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

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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 2

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INTRODUCTION DECISION MAKING 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 Aggregation phase Exploitation phase –Solution set of alternative/s AGGREGATIONEXPLOITATION PREFERENCESSOLUTION SET 3

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INTRODUCTION Basic Elements of a Classical Decision Problem A set of alternatives or available decisions: A set of states of nature that defines the framework of the problem: A set of utility values,, each one associated to a pair composed of an alternative and a state of nature: A function that establishes the expert’s preferences regarding the plausible results. s1s1... s N Alternative 1u 11 u 12...u1Nu1N Alternative 2u 21 u 22...u2Nu2N … Alternative MuM1uM1 uM2uM2...u MN 4

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INTRODUCTION DECISION PROBLEMS –EXPERTS PREFERENCES –ASPECTS OR CRITERIA NATURE –QUANTITATIVE »How tall is John ? –QUALITATIVE »How comfortable is that chair ? 5

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

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

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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. 8

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

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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). Very low Low Medium HighVery high Linguistic terms Semantic rule Variable Linguistic variable 10

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INTRODUCTION COMPUTING WITH WORDS –LINGUISTIC COMPUTING MODELS Based on Membership Functions Based on Ordinal Scales 2-Tuple based computational model –INTERPRETABILITY –ACCURACY –EXTENSIONS LACK OF ACCURACY IN RETRANSLATION 11

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FOUNDATIONS: LINGUISTIC 2-TUPLE REPRESENTATION MODEL 12

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LINGUISTIC 2-Tuple BIBLIOGRAPHY –F. Herrera and L. Martínez. A 2-tuple fuzzy linguistic representation model for computing with words. IEEE Transactions on Fuzzy Systems, 8(6): , 2000 –F. Herrera, L. Martínez. An Approach for Combining Numerical and Linguistic Information based on the 2-tuple fuzzy linguistic representation model in Decision Making. International Journal of Uncertainty, Fuzziness and Knowledge -Based Systems. 8.5 (2000) –F. Herrera, L. Martínez. The 2-tuple Linguistic Computational Model. Advantages of its linguistic description, accuracy and consistency. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems. 2001, Vol 9 pp –F. Herrera, L. Martínez. 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, 2001, Vol 31 Num 2 pp –F. Herrera, L. Martínez. P.J. Sánchez. Managing non-homogeneous information in group decision making. European Journal of Operational Research 166:1(2005) pp –F. Herrera, E. Herrera-Viedma, L. Martínez, A Fuzzy Linguistic Methodology To Deal With Unbalanced Linguistic Term Sets. IEEE Transactions on Fuzzy Systems Page(s): Volume: 16, Issue: 2. –M. Espinilla, J. Liu, L. Martínez. An extended hierarchical linguistic model for decision-making problems. Computational Intelligence. In press

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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)=( )/4=3,25 14

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

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

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LINGUISTIC 2-Tuple 17 2-tuple Functions –It inverse –Example

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

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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 LINGUISTIC 2-Tuple 19

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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 LINGUISTIC 2-Tuple 20

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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 21

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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 22

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LINGUISTIC 2-TUPLE EXTENSIONS 23

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

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MULTI-GRANULAR LINGUISTIC INFORMATION 25

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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 26

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MULTI-GRANULAR LINGUISTIC INFORMATION BIBLIOGRAPHY 1.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), 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), 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), 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), 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), M. Espinilla, J. Liu, L. Martínez. An extended hierarchical linguistic model for decision-making problems. Computational Intelligence. In press

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MULTI-GRANULAR LINGUISTIC FUSION APPROACH 28

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

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FUSION APPROACH FEATURES –MEMBERSHIP BASED COMPUTATIONS –LACK OF ACCURACY 30

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FUSION APPROACH –MULTIPLE EXPERTS –DIFFERENT LINGUISTIC TERM SETS 31

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FUSION APPROACH COMPUTATIONAL MODEL –SELECTING A BASIC LINGUISTIC TERM SET S T –UNIFICATION PHASE –COMPUTATIONAL PHASE –FUZZY ARITHMETIC 32

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FUSION APPROACH COMPUTATIONAL MODEL –LINGUISTIC RESULTS 33

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LINGUISTIC HIERARCHIES 34

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

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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: 36

