1. © J. Christopher Beck 20081 Lecture 4: Program Evaluation and Review Technique (PERT)

2. © J. Christopher Beck 2008 2 Outline Quick CPM Review Program Evaluation and Review Technique (PERT)

3. © J. Christopher Beck 2008 3 Readings P Ch 4.2, 4.3 Slides borrowed from Twente & Iowa See Pinedo CD

4. © J. Christopher Beck 2008 4 A Small Example (again) “job on node”-representation: 1 2 3 64 5

5. © J. Christopher Beck 2008 5 Forward Procedure STEP1: For each job that has no predecessors: STEP2: compute for each job j: STEP3: 1 2 3 64 5 S’ 1 = 0 S’ 2 = 0 S’ 3 = 0 C’ 1 = 2 C’ 2 = 3 C’ 3 = 1 S’ 4 = 3 S’ 5 = 3 S’ 6 = 7 C’ 6 = 8 C’ 5 = 5 C’ 4 = 7 C max = 8

6. © J. Christopher Beck 2008 6 Backward Procedure STEP1: For each job that has no successors: STEP2: compute for each job j: STEP3: Verify that: 1 2 3 64 5 S’’ 1 = 1 S’’ 2 = 0 C’’ 1 = 3 C’’ 2 = 3 S’’ 3 = 7S’’ 5 = 6 S’’ 6 = 7 C’’ 3 = 8 C’’ 6 = 8 C’’ 5 = 8 S’’ 4 = 3C’’ 4 = 7 C max = 8

7. © J. Christopher Beck 2008 7 OK so …

8. © J. Christopher Beck 2008 8 Uncertain Processing Times Great, project scheduling is easy! In the real world, do we really know the duration of a job? What if we have estimates of duration? What if we have a distribution: p j = (μ j, δ j )?

9. © J. Christopher Beck 2008 9 Program Evaluation & Review Technique (PERT) Idea: estimate p j and use CPM to estimate: Ê(C max ) – expected makespan Ṽ(C max ) – variance of makespan

10. © J. Christopher Beck 2008 10 Simplest Approach Given p j = (μ j, δ j ), let p j = μ j Use CPM to find critical path Estimate the expected makespan Ê(C max ) = Σ μ j, j in critical path Ṽ(C max ) = Σ (δ j 2 ), j in critical path This is a very crude approximation! See Example 4.3.2 Q: What if there are two CPs?

11. © J. Christopher Beck 2008 11 Estimating (μ j, δ j ) Assume you have 3 estimates of p j Optimistic: p a j Most likely: p m j Pessimistic: p b j Reasonable estimates: μ j = (p a j +4p m j +p b j ) / 6 δ j = (p b j -p a j ) / 6 “No battle plan survives the first encounter with the enemy.”

12. © J. Christopher Beck 2008 12 PERT Steps 1. Find μ j, δ j 2 i.e., using estimates on previous slide 2. Use CPM to find critical path(s) with p j = μ j 3. Estimated expected value and variance of C max Assume makespan is normally distributed

13. © J. Christopher Beck 2008 13 PERT Problems More than one CP? non-CP with high variance? expected makespan must be larger than single CP estimate (why?) Assumption of normal distribution

14. © J. Christopher Beck 2008 14 PERT Practice Jobpajpaj pmjpmj pbjpbj Predecessors 12412- 21015201 368221 4816181 5210182,3,4 6812242 72585 834115 948246,7 101598 Draw precedence graph Find μ j, δ j 2 Find Critical Path(s) Estimate expected value and variance of C max

15. © J. Christopher Beck 2008 15 More PERT Practice Example 4.3.1 Jobs1234567891011121314 pajpaj 44810612451076672 pmjpmj 568117121161087875 pbjpbj 681418812 7101581078 1 24 3 5 7 8 12 14 13 69 11 Find expected makespan and variance Hint: same graph as 4.2.3