Topic 11

1. Topic 11 Basic Project Scheduling IE 514

2. Project Scheduling • Jobs subject to precedence constraints • Job on arc format (most common) 2 5 6 0 1 4 7 2 3 IE 514

3. Overview Critical Path Method (CPM) Program Evaluation and Review Technique (PERT) No resource Constraints Project Scheduling Heuristic Resource Leveling Integer Programming Formulations Resource Constraints IE 514

4. Planning a Concert IE 514

5. Job on Arc Network • Not allowed: no two jobs can have the same starting and ending node! • Need to introduce a dummy job. B A D C B A D C IE 514

6. Changing a Tire IE 514

7. Job on Arc Network A E B D C Is this correct? IE 514

8. Job on Arc Network A B E C D IE 514

9. A B E D C Job on Node Network • No need for a dummy node • Less used traditionally • Recently more popular IE 514

10. Critical Path Method (CPM) • Think of unlimited machines in parallel • … and n jobs with precedence constraints • Processing times pj as before • Objective to minimize makespan IE 514

11. Critical Path Method • Forward procedure: • Starting at time zero, calculate the earliest each job can be started • The completion time of the last job is the makespan • Backward procedure • Starting at time equal to the makespan, calculate the latest each job can be started so that this makespan is realized IE 514

12. Forward Procedure Step 1: Set at time t = 0 for all jobs j with no predecessors, Sj’=0 and set Cj’ = pj. Step 2: Compute for each job j Cj’ = Sj’ + pj. Step 3: The optimal makespan is STOP IE 514

13. Backward Procedure Step 1: Set at time t = Cmax for all jobs j with no successors, Cj’’= Cmax and set Sj’’ = Cmax - pj. Step 2: Compute for each job j Sj’’ = Cj’’ - pj. Step 3: Verify that STOP IE 514

14. Comments • The forward procedure gives the earliest possible starting time for each job • The backwards procedures gives the latest possible starting time for each job • If these are equal the job is a critical job. • If these are different the job is a slack job, and the difference is the float. • A critical path is a chain of jobs starting at time 0 and ending at Cmax. IE 514

15. 1 2 3 6 9 5 8 4 7 11 10 12 14 13 Example IE 514

16. 1 2 3 6 9 5 8 4 7 14 13 12 11 10 Forward Procedure 5+6=11 11+12=23 23+10=33 33+9=42 5 43+8=51 14+12=26 26+10=36 51+5=56 5+9=14 43+7=50 36+7=43 14+7=21 26+6=32 IE 514

17. 1 2 3 6 9 5 8 4 7 14 13 12 11 10 Backwards Procedure 24-12=12 34-10=24 43-9=34 51-8=43 14-9=5 56-5=51 36-10=26 43-7=36 56 26-12=14 56-5=51 51-8=43 35-10=26 43-7=36 IE 514

18. 2 5 8 4 7 10 13 Critical Path 1 6 9 12 14 3 11 IE 514

19. Topic 12 Variable Processing Times IE 514

20. Time/Cost Trade-Offs • Assumed the processing times were fixed • More money  shorter processing time • Start with linear costs • Processing time • Marginal cost IE 514

21. Linear Costs Resources (money) Processing time IE 514

22. Solution Methods • Objective: minimum cost of project • Time/Cost Trade-Off Heuristic • Good schedules • Works also for non-linear costs • Linear programming formulation • Optimal schedules • Non-linear version not easily solved IE 514

23. Sources, Sinks, and Cuts Cut set Sink node Source (dummy) node Minimal cut set IE 514

24. Time/Cost Trade-Off Heuristic • Step 1: • Set all processing times at their maximum • Determine all critical paths with these processing times • Construct the graph Gcp of critical paths • Continue to Step 2 IE 514

25. Time/Cost Trade-Off Heuristic • Step 2: • Determine all minimum cut sets in Gcp • Consider those sets where all processing times are larger than their minimum • If no such set STOP; otherwise continue to Step 2 IE 514

26. Time/Cost Trade-Off Heuristic • Step 3: • For each minimum cut set: • Compute the cost of reducing all processing times by one time unit. • Take the minimum cut set with the lowest cost • If this is less than the overhead per time unit go on to Step 4; otherwise STOP IE 514

