# Impact of the Orientation of the Ordered Fuzzy Assessment on the Simple Additive Weighted Method

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

- Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) [4].

## 2. Elements of Fuzzy Sets Theory

#### 2.1. Fuzzy Numbers

**Definition**

**1.**

#### 2.2. Ordered Fuzzy Numbers

**Definition**

**2**

**.**For any monotonic sequence$(a,b,c,d)\subset \mathbb{R}$, the trapezoidal ordered fuzzy number (TrOFN) $\overleftrightarrow{Tr}(a,b,c,d)=\overleftrightarrow{\mathcal{T}}$ is the pair of the orientation $\overrightarrow{a,d}=(a,d)$ and fuzzy subset $\mathcal{T}\in \mathcal{F}(\mathbb{R})$ determined explicitly by its membership functions ${\mu}_{T}\in {\left[0,1\right]}^{\mathbb{R}}$ as follows

#### 2.3. Disorientation Map

**Example**

**1.**

**Theorem**

**1.**

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

**Theorem**

**2**

**.**For any TrFNs $Tr(a,b,c,d),Tr(e,f,g,h)\in {\mathbb{F}}_{Tr}$ we have

**Theorem**

**3.**

#### 2.5. Relation “Greater than or Equal to” for Trapezoidal Oriented Fuzzy Numbers

- for any pair $(\overleftrightarrow{\mathcal{K}},\overleftrightarrow{\mathcal{L}})\in {({\mathbb{K}}_{Tr}^{+}{{\displaystyle \cup}}^{\text{}}\mathbb{R})}^{2}$, the extension principle$${\nu}_{GE}(\overleftrightarrow{\mathcal{K}},\overleftrightarrow{\mathcal{L}})={\nu}_{\left[GE\right]}(\Psi (\overleftrightarrow{\mathcal{K}}),\Psi (\overleftrightarrow{\mathcal{L}})),$$
- for any pair $(\overleftrightarrow{\mathcal{K}},\overleftrightarrow{\mathcal{L}})\in {({\mathbb{K}}_{Tr}^{-}{{\displaystyle \cup}}^{\text{}}\mathbb{R})}^{2}$, the sign exchange law$${\nu}_{GE}(\overleftrightarrow{\mathcal{K}},\overleftrightarrow{\mathcal{L}})={\nu}_{GE}(\boxminus \overleftrightarrow{\mathcal{L}},\boxminus \overleftrightarrow{\mathcal{K}}),$$
- for any pair $(\overleftrightarrow{\mathcal{K}},\overleftrightarrow{\mathcal{L}})\in ({\mathbb{K}}_{Tr}^{+}{{\displaystyle \cup}}^{\text{}}\mathbb{R})\times ({\mathbb{K}}_{Tr}^{-}{{\displaystyle \cup}}^{\text{}}\mathbb{R})$, the law of subtraction of parties of inequality$${\nu}_{GE}(\overleftrightarrow{\mathcal{K}},\overleftrightarrow{\mathcal{L}})={\nu}_{GE}(\overleftrightarrow{\mathcal{K}}\boxminus \overleftrightarrow{\mathcal{L}},\u301a0\u301b).$$

**Theorem**

**4**

**.**For any pair $(\overleftrightarrow{\mathcal{K}},\overleftrightarrow{\mathcal{L}})\in {({\mathbb{K}}_{Tr})}^{2}$ we have

## 3. Oriented Fuzzy SAW vs. Disoriented Fuzzy SAW

**Procedure 1:**

**Step 1:**Define the set $\mathbb{D}=\left\{{\mathcal{C}}_{1},{\mathcal{C}}_{2},\dots ,{\mathcal{C}}_{n}\right\}$ of evaluation criteria;

**Step 2:**Define the weight vector $({w}_{1},{w}_{2},\dots ,{w}_{n})\in {({\mathbb{R}}_{0}^{+})}^{n}$ fulfilling the condition

**Step 3:**For each evaluation criterion ${\mathcal{C}}_{j}(j=1,2,\dots ,n)$, determine its scope ${Y}_{j}$;

**Step 4:**Determine the evaluation template $\mathbb{Y}={Y}_{1}\times {Y}_{2}\times \dots .\times {Y}_{n}$;

**Step 5:**Define the o-NOS $\mathbb{O}\subset {\mathbb{K}}_{tr}$;

**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

**Step 7:**Determine the set $\mathbb{A}=\left\{{\mathcal{A}}_{1},{\mathcal{A}}_{2},\dots ,{\mathcal{A}}_{m}\right\}\subset \mathbb{Y}$ of evaluated decision alternatives;

