# A Fuzzy Trade-Off Ranking Method for Multi-Criteria Decision-Making

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## Abstract

**:**

## 1. Introduction

## 2. Ranking Alternatives in Fuzzy Multi-Criteria Decision-Making

#### 2.1. Fuzzy Multi-Criteria Decision-Making

Criterion | ||||||

Alternative | ${Y}_{1}$ | ${Y}_{2}$ | ${Y}_{3}$ | … | ${Y}_{n}$ | , |

${A}_{1}$ | ${Y}_{11}$ | ${Y}_{12}$ | ${Y}_{13}$ | … | ${Y}_{1n}$ | |

${A}_{2}$ | ${Y}_{21}$ | ${Y}_{22}$ | ${Y}_{23}$ | … | ${Y}_{2n}$ | |

${A}_{3}$ | ${Y}_{31}$ | ${Y}_{32}$ | ${Y}_{33}$ | … | ${Y}_{3n}$ | |

⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | |

${A}_{q}$ | ${Y}_{q1}$ | ${Y}_{q2}$ | ${Y}_{q3}$ | … | ${Y}_{qn}$ |

Criterion | ||||||

Alternative | ${\tilde{Y}}_{1}$ | ${\tilde{Y}}_{2}$ | ${\tilde{Y}}_{3}$ | … | ${\tilde{Y}}_{n}$ | , |

${\tilde{A}}_{1}$ | ${\tilde{Y}}_{11}$ | ${\tilde{Y}}_{12}$ | ${\tilde{Y}}_{13}$ | … | ${\tilde{Y}}_{1n}$ | |

${\tilde{A}}_{2}$ | ${\tilde{Y}}_{21}$ | ${\tilde{Y}}_{22}$ | ${\tilde{Y}}_{23}$ | … | ${\tilde{Y}}_{2n}$ | |

${\tilde{A}}_{3}$ | ${\tilde{Y}}_{31}$ | ${\tilde{Y}}_{32}$ | ${\tilde{Y}}_{33}$ | … | ${\tilde{Y}}_{3n}$ | |

⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | |

${\tilde{A}}_{q}$ | ${\tilde{Y}}_{q1}$ | ${\tilde{Y}}_{q2}$ | ${\tilde{Y}}_{q3}$ | … | ${\tilde{Y}}_{qn}$ |

#### 2.2. Arithmetic Operations on Triangular Fuzzy Numbers

#### 2.3. The Fuzzy TOPSIS Method

#### 2.4. The Fuzzy VIKOR Method

- Determination of the ideal ${\tilde{Y}}_{j}^{+}=({a}_{j}^{+},{b}_{j}^{+},{c}_{j}^{+})$ and the anti-ideal ${\tilde{Y}}_{j}^{-}=({a}_{j}^{-},{b}_{j}^{-},{c}_{j}^{-})$ for $j=1,\dots ,n$, where
- (a)
- ${\tilde{Y}}_{j}^{+}=\underset{i}{\mathrm{MAX}\phantom{\rule{4.pt}{0ex}}}{\tilde{Y}}_{ij}$ and ${\tilde{Y}}_{j}^{-}=\underset{i}{\mathrm{MIN}\phantom{\rule{4.pt}{0ex}}}{\tilde{Y}}_{ij}$, if the j-th criteria represents the benefit,
- (b)
- ${\tilde{Y}}_{j}^{+}=\underset{i}{\mathrm{MIN}\phantom{\rule{4.pt}{0ex}}}{\tilde{Y}}_{ij}$ and ${\tilde{Y}}_{j}^{-}=\underset{i}{\mathrm{MAX}\phantom{\rule{4.pt}{0ex}}}{\tilde{Y}}_{ij}$, if the j-th criteria represents the cost.

