*Preorders and Topologies*

In this subsection, we present some basic properties about the connection between preorders and topologies proposed by [47].

Topological structures and classical order structures are well recognised to have close relationships, which can be summarised as follows:


#### **3. Construction Tow Preorderings in Lower and Upper Spaces**

By using the notion of [*<sup>α</sup>*1, *<sup>α</sup>*2]-level sets of interval-valued fuzzy soft open sets in (*<sup>X</sup>*, *E*, *<sup>τ</sup>*), this section, introduces two topological spaces, known as lower and upper spaces, by which two preordering relations over the universal set *X* are investigated.

**Definition 4.** *Let fE be an IVFS set over X. Corresponding to each parameter e* ∈ *E*,*, we define two crisp sets, called α-upper-e crisp set and β-lower-e crisp set, where α* = [*<sup>α</sup>*1, *<sup>α</sup>*2] ⊂ I, *β* = [*β*1, *β*2] ⊂ I *as the following:*

$$\begin{array}{rcl} \text{ULC.S}\_{\mathfrak{a}}^{f}(\mathfrak{e}) &=& \{ \mathbf{x} \in \mathbf{X} : [f\_{\mathfrak{e}}^{-}(\mathbf{x}), f\_{\mathfrak{e}}^{+}(\mathbf{x})] > \mathfrak{a}, \mathfrak{a} \subseteq [0, 1) \} \\ &=& \{ \mathbf{x} \in \mathbf{X} : f\_{\mathfrak{e}}^{-}(\mathbf{x}) > \mathfrak{a}\_{1}, f\_{\mathfrak{e}}^{+}(\mathbf{x}) > \mathfrak{a}\_{2}, \mathfrak{a}\_{1}, \mathfrak{a}\_{2} \in [0, 1) \} \\ \text{L.C.S}\_{\mathfrak{f}}^{f}(\mathfrak{e}) &=& \{ \mathbf{x} \in \mathbf{X} : [f\_{\mathfrak{e}}^{-}(\mathbf{x}), f\_{\mathfrak{e}}^{+}(\mathbf{x})] < \mathfrak{f}, \mathfrak{f} \subseteq (0, 1] \} \\ &=& \{ \mathbf{x} \in \mathbf{X} : f\_{\mathfrak{e}}^{-}(\mathbf{x}) < \mathfrak{f}\_{1}, f\_{\mathfrak{e}}^{+}(\mathbf{x}) < \mathfrak{f}\_{2}, \mathfrak{f}\_{1}, \mathfrak{f}\_{2} \in (0, 1] \} \end{array}$$

**Proposition 1.** *Let X be the set of objects, E be the set of parameters and fE, gE be two IVFSs over X. Suppose that the threshold intervals α*1, *α*2, ⊆ [0, <sup>1</sup>), *and β*1, *β*2, ⊆ (0, 1] *are given such that α*1 = [*α*-1, *α*--1 ], *α*2 = [*α*-2, *α*--2 ], *β*1 = [*β*-1, *β*--1 ] *and β*2 = [*β*-2, *β*--2 ]*. Consider the parameter e* ∈ *E.*


*3. If fE* = *XE, then <sup>U</sup>*.*C*.*Sf<sup>α</sup>*1 (*e*) = *X and <sup>L</sup>*.*C*.*Sfβ*1 (*e*) = ∅. *Moreover, if fE* = ∅*E then, <sup>U</sup>*.*CSf<sup>α</sup>*1(*e*) = ∅ *and <sup>L</sup>*.*CSfβ*1(*e*) = *X*.

*4. <sup>U</sup>*.*C*.*Sf<sup>α</sup>*1 (¬*<sup>e</sup>*) = *<sup>L</sup>*.*CSf*[<sup>1</sup>−*α*--1,1−*α*-1](*e*) *and <sup>L</sup>*.*CSf<sup>α</sup>*1 (¬*<sup>e</sup>*) = *<sup>U</sup>*.*CSf*[<sup>1</sup>−*α*--1,1−*α*-1](*e*).

