1. Critical Path Method Farrokh Alemi, Ph.D.
This is the lecture on critical path method. This lecture is part of the open online course on project management organized by Professor Alemi.

2. Purpose of CPM Finish fast Quantify progress Communicate effectively
The purpose of Critical Path Method is to finish a project in the shortest amount of time. The CPM method allows us to quantify progress towards the project’s end and to be able to report how various events will lead to changes in project end date.

3. Advantages and Disadvantages of Formal Schedules
Saves time and money on large complex projects
Effective communication tool
Too much work
Assessment of duration of tasks
Assessment of interdependencies
The CPM method is a formal and mathematical analysis of the shortest way of finishing a project. It is not appropriate for all project. The Critical Path Method is specially good for complex, multi-task projects in which there are many ways of sequencing the tasks. In these situations a formal analysis of sequence of the tasks could save a great deal from the end date. In additional formal analysis and presentation of the findings, would go a long way in communicating the details of the project to various stockholders. The main disadvantage of the approach is the extent to which various details have to be looked after. CPM requires assessment of duration of each task and the interdependencies among the tasks. Some project managers, specially those who are dealing with small scale projects prefer to muddle through the tasks without formally assessing the duration of each task. They may prefer to act spontaneously and decide what to do next as opposed to plan activities ahead of time. But as projects get more complex, the number of people involved increases and the importance of formal planning and optimal scheduling increases.

4. Resource Oriented and Time Oriented Schedules
Resource oriented schedules sequence tasks because of limited access to resources needed to complete a task. For example, construction of a wing of the hospital maybe delayed because the large equipment needed for doing so is in use in another part of the construction. Time oriented schedules focus only on precedents relationships among tasks. A precedence relationship indicates that a task cannot start until other tasks are done. For example, training of staff on how to use an EHR cannot start until the EHR system has been implemented and configured.

5. Elements of a Critical Path Chart
Activity
Precedence
Duration/cost
A critical path chart has several elements. First, it includes a set of well defined activities. By well defined we mean activities that have clear start and end characteristics. These activities should be mutually exclusive with no overlap. The description of activities must be clear to project team members.

6. Elements of a Critical Path Chart
Activity
Precedence
Duration/cost
Activity
Description A
Select satisfaction survey tool B
Print survey tool C
Obtaining list of patients D
Sample from list E
Mail surveys to selected patients F
Collect responses G
Analyze data
Here is a list of activities one might engage in to survey patients and find out their satisfaction with care. These activities include selecting the survey tool, printing the tool, obtaining list of patients, sampling from the list, mailing the surveys to selected patients, collecting responses, and analyzing the data. Each activity is assigned a unique letter so as to easily refer to it within a network structure.

7. Elements of a Critical Path Chart
Activity
Precedence
Duration/cost
For each activity we also need to know what must be accomplished before the activity can start, this is referred to as a precedence. A critical path chart shows both the activities and their precedents.

8. Elements of a Critical Path Chart
Activity
Precedence
Duration/cost
Activity
Description
Precedents A
Select satisfaction survey tool B
Print survey tool C
Select list of patients D
Sample from list E
Mail surveys to selected patients F
Collect responses G
Analyze data
Not that in this example, we are saying that we cannot print the survey until we have selected the survey tool. We cannot sample from the list of patients until we have obtained the list. We cannot mail the survey until we know who should receive it. Finally the precedent structure described in this example says that we cannot analyze the data until we have collected it. Of course, we cannot analyze the data if the surveys have not been mailed to the patients. But we do not need to show this as a precedence for analyzing the data as it is a precedence for collecting responses. In a precedence table we show only the immediate precedents for a task.

9. Elements of a Critical Path Chart
Activity
Precedence
Duration/cost
For each activity we also need to know its duration and the resources needed to accomplish it. The Critical Path Method uses information on precedents and duration of activities to arrive at an optimal schedule for the tasks.

10. Elements of a Critical Path Chart
Activity
Precedence
Duration/cost
Activity
Description
Precedents
Duration A
Select satisfaction survey tool
2 weeks B
Print survey tool
3 weeks C
Select list of patients
1 week D
Sample from list E
Mail surveys to selected patients F
Collect responses
4 weeks G
Analyze data
Here we see all the elements that go into a Critical Path Chart in the form a precedence table. In this example, we show the duration of each task on the right column of the table. These data, for example, tell us that it may take a month to receive responses back from patients. A Critical Path Chart, of course, is not a listing or a table. It is a visual chart, often in the form of a network. We now turn to how these charts are displayed.

11. Branch and Node Charts Activity on branch Activity on node
There are two visual ways for displaying Critical Path Charts: activity on branch and activity on node charts.

