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)
Raise Inflation Rate 0.05 Raise 0.01 0.09 Teaching_performance 10 0.09 Teaching_Performance Research_Level Raise 0.01 20
Raise 0.01 0.09 Teach_Ratio Teaching_Performance 10 0.09 1 Teach_Ratio Research_L evel 10 Note : Teach_Ratio 0.8
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
if
Teaching_P erformance 3, 0.04 Raise 0.01 Teaching_P erf xTeach_Rati o 3 0.09 0.01 Res_Lev x1 Teach_Rati o ; 10
else if Teaching_P erformance 7, Raise 0.05xTeach _Ratio 0.09 1 - Teach_Rati o xRes_Lev 0.01; 10 else if Teaching_P erformance 10 0.05 Teaching_P erf - 7 0.05 xTeach_Rati o Raise 3 0.09 1 - Teach_Rati o Res_Lev 0.01 10
FLAT INFLATION RAISE
LINEAR RAISE
SALARY RAISE
0.1 0.05 0
0.1 0.05 0
0 5 10 MIDDLE FLAT REGION
0
5
10
SALARY RAISE
0.1 0.05 0
0 5 10 TEACHING PERFORMACE
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);
Fuzzy Logic Model For Salary Raises
• COMMON SENSE RULES
1. 2. 3. 4. 5.
If teaching quality is bad, raise is low. If teaching quality is good, raise is average. If teaching quality is excellent, raise is generous If research level is bad, raise is low 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
(interpreted)
(assigned)
Fuzzy Logic Model: General Case
(interpreted)
(assigned)
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
(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 => raise = average BINARY LOGIC p -->q FUZZY LOGIC 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
Membership Functions
• FUZZY SETS deal with MFs (membership functions) – CLASSICAL (crisp)SET:
A x x G
– FUZZY SET:
A x, A x x X A is the MF
• FUZZY SETS DESCRIBE VAGUE CONCEPTS (e.g., fast runner, old man, hot weather, good student) • FUZZY SETS ALLOW PARTIAL MEMBERSHIP
0 A x 1
• FUZZY LOGICAL OPERATORS
AND min(A, B) OR max(A, B) NOT (1 - A)
• T-NORM OPERATOR for FUZZY Intersection & Union
AB T A x , B x A x B x AB T A x , B x A x B x
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
x X A xi / xi if X is collection discrete objects A A x / x if X is a continuous space X
i
(a) MF on a Discrete Universe
(b) MF on a Continuous Universe
Membership Grades
1
Membership Grades
0 2 4 6 X = Number of Children
1
0.5
0.5
0
0
0
50 X = Age
100
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‖
B x, B x | x X
B x
1 x 50 1 10
4
Membership Functions of Linguistic Variables
―Young‖ ―Middle Aged‖ ―Old‖
Membership Grades
Young 1 0.5 0 Middle Aged Old
0
10
20
30
40 50 X = Age
60
70
80
Definitions: Core Cross-Over Points and bandwidth Support Fuzzy Singleton Normality alpha-Cut (strong alpha-cut) Fuzzy Numbers Symmetry
(a) Two Convex Fuzzy Sets (b) A Nonconvex Fuzzy Set
Membership Grades
1
Membership Grades
1
0.5
0.5
0
0
Figure 2.4
a) Two convex membership functions b) A non-convex membership function
Set-theoretic Operations:
Fuzzy Union, Fuzzy Intersection, Fuzzy Complement
A Is Contained in B
1
B
Membership Grades
0.8
A
0.6
0.4
0.2
0
(a) Fuzzy Sets A and B 1 A B 1
(b) Fuzzy Set "not A"
0.5
0.5
0 (c) Fuzzy Set "A OR B" 1
0 (d) Fuzzy Set "A AND B" 1
0.5
0.5
0
0
Parameterized Membership Functions
(a) Triangular MF (b) Trapezoidal MF
Membership Grades
1
Membership Grades
0 20 40 60 80 100 (c) Gaussian MF
1
0.5
0.5
0
0
0
20 40 60 80 100 (d) Generalized Bell MF
Membership Grades
1
Membership Grades
0 20 40 60 80 100
1
0.5
0.5
0
0
0
20
40
60
80 100
Gaussian x; c, e
1 x c 2
2
0 x a Triangle x; a, b, c b a cx c b 0
xa a xb b xc cx
Parameterized Mfs - BELL
(a) Changing 'a' 1 1
(b) Changing 'b'
0.5
0.5
0 -10
0 (c) Changing 'c'
10
0 -10
0 10 (d) Changing 'a' and 'b'
1
1
0.5
0.5
0 -10
0
10
0 -10
0
10
Bell ( x; a, b, c)
1 xc 1 a
2b
Mfs of Two Dimensions
(a) Base Fuzzy Set A (b) Cylindrical Extension of A 1 0.5 0
Membership Grades
1
0.5
0 X
Membership Grades
y
X
(a) A Two-dimensional MF Projection onto X (b) 1 0.5 0 Y X 1 0.5 0 Y X
(c) Projection onto Y 1 0.5 0 Y X
Cylindrical extension in XxY of a fuzzy set C(A)
c A
A x /( x, y)
XxY
Projections of a 2-D fuzzy set
RX max x R x, y / x
X
RY max y R x, y / y
Y
Fuzzy Complement
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:
(a) Sugeno's Complements 1 s = -0.95 s = -0.7 (b) Yager's Complements 1 w=3 w = 1.5
N(a)
0.5
s=0 s=2
N(a)
0.5
w=1 w = 0.7
0
s = 20 0 0.5 X=a 1
0
w = 0.4 0 0.5 X=a 1
N S a
1 a 1 sa
1 w w a
Sugeno' s complement Yeater' s complement
N Y a 1
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:
AB ( x) T A x , B x A x ~ B x
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)
Algebraic product : Tap a, b ab Minimum : Tmin a, b min a, b a b
Bounded product : Tbp a, b 0 a b 1 a Drastic Product : Tdp a, b b 0
(a) Two fuzzy sets A and B A 1 0.5 0 1 0.5 0 1 0.5 0 (c) T-conorm (S-norm) of A and B (b) T-norm of A and B B
if b 1 if a 1 if a, b 1
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:
AB ( x) S A x , B x A x ~ B x
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)
Algebraic sum : S as a, b a b ab Bounded sum : S bs a, b 1 a b a Drastic sum : S ds a, b b 1 Maximum : S max a, b max a, b a b
if b 0 if a 0 if a, b 0