{\left\{v_{i_{\Theta }}\right\}}^{{\lambda }_2}\right)\hfill \\ & ={\lambda }_1\Theta \oplus {\lambda }_2\Theta \hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill {(\Theta \otimes \Phi )}^{\lambda }& ={\left(\bigcup _{{\mu }_{i_{\Theta }}\in U_{\Theta },{\mu }_{i_{\Phi }}\in U_{\Phi }}\left\{{\mu }_{i_{\Theta }}{\mu }_{i_{\Phi }}\right\},\bigcup _{v_{i_{\Theta }}\in V_{\Theta },v_{i_{\Phi }}\in V_{\Phi }}\left\{\sqrt[3]{v_{i_{\Theta }}^3+v_{i_{\Phi }}^3-v_{i_{\Theta }}^3{v}_{i_{\Phi }}^3}\right\}\right)}^{\lambda }\hfill \\ & =\left(\bigcup _{{\mu }_{i_{\Theta }}\in U_{\Theta },{\mu }_{i_{\Phi }}\in U_{\Phi }}{\left\{{\mu }_{i_{\Theta }}{\mu }_{i_{\Phi }}\right\}}^{\lambda },\bigcup _{v_{i_{\Theta }}\in V_{\Theta },v_{i_{\Phi }}\in V_{\Phi }}\left\{\sqrt[3]{1-{(1-v_{i_{\Theta }}^3-v_{i_{\Phi }}^3+v_{i_{\Theta }}^3{v}_{i_{\Phi }}^3)}^{\lambda }}\right\}\right)\hfill \\ & =\left(\bigcup _{{\mu }_{i_{\Theta }}\in U_{\Theta },{\mu }_{i_{\Phi }}\in U_{\Phi }}\left\{{\mu }_{i_{\Theta }}^{\lambda }{\mu }_{i_{\Phi }}^{\lambda }\right\},\bigcup _{v_{i_{\Theta }}\in V_{\Theta },v_{i_{\Phi }}\in V_{\Phi }}\left\{\sqrt[3]{1-{(1-v_{i_{\Theta }}^3)}^{\lambda }{(1-v_{i_{\Phi }}^3)}^{\lambda }}\right\}\right)\hfill \\ & =\left(\bigcup _{{\mu }_{i_{\Theta }}\in U_{\Theta }}\left\{{\mu }_{i_{\Theta }}^{\lambda }\right\},\bigcup _{v_{i_{\Theta }}\in V_{\Theta }}\left\{\sqrt[3]{1-{(1-v_{i_{\Theta }}^3)}^{\lambda }}\right\}\right)\otimes \hfill \\ & \left(\bigcup _{{\mu }_{i_{\Phi }}\in U_{\Phi }}\left\{{\mu }_{i_{\Phi }}^{\lambda }\right\},\bigcup _{v_{i_{\Phi }}\in V_{\Phi }}\left\{\sqrt[3]{1-{(1-v_{i_{\Phi }}^3)}^{\lambda }}\right\}\right)={\Theta }^{\lambda }\otimes {\Phi }^{\lambda }\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill {\Theta }^{{\lambda }_1}\otimes {\Theta }^{{\lambda }_2}& =\left(\bigcup _{{\mu }_{i_{\Theta }}\in U_{\Theta }}\left\{{\mu }_{i_{\Theta }}^{{\lambda }_1}\right\},\bigcup _{v_{i_{\Theta }}\in V_{\Theta }}\left\{\sqrt[3]{1-{(1-v_{i_{\Theta }}^3)}^{{\lambda }_1}}\right\}\right)\otimes \hfill \\ & \left(\bigcup _{{\mu }_{i_{\Theta }}\in U_{\Theta }}\left\{{\mu }_{i_{\Theta }}^{{\lambda }_2}\right\},\bigcup _{v_{i_{\Theta }}\in V_{\Theta }}\left\{\sqrt[3]{1-{(1-v_{i_{\Theta }}^3)}^{{\lambda }_2}}\right\}\right)\hfill \\ & =\left(\bigcup _{{\mu }_{i_{\Theta }}\in U_{\Theta }}\left\{{\mu }_{i_{\Theta }}^{{\lambda }_1+{\lambda }_2}\right\},\bigcup _{v_{i_{\Theta }}\in V_{\Theta }}\left\{\sqrt[3]{1-{(1-v_{i_{\Theta }}^3)}^{{\lambda }_1+{\lambda }_2}}\right\}\right)\hfill \\ & ={\Theta }^{{\lambda }_1+{\lambda }_2}\hfill \end{array}$$

□

Theorem 2.

1 

$A_1\cap A_2=A_2\cap A_1$

2 

$A_1\cup A_2=A_2\cup A_1$

3 

$A_1\cap (A_2\cap A_3)=(A_1\cap A_2)\cap A_3$

4 

$A_1\cup (A_2\cup A_3)=(A_1\cup A_2)\cup A_3$

5 

$\lambda (A_1\cup A_2)=\lambda A_1\cup \lambda A_2$

6 

${(A_1\cup A_2)}^{\lambda }=A_1^{\lambda }\cup A_2^{\lambda }$

Proof. 

We will prove (1), (3), and (5).

(1)

$$\begin{array}{cc}\hfill A_1\cap A_2& =\left\{⟨(x,\varsigma ),min({\mu }_{i{A}_1}(x,\varsigma ),{\mu }_{i{A}_2}(x,\varsigma )),max(v_{i{A}_1}(x,\varsigma ),v_{i{A}_2}(x,\varsigma ))⟩\right\}\hfill \\ & =\left\{⟨(x,\varsigma ),min({\mu }_{i{A}_2}(x,\varsigma ),{\mu }_{i{A}_1}(x,\varsigma )),max(v_{i{A}_2}(x,\varsigma ),v_{i{A}_1}(x,\varsigma ))⟩\right\}\hfill \\ & =A_2\cap A_1\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill A_1\cap (A_2\cap A_3)& =A_1\cap \{⟨(x,\varsigma ),min({\mu }_{i{A}_2}(x,\varsigma ),{\mu }_{i{A}_3}(x,\varsigma )),\hfill \\ & max(v_{i{A}_2}(x,\varsigma ),v_{i{A}_3}(x,\varsigma ))⟩\}\hfill \\ & =\{⟨(x,\varsigma ),min\left({\mu }_{i{A}_1}(x,\varsigma ),min({\mu }_{i{A}_2}(x,\varsigma ),{\mu }_{i{A}_3}(x,\varsigma ))\right),\hfill \\ & max\left(v_{i{A}_1}(x,\varsigma ),max(v_{i{A}_2}(x,\varsigma ),v_{i{A}_3}(x,\varsigma ))\right)⟩\}\hfill \\ & =\{⟨(x,\varsigma ),min\left(min({\mu }_{i{A}_1}(x,\varsigma ),{\mu }_{i{A}_2}(x,\varsigma )),{\mu }_{i{A}_3}(x,\varsigma )\right),\hfill \\ & max\left(max(v_{i{A}_1}(x,\varsigma ),v_{i{A}_2}(x,\varsigma )),v_{i{A}_3}(x,\varsigma )\right)⟩\}\hfill \\ & =\{⟨(x,\varsigma ),min({\mu }_{i{A}_1}(x,\varsigma ),{\mu }_{i{A}_2}(x,\varsigma )),\hfill \\ & max(v_{i{A}_1}(x,\varsigma ),v_{i{A}_2}(x,\varsigma ))⟩\}\hfill \\ & \cap \{⟨(x,\varsigma ),min({\mu }_{i{A}_3}(x,\varsigma )),\hfill \\ & max\left(v_{i{A}_3}(x,\varsigma )\right)⟩\}\hfill \\ & =(A_1\cap A_2)\cap A_3\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill \lambda (A_1\cup A_2)& =\lambda ⟨\left\{(x,\varsigma ),max\left({\mu }_{i{A}_1}(x,\varsigma ),{\mu }_{i{A}_2}(x,\varsigma )\right),min\left(v_{i{A}_1}(x,\varsigma ),v_{i{A}_2}(x,\varsigma )\right)\right\}⟩\hfill \\ & =⟨\bigcup _{\begin{array}{c}{\mu }_{i{A}_1}\in U_{A_1}\\ {\mu }_{i{A}_2}\in U_{A_2}\end{array}}\left\{\sqrt[3]{1-{\left(max\left({\mu }_{i{A}_1}^3,{\mu }_{i{A}_2}^3\right)\right)}^{\lambda }}\right\},\bigcup _{\begin{array}{c}{v}_{i{A}_1}\in V_{A_1}\\ v_{i{A}_2}\in V_{A_2}\end{array}}\left\{min\left(v_{i{A}_1}^{\lambda },v_{i{A}_2}^{\lambda }\right)\right\}⟩\hfill \\ \hfill \lambda A_1\cup \lambda A_2& =⟨\bigcup _{{\mu }_{i{A}_1}\in U_{A_1}}\left\{\sqrt[3]{1-{\left(1-{\mu }_{i{A}_1}^3\right)}^{\lambda }}\right\},\bigcup _{v_{i{A}_1}\in V_{A_1}}\left\{v_{i{A}_1}^{\lambda }\right\}⟩\hfill \\ & \cup ⟨\bigcup _{{\mu }_{i{A}_2}\in U_{A_2}}\left\{\sqrt[3]{1-{\left(1-{\mu }_{i{A}_2}^3\right)}^{\lambda }}\right\},\bigcup _{v_{i{A}_2}\in V_{A_2}}\left\{v_{i{A}_2}^{\lambda }\right\}⟩\hfill \\ & =⟨max\left(\bigcup _{\begin{array}{c}{\mu }_{i{A}_1}\in U_{A_1}\\ {\mu }_{i{A}_2}\in U_{A_2}\end{array}}\left\{\sqrt[3]{1-{\left(1-{\mu }_{i{A}_1}^3\right)}^{\lambda }},\sqrt[3]{1-{\left(1-{\mu }_{i{A}_2}^3\right)}^{\lambda }}\right\}\right),\hfill \\ & min\left(\bigcup _{\begin{array}{c}{v}_{i{A}_1}\in V_{A_1}\\ v_{i{A}_2}\in V_{A_2}\end{array}}\left\{v_{i{A}_1}^{\lambda },v_{i{A}_2}^{\lambda }\right\}\right)⟩\hfill \end{array}$$

$$\begin{array}{cc}\hfill & =⟨\bigcup _{\begin{array}{c}{\mu }_{i{A}_1}\in U_{A_1}\\ {\mu }_{i{A}_2}\in U_{A_2}\end{array}}\left\{\sqrt[3]{1-{\left(max\left({\mu }_{i{A}_1}^3,{\mu }_{i{A}_2}^3\right)\right)}^{\lambda }}\right\},\bigcup _{\begin{array}{c}{v}_{i{A}_1}\in V_{A_1}\\ v_{i{A}_2}\in V_{A_2}\end{array}}\left\{min\left(v_{i{A}_1}^{\lambda },v_{i{A}_2}^{\lambda }\right)\right\}⟩\hfill \\ & =\lambda (A_1\cup A_2)\hfill \end{array}$$

The other items can be proved analogously.    □

Theorem 3.

