Critical path analysis

Represent and analyse projects as activity networks; perform forward and backward scans to find earliest and latest start times; identify the critical path and minimum project completion time.

Networks · MS-N3

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Open Math

Critical path analysis

Networks · MS-N3

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Activity networks

Q1Straightforward

A project has four activities with the following immediate predecessors:

Activity	Immediate predecessors
A	—
B	A
C	A
D	B, C

How many immediate predecessors does activity D have?

Q2Straightforward

A project has five activities with the following immediate predecessors:

Activity	Immediate predecessors
A	—
B	—
C	—
D	A, B
E	B, C

How many activities have no immediate predecessor (i.e., can start at the beginning of the project)?

Q3Straightforward

In an activity network, a dummy activity is sometimes added to correctly represent dependencies without creating a false link. What is the duration (in days) of a dummy activity?

Q4Moderate

A project has six activities:

Activity	Immediate predecessors
A	—
B	—
C	A
D	A, B
E	C
F	D, E

How many immediate predecessors does activity D have?

Q5Moderate

Using the same project as the previous question:

Activity	Immediate predecessors
A	—
B	—
C	A
D	A, B
E	C
F	D, E

How many activities have B as an immediate predecessor?

Q6Moderate

A project has six activities:

Activity	Immediate predecessors
A	—
B	—
C	A, B
D	B
E	C, D
F	C

How many immediate predecessors does activity E have?

Q7Moderate

A project has seven activities:

Activity	Immediate predecessors
A	—
B	A
C	A
D	B, C
E	D
F	B
G	E, F

How many activities lead directly to the finish of the project (i.e., have no successors — nothing depends on them)?

Q8Moderate

In a project network, dummy activities are used to keep the diagram logically correct. Each dummy activity has a duration of 0 days. If a project requires 3 dummy activities, what is the total number of days contributed by all dummy activities combined?

Q9Challenging

A project has seven activities:

Activity	Immediate predecessors
A	—
B	—
C	A, B
D	A
E	C, D
F	C
G	E, F

How many immediate predecessors does activity G have?

Q10Challenging

A project has eight activities:

Activity	Immediate predecessors
A	—
B	—
C	A
D	A, B
E	C
F	D, E
G	B
H	F, G

How many activities have exactly 2 immediate predecessors?

Q11Challenging

A project has the following activities:

Activity	Immediate predecessors
A	—
B	—
C	A, B
D	B
E	C, D
F	A
G	E, F

Activity B is an immediate predecessor of how many activities?

Q12Challenging

A project manager creates the following precedence table:

Activity	Immediate predecessors
A	—
B	—
C	A
D	A, B
E	C
F	D, E, B
G	F

How many immediate predecessors does activity F have?

Forward and backward scanning

Q13Straightforward

Activity A has no predecessors and a duration of 6 days. Using a forward scan (earliest start time = 0 for activities with no predecessors), what is the earliest finish time (EFT) of activity A?

Earliest finish time = EST + duration.

Q14Straightforward

Activity B has an earliest start time (EST) of 8 days and a latest start time (LST) of 11 days. What is the float (slack) of activity B?

Float = LST − EST.

Q15Straightforward

A project must be completed in 20 days. Activity D is the last activity and has a duration of 4 days. Using a backward scan, what is the latest start time (LST) of activity D?

For the last activity: LST = project duration − duration of the activity.

Q16Moderate

Activity A (duration 3 days) and Activity B (duration 5 days) both have no predecessors, so each has EST = 0. Activity C cannot start until both A and B are finished. What is the earliest start time (EST) of activity C?

Q17Moderate

A project has the following activities (duration in days):

Activity	Predecessors	Duration
A	—	4
B	—	7
C	A	3
D	B	2
E	C, D	5

Perform a forward scan and find the EST of activity E.

Q18Moderate

A project has the following activities (duration in days):

Activity	Predecessors	Duration
A	—	3
B	—	6
C	A, B	4
D	A	5
E	C, D	2

Perform a forward scan and find the EST of activity C.

Q19Moderate

A project has a minimum completion time of 18 days. Activity F has a duration of 4 days and is the last activity before the finish. Using a backward scan, what is the latest start time (LST) of activity F?

