Fuzzy System

Fuzzy expert systems

30 October 2014

MMFS 5013 Intelligent Manufacturing 1

Boolean logic, Fuzzy logic


Fuzzy Sets

A generalization of an ordinary set by allowing a degree of membership for each element. Fuzzy set is represented in the format <element>/<degree>. The membership function of a set maps each element to its degree. The membership degree is represented in the format (0 degree 1), which means plausibility.

3 MMFS 5013 Intelligent Manufacturing

Crisp, Fuzzy


Crisp, Fuzzy


Fuzzy sets

Crisp set of : o Characteristic function,

o

Fuzzy set of : Membership function,


Crisp, Fuzzy


Crisp, Fuzzy subset

If is the reference super set and is a subset of if, and only if (iff),


Linguistic variables and hedges

A linguistic variable is a fuzzy variable. Kak Mah

Kak Mah: linguistic variable Beautiful: linguistic value Linguistic variables and values are used in fuzzy rules.


Fuzzy rules


Hedges



Mathematical expression for hedges


Representation of hedges


Example 1: Fuzzy sets

Universe U = set of five specific men {Hishammuddin (52), Tok Janggut Bomoh Tauhid (74), Ahmad Jauhari Yahya (59), Bomoh Warisan (54), Kapten Zaharie (53)} = {a, b, c, d, e} Fuzzy set Fuzzy set


Example 2: Membership function Membership function function of age, .

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Age Youngness

1 1.00

25 1.00

30 0.50

40 0.10

50 0.00

65 0.00

70 0.00

MMFS 5013 Intelligent Manufacturing

Fuzzy sets relation

Equality: iff for Subset: iff for Fuzzy support: The support of fuzzy set A is the crisp set of every element in for which

in Support of If Then support


Fuzzy Sets Relation

Fuzzy singleton: A fuzzy set whose support is a set of a single element . If Then is a fuzzy singleton since its support is

.


Operations of fuzzy sets

Union:

Intersection:

Complement:


Operations of fuzzy sets


Example 3: Operation

Membership function to be young and old and their union and intersection.

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x (Age) y (Young) z (Old) Union Intersection

1 1.00 0.00 1.00 0.00

25 1.00 0.0 1.00 0.00

30 0.50 0.0 0.50 0.00

40 0.10 0.2 0.20 0.10

50 0.00 0.6 0.60 0.00

60 0.00 0.8 0.80 0.00

65 0.00 1.0 1.00 0.00

70 0.00 1.0 1.00 0.00


Union of singletons

The union of its constituent singletons:

If is finite: where

If is continuous:


Unique Operations

Concentration: , where . Reduction of membership function.


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Age Youngness Oldness 1 1.00 0.00

25 1.00 0.00 30 0.50 0.00 40 0.10 0.20 50 0.00 0.60 60 0.00 0.80 65 0.00 1.00 70 0.00 1.00

CON(Youngness) CON(Oldness) 1.00 0.00 1.00 0.00 0.25 0.00 0.01 0.04 0.00 0.36 0.00 0.64 0.00 1.00 0.00 1.00


Unique Operations

Dilation:

Increment of membership function.


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Age Youngness Oldness 1 1.00 0.00

25 1.00 0.00 30 0.50 0.00 40 0.10 0.20 50 0.00 0.60 60 0.00 0.80 65 0.00 1.00 70 0.00 1.00

DIL(Youngness) DIL(Oldness) 1.00 0.00 1.00 0.00 0.71 0.00 0.32 0.45 0.00 0.77 0.00 0.89 0.00 1.00 0.00 1.00


Fuzzy rules

A fuzzy rule can be defined as a conditional statement in the form:


The difference between classical and fuzzy rules


To reason with fuzzy rules


To reason with fuzzy rules


The antecedent of a fuzzy rule with multiple parts


The consequent of a fuzzy rule with multiple parts


Fuzzy inference

Mamdani-style Sugeno-style


Mamdani-style

fuzzification of the input variables rule evaluation aggregation of the rule outputs defuzzification


Mamdani-style





Rule evaluation



Defuzzification


Defuzzification



Sugeno-style

Use a single spike, a singleton, as the membership function of the rule consequent. Use a mathematical function of the input variable. Use weighted average (WA) of all the singletons.






Mamdani versus Sugeno

Mamdani: For capturing expert knowledge Mamdani: Describes the expertise in more intuitive, more human-like manner Mamdani: Fuzzy inference, computational burden Sugeno: Computationally effective and works well with optimization and adaptive techniques Sugeno: Control problems, particularly for dynamic nonlinear systems


Building a fuzzy expert system

Specify the problem and define linguistic variables. Determine fuzzy sets. Produce and construct fuzzy rules. Encode the fuzzy sets, fuzzy rules and procedures to perform fuzzy inference into the expert system. Evaluate and tune the system.


Operating a service center of spare parts

There are four linguistic variables: o average waiting time, . o repair utilization factor of the service

center, . o number of servers, . o initial number of spare parts, .


inputs

output


The objectives are to: o keep as high as possible. o reduce by increasing and .








Fuzzy rules

2 input, 1 output schema: 3 inputs, 1 output schema: cube

rules rules

Add more






The objectives are to: o keep as high as possible. o reduce by increasing and .


Example: Dinner for two


Fuzzy inputs


Apply fuzzy operator


Apply implication


Aggregate all outputs


Defuzzification


References 1) Michael Negnevitsky (2005). Artificial Intelligence: A

Guide to Intelligent Systems, 2th Edition, China: Addison Wesley.

2) Toshinori Munakata (2008). Fundamentals of New Artificial Intelligence, 2nd Edition, Springer, ISBN: 978-1-84628-838-8.

3) M. Tim Jones (2008). Artificial Intelligence: A System Approach, Infinity Science Press, ISBN: 978-0-9778582-3-1

4) Lotfi A. Zadeh original papers (1965, 1973, 1976). 5) Dinner for two, reprise. http://www-

rohan.sdsu.edu/doc/matlab/toolbox/fuzzy/fuzzytu7.html


Documents

Fuzzy System