# A MAGDM Algorithm with Multi-Granular Probabilistic Linguistic Information

^{1}

^{2}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

## 3. Main Results

#### 3.1. Definitions of Multi-Granular Probabilistic Linguistic Term Sets

**Definition**

**4.**

**Definition**

**5.**

- (1)
- If ${{\displaystyle \sum}}_{{k}_{i}=1}^{\#{L}_{i}(P)}{P}_{i}{}^{({k}_{i})}<1$, then by Equation (2), we calculate $\dot{{L}_{i}}(P),i=1,2.$
- (2)
- If $\#{L}_{1}(P)\ne \#{L}_{2}(P)$, then by Definition 4, we add some elements to the one with the smaller number of elements.

**Definition**

**6.**

#### 3.2. Distance Measures between Multi-Granular PLTSs

**Definition**

**7.**

- (1)
- 0 $\le $ $d({L}_{1}(P),{L}_{2}\text{}(P))\le $ 1;
- (2)
- $d({L}_{1}(P),{L}_{2}\text{}(P))=$ 0, if and only if ${L}_{1}(P)={L}_{2}\text{}(P)$; and
- (3)
- $d({L}_{1}(P),{L}_{2}\text{}(P))=\text{}d({L}_{2}\text{}(P),{L}_{1}(P))$.

**Definition**

**8.**

**Definition**

**9.**

#### 3.3. A MAGDM Algorithm Based on PT

**Definition**

**10.**

**Definition**

**11.**

**Step 1.**Information gathering process: individual reference points (RPs) over the alternatives on different attributes provided by experts are gathered as $\mathrm{R}={\left[{\mathrm{L}}_{\mathrm{ij}}(\mathrm{P})\right]}_{\mathrm{m}\times \mathrm{n}}$ by Equation (9).

**Step 2.**Compute the weight vector $\mathrm{w}={({\mathrm{w}}_{1},{\mathrm{w}}_{2},\dots ,{\mathrm{w}}_{\mathrm{n}})}^{\mathrm{T}}$ of $\mathrm{n}$ attributes $\mathrm{X}={({\mathrm{x}}_{1},{\mathrm{x}}_{2},\dots ,{\mathrm{x}}_{\mathrm{n}})}^{\mathrm{T}}$ as follows. See Equation (12).

**Step 3.**Aggregation process: get weighted DM matrix ${R}^{*}$. See Equation (13).

**Step 4.**Calculate the positive ideal solution and the negative ideal solution respectively.

**Step 5.**Calculate the gains value and the losses value: gains and losses are calculated with respect to the group reference points of the different alternatives.

**Step 6.**Calculate the ration between the gains value and the losses value of each alternative.

## 4. Case Studies

#### 4.1. The Applications of the Algorithm

**Step 1.**Collect the users’ assessment information on the “Auto Home” website until May 20 in 2018. See Table 1, Table 2 and Table 3.

**Step 2.**Calculate the weight vector $w={({w}_{1},{w}_{2},{w}_{3},{w}_{4},{w}_{5},{w}_{6},{w}_{7},{w}_{8})}^{T}$ on the eight attributes $X={({x}_{1},{x}_{2},{x}_{3},{x}_{4},{x}_{5},{x}_{6},{x}_{7},{x}_{8})}^{T}$ by Equation (12). See Table 6.

**Step 3.**Get a weighted DM matrix. See Table 7.

**Step 4**. Calculate the PLPIS and PLNIS respectively.

**Step 5.**The results of $V({A}_{i},{L}^{-})$ and $V({A}_{i},{L}^{+})(i=1,2,\dots ,7)$ are as follows. See Table 10 and Table 11.