(1) 

${(A_1\cap A_2)}^c=A_1^c\cup A_2^c$

(2) 

${(A_1\cup A_2)}^c=A_1^c\cap A_2^c$

(3) 

${(A_1\oplus A_2)}^c=A_1^c\otimes A_2^c$

(4) 

${(A_1\otimes A_2)}^c=A_1^c\oplus A_2^c$

(5) 

${\left(A_1^c\right)}^{\lambda }={\left(\lambda A_1\right)}^c$

(6) 

$\lambda \left(A_1^c\right)={\left(A_1^{\lambda }\right)}^c$

Proof. 

We will prove (1), (3) and (5)

(1)

$$\begin{array}{cc}\hfill {(A_1\cap A_2)}^c& ={⟨\left\{(x,\varsigma ),min\left({\mu }_{i{A}_1}(x,\varsigma ),{\mu }_{i{A}_2}(x,\varsigma )\right),max\left(v_{i{A}_1}(x,\varsigma ),v_{i{A}_2}(x,\varsigma )\right)\right\}⟩}^c\hfill \\ & =⟨\left\{(x,\varsigma ),max\left(v_{i{A}_1}(x,\varsigma ),v_{i{A}_2}(x,\varsigma )\right),min\left({\mu }_{i{A}_1}(x,\varsigma ),{\mu }_{i{A}_2}(x,\varsigma )\right)\right\}⟩\hfill \\ & =A_1^c\cup A_2^c\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill {(A_1\oplus A_2)}^c& ={⟨\bigcup _{\begin{array}{c}{\mu }_{i{A}_1}\in U_{A_1}\\ {\mu }_{i{A}_2}\in U_{A_2}\end{array}}\left\{\sqrt[3]{{\mu }_{i{A}_1}^3+{\mu }_{i{A}_2}^3-{\mu }_{i{A}_1}^3{\mu }_{i{A}_2}^3}\right\},\bigcup _{\begin{array}{c}{v}_{i{A}_1}\in V_{A_1}\\ v_{i{A}_2}\in V_{A_2}\end{array}}\left\{v_{i{A}_1}{v}_{i{A}_2}\right\}⟩}^c\hfill \\ & =⟨\bigcup _{\begin{array}{c}{v}_{i{A}_1}\in V_{A_1}\\ v_{i{A}_2}\in V_{A_2}\end{array}}\left\{v_{i{A}_1}{v}_{i{A}_2}\right\},\bigcup _{\begin{array}{c}{\mu }_{i{A}_1}\in U_{A_1}\\ {\mu }_{i{A}_2}\in U_{A_2}\end{array}}\left\{\sqrt[3]{{\mu }_{i{A}_1}^3+{\mu }_{i{A}_2}^3-{\mu }_{i{A}_1}^3{\mu }_{i{A}_2}^3}\right\}⟩\hfill \\ & =A_1^c\otimes A_2^c\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill {\left(A_1^c\right)}^{\lambda }& =⟨\bigcup _{{\mu }_{i{A}_1}\in U_{A_1}}\left\{{\mu }_{i{A}_1}^{\lambda }\right\},\bigcup _{v_{i{A}_1}\in V_{A_1}}\left\{\sqrt[3]{1-{(1-v_{i{A}_1}^3)}^{\lambda }}\right\}⟩\hfill \\ & ={⟨\bigcup _{v_{i{A}_1}\in V_{A_1}}\left\{\sqrt[3]{1-{(1-v_{i{A}_1}^3)}^{\lambda }}\right\},\bigcup _{{\mu }_{i{A}_1}\in U_{A_1}}\left\{{\mu }_{i{A}_1}^{\lambda }\right\}⟩}^c\hfill \\ & ={\left(\lambda A_1\right)}^c\hfill \end{array}$$

The other items can be proved analogously.    □

Theorem 4.

Let

$$A_i=\left\{⟨\mathfrak{w},\varsigma ⟩,U_{A_i}(\mathfrak{w},\varsigma ),V_{A_i}(\mathfrak{w},\varsigma )\right\},\mathit{for}i=1,2$$

be two multi-Q $\mathcal{FHFS}ʃ$. Then:

(1) 

$A_1=A_2$ if

$$\begin{array}{cc}\hfill \{{\mu }_{i{A}_1}(\mathfrak{w},\varsigma ),v_{i{A}_1}(\mathfrak{w},\varsigma )\}& =\{{\mu }_{i{A}_2}(\mathfrak{w},\varsigma ),v_{i{A}_2}(\mathfrak{w},\varsigma )\}\hfill \\ \hfill {\mu }_{i{A}_1}(\mathfrak{w},\varsigma )& ={\mu }_{i{A}_2}(\mathfrak{w},\varsigma )\hfill \\ \hfill v_{i{A}_1}(\mathfrak{w},\varsigma )& =v_{i{A}_2}(\mathfrak{w},\varsigma )\hfill \end{array}$$

(2) 

$A_1\subseteq A_2$ if

$$\begin{array}{cc}\hfill \{{\mu }_{i{A}_1}(\mathfrak{w},\varsigma ),v_{i{A}_1}(\mathfrak{w},\varsigma )\}& \le \{{\mu }_{i{A}_2}(\mathfrak{w},\varsigma ),v_{i{A}_2}(\mathfrak{w},\varsigma )\}\hfill \\ \hfill {\mu }_{i{A}_1}(\mathfrak{w},\varsigma )& \le {\mu }_{i{A}_2}(\mathfrak{w},\varsigma )\hfill \\ \hfill v_{i{A}_1}(\mathfrak{w},\varsigma )& \le v_{i{A}_2}(\mathfrak{w},\varsigma )\hfill \end{array}$$

(3) 

$A_2\subseteq A_1$ if

$$\begin{array}{cc}\hfill \left\{{\mu }_{i_{A_1}}(\mathfrak{w},\varsigma ),v_{i_{A_1}}(\mathfrak{w},\varsigma )\right\}& \ge \left\{{\mu }_{i_{A_2}}(\mathfrak{w},\varsigma ),v_{i_{A_2}}(\mathfrak{w},\varsigma )\right\}\hfill \\ \hfill {\mu }_{i_{A_1}}(\mathfrak{w},\varsigma )& \ge {\mu }_{i_{A_2}}(\mathfrak{w},\varsigma )\hfill \\ \hfill v_{i_{A_1}}(\mathfrak{w},\varsigma )& \ge v_{i_{A_2}}(\mathfrak{w},\varsigma )\hfill \end{array}$$

4. Multi-Q Fermatean Hesitant FSS

4.1. Fermatean Hesitant FSS

Definition 6.

Assume that the set of discourse U and E is a set of parameters in which ${\tilde{F}}_H\left(\sigma \right)$ is the collection of all $\mathcal{FHFSS}$ on U. A pair $(\tilde{F},E)$ is known as a $\mathcal{FHFSS}$ over U, given that $\tilde{F}:E\to {\tilde{F}}_H\left(\sigma \right)$ with $\tilde{F}\left(e\right)=\{⟨u,{\mu }_e\left(\sigma \right),v_e\left(\sigma \right)⟩\mid \sigma \in U\}\in {\tilde{F}}_H\left(\sigma \right)$ and $(\tilde{F},E)=\{⟨e,\tilde{F}\left(e\right)⟩\mid e\in E\}$.

4.2. Multi-Q Fermatean Hesitant FSS

Definition 7.

Let W be a set of discourse, $E\ne \varnothing$ be a set of parameters, and $A\subseteq E$.

Define ${\tilde{F}}_Q:A\to M^k\mathcal{QFHFSS}$, where $M^k\mathcal{QFHFSS}$ denotes the set of all multi-$Q\mathcal{FHFSS}$.

The pair $({\tilde{F}}_Q,A)$ represents ($M^k$QFHFSS) over w, and is described by

$$({\tilde{F}}_Q,A)=\left\{⟨e,U(x,q),V(x,q)⟩\mid x\in W,q\in Q,e\in E\right\}.$$

Example 2.

Assume that a corporation wants to purchase three types of products from two distinct businesses, and is considering consulting with two experts on these products ($k=2$). Let $W=\{w_1,w_2,w_3\}$ denote the set of all products, $Q=\{q_1,q_2\}$ the set of brands, and $E=\{e_1=\mathit{color},e_2=\mathit{price},e_3=\mathit{ease}\mathit{to}\mathit{use}\}$ the set of parameters.