**Step 8:**For the OF-SAW method, determine the scoring function $\overleftrightarrow{saw}:{\mathbb{O}}^{n}\to {\mathbb{K}}_{Tr}$ given for any as an aggregated evaluation index

**Step 9:**Using OF-SAW method, evaluate each alternative ${\mathcal{A}}_{i}\in \mathbb{A}$$(i=1,2,\dots ,m)$ by a scoring value

**Step 10:**For the d-SAW method, determine the scoring function $\stackrel{=}{saw}:{\mathbb{O}}^{n}\to {\mathbb{F}}_{Tr}$ given for any as an aggregated evaluation index

**Step 11:**Using d-SAW method, evaluate each alternative ${\mathcal{A}}_{i}\in \mathbb{A}$$(i=1,2,\dots ,m)$ by a scoring value

## 4. Disorientation Impact on Decisions Ordering

**Theorem**

**5.**

**Proof.**

**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

**Step 4**: If the condition

**Step 5**: We have a sequence ${({B}^{(k)}(\gamma ))}_{k=1}^{m(\gamma )}$ in which each recommendation ${\mathcal{A}}_{i}\in \mathbb{A}$ belongs only to one set ${B}^{(k)}(\gamma )$. For any given numbers $l>k$, the value $\gamma ({\mathcal{A}}_{i})\in {B}^{(k)}(\gamma )$ dominates the values $\gamma ({\mathcal{A}}_{j})\in {B}^{(l)}(\gamma )$ Thanks to that, any pair $({\mathcal{A}}_{i},{\mathcal{A}}_{j})\in {B}^{(k)}(\gamma )\times {B}^{(l)}(\gamma )$ is ordered as follows:

## 5. Case Study

- ${\mathcal{C}}_{1}$—unit price expressed in €,
- ${\mathcal{C}}_{2}$—complaint conditions described verbally,
- ${\mathcal{C}}_{3}$—time of payment determined in days.

## 6. Final Conclusions

- omitting information about criterion rating orientation noticeable increases ambiguity risk of choosing the right negotiation offer;
- the orientation omission noticeable decreases the amount of information about negotiating Seller’s preferences;
- the orientation omission noticeably blurs the best approximation of a real order of negotiation packages.

- If we use the OF-SAW method then omitting information about criterion rating orientation significantly increases in risk burdening decision making process.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Nomenclature

d-NOS | disoriented NOS introduced in Section 3 |

d-SAW | disoriented SAW method introduced in Section 3 |

FN | Fuzzy Number defined in [9] |

g-SAW | generalized SAW method introduced in Section 4 |

LOS | Linguistic Order Scale mentioned in Section 3 |

NOS | Numerical Order Scale mentioned in Section 3 |

OFN | Ordered Fuzzy Number defined in [10] |

o-NOS | oriented NOS introduced in Section 3 |

OF-SAW | Oriented Fuzzy SAW method described in Section 3 |

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

TrFN | Trapezoidal FN determined by its membership function (3) |

TrOFN | Trapezoidal OFN determined by its membership function (9) |

## Appendix A

**Theorem**

**A1.**

**Proof.**

- if (A4) or (A6) then$${\overleftrightarrow{\mathcal{K}}}_{1}\u229e{\overleftrightarrow{\mathcal{K}}}_{2}=\overleftrightarrow{Tr}(\mathrm{min}\left\{a+e,b+f\right\},b+f,c+g,\mathrm{max}\left\{d+h,c+g\right\}),$$
- if (A5) or (A7) then$${\overleftrightarrow{\mathcal{K}}}_{1}\u229e{\overleftrightarrow{\mathcal{K}}}_{2}=\overleftrightarrow{Tr}(\mathrm{max}\left\{a+e,b+f\right\},b+f,c+g,\mathrm{min}\left\{d+h,c+g\right\}).$$

