# Simple Additive Weighting Method Equipped with Fuzzy Ranking of Evaluated Alternatives

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## Abstract

**:**

## 1. Introduction

## 2. Elements of Fuzzy Sets Theory

#### 2.1. Fuzzy Numbers

**Definition**

**1.**

#### 2.2. Ordered Fuzzy Numbers

**Definition**

**2.**

#### 2.3. Defuzzification of Ordered Fuzzy Numbers

**Definition**

**3.**

- The weighted maximum (WM) functional$${\varphi}_{WM}(\overleftrightarrow{Tr}(a,b,c,d)|\lambda )=\lambda \xb7b+(1-\lambda )\xb7c,\text{}\lambda \in \left[0;1\right],$$
- The first maximum (FM) functional$${\varphi}_{FM}(\overleftrightarrow{Tr}(a,b,c,d))={\varphi}_{WM}(\overleftrightarrow{Tr}(a,b,c,d)|1)=b,$$
- The last maximum (LM) functional$${\varphi}_{LM}(\overleftrightarrow{Tr}(a,b,c,d))={\varphi}_{WM}(\overleftrightarrow{Tr}(a,b,c,d)|0)=c,$$
- The middle maximum (MM) functional$${\varphi}_{MM}(\overleftrightarrow{Tr}(a,b,c,d))={\varphi}_{WM}(\overleftrightarrow{Tr}(a,b,c,d)|\frac{1}{2})=\frac{1}{2}\xb7(b+c),$$
- The gravity center (GC) functional$${\varphi}_{CG}(\overleftrightarrow{Tr}(a,b,c,d))=\{\begin{array}{c}\frac{{a}^{2}+a\xb7b+{b}^{2}-{c}^{2}-c\xb7d-{d}^{2}}{3(a+b-c-d)}\text{}a\ne d,\\ a\text{}a=d,\end{array}$$
- The geometrical mean (GM) functional$${\varphi}_{GM}(\overleftrightarrow{Tr}(a,b,c,d))=\{\begin{array}{c}\frac{a\xb7b-c\xb7d}{a+b-c-d}\text{}a\ne d,\\ a\text{}a=d.\end{array}$$

#### 2.4. Relation of “Greater than or Equal to” for Trapezoidal Ordered Fuzzy Numbers

- For any pair $(\overleftrightarrow{\mathcal{K}},\overleftrightarrow{\mathcal{L}})\in ({\mathbb{K}}_{Tr}^{+}{{\displaystyle \cup}}^{}\mathbb{R})\times ({\mathbb{K}}_{Tr}^{+}{{\displaystyle \cup}}^{}\mathbb{R})$, the extension law$${\nu}_{GE}(\overleftrightarrow{\mathcal{K}},\overleftrightarrow{\mathcal{L}})={\nu}_{\left[GE\right]}(\mathsf{\Psi}(\overleftrightarrow{\mathcal{K}}),\mathsf{\Psi}(\overleftrightarrow{\mathcal{L}})),$$
- For any pair $(\overleftrightarrow{\mathcal{K}},\overleftrightarrow{\mathcal{L}})\in ({\mathbb{K}}_{Tr}^{-}{{\displaystyle \cup}}^{}\mathbb{R})\times ({\mathbb{K}}_{Tr}^{-}{{\displaystyle \cup}}^{}\mathbb{R})$, the sign exchange law$${\nu}_{GE}(\overleftrightarrow{\mathcal{K}},\overleftrightarrow{\mathcal{L}})={\nu}_{GE}(\ominus \overleftrightarrow{\mathcal{L}},\text{}\ominus \overleftrightarrow{\mathcal{K}}),$$
- For any pair $(\overleftrightarrow{\mathcal{K}},\overleftrightarrow{\mathcal{L}})\in ({\mathbb{K}}_{Tr}^{-}{{\displaystyle \cup}}^{}\mathbb{R})\times ({\mathbb{K}}_{Tr}^{+}{{\displaystyle \cup}}^{}\mathbb{R}){{\displaystyle \cup}}^{}({\mathbb{K}}_{Tr}^{+}{{\displaystyle \cup}}^{}\mathbb{R})\times ({\mathbb{K}}_{Tr}^{-}{{\displaystyle \cup}}^{}\mathbb{R})$, the law of parties’ subtraction of inequality$${\nu}_{GE}(\overleftrightarrow{\mathcal{K}},\overleftrightarrow{\mathcal{L}})={\nu}_{GE}(\overleftrightarrow{\mathcal{K}\text{}}\overline{)-}\overleftrightarrow{\text{}\mathcal{L}},\overleftrightarrow{Tr}(0,0,0,0)).$$

