Lecture 4:
Program Evaluation and
Review Technique (PERT)
© J. Christopher Beck 2008
1
Outline


Quick CPM Review
Program Evaluation and Review
Technique (PERT)
© J. Christopher Beck 2008
2
Readings


P Ch 4.2, 4.3
Slides borrowed
from Twente &
Iowa

See Pinedo CD
© J. Christopher Beck 2008
3
A Small Example (again)
Job p(j) Predecessors
1 2
2 3
3 1
4 4
1,2
5 2
2
6 1
4
“job on node”-representation:
1
2
3
© J. Christopher Beck 2008
4
6
5
4
Forward Procedure
STEP1: For each job that has no predecessors:
'
'
S

max
C
k
STEP2: compute for each job j: j
S 'j  0
C 'j  p j
kj
C
STEP3: Cmax  max
j
C 'j  S 'j  p j
'
j
C’1 = 2
S’1 = 0
S’6 = 7
1
C’2 = 3
S’2 = 0
S’4 = 3
2
C’4 = 7
4
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
C ''j  Cmax
STEP1: For each job that has no successors: S ''j  C max  p j
''
''
STEP2: compute for each job j: C j  min S k
jall k
STEP3: Verify that: 0  min S
j
''
j
S ''j  C ''j  p j
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




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:

pj = (μj, δj)?
© J. Christopher Beck 2008
8
Program Evaluation & Review
Technique (PERT)

Idea: estimate pj and use CPM to
estimate:


Ê(Cmax) – expected makespan
Ṽ(Cmax) – variance of makespan
© J. Christopher Beck 2008
9
Simplest Approach



Given pj = (μj , δj), let pj = μj
Use CPM to find critical path
Estimate the expected makespan



This is a very crude approximation!


Ê(Cmax) = Σ μj, j in critical path
Ṽ(Cmax) = Σ (δj2), j in critical path
See Example 4.3.2
Q: What if there are two CPs?
© J. Christopher Beck 2008
10
Estimating (μj , δj)

Assume you have 3 estimates of pj




Optimistic: paj
Most likely: pmj
Pessimistic: pbj
Reasonable estimates:


μj = (paj+4pmj+pbj) / 6
δj = (pbj-paj) / 6
© J. Christopher Beck 2008
“No battle plan
survives the first
encounter with the
enemy.”
11
PERT Steps

1. Find μj , δj2


2. Use CPM to find critical path(s)


i.e., using estimates on previous slide
with pj = μj
3. Estimated expected value and
variance of Cmax
Eˆ (Cmax )   ˆ j

Assume makespan is
normally distributed Vˆ (C ) 
max
© J. Christopher Beck 2008
jJ cp
2

 j
jJ cp
12
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 2008
13
PERT Practice




Draw precedence
graph
Find μj , δj2
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
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
pa j
4 4 8 10 6 12 4
5 10 7
6
6
7
2
pmj
5 6 8 11 7 12 11 6 10 8
7
8
7
5
pbj
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