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Topic 11

Topic 11. Basic Project Scheduling. Project Scheduling. Jobs subject to precedence constraints Job on arc format (most common). 2. 5. 6. 0. 1. 4. 7. 2. 3. Overview. Critical Path Method (CPM) Program Evaluation and Review Technique (PERT). No resource Constraints. Project

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Topic 11

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  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

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