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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
• 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 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
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, 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
LINGUISTIC HIERARCHIES
)1)(2,1())(,( tntltntl
l (1,3)
l (2,5)= l (2,(2*3)-1)
l (3,9)= l (3,(2*5)-1)
39
LINGUISTIC HIERARCHIES
)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
LINGUISTIC HIERARCHIES
• 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
LINGUISTIC HIERARCHIES
• Computational Model– Example
43
LINGUISTIC HIERARCHIES
• Computational Model
– Translation• Unification phase
– Computations– Retranslation
• Transformation• Different levels
44
LINGUISTIC HIERARCHIES
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
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
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 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)+1t={1,…,m}
51
l(1,3)
l(2,5)
l(3,7)
LCM(2,4,6)+1=13
l(4,13)
EXTENDED LINGUISTIC HIERARCHIES
52
l(1,3)
l(2,5)
l(3,7)
l(4,13) LCM(2,4,6)+1=13
EXTENDED LINGUISTIC HIERARCHIES
53
l(1,3)
l(2,5)
l(3,7)
l(4,13) LCM(2,4,6)+1=13
EXTENDED LINGUISTIC HIERARCHIES
54
l(1,3)
l(2,5)
l(3,7)
l(4,13) LCM(2,4,6)+1=13
EXTENDED LINGUISTIC HIERARCHIES
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
EXTENDED LINGUISTIC HIERARCHIES
56
• Unification of the information
• Transformation Functions• Level t’
EXTENDED LINGUISTIC HIERARCHIES
57
• Computations and Results
– Aggregation of the information
– Results• Initial linguistic term sets
EXTENDED LINGUISTIC HIERARCHIES
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
HETEROGENEOUS INFORMATION
• 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)
HETEROGENEOUS INFORMATION
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
HETEROGENEOUS INFORMATION
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
HETEROGENEOUS INFORMATION
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
HETEROGENEOUS INFORMATION
65
HETEROGENEOUS INFORMATION
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
TRANSFORMATION FUNCTIONS
HETEROGENEOUS INFORMATION
67
HETEROGENEOUS INFORMATION
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
)(
TRANSFORMATION FUNCTIONS
HETEROGENEOUS INFORMATION
69
– Its transformation into a FS in ST is carried out as:
)(: TIS SFIT
,/)(0
ikk
g
kIS cI
T
)}(),(min{max yykcIy
ik
TRANSFORMATION FUNCTIONS
HETEROGENEOUS INFORMATION
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
HETEROGENEOUS INFORMATION
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
TRANSFORMATION FUNCTIONS
HETEROGENEOUS INFORMATION
72
•Example
HETEROGENEOUS INFORMATION
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
HETEROGENEOUS INFORMATION
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
UNBALANCED LINGUISTIC INFORMATION
77
•OUTLINE
– Basic Ideas
– Algorithm• Representation
– Computational model• Computing with Words
UNBALANCED LINGUISTIC INFORMATION
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
UNBALANCED LINGUISTIC INFORMATION Semantic Representation Algorithm
•Basic ideas: Total
AbsenceBarely
PerceptibleSlight
Average Great
LS RSCS
TotalAbsence
BarelyPerceptibl
e
Slight
AverageGreat
One level
Two levels
80
UNBALANCED LINGUISTIC INFORMATION Semantic Representation Algorithm
•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
UNBALANCED LINGUISTIC INFORMATION Semantic Representation Algorithm
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
UNBALANCED LINGUISTIC INFORMATION Semantic Representation Algorithm
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
UNBALANCED LINGUISTIC INFORMATION Semantic Representation Algorithm
},...,{ )(1)((
)(1)2/)1)(((
)()( tntn
tntn
tnR
tnR ssSAS
},{ 54
53
5 ssASR
84
• Representation using two levels:
LS RSCS
MiddleDensityRS
UNBALANCED LINGUISTIC INFORMATION Semantic Representation Algorithm
labels close to the centre labels close to the extreme
RCS
RES
},{}{ ABCSSS RERCR LS RS
RESRCS
CS
LS RSCS
ExtremeDensityRS
85
UNBALANCED LINGUISTIC INFORMATION Semantic Representation Algorithm
•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
UNBALANCED LINGUISTIC INFORMATION Semantic Representation Algorithm
•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
UNBALANCED LINGUISTIC INFORMATION Semantic Representation Algorithm
•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
UNBALANCED LINGUISTIC INFORMATION Semantic Representation 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
UNBALANCED LINGUISTIC INFORMATION Computing with Words
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
UNBALANCED LINGUISTIC INFORMATION Computing with Words
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
UNBALANCED LINGUISTIC INFORMATION Computing with Words
93
• Define transformation functions
F D C B A
31
)1(0)0,( ssF n LH
UNBALANCED LINGUISTIC INFORMATION Computing with Words
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[(),()()(
)(
UNBALANCED LINGUISTIC INFORMATION Computing with Words
95
• Define transformation functions
F D C B A
)0,()( )1(0
30 Fss n -1-1 LHLH
UNBALANCED LINGUISTIC INFORMATION Computing with Words
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
UNBALANCED LINGUISTIC INFORMATION Computing with Words
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
UNBALANCED LINGUISTIC INFORMATION Computing with Words
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
104
Context-free Grammar
• Transformation function, – Obtain HFLTS from the linguistic expressions
– Linguistic expressions are transformed:
105
•Linguistic Information– Computing with words– Symbolic Approaches
•Linguistic 2-tuple– Accuracy– Interpretability
•Extensions•Hesitant information
CONCLUSIONS
106
107
THANKS A LOTFOR YOUR ATTENTION
QUESTIONS