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Compensatory Fuzzy LogicCompensatory Fuzzy Logic
Discovery of strategically useful Discovery of strategically useful knowledgeknowledge
Prof. Dr. Rafael Alejandro Espin AndradeManagement Technology Studies Center
Industrial Engineering FacultyTechnical University of Havana
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1.InteligenceDiscovering of Problems and
useful knowledge
3.Internal and External Control
and analysis
2.DesignElaboration of
Course of Actions
1.Planning
3.SelectionDecision itself
2.Implementation
Enterprise Activities
Decision Making Process
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Expert KnowledgeLiterature
Knowledge
Knowledge Engineering Knowledge Discovering
Fuzzy Integrated
Models
EnterpriseKnowledge Data
EmergentStrategicProject
DecisionCoherency
KnowledgeManagement Decision Making
Data
Strategic Integration
FBIA
Why a new multivalued fuzzy logicWhy a new multivalued fuzzy logic• Learning, Judgment, Reasoning and Decision Making are parts of a same process of thinking, and have to be studied and modeled as a hole.
• No compensation among truth value of basic predicates are an obstacle to model human judgment and decision making.
• Associativity is an obstacle to get compensatory operators with sensitivity to changes in truth values of basic predicates, and possibilities of interpretation of composed predicates truth values according a scale
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It is not yet enough a formal fieldBad behavior of multi-valued logic systemsPragmatic Combination of operators
without axiomatic formalizationConfluence of Objectives using only one
operator
Fuzzy Logic based Decision Making Modeling
Compensatory LogicCompensatory Logic It allows compensation among truth value of basic
predicates inside the composed predicate. It is a not associative system. It is a sensitive and interpretable system It generalizes Classic Logic in a new and complete way. It is possible to model decision making problems under
risk, in a compatible way with utility theory. It explains the experimental results of descriptive
prospect theory as a rational way to think It allows a new mixed inference way using statistical
and logical inference Its properties allows a better way to deal with modeling
from natural and professional languages
Existing efforts to create fuzzy Existing efforts to create fuzzy semantics standards semantics standards
• Using min-max logic
• Using a pragmatic combination of operators
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ModelsModels
• Competitive Enterprises Evaluation from Secondary Sources. (*)
• Analysis SWOT-OA (SWOT+BSC) (*)
•Competences Analysis
•Composed Inference from Compensatory Logic (Useful for Data Mining, Knowledge Discovering, Simulation) (*)
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ModelsModels
Integral Project EvaluationNegotiation:
New Theoretical Treatment of Cooperative n-person Games Theory: Quantitative Indexes for Decision Making in Business Negotiation (Good Deal Index, Convenience Counterpart Index)
SDI’Readiness
nnn xxxxxxc /1
2121 )......(),....,,(
Compensatory ConjunctionCompensatory Conjunction
Geometric Mean Geometric Mean
NegaNegattiioonn
n(x) = 1-x.
nn
n
xxx
xxxd/1
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21
))1)....(1)(1((1
),....,,(
Compensatory DisjunctionCompensatory DisjunctionDual of Geometric Mean
ZadeZadeh Implicationh Implication
i(x,y)=d(n(x),c(x,y))i(x,y)=d(n(x),c(x,y))
Rule: Definition of AndRule: Definition of And
),1(var)],1,(
),,1(var),(),([
)3var,2var,1(var
)],2,1(var)3var,2var,2([
listclandnmdiv
mxpowlistxmultlistnlenght
cland
nxpowxmult
Operators of CFL using SWRLOperators of CFL using SWRL
Rule: Definition of Negation)2var,1(var)2var,1,1(var cldenysubstract
Rule Definition of Or
Operators of CFL using SWRLOperators of CFL using SWRL
)3var,2var,1(var2
)3,2,2(2)3,3(var
)2,2(var)2,1(var
clor
yyzclandycldeny
ycldenyzcldeny
Operators of CFL using SWRLOperators of CFL using SWRL
Rule Definition of Implication
)3var,2var,1(var),,1(var2
)2var,()3var,2var,(2
clipvuclor
vcldenyucland
Operators of CFL using SWRLOperators of CFL using SWRL
Rule Definition of Equivalence
)3var,2var,1(var)3var,2var,(
)3var,2var,(),,1(var2
clequivuclip
uclipvucland
Creating Ontologies from fuzzy Creating Ontologies from fuzzy treestrees
• Create the tree from formulation in natural language
• Create classes using OWL or SWRL (using built ins for membership functions)
• Use the created built ins to create the new classes inside SWRL
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)z,y,x(cx))z),y,x(c(cx.16
)z,y,x(dx))z),y,x(d(dx.17
No Associativity No Associativity
Level PropertiesLevel Properties
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c(x,y,z)c(x,y,z)
xxyy
zz
c(c(x,y),z)c(c(x,y),z)
c(x,yc(x,y) zz
xx yy
)z,y,x(cx))z),y,x(c(cx.16
As higher level a basic predicate be, more influence it will has in the truth value of the composed predicate.
