Modelo de Representación 2-tupla. Un enfoque computacional simbólico

1. Modelo de Representación 2-tupla.Un enfoque computacional simbólico EXTENSIONES Y APLICACIONES EN TOMA DE DECISION LINGÜÍSTICA Charlas Sinbad2

2. OUTLINE • INTRODUCTION • DECISION MAKING AND PREFERENCE MODELLING • FUZZY LINGUISTIC APPROACH AND CWW • LINGUISTIC 2-TUPLE MODEL • EXTENSIONS • MULTIGRANULAR LINGUISTIC INFORMATION • HETEROGENOUS INFORMATION • UNBALANCED LINGUISTIC INFORMATION • HESITANT FUZZY LINGUISTIC TERM SETS • CONCLUSIONS

3. INTRODUCTION • DECISION MAKING Decision making is a core area of different research pursuits such as engineering, both theory and practice, management, medicine and alike. It tries to make the best selection among a set of feasible solutions • SELECTION PROCESS • Aggregationphase • Exploitationphase • Solution set of alternative/s PREFERENCES SOLUTION SET AGGREGATION EXPLOITATION

4. INTRODUCTION • Basic Elements of a ClassicalDecisionProblem • A set of alternativesoravailabledecisions: • A set of states of naturethat defines theframework of theproblem: • A set of utilityvalues, , eachoneassociatedto a paircomposed of analternative and a state of nature: • A functionthatestablishestheexpert’spreferencesregardingthe plausible results.

5. INTRODUCTION • DECISION PROBLEMS • EXPERTS PREFERENCES • ASPECTS OR CRITERIA • NATURE • QUANTITATIVE • Howtallis John ? • QUALITATIVE • Howcomfortableisthatchair?

6. INTRODUCTION • DECISION PROBLEMS • QUANTITATIVE • NUMERICAL INFORMATION • CRISP • INTERVALS • QUALITATIVE ASPECTS • SUBJECTIVITY • VAGUENESS • IMPRECISION NUMBERS ARE NOT ADEQUATED HARD TO EXPRESS NUMERICALLY

7. INTRODUCTION • REAL WORLD DECISION PROBLEMS • UNCERTAINTY • PROBABILISTIC • PROBABILITY BASED MODELS • DECISION THEORY • NON PROBABILISTIC • CHALLENGE • EXPERTS: LINGUISTIC DESCRIPTORS

8. INTRODUCTION • DECISION MAKING Issues related to decision making have been traditionally handled either by deterministic or by probabilistic approaches. The first one completely ignores uncertainty, while the second one assumes that any uncertainty can be represented as a probability distribution. However in real-world problems (say, engineering, scheduling, and planning) decisions should be made under circumstances with vague, imprecise and uncertain information. Commonly, the uncertainty could be of non-probabilistic nature. Among the appropriate tools to overcome these difficulties are fuzzy logic and fuzzy linguistic approach. The use of linguistic information enhances the reliability and flexibility of classical decision models.

9. INTRODUCTION • DECISION PROBLEMS • NON-PROBABLISTIC UNCERTAINTY • LINGUISTIC INFORMATION • FUZZY LOGIC • FUZZY LINGUISTIC APPROACH

10. Linguistic variable Linguistic terms Variable Very low Low Medium High Very high Semantic rule INTRODUCTION FUZZY LINGUISTIC APPROACH Linguistic variables differ from numerical variables in that their values are not numbers but are words or phrases in a natural or artificial language (Zadeh, 1975).

11. INTRODUCTION • COMPUTING WITH WORDS • LINGUISTIC COMPUTING MODELS • BasedonMembershipFunctions • Basedon Ordinal Scales • 2-Tuple basedcomputationalmodel • INTERPRETABILITY • ACCURACY • EXTENSIONS LACK OF ACCURACY IN RETRANSLATION

12. FOUNDATIONS:LINGUISTIC 2-TUPLE REPRESENTATION MODEL

13. LINGUISTIC 2-Tuple • BIBLIOGRAPHY • F. Herrera and L. Martínez. A 2-tuple fuzzylinguisticrepresentationmodelforcomputingwithwords. IEEE TransactionsonFuzzySystems, 8(6):746-752, 2000 • F. Herrera, L. Martínez.AnApproachforCombiningNumerical and LinguisticInformationbasedonthe 2-tuple fuzzylinguisticrepresentationmodel in DecisionMaking. International Journal of Uncertainty , Fuzziness and Knowledge -BasedSystems. 8.5 (2000) 539-562 • F. Herrera, L. Martínez. The 2-tuple LinguisticComputationalModel. Advantages of itslinguisticdescription, accuracy and consistency. International Journal of Uncertainty , Fuzziness and Knowledge-BasedSystems. 2001, Vol 9 pp. 33-48 • F. Herrera, L. Martínez. A modelbasedonlinguistic 2-tuples fordealingwithmultigranularityhierarchicallinguisticcontexts in MultiexpertDecision-Making. IEEE TransactionsonSystems, Man and Cybernetics. Part B: Cybernetics, 2001, Vol 31 Num 2 pp. 227.234. • F. Herrera, L. Martínez. P.J. Sánchez. Managing non-homogeneousinformation in groupdecisionmaking. EuropeanJournal of OperationalResearch 166:1(2005) pp. 115-132 • F. Herrera, E. Herrera-Viedma, L. Martínez, A FuzzyLinguisticMethodologyToDealWithUnbalancedLinguisticTerm Sets. IEEE TransactionsonFuzzySystems 2008. Page(s): 354-370. Volume: 16, Issue: 2. • M. Espinilla, J. Liu, L. Martínez. An extended hierarchical linguistic model for decision-making problems. Computational Intelligence. In press. 2011