12. Activity In Node A B E F G C D Activity Description Precedents A
Select satisfaction survey tool B
Print survey tool C
Obtain list of patients D
Sample from list E
Mail surveys to selected patients
D,B F
Collect responses G
Analyze data
This Figure shows a chart where activities are in the nodes. The arrows show the precedence structure. Each node is connected to its immediate precedents. One could follow the arrows backwards to find what is required for each task and follow it forwards to see what task is next. Note that in this example, task C which is “mailing the surveys” requires us to have completed two other tasks: task B “printing of surveys” and task D “selection of recipients.” Tasks B and D are not linked, they can occur simultaneously.

13. Activity On Branches A B E F G 1 3 5 6 7 End Start C D 2 4 Activity
Description
Precedents A
Select satisfaction survey tool B
Print survey tool C
Obtain list of patients D
Sample from list E
Mail surveys to selected patients
D,B F
Collect responses G
Analyze data
This Figure shows a chart where activities are on the branches. The arrows show the completion of an activity. Each node shows the sequence of start or end of the activity. One could follow the nodes to identify which activities have to start when. In this example, task E “Mailing of the surveys” is the fifth task to start. All other four tasks need to be completed before it can start.

14. Earliest and Latest Start and Finish Times2 B 3 E 2 F 4 G 3
End
Start C 1 D 1
Activity
Description A
Select satisfaction survey tool B
Print survey tool C
Select list of patients D
Sample from list E
Mail surveys to patients F
Collect responses G
Analyze data
Lets return to the activity in node chart we had presented earlier. This time we show the duration of each activity on the arrows from each node.

15. Earliest and Latest Start and Finish Times
Forward pass calculates early start and finish A 2 B 3 E 2 F 4 G 3
End
Start C 1 D 1
Activity
Description A
Select satisfaction survey tool B
Print survey tool C
Select list of patients D
Sample from list E
Mail surveys to patients F
Collect responses G
Analyze data
The discovery of the critical path is done through calculating earliest start and finish times for each node, beginning from the start point and moving forward. This is called the forward pass.

16. Earliest and Latest Start and Finish Times
Backward pass calculates latest start and end times A 2 B 3 E 2 F 4 G 3
End
Start C 1 D 1
Activity
Description A
Select satisfaction survey tool B
Print survey tool C
Select list of patients D
Sample from list E
Mail surveys to patients F
Collect responses G
Analyze data
The backward pass calculates latest start and finish times for each activity.

17. Earliest and Latest Start and Finish Times
Critical Path includes activities where earliest start time and latest finish time are equal A 2 B 3 E 2 F 4 G 3
End
Start C 1 D 1
Activity
Description A
Select satisfaction survey tool B
Print survey tool C
Select list of patients D
Sample from list E
Mail surveys to patients F
Collect responses G
Analyze data
The critical path is then identified from the difference of the earliest time it can start and the latest time it can finish. This difference is called slack time. The critical path is the path where the earliest start and the latest finish time are the same and therefore there is no slack in these tasks, a delay in these tasks leads to a delay in the entire project.

18. Earliest and Latest Start and Finish Times
Activities not on the critical path have “slack.” The latest finish time and the earliest start date are different A 2 B 3 E 2 F 4 G 3
End
Start C 1 D 1
Activity
Description A
Select satisfaction survey tool B
Print survey tool C
Select list of patients D
Sample from list E
Mail surveys to patients F
Collect responses G
Analyze data
Slack refers to the difference of earliest start and latest finish time. Activities that have slack time can be delayed without causing a delay in the entire project. These activities are not on the critical path.

19. Earliest and Latest Start and Finish Times
Early
Late A 2 B 3 E 2 F 4 G 3
End
Start C 1 D 1
Activity
Description A
Select satisfaction survey tool B
Print survey tool C
Select list of patients D
Sample from list E
Mail surveys to patients F
Collect responses G
Analyze data
The determination of slack and critical path depends on calculation of earliest start in the forward pass and latest finish time in the backward pass. We now show you how the forward and backward passes work. For the forward pass, we start at the first node and calculate the earliest start time. The earliest start time is usually the maximum of the earliest finish time of the previous activities. The earliest start time for activity A, selection of satisfaction surveys, is at week zero.

20. Earliest and Latest Start and Finish Times
Early 2
Late A 2 B 3 E 2 F 4 G 3
End
Start C 1 D 1
Activity
Description A
Select satisfaction survey tool B
Print survey tool C
Select list of patients D
Sample from list E
Mail surveys to patients F
Collect responses G
Analyze data
Earliest finish for activity A is the earliest start time plus the duration, in this case 2 weeks. So the earliest finish time for activity A is on week 2.