- if (A2) then$$\stackrel{=}{\Psi}({\overleftrightarrow{\mathcal{K}}}_{1})\oplus \stackrel{=}{\Psi}({\overleftrightarrow{\mathcal{K}}}_{2})=Tr(a+e,b+f,c+g,d+h),$$
- if (A3) then$$\stackrel{=}{\Psi}({\overleftrightarrow{\mathcal{K}}}_{1})\oplus \stackrel{=}{\Psi}({\overleftrightarrow{\mathcal{K}}}_{2})=Tr(d+h,c+g,b+f,a+e),$$
- if (A4) or (A5) then$$\stackrel{=}{\Psi}({\overleftrightarrow{\mathcal{K}}}_{1})\oplus \stackrel{=}{\Psi}({\overleftrightarrow{\mathcal{K}}}_{2})=Tr(a+h,b+g,c+f,d+e),$$
- if (A6) or (A7) then$$\stackrel{=}{\Psi}({\overleftrightarrow{\mathcal{K}}}_{1})\oplus \stackrel{=}{\Psi}({\overleftrightarrow{\mathcal{K}}}_{2})=Tr(d+e,c+f,b+g,a+h).$$

## Appendix B

**Theorem**

**A2.**

**Proof:**

## Appendix C

**Table A1.**Membership function of initial preorder $BE(\stackrel{=}{\Psi}\circ \overleftrightarrow{SAW})$ (Source: Own elaboration).

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

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

**Table A2.**Membership function of initial preorder $BE(\stackrel{=}{SAW})$ (Source: Own elaboration).

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

${\mathcal{A}}_{1}$ | 1 | 0 | 0 | 0.50 | 0 | 0 | 0 | 0 | 1 | 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 | 0.50 | 0.33 | 0 | 0.40 | 1 | 1 | 0.50 | 0.50 | 0 | 0 | 0 |

${\mathcal{A}}_{4}$ | 1 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 0.67 | 0 | 0 | 0 | 0 | 0 |

${\mathcal{A}}_{5}$ | 1 | 0 | 1 | 1 | 1 | 0.67 | 0 | 1 | 1 | 1 | 0.67 | 1 | 0 | 0 | 0 |

${\mathcal{A}}_{6}$ | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0.50 | 1 | 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 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 1 | 0 |

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

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

${\mathcal{A}}_{11}$ | 1 | 1 | 1 | 1 | 1 | 1 | 0.50 | 1 | 1 | 1 | 1 | 1 | 0 | 0.80 | 0 |

${\mathcal{A}}_{12}$ | 1 | 1 | 1 | 1 | 1 | 1 | 0.80 | 0.89 | 1 | 1 | 1 | 1 | 0 | 1 | 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 | 1 | 1 | 0 |