14. LINGUISTIC 2-Tuple • Linguistic representation: • Model based on the symbolic approach. • Linguistic Domain: Continuous • Linguistic representation any symbolic computation Arith_Mean(L,VL,VH,P)=(2+1+5+6)/4=3,25

15. LINGUISTIC 2-Tuple • Linguistic Representation based on pair of values • Symbolic Translation

16. LINGUISTIC 2-Tuple • 2-tuple Functions • From a numerical value in the interval of granularity into a 2-tuple • Example

17. LINGUISTIC 2-Tuple • 2-tuple Functions • It inverse • Example

18. LINGUISTIC 2-Tuple 2-Tuple Computational Model • Negation Operator • Example

19. LINGUISTIC 2-Tuple 2-tuple Computational Model • Aggregation 2-tuple operators • To use and compute as in the numerical models • To use and transform in a 2-tuple • Aggregation operators • Arithmetic mean • Weighting average • OWA operator

20. LINGUISTIC 2-Tuple 2-tuple Computational Model • Comparison: Lexico-graphic order Let and be two 2-tuples If k < l then is less than If k = l then: • If then and are equal • If then is less than • If then is greater than

21. LINGUISTIC 2-Tuple • Applications • Decision Making and Decision Analysis • Multi-Criteria Decision Making • Group Decision Making • Consensus Reaching Processes • Evaluation • Sensory Evaluation • Performance Appraisal • Internet Based Services • Recommender Systems • Information Retrieval • Genetic Fuzzy Systems

22. LINGUISTIC 2-Tuple • Problems • Complex frameworks • Different degrees of knowledge • Multiple linguistic scales • Information of different nature • Quantitative aspects • Qualitative aspects • Non-symmetrically distributed linguistic information • Unbalanced Linguistic Information 2-tuple EXTENSIONS

23. LINGUISTIC 2-TUPLE EXTENSIONS

24. 2-Tuple EXTENSIONS • MULTIGRANULAR LINGUISTIC INFORMATION • FUSION APPROACH • LINGUISTIC HIERARCHIES • EXTENDED LINGUISTIC HIERARCHIES • HETEROGENOUS INFORMATION • UNBALANCED LINGUISTIC INFORMATION

25. MULTI-GRANULARLINGUISTIC INFORMATION

26. MULTI-GRANULAR LINGUISTIC INFORMATION • Real World Problems • Multiple Sources of information • Different degree of uncertainty • Different degree of knowledge • Linguistic Information • Necessity of Multiple scales • Different Approaches • Based on membership functions • Probabilistic • Symbolic

27. MULTI-GRANULAR LINGUISTIC INFORMATION • BIBLIOGRAPHY • Herrera, F., Herrera-Viedma, E., and Martínez, L. (2000). A fusion approach for managing multi-granularity linguistic term sets in decision making. Fuzzy Sets and Systems, 114(1), 43-58. • Herrera, F. and Martínez, L. (2001). A model based on linguistic 2-tuples for dealing with multigranularity hierarchical linguistic contexts in multiexpert decision-making. IEEE Transactions on Systems, Man and Cybernetics. Part B: Cybernetics, 31(2), 227-234. • Huynh, V. and Nakamori, Y. (2005). A satisfactory-oriented approach to multiexpert decision-making with linguistic assessments. IEEE Transactions On Systems Man And Cybernetics Part B-Cybernetics, 35(2), 184-196. • Chen, Z. and Ben-Arieh, D. (2006). On the fusion of multi-granularity linguistic label sets in group decision making. Computers and Industrial Engineering, 51(3), 526-541. • Chang, S., Wang, R., and Wang, S. (2007). Applying a direct multi-granularity linguistic and strategy-oriented aggregation approach on the assessment of supply performance. European Journal of Operational Research, 117(2), 1013-1025. • M. Espinilla, J. Liu, L. Martínez. An extended hierarchical linguistic model for decision-making problems. Computational Intelligence. In press. 2011

