# Valuation Fuzzy Soft Sets: A Flexible Fuzzy Soft Set Based Decision Making Procedure for the Valuation of Assets

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Notation and Definitions

**FS**(X).

#### 2.1. Soft Sets and Fuzzy Soft Sets

**Definition**

**1**

**Definition**

**2**

**Example**

**1.**

- 1.
- ${h}_{1}\in F\left({e}_{1}\right)\cap F\left({e}_{4}\right)$, ${h}_{3}\notin F\left({e}_{2}\right)\cup F\left({e}_{3}\right)$.
- 2.
- ${h}_{2}\in F\left({e}_{1}\right)\cap F\left({e}_{3}\right)$, ${h}_{1}\notin F\left({e}_{2}\right)\cup F\left({e}_{4}\right)$.
- 3.
- ${h}_{3}\in F\left({e}_{2}\right)$, ${h}_{2}\notin F\left({e}_{1}\right)\cup F\left({e}_{3}\right)\cup F\left({e}_{4}\right)$.

#### 2.2. Basic Operations

**Definition**

**3**

**Definition**

**4**

**Definition**

**5**

**Definition**

**6**

**Definition**

**7**

**Definition**

**8**

## 3. Some Novel Concepts Related to Valuation Fuzzy Soft Sets

#### Valuation and Partial Valuation Fuzzy Soft Sets

**Definition**

**9.**

**V**(U) the set of all valuation fuzzy soft sets over U.

**Definition**

**10.**

**Definition**

**11.**

**Definition**

**12.**

**Definition**

**13.**

**Example**

**2.**

**Definition**

**14.**

## 4. Data Filling in Partial Valuation Fuzzy Soft Sets

**Remark**

**1.**

- Let us input our PVFSS, namely, $(F,A,{V}^{*})$.
- We use the rating procedure in order to associate a unique number with each alternative. In this way, we obtain a VFSS $(F,A,W)$ associated with the same FSS $(F,A)$ as the original PVFSS.
- Now, as long as there are two values in ${V}^{*}$ that belong to $\mathbb{R}$ (i.e., two valuations that are not missing in the input data), we calculate a regression equation to fill the missing valuation data.In order to run the regression, the independent variables (or abscissas) are the values ${W}_{i}$ given by the rating procedure that has been singled out, and the dependent variables (their respective ordinates) are the corresponding ${V}_{i}\in \mathbb{R}$ valuations.
- Once the regression function has been calculated, we can estimate the real values of the missing valuations ${V}_{i}=*\phantom{\rule{4pt}{0ex}}$ by its evaluations in the corresponding ${W}_{i}$ values.

## 5. Valuation of Goods: An Example

## 6. A Real Case Study

- The maximum surface in our sample of seven apartments is 114.44 square meters. We have divided the surface of each apartment by this maximum figure.
- We have divided the number of bathrooms of each apartment by two, the maximum number of bathrooms per apartment in our sample.
- In order to rank the attribute “quality”, we have considered four levels of quality: bad, normal, good, and luxury. We assign the values $0,1/3,2/3$ and 1 to each level, respectively.
- For the attribute “number of bedrooms”, we have divided the actual number of bedrooms by the maximum number of bedrooms, which, in our sample, is four.

#### 6.1. Evaluation of the Apartment

#### 6.2. Sensitivity Analysis

- When the first apartment is suppressed from the analysis, the remaining data produce a new comparison table and scores. With such data, we obtain $\mathcal{V}=(5.8693,-2.5846,3.3923,1.6423,4.1193,4.2807)$. The regression line equation for the observations$$\left(\right(5.8693,157),(-2.5846,115),(3.3923,132),(1.6423,132),(4.1193,157\left)\right)$$
- When the second apartment is suppressed from the analysis, the remaining data produce $\mathcal{V}=(-2.1462,-3.6462,1.4769,-0.0231,2.100,3.0114)$. The regression line equation for the observations$$\left(\right(-2.1462,95),(-3.6462,115),(1.4769,132),(-0.0231,132),(2.100,157\left)\right)$$
- When the sixth apartment is suppressed from the analysis, the remaining data produce $\mathcal{V}=(-2.3962,3.6000,-3.8962,1.2269,-0.2731,2.7614)$. The regression line equation for the observations$$\left(\right(-2.3962,95),(3.6000,157),(-3.8962,115),(1.2269,132),(-0.2731,132\left)\right)$$

