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)

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;

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

%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); 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 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 (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 low 2. If teaching is good, raise is average 3. If teaching or research is excellent, raise is excellent IF-THEN RULES if x is A the y is B if teaching = good => raise = average BINARY LOGIC FUZZY 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

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” Fig. 2.1 A = “sensible number of children in a family” B = “about 50 years old General Notation: A fuzzy set A in X is defined as a set of ordered pairs and can also be denoted as Fuzzy Set Definition And Notation

Membership Functions of Linguistic Variables “Young” “Middle Aged” “Old” Definitions: Core Cross-Over Points and bandwidth Support Fuzzy Singleton Normality alpha-Cut (strong alpha-cut) Fuzzy Numbers Symmetry Figure 2.4a) Two convex membership functions b) A non-convex membership function

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

Parameterized Membership Functions

Parameterized Mfs - BELL

Mfs of Two Dimensions Cylindrical extension in XxY of a fuzzy set C(A) Projections of a 2-D fuzzy set

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:

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)

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)