#### 4.2. Sensitivity Analysis

#### 4.3. Comparative Analysis

#### 4.4. The Second Case Study

## Funding

## Conflicts of Interest

## References

- Zhou, L.G.; Chen, H.Y. Continous ordered linguistic distance measure and its application to multiple attribute group decision making. Group Decis. Negot.
**2013**, 22, 739–758. [Google Scholar] [CrossRef] - Parreiras, R.O.; Ekel, P.Y.; Martni, J.S.C.; Palhares, R.M. A flexible consensus scheme for multicriteria group decision making under linguistiic assessments. Inf. Sci.
**2010**, 180, 1075–1089. [Google Scholar] [CrossRef] - Grossglauser, M.; Saner, H. Data–driven healthcare: From patterns to actions. Eur. J. Prev. Cardiol.
**2014**, 21, 14–17. [Google Scholar] [CrossRef] - Celotto, A.; Loia, V.; Senatore, S. Fuzzy linguistic approach to quality assessment model for electricity network infrastructure. Inf. Sci.
**2015**, 304, 1–15. [Google Scholar] [CrossRef] - Tao, Z.F.; Chen, H.Y.; Zhou, L.G.; Liu, J.P. 2-Tuple linguistic soft set and its application to group decision making. Soft Comput.
**2015**, 19, 1201–1213. [Google Scholar] [CrossRef] - Sengupta, A.T.; Pal, K.; Zhou, L.G.; Chen, H.Y. Fuzzy Preference Ordering of Interval Numbers in Decision Problems; Springer: Heidelberg, Germany, 2009; Volume 238, pp. 140–143. [Google Scholar]
- Zadeh, L.A. Fuzzy sets. Inf. Control
**1965**, 3, 338–353. [Google Scholar] [CrossRef] - Xu, Z.S. Linguistic Decision Making: Theory and Methods; Springer: Berlin/Heidelberg, Germany, 2012. [Google Scholar]
- Herrera, F.; Verdegay, J.L. Linguistic assessments in group decision. In Proceedings of the First European Congress on Fuzzy and Intelligent Technologies, Aachen, Germany, 7–10 September 1993; pp. 941–948. [Google Scholar]
- Herrera, F.; Herrera-Viedma, E.; Verdegay, J.L. A model of consensus in group decision making under linguistic assessments. Fuzzy Sets Syst.
**1996**, 78, 73–87. [Google Scholar] [CrossRef] - Herrera, F.; Herrera-Viedma, E.; Verdegay, J.L. A rational consensus model in group decision making using linguistic assessments. Fuzzy Sets Syst.
**1997**, 88, 31–49. [Google Scholar] [CrossRef] - Xu, Z.S. A method for multiple attribute decision making with incomplete weight information in linguistic setting. Knowl.-Based Syst.
**2007**, 20, 719–725. [Google Scholar] [CrossRef] - Arieh, D.B.; Chen, Z. Linguistic group decision-making: Opinion aggregation and measures of consensus. Fuzzy Opt. Decis. Mak.
**2006**, 5, 371–386. [Google Scholar] [CrossRef] - Xu, Z.S. An approach based on the uncertain LOWG and induced uncertain LOWG operators to group decision making with uncertain multiplicative linguistic preference relations. Decis. Support Syst.
**2006**, 41, 488–499. [Google Scholar] [CrossRef] - Liu, P.D.; Mahmood, T.; Khan, Q. Multi-Attribute Decision-Making Based on Prioritized Aggregation Operator under Hesitant. Symmetry
**2017**, 9, 11. [Google Scholar] [CrossRef] - Liao, H.C.; Zhang, C.; Luo, L. A multiple attribute group decision making method based on two novel intuitionistic multiplicative distance measures. Inf. Sci.
**2018**, 467, 766–783. [Google Scholar] [CrossRef] - Rodríguez, R.M.; Martínez, L.; Herrera, F. Hesitant fuzzy linguistic term sets for decision making. IEEE Trans. Fuzzy Syst.
**2012**, 20, 109–119. [Google Scholar] [CrossRef] - Torra, V. Hesitant fuzzy sets. Int. J. Intell. Syst.
**2010**, 25, 529–539. [Google Scholar] [CrossRef] - Zadeh, L.A. The concept of a linguistic variable and its application to approximate reasoning—I. Inf. Sci.
**1975**, 8, 199–249. [Google Scholar] [CrossRef] - Dong, Y.C.; Wu, Y.Z.; Zhang, H.J.; Zhang, G.Q. Multi-granular unbalanced linguistic distribution assessments with interval symbolic proportions. Knowl.-Based Syst.
**2015**, 82, 139–151. [Google Scholar] [CrossRef] - Wu, Z.B.; Xu, J.P. Possibility distribution-based approach for MAGDM with hesitant fuzzy linguistic information. IEEE Trans. Cybern.
**2016**, 46, 694–705. [Google Scholar] [CrossRef] - Zhang, G.Q.; Dong, Y.C.; Xu, Y.F. Consistency and consensus measures for linguistic preference relations based on distribution assessments. Inf. Fusion
**2014**, 17, 46–55. [Google Scholar] [CrossRef] - Liu, H.B.; Rodriguez, R.M. A fuzzy envelope for hesitant fuzzy linguistic term set and its application to multi-criteria decision making. Inf. Sci.
**2014**, 258, 220–238. [Google Scholar] [CrossRef] - Yang, J.B. Rule and utility based evidential reasoning approach for multi-attribute decision analysis under uncertainties. Eur. J. Oper. Res.
**2001**, 131, 31–61. [Google Scholar] [CrossRef] - Yang, J.B.; Xu, D.L. On the evidential reasoning algorithm for multiple attribute decision analysis under uncertainty. IEEE Trans. Syst. Man Cybern. Part A Syst. Hum.
**2002**, 32, 289–304. [Google Scholar] [CrossRef] - Wang, J.H.; Hao, J.Y. A new version of 2-tuple fuzzy linguistic representation model for computing with words. IEEE Trans. Fuzzy Syst.
**2006**, 14, 435–445. [Google Scholar] [CrossRef] - Pang, Q.; Wang, H.; Xu, Z.S. Probabilistic linguistic term sets in multi-attribute group decision making. Inf. Sci.
**2016**, 369, 128–143. [Google Scholar] [CrossRef] - Lin, M.W.; Xu, Z.S.; Zhai, Y.L.; Yao, Z.Q. Multi-attribute group decision-making under probabilistic uncertain linguistic environment. J. Oper. Res. Soc.
**2017**, 1–15. [Google Scholar] [CrossRef] - Xu, Z.S.; Zhou, W. Consensus building with a group of decision makers under the hesitant probabilistic fuzzy environment. Fuzzy Opt. Decis. Mak.
**2017**, 16, 481–503. [Google Scholar] [CrossRef] - Lin, M.W.; Xu, Z.S. Probabilistic Linguistic Distance Measures and Their Applications in Multi-criteria Group Decision Making. In Soft Computing Applications for Group Decision-Making and Consensus Modeling; Springer: Cham, Switzerland, 2018; Volume 357, pp. 411–440. [Google Scholar]
- Kobina, A.; Liang, D.C. Probabilistic linguistic power aggregation operators for multi-criteria group decision making. Symmetry
**2017**, 9, 12. [Google Scholar] [CrossRef] - Xu, Z.S.; Wang, H. Managing multi-granularity linguistic information in qualitative group decision making: An overview. Granul. Comput.
**2016**, 1, 21–35. [Google Scholar] [CrossRef] - Wang, H.; Xu, Z.S.; Zeng, X.J. Hesitant fuzzy linguistic term sets for linguistic decision making: Current developments, issues and challenges. Inf. Fusion
**2018**, 43, 1–12. [Google Scholar] [CrossRef] - Gorzalczany, M.B. An interval-valued fuzzy inference method-some basic properties. Fuzzy Sets Syst.
**1989**, 31, 243–251. [Google Scholar] [CrossRef] - Atanassov, K.T. Intuitionistic fuzzy sets. Fuzzy Sets Syst.
**1986**, 20, 87–96. [Google Scholar] [CrossRef] - Atanassov, K.T.; Gargov, G. Interval–valued intuitionistic fuzzy sets. Fuzzy Sets Syst.
**1989**, 31, 343–349. [Google Scholar] [CrossRef] - Chen, N.; Xu, Z.S.; Xia, M.M. Interval-valued hesitant preference relations and their applications to group decision making. Knowl-Based Syst.
**2013**, 37, 528–540. [Google Scholar] [CrossRef] - W, L.; W, Y.M.; Martinez, L. A group decision method based on prospect theory for emergency situations. Inf. Sci.
**2017**, 418, 119–135. [Google Scholar] - Yao, S.; Yu, D.; Song, Y.; Yao, H.; Hu, Y.; Guo, B. Dry bulk carrier investment selection through a dual group decision fusing mechanism in the green supply chain. Sustainability
**2018**, 10, 4528. [Google Scholar] [CrossRef] - Herrera, F.; Herrera-Viedma, E.; Verdegay, J.L. A sequential selection process in group decision making with a linguistic assessment approach. Inf. Sci.
**1995**, 85, 223–239. [Google Scholar] [CrossRef] - Rodríguez, R.M.; Martínez, L.; Herrera, F. A group decision making model dealing with comparative linguistic expressions based on hesitant fuzzy linguistic term sets. Inf. Sci.
**2013**, 241, 28–42. [Google Scholar] [CrossRef] - Yager, R.R. Generalized OWA aggregation operators. Fuzzy Opt. Decis. Mak.
**2004**, 3, 93–107. [Google Scholar] [CrossRef] - Ma, J.; Fan, Z.P.; Huang, L.H. A subjective and objective integrated approach to determine attribute weights. Eur. J. Oper. Res.
**1999**, 112, 397–404. [Google Scholar] [CrossRef] - Kahneman, D.; Tversky, A. Prospect theory: An analysis of decision under risk. Econometrica
**1979**, 47, 263–292. [Google Scholar] [CrossRef] - Xu, Z.S.; Xia, M.M. Hesitant Fuzzy entropy and cross-entropy and their use in multi-attribute decision-making. Int. J. Intell. Syst.
**2012**, 27, 799–822. [Google Scholar] [CrossRef] - Wang, L.Z.; Zhang, X.; Wang, Y.M. A prospect theory-based interval dynamic reference point method for emergency decision making. Expert Syst. Appl.
**2015**, 42, 9379–9388. [Google Scholar] [CrossRef] - Zhang, X.L.; Xu, Z.S. The TODIM analysis approach based on novel measured functions under hesitant fuzzy environment. Knowl.-Based Syst.
**2014**, 61, 48–58. [Google Scholar] [CrossRef]

Alternative | ${\mathit{x}}_{1}$ | ${\mathit{x}}_{2}$ | ${\mathit{x}}_{3}$ | ${\mathit{x}}_{4}$ | ${\mathit{x}}_{5}$ | ${\mathit{x}}_{6}$ | ${\mathit{x}}_{7}$ | ${\mathit{x}}_{8}$ |
---|---|---|---|---|---|---|---|---|

${A}_{1}$ | ${S}_{5}^{5}$ | ${S}_{5}^{5}$ | ${S}_{5}^{5}$ | ${S}_{4}^{5}$ | ${S}_{5}^{5}$ | ${S}_{5}^{5}$ | ${S}_{4}^{5}$ | ${S}_{5}^{5}$ |

${A}_{2}$ | ${S}_{4.79}^{5}$ | ${S}_{4.76}^{5}$ | $\text{}{S}_{4.78}^{5}$ | ${S}_{4.93}^{5}$ | ${S}_{4.45}^{5}$ | $\text{}{S}_{4.87}^{5}$ | $\text{}{S}_{4.15}^{5}$ | $\text{}{S}_{4.88}^{5}$ |

${A}_{3}$ | ${S}_{4.59}^{5}$ | $\text{}{S}_{4.33}^{5}$ | $\text{}{S}_{4.39}^{5}$ | $\text{}{S}_{4.90}^{5}$ | ${S}_{4.04}^{5}$ | ${S}_{4.34}^{5}$ | ${S}_{3.68}^{5}$ | ${S}_{4.35}^{5}$ |

${A}_{4}$ | ${S}_{4.83}^{5}$ | ${S}_{4.89}^{5}$ | $\text{}{S}_{4.13}^{5}$ | $\text{}{S}_{4.79}^{5}$ | $\text{}{S}_{3.41}^{5}$ | $\text{}{S}_{4.64}^{5}$ | $\text{}{S}_{3.56}^{5}$ | ${S}_{4.42}^{5}$ |

${A}_{5}$ | ${S}_{4.60}^{5}$ | ${S}_{4.04}^{5}$ | ${S}_{4.06}^{5}$ | ${S}_{4.15}^{5}$ | ${S}_{3.68}^{5}$ | ${S}_{4.85}^{5}$ | ${S}_{3.79}^{5}$ | ${S}_{4.49}^{5}$ |