*5. <sup>U</sup>*.*CS*<sup>¬</sup>*<sup>f</sup> α*1 (*e*) = *<sup>L</sup>*.*CSf*[<sup>1</sup>−*α*--1 ,1−*α*-1 ](*e*) *and <sup>L</sup>*.*CS*<sup>¬</sup>*<sup>f</sup> α*1 (¬*<sup>e</sup>*) = *<sup>U</sup>*.*Desf*[<sup>1</sup>−*α*--1 ,1−*α*-1 ](*e*).

**Proof.** It is straightforward.

**Theorem 1.** *Let* (*<sup>X</sup>*, *E*, *τ*) *be an IVFSTS*. *Suppose that the threshold intervals α*1, *α*2, ⊆ [0, <sup>1</sup>), *and β*1, *β*2, ⊆ (0, 1] *are given such that α* = [*<sup>α</sup>*1, *<sup>α</sup>*2] *and β* = [*β*1, *β*2]*, then*

	- (b) Let {*<sup>U</sup>*.*C*.*Sfi α* (*e*)}*<sup>i</sup>*∈*<sup>I</sup>* be a subfamily of *<sup>τ</sup>ue*,*α*. Then, we have *iU*.*C*.*Sfi α* (*e*) = *<sup>U</sup>*.*C*.*S*(∨˜ *i*∈*I fi*) *α*(*e*) ∈ *<sup>τ</sup>ue*,*α*, since ∨˜ *i*∈*IfiE* ∈ *τ*.

(c) Let *<sup>U</sup>*.*C*.*Sfα*(*e*) and *<sup>U</sup>*.*C*.*Sgα*(*e*) be two open sets in *<sup>τ</sup>ue*,*α*. Then, we have *<sup>U</sup>*.*CSfα*(*e*) ∩ *<sup>U</sup>*.*C*.*Sgα*(*e*) = *<sup>U</sup>*.*C*.*S*(*f*∧˜ *g*) *α*(*e*) ∈ *<sup>τ</sup>ue*,*α*, since *fE*∧˜ *gE*∈ *τ*. This completes the proof.

2. (a) That *X* ∈ <sup>B</sup>*lβ*(*e*) is implied from ∅*E* is in *τ*.

*y* (b) Let *<sup>L</sup>*.*C*.*Sfβ*(*e*) and *<sup>L</sup>*.*C*.*Sgβ*(*e*) in <sup>B</sup>*lβ*(*e*). Then, we have *<sup>L</sup>*.*C*.*Sfβ*(*e*) ∩ *<sup>L</sup>*.*C*.*Sgβ*(*e*) = *<sup>L</sup>*.*C*.*Sf*∨˜ *g β* (*e*) ∈ <sup>B</sup>*lβ*(*e*) that is implied form *f*∨˜ *g* ∈ *τ*.

**Theorem 2.** *Let* (*<sup>X</sup>*, *E*, *τ*) *be an IVFSTS*. *Suppose that the threshold intervals α*1, *α*2, ⊆ [0, <sup>1</sup>), *and β*1, *β*2, ⊆ (0, 1] *are given such that α* = [*<sup>α</sup>*1, *<sup>α</sup>*2] *and β* = [*β*1, *β*2]. *1.Thebinaryrelation<sup>τ</sup>e*,*<sup>α</sup>onXdefinedby*

$$\cdots \sim \varkappa\_{\mathcal{A}} \mathbf{x} \text{ -- } \cdots \text{ -- } \mathbf{m}\_{j} \text{--- } \mathbf{r}\_{\mathcal{I}}$$

$$\mathcal{Y} \succeq\_{\mathbf{c}, \mathcal{A}}^{\mathsf{T}} \mathbf{x} \Leftrightarrow [\forall V \in \mathsf{T}\_{\mathsf{c}, \mathcal{A}}^{\mathsf{u}} : \mathsf{x} \in V \Rightarrow \mathsf{y}]$$

 *is a preorder relation called α-upper-e preorder relation on X*.

*x*

*2. The binary relation <sup>τ</sup>*,*<sup>β</sup> e on X defined by*

$$y \preceq\_{\varepsilon}^{\tau, \emptyset} x \Leftrightarrow [\forall \mathcal{U} \in \pi^I\_{\varepsilon, \emptyset} : x \in \mathcal{U} \Rightarrow y \in \mathcal{U}]$$

 : *x*

 *y* ∈ *V*]

*is a preorder relation called β-lower-e preorder relation on X*.


