armin shmilovici ben-gurion university, israel armin@bgu.ac.il

Armin ShmiloviciBen-Gurion University, Israel

armin@bgu.ac.il

Credits for some slidesS.Natarajan, Fuzzy Decision MakingJ.-S.R. Jang, Neuro-Fuzzy and Soft

ComputingM. Casey, Lectures on Artificial

Intelligence – CS364

Fuzzy Vs. ProbabilityWalking in the desert, close to being

dehydrated, you find two sealed bottles:The first labeled water - 95% probabilityThe second labeled water - 0.95% similarityWhich one will you choose to drink from???

UncertaintyInformation can be incomplete inconsistent,

uncertain, or all three. Uncertainty is defined as the lack of the exact

knowledge that would enable us to reach a perfectly reliable conclusion.

Some Sources of Uncertain Knowledge:Weak implicationsImprecise language. Our natural language is

ambiguous and imprecise.Unknown data. When the data is incomplete or

missingviews of different experts.

DefinitionMany decision-making and problem-solving tasks

are too complex to be understood quantitatively, however, people succeed by using knowledge that is imprecise rather than precise.

Fuzzy set theory resembles human reasoning in its use of approximate information and uncertainty to generate decisions.

It was specifically designed to mathematically represent uncertainty and vagueness and provide formalized tools for dealing with the imprecision intrinsic to many problems.

More DefinitionsFuzzy logic is a set of mathematical principles for knowledge

representation based on degrees of membership.

Unlike two-valued Boolean logic, fuzzy logic is multi-valued. It deals with degrees of membership and degrees of truth.

Fuzzy logic uses the continuum of logical values between 0 (completely false) and 1 (completely true). Instead of just black and white, it employs the spectrum of colours, accepting that things can be partly true and partly false at the same time.

(a) Boolean Logic. (b) Multi-valued Logic.0 1 10 0.2 0.4 0.6 0.8 100 1 10

Crisp Vs Fuzzy Sets

150 210170 180 190 200160

Height, cmDegree ofMembership

Tall Men

150 210180 190 200

Degree ofMembership

Height, cm

Fuzzy Sets

Crisp Sets

The x-axis represents the universe of discourse – the range of all possible values applicable to a chosen variable. In our case, the variable is the man height. According to this representation, the universe of men’s heights consists of all tall men.

The y-axis represents the membership value of the fuzzy set. In our case, the fuzzy set of “tall men” maps height values into corresponding membership values.

A Fuzzy Set has Fuzzy Boundaries In the fuzzy theory, fuzzy set A of universe X is defined by

function µA(x) called the membership function of set A

µA(x) : X {0, 1}, where µA(x) = 1 if x is totally in A;

µA(x) = 0 if x is not in A;

0 < µA(x) < 1 if x is partly in A.

This set allows a continuum of possible choices. For any element x of universe X, membership function µA(x) equals the degree to which x is an element of set A. This degree, a value between 0 and 1, represents the degree of membership, also called membership value, of element x in set A.

Fuzzy Set RepresentationFirst, we determine the membership functions. In

our “tall men” example, we can obtain fuzzy sets of tall, short and average men.

The universe of discourse – the men’s heights – consists of three sets: short, average and tall men. As you will see, a man who is 184 cm tall is a member of the average men set with a degree of membership of 0.1, and at the same time, he is also a member of the tall men set with a degree of 0.4. (see graph on next page)

Fuzzy Set Representation

150 210170 180 190 200160

Height, cmDegree ofMembership

Tall Men

150 210180 190 200

Degree ofMembership

Short Average ShortTall

Fuzzy Sets

Crisp Sets

Short Average

Fuzzy SetsFuzzy SetsFormal definition:

A fuzzy set A in X is expressed as a set of ordered pairs:

19/04/23 11

A x x x XA {( , ( ))| }

Universe oruniverse of discourse

Fuzzy setMembership

function(MF)

A fuzzy set is totally characterized by aA fuzzy set is totally characterized by amembership function (MF).membership function (MF).

