[EN]This paper introduces a model where the options are characterized by one fuzzy soft set in each of an indefinite number of periods. This model extends the standard case of fuzzy soft sets. We explain how to associate a characteristic fuzzy soft set with each model. Finally, a decision making procedure for the selection of alternatives is proposed.
The target applications include portfolio selection in finance, environmental issues, et cetera

Muñoz Torrecillas, María

Rodríguez Alcantud, José Carlos

English

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FUZZY SOFT SETS: A MODEL FOR MAKING CHOICES IN AN INTERTEMPORAL FRAMEWORK José Carlos Rodríguez Alcantud Universidad de Salamanca (Spain) jcr@usal.es María José Muñoz Torrrecillas Universidad de Almería (Spain) mjmtorre@ual.es XXth Edition of the Mathematical Modelling Conference Series at the Institute for Multidisciplinary Mathematics Universitat Politècnica de València Valencia, Spain, 16-18 July 2018  Mathematical Modelling In Engineering & Human Behaviour 2018 2 FUZZY SOFT SETS: A MODEL FOR MAKING CHOICES IN AN INTERTEMPORAL FRAMEWORK Abstract This paper introduces a model where the options are characterized by one fuzzy soft set in each of an indefinite number of periods. This model extends the standard case of fuzzy soft sets. We explain how to associate a characteristic fuzzy soft set with each model. Finally, a decision making procedure for the selection of alternatives is proposed. The target applications include portfolio selection in finance, environmental issues, et cetera. Key-words: Fuzzy soft set, intergenerational choice, decision making.  Mathematical Modelling In Engineering & Human Behaviour 2018 3 FUZZY SOFT SETS: A MODEL FOR MAKING CHOICES IN AN INTERTEMPORAL FRAMEWORK Notation and definitions •  Let X denote a set. Then 𝒫(X) is the set of all non-empty subsets of X.  •  A fuzzy subset (also, FS) A of X is a function µA : X → [0, 1].   For each 𝑥 ∊ X, µA(𝑥) ∊ [0, 1] is the degree of membership of 𝑥 in that  subset.  •  The set of all fuzzy subsets of X will be denoted by FS(X). •  Universe of objects: U ; universal set of parameters: E. Definition 1. (Molodtsov, 1999). Let A be a subset of E. The pair (F, A) is a soft set over U  if F : A → 𝒫(U ).  Mathematical Modelling In Engineering & Human Behaviour 2018 4 FUZZY SOFT SETS: A MODEL FOR MAKING CHOICES IN AN INTERTEMPORAL FRAMEWORK Notation and definitions • Soft set based decision making was pioneered by Maji, Biswas and Roy (2002), and further applications of soft sets in decision making were given in Alcantud, Cruz and Muñoz (2017), Alcantud and Santos-García (2015, 2017), and Roy and Maji (2007). Definition 2. (Maji, Biswas and Roy, 2002). The pair (F, A) is a fuzzy soft set (FSS) over U when A  ⊆ E  and F : A → FS(U). The set of all fuzzy soft sets over U will be denoted as 𝓕𝓢(U). ! Example: Films parameterized by attributes.  Mathematical Modelling In Engineering & Human Behaviour 2018 5 FUZZY SOFT SETS: A MODEL FOR MAKING CHOICES IN AN INTERTEMPORAL FRAMEWORK Notation and definitions • In real practice both U and A use to be finite. Then let k and n denote the respective number of elements of U and A. These soft sets can be represented either by k  x n  matrices or by a tabular form.  • Both practical representations are binary, that is to say, all cells are either 0 or 1.  • One can proceed in a similar way in fuzzy soft sets, but now the possible values in the cells lie in [0, 1].  