# A New Evaluation for Solving the Fully Fuzzy Data Envelopment Analysis with Z-Numbers

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

#### 2.1. Definitions

**Definition**

**1**

**.**A fuzzy subset $\tilde{A}$ of a set $X$ is defined by its membership function ${\mu}_{\tilde{A}}:X\to [0,1]$, where the value of ${\mu}_{\tilde{A}}(x)$ at $x$ shows the grade of membership of $x$ in $\tilde{A}$.

**Definition**

**2**

**.**A triangular fuzzy number (TFNs) $\tilde{A}$ can be defined by $(a,b,c)$, where $c\ge b\ge a$. The membership function ${\mu}_{\tilde{A}}(x)$ is given by (1):

**Definition**

**3**

**.**Let $\tilde{A}=(a,b,c)$ be a triangular fuzzy number. Then $\tilde{A}$ is called a non-negative fuzzy number if and only if $a\ge 0$.

**Definition**

**4**

**.**Let $\tilde{A}=(a,b,c)$ be a triangular fuzzy number. Then $\tilde{A}$ is called an unrestricted fuzzy number if $a,b,c\in R$.

**Definition**

**5**

**.**Consider $\tilde{A}=(a,b,c)$ and $\tilde{B}=(d,e,f)$ as two triangular fuzzy numbers, then we have:

- (i)
- $\tilde{A}\oplus \tilde{B}=(a,b,c)\oplus (d,e,f)=(a+d,b+e,c+f),$
- (ii)
- $\tilde{A}-\tilde{B}=(a,b,c)-(d,e,f)=(a-f,b-e,c-d),$
- (iii)
- $\tilde{A}\otimes \tilde{B}=(\mathrm{min}(\gamma ),be,\mathrm{max}(\gamma ))$where,$\gamma =\{ad,af,cd,cf\}$.

**Definition**

**6**

**.**Consider $\tilde{A}=(a,b,c)$ and $\tilde{B}=(d,e,f)$ as two triangular fuzzy numbers. Then these numbers are equal if and only if $a=d,b=e$ and $c=f$.

**Definition**

**7**

**.**Consider $\tilde{A}=(a,b,c)$ as a triangular fuzzy number. Then the ranking function of $\tilde{A}$ is defined as follows:

**Definition**

**8**

- (i)
- $\tilde{A}\le \tilde{B}$if and only if$R(\tilde{A})\le R(\tilde{B})$.
- (ii)
- $\tilde{A}<\tilde{B}$if and only if$R(\tilde{A})<R(\tilde{B})$.

**Definition**

**9**

**.**A triangular fuzzy number can also be defined as $\tilde{A}=(M,\alpha ,\beta )$ which is referred to as a left right (L-R) fuzzy number. $M$ Is the central value, $\alpha $ is the left width (spread) and $\beta $ is the right width (spread). The membership function also has the following form:

**Remark**

**1.**

**Definition**

**10.**

**Definition**

**11.**

**Definition**

**12.**

**Definition**

**13**

**.**A Z-number is an ordered pair of fuzzy numbers indicated as $Z=(\tilde{A},\tilde{B})$. The first component, $\tilde{A}$, is a fuzzy restriction and the second component, $\tilde{B}$, is a level of reliability of the first component. For ease, A and B are supposed to be triangular fuzzy numbers.

#### 2.2. Converting a Z-Number into a Fuzzy Number

**Step****1:**- Convert the second part $(\tilde{B})$ into a crisp number. Defuzzification transforms a fuzzy number into a crisp value. The most commonly used defuzzification technique is the centroid defuzzification approach. This calculation is completed by using Equation (6).$$\alpha =\frac{{\displaystyle \int x{\mu}_{\tilde{B}}(x)dx}}{{\displaystyle \int {\mu}_{\tilde{B}}(x)dx}}$$We noted that when $\tilde{B}=({b}_{1},{b}_{2},{b}_{3})$, Equation (6) becomes as follows:$$\alpha =\frac{{b}_{1}+{b}_{2}+{b}_{3}}{3}$$
**Step****2:**- The weighted Z-number can be defined as:$${\tilde{Z}}^{\alpha}=\{(x,{\mu}_{{\tilde{A}}^{\alpha}})|{\mu}_{{\tilde{A}}^{\alpha}}=\alpha {\mu}_{\tilde{A}}(x),x\in [0,1]\}$$
**Step****3:**- By multiplying $\sqrt{\alpha}$, convert the weighted Z-number into the following classical fuzzy number:$${\tilde{Z}}^{\prime}=\sqrt{\alpha}\times {\tilde{A}}^{\alpha}=(\sqrt{\alpha}\times a,\sqrt{\alpha}\times b,\sqrt{\alpha}\times c,\sqrt{\alpha}\times d)$$