28. MULTI-GRANULAR LINGUISTIC FUSION APPROACH

29. MULTI-GRANULAR LINGUISTIC CONTEXTS LINGUISTIC HIERARCHIES • PROBLEMS • Multiple Experts or criteria • Different degree of Knowledge • Linguistic modelling • Multiple Linguistic term sets • INTERPRETABILITY • LINGUISTIC RESULTS

30. FUSION APPROACH • FEATURES • MEMBERSHIP BASED COMPUTATIONS • LACK OF ACCURACY

31. FUSION APPROACH MULTIPLE EXPERTS DIFFERENT LINGUISTIC TERM SETS

32. FUSION APPROACH • COMPUTATIONAL MODEL • SELECTING A BASIC LINGUISTIC TERM SET ST • UNIFICATION PHASE • COMPUTATIONAL PHASE • FUZZY ARITHMETIC

33. FUSION APPROACH • COMPUTATIONAL MODEL • LINGUISTIC RESULTS

34. LINGUISTIC HIERARCHIES

35. MULTI-GRANULAR LINGUISTIC CONTEXTS LINGUISTIC HIERARCHIES • PROBLEMS • Multiple Experts or criteria • Different degree of Knowledge • Linguistic modelling • Multiple Linguistic term sets • INTERPRETABILITY • ACCURACY • AVOID LOSS OF INFORMATION

36. LINGUISTIC HIERARCHIES Linguistic Hierarchies LH: A set of levels Level: A linguistic term set with different granularity to the remaining ones  l(t,n(t)) The linguistic term set of a LH of the level t:

37. LINGUISTIC HIERARCHIES • Linguistic Hierarchy • The label sets of a hierarchy • Semantics: triangular membership functions • Uniformly and symmetrically distributed in [0,1] • Odd granularity • Middle label stands for indifference

38. LINGUISTIC HIERARCHIES LinguisticHierarchyBasicRules Rule 1: To preserve all former modal pointsof the membership functions of each linguistic term from one level to the following one. Rule 2: To make smooth transitions between successive levels. The aim is to build a new linguistic term set, Sn(t+1). A new linguistic term will be added between each pair of terms belonging to the term set of the previous level t. To carry out this insertion, we shall reduce the support of the linguistic labels in order to keep place for the new one located in the middle of them.

39. LINGUISTIC HIERARCHIES l (1,3) l (2,5)= l (2,(2*3)-1) l (3,9)= l (3,(2*5)-1)

40. LINGUISTIC HIERARCHIES l (1,3) l (2,5)= l (2,(2*3)-1) l (3,9)= l (3,(2*5)-1) F. Herrera and L. Martínez. A Model Based on Linguistic 2-Tuples for Dealing with Multigranular Hierarchical Linguistic Context in Multi-Expert Decision Making. IEEE Transactions on SMC - Part B: Cybernetics 31 (2001) 227-234.

41. LINGUISTIC HIERARCHIES • Computational Model COMPUTING WITH WORDS MULTIPLE LINGUISTIC SCALES

42. LINGUISTIC HIERARCHIES Transformation functions One to One mapping Without loss of information Computing based on: 2-tuple computational model Transformation functions

43. LINGUISTIC HIERARCHIES • Computational Model • Example

44. LINGUISTIC HIERARCHIES • Computational Model • Translation • Unification phase • Computations • Retranslation • Transformation • Different levels

45. LINGUISTIC HIERARCHIES Strong limitation!! To deal with some linguistic term sets

46. LIMITATIONS Definition framework It is not possible the use of any linguistic term set 5 and 7 linguistic term sets are not possible with a LH CHALLENGE New structure able to deal with any linguistic term set EXTENDED LINGUISTIC HIERARCHIES LINGUISTIC HIERARCHIES

47. EXTENDED LINGUISTIC HIERARCHIES

48. EXTENDED LINGUISTIC HIERARCHIES Extended Linguistic Hierarchies (ELH) Flexible evaluation framework Accuracy Desirable Features! Results in the framework Flexible evaluation framework 3,5,7 5,7,9 Etc. l(1,3) l(2,5) l(3,7) M. Espinilla, J. Liu, L. Martínez. An extended hierarchical linguistic model for decision-making problems. Computational Intelligence. ComputationalIntelligence, Vol. 27, Issue 3, pp. 489-512

49. EXTENDED LINGUISTIC HIERARCHIES Extended Hierarchical Rules • Extended Rule 1 • Include a finite number of the levels t={1,…,m} • Not necessary to keep the former modal points one to another. • Extended Rule 2 • Add a new level t’ that keeps all the former modal points of all the previous levels • Granularity level t’ • n(t’) = (LCM( n(t)-1, n(t)-1, …., n(t)-1)+1 • t={1,…,m}

50. EXTENDED LINGUISTIC HIERARCHIES l(1,3) l(2,5) l(3,7) LCM(2,4,6)+1=13 l(4,13)