1. Chapter 8 - Project Management Chapter Topics
The Elements of Project Management
The Project Network
Probabilistic Activity Times
Project Crashing and Time-Cost Trade-Off
Formulating the CPM/PERT Network as a Linear Programming Model
Chapter 8 - Project Management

2. Project Management Overview
Uses networks for project analysis.
Networks show how projects are organized and are used to determine time duration for completion.
Network techniques used are
- CPM (Critical Path Method)
- PERT (Project Evaluation and Review Technique)
Developed during late 1950s.
Chapter 8 - Project Management

3. The Elements of Project Management
Management is generally perceived as concerned with planning, organizing, and control of an ongoing process or activity.
Project Management is concerned with control of an activity for a relatively short period of time after which management effort ends
Primary elements of Project Management to be discussed:
- Project team
- Project planning
- Project control.
Chapter 8 - Project Management

4. The Elements of Project Management The Project team
Project team typically consists of a group of individuals from various areas in an organization and often includes outside consultants.
Members of engineering staff often assigned to project work.
Most important member of project team is the project manager.
Project manager is often under great pressure because of uncertainty inherent in project activities and possibility of failure.
Project manager must be able to coordinate various skills of team members into a single focused effort.
Chapter 8 - Project Management

5. Project Planning: PERT/CPM
Program Evaluation and Review Technique
Developed by U.S. Navy for Polaris missile project
Developed to handle uncertain activity times
CPM
Critical Path Method
Developed by Du Pont & Remington Rand
Developed for industrial projects for which activity times generally were known
Today’s project management software packages have combined the best features of both approaches.
Chapter 8 - Project Management

6. Chapter 8 - Project Management
PERT/CPM
PERT and CPM have been used to plan, schedule, and control a wide variety of projects:
R&D of new products and processes
Construction of buildings and highways
Maintenance of large and complex equipment
Design and installation of new systems
Chapter 8 - Project Management

7. Chapter 8 - Project Management
PERT/CPM
Project managers rely on PERT/CPM to help them answer questions such as:
What is the total time to complete the project?
What are the scheduled start and finish dates for each specific activity?
Which activities are critical and must be completed exactly as scheduled to keep the project on schedule?
How long can noncritical activities be delayed before they cause an increase in the project completion time?
Chapter 8 - Project Management

8. PERT Network: Activity-on-Node Approach
A PERT network can be constructed to model the precedence of the activities.
The arcs of the network represent the precedence relationships of the activities.
The nodes (rectangles) of the network represent activities.
You will need to add a “Start” and a “Finish” nodes.
Chapter 8 - Project Management

9. PERT Network: Activity-on-Node Approach
Activity time estimates usually can not be made with certainty.
In the three-time estimate approach, the time to complete an activity is assumed to follow a Beta distribution.
An activity i’s mean completion time is:
ti = (ai + 4mi + bi)/6
An activity’s completion time variance is:
s2i = ((bi-ai)/6)2
ai = activitity i’s optimistic completion time estimate
bi = activitity i’s pessimistic completion time estimate
mi = activitity i’s most likely completion time estimate
Chapter 8 - Project Management

10. PERT Network: Activity-on-Node Approach
In the three-time estimate approach, the critical path is determined as if the mean times for the activities were fixed times.
The expected project time is the sum of the expected times of the critical path activities.
The project variance is the sum of the variances of the critical path activities.
The expected project time is assumed to be normally distributed (based on central limit theorem).
Chapter 8 - Project Management

11. PERT Analysis Algorithm
Step 1: Make a forward pass through the network as follows: For each of these activities, i, compute:
Earliest Start (ES) Time = the maximum of all earliest finish times for all its immediate predecessors. (For node “START”, this is 0.)
ESi= Maximum (EFj) for all immediate proceeding activities j.
Earliest Finish (EF) Time = (Earliest Start Time) + (Time to complete activity i).
EFi= ESi+ ti
The project completion time is the of the Earliest Finish Times at the “FINISH” node.
This will also be used as Latest Finish Time at “FINISH” node in the next step.
Chapter 8 - Project Management

