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

Teacher Salary Raise Model

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

0.8 oTeach_Rati :Note

evelResearch_L10

09.0oTeach_Rati1

erformanceTeaching_P10

0.09oTeach_Rati

01.0Raise

evelResearch_LerformanceTeaching_P20

0.090.01Raise

erformanceTeaching_p10

0.090.01 Raise

0.05 RateInflation Raise

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 < 3, Raise = 0.01 + 0.04/3(Teaching_Performance);

else if Teaching_Performance < 7, Raise = 0.05;

else if Teaching_Performance <= 10, Raise = 0.05/3(Teaching_Performance-7)+0.05;

TEACHER SALARY RAISE MODEL - Revisited I ctd.

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

0.01Res_Lev10

09.0oTeach_Rati-1

oTeach_Rati05.07-erfTeaching_P3

0.05Raise

10erformanceTeaching_P

;0.01Res_Lev10

09.0oTeach_Rati-1

_Ratio0.05xTeachRaise

7, erformanceTeaching_P

;oTeach_Rati1Res_Lev10

0.090.01

oTeach_RatierfTeaching_P3

0.040.01Raise

3,erformanceTeaching_P

x

ifelse

x

ifelse

x

x

if

0 5 100

0.05

0.1

LINEAR RAISE

0 5 100

0.05

0.1

MIDDLE FLAT REGION

TEACHING PERFORMACE

SA

LA

RY

RA

ISE

0 5 100

0.05

0.1

FLAT INFLATION RAISE

SA

LA

RY

RA

ISE

%Establish constantsTeach_Ratio = 0.8Lo_Raise =0.01;Avg_Raise=0.05;Hi_Raise=0.1;Raise_Range=Hi_Raise-Lo_Raise;Bad_Teach = 0;OK_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 lowif teaching < OK_Teach raise=((Avg_Raise - Lo_Rasie)/(OK_Teach - Bad_Teach) *teaching + Lo_Raise)*Teach_Ratio + (1 - Teach_ratio)(Raise_Range/Res_Range*research + Lo_Raise);%If teaching is good, raise is goodelseif teaching < Good_Teach raise=Avg_raise*Teach_ratio + (1 - Teach_ratio)*(Raise_Range/res_range*research + Lo_Raise);%If teaching or research is excellent, raise is excellentelse raise = ((Hi_Raise - Avg_Raise)/(Great_Teach - Good_teach) *(teach - Good_teach + Avg_Raise)*Teach_Ratio + (1 - Teach_Ratio) *(Raise_Range/Res_Range*research+Lo_Raise);

Generic MATLAB Code For Salary Raises

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 generous4. If research level is bad, raise is low5. If research level is excellent, raise is generous

• COMBINE RULES

1. If teaching is poor or research is poor, raise is low2. If teaching is good, raise is average3. If teaching or research is excellent, raise is excellent

(interpreted) (assigned)

(interpreted) (assigned)

Fuzzy Logic Model: General Case

(assigned to be: low, average, generous)

(interpreted as good, poor,excellent)

1. If teaching is poor or research is poor, raise is low2. If teaching is good, raise is average3. If teaching or research is excellent, raise is excellent

IF-THEN RULES

if x is A the y is Bif teaching = good => raise = average

BINARY LOGICFUZZY LOGIC

p -->q 0.5 p --> 0.5 q

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• T-co-norm operator

• 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

Membership Functions

GxxA

MF theis, AA XxxxA

10 xA

A)-(1NOT

B)max(A,OR

B)min(A,AND

xxxxT

xxxxT

BABABA

BABABA

,

,

0 2 4 60

0.5

1

X = Number of Children

Me

mb

ers

hip

Gra

de

s

(a) MF on a Discrete Universe

0 50 1000

0.5

1

X = Age

Me

mb

ers

hip

Gra

de

s

(b) MF on a Continuous Universe

X = {0, 1, 2, 3, 4, 5, 6} is the set of # children in a familyFuzzy 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 = 0.1/0+0.3/1+0.7/2+1.0/3+0.7/4+0.3/5+0.1/6

X = R+ is set of possible ages for human beingsFuzzy set B = “about 50 years old”

Fig. 2.1 A = “sensible number of children in a family”B = “about 50 years old

4

10

501

1

|,

xx

XxxxB

B

B

General Notation: A fuzzy set A in X is defined as a set of ordered pairs and can also be denoted as

space continuous a is X if /

objects discrete collection is X if /

XA xx

xxA

Xx iiAi

Fuzzy Set Definition And Notation

0

0.5

1

Me

mb

ers

hip

Gra

de

s

(a) Two Convex Fuzzy Sets

0

0.5

1

Me

mb

ers

hip

Gra

de

s

(b) A Nonconvex Fuzzy Set

0 10 20 30 40 50 60 70 800

0.5

1

X = Age

Me

mb

ers

hip

Gra

de

s

Young Middle Aged Old

Membership Functions of Linguistic Variables “Young” “Middle Aged” “Old”