**Theorem 3.** *Let* (*<sup>X</sup>*, *E*, *τ*) *be an IVFSTS*. *Suppose that the threshold intervals α*1, *α*2, ⊆ [0, <sup>1</sup>), *and β*1, *β*2, ⊆ (0, 1] *are given such that α* = [*<sup>α</sup>*1, *<sup>α</sup>*2] *and β* = [*β*1, *β*2].

*1. The binary relation τe*,*α*,*defined by*

$$y \simeq\_{\mathfrak{c}, \mathfrak{a}}^{\mathsf{T}} \mathfrak{x} \Leftrightarrow [y \succeq\_{\mathfrak{c}, \mathfrak{a}}^{\mathsf{T}} \mathfrak{x}, \mathfrak{x} \succeq\_{\mathfrak{c}, \mathfrak{a}}^{\mathsf{T}} y]]$$

*is an equivalence relation over X*. *If y τe*,*<sup>α</sup> x*, *then we say x and y are α-upper equivalent with to respect to the parameter e*.

*The equivalence relation τe*,*α*, *generates the partition <sup>P</sup><sup>τ</sup>e*,*<sup>α</sup> of X where the equivalence classes are defined as* [*x*]*<sup>τ</sup>e*,*<sup>α</sup>* = {*z* ∈ *X* : *z τe*,*<sup>α</sup> x and are called α-upper-e equivalence classes.*

*2. The binary relation τ*,*β e* , *where β* = [*β*1, *β*2],

$$y \asymp\_{\mathfrak{c}}^{\mathsf{r}, \mathfrak{c}} \mathfrak{x} \Leftrightarrow [y \asymp\_{\mathfrak{c}}^{\mathsf{r}, \mathfrak{c}} \mathsf{x}, \mathsf{x} \xleftarrow{\mathfrak{c}}\_{\mathfrak{c}}^{\mathsf{r}, \mathfrak{c}} \ y]$$

*is an equivalence relation over X*. *If y <sup>τ</sup><sup>e</sup>*,*β x*, *then we say x and y are* [*β*1, *β*2]*-lower equivalent with to respect to the parameter e*. *The equivalence relation τ*,*β e* , *generates the partition <sup>P</sup><sup>τ</sup>*,*<sup>β</sup> e of X where the equivalence classes are defined as* [*x*]*<sup>τ</sup>*,*<sup>β</sup> e* = {*z* ∈ *X* : *z τ*,*β e x and are called β-lower-e equivalence classes.*

#### **Proof.** It is straightforward.

#### *Preorder and Equivalence Matrices*

Now, let the finite sets *X* = {*<sup>x</sup>*1, ··· , *xm*} and *E* = {*<sup>e</sup>*1, ··· ,*en*} be given as the sets of objects and parameters. Then, the previous properties can be represented by using the matrix form of *IVFS* sets as the following.

Take an *IVFS* set *fE* over *X*. First, for any 1 ≤ *i* ≤ *m* and 1 ≤ *t* ≤ *n*, the concepts of *α*-upper-*et* and *β*-lower-*et* matrices of *fE*, where *α*, *β* ⊆ I, can be formulated as the following two matrices (or row vectors)

$$\text{LI.C.}\,S\_{\mathfrak{a}}\mathbf{c}\_{t}^{f} = [\mathfrak{u}\_{j}^{f}(\mathbf{c}\_{t},\mathbf{a})]\_{1\times m} = \begin{cases} 1 & \text{if } f\_{\mathfrak{e}\_{l}}^{-}(\mathbf{x}\_{i}) > a\_{1\prime}f\_{\mathfrak{e}\_{l}}^{+}(\mathbf{x}\_{i}) > a\_{2} \\ 0 & \text{if } f\_{\mathfrak{e}\_{l}}^{-}(\mathbf{x}\_{i}) \le a\_{1\prime}f\_{\mathfrak{e}\_{l}}^{+}(\mathbf{x}\_{i}) \le a\_{2} \end{cases} \tag{1}$$

and

$$L.C.S\_{\beta}e\_{t}^{f} = [l\_{i}^{f}(e\_{t}, \beta)]\_{1 \times m} = \begin{cases} \begin{array}{l} 0 & \text{if } f\_{e\_{l}}^{-}(\mathbf{x}\_{i}) \ge \beta\_{1}, f\_{e\_{l}}^{+}(\mathbf{x}\_{i}) \ge \beta\_{2} \\ 1 & \text{if } f\_{e\_{l}}^{-}(\mathbf{x}\_{i}) < \beta\_{1}, f\_{e\_{l}}^{+}(\mathbf{x}\_{i}) < \beta\_{2} \end{array} \end{cases} \tag{2}$$

where *α* = [*<sup>α</sup>*1, *<sup>α</sup>*1] and *β* = [*β*1, *β*2] are the given threshold vectors.