21. Earliest and Latest Start and Finish Times
Early 2 5
Late S F E 2 L A 2 B 3 E 2 F 4 G 3
End
Start C 1 D 1
Activity
Description A
Select satisfaction survey tool B
Print survey tool C
Select list of patients D
Sample from list E
Mail surveys to patients F
Collect responses G
Analyze data
Lets now look at the earliest start and finish times for node B, printing of survey tool. As before, the earliest start time of an activity is the maximum of the earliest finish time of all of its immediate preceding activities. Activity B has only one precedent and it finishes on week 2. Therefore the earliest start of activity B is on week 2. The earliest finish time is calculated as the earliest start time plus the duration. The earliest finish time for activity B is 5 weeks.

22. Earliest and Latest Start and Finish TimesS F E 2 L S F E 2 5 L A 2 B 3 E 2 F 4 G 3
End
Start C 1 D 1
Start
Finish
Early 2 5
Late
Activity
Description A
Select satisfaction survey tool B
Print survey tool C
Select list of patients D
Sample from list E
Mail surveys to patients F
Collect responses G
Analyze data
The earliest start and earliest finish for activity C is calculated to be on weeks 0 and 1.

23. Earliest and Latest Start and Finish TimesS F E 2 L S F E 2 5 L A 2 B 3 E 2 F 4 G 3
End
Start C 1 D 1 S F E 1 L
Start
Finish
Early 1 2
Late
Activity
Description A
Select satisfaction survey tool B
Print survey tool C
Select list of patients D
Sample from list E
Mail surveys to patients F
Collect responses G
Analyze data
The earliest start and earliest finish for activity D is calculated to be on weeks 1 and 2.

24. Earliest and Latest Start and Finish Times
Early 5 7
Late S F E 2 L S F E 2 5 L A 2 B 3 E 2 F 4 G 3
End
Start C 1 D 1 S F E 1 L S F E 1 2 L
Activity
Description A
Select satisfaction survey tool B
Print survey tool C
Select list of patients D
Sample from list E
Mail surveys to patients F
Collect responses G
Analyze data
The calculation of early start and early finish for activity E, mailing of the surveys to the patients, is more complicated as there are two precedents for this activity. The earliest finish for activity B is on week 5 and for activity D is on week 2. We select the maximum which is 5 for the earliest start date of activity E. The earliest finish date is then calculated as 5 plus the duration of the activity. The earliest finish date for activity E is week 7.

25. Earliest and Latest Start and Finish TimesS F E 2 L S F E 2 5 L S F E 5 7 L S F E 7 11 L S F E 11 14 L A 2 B 3 E 2 F 4 G 3
End
Start C 1 D 1 S F E 1 L S F E 1 2 L
Activity
Description A
Select satisfaction survey tool B
Print survey tool C
Select list of patients D
Sample from list E
Mail surveys to patients F
Collect responses G
Analyze data
Here we show the earliest start and earliest finish dates for activities F and G. Since G is the last activity, it follows that the earliest time this project can finish is in 14 weeks.

26. Earliest and Latest Start and Finish TimesS F E 11 14 L S F E 2 L S F E 2 5 L S F E 5 7 L S F E 7 11 L A 2 B 3 E 2 F 4 G 3
End
Start C 1 D 1 S F E 1 L S F E 1 2 L
In the backward pass, we calculate the latest finish and start time. The latest finish is calculated from the minimum of the latest start of succeeding task. Since there is no task succeeding task G, we start with the earliest finish time of this task. In task G, the latest finish is week 14.
Latest finish is the minimum of latest start of succeeding tasks. Latest start is the latest finish minus duration.

27. Earliest and Latest Start and Finish TimesS F E 2 L S F E 2 5 L S F E 5 7 L S F E 7 11 L S F E 11 14 L A 2 B 3 E 2 F 4 G 3
End
Start C 1 D 1 S F E 1 L S F E 1 2 L
The latest start is the latest finish minus the duration, in task G, the duration is 3 weeks and therefore the latest start is week 11.
Latest finish is the minimum of latest start of succeeding tasks. Latest start is the latest finish minus duration.

28. Earliest and Latest Start and Finish TimesS F E 7 11 L S F E 2 L S F E 2 5 L S F E 5 7 L S F E 11 14 L A 2 B 3 E 2 F 4 G 3
End
Start C 1 D 1 S F E 1 L S F E 1 2 L
Activity
Description A
Select satisfaction survey tool B
Print survey tool C
Select list of patients D
Sample from list E
Mail surveys to patients F
Collect responses G
Analyze data
For task F, the task of collecting responses, there is one succeeding task. The latest start of task G is week 11. Therefore, the latest finish of task F has to be week 11. Given that the latest finish is week 14 then the latest start has to be week 7.