The multi-Q $\mathcal{FHFSS}$ $({\tilde{F}}_Q,A)$ is defined as follows:

$$\begin{array}{cc}\hfill ({\tilde{F}}_Q,A)=\{& ⟨e_1,\left({\displaystyle \frac{w_1{q}_1}{\{0.2,0.5\},\{0.1,0.3\}}}\right),\left({\displaystyle \frac{w_1{q}_2}{\{0.8,0.1\},\{0.4,0.5\}}}\right),\left({\displaystyle \frac{w_3{q}_1}{\{0.3,0.7\},\{0.1,0.2\}}}\right),\left({\displaystyle \frac{w_3{q}_2}{\{0.8,0.5\},\{0.6,0.2\}}}\right)⟩\hfill \\ & ⟨e_2,\left({\displaystyle \frac{w_2{q}_1}{\{0.7,0.6\},\{0.2,0.3\}}}\right),\left({\displaystyle \frac{w_2{q}_2}{\{0.5,0.2\},\{0.7,0.3\}}}\right),\left({\displaystyle \frac{w_3{q}_1}{\{0.4,0.3\},\{0.1,0.8\}}}\right),\left({\displaystyle \frac{w_3{q}_2}{\left\{0.1\right\},\{0.2,0.3\}}}\right)⟩\hfill \\ & ⟨e_3,\left({\displaystyle \frac{w_1{q}_1}{\{0.4,0.6\},\{0.2,0.4\}}}\right),\left({\displaystyle \frac{w_1{q}_2}{\{0.3,0.7\},\{0.4,0.5\}}}\right),\left({\displaystyle \frac{w_2{q}_1}{\{0.5,0.4\},\{0.6,0.7\}}}\right),\left({\displaystyle \frac{w_2{q}_2}{\{0.3,0.4\},\{0.7,0.2\}}}\right)⟩\}.\hfill \end{array}$$

Definition 8.

We call $({\tilde{H}}_Q,A)\in M^k\mathrm{QFHFSS}$ a null multi-Q $\mathcal{FHFSS}$ if

$${\tilde{H}}_Q\left(f\right)=\varnothing \mathit{for}\mathit{all}f\in A,$$

represented by $(\varnothing ,A)$.

Definition 9.

We call $({\tilde{H}}_Q,A)\in M^k\mathrm{QFHSS}$ an absolute multi-Q FHFSS if

$${\tilde{H}}_Q\left(f\right)=W\mathit{for}\mathit{all}f\in A,$$

represented by $(W,A)$.

Definition 10.

Let $({\tilde{H}}_Q,A)$, $({\tilde{B}}_Q,B)\in M^k\mathrm{QFHFSS}$. Consequently, we assert that $({\tilde{H}}_Q,A)$ is a multi-Q $\mathcal{FHFS}$ subset of $({\tilde{B}}_Q,B)$, represented by $({\tilde{H}}_Q,A)\subseteq ({\tilde{B}}_Q,B)$, if 

$${\tilde{H}}_Q\subseteq {\tilde{B}}_Q\mathit{and}A\subseteq B.$$

Definition 11.

Let $({\tilde{H}}_Q,A)$, $({\tilde{B}}_Q,B)\in M^k\mathrm{QFHFS}$; then,

$$({\tilde{H}}_Q,A)=({\tilde{B}}_Q,B)\mathit{when}({\tilde{H}}_Q,A)\subseteq ({\tilde{B}}_Q,B)$$

and

$$({\tilde{B}}_Q,B)\subseteq ({\tilde{H}}_Q,A).$$

Proposition 2.

Let $({\tilde{H}}_Q,A)$, $({\tilde{B}}_Q,B)\in M^k\mathrm{QFHFSS}$; then:

(i) 

$({\tilde{H}}_Q,A)\subseteq (W,E)$

(ii) 

$(\varnothing ,A)\subseteq ({\tilde{H}}_Q,A)$

(iii) 

If $({\tilde{H}}_Q,A)\subseteq ({\tilde{B}}_Q,B)$ and $({\tilde{B}}_Q,B)\subseteq ({\tilde{K}}_Q,C)$, then $({\tilde{H}}_Q,A)\subseteq ({\tilde{K}}_Q,C)$.

Proof. 

The proof can be obtained simply from Definition 10.    □

Proposition 3.

Let $({\tilde{H}}_Q,A)$, $({\tilde{B}}_Q,B)\in M^k\mathrm{QFHFSS}$ if $({\tilde{H}}_Q,A)=({\tilde{B}}_Q,B)$ and $({\tilde{B}}_Q,B)=({\tilde{K}}_Q,C)$; then, $({\tilde{H}}_Q,A)=({\tilde{K}}_Q,C)$.

Proof. 

The proof can be obtained simply from Definition 10.    □

4.3. Operation of Multi-Q Fermatean Hesitant Fuzzy Soft Sets

1.

Union

Let $({\tilde{H}}_Q,\Theta )$ and $({\tilde{\Phi }}_Q,\Phi )\in M^K\mathrm{QFHFS}\left(W\right)$.

The union of two multi-Q $\mathcal{FHFSS}$ $({\tilde{H}}_Q,\Theta )$ and $({\tilde{\Phi }}_Q,\Phi )$ is written as $({\tilde{H}}_Q,\Theta )\cup ({\tilde{\Phi }}_Q,\Phi )=({\tilde{K}}_Q,\mathbb{C})$, where $\mathbb{C}=\Theta \cup \Phi$ for all $e\in \mathbb{C}$ and 

$${\tilde{K}}_Q\left(e\right)=\left\{\begin{array}{cc}{\tilde{H}}_Q\left(e\right)\hfill & \mathrm{if}e\in \Theta -\Phi \hfill \\ {\tilde{\Phi }}_Q\left(e\right)\hfill & \mathrm{if}e\in \Phi -\Theta \hfill \\ {\tilde{H}}_Q\left(e\right)\cup {\tilde{\Phi }}_Q\left(e\right)\hfill & \mathrm{if}e\in \Theta \cap \Phi .\hfill \end{array}\right.$$

2.

Intersection

Let $({\tilde{H}}_Q,\Theta ),({\tilde{\Phi }}_Q,\Phi )\in M^K\mathrm{QFHFS}\left(W\right)$. Then, the intersection of two multi-QFHFSS $({\tilde{H}}_Q,\Theta )$ and $({\tilde{\Phi }}_Q,\Phi )$ is written as

$$({\tilde{H}}_Q,\Theta )\cap ({\tilde{\Phi }}_Q,\Phi )=({\tilde{K}}_Q,\mathbb{C}),$$

where $\mathbb{C}=\Theta \cap \Phi \forall e\in \mathbb{C}$,

$$({\tilde{K}}_Q,\mathbb{C})=\left\{⟨e,min\left\{{\mu }_{i{\tilde{H}}_Q}(x,q),{\mu }_{i{\tilde{\Phi }}_Q}(x,q)\right\},max\left\{V_{i{\tilde{H}}_Q}(x,q),V_{i{\tilde{\Phi }}_Q}(x,q)\right\}⟩:x\in W,q\in Q\right\}$$

for $i=1,2,\dots ,k$.

3.

Complement

Let $({\tilde{H}}_Q,\Theta )\in M^K\mathrm{QFHFS}\left(W\right)$. Then, the complement of the multi-$\mathcal{QFHFSS}$ represented by ${({\tilde{H}}_Q,\Theta )}^c$ is described by ${({\tilde{H}}_Q,\Theta )}^c=({\tilde{H}}_Q^c,\neg \Theta )$, where

$${\tilde{H}}_Q:\neg \Theta ⟶M^K\mathrm{QFHFSS}$$

is the transformation provided by ${\tilde{H}}_Q^c\left(e\right)={\left({\tilde{H}}_Q\left(e\right)\right)}^c$ $\forall e\in \neg \Theta$.

Proposition 4.

Let $({\tilde{H}}_Q,\Theta )\in M^K\mathrm{QFHFS}\left(W\right)$; then:

(i) 

${\left({({\tilde{H}}_Q,\Theta )}^c\right)}^c=({\tilde{H}}_Q,\Theta )$

(ii) 

${(\varnothing ,\Theta )}^c=(W,\Theta )$

(iii) 

${(W,\Theta )}^c=(\varnothing ,\Theta )$.

Proof. 

The proof follows directly from the definition of complement (Operation 3).    □

Proposition 5.

Let $({\tilde{H}}_Q,\Theta )$, $({\tilde{\Phi }}_Q,\Phi )$ and $({\tilde{K}}_Q,\mathbb{C})\in M^K\mathrm{QFHFSS}$; then:

(i) 

$({\tilde{H}}_Q,\Theta )\cup ({\tilde{H}}_Q,\Theta )=({\tilde{H}}_Q,\Theta )$

(ii) 

$({\tilde{H}}_Q,\Theta )\cup ({\tilde{\Phi }}_Q,\Phi )=({\tilde{\Phi }}_Q,\Phi )\cup ({\tilde{H}}_Q,\Theta )$

(iii) 

$({\tilde{H}}_Q,\Theta )\cup \left(({\tilde{\Phi }}_Q,\Phi )\cup ({\tilde{K}}_Q,\mathbb{C})\right)=\left(({\tilde{H}}_Q,\Theta )\cup ({\tilde{\Phi }}_Q,\Phi )\right)\cup ({\tilde{K}}_Q,\mathbb{C})$

(iv) 

$({\tilde{H}}_Q,\Theta )\cup (W,\Theta )=(W,\Theta )$

(v) 

$({\tilde{H}}_Q,\Theta )\cup (\varnothing ,\Theta )=({\tilde{H}}_Q,\Theta )$.

Proof. 

i.