Then, obviously, for any *et* ∈ *E*, the topologies *<sup>τ</sup>uet*,*<sup>α</sup>* and *<sup>τ</sup>let*,*<sup>β</sup>* can be represented by the collections

$$\tau\_{\mathfrak{e}\_t,\alpha}^{\mathfrak{u}} = \{ [\mathfrak{u}\_i^f(\mathfrak{e}\_t,\alpha)]\_{1\times m} : \mathfrak{a} \subseteq [0,1), f\_E \in \mathfrak{r}, 1 \le i \le m \} $$

and

$$\pi\_{\mathfrak{e}\_{t\cdot\mathfrak{F}}}^{l} = \{ [l\_i^f(\mathfrak{e}\_{t\prime}\beta)]\_{1\times m} : \beta \subseteq (0,1], f\_{\mathcal{E}} \in \mathfrak{r}, 1 \le i \le m \} $$

where *τ* is the *IVFST* on *X*.

Accordingly, the preorderings *<sup>τ</sup>et*,*<sup>α</sup>* and *<sup>τ</sup>*,*<sup>β</sup> et* can be represented by

$$\iota \ge\_{\iota} \succeq\_{\iota\_t, \alpha}^{\pi} \mathfrak{x}\_{\jmath} \Leftrightarrow \left[ \forall f\_E \in \pi : \iota\_{\jmath}^f(\mathfrak{e}\_{t\iota}\alpha) = 1 \Rightarrow \iota\_{\iota}^f(\mathfrak{e}\_{t\iota}\alpha) = 1 \right]$$

and

$$\mathbf{x}\_{i} \preceq\_{c\_{t}}^{\tau,\beta} \mathbf{x}\_{j} \Leftrightarrow [\forall f\_{E} \in \tau: l\_{j}^{f}(\mathfrak{e}\_{t},\beta) = 1 \Rightarrow l\_{i}^{f}(\mathfrak{e}\_{t},\beta) = 1]$$

where *xi*, *xj* ∈ *X*.

The matrix forms of the preorderings *<sup>τ</sup>et*,*<sup>α</sup>* and *<sup>τ</sup>*,*<sup>β</sup> et* are used to define two comparison matrices *<sup>G</sup>α*(*et*)=[*gα*(*et*)*ij*]*m*×*m* and *<sup>S</sup>β*(*et*)=[*<sup>s</sup>β*(*et*)*ij*]*m*×*m*, which are two square matrices whose rows and columns are labeled by the objects of *X*, as below.

**Definition 5.** *Consider the binary relations <sup>τ</sup>et*,*<sup>α</sup> and <sup>τ</sup>*,*<sup>β</sup> et and threshold intervals α* = [*<sup>α</sup>*1, *<sup>α</sup>*2], *β* = [*β*1, *β*2 ⊆ I*. Then, we define*

$$\mathbf{G}\_{\mathfrak{a}}(\mathfrak{e}\_{l}) = [\mathbf{g}\_{\mathfrak{a}}(\mathfrak{e}\_{l})\_{\vec{ij}}]\_{\mathfrak{m}\times\mathfrak{m}} : \mathbf{g}\_{[a\_{1},a\_{2}]}(\mathfrak{e}\_{l})\_{\vec{ij}} = \begin{cases} 1 & \text{if } \mathfrak{x}\_{i} \succeq\_{\mathfrak{e}\mathfrak{e}\_{l},\mathfrak{a}}^{\mathsf{T}} \mathbf{x}\_{\mathfrak{j}} \\ 0 & \text{otherwise} \end{cases} \tag{3}$$