29. Earliest and Latest Start and Finish TimesS F E 2 L S F E 2 5 L S F E 5 7 L S F E 7 11 L S F E 11 14 L A 2 B 3 E 2 F 4 G 3
End
Start C 1 D 1 S F E 1 L 3 4 S F E 1 2 L 4 5
Activity
Description A
Select satisfaction survey tool B
Print survey tool C
Select list of patients D
Sample from list E
Mail surveys to patients F
Collect responses G
Analyze data
The same calculations can be done for remaining tasks.

30. Slack time = Latest start – Earliest startS F E 2 L S F E 2 5 L S F E 5 7 L S F E 7 11 L S F E 11 14 L A 2 B 3 E 2 F 4 G 3
End
Start C 1 D 1 S F E 1 L 3 4 S F E 1 2 L 4 5
Having calculated the start and finish time for each activity, we are now ready to calculate the slack time associated with each activity. Slack is the difference of earliest and latest start times. In most of the activities in this example, the earliest and the latest start times are equal and therefore the slack is 0.
Slack time = Latest start – Earliest start

31. Critical path includes activities with no slack.S F E 2 L S F E 2 5 L S F E 5 7 L S F E 7 11 L S F E 11 14 L A 2 B 3 E 2 F 4 G 3
End
Start C 1 D 1 S F E 1 L 3 4 S F E 1 2 L 4 5
In activity C and D, the slack is not zero. These activities each have a 3 week slack. Therefore these activities are not on the critical path. Delays in these two activities of up to 3 weeks will not delay the entire project. But the activities in green have no slack time. These activities are on the critical path and delays in these activities will delay the entire project.
Critical path includes activities with no slack.

32. Communicating Project Schedule
Time scaled network of activities
Gantt chart
Time an activity graphs and completion graphs
It is important that project team members understand the impact of their activities on the entire project. For this reason it is necessary to communicate to the project team the network of activities and the critical path through them. This can be done effectively through a timed scale network of activities on the node.

33. Communicating Project Schedule
Time scaled network of activities
Gantt chart
Time an activity graphs and completion graphs
In addition, it helps if the team members can see a Gantt chart of the activities.

34. Communicating Project Schedule
Time scaled network of activities
Gantt chart
Time an activity graphs and completion graphs
Communication can be further facilitated if the activity completion and time in an activity are marked on the charts by changing the color proportional to extent of activity completion.

35. Activity Splitting A B E F G C D Activity Description Precedents A
Select satisfaction survey tool B
Print survey tool C
Obtain list of patients D
Sample from list E
Mail surveys to selected patients
D,B F
Collect responses G
Analyze data
Activity splitting can shorten project completion if it allows earlier start of subsequent activities. Recall from our previous example that it would take us 14 weeks to start a survey of patient satisfaction to completion of analysis of the data received from the patients. This time can be shortened by splitting some of the activities.

36. Activity Splitting A B E F G C D Activity Description Precedents A
Select satisfaction survey tool B
Print survey tool C
Obtain list of patients D
Sample from list E
Mail surveys to selected patients
D,B F
Collect responses G
Analyze data
Collecting patient responses is projected to take 4 weeks. We are planning as if we have to wait the entire time for all the responses to arrive. But it is possible to start some analysis before hand. In particular, we can analyze early responses to predict the complete response of the patients.

37. Activity Splitting A B E F1 G C D F2 Activity Description Precedents A
Select satisfaction survey tool B
Print survey tool C
Obtain list of patients D
Sample from list E
Mail surveys to selected patients
D,B F
Collect responses (F1=early, F2=Late) G
Analyze data
If we break collecting data into two activities, one name collecting early responses and the other late responses, then we can start the analysis sooner. As late responses arrive they are added to the analysis but the core analysis starts after the early data are available. This way the project can finish several weeks earlier. Before analysis would only start after the collection of late responses, now analysis and collection of late responses should finish together, reducing significant waiting time.

38. Software Discovery of Critical Path
The discovery of critical path has been significantly simplified in modern project management software. There is no longer any need for calculation of early and late start times. These values are calculated by the software.

39. Crashing Schedules Cost time tradeoff Order of crashing
It is possible to speed up schedules by making more resources available. Most activities can be done faster if more people are hired to do it. When activities are speeded up by spending more money, it is called crashing the schedule. The critical path method allows us to detect how much time is saved by crashing a schedule. Then managers can decide if it is worth while to spend the initial money to speed up a task to save a certain number of days in terms of project completion date. Since different activities lead to different time savings, activities are crashed in the order of time they shave off the project’s end.

40. The critical path includes activities with no slack
Take Home Lesson
The critical path includes activities with no slack
This lecture has shown you how to calculate the critical path for a project through calculation of slack time for the activity, which is the difference of the earliest and latest start time of the activity.