**Theorem**

**1.**

- if $\overleftrightarrow{\mathcal{M}}\in {\mathbb{K}}_{Tr}^{+}{{\displaystyle \cup}}^{}\mathbb{R}$ then$${\nu}_{GE}(\overleftrightarrow{\mathcal{K}},\overleftrightarrow{\mathcal{L}})=\{\begin{array}{c}0,\text{}0{d}_{\mathcal{M}},\\ \frac{-\text{}{d}_{\mathcal{M}}}{{c}_{\mathcal{M}}-\text{}{d}_{\mathcal{M}}},\text{}{d}_{\mathcal{M}}\ge 0{c}_{\mathcal{M}},\\ 1,\text{}{c}_{\mathcal{M}}\ge 0,\end{array}$$
- if $\overleftrightarrow{\mathcal{M}}\in {\mathbb{K}}_{Tr}^{-}$ then$${\nu}_{GE}(\overleftrightarrow{\mathcal{K}},\overleftrightarrow{\mathcal{L}})=\{\begin{array}{c}0,\text{}0{a}_{\mathcal{M}},\\ \frac{-\text{}{a}_{\mathcal{M}}}{{b}_{\mathcal{M}}-\text{}{a}_{\mathcal{M}}},\text{}{a}_{\mathcal{M}}\ge 0{b}_{\mathcal{M}}\\ 1,\text{}{b}_{\mathcal{M}}\ge \text{}0.\end{array}.$$

**Theorem**

**2.**

**Proof.**

## 3. The Oriented Fuzzy SAW with Fuzzy Order

**Procedure 1:**

**Step 1:**Define the set $\mathbb{D}=\{{\mathcal{C}}_{1},{\mathcal{C}}_{2},\dots ,{\mathcal{C}}_{n}\}$ of evaluation criteria;**Step 2:**For each evaluation ${\mathcal{C}}_{j}\text{}(j=1,2,\dots ,n)$, determine its scope ${Y}_{j}$;**Step 3:**Determine the evaluation template$$\mathbb{Y}={Y}_{1}\times {Y}_{2}\times \dots .\times {Y}_{n},$$**Step 4:**Define a NOS-TrOFN $\mathbb{O}\subset {\mathbb{K}}_{tr}$;**Step 5:**Determine the scoring function $\overleftrightarrow{saw}:{\mathbb{O}}^{n}\to {\mathbb{K}}_{Tr}$ given for any $\mathcal{Z}=({\overleftrightarrow{Z}}_{1},{\overleftrightarrow{Z}}_{2},\dots ,{\overleftrightarrow{Z}}_{n})\in {\mathbb{O}}^{n}$ as an aggregated evaluation index$$\overleftrightarrow{saw}(\mathcal{Z})=({w}_{1}\odot {\overleftrightarrow{Z}}_{1})\u229e({w}_{2}\odot {\overleftrightarrow{Z}}_{2})\u229e\dots \u229e({w}_{n}\odot {\overleftrightarrow{Z}}_{n})\in {\mathbb{K}}_{tr},$$**Step 6:**Define the evaluation function $\mathcal{X}:\mathbb{Y}\times \mathbb{D}\to \mathbb{O}\subset {\mathbb{K}}_{tr}$ fulfilling for each $(j=1,2,\dots ,n)$ the condition$$\mathcal{X}(\mathcal{A},{\mathcal{C}}_{j})\text{}\tilde{GE}\text{}\mathcal{X}(\mathcal{B},{\mathcal{C}}_{j})\iff \u201cFrom\text{}the\text{}perspective\text{}of\text{}the\text{}criterion\text{}{\mathcal{C}}_{j},\text{}\phantom{\rule{0ex}{0ex}}the\text{}decision\text{}alternative\text{}\mathcal{A}\text{}is\text{}not\text{}worse\text{}than\text{}the\text{}decision\text{}alternative\text{}\mathcal{B}.\u201d;$$**Step 7:**Determine the set $\mathbb{A}=\{{\mathcal{A}}_{1},\text{}{\mathcal{A}}_{2},\dots ,\text{}{\mathcal{A}}_{m}\}\subset \mathbb{Y}$ of evaluated decision alternatives;**Step 8:**Evaluate each alternative ${\mathcal{A}}_{i}\in \mathbb{A}$ $(i=1,2,\dots ,m)$ by the value$$\overleftrightarrow{SAW}({\mathcal{A}}_{i})=\overleftrightarrow{saw}(\mathcal{X}({\mathcal{A}}_{i})),$$$$\mathcal{X}({\mathcal{A}}_{i})=(\mathcal{X}({\mathcal{A}}_{i},{\mathcal{C}}_{1}),\mathcal{X}({\mathcal{A}}_{i},{\mathcal{C}}_{2}),\dots ,\mathcal{X}({\mathcal{A}}_{i},{\mathcal{C}}_{n}))\in {\mathbb{O}}^{n};$$**Step 9:**Determine a scoring order ${\mathcal{A}}_{i}\text{}\overline{BE}\text{}{\mathcal{A}}_{k}\text{}(i=1,2,\dots ,m;k=1,2,\dots ,m)$ by means of obtained values $\overleftrightarrow{SAW}({\mathcal{A}}_{i})$.