*and*

$$S\_{\beta}(\mathbf{c}\_{t}) = [\mathbf{s}\_{\beta}(\mathbf{c}\_{t})\_{ij}]\_{\mathbf{m} \times \mathbf{m}} : s\_{[\beta\_{1}, \beta\_{2}]}(\mathbf{c}\_{t})\_{ij} = \begin{cases} 1 & \text{if } \mathbf{x}\_{i} \stackrel{\precsim\_{t}^{\pi}\mathcal{C}}{\sim}\_{c\_{t}} \mathbf{x}\_{j} \\\ 0 & \text{otherwise} \end{cases} \tag{4}$$

**Proposition 2.** *Let* (*<sup>X</sup>*, *E*, *τ*) *be an IVFST and <sup>G</sup>α*(*e*) *and <sup>S</sup>β*(*e*) *be two matrices defined in Equations* (3) *and* (4)*. Then,*


*where i*, *j*, *k* ∈ {1, . . . , *m*}

**Proof.** It is straightforward.

**Proposition 3.** *Let* (*<sup>X</sup>*, *E*, *τ*) *be an IVFSTS and α*, *β* ⊆ I, *where α* = [*<sup>α</sup>*1, *<sup>α</sup>*1] *and β* = [*β*1, *β*2] *are the threshold intervals, then*

*1. <sup>G</sup>α*(*et*) *is an identity matrix if and only if* ¬(*xi <sup>τ</sup>etα xj*), ∀*i*, *j* = 1, . . . , *m and i* = *j*.


**Proof.** It is straightforward.

**Proposition 4.** *Let* (*<sup>X</sup>*, *E*, *τ*) *be an IVFSTS and α*, *β* ⊆ I, *where α* = [*<sup>α</sup>*1, *<sup>α</sup>*1], *β* = [*β*1, *β*2] *are the threshold intervals, then*


*where IUm* , *ILm are the upper and lower triangular matrix, respectively.*

**Proof.** It is straightforward.

Analogously, the equivalence relations *τet*,*<sup>α</sup>* and *τ*,*β et* can be applied to compute the following two square matrices

*<sup>E</sup>Uα* (*et*)=[*euα*(*et*)*ij*]*m*×*<sup>m</sup>* and *ELβ* (*et*)=[*elβ*(*et*)*ij*]*m*×*m*, respectively, where *α* = [*<sup>α</sup>*1, *<sup>α</sup>*2], *β* = [*β*1, *β*2] ⊆ I.

**Definition 6.** *Consider the binary relations τet*,*<sup>α</sup> and τ*,*β et and threshold intervals α* = [*<sup>α</sup>*1, *<sup>α</sup>*2], *β* = [*β*1, *β*2 ⊆ I*. We define*

$$E\_a^{II}(e\_l) = [e\_a^{\mu}(e\_l)\_{ij}]\_{m \times m} : e\_a^{\mu}(e\_l)\_{ij} = \begin{cases} 1 & \text{if } \mathbf{x}\_i \xrightleftarrow{\square\_{\mathbf{f}\_l,a}^{\mathsf{T}}} \mathbf{x}\_j\\ 0 & \text{otherwise} \end{cases} \tag{5}$$

*and*

$$E^L\_{\beta}(e\_t) = [\mathbf{e}^l\_{\beta}(e\_t)\_{ij}]\_{m \times m} : \mathbf{e}^l\_{\beta}(e\_t)\_{ij} = \begin{cases} 1 & \text{if } \mathbf{x}\_i \xleftarrow{\mathbf{r}\_t^\eta \delta} \mathbf{x}\_j \\ 0 & \text{otherwise} \end{cases} \tag{6}$$

**Proposition 5.** *Let* (*<sup>X</sup>*, *E*, *τ*) *be an IVFST and <sup>E</sup>Uα* (*et*) *and ELβ* (*et*) *be the comparison matrices defined in Equations* (5) *and* (6)*. Then,*


*where i*, *j*, *k* ∈ {1, ··· , *m*}

**Proof.** It is straightforward.

#### **4. An Application in Decision-Making Problems**

The main task in decision making methods is to rank the given candidates to find the optimum choice. Since the proposed preorderings, given in Section 3, are not total or linear, we define a score function *S* based on the entries of defined comparison matrices to obtain a new ranking system of objects according to preorderings *<sup>τ</sup>et*,*<sup>α</sup>* and *<sup>τ</sup>*,*<sup>β</sup> et*.