#### 6.3. A Description of Existing Methodologies

- Quantitative and continuous variables, such as the surface of a house.
- Quantitative and discrete variables, such as:
- Number of complete bathrooms.
- Number of incomplete bathrooms.
- Age of the building.
- Number of rooms.
- Level in which is the apartment situated in the building (floor).
- Number of outward-facing rooms, etc.

- Qualitative variables, such as quality of the construction.
- Dummy variables, such as:
- The building has a garage.
- The house has a balcony.
- Repairing and renovation works were made in the house, etc.

- ${X}_{1}$: “Surface”.
- ${X}_{2}$: “Number of bathrooms”.
- ${X}_{3}$: “Quality”.
- ${X}_{4}$: “Number of bedrooms”.
- Etcetera.

#### 6.4. Evaluation with Alternative Procedures and Discussion

- The possible existence of coefficients with the wrong sign (in our case, the coefficient of variable ${x}_{4}$).
- The possibility that a coefficient vanishes (in this section, the coefficient of variable ${x}_{3}$ is zero). In such case, the characteristic associated with the corresponding variable is of no use for evaluation purposes.
- The coefficient of determination may be small (although, in the example in this section, ${R}^{2}$ is pretty high).

## 7. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

MDPI | Multidisciplinary Digital Publishing Institute |

DOAJ | Directory of Open Access Journals |

FS | Fuzzy set |

PVFSS | Partial valuation fuzzy soft set |

VFSS | Valuation fuzzy soft set |

VIKOR | VIsekriterijumska optimizacija i KOmpromisno Resenje |

## References

- Zadeh, L. Fuzzy sets. Inf. Control
**1965**, 8, 338–353. [Google Scholar] [CrossRef] - Molodtsov, D. Soft set theory–first results. Comput. Math. Appl.
**1999**, 37, 19–31. [Google Scholar] [CrossRef] - Maji, P.K.; Biswas, R.; Roy, A.R. An application of soft sets in a decision making problem. Comput. Math. Appl.
**2002**, 44, 1077–1083. [Google Scholar] [CrossRef] - Maji, P.K.; Biswas, R.; Roy, A.R. Soft set theory. Comput. Math. Appl.
**2003**, 45, 555–562. [Google Scholar] [CrossRef] - Alcantud, J.C.R. Some formal relationships among soft sets, fuzzy sets, and their extensions. Int. J. Approx. Reason.
**2016**, 68, 45–53. [Google Scholar] [CrossRef] - Ali, M.I. A note on soft sets, rough soft sets and fuzzy soft sets. Appl. Soft Comput.
**2011**, 11, 3329–3332. [Google Scholar] [CrossRef] - Ali, M.I.; Shabir, M. Logic connectives for soft sets and fuzzy soft sets. IEEE Trans. Fuzzy Syst.
**2014**, 22, 1431–1442. [Google Scholar] [CrossRef] - Maji, P.K.; Biswas, R.; Roy, A.R. Fuzzy soft sets. J. Fuzzy Math.
**2001**, 9, 589–602. [Google Scholar] - Alcantud, J.C.R.; de Andrés Calle, R.; Torrecillas, M.J.M. Hesitant fuzzy worth: An innovative ranking methodology for hesitant fuzzy subsets. Appl. Soft Comput.
**2016**, 38, 232–243. [Google Scholar] [CrossRef] - Alcantud, J.C.R.; de Andrés Calle, R. A segment-based approach to the analysis of project evaluation problems by hesitant fuzzy sets. Int. J. Comput. Intell. Syst.
**2016**, 29, 325–339. [Google Scholar] [CrossRef] - Faizi, S.; Rashid, T.; Sałabun, W.; Zafar, S.; Wątróbski, J. Decision making with uncertainty using hesitant fuzzy sets. Int. J. Fuzzy Syst.