The MAX and MIN are fuzzy operators as in Formulae (9) and (10), respectively. - Compute ${\tilde{S}}_{i}=({S}_{i}^{a},{S}_{i}^{b},{S}_{i}^{c})$ and ${\tilde{R}}_{i}=({R}_{i}^{a},{R}_{i}^{b},{R}_{i}^{c})$, $i=1,\dots ,q$ by the equations$${\tilde{S}}_{i}=\sum _{j=1}^{n}\oplus ({\tilde{w}}_{j}\otimes {\tilde{d}}_{ij}),$$$${\tilde{R}}_{i}=\underset{j}{\mathrm{MAX}\phantom{\rule{4.pt}{0ex}}}({\tilde{w}}_{j}\otimes {\tilde{d}}_{ij}),\phantom{\rule{3.33333pt}{0ex}}j=1,\dots ,n,$$
- (a)
- ${\tilde{d}}_{ij}=({\tilde{Y}}_{j}^{+}\ominus {\tilde{Y}}_{ij})/({c}_{j}^{+}-{a}_{j}^{-})$, if the j-th criteria represents the benefit,
- (b)
- ${\tilde{d}}_{ij}=({\tilde{Y}}_{j}^{+}\ominus {\tilde{Y}}_{ij})/({c}_{j}^{-}-{a}_{j}^{+})$, if the j-th criteria represents the cost,

where ${\tilde{d}}_{ij}$ is a normalized fuzzy difference, $\tilde{S}$ is a fuzzy weighted sum as in Equation (3) and $\tilde{R}$ is a fuzzy operator MAX (9). - Compute ${\tilde{Q}}_{i}=({Q}_{i}^{a},{Q}_{i}^{b},{Q}_{i}^{c})$, $i=1,\dots ,q$, by the equation$${\tilde{Q}}_{i}=v({\tilde{S}}_{i}\ominus {\tilde{S}}^{+})/({S}^{-c}-{S}^{+a})\oplus (1-v)({\tilde{R}}_{i}\ominus {\tilde{R}}^{+})/({R}^{-c}-{R}^{+a}),$$
- “Core” ranking.Rank the alternatives by sorting the values of ${Q}_{i}^{b}$, $i=1,\dots ,q$. A lower value implies a higher ranking. The obtained ranking is denoted by ${\{Rank\}}_{{Q}^{b}}$.
- Fuzzy ranking.The i-th ranking position in ${\{Rank\}}_{{Q}^{b}}$ is confirmed if $\underset{k\in \ell}{\mathrm{MIN}\phantom{\rule{4.pt}{0ex}}}{\tilde{Q}}^{(k)}={\tilde{Q}}^{(i)}$, where $\ell =\{i,i+1,\dots ,q\}$ and ${\tilde{Q}}^{(k)}$ is the fuzzy numbers for alternative ${A}^{(k)}$ at the k-th position in ${\{Rank\}}_{{Q}^{b}}$. Confirmed ordering represents fuzzy ranking ${\{Rank\}}_{\tilde{Q}}$.
- Defuzzification of ${\tilde{S}}_{i},{\tilde{R}}_{i},{\tilde{Q}}_{i},i=1,\dots ,q$ to convert the fuzzy numbers into crisp value using Equation (12).
- Defuzzification ranking.Rank the alternatives by sorting the crisp values of $S,R$ and Q in Step 6. A lower value implies a higher ranking. The results of the ranking lists are denoted by ${\{Rank\}}_{S}$, ${\{Rank\}}_{R}$ and ${\{Rank\}}_{Q}$, respectively.
- The best solution $({A}^{(1)})$ ranked in ${\{Rank\}}_{Q}$ is regarded as the best compromise solution if the following two conditions are satisfied:
- (a)
**C1.**Suppose ${A}^{(1)}$ is the first rank alternative and ${A}^{(2)}$ is the second rank in ${\{Rank\}}_{Q}$, $Adv\ge DQ,$ where $DQ=1/(q-1)$ and $Adv=(Q({A}^{(2)})-Q({A}^{(1)}))/(Q({A}^{(q)})-Q({A}^{(1)}))$.- (b)
**C2.**The alternative ${A}^{(1)}$ is also the best solution ranked by S and/or R.