## 3. DEA, FDEA and FFDEA

## 4. Sotoudeh-Anvari et al.’s Algorithm

**Method 1.**

**Step****1:**- Consider the inputs and outputs of each DMU as well as their weights by using Z-numbers. Using the Charnes and Cooper transformation we have:$$\begin{array}{ll}& {\theta}_{p}{}^{\approx}=\mathrm{max}{\displaystyle \sum _{r=1}^{s}{\tilde{\tilde{u}}}_{r}{\tilde{\tilde{y}}}_{rp}}\\ s.t:& \\ & {\displaystyle \sum _{i=1}^{m}{\tilde{\tilde{v}}}_{i}{\tilde{\tilde{x}}}_{ip}}=1\\ & {\displaystyle \sum _{r=1}^{s}{\tilde{\tilde{u}}}_{r}{\tilde{\tilde{y}}}_{rj}}-{\displaystyle \sum _{i=1}^{m}{\tilde{\tilde{v}}}_{i}{\tilde{\tilde{x}}}_{ij}}\le 0,\phantom{\rule{4em}{0ex}}\forall j\\ & {\tilde{\tilde{u}}}_{r},{\tilde{\tilde{v}}}_{i}\ge 0\phantom{\rule{4em}{0ex}}\forall r,i\end{array}$$
**Step****2:**- Using the Kang et al. [50] model, convert Z-numbers into usual fuzzy numbers. Then the inputs and outputs of each DMU convert into ${\tilde{x}}_{ij}=({x}_{ij}^{M},{x}_{ij}^{\alpha},{x}_{ij}^{\beta})$ and ${\tilde{y}}_{ij}=({y}_{ij}^{M},{y}_{ij}^{\alpha},{y}_{ij}^{\beta})$. Furthermore, their weights will be ${\tilde{u}}_{r}=({u}_{r}^{M},{u}_{r}^{\alpha},{u}_{r}^{\beta})$, ${\tilde{v}}_{r}=({v}_{r}^{M},{v}_{r}^{\alpha},{v}_{r}^{\beta})$ and we have:$$\begin{array}{l}\mathrm{max}\phantom{\rule{2em}{0ex}}{\tilde{\theta}}_{p}={\displaystyle \sum _{r=1}^{s}({u}_{r}^{M}},{u}_{r}^{\alpha},{u}_{r}^{\beta})\otimes ({y}_{rp}^{M},{y}_{rp}^{\alpha},{y}_{rp}^{\beta})\\ s.t.\\ {\displaystyle \sum _{i=1}^{m}({v}_{i}^{M},{v}_{i}^{\alpha},{v}_{i}^{\beta})\otimes ({x}_{ip}^{M},{y}_{ip}^{\alpha},{y}_{ip}^{\beta})\approx (1,0,0)}\\ {\displaystyle \sum _{r=1}^{s}({u}_{r}^{M},{u}_{r}^{\alpha},{u}_{r}^{\beta})\otimes ({y}_{rj}^{M},{y}_{rj}^{\alpha},{y}_{rj}^{\beta})}-{\displaystyle \sum _{i=1}^{m}({u}_{r}^{M},{u}_{r}^{\alpha},{u}_{r}^{\beta})\otimes ({x}_{ij}^{M},{y}_{ij}^{\alpha},{y}_{ij}^{\beta})\le (0,0,0)\phantom{\rule{1em}{0ex}}\forall j}\\ ({u}_{r}^{M},{u}_{r}^{\alpha},{u}_{r}^{\beta}),({v}_{i}^{M},{v}_{i}^{\alpha},{v}_{i}^{\beta})\ge 0\phantom{\rule{4em}{0ex}}\forall r,i.\end{array}$$
**Step****3:**- The fuzzy DEA Equation (16) can be transformed into the following DEA model:$$\begin{array}{l}\mathrm{max}\phantom{\rule{2em}{0ex}}{\tilde{\theta}}_{p}={\displaystyle \sum _{r=1}^{s}({u}_{r}^{M}}({y}_{rp}^{M}+{y}_{rp}^{\beta}-{y}_{rp}^{\alpha}),{u}_{r}^{\beta}{y}_{rp}^{M},{u}_{r}^{\alpha}{y}_{rp}^{M})\\ s.t.\\ {\displaystyle \sum _{i=1}^{m}({v}_{i}^{M}}({x}_{ip}^{M}+{x}_{ip}^{\beta}-{x}_{ip}^{\alpha}),{v}_{i}^{\beta}{x}_{ip}^{M},{v}_{i}^{\alpha}{x}_{ip}^{M})=1,\\ {\displaystyle \sum _{r=1}^{s}({u}_{r}^{M}}({y}_{rj}^{M}+{y}_{rj}^{\beta}-{y}_{rj}^{\alpha}),{u}_{r}^{\beta}{y}_{rj}^{M},{u}_{r}^{\alpha}{y}_{rj}^{M})-{\displaystyle \sum _{i=1}^{m}({v}_{i}^{M}}({x}_{ij}^{M}+{x}_{ij}^{\beta}-{x}_{ij}^{\alpha}),{v}_{i}^{\beta}{x}_{ij}^{M},{v}_{i}^{\alpha}{x}_{ij}^{M})\le (0,0,0)\phantom{\rule{4em}{0ex}}\forall j\\ ({u}_{r}^{M},{u}_{r}^{\alpha},{u}_{r}^{\beta}),({v}_{i}^{M},{v}_{i}^{\alpha},{v}_{i}^{\beta})\ge 0\phantom{\rule{4em}{0ex}}\forall r,i.\end{array}$$
**Step****4:**- Convert the fuzzy DEA Equation (17) into the following LP model:$$\begin{array}{l}\mathrm{max}\phantom{\rule{2em}{0ex}}{\theta}_{P}=R({\tilde{\theta}}_{P})={\displaystyle \sum _{r=1}^{s}\left[{u}_{r}^{M}\left({y}_{rp}^{M}+(\frac{1}{4}){y}_{rp}^{\beta}-(\frac{1}{4}){y}_{rp}^{\alpha}\right)+{u}_{r}^{\beta}\left(\left(\frac{1}{4}\right){y}_{rp}^{M}\right)-{u}_{r}^{\alpha}\left(\left(\frac{1}{4}\right){y}_{rp}^{M}\right)\right]}\\ s.t.\\ {\displaystyle \sum _{i=1}^{m}\left[{v}_{i}^{M}\left({x}_{ip}^{M}+\left(\frac{1}{4}\right){x}_{ip}^{\beta}-\left(\frac{1}{4}\right){x}_{ip}^{\alpha}\right)+{v}_{i}^{\beta}\left(\left(\frac{1}{4}\right){x}_{ip}^{M}\right)-{v}_{i}^{\alpha}\left(\left(\frac{1}{4}\right){x}_{ip}^{M}\right)\right]}=1\end{array}\phantom{\rule{0ex}{0ex}}{\displaystyle \sum _{r=1}^{s}\left[{u}_{r}^{M}\left({y}_{rj}^{M}+\left(\frac{1}{4}\right){y}_{rj}^{\beta}-\left(\frac{1}{4}\right){y}_{rj}^{\alpha}\right)\right]}+{u}_{r}^{\beta}\left(\left(\frac{1}{4}\right){y}_{rj}^{M}\right)-{u}_{r}^{\alpha}\left(\left(\frac{1}{4}\right){y}_{rj}^{M}\right)\le \phantom{\rule{0ex}{0ex}}{\displaystyle \sum _{r=1}^{s}\left[{v}_{i}^{M}\left({x}_{ij}^{M}+(\frac{1}{4}){x}_{ij}^{\beta}-(\frac{1}{4}){x}_{ij}^{\alpha}\right)+{v}_{i}^{\beta}\left(\left(\frac{1}{4}\right){x}_{ij}^{M}\right)-{v}_{i}^{\alpha}\left(\left(\frac{1}{4}\right){x}_{ij}^{M}\right)\right]},\phantom{\rule{1em}{0ex}}\forall j\phantom{\rule{0ex}{0ex}}{u}_{r}^{M}-{u}_{r}^{\alpha}\ge 0,\phantom{\rule{6em}{0ex}}\forall r\phantom{\rule{0ex}{0ex}}{u}_{r}^{M}-\left(\frac{1}{4}\right){u}_{r}^{\alpha}+\left(\frac{1}{4}\right){u}_{r}^{\beta}\ge 0\phantom{\rule{2em}{0ex}}\forall r\phantom{\rule{0ex}{0ex}}{v}_{i}^{M}-{u}_{i}^{\alpha}\ge 0,\phantom{\rule{6em}{0ex}}\forall i\phantom{\rule{0ex}{0ex}}\begin{array}{l}{v}_{i}^{M}-\left(\frac{1}{4}\right){v}_{i}^{\alpha}+\left(\frac{1}{4}\right){v}_{i}^{\beta}\ge 0\phantom{\rule{2em}{0ex}}\forall i\\ {u}_{r}^{\alpha}\ge 0,{u}_{r}^{\beta}\ge 0,\begin{array}{cc}& \end{array}\forall r\\ {v}_{i}^{\alpha}\ge 0,{v}_{i}^{\beta}\ge 0,\begin{array}{cc}& \end{array}\forall i.\end{array}$$
**Step****5:**- Run Equation (18) and obtain the optimal solutions of ${u}_{r}^{{M}^{\ast}},{u}_{r}^{{\alpha}^{\ast}},{u}_{r}^{{\beta}^{\ast}},{v}_{i}^{{M}^{\ast}},{v}_{i}^{{\alpha}^{\ast}}$ and ${v}_{i}^{{\beta}^{\ast}}$.