**Definition 7.** *Let X and E be the universal sets of objects and parameters, respectively, and α*, *β* ⊆ I, *where α* = [*<sup>α</sup>*1, *<sup>α</sup>*1] *and β* = [*β*1, *β*2]*, are the threshold intervals. The mapping S* = *X* → R *defined by*

$$S(\mathbf{x}\_i) = S\_i = \sum\_{t=1}^n \left( \left[ \sum\_{j=1}^m g\_a(e\_t)\_{ij} - \sum\_{j=1}^m e\_a^\mu(e\_t)\_{ij} \right] - \left[ \sum\_{j=1}^m s\_{\not\!p}(e\_t)\_{ij} - \sum\_{j=1}^m e\_{\not\!p}^l(e\_t)\_{ij} \right] \right)$$

*where xi* ∈ *X and Si is score value of object xi.*

**Example 1.** *Suppose that X* = {*<sup>o</sup>*1, *o*2, *o*3, *<sup>o</sup>*5} *be a set of 5 hotels in Langkawi and E* = {*<sup>e</sup>*1, ... ,*e*4} *be a set of parameters where for any t* = 1, ... , 4 *the parameter et stands for "location", "cleanliness", "facilities", and " food", respectively. Reviewers are classified into three groups: couples, solo travelers, and a group of friends. We consider these groups of reviewers as three different decision-makers, f*1, *f*2, *f*<sup>3</sup>*, characterized based on the criteria et* ∈ *E. These three groups provide the following three IVFS matrices f*<sup>1</sup>*E*, *f*2*E*, *f*3*E*.

Step 1. The following three interval-valued fuzzy soft set *fsE*(*s* = 1, 2, 3) that are given in Tables 1–3.

**Table 1.** *f*<sup>1</sup>*E*.


**Table 2.** *f*2*E*.


**Table 3.** *f*3*E*.


Step 2. Assume that [*<sup>α</sup>*1, *<sup>α</sup>*2]=[0.3, 0.6] and [*β*, *β*2 = [0.2, 0.4]. Step3.Theuppercrispmatrices andlowercrispmatrices,asbelow:


Step 4. The upper topology and lower topology are shown in Tables 4 and 5.


**Table 4.** *α*-Upper-*et* topology; *α* = [*<sup>α</sup>*1, *<sup>α</sup>*2], *t* = 1, . . . , 4.

**Table 5.** *β*-Lower-*et* topology; *β* = [*β*1, *β*2], *t* = 1, . . . , 4.


Step 5. The comparison matrices *<sup>G</sup>*(*et*, *<sup>α</sup>*), *<sup>S</sup>*(*et*, *β*), *<sup>E</sup><sup>U</sup>*(*et*, *α*) and *<sup>E</sup><sup>U</sup>*(*et*, *<sup>α</sup>*), over *X* where *α* = [*<sup>α</sup>*1, *<sup>α</sup>*2], *β* = [*β*1, *β*2], *t* = 1, . . . , 4 as below:


*<sup>E</sup><sup>U</sup>*(*<sup>e</sup>*2, [0.3, 0.6]) = ⎡ ⎢⎢⎢⎢⎣ 10001 01100 01100 00001 10011 ⎤ ⎥⎥⎥⎥⎦*EL*(*e*2, [0.2, 0.4) = ⎡ ⎢⎢⎢⎢⎣ 11111 11111 11111 11111 11111 ⎤ ⎥⎥⎥⎥⎦ *<sup>E</sup><sup>U</sup>*(*<sup>e</sup>*3, [0.3, 0.6]) = ⎡ ⎢⎢⎢⎢⎣ 10010 01000 00101 10010 00101 ⎤ ⎥⎥⎥⎥⎦*EL*(*e*3, [0.2, 0.4) = ⎡⎢⎢⎢⎢⎣ 11000 11000 00111 00111 00111 ⎤⎥⎥⎥⎥⎦ *<sup>E</sup><sup>U</sup>*(*<sup>e</sup>*4, [0.3, 0.6]) = ⎡ ⎢⎢⎢⎢⎣ 10000 01000 00110 00110 00001 ⎤⎥⎥⎥⎥⎦*EL*(*e*4, [0.2, 0.4) = ⎡⎢⎢⎢⎢⎣ 11010 11010 00101 11010 00101 ⎤⎥⎥⎥⎥⎦

Step 6. By using Definition (2), we have,

$$S\_1 = r\_1(\varepsilon\_1; [0.3, 0.6], [0.2, 0.4]) + r\_1(\varepsilon\_2; [0.3, 0.6], [0.2, 0.4]) + r\_1(\varepsilon\_3; [0.3, 0.6], [0.2, 0.4])$$

$$+ r\_1(\varepsilon\_4; [0.3, 0.6], [0.2, 0.4] = 1.$$

Similarly, *S*2 = 5, *S*3 = 0, *S*4 = −1, *S*6 = 5.