**2017**. [Google Scholar] [CrossRef] - Faizi, S.; Sałabun, W.; Rashid, T.; Wątróbski, J.; Zafar, S. Group decision-making for hesitant fuzzy sets based on characteristic objects method. Symmetry
**2017**, 9, 136. [Google Scholar] [CrossRef] - Zhang, X.; Xu, Z. Consensus model-based hesitant fuzzy multiple criteria group decision analysis. In Hesitant Fuzzy Methods for Multiple Criteria Decision Analysis, Studies in Fuzziness and Soft Computing; Zhang, X., Xu, Z., Eds.; Springer: Berlin, Germany, 2017; Volume 345, pp. 143–157. [Google Scholar]
- Zhan, J.; Zhu, K. Reviews on decision making methods based on (fuzzy) soft sets and rough soft sets. J. Intell. Fuzzy Syst.
**2015**, 29, 1169–1176. [Google Scholar] [CrossRef] - Alcantud, J.C.R. A novel algorithm for fuzzy soft set based decision making from multiobserver input parameter data set. Inf. Fusion
**2016**, 29, 142–148. [Google Scholar] [CrossRef] - Ma, X.; Liu, Q.; Zhan, J. A survey of decision making methods based on certain hybrid soft set models. Artif. Intell. Rev.
**2017**, 47, 507–530. [Google Scholar] [CrossRef] - Chen, J.; Ye, J. Some single-valued neutrosophic Dombi weighted aggregation operators for multiple attribute decision-making. Symmetry
**2017**, 9, 82. [Google Scholar] [CrossRef] - Ye, J. Multiple Attribute Decision-Making Method Using Correlation Coefficients of Normal Neutrosophic Sets. Symmetry
**2017**, 9, 80. [Google Scholar] [CrossRef] - Peng, X.; Dai, J.; Yuan, H. Interval-valued fuzzy soft decision making methods based on MABAC, similarity measure and EDAS. Fundam. Inform.
**2017**, 152, 373–396. [Google Scholar] [CrossRef] - Peng, X.; Yang, Y. Algorithms for interval-valued fuzzy soft sets in stochastic multi-criteria decision making based on regret theory and prospect theory with combined weight. Appl. Soft Comput.
**2017**, 54, 415–430. [Google Scholar] [CrossRef] - Fatimah, F.; Rosadi, D.; Hakim, R.B.F.; Alcantud, J.C.R. Probabilistic soft sets and dual probabilistic soft sets in decision-making. Neural Comput. Appl.
**2017**. [Google Scholar] [CrossRef] - Chang, T.-H. Fuzzy VIKOR method: A case study of the hospital service evaluation in Taiwan. Inf. Sci.
**2014**, 271, 196–212. [Google Scholar] [CrossRef] - Espinilla, M.; Medina, J.; García-Fernández, Á.L.; Campaña, S.; Londoño, J. Fuzzy intelligent system for patients with preeclampsia in wearable devices. Mob. Inf. Syst.
**2017**, 2017, 7838464. [Google Scholar] [CrossRef] - Alcantud, J.C.R.; Santos-García, G.; Galilea, E.H. Glaucoma diagnosis: A soft set based decision making procedure. In Advances in Artificial Intelligence; Puerta, J.M., Ed.; Springer: Cham, Switzerland, 2015; Volume 9422, pp. 49–60. [Google Scholar]
- Xu, Z.S.; Xia, M.M. Distance and similarity measures for hesitant fuzzy sets. Inf. Sci.