${A}_{6}$ | ${S}_{4.25}^{5}$ | ${S}_{3.80}^{5}$ | ${S}_{4.38}^{5}$ | ${S}_{4.37}^{5}$ | ${S}_{3.63}^{5}$ | ${S}_{4.63}^{5}$ | ${S}_{4.05}^{5}$ | ${S}_{4.44}^{5}$ |

${A}_{7}$ | ${S}_{3.55}^{5}$ | ${S}_{3.60}^{5}$ | ${S}_{4.38}^{5}$ | ${S}_{4.28}^{5}$ | ${S}_{3.53}^{5}$ | $\text{}{S}_{4.62}^{5}$ | ${S}_{3.34}^{5}$ | ${S}_{4.40}^{5}$ |

Alternative | ${\mathit{x}}_{1}$ | ${\mathit{x}}_{2}$ | ${\mathit{x}}_{3}$ | ${\mathit{x}}_{4}$ | ${\mathit{x}}_{5}$ | ${\mathit{x}}_{6}$ | ${\mathit{x}}_{7}$ | ${\mathit{x}}_{8}$ |
---|---|---|---|---|---|---|---|---|

${A}_{1}$ | ${S}_{6}^{7}$ | ${S}_{3}^{7}$ | $-$ | ${S}_{5}^{7}$ | ${S}_{2}^{7}$ | ${S}_{7}^{7}$ | ${S}_{2}^{7}$ | ${S}_{4}^{7}$ |

${A}_{2}$ | ${S}_{6}^{7}$ | ${S}_{3}^{7}$ | ${S}_{7}^{7}$ | ${S}_{5}^{7}$ | ${S}_{4}^{7}$ | ${S}_{7}^{7}$ | ${S}_{2}^{7}$ | ${S}_{5}^{7}$ |

${A}_{3}$ | ${S}_{5}^{7}$ | ${S}_{3}^{7}$ | $-$ | ${S}_{5}^{7}$ | ${S}_{4}^{7}$ | ${S}_{6}^{7}$ | ${S}_{4}^{7}$ | ${S}_{6}^{7}$ |

${A}_{4}$ | ${S}_{7}^{7}$ | ${S}_{3}^{7}$ | $\text{}{S}_{5}^{7}$ | ${S}_{5}^{7}$ | ${S}_{2}^{7}$ | ${S}_{7}^{7}$ | $-$ | $\text{}{S}_{5}^{7}$ |

${A}_{5}$ | ${S}_{5}^{7}$ | ${S}_{4}^{7}$ | ${S}_{6}^{7}$ | ${S}_{5}^{7}$ | ${S}_{3}^{7}$ | ${S}_{7}^{7}$ | ${S}_{2}^{7}$ | ${S}_{6}^{7}$ |

${A}_{6}$ | ${S}_{6}^{7}$ | ${S}_{3}^{7}$ | ${S}_{7}^{7}$ | ${S}_{6}^{7}$ | ${S}_{5}^{7}$ | ${S}_{6}^{7}$ | ${S}_{3}^{7}$ | ${S}_{4}^{7}$ |

${A}_{7}$ | ${S}_{5}^{7}$ | ${S}_{5}^{7}$ | ${S}_{4}^{7}$ | $-$ | ${S}_{3}^{7}$ | $\text{}{S}_{5}^{7}$ | ${S}_{3}^{7}$ | ${S}_{5}^{7}$ |

Alternative | ${\mathit{x}}_{1}$ | ${\mathit{x}}_{2}$ | ${\mathit{x}}_{3}$ | ${\mathit{x}}_{4}$ | ${\mathit{x}}_{5}$ | ${\mathit{x}}_{6}$ | ${\mathit{x}}_{7}$ | ${\mathit{x}}_{8}$ |
---|---|---|---|---|---|---|---|---|

${A}_{1}$ | ${S}_{8}^{9}$ | $\text{}{S}_{6}^{9}$ | ${S}_{7}^{9}$ | ${S}_{8}^{9}$ | ${S}_{4}^{9}$ | $\text{}{S}_{8}^{9}$ | $\text{}{S}_{7}^{9}$ | $\text{}{S}_{7}^{9}$ |

${A}_{2}$ | ${S}_{7}^{9}$ | $\text{}{S}_{7}^{9}$ | $\text{}{S}_{7}^{9}$ | ${S}_{8}^{9}$ | $\text{}{S}_{6}^{9}$ | $\text{}{S}_{7}^{9}$ | $\text{}{S}_{6}^{9}$ | $\text{}{S}_{9}^{9}$ |

${A}_{3}$ | ${S}_{9}^{9}$ | $\text{}{S}_{6}^{9}$ | $\text{}{S}_{9}^{9}$ | $\text{}{S}_{8}^{9}$ | $\text{}{S}_{4}^{9}$ | $\text{}{S}_{4}^{9}$ | $\text{}{S}_{1}^{9}$ | $\text{}{S}_{7}^{9}$ |

${A}_{4}$ | ${S}_{9}^{9}$ | ${S}_{4}^{9}$ | ${S}_{4}^{9}$ | ${S}_{9}^{9}$ | ${S}_{3}^{9}$ | ${S}_{6}^{9}$ | ${S}_{1}^{9}$ | ${S}_{6}^{9}$ |

${A}_{5}$ | ${S}_{8}^{9}$ | ${S}_{5}^{9}$ | ${S}_{3}^{9}$ | ${S}_{3}^{9}$ | ${S}_{4}^{9}$ | ${S}_{7}^{9}$ | ${S}_{4}^{9}$ | ${S}_{7}^{9}$ |

${A}_{6}$ | ${S}_{4}^{9}$ | ${S}_{6}^{9}$ | ${S}_{9}^{9}$ | ${S}_{9}^{9}$ | ${S}_{2}^{9}$ | ${S}_{8}^{9}$ | ${S}_{4}^{9}$ | ${S}_{8}^{9}$ |

${A}_{7}$ | ${S}_{6}^{9}$ | ${S}_{5}^{9}$ | ${S}_{8}^{9}$ | ${S}_{5}^{9}$ | ${S}_{6}^{9}$ | ${S}_{8}^{9}$ | ${S}_{8}^{9}$ | ${S}_{9}^{9}$ |

Alternative | ${\mathit{x}}_{1}$ | ${\mathit{x}}_{2}$ | ${\mathit{x}}_{3}$ | ${\mathit{x}}_{4}$ |

${A}_{1}$ | $\left\{{S}_{5}^{5}(\frac{1}{3}),{S}_{6}^{7}(\frac{1}{3}),{S}_{8}^{9}(\frac{1}{3})\right\}\text{}$ | $\left\{{S}_{1}^{5}(\frac{1}{3}),{S}_{3}^{7}(\frac{1}{3}),{S}_{6}^{9}(\frac{1}{3})\right\}\text{}$ | $\left\{{S}_{5}^{5}(\frac{1}{2}),{S}_{7}^{9}(\frac{1}{2})\right\}\text{}$ | $\left\{{S}_{4}^{5}(\frac{1}{3}),{S}_{5}^{7}(\frac{1}{3}),{S}_{8}^{9}(\frac{1}{3})\right\}$ |

${A}_{2}$ | $\left\{{S}_{4.79}^{5}(\frac{1}{3}),{S}_{6}^{7}(\frac{1}{3}),{S}_{7}^{9}(\frac{1}{3})\right\}$ | $\left\{{S}_{4.76}^{5}(\frac{1}{3}),{S}_{3}^{7}(\frac{1}{3}),{S}_{7}^{9}(\frac{1}{3})\right\}$ | $\left\{{S}_{4.78}^{5}(\frac{1}{3}),{S}_{7}^{7}(\frac{1}{3}),{S}_{7}^{9}(\frac{1}{3})\right\}$ | $\left\{{S}_{4.93}^{5}(\frac{1}{3}),{S}_{5}^{7}(\frac{1}{3}),{S}_{8}^{9}(\frac{1}{3})\right\}$ |

${A}_{3}$ | $\left\{{S}_{4.59}^{5}(\frac{1}{3}),{S}_{5}^{7}(\frac{1}{3}),{S}_{9}^{9}(\frac{1}{3})\right\}$ | $\left\{{S}_{4.33}^{5}(\frac{1}{3}),{S}_{3}^{7}(\frac{1}{3}),{S}_{6}^{9}(\frac{1}{3})\right\}$ | $\left\{{S}_{4.39}^{5}(\frac{1}{2}),{S}_{9}^{9}(\frac{1}{2})\right\}$ | $\text{}\left\{{S}_{4.90}^{5}(\frac{1}{3}),{S}_{5}^{7}(\frac{1}{3}),{S}_{8}^{9}(\frac{1}{3})\right\}$ |