Let $({\tilde{H}}_Q,\Theta )\cup ({\tilde{H}}_Q,\Theta )=({\tilde{Z}}_Q,\mathbb{C})$, where $\mathbb{C}=\Theta \cup \Theta =\Theta$. Hence,

$${\tilde{Z}}_Q\left(e\right)=\left\{\begin{array}{cc}{\tilde{H}}_Q\left(e\right)\hfill & \mathrm{if}e\in \Theta -\Theta =\varnothing \hfill \\ {\tilde{H}}_Q\left(e\right)\hfill & \mathrm{if}e\in \Theta -\Theta =\varnothing \hfill \\ {\tilde{H}}_Q\left(e\right)\cup {\tilde{H}}_Q\left(e\right)\hfill & \mathrm{if}e\in \Theta \cap \Theta =\Theta .\hfill \end{array}\right.$$

Thus, ${\tilde{Z}}_Q\left(e\right)={\tilde{H}}_Q\left(e\right)$ if $e\in \Theta -\Theta =\varnothing$. In addition,

$$\begin{array}{cc}\hfill {\tilde{Z}}_Q\left(e\right)& ={\tilde{H}}_Q\left(e\right)\cup {\tilde{L}}_Q\left(e\right)\mathrm{if}e\in \Theta \cap \Theta =\Theta \hfill \\ & ={\tilde{H}}_Q\left(e\right)\mathrm{if}e\in \Theta .\hfill \end{array}$$

Therefore, ${\tilde{Z}}_Q\left(e\right)={\tilde{H}}_Q\left(e\right)$ for all $e\in \Theta$; hence,

$$({\tilde{H}}_Q,\Theta )\cup ({\tilde{H}}_Q,\Theta )=({\tilde{H}}_Q,\Theta ).$$

ii.

Let $({\tilde{H}}_Q,\Theta )\cup ({\tilde{\Phi }}_Q,\Phi )=({\tilde{Z}}_Q,\mathbb{C})$, where $\mathbb{C}=\Theta \cup \Phi$. Thus,

$${\tilde{Z}}_Q\left(e\right)=\left\{\begin{array}{cc}{\tilde{H}}_Q\left(e\right)\hfill & \mathrm{if}e\in \Theta -\Phi \hfill \\ {\tilde{\Phi }}_Q\left(e\right)\hfill & \mathrm{if}e\in \Phi -\Theta \hfill \\ {\tilde{H}}_Q\left(e\right)\cup {\tilde{\Phi }}_Q\left(e\right)\hfill & \mathrm{if}e\in \Theta \cap \Phi .\hfill \end{array}\right.$$

Therefore,

$${\tilde{Z}}_Q\left(e\right)=\left\{\begin{array}{cc}{\tilde{H}}_Q\left(e\right)\hfill & \mathrm{if}e\in \Theta -\Phi \hfill \\ {\tilde{\Phi }}_Q\left(e\right)\hfill & \mathrm{if}e\in \Phi -\Theta \hfill \\ {\tilde{H}}_Q\left(e\right)\cup {\tilde{\Phi }}_Q\left(e\right)\hfill & \mathrm{if}e\in \Theta \cap \Phi \hfill \end{array}\right.=\left\{\begin{array}{cc}{\tilde{\Phi }}_Q\left(e\right)\hfill & \mathrm{if}e\in \Phi -\Theta \hfill \\ {\tilde{H}}_Q\left(e\right)\hfill & \mathrm{if}e\in \Theta -\Phi \hfill \\ {\tilde{\Phi }}_Q\left(e\right)\cup {\tilde{H}}_Q\left(e\right)\hfill & \mathrm{if}e\in \Phi \cap \Theta ,\hfill \end{array}\right.$$

as ${\tilde{H}}_Q\left(e\right)\cup {\tilde{\Phi }}_Q\left(e\right)={\tilde{\Phi }}_Q\left(e\right)\cup {\tilde{H}}_Q\left(e\right)$. Hence,

$$({\tilde{H}}_Q,\Theta )\cup ({\tilde{\Phi }}_Q,\Phi )=({\tilde{\Phi }}_Q,\Phi )\cup ({\tilde{H}}_Q,\Theta ).$$

iii.

Let $({\tilde{\Phi }}_Q,\Phi )\cup ({\tilde{K}}_Q,\mathbb{C})=({\tilde{Z}}_Q,D)$, where

$$D=\Phi \cup \mathbb{C}.$$

Here, ${\tilde{Z}}_Q\left(e\right)={\tilde{\Phi }}_Q\left(e\right)\cup {\tilde{K}}_Q\left(e\right)$ if $e\in D=\Phi \cup \mathbb{C}$.

In addition, let $({\tilde{H}}_q,\Theta )\cup ({\tilde{Z}}_Q,D)=({\tilde{L}}_Q,E)$, where

$$E=\Theta \cup D.$$

Here, ${\tilde{L}}_Q\left(e\right)={\tilde{H}}_Q\left(e\right)\cup {\tilde{Z}}_Q\left(e\right)$ if $e\in E=\Theta \cup D$.

Now,

$$\begin{array}{cccc}\hfill {\tilde{L}}_Q\left(e\right)& ={\tilde{H}}_Q\left(e\right)\cup {\tilde{Z}}_Q\left(e\right)\hfill & & \mathrm{if}e\in E=\Theta \cup D\hfill \\ & ={\tilde{H}}_Q\left(e\right)\cup ({\tilde{\Phi }}_Q\left(e\right)\cup {\tilde{K}}_Q\left(e\right))\hfill & & \mathrm{if}e\in E=\Theta \cup D=\Theta \cup (\Phi \cup \mathbb{C})\hfill \\ & =({\tilde{H}}_Q\left(e\right)\cup {\tilde{\Phi }}_Q\left(e\right))\cup {\tilde{K}}_Q\left(e\right)\hfill & & \mathrm{if}e\in E=\Theta \cup D=\Theta \cup (\Phi \cup \mathbb{C}).\hfill \end{array}$$

Therefore, $({\tilde{L}}_Q,E)=(({\tilde{H}}_Q,\Theta )\cup ({\tilde{\Phi }}_Q,\Phi ))\cup ({\tilde{K}}_Q,\mathbb{C})$.

Hence,

$$({\tilde{H}}_Q,\Theta )\cup (({\tilde{\Phi }}_Q,\Phi )\cup ({\tilde{K}}_Q,\mathbb{C}))=(({\tilde{H}}_Q,\Theta )\cup ({\tilde{\Phi }}_Q,\Phi ))\cup ({\tilde{K}}_Q,\mathbb{C}).$$

In the above, (iv) and (v) follow directly from the definitions of the union and intersection of multi-$\mathcal{QFHFSS}$.

□

Proposition 6.

Let $({\tilde{\mathcal{H}}}_Q,\Theta )$, $({\tilde{\Phi }}_Q,\Phi )$, and $({\tilde{K}}_Q,C)\in M^K\mathrm{QF}\mathcal{H}\mathrm{FSS}$; then:

1. 

$({\tilde{\mathcal{H}}}_Q,\Theta )\cap (\varnothing ,\Theta )=(\varnothing ,\Theta )$

2. 

$({\tilde{\mathcal{H}}}_Q,\Theta )\cap (W,\Theta )=({\tilde{\mathcal{H}}}_Q,\Theta )$

3. 

$({\tilde{\mathcal{H}}}_Q,\Theta )\cap ({\tilde{\mathcal{H}}}_Q,\Theta )=({\tilde{\mathcal{H}}}_Q,\Theta )$

4. 

$({\tilde{\mathcal{H}}}_Q,\Theta )\cap ({\tilde{\Phi }}_Q,\Phi )=({\tilde{\Phi }}_Q,\Phi )\cap ({\tilde{\mathcal{H}}}_Q,\Theta )$

5. 

$({\tilde{\mathcal{H}}}_Q,\Theta )\cap \left(({\tilde{\Phi }}_Q,\Phi )\cap ({\tilde{K}}_Q,C)\right)=\left(({\tilde{\mathcal{H}}}_Q,\Theta )\cap ({\tilde{\Phi }}_Q,\Phi )\right)\cap ({\tilde{K}}_Q,C)$.

Proof. 

The proofs are straightforward.    □

Proposition 7.

Let $({\tilde{\mathcal{H}}}_Q,\Theta )$, $({\tilde{\Phi }}_Q,\Phi )$ and $({\tilde{K}}_Q,C)\in M^k\mathcal{QF}\mathcal{H}\mathcal{FSS}$; then:

i. 

${\left(({\tilde{\mathcal{H}}}_Q,\Theta )\cup ({\tilde{\Phi }}_Q,\Phi )\right)}^c={({\tilde{\mathcal{H}}}_Q,\Theta )}^c\cap {({\tilde{\Phi }}_Q,\Phi )}^c$

ii. 

${\left(({\tilde{\mathcal{H}}}_Q,\Theta )\cap ({\tilde{\Phi }}_Q,\Phi )\right)}^c={({\tilde{\mathcal{H}}}_Q,\Theta )}^c\cup {({\tilde{\Phi }}_Q,\Phi )}^c$.

Proof. 

   

i.

Let $({\tilde{\mathcal{H}}}_Q,\Theta )\cup ({\tilde{\Phi }}_Q,\Phi )=({\tilde{K}}_Q,C)$, where $C=\Theta \cup \Phi$.