Mathematical Modelling In Engineering & Human Behaviour 2018 6 FUZZY SOFT SETS: A MODEL FOR MAKING CHOICES IN AN INTERTEMPORAL FRAMEWORK Notation and definitions [12]: Roy and Maji (2007) [7]: Fen, Jun, Liu and Li (2010) [1]: Alcantud (2016) [8]: Liu, Qin and Pei (2017)  Mathematical Modelling In Engineering & Human Behaviour 2018 7 FUZZY SOFT SETS: A MODEL FOR MAKING CHOICES IN AN INTERTEMPORAL FRAMEWORK An intertemporal model for FSSs • Let 𝒮 denote the set of infinite sequences of the interval [0, 1] (also called infinite utility streams).  • Our intertemporal model of fuzzy soft sets over U is defined by   F : A → S (U ) where S (U ) represents the mappings U → 𝒮.  • For each attribute and each alternative, we express the degree of belongingness of such an alternative in each period of time. • In practical terms, where both U = {o1, … , om} and A = {e1, … , em} are finite, we can represent this information in a table where the cells are either finite or infinite sequences of membership degrees:  Mathematical Modelling In Engineering & Human Behaviour 2018 8 FUZZY SOFT SETS: A MODEL FOR MAKING CHOICES IN AN INTERTEMPORAL FRAMEWORK An intertemporal model for FSSs  Mathematical Modelling In Engineering & Human Behaviour 2018 9 FUZZY SOFT SETS: A MODEL FOR MAKING CHOICES IN AN INTERTEMPORAL FRAMEWORK Decision making in intertemporal FSSs. Algorithm for decision making • Inputs: An intertemporal table of fuzzy soft sets, a reduction mechanism, and a fuzzy soft set decision making procedure (e.g., from Table 1). • Procedure for prioritizing the alternatives in Table 2: 1.  Associate a FSS with the original intertemporal information by means of the selected reduction mechanism. 2.  Prioritize the alternatives in the reduced FSS by the selected decision making procedure (Alcantud, 2016). 3.  The result of the decision is any object ok that is at the top of the ranking in the previous step.  Mathematical Modelling In Engineering & Human Behaviour 2018 10 FUZZY SOFT SETS: A MODEL FOR MAKING CHOICES IN AN INTERTEMPORAL FRAMEWORK Decision making in intertemporal FSSs. Algorithm for decision making  Mathematical Modelling In Engineering & Human Behaviour 2018 11 FUZZY SOFT SETS: A MODEL FOR MAKING CHOICES IN AN INTERTEMPORAL FRAMEWORK Decision making in intertemporal FSSs. Example 1  Mathematical Modelling In Engineering & Human Behaviour 2018 12 FUZZY SOFT SETS: A MODEL FOR MAKING CHOICES IN AN INTERTEMPORAL FRAMEWORK Decision making in intertemporal FSSs. Example 1  Mathematical Modelling In Engineering & Human Behaviour 2018 13 FUZZY SOFT SETS: A MODEL FOR MAKING CHOICES IN AN INTERTEMPORAL FRAMEWORK Decision making in intertemporal FSSs. Intertemporal choices • “Most choices require decision-makers to trade-off costs and benefits at different points in time. Decisions with consequences in multiple time periods are referred to as intertemporal choices. Decisions about savings, work effort, education, nutrition, exercise, and health care are all intertemporal choices.” (Chavris et al. 2008) • Discounted utility models (Ramsey, 1928; Samuelson, 1937; Koopmans, 1960) consider decision-makers facing the selection of alternatives based on the weighted addition of utilities, being these weights discount factors based on temporal delays.   