Step 7. Then, the ordering is obtained as below

$$o\_2 \asymp o\_5 \succeq o\_1 \succeq o\_3 \succeq o\_4$$

Steps 8 and 9. Accordingly, *o*2 and *o*5 can be the best objects (Acceptance region), while *o*4 not be selected(Rejection region), and *o*1, *o*3 cannot be judged(Boundary region).

#### *4.1. Comparison with Existing Methods*

In this section, we will apply and compare present method and other methods [25,43,44] using real-life example via datasets given in [47] Table 8 from the www.weather.com.cn website. (accessed on 15 May 2021).

**Example 2.** *Let an IFVSs fE describes a family who wants to go to a city in China. Suppose that the weather provides a forecast for fifteen cities in China during the holiday, X* = {*<sup>o</sup>*1, ... , *<sup>o</sup>*15}, *which is shown in Table 6. Suppose that the data of weather forecast describes five parameters E* = {*<sup>e</sup>*1,*e*2,*e*3,*e*4,*e*5}. *Parameters et*, *t* = 1, ... , 5, *stand for "temperature", "air quality index", "levels of ultraviolet radiation", "wind speed", "precipitation", respectively.*

Step 1. The *IVFSs fE* is given in Table 6. Step 2. Suppose that *α* = [0.67, 0.92], [0.75, 0.94], [0.66, 0.92], [0.49, 0.75], [0.96, 0.99] *β* = [0.14, 0.8], [0.37, 0.77], [0.25, 0.76], [0.26, 0.76], [0.67, 1], where *α* = [*<sup>α</sup>*1, *<sup>α</sup>*2], *β* = [*β*1, *β*2]


**Table 6.** Table for *fE*.

Steps 3 and 4. The *α*-Upper-*et* Crisp and *β*-Lower-*et* Crisp; the *α*-Upper-*et* Topology and *β*-Lower-*et* Topology (where (*t* = 1, . . . , 5)) as shown in Tables 7–10.

**Table 7.** *α*-Upper-*et*; *t* = 1, . . . , 5.


**Table 8.** *β*-Lower-*et*; *t* = 1, . . . , 5.


**Table 9.** *α*-Upper-*et* topology; *t* = 1, . . . , 5.


**Table 10.** *β*-Lower-*et* topology; *t* = 1, . . . , 5.


Step 5. The comparison matrices *<sup>G</sup>*(*et*, *<sup>α</sup>*), *<sup>S</sup>*(*et*, *β*), *<sup>E</sup><sup>U</sup>*(*et*, *α*) and *<sup>E</sup><sup>L</sup>*(*et*, *β*), where *α* = [*<sup>α</sup>*1, *<sup>α</sup>*2], *β* = [*β*1, *β*2], *t* = 1, . . . , 5 are below: *<sup>G</sup>*(*<sup>e</sup>*1, [0.67,0.92]) and *<sup>L</sup>*(*<sup>e</sup>*1, [0.13,0.8])


*<sup>G</sup>*(*<sup>e</sup>*2, [0.75, 0.94]) and *<sup>L</sup>*(*<sup>e</sup>*2, [0.37, 0.77])


*<sup>G</sup>*(*<sup>e</sup>*3, [0.66, 0.92]) and *<sup>L</sup>*(*<sup>e</sup>*3, [0.25, 0.76])

Now, we compute matrices *<sup>E</sup><sup>U</sup>*(*et*, *<sup>α</sup>*), *<sup>E</sup><sup>L</sup>*(*et*, *β*)*α* = [*<sup>α</sup>*1, *<sup>α</sup>*2], *β* = [*β*1, *β*2], *t* = 1, . . . , 5 *<sup>E</sup><sup>U</sup>*(*<sup>e</sup>*1, [0.67, 0.92]) and *<sup>E</sup><sup>L</sup>*(*<sup>e</sup>*1, [0.13, 0.8])