Thus,

$${\tilde{K}}_Q\left(\varrho \right)=\left\{\begin{array}{cc}{\tilde{\mathcal{H}}}_Q\left(\varrho \right)\hfill & \mathrm{if}\varrho \in \Theta -\Phi \hfill \\ {\tilde{\Phi }}_Q\left(\varrho \right)\hfill & \mathrm{if}\varrho \in \Phi -\Theta \hfill \\ {\tilde{\mathcal{H}}}_Q\left(\varrho \right)\cup {\tilde{\Phi }}_Q\left(\varrho \right)\hfill & \mathrm{if}\varrho \in \Theta \cap \Phi .\hfill \end{array}\right.$$

Now, ${\left(({\tilde{\mathcal{H}}}_Q,\Theta )\cup ({\tilde{\Phi }}_Q,\Phi )\right)}^c={({\tilde{K}}_Q,C)}^c=({\tilde{K}}_Q^c,C)$ and

$${\tilde{K}}_Q^c\left(c\right)=\left\{\begin{array}{cc}{\tilde{\mathcal{H}}}_Q^c\left(\varrho \right)\hfill & \mathrm{if}\varrho \in \Theta -\Phi \hfill \\ {\tilde{\Phi }}_Q^c\left(\varrho \right)\hfill & \mathrm{if}\varrho \in \Phi -\Theta \hfill \\ {\tilde{\mathcal{H}}}_Q^c\left(\varrho \right)\cap {\tilde{\Phi }}_Q^c\left(\varrho \right)\hfill & \mathrm{if}\varrho \in \Theta \cap \Phi .\hfill \end{array}\right.$$

Taking $\Phi =\Theta$, we have $C=\Theta \cup \Theta =\Theta$. Thus,

$$\begin{array}{cc}\hfill {\tilde{K}}_Q^c\left(\varrho \right)& =\left\{\begin{array}{cc}{\tilde{\mathcal{H}}}_Q^c\left(\varrho \right)\hfill & \mathrm{if}\varrho \in \Theta -\Theta \hfill \\ {\tilde{\Phi }}_Q^c\left(\varrho \right)\hfill & \mathrm{if}\varrho \in \Theta -\Theta \hfill \\ {\tilde{\mathcal{H}}}_Q^c\left(\varrho \right)\cap {\tilde{\Phi }}_Q^c\left(\varrho \right)\hfill & \mathrm{if}\varrho \in \Theta \cap \Theta \hfill \end{array}\right.\hfill \\ & ={\tilde{\mathcal{H}}}_Q^c\left(\varrho \right)\cap {\tilde{\Phi }}_Q^c\left(\varrho \right)\mathrm{if}\varrho \in \Theta .\hfill \end{array}$$

This proves that ${({\tilde{K}}_Q,C)}^c={({\tilde{\mathcal{H}}}_Q,\Theta )}^c\cap {({\tilde{\Phi }}_Q,\Phi )}^c$.

Hence, ${\left(({\tilde{\mathcal{H}}}_Q,\Theta )\cup ({\tilde{\Phi }}_Q,\Phi )\right)}^c={({\tilde{\mathcal{H}}}_Q,\Theta )}^c\cap {({\tilde{\Phi }}_Q,\Phi )}^c$.

ii.

We can prove this similarly to (i).

□

Definition 12 (AND, OR operations). 

Let $({\tilde{\mathcal{H}}}_Q,\Theta )$ and $({\tilde{\Phi }}_Q,\Phi )\in M^K\mathcal{QF}\mathcal{H}\mathcal{FS}\left(W\right)$.

We define two operations on such $M^K\mathcal{QF}\mathcal{H}\mathcal{FSS}$ as follows:

1. 

$({\tilde{\mathcal{H}}}_Q,\Theta )\wedge ({\tilde{\Phi }}_Q,\Phi )=({\tilde{D}}_Q,\Theta \times \Phi )$, where

$${\tilde{D}}_Q(a,b)={\tilde{\mathcal{H}}}_Q\left(a\right)\cap {\tilde{\Phi }}_Q\left(b\right),\forall a\in \Theta ,b\in \Phi .$$

2. 

$({\tilde{\mathcal{H}}}_Q,\Theta )\vee ({\tilde{\Phi }}_Q,\Phi )=({\tilde{D}}_Q,\Theta \times \Phi )$, where

$${\tilde{D}}_Q(a,b)={\tilde{\mathcal{H}}}_Q\left(a\right)\cup {\tilde{\Phi }}_Q\left(b\right),\forall a\in \Theta ,b\in \Phi .$$

Then, we obtain the following properties of $M^K\mathcal{QF}\mathcal{H}\mathcal{FSS}$.

Theorem 5.

Let $({\tilde{\mathcal{H}}}_Q,\Theta )$ and $({\tilde{\Phi }}_Q,\Phi )\in M^K\mathcal{QF}\mathcal{H}\mathcal{FSS}$; then, the following De Morgan’s laws hold:

i. 

${\left(({\tilde{\mathcal{H}}}_Q,\Theta )\wedge ({\tilde{\Phi }}_Q,\Phi )\right)}^c={({\tilde{\mathcal{H}}}_Q,\Theta )}^c\vee {({\tilde{\Phi }}_Q,\Phi )}^c$

ii. 

${\left(({\tilde{\mathcal{H}}}_Q,\Theta )\vee ({\tilde{\Phi }}_Q,\Phi )\right)}^c={({\tilde{\mathcal{H}}}_Q,\Theta )}^c\wedge {({\tilde{\Phi }}_Q,\Phi )}^c$.

Proof. 

We use Operation (3), Definition 12 (1), and Proposition 7 (i) to prove Theorem 5 (i):

$${\left(({\tilde{\mathcal{H}}}_Q,\Theta )\wedge ({\tilde{\Phi }}_Q,\Phi )\right)}^c={({\tilde{D}}_Q,\Theta \times \Phi )}^c=({\tilde{D}}_Q^c,\Theta \times \Phi )$$

where

$${\tilde{D}}_Q^c(a,b)={\left({\tilde{\mathcal{H}}}_Q\left(a\right)\cap {\tilde{\Phi }}_Q\left(b\right)\right)}^c={\tilde{\mathcal{H}}}_Q^c\left(a\right)\cup {\tilde{\Phi }}_Q^c\left(b\right)={\tilde{C}}_Q(a,b)$$

for all $a\in \Theta$, $b\in \Phi$, where $({\tilde{C}}_Q,\Theta \times \Phi )={({\tilde{\mathcal{H}}}_Q,\Theta )}^c\vee {({\tilde{\Phi }}_Q,\Phi )}^c$.

Similarly, we can apply Operation (3), Definition 12 (2), and Proposition 7 (ii) to prove Theorem 5 (ii).    □

Theorem 6.

Let $({\tilde{\mathcal{H}}}_Q,\Theta )$, $({\tilde{\Phi }}_Q,\Phi )$, and $({\tilde{K}}_Q,C)\in M^K\mathcal{QFHFS}\left(W\right)$. Then, the following associative laws hold:

i. 

$({\tilde{\mathcal{H}}}_Q,\Theta )\wedge \left(({\tilde{\Phi }}_Q,\Phi )\wedge ({\tilde{K}}_Q,C)\right)=\left(({\tilde{\mathcal{H}}}_Q,\Theta )\wedge ({\tilde{\Phi }}_Q,\Phi )\right)\wedge ({\tilde{K}}_Q,C)$

ii. 

$({\tilde{\mathcal{H}}}_Q,\Theta )\vee \left(({\tilde{\Phi }}_Q,\Phi )\vee ({\tilde{K}}_Q,C)\right)=\left(({\tilde{\mathcal{H}}}_Q,\Theta )\vee ({\tilde{\Phi }}_Q,\Phi )\right)\vee ({\tilde{K}}_Q,C)$.

Proof. 

Using Definition 12 (1) and Proposition 7 (i), we have

$$({\tilde{\mathcal{H}}}_Q,\Theta )\wedge \left(({\tilde{\Phi }}_Q,\Phi )\wedge ({\tilde{K}}_Q,C)\right)=\left(({\tilde{\mathcal{H}}}_Q,\Theta )\wedge ({\tilde{D}}_Q,\Phi \times C)\right)=({\tilde{N}}_Q,\Theta \times \Phi \times C),$$

where $\forall b\in \Phi$, $c\in C$:

$${\tilde{D}}_Q(b,c)={\tilde{\Phi }}_Q\left(b\right)\cap {\tilde{K}}_Q\left(c\right)$$

and $\forall \sigma \in \Theta$, $b\in \Phi$, $c\in C$:

$$\begin{array}{cc}\hfill {\tilde{N}}_Q\varrho ,b,c)& ={\tilde{\mathcal{H}}}_Q\varrho )\cap {\tilde{D}}_Q(b,c)\hfill \\ & ={\tilde{\mathcal{H}}}_Q\varrho )\cap \left({\tilde{\Phi }}_Q\left(b\right)\cap {\tilde{K}}_Q\left(c\right)\right)\hfill \\ & =\left({\tilde{\mathcal{H}}}_Q\varrho )\cap {\tilde{\Phi }}_Q\left(b\right)\right)\cap {\tilde{K}}_Q\left(c\right)\hfill \\ & ={\tilde{M}}_Q\varrho ,b)\cap {\tilde{K}}_Q\left(c\right)\hfill \end{array}$$

with

$${\tilde{M}}_Q\varrho ,b)={\tilde{\mathcal{H}}}_Q\varrho )\cap {\tilde{\Phi }}_Q\left(b\right).$$

Note that

$$({\tilde{M}}_Q,\Theta \times \Phi )\wedge ({\tilde{K}}_Q,C)=\left(({\tilde{\mathcal{H}}}_Q,\Theta )\wedge ({\tilde{\Phi }}_Q,\Phi )\right)\wedge ({\tilde{K}}_Q,C).$$

Thus, Theorem 6 (i) is proved.

Similarly, (ii) can be proved using Definition 12 (2) and Proposition 7 (ii).    □

Theorem 7.

Let $({\tilde{\mathcal{H}}}_Q,\Theta )$ and $({\tilde{\Phi }}_Q,\Phi )\in M^K\mathcal{QF}\mathcal{H}\mathcal{FSS}$; then:

1. 

${\left(({\tilde{\mathcal{H}}}_Q,\Theta )\cup ({\tilde{\Phi }}_Q,\Phi )\right)}^c\subseteq {({\tilde{\mathcal{H}}}_Q,\Theta )}^c\cup {({\tilde{\Phi }}_Q,\Phi )}^c$

2. 

${({\tilde{\mathcal{H}}}_Q,\Theta )}^c\cap {({\tilde{\Phi }}_Q,\Phi )}^c\subseteq {\left(({\tilde{\mathcal{H}}}_Q,\Theta )\cap ({\tilde{\Phi }}_Q,\Phi )\right)}^c$.

Proof. 

1.

