# A Generalization of Trapezoidal Fuzzy Numbers Based on Modal Interval Theory

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

#### 2.1. Modal Intervals

#### 2.2. Fuzzy Numbers

## 3. Modal Interval Trapezoidal Fuzzy Numbers

**Definition**

**1.**

**Definition**

**2.**

- A is a proper-improper MITFN (MITFN${}_{P}^{I}$) if $supp\left(A\right)$ is a proper interval and $core\left(A\right)$ is an improper interval. We denote it by $A=\left(set\left(A\right),\exists ,\forall \right)$;
- A is an improper-proper MITFN (MITFN${}_{I}^{P}$) if $supp\left(A\right)$ is an improper interval and $core\left(A\right)$ is a proper interval. We denote it by $A=\left(set\left(A\right),\forall ,\exists \right)$;
- A is a proper-proper MITFN (MITFN${}_{P}^{P}$) if both the support and core of A are proper intervals. We denote it by $A=\left(set\left(A\right),\exists ,\exists \right)$;
- A is an improper-improper MITFN (MITFN${}_{I}^{I}$) if both the support and core of A are improper intervals. We denote it by $A=\left(set\left(A\right),\forall ,\forall \right)$.

**Proposition**

**1.**

**Proof.**

**Proposition**

**2.**

- A is an MITFN${}_{P}^{I}\iff {a}_{1}\le {a}_{3}\le {a}_{2}\le {a}_{4}$;
- A is an MITFN${}_{I}^{P}\iff {a}_{4}\le {a}_{2}\le {a}_{3}\le {a}_{1}$;
- A is an MITFN${}_{P}^{P}\iff {a}_{1}\le {a}_{2}\le {a}_{3}\le {a}_{4}$;
- A is an MITFN${}_{I}^{I}\iff {a}_{4}\le {a}_{3}\le {a}_{2}\le {a}_{1}$.

**Proof.**

- If A is an MITFN${}_{P}^{I}$, then $supp\left(A\right)=\left[{a}_{1},{a}_{4}\right]$ is proper, that is, ${a}_{1}\le {a}_{4}$ and $core\left(A\right)=\left[{a}_{2},{a}_{3}\right]$ is improper, that is, ${a}_{3}\le {a}_{2}.$As Definition 1 establishes $set\left(\left[{a}_{2},{a}_{3}\right]\right)\subseteq set\left(\left[{a}_{1},{a}_{4}\right]\right)$, in this case, $set\left(\left[{a}_{2},{a}_{3}\right]\right)=\left[{a}_{3},{a}_{2}\right]$ and $set\left(\left[{a}_{1},{a}_{4}\right]\right)=\left[{a}_{1},{a}_{4}\right]$ which means $\left[{a}_{3},{a}_{2}\right]\subseteq $ $\left[{a}_{1},{a}_{4}\right]$, that is ${a}_{3}\ge {a}_{1}$ and ${a}_{2}\le {a}_{4}.$From ${a}_{1}\le {a}_{4},$ ${a}_{3}\le {a}_{2},{a}_{3}\ge {a}_{1},{a}_{2}\le {a}_{4}$, it follows that ${a}_{1}\le {a}_{3}\le {a}_{2}\le {a}_{4}.$
- If A is an MITFN${}_{I}^{P}$, $supp\left(A\right)=\left[{a}_{1},{a}_{4}\right]$ is an improper interval, that is, ${a}_{4}\le {a}_{1}$ and $core\left(A\right)=\left[{a}_{2},{a}_{3}\right]$ is a proper interval, that is, ${a}_{2}\le {a}_{3}.$As it must be that $set\left(\left[{a}_{2},{a}_{3}\right]\right)\subseteq set\left(\left[{a}_{1},{a}_{4}\right]\right)$ and $set\left(\left[{a}_{2},{a}_{3}\right]\right)=\left[{a}_{2},{a}_{3}\right],$ $set\left(\left[{a}_{1},{a}_{4}\right]\right)=\left[{a}_{4},{a}_{1}\right],$ so it follows that $\left[{a}_{2},{a}_{3}\right]\subseteq $ $\left[{a}_{4},{a}_{1}\right]$, that is, ${a}_{2}\ge {a}_{4}$ and ${a}_{3}\le {a}_{1}.$From ${a}_{4}\le {a}_{1},$ ${a}_{2}\le {a}_{3},{a}_{2}\ge {a}_{4},{a}_{3}\le {a}_{1}$, it follows that ${a}_{4}\le {a}_{2}\le {a}_{3}\le {a}_{1}.$