**2011**, 181, 2128–2138. [Google Scholar] [CrossRef] - Taş, N.; Yilmaz Özgür, N.; Demir, P. An application of soft set and fuzzy soft set theories to stock management. J. Nat. Appl. Sci.
**2017**. [Google Scholar] [CrossRef] - Xu, W.; Xiao, Z.; Dang, X.; Yang, D.; Yang, X. Financial ratio selection for business failure prediction using soft set theory. Knowl.-Based Syst.
**2014**, 63, 59–67. [Google Scholar] [CrossRef] - Kalaichelvi, A.; Haritha Malini, P. Application of fuzzy soft sets to investment decision making problem. Int. J. Math. Sci. Appl.
**2011**, 1, 1583–1586. [Google Scholar] - Özgür, Y.; Taş, N. A note on “Application of fuzzy soft sets to investment decision making problem”. J. New Theory
**2015**, 1, 1–10. [Google Scholar] - Pagourtzi, E.; Assimakopoulos, V.; Hatzichristos, T.; French, N. Real estate appraisal: A review of valuations methods. J. Prop. Invest. Finance
**2003**, 21, 383–401. [Google Scholar] [CrossRef] - Ministerio de Economía (Spain), Orden ECO/805/2003, de 27 De Marzo, Sobre Normas De Valoración De Bienes Inmuebles Y De Determinados Derechos Para Ciertas Finalidades Financieras. Available online: https://www.boe.es/buscar/doc.php?id=BOE-A-2003-7253 (accessed on 4 October 2016).
- González, M.A.S.; Formoso, C.T. Mass appraisal with genetic fuzzy rule-based systems. Prop. Manag.
**2006**, 24, 20–30. [Google Scholar] [CrossRef] - Zurada, J.; Levitan, A.; Guan, J. A comparison of regression and artificial intelligence methods in a mass appraisal context. J. Real Estate Res.
**2011**, 33, 349–387. [Google Scholar] [CrossRef] - Zhang, R.; Du, Q.; Geng, J.; Liu, B.; Huang, Y. An improved spatial error model for the mass appraisal of commercial real estate based on spatial analysis: Shenzhen as a case study. Habitat Int.
**2015**, 46, 196–205. [Google Scholar] [CrossRef] - Feng, F.; Li, Y. Soft subsets and soft product operations. Inf. Sci.
**2013**, 232, 44–57. [Google Scholar] [CrossRef] - Qin, H.; Ma, X.; Herawan, T.; Zain, J. Data filling approach of soft sets under incomplete information. In Intelligent Information and Database Systems; Nguyen, N., Kim, C.-G., Janiak, A., Eds.; Springer: Berlin/Heidelberg, Germany, 2011; pp. 302–311. [Google Scholar]
- Qin, H.; Ma, X.; Zain, J.M.; Herawan, T. A novel soft set approach in selecting clustering attribute. Knowl.-Based Syst.
**2012**, 36, 139–145. [Google Scholar] [CrossRef] - Yao, Y.Y. Relational interpretations of neighbourhood operators and rough set approximation operators. Inf. Sci.
**1998**, 111, 239–259. [Google Scholar] [CrossRef] - Roy, A.R.; Maji, P.K. A fuzzy soft set theoretic approach to decision making problems. J. Comput. Appl. Math.
**2007**, 203, 412–418. [Google Scholar] [CrossRef] - Feng, F.; Jun, Y.; Liu, X.; Li, L. An adjustable approach to fuzzy soft set based decision making. J. Comput. Appl. Math.
**2010**, 234, 10–20. [Google Scholar] [CrossRef] - Alcantud, J.C.R. Fuzzy soft set based decision making: A novel alternative approach. In Proceedings of the 2015 Conference of the International Fuzzy Systems Association and the European Society for Fuzzy Logic and Technology, Gijón, Spain, 30 June–3 July 2015; Atlantics Press, 2015; pp. 106–111. [Google Scholar] [CrossRef]
- Ali, M.I.; Feng, F.; Liu, X.; Min, W.K.; Shabir, M. On some new operations in soft set theory. Comput. Math. Appl.