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

## References

- Herrera, F.; Alonso, S.; Chiclana, F.; Herrera-Viedma, E. Computing with words in decision making: Foundations, trends and prospects. Fuzzy Optim. Decis. Mak.
**2009**, 8, 337–364. [Google Scholar] [CrossRef] - Schoop, M.; Jertila, A.; List, T. Egoisst: A negotiation support system for electronic business-to-business negotiations in e-commerce. Data Knowl. Eng.
**2003**, 47, 371–401. [Google Scholar] [CrossRef] - Kersten, G.E.; Noronha, S.J. WWW-based negotiation support: Design, implementation, and use. Decis. Support Syst.
**1999**, 25, 135–154. [Google Scholar] [CrossRef] - Wachowicz, T.; Błaszczyk, P. TOPSIS based approach to scoring negotiating offers in negotiation support systems. Group. Decis. Negotiat.
**2013**, 22, 1021–1050. [Google Scholar] [CrossRef] - Zadeh, L. The concept of a linguistic variable and its application to approximate reasoning—I. Inf. Sci.
**1975**, 8, 199–249. [Google Scholar] [CrossRef] - Zadeh, L. The concept of a linguistic variable and its application to approximate reasoning—II. Inf. Sci.
**1975**, 8, 301–357. [Google Scholar] [CrossRef] - Zadeh, L. The concept of a linguistic variable and its application to approximate reasoning—III. Inf. Sci.
**1975**, 9, 43–80. [Google Scholar] [CrossRef] - Chou, S.; Chang, Y. A decision support system for supplier selection based on a strategy-aligned fuzzy SMART approach. Expert Syst. Appl.
**2008**, 34, 2241–2253. [Google Scholar] [CrossRef] - Dubois, D.; Prade, H. Fuzzy real algebra: Some results. Fuzzy Sets Syst.
**1979**, 2, 327–348. [Google Scholar] [CrossRef] - Kosiński, W.; Prokopowicz, P.; Ślęzak, D. Drawback of Fuzzy Arithmetics—New Intuitions and Propositions. In Methods of Artificial Intelligence; Burczyński, T., Cholewa, W., Moczulski, W., Eds.; PACM: Gliwice, Poland, 2002; pp. 231–237. [Google Scholar]
- Kosiński, W. On fuzzy number calculus. Int. J. Appl. Math. Comput. Sci.
**2006**, 16, 51–57. [Google Scholar] - Piasecki, K. Revision of the Kosiński’s Theory of Ordered Fuzzy Numbers. Axioms
**2018**, 7, 16. [Google Scholar] [CrossRef] - Roszkowska, E.; Kacprzak, D. The fuzzy saw and fuzzy TOPSIS procedures based on ordered fuzzy numbers. Inf. Sci.
**2016**, 369, 564–584. [Google Scholar] [CrossRef] - Piasecki, K.; Roszkowska, E. On Application of Ordered Fuzzy Numbers in Ranking Linguistically Evaluated Negotiation Offers. Adv. Fuzzy Syst.
**2018**, 2018, 1–12. [Google Scholar] [CrossRef] - Piasecki, K.; Roszkowska, E.; Łyczkowska-Hanćkowiak, A. Simple Additive Weighting Method Equipped with Fuzzy Ranking of Evaluated Alternatives. Symmetry
**2019**, 11, 482. [Google Scholar] [CrossRef] - Zadeh, L. Fuzzy sets. Inf. Control
**1965**, 8, 338–353. [Google Scholar] [CrossRef] [Green Version] - Klir, G.J. Developments in uncertainty-based information. Adv. Comp.
**1993**, 36, 255–332. [Google Scholar] [CrossRef] - Goetschel, R.; Voxman, W. Elementary fuzzy calculus. Fuzzy Set. Syst.
**1986**, 18, 31–43. [Google Scholar] [CrossRef] - Piasecki, K. Relation “Greater than or Equal to” between Ordered Fuzzy Numbers. Appl. Syst. Innov.
**2019**, 2, 26. [Google Scholar] [CrossRef] - Orlovsky, S. Decision-making with a fuzzy preference relation. Fuzzy Sets Syst.
**1978**, 1, 155–167. [Google Scholar] [CrossRef] - Herrera, F.; Herrera-Viedma, E. Linguistic decision analysis: Steps for solving decision problems under linguistic information. Fuzzy Sets Syst.
**2000**, 115, 67–82. [Google Scholar] [CrossRef] - Martínez, L.; Ruan, D.; Herrera, F. Computing with Words in Decision support Systems: An overview on Models and Applications. Int. J. Comput. Intell. Syst.
**2010**, 3, 382–395. [Google Scholar] [CrossRef] - Brzostowski, J.; Wachowicz, T.; Roszkowska, E. Reference points-based methods in supporting the evaluation of negotiation offers. Oper. Res. Decis.
**2012**, 22, 21–40. [Google Scholar] [CrossRef] - Oxford Dictionaries, British and World English Dictionary. Available online: http://www.oxforddictionaries.com/definition/english/ (accessed on 26 July 2019).
- Oxford Dictionaries, US English Dictionary. Available online: http://www.oxforddictionaries.com/definition/american_english/ (accessed on 26 July 2019).
- Kosiński, W.; Wilczyńska-Sztyma, D. Defuzzification and Implication within Ordered Fuzzy Numbers. In Proceedings of the International Conference on Fuzzy Systems, Barcelona, Spain, 18–23 July 2010; pp. 1073–1079. [Google Scholar]
- Kendall, M.G. Rank Correlation Methods; Charles Griffin & Company Limited: London, UK, 1955. [Google Scholar]

**Figure 1.**The membership functions of ${\stackrel{=}{\Psi}(\overleftrightarrow{Tr}(1,2,\text{}3,4)\u229e\overleftrightarrow{Tr}(12,10,\text{}8,6))}$ and ${\stackrel{=}{\Psi}(\overleftrightarrow{Tr}(1,2,\text{}3,4))\oplus \stackrel{=}{\Psi}(\overleftrightarrow{Tr}(12,10,\text{}8,6))}$.

