1. EMIS 8374 Max-Flow Applications: Job Shop Scheduling Updated 18 March 2008

2. Slide 2 Scheduling Problem Job(j)1234 Processing time (p j )1.51.252.13.6 Release date (r j )3135 Due date (d j )5479

3. Slide 3 Assumptions 3 identical machines are available. We can start processing a job on any machine on or after its release date. Each job must be completed on or before its due date. Each machine can work on at most one job at a time. Each job can be run on at most one machine at a time. Preemption: jobs may be interrupted and moved from machine to machine.

4. Slide 4 Feasible Scheduling via Max Flow jobs 1 2 4 3 s 1.5 1.25 2.1 3.6 capacity = processing times in machine-days. We must assign a total of 8.45 days of processing time.

5. Slide 5 Independent Time Slots A job j is available during a time slot a-b if r j  a and d j  b. Divide the planning horizon into independent time slots such that the set of available jobs does not change within a given time slot.

6. Slide 6 Independent Time Slots 123456789123456789 Job 2 Job 1 Job 3 Job 4 Timeline indicates beginning of day i.

7. Slide 7 Adding Time-Slot Nodes jobs 1 2 4 3 s 1.5 1.25 2.1 3.6 1-3 3-4 5-7 4-5 7-9 2 1 1 1 1 1 2 2 2 capacity = machine-days we can allot to job j in the time slot One machine can work on job 2 for days one and two.

8. Slide 8 Adding the Sink Node jobs 1 2 4 3 s 1.5 1.25 2.1 3.6 1-3 3-4 5-7 4-5 7-9 2 1 1 1 1 1 2 2 2 t 6 machine-days 3 3 6 6

9. Slide 9 Solution jobs 1 2 4 3 s 1.5 1.25 2.1 3.6 1-3 3-4 5-7 4-5 7-9 1.25 1 0.5 1 1 0.1 2.1 1.6 t 1.25 1.6 2 2 1.5

10. Slide 10 Job 2: Days 1 and 2 jobs 1 2 4 3 s 1.5 1.25 2.1 3.6 1-3 3-4 5-7 4-5 7-9 1.25 1 0.5 1 1 0.1 2.1 1.6 t 1.25 1.6 2 2 1.5

11. Slide 11 Job 1: Days 3 and 4 jobs 1 2 4 3 s 1.5 1.25 2.1 3.6 1-3 3-4 5-7 4-5 7-9 1.25 1 0.5 1 1 0.1 2.1 1.6 t 1.25 1.6 2 2 1.5

12. Slide 12 Job 3: Days 3, 4, and 5 jobs 1 2 4 3 s 1.5 1.25 2.1 3.6 1-3 3-4 5-7 4-5 7-9 1.25 1 0.5 1 1 0.1 2.1 1.6 t 1.25 1.6 2 2 1.5

13. Slide 13 Job 4: Days 5, 6, 7, and 8 jobs 1 2 4 3 s 1.5 1.25 2.1 3.6 1-3 3-4 5-7 4-5 7-9 1.25 1 0.5 1 1 0.1 2.1 1.6 t 1.25 1.6 2 2 1.5

14. Slide 14 Feasible Schedule with 2 Machines 123456789123456789 Job 2 Job 1 Job 3 Job 4 M1 M2

15. Slide 15 Can We Get By with 1 Machine? jobs 1 2 4 3 s 1.5 1.25 2.1 3.6 1-3 3-4 5-7 4-5 7-9 2 1 1 1 1 1 2 2 2 t 2 machine-days 1 1 2 2

16. Slide 16 Max Flow with 1 Machine = 7.25 jobs 1 2 4 3 s 1 1.25 1.4 3.6 1-3 3-4 5-7 4-5 7-9 1.25 1 1 0.4 2 2 t 1.25 2 1.6 1 1 Not feasible.