**Procedure 2:**

**Step 1**: Perform the substitutions $k:=0$, ${A}^{(1)}:=\mathbb{A}$;**Step 2**: Perform the substitutions $k:=k+1$;**Step 3**: Perform the substitutions$${B}^{(k)}:=\mathrm{Best}({A}^{(k)}),$$$${A}^{(k+1)}:={A}^{(k)}\backslash \mathbb{S}({B}^{(k)});$$**Step 4**: If the condition$${A}^{(k+1)}\ne \varnothing ,$$**Step 5**: Each recommendation ${\mathcal{A}}_{i}\in \mathbb{A}$ belongs only to one set $\mathbb{S}({B}^{(k)})$. For any given number $l>k$ and any recommendation ${A}_{j}\in \mathbb{S}({B}^{(l)})$, the value $\overleftrightarrow{SAW}({\mathcal{A}}_{i})$ dominates the values $\overleftrightarrow{SAW}({\mathcal{A}}_{j})$. Thanks to that, any pair $({\mathcal{A}}_{i},\text{}{\mathcal{A}}_{j})\in \mathbb{S}({B}^{(k)})\times \mathbb{S}({B}^{(l)})$ is ordered as follows:$$l>k\text{}\Rightarrow {\mathcal{A}}_{i}\text{}{\overline{BE}}_{IND}\text{}{\mathcal{A}}_{j},$$$$k\le l\text{}\Rightarrow {\mathcal{A}}_{j}\text{}{\overline{BE}}_{IND}\text{}{\mathcal{A}}_{i}.$$

## 4. Illustrative Example

**Example**

**1.**

## 5. Generalization of Conclusions

**Theorem**

**3.**

**Proof.**

**Example**

**2.**

## 6. Recapitulation

**Hypothesis:**

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Nomenclature

FM | First Maximum functional determined by identity Equation (18) |

FN | Fuzzy Number defined in [11] |

F-SAW | Fuzzy Simple Additive Weighting method described in [9] |

GC | Gravity Center functional determined by identity Equation (21) |

$\tilde{\mathrm{GE}}$ | fuzzy relation “greater than or equal to” for TrOFNs by |

GM | Geometrical Mean functional determined by identity Equation (22) |

LM | Last Maximum functional determined by identity Equation (19) |

MM | Middle Maximum functional determined by identity Equation (20) |

NOS | Numerical Order Scale mentioned in Section 3 |

NOS-TrOFN | NOS given as a sequence of TrOFN introduced in Section 3 |

OFN | Ordered Fuzzy Number defined in [10,14] |

OF-R | Oriented Fuzzy Ranking (OF-R) method introduced in Section 5 |

OF-SAW | Oriented Fuzzy Simple Additive Weighting method described in Section 3 |