**Table 1.**Applied order scales (Source: [13] and own elaboration).

Linguistic Variable | LOS | o-NOS | d-NOS |
---|---|---|---|

Very Bad | $VB$ | $\overleftrightarrow{Tr}(1,1,1,1)$ | $Tr(1,1,1,1)$ |

at least Very Bad | $\mathcal{L}.VB$ | $\overleftrightarrow{Tr}(1,1,1.5,2)$ | $Tr(1,1,1.5,2)$ |

at most Bad | $\mathcal{M}.B$ | $\overleftrightarrow{Tr}(2,2,1.5,1)$ | $Tr(1,1.5,2,2)$ |

Bad | $B$ | $\overleftrightarrow{Tr}(2,2,2,2)$ | $Tr(2,2,2,2)$ |

at least Bad | $\mathcal{L}.B$ | $\overleftrightarrow{Tr}(2,2,2.5,3)$ | $Tr(2,2,2.5,3)$ |

at most Average | $\mathcal{M}.AV$ | $\overleftrightarrow{Tr}(3,3,2.5,2)$ | $Tr(2,2.5,3,3)$ |

Average | $AV$ | $\overleftrightarrow{Tr}(3,3,3,3)$ | $Tr(3,3,3,3)$ |

at least Average | $\mathcal{L}.AV$ | $\overleftrightarrow{Tr}(3,3,3.5,4)$ | $Tr(3,3,3.5,4)$ |

at most Good | $\mathcal{M}.G$ | $\overleftrightarrow{Tr}(4,4,3.5,3)$ | $Tr(3,3.5,4,4)$ |

Good | $G$ | $\overleftrightarrow{Tr}(4,4,4,4)$ | $Tr(4,4,4,4)$ |

at least Good | $\mathcal{L}.G$ | $\overleftrightarrow{Tr}(4,4,4.5,5)$ | $Tr(4,4,4.5,5)$ |

at most Very Good | $\mathcal{M}.VG$ | $\overleftrightarrow{Tr}(5,5,4.5,4)$ | $Tr(4,4.5,5,5)$ |