Suppose that $({\tilde{\mathcal{H}}}_Q,\Theta )\cup ({\tilde{\Phi }}_Q,\Phi )=({\tilde{K}}_Q,C)$, where $C=\Theta \cup \Phi$ and $\forall \varrho \in C$,

$${\tilde{K}}_Q\left(\varrho \right)=\left\{\begin{array}{cc}{\tilde{\mathcal{H}}}_Q\left(\varrho \right)\hfill & \mathrm{if}\varrho \in \Theta -\Phi \hfill \\ {\tilde{\Phi }}_Q\left(\varrho \right)\hfill & \mathrm{if}\varrho \in \Phi -\Theta \hfill \\ {\tilde{\mathcal{H}}}_Q\left(\varrho \right)\cup {\tilde{\Phi }}_Q\left(\varrho \right)\hfill & \mathrm{if}\varrho \in \Theta \cap \Phi .\hfill \end{array}\right.$$

Thus,

$${\left(({\tilde{\mathcal{H}}}_Q,\Theta )\cup ({\tilde{\Phi }}_Q,\Phi )\right)}^c={({\tilde{K}}_Q,C)}^c=({\tilde{K}}_Q^c,C)$$

and

$${\tilde{K}}_Q^c\left(\varrho \right)=\left\{\begin{array}{cc}{\tilde{\mathcal{H}}}_Q^c\left(\varrho \right)\hfill & \mathrm{if}\varrho \in \Theta -\Phi \hfill \\ {\tilde{\Phi }}_Q^c\left(\varrho \right)\hfill & \mathrm{if}\varrho \in \Phi -\Theta \hfill \\ {\tilde{\mathcal{H}}}_Q^c\left(\varrho \right)\cap {\tilde{\Phi }}_Q^c\left(\varrho \right)\hfill & \mathrm{if}\varrho \in \Theta \cap \Phi .\hfill \end{array}\right.$$

Again, suppose that

$${({\tilde{\mathcal{H}}}_Q,\Theta )}^c\cup {({\tilde{\Phi }}_Q,\Phi )}^c=({\tilde{\mathcal{H}}}_Q^c,\Theta )\cup ({\tilde{\Phi }}_Q^c,\Phi )=({\tilde{N}}_Q,F),$$

where $F=\Theta \cap \Phi$ and $\forall \varrho \in F$,

$${\tilde{N}}_Q\left(\varrho \right)={\tilde{\mathcal{H}}}_Q^c\left(\varrho \right)\cap {\tilde{\Phi }}_Q^c\left(\varrho \right).$$

It can be seen that $F\subseteq C$ and $\forall \varrho \in F$, ${\tilde{N}}_Q\left(\varrho \right)={\tilde{\mathcal{H}}}_Q^c\left(\varrho \right)$.

2.

The result can be shown in the same way.

□

Theorem 8.

Let $({\tilde{\mathcal{H}}}_Q,\Theta )$ and $({\tilde{\Phi }}_Q,\Phi )\in M^K\mathcal{QF}\mathcal{H}\mathcal{FSS}$; then:

(i) 

$({\tilde{\mathcal{H}}}_Q,\Theta )\cap \left(({\tilde{\Phi }}_Q,\Phi )\cup ({\tilde{K}}_Q,C)\right)=\left(({\tilde{\mathcal{H}}}_Q,\Theta )\cap ({\tilde{\Phi }}_Q,\Phi )\right)\cup \left(({\tilde{\mathcal{H}}}_Q,\Theta )\cap ({\tilde{K}}_Q,C)\right)$

(ii) 

$({\tilde{\mathcal{H}}}_Q,\Theta )\cup \left(({\tilde{\Phi }}_Q,\Phi )\cap ({\tilde{K}}_Q,C)\right)=\left(({\tilde{\mathcal{H}}}_Q,\Theta )\cup ({\tilde{\Phi }}_Q,\Phi )\right)\cap \left(({\tilde{\mathcal{H}}}_Q,\Theta )\cup ({\tilde{K}}_Q,C)\right).$

Proof. 

1.

Let $({\tilde{\Phi }}_Q,\Phi )\cup ({\tilde{K}}_Q,C)=({\tilde{L}}_Q,D)$, where $D=\Phi \cup C$.

Thus,

$${\tilde{L}}_Q\left(\varrho \right)=\left\{\begin{array}{cc}{\tilde{\Phi }}_Q\left(\varrho \right)\hfill & \mathrm{if}\varrho \in \Phi -C\hfill \\ {\tilde{K}}_Q\left(\varrho \right)\hfill & \mathrm{if}\varrho \in C-\Phi \hfill \\ {\tilde{\Phi }}_Q\left(\varrho \right)\cup {\tilde{K}}_Q\left(\varrho \right)\hfill & \mathrm{if}\varrho \in \Phi \cap C.\hfill \end{array}\right.$$

Additionally, let

$$({\tilde{\mathcal{H}}}_Q,\Theta )\cap ({\tilde{L}}_Q,D)=({\tilde{M}}_Q,V),$$

where

$$V=\Theta \cap D.$$

Then,

$${\tilde{M}}_Q\left(\varrho \right)={\tilde{\mathcal{H}}}_Q\left(\varrho \right)\cap {\tilde{L}}_Q\left(\varrho \right)\mathrm{if}\varrho \in V=\Theta \cap D.$$

Now,

$$\begin{array}{cc}\hfill {\tilde{M}}_Q\left(\varrho \right)& ={\tilde{\mathcal{H}}}_Q\left(\varrho \right)\cap {\tilde{L}}_Q\left(\varrho \right)\mathrm{if}\varrho \in V=\Theta \cap D\hfill \\ & ={\tilde{\mathcal{H}}}_Q\left(\varrho \right)\cap {\tilde{L}}_Q\left(\varrho \right)\mathrm{if}\varrho \in \Theta \mathrm{and}\varrho \in D.\hfill \end{array}$$

If $e\in D=\Phi \cup C$, then there are three cases:

(a)

Case I: Assume that $e\in \Phi -C$, then ${\tilde{L}}_Q\left(\varrho \right)={\tilde{\Phi }}_Q\left(\varrho \right)$ if $\Phi -C$

(b)

Case II: Assume that $e\in C-\Phi$, then ${\tilde{L}}_Q\left(\varrho \right)={\tilde{K}}_Q\left(\varrho \right)$ if $C-\Phi$

(c)

Case III: Assume that $e\in \Phi \cap C$, then ${\tilde{L}}_Q\left(\varrho \right)={\tilde{\Phi }}_Q\left(\varrho \right)\cup {\tilde{K}}_Q\left(\varrho \right)$ if $\Phi \cap C$.

Now,

$${\tilde{M}}_Q\left(\varrho \right)={\tilde{\mathcal{H}}}_Q\left(\varrho \right)\cap {\tilde{L}}_Q\left(\varrho \right)\mathrm{if}\varrho \in \Theta \mathrm{and}\varrho \in D$$

$$=\left\{\begin{array}{cc}{\tilde{\mathcal{H}}}_Q\left(\varrho \right)\cap {\tilde{\Phi }}_Q\left(\varrho \right)\hfill & \mathrm{if}\varrho \in \Theta \mathrm{and}\varrho \in \Phi -C,\mathrm{by}\mathrm{case}1\hfill \\ {\tilde{\mathcal{H}}}_Q\left(\varrho \right)\cap {\tilde{K}}_Q\left(\varrho \right)\hfill & \mathrm{if}\varrho \in \Theta \mathrm{and}\varrho \in C-\Phi \mathrm{by}\mathrm{case}2\hfill \\ {\tilde{\mathcal{H}}}_Q\left(\varrho \right)\cap ({\tilde{\Phi }}_Q\left(\varrho \right)\cup {\tilde{K}}_Q\left(\varrho \right))\hfill & \mathrm{if}\varrho \in \Theta \mathrm{and}\varrho \in \Phi \cap C\mathrm{by}\mathrm{case}3\hfill \end{array}\right.$$

$$=\left\{\begin{array}{cc}({\tilde{\mathcal{H}}}_Q\left(\varrho \right)\cap {\tilde{\Phi }}_Q\left(\varrho \right))\hfill & \mathrm{if}\varrho \in \Theta \mathrm{and}\varrho \in (\Phi -C)\hfill \\ ({\tilde{\mathcal{H}}}_Q\left(\varrho \right)\cap {\tilde{K}}_Q\left(\varrho \right))\hfill & \mathrm{if}\varrho \in \Theta \mathrm{and}\varrho \in (C-\Phi )\hfill \\ ({\tilde{\mathcal{H}}}_Q\left(\varrho \right)\cap {\tilde{\Phi }}_Q\left(\varrho \right))\cup ({\tilde{\mathcal{H}}}_Q\left(\varrho \right)\cap {\tilde{K}}_Q\left(\varrho \right))\hfill & \mathrm{if}\varrho \in \Theta \mathrm{and}\varrho \in (\Phi \cap C).\hfill \end{array}\right.$$

Therefore,

$$({\tilde{M}}_Q,V)=(({\tilde{\mathcal{H}}}_Q,\Theta )\cap ({\tilde{\Phi }}_Q,\Phi ))\cup (({\tilde{\mathcal{H}}}_Q,\Theta )\cap ({\tilde{K}}_Q,C)).$$

2.

The proof is complete. Part (ii) can be proved similarly.

□

Theorem 9.

Let $({\mathcal{H}}_Q,\Theta )$ and $({\tilde{\Phi }}_Q,\Phi )\in M^K\mathcal{QF}\mathcal{H}\mathcal{FSS}$; then:

(i) 

$({\mathcal{H}}_Q,\Theta )\cup \left(({\mathcal{H}}_Q,\Theta )\cap ({\tilde{\Phi }}_Q,\Phi )\right)=({\tilde{\mathcal{H}}}_Q,\Theta )$

(ii) 

$({\tilde{\mathcal{H}}}_Q,\Theta )\cap \left(({\tilde{\mathcal{H}}}_Q,\Theta )\cup ({\tilde{\Phi }}_Q,\Phi )\right)=({\tilde{\mathcal{H}}}_Q,\Theta ).$

Proof. 

i.