**2009**, 57, 1547–1553. [Google Scholar] [CrossRef] - Han, B.-H.; Li, Y.; Liu, J.; Geng, S.; Li, H. Elicitation criterions for restricted intersection of two incomplete soft sets. Knowl.-Based Syst.
**2014**, 59, 121–131. [Google Scholar] [CrossRef] - Alcantud, J.C.R.; Santos-García, G. Incomplete soft sets: New solutions for decision making problems. In Advances in Intelligent Systems and Computing; Bucciarelli, E., Ed.; Springer: Cham, Switzerland, 2016; Volume 475, pp. 9–17. [Google Scholar]
- Alcantud, J.C.R.; Santos-García, G. A new criterion for soft set based decision making problems under incomplete information. Int. J. Comput. Intell. Syst.
**2017**, 10, 394–404. [Google Scholar] [CrossRef] - Zou, Y.; Xiao, Z. Data analysis approaches of soft sets under incomplete information. Knowl.-Based Syst.
**2008**, 21, 941–945. [Google Scholar] [CrossRef] - Aiken, L.S.; West, S.G. Testing and Interpreting Interactions; SAGE Publications: Thousand Oaks, CA, USA, 1991. [Google Scholar]
- Bagnoli, C.; Smith, H.C. The theory of fuzzy logic and its application to real estate valuation. J. Real Estate Res.
**1998**, 16, 169–200. [Google Scholar] - Saltelli, A. Sensitivity analysis for importance assessment. Risk Anal.
**2002**, 22, 1–12. [Google Scholar] [CrossRef] - Aznar, J.; Guijarro, F. Housing valuation in Spain. Homogenization method and alternative methodologies. Finance Markets Valuat.
**2016**, 2, 91–125. [Google Scholar] - González-Nebreda, P.; Turmo-de-Padura, J.; Villaronga-Sánchez, E. La Valoración Inmobiliaria. Teoría y Práctica; Wolters Kluwer España, S.A.: Madrid, Spain, 2006. [Google Scholar]
- Alcantud, J.C.R.; Torra, V. Decomposition theorems and extension principles for hesitant fuzzy sets. Inf. Fusion
**2018**, 41, 48–56. [Google Scholar] [CrossRef] - Torra, V. Hesitant fuzzy sets. Int. J. Intell. Syst.
**2010**, 25, 529–539. [Google Scholar] [CrossRef] - Kobza, V.; Janiš, V.; Montes, S. Divergence measures on hesitant fuzzy sets. J. Intell. Fuzzy Syst.
**2017**, 33, 1589–1601. [Google Scholar] [CrossRef] - Torra, V.; Narukawa, Y. On hesitant fuzzy sets and decision. In Proceedings of the 2009 IEEE International Conference on Fuzzy Systems, Jeju Island, Korea, 20–24 August 2009; pp. 1378–1382. [Google Scholar]
- Xia, M.; Xu, Z. Hesitant fuzzy information aggregation in decision making. Int. J. Approx. Reason.
**2011**, 52, 395–407. [Google Scholar] [CrossRef] - Farhadinia, B. A series of score functions for hesitant fuzzy sets. Inf. Sci.
**2014**, 277, 102–110. [Google Scholar] [CrossRef] - Farhadinia, B. A novel method of ranking hesitant fuzzy values for multiple attribute decision-making problems. Int. J. Intell. Syst.
**2013**, 28, 752–767. [Google Scholar] [CrossRef] - Ali, M.I.; Mahmood, T.; Rehman, M.M.U.; Aslam, M.F. On lattice ordered soft sets. Appl. Soft Comput.
**2015**, 36, 499–505. [Google Scholar] [CrossRef] - Fatimah, F.; Rosadi, D.; Hakim, R.B.F.; Alcantud, J.C.R. N-soft sets and their decision making algorithms. Soft Comput.
**2017**. [Google Scholar] [CrossRef] - Zhan, J.; Ali, M.I.; Mehmood, N. On a novel uncertain soft set model: Z-soft fuzzy rough set model and corresponding decision making methods. Appl. Soft Comput.
**2017**, 56, 446–457. [Google Scholar] [CrossRef]