**Proposition**

**3.**

**Proof.**

- If both $supp\left(A\right)$ and $core\left(A\right)$ are proper intervals, then ${a}_{1}\le {a}_{4}$ and ${a}_{2}\le {a}_{3}$ so ${a}_{1}+{a}_{2}\le {a}_{3}+{a}_{4}$ and $\left[\frac{{a}_{1}+{a}_{2}}{2},\frac{{a}_{3}+{a}_{4}}{2}\right]$ is a proper interval.
- If both $supp\left(A\right)$ and $core\left(A\right)$ are improper intervals, then ${a}_{1}\ge {a}_{4}$ and ${a}_{2}\ge {a}_{3}$ so ${a}_{1}+{a}_{2}\ge {a}_{3}+{a}_{4}$ and $\left[\frac{{a}_{1}+{a}_{2}}{2},\frac{{a}_{3}+{a}_{4}}{2}\right]$ is an improper interval.
- If $supp\left(A\right)$ is a proper interval and $core\left(A\right)$ is an improper one, then $set\left(core\left(A\right)\right)=\left[{a}_{3},{a}_{2}\right]$ and $set\left(supp\left(A\right)\right)=\left[{a}_{1},{a}_{4}\right]$. As $set\left(core\left(A\right)\right)\subseteq set\left(supp\left(A\right)\right),$ it holds that ${a}_{3}\ge {a}_{1}$ and ${a}_{2}\le {a}_{4}.$ Thus, $\left[\frac{{a}_{1}+{a}_{2}}{2},\frac{{a}_{3}+{a}_{4}}{2}\right]$ is a proper interval.
- If $supp\left(A\right)$ is an improper interval and $core\left(A\right)$ is a proper interval, then $set\left(core\left(A\right)\right)=\left[{a}_{2},{a}_{3}\right]$ and $set\left(supp\left(A\right)\right)=\left[{a}_{4},{a}_{1}\right]$. As $set\left(core\left(A\right)\right)\subseteq set\left(supp\left(A\right)\right),$ it holds that ${a}_{2}\ge {a}_{4}$ and ${a}_{3}\le {a}_{1}.$ Thus, $\left[\frac{{a}_{1}+{a}_{2}}{2},\frac{{a}_{3}+{a}_{4}}{2}\right]$ is an improper interval.

**Dual operator**. If $A=\left({A}^{\prime},{Q}_{1},{Q}_{2}\right)\in $ $T{I}^{\ast}\left(\mathbb{R}\right)$ the dual operator on A, $dual\left(A\right)$ is defined as:

**Proper and improper operators**. If $A=\left({A}^{\prime},{Q}_{1},{Q}_{2}\right)\in $ $T{I}^{\ast}\left(\mathbb{R}\right)$ we define the proper operator on $A,$ as $prop\left(A\right)=\left({A}^{\prime},\exists ,\exists \right)$ and the improper operator on A as $impr\left(A\right)=\left({A}^{\prime},\forall ,\forall \right)$. Using the canonical notation, if $A=\left(\left[{a}_{1},{a}_{4}\right],\left[{a}_{2},{a}_{3}\right]\right)$, then:

#### 3.1. Graphical Representation of an MITFN in the Interval Plane

#### 3.2. The Lattice of MITFNs

**Lemma**

**1.**

**Proof.**

**Proposition**

**4.**

- If $A=\left({A}^{\prime},\exists ,\exists \right)$ and $B=\left({B}^{\prime},\exists ,\exists \right)$, then$A\subseteq B\iff set\left(supp\left(A\right)\right)\subseteq set\left(supp\left(B\right)\right)$ and $set\left(core\left(A\right)\right)\subseteq set\left(core\left(B\right)\right)$;
- If $A=\left({A}^{\prime},\exists ,\exists \right)$ and $B=\left({B}^{\prime},\forall ,\forall \right)$, then$A\subseteq B\iff A=B=\left(p,p,p,p\right)$;
- If $A=\left({A}^{\prime},\exists ,\exists \right)$ and $B=\left({B}^{\prime},\exists ,\forall \right)$, then$A\subseteq B\iff set\left(supp\left(A\right)\right)\subseteq set\left(supp\left(B\right)\right)$ and $core\left(A\right)=core\left(B\right)=\left[p,p\right]$;
- If $A=\left({A}^{\prime},\exists ,\exists \right)$ and $B=\left({B}^{\prime},\forall ,\exists \right)$, then$A\subseteq B\iff A=B=\left(p,p,p,p\right)$;
- If $A=\left({A}^{\prime},\forall ,\forall \right)$ and $B=\left({B}^{\prime},\exists ,\exists \right)$, then$A\subseteq B\iff set\left(supp\left(A\right)\right)\cap set\left(supp\left(B\right)\right)\ne \varnothing $ and $set\left(core\left(A\right)\right)\cap set\left(core\left(B\right)\right)\ne \varnothing $;
- If $A=\left({A}^{\prime},\forall ,\forall \right)$ and $B=\left({B}^{\prime},\exists ,\forall \right)$, then$A\subseteq B\iff $ $set\left(supp\left(A\right)\right)\cap set\left(supp\left(B\right)\right)\ne \varnothing $ and $set\left(core\left(A\right)\right)\supseteq set\left(core\left(B\right)\right)$;
- If $A=\left({A}^{\prime},\forall ,\forall \right)$ and $B=\left({B}^{\prime},\forall ,\exists \right)$, then$A\subseteq B\iff set\left(supp\left(A\right)\right)\supseteq set\left(supp\left(B\right)\right)$ and $set\left(core\left(A\right)\right)\cap set\left(core\left(B\right)\right)\ne \varnothing $;
- If $A=\left({A}^{\prime},\exists ,\forall \right)$ and $B=\left({B}^{\prime},\exists ,\forall \right)$, then$A\subseteq B\iff set\left(supp\left(A\right)\right)\subseteq set\left(supp\left(B\right)\right)$ and $set\left(core\left(A\right)\right)\supseteq set\left(core\left(B\right)\right)$;
- If $A=\left({A}^{\prime},\exists ,\forall \right)$ and $B=\left({B}^{\prime},\forall ,\exists \right)$, then$A\subseteq B\iff A=B=\left(p,p,p,p\right)$;
- If $A=\left({A}^{\prime},\forall ,\exists \right)$ and $B=\left({B}^{\prime},\exists ,\forall \right)$, then$A\subseteq B\iff set\left(supp\left(A\right)\right)\cap set\left(supp\left(B\right)\right)\ne \varnothing $ and $core\left(A\right)=core\left(B\right)=\left[p,p\right].$

