OVERVIEW

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OVERVIEW

• Definitions
• Teacher Salary Raise Model
• Teacher Salary Raise Model (revisited)
• Fuzzy Teacher Salary Model
• Extension Principle
• one to one
• many to one
• n-D Carthesian product to y

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TEACHER SALARY RAISE MODEL

• Naïve model
• Base salary raise Teaching performance
• Base Teaching research performance (linear)
• Base 80 teaching and 20 research (linear)

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TEACHER SALARY RAISE MODEL - Revisited I

• More sophistication desired
• Flat response in middle
• Raise is going to be inflation level in general
• We will depart from this only if teaching is
• exceptionally good or bad
• Ignore research for the time being

if Teaching_Performance lt 3, Raise
0.01 0.04/3(Teaching_Performance) else if
Teaching_Performance lt 7, Raise 0.05 else
if Teaching_Performance lt 10, Raise
0.05/3(Teaching_Performance-7)0.05
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TEACHER SALARY RAISE MODEL - Revisited I ctd.

• 2-D model for both research and teaching
• Teach_Ratio 0.8

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GENERIC MATLAB CODE FOR SALARY RAISES.
Establish constants Teach_Ratio 0.8 Lo_Raise
0.01Avg_Raise0.05Hi_Raise0.1 Raise_RangeHi_
Raise-Lo_Raise Bad_Teach 0OK_Teach 3
Good_Teach 7 Great_Teach 10 Teach_Range
Great_Teach-Bad_Teach Bad_Res 0 Great_Res
10 Res_Range Great_Res-Bad_res If teaching
is poor or research is poor, raise is low if
teaching lt OK_Teach raise((Avg_Raise -
Lo_Rasie)/(OK_Teach - Bad_Teach) teaching
Lo_Raise)Teach_Ratio (1 -
Teach_ratio)(Raise_Range/Res_Rangeresearch
Lo_Raise) If teaching is good, raise is
good elseif teaching lt Good_Teach
raiseAvg_raiseTeach_ratio (1 -
Teach_ratio)(Raise_Range/res_rangeresearch
Lo_Raise) If teaching or research is
excellent, raise is excellent else raise
((Hi_Raise - Avg_Raise)/(Great_Teach -
Good_teach) (teach - Good_teach
Avg_Raise)Teach_Ratio (1 - Teach_Ratio)
(Raise_Range/Res_RangeresearchLo_Raise)
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FUZZY LOGIC MODEL FOR SALARY RAISES

• COMMON SENSE RULES

1. If teaching quality is bad, raise is low. 2.
If teaching quality is good, raise is average. 3.
If teaching quality is excellent, raise is
generous 4. If research level is bad, raise is
low 5. If research level is excellent, raise is
generous

• COMBINE RULES

1. If teaching is poor or research is poor,
raise is low 2. If teaching is good, raise is
average 3. If teaching or research is excellent,
raise is excellent
INPUT
OUTPUT
RULES
OUTPUT TERMS
INPUT TERMS
(assigned)
(interpreted)
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FUZZY LOGIC MODEL GENERAL CASE
TEACHING RESEARCH
RAISE
1. If teaching is poor or research is poor,
raise is low 2. If teaching is good, raise is
average 3. If teaching or research is excellent,
raise is excellent
TEACHING QUALITY RESEARCH QUALITY
RAISE
(assigned to be low, average, generous)
(interpreted as good, poor,excellent)
IF-THEN RULES
if x is A the y is B if teaching good then
raise is average
BINARY LOGIC FUZZY LOGIC
p ---gtq 0.5 p ---gt 0.5 q
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DEFINITIONS

• Fuzzy set
• Support
• Core
• Normality
• Fuzzy singleton
• Cross-over point
• Alpha-cut (strong alpha-cut)
• Convexity
• Fuzzy number
• Bandwidth
• Fuzzy membership function
• Linguistic variable
• Set theoretic operations (fuzzy union,
• fuzzy intersection, fuzzy complement)
• Open-left, open-right closed fuzzy sets
• Symmetry
• Cylindrical extension in XxY of a set C(A)
• Projection of fuzzy sets
• T and S-norm operators

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MEMBERSHIP FUNCTIONS

• FUZZY SETS deal with MFs (membership functions)
• CLASSICAL (crisp)SET
• FUZZY SET
• FUZZY SETS DESCRIBE VAGUE CONCEPTS
• (e.g., fast runner, old man, hot weather,
  good student)
• FUZZY SETS ALLOW PARTIAL MEMBERSHIP
• FUZZY LOGICAL OPERATORS
• T-NORM OPERATOR for FUZZY Intersection Union

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FUZZY SET DEFINITION AND NOTATION
General Notation A fuzzy set A in X is defined
as a set of ordered pairs
and can also be denoted as
Fig. 2.1 A sensible number of children in a
family B about 50 years old
X 0, 1, 2, 3, 4, 5, 6 is the set of
children in a family Fuzzy set A sensible
number of children in a family A
(0,0.1),(1,0.3),(2,0.7),(3,1),(4,0.7),(5,0.3),(6,
0.1) A 0.1/00.3/10.7/21.0/30.7/40.3/50.1/
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X R is set of possible ages for human
beings Fuzzy set B about 50 years old
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MEMBERSHIP FUNCTIONS of LINGUISTIC VALUES
young middle aged old
Definitions Core Cross-Over Points and
bandwidth Support Fuzzy Singleton Normality al
pha-cut (strong alpha-cut) Fuzzy
Numbers Symmetry
Figure 2.4 a) Two convex membership functions b)
A nonconvex membership function
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SET-THEORETIC OPERATIONS fuzzy union, fuzzy
intersection, fuzzy complement
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PARAMETERIZED MEMBERSHIP FUNCTIONS
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PARAMETERIZED MFs - BELL
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MFs of TWO DIMENSIONS
Cylindrical extension in XxY of a fuzzy set
C(A) Projections of a 2-D fuzzy set
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FUZZY COMPLEMENT
The fuzzy complement operator is a continuous
function N 0, 1 gt 1, 0 which meets
following requirements
N(0) 1 and N(1) 0 (boundary) N(a) gt N(b)
if a lt b (montonicity)
Examples
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FUZZY INTERSECTION or T-norm
The intersection of two fuzzy sets A and B is
specified in general by a function
T0,1x0,1gt0,1 which aggregates the two
membership grades as follows
The T-norm operator is a two-place function
T(.,.) satisfying T(0,0) 0 T(a,1) T(1,a)
a (boundary) T(a,b) ltT(c,d) if altc and
bltd (monotonicity) T(a,b) T(b,a) (cummutativit
y) T(a,T(b,c))T(T(a,b),c) (associativity)
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FUZZY UNION or T-conorm (S-norm)
The union of two fuzzy sets A and B is specified
in general by a function T0,1x0,1gt0,1
which aggregates the two membership grades as
follows
The S-norm operator is a two-place function
S(.,.) satisfying S(1,1) 1 S(a,0) S(0,a)
a (boundary) S(a,b) ltS(c,d) if altc and
bltd (monotonicity) S(a,b) S(b,a) (cummutativit
y) S(a,S(b,c))S(S(a,b),c) (associativity)