Let $({\tilde{\mathcal{H}}}_Q,\Theta )\cap ({\tilde{\Phi }}_Q,\Phi )=({\tilde{Z}}_Q,C)$, where $C=\Theta \cap \Phi ;\mathrm{thus},$

$${\tilde{Z}}_Q\left(\varrho \right)={\tilde{\mathcal{H}}}_Q\left(\varrho \right)\cap {\tilde{\Phi }}_Q\left(\varrho \right)\mathrm{if}\varrho \in \Theta \cap \Phi .$$

Let $({\tilde{\mathcal{H}}}_Q,\Theta )\cup ({\tilde{K}}_Q,C)=({\tilde{L}}_Q,D)$, where $D=\Theta \cup C$; thus,

$${\tilde{L}}_Q\left(\varrho \right)=\left\{\begin{array}{cc}{\tilde{\mathcal{H}}}_Q\left(\varrho \right)\hfill & \mathrm{if}\varrho \in \Theta -C\hfill \\ {\tilde{K}}_Q\left(\varrho \right)\hfill & \mathrm{if}\varrho \in C-\Theta \hfill \\ {\tilde{\mathcal{H}}}_Q\left(\varrho \right)\cup {\tilde{K}}_Q\left(\varrho \right)\hfill & \mathrm{if}\varrho \in \Theta \cap C.\hfill \end{array}\right.$$

Now, there are three cases:

(a)

Case 1: Let $e\in \Theta -C$; then,

$$\begin{array}{cc}\hfill {\tilde{L}}_Q\left(\varrho \right)& ={\tilde{\mathcal{H}}}_Q\left(\varrho \right)\mathrm{if}\varrho \in \Theta -C\hfill \\ & ={\tilde{\mathcal{H}}}_Q\left(\varrho \right)\mathrm{if}\varrho \in \Theta .\hfill \end{array}$$

Thus, $({\tilde{L}}_Q,D)=({\tilde{\mathcal{H}}}_Q,\Theta )$.

(b)

Case 2: Let $e\in C-\Theta$. Now, $C-\Theta =(\Theta \cap \Phi )-\Theta =\varnothing$, meaning that

$$\begin{array}{cc}\hfill {\tilde{L}}_Q\left(\varrho \right)& ={\tilde{\mathcal{H}}}_Q\left(\varrho \right)\mathrm{if}\varrho \in C-\Theta =\varnothing \hfill \\ & =\varnothing (\mathrm{the}\mathrm{null}{M}^K\mathcal{QF}\mathcal{H}\mathcal{FSS}).\hfill \end{array}$$

Thus, $({\tilde{L}}_Q,D)$ is the null $M^K\mathcal{QF}\mathcal{H}\mathcal{FSS}$.

(c)

Case 3: Let $e\in \Theta \cap C$. Now, $\Theta \cap C=\Theta \cap (\Theta \cap \Phi )=\Theta \cap \Phi =C$, meaning that

$$\begin{array}{cc}\hfill {\tilde{L}}_Q\left(\varrho \right)& ={\tilde{\mathcal{H}}}_Q\left(\varrho \right)\cup {\tilde{Z}}_Q\left(\varrho \right)\mathrm{if}\varrho \in \Theta \cap C=C\hfill \\ & ={\tilde{\mathcal{H}}}_Q\left(\varrho \right)\cup ({\tilde{\mathcal{H}}}_Q\left(\varrho \right)\cap {\tilde{\Phi }}_Q\left(\varrho \right))\mathrm{if}\varrho \in C\hfill \\ & ={\tilde{\mathcal{H}}}_Q\left(\varrho \right)\mathrm{if}\varrho \in C.\hfill \end{array}$$

Thus, $({\tilde{L}}_Q,D)=({\tilde{\mathcal{H}}}_Q,\Theta )$.

In all cases, we get

$$({\tilde{\mathcal{H}}}_Q,\Theta )\cup \left(({\tilde{\mathcal{H}}}_Q,\Theta )\cap ({\tilde{\Phi }}_Q,\Phi )\right)=({\tilde{\mathcal{H}}}_Q,\Theta ).$$

ii.

We can prove this similarly.

□

5. Application of Multi-$\mathcal{QF}\mathcal{H}\mathcal{FSS}$ in Decision-Making

The following section illustrates an instance of multi-$\mathcal{QF}\mathcal{H}\mathcal{FSS}$. We introduce a method to address the DM problem that relies on evaluating the value of several options; specifically, a greater value indicates a superior object.

Definition 13.

Let $X=\{e_1,e_2,\dots ,e_i\}$. Take any two $\mathcal{F}\mathcal{H}\mathcal{FSS}$ $\Theta =⟨e_i,{\mu }_{\Theta }\left(e_i\right),v_{\Theta }\left(e_i\right):e_i\in X⟩$ and $\Phi =⟨e_i,{\mu }_{\Phi }\left(e_i\right),v_{\Phi }\left(e_i\right):e_i\in X⟩$. The cosine similarity is provided by

$$C_{\mathrm{F}\mathcal{H}\mathrm{S}}={\displaystyle \frac{1}{n}}\sum _{i=1}^n{\displaystyle \frac{{\mu }_{\Theta }^3\left(e_i\right){\mu }_{\Phi }^3\left(e_i\right)+v_{\Theta }^3\left(e_i\right)v_{\Phi }^3\left(e_i\right)}{\sqrt[3]{{\mu }_{\Theta }^6\left(e_i\right)+v_{\Theta }^6\left(e_i\right)}+\sqrt[3]{{\mu }_{\Phi }^6\left(e_i\right)+v_{\Phi }^6\left(e_i\right)}.}}$$

The step by step procedure of the proposed model is illustrated in Section Algorithm Algorithm 1.

Algorithm

Algorithm 1: The step by step procedure of the proposed mode.

Step i: Collect information for $\mathcal{MQFHFSS}$ Step ii: Write $\mathcal{MQFHFSS}$ Step iii: Consider the perfect package recommended by experts as a fantastic alternative. Step iv: Apply cosine similarity. Step v: Evaluate the production rate of all goods. Step vi: Select the greatest value. Step vii: End.

Example 3.

Vision 2030 aims to achieve a comprehensive transformation in the Kingdom of Saudi Arabia, with artificial intelligence playing a pivotal role in achieving many of its goals.

The role of the Saudi Data and Artificial Intelligence Authority (SDAIA) is to lead the national data and artificial intelligence drive and advance the Kingdom to become a leading data-driven economy.

If we are presented with two types of programs to help families make smart decisions within two different systems, taking into account the opinions of two experts, we can assume that $U_1$ and $U_2$ represent the set of programs, while $Q_1$ and $Q_2$ represent the set of systems. The decision attributes are defined by $e_1$ (efficiency), $e_2$ (ease of use), and $e_3$ (reliability).

• Step i:

  -

  $U_1$ and $U_2$ denote 2 distinct programs

  -

  $Q_1$ and $Q_2$ denote 2 distinct types of systems

  -

  $e_1,e_2$, and $e_3$ respectively denote the efficiency, ease of use, and reliability

  	$e_1$	$e_2$	$e_3$
  $U_1{Q}_1$	$\{0.30,0.12\},\{0.23,0.14\}$	$\{0.20,0.17\},\{0.33,0.13\}$	$\{0.35,0.32\},\{0.20,0.10\}$
  $U_1{Q}_2$	$\{0.6,0.24\},\{0.13,0.15\}$	$\{0.28,0.22\},\{0.14,0.25\}$	$\{0.50,0.18\},\{0.30,0.40\}$

  	$e_1$	$e_2$	$e_3$
  $U_2{Q}_1$	$\{0.11,0.28\},\{0.30,0.16\}$	$\{0.15,0.20\},\{0.33,0.27\}$	$\{0.12,0.35\},\{0.23,0.29\}$
  $U_2{Q}_2$	$\{0.24,0.37\},\{0.50,0.20\}$	$\{0.13,0.30\},\{0.18,0.29\}$	$\{0.19,0.14\},\{0.37,0.26\}$

• Step ii: Write $\mathcal{MQFHFS}\mathcal{s}$

  $$F(U_1,U_2)=\left\{\begin{array}{ccccc}⟨e_1,& {\displaystyle \frac{U_1{Q}_1}{\{0.30,0.12\},\{0.23,0.14\}}}& {\displaystyle \frac{U_1{Q}_2}{\{0.6,0.24\},\{0.13,0.15\}}}& {\displaystyle \frac{U_2{Q}_1}{\{0.11,0.28\},\{0.30,0.16\}}}& {\displaystyle \frac{U_2{Q}_2}{\{0.24,0.37\},\{0.50,0.20\}}}⟩\\ & & & & \\ ⟨e_2,& {\displaystyle \frac{U_1{Q}_1}{\{0.20,0.17\},\{0.33,0.13\}}},& {\displaystyle \frac{U_1{Q}_2}{\{0.28,0.22\},\{0.14,0.25\}}}& {\displaystyle \frac{U_2{Q}_1}{\{0.15,0.20\},\{0.33,0.27\}}},& {\displaystyle \frac{U_2{Q}_2}{\{0.13,0.30\},\{0.18,0.29\}}}⟩\\ & & & & \\ ⟨e_3,& {\displaystyle \frac{U_1{Q}_1}{\{0.35,0.32\},\{0.20,0.10\}}},& {\displaystyle \frac{U_1{Q}_2}{\{0.50,0.18\},\{0.30,0.40\}}}& {\displaystyle \frac{U_2{Q}_1}{\{0.12,0.35\},\{0.23,0.29\}}},& {\displaystyle \frac{U_2{Q}_2}{\{0.19,0.14\},\{0.37,0.26\}}}⟩\end{array}\right\}$$
• Step iii: Take the perfect set