If one of the conditions is not satisfied, a set of compromise solutions is then proposed compromising the following:- (a)
- Alternatives ${A}^{(1)}$ and ${A}^{(2)}$ if only condition
**C2**is not satisfied; or - (b)
- Alternatives ${A}^{(1)},{A}^{(2)},\dots ,{A}^{(M)}$ if condition
**C1**is not satisfied; ${A}^{(M)}$ is determined by the relation $Q({A}^{(M)})-Q({A}^{(1)})<DQ$ for maximum M.

## 3. Trade-Off Ranking Method

#### 3.1. Trade-Off Ranking Method with Defuzzification

#### 3.2. Fuzzy Trade-Off Ranking Method

- Normalization of the performance of criterion j in alternative i,${\tilde{Y}}_{ij}=({a}_{ij},{b}_{ij},{c}_{ij})$, by equation:$${\tilde{f}}_{ij}=\frac{{\tilde{Y}}_{ij}\ominus \tilde{a}}{\underset{j}{\mathrm{max}\phantom{\rule{4.pt}{0ex}}}{c}_{ij}-\underset{j}{\mathrm{min}\phantom{\rule{4.pt}{0ex}}}{a}_{ij}},i=1,\dots ,q,\phantom{\rule{3.33333pt}{0ex}}j=1,\dots ,n,$$
- Determination of the extreme solutions, ${\tilde{A}}_{k}^{*},\phantom{\rule{3.33333pt}{0ex}}k=1,\dots ,n$, by formula:$$\begin{array}{cc}\hfill {\tilde{A}}_{k}^{*}& =\{\underset{1\le i\le q}{\mathrm{min}\phantom{\rule{4.pt}{0ex}}}{f}_{ij1}\},j=1,\dots ,n,\phantom{\rule{4.pt}{0ex}}\mathrm{for}\phantom{\rule{4.pt}{0ex}}\mathrm{the}\phantom{\rule{4.pt}{0ex}}\mathrm{cost}\phantom{\rule{4.pt}{0ex}}\mathrm{criteria},\phantom{\rule{4.pt}{0ex}}\mathrm{or}\phantom{\rule{4.pt}{0ex}}\hfill \\ \hfill {\tilde{A}}_{k}^{*}& =\{\underset{1\le i\le q}{\mathrm{max}\phantom{\rule{4.pt}{0ex}}}{f}_{ij3}\},j=1,\dots ,n,\phantom{\rule{4.pt}{0ex}}\mathrm{for}\phantom{\rule{4.pt}{0ex}}\mathrm{the}\phantom{\rule{4.pt}{0ex}}\mathrm{benefit}\phantom{\rule{4.pt}{0ex}}\mathrm{criteria}.\hfill \end{array}$$
- Calculation of the distance of an alternative $\alpha $ to an extreme solution ${\tilde{A}}_{k}^{*}$, denoted as ${d}_{FTOR1}({\tilde{A}}_{k}^{*},{\tilde{A}}_{\alpha})$, using equation:$$\begin{array}{c}\hfill {d}_{FTOR1}({\tilde{A}}_{k}^{*},{\tilde{A}}_{\alpha})=\sum _{j=1}^{n}[d({\tilde{f}}_{kj},{\tilde{f}}_{\alpha j})],\phantom{\rule{3.33333pt}{0ex}}\alpha =1,\dots ,q,\phantom{\rule{3.33333pt}{0ex}}k=1,\dots ,n.\end{array}$$
- Calculation of the first level of fuzzy trade-off, which is the trade-off between an alternative with all the extreme solutions, is given by formula:$$\begin{array}{cc}\hfill DFT{1}_{{\tilde{A}}_{\alpha}}& =\sum _{j=1}^{n}[{w}_{j}^{\prime}\times {d}_{FTOR1}({\tilde{A}}_{k}^{*},{\tilde{A}}_{\alpha})],\alpha =1,\dots ,q,\phantom{\rule{3.33333pt}{0ex}}k=1,\dots ,n,\hfill \end{array}$$$$\begin{array}{cc}\hfill {w}_{j}^{\prime}& =\frac{{w}_{j}}{{\sum}_{j=1}^{n}{w}_{j}},\phantom{\rule{3.33333pt}{0ex}}j=1,\dots ,n,\hfill \\ \hfill {w}_{j}& =\mathrm{Crisp}({\tilde{w}}_{j}),j=1,\dots ,n.\hfill \end{array}$$
- Calculation of the distance of an alternative to the other alternatives is determined by the formula:$${d}_{FTOR2}({\tilde{A}}_{\alpha},{\tilde{A}}_{\beta})=\sum _{j=1}^{n}\left(\right)open="["\; close="]">d({\tilde{P}}_{\alpha j},{\tilde{P}}_{\beta j})$$
- Calculation of the second level of fuzzy trade-off, which is the trade-off among the alternatives, is given by equation:$$DFT{2}_{{\tilde{A}}_{\alpha}}=\sum _{i=1}^{q}\left(\right)open="["\; close="]">{d}_{FTOR2}({\tilde{A}}_{\alpha},{\tilde{A}}_{i})$$