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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 37

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LINGUISTIC HIERARCHIES Linguistic Hierarchy Basic Rules Rule 1: To preserve all former modal points of 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, S n(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. 38

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

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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)

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LINGUISTIC HIERARCHIES Computational Model COMPUTING WITH WORDS MULTIPLE LINGUISTIC SCALES 41

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

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LINGUISTIC HIERARCHIES Computational Model – Example 43

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

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LINGUISTIC HIERARCHIES Strong limitation!! To deal with some linguistic term sets LHl (t,n(t)) t=1l (1,3)l (1,7) t=2l (2,5)l (2,13) t=3l (3,9) 45

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LINGUISTIC HIERARCHIES 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 46

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Comparative Analysis –Fusion Approach for Managing MLI (Fusion) –A New Fusion Approach for Managing MLI (New Fusion) –Linguistic Hierarchies (LH) –Hierarchical Tree ( T LH ) New Approach Based on LH: Extended Linguistic Hierarchies FeaturesFusionNew FusionLHT LH AccuracyNo YesNo Results in the framework No YesNo Domain of final results Fuzzy Sets Linguistic TermNumerical Computational Model Extension Principle 2-tupleSatisfactory Principle Term SetsAny LimitedAny MULTI-GRANULAR LINGUISTIC INFORMATION 47

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EXTENDED LINGUISTIC HIERARCHIES 48

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EXTENDED LINGUISTIC HIERARCHIES MULTI-GRANULAR LINGUISTIC CONTEXTS Based on Linguistic Hierarchies Offer a greater flexibility –Definition Framework Keep LH properties –Computational model –Accuracy 49

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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. Computational Intelligence, Vol. 27, Issue 3, pp

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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} 51

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l(1,3) l(2,5) l(3,7) LCM(2,4,6)+1=13 l(4,13) EXTENDED LINGUISTIC HIERARCHIES 52

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l(1,3) l(2,5) l(3,7) l(4,13) LCM(2,4,6)+1=13 EXTENDED LINGUISTIC HIERARCHIES 53

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l(1,3) l(2,5) l(3,7) l(4,13) LCM(2,4,6)+1=13 EXTENDED LINGUISTIC HIERARCHIES 54

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l(1,3) l(2,5) l(3,7) l(4,13) LCM(2,4,6)+1=13 EXTENDED LINGUISTIC HIERARCHIES 55

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CW in ELH –Without loss of information –Use 2-tuple computational model Transformation functions –The information cannot be unified in any level. –t={1,…,m} and t’=m+1 EXTENDED LINGUISTIC HIERARCHIES 56

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Unification of the information Transformation Functions Level t’ EXTENDED LINGUISTIC HIERARCHIES 57

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Computations and Results –Aggregation of the information –Results Initial linguistic term sets EXTENDED LINGUISTIC HIERARCHIES 58

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HETEROGENEOUS INFORMATION 59

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HETEROGENEOUS INFORMATION Non Homogeneous contexts –Representation Structures point of view Preference Relations Utility Vectors Ordered preferences –Representation Models point of view Numerical Interval-Valued Linguistic 60

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HETEROGENEOUS INFORMATION Heterogenous framework: –Numerical –Linguistic –Linguistic-MG –Interval-valued –operate directly Different domains 61

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To Operate with Non Homogenous Information –To Make information Uniform –Basic Linguistic Term Set (BLTS) –Fuzzy Sets (FSs) HETEROGENEOUS INFORMATION 62

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Transformation Functions –Numerical Information into a FS in the BLTS –Linguistic Information into a FS in the BLTS –Interval-valued Information into a FS in the BLTS –FS in the BLTS to a 2-tuple in BLTS HETEROGENEOUS INFORMATION 63

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SELECTING THE BASIC LINGUISTIC TERM SET It context dependent –It should keep the level of discrimination used by the experts Granularity: Maximum –Transformation Functions without loss of information: Fuzzy Partition Semantics: Triangular fuzzy membership functions –To make the information uniform in the BLTS that we note as S T Measures of comparison HETEROGENEOUS INFORMATION 64

