modelo de representación 2-tupla. un enfoque computacional simbólico charlas sinbad 2 extensiones...

Modelo de Representación 2-tupla.Un enfoque computacional simbólico

Charlas Sinbad2

EXTENSIONES Y APLICACIONES EN TOMA DE

DECISION LINGÜÍSTICA

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• Aggregation phase• Exploitation phase

– Solution set of alternative/s

AGGREGATION EXPLOITATIONPREFERENCES SOLUTION

SET

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.

},...,{ 1 maaA

},...,{ 1 nssS iju

s1 ... sN

Alternative 1 u11 u12 ... u1N

Alternative 2 u21 u22 ... u2N

... ... ... … ...

Alternative M uM1 uM2 ... uMN

jiij sau ,:

4

5

INTRODUCTION

• DECISION PROBLEMS

– EXPERTS PREFERENCES – ASPECTS OR CRITERIA• NATURE

– QUANTITATIVE» How tall is John ?

– QUALITATIVE» How comfortable is that chair ?

6

INTRODUCTION

• DECISION PROBLEMS

– QUANTITATIVE• NUMERICAL INFORMATION

– CRISP– INTERVALS

– QUALITATIVE ASPECTS• SUBJECTIVITY• VAGUENESS• IMPRECISION

NUMBERS ARE NOT ADEQUATEDHARD 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

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 High Very high

Linguistic terms

Semantic rule

VariableLinguistic variable

11

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

12

FOUNDATIONS:LINGUISTIC 2-TUPLE REPRESENTATION

MODEL

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):746-752, 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) 539-562

– 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. 33-48

– 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. 227.234.

– F. Herrera, L. Martínez. P.J. Sánchez. Managing non-homogeneous information in group decision making. European Journal of Operational Research 166:1(2005) pp. 115-132

– F. Herrera, E. Herrera-Viedma, L. Martínez, A Fuzzy Linguistic Methodology To Deal With Unbalanced Linguistic Term Sets. IEEE Transactions on Fuzzy Systems 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

13

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

14

LINGUISTIC 2-Tuple

15

• Linguistic Representation based on pair of values

• Symbolic Translation

Sss iii ),,( )5.0,5.0[i

LINGUISTIC 2-Tuple

16

• 2-tuple Functions– From a numerical value in the interval of granularity

into a 2-tuple

– Example

)5.0,.5.0(,0: Sg

)5,0,5.0[

)(),,()(

i

roundiswiths i

i

)25.0,()75.2( 3 s

)25.0,()25.2( 2s

LINGUISTIC 2-Tuple

17

• 2-tuple Functions– It inverse

– Example

gS ,05.0,5,0:1

isi ),(1

75.2)25.0,( 31 s

2-Tuple Computational Model

• Negation Operator

• Example)),((),( 1 ii sgsNeg

)25.0,()25.5())25.0,(6()25.0,( 1 VHVLVLNeg

LINGUISTIC 2-Tuple

18

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

1

LINGUISTIC 2-Tuple

19

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

),( 1ks ),( 2ls

),( 1ks ),( 2ls

21

),( 1ks),( 2ls

21

),( 1ks

),( 1ks),( 2ls

),( 2ls

21

LINGUISTIC 2-Tuple

20

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

• BIBLIOGRAPHY1. 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.

2. 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.

3. 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.

4. 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.

5. 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.

6. 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

LINGUISTIC HIERARCHIES•MULTI-GRANULAR LINGUISTIC CONTEXTS

• PROBLEMS

– Multiple Experts or criteria– Different degree of Knowledge– Linguistic modelling– Multiple Linguistic term sets

•INTERPRETABILITY

– LINGUISTIC RESULTS

FUSION APPROACH•FEATURES

– MEMBERSHIP BASED COMPUTATIONS– LACK OF ACCURACY

30

FUSION APPROACH

– MULTIPLE EXPERTS

– DIFFERENT LINGUISTIC TERM SETS

31

FUSION APPROACH

• COMPUTATIONAL MODEL

– SELECTING A BASIC LINGUISTIC TERM SET ST

– UNIFICATION PHASE

– COMPUTATIONAL PHASE– FUZZY ARITHMETIC

32

FUSION APPROACH

• COMPUTATIONAL MODEL

– LINGUISTIC RESULTS

g

jj

g

jj

jj

g

jT

j

sSF

0

0

0

·

)/())(((

],0[)(: gSF T

33

))((( TSF ( )47.,()47.0,(53.0)) 1 VLs

34

LINGUISTIC HIERARCHIES

35

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


• 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


• 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


• 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, 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.

)1)(2,1())(,( tntltntl

38


)1)(2,1())(,( tntltntl

l (1,3)

l (2,5)= l (2,(2*3)-1)

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

39


)1)(2,1())(,( tntltntl

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.