**Proof.**

**Definition**

**3.**

- $inf\left\{A,B\right\}=X$ if $X\subseteq A,X\subseteq B$ and if there exists a $D\in T{I}^{\ast}\left(\mathbb{R}\right)$ such that $D\subseteq A\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{4pt}{0ex}}$and $D\subseteq B$, then $D\subseteq X$;
- $sup\left\{A,B\right\}=Y$ if $A\subseteq Y,B\subseteq Y$ and if there exists a $D\in T{I}^{\ast}\left(\mathbb{R}\right)$ such that $A\subseteq D\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{4pt}{0ex}}$and $B\subseteq D$, then $Y\subseteq D.$

**Proposition**

**5.**

**Proof.**

- ${L}^{0}$ and ${L}^{1}$ are proper intervals.As ${L}^{0}$ and ${L}^{1}$ are proper intervals, then A and B are MITFNs${}_{P}^{P}$. Thus, ${A}^{1}\subseteq {A}^{0}$ and ${B}^{1}\subseteq {B}^{0}$ so ${L}^{1}\subseteq {L}^{0}$ and as ${L}^{1}$ and ${L}^{0}$ are proper intervals, $set\left({L}^{1}\right)\subseteq set\left({L}^{0}\right)$. If $X=\left({L}^{0},{L}^{1}\right)$, then $X\subseteq A$, $X\subseteq B$ and $X\in T{I}^{\ast}\left(\mathbb{R}\right)$. Moreover, if $D\in T{I}^{\ast}\left(\mathbb{R}\right)$ conforms to $D\subseteq A\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{4pt}{0ex}}$and $D\subseteq B$, then ${D}^{0}\subseteq {A}^{0}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{4pt}{0ex}},$ ${D}^{0}\subseteq {B}^{0}$, ${D}^{1}\subseteq {A}^{1}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{4pt}{0ex}}$and ${D}^{1}\subseteq {B}^{1}$ so ${D}^{0}\subseteq {A}^{0}\wedge {B}^{0}$ and ${D}^{1}\subseteq {A}^{1}\wedge {B}^{1}$. That is, $D\subseteq X$; thus, $\left({L}^{0},{L}^{1}\right)=$ Inf$\left\{A,B\right\}.$
- ${L}^{0}$ is a proper interval and ${L}^{1}$ an improper interval.As ${L}^{0}={A}^{0}\wedge {B}^{0}$ is a proper interval, ${L}^{1}={A}^{1}\wedge {B}^{1}$ is an improper interval and $set\left({A}^{1}\right)\subseteq set\left({A}^{0}\right),$ $set\left({B}^{1}\right)\subseteq set\left({B}^{0}\right),$ and it follows that ${L}^{1}\subseteq {L}^{0}.$ We distinguish the following two cases according to the inclusion set of ${L}^{1}$ and ${L}^{0}$:
- (a)
- If $set\left({L}^{1}\right)\subseteq set\left({L}^{0}\right)$, we can proceed as in the first case.
- (b)
- If $set\left({L}^{1}\right)\u2288set\left({L}^{0}\right)$, then let us prove that $X=\left({L}^{0}\wedge {L}^{1},{L}^{1}\right)$ corresponds to $inf\left\{A,B\right\}$. Notice that, if ${L}^{1}\subseteq {L}^{0}$, then ${L}^{0}\wedge {L}^{1}={L}^{1}$; thus, $X=\left({L}^{1},{L}^{1}\right)$. It is obvious that $X\in T{I}^{\ast}\left(\mathbb{R}\right)$. Moreover, ${X}^{0}\subseteq {A}^{0}$ and ${X}^{0}\subseteq {B}^{0}$ as ${X}^{0}={L}^{1}\subseteq {L}^{0}$. In a similar way, ${X}^{1}\subseteq {A}^{1}$ and ${X}^{1}\subseteq {B}^{1}$. Therefore, $X\subseteq A$ and $X\subseteq B$.If $D\in T{I}^{\ast}\left(\mathbb{R}\right)$ conforms to $D\subseteq A\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{4pt}{0ex}}$and $D\subseteq B$, then ${D}^{1}\subseteq {A}^{1}\wedge {B}^{1}={L}^{1}.$ Notice that, as ${L}^{1}$ is an improper interval, ${D}^{1}$ will also be an improper interval and then $set\left({L}^{1}\right)\subseteq set\left({D}^{1}\right)$.We must prove that ${D}^{0}\subseteq {L}^{1}$ and, consequently, $D\subseteq X$.
- If ${D}^{0}$ is an improper interval, as $set\left({D}^{1}\right)\subseteq set\left({D}^{0}\right)$, it follows that ${D}^{0}\subseteq {D}^{1}.$ As ${D}^{1}\subseteq {L}^{1}$, then ${D}^{0}\subseteq {L}^{1}$.
- If ${D}^{0}$ is a proper interval, as $set\left({D}^{1}\right)\subseteq set\left({D}^{0}\right)={D}^{0}$ and ${D}^{0}\subseteq {L}^{0}$, it follows that $set\left({D}^{1}\right)\subseteq {L}^{0}.$ Using the inclusion $set\left({L}^{1}\right)\subseteq set\left({D}^{1}\right)$, we obtain $set\left({L}^{1}\right)\subseteq set\left({L}^{0}\right),$ which contradicts the hypothesis $set\left({L}^{1}\right)\u2288set\left({L}^{0}\right)$.