Both trees are the same for Associative Logic Systems.
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NaturalNatural Implication Implication
i(x,y)=d(n(x),y)i(x,y)=d(n(x),y)
22Natural Implication
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ZadeZadeh Implicationh Implication
i(x,y)=d(n(x),c(x,y))i(x,y)=d(n(x),c(x,y))
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Universal and Existential Universal and Existential QuantifiersQuantifiers
casootrocualquieren0
0)(si)))(ln(1
exp(
)()()(
Ux
nUxUxUx
xpxxpn
xpxpxp
casootrocualquieren0
0)(si)))(1ln(1
exp(1
))(1(1)()(
Ux
nUxUxUx
xpxxpn
xpxpxp
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Universal and Existential Quantifiers over Universal and Existential Quantifiers over bounded universes of Rbounded universes of Rnn
caseanother
XxallforxpifexpxX
X
dx
dxxp
0
0)()(
))(ln(
caseanother
XxallforxpifexpxX
X
dx
dxxp
1
0)(1)(
))(1ln(
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Compatibility with Propositional Classical Compatibility with Propositional Classical CalculusCalculus
1,01,0: nf
2
1e 1,0
1,0
dx
dx))x(fln(
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Natural ZadehAx 1 0.5859 0.5685Ax 2 0.5122 0.5073Ax 3 0.5556 0.5669Ax 4 0.5859 0.5661Ax 5 0.8533 0.5859Ax 6 0.5026 0.5038Ax 7 0.5315 0.5137Ax 8 0.5981 0.5981
Compatibility with Propositional Classical Compatibility with Propositional Classical Calculus (Kleene Axioms)Calculus (Kleene Axioms)
Theorem of Compatibility: Theorem of Compatibility: Exclusive Property of CFL useful to get Exclusive Property of CFL useful to get fuzzy ontologies and connected it with fuzzy ontologies and connected it with
non fuzzy onesnon fuzzy ones
p is an only is a correct formula (tautology) of Propositional Calculus according to bivalued logic if it has truth value greater than 0.5 in CFL
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Inference
LogicInference
StatisticalInference
ComposedInference
Composed Inference: It allows to make and to model hypothesis using ‘Background Knowledge, to estimate truth value of hypothesis using a sample and search in parameters space of the model increasing truth
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HypothesisHypothesis
1. If past time t from t0 is short, PIB at t0 is high, and exchange rate peso-dollar is good, and inflation too, then inflation at t0+t will be good. (sufficient condition for goodness of future inflation)
2. If past time t from t0 is short, PIB at t0 is high, and exchange rate peso-dollar is good, and inflation too, then exchange rate at t0+t will be good. (sufficient condition for goodness of future exchange rate)
3. If past time t from t0 is short, PIB at t0 is high, and exchange rate peso-dollar is good, and inflation too, then PIB at t0+t will be high. (sufficient condition for goodness of future PIB)
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Hypothesis 1
Hypothesis 2
Hypothesis 3
Hypothesis 1'
Hypothesis 2'
Hypothesis 3'
0,576160086 0,620615758 0,319922171 0,237583737 0,539710004 0,548489528
0,104861817 0,109086145 0,278621583 0,999987824 0,988994364 0,194511245
0,954676527 0,955654118 0,966207572 0,997833826 0,036824939 0,256004496
0,619516745 0,682054585 0,704510956 0,843278654 0,236556481 0,349107639
0,360116603 0,596395877 0,681417533 0,905856942 0,449708517 0,583310592
0,503173195 0,601806401 0,806558599 0,875167679 0,375314384 0,675469125
0,240645922 0,243684437 0,388070789 0,999988786 0,988986017 0,194481187