40


• Computational Model

COMPUTING WITH WORDS

MULTIPLE LINGUISTIC SCALES

41

LINGUISTIC HIERARCHIES• Transformation functions

– One to One mapping– Without loss of information– Computing based on:

• 2-tuple computational model• Transformation functions

42


• Computational Model– Example

43


• Computational Model

– Translation• Unification phase

– Computations– Retranslation

• Transformation• Different levels

44


Strong limitation!!To deal with

some linguistic term sets

LH l (t,n(t)) l (t,n(t))

t=1 l (1,3) l (1,7)

t=2 l (2,5) l (2,13)

t=3 l (3,9)

)1)(2,1())(,( tntltntl

45

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

48

EXTENDED LINGUISTIC

HIERARCHIES

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. 489-512 50


• 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)+1t={1,…,m}

51

l(1,3)

l(2,5)

l(3,7)

LCM(2,4,6)+1=13

l(4,13)


52

l(1,3)

l(2,5)

l(3,7)

l(4,13) LCM(2,4,6)+1=13


53

l(1,3)

l(2,5)

l(3,7)

l(4,13) LCM(2,4,6)+1=13


54

l(1,3)

l(2,5)

l(3,7)

l(4,13) LCM(2,4,6)+1=13


55

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


56

• Unification of the information

• Transformation Functions• Level t’


57

• Computations and Results

– Aggregation of the information

– Results• Initial linguistic term sets


58

59

HETEROGENEOUSINFORMATION

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

61


• Heterogenous framework:– Numerical– Linguistic– Linguistic-MG– Interval-valued

– operate directly Different domains

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


62

•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


63

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 ST

• Measures of comparison


64

•Numerical Information into a FS in the BLTS– Let a numerical value in [0,1]– Its transformation into a FS in ST is carried out as:

Nijs

)(1,0: TNS SFT

1,0,,/)(0

iTi

g

iiiNS Sss

T

iNiji

iNiji

ii

Niji

iNiji

ii

iNij

Nijs

Nij

si

csd

dsc

if

if

dc

sc

bsaifcb

as

sSupportsifi

i

1

))((0

)(

TRANSFORMATION FUNCTIONS


65


66

Linguistic Information into a FS in the BLTS– Let a linguistic value in S– Its transformation into a FS in ST is carried out as :

Lijs

)(: TSS SFST

TkLi

ikk

g

k

LiSS ScSscs

T

,,/)(0

)}(),(min{max yyk

Li

csyik



67


68

•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:

Iijs

iif

iiif

iif

I

0

1

0

)(



69

– Its transformation into a FS in ST is carried out as:

)(: TIS SFIT

,/)(0

ikk

g

kIS cI

T

)}(),(min{max yykcIy

ik



70

•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


71

•To Transform a Fuzzy set into a 2-tuple

•Information unified by means of 2-tuples•2-tuple computational model

gSF T ,0)(:

g

jj

g

jj

jj

g

j

j

s

0

0

0

·

)/())((

),()( is



72

•Example


INPUTS

UNIFICATION

AGGREGATION AND

2-TUPLE73

•Numerical and Linguistic

•Linguistic Multi-granular

•Numerical, Interval-Valued and Linguistic

•Numerical, Interval-Valued and Linguistic Multi-Granular

CONTEXTS


74

75

UNBALANCED LINGUISTIC

INFORMATION

•Linguistic Scales– Usually: Symmetrical

– Sometimes: non-symmetrical Unbalanced

• How to manage, Representation and computations ?

UNBALANCED LINGUISTIC INFORMATION

TotalAbsence

BarelyPerceptible Good

Average Great

TotalAbsence

BarelyPerceptible

Slight

Average Great

F. Herrera, E. Herrera-Viedma, L. Martínez, A Fuzzy Linguistic Methodology To Deal With Unbalanced Linguistic Term Sets. IEEE Transactions on Fuzzy Systems 2008. Page(s): 354-370. Volume: 16, Issue: 2

76

•Methodology

– Representation• Semantic Algorithm– Linguistic 2-tuple– Linguistic Hierarchies

– Computations: CW• Symbolic• Accurate• Interpretable


77

•OUTLINE

– Basic Ideas

– Algorithm• Representation

– Computational model• Computing with Words


78

UNBALANCED LINGUISTIC INFORMATION Semantic Representation Algorithm

•Basic ideas:

1)(#

1)(#

3)(#

R

C

L

S

S

S

TotalAbsence

BarelyPerceptible

Slight

Average Great

LS RSCS

RCL SSSS

TotalAbsence

BarelyPerceptible

Slight

Average Great

79


•Basic ideas: Total

AbsenceBarely

PerceptibleSlight

Average Great

LS RSCS

TotalAbsence

BarelyPerceptibl

e

Slight

AverageGreat

One level

Two levels

80


•Basic ideas: One level in the hierarchy

1)(#2

1)1(

RS

n

TotalAbsence

BarelyPerceptible

Slight Average Great

RS

What side?

In Level (1,3)

Represent andRS

)(tnRR SS

Cs

RS

)(tnCC ss

81

• Representation using one level:– It is analogous to :

)(tnLL SS

)(tnCC ss


LS

RS

LS RSCS

F D C B A

82

• Representation using two levels:

–What levels?