Definitions: CoreCross-Over Points and bandwidthSupportFuzzy SingletonNormalityalpha-Cut (strong alpha-cut)Fuzzy NumbersSymmetry

Figure 2.4 a) Two convex membership functionsb) A non-convex membership function

Set-theoretic Operations:Fuzzy Union, Fuzzy Intersection, Fuzzy Complement

0

0.2

0.4

0.6

0.8

1

A Is Contained in B

Me

mb

ers

hip

Gra

de

s

B

A

0

0.5

1

(a) Fuzzy Sets A and B

A B

0

0.5

1

(c) Fuzzy Set "A OR B"

0

0.5

1

(d) Fuzzy Set "A AND B"

0

0.5

1

(b) Fuzzy Set "not A"

Parameterized Membership Functions

0 20 40 60 80 1000

0.5

1

Me

mb

ers

hip

Gra

de

s(a) Triangular MF

0 20 40 60 80 1000

0.5

1

Me

mb

ers

hip

Gra

de

s

(b) Trapezoidal MF

0 20 40 60 80 1000

0.5

1

Me

mb

ers

hip

Gra

de

s

(c) Gaussian MF

0 20 40 60 80 1000

0.5

1M

em

be

rshi

p G

rad

es

(d) Generalized Bell MF

xc

cxbbc

xc

bxaab

axax

cbaxTriangle

ecxGaussiancx

0

0

,,;

,;

2

2

1

Parameterized Mfs - BELL

b

a

cxcbaxBell

2

1

1),,;(

-10 0 100

0.5

1

(a) Changing 'a'

-10 0 100

0.5

1

(b) Changing 'b'

-10 0 100

0.5

1

(c) Changing 'c'

-10 0 100

0.5

1

(d) Changing 'a' and 'b'

0

0.5

1

X

Me

mb

ers

hip

Gra

de

s

(a) Base Fuzzy Set A

0

0.5

1

Xy

Me

mb

ers

hip

Gra

de

s

(b) Cylindrical Extension of A

Mfs of Two Dimensions

0

0.5

1

XY

(a) A Two-dimensional MF

0

0.5

1

XY

(b) Projection onto X

0

0.5

1

XY

(c) Projection onto Y

Cylindrical extension in XxY of a fuzzy set C(A)

Projections of a 2-D fuzzy set

),/( yxxAcXxY

A

yyxR

xyxR

Y

RyY

X

RxX

/,max

/,max

0 0.5 10

0.5

1(a) Sugeno's Complements

X = a

N(a

)

s = 20

s = 2

s = 0

s = -0.7

s = -0.95

0 0.5 10

0.5

1(b) Yager's Complements

X = a

N(a

)

w = 0.4

w = 0.7

w = 1

w = 1.5

w = 3

Fuzzy Complement

The fuzzy complement operator is a continuous functionN: [0, 1] [1, 0] which meets following requirements:

N(0) = 1 and N(1) = 0 (boundary)N(a) N(b) if a <= b (montonicity)

complement sYeater'1

complement sSugeno'1

1

1

wwY

S

aaN

sa

aaN

Examples:

Fuzzy Intersection or T-Norm

The intersection of two fuzzy sets A and B is specified ingeneral by a function T:[0,1]x[0,1] [0,1] whichaggregates the two membership grades as follows:

xxxxTx BABABA ~,)(

The T-norm operator is a two-place function T(.,.) satisfyingT(0,0) = 0; T(a,1) = T(1,a) = a (boundary)T(a,b) <= T(c,d) if a <=c and b <=d (monotonicity)T(a,b) = T(b,a) (cummutativity)T(a,T(b,c)) =T(T(a,b),c) (associativity)

1 ba, if 0

1 a if b

1b if a

,:Product Drastic

10,:product Bounded

,:product Algebraic

,min,:Minimum min

baT

babaT

abbaT

bababaT

dp

bp

ap

0

0.5

1

(a) Two fuzzy sets A and B

A B

0

0.5

1

(b) T-norm of A and B

0

0.5

1

(c) T-conorm (S-norm) of A and B

Fuzzy Union or T-conorm (S-norm)

The union of two fuzzy sets A and B is specified ingeneral by a function T:[0,1]x[0,1]——>[0,1] whichaggregates the two membership grades as follows:

xxxxSx BABABA ~,)(

The S-norm operator is a two-place function S(.,.) satisfyingS(1,1) = 1; S(a,0) = S(0,a) = a (boundary)S(a,b) <=S(c,d) if a<=c and b<=d (monotonicity)S(a,b) = S(b,a) (cummutativity)S(a,S(b,c))=S(S(a,b),c) (associativity)

0 ba, if 1

0 a if b

0b if a

,:sum Drastic

1,:sum Bounded

,:sum Algebraic

,max,:Maximum max

baS

babaS

abbabaS

bababaS

ds

bs

as