**Figure 1.**Flow diagram with the solution to the data filling problem in Section 4.

**Figure 2.**The regression line in Section 5. The black square shows the valuation of the missing option ${o}_{4}$, with score $-2.24$ at the horizontal axis.

**Figure 3.**The regression line in the case study. The black square shows the valuation of the missing option ${h}_{7}$, with score $2.0114$ at the horizontal axis.

${\mathit{e}}_{1}$ | ${\mathit{e}}_{2}$ | ${\mathit{e}}_{3}$ | ${\mathit{e}}_{4}$ | |
---|---|---|---|---|

${h}_{1}$ | 1 | 0 | 0 | 1 |

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

${h}_{3}$ | 0 | 1 | 0 | 0 |

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

${o}_{1}$ | 0.9 | 0.1 | 0.2 | 0.1 | 0.3 |

${o}_{2}$ | 0.19 | 0.3 | 0.4 | 0.3 | 0.4 |

${\mathit{e}}_{1}$ | ${\mathit{e}}_{2}$ | ${\mathit{e}}_{3}$ | ${\mathit{e}}_{4}$ | ${\mathit{e}}_{5}$ | ${\Pi}_{\mathit{c}}^{\mathit{i}}={\mathit{c}}_{\mathit{i}}$ | ${\Pi}_{\mathit{r}}^{\mathit{i}}={\mathit{s}}_{\mathit{i}}$ | ${\Pi}_{\mathit{a}}^{\mathit{i}}={\mathit{S}}_{\mathit{i}}$ | |
---|---|---|---|---|---|---|---|---|

${o}_{1}$ | 0.9 | 0.1 | 0.2 | 0.1 | 0.3 | 1.60 | −3 | −1.29 |

${o}_{2}$ | 0.19 | 0.3 | 0.4 | 0.3 | 0.4 | 1.59 | 3 | 1.29 |

**Table 4.**Tabular representation of the partial valuation fuzzy soft set $(F,A,{V}^{*})$ in Section 5. All ${V}_{i}$s are expressed in thousands of euros.

${\mathit{p}}_{1}$ | ${\mathit{p}}_{2}$ | ${\mathit{p}}_{3}$ | ${\mathit{p}}_{4}$ | ${\mathit{p}}_{5}$ | ${\mathit{p}}_{6}$ | ${\mathit{p}}_{7}$ | ${\mathit{V}}_{\mathit{i}}$ | |
---|---|---|---|---|---|---|---|---|

${o}_{1}$ | $0.036$ | $0.015$ | $0.064$ | $0.216$ | $0.048$ | $0.054$ | $0.405$ | 137 |

${o}_{2}$ | $0.144$ | $0.084$ | $0.360$ | $0.045$ | $0.036$ | $0.020$ | $0.175$ | 109 |

${o}_{3}$ | $0.120$ | $0.084$ | $0.180$ | $0.030$ | $0.096$ | $0.021$ | $0.294$ | 97 |

${o}_{4}$ | $0.504$ | $0.192$ | $0.108$ | $0.006$ | $0.048$ | $0.048$ | $0.096$ | * |

${o}_{5}$ | $0.084$ | $0.245$ | $0.036$ | $0.096$ | $0.270$ | $0.200$ | $0.140$ | 192 |

${o}_{6}$ | $0.216$ | $0.315$ | $0.042$ | $0.108$ | $0.224$ | $0.126$ | $0.135$ | 198 |

**Table 5.**Attributes of the comparable six apartments. Source: Real data from the Spanish real estate market (Almería, Spain, 2016).