## 5. Main Results

#### 5.1. The Shortcoming of the Existing Algorithm

#### 5.2. Improvement Model for FFDEA with Z-Numbers

**Method 2.**

**Step****1:**- Consider the DEA model that the inputs and outputs of each DMU as well as their weights are Z-numbers.
**Step****2:**- Using Kang et al.’s [50] model, convert Z-numbers into usual fuzzy numbers and obtain a fully fuzzy DEA model with triangular fuzzy numbers.
**Step****3:**- Using Definition 11, the fully fuzzy DEA model of Step 2 can be transformed into the following model:$$\begin{array}{l}\mathrm{max}\phantom{\rule{2em}{0ex}}{\tilde{\theta}}_{p}={\displaystyle \sum _{r=1}^{s}({u}_{r}^{M}}{y}_{rp}^{M},{u}_{r}^{M}{y}_{rp}^{\alpha}+{y}_{rp}^{M}{u}_{r}^{\alpha}-{u}_{r}^{\alpha}{y}_{rp}^{\alpha},{u}_{r}^{M}{y}_{rp}^{\beta}+{y}_{rp}^{M}{u}_{r}^{\beta}+{u}_{r}^{\beta}{y}_{rp}^{\beta})\\ s.t.\\ {\displaystyle \sum _{i=1}^{m}({v}_{i}^{M}}{x}_{ip}^{M},{v}_{i}^{M}{x}_{ip}^{\alpha}+{x}_{ip}^{M}{v}_{i}^{\alpha}-{v}_{i}^{\alpha}{x}_{ip}^{\alpha},{v}_{i}^{M}{x}_{ip}^{\beta}+{x}_{ip}^{M}{v}_{i}^{\beta}+{v}_{i}^{\beta}{x}_{ip}^{\beta})\approx 1,\\ {\displaystyle \sum _{r=1}^{s}({u}_{r}^{M}}{y}_{rj}^{M},{u}_{r}^{M}{y}_{rj}^{\alpha}+{y}_{rj}^{M}{u}_{r}^{\alpha}-{u}_{r}^{\alpha}{y}_{rj}^{\alpha},{u}_{r}^{M}{y}_{rj}^{\beta}+{y}_{rj}^{M}{u}_{r}^{\beta}+{u}_{r}^{\beta}{y}_{rj}^{\beta})\le \\ \phantom{\rule{4em}{0ex}}{\displaystyle \sum _{i=1}^{m}({v}_{i}^{M}}{x}_{ij}^{M},{v}_{i}^{M}{x}_{ij}^{\alpha}+{x}_{ij}^{M}{v}_{i}^{\alpha}-{v}_{i}^{\alpha}{x}_{ij}^{\alpha},{v}_{i}^{M}{x}_{ij}^{\beta}+{x}_{ij}^{M}{v}_{i}^{\beta}+{v}_{i}^{\beta}{x}_{ij}^{\beta})\begin{array}{cc}& \end{array}\forall j\\ ({u}_{r}^{M},{u}_{r}^{\alpha},{u}_{r}^{\beta}),({v}_{i}^{M},{v}_{i}^{\alpha},{v}_{i}^{\beta})\ge 0\phantom{\rule{4em}{0ex}}\forall r,i\end{array}$$
**Step****4:**- Based on Definitions 7 and 8, convert the fuzzy DEA Equation (19) into the following model:$$\begin{array}{l}\mathrm{max}\phantom{\rule{2em}{0ex}}{\theta}_{P}=R({\tilde{\theta}}_{P})={\displaystyle \sum _{r=1}^{s}R(({u}_{r}^{M}}{y}_{rp}^{M},{u}_{r}^{M}{y}_{rp}^{\alpha}+{y}_{rp}^{M}{u}_{r}^{\alpha}-{u}_{r}^{\alpha}{y}_{rp}^{\alpha},{u}_{r}^{M}{y}_{rp}^{\beta}+{y}_{rp}^{M}{u}_{r}^{\beta}+{u}_{r}^{\beta}{y}_{rp}^{\beta}))\\ {\displaystyle \sum _{i=1}^{m}R(({v}_{i}^{M}}{x}_{ip}^{M},{v}_{i}^{M}{x}_{ip}^{\alpha}+{x}_{ip}^{M}{v}_{i}^{\alpha}-{v}_{i}^{\alpha}{x}_{ip}^{\alpha},{v}_{i}^{M}{x}_{ip}^{\beta}+{x}_{ip}^{M}{v}_{i}^{\beta}+{v}_{i}^{\beta}{x}_{ip}^{\beta}))=R(1,1,1),\\ {\displaystyle \sum _{r=1}^{s}R(({u}_{r}^{M}}{y}_{rj}^{M},{u}_{r}^{M}{y}_{rj}^{\alpha}+{y}_{rj}^{M}{u}_{r}^{\alpha}-{u}_{r}^{\alpha}{y}_{rj}^{\alpha},{u}_{r}^{M}{y}_{rj}^{\beta}+{y}_{rj}^{M}{u}_{r}^{\beta}+{u}_{r}^{\beta}{y}_{rj}^{\beta}))\le \\ \begin{array}{l}\phantom{\rule{4em}{0ex}}{\displaystyle \sum _{i=1}^{m}R(({v}_{i}^{M}}{x}_{ij}^{M},{v}_{i}^{M}{x}_{ij}^{\alpha}+{x}_{ij}^{M}{v}_{i}^{\alpha}-{v}_{i}^{\alpha}{x}_{ij}^{\alpha},{v}_{i}^{M}{x}_{ij}^{\beta}+{x}_{ij}^{M}{v}_{i}^{\beta}+{v}_{i}^{\beta}{x}_{ij}^{\beta}))\begin{array}{cc}& \end{array}\forall j\end{array}\\ R({u}_{r}^{M},{u}_{r}^{\alpha},{u}_{r}^{\beta})\ge (0,0,0),R({v}_{i}^{M},{v}_{i}^{\alpha},{v}_{i}^{\beta})\ge (0,0,0),\phantom{\rule{4em}{0ex}}\forall r,i\\ {u}_{r}^{M}-{u}_{r}^{\alpha}\ge 0,\begin{array}{cc}& \forall r,\end{array}\\ {v}_{i}^{M}-{v}_{i}^{\alpha}\ge 0,\begin{array}{cc}& \forall i,\end{array}\\ {u}_{r}^{\alpha}\ge 0,{u}_{r}^{\beta}\ge 0,\begin{array}{cc}& \end{array}\forall r\\ {v}_{i}^{\alpha}\ge 0,{v}_{i}^{\beta}\ge 0,\begin{array}{cc}& \end{array}\forall i.\end{array}$$
**Step****5:**- Based on Definition 12, convert Equation (20) into the following model:$$\mathrm{max}\phantom{\rule{2em}{0ex}}{\theta}_{P}=R({\tilde{\theta}}_{P})={\displaystyle \sum _{r=1}^{s}\left[{u}_{r}^{M}{y}_{rp}^{M}+\frac{1}{4}[{u}_{r}^{M}{y}_{rp}^{\beta}+{y}_{rp}^{M}{u}_{r}^{\beta}+{u}_{r}^{\beta}{y}_{rp}^{\beta}]-\frac{1}{4}[{u}_{r}^{M}{y}_{rp}^{\alpha}+{y}_{rp}^{M}{u}_{r}^{\alpha}-{u}_{r}^{\alpha}{y}_{rp}^{\alpha}]\right]}\phantom{\rule{0ex}{0ex}}s.t.\phantom{\rule{0ex}{0ex}}{\displaystyle \sum _{i=1}^{m}\left[{v}_{i}^{M}{x}_{ip}^{M}+\frac{1}{4}[{v}_{i}^{M}{x}_{ip}^{\beta}+{x}_{ip}^{M}{v}_{i}^{\beta}+{v}_{i}^{\beta}{x}_{ip}^{\beta}]-\frac{1}{4}[{v}_{i}^{M}{x}_{ip}^{\alpha}+{x}_{ip}^{M}{v}_{i}^{\alpha}-{v}_{i}^{\alpha}{x}_{ip}^{\alpha}]\right]}=1,\phantom{\rule{0ex}{0ex}}\begin{array}{l}{\displaystyle \sum _{r=1}^{s}\left[{u}_{r}^{M}{y}_{rj}^{M}+\frac{1}{4}[{u}_{r}^{M}{y}_{rj}^{\beta}+{y}_{rj}^{M}{u}_{r}^{\beta}+{u}_{r}^{\beta}{y}_{rj}^{\beta}]-\frac{1}{4}[{u}_{r}^{M}{y}_{rj}^{\alpha}+{y}_{rj}^{M}{u}_{r}^{\alpha}-{u}_{r}^{\alpha}{y}_{rj}^{\alpha}]\right]}\le \\ {\displaystyle \sum _{i=1}^{m}\left[{v}_{i}^{M}{x}_{ij}^{M}+\frac{1}{4}[{v}_{i}^{M}{x}_{ij}^{\beta}+{x}_{ij}^{M}{v}_{i}^{\beta}+{v}_{i}^{\beta}{x}_{ij}^{\beta}]-\frac{1}{4}[{v}_{i}^{M}{x}_{ij}^{\alpha}+{x}_{ij}^{M}{v}_{i}^{\alpha}-{v}_{i}^{\alpha}{x}_{ij}^{\alpha}]\right]},\phantom{\rule{2em}{0ex}}\forall j\\ {u}_{r}^{M}-{u}_{r}^{\alpha}\ge 0,\phantom{\rule{8em}{0ex}}\forall r\\ {u}_{r}^{M}-\left(\frac{1}{4}\right){u}_{r}^{\alpha}+\left(\frac{1}{4}\right){u}_{r}^{\beta}\ge 0\phantom{\rule{2em}{0ex}}\forall r\\ {v}_{i}^{M}-{u}_{i}^{\alpha}\ge 0,\phantom{\rule{8em}{0ex}}\forall i\\ {v}_{i}^{M}-\left(\frac{1}{4}\right){v}_{i}^{\alpha}+\left(\frac{1}{4}\right){v}_{i}^{\beta}\ge 0\phantom{\rule{2em}{0ex}}\forall i\\ {u}_{r}^{\alpha}\ge 0,{u}_{r}^{\beta}\ge 0,\begin{array}{cc}& \end{array}\forall r\\ {v}_{i}^{\alpha}\ge 0,{v}_{i}^{\beta}\ge 0,\begin{array}{cc}& \end{array}\forall i.\end{array}$$
**Step****6:**- Obtain the optimal solutions of ${u}_{r}^{{M}^{\ast}},{u}_{r}^{{\alpha}^{\ast}},{u}_{r}^{{\beta}^{\ast}},{v}_{i}^{{M}^{\ast}},{v}_{i}^{{\alpha}^{\ast}}$ and ${v}_{i}^{{\beta}^{\ast}}$.

## 6. Numerical Example

**Problem**

**1.**

_{A}can be used as follows:

