Koç University
OPSM 301 Operations Management
Class 9:
Project Management:
PERT and project crashing
Zeynep Aksin
zaksin@ku.edu.tr
Announcements
 Change in syllabus plan as follows:
– Will swap last session on project
management with decision trees
– Last session of project management will be
after the bayram on 8/11
• Class will be held in the lab (TBA)
• Second group assignment will be due
• We will have quiz 2 on Project Management
– Decision Trees will be on 1/11
• Quiz 3 on 10/11 Thursday
Example
Suppose you are an advertising manager responsible
for the launch of a new media advertising campaign.
The campaign (project) has the following activities:
Activity
A. Media bids
B. Ad concept
C. Pilot layouts
D. Select media
E. Client approval
F. Pre-production
G. Final production
H. Launch campaign
Predecessors
none
none
B
A
A,C
B
E,F
D,G
Time
2 wks
6
3
8
6
8
5
0
CPM with
Three Activity Time Estimates
Task
Immediate
Predecesors
Optimistic
Most Likely
Pessimistic
A
B
C
D
E
F
G
H
I
None
None
A
A
C
D
B
E,F
G,H
3
2
6
2
5
3
3
1
4
6
4
12
5
11
6
9
4
19
15
14
30
8
17
15
27
7
28
Some CPM/PERT Assumptions
 Project control should focus on the critical
path
 The activity times in PERT follow the beta
distribution, with the variance of the project
assumed to equal the sum of the
variances along the critical path
Beta Distribution Assumption
density
Assume a “Beta” distribution
activity
duration
density
Beta Distribution Assumption
activity
duration
a
m
b
Expected Time and Variance
Expected Time =
Variance =
a + 4m + b
6
(b - a)2
36
Expected Times
Task
A
B
C
D
E
F
G
H
I
Immediate Expected
Predecesors
Time
None
None
A
A
C
D
B
E,F
G,H
7
5.333
14
5
11
7
11
4
18
7
0
21
C, 14
7
21
7
21
32
E, 11
21
32
0
Start
0
0 0
36
EF
LS
LF
H, 4
A, 7
0
0
32
ES
7
7
12
12
D, 5
20
0
5.33
B
5.33
19.67 25
19
F, 7
25
25
32
5.33
16.33
G, 11
25 36
32 36
36 54
I, 18
36
54
Expected Completion Time = 54 Days
C, 14
E, 11
H, 4
A, 7
D, 5
F, 7
Start
0
I, 18
B
5.33
G, 11
What is the probability of finishing this project in
less than 53 days? We need the variance also!
Pr(t < D)
t
TE = 54
D=53
z=
D - TE
  cp
2
Task
A
B
C
D
E
F
G
H
I
Optimistic Most Likely Pessimistic
3
2
6
2
5
3
3
1
4
6
4
12
5
11
6
9
4
19
Variance
15
14
30
8
17
15
27
7
28
Sum the variance along the critical path
=
4
16
4
1
16

2
= 41
Pr(t < D)
t
D=53 TE = 54
z=
D - TE

2
cp
=
53 - 54
41
= - .156
p(z < -.156) =.436, or 43.6%
There is a 43.6% probability that this project will be
completed in less than 53 weeks.
PERT Probability Example
You’re a project planner for
General Dynamics. A
submarine project has an
expected completion time
of 40 weeks, with a
standard deviation of 5
weeks. What is the
probability of finishing the
sub in 50 weeks or less?
© 1995
Corel Corp.
Converting to Standardized Variable
X - T 50 - 40
=
= 2 .0
Z =
s
5
Normal
Distribution
Standardized Normal
Distribution
sZ = 1
s =5
T = 40 50
X
mz = 0 2.0
Z
Obtaining the Probability
Standardized Normal Probability
Table (Portion)
Z
.00
.01
.02
sZ =1
0.0 .50000 .50399 .50798
:
:
:
:
2.0 .97725 .97784 .97831
.97725
2.1 .98214 .98257 .98300
mz = 0 2.0
Probabilities in body
Z
Variability of Completion Time for Noncritical
Paths
 Variability of times for activities on
noncritical paths must be considered when
finding the probability of finishing in a
specified time.
 Variation in noncritical activity may cause
change in critical path.
Time-Cost Models
 Basic Assumption: Relationship between
activity completion time and project cost
 Time Cost Models: Determine the
optimum point in time-cost tradeoffs
– Activity direct costs
– Project indirect costs
– Activity completion times
Project Costs vs. Project Duration
Cost Analysis
 We assume a linear relation between
activity duration and activity cost
– Regulate activity durations to minimize the
total project cost
Cost Analysis
 We require two time estimates and two associated cost
estimates:
– Normal Time: Time required if a usual amount of
resources are applied to the activity.
– Normal Cost: Cost of completing an activity in normal
time.
– Crash Time: Least time that an activity can be performed
in if all available resources are applied to it.
– Crash Cost: Cost of completing an activity in crash time.
 Incremental Cost:
 I = (Crash Cost - Normal Cost)/(Normal Time - Crash Time)
 I=Cost of reducing duration of an acitivity by 1 unit of time
Crash and Normal Times and Costs for
Activity B
Steps for Solution
1. Perform PERT analysis using normal times and
calculate I for all critical activities
2. Pick the activity (critical) with the smallest I and
shorten its duration as much as possible. That is,
until
– a. Duration reaches crash time
– b. Another path becomes critical
3. If duration of the project cannot be reduced any
more, then stop; otherwise return to the second step
The above process results in the modified crash
program.
NT,CT
Example
4,2
(normal time, crash time)
4,2
0 A 4
0 4 4
70,10
(crash cost, normal cost)
4 B 8
5 4 9
50,30
5, 2
4 C 9
4 5 9
65,50
ES
EF
LS
NT LF
CC,NC
2,1
9 D
9 2
100,10
11
11
Solution Procedure
 Crash the activity with smallest I (least
cost)
 Check if critical path changed at each step
 Continue crashing until satisfied or not
possible
 Total Cost = Indirect cost + direct cost,
 Minimum Cost schedule is the one that has
minimum total cost
Advantages of PERT/CPM
 Especially useful when scheduling and
controlling large projects.
 Straightforward concept and not
mathematically complex.
 Graphical networks aid perception of
relationships among project activities.
 Critical path & slack time analyses help
pinpoint activities that need to be closely
watched.
 Project documentation and graphics point out
who is responsible for various activities.
 Applicable to a wide variety of projects.
 Useful in monitoring schedules and costs.
Limitations of PERT/CPM
 Assumes clearly defined, independent,
& stable activities
 Specified precedence relationships
 Activity times (PERT) follow
beta distribution
 Subjective time estimates
 Over-emphasis on critical path