1. Teacher Salary Raise Model Teacher Salary Raise Model (revisited)
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

2. TEACHER SALARY RAISE MODEL
Naïve model
Base salary raise + Teaching performance
Base + Teaching & research performance (linear)
Base + 80% teaching and 20% research (linear)

3. 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 = /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;

4. TEACHER SALARY RAISE MODEL - Revisited I ctd.
2-D model for both research and teaching
Teach_Ratio = 0.8

6. GENERIC MATLAB CODE FOR SALARY RAISES.
%Establish constants
Teach_Ratio = 0.8
Lo_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 low
if 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 good
elseif 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 excellent
else
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);

7. 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)

11. FUZZY LOGIC MODEL: GENERAL CASE
INPUT
OUTPUT
TERMS
RULES
(interpreted)
(assigned)
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
(interpreted as
good, poor,excellent)
(assigned to be:
low, average, generous)
IF-THEN RULES
if x is A the y is B
if teaching == good then raise is average
BINARY LOGIC
FUZZY LOGIC
p --->q
0.5 p ---> 0.5 q

12. Alpha-cut (strong alpha-cut) Convexity Fuzzy number Bandwidth
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

13. 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

14. 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 = 0.1/0+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 beings
Fuzzy set B = “about 50 years old”

15. MEMBERSHIP FUNCTIONS of LINGUISTIC VALUES “young” “middle aged” “old”
Definitions: Core
Cross-Over Points and bandwidth
Support
Fuzzy Singleton
Normality
alpha-cut (strong alpha-cut)
Fuzzy Numbers
Symmetry
Figure 2.4 a) Two convex membership functions
b) A nonconvex membership function

16. SET-THEORETIC OPERATIONS:
fuzzy union, fuzzy intersection, fuzzy complement

17. PARAMETERIZED MEMBERSHIP FUNCTIONS

18. PARAMETERIZED MFs - BELL

19. Cylindrical extension in XxY of a fuzzy set C(A)
MFs of TWO DIMENSIONS
Cylindrical extension in XxY of a fuzzy set C(A)
Projections of a 2-D fuzzy set

20. The fuzzy complement operator is a continuous function
N: [0, 1] —> [1, 0] which meets following requirements:
N(0) = 1 and N(1) = 0 (boundary)
N(a) >= N(b) if a <= b (montonicity)
Examples:

21. FUZZY INTERSECTION or T-norm
The intersection of two fuzzy sets A and B is specified in
general by a function T:[0,1]x[0,1]——>[0,1] 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) <=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)

22. FUZZY UNION or T-conorm (S-norm)
The union of two fuzzy sets A and B is specified in
general by a function T:[0,1]x[0,1]——>[0,1] 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) <=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)