Valuation Fuzzy Soft Sets: A Flexible Fuzzy Soft Set Based Decision Making Procedure for the Valuation of Assets
Abstract
:1. Introduction
2. Notation and Definitions
2.1. Soft Sets and Fuzzy Soft Sets
- 1.
- , .
- 2.
- , .
- 3.
- , .
2.2. Basic Operations
3. Some Novel Concepts Related to Valuation Fuzzy Soft Sets
Valuation and Partial Valuation Fuzzy Soft Sets
4. Data Filling in Partial Valuation Fuzzy Soft Sets
- Let us input our PVFSS, namely, .
- We use the rating procedure in order to associate a unique number with each alternative. In this way, we obtain a VFSS associated with the same FSS as the original PVFSS.
- Now, as long as there are two values in that belong to (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 given by the rating procedure that has been singled out, and the dependent variables (their respective ordinates) are the corresponding valuations.
- Once the regression function has been calculated, we can estimate the real values of the missing valuations by its evaluations in the corresponding 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 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 . The regression line equation for the observations
- When the second apartment is suppressed from the analysis, the remaining data produce . The regression line equation for the observations
- When the sixth apartment is suppressed from the analysis, the remaining data produce . The regression line equation for the observations
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.
- : “Surface”.
- : “Number of bathrooms”.
- : “Quality”.
- : “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 ).
- The possibility that a coefficient vanishes (in this section, the coefficient of variable 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, 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]
1 | 0 | 0 | 1 | |
1 | 0 | 1 | 0 | |
0 | 1 | 0 | 0 |
0.9 | 0.1 | 0.2 | 0.1 | 0.3 | |
0.19 | 0.3 | 0.4 | 0.3 | 0.4 |
0.9 | 0.1 | 0.2 | 0.1 | 0.3 | 1.60 | −3 | −1.29 | |
0.19 | 0.3 | 0.4 | 0.3 | 0.4 | 1.59 | 3 | 1.29 |
137 | ||||||||
109 | ||||||||
97 | ||||||||
* | ||||||||
192 | ||||||||
198 |
Item | Surface (sq. m.) | No. of Bathrooms | Quality | No. of Bedrooms |
---|---|---|---|---|
75 | 1 | Normal | 3 | |
105 | 2 | Normal | 4 | |
75 | 1 | Normal | 2 | |
90 | 2 | Normal | 3 | |
90 | Normal | 3 | ||
105 | 2 | Normal | 3 |
Item | Surface (sq. m.) | No. of Bathrooms | Quality | No. of Bedrooms | Price |
---|---|---|---|---|---|
95 | |||||
1 | 1 | 157 | |||
115 | |||||
1 | 132 | ||||
132 | |||||
1 | 157 | ||||
1 | * |
0 | 0 | 0 | 0 | 0 | ||||
0 | 1 | |||||||
0 | 0 | 0 | 0 | 0 | 0 | 0 | ||
0 | 0 | 0 | ||||||
0 | 0 | 0 | 0 | |||||
0 | 0 | |||||||
0 |
Quantitative | Qualitative | “Dummy” | |
---|---|---|---|
Discrete | Continuous | ||
0 | |||
1 | |||
2 | |||
⋮ | ⋮ | ⋮ | ⋮ |
Surface | Bathrooms | Quality | Bedrooms | ||||
---|---|---|---|---|---|---|---|
Interval | Weight | Number | Weight | Level | Weight | Number | Weight |
0.00 | 0 | 0.00 | Bad | 0.04 | 1 | 0.03 | |
0.06 | 1 | 0.04 | Low | 0.08 | 2 | 0.06 | |
0.08 | 2 | 0.08 | Normal | 0.12 | 3 | 0.09 | |
0.10 | 3 | 0.12 | Good | 0.16 | 4 | 0.12 | |
0.12 | 4 | 0.16 | Luxury | 0.20 | 5 | 0.14 | |
0.14 | 5 | 0.20 | − | − | 6 | 0.16 | |
0.16 | − | − | − | − | 7 | 0.17 | |
0.18 | − | − | − | − | 8 | 0.18 | |
0.20 | − | − | − | − | 9 | 0.19 | |
0.22 | − | − | − | − | 10 | 0.20 | |
0.24 | − | − | − | − | − | − | |
0.26 | − | − | − | − | − | − | |
0.28 | − | − | − | − | − | − | |
0.30 | − | − | − | − | − | − |
Item | Price | Surface | Bathrooms | Quality | Bedrooms | ||||
---|---|---|---|---|---|---|---|---|---|
Value | Weight | Number | Weight | Level | Weight | Number | Weight | ||
95,000 | 75 | 0.18 | 1 | 0.04 | Normal | 0.12 | 3 | 0.09 | |
157,000 | 105 | 0.24 | 2 | 0.04 | Normal | 0.12 | 4 | 0.12 | |
115,000 | 75 | 0.18 | 1 | 0.04 | Normal | 0.12 | 2 | 0.06 | |
132,000 | 90 | 0.20 | 2 | 0.08 | Normal | 0.12 | 3 | 0.09 | |
132,000 | 90 | 0.20 | 1.5 | 0.06 | Normal | 0.12 | 3 | 0.09 | |
157,000 | 105 | 0.24 | 2 | 0.08 | Normal | 0.12 | 3 | 0.09 | |
* | 114.44 | 0.26 | 1 | 0.04 | Good | 0.16 | 2 | 0.06 |
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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
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 StyleAlcantud, 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