SAW | Simple Additive Weighting method [2,3] |

SMART | The Simple Multi Attribute Rating Technique [2,3] |

TOPSIS | Technique for Order of Preference by Similarity to Ideal Solution [5] |

TrFN | Trapezoidal Fuzzy Number determined by membership function Equation (3) |

TrOFN | Trapezoidal Oriented Fuzzy Number determined by membership function Equation (6) |

WM | weighted maximum functional determined by identity Equation (17) |

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**Figure 1.**Impact of defuzzification method MM on the creation of deceptive information (Source: Own elaboration).

**Figure 2.**Impact of defuzzification method GC on the creation of deceptive information (Source: Own elaboration).

**Table 1.**The values of the Simple Additive Weighting method (SAW) scoring function and their defuzzification [15].

Decision Alternatives | Scoring Function Values $\overleftrightarrow{\mathit{S}\mathit{A}\mathit{W}}({\mathcal{A}}_{\mathit{i}})$ | Defuzzified Values ${\mathit{\varphi}}_{\mathit{D}}(\overleftrightarrow{\mathit{S}\mathit{A}\mathit{W}}({\mathcal{A}}_{\mathit{i}}))$ | |||||
---|---|---|---|---|---|---|---|

FM | LM | MM | WM | CG | GM | ||

${\mathcal{A}}_{1}$ | $\overleftrightarrow{Tr}(1.8,1.8,1.8,1.8)$ | 1.80 | 1.80 | 1.80 | 1.80 | 1.80 | 1.80 |

${\mathcal{A}}_{2}$ | $\overleftrightarrow{Tr}(3.0,3.0,2.9,2.8)$ | 3.00 | 2.90 | 2.95 | 2.91 | 2.92 | 2.93 |

${\mathcal{A}}_{3}$ | $\overleftrightarrow{Tr}(2.0,2.0,2.4,2.8)$ | 2.00 | 2.40 | 2.20 | 2.36 | 2.31 | 2.27 |

${\mathcal{A}}_{4}$ | $\overleftrightarrow{Tr}(2.4,2.4,2.2,2.0)$ | 2.40 | 2.20 | 2.30 | 2.22 | 2.24 | 2.27 |

${\mathcal{A}}_{5}$ | $\overleftrightarrow{Tr}(2.6,2.6,2.7,2.8)$ | 2.60 | 2.70 | 2.65 | 2.69 | 2.68 | 2.67 |

${\mathcal{A}}_{6}$ | $\overleftrightarrow{Tr}(3.0,3.0,3.1,3.2)$ | 3.00 | 3.10 | 3.05 | 3.09 | 3.08 | 3.07 |

${\mathcal{A}}_{7}$ | $\overleftrightarrow{Tr}(3.2,3.2,3.2,3.2)$ | 3.20 | 3.20 | 3.20 | 3.20 | 3.20 | 3.20 |

${\mathcal{A}}_{8}$ | $\overleftrightarrow{Tr}(2.8,2.8,3.0,3.2)$ | 2.80 | 3.00 | 2.90 | 2.98 | 2.96 | 2.93 |

${\mathcal{A}}_{9}$ | $\overleftrightarrow{Tr}(2.2,2.2,2.0,1.8)$ | 2.20 | 2.00 | 2.10 | 2.02 | 2.04 | 2.07 |

${\mathcal{A}}_{10}$ | $\overleftrightarrow{Tr}(2.4,2.4,2.4,2.4)$ | 2.40 | 2.40 | 2.40 | 2.40 | 2.40 | 2.40 |

${\mathcal{A}}_{11}$ | $\overleftrightarrow{Tr}(2.6,2.6,2.9,3.2)$ | 2.60 | 2.90 | 2.75 | 2.87 | 2.83 | 2.80 |

${\mathcal{A}}_{12}$ | $\overleftrightarrow{Tr}(2.6,2.6,3.0,3.4)$ | 2.60 | 3.00 | 2.80 | 2.96 | 2.91 | 2.87 |