12. PERT Analysis Algorithm
Step 2: Make a backwards pass through the network as follows: Move sequentially backwards from the last node, “FINISH” to its immediate predecessors, etc. At a given node, j, consider all activities immediately following it and compute:
Latest Finish (LF) Time = the minimum of the latest start times for all activities that immediately follow j. (For node “FINISH”, this is the project completion time.)
LFj= Minimum (LSi) for all immediate following activities i.
Latest Start (LS) Time = (Latest Finish Time) - (Time to complete activity j).
LSj= LFj - tj
Chapter 8 - Project Management

13. PERT Analysis Algorithm
Step 3: Calculate the slack time for each activity by:
Slack = (Latest Start) - (Earliest Start) or
= (Latest Finish) - (Earliest Finish).
A critical path is a path of activities, from node “START” to “FINISH”, with 0 slack times.
Shared slack is slack available for a sequence of activities.
Chapter 8 - Project Management

14. Example: Riverwalk Associates
Riverwalk Associates is in the business of building elaborate parade floats. Its crew has a new float to build and want to use PERT/CPM to help them manage the project .
The table on the next slide shows the activities that comprise the project. Each activity’s estimated completion time (in weeks) and immediate predecessors are listed as well.
The project manager wants to know the total time to complete the project, which activities are critical, and the earliest and latest start and finish dates for each activity.
Chapter 8 - Project Management

15. Example: Riverwalk Associates
Project activity initial information:
Immed. Optimistic Most Likely Pessimistic
Activity (i) Predec. Time (weeks) Time (wk.) Time (wk.)
A —
B —
C A
D A
E A
F B,C
G B,C
H E,F
I E,F
J D,H
K G,I
Chapter 8 - Project Management

16. Example: Riverwalk Associates
Activity Expected Time and Variances
ti = (ai + 4mi + bi)/6 s2i = ((bi-ai)/6)2
Activity (i) Expected Time Variance (week2)
A /9
B /9 C
D /9
E /36
F /9
G /9
H /9 I
J /9
K /9
Chapter 8 - Project Management

17. Example: Riverwalk Associates
PERT Activity Node Representation
Earliest Start
Earliest Finish A 6
ES EF
LS LF
Expected
Duration of
the activity
Latest Start
Latest Finish
Chapter 8 - Project Management

18. Example: Riverwalk Associates
PERT Network Representation J 3 C 3 A 6 H 6 D 5
FINISH
START I 5 E 1 F 4 K 5 B 4 G 2
Chapter 8 - Project Management

19. Example: Riverwalk Associates
Earliest/Latest Times
Activity ES EF LS LF Slack
A *critical B
C * D E
F * G H
I * J
K *
The estimated project completion time is t0 = 23 (weeks) at FINISH.
Chapter 8 - Project Management

20. Riverwalk Associates – Linear Programming Form
Define variables for each activity in the following manner:
ES_i = Earliest Start time for activity i
EF_i = Earliest Finish time for activity i
LS_i = Latest Start time for activity i
LF_i = Latest Finish time for activity i
where i=A, B…K;
and FINISH = the earliest and also latest completion time of the project.
Chapter 8 - Project Management