**Step****1:**- Obtain a fully fuzzy DEA model with triangular fuzzy numbers:$$\begin{array}{l}\mathrm{max}\phantom{\rule{2em}{0ex}}{\tilde{\theta}}_{A}\approx (2.6,0.2,0.2)\otimes ({u}_{1}^{M},{u}_{1}^{\alpha},{u}_{1}^{\beta})+(4.1,0.3,0.3)\otimes ({u}_{2}^{M},{u}_{2}^{\alpha},{u}_{2}^{\beta})\\ s.t.\\ (4,0.5,0.5)\otimes ({v}_{1}^{M},{v}_{1}^{\alpha},{v}_{1}^{\beta})+(2.1,0.2,0.2)\otimes ({v}_{2}^{M},{v}_{2}^{\alpha},{v}_{2}^{\beta})\approx 1,\\ \left[(2.6,0.2,0.2)\otimes ({u}_{1}^{M},{u}_{1}^{\alpha},{u}_{1}^{\beta})+(4.1,0.3,0.3)\otimes ({u}_{2}^{M},{u}_{2}^{\alpha},{u}_{2}^{\beta})\right]-\\ \begin{array}{cc}& \end{array}\left[(4,0.5,0.5)\otimes ({v}_{1}^{M},{v}_{1}^{\alpha},{v}_{1}^{\beta})+(2.1,0.2,0.2)\otimes ({v}_{2}^{M},{v}_{2}^{\alpha},{v}_{2}^{\beta})\right]\le (0,0,0),\end{array}\phantom{\rule{0ex}{0ex}}\begin{array}{l}\left[(2.2,0,0)\otimes ({u}_{1}^{M},{u}_{1}^{\alpha},{u}_{1}^{\beta})+(3.5,0.2,0.2)\otimes ({u}_{2}^{M},{u}_{2}^{\alpha},{u}_{2}^{\beta})\right]-\\ \begin{array}{cc}& \end{array}\left[(2.9,0,0)\otimes ({v}_{1}^{M},{v}_{1}^{\alpha},{v}_{1}^{\beta})+(1.5,0.1,0.1)\otimes ({v}_{2}^{M},{v}_{2}^{\alpha},{v}_{2}^{\beta})\right]\le (0,0,0),\end{array}\phantom{\rule{0ex}{0ex}}\begin{array}{l}\left[(3.2,0.5,0.5)\otimes ({u}_{1}^{M},{u}_{1}^{\alpha},{u}_{1}^{\beta})+(5.1,0.8,0.8)\otimes ({u}_{2}^{M},{u}_{2}^{\alpha},{u}_{2}^{\beta})\right]-\\ \begin{array}{cc}& \end{array}\left[(4.9,0.5,0.5)\otimes ({v}_{1}^{M},{v}_{1}^{\alpha},{v}_{1}^{\beta})+(2.6,0.4,0.4)\otimes ({v}_{2}^{M},{v}_{2}^{\alpha},{v}_{2}^{\beta})\right]\le (0,0,0),\end{array}\phantom{\rule{0ex}{0ex}}\begin{array}{l}\left[(2.5,0.2,0.4)\otimes ({u}_{1}^{M},{u}_{1}^{\alpha},{u}_{1}^{\beta})+(5.7,0.9,0.9)\otimes ({u}_{2}^{M},{u}_{2}^{\alpha},{u}_{2}^{\beta})\right]-\\ \begin{array}{cc}& \end{array}\left[(4.1,0.7,0.7)\otimes ({v}_{1}^{M},{v}_{1}^{\alpha},{v}_{1}^{\beta})+(2.3,0.1,0.1)\otimes ({v}_{2}^{M},{v}_{2}^{\alpha},{v}_{2}^{\beta})\right]\le (0,0,0),\\ \left[(5.1,0.7,0.7)\otimes ({u}_{1}^{M},{u}_{1}^{\alpha},{u}_{1}^{\beta})+(7.4,0.9,0.9)\otimes ({u}_{2}^{M},{u}_{2}^{\alpha},{u}_{2}^{\beta})\right]-\\ \begin{array}{cc}& \end{array}\left[(6.1,0.2,1)\otimes ({v}_{1}^{M},{v}_{1}^{\alpha},{v}_{1}^{\beta})+(4.1,0.5,0.5)\otimes ({v}_{2}^{M},{v}_{2}^{\alpha},{v}_{2}^{\beta})\right]\le (0,0,0),\\ ({u}_{r}^{M},{u}_{r}^{\alpha},{u}_{r}^{\beta}),({v}_{i}^{M},{v}_{i}^{\alpha},{v}_{i}^{\beta})\ge 0\phantom{\rule{4em}{0ex}}\forall r,i\end{array}$$
**Step****2:**- Using the Definition 11, the fully fuzzy DEA model of Step 2 can be transformed into the following model:$$\begin{array}{l}\mathrm{max}\phantom{\rule{2em}{0ex}}{\tilde{\theta}}_{A}\approx (2.6{u}_{1}^{M},0.2{u}_{1}^{M}+2.4{u}_{1}^{\alpha},0.2{u}_{1}^{M}+2.8{u}_{1}^{\beta})+(4.1{u}_{2}^{M},0.3{u}_{2}^{M}+3.8{u}_{2}^{\alpha},0.3{u}_{2}^{M}+4.4{u}_{2}^{\beta})\\ s.t.\\ (4{v}_{1}^{M},0.5{v}_{1}^{M}+3.5{v}_{1}^{\alpha},0.5{v}_{1}^{M}+4.5{v}_{1}^{\beta})+(2.1{v}_{2}^{M},0.2{v}_{2}^{M}+1.9{v}_{2}^{\alpha},0.2{v}_{2}^{M}+2.3{v}_{2}^{\beta})\approx (1,1,1),\\ (2.6{u}_{1}^{M},0.2{u}_{1}^{M}+2.4{u}_{1}^{\alpha},0.2{u}_{1}^{M}+2.8{u}_{1}^{\beta})+(4.1{u}_{2}^{M},0.3{u}_{2}^{M}+3.8{u}_{2}^{\alpha},0.3{u}_{2}^{M}+4.4{u}_{2}^{\beta})\\ \begin{array}{ccc}& & \end{array}-(4{v}_{1}^{M},0.5{v}_{1}^{M}+3.5{v}_{1}^{\alpha},0.5{v}_{1}^{M}+4.5{v}_{1}^{\beta})-(2.1{v}_{2}^{M},0.2{v}_{2}^{M}+1.9{v}_{2}^{\alpha},0.2{v}_{2}^{M}+2.3{v}_{2}^{\beta})\le (0,0,0),\\ (2.2{u}_{1}^{M},2.2{u}_{1}^{\alpha},2.2{u}_{1}^{\beta})+(3.5{u}_{2}^{M},0.2{u}_{2}^{M}+3.3{u}_{2}^{\alpha},0.2{u}_{2}^{M}+3.7{u}_{2}^{\beta})\\ \begin{array}{ccc}& & \end{array}-(2.9{v}_{1}^{M},2.9{v}_{1}^{\alpha},2.9{v}_{1}^{\beta})-(1.5{v}_{2}^{M},0.1{v}_{2}^{M}+1.4{v}_{2}^{\alpha},0.1{v}_{2}^{M}+1.6{v}_{2}^{\beta})\le (0,0,0),\\ (3.2{u}_{1}^{M},0.5{u}_{1}^{M}+2.7{u}_{1}^{\alpha},0.5{u}_{1}^{M}+3.7{u}_{1}^{\beta})+(5.1{u}_{2}^{M},0.8{u}_{2}^{M}+4.3{u}_{2}^{\alpha},0.8{u}_{2}^{M}+5.9{u}_{2}^{\beta})\\ \begin{array}{ccc}& & \end{array}-(4.9{v}_{1}^{M},0.5{v}_{1}^{M}+4.4{v}_{1}^{\alpha},0.5{v}_{1}^{M}+5.4{v}_{1}^{\beta})-(2.6{v}_{2}^{M},0.4{v}_{2}^{M}+2.2{v}_{2}^{\alpha},0.4{v}_{2}^{M}+3{v}_{2}^{\beta})\le (0,0,0),\\ (2.5{u}_{1}^{M},0.2{u}_{1}^{M}+2.3{u}_{1}^{\alpha},0.4{u}_{1}^{M}+2.9{u}_{1}^{\beta})+(5.7{u}_{2}^{M},0.2{u}_{2}^{M}+5.5{u}_{2}^{\alpha},0.2{u}_{2}^{M}+5.9{u}_{2}^{\beta})\\ \begin{array}{ccc}& & \end{array}-(4.1{v}_{1}^{M},0.7{v}_{1}^{M}+3.4{v}_{1}^{\alpha},0.7{v}_{1}^{M}+4.8{v}_{1}^{\beta})-(2.3{v}_{2}^{M},0.1{v}_{2}^{M}+2.2{v}_{2}^{\alpha},0.1{v}_{2}^{M}+2.4{v}_{2}^{\beta})\le (0,0,0),\\ (5.1{u}_{1}^{M},0.7{u}_{1}^{M}+4.4{u}_{1}^{\alpha},0.7{u}_{1}^{M}+5.8{u}_{1}^{\beta})+(7.4{u}_{2}^{M},0.9{u}_{2}^{M}+6.5{u}_{2}^{\alpha},0.9{u}_{2}^{M}+8.3{u}_{2}^{\beta})\\ \begin{array}{ccc}& & \end{array}-(6.1{v}_{1}^{M},0.2{v}_{1}^{M}+5.9{v}_{1}^{\alpha},{v}_{1}^{M}+7.1{v}_{1}^{\beta})-(4.1{v}_{2}^{M},0.5{v}_{2}^{M}+3.6{v}_{2}^{\alpha},0.5{v}_{2}^{M}+4.6{v}_{2}^{\beta})\le (0,0,0),\\ ({u}_{r}^{M},{u}_{r}^{\alpha},{u}_{r}^{\beta}),({v}_{i}^{M},{v}_{i}^{\alpha},{v}_{i}^{\beta})\ge (0,0,0)\phantom{\rule{4em}{0ex}}\forall r,i\end{array}$$
**Step****3:**- Based on Definitions 7 and 8, convert the above fuzzy DEA model to the following model:$$\begin{array}{l}\mathrm{max}\phantom{\rule{2em}{0ex}}{\theta}_{A}=R({\tilde{\theta}}_{A})=R\left((2.6{u}_{1}^{M},0.2{u}_{1}^{M}+2.4{u}_{1}^{\alpha},0.2{u}_{1}^{M}+2.8{u}_{1}^{\beta})+(4.1{u}_{2}^{M},0.3{u}_{2}^{M}+3.8{u}_{2}^{\alpha},0.3{u}_{2}^{M}+4.4{u}_{2}^{\beta})\right)\\ s.t.\\ R\left((4{v}_{1}^{M},0.5{v}_{1}^{M}+3.5{v}_{1}^{\alpha},0.5{v}_{1}^{M}+4.5{v}_{1}^{\beta})+(2.1{v}_{2}^{M},0.2{v}_{2}^{M}+1.9{v}_{2}^{\alpha},0.2{v}_{2}^{M}+2.3{v}_{2}^{\beta})\right)=R(1,1,1),\\ R\left(\begin{array}{l}(2.6{u}_{1}^{M},0.2{u}_{1}^{M}+2.4{u}_{1}^{\alpha},0.2{u}_{1}^{M}+2.8{u}_{1}^{\beta})+(4.1{u}_{2}^{M},0.3{u}_{2}^{M}+3.8{u}_{2}^{\alpha},0.3{u}_{2}^{M}+4.4{u}_{2}^{\beta})\\ \begin{array}{ccc}& & \end{array}-(4{v}_{1}^{M},0.5{v}_{1}^{M}+3.5{v}_{1}^{\alpha},0.5{v}_{1}^{M}+4.5{v}_{1}^{\beta})-(2.1{v}_{2}^{M},0.2{v}_{2}^{M}+1.9{v}_{2}^{\alpha},0.2{v}_{2}^{M}+2.3{v}_{2}^{\beta})\end{array}\right)\le R(0,0,0),\\ R\left(\begin{array}{l}(2.2{u}_{1}^{M},2.2{u}_{1}^{\alpha},2.2{u}_{1}^{\beta})+(3.5{u}_{2}^{M},0.2{u}_{2}^{M}+3.3{u}_{2}^{\alpha},0.2{u}_{2}^{M}+3.7{u}_{2}^{\beta})\\ \begin{array}{ccc}& & \end{array}-(2.9{v}_{1}^{M},2.9{v}_{1}^{\alpha},2.9{v}_{1}^{\beta})-(1.5{v}_{2}^{M},0.1{v}_{2}^{M}+1.4{v}_{2}^{\alpha},0.1{v}_{2}^{M}+1.6{v}_{2}^{\beta})\end{array}\right)\le R(0,0,0),\end{array}\phantom{\rule{0ex}{0ex}}\begin{array}{l}R\left(\begin{array}{l}(3.2{u}_{1}^{M},0.5{u}_{1}^{M}+2.7{u}_{1}^{\alpha},0.5{u}_{1}^{M}+3.7{u}_{1}^{\beta})+(5.1{u}_{2}^{M},0.8{u}_{2}^{M}+4.3{u}_{2}^{\alpha},0.8{u}_{2}^{M}+5.9{u}_{2}^{\beta})\\ \begin{array}{ccc}& & \end{array}-(4.9{v}_{1}^{M},0.5{v}_{1}^{M}+4.4{v}_{1}^{\alpha},0.5{v}_{1}^{M}+5.4{v}_{1}^{\beta})-(2.6{v}_{2}^{M},0.4{v}_{2}^{M}+2.2{v}_{2}^{\alpha},0.4{v}_{2}^{M}+3{v}_{2}^{\beta})\end{array}\right)\le R(0,0,0),\\ R\left(\begin{array}{l}(2.5{u}_{1}^{M},0.2{u}_{1}^{M}+2.3{u}_{1}^{\alpha},0.4{u}_{1}^{M}+2.9{u}_{1}^{\beta})+(5.7{u}_{2}^{M},0.2{u}_{2}^{M}+5.5{u}_{2}^{\alpha},0.2{u}_{2}^{M}+5.9{u}_{2}^{\beta})\\ \begin{array}{ccc}& & \end{array}-(4.1{v}_{1}^{M},0.7{v}_{1}^{M}+3.4{v}_{1}^{\alpha},0.7{v}_{1}^{M}+4.8{v}_{1}^{\beta})-(2.3{v}_{2}^{M},0.1{v}_{2}^{M}+2.2{v}_{2}^{\alpha},0.1{v}_{2}^{M}+2.4{v}_{2}^{\beta})\end{array}\right)\le R(0,0,0),\\ R\left(\begin{array}{l}(5.1{u}_{1}^{M},0.7{u}_{1}^{M}+4.4{u}_{1}^{\alpha},0.7{u}_{1}^{M}+5.8{u}_{1}^{\beta})+(7.4{u}_{2}^{M},0.9{u}_{2}^{M}+6.5{u}_{2}^{\alpha},0.9{u}_{2}^{M}+8.3{u}_{2}^{\beta})\\ \begin{array}{ccc}& & \end{array}-(6.1{v}_{1}^{M},0.2{v}_{1}^{M}+5.9{v}_{1}^{\alpha},{v}_{1}^{M}+7.1{v}_{1}^{\beta})-(4.1{v}_{2}^{M},0.5{v}_{2}^{M}+3.6{v}_{2}^{\alpha},0.5{v}_{2}^{M}+4.6{v}_{2}^{\beta})\end{array}\right)\le R(0,0,0),\\ R({u}_{r}^{M},{u}_{r}^{\alpha},{u}_{r}^{\beta})\ge R(0,0,0),\phantom{\rule{4em}{0ex}}\forall r\\ R({v}_{i}^{M},{v}_{i}^{\alpha},{v}_{i}^{\beta})\ge R(0,0,0)\phantom{\rule{4em}{0ex}}\forall i\\ {u}_{r}^{M}-{u}_{r}^{\alpha}\ge 0,\begin{array}{cc}& \forall r,\end{array}\\ {v}_{i}^{M}-{v}_{i}^{\alpha}\ge 0,\begin{array}{cc}& \forall i,\end{array}\\ {u}_{r}^{\alpha}\ge 0,{u}_{r}^{\beta}\ge 0,\begin{array}{cc}& \end{array}\forall r\\ {v}_{i}^{\alpha}\ge 0,{v}_{i}^{\beta}\ge 0,\begin{array}{cc}& \end{array}\forall i.\end{array}$$
**Step****4:**- Based on Definition 12, convert the above model to the following model:$$\begin{array}{l}\mathrm{max}\phantom{\rule{2em}{0ex}}{\theta}_{A}=\left((2.6{u}_{1}^{M}-0.6{u}_{1}^{\alpha}+0.7{u}_{1}^{\beta})+(4.1{u}_{2}^{M}-0.95{u}_{2}^{\alpha}+1.1{u}_{2}^{\beta})\right)\\ s.t.\\ \left((4{v}_{1}^{M}-0.875{v}_{1}^{\alpha}+1.125{v}_{1}^{\beta})+(2.1{v}_{2}^{M}-0.475{v}_{2}^{\alpha}+0.575{v}_{2}^{\beta})\right)=1,\\ (2.6{u}_{1}^{M}-0.6{u}_{1}^{\alpha}+0.7{u}_{1}^{\beta})+(4.1{u}_{2}^{M}-0.95{u}_{2}^{\alpha}+1.1{u}_{2}^{\beta})-\\ \left((4{v}_{1}^{M}-0.875{v}_{1}^{\alpha}+1.125{v}_{1}^{\beta})+(2.1{v}_{2}^{M}-0.475{v}_{2}^{\alpha}+0.575{v}_{2}^{\beta})\right)\le 0,\\ (2.2{u}_{1}^{M}-0.55{u}_{1}^{\alpha}+0.55{u}_{1}^{\beta})+(3.5{u}_{2}^{M}-0.825{u}_{2}^{\alpha}+0.925{u}_{2}^{\beta})-\\ \left((2.9{v}_{1}^{M}-0.725{v}_{1}^{\alpha}+0.725{v}_{1}^{\beta})+(1.5{v}_{2}^{M}-0.35{v}_{2}^{\alpha}+0.4{v}_{2}^{\beta})\right)\le 0,\\ (3.2{u}_{1}^{M}-0.675{u}_{1}^{\alpha}+0.925{u}_{1}^{\beta})+(5.1{u}_{2}^{M}-1.075{u}_{2}^{\alpha}+1.475{u}_{2}^{\beta})-\\ \left((4.9{v}_{1}^{M}-1.1{v}_{1}^{\alpha}+1.35{v}_{1}^{\beta})+(2.6{v}_{2}^{M}-0.55{v}_{2}^{\alpha}+0.75{v}_{2}^{\beta})\right)\le 0,\end{array}\phantom{\rule{0ex}{0ex}}\begin{array}{l}(2.55{u}_{1}^{M}-0.575{u}_{1}^{\alpha}+0.725{u}_{1}^{\beta})+(5.7{u}_{2}^{M}-1.2{u}_{2}^{\alpha}+1.65{u}_{2}^{\beta})-\\ \left((4.1{v}_{1}^{M}-0.85{v}_{1}^{\alpha}+1.2{v}_{1}^{\beta})+(2.3{v}_{2}^{M}-0.55{v}_{2}^{\alpha}+0.6{v}_{2}^{\beta})\right)\le 0,\\ (5.1{u}_{1}^{M}-1.1{u}_{1}^{\alpha}+1.45{u}_{1}^{\beta})+(7.4{u}_{2}^{M}-1.625{u}_{2}^{\alpha}+2.075{u}_{2}^{\beta})-\\ \left((6.3{v}_{1}^{M}-1.475{v}_{1}^{\alpha}+1.775{v}_{1}^{\beta})+(4.1{v}_{2}^{M}-0.9{v}_{2}^{\alpha}+1.15{v}_{2}^{\beta})\right)\le 0,\end{array}\phantom{\rule{0ex}{0ex}}{u}_{1}^{M}-0.25{u}_{1}^{\alpha}+0.25{u}_{1}^{\beta}\ge 0,{u}_{2}^{M}-0.25{u}_{2}^{\alpha}+0.25{u}_{2}^{\beta}\ge 0,\phantom{\rule{0ex}{0ex}}{v}_{1}^{M}-0.25{v}_{1}^{\alpha}+0.25{v}_{1}^{\beta}\ge 0,{v}_{2}^{M}-0.25{v}_{2}^{\alpha}+0.25{v}_{2}^{\beta}\ge 0,\phantom{\rule{0ex}{0ex}}{u}_{1}^{M}-{u}_{1}^{\alpha}\ge 0,{u}_{2}^{M}-{u}_{2}^{\beta}\ge 0,\phantom{\rule{0ex}{0ex}}\begin{array}{l}{v}_{1}^{M}-{v}_{1}^{\alpha}\ge 0,{v}_{2}^{M}-{v}_{2}^{\alpha}\ge 0,\\ {u}_{r}^{\alpha}\ge 0,{u}_{r}^{\beta}\ge 0,\begin{array}{cc}& \end{array}r=1,2,\\ {v}_{i}^{\alpha}\ge 0,{v}_{i}^{\beta}\ge 0,\begin{array}{cc}& \end{array}i=1,2.\end{array}$$