${\mathcal{A}}_{13}$ | $\overleftrightarrow{Tr}(3.6,3.6,3.5,3.4)$ | 3.60 | 3.50 | 3.55 | 3.51 | 3.52 | 3.53 |

${\mathcal{A}}_{14}$ | $\overleftrightarrow{Tr}(3.2,3.2,3.3,3.4)$ | 3.20 | 3.30 | 3.25 | 3.29 | 3.28 | 3.27 |

${\mathcal{A}}_{15}$ | $\overleftrightarrow{Tr}(4.0,4.0,3.9,3.8)$ | 4.00 | 3.90 | 3.95 | 3.91 | 3.92 | 3.93 |

**Table 2.**The decision alternatives’ rankings determined by SAW methods (Source: [15] and own elaboration).

Decision Alternatives | Order with Applied Defuzzification Method | Faithful Order | |||||
---|---|---|---|---|---|---|---|

FM | LM | MM | WM | CG | GM | ||

${\mathcal{A}}_{15}$ | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

${\mathcal{A}}_{13}$ | 2 | 2 | 2 | 2 | 2 | 2 | 2 |

${\mathcal{A}}_{14}$ | 3.5 | 3 | 3 | 3 | 3 | 3 | 3.5 |

${\mathcal{A}}_{7}$ | 3.5 | 4 | 4 | 4 | 4 | 4 | 3.5 |

${\mathcal{A}}_{6}$ | 5.5 | 5 | 5 | 5 | 5 | 5 | 5 |

${\mathcal{A}}_{8}$ | 7 | 6.5 | 7 | 6 | 6 | 6.5 | 7 |

${\mathcal{A}}_{2}$ | 5.5 | 8.5 | 6 | 8 | 7 | 6.5 | 7 |

${\mathcal{A}}_{12}$ | 9 | 6.5 | 8 | 7 | 8 | 8 | 7 |

${\mathcal{A}}_{11}$ | 9 | 8.5 | 9 | 9 | 9 | 9 | 9 |

${\mathcal{A}}_{10}$ | 11.5 | 11.5 | 11 | 11 | 11 | 11 | 11.5 |

${\mathcal{A}}_{5}$ | 9 | 10 | 10 | 10 | 10 | 10 | 11.5 |

${\mathcal{A}}_{3}$ | 14 | 11.5 | 13 | 12 | 12 | 12.5 | 11.5 |

${\mathcal{A}}_{4}$ | 11.5 | 13 | 12 | 13 | 13 | 12.5 | 11.5 |

${\mathcal{A}}_{9}$ | 13 | 14 | 14 | 14 | 14 | 14 | 14 |

${\mathcal{A}}_{1}$ | 15 | 15 | 15 | 15 | 15 | 15 | 15 |

Number of equivalence classes | 12 | 12 | 15 | 15 | 15 | 13 | 9 |

Decision Alternatives | ${\mathcal{A}}_{1}$ | ${\mathcal{A}}_{2}$ | ${\mathcal{A}}_{3}$ | ${\mathcal{A}}_{4}$ | ${\mathcal{A}}_{5}$ | ${\mathcal{A}}_{6}$ | ${\mathcal{A}}_{7}$ | ${\mathcal{A}}_{8}$ | ${\mathcal{A}}_{9}$ | ${\mathcal{A}}_{10}$ | ${\mathcal{A}}_{11}$ | ${\mathcal{A}}_{12}$ | ${\mathcal{A}}_{13}$ | ${\mathcal{A}}_{14}$ | ${\mathcal{A}}_{15}$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

${\mathcal{A}}_{1}$ | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

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

${\mathcal{A}}_{3}$ | 1 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 0 |

${\mathcal{A}}_{4}$ | 1 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 0 |

${\mathcal{A}}_{5}$ | 1 | 0 | 1 | 1 | 1 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 0 |

${\mathcal{A}}_{6}$ | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 |

${\mathcal{A}}_{7}$ | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 1 | 0 |

${\mathcal{A}}_{8}$ | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 |

${\mathcal{A}}_{9}$ | 1 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |

${\mathcal{A}}_{10}$ | 1 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 0 |

${\mathcal{A}}_{11}$ | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 0 | 0 | 0 |