${A}_{4}$ | $\left\{{S}_{4.83}^{5}(\frac{1}{3}),{S}_{7}^{7}(\frac{1}{3}),{S}_{9}^{9}(\frac{1}{3})\right\}$ | $\left\{{S}_{4.89}^{5}(\frac{1}{3}),{S}_{3}^{7}(\frac{1}{3}),{S}_{4}^{9}(\frac{1}{3})\right\}$ | $\left\{{S}_{4.13}^{5}(\frac{1}{3}),{S}_{5}^{7}(\frac{1}{3}),{S}_{4}^{9}(\frac{1}{3})\right\}$ | $\left\{{S}_{4.79}^{5}(\frac{1}{3}),{S}_{5}^{7}(\frac{1}{3}),{S}_{9}^{9}(\frac{1}{3})\right\}$ |

${A}_{5}$ | $\left\{{S}_{4.60}^{5}(\frac{1}{3}),{S}_{5}^{7}(\frac{1}{3}),{S}_{8}^{9}(\frac{1}{3})\right\}$ | $\left\{{S}_{4.04}^{5}(\frac{1}{3}),{S}_{4}^{7}(\frac{1}{3}),{S}_{5}^{9}(\frac{1}{3})\right\}$ | $\left\{{S}_{4.06}^{5}(\frac{1}{3}),{S}_{6}^{7}(\frac{1}{3}),{S}_{3}^{9}(\frac{1}{3})\right\}$ | $\left\{{S}_{4.15}^{5}(\frac{1}{3}),{S}_{5}^{7}(\frac{1}{3}),{S}_{3}^{9}(\frac{1}{3})\right\}$ |

${A}_{6}$ | $\left\{{S}_{4.25}^{5}(\frac{1}{3}),{S}_{6}^{7}(\frac{1}{3}),{S}_{4}^{9}(\frac{1}{3})\right\}$ | $\left\{{S}_{3.80}^{5}(\frac{1}{3}),{S}_{3}^{7}(\frac{1}{3}),{S}_{6}^{9}(\frac{1}{3})\right\}$ | $\left\{{S}_{4.38}^{5}(\frac{1}{3}),{S}_{7}^{7}(\frac{1}{3}),{S}_{9}^{9}(\frac{1}{3})\right\}$ | $\left\{{S}_{4.37}^{5}(\frac{1}{3}),{S}_{6}^{7}(\frac{1}{3}),{S}_{9}^{9}(\frac{1}{3})\right\}$ |

${A}_{7}$ | $\left\{{S}_{3.55}^{5}(\frac{1}{3}),{S}_{5}^{7}(\frac{1}{3}),{S}_{6}^{9}(\frac{1}{3})\right\}$ | $\left\{{S}_{3.60}^{5}(\frac{1}{3}),{S}_{5}^{7}(\frac{1}{3}),{S}_{5}^{9}(\frac{1}{3})\right\}$ | $\left\{{S}_{4.38}^{5}(\frac{1}{3}),{S}_{4}^{7}(\frac{1}{3}),{S}_{8}^{9}(\frac{1}{3})\right\}$ | $\left\{{S}_{4.28}^{5}(\frac{1}{2}),{S}_{5}^{9}(\frac{1}{2})\right\}$ |

Alternative | ${\mathit{x}}_{5}$ | ${\mathit{x}}_{6}$ | ${\mathit{x}}_{7}$ | ${\mathit{x}}_{8}$ |

${A}_{1}$ | $\left\{{S}_{5}^{5}(\frac{1}{3}),{S}_{2}^{7}(\frac{1}{3}),{S}_{4}^{9}(\frac{1}{3})\right\}\text{}$ | $\left\{{S}_{5}^{5}(\frac{1}{3}),{S}_{7}^{7}(\frac{1}{3}),{S}_{8}^{9}(\frac{1}{3})\right\}$ | $\text{}\left\{{S}_{4}^{5}(\frac{1}{3}),{S}_{2}^{7}(\frac{1}{3}),{S}_{7}^{9}(\frac{1}{3})\right\}$ | $\left\{{S}_{5}^{5}(\frac{1}{3}),{S}_{4}^{7}(\frac{1}{3}),{S}_{7}^{9}(\frac{1}{3})\right\}$ |

${A}_{2}$ | $\left\{{S}_{4.45}^{5}(\frac{1}{3}),{S}_{4}^{7}(\frac{1}{3}),{S}_{6}^{9}(\frac{1}{3})\right\}$ | $\left\{{S}_{4.87}^{5}(\frac{1}{3}),{S}_{7}^{7}(\frac{1}{3}),{S}_{7}^{9}(\frac{1}{3})\right\}$. | $\left\{{S}_{4.15}^{5}(\frac{1}{3}),{S}_{2}^{7}(\frac{1}{3}),{S}_{6}^{9}(\frac{1}{3})\right\}$ | $\left\{{S}_{4.88}^{5}(\frac{1}{3}),{S}_{5}^{7}(\frac{1}{3}),{S}_{9}^{9}(\frac{1}{3})\right\}$ |

${A}_{3}$ | $\left\{{S}_{4.04}^{5}(\frac{1}{3}),{S}_{4}^{7}(\frac{1}{3}),{S}_{4}^{9}(\frac{1}{3})\right\}$ | $\left\{{S}_{4.34}^{5}(\frac{1}{3}),{S}_{6}^{7}(\frac{1}{3}),{S}_{4}^{9}(\frac{1}{3})\right\}$ | $\left\{{S}_{3.68}^{5}(\frac{1}{3}),{S}_{4}^{7}(\frac{1}{3}),{S}_{1}^{9}(\frac{1}{3})\right\}$ | $\left\{{S}_{4.35}^{5}(\frac{1}{3}),{S}_{6}^{7}(\frac{1}{3}),{S}_{7}^{9}(\frac{1}{3})\right\}$ |

${A}_{4}$ | $\left\{{S}_{3.41}^{5}(\frac{1}{3}),{S}_{2}^{7}(\frac{1}{3}),{S}_{3}^{9}(\frac{1}{3})\right\}$ | $\left\{{S}_{4.64}^{5}(\frac{1}{3}),{S}_{7}^{7}(\frac{1}{3}),{S}_{6}^{9}(\frac{1}{3})\right\}$ | $\left\{{S}_{3.56}^{5}(\frac{1}{2}),{S}_{1}^{9}(\frac{1}{2})\right\}$ | $\left\{{S}_{4.42}^{5}(\frac{1}{3}),{S}_{5}^{7}(\frac{1}{3}),{S}_{6}^{9}(\frac{1}{3})\right\}$ |

${A}_{5}$ | $\left\{{S}_{3.68}^{5}(\frac{1}{3}),{S}_{3}^{7}(\frac{1}{3}),{S}_{4}^{9}(\frac{1}{3})\right\}$ | $\left\{{S}_{4.85}^{5}(\frac{1}{3}),{S}_{7}^{7}(\frac{1}{3}),{S}_{7}^{9}(\frac{1}{3})\right\}$ | $\left\{{S}_{3.79}^{5}(\frac{1}{3}),{S}_{2}^{7}(\frac{1}{3}),{S}_{4}^{9}(\frac{1}{3})\right\}$ | $\left\{{S}_{4.49}^{5}(\frac{1}{3}),{S}_{6}^{7}(\frac{1}{3}),{S}_{7}^{9}(\frac{1}{3})\right\}$ |

${A}_{6}$ | $\left\{{S}_{3.63}^{5}(\frac{1}{3}),{S}_{5}^{7}(\frac{1}{3}),{S}_{2}^{9}(\frac{1}{3})\right\}$ | $\left\{{S}_{4.63}^{5}(\frac{1}{3}),{S}_{6}^{7}(\frac{1}{3}),{S}_{8}^{9}(\frac{1}{3})\right\}$ | $\left\{{S}_{4.05}^{5}(\frac{1}{3}),{S}_{3}^{7}(\frac{1}{3}),{S}_{4}^{9}(\frac{1}{3})\right\}$ | $\left\{{S}_{4.44}^{5}(\frac{1}{3}),{S}_{4}^{7}(\frac{1}{3}),{S}_{8}^{9}(\frac{1}{3})\right\}$ |