Very Good | $VG$ | $\overleftrightarrow{Tr}(5,5,5,5)$ | $Tr(5,5,5,5)$ |

**Table 2.**Linguistic rating of negotiations packages (Source; [12]).

Package | Negotiation Issues | ||
---|---|---|---|

${\mathcal{C}}_{1}$ | ${\mathcal{C}}_{2}$ | ${\mathcal{C}}_{3}$ | |

${\mathcal{A}}_{1}$ | $VB$ | $\mathcal{L}.B$ | $\mathcal{M}.G$ |

${\mathcal{A}}_{2}$ | $B$ | $\mathcal{M}.VG$ | $G$ |

${\mathcal{A}}_{3}$ | $\mathcal{L}.VB$ | $G$ | $\mathcal{L}.AV$ |

${\mathcal{A}}_{4}$ | $\mathcal{M}.B$ | $AV$ | $\mathcal{L}.AV$ |

${\mathcal{A}}_{5}$ | $B$ | $\mathcal{L}.AV$ | $G$ |

${\mathcal{A}}_{6}$ | $\mathcal{L}.B$ | $\mathcal{M}.G$ | $\mathcal{M}.VG$ |

${\mathcal{A}}_{7}$ | $AV$ | $\mathcal{M}.G$ | $\mathcal{L}.AV$ |

${\mathcal{A}}_{8}$ | $\mathcal{L}.B$ | $AV$ | $\mathcal{M}.VG$ |

${\mathcal{A}}_{9}$ | $\mathcal{M}.AV$ | $\mathcal{L}.VB$ | $VB$ |

${\mathcal{A}}_{10}$ | $AV$ | $\mathcal{M}.B$ | $\mathcal{L}.VB$ |

${\mathcal{A}}_{11}$ | $\mathcal{L}.AV$ | $B$ | |

${\mathcal{A}}_{12}$ | $\mathcal{L}.AV$ | $\mathcal{L}.B$ | $B$ |

${\mathcal{A}}_{13}$ | $G$ | $AV$ | $\mathcal{M}.AV$ |

${\mathcal{A}}_{14}$ | $\mathcal{L}.AV$ | $\mathcal{M}.G$ | $\mathcal{M}.AV$ |

${\mathcal{A}}_{15}$ | $VG$ | $B$ | $\mathcal{M}.AV$ |

**Table 3.**Numeric rating of negotiations packages (Source: [13] and own elaboration).

Packages | Negotiation Issues | |||||
---|---|---|---|---|---|---|

${\mathcal{C}}_{1}$ | ${\mathcal{C}}_{2}$ | ${\mathcal{C}}_{3}$ | ||||

o-NOS | d-NOS | o-NOS | d-NOS | o-NOS | d-NOS | |

${\mathcal{A}}_{1}$ | $\overleftrightarrow{Tr}(1,1,1,1)$ | $Tr(1,1,1,1)$ | $\overleftrightarrow{Tr}(2,2,2.5,3)$ | $Tr(2,2,2.5,3)$ | $\overleftrightarrow{Tr}(4,4,3.5,3)$ | $Tr(3,3.5,4,4)$ |

${\mathcal{A}}_{2}$ | $\overleftrightarrow{Tr}(2,2,2,2)$ | $Tr(2,2,2,2)$ | $\overleftrightarrow{Tr}(5,5,4.5,4)$ | $Tr(4,4.5,5,5)$ | $\overleftrightarrow{Tr}(4,4,4,4)$ | $Tr(4,4,4,4)$ |

${\mathcal{A}}_{3}$ | $\overleftrightarrow{Tr}(1,1,1.5,2)$ | $Tr(1,1,1.5,2)$ | $\overleftrightarrow{Tr}(4,4,4,4)$ | $Tr(4,4,4,4)$ | $\overleftrightarrow{Tr}(3,3,3.5,4)$ | $Tr(3,3,3.5,4)$ |

${\mathcal{A}}_{4}$ | $\overleftrightarrow{Tr}(2,2,1.5,1)$ | $Tr(1,1.5,2,2)$ | $\overleftrightarrow{Tr}(3,3,3,3)$ | $Tr(3,3,3,3)$ | $\overleftrightarrow{Tr}(3,3,3.5,4)$ | $Tr(3,3,3.5,4)$ |

${\mathcal{A}}_{5}$ | $\overleftrightarrow{Tr}(2,2,2,2)$ | $Tr(2,2,2,2)$ | $\overleftrightarrow{Tr}(3,3,3.5,4)$ | $Tr(3,3,3.5,4)$ | $\overleftrightarrow{Tr}(4,4,4,4)$ | $Tr(4,4,4,4)$ |

${\mathcal{A}}_{6}$ | $\overleftrightarrow{Tr}(2,2,2.5,3)$ | $Tr(2,2,2.5,3)$ | $\overleftrightarrow{Tr}(4,4,3.5,3)$ | $Tr(3,3.5,4,4)$ | $\overleftrightarrow{Tr}(5,5,4.5,4)$ | $Tr(4,4.5,5,5)$ |

${\mathcal{A}}_{7}$ | $\overleftrightarrow{Tr}(3,3,3,3)$ | $Tr(3,3,3,3)$ | $\overleftrightarrow{Tr}(4,4,3.5,3)$ | $Tr(3,3.5,4,4)$ | $\overleftrightarrow{Tr}(3,3,3.5,4)$ | $Tr(3,3,3.5,4)$ |

${\mathcal{A}}_{8}$ | $\overleftrightarrow{Tr}(2,2,2.5,3)$ | $Tr(2,2,2.5,3)$ | $\overleftrightarrow{Tr}(3,3,3,3)$ | $Tr(3,3,3,3)$ | $\overleftrightarrow{Tr}(5,5,4.5,4)$ | $Tr(4,4.5,5,5)$ |

${\mathcal{A}}_{9}$ | $\overleftrightarrow{Tr}(3,3,2.5,2)$ | $Tr(2,2.5,3,3)$ | $\overleftrightarrow{Tr}(1,1,1.5,2)$ | $Tr(1,1,1.5,2)$ | $\overleftrightarrow{Tr}(1,1,1,1)$ | $Tr(1,1,1,1)$ |