Mathematical Modelling In Engineering & Human Behaviour 2018 14 FUZZY SOFT SETS: A MODEL FOR MAKING CHOICES IN AN INTERTEMPORAL FRAMEWORK Decision making in intertemporal FSSs. Intertemporal choices • The discounted utility theory states that individuals discount future events at a constant rate, so the value of an experience during a period of time [0,T] is given by:  • The discounted utility model is used in project appraisals where the expected cash-flows of a project are discounted to obtain its Net Present Value (NPV) at instant 0. • The European Commission (2014) recommends the use of the exponential discounting model and a constant 5% European social discount rate for the Cohesion Fund eligible countries.   Mathematical Modelling In Engineering & Human Behaviour 2018 15 FUZZY SOFT SETS: A MODEL FOR MAKING CHOICES IN AN INTERTEMPORAL FRAMEWORK Decision making in intertemporal FSSs. Example 2 • Evaluation of two alternative portfolios of projects that a public administration may undertake.  • Objective: to choose the best alternative, in terms of utility.  • Projects with an unlimited time horizon that could be enjoyed by different generations.  • Each portfolio is composed by four elements, being:  !  different type of projects (e.g. city park, urban infrastructure, sports facilities, etc.) which are addressed to the population in general,  !  different projects which are addressed to different groups of population, depending on their special needs,  !  different/similar projects that will be carried out in different geographical areas.  Mathematical Modelling In Engineering & Human Behaviour 2018 16 FUZZY SOFT SETS: A MODEL FOR MAKING CHOICES IN AN INTERTEMPORAL FRAMEWORK Decision making in intertemporal FSSs. Example 2 Period	 Attribute 1	 Attribute 2	 Attribute 3	 Attribute 4	1	 0.05	 1.00	 0.70	 1.00	2	 0.10	 0.95	 0.70	 0.95	3	 0.15	 0.90	 0.70	 0.90	4	 0.20	 0.85	 0.70	 0.85	5	 0.25	 0.80	 0.70	 0.80	6	 0.30	 0.75	 0.70	 0.75	7	 0.35	 0.70	 0.70	 0.70	8	 0.40	 0.65	 0.70	 0.65	9	 0.45	 0.60	 0.70	 0.70	10	 0.50	 0.55	 0.70	 0.75	11	 0.55	 0.50	 0.70	 0.80	12	 0.60	 0.45	 0.70	 0.85	13	 0.65	 0.40	 0.70	 0.90	14	 0.70	 0.35	 0.70	 0.95	15	 0.75	 0.35	 0.70	 1.00	16	 0.80	 0.35	 0.70	 1.00	17	 0.85	 0.35	 0.70	 1.00	18	 0.90	 0.35	 0.70	 1.00	19	 0.95	 0.35	 0.70	 1.00	20	 1.00	 0.35	 0.70	 1.00	21	 1.00	 0.35	 0.70	 1.00	22	 1.00	 0.35	 0.70	 1.00	23 onwards	 1.00	 0.35	 0.70	 1.00	Ai	 13.09	 10.61	 14.00	 18.33	Ai/T	 0.65	 0.53	 0.70	 0.92	Table 3: Portfolio 1  Mathematical Modelling In Engineering & Human Behaviour 2018 17 FUZZY SOFT SETS: A MODEL FOR MAKING CHOICES IN AN INTERTEMPORAL FRAMEWORK Decision making in intertemporal FSSs. Example 2 Table 4: Portfolio 2 Period	 Attribute 1	 Attribute 2	 Attribute 3	 Attribute 4	1	 0.00	 1.00	 0.60	 1.00	2	 0.10	 0.94	 0.60	 0.90	3	 0.20	 0.88	 0.60	 0.80	4	 0.30	 0.82	 0.60	 0.70	5	 0.40	 0.76	 0.60	 0.60	6	 0.50	 0.70	 0.60	 0.50	7	 0.60	 0.64	 0.60	 0.40	8	 0.70	 0.58	 0.60	 0.30	9	 0.80	 0.52	 0.60	 0.40	10	 0.90	 0.52	 0.60	 0.50	11	 1.00	 0.52	 0.60	 0.60	12	 1.00	 0.52	 0.60	 0.70	13	 1.00	 0.52	 0.60	 0.80	14	 1.00	 0.52	 0.60	 0.90	15	 1.00	 0.52	 0.60	 1.00	16	 1.00	 0.52	 0.60	 1.00	17	 1.00	 0.52	 0.60	 1.00	18	 1.00	 0.52	 0.60	 1.00	19	 1.00	 0.52	 0.60	 1.00	20	 1.00	 0.52	 0.60	 1.00	21	 1.00	 0.52	 0.60	 1.00	22	 1.00	 0.52	 0.60	 1.00	23 onwards	 1.00	 0.52	 0.60	 1.00	Ai	 15.44	 12.24	 12.00	 16.65	Ai/T	 0.77	 0.61	 0.60	 0.83	 Mathematical Modelling In Engineering & Human Behaviour 2018 18 FUZZY SOFT SETS: A MODEL FOR MAKING CHOICES IN AN INTERTEMPORAL FRAMEWORK Decision making in intertemporal FSSs. Example 2 Table 5: Intertemporal fuzzy soft sets	 	 e1	 e2	 e3	 e4	o1	 (0.05. 0.10. 0.15. 0.20. 0.25. … )	 (1.00. 0.95. 0.90. 0.85. 0.80. … )	 (0.70. 0.70. 0.70. 0.70. 0.70. … )	 (1.00. 0.95. 0.90. 0.85. 0.80. … )	o2	 (0.00. 0.10. 0.20. 