- ${L}^{0}$ is an improper interval and ${L}^{1}$ a proper one.As ${L}^{0}={A}^{0}\wedge {B}^{0}$ is an improper interval, ${L}^{1}={A}^{1}\wedge {B}^{1}$ is a proper interval and $set\left({A}^{1}\right)\subseteq set\left({A}^{0}\right),$ $set\left({B}^{1}\right)\subseteq set\left({B}^{0}\right)$, and it follows that $set\left({L}^{1}\right)\subseteq set\left({L}^{0}\right)$; thus, the demonstration follows as in the first case.
- ${L}^{0}$ and ${L}^{1}$ are improper intervals.
- (a)
- If ${L}^{1}\subseteq {L}^{0}$, then $set\left({L}^{0}\right)\subseteq set\left({L}^{1}\right)$. Let us prove that $X=\left({L}^{0}\wedge {L}^{1},{L}^{1}\right)$ corresponds to $inf\left\{A,B\right\}$. Notice that if ${L}^{1}\subseteq {L}^{0}$, then ${L}^{0}\wedge {L}^{1}={L}^{1}$ and thus $X=\left({L}^{1},{L}^{1}\right)$. It is obvious that $X\in T{I}^{\ast}\left(\mathbb{R}\right)$. Moreover, ${X}^{0}\subseteq {A}^{0}$ and ${X}^{0}\subseteq {B}^{0}$ as ${X}^{0}={L}^{1}\subseteq {L}^{0}$. In a similar way, ${X}^{1}\subseteq {A}^{1}$ and ${X}^{1}\subseteq {B}^{1}$. Therefore, $X\subseteq A$ and $X\subseteq B$.If $D\in T{I}^{\ast}\left(\mathbb{R}\right)$ conforms to $D\subseteq A\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{4pt}{0ex}}$and $D\subseteq B$, then ${D}^{1}\subseteq {A}^{1}\wedge {B}^{1}={L}^{1}$ and ${D}^{0}\subseteq {A}^{0}\wedge {B}^{0}={L}^{0}$. This implies that ${D}^{0}$ and ${D}^{1}$ are improper intervals.Moreover, as $set\left({D}^{1}\right)\subseteq set\left({D}^{0}\right)$, then due to the modality of ${D}^{0}$ and ${D}^{1}$ it will be the case that ${D}^{0}\subseteq {D}^{1}$ and as ${D}^{1}\subseteq {L}^{1}$, it follows that ${D}^{0}\subseteq {L}^{1}$. Thus, $X=\left({L}^{0}\wedge {L}^{1},{L}^{1}\right)=inf\left\{A,B\right\}.$
- (b)
- If ${L}^{0}\subseteq {L}^{1}$, then $set\left({L}^{1}\right)\subseteq set\left({L}^{0}\right)$. Let us prove that $X=\left({L}^{0},{L}^{1}\right)$ corresponds to $inf\left\{A,B\right\}$. If $X=\left({L}^{0},{L}^{1}\right)$, then $X\subseteq A$, $X\subseteq B$ and $X\in T{I}^{\ast}\left(\mathbb{R}\right)$. Moreover, if $D\in T{I}^{\ast}\left(\mathbb{R}\right)$ conforms to $D\subseteq A\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{4pt}{0ex}}$and $D\subseteq B$, then ${D}^{0}\subseteq {A}^{0}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{4pt}{0ex}},$ ${D}^{0}\subseteq {B}^{0}$, ${D}^{1}\subseteq {A}^{1}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{4pt}{0ex}}$and ${D}^{1}\subseteq {B}^{1}$ so ${D}^{0}\subseteq {A}^{0}\wedge {B}^{0}$ and ${D}^{1}\subseteq {A}^{1}\wedge {B}^{1}$. That is, $D\subseteq X$; thus, $\left({L}^{0},{L}^{1}\right)=$ Inf$\left\{A,B\right\}.$
- (c)
- If ${L}^{0}\le {L}^{1}$ or ${L}^{1}\le {L}^{0}$, then let us prove that $X=\left({L}^{0}\wedge {L}^{1},{L}^{1}\right)$ corresponds to $inf\left\{A,B\right\}$. Notice that ${L}^{0}\wedge {L}^{1}\subseteq {L}^{1}$ as ${L}^{0}\wedge {L}^{1}$ and ${L}^{1}$ are improper intervals, $set\left({L}^{1}\right)\subseteq set\left({L}^{0}\wedge {L}^{1}\right),$ thus $X\in T{I}^{\ast}\left(\mathbb{R}\right)$ and obviously $X\subseteq A$ and $X\subseteq B$.If $D\in T{I}^{\ast}\left(\mathbb{R}\right)$ conforms to $D\subseteq A\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{4pt}{0ex}}$and $D\subseteq B$, then ${D}^{1}\subseteq {A}^{1}\wedge {B}^{1}={L}^{1}$ and ${D}^{0}\subseteq {A}^{0}\wedge {B}^{0}={L}^{0}$. As ${L}^{1}$ and ${L}^{0}$ are both improper, ${D}^{1}$ and ${D}^{0}$ will also be improper intervals.Moreover, as $set\left({D}^{1}\right)\subseteq set\left({D}^{0}\right)$, then due to the modality of ${D}^{0}$ and ${D}^{1}$, it will be the case that ${D}^{0}\subseteq {D}^{1}$ and as ${D}^{1}\subseteq {L}^{1}$, it follows that ${D}^{0}\subseteq {L}^{1}$ and consequently ${D}^{0}\subseteq {L}^{0}\wedge {L}^{1}$. That is $D\subseteq X$, thus $X=\left({L}^{0}\wedge {L}^{1},{L}^{1}\right)=inf\left\{A,B\right\}$.