${\mathcal{A}}_{12}$ | 1 | 1 | 1 | 1 | 1 | 0.67 | 0.50 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 |

${\mathcal{A}}_{13}$ | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0 |

${\mathcal{A}}_{14}$ | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 1 | 0 |

${\mathcal{A}}_{15}$ | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

Decision Alternatives | Scoring Function Values $\overleftrightarrow{\mathit{S}\mathit{A}\mathit{W}}({\mathcal{A}}_{\mathit{j}})$ | Defuzzified Values ${\mathit{\varphi}}_{\mathit{D}}(\overleftrightarrow{\mathit{S}\mathit{A}\mathit{W}}({\mathcal{A}}_{\mathit{j}}))$ | |||||
---|---|---|---|---|---|---|---|

FM | LM | MM | WM | CG | GM | ||

${\mathcal{A}}_{1}$ | $\overleftrightarrow{Tr}(5.0,15.0,25.0,35.0)$ | 24 | 15 | 25 | 20 | 20 | 20 |

${\mathcal{A}}_{2}$ | $\overleftrightarrow{Tr}(35.0,24.9,15.0,5.0)$ | 15.99 | 24.9 | 15 | 19.95 | 19.9792 | 19.96241 |

${\mathcal{A}}_{3}$ | $\overleftrightarrow{Tr}(5.0,15.1,20.0,35.0)$ | 19.51 | 15.1 | 20 | 17.55 | 19.06867 | 17.89398 |

${\mathcal{A}}_{4}$ | $\overleftrightarrow{Tr}(35.0,20.0,14.9,5.0)$ | 15.41 | 20 | 14.9 | 17.45 | 19.0265 | 17.82051 |

${\mathcal{A}}_{5}$ | $\overleftrightarrow{Tr}(5.0,15.1,24.9,30.0)$ | 23.91 | 15 | 24.9 | 19.95 | 18.54833 | 19.25501 |

${\mathcal{A}}_{6}$ | $\overleftrightarrow{Tr}(35.0,20.0,14.8,5.0)$ | 15.82 | 25 | 14.8 | 19.9 | 18.53182 | 19.20455 |

${\mathcal{A}}_{7}$ | $\overleftrightarrow{Tr}(5.0,10.2,20.0,35.0)$ | 19.02 | 10.2 | 20 | 15.1 | 17.96449 | 16.30653 |

${\mathcal{A}}_{8}$ | $\overleftrightarrow{Tr}(35.0,20.0,9.9,5.0)$ | 10.91 | 20 | 9.9 | 14.95 | 17.89268 | 16.22195 |

${\mathcal{A}}_{9}$ | $\overleftrightarrow{Tr}(5.0,14.9,20.0,30.0)$ | 19.49 | 14.9 | 20 | 17.45 | 17.48051 | 17.45847 |

${\mathcal{A}}_{10}$ | $\overleftrightarrow{Tr}(30.0,19.9,15.0,5.0)$ | 15.49 | 19.9 | 15 | 17.45 | 17.4806 | 17.45819 |

${\mathcal{A}}_{11}$ | $\overleftrightarrow{Tr}(1.0,14.9,25.2,30.0)$ | 24.17 | 14.9 | 25.2 | 20.05 | 17.41416 | 18.85751 |

${\mathcal{A}}_{12}$ | $\overleftrightarrow{Tr}(30.0,25.1,15.0,1.0)$ | 16.01 | 25.1 | 15 | 20.05 | 17.40844 | 18.87468 |

${\mathcal{A}}_{13}$ | $\overleftrightarrow{Tr}(5.0,9.9,21.1,30.0)$ | 19.98 | 9.9 | 21.1 | 15.5 | 16.62707 | 16.11878 |

${\mathcal{A}}_{14}$ | $\overleftrightarrow{Tr}(5.0,10.0,20.0,30.0)$ | 19 | 10 | 20 | 15 | 16.42857 | 15.71429 |

${\mathcal{A}}_{15}$ | $\overleftrightarrow{Tr}(1.0,15.0,19.9,30.0)$ | 19.41 | 15 | 19.9 | 17.45 | 16.24395 | 17.16814 |

