Operations Management
Session 27: Project Management

From Action Plan and WBS to Gantt chart and project network.
 Gantt Chart
 Project Network


Activity-on-arrow
Activity-on-node
 CPM and PERT
 Risk analysis involves determining the likelihood that a project can be completed on time


Statistics
Simulation
Session 27 Operations Management 2

Session 27 Operations Management 3

Late 1950s

Critical Path Method (CPM)


Dupont De Nemours Inc. developed the method
Deterministic activity durations

Program Evaluation and Review Technique (PERT)

U.S. Navy, Booz-Allen Hamilton, and Lockeheed Aircraft

Probabilistic activity durations
Session 27 Operations Management 4

The Language of
PERT/CPM
 Activity


Task or set of tasks
Takes time and needs resources
 Precedence Relationships

The immediate predecessor activities
 Event


Completion of one or more activities (to allow the next activity or activities to start)
Zero duration, zero resource
 Milestones

Significant events – showing completion of a significant portion of the project
Session 27 Operations Management 5

The Language of PERT/CPM
 Network


Diagram of nodes connected by directed arcs
Shows technological relationships among activities
 Path

A set of connected activities such that each activity on both sides is connected to one and only one other activity (with exception!) .
 Critical Path



A path where a delay in any of its activities will delay the project
The longest path on the network
The shortest time to complete the project
 Critical Time

The total time to complete all activities on the critical path
Session 27 Operations Management 6

Two Types of Network
Diagrams
 Activity-on-Arrow Network (Arrow Diagramming
Method)



Easier to show events and milestones
More compatible with network theory techniques
Sometimes requires dummy (artificial) activities
 Activity-on-Node Network (Precedence
Diagramming Method)


Easier representation
No dummy activity
Session 27 Operations Management 7

Activity on Arrow
Network
An Activity is an arc with two nodes at its beginning and its end c d a b
Session 27 Operations Management 8

AoA: Activity
Predecessors
A list of immediate predecessors is needed.
Task Predecessor c d a b a b
-a c b
Task Predecessor
---
-b a a b a b
Session 27 Operations Management d c c
9

Task Predecessor c d a b
--a a a
Task Predecessor a b c d
Session 27
---
--
-a,b,c
AoA: Activity
Predecessors a b c a
Operations Management b d c d
10

AoA May Need Dummy
Activity
 Two activities have the same starting and ending nodes
 A single activity connects to two or more nodes
Task Predecessor c d a b
--a a b,c a
 Try this: a,b  c and a,d  e b c d
Session 27 Operations Management 11

Task f e c d g a b i h j
Session 27
AoA: A Power Plant
Construction Project
Description Predecessor
Design & engineering
Select site
Select vendor
Select personnel
Prepare site
Manufacture generator
Prepare operation manual
Install generator
Train operators
Obtain license
Operations Management b c a a c
--a e,f d,g h,i
12

a
AoA: A Power Plant
Construction Project
Session 27 b c d e f g i h
Operations Management j a:b:a c:a d:a e: b f:c g:c h:e,f i: d,g j: h,i
13

Draw AoN Network
Session 27 Operations Management a:b:a c:a d:a e: b f:c g:c h:e,f i: d,g j: h,i
14

Session 27
Draw AOA
Activity Predecessor Duration c d a b e f k j i g h
-
-
a a c b,e,f c b,e,f h d,i
5
4
6
2
3
8
4
9
9
7
6
Operations Management 15

1
Session 27
Transform into AON
2 b = 2
3 d = 4
4
6 g = 9
5 h = 9
Operations Management
7 a:b:c:d:a e: a f:c g:b,e,f h:c i: b,e,f j: h k:d,i
16

Session 27
Draw AoN Network a:b:c:d:a e: a f:c g:b,e,f h:c i: b,e,f j: h k:d,i
17 Operations Management

Critical Path and Critical time
 The critical path is the shortest time in which a project can be completed
 If a critical activity is delayed, the entire project will be delayed.
 There may be more than one critical path.
 Brute force approach to finding critical path:
1.
2.
3.
identify all possible paths from start to finish sum up duration of activities on each path largest total indicates critical path
Session 27 Operations Management 18

S
4
A1
6
A3 E
3
A2
4
A4
2
A5
3
A6
Find the Critical Path.
Session 27 Operations Management 19

4
A1
6 How many path?
A3 E
S 4
A4
3
A6
3
2
10 11 8
A2
A5
Critical Path is the longest path. It is the shortest time to complete the project
Session 27 Operations Management 20

S
0
0
6 4
4
A1 A3
10 E
11 11
0 0 4 4 10
3
4
4
4
A4
8
8
5
8
A6
11
3
2
0
A2
A5
3
0 3
3 5
Session 27 Operations Management 21

Forward Path
10
30
20
Max = 30
5
35
35
35
Session 27 Operations Management 22

S
0
0
0 3
0
0 4
A1
4
5
4 4
5 6
A3
11
11
10
11
11
E
11
0 0
3
3 6
4
4
4
4
4
6
4
A4
2
8
10
8
8
8
8
8
5
8
8 3 11 11
A6
11
A2
6
A5
3
0 3
3 5
Session 27 Operations Management 23

Backward Path
30
30
30
30
5
Min = 35
35
45
Session 27 Operations Management 24

Activity Slack
 Slack, or float: The amount of time a noncritical task can be delayed without delaying the project
 Slack—LFT – EFT or LST – EST
 EST—Earliest Start Time