0,166064725 0,658210493 0,366102952 0,990618051 0,604258741 0,27023985
0,957234283 0,957808848 0,969335758 0,949513696 0,023763872 0,295017189
0,623564594 0,688110949 0,820055235 0,929246656 0,24713002 0,566004987
0,406582528 0,562805296 0,763656353 0,862017409 0,438632042 0,6827389
0,315003135 0,609940962 0,481272314 0,991513016 0,448097639 0,267486414
0,348881932 0,632987705 0,445100645 0,785234473 0,530719918 0,406855462
0,960632803 0,96120307 0,981985893 0,977613627 0,064500183 0,546627412
0,658386683 0,698912399 0,87122133 0,898146889 0,29047844 0,664809821
0,459222487 0,617040225 0,55840803 0,808412114 0,384684691 0,380968904
0,36295827 0,596407224 0,682984162 0,906088702 0,447004634 0,583111022
0,964920965 0,965265218 0,987462709 0,968219979 0,159229346 0,646455267
0,489849965 0,62196448 0,751265713 0,916814967 0,339829639 0,57450373
0,448296532 0,577103867 0,782469741 0,867623112 0,410028953 0,679530673
0,55878818 0,632516495 0,830278389 0,883071359 0,342921337 0,671518873
0,439202131 0,574390855 0,604586702 0,846168613 0,281018624 0,395033281
0,519845804 0,657303193 0,754040445 0,890357379 0,330713346 0,578926238
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Hypothesis 1
Hypothesis 2
Hypothesis 3
Hypothesis 1'
Hypothesis 2'
Hypothesis 3'
0,129937234 0,786977784 0,163182976 0,897313418 0,78995843 0,246740413
0,040471107 0,045576384 0,226920195 1 0,98899831 0,194313157
0,96601297 0,966586912 0,974686783 0,999999764 0,031185555 0,255212475
0,316375835 0,614728727 0,512356471 0,999226656 0,439429951 0,291353006
0,099650901 0,662430526 0,589342664 0,999838473 0,642938808 0,545153518
0,218548001 0,549156052 0,721617082 0,999637465 0,508526254 0,645300844
0,031375461 0,036578117 0,219591953 1 0,988998869 0,194313157
0,040525407 0,776150562 0,285353418 0,999998745 0,770101561 0,25525313
0,966018421 0,966594251 0,975835847 0,99982785 0,019121177 0,28913524
0,314847536 0,596608509 0,68759951 0,999859116 0,439114653 0,545002502
0,100646896 0,567339971 0,679511054 0,999611013 0,5903147 0,645417242
0,031430795 0,793473197 0,278579081 0,999998739 0,789884048 0,255253529
0,04487921 0,802329191 0,317793959 0,999085299 0,794414393 0,292286066
0,966014676 0,966568235 0,984519648 0,999968639 0,061925026 0,544546456
0,315423665 0,558446985 0,756193558 0,999660725 0,450390363 0,64520529
0,035867632 0,823077109 0,311327136 0,999080973 0,817720938 0,2923171
0,041885488 0,720353211 0,562956364 0,999833364 0,722064447 0,545194081
0,966016088 0,966509609 0,987918683 0,999924476 0,158754822 0,644565563
0,032816816 0,732504923 0,558813514 0,999832576 0,736638274 0,54520045
0,043013751 0,591969844 0,658918424 0,99959871 0,63696319 0,645474182
0,033966772 0,597042779 0,655685287 0,999596813 0,64484329 0,645483123
0,15293434 0,493060755 0,476469006 0,992585058 0,361083333 0,353165336
0,064133476 0,727586495 0,58911645 0,999644446 0,552232295 0,534562806
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)(11
)( xe
xu
)1.0ln()9.0ln(
: as true as false : almost false
Membership Functions
35 As true as false:10; Almost false:5
Membership function
36 As true as false:40; Almost false:15
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Gamma Beta Gamma Beta
Inflation 11 5 10.3022482 5.30220976
GIP 2 0 2.70067599 0.12747186
Money Value 7 12 6.8657769 12.0650321
Future Inflation 11 5 6.39146712 6.35080391
Future GIP 2 0 2 0
Future Money Value 7 12 7 12
Time 2 4 1.19916547 4.14284971