*<sup>E</sup><sup>U</sup>*(*<sup>e</sup>*2, [0.75, 0.94]) and *<sup>E</sup><sup>L</sup>*(*<sup>e</sup>*2, [0.37, 0.77])


## *<sup>E</sup><sup>U</sup>*(*<sup>e</sup>*3, [0.66, 0.92]) and *<sup>E</sup><sup>L</sup>*(*<sup>e</sup>*3, [0.25, 0.76])


*<sup>E</sup><sup>U</sup>*(*<sup>e</sup>*4, [0.48, 0.74]) and *<sup>E</sup><sup>L</sup>*(*<sup>e</sup>*4, [0.26, 0.76])


010011 1 11010101 101100 0 00101010 010011 1 11010101 101100 0 00101010 010011 1 11010101 111110110111111111110110111111111110110111111111110110111111111110110111111

> Step 6. By using Definition (7), we have:

*S*1 = *<sup>r</sup>*1(*<sup>e</sup>*1; [0.67, 0.92], [0.13, 0.8]) + *<sup>r</sup>*1(*<sup>e</sup>*2, [0.75, 0.94], [0.37, 0.77]) + *r*1 (*<sup>e</sup>*3, [0.66, 0.92], [0.25, 0.76]) + *<sup>r</sup>*1(*<sup>e</sup>*4, [0.48, 0.74], [0.26, 0.76] + *<sup>r</sup>*1(*<sup>e</sup>*5, [0.96, 0.99], [0.67, 1] = 0 + 0 − 3 + 0 + 0 = −3

Similarly, we have:

$$S\_2 = -12, S\_3 = -5, S\_4 = -9, S\_5 = 6, S\_6 = 7, S\_7 = 15, S\_8 = 42, S\_9 = 2,$$

$$S\_8 = 42, S\_9 = 2, S\_{10} = 8, S\_{11} = -16, S\_{12} = 6, S\_{13} = -3, S\_{14} = -9, S\_{15} = 0.$$

Step 7. We have the following ordering system on *X* :

*o*8 *o*7 *o*10 *o*6 *o*12 ! *o*5 *o*9 *o*15 *o*13 ! *o*1 *o*14 ! *o*4 *o*2 *o*11.

Steps 8 and 9. Then, from the corresponding object, we obtain, *o*8 to be the best object (Acceptance region), while *o*11 is not selected (Rejection region) and others options (*<sup>o</sup>*7, *o*10, *o*6, *o*12, *o*5, *o*9, *o*15,

*o*13, *o*1, *o*14, *o*4, *<sup>o</sup>*2) cannot be judged(Boundary region).

**Example 3.** *(Example 2) Let us discuss Example 2 compared to existing methods proposed in [25,43,44] according to the ranking of objects.*

*Yang et al. [25] defined the function score value as simply the total of lower and upper membership degrees of objects concerning each parameter. Ma et al. [44] applied Yang's Algorithm 1, which is given in [25] to solve Example 2 and showed the score value as follows: o*8 *o*6 *o*14 *o*5 *o*4 *o*10 *o*12 *o*3 *o*7 *o*13 *o*15 *o*11 *o*9 *o*2 *o*1.

*Ma et al. [44] proposed a new efficient decision-making algorithm by using added objects. By using Algorithm 3 Section 4 in [44], Example 2 was solved and the score value for all objects was obtained as follows o*8 *o*6 *o*14 *o*5 *o*4 *o*10 *o*12 *o*3 *o*7 *o*13 *o*15 *o*11 *o*9 *o*2 *o*1.

*Ma et al. [43] applied a new decision-making algorithm, based on the average table and the antithesis table—the antithesis the table has symmetry between the objects. Applying Algorithm in [43], Section 3, to solve the Example 2, the following ranking of objects is obtained o*8 *o*6 *o*5 *o*14 *o*4 *o*12 *o*10 *o*3 *o*13 *o*7 *o*15 *o*11 *o*2 *o*9 *o*1.

*The comparison results among the present method and methods in [25,43,44] are given in Figure 1.*

**Figure 1.** Comparison methods.