${A}_{7}$ | $\left\{{S}_{3.53}^{5}(\frac{1}{3}),{S}_{3}^{7}(\frac{1}{3}),{S}_{6}^{9}(\frac{1}{3})\right\}$ | $\left\{{S}_{4.62}^{5}(\frac{1}{3}),{S}_{5}^{7}(\frac{1}{3}),{S}_{8}^{9}(\frac{1}{3})\right\}$ | $\left\{{S}_{3.34}^{5}(\frac{1}{3}),{S}_{3}^{7}(\frac{1}{3}),{S}_{8}^{9}(\frac{1}{3})\right\}$ | $\left\{{S}_{4.40}^{5}(\frac{1}{3}),{S}_{5}^{7}(\frac{1}{3}),{S}_{9}^{9}(\frac{1}{3})\right\}$ |

Alternative | ${\mathit{x}}_{1}$ | ${\mathit{x}}_{2}$ | ${\mathit{x}}_{3}$ | ${\mathit{x}}_{4}$ |

${A}_{1}$ | $\text{}\left\{{S}_{\frac{1}{3}},{S}_{\frac{8}{27}},{S}_{\frac{2}{7}}\right\}$ | $\left\{{S}_{\frac{1}{3}},{S}_{\frac{2}{9}},{S}_{\frac{1}{7}}\right\}$ | $\left\{{S}_{\frac{1}{2}},{S}_{\frac{7}{18}},{S}_{0}\right\}$ | $\left\{{S}_{\frac{8}{27}},{S}_{\frac{4}{15}},{S}_{\frac{5}{21}}\right\}\text{}$ |

${A}_{2}$ | $\left\{{S}_{\frac{4.79}{15}},{S}_{\frac{2}{7}},{S}_{\frac{7}{27}}\right\}$ | $\left\{{S}_{\frac{4.76}{15}},{S}_{\frac{7}{27}},{S}_{\frac{1}{7}}\right\}$ | $\left\{{S}_{\frac{1}{3}},{S}_{\frac{4.78}{15}},{S}_{\frac{7}{27}}\right\}$ | $\left\{{S}_{\frac{4.93}{15}},{S}_{\frac{8}{27}},{S}_{\frac{5}{21}}\right\}$ |

${A}_{3}$ | $\left\{{S}_{\frac{1}{3}},{S}_{\frac{4.59}{15}},{S}_{\frac{5}{21}}\right\}$ | $\left\{{S}_{\frac{4.33}{15}},{S}_{\frac{2}{9}},{S}_{\frac{1}{7}}\right\}$ | $\left\{{S}_{\frac{1}{2}},{S}_{\frac{4.39}{10}},{S}_{0}\right\}$ | $\left\{{S}_{\frac{4.90}{15}},{S}_{\frac{8}{27}},{S}_{\frac{5}{21}}\right\}$ |

${A}_{4}$ | $\left\{{S}_{\frac{2}{3}},{S}_{\frac{4.83}{15}},{S}_{0}\right\}$ | $\left\{{S}_{\frac{4.89}{15}},{S}_{\frac{4}{27}},{S}_{\frac{1}{7}}\right\}$ | $\left\{{S}_{\frac{4.13}{15}},{S}_{\frac{5}{21}},{S}_{\frac{4}{27}}\right\}$ | $\left\{{S}_{\frac{1}{3}},{S}_{\frac{4.79}{15}},{S}_{\frac{5}{21}}\right\}$ |

${A}_{5}$ | $\left\{{S}_{\frac{4.60}{15}},{S}_{\frac{8}{27}},{S}_{\frac{5}{21}}\right\}$ | $\left\{{S}_{\frac{4.04}{15}},{S}_{\frac{4}{21}},{S}_{\frac{5}{27}}\right\}$ | $\left\{{S}_{\frac{2}{7}},{S}_{\frac{4.06}{15}},{S}_{\frac{1}{9}}\right\}$ | $\left\{{S}_{\frac{4.15}{15}},{S}_{\frac{5}{21}},{S}_{\frac{1}{9}}\right\}$ |

${A}_{6}$ | $\left\{{S}_{\frac{2}{7}},{S}_{\frac{4.25}{15}},{S}_{\frac{4}{27}}\right\}$ | $\left\{{S}_{\frac{3.80}{15}},{S}_{\frac{2}{9}},{S}_{\frac{1}{7}}\right\}$ | $\left\{{S}_{\frac{2}{3}},{S}_{\frac{4.38}{15}},{S}_{0}\right\}$ | $\left\{{S}_{\frac{1}{3}},{S}_{\frac{4.37}{15}},{S}_{\frac{2}{7}}\right\}$ |

${A}_{7}$ | $\left\{{S}_{\frac{5}{21}},{S}_{\frac{3.55}{15}},{S}_{\frac{2}{9}}\right\}$ | $\left\{{S}_{\frac{3.60}{15}},{S}_{\frac{5}{21}},{S}_{\frac{5}{27}}\right\}$ | $\left\{{S}_{\frac{8}{27}},{S}_{\frac{4.38}{15}},{S}_{\frac{4}{21}}\right\}$ | $\left\{{S}_{\frac{4.28}{10}},{S}_{\frac{5}{18}},{S}_{0}\right\}$ |

Alternative | ${\mathit{x}}_{5}$ | ${\mathit{x}}_{6}$ | ${\mathit{x}}_{7}$ | ${\mathit{x}}_{8}$ |

${A}_{1}$ | $\left\{{S}_{\frac{1}{3}},{S}_{\frac{4}{27}},{S}_{\frac{2}{21}}\right\}$ | $\left\{{S}_{\frac{2}{3}},{S}_{\frac{8}{27}},{S}_{0}\right\}$ | $\left\{{S}_{\frac{4}{15}},{S}_{\frac{7}{27}},{S}_{\frac{2}{21}}\right\}$ | $\left\{{S}_{\frac{1}{3}},{S}_{\frac{7}{27}},{S}_{\frac{4}{21}}\right\}$ |

${A}_{2}$ | $\left\{{S}_{\frac{4.45}{15}},{S}_{\frac{2}{9}},{S}_{\frac{4}{21}}\right\}$ | $\left\{{S}_{\frac{1}{3}},{S}_{\frac{4.87}{15}},{S}_{\frac{7}{27}}\right\}$ | $\left\{{S}_{\frac{4.15}{15}},{S}_{\frac{2}{9}},{S}_{\frac{2}{21}}\right\}$ | $\left\{{S}_{\frac{1}{3}},{S}_{\frac{4.88}{15}},{S}_{\frac{5}{21}}\right\}$ |

${A}_{3}$ | $\left\{{S}_{\frac{4.04}{15}},{S}_{\frac{4}{21}},{S}_{\frac{4}{27}}\right\}$ | $\left\{{S}_{\frac{4.34}{15}},{S}_{\frac{2}{7}},{S}_{\frac{4}{27}}\right\}$ | $\left\{{S}_{\frac{3.68}{15}},{S}_{\frac{4}{21}},{S}_{\frac{1}{27}}\right\}$ | $\left\{{S}_{\frac{4.35}{15}},{S}_{\frac{2}{7}},{S}_{\frac{7}{27}}\right\}$ |

${A}_{4}$ | $\left\{{S}_{\frac{3.41}{15}},{S}_{\frac{1}{9}},{S}_{\frac{2}{21}}\right\}$ | $\left\{{S}_{\frac{1}{3}},{S}_{\frac{4.64}{15}},{S}_{\frac{2}{9}}\right\}$ | $\left\{{S}_{\frac{3.56}{10}},{S}_{\frac{1}{18}},{S}_{0}\right\}$ | $\left\{{S}_{\frac{4.42}{15}},{S}_{\frac{5}{21}},{S}_{\frac{2}{9}}\right\}$ |

${A}_{5}$ | $\left\{{S}_{\frac{3.68}{15}},{S}_{\frac{4}{27}},{S}_{\frac{1}{7}}\right\}$ | $\left\{{S}_{\frac{1}{3}},{S}_{\frac{4.85}{15}},{S}_{\frac{7}{27}}\right\}$ | $\left\{{S}_{\frac{3.79}{15}},{S}_{\frac{4}{27}},{S}_{\frac{2}{21}}\right\}$ | $\left\{{S}_{\frac{4.49}{15}},{S}_{\frac{2}{7}},{S}_{\frac{7}{27}}\right\}$ |

