PERT/CPM-1

1
Project Management with PERT/CPM
2
PERT/CPM

• PERT program evaluation and review technique
• CPM critical path method
• Use a project network, Activity-on-Node (AON)
• Nodes activities, or tasks, to be performed
• Arcs show immediate predecessors to an activity
• Times duration times of activities are written
  next to the node

3
Reliable Construction Company Example

• Activity list for the Reliable Construction Co.
  project

Activity Activity Description Immediate Predecessors Estimated Duration
A Excavate - 2 weeks
B Lay the foundation A 4 weeks
C Put up the rough wall B 10 weeks
D Put up the roof C 6 weeks
E Install the exterior plumbing C 4 weeks
F Install the interior plumbing E 5 weeks
G Put up the exterior siding D 7 weeks
H Do the exterior painting E,G 9 weeks
I Do the electrical work C 7 weeks
J Put up the wallboard F,I 8 weeks
K Install the flooring J 4 weeks
L Do the interior painting J 5 weeks
M Install the exterior fixtures H 2 weeks
N Install the interior fixtures K,L 6 weeks


4
The project network for the Reliable Construction
Co. project
START
0
Activity Code A. Excavate B. Foundation C. Rough
wall D. Roof E. Exterior plumbing F. Interior
plumbing G. Exterior siding H. Exterior
painting I. Electrical work J. Wallboard K.
Flooring L. Interior painting M. Exterior
fixtures N. Interior fixtures
A
2
B
4
10
C
I
E
D
7
4
6
5
F
G
7
J
8
H
9
K
L
4
5
M
N
2
6
FINISH
0
5
Bake a Cake Example
Task Immediate predecessors Task Time
A Buy frosting ingredients ½ hr
B Clean up kitchen 1 hr
C Buy cake ingredients ½ hr
D Prepare frosting A,B ¼ hr
E Prepare batter bake B,C 2 hrs
F Frost cake D,E ½ hr
6
Critical Path

• A path through a project network is a route from
  START to FINISH.
• The length of path is the sum of the task times
  (durations) of the nodes (activities) on the
  path.
• The critical path is the longest path.
• The project duration is the length of the longest
  path.

7

• The paths and lengths through Reliables project
  network

Path Length
START ?A ?B ?C ?D ?G ?H ?M?FINISH START ?A ?B ?C ?E ?H ?M ?FINISH START ?A ?B ?C ?E ?F ?J ?K ?N ?FINISH START ?A ?B ?C ?E ?F ?J ?L ?N ?FINISH START ?A ?B ?C ?I ?J ?K ?N ?FINISH START ?A ?B ?C ?I ?J ?L ?N ?FINISH 2 4 10 6 7 9 2 40 weeks 2 4 10 4 9 2 31 weeks 2 4 10 4 5 8 4 6 43 weeks 2 4 10 4 5 8 5 6 44 weeks 2 4 10 7 8 4 6 41 weeks 2 4 10 7 8 5 6 42 weeks
8
To Find the Critical Path and Slacks

• Work from top to bottom in the network,
  calculating
• ES earliest start time for an activity
• EF earliest finish time for an activity
• ES for activity i largest EF of the immediate
  predecessors
• ES 0 if no immediate predecessors
• EF ES activity duration time
• Work from bottom to top in the network,
  calculating
• LS latest start time for an activity
• LF latest finish time for an activity
• LS LF activity duration time
• LF for activity i smallest LS of the immediate
  successors
• LF at Finish EF at Finish if no immediate
  successors
• Slack LF - EF LS - ES
• If slack is zero, the activity is on the
  critical path.

9
The complete project network showing ES, LS, EF
and LF for each activity of the baking example
S (ES, LS) F (EF, LF) Slack LS ES LF -
EF
S( ) F( ) Slack
0
Start
S( ) F( ) Slack
1
S( ) F( ) Slack
B
½
S( ) F( ) Slack
A
½
C
S( ) F( ) Slack
S( ) F( ) Slack
2
¼
D
E
F
S( ) F( ) Slack
½
S( ) F( ) Slack
0
Finish

1. Work down the network calculating ES and EF (ES
   at Start 0, EF at Start 0)
2. Work backward up the network calculating LS and
   LF (LF and LS at Finish is EF and ES at Finish)

