1. Project Scheduling: Lagging, Crashing and Activity Networks
Chapter 10

2. Learning Goals Apply lag relationships to project activities.
Construct and comprehend Gantt charts.
Understand the trade-offs required in the decision to crash project activities.
Develop activity networks using Activity- on-Arrow techniques.
Understand the differences in AON and AOA and recognize the advantages and disadvantages of each technique.

3. Lags in Precedence Relationships
The logical relationship between the start and finish of one activity and the start and finish of another activity.
Four logical relationships between tasks
Finish to Start
Finish to Finish
Start to Start
Start to Finish

4. Finish-to-Start (FS) Lag
Most common type of sequencing
Shown on the line joining the nodes
Added during forward pass
Subtracted during backward pass
This lag is not the same as activity slack A
Spec Design 6 B
Design Check 5 C
Blueprinting 7
Lag 4 days

5. Finish-to-Finish (FF) Lag
Two activities share a similar completion point
The interior construction cannot happen until wiring, plumbing, and HVAC installation are complete S
Plumbing 2 T
HVAC 3 U
Interior Const. 6 R
Wiring
Lag 3 days

6. Start-to-Start (SS) Lag
Often two or more activities must start at the same time S
Plumbing 2 T
HVAC 3 U
Inspection 6 R
Wiring
Lag 3
Logic must be maintained by both forward and backward pass

7. Start-to-Finish (SF) Lag
Least common type of lag relationship
Successor’s finish dependent on predecessor’s start R
Wiring 6
Lag 3 S
Plumbing 6 T
HVAC 5 U
Inspection 1

8. Gantt Charts
A graphical representation of the project schedule that illustrates how work flows over time
Shows activity start and end dates and durations
Links project activities to a project schedule baseline
Can be used as a tracking tool
Benefits of Gantt charts
Easy to create, read, and comprehend
Identify the project network and schedule baseline
Allows for updating and control
Useful to identify resource needs and assigning resources to tasks

9. Example
Task
Time
Predecessor Z 8 -- Y 5 X W 4
Y,X V U 3 T 6 S 7
U,T R 9
Create a Gantt chart based on the activities listed in the table.
Task ES EF LS LF Z Y X W V U T S R

10. Gantt Chart Example from MSExcel
Task
Start
Duration Z 8 Y 5 X W 16 4 V 20 U 3 T 25 6 S 31 7 R 38 9

11. Sample Tracking Gantt Chart With Critical Path A-C-D-F-H
Critical path in red

12. Principal options for crashing
Crashing Projects
Accelerating a project by committing more resources.
Principal options for crashing
Improving existing resources’ productivity
Changing work methods used
Increasing the quantity of resources

13. Crashing Projects – Under What Conditions?
The initial schedule may be too optimistic.
Market needs change and the project is in demand earlier than anticipated.
The project has slipped considerably behind schedule.
The contractual situation provides even more incentive to avoid schedule slippage.

14. Steps in Project Crashing
Compute the crash cost per time period. If crash costs are linear over time:
Crash cost
per period =
(Crash cost – Normal cost)
(Normal time – Crash time)
Using current activity times, find the critical path and identify the critical activities.

15. Steps in Project Crashing
If there is only one critical path, then select the activity on this critical path that
can still be crashed, and
has the smallest crash cost per period.
If there is more than one critical path, then select one activity from each critical path such that
each selected activity can still be crashed, and
the total crash cost of all selected activities is the smallest.
Note: the same activity may be common to more than one critical path.

16. Steps in Project Crashing
Update all activity times.
If the desired due date has been reached, stop. If not, return to Step 2.

17. Time-Cost Tradeoffs for Crashing Activities
| | |
1 2 3 Time (Weeks)
$34,000 —
$33,000 —
$32,000 —
$31,000 —
$30,000 — —
Activity Cost
Crash Point
Normal Point
Crash Cost/Wk =
Crash Cost – Normal Cost
Normal Time – Crash Time =
$34,000 – $30,000
3 – 1
= = $2,000/Wk
$4,000
2 Wks
Crash Time
Normal Time
Crash Cost
Normal Cost
How far do you need to speed
up the process?