**Proposition**

**6.**

**Proof.**

- If $set\left({L}^{1}\right)\subseteq set\left({L}^{0}\right)$, then $set\left({M}^{1}\right)\subseteq set\left({M}^{0}\right)$ and so:$$\begin{array}{ccc}\hfill sup\left\{A,B\right\}& =& dual\left(du\left({A}^{0}\right)\wedge du\left({B}^{0}\right),du\left({A}^{1}\right)\wedge du\left({B}^{1}\right)\right)\hfill \\ & =& \left(du\left(du\left({A}^{0}\right)\wedge du\left({B}^{0}\right)\right),du\left(du\left({A}^{1}\right)\wedge du\left({B}^{1}\right)\right)\right)\hfill \\ & =& \left({A}^{0}\vee {B}^{0},{A}^{1}\vee {B}^{1}\right)=\left({M}^{0},{M}^{1}\right).\hfill \end{array}$$
- If $set\left({L}^{1}\right)\u2288set\left({L}^{0}\right)$, then$$\begin{array}{ccc}\hfill sup\left\{A,B\right\}& =& dual\left(du\left({A}^{0}\right)\wedge du\left({B}^{0}\right)\wedge du\left({A}^{1}\right)\wedge du\left({B}^{1}\right),du\left({A}^{1}\right)\wedge du\left({B}^{1}\right)\right)\hfill \\ & =& \left(du\left(du\left({A}^{0}\right)\wedge du\left({B}^{0}\right)\right),du\left(du\left({A}^{1}\right)\wedge du\left({B}^{1}\right)\right)\right)\hfill \\ & =& \left({A}^{0}\vee {B}^{0}\vee {A}^{1}\vee {B}^{1},{A}^{1}\vee {B}^{1}\right)=\left({M}^{0}\vee {M}^{1},{M}^{1}\right).\hfill \end{array}$$