## 4. Numerical Example: Personnel Selection Problem

- Emotional steadiness, ${Y}_{1}$;
- Oral communication skill, ${Y}_{2}$;
- Personality, ${Y}_{3}$;
- Past experience, ${Y}_{4}$;
- Self-confidence, ${Y}_{5}$.

## 5. Conclusions

## Supplementary Materials

Supplementary File 1## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 4.**Triangular fuzzy numbers and their crisp values of each criterion for alternative ${A}_{2}$.

Meaning of Linguistic Scale | Numerical Scale |
---|---|

Very low (VL) | (0, 0, 0.1) |

Low (L) | (0, 0.1, 0.3) |

Medium low (ML) | (0.1, 0.3, 0.5) |

Medium (M) | (0.3, 0.5, 0.7) |

Medium high (MH) | (0.5, 0.7, 0.9) |

High (H) | (0.7, 0.9, 1.0) |

Very high (VH) | (0.9, 1.0, 1.0) |

Meaning of Linguistic Scale | Numerical Scale |
---|---|

Very poor (VP) | (0, 0, 1) |

Poor (P) | (0, 1, 3) |

Medium poor (MP) | (1, 3, 5) |

Fair (F) | (3, 5, 7) |

Medium good (MG) | (5, 7, 9) |

Good (G) | (7, 9, 10) |

Very good (VG) | (9, 10, 10) |

Criterion | ${\mathit{DM}}_{1}$ | ${\mathit{DM}}_{2}$ | ${\mathit{DM}}_{3}$ |
---|---|---|---|

${Y}_{1}$ | H | H | H |

${Y}_{2}$ | VH | VH | VH |

${Y}_{3}$ | VH | H | H |

${Y}_{4}$ | VH | VH | VH |

${Y}_{5}$ | M | MH | MH |

Criterion | ${\mathit{A}}_{1}$ | ${\mathit{A}}_{2}$ | ${\mathit{A}}_{3}$ | ||||||
---|---|---|---|---|---|---|---|---|---|

${\mathit{DM}}_{\mathbf{1}}$ | ${\mathit{DM}}_{\mathbf{2}}$ | ${\mathit{DM}}_{\mathbf{3}}$ | ${\mathit{DM}}_{\mathbf{1}}$ | ${\mathit{DM}}_{\mathbf{2}}$ | ${\mathit{DM}}_{\mathbf{3}}$ | ${\mathit{DM}}_{\mathbf{1}}$ | ${\mathit{DM}}_{\mathbf{2}}$ | ${\mathit{DM}}_{\mathbf{3}}$ | |

${Y}_{1}$ | MG | G | MG | G | G | MG | VG | G | F |

${Y}_{2}$ | VG | VG | VG | MG | MG | MG | G | G | G |

${Y}_{3}$ | G | G | G | F | G | G | VG | VG | G |

${Y}_{4}$ | G | G | G | VG | VG | VG | VG | G | VG |

${Y}_{5}$ | G | G | G | F | F | F | G | G | MG |

Weight | ${\tilde{\mathit{Y}}}_{1}$ | ${\tilde{\mathit{Y}}}_{2}$ | ${\tilde{\mathit{Y}}}_{3}$ | ${\tilde{\mathit{Y}}}_{4}$ | ${\tilde{\mathit{Y}}}_{5}$ |
---|---|---|---|---|---|