Q20Moderate

A project has the following activities:

Activity	Predecessors	Duration (days)
A	—	5
B	A	3
C	A	4
D	B, C	2
E	D	6

After performing a complete forward and backward scan, the project duration is 17 days. The LST of D is 9 days and its EST is 9 days. What is the float of activity B?

(Hint: EFT of A = 5. EST of B = 5. EFT of B = 8. LST of D = 9, so LST of B = 9 − 3 = 6.)

Q21Challenging

A project has the following activities:

Activity	Predecessors	Duration (days)
A	—	3
B	—	5
C	A	4
D	A, B	2
E	A	6
F	C, D, E	3

Perform a forward scan. What is the EST of activity D?

Q22Challenging

Using the same project as the previous question:

Activity	Predecessors	Duration (days)
A	—	3
B	—	5
C	A	4
D	A, B	2
E	A	6
F	C, D, E	3

Complete the forward scan and find the minimum project completion time.

Q23Challenging

A project has the following activities:

Activity	Predecessors	Duration (days)
A	—	3
B	—	5
C	A	4
D	A, B	2
E	A	6
F	C, D, E	3

The minimum project completion time is 12 days. Perform a backward scan and find the float of activity D.

Q24Challenging

A project has the following activities:

Activity	Predecessors	Duration (days)
A	—	3
B	—	5
C	A	4
D	A, B	2
E	A	6
F	C, D, E	3

The minimum project completion time is 12 days. Perform a complete backward scan and find the latest start time (LST) of activity B.

The critical path

Q25Straightforward

An activity has an earliest start time (EST) of 4 days and a latest start time (LST) of 4 days. What is the float of this activity?

Float = LST − EST.

Q26Straightforward

A project has the following activities with durations (in days):

Activity	Predecessors	Duration
A	—	5
B	A	3
C	A	6
D	B, C	4

The minimum completion time of the project is the EFT of the last activity. Perform a forward scan and find the minimum completion time.

Q27Straightforward

Which statement correctly describes the critical path in a project network?

A.The shortest path from start to finish
B.The path of activities with the most float
C.The longest path from start to finish, determining the minimum project duration
D.The path with the fewest activities

Q28Moderate

A project has the following activities:

Activity	Predecessors	Duration (days)
A	—	3
B	—	7
C	A	5
D	A, B	2
E	C, D	4

Perform a forward scan and find the minimum project completion time.

Q29Moderate

Using the same project as the previous question (minimum completion time = 13 days):

Activity	Predecessors	Duration (days)
A	—	3
B	—	7
C	A	5
D	A, B	2
E	C, D	4

After completing a forward and backward scan, the float values are:
- A: float = 4, B: float = 0, C: float = 4, D: float = 0, E: float = 0

How many activities are on the critical path?

Q30Moderate

A project has minimum completion time of 20 days. Activity G is on the critical path and has a duration of 3 days. If activity G is delayed by 2 days, what is the new minimum project completion time (in days)?

Q31Moderate

Activity H has float of 5 days. If activity H is delayed by 3 days, by how many days is the project completion time extended?

Q32Moderate

A project has the following activities:

Activity	Predecessors	Duration (days)
A	—	4
B	—	2
C	A	3
D	B	6
E	C, D	5

The critical path runs through A → C → E or B → D → E. Determine the minimum project completion time.

Q33Challenging

A project has the following activities:

Activity	Predecessors	Duration (days)
A	—	2
B	—	5
C	A	4
D	A, B	3
E	C	6
F	D, E	2

Perform a complete forward scan and find the minimum project completion time.

Q34Challenging

Using the same project as the previous question (minimum completion time = 14 days):

Activity	Predecessors	Duration (days)
A	—	2
B	—	5
C	A	4
D	A, B	3
E	C	6
F	D, E	2

Perform a backward scan. What is the float of activity B?

Q35Challenging

Using the same project (completion time = 14 days):

Activity	Predecessors	Duration (days)
A	—	2
B	—	5
C	A	4
D	A, B	3
E	C	6
F	D, E	2

Activity B has float of 4 days. If activity B is delayed by 5 days, by how many days is the minimum project completion time extended?

Q36Challenging

A project has the following activities:

Activity	Predecessors	Duration (days)
A	—	6
B	—	4
C	A	3
D	A, B	5
E	B	8
F	C, D	2
G	E, F	1

Find the minimum project completion time.

Worked solutions and answers at openmath.au/year-12/standard-2/critical-path-analysis