  	$e_1$	$e_2$	$e_3$
  $U\times Q$	$\{0.31,0.25\},\{0.29,0.16\}$	$\{0.19,0.22\},\{0.24,0.23\}$	$\{0.29,0.25\},\{0.27,0.26\}$

  $$U_2{Q}_1{e}_1={\displaystyle \frac{1}{2}}{\sum }_{i=1}^2\left\{{\displaystyle \frac{{(0.30\times 0.31)}^3+{(0.12\times 0.25)}^3+{(0.23\times 0.29)}^3+{(0.14\times 0.16)}^3}{\sqrt[3]{{\left(0.30\right)}^6+{\left(0.12\right)}^6+{\left(0.23\right)}^6+{\left(0.14\right)}^6}\times \sqrt[3]{{\left(0.31\right)}^6+{\left(0.25\right)}^6+{\left(0.29\right)}^6+{\left(0.16\right)}^6}}}\right\}=0.0491$$

  $$U_1{Q}_1{e}_2={\displaystyle \frac{1}{2}}{\sum }_{i=1}^2\left\{{\displaystyle \frac{{(0.20\times 0.19)}^3+{(0.17\times 0.22)}^3+{(0.33\times 0.24)}^3+{(0.13\times 0.23)}^3}{\sqrt[3]{{\left(0.20\right)}^6+{\left(0.17\right)}^6+{\left(0.33\right)}^6+{\left(0.13\right)}^6}\times \sqrt[3]{{\left(0.19\right)}^6+{\left(0.22\right)}^6+{\left(0.24\right)}^6+{\left(0.23\right)}^6}}}\right\}=0.0357$$

  $$U_1{Q}_1{e}_3={\displaystyle \frac{1}{2}}{\sum }_{i=1}^2\left\{{\displaystyle \frac{{(0.33\times 0.29)}^3+{(0.32\times 0.25)}^3+{(0.20\times 0.27)}^3+{(0.10\times 0.26)}^3}{\sqrt[3]{{\left(0.35\right)}^6+{\left(0.32\right)}^6+{\left(0.20\right)}^6+{\left(0.10\right)}^6}\times \sqrt[3]{{\left(0.29\right)}^6+{\left(0.25\right)}^6+{\left(0.27\right)}^6+{\left(0.26\right)}^6}}}\right\}=0.0524$$

  $$U_1{Q}_2{e}_1={\displaystyle \frac{1}{2}}{\sum }_{i=1}^2\left\{{\displaystyle \frac{{(0.6\times 0.31)}^3+{(0.24\times 0.25)}^3+{(0.13\times 0.29)}^3+{(0.15\times 0.16)}^3}{\sqrt[3]{{\left(0.6\right)}^6+{\left(0.24\right)}^6+{\left(0.13\right)}^6+{\left(0.15\right)}^6}\times \sqrt[3]{{\left(0.31\right)}^6+{\left(0.25\right)}^6+{\left(0.29\right)}^6+{\left(0.16\right)}^6}}}\right\}=0.0774$$

  $$U_1{Q}_2{e}_2={\displaystyle \frac{1}{2}}{\sum }_{i=1}^2\left\{{\displaystyle \frac{{(0.28\times 0.19)}^3+{(0.22\times 0.22)}^3+{(0.14\times 0.24)}^3+{(0.25\times 0.23)}^3}{\sqrt[3]{{\left(0.28\right)}^6+{\left(0.22\right)}^6+{\left(0.14\right)}^6+{\left(0.25\right)}^6}\times \sqrt[3]{{\left(0.19\right)}^6+{\left(0.22\right)}^6+{\left(0.24\right)}^6+{\left(0.23\right)}^6}}}\right\}=0.0326$$

  $$U_1{Q}_2{e}_3={\displaystyle \frac{1}{2}}{\sum }_{i=1}^2\left\{{\displaystyle \frac{{(0.50\times 0.29)}^3+{(0.18\times 0.25)}^3+{(0.30\times 0.27)}^3+{(0.4\times 0.26)}^3}{\sqrt[3]{{\left(0.50\right)}^6+{\left(0.18\right)}^6+{\left(0.30\right)}^6+{\left(0.4\right)}^6}\times \sqrt[3]{{\left(0.29\right)}^6+{\left(0.25\right)}^6+{\left(0.27\right)}^6+{\left(0.26\right)}^6}}}\right\}=0.0760$$

  $$U_2{Q}_1{e}_1={\displaystyle \frac{1}{2}}{\sum }_{i=1}^2\left\{{\displaystyle \frac{{(0.11\times 0.31)}^3+{(0.28\times 0.25)}^3+{(0.30\times 0.29)}^3+{(0.16\times 0.16)}^3}{\sqrt[3]{{\left(0.11\right)}^6+{\left(0.28\right)}^6+{\left(0.30\right)}^6+{\left(0.16\right)}^6}\times \sqrt[3]{{\left(0.31\right)}^6+{\left(0.25\right)}^6+{\left(0.29\right)}^6+{\left(0.16\right)}^6}}}\right\}=0.0410$$

  $$U_2{Q}_1{e}_2={\displaystyle \frac{1}{2}}{\sum }_{i=1}^2\left\{{\displaystyle \frac{{(0.15\times 0.19)}^3+{(0.20\times 0.22)}^3+{(0.33\times 0.24)}^3+{(0.27\times 0.23)}^3}{\sqrt[3]{{\left(0.15\right)}^6+{\left(0.20\right)}^6+{\left(0.33\right)}^6+{\left(0.27\right)}^6}\times \sqrt[3]{{\left(0.19\right)}^6+{\left(0.22\right)}^6+{\left(0.24\right)}^6+{\left(0.23\right)}^6}}}\right\}=0.0442$$

  $$U_2{Q}_1{e}_3={\displaystyle \frac{1}{2}}{\sum }_{i=1}^2\left\{{\displaystyle \frac{{(0.12\times 0.29)}^3+{(0.35\times 0.25)}^3+{(0.23\times 0.27)}^3+{(0.29\times 0.26)}^3}{\sqrt[3]{{\left(0.12\right)}^6+{\left(0.35\right)}^6+{\left(0.23\right)}^6+{\left(0.29\right)}^6}\times \sqrt[3]{{\left(0.29\right)}^6+{\left(0.25\right)}^6+{\left(0.27\right)}^6+{\left(0.26\right)}^6}}}\right\}=0.0436$$

  $$U_2{Q}_2{e}_1={\displaystyle \frac{1}{2}}{\sum }_{i=1}^2\left\{{\displaystyle \frac{{(0.24\times 0.31)}^3+{(0.37\times 0.25)}^3+{(0.50\times 0.29)}^3+{(0.20\times 0.16)}^3}{\sqrt[3]{{\left(0.24\right)}^6+{\left(0.37\right)}^6+{\left(0.50\right)}^6+{\left(0.20\right)}^6}\times \sqrt[3]{{\left(0.31\right)}^6+{\left(0.25\right)}^6+{\left(0.29\right)}^6+{\left(0.16\right)}^6}}}\right\}=0.0675$$

  $$U_2{Q}_2{e}_2={\displaystyle \frac{1}{2}}{\sum }_{i=1}^2\left\{{\displaystyle \frac{{(0.13\times 0.19)}^3+{(0.30\times 0.22)}^3+{(0.18\times 0.24)}^3+{(0.29\times 0.23)}^3}{\sqrt[3]{{\left(0.13\right)}^6+{\left(0.30\right)}^6+{\left(0.18\right)}^6+{\left(0.29\right)}^6}\times \sqrt[3]{{\left(0.19\right)}^6+{\left(0.22\right)}^6+{\left(0.24\right)}^6+{\left(0.23\right)}^6}}}\right\}=0.0385$$

  $$U_2{Q}_2{e}_3={\displaystyle \frac{1}{2}}{\sum }_{i=1}^2\left\{{\displaystyle \frac{{(0.19\times 0.29)}^3+{(0.14\times 0.25)}^3+{(0.37\times 0.27)}^3+{(0.26\times 0.26)}^3}{\sqrt[3]{{\left(0.19\right)}^6+{\left(0.14\right)}^6+{\left(0.37\right)}^6+{\left(0.26\right)}^6}\times \sqrt[3]{{\left(0.29\right)}^6+{\left(0.25\right)}^6+{\left(0.27\right)}^6+{\left(0.26\right)}^6}}}\right\}=0.0457$$
• Step v: Calculate cosine similarity values

  	$U_1{Q}_1$	$U_1{Q}_2$	$U_2{Q}_1$	$U_2{Q}_2$
  $e_1$	0.0491	0.0774	0.0410	0.0675
  $e_2$	0.0357	0.0326	0.0442	0.0385
  $e_3$	0.0524	0.0760	0.0436	0.0457
  	0.0457	0.0620	0.0429	0.0505

• Step vi: The item with the highest mean cosine similarity is the optimal choice.
  Therefore, $(U_1{Q}_2=0.0620)$ is the optimal choice.

6. Conclusions

This manuscript has combined the multi Q-FSS and $\mathcal{FHFS}$ concepts by establishing the idea of $\mathcal{MQFHFSS}$, which brings together the best characteristics of both to effectively solve complicated DM problems with multiple criteria. In this study, we investigate the novel concept of $\mathcal{MQFHFSS}$ along with the associated operations and address multi-parameter DM issues within this framework. Through a rigorous exploration of the structure and operations associated with $\mathcal{MQFHFSS}$, we establish a solid theoretical foundation for the concept. We examine essential aggregation operators and demonstrate how these can effectively capture hesitancy, uncertainty, and degree-based preferences in DM environments. The proposed model enables decision-makers to evaluate multiple conflicting criteria with a higher degree of flexibility and precision. Overall, $\mathcal{MQFHFSS}$ presents a robust and versatile extension to existing fuzzy set models, opening new possibilities for applications in fields such as expert systems, data analysis, engineering, and economics. In future work, we intend to integrate our proposed method with complex Fermatean fuzzy sets [26] to address two-dimensional multipolarity uncertainty and information.