${A}_{6}$ | $\left\{{S}_{\frac{3.63}{15}},{S}_{\frac{5}{21}},{S}_{\frac{2}{27}}\right\}$ | $\left\{{S}_{\frac{4.63}{15}},{S}_{\frac{8}{27}},{S}_{\frac{2}{7}}\right\}$ | $\left\{{S}_{\frac{4.05}{15}},{S}_{\frac{4}{27}},{S}_{\frac{1}{7}}\right\}$ | $\left\{{S}_{\frac{8}{27}},{S}_{\frac{4.44}{15}},{S}_{\frac{4}{21}}\right\}$ |

${A}_{7}$ | $\left\{{S}_{\frac{3.53}{15}},{S}_{\frac{2}{9}},{S}_{\frac{1}{7}}\right\}$ | $\left\{{S}_{\frac{4.62}{15}},{S}_{\frac{8}{27}},{S}_{\frac{5}{21}}\right\}$ | $\left\{{S}_{\frac{8}{27}},{S}_{\frac{3.34}{15}},{S}_{\frac{1}{7}}\right\}$. | $\left\{{S}_{\frac{1}{3}},{S}_{\frac{4.40}{15}},{S}_{\frac{5}{21}}\right\}\text{}$ |

Attribute | ${x}_{1}$ | ${x}_{2}$ | ${x}_{3}$ | ${x}_{4}$ | ${x}_{5}$ | ${x}_{6}$ | ${x}_{7}$ | ${x}_{8}$ |

Weight | 0.0922 | 0.1481 | 0.1022 | 0.1023 | 0.1815 | 0.0832 | 0.1948 | 0.0955 |

Alternative | ${\mathit{x}}_{1}$ | ${\mathit{x}}_{2}$ | ${\mathit{x}}_{3}$ | ${\mathit{x}}_{4}$ |

${A}_{1}$ | (0.0307,0.0273,0.0263) | (0.0494,0.0329,0.0212) | (0.0511,0.0397,0.0000) | (0.0303,0.0273,0.0244) |

${A}_{2}$ | (0.0294,0.0263,0.0239) | (0.0470,0.0384,0.0212) | (0.0341,0.0326,0.0265) | (0.0336,0.0303,0.0244) |

${A}_{3}$ | (0.0307,0.0282,0.0220) | (0.0428,0.0329,0.0212) | (0.0511,0.0449,0.0000) | (0.0334,0.0303,0.0244) |

${A}_{4}$ | (0.0615,0.0297,0.0000) | (0.0483,0.0219,0.0212) | (0.0281,0.0243,0.0151) | (0.0341,0.0327,0.0244) |

${A}_{5}$ | (0.0283,0.0273,0.0220) | (0.0399,0.0282,0.0274) | (0.0292,0.0277,0.0114) | (0.0283,0.0244,0.0114) |

${A}_{6}$ | (0.0263,0.0261,0.0137) | (0.0375,0.0329,0.0212) | (0.0681,0.0298,0.0000) | (0.0341,0.0298,0.0292) |

${A}_{7}$ | (0.0220,0.0218,0.0205) | (0.0355,0.0353,0.0274) | (0.0303,0.0298,0.0195) | (0.0438,0.0284,0.0000) |

Alternative | ${\mathit{x}}_{5}$ | ${\mathit{x}}_{6}$ | ${\mathit{x}}_{7}$ | ${\mathit{x}}_{8}$ |

${A}_{1}$ | (0.0605,0.0269,0.0173) | (0.0555,0.0247,0.0000) | (0.0519,0.0505,0.0186) | (0.0318,0.0248,0.0182) |

${A}_{2}$ | (0.0538,0.0403,0.0346) | (0.0277,0.0270,0.0216) | (0.0539,0.0433,0.0186) | (0.0318,0.0311,0.0227) |

${A}_{3}$ | (0.0489,0.0346,0.0269) | (0.0241,0.0238,0.0123) | (0.0478,0.0371,0.0072) | (0.0277,0.0273,0.0248) |

${A}_{4}$ | (0.0413,0.0202,0.0173) | (0.0277,0.0257,0.0185) | (0.0693,0.0108,0.0000) | (0.0281,0.0227,0.0212) |

${A}_{5}$ | (0.0445,0.0269,0.0259) | (0.0277,0.0269,0.0216) | (0.0492,0.0289,0.0186) | (0.0286,0.0273,0.0248) |

${A}_{6}$ | (0.0439,0.0432,0.0134) | (0.0257,0.0247,0.0238) | (0.0526,0.0289,0.0278) | (0.0283,0.0283,0.0182) |

${A}_{7}$ | (0.0427,0.0403,0.0259) | (0.0256,0.0247,0.0198) | (0.0577,0.0434,0.0278) | (0.0318,0.0280,0.0227) |

Attribute | ${\mathit{x}}_{1}$ | ${\mathit{x}}_{2}$ | ${\mathit{x}}_{3}$ | ${\mathit{x}}_{4}$ |

PLPIS | (0.0615,0.0297,0.0263) | (0.0494,0.0384,0.0274) | (0.0681,0.0449,0.0265) | (0.0438,0.0327,0.0292) |

Attribute | ${\mathit{x}}_{5}$ | ${\mathit{x}}_{6}$ | ${\mathit{x}}_{7}$ | ${\mathit{x}}_{8}$ |

PLPIS | (0.0605,0.0432,0.0346) | (0.0555,0.0270,0.0238) | (0.0693,0.0505,0.0278) | (0.0318,0.0311,0.0248) |

Attribute | ${\mathit{x}}_{1}$ | ${\mathit{x}}_{2}$ | ${\mathit{x}}_{3}$ | ${\mathit{x}}_{4}$ |

PLNIS | (0.0220,0.0218,0.0000) | (0.0355,0.0219,0.0212) | (0.0281,0.0243,0.0000) | (0.0283,0.0244,0.0000) |

Attribute | ${\mathit{x}}_{5}$ | ${\mathit{x}}_{6}$ | ${\mathit{x}}_{7}$ | ${\mathit{x}}_{8}$ |

PLNIS | (0.0413,0.0202,0.0134) | (0.0241,0.0238,0.0000) | (0.0478,0.0108,0.0000) | (0.0277,0.0227,0.0182) |

Attribute | ${\mathit{A}}_{1}$ | ${\mathit{A}}_{2}$ | ${\mathit{A}}_{3}$ | ${\mathit{A}}_{4}$ | ${\mathit{A}}_{5}$ | ${\mathit{A}}_{6}$ | ${\mathit{A}}_{7}$ | |
---|---|---|---|---|---|---|---|---|

Distance Parameter | ||||||||

$\lambda =1$ | 2.2775 | 2.3581 | 2.1791 | 2.0516 | 2.0394 | 2.1872 | 2.1285 | |

$\lambda =2$ | 2.4984 | 2.3883 | 2.2884 | 2.2535 | 2.0766 | 2.3228 | 2.2020 |

Attribute | $\text{}{\mathit{A}}_{1}$ | $\text{}{\mathit{A}}_{2}$ | $\text{}{\mathit{A}}_{3}$ | $\text{}{\mathit{A}}_{4}$ | $\text{}{\mathit{A}}_{5}$ | $\text{}{\mathit{A}}_{6}$ | $\text{}{\mathit{A}}_{7}$ | |
---|---|---|---|---|---|---|---|---|

Distance Parameter | ||||||||

$\lambda =1$ | 4.8225 | 4.9277 | 4.5605 | 4.3157 | 4.2010 | 4.5976 | 4.4524 | |

$\lambda =2$ | 5.2726 | 5.0031 | 4.7956 | 4.7448 | 4.3088 | 4.8694 | 4.5975 |

Attribute | $\text{}{\mathit{A}}_{1}$ | $\text{}{\mathit{A}}_{2}$ | $\text{}{\mathit{A}}_{3}$ | $\text{}{\mathit{A}}_{4}$ | $\text{}{\mathit{A}}_{5}$ | $\text{}{\mathit{A}}_{6}$ | $\text{}{\mathit{A}}_{7}$ | |
---|---|---|---|---|---|---|---|---|