21. Riverwalk Associates – LP Model 1 for Earliest Times
Minimize ES_A + EF_A + ES_B + EF_B + … + EF_K + FINISH
S.t.
EF_A - ES_A >= 6 (“=“ OK)
EF_B - ES_B >= 4
EF_C - ES_C >= 3
EF_D - ES_D >= 5
EF_E - ES_E >= 1
EF_F - ES_F >= 4
EF_G - ES_G >= 2
EF_H - ES_H >= 6
EF_I - ES_I >= 5
EF_J - ES_J >= 3
EF_K - ES_K >= 5 (“=“ OK)
ES_C - EF_A >= 0 (Not “=“ )
ES_D - EF_A >= 0
ES_E - EF_A >= 0
ES_F - EF_B >= 0
ES_G - EF_B >= 0
ES_F - EF_C >= 0
ES_G - EF_C >= 0
ES_J - EF_D >= 0
ES_J - EF_H >= 0
ES_H - EF_E >= 0
ES_H - EF_F >= 0
ES_I - EF_E >= 0
ES_I - EF_F >= 0
ES_K - EF_G >= 0
ES_K - EF_I >= 0
FINISH - EF_J >= 0
FINISH - EF_K >= 0
Chapter 8 - Project Management

22. Riverwalk Associates – LP Model 2 for Latest Times
Maximize LS_A + LF_A + LS_B + LF_B + … + LF_K
S.t.
LF_A - LS_A >= 6 (“=“ OK) LF_B - LS_B >= 4
LF_C - LS_C >= 3
LF_D - LS_D >= 5
LF_E - LS_E >= 1
LF_F - LS_F >= 4
LF_G - LS_G >= 2
LF_H - LS_H >= 6
LF_I - LS_I >= 5
LF_J - LS_J >= 3
LF_K - LS_K >= 5 (“=“ OK)
LS_C - LF_A >= 0 (Not just “=“)
LS_D - LF_A >= 0
LS_E - LF_A >= 0
LS_F - LF_B >= 0
LS_G - LF_B >= 0
LS_F - LF_C >= 0
LS_G - LF_C >= 0
LS_J - LF_D >= 0
LS_J - LF_H >= 0
LS_H - LF_E >= 0
LS_H - LF_F >= 0
LS_I - LF_E >= 0
LS_I - LF_F >= 0
LS_K - LF_G >= 0
LS_K - LF_I >= 0
FINISH - LF_J >= 0
FINISH - LF_K >= 0
FINISH = 23 (from the optimal results of Model 1) – this is the main diff.)
Chapter 8 - Project Management

23. Example: Riverwalk Associates
Probability the project will be completed within t1=24 weeks
project time variance s2 = s2A + s2C + s2F + s2I + s2K
= 4/ / /9
= 2 (weeks-squared)
project time standard deviation s = (weeks).
z1 = ( )/ s = (24-23)weeks/1.414weeks = .71
From the Standard Normal Distribution table:
P(z < z1=.71) = =
More precisely,
P(t < t1) = P(t-t0 < t1-t0) = P[(t-t0)/s < (t1-t0)/s]
= P(z < z1=.71) = .7611
if we define z= (t-t0)/s and z1 = (t1-t0)/s.
Chapter 8 - Project Management

24. Probability Analysis of the Project: Example 2
Question: If the mean project completion time is X0 = 25, what is the probability that the project will be completed within X1=30 weeks?
2 = 6.9,  = Z1 = (X1 - X0)/  = (30 -25)/2.63 = 1.90
Z1 value of 1.90 corresponds to probability of in Table A.1, appendix A. Probability of completing project in 30 weeks or less : ( ) =
More precisely, P(x < X1) = P(x- X0 < X1 - X0) = P[(x- X0)/  < (X1 - X0)/  ]
= P(z < Z1=1.90) = .9713
if we define
z= (x - X0)/  (new variable)
and
Z1 = (X1 - X0)/  (constant).
Figure 8.14
Probability the network will be completed in 30 weeks or less
Chapter 8 - Project Management

25. Probability Analysis of the Project: Example 3
Question: If the mean project completion time is X0 = 25, what is the probability that the project will be completed within X1=22 weeks?
Z1 = ( )/2.63 = -1.14
Where Z1 value of 1.14 (ignore negative) corresponds to probability of in Table A.1, appendix A. Probability that customer will be retained is .1271
Figure 8.15
Probability the network will be completed in 22 weeks or less
Chapter 8 - Project Management