– In • Level (2,5) • Level (3,9)

LS RSCS

2

1)1()(#

2

1)(

tnS

tnR

432


RS

83

• Representation using two levels:– The right side of the levels contains the assignable labels to

LS RSCS

RS

},...,{ )1(1)1((

)1(1)2/)1)1(((

)1()1(

tntn

tntn

tnR

tnR ssSAS

},,,{ 98

97

96

95

9 ssssASR


},...,{ )(1)((

)(1)2/)1)(((

)()( tntn

tntn

tnR

tnR ssSAS

},{ 54

53

5 ssASR

84

• Representation using two levels:

LS RSCS

MiddleDensityRS


labels close to the centre labels close to the extreme

RCS

RES

},{}{ ABCSSS RERCR LS RS

RESRCS

CS

LS RSCS

ExtremeDensityRS

85


•Assigning labels from two levels in the LH

)(#1 Rtt Slablab

)(#)2/)1)1((( Rt Stnlab

1tlab

3)(# RS

21 tlab

30s

31s

32s

CS RESRCS

50s

51s

52s

53s

54s

90s

91s

92s

93s

94s

95s

96s

97s

98s

LS

30s

CS RESRCS

31s

32s

50s

51s

52s

53s

54s

90s

91s

92s

93s

94s

95s

96s

97s

98s

LS

30s

CS RESRCS

31s

32s

50s

51s

52s

53s

54s

90s

91s

92s

93s

94s

95s

96s

97s

98s

LS

IF

THENis represented onis represented on ELSE is represented onis represented on

extremedensityRS

RES

RES

RCS

RCS

)1( tnRAS

)1( tnRAS

)(tnRAS

)(tnRAS

86


•Assigning from two levels: Brigdes

–S must be a fuzzy partition:• Bridging Gaps

LS RSCS

30s

CS RESRCS

31s

32s

50s

51s

52s

53s

54s

90s

91s

92s

93s

94s

95s

96s

97s

98s

LS

Brigde

IF

THEN

ELSE

extremedensityRS

ik

ssss tnkjump

tnijump

*2

; )1()(

ik

ssss tnkjump

tnijump

*2

; )1()(

LS RSCS

87


•Assigning from two levels: Central label

–S must be a fuzzy partition:

IF

THEN

ELSE

extremedensityRS

)(tnCC ss

)1( tnCC ss

LS RSCS

88

•Algorithm


89

UNBALANCED LINGUISTIC INFORMATION Computing with Words

Outputs: Five subsets :

Sets of levels: Table:

{tLE, tLC, tRC ,tRE} = {1,1,2,3}

SLE = SLC= SL= {F}Sc = {D}SRC = {C,B}SRE = {A}

F D C B A

I(i) label index G(i) level in the LH n(t)

90

•Ouputs– Semantics and representation

TotalAbsence

BarelyPerceptible

Slight

AverageGreat


91

• 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


92

• Define transformation functions:– Unbalanced term Linguistic term into LH

)5.0,5.0[)5.0,5.0[: LHSLH

LHsss

siGiIi

iGiIii

ii

)(

)()(

)( ),,(),(

))5.0,5.0[(),(

LH

S


93

• Define transformation functions

F D C B A

31

)1(0)0,( ssF n LH


94

• Define transformation functions:– Unbalanced term Linguistic term into LH

)5.0,5.0[)5.0,5.0[: SLH-1LH

LHtntlSs

LHstntn

k

ktn

k

))(,(,|

))5.0,5.0[(),()()(

)(


95

• Define transformation functions

F D C B A

)0,()( )1(0

30 Fss n -1-1 LHLH


96

• Computational Model Scheme1. Unbalanced Linguistic Assessments in S

2. Unbalanced Linguistic Assessments in LH

3. Unbalanced Linguistic Assessments in Sn(t’)

4. Result in Sn(t’)

5. Result in S

Sss iii ),,(

),( )()( iiGiIs

),( )'(j

tnjs

),( )'(f

tnfs

),( hhs

),( iis LH

),( )(' k

tnk

tt sTF

torFtupleOpera2

),( )'(f

tnfs -1LH


97

• Computational Model Scheme)0,(),0,( FA

),( )()( iiGiIs

),( )'(j

tnjs

),( )'(f

tnfs

),( hhs

),( iis LH

),( )(' k

tnk

tt sTF

),( )'(f

tnfs -1LHF D C B A

Arithmetic Mean 2T


98

99

HESITANT SITUATIONS INDECISION MAKING

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

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

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={s0,…,sg} be a linguistic term set, a HFLTS, Hs, is an ordered finite subset of consecutive linguistic terms of S

Example:S={s0:nothing, s1:very_low, s2:low, s3:medium, s4: high, s5:very_high, s6:perfect}

HS={high, very_high, perfect}

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

Context-free Grammar

• Let GH be a context-free grammar and S={s0,…,sg} a linguistic term set. The elemets of

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Context-free Grammar

• Transformation function, – Obtain HFLTS from the linguistic expressions

– Linguistic expressions are transformed:

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•Linguistic Information– Computing with words– Symbolic Approaches

•Linguistic 2-tuple– Accuracy– Interpretability

•Extensions•Hesitant information

CONCLUSIONS

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QUESTIONS

modelo de representación 2-tupla. un enfoque computacional simbólico charlas sinbad 2 extensiones...

Documents

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