27. Time/Cost Trade-Off Heuristic • Step 4: • Reduce all processing times in the minimum cut set by one time units • Determine the new set of critical paths • Revise graph Gcp and go back to Step 2 IE 514

28. Example IE 514

29. 1 2 3 6 9 5 8 4 7 11 10 12 14 13 Maximum Processing Times IE 514

30. 1 2 3 6 9 5 8 4 7 11 10 12 14 13 Maximum Processing Times IE 514

31. 1 3 6 9 14 11 12 Critical Path Subgraph (Gcp) C1=7 C12=2 C6=3 C9=4 C14=8 C11=2 C3=4 Cut sets: {1},{3},{6},{9}, {11},{12},{14}. Minimum cut set with lowest cost IE 514

32. 1 3 6 9 14 11 12 13 Critical Path Subgraph (Gcp) C1=7 C12=2 C6=3 C9=4 C14=8 C11=2 C13=4 C3=4 Cut sets: {1},{3},{6},{9}, {11},{12,13},{14}. Minimum cut set with lowest cost IE 514

33. 1 3 6 9 14 12 13 11 Critical Path Subgraph (Gcp) C1=7 C12=2 C6=3 C9=4 C14=8 C11=2 C13=14 C3=4 Reduce processing time until = 4 Reduce processing time next on job 6 IE 514

34. 4 1 3 6 9 7 2 12 13 11 14 10 Critical Path Subgraph (Gcp) C2=2 C4=3 C7=4 C10=5 C1=7 C12=2 C6=3 C9=4 C14=8 C11=2 C13=4 C3=4 IE 514

35. Linear Programming Formulation • The heuristic does not guaranteed optimum • Here total cost is linear • Want to minimize IE 514

36. Linear Program Minimize subject to IE 514

37. Topic 13 PERT IE 514

38. Program Evaluation and Review Technique (PERT) • Assumed processing times deterministic • Processing time of j random with mean mj and variance sj2. • Want to determine the expected makespan • Assume we have pja = most optimistic processing time pjm = most likely processing time (mode) pjb = most pessimistic processing time IE 514

39. Expected Makespan • Estimate expected processing time • Apply CPM with expected processing times • Let Jcp be a critical path • Estimate expected makespan IE 514

40. Distribution of Makespan • Estimate the variance of processing times • and the variance of the makespan • Assume it is normally distributed IE 514

41. Discussion • Potential problems with PERT: • Always underestimates project duration • other paths may delay the project • Non-critical paths ignored • critical path probability • critical activity probability • Activities are not always independent • same raw material, weather conditions, etc. • Estimates by be inaccurate IE 514

42. Discussion • No resource constraints: • Critical Path Method (CPM) • Simple deterministic • Time/cost trade-offs • Linear cost (heuristic or exact) • Non-linear cost (heuristic) • Accounting for randomness (PERT) IE 514

43. Topic 14 Adding Resource Constraints IE 514

44. Resource Constraints • Renewable resources • Very hard problem • No LP • Can develop an IP • Let job n+1 be dummy job (sink) and IE 514

45. Makespan • Let H bound the makespan, e.g. • Completion time of job j is and the makespan IE 514

46. IP Formulation Minimize Subject to IE 514

47. In Practice • The IP cannot be solved • (Almost) always resource constraints • Heuristic: Resource constraint  Precedence constraint • Example: pouring foundation and sidewalk • both require same cement mixer • delaying foundation delays building • precedence constraint: pour foundation first • not a logical constraint IE 514

48. Optimality of Heuristic • Say n jobs need the same resource • Could otherwise all be done simultaneously • Add (artificial) precedence constraints • Have n! possibilities • Will we select the optimal sequence? IE 514

49. Decision Support • Resource leveling • Solve with no resource constraints • Plot the resource use as a function of time • If infeasible suggest precedence constraints • Longest job • Minimum slack • User adds constraints • Start over IE 514

50. Example: One Resource 2 Uses resource 6 9 6 3 6 11 Resource Profile Capacity Time 10 20 30 IE 514