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Numerical Information into a FS in the BLTS –Let a numerical value in [0,1] –Its transformation into a FS in S T is carried out as: TRANSFORMATION FUNCTIONS HETEROGENEOUS INFORMATION 65

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HETEROGENEOUS INFORMATION 66

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Linguistic Information into a FS in the BLTS –Let a linguistic value in S –Its transformation into a FS in S T is carried out as : TRANSFORMATION FUNCTIONS HETEROGENEOUS INFORMATION 67

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HETEROGENEOUS INFORMATION 68

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Interval-Valued Information into a FS in the BLTS –Let be an interval valued in I([0,1]) –Before transforming in a FS. The interval-value will be represented as: TRANSFORMATION FUNCTIONS HETEROGENEOUS INFORMATION 69

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–Its transformation into a FS in S T is carried out as: TRANSFORMATION FUNCTIONS HETEROGENEOUS INFORMATION 70

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Unified Information –Fuzzy sets in the BLTS To operate over the FSs by means of the Extension Principle –Membership functions –Limitations, difficulties To Transform FSs into: –2-tuples HETEROGENEOUS INFORMATION 71

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To Transform a Fuzzy set into a 2-tuple Information unified by means of 2-tuples 2-tuple computational model TRANSFORMATION FUNCTIONS HETEROGENEOUS INFORMATION 72

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Example HETEROGENEOUS INFORMATION INPUTS UNIFICATION AGGREGATION AND 2-TUPLE 73

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Numerical and Linguistic Linguistic Multi-granular Numerical, Interval-Valued and Linguistic Numerical, Interval-Valued and Linguistic Multi-Granular CONTEXTS HETEROGENEOUS INFORMATION 74

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UNBALANCED LINGUISTIC INFORMATION 75

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Linguistic Scales –Usually: Symmetrical –Sometimes: non-symmetrical Unbalanced How to manage, Representation and computations ? UNBALANCED LINGUISTIC INFORMATION Total Absence Barely Perceptible Good AverageGreat Total Absence Barely Perceptible Slight AverageGreat F. Herrera, E. Herrera-Viedma, L. Martínez, A Fuzzy Linguistic Methodology To Deal With Unbalanced Linguistic Term Sets. IEEE Transactions on Fuzzy Systems Page(s): Volume: 16, Issue: 2 76

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Methodology –Representation Semantic Algorithm – Linguistic 2-tuple – Linguistic Hierarchies –Computations: CW Symbolic Accurate Interpretable UNBALANCED LINGUISTIC INFORMATION 77

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OUTLINE –Basic Ideas –Algorithm Representation –Computational model Computing with Words UNBALANCED LINGUISTIC INFORMATION 78

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UNBALANCED LINGUISTIC INFORMATION Semantic Representation Algorithm Basic ideas: Total Absence Barely Perceptible Slight Average Great Total Absence Barely Perceptible Slight Average Great 79

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UNBALANCED LINGUISTIC INFORMATION Semantic Representation Algorithm Basic ideas: Total Absence Barely Perceptible Slight Average Great Total Absence Barely Perceptible Slight Average Great One level Two levels 80

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UNBALANCED LINGUISTIC INFORMATION Semantic Representation Algorithm One level in the hierarchyBasic ideas: One level in the hierarchy Total Absence Barely Perceptible SlightAverageGreat What side? In Level (1,3) Represent and 81

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Representation using one level: –It is analogous to : UNBALANCED LINGUISTIC INFORMATION Semantic Representation Algorithm FDC B A 82

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Representation using two levels: –What levels? –In Level (2,5) Level (3,9) UNBALANCED LINGUISTIC INFORMATION Semantic Representation Algorithm 83

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Representation using two levels: –The right side of the levels contains the assignable labels to UNBALANCED LINGUISTIC INFORMATION Semantic Representation Algorithm 84

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Representation using two levels: UNBALANCED LINGUISTIC INFORMATION Semantic Representation Algorithm labels close to the centre labels close to the extreme 85

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UNBALANCED LINGUISTIC INFORMATION Semantic Representation Algorithm Assigning labels from two levels in the LH IF THEN is represented on ELSE is represented on 86

