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

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

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

© J. Christopher Beck A Small Example (again) “job on node”-representation:

© J. Christopher Beck Forward Procedure STEP1: For each job that has no predecessors: STEP2: compute for each job j: STEP3: 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

© J. Christopher Beck Backward Procedure STEP1: For each job that has no successors: STEP2: compute for each job j: STEP3: Verify that: 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

© J. Christopher Beck OK so …

© J. Christopher Beck 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 )?

© J. Christopher Beck 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

© J. Christopher Beck 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 Q: What if there are two CPs?

© J. Christopher Beck 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.”

© J. Christopher Beck 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

© J. Christopher Beck 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

© J. Christopher Beck PERT Practice Jobpajpaj pmjpmj pbjpbj Predecessors ,3, , Draw precedence graph Find μ j, δ j 2 Find Critical Path(s) Estimate expected value and variance of C max

© J. Christopher Beck More PERT Practice Example Jobs pajpaj pmjpmj pbjpbj Find expected makespan and variance Hint: same graph as 4.2.3