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Relation between CFL Relation between CFL and Utility Theoryand Utility Theory
Two possible outlooks of Decision Making problem under risk using Compensatory Fuzzy Logic are possible
First one
Security: All scenarios are convenient in correspondence with its probabilities of occurrence (Its is equivalent to be risk adverse)
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HedgesHedges
Operators which models words like very, more or least, enough, etc. They modifies the truth value intensifying or un-intensifying judgments.
More used functions to define hedges are functions f(x)=xa , a is an exponent greater or equal to cero. It is used to use 2 and 3 as exponents to define the words very and hyper respectively, and ½ for more or less.
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))(ln()(
))],,...,,((exp[)](exp[
]))(ln(1
exp[]))((ln1
exp[
)]))((ln1
exp[))((exp(ln
))(())((),,...,,(
1
1111
1
11
11
111
(
xvxudonde
xpxpuxup
xvpnn
xvn
xvn
xv
xvxvixpxpv
nn
n
iii
n
iii
npi
n
i
npi
n
i
n npi
n
i
n npi
n
i
npinn
i
ii
ii
Function u(x)=ln(v(x) have second diferential positive (risk averse) when v es sigmoidal.
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Relation between CFL Relation between CFL and Utility Theoryand Utility Theory
Second outlook
Opportunity: There are convenient scenarios according with their probabilities (It is equivalent to be risk prone)
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n
iii
npi
n
i
n npi
n
i
npinn
xvp
xvn
xv
xvixpxpv
i
i
i
1
1
1
112
))](1ln(exp[1
)]))(1(ln1
exp[1
))(1(1
))((),,...,,(
(
These preferences are represented by u(x)=-ln(1-v(x) (It is proved by increasing transformations). This function have negative second differential (risk prone) when v is sigmoidal.
TeoremasTeoremas Teorema 1:
Si f es un predicado difuso que representa la conveniencia de los premios. El punto de vista de la seguridad usando LDC representa las preferencias de un decisor averso al riesgo con función de utilidad u(x)=ln(f(x)).
El punto de vista de la oportunidad usando LDC representa las preferencias de un decisor propenso al riesgo con función utilidad u(x)=-ln(1-f(x))
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TeoremasTeoremasTeorema 2:
Dada un decisor con función utilidad u acotada en el intervalo (m,M).
Si el decisor es averso al riesgo, el predicado de la Lógica Difusa Compensatoria que representa la conveniencia de los precios es v(x)=exp(u(x)-M). Si es propenso al riesgo, el predicado de la Lógica Difusa Compensatoria que representa la conveniencia de los premios es v(x)=1-exp(1-u(x)-m).
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Prospect TheoryProspect Theory
It is a descriptive decision making theory of decision making under risks, based on experiments. It deserved the nobel prize of Economy for Kahnemann and Tervsky in 2003.
Individual decision makers are used to be risk averse attitude about benefits and risk prone attitude about loses
More general
There is a reference value a, satisfying for x<a that utility function is convex and for x>a is concave.
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Prospect TheoryProspect Theory
Differential of the function for loses is great than differential for benefits.
Individual decision makers are used to attribute not linear weights to utilities using probabilities of the correspondent scenarios.
That function are used to be concave in certain interval [0,b] and convex in [b,1]; b is a real number greater than 0 and less than 1.