${\mathcal{A}}_{10}$ | $\overleftrightarrow{Tr}(3,3,3,3)$ | $Tr(3,3,3,3)$ | $\overleftrightarrow{Tr}(2,2,1.5,1)$ | $Tr(1,1.5,2,2)$ | $\overleftrightarrow{Tr}(1,1,1.5,2)$ | $Tr(1,1,1.5,2)$ |

${\mathcal{A}}_{11}$ | $\overleftrightarrow{Tr}(3,3,3.5,4)$ | $Tr(3,3,3.5,4)$ | $\overleftrightarrow{Tr}(2,2,2,2)$ | $Tr(2,2,2,2)$ | $\overleftrightarrow{Tr}(2,2,2,2)$ | $Tr(2,2,2,2)$ |

${\mathcal{A}}_{12}$ | $\overleftrightarrow{Tr}(3,3,3.5,4)$ | $Tr(3,3,3.5,4)$ | $\overleftrightarrow{Tr}(2,2,2.5,3)$ | $Tr(2,2,2.5,3)$ | $\overleftrightarrow{Tr}(2,2,2,2)$ | $Tr(2,2,2,2)$ |

${\mathcal{A}}_{13}$ | $\overleftrightarrow{Tr}(4,4,4,4)$ | $Tr(4,4,4,4)$ | $\overleftrightarrow{Tr}(3,3,3,3)$ | $Tr(3,3,3,3)$ | $\overleftrightarrow{Tr}(3,3,2.5,2)$ | $Tr(2,2.5,3,3)$ |

${\mathcal{A}}_{14}$ | $\overleftrightarrow{Tr}(3,3,3.5,4)$ | $Tr(3,3,3.5,4)$ | $\overleftrightarrow{Tr}(4,4,3.5,3)$ | $Tr(3,3.5,4,4)$ | $\overleftrightarrow{Tr}(3,3,2.5,2)$ | $Tr(2,2.5,3,3)$ |

${\mathcal{A}}_{15}$ | $\overleftrightarrow{Tr}(5,5,5,5)$ | $Tr(5,5,5,5)$ | $\overleftrightarrow{Tr}(2,2,2,2)$ | $Tr(2,2,2,2)$ | $\overleftrightarrow{Tr}(3,3,2.5,2)$ | $Tr(2,2.5,3,3)$ |

Scoring Evaluations | Induced Partial Orders | ||||
---|---|---|---|---|---|

Packages | $\overleftrightarrow{\mathit{S}\mathit{A}\mathit{W}}({\mathcal{A}}_{\mathit{i}})$ | $\stackrel{\mathit{=}}{\mathit{\Psi}}(\overleftrightarrow{\mathit{S}\mathit{A}\mathit{W}}({\mathcal{A}}_{\mathit{i}}))$ | $\stackrel{\mathit{=}}{\mathit{S}\mathit{A}\mathit{W}}({\mathcal{A}}_{\mathit{i}})$ | ${\overline{\mathit{B}\mathit{E}}}_{\mathit{I}\mathit{N}\mathit{D}}$ ${(\stackrel{\mathit{=}}{\mathit{\Psi}}\circ \overleftrightarrow{\mathit{S}\mathit{A}\mathit{W}})}_{\text{}}$ | ${\overline{\mathit{B}\mathit{E}}}_{\mathit{I}\mathit{N}\mathit{D}}{(\stackrel{\mathit{=}}{\mathit{S}\mathit{A}\mathit{W}})}_{\text{}}$ |

${\mathcal{A}}_{15}$ | $\overleftrightarrow{Tr}(4.0,4.0,3.9,3.8)$ | $Tr(3.8,3.9,4.0,4.0)$ | $Tr(3.8,3.9,4.0,4.0)$ | 1 | 1 |

${\mathcal{A}}_{13}$ | $\overleftrightarrow{Tr}(3.6,3.6,3.5,3.4)$ | $Tr(3.4,3.5,3.6,3.6)$ | $Tr(3.4,3.5,3.6,3.6)$ | 2 | 3 |

${\mathcal{A}}_{7}$ | $\overleftrightarrow{Tr}(3.2,3.2,3.2,3.2)$ | $Tr(3.2,3.2,3.2,3.2)$ | $Tr(3.0,3.1,3.3,3.4)$ | 3.5 | 6.5 |