10
The complete project network showing ES, LS, EF
and LF for each activity of the baking example
S (ES, LS) F (EF, LF) Slack LS ES LF -
EF
S(0, 0) F(0, 0) Slack0
0
Start
S(0, 0) F(1,1) Slack0
1
S(0, 2 ¼ ) F(½, 2 ¾ ) Slack2 ¼
B
½
S(0, ½) F(½, 1) Slack ½
A
½
C
S(1, 2 ¾) F(1 ¼, 3) Slack1 ¾
S(1, 1) F(3, 3) Slack0
2
¼
D
E
F
S(3, 3) F(3 ½, 3 ½) Slack0
½
S(3 ½, 3 ½) F(3 ½, 3 ½) Slack0
0
Finish
Critical Path is Start ?B?E ?F ?Finish Activity D
has slack of 1¾ hours (Start of D could be
delayed without affecting total project duration)
Also, activities A and C have slack of 2 ¼ and ½
respectively.
11
The complete project network showing ES, LS, EF
and LF for each activity of the Reliable
Construction Co. project
S (0, 0) F (0, 0)
0
START
S (ES, LS) F (EF, LF)
S (0, 0) F (2, 2)
A
2
S (2, 2) F (6, 6)
B
4
S (6, 6) F (16, 16)
10
C
S (16,20) F (22,26)
S (16, 16) F (20, 20)
D
S (16,18) F (23,25)
I
4
E
6
7
S (20,20) F (25,25)
S (22,26) F (29,33)
5
G
7
S (25,25) F (33,33)
F
J
8
S (29,33) F (38,42)
H
9
S (33,34) F (37,38)
S (33,33) F (38,38)
K
4
L
5
N
S (38,42) F (40,44)
M
2
S (38,38) F (44,44)
6
S (44,44) F (44,44)
0
FINISH
12

• Slack for Reliables activities

Activity Slack (LF - EF) On Critical Path?
A 0 Yes
B 0 Yes
C 0 Yes
D 4 No
E 0 Yes
F 0 Yes
G 4 No
H 4 No
I 2 No
J 0 Yes
K 1 No
L 0 Yes
M 4 No
N 0 Yes
13
The spreadsheet used by MS project for entering
the activity list for the Reliable Construction
Co. project
14
Incorporate Uncertain Activity Duration Times
(Probabilistic)

• PERT Three-Estimate Approach
• m most likely estimate of activity duration
  time
• o optimistic estimate of activity duration time
• p pessimistic estimate of activity duration
  time
• Assume Beta distribution of activity time
• Approximately

15
Expected value and variance of the duration of
each activity for Reliables project
Activity Optimistic Estimate o Most Likely Estimate m Pessimistic Estimate p
A 1 2 3 2 1/9
B 2 3 ½ 8 4 1
C 6 9 18 10 4
D 4 5 ½ 10 6 1
E 1 4 ½ 5 4 4/9
F 4 4 10 5 1
G 5 6 ½ 11 7 1
H 5 8 17 9 4
I 3 7 ½ 9 7 1
J 3 9 9 8 1
K 4 4 4 4 0
L 1 5 ½ 7 5 1
M 1 2 3 2 1/9
N 5 5 ½ 9 6 4/9
16
The paths and path lengths through Reliables
project network when the duration of each
activity equals its pessimistic estimate
Path Length
START ?A ?B ?C ?D ?G ?H ?M?FINISH START ?A ?B ?C ?E ?H ?M ?FINISH START ?A ?B ?C ?E ?F ?J ?K ?N ?FINISH START ?A ?B ?C ?E ?F ?J ?L ?N ?FINISH START ?A ?B ?C ?I ?J ?K ?N ?FINISH START ?A ?B ?C ?I ?J ?L ?N ?FINISH 3 8 18 10 11 17 3 70 weeks 3 8 18 5 17 3 54 weeks 3 8 18 5 10 9 4 9 66 weeks 3 8 18 5 10 9 7 9 69 weeks 3 8 18 9 9 4 9 60 weeks 3 8 18 9 9 7 9 63 weeks
17