Fuzzy Sets with Discrete UniversesFuzzy Sets with Discrete UniversesFuzzy set C = “desirable city to live in”X = {SF, Boston, LA} (discrete and nonordered)C = {(SF, 0.9), (Boston, 0.8), (LA, 0.6)}

Fuzzy set A = “sensible number of children”X = {0, 1, 2, 3, 4, 5, 6} (discrete universe)A = {(0, .1), (1, .3), (2, .7), (3, 1), (4, .6), (5, .2),

(6, .1)}

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Fuzzy Sets with Cont. UniversesFuzzy Sets with Cont. Universes

Fuzzy set B = “about 50 years old”X = Set of positive real numbers (continuous)B = {(x, mB(x)) | x in X}

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Alternative NotationAlternative NotationA fuzzy set A can be alternatively denoted as

follows:

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A x xAx X

A x xA

X is discrete

X is continuous

Note that and integral signs stand for the union of membership grades; “/” stands for a marker and does not imply division.

Fuzzy PartitionFuzzy Partition

Fuzzy partitions formed by the linguistic values “young”, “middle aged”, and “old”:

19/04/23 15

lingmf.m

Linguistic Variables and HedgesThe range of possible values of a linguistic

variable represents the universe of discourse of that variable. For example, the universe of discourse of the linguistic variable speed might have the range between 0 and 220 km/h and may include such fuzzy subsets as very slow, slow, medium, fast, and very fast.

A linguistic variable carries with it the concept of fuzzy set qualifiers, called hedges.

Hedges are terms that modify the shape of fuzzy sets. They include adverbs such as very, somewhat, quite, more or less and slightly.

Linguistic Variables and Hedges

Very Tall

Short Tall

Degree ofMembership

150 210180 190 200

160 170

Height, cm

Average

TallVery Short Very Tall

Linguistic Variables and HedgesHedge Mathematical

Expression

A little

Slightly

Extremely

Hedge MathematicalExpression Graphical Representation

[A ( x )]1.3

[A ( x )]1.7

[A ( x )]2

[A ( x )]3

Operations on Fuzzy Sets

Complement

Containment

Intersection

Union0

A BA B

Note: Membership FunctionsFor the sake of convenience, usually a fuzzy set is

denoted as:

A = A(xi)/xi + …………. + A(xn)/xn

where A(xi)/xi (a singleton) is a pair “grade of membership” element, that belongs to a finite universe of discourse:

A = {x1, x2, .., xn}

Basic definitions & Terminology

Support(A) = {x X | A(x) > 0}

Core(A) = {x X | A(x) = 1}

Normality: core(A) A is a normal fuzzy set

Crossover(A) = {x X | A(x) = 0.5}

- cut: A = {x X | A(x) }

Strong - cut: A’ = {x X | A(x) > }

19/04/23 21

Set-Theoretic OperationsSet-Theoretic OperationsSubset:

Complement:

Union:

Intersection:

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A B A B

C A B x x x x xc A B A B ( ) max( ( ), ( )) ( ) ( )

C A B x x x x xc A B A B ( ) min( ( ), ( )) ( ) ( )

A X A x xA A ( ) ( )1

Set-Theoretic OperationsSet-Theoretic Operations

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subset.m

fuzsetop.m

MF FormulationMF FormulationTriangular MF:

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trimf x a b cx ab a

c xc b

( ; , , ) max min , ,

Trapezoidal MF: trapmf x a b c dx ab a

d xd c

( ; , , , ) max min , , ,

Generalized bell MF: gbellmf x a b cx cb

b( ; , , )

Gaussian MF: gaussmf x a b c ex c

( ; , , )

MF FormulationMF Formulation

19/04/23 25

disp_mf.m

EqualityFuzzy set A is considered equal to a fuzzy set B,

IF AND ONLY IF (iff): A(x) = B(x), xX

A = 0.3/1 + 0.5/2 + 1/3B = 0.3/1 + 0.5/2 + 1/3

therefore A = B

InclusionInclusion of one fuzzy set into another fuzzy set.