## 4. Interpretability of the Calculations

**Definition**

**4.**

**Proposition**

**7.**

**Proof.**

**Theorem**

**1.**

**Proof.**

**Corollary**

**1.**

**Corollary**

**2.**

**Definition**

**5.**

**Definition**

**6.**

**Example**

**1.**

- $\forall \alpha \in \left[0,\frac{3}{4}\right]\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\left(\forall a\in set\left({A}^{\alpha}\right)\right)\left(\forall x\in set\left({X}^{\alpha}\right)\right)\left(\exists b\in set\left({B}^{\alpha}\right)\right)$$a+x=b;$
- $\forall \alpha \in \left[\frac{3}{4},1\right]\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\left(\forall a\in set\left({A}^{\alpha}\right)\right)\left(\exists b\in set\left({B}^{\alpha}\right)\right)\left(\exists x\in set\left({X}^{\alpha}\right)\right)$$a+x=b.$

- $\forall \alpha \in \left[0,0.5\right]\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\left(\forall a\in set\left({A}^{\alpha}\right)\right)\left(\forall b\in set\left({B}^{\alpha}\right)\right)\left(\exists x\in set\left({X}^{\alpha}\right)\right)$$a+x=b;$
- $\forall \alpha \in \left[0.5,0.75\right]\phantom{\rule{4pt}{0ex}}\left(\forall a\in set\left({A}^{\alpha}\right)\right)\left(\exists b\in set\left({B}^{\alpha}\right)\right)\left(\exists x\in set\left({X}^{\alpha}\right)\right)$$a+x=b;$
- $\forall \alpha \in \left[0.75,1\right]\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\left(\forall a\in set\left({A}^{\alpha}\right)\right)\left(\forall x\in set\left({X}^{\alpha}\right)\right)\left(\exists b\in set\left({B}^{\alpha}\right)\right)$$a+x=b.$