26. Chapter 8 - Project Management
Probability Analysis of the Project Network CPM/PERT Analysis with QM for Windows
Exhibit 8.1
Chapter 8 - Project Management

27. Project Crashing and Time-Cost Trade-Off: Definition
Project duration can be reduced by assigning more resources to project activities.
Doing this however increases project cost.
Decision is based on analysis of trade-off between time and cost.
Project crashing is a method for shortening project duration by reducing one or more critical activities to a time less than normal activity time.
Crashing achieved by devoting more resources to crashed activities.
Chapter 8 - Project Management

28. Crashing Activity Times
In the Critical Path Method (CPM) approach to project scheduling, it is assumed that the normal time to complete an activity, tj , which can be met at a normal cost, cj , can be crashed to a reduced time, tj’, under maximum crashing for an increased cost, cj’.
It is assumed that its cost per unit reduction, Kj , is linear and can be calculated by:
Kj = (cj' - cj)/(tj - tj').
E.g.: in the example on the right,
Kj = total crash cost/total crash time
= $2000/5 = $400/wk
Chapter 8 - Project Management

29. Crashing Example for Riverwalk Associates
Normal Costs and Crash Costs
Normal Crash
Activity Time Cost Time Cost
A) Study Feasibility $ 80, $100,000
B) Purchase Building , ,000
C) Hire Project Leader , ,000
D) Select Advertising Staff , ,000
E) Purchase Materials , ,000
F) Hire Manufacturing Staff , ,000
G) Manufacture Prototype , ,000
H) Produce First 50 Units , ,000
I) Advertising Product , ,000
J) Assessing User Feedback , ,000
K) Distributing Product , ,000
Chapter 8 - Project Management

30. Crashing Example for Riverwalk Associates
Crashing: The completion time for this project using normal times is 23 weeks. Which activities should be crashed, and by how many weeks, in order for the project to be completed in a Target of 20 weeks?
Let: Yi = the amount of time activity i is crashed.
Then borrow from the LP formulation for the Earliest Times, we have the following…
Chapter 8 - Project Management

31. Riverwalk Associates – LP Model for Crashing
Min 20YA + 50YC + 50YD + 60YF + 350YH + 75YI
S.t.
EF_A - ES_A >= 6 - YA
EF_B - ES_B >= 4
EF_C - ES_C >= 3 - YC
EF_D - ES_D >= 5 - YD
EF_E - ES_E >= 1
EF_F - ES_F >= 4 - YF
EF_G - ES_G >= 2
EF_H - ES_H >= 6 - YH
EF_I - ES_I >= 5 - YI
EF_J - ES_J >= 3
EF_K - ES_K >= 5
ES_C - EF_A >= 0 (Not “=“ )
ES_D - EF_A >= 0
ES_E - EF_A >= 0
ES_F - EF_B >= 0
ES_G - EF_B >= 0
ES_F - EF_C >= 0
ES_G - EF_C >= 0
ES_J - EF_D >= 0
ES_J - EF_H >= 0
ES_H - EF_E >= 0
ES_H - EF_F >= 0
ES_I - EF_E >= 0
ES_I - EF_F >= 0
ES_K - EF_G >= 0
ES_K - EF_I >= 0
FINISH - EF_J >= 0
FINISH - EF_K >= 0
FINISH <= 20 (Target)
YA <=1
YC <=1
YD <=3
YF <=3
YH <=1
YI <=4
Chapter 8 - Project Management

32. Chapter 8 - Project Management
Project Crashing and Time-Cost Trade-Off Project Crashing with QM for Windows
Exhibit 8.2
Chapter 8 - Project Management

33. Chapter 8 - Project Management
Project Crashing and Time-Cost Trade-Off General Relationship of Time and Cost
Project crashing costs and indirect costs have an inverse relationship.
Crashing costs are highest when the project is shortened.
Indirect costs increase as the project duration increases.
Optimal project time is at minimum point on the total cost curve.
Figure 8.20
The time–cost trade-off
Chapter 8 - Project Management