Fuzzy set A X is included in (is a subset of) another fuzzy set, B X:

A(x) B(x), xX

Consider X = {1, 2, 3} and sets A and B

A = 0.3/1 + 0.5/2 + 1/3;B = 0.5/1 + 0.55/2 + 1/3

then A is a subset of B, or A B

CardinalityCardinality of a non-fuzzy set, Z, is the number of elements in

Z. BUT the cardinality of a fuzzy set A, the so-called SIGMA COUNT, is expressed as a SUM of the values of the membership function of A, A(x):

cardA = A(x1) + A(x2) + … A(xn) = ΣA(xi), for i=1..n

Consider X = {1, 2, 3} and sets A and B

A = 0.3/1 + 0.5/2 + 1/3;B = 0.5/1 + 0.55/2 + 1/3

cardA = 1.8

cardB = 2.05

Empty Fuzzy SetA fuzzy set A is empty, IF AND ONLY IF:

A(x) = 0, xX

Consider X = {1, 2, 3} and set A

A = 0/1 + 0/2 + 0/3

then A is empty

Alpha-cutAn -cut or -level set of a fuzzy set A X is an ORDINARY

SET A X, such that:

A={A(x), xX}.

Consider X = {1, 2, 3} and set A

A = 0.3/1 + 0.5/2 + 1/3

then A0.5 = {2, 3},

A0.1 = {1, 2, 3},

A1 = {3}

Normalized Fuzzy SetsA fuzzy subset of X is called normal if there exists at least

one element xX such that A(x) = 1.

A fuzzy subset that is not normal is called subnormal.

All crisp subsets except for the null set are normal. In fuzzy set theory, the concept of nullness essentially generalises to subnormality.

The height of a fuzzy subset A is the large membership grade of an element in A

height(A) = maxx(A(x))

Fuzzy Sets Core and SupportAssume A is a fuzzy subset of X:

the support of A is the crisp subset of X consisting of all elements with membership grade:

supp(A) = {x A(x) 0 and xX}

the core of A is the crisp subset of X consisting of all elements with membership grade:

core(A) = {x A(x) = 1 and xX}

Fuzzy Set Math OperationsaA = {aA(x), xX}

Let a =0.5, and A = {0.5/a, 0.3/b, 0.2/c, 1/d}

thenAa = {0.25/a, 0.15/b, 0.1/c, 0.5/d}

Aa = {A(x)a, xX}Let a =2, and

A = {0.5/a, 0.3/b, 0.2/c, 1/d}then

Aa = {0.25/a, 0.09/b, 0.04/c, 1/d}…

Classical Decision MakingAlternative actions {drill/not drill}Alternative states of nature {oil in

ground}Relations between action/state/outcomeUtility or objective function(s)Weakness:

Assume utility is known and accurateAssume relations are known and accurateAssumptions not correct ->solution not

accurate!

Fuzzy LogicFuzzy LogicSimulating the process of human Simulating the process of human

reasoningreasoning framework to computing with words and framework to computing with words and

perception, using linguistics variables.perception, using linguistics variables. Deals with uncertaintiesDeals with uncertainties Dreative decision-making processDreative decision-making process

Fuzzy Fuzzy Decision MakingDecision MakingA model for decision making in a fuzzy

environment.Object function and constraints are

characterized as their membership functions.The intersection of fuzzy constraints and

fuzzy objection function.

ExampleObjective function:

x should be larger than 10Constraint:

x should be in the vicinity of 11

Fuzzy decision-making method

Representation of the decision problem

Fuzzy set evaluation of the decision alternatives

Selection of the optimal alternative

The method consists of three main steps:

Different classical decision-making

Fuzzy Decision-Solution

Example-Job SelectionChoose one of four alternative job offers

{a1,a2,a3,a4}Following criteria

High Salary (G1)Interesting Job (C1/G2)Close driving time (C2)

Salaries in Eu {40k,45k,50k,60k}Time in minutes {42, 12, 18, 4}

Job Selection – Cont.Fuzzify the goals and constraints according to your personal utility functions. For example, for a linear salary utility function (membership function) such as 37K is a 0.0 membership in “High Salary”, and 64k is a 1.0 membership in “High Salary”,then

G1=0.11/a1+0.3/a2+0.48/a3+0.8/a4C1 is fuzzy by its nature, for exampleC1=0.4/a1+0.6/a2+0.2/a3+0.2/a4For a linear driving time utility function (membership

function) such as 8 minutes is a 0.0 membership in “Close driving time ”, and 50 minutes is a 1.0 membership in “Close driving time ”,then