Distance Parameter | ||||||||

$\lambda =1$ | 0.4723 | 0.4785 | 0.4778 | 0.4754 | 0.4854 | 0.4776 | 0.4781 | |

$\lambda =2$ | 0.4739 | 0.4774 | 0.4772 | 0.4779 | 0.4820 | 0.4770 | 0.4789 |

Distance Parameter | Rank |
---|---|

$\lambda =1$ | ${A}_{5}\succ {A}_{2}\succ {A}_{7}\succ {A}_{3}\succ {A}_{6}\succ {A}_{4}\succ {A}_{1}$ |

$\lambda =2$ | ${A}_{5}\succ {A}_{7}\succ {A}_{2}\succ {A}_{3}\succ {A}_{6}\succ {A}_{4}\succ {A}_{1}$ |

Distance Parameter | Parameters | ${\mathit{A}}_{1}$ | ${\mathit{A}}_{2}$ | ${\mathit{A}}_{3}$ | ${\mathit{A}}_{4}$ | ${\mathit{A}}_{5}$ | ${\mathit{A}}_{6}$ | ${\mathit{A}}_{7}$ | ||

$\lambda =1$ | $\alpha =0.85,$ | $\beta =0.85$ | $\theta =4.1$ | 0.2587 | 0.2620 | 0.2616 | 0.2604 | 0.2657 | 0.2615 | 0.2617 |

$\alpha =0.725$ | $\beta =0.717$ | $\theta =2.04$ | 0.5097 | 0.5152 | 0.5146 | 0.5123 | 0.5210 | 0.5143 | 0.5142 | |

$\alpha =0.89$ | $\beta =0.92$ | $\theta =2.25$ | 0.4939 | 0.5003 | 0.5005 | 0.4985 | 0.5101 | 0.5003 | 0.5016 | |

Distance Parameter | Parameters | ${\mathit{A}}_{1}$ | ${\mathit{A}}_{2}$ | ${\mathit{A}}_{3}$ | ${\mathit{A}}_{4}$ | ${\mathit{A}}_{5}$ | ${\mathit{A}}_{6}$ | ${\mathit{A}}_{7}$ | ||

$\lambda =2$ | $\alpha =0.85$ | $\beta =0.85$ | $\theta =4.1$ | 0.2595 | 0.2614 | 0.2613 | 0.2601 | 0.2638 | 0.2612 | 0.2622 |

$\alpha =0.725$ | $\beta =0.717$ | $\theta =2.04$ | 0.5114 | 0.5141 | 0.5140 | 0.5121 | 0.5178 | 0.5139 | 0.5152 | |

$\alpha =0.89$ | $\beta =0.92$ | $\theta =2.25$ | 0.4939 | 0.4988 | 0.4991 | 0.4965 | 0.5060 | 0.4985 | 0.5019 |

Distance Parameter | Parameters | Rank | ||
---|---|---|---|---|

$\lambda =1$ | $\alpha =0.85,$ | $\beta =0.85$ | $\theta =4.1$ | ${A}_{5}\succ {A}_{2}\succ {A}_{7}\succ {A}_{3}\succ {A}_{6}\succ {A}_{4}\succ {A}_{1}$ |

$\alpha =0.725$ | $\beta =0.717$ | $\theta =2.04$ | ${A}_{5}\succ {A}_{2}\succ {A}_{3}\succ {A}_{6}\succ {A}_{7}\succ {A}_{4}\succ {A}_{1}$ | |

$\alpha =0.89$ | $\beta =0.92$ | $\theta =2.25$ | ${A}_{5}\succ {A}_{7}\succ {A}_{3}\succ {A}_{2}={A}_{6}\succ {A}_{4}\succ {A}_{1}$ | |

$\lambda =2$ | $\alpha =0.85$ | $\beta =0.85$ | $\theta =4.1$ | ${A}_{5}\succ {A}_{7}\succ {A}_{2}\succ {A}_{3}\succ {A}_{6}\succ {A}_{4}\succ {A}_{1}$ |

$\alpha =0.725$ | $\beta =0.717$ | $\theta =2.04$ | ${A}_{5}\succ {A}_{7}\succ {A}_{2}\succ {A}_{3}\succ {A}_{6}\succ {A}_{4}\succ {A}_{1}$ | |

$\alpha =0.89$ | $\beta =0.92$ | $\theta =2.25$ | ${A}_{5}\succ {A}_{7}\succ {A}_{3}\succ {A}_{2}\succ {A}_{6}\succ {A}_{4}\succ {A}_{1}$ |

Attribute | ${\mathit{A}}_{1}$ | ${\mathit{A}}_{2}$ | ${\mathit{A}}_{3}$ | ${\mathit{A}}_{4}$ | ${\mathit{A}}_{5}$ | ${\mathit{A}}_{6}$ | ${\mathit{A}}_{7}$ | |
---|---|---|---|---|---|---|---|---|

Distance Parameter | ||||||||

$\lambda =1$ | 0.5170 | 0.5209 | 0.5202 | 0.5187 | 0.5247 | 0.5201 | 0.5206 | |

$\lambda =2$ | 0.5181 | 0.5202 | 0.5200 | 0.5186 | 0.5228 | 0.5199 | 0.5211 |

Distance Parameter | Rank |
---|---|

$\lambda =1$ | ${A}_{5}\succ {A}_{2}\succ {A}_{7}\succ {A}_{3}\succ {A}_{6}\succ {A}_{4}\succ {A}_{1}$ |

$\lambda =2$ | ${A}_{5}\succ {A}_{7}\succ {A}_{2}\succ {A}_{3}\succ {A}_{6}\succ {A}_{4}\succ {A}_{1}$ |

Attribute | ${\mathit{A}}_{1}$ | ${\mathit{A}}_{2}$ | ${\mathit{A}}_{3}$ | ${\mathit{A}}_{4}$ | ${\mathit{A}}_{5}$ | ${\mathit{A}}_{6}$ | ${\mathit{A}}_{7}$ | |
---|---|---|---|---|---|---|---|---|

Distance Parameter | ||||||||

$\lambda =1$ | 0.6045 | 0.0000 | 0.5221 | 1.0000 | 0.6663 | 0.7571 | 0.4424 | |

$\lambda =2$ | 0.3547 | 0.0088 | 0.4947 | 0.8607 | 0.7075 | 0.6459 | 0.4109 |

Distance Parameter | Rank |
---|---|

$\lambda =1$ | ${A}_{4}\succ {A}_{6}\succ {A}_{5}\succ {A}_{1}\succ {A}_{3}\succ {A}_{7}\succ {A}_{2}$ |

$\lambda =2$ | ${A}_{4}\succ {A}_{5}\succ {A}_{6}\succ {A}_{3}\succ {A}_{7}\succ {A}_{1}\succ {A}_{2}$ |

Attribute | $\text{}{\mathit{A}}_{1}$ | $\text{}{\mathit{A}}_{2}$ | $\text{}{\mathit{A}}_{3}$ | $\text{}{\mathit{A}}_{4}$ | $\text{}{\mathit{A}}_{5}$ | $\text{}{\mathit{A}}_{6}$ | $\text{}{\mathit{A}}_{7}$ | |
---|---|---|---|---|---|---|---|---|

Distance Parameter | ||||||||

$\lambda =1$ | −0.2062 | −0.1958 | −0.1821 | −0.1767 | −0.1506 | −0.1832 | −0.1754 | |

$\lambda =2$ | −0.2585 | −0.2349 | −0.2251 | −0.2280 | −0.1903 | −0.2297 | −0.2103 |

Distance Parameter | Rank |
---|---|

$\lambda =1$ | ${A}_{5}\succ {A}_{7}\succ {A}_{4}\succ {A}_{3}\succ {A}_{6}\succ {A}_{2}\succ {A}_{1}$ |

$\lambda =2$ | ${A}_{5}\succ {A}_{7}\succ {A}_{3}\succ {A}_{4}\succ {A}_{6}\succ {A}_{2}\succ {A}_{1}$ |

Alternative | ${\mathit{x}}_{1}$ | ${\mathit{x}}_{2}$ | ${\mathit{x}}_{3}$ | ${\mathit{x}}_{4}$ | ${\mathit{x}}_{5}$ | ${\mathit{x}}_{6}$ | ${\mathit{x}}_{7}$ | ${\mathit{x}}_{8}$ |
---|---|---|---|---|---|---|---|---|