${\mathcal{A}}_{14}$ | $\overleftrightarrow{Tr}(3.2,3.2,3.3,3.4)$ | $Tr(3.2,3.2,3.3,3.4)$ | $Tr(2.8,3.0,3.5,3.8)$ | 3.5 | 3 |

${\mathcal{A}}_{6}$ | $\overleftrightarrow{Tr}(3.0,3.0,3.1,3.2)$ | $Tr(3.0,3.0,3.1,3.2)$ | $Tr(2.6,2.8,3.3,3.6)$ | 5 | 3 |

${\mathcal{A}}_{2}$ | $\overleftrightarrow{Tr}(3.0,3.0,2.9,2.8)$ | $Tr(2.8,2.9,3.0,3.0)$ | $Tr(2.8,2.9,3.0,3.0)$ | 7 | 9 |

${\mathcal{A}}_{8}$ | $\overleftrightarrow{Tr}(2.8,2.8,3.0,3.2)$ | $Tr(2.8,2.8,3.0,3.2)$ | $Tr(2.6,2.7,3.1,3.4)$ | 7 | 6.5 |

${\mathcal{A}}_{12}$ | $\overleftrightarrow{Tr}(2.6,2.6,3.0,3.4)$ | $Tr(2.6,2.6,3.0,3.4)$ | $Tr(2.6,2.6,3.0,3.4)$ | 7 | 6.5 |

${\mathcal{A}}_{11}$ | $\overleftrightarrow{Tr}(2.6,2.6,2.9,3.2)$ | $Tr(2.6,2.6,2.9,3.2)$ | $Tr(2.6,2.6,2.9,3.2)$ | 9 | 6.5 |

${\mathcal{A}}_{5}$ | $\overleftrightarrow{Tr}(2.6,2.6,2.7,2.8)$ | $Tr(2.6,2.6,2.7,2.8)$ | $Tr(2.6,2.6,2.7,2.8)$ | 10 | 10.5 |

${\mathcal{A}}_{3}$ | $\overleftrightarrow{Tr}(2.0,2.0,2.4,2.8)$ | $Tr(2.0,2.0,2.4,2.8)$ | 11.5 | 10.5 | |

${\mathcal{A}}_{4}$ | $\overleftrightarrow{Tr}(2.4,2.4,2.2,2.0)$ | $Tr(2.0,2.2,2.4,2.4)$ | $Tr(1.8,2.1,2.5,2.6)$ | 11.5 | 13 |

${\mathcal{A}}_{10}$ | $\overleftrightarrow{Tr}(2.4,2.4,2.4,2.4)$ | $Tr(2.2,2.3,2.5,2.6)$ | 13 | 13 | |

${\mathcal{A}}_{9}$ | $\overleftrightarrow{Tr}(2.2,2.2,2.0,1.8)$ | $Tr(1.8,2.0,2.2,2.2)$ | $Tr(1.6,1.9,2.3,2.4)$ | 14 | 13 |

${\mathcal{A}}_{1}$ | $\overleftrightarrow{Tr}(1.8,1.8,1.8,1.8)$ | $Tr(1.8,1.8,1.8,1.8)$ | $Tr(1.6,1.7,1.9,2.0)$ | 15 | 15 |

Amount of equivalence classes | 11 | 7 |

© 2019 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**

Piasecki, K.; Roszkowska, E.; Łyczkowska-Hanćkowiak, A.
Impact of the Orientation of the Ordered Fuzzy Assessment on the Simple Additive Weighted Method. *Symmetry* **2019**, *11*, 1104.
https://doi.org/10.3390/sym11091104

**AMA Style**

Piasecki K, Roszkowska E, Łyczkowska-Hanćkowiak A.
Impact of the Orientation of the Ordered Fuzzy Assessment on the Simple Additive Weighted Method. *Symmetry*. 2019; 11(9):1104.
https://doi.org/10.3390/sym11091104

**Chicago/Turabian Style**

Piasecki, Krzysztof, Ewa Roszkowska, and Anna Łyczkowska-Hanćkowiak.
2019. "Impact of the Orientation of the Ordered Fuzzy Assessment on the Simple Additive Weighted Method" *Symmetry* 11, no. 9: 1104.
https://doi.org/10.3390/sym11091104