Item | Surface (sq. m.) | No. of Bathrooms | Quality | No. of Bedrooms |
---|---|---|---|---|

${h}_{1}$ | 75 | 1 | Normal | 3 |

${h}_{2}$ | 105 | 2 | Normal | 4 |

${h}_{3}$ | 75 | 1 | Normal | 2 |

${h}_{4}$ | 90 | 2 | Normal | 3 |

${h}_{5}$ | 90 | $1.5$ | Normal | 3 |

${h}_{6}$ | 105 | 2 | Normal | 3 |

**Table 6.**The PVFSS in the real case study in Section 6. Sale prices are given in thousands of euros.

Item | Surface (sq. m.) | No. of Bathrooms | Quality | No. of Bedrooms | Price |
---|---|---|---|---|---|

${h}_{1}$ | $0.52$ | $0.5$ | $1/3$ | $0.75$ | 95 |

${h}_{2}$ | $0.73$ | 1 | $1/3$ | 1 | 157 |

${h}_{3}$ | $0.52$ | $0.5$ | $1/3$ | $0.5$ | 115 |

${h}_{4}$ | $0.62$ | 1 | $1/3$ | $0.75$ | 132 |

${h}_{5}$ | $0.62$ | $0.75$ | $1/3$ | $0.75$ | 132 |

${h}_{6}$ | $0.73$ | 1 | $1/3$ | $0.75$ | 157 |

${h}_{7}$ | 1 | $0.5$ | $2/3$ | $0.5$ | * |

**Table 7.**Comparison table and scores associated with Table 6. The values have been rounded off.

${\mathit{h}}_{1}$ | ${\mathit{h}}_{2}$ | ${\mathit{h}}_{3}$ | ${\mathit{h}}_{4}$ | ${\mathit{h}}_{5}$ | ${\mathit{h}}_{6}$ | ${\mathit{h}}_{7}$ | ${\mathit{S}}_{\mathit{i}}$ | |
---|---|---|---|---|---|---|---|---|

${h}_{1}$ | 0 | 0 | $0.2500$ | 0 | 0 | 0 | $0.2500$ | $-3.1038$ |

${h}_{2}$ | $0.9577$ | 0 | $1.2077$ | $0.3538$ | $0.6038$ | $0.2500$ | 1 | $3.6000$ |

${h}_{3}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | $-4.8538$ |

${h}_{4}$ | $0.6038$ | 0 | $0.8538$ | 0 | $0.2500$ | 0 | $0.7500$ | $1.1231$ |

${h}_{5}$ | $0.3538$ | 0 | $0.6038$ | 0 | 0 | 0 | $0.5000$ | $-0.6269$ |

${h}_{6}$ | $0.7077$ | 0 | $0.9577$ | $0.1038$ | $0.3538$ | 0 | $0.7500$ | $1.8500$ |

${h}_{7}$ | $0.9808$ | $0.7731$ | $0.9808$ | $0.8769$ | $0.8769$ | $0.7731$ | 0 | $2.0114$ |

Quantitative | Qualitative | “Dummy” | |
---|---|---|---|

Discrete | Continuous | ||

${x}_{k1}$ | $[0,{x}_{k1}]$ | ${q}_{k1}$ | 0 |

${x}_{k2}$ | $[{x}_{k1},{x}_{k2}]$ | ${q}_{k2}$ | 1 |

${x}_{k3}$ | $[{x}_{k2},{x}_{k3}]$ | ${q}_{k3}$ | 2 |

⋮ | ⋮ | ⋮ | ⋮ |

${x}_{k{m}_{k}}$ | $[{x}_{k{m}_{k-1}},{x}_{k{m}_{k}}]$ | ${q}_{k{m}_{k}}$ | ${m}_{k}$ |