${\mathcal{A}}_{16}$ | $\overleftrightarrow{Tr}(30.0,20.0,15.0,1.0)$ | 15.5 | 20 | 15 | 17.5 | 16.26471 | 17.20588 |

${\mathcal{A}}_{17}$ | $\overleftrightarrow{Tr}(1.0,10.0,20.3,30.0)$ | 19.27 | 10 | 20.3 | 15.15 | 15.35276 | 15.24173 |

${\mathcal{A}}_{18}$ | $\overleftrightarrow{Tr}(30.0,20.0,10.0,1.0)$ | 11 | 20 | 10 | 15 | 15.2906 | 15.12821 |

Decision Alternatives | Order with Applied Defuzzification Method | Faithful Order | |||||
---|---|---|---|---|---|---|---|

FM | LM | MM | WM | CG | GM | ||

${\mathcal{A}}_{1}$ | 2 | 11 | 2 | 3 | 1 | 1 | 9.5 |

${\mathcal{A}}_{2}$ | 12 | 3 | 12.5 | 4.5 | 2 | 2 | 9.5 |

${\mathcal{A}}_{3}$ | 6 | 9 | 7.5 | 7 | 3 | 7 | 9.5 |

${\mathcal{A}}_{4}$ | 16 | 5.5 | 15 | 10.5 | 4 | 8 | 9.5 |

${\mathcal{A}}_{5}$ | 3 | 11 | 3 | 4.5 | 5 | 4 | 9.5 |

${\mathcal{A}}_{6}$ | 13 | 2 | 15 | 6 | 6 | 3 | 9.5 |

${\mathcal{A}}_{7}$ | 9 | 15 | 7.5 | 16 | 7 | 13 | 9.5 |

${\mathcal{A}}_{8}$ | 18 | 5.5 | 18 | 18 | 8 | 14 | 9.5 |

${\mathcal{A}}_{9}$ | 7 | 13.5 | 12.5 | 10.5 | 9 | 9 | 9.5 |

${\mathcal{A}}_{10}$ | 15 | 8 | 12 | 10.5 | 11 | 10 | 9.5 |

${\mathcal{A}}_{11}$ | 1 | 13.5 | 1 | 1.5 | 10 | 6 | 9.5 |

${\mathcal{A}}_{12}$ | 11 | 1 | 12.5 | 1.5 | 12 | 5 | 9.5 |

${\mathcal{A}}_{13}$ | 4 | 18 | 4 | 13 | 13 | 15 | 9.5 |

${\mathcal{A}}_{14}$ | 10 | 16.5 | 7.5 | 16.5 | 14 | 16 | 9.5 |

${\mathcal{A}}_{15}$ | 8 | 11 | 10 | 10.5 | 15 | 11 | 9.5 |

${\mathcal{A}}_{16}$ | 14 | 5.5 | 12.5 | 8 | 16 | 12 | 9.5 |

${\mathcal{A}}_{17}$ | 9 | 16.5 | 5 | 14 | 17 | 18 | 9.5 |

${\mathcal{A}}_{18}$ | 17 | 5.5 | 17 | 16.5 | 18 | 17 | 9.5 |

Number of equivalence classes | 18 | 15 | 13 | 13 | 18 | 18 | 1 |

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**MDPI and ACS Style**

Piasecki, K.; Roszkowska, E.; Łyczkowska-Hanćkowiak, A.
Simple Additive Weighting Method Equipped with Fuzzy Ranking of Evaluated Alternatives. *Symmetry* **2019**, *11*, 482.
https://doi.org/10.3390/sym11040482

**AMA Style**

Piasecki K, Roszkowska E, Łyczkowska-Hanćkowiak A.
Simple Additive Weighting Method Equipped with Fuzzy Ranking of Evaluated Alternatives. *Symmetry*. 2019; 11(4):482.
https://doi.org/10.3390/sym11040482

**Chicago/Turabian Style**

Piasecki, Krzysztof, Ewa Roszkowska, and Anna Łyczkowska-Hanćkowiak.
2019. "Simple Additive Weighting Method Equipped with Fuzzy Ranking of Evaluated Alternatives" *Symmetry* 11, no. 4: 482.
https://doi.org/10.3390/sym11040482