_{A}:

_{A}is inefficient, and the others are efficient. Further, we have used Wang et al.’s model [28] for comparing and ranking fuzzy efficiencies.

_{B}, DMU

_{C}and DMU

_{E}are efficient. However, by the model of [48], we can see that DMU

_{B}, DMU

_{D}and DMU

_{E}are efficient. Even if the results were the same, because the model of [48] uses the wrong strategy, it would still not be valid. We take note that this example is utilized to demonstrate the computational procedure of the proposed technique and such a comparison is insignificant. Although the suggested procedure has been employed to a numerical case, the same frames could be used, with some adjustment, to other benchmarking problems.

## 7. Conclusions and Future Work

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Table 1.**Five decision-making units (DMUs) with two Z-number inputs and two Z-number outputs. M = medium. MH = medium high. H = high. VH = very high.

DMU | Inputs 1 | Inputs 2 | Outputs 1 | Outputs 2 |
---|---|---|---|---|

A | ((4.51, 5.16, 5.80), MH) | ((2.125, 2.348, 2.572), VH) | ((2.870, 3.110, 3.349), H) | ((4.545, 4.904, 5.263), H) |

B | ((3.46, 3.46, 3.46), H) | ((1.674, 1.794, 1.913), H) | ((2.460, 2.460, 2.460), VH) | ((4.263, 4.521, 4.780), MH) |

C | ((4.92, 5.48, 6.04), VH) | ((2.631, 3.11, 3.588), H) | ((3.229, 3.827, 4.425), H) | ((4.809, 5.704, 6.599), VH) |

D | ((4.80, 5.80, 6.79), M) | ((2.460, 2.572, 2.684), VH) | ((2.971, 3.229, 3.746), MH) | ((7.779, 8.062, 8.345), M) |

E | ((7.05, 7.77, 8.49), H) | ((4.647, 5.293, 5.938), MH) | ((6.223, 7.213, 8.203), M) | ((7.775, 8.851, 9.928), H) |

Linguistic Term | Abbreviation | Corresponding TFNs |
---|---|---|

Very Low | VL | (0.1, 0.2, 0.3) |

Low | L | (0.2, 0.3, 0.4) |

Medium Low | ML | (0.3, 0.4, 0.5) |

Medium | M | (0.4, 0.5, 0.6) |

Medium High | MH | (0.5, 0.6, 0.7) |

High | H | (0.6, 0.7, 0.8) |

Very High | VH | (0.7, 0.8, 0.9) |

DMU | Inputs 1 | Inputs 2 | Outputs 1 | Outputs 2 |
---|---|---|---|---|

A | (4, 0.5, 0.5) | (2.1, 0.2, 0.2) | (2.6, 0.2, 0.2) | (4.1, 0.3, 0.3) |

B | (2.9, 0, 0) | (1.5, 0.1, 0.1) | (2.2, 0, 0) | (3.5, 0.2, 0.2) |

C | (4.9, 0.5, 0.5) | (2.6, 0.4, 0.4) | (3.2, 0.5, 0.5) | (5.1, 0.8, 0.8) |

D | (4.1, 0.7, 0.7) | (2.3, 0.1, 0.1) | (2.5, 0.2, 0.4) | (5.7, 0.2, 0.2) |

E | (6.1, 0.2, 1) | (4.1, 0.5, 0.5) | (5.1, 0.7, 0.7) | (7.4, 0.9, 0.9) |

DMU_{B} | DMU_{C} | DMU_{D} | DMU_{E} | |
---|---|---|---|---|

${u}_{1}^{\ast}$ | $(0,0,0)$ | $(0,0,1.08108)$ | $(0,0,0)$ | $(0,0,0.68965)$ |

${u}_{2}^{\ast}$ | $(0,0,1.08108)$ | $(0,0,0)$ | $(0.17544,0,0)$ | $(0,0,0)$ |

${v}_{1}^{\ast}$ | $(0,0,1.158997)$ | $(0,0,0.45747)$ | $(0.24390,0,0)$ | $(0.16949,0,0)$ |

${v}_{2}^{\ast}$ | $(0,0,0.39932)$ | $(0.14282,0,0.14783)$ | $(0,0,0)$ | $(0,0,0)$ |

${\tilde{\theta}}^{\ast}$ | $(0,0,4)$ | $(0,0,4)$ | $(1,0.1579,0.1579)$ | $(0,0,4)$ |

**Table 5.**The matrix of the degree of preference for fuzzy efficiencies obtained by model [28].

DMUs | A | B | C | D | E |
---|---|---|---|---|---|

A | - | 0.4308 | 0.4308 | 0.3943 | 0.4308 |

B | 0.5692 | - | 0.5 | 0.5557 | 0.5 |

C | 0.5692 | 0.5 | - | 0.5557 | 0.5 |

D | 0.6257 | 0.4443 | 0.4443 | - | 0.4443 |

E | 0.5692 | 0.5 | 0.5 | 0.5557 | - |

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**MDPI and ACS Style**

Namakin, A.; Najafi, S.E.; Fallah, M.; Javadi, M.
A New Evaluation for Solving the Fully Fuzzy Data Envelopment Analysis with Z-Numbers. *Symmetry* **2018**, *10*, 384.
https://doi.org/10.3390/sym10090384

**AMA Style**

Namakin A, Najafi SE, Fallah M, Javadi M.
A New Evaluation for Solving the Fully Fuzzy Data Envelopment Analysis with Z-Numbers. *Symmetry*. 2018; 10(9):384.
https://doi.org/10.3390/sym10090384

**Chicago/Turabian Style**

Namakin, Ali, Seyyed Esmaeil Najafi, Mohammad Fallah, and Mehrdad Javadi.
2018. "A New Evaluation for Solving the Fully Fuzzy Data Envelopment Analysis with Z-Numbers" *Symmetry* 10, no. 9: 384.
https://doi.org/10.3390/sym10090384