Surface | Bathrooms | Quality | Bedrooms | ||||
---|---|---|---|---|---|---|---|

Interval | Weight | Number | Weight | Level | Weight | Number | Weight |

$[0,10]$ | 0.00 | 0 | 0.00 | Bad | 0.04 | 1 | 0.03 |

$[10,20]$ | 0.06 | 1 | 0.04 | Low | 0.08 | 2 | 0.06 |

$[20,30]$ | 0.08 | 2 | 0.08 | Normal | 0.12 | 3 | 0.09 |

$[30,40]$ | 0.10 | 3 | 0.12 | Good | 0.16 | 4 | 0.12 |

$[40,50]$ | 0.12 | 4 | 0.16 | Luxury | 0.20 | 5 | 0.14 |

$[50,60]$ | 0.14 | 5 | 0.20 | − | − | 6 | 0.16 |

$[60,70]$ | 0.16 | − | − | − | − | 7 | 0.17 |

$[70,80]$ | 0.18 | − | − | − | − | 8 | 0.18 |

$[80,90]$ | 0.20 | − | − | − | − | 9 | 0.19 |

$[90,100]$ | 0.22 | − | − | − | − | 10 | 0.20 |

$[100,110]$ | 0.24 | − | − | − | − | − | − |

$[110,120]$ | 0.26 | − | − | − | − | − | − |

$[120,130]$ | 0.28 | − | − | − | − | − | − |

$[130,140]$ | 0.30 | − | − | − | − | − | − |

Item | Price | Surface | Bathrooms | Quality | Bedrooms | ||||
---|---|---|---|---|---|---|---|---|---|

Value | Weight | Number | Weight | Level | Weight | Number | Weight | ||

${h}_{1}$ | 95,000 | 75 | 0.18 | 1 | 0.04 | Normal | 0.12 | 3 | 0.09 |

${h}_{2}$ | 157,000 | 105 | 0.24 | 2 | 0.04 | Normal | 0.12 | 4 | 0.12 |

${h}_{3}$ | 115,000 | 75 | 0.18 | 1 | 0.04 | Normal | 0.12 | 2 | 0.06 |

${h}_{4}$ | 132,000 | 90 | 0.20 | 2 | 0.08 | Normal | 0.12 | 3 | 0.09 |

${h}_{5}$ | 132,000 | 90 | 0.20 | 1.5 | 0.06 | Normal | 0.12 | 3 | 0.09 |

${h}_{6}$ | 157,000 | 105 | 0.24 | 2 | 0.08 | Normal | 0.12 | 3 | 0.09 |

${h}_{7}$ | * | 114.44 | 0.26 | 1 | 0.04 | Good | 0.16 | 2 | 0.06 |

© 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**

Alcantud, J.C.R.; Rambaud, S.C.; Torrecillas, M.J.M.
Valuation Fuzzy Soft Sets: A Flexible Fuzzy Soft Set Based Decision Making Procedure for the Valuation of Assets. *Symmetry* **2017**, *9*, 253.
https://doi.org/10.3390/sym9110253

**AMA Style**

Alcantud JCR, Rambaud SC, Torrecillas MJM.
Valuation Fuzzy Soft Sets: A Flexible Fuzzy Soft Set Based Decision Making Procedure for the Valuation of Assets. *Symmetry*. 2017; 9(11):253.
https://doi.org/10.3390/sym9110253

**Chicago/Turabian Style**

Alcantud, José Carlos R., Salvador Cruz Rambaud, and María J. Muñoz Torrecillas.
2017. "Valuation Fuzzy Soft Sets: A Flexible Fuzzy Soft Set Based Decision Making Procedure for the Valuation of Assets" *Symmetry* 9, no. 11: 253.
https://doi.org/10.3390/sym9110253