## 5. Conclusions

## Author Contributions

## Conflicts of Interest

## References

- Zadeh, L.A. Fuzzy sets. Inf. Control
**1965**, 8, 338–353. [Google Scholar] [CrossRef] - Bashir, Z.; Wątróbski, J.; Rashid, T.; Sałabun, W.; Ali, J. Intuitionistic-fuzzy goals in zero-sum multi criteria matrix games. Symmetry
**2017**, 9, 158. [Google Scholar] [CrossRef] - Faizi, S.; Sałabun, W.; Rashid, T.; Wątróbski, J.; Zafar, S. Group decision-making for hesitant fuzzy sets based on characteristic objects method. Symmetry
**2017**, 9, 136. [Google Scholar] [CrossRef] - Faizi, S.; Rashid, T.; Sałabun, W.; Zafar, S.; Wątróbski, J. Decision making with uncertainty using hesitant fuzzy sets. Int. J. Fuzzy Syst.
**2017**, 1–11. [Google Scholar] [CrossRef] - Chen, J.; Huang, X. Dual hesitant fuzzy probability. Symmetry
**2017**, 9, 52. [Google Scholar] [CrossRef] - Hou, S.; Wang, H.; Feng, S. Attribute control chart construction based on fuzzy score number. Symmetry
**2016**, 8, 139. [Google Scholar] [CrossRef] - Sałabun, W.; Piegat, A. Comparative analysis of MCDM methods for the assessment of mortality in patients with acute coronary syndrome. Artif. Intell. Rev.
**2016**, 1–15. [Google Scholar] [CrossRef] - Bucolo, M.; Fortuna, L.; La Rosa, M. Complex dynamics through fuzzy chains. IEEE Trans. Fuzzy Syst.
**2004**, 12, 289–295. [Google Scholar] [CrossRef] - Gardeñes, E.; Mielgo, H. Modal intervals: Functions. In Polish Symposium on Interval and Fuzzy Mathematics; Zamenkhov’s University of Poznan: Poznan, Poland, 1986. [Google Scholar]
- Adillon, R.; Jorba, L. Quantified trapezoidal fuzzy numbers. J. Intell. Fuzzy Syst.
**2017**, 33, 601–611. [Google Scholar] [CrossRef] - Makó, Z. Real vector space of LR-fuzzy intervals with respect to the shape-preserving t-norm-based addition. Fuzzy Sets Syst.
**2012**, 200, 136–149. [Google Scholar] [CrossRef] - Warmus, M. Calculus of Approximations. Bull. Acad. Pol. Sci.
**1956**, 4, 253–257. [Google Scholar] - Sunaga, T. Theory of interval algebra and its application to numerical analysis. In Research Association of Applied Geometry (RAAG) Memoirs; Ggujutsu Bunken Fukuy-kai: Tokyo, Japan, 1958. [Google Scholar]
- Moore, R.E. Interval Analysis; Prentice Hall: Englewood, IL, USA, 1966. [Google Scholar]
- Nickel, K. Verbandtheoretische grundlagen der intervallmathematik. In Lecture Notes in Computer Science 29; Springer: Heildelberg, Germany, 1975; pp. 251–262. [Google Scholar]
- Alefeld, G.; Herzberger, J. Introduction to Interval Computations; Academic: New York, NY, USA, 1983. [Google Scholar]
- Kaucher, E. Algebraische erweiterungen der intervallrechnung unter erhaltung der ordnungs und verbandsstrukturen. Comput. Suppl.
**1977**, 1, 65–79. [Google Scholar] - Gardeñes, E.; Mielgo, H.; Trepat, A. Modal intervals: Reasons and ground semantics, interval mathematics. In Lecture Notes in Computer Science 212; Springer: Heildelberg, Germany, 1985; pp. 27–35. [Google Scholar]
- Wang, Y. Stochastiics dynamics simulation with generalized interval probability. Int. J. Comput. Math.
**2015**, 92, 623–642. [Google Scholar] [CrossRef] - Gardeñes, E.; Sainz, M.; Jorba, L.; Calm, R.; Estela, R.; Mielgo, H.; Trepat, A. Modal intervals. Reliab. Comput.
**2001**, 7, 77–111. [Google Scholar] [CrossRef] - Pawlak, Z. Rough sets. Int. J. Comput. Inf. Sci.
**1982**, 11, 341–356. [Google Scholar] [CrossRef] - Yao, Y.Y. A comparative study of fuzzy sets and rough sets. Inf. Sci.
**1998**, 109, 227–242. [Google Scholar] [CrossRef] - Zhan, J.; Liu, Q.; Davvaz, B. A new rough set theory: rough soft hemirings. J. Intell. Fuzzy Syst.
**2015**, 28, 1687–1697. [Google Scholar] - Sainz, M.; Armengol, J.; Calm, R.; Herrero, P.; Jorba, L.; Vehi, J. Modal interval analysis: New tools for numerical information. In Lecture Notes in Mathematics 2091; Springer: Cham, Switzerland, 2014. [Google Scholar]
- Adillon, R.; Jorba, L. Numerical clouds. A treatment for indiscernibility. Int. J. Fuzzy Syst.
**2013**, 15, 274–278. [Google Scholar] - Dubois, D.; Prade, H. Operations on fuzzy numbers. Int. J. Syst. Sci.
**1987**, 9, 613–626. [Google Scholar] [CrossRef] - Dubois, D.; Prade, H. The mean value of a fuzzy number. Fuzzy Sets Syst.
**1987**, 24, 279–300. [Google Scholar] [CrossRef] - Grzegorzewski, P. Nearest interval approximation of a fuzzy number. Fuzzy Sets Syst.
**2002**, 130, 321–330. [Google Scholar] [CrossRef] - Heilpern, S. The expected value of a fuzzy number. Fuzzy Sets Syst.
**1992**, 47, 81–86. [Google Scholar] [CrossRef] - Adillon, R.; Jorba, L. A new point of view for fuzzy numbers and their defuzzification. Int. J. Uncertain. Fuzziness Knowl.-Based Syst.
**2015**, 23, 909–926. [Google Scholar] [CrossRef] - Sainz, M.A.; Herrero, P.; Armengol, J.; Vehí, J. Continuous minimax optimization using modal intervals. J. Math. Anal. Appl.
**2008**, 339, 18–30. [Google Scholar] [CrossRef] - Abbasbandy, S.; Asadi, B. The nearest trapezoidal fuzzy number to a fuzzy quantity. Appl. Math. Comput.
**2004**, 156, 381–386. [Google Scholar] [CrossRef] - Grzegorzewski, P.; Mrówka, E. Trapezoidal approximation of fuzzy numbers-revisited. Fuzzy Sets Syst.
