Lecture 4 Program Evaluation and Review Technique PERT

Lecture 4: Program Evaluation and Review Technique (PERT) © J. Christopher Beck 2008 1

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

Readings n n P Ch 4. 2, 4. 3 Slides borrowed from Twente & Iowa n See Pinedo CD © J. Christopher Beck 2008 3

A Small Example (again) “job on node”-representation: 1 2 3 © J. Christopher Beck 2008 4 6 5 4

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

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

OK so … © J. Christopher Beck 2008 7

Uncertain Processing Times n n 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: n pj = (μj, δj)? © J. Christopher Beck 2008 8

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

Simplest Approach n n n Given pj = (μj , δj), let pj = μj Use CPM to find critical path Estimate the expected makespan n This is a very crude approximation! n n Ê(Cmax) = Σ μj, j in critical path Ṽ(Cmax) = Σ (δj 2), j in critical path See Example 4. 3. 2 Q: What if there are two CPs? © J. Christopher Beck 2008 10

Estimating (μj , δj) n Assume you have 3 estimates of pj n n Optimistic: paj Most likely: pmj Pessimistic: pbj Reasonable estimates: n n μj = (paj+4 pmj+pbj) / 6 δj = (pbj-paj) / 6 © J. Christopher Beck 2008 “No battle plan survives the first encounter with the enemy. ” 11

PERT Steps n 1. Find μj , δj 2 n n 2. Use CPM to find critical path(s) n n i. e. , using estimates on previous slide with pj = μj 3. Estimated expected value and variance of Cmax n Assume makespan is normally distributed © J. Christopher Beck 2008 12

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

PERT Practice n n Draw precedence graph Find μj , δj 2 Find Critical Path(s) Estimate expected value and variance of Cmax © J. Christopher Beck 2008 Job paj pmj pbj Predecessors 1 2 4 12 - 2 10 15 20 1 3 6 8 22 1 4 8 16 18 1 5 2 10 18 2, 3, 4 6 8 12 24 2 7 2 5 8 3 4 11 5 9 4 8 24 6, 7 10 1 5 9 8 14

More PERT Practice Example 4. 3. 1 Jobs 1 2 3 4 5 6 7 8 9 10 11 12 13 14 paj 4 4 8 10 6 12 4 5 10 7 6 6 7 2 p mj 5 6 8 11 7 12 11 6 10 8 7 5 p bj 6 8 14 18 8 12 12 7 10 15 8 10 7 8 Hint: same graph as 1 4. 2. 3 2 3 © J. Christopher Beck 2008 4 6 7 9 10 11 5 8 Find expected makespan and variance 12 14 13 15