0.30. 0.40. … )	 (1.00. 0.94. 0.88. 0.82. 0.76. … )	 (0.60. 0.60. 0.60. 0.60. 0.60. … )	 (1.00. 0.90. 0.80. 0.70. 0.60. … )	Table 6: Tabular representations of the reduced FSS (by discounted utility) associated with Table 5	 	 Attribute 1	 Attribute 2	 Attribute 3	 Attribute 4	P1	 0.65	 0.53	 0.70	 0.92	P2	 0.77	 0.61	 0.60	 0.83	 Mathematical Modelling In Engineering & Human Behaviour 2018 19 FUZZY SOFT SETS: A MODEL FOR MAKING CHOICES IN AN INTERTEMPORAL FRAMEWORK Decision making in intertemporal FSSs. Example 2  Mathematical Modelling In Engineering & Human Behaviour 2018 20 FUZZY SOFT SETS: A MODEL FOR MAKING CHOICES IN AN INTERTEMPORAL FRAMEWORK References 1.  J. C. R. Alcantud, A novel algorithm for fuzzy soft set based decision making from multiobserver input parameter data set, Information Fusion 29 (2016) 142-148. 2.  J. C. R. Alcantud, G. Bosi, M. J. Campión, J. C. Candeal, E. Induráin, C. Rodríguez Palmero, Continuous utility functions through scales, Theory and Decision 64(4) (2008), 479-494. 3.  J. C. R. Alcantud, S. Cruz Rambaud, M. J. Muñoz Torrecillas, Valuation fuzzy soft sets: A fexible fuzzy soft set based decision making procedure for the valuation of assets, Symmetry 9 (2017), 253. 4.  J. C. R. Alcantud, G. Santos-García, A new criterion for soft set based decision making problems under incomplete information. International Journal of Computational Intelligence Systems 10 (2017) 394-404. 5.  J. C. R. Alcantud, G. Santos-García, E. H. Galilea, Glaucoma diagnosis: A soft set based decision making procedure, in: J. M. Puerta et al. (eds.), Advances in Artificial Intelligence, vol. 9422 of LNCS, Springer, 2015, pp. 49-60. 6.  C. Chabris, D. Laibson, J. Schuldt, Intertemporal Choice, Palgrave Dictionary of Economics (2008).  Mathematical Modelling In Engineering & Human Behaviour 2018 21 FUZZY SOFT SETS: A MODEL FOR MAKING CHOICES IN AN INTERTEMPORAL FRAMEWORK References 7  European Commission Guide to cost benefit analysis of investment projects: Economic appraisal tool for Cohesion Policy 2014-2020 (2014). Available at: http://ec.europa.eu/regional_policy/sources/docgener/studies/pdf/cba_guide.pdf 8  F. Feng, Y. Jun, X. Liu, L. Li, An adjustable approach to fuzzy soft set based decision making, Journal of Computational and Applied Mathematics 234 (2010) 10-20. 9  T.C. Koopmans, Stationary ordinal utility and impatience, Econometrica 28 (1960) 287-309. 10  Z. Liu, K. Qin, Z. Pei, A method for fuzzy soft sets in decision-making based on an ideal solution, Symmetry 9(10) (2017) 246. 11  P. Maji, R. Biswas, A. Roy, An application of soft sets in a decision making problem, Computers & Mathematics with Applications 44 (2002) 1077-1083. 12 D. Molodtsov, Soft set theory-first results, Computers & Mathematics with Applications 37 (1999) 19-31. 13  F. Ramsey, A mathematical theory of saving, The Economic Journal 38 (1928) 543-559.  Mathematical Modelling In Engineering & Human Behaviour 2018 22 FUZZY SOFT SETS: A MODEL FOR MAKING CHOICES IN AN INTERTEMPORAL FRAMEWORK References 14. A. Roy, P. Maji, A fuzzy soft set theoretic approach to decision making problems, Journal of Computational and Applied Mathematics 203 (2007) 412-418. 15. P. Samuelson, A note on measurement of utility, Review of Economic Studies 4 (1937)  155-161. 

Fuzzy soft sets: a model for making choices in an intertemporal framework

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Fuzzy soft sets: a model for making choices in an intertemporal framework

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