**2007**, 158, 757–768. [Google Scholar] [CrossRef] - Grzegorzewski, P. Trapezoidal approximations of fuzzy numbers preserving the expected interval—Algorithms and properties. Fuzzy Sets Syst.
**2008**, 159, 1354–1364. [Google Scholar] [CrossRef] - Yeh, C.T. A note on trapezoidal approximations of fuzzy numbers. Fuzzy Sets Syst.
**2007**, 158, 747–754. [Google Scholar] [CrossRef] - Yeh, C.T. Trapezoidal and triangular approximations preserving the expected interval. Fuzzy Sets Syst.
**2008**, 159, 1345–1353. [Google Scholar] [CrossRef] - Veeramani, C.; Duraisami, C.; Sumathi, M. Nearest symmetric trapezoidal fuzzy number approximation preserving expected interval. Int. J. Uncertain. Fuzziness Knowl.-Based Syst.
**2013**, 21, 777–794. [Google Scholar] [CrossRef] - Ban, A.I.; Coroianu, L.; Khastan, A. Conditioned weighted L-R approximations of fuzzy numbers. Fuzzy Sets Syst.
**2016**, 283, 56–82. [Google Scholar] [CrossRef] - Eslamipoor, R.; Hosseini-nasab, H.; Sepehriar, A. An improved ranking method for generalized fuzzy numbers based on Euclidean distance concept. Afrik. Mat.
**2015**, 26, 1291–1297. [Google Scholar] [CrossRef] - Janizade-Haji, M.; Zare, H.K.; Eslamipoor, R.; Sepehriar, A. A developed distance method for ranking generalized fuzzy numbers. Neural Comput. Appl.
**2014**, 25, 727–731. [Google Scholar] [CrossRef] - Wagenknecht, M.; Schneider, V. Inclusion of fuzzy numbers. The variable Plateau Case. In Proceedings of the Joint 4th Conference of the European Society for Fuzzy Logic and Technology and the 11th Rencontres Francophones sur la Logique Floue et ses Applications, Barcelona, Spain, 7–9 September 2005; pp. 893–897. [Google Scholar]
- Ghorabaee, M.K.; Amiri, M.; Zavadskas, E.K.; Hooshmand, R.; Antuchevičiené, J. Fuzzy extension of the CODAS method for multi-criteria market segment evaluation. J. Bus. Econ. Manag.
**2017**, 18, 1–19. [Google Scholar] [CrossRef] - Zhang, X.; Xu, Z.; Liu, M. Hesitant trapezoidal fuzzy QUALIFLEX method and its application in the evaluation of green supply chain initiatives. Sustainability
**2016**, 8, 952. [Google Scholar] [CrossRef] - Veeramachaneni, S.; Kandikonda, H. An ELECTRE approach for multicriteria interval-valued intuitionistic trapezoidal fuzzy group decision making problems. Adv. Fuzzy Syst.
**2016**, 2016, 1–17. [Google Scholar] [CrossRef] - Chen, T.Y. An ELECTRE-based outranking method for multiple criteria group decision making using interval type-2 fuzzy sets. Inf. Sci.
**2014**, 263, 1–21. [Google Scholar] [CrossRef] - Liu, H.C.; You, J.X.; Chen, Y.Z.; Fan, X.J. Site selection in municipal solid waste management with extended VIKOR method under fuzzy environment. Environ. Earth Sci.
**2014**, 72, 4179–4189. [Google Scholar] [CrossRef] - Mardani, A.; Zavadskas, E.K.; Govindan, K.; Senin, A.A.; Jusoh, A. VIKOR technique: A systematic review of the state of the art literature on methodologies and applications. Sustainability
**2016**, 8, 37. [Google Scholar] [CrossRef] [Green Version] - Baležentis, T.; Zeng, S.H. Group multi-criteria decision making based upon interval-valued fuzzy numbers: An extension of the MULTIMOORA method. Expert Syst. Appl.
**2013**, 40, 543–550. [Google Scholar] [CrossRef] - Ghorabaee, M.K.; Zavadskas, E.K.; Amiri, M.; Antuchevičiené, J. Evaluation by an area-based method of ranking interval type-2 fuzzy sets (EAMRIT-2F) for multi-criteria group decision-making. Transform. Bus. Econ.
**2016**, 15, 76–95. [Google Scholar] - Celik, E.; Erdogan, M.; Gumus, A.T. An extended fuzzy TOPSIS–GRA method based on different separation measures for green logistics service provider selection. Int. J. Environ. Sci. Technol.
**2016**, 13, 1377–1392. [Google Scholar] [CrossRef] - Ghorabaee, M.K.; Amiri, M.; Zavadskas, E.K.; Turskis, Z. Multi-criteria group decision-making using an extended EDAS method with internal type-2 fuzzy sets. Ekon. Manag.
**2017**, 20, 48–68. [Google Scholar] - Ghorabaee, N.K.; Zavadskas, E.K.; AMIRI, M.; Antuchevičiené, J. A new method of assessment based on fuzzy ranking and aggregated weights (AFRAW) for MCDM problems under type-2 fuzzy environment. Econ. Comput. Econ. Cybern. Stud. Res.
**2016**, 50, 39–68. [Google Scholar] - Liu, P.; Li, Y.; Antuchevičiené, J. Multi-criteria decision-making method based on intuitionistic trapezoidal fuzzy prioritised owa operator. Technol. Econ. Dev. Econ.
**2016**, 22, 453–469. [Google Scholar] [CrossRef] - Ghorabaee, M.K.; Zavadskas, E.K.; Amiri, M.; Esmaeili, A. Multi-criteria evaluation of green suppliers using an extended WASPAS method with interval type-2 fuzzy sets. J. Clean. Prod.
**2016**, 137, 213–229. [Google Scholar] [CrossRef]

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Jorba, L.; Adillon, R.
A Generalization of Trapezoidal Fuzzy Numbers Based on Modal Interval Theory. *Symmetry* **2017**, *9*, 198.
https://doi.org/10.3390/sym9100198

**AMA Style**

Jorba L, Adillon R.
A Generalization of Trapezoidal Fuzzy Numbers Based on Modal Interval Theory. *Symmetry*. 2017; 9(10):198.
https://doi.org/10.3390/sym9100198

**Chicago/Turabian Style**

Jorba, Lambert, and Romà Adillon.
2017. "A Generalization of Trapezoidal Fuzzy Numbers Based on Modal Interval Theory" *Symmetry* 9, no. 10: 198.
https://doi.org/10.3390/sym9100198