(0.7,0.9,1) | (0.9,1,1) | (0.77,0.93,1) | (0.9,1,1) | (0.43,0.63,0.83) | |

${\tilde{A}}_{1}$ | (5.7,7.7,9.3) | (9,10,10) | (7,9,10) | (7,9,10) | (7,9,10) |

${\tilde{A}}_{2}$ | (6.3,8.3,9.7) | (5,7,9) | (5.7,7.7,9) | (9,10,10) | (3,5,7) |

${\tilde{A}}_{3}$ | (6.3,8,9) | (7,9,10) | (8.3,9.7,10) | (8.3,9.7,10) | (6.3,8.3,9.7) |

Weight | ${\mathit{Y}}_{1}$ | ${\mathit{Y}}_{2}$ | ${\mathit{Y}}_{3}$ | ${\mathit{Y}}_{4}$ | ${\mathit{Y}}_{5}$ |
---|---|---|---|---|---|

0.875 | 0.975 | 0.908 | 0.975 | 0.630 | |

${A}_{1}$ | 7.60 | 9.75 | 8.75 | 8.75 | 8.75 |

${A}_{2}$ | 8.15 | 7.00 | 7.53 | 9.75 | 5.00 |

${A}_{3}$ | 7.83 | 8.75 | 9.43 | 9.42 | 8.15 |

Weight | ${\mathit{Y}}_{1}$ | ${\mathit{Y}}_{2}$ | ${\mathit{Y}}_{3}$ | ${\mathit{Y}}_{4}$ | ${\mathit{Y}}_{5}$ |
---|---|---|---|---|---|

0.201 | 0.223 | 0.208 | 0.223 | 0.144 | |

${A}_{1}$ | 0 | 1 | 0.64 | 0 | 1 |

${A}_{2}$ | 1 | 0 | 0 | 1 | 0 |

${A}_{3}$ | 0.42 | 0.64 | 1 | 0.68 | 0.84 |

Ranking | 1 | 2 | 3 |
---|---|---|---|

Fuzzy trade-off | ${A}_{3}$ | ${A}_{1}$ | ${A}_{2}$ |

Defuzzification | ${A}_{3}$ | ${A}_{1}$ | ${A}_{2}$ |

Trade-Off | ${\mathit{A}}_{1}$ | ${\mathit{A}}_{2}$ | ${\mathit{A}}_{3}$ |
---|---|---|---|

$DFT1$ | 1.027 | 1.039 | 0.984 |

$DT1$ | 1.090 | 1.105 | 1.030 |

Ordering | 1 | 2 | 3 | |
---|---|---|---|---|

${\{Rank\}}_{{Q}^{b}}$ | ${A}_{3}$ | ${A}_{1}$ | ${A}_{2}$ | |

${\{Rank\}}_{\tilde{Q}}$ | ${A}_{1}$ | ${A}_{2}$ | ||

Defuzzification | ${\{Rank\}}_{Q}$ | ${A}_{3}$ | ${A}_{1}$ | ${A}_{2}$ |

${\{Rank\}}_{S}$ | ${A}_{3}$ | ${A}_{1}$ | ${A}_{2}$ | |

${\{Rank\}}_{R}$ | ${A}_{3}$ | ${A}_{1}$ | ${A}_{2}$ |

${\mathit{A}}_{1}$ | ${\mathit{A}}_{2}$ | ${\mathit{A}}_{3}$ | ||
---|---|---|---|---|