• For a path (typically the critical path), find
  the mean length (time) µp and the variance sp2
• mean length of path µp sum of mean activity
  times on the path
• (because EXY EX EY)
• Variance of length of path sp2 sum of variances
  of activity times on path
• (because we assume independence
• Var XY Var X Var Y if X,Y
  independent)
• Example
• critical path Start ?A ?B ?C ?E ?F ?J ?L ?N
  ?Finish
• µp 2 4 10 4 5 8 5 6 44
• sp2 1/9 1 4 4/9 1 1 1 4/9 9

18
Find the probability the project is completed in
47 weeks
using an assumption of a Normal distribution
19
Time-Cost Trade-offs

• If one can expedite the project, use
  money/resources to reduce task times, what is the
  best way to allocate money?
• For an activity, could pay extra to reduce time
  (crash)
• How much would it cost to reduce total project
  duration from 44 weeks to 40 weeks? Which
  activities should be crashed?
• Could calculate crash cost and crash time for all
  possible paths but can also apply LP!

20

• Time-cost trade-off data for the activities of
  Reliables project

Activity Time Time Cost Cost Maximum Reduction in Time Crash Cost per Week Saved
Activity Normal Crash Normal Crash Maximum Reduction in Time Crash Cost per Week Saved
A 2 weeks 1 week 180,000 280,000 1 week 100,000
B 4 weeks 2 weeks 320,000 420,000 2 weeks 50,000
C 10 weeks 7 weeks 620,000 860,000 3 weeks 80,000
D 6 weeks 4 weeks 260,000 340,000 2 weeks 40,000
E 4 weeks 3 weeks 410,000 570,000 1 week 160,000
F 5 weeks 3 weeks 180,000 260,000 2 weeks 40,000
G 7 weeks 4 weeks 900,000 1,020,000 3 weeks 40,000
H 9 weeks 6 weeks 200,000 380,000 3 weeks 60,000
I 7 weeks 5 weeks 210,000 270,000 2 weeks 30,000
J 8 weeks 6 weeks 430,000 490,000 2 weeks 30,000
K 4 weeks 3 weeks 160,000 200,000 1 week 40,000
L 5 weeks 3 weeks 250,000 350,000 2 weeks 50,000
M 2 weeks 1 week 100,000 200,000 1 week 100,000
N 6 weeks 3 weeks 330,000 510,000 3 weeks 60,000
If do all tasks normal, 44 weeks, cost is 4.55
million. If do all tasks crash, 28 weeks, cost is
6.15 million.
21
Marginal Cost Analysis

• For small networks, may reduce the project 1 week
  at a time, and observe the changes.

Activity to Crash Crash Cost Length of Path Length of Path Length of Path Length of Path Length of Path Length of Path
Activity to Crash Crash Cost ABCDGHM ABCEHM ABCEFJKN ABCEFJLN ABCIJKN ABCIJLN
40 31 43 44 41 42
J 30,000 40 31 42 43 40 41
J 30,000 40 31 41 42 39 40
F 40,000 40 31 40 41 39 40
F 40,000 40 31 39 40 39 40
22
Linear Programming to Make Crashing Decisions

• Let
• Z total cost of crashing on any activity
• xj reduction in the duration of activity j due
  to crashing
• j A,B,C,,N
• xj maximum reduction time normal time
  crash time
• yFINISH project duration, time at which FINISH
  node is reached
• yj start time of activity j
• yj yi normal timei xi i is an immediate
  predecessor of j

23
Linear Programming Model
24
The schedule of cumulative project costs when all
activities begin at their earliest start times or
at their latest start times
25
Time-cost trade-off data for the activities of
the baking project
Activity Normal Time Crash Time Normal Cost Crash Cost
A Buy frosting 0.5 0.25 0 5
B Clean kitchen 1.0 0.25 0 10
C Buy cake 0.5 0.25 0 5
D Prepare frosting 0.25 0.25 0 0
E Prepare batter bake 2.0 1.5 0 5
F Frost cake 0.5 0.25 0 5
26
The project network showing Normal Time and Crash
Time for each activity of the baking project
N Normal time C Crash time
Start
If all are normal ADF 1 ¼ BDF 1 ¾ BEF 3
½ CEF 3 Total time is 3 ½. If all are
crashed ADF ¾ BDF ¾ BEF 2 CEF
2 Total time is 2.
N C 1 , 1/4
N C 1/2 , 1/4
C
A
N C 1/2 , 1/4
B
E
D
N C 2 , 1 1/2
N C 1/4 , 1/4
F
N C 1/2, 1/4
Finish
27