Largest EFT of all predecessors
 EFT—Earliest Finish Time

EST + duration for this task
 LFT—Latest Finish Time

Smallest LST of following tasks
 LST—Latest Start Time

LFT – duration for this task
Session 27 Operations Management 25

Session 27
Computing Slack Times
EST EFT
LST
Task = duration slack = xxxx
LFT
Operations Management 26

S
0 4
A1
4
0 4
0
3 3
6
A2
3
Session 27
4
4
5 6
A3
11
4
10
8
A4
4
6 2
A5
8
8
3 5
Operations Management
8 3 11
A6
8 11
27
11
E
11

Slack Times Example
Task Pred. Dur.
a -4 b -c a d a 2
3
3
Task Pred. Dur.
g c,d 1 j i h f e e,g 6
4
5 f e b b
6
4 k h,i 1
For each task, compute ES, EF, LF, LS, slack
Session 27 Operations Management 28

Start
Session 27
Slack Times Example c=3 slack=
LST LFT
Task=dur slack=xxx
EST EFT a=4 slack= g=1 slack= d=2 slack= j=6 slack=
Finish e=6 slack= h=4 slack= b=3 slack= k=1 slack= f=4 i=5 slack= slack=
Operations Management 29

Activity Times in PERT
 Optimistic ( a )

Activity duration to be ≤ a has 1% probability.  ≥ a has 99% probability
 Pessimistic ( b)

Activity duration to be ≥ b has 1% probability  ≤ b has
99% probability
 Most likely ( m )

The mode of the distribution
 All possible task durations (or task costs) can be represented by statistical distributions
Session 27 Operations Management 30

Beta Distribution: The Probability
Distribution of Activity Times
Session 27 Operations Management 31

Activity Expected Time and
Variance
 Mean, “expected time”
T
E
= ( a + 4 m + b )/6

Standard deviation,


= ( b a )/6

Variance

2 = [( b-a )/6] 2
Session 27 Operations Management 32

95% & 90% Levels
 If we replace 99% with 95% or 90% levels


Activity duration to be ≤ a has 5% probability
Activity duration to be ≥ b has 5% probability
 
( b
 a )
3 .
3


Activity duration to be ≤ a has 10% probability
Activity duration to be ≥ b has 10% probability
 
( b
 a
2 .
6
)
Session 27 Operations Management 33

Probability of Completing the Critical Path on Time
 We assume the various activities are statistically independent of each other
 Individual variances (and mean) of the activities on a path can then be summed to find the variance (mean) of the path
 Determine the mean and standard deviation of the critical path
 Compute the probability of critical path being ≤ a
Session 27 Operations Management 34

The Probability of Completing the
Critical Path on Time
Z

D
CP


2
CP m
CP
D
CP m
CP
= the desired completion date of the critical path
= the sum of the
T
E for the activities on the critical path
 2
CP
= the sum of the variances of the activities on the critical path
Given Z, the probability of having the standard normal variable being ≤ Z is the probability of completing the project in a time ≤
D
Session 27
Ardavan Asef-Vaziri
Operations Management 35

Selecting Risk and
Finding D
Select the probability of meeting the completion date and solve for the desired date, D
D
CP
 m 
CP
Z

CP
Using the probability, you can compute Z and then solve for D
4/15/2020
Session 27 Operations Management 36

Probability of Completing a
Project on Time
 Find all paths in the network
 Compute mean and standard deviation of each path
 Compute the probability of completing each path in ≤ the given time
 Calculate the probability that the entire project is completed within the specified time by multiplying these probabilities together
Session 27 Operations Management 37

Critical Path Method: Paths
Suppose all activities have beta distribution
4,1 6,2
A1 A3 E
S
3,0.5
A2
4,1
A4
2,0.5
A5
10 11
3,1
A6
8
The first number is the mean; the second is standard deviation.
Session 27 Operations Management 38

Probability of Completing
CP in 12 days
What is the probability of competing the critical path in a maximum of 12 days?
D
CP m
CP
= the desired completion date of the critical path
= 4+4+3 = 11
 2
CP

CP
= 1 2 +1
= 1.73
2 +1 2 = 3
Z

D
CP


CP m
CP

12

11
1 .
73

0 .
58
Z= 0.58  P(z≤0.58) = 0.72
Session 27 Operations Management 39

Selecting Risk and
Finding CP Time
With a probability of 90%, in how many days will the CP be completed?
From Standard Normal Table

Z
90%
= 1.28
D
CP
 m
CP

Z

CP
D
CP m
CP

11 ,

CP

1 .
73 , Z
% 90

1 .
28

11

1 .
28

1 .
73

11

2 .
21

13 .
21
Session 27 Operations Management 40

Probability of Completing
The Project in 12 days
The probability of completing the critical path in not more than 12 days was 0.72. We need to compute this probability for blue path and green path too, and then multiply these probabilities m
CP
= 6+4 = 10
 2
CP
 m
CP
CP
 2
CP

CP
= 1 2
= 1.1
+2
= 0.5 2
2 = 5
= 2.24
= 3+2+3 = 8
+0.5 2
Z= (12-10)/2.24 = 0.89
+1 2 = 1.22
 P(z≤0.89) = 0.81
Z= (12-8)/1.1 = 3.63  P(z≤3.63) ≈ 1
The probability of competing the Project in not more than 12 days is
0.72
×
0.81
×
1 = 0.58
Session 27 Operations Management 41