$\tilde{S}$ | ${S}^{a}$ | $-1.49$ | $-1.02$ | $-1.42$ |

${S}^{b}$ | 0.62 | 1.39 | 0.43 | |

${S}^{c}$ | 3.25 | 4.01 | 2.85 | |

Crisp$(S)$ | 0.75 | 1.44 | 0.57 | |

$\tilde{R}$ | ${R}^{a}$ | $-0.18$ | 0 | $-0.17$ |

${R}^{b}$ | 0.33 | 0.6 | 0.2 | |

${R}^{c}$ | 1 | 1 | 0.85 | |

Crisp$(R)$ | 0.37 | 0.55 | 0.27 | |

$\tilde{Q}$ | ${Q}^{a}$ | 0 | 0.11 | 0.01 |

${Q}^{b}$ | 0.066 | 0.24 | 0 | |

${Q}^{c}$ | 0.095 | 0.18 | 0 | |

Crisp$(Q)$ | 0.057 | 0.193 | 0.003 |

Ranking | 1 | 2 | 3 |
---|---|---|---|

${\mathit{A}}_{\mathbf{3}}$ | ${\mathit{A}}_{\mathbf{1}}$ | ${\mathit{A}}_{\mathbf{2}}$ | |

${d}^{+}$ | 1.45 | 1.48 | 1.87 |

${d}^{-}$ | 3.93 | 3.95 | 3.52 |

$CC$ | 0.731 | 0.728 | 0.653 |

Criterion | ${\mathit{DM}}_{1}$ | ${\mathit{DM}}_{2}$ | ${\mathit{DM}}_{3}$ |
---|---|---|---|

${Y}_{1}$ | MH | H | H |

${Y}_{2}$ | VL | L | VL |

${Y}_{3}$ | ML | ML | ML |

${Y}_{4}$ | H | H | VH |

${Y}_{5}$ | L | VL | L |

${\tilde{\mathit{Y}}}_{\mathit{j}}$ | ${\tilde{\mathit{Y}}}_{1}$ | ${\tilde{\mathit{Y}}}_{2}$ | ${\tilde{\mathit{Y}}}_{3}$ | ${\tilde{\mathit{Y}}}_{4}$ | ${\tilde{\mathit{Y}}}_{5}$ |
---|---|---|---|---|---|

${\tilde{w}}_{j}$ | (0.63,0.83,0.97) | (0,0.03,0.17) | (0.1,0.3,0.5) | (0.77,0.93,1) | (0,0.07,0.23) |

${w}_{j}$ | 0.82 | 0.06 | 0.3 | 0.91 | 0.09 |

MCDM Method | ${\mathit{A}}_{1}$ | ${\mathit{A}}_{2}$ | ${\mathit{A}}_{3}$ |
---|---|---|---|

Fuzzy Trade-off | |||

$DFT1$ | 1.700 | 0.354 | 1.272 |

$DT1$ | 1.799 | 0.364 | 1.334 |

Fuzzy VIKOR | |||

Crisp$(S)$ | 0.662 | 0.411 | 0.357 |

Crisp$(R)$ | 0.405 | 0.276 | 0.253 |

${Q}^{a}$ | 0.007 | 0 | 0.016 |

${Q}^{b}$ | 0.145 | 0.024 | 0 |

${Q}^{c}$ | 0.188 | 0.045 | 0 |

Crisp$(Q)$ | 0.121 | 0.023 | 0.004 |

Fuzzy TOPSIS | |||

${d}^{+}$ | 3.283 | 3.215 | 3.184 |

${d}^{-}$ | 2.074 | 2.096 | 2.125 |

$CC$ | 0.387 | 0.395 | 0.400 |

© 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**

Jaini, N.I.; Utyuzhnikov, S.V.
A Fuzzy Trade-Off Ranking Method for Multi-Criteria Decision-Making. *Axioms* **2018**, *7*, 1.
https://doi.org/10.3390/axioms7010001

**AMA Style**

Jaini NI, Utyuzhnikov SV.
A Fuzzy Trade-Off Ranking Method for Multi-Criteria Decision-Making. *Axioms*. 2018; 7(1):1.
https://doi.org/10.3390/axioms7010001

**Chicago/Turabian Style**

Jaini, Nor Izzati, and Sergey V. Utyuzhnikov.
2018. "A Fuzzy Trade-Off Ranking Method for Multi-Criteria Decision-Making" *Axioms* 7, no. 1: 1.
https://doi.org/10.3390/axioms7010001