• Could solve the crash LP for finish times between
  3.5 and 2 to evaluate alternatives

25
16.67
Cost
11.67
8.33
5
2.5
0
2
2.25
2.5
2.75
3
3.25
3.5
T y
Finish
28
Lasagna Dinner Example
Task Tasks that must precede Time
A Buy the mozzarella cheese 30 mins
B Slice the mozzarella A 5 mins
C Beat 2 eggs 2 mins
D Mix eggs and ricotta cheese C 3 mins
E Cut up onions and mushrooms 7 mins
F Cook the tomato sauce E 25 mins
G Boil large quantity of water 15 mins
H Boil the lasagna noodles G 10 mins
I Drain the lasagna noodles H 2 mins
J Assemble all the ingredients I, F, D, B 10 mins
K Preheat the oven 15 mins
L Bake the lasagna J, K 30 mins
29
Construct project network
Start
G
15
E
7
K
C
15
A
2
30
H
10
F
25
D
3
B
5
I
2
J
A?B ?J ?L 75 C?D ?J ?L 45 E?F ?J
?L 73 G?H ?I ?J ?L 45
10
L
30
Finish
30
EF ES activity time (or duration) if no
predecessors, ES 0 otherwise ES max (EF)
(immediate predecessors) work forward through
network
Start
ES0EF15
ES0EF7
ES0EF2
ES0EF30
G
15
E
7
ES0EF15
A
C
K
30
15
2
ES15EF25
H
10
ES7EF32
F
25
ES30EF35
B
5
ES25EF27
I
2
J
ES35EF45
10
ES45EF75
L
30
Finish
31
The complete project network showing ES, LS, EF,
LF and Slack for each activity of the Lasagna
dinner example
S (ES, LS) F (EF, LF) Slack LF-EFLS-ES
Start
S(0,0) F(30,30) Slack0
S(0,30) F(15,45) Slack30
S(0,3) F(7,10) Slack3
S(0,10) F(15,25) Slack10
A
S(0,30)F(2,32) slack30
K
G
15
30
C
15
E
7
2
S(15,25) F(25,35) Slack10
S(30,30) F(35,35) Slack0
S(7,10) F(32,35) Slack3
H
S(2,32) F(5,35) slack30
10
B
5
F
25
D
3
S(25,33) F(27,35) Slack8
I
2
J
S(35,35) F(45,45) Slack0
10
S(45,45) F(75,75) Slack0
L
30
Critical path has zero slack A?B ?J ?L
Finish
32

• Because of a phone call, you will delayed by 6
  minutes to
• cut onions and mushrooms (Task E) . By how much
  will
• dinner be delayed?
• slack is 3, delay of 6 minutes will delay dinner
  by 6-33
• If you use your food processor instead to reduce
  cutting
• time from 7 minutes to 2 minutes, will dinner
  still be
• delayed?

ES0 LS8 LS2 LF10
E
2
slack 8, so phone of 6 wont delay dinner
33

• All of the critical path, ES, LS, EF, LF, are
  based on estimates of the activity times.
• How can you incorporate uncertainty into
  planning?
• PERT 3 estimate approach
• Most Likely Estimate (m) most probable
  event
• Optimistic Estimate (s) if everything
  goes perfectly
• Pessimistic Estimate (p) if everything
  goes wrong

34
What is the probability of meeting your deadline?

• Assume the distribution for activity time is a
  Beta distribution

density f(t)
t time
µ - 3s
µ 3s
µ 3s interval captures 99.73 of distribution
35
Consider Critical Path
o10 m30 p50
A
µ 30 Buy mozzarella cheese s2
(6 2/3)2
B
µ 5 Slice cheese s2 (11/3)2
o3 m4 p11
J
µ 10 Assemble s2 (2)2 4
L
µ 30 Bake s2 (3 2/3)2
o20 m30 p40
36

• Could calculate pessimistic length
• 50 11 14 40 115
• Longest pessimistic path may not be the critical
  mean path
• The mean length is sum of means
• 30 5 10 30 75 µp
• Assume all task times are independent, variance
  for path is sum of variances

37

• Assume distribution of path time is normal
  (central limit theorem if lots of tasks on a path)