${A}_{1}$ | ${S}_{4.45}^{5}$ | ${S}_{4.82}^{5}$ | ${S}_{4.93}^{5}$ | ${S}_{4.72}^{5}$ | ${S}_{4.13}^{5}$ | ${S}_{4.92}^{5}$ | ${S}_{4.52}^{5}$ | ${S}_{4.78}^{5}$ |

${A}_{2}$ | ${S}_{4.65}^{5}$ | ${S}_{4.53}^{5}$ | $\text{}{S}_{4.70}^{5}$ | $\text{}{S}_{4.75}^{5}$ | ${S}_{4.61}^{5}$ | $\text{}{S}_{4.88}^{5}$ | $\text{}{S}_{4.57}^{5}$ | $\text{}{S}_{4.56}^{5}$ |

${A}_{3}$ | ${S}_{4.88}^{5}$ | $\text{}{S}_{4.71}^{5}$ | $\text{}{S}_{4.61}^{5}$ | $\text{}{S}_{4.60}^{5}$ | ${S}_{4.26}^{5}$ | ${S}_{4.84}^{5}$ | ${S}_{4.4}^{5}$ | ${S}_{4.54}^{5}$ |

${A}_{4}$ | ${S}_{3.96}^{5}$ | ${S}_{4.77}^{5}$ | $\text{}{S}_{4.77}^{5}$ | $\text{}{S}_{4.63}^{5}$ | $\text{}{S}_{4.34}^{5}$ | $\text{}{S}_{5}^{5}$ | $\text{}{S}_{4.03}^{5}$ | ${S}_{4.27}^{5}$ |

${A}_{5}$ | ${S}_{4.53}^{5}$ | ${S}_{4.41}^{5}$ | ${S}_{4.65}^{5}$ | $\text{}{S}_{4.42}^{5}$ | ${S}_{4.26}^{5}$ | ${S}_{4.65}^{5}$ | ${S}_{4.38}^{5}$ | ${S}_{4.43}^{5}$ |

${A}_{6}$ | ${S}_{4.7}^{5}$ | ${S}_{3.96}^{5}$ | ${S}_{4.36}^{5}$ | ${S}_{4.22}^{5}$ | ${S}_{4.1}^{5}$ | ${S}_{4.86}^{5}$ | ${S}_{3.79}^{5}$ | ${S}_{4.1}^{5}$ |

${A}_{7}$ | ${S}_{4.52}^{5}$ | ${S}_{3.64}^{5}$ | ${S}_{4.40}^{5}$ | ${S}_{4.25}^{5}$ | ${S}_{4.12}^{5}$ | $\text{}{S}_{4.75}^{5}$ | ${S}_{3.82}^{5}$ | $\text{}{S}_{3.99}^{5}$ |

Alternative | ${\mathit{x}}_{1}$ | ${\mathit{x}}_{2}$ | ${\mathit{x}}_{3}$ | ${\mathit{x}}_{4}$ | ${\mathit{x}}_{5}$ | ${\mathit{x}}_{6}$ | ${\mathit{x}}_{7}$ | ${\mathit{x}}_{8}$ |
---|---|---|---|---|---|---|---|---|

${A}_{1}$ | ${S}_{5}^{7}$ | $-$ | ${S}_{7}^{7}$ | ${S}_{6}^{7}$ | ${S}_{7}^{7}$ | ${S}_{5}^{7}$ | ${S}_{2}^{7}$ | ${S}_{4}^{7}$ |

${A}_{2}$ | ${S}_{7}^{7}$ | ${S}_{2}^{7}$ | ${S}_{3}^{7}$ | ${S}_{6}^{7}$ | ${S}_{6}^{7}$ | ${S}_{6}^{7}$ | ${S}_{7}^{7}$ | ${S}_{5}^{7}$ |

${A}_{3}$ | ${S}_{7}^{7}$ | ${S}_{7}^{7}$ | ${S}_{4}^{7}$ | ${S}_{5}^{7}$ | ${S}_{6}^{7}$ | ${S}_{4}^{7}$ | ${S}_{5}^{7}$ | ${S}_{6}^{7}$ |

${A}_{4}$ | ${S}_{3}^{7}$ | ${S}_{7}^{7}$ | $-$ | ${S}_{6}^{7}$ | ${S}_{4}^{7}$ | ${S}_{7}^{7}$ | ${S}_{5}^{7}$ | $\text{}{S}_{6}^{7}$ |

${A}_{5}$ | ${S}_{5}^{7}\text{}$ | ${S}_{2}^{7}$ | ${S}_{5}^{7}$ | ${S}_{6}^{7}$ | ${S}_{5}^{7}$ | ${S}_{5}^{7}$ | ${S}_{7}^{7}$ | ${S}_{7}^{7}$ |

${A}_{6}$ | ${S}_{4}^{7}$ | ${S}_{1}^{7}$ | ${S}_{7}^{7}$ | ${S}_{6}^{7}$ | ${S}_{6}^{7}$ | ${S}_{7}^{7}$ | ${S}_{1}^{7}$ | ${S}_{5}^{7}$ |

${A}_{7}$ | ${S}_{7}^{7}$ | ${S}_{1}^{7}$ | $-$ | $-$ | ${S}_{5}^{7}$ | $\text{}{S}_{6}^{7}$ | $-$ | ${S}_{4}^{7}$ |

Alternative | ${\mathit{x}}_{1}$ | ${\mathit{x}}_{2}$ | ${\mathit{x}}_{3}$ | ${\mathit{x}}_{4}$ | ${\mathit{x}}_{5}$ | ${\mathit{x}}_{6}$ | ${\mathit{x}}_{7}$ | ${\mathit{x}}_{8}$ |
---|---|---|---|---|---|---|---|---|

${A}_{1}$ | ${S}_{7}^{9}$ | $\text{}{S}_{7}^{9}$ | ${S}_{8}^{9}$ | ${S}_{4}^{9}$ | ${S}_{4}^{9}$ | $\text{}{S}_{9}^{9}$ | $\text{}{S}_{3}^{9}$ | $\text{}{S}_{4}^{9}$ |

${A}_{2}$ | ${S}_{7}^{9}$ | $\text{}{S}_{4}^{9}$ | $\text{}{S}_{7}^{9}$ | ${S}_{6}^{9}$ | $\text{}{S}_{9}^{9}$ | $\text{}{S}_{7}^{9}$ | $\text{}{S}_{7}^{9}$ | $\text{}{S}_{7}^{9}$ |

${A}_{3}$ | ${S}_{8}^{9}$ | $\text{}{S}_{7}^{9}$ | $\text{}{S}_{6}^{9}$ | $\text{}{S}_{3}^{9}$ | $\text{}{S}_{7}^{9}$ | $\text{}{S}_{8}^{9}$ | $\text{}{S}_{6}^{9}$ | $\text{}{S}_{7}^{9}$ |

${A}_{4}$ | ${S}_{2}^{9}$ | ${S}_{3}^{9}$ | ${S}_{8}^{9}$ | ${S}_{8}^{9}$ | ${S}_{9}^{9}$ | ${S}_{6}^{9}$ | ${S}_{8}^{9}$ | ${S}_{5}^{9}$ |

${A}_{5}$ | ${S}_{8}^{9}$ | ${S}_{4}^{9}$ | ${S}_{8}^{9}$ | ${S}_{3}^{9}$ | ${S}_{2}^{9}$ | ${S}_{9}^{9}$ | ${S}_{9}^{9}$ | ${S}_{5}^{9}$ |

${A}_{6}$ | ${S}_{7}^{9}$ | ${S}_{2}^{9}$ | ${S}_{9}^{9}$ | ${S}_{7}^{9}$ | ${S}_{9}^{9}$ | ${S}_{3}^{9}$ | ${S}_{5}^{9}$ | ${S}_{4}^{9}$ |

${A}_{7}$ | ${S}_{1}^{9}$ | ${S}_{5}^{9}$ | ${S}_{8}^{9}$ | ${S}_{7}^{9}$ | ${S}_{7}^{9}$ | ${S}_{7}^{9}$ | ${S}_{7}^{9}$ | ${S}_{7}^{9}$ |

Attribute | ${x}_{1}$ | ${x}_{2}$ | ${x}_{3}$ | ${x}_{4}$ | ${x}_{5}$ | ${x}_{6}$ | ${x}_{7}$ | ${x}_{8}$ |

Weight | 0.1258 | 0.1774 | 0.1025 | 0.1222 | 0.1137 | 0.0953 | 0.1409 | 0.1222 |

Alternative |