C2=0.1/a1+0.9/a2+0.7/a3+1.0/a4Find D by minimizing across alternativesD=0.1/a1+0.3/a2+0.2/a3+0.2/a4Choose alternative which maximize D -> a2Note that a2 is the best, but its not good (only 0.2)

MULTIOBJECTIVE DECISION MAKINGMULTIOBJECTIVE DECISION MAKING

Most simple decision processes are based Most simple decision processes are based on a single objective, such as minimizing on a single objective, such as minimizing cost, maximizing profit, minimizing run cost, maximizing profit, minimizing run time, and so forth. Often, however, time, and so forth. Often, however, decisions must be made in an environment decisions must be made in an environment where more than one objective function where more than one objective function governs constraints on the problem, and governs constraints on the problem, and the relative value of each of these the relative value of each of these objectives is different. objectives is different.

For example, we are designing a new For example, we are designing a new computer, and we want simultaneously to computer, and we want simultaneously to minimize the cost, maximize CPU, minimize the cost, maximize CPU, maximize Random Access Memory (RAM), maximize Random Access Memory (RAM), and maximize reliability.and maximize reliability.

Multi-objective Decision Making

A = {a1,a2,…,an}: set of alternatives

O = {o1,o2,…,or}: set of objectives

The degree of membership of alternative a in Oj is given below.

Decision function:

The optimum decision a*

,...,min

aa DAa

bbii is a parameter measuring how important is a parameter measuring how important is the objective to a decision maker for a is the objective to a decision maker for a given decisiongiven decision

D = M(OD = M(O11,b,b11) ∩ M(O) ∩ M(O22,b,b22) ∩ .. ∩ M(O) ∩ .. ∩ M(Orr,b,brr))This function is represented as the This function is represented as the

intersection of r tuples of a decision intersection of r tuples of a decision measure, M(Omeasure, M(Oii,b,bii), involving the objectives ), involving the objectives and preferences,and preferences,

The classical implication satisfies the The classical implication satisfies the requirements. Of preserving the linear requirements. Of preserving the linear ordering the preference set, and at the same ordering the preference set, and at the same time relates the two quantities in a logical time relates the two quantities in a logical way where negation is also accommodated. way where negation is also accommodated.

The decision measure for a particular The decision measure for a particular alternative, a, can be replaced with a alternative, a, can be replaced with a classical implication of the form M(Oclassical implication of the form M(Oii(a(aii), b), bii) ) = b= bi i ----> O> Oii(a) = b¯(a) = b¯ii V O V Oii(a)(a)

Multi-objective Decision Making

Define a set of preferences {P}Parameter bi is contained on set {P}

ObCLet

aObbaOM

bOMbOMbOMD

,...,,minmax

,...,,

EXAMPLE – Multi-Objective DecisionEXAMPLE – Multi-Objective DecisionA construction engineer on a construction A construction engineer on a construction

project must retain the mass of soil form project must retain the mass of soil form sliding into a building site. Alternatives : sliding into a building site. Alternatives :

MSE - mechanically stabilized embankmentMSE - mechanically stabilized embankmentConc - a mass concrete spread wallConc - a mass concrete spread wallGab -a gabion wallGab -a gabion wallFour objectives that impact his decision:Four objectives that impact his decision:Cost Cost CCMaintainability MMaintainability MStandard S Standard S Environment E Environment E A = { MSE, Conc, Gab} = {aA = { MSE, Conc, Gab} = {a11, a, a22, a, a33}}O = {C, M, S, E} = {OO = {C, M, S, E} = {O11, O, O22, O, O33, O, O44}}P = {bP = {b11, b, b22, b, b33, b, b44} [0,1]} [0,1]

OO11 = { 0.4/MSE + 1/Conc + 0.1/Gab } = { 0.4/MSE + 1/Conc + 0.1/Gab }

OO2 2 = { 0.7/MSE + .8/Conc + 0.4/Gab} = { 0.7/MSE + .8/Conc + 0.4/Gab}

OO33 = {0.2/MSE + .4/Conc + 1/Gab } = {0.2/MSE + .4/Conc + 1/Gab }

OO44 = {1/MSE + 0.5/Conc + 0.5/Gab } = {1/MSE + 0.5/Conc + 0.5/Gab }

Mse conc Gab

0O1 O2 O3 O4

Multiple-Choice Multiple-Criteria

niAA i ,...,2,1

ktCC t ,...,2,1

Example – Cooling an nuclear Reactor(Step1:representation of decision problem)