18. Crashing Activities – An Example
Formula Slope = crash cost – normal cost normal time – crash time Example – calculating the cost of crashing Suppose: Normal activity duration = 8 weeks Normal cost = $14,000 Crashed activity duration = 5 weeks Crashed cost = $23,000 The activity cost slope = 23,000 – 14,000 or, $9,000 = $3,000 per week 8 – 5 3

19. Managerial Considerations
Determine activity fixed and variable costs
The crash point is the fully expedited activity
Optimize time-cost tradeoffs
Shorten activities on the critical path
Cease crashing when
the target completion time is reached
the crash cost exceeds the penalty cost

20. Crash Example Activity Pred Normal Time Min Time Normal Cost
Crash Cost A -- 14 9
500
1500 B 5 2
1000
1600 C 10 8
2000
2900 D
B, C
2500 E 6
1900 F
3000 G
E, F 7 4
600
1800 H 15 11
3600
Initial early and late start and finish times
Task ES EF LS LF Slack A B C D E F G H
Project 63
Reduce A by 5 weeks at $200/week = $1000 Project length is now 58 weeks
Reduce G by 3 weeks at $400/week = $1200 Project length is now 55 weeks
Reduce C by 2 weeks at $450/week = $900 Project length is now 53 weeks
Total cost to finish project in 53 weeks is $3100 extra

21. Crash Example
What is the lowest cost to complete this project in 53 weeks? Times are in weeks and costs in dollars.

22. Activity on Arrow (AOA) Networks
Activities represented by arrows
Event nodes easy to flag
Forward and backward pass logic similar to AON
Two activities may not begin and end at common nodes
Dummy activities may be required
Earliest Event Time (ES)
Event label
Latest Event Time (LF)
Task Description
Duration

23. Sample AOA Network Diagram
Figure 10.18

24. Activity on Arrow (AOA) NetworkB H A F C D K G
Task ES EF LS LF A B C D E F G H K

25. A Comparison of AON and AOA Network Conventions
Activity on Activity Activity on
Node (AON) Meaning Arrow (AOA)
A comes before B, which comes before C
(a) A B C
A and B must both be completed before C can start
(b) A C B
B and C cannot begin until A is completed
(c) B A C

26. A Comparison of AON and AOA Network Conventions
Activity on Activity Activity on
Node (AON) Meaning Arrow (AOA)
C and D cannot begin until both A and B are completed
(d) A B C D
C cannot begin until both A and B are completed; D cannot begin until B is completed. A dummy activity is introduced in AOA
(e) C A B D
Dummy activity

27. A Comparison of AON and AOA Network Conventions
Activity on Activity Activity on
Node (AON) Meaning Arrow (AOA)
B and C cannot begin until A is completed. D cannot begin until both B and C are completed. A dummy activity is again introduced in AOA.
(f) A C D B
Dummy activity

28. Discussion Questions
Please give examples of circumstances in which a project would employ lag relationships between activities using:
Finish to start
Finish to finish
Start to start
Start to finish
The advantage of Gantt Charts lies in their linkage to the project schedule baseline. Explain this concept.
What are the advantages in the use of Gantt charts over PERT diagrams? In what ways might PERT diagrams be advantageous?
Under what circumstances might you wish to crash a project?

29. Discussion Questions
In crashing a project, we routinely focus on those activities that lie on the critical path, not activities with slack time. Explain why this is the case.
What are some of the advantages in the use of AOA notation as opposed to AON? Under what circumstances does it seem better to apply AON methodology in network development? Explain the concept of a “dummy variable.” Why are they employed in AOA notation? Why is there no need to use dummy variables in an AON network?
Explain the concept of a “dummy variable.” Why are they employed in AOA notation? Why is there no need to use dummy variables in an AON network?