1. Authors ： S.S. Lam, X. Cai Public ： Nonlinear Analysis: Real World Applications 3(2002) 307-316 Adviser ： RC. Chen Speaker : CC. Lin Date ： 2005/12/27 single machine scheduling with nonlinear lateness cost functions and fuzzy due dates 1

2. Outline Introduction Motivation and Goals Problem definition Fuzzy Basic concepts Fuzzy due date implement Fuzzy due date with Genetic Algorithms Conclusions Outline 2

3. Introduction scheduling a number of jobs Single machine Parallel machine Flow shop machine Job shop machine due date and cost function Minimize earliness Minimize lateness Minimize earliness and lateness 3

4. Motivation and Goals Motivation Due dates are often crisp number Due date may be vague in many practical situations Goals Fuzzy lateness function with triangular fuzzy number Minimize the maximum lateness 3

5. Problem definition n number of independent jobs p j processing time of job j r j released for processing at time α j the weights of independent jobs j β j the weights of independent jobs j fuzzy due date of jobs j λ sequence of the jobs C j completion time of job j under λ membership function of 3

6. Fuzzy concepts(1) – fuzzy number 3 are increasing functions are decreasing functions is a convex normalized fuzzy set A of real line with a membership function

7. Fuzzy concepts(2) – Expected value of fuzzy number 3 Expected value EV ：

8. Fuzzy due date implement(1) 3 triangular fuzzy number, TFN membership function of

9. Fuzzy due date implement(2) 3 The Crispy number d j, and Cost function ： The fuzzy number, and Cost function ：

10. Fuzzy due date with Genetic Algorithms 3 Pigeon-hole coding scheme Roulette wheel selection with Elitist model

11. Conclusions Improving the result of HMAX 3