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Rational explanation of Experimental Rational explanation of Experimental Results of Kahnemann and TervskyResults of Kahnemann and Tervsky
56 lotteries and its experimental equivalents were used from experiments of Kahnemann and Tervsky.
We estimated the truth value of the statement: ‘Every lottery is equivalent (in preference) to its experimental equivalent’, according CFL for each preference model: Universal (Risk Averse), Existential (Risk Prone), Conjunction Rule and Disjunction Rule. Best parameters of membership functions maximizing the statement truth value for all the models.
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Rational explanation of Experimental Rational explanation of Experimental Results of Kahnemann and TervskyResults of Kahnemann and Tervsky
Result: Experimental results of Kahnemann and Tervsky can be explained as result of a new based-CFL rationality working with no linear membership functions of probabilities and considering that security and opportunity are both desirables for individual decision makers.
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Prize1 Prize2 Prob1 Prob2 Equiv Universal Existential Conjunction Disjunction
50 150 0.05 0.95 128 0.81046267 0.81043085 0.84579078 0.86240609
-50 -150 0.95 0.05 -60 0.98729588 0.98699553 0.98830472 0.988450983
-50 -150 0.75 0.25 -71 0.98987688 0.9898564 0.99040911 0.990496525
-50 -150 0.5 0.5 -92 0.99282463 0.99193467 0.99355369 0.993616354
-50 -150 0.25 0.75 -113 0.99547171 0.9930822 0.99578486 0.995822571
-50 -150 0.05 0.95 -132 0.99680682 0.98931435 0.99720726 0.997233981
100 200 0.95 0.05 118 0.80956738 0.76106093 0.85650544 0.874712043
100 200 0.75 0.25 130 0.79924863 0.77146557 0.8415966 0.86176461
100 200 0.5 0.5 141 0.77419623 0.76244496 0.82436946 0.845652063
100 200 0.25 0.75 162 0.75472897 0.75240339 0.79542526 0.822098935
100 200 0.05 0.95 178 0.72808136 0.72808118 0.77001401 0.801244793
-100 -200 0.95 0.05 -112 0.99547158 0.99535134 0.99569579 0.995718909
-100 -200 0.75 0.25 -121 0.99634217 0.99633821 0.99644367 0.996456576
-100 -200 0.5 0.5 -142 0.99762827 0.99734518 0.99777768 0.997786734
-100 -200 0.25 0.75 -158 0.99843469 0.99764225 0.99848705 0.998493062
-100 -200 0.05 0.95 -179 0.99905141 0.99589186 0.9991227 0.999126427
0.91683692 0.89594007 0.93079936 0.942636789
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Expert KnowledgeLiterature
Knowledge
Knowledge Engineering Knowledge Discovering
Fuzzy Integrated
Models
EnterpriseKnowledge Data
EmergentStrategicProject
DecisionCoherency
KnowledgeManagement Decision Making
Data
Strategic Integration
FBIA
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Users and Groups
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Organizations and Matrices
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Organizations and Matrices
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Organization Matrices
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Job Parameters
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Parameters Configuration
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Parameters Configuration
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Comparación Ponderada de los Parámetros
Compensatory LogicCompensatory Logic It allows compensation among truth value of basic
predicates inside the composed predicate. It is a not associative system. It is a sensitive and interpretable system It generalizes Classic Logic in a new and complete way. It is possible to model decision making problems under
risk, in a compatible way with utility theory. It explains the experimental results of descriptive
prospect theory as a rational way to think It allows a new mixed inference way using statistical
and logical inference Its properties allows a better way to deal with modeling
from natural and professional languages
60
Some Scientific PerspectivesSome Scientific PerspectivesDevelopment of a new fuzzy framework of
Cooperative Games Theory CFL-BasedCFL-based Ontologies using OWL-SWRL-
Protégé OntologiesCreation of mathematically formal hybrid
frameworks mixing Neural networks, Evolutionary algorithms, Trees and CFL
Experimental research line about judgment, election and reasoning from CFL
