1. Machine scheduling Job 1Job 3 Job 4 Job 5Machine 1 Machine 2 time 0C max Job 2

2. Machine scheduling Job 1Job 3 Job restrictions: Job 4 Job 5Machine 1 Machine 2 time 0C max Objective functions:Machine environments: Job 2

3. Machine scheduling Job 1Job 3 Job restrictions: Job 4 Job 5Machine 1 Machine 2 time 0C max Objective functions:Machine environments: - release dates - deadlines - precedence constraints - preemptions Job 2

4. Machine scheduling Job 1Job 3 Job restrictions: Job 4 Job 5Machine 1 Machine 2 time 0C max Objective functions:Machine environments: - release dates - deadlines - precedence constraints - preemptions - makespan - average completion time - F(C 1,C 2,...,C n ) Job 2

5. Machine scheduling Job 1Job 3 Job restrictions: Job 4 Job 5Machine 1 Machine 2 time 0C max Objective functions:Machine environments: - release dates - deadlines - precedence constraints - preemptions - makespan - average completion time - F(C 1,C 2,...,C n ) -Identical machines: p j for all jobs j - Unrelated machines: p ij for all mach. i and jobs j Job 2

6. On-line optimization M1M1 M2M2 0 1 2 3 4 5 t = 0

7. On-line optimization M1M1 M2M2 0 1 2 3 4 5

8. On-line optimization M1M1 M2M2 0 1 2 3 4 5 t = 1

9. On-line optimization M1M1 M2M2 0 1 2 3 4 5 On-line solution

10. On-line optimization M1M1 M2M2 0 1 2 3 4 5 M1M1 M2M2 On-line solution Optimal solution

11. On-line optimization M1M1 0 1 2 3 4 5 Single machine, preemption allowed, min. total completion time: Σ j C j Shortest remaining processing time rule gives optimal schedule. t = 0

12. On-line optimization M1M1 0 1 2 3 4 5 Single machine, preemption allowed, min. total completion time: Σ j C j Shortest remaining processing time rule gives optimal schedule. t = 1

13. On-line optimization M1M1 0 1 2 3 4 5 Single machine, preemption allowed, min. total completion time: Σ j C j Shortest remaining processing time rule gives optimal schedule. t = 1

14. On-line optimization M1M1 0 1 2 3 4 5 Single machine, preemption allowed, min. total completion time: Σ j C j Shortest remaining processing time rule gives optimal schedule. t = 3

15. On-line optimization M1M1 0 1 2 3 4 5 Single machine, preemption allowed, min. total completion time: Σ j C j Shortest remaining processing time rule gives optimal schedule.

16. On-line optimization Competitive analysis An algorithm A is called α-competitive (α>= 1) if for every instance I and feasible solution X An algorithm A is called competitive if there exists a constant α>=1 such that A is α-competitive.

17. On-line optimization Metrical service systems - k servers in metric space M - Requests arrive one by one - A request r is a subset of M - Move one server to r - Goal: minimize tot. travelled distance

18. Metrical service systems Caching problem 17 59 123 n main memorycache...., 23, 2, 17, 4 requests 3

19. Metrical service systems Caching problem 3 17 59 123 n main memorycache...., 23, 2, 17, 4 requests Algorithms: -FIFO -LIFO -LFU 12 n 9 5 3 3 17

20. Metrical service systems Scheduling -Single machine with k possible states -Switching between state x and state y costs d xy -processing job j in state x costs p xj