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UNBALANCED LINGUISTIC INFORMATION Semantic Representation Algorithm Assigning from two levels: Brigdes –S must be a fuzzy partition: Bridging Gaps Brigde IF THEN ELSE 87

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UNBALANCED LINGUISTIC INFORMATION Semantic Representation Algorithm Central labelAssigning from two levels: Central label –S must be a fuzzy partition: IF THEN ELSE 88

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Algorithm UNBALANCED LINGUISTIC INFORMATION Semantic Representation Algorithm 89

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UNBALANCED LINGUISTIC INFORMATION Computing with Words Outputs: Five subsets : Sets of levels: Table: {t LE, t LC, t RC,t RE } = {1,1,2,3} S LE = S LC = S L = {F} S c = {D} S RC = {C,B} S RE = {A} F D C B A I(i) label index G(i) level in the LH n(t) 90

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Ouputs –Semantics and representation Total Absence Barely Perceptible Slight Average Great UNBALANCED LINGUISTIC INFORMATION Computing with Words 91

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Representation Model: –Semantics is represented using different levels of the LH Computational Model –Operate with unbalanced linguistic term sets Without loss of information. Context: –Linguistic Hierarchy –2-tuple Linguistic Accurate Computational Model UNBALANCED LINGUISTIC INFORMATION Computing with Words 92

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Define transformation functions: –Unbalanced term Linguistic term into LH UNBALANCED LINGUISTIC INFORMATION Computing with Words 93

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Define transformation functions F DC B A UNBALANCED LINGUISTIC INFORMATION Computing with Words 94

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Define transformation functions: –Unbalanced term Linguistic term into LH UNBALANCED LINGUISTIC INFORMATION Computing with Words 95

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Define transformation functions F DC B A UNBALANCED LINGUISTIC INFORMATION Computing with Words 96

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Computational Model Scheme 1. Unbalanced Linguistic Assessments in S 2. Unbalanced Linguistic Assessments in LH 3. Unbalanced Linguistic Assessments in S n(t’) 4. Result in S n(t’) 5. Result in S UNBALANCED LINGUISTIC INFORMATION Computing with Words 97

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Computational Model Scheme F DC B A Arithmetic Mean 2T UNBALANCED LINGUISTIC INFORMATION Computing with Words 98

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HESITANT SITUATIONS IN DECISION MAKING 99

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Hesitant Fuzzy Sets (HFS) Hesitant fuzzy sets (Torra 2010) –Fulfil the management of decision situations –Quantitative contexts Decision makers Among different values Assess criteria or alternatives –HFS A function that returns a subset of values in [0,1] In terms of the union of their membership degree to set a fuzzy sets 100

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Linguistic Hesitant Situations Qualitative Setting –Hesitant Fuzzy Linguistic Term Sets (HFLTS) Objectives –Improve the flexibility of the elicitation –Experts hesitate among different linguistic values New linguistic expressions –Closer to human beings expressions –Context-free grammar 101

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Hesitant Fuzzy Linguistic Term Sets (HFLTS) Similarly to the HFS Qualitative context –Decision makers –Among different linguistic values To manage such situations –HFLTS FLA and HFS 102 Let S={s 0,…, s g } be a linguistic term set, a HFLTS, Hs, is an ordered finite subset of consecutive linguistic terms of S Example: S={s 0 :nothing, s 1 :very_low, s 2 :low, s 3 :medium, s 4 : high, s 5 :very_high, s 6 :perfect} H S ={high, very_high, perfect}

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Hesitant Fuzzy Linguistic Term Sets (HFLTS) Define two operators –Maximum and minimum bounds of a HFLTS –Compare two HFLTS Envelope of a HFLTS It is a linguistic interval 103 Upper bound Lower bound

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Context-free Grammar Let G H be a context-free grammar and S={s 0,…, s g } a linguistic term set. The elemets of 104

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Context-free Grammar Transformation function, –Obtain HFLTS from the linguistic expressions –Linguistic expressions are transformed: 105

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Linguistic Information –Computing with words –Symbolic Approaches Linguistic 2-tuple –Accuracy –Interpretability Extensions Hesitant information CONCLUSIONS 106

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THANKS A LOT FOR YOUR ATTENTION QUESTIONS 107

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