Decision Alternatives A = ｛ A1,A2,A3 ｝A1 ： flooding the reactor cavity)

A2 ： depressurizing the primary system

A3 ： doing nothing

Decision Criteria C= ｛ C1,C2,C3,C4,C5 ｝

C1 ： feasibility of the strategy C2 ：effectiveness of the strategy

C3 ： possibility of adverse effects C4 ： amount

of information needs

C5 ： compatibility with existing procedures

Step1:representation of decision problem(Cont.)

C1C2C3C4C5

Step1:representation of decision problem (Cont.)

C1 C2 C3 C4 C5

A1A2 A

Hierarchical structure of an example problem

Fuzzy set evaluation of decision alternatives

The step consist of three activities

(1)Choosing sets of the preference ratings for the importance weights of the decision

Preference ratings include Linguistic variable and triangular fuzzy number

Example ： The preference ratings for the importance weights of the decision criteria can be defined as follows:T importance= ｛ very low, low, medium, high,very high ｝

Fuzzy set evaluation of decision alternatives(Cont.)

Where a,b and c are real numbers and a ≦ b ≦ c

The triangular fuzzy numbers are used as membership functions corresponding to the elements in term set

0 0.25 0.5 0.75 1

Linguistic variable and triangular fuzzy number

VL L M H VH

Fuzzy set evaluation of decision alternatives

Choosing sets of the preference ratings for the importance weights of the decision

(2) Evaluating the importance weights of the criteria and the degrees of appropriateness of the decision alternatives

(3) Aggregating the weights of the decision criteria

Substituting Sit and Wt with triangular fuzzy numbers, that is, Sit = (oit, pit, qit) and Wt =(at, bt, ct), Fi is approximated as

For i =1,2,…,n and t = 1,2,..,k

Evaluating and Aggregating

0.2375

0.5375

0.8625

0.2125

0.50.7625

0.2750.5625

This step includes two activities Prioritization of the decision alternatives

using the aggregated assessments Choice of the decision alternative with

highest priority as the optimal

Selection of the optimal alternative(Cont.)

α = index, the degree of optimism of the decision-maker

0≦ α ≦ 1

Larger Fi means the higher appropriateness of the decision alternative

The total integral value methodLet the total integral value for triangular fuzzy number F=(a, b, c)

If α =0 then A3 is Best!If α =0.5 then A3 is Best!If α =1.0 then A1 is Best!

0.54375) 2)0.3875) 2)0.7) 1)

0.49375) 3)0.35625) 3)0.63125) 3)

0.55) 1)0.41875) 1)0.68125) 2)

Fuzzy Ordering- contdy1 = 5 , y2 = 2 , y1 >= y2, usually no ambiguity in the ranking, so we have CRISP ordering

Situations where issues or actions are associated with uncertainty, random or fuzzy , ranking is necessary

Uncertainty in rank is random – we can use pdf ( probability density function), use Gaussian normal function involving standard deviation and solve

Plot of pdf function is there in the previous slide

Example of uncertain ranking – x1 height of Italians and x2 is the height of Swedes

Are Swedes taller than Italians ? Assign membership function µ1 for

Swedes and membership function µ2 for Italians.

Fuzzy Ordering

Fuzzy Expert SystemsUsed to capture human procedural

knowledge in the form of linguistic variableFuzzy rules: If oven’s temperature=HIGH

then cooking time=SHORT A fuzzy expert system can chain rules

together via the process of inferencing.Many successful application, especially for

control of nonlinear systems.

Fuzzy ReasoningFuzzy ReasoningSingle rule with multiple antecedentRule: if x is A and y is B then z is CFact: x is A’ and y is B’Conclusion: z is C’

Graphic Representation:

19/04/23 69

A B T-norm

A’ B’ C2

A’ B’

x is A’ y is B’ z is C’

armin shmilovici ben-gurion university, israel armin@bgu.ac.il

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