What is critical path analysis (CPA)?

CPA uses an activity network to find the **minimum project duration**, **critical activities** (zero float), the **critical path** (source to sink through critical activities), and **total float** for non-critical activities.

How do you calculate earliest event times?

**Forwards pass** from source (0). For each node j: EET(j) = max over all arcs (i,j) of (EET(i) + duration(i,j)). Take the **maximum** because all predecessors must finish before we can proceed.

How do you calculate latest event times?

**Backwards pass** from sink (set = EET at sink). For each node i: LET(i) = min over all arcs (i,j) of (LET(j) - duration(i,j)). Take the **minimum** to avoid delaying the project.

What is the formula for total float of activity (i,j)?

F(i,j) = LET(j) - EET(i) - duration(i,j). Where e_i is earliest time at start, l_j is latest time at end. Total float = latest finish - earliest start - duration.

What defines a critical activity?

A **critical activity** has **total float zero**. It must start at its earliest event time for the project to finish in minimum duration. Its start and end events are critical (EET = LET).

What is the critical path?

The **critical path** is the path from **source node to sink node** that passes only through activities with **total float zero**. It is written as a sequence, e.g. A-C-G-I. There may be more than one critical path.

How do you find the minimum project duration?

The **minimum project duration** is the **earliest event time at the sink node** (or equivalently the latest event time at the sink, since they are equal there).

Why is the earliest event time at a node the maximum of (EET + duration) over incoming arcs?

All activities leading to a node must finish before we can proceed. The **last** one to finish determines when we can leave, so we take the **maximum**.

Why is the latest event time at a node the minimum of (LET - duration) over outgoing arcs?

We must leave early enough so that every following activity can finish by its latest time. The **tightest** constraint gives the latest we can leave, so we take the **minimum**.

Can a non-critical activity lie between two critical events?

Yes. If an activity is critical, its start and end events must be critical. But the **converse** is false: an activity between two critical events can be non-critical (have positive float). Always check the float.

Critical Paths | Pearson Edexcel International AS Maths

Critical Path Analysis

Critical Paths

Summary

Critical path analysis (CPA) uses activity networks to determine the minimum project duration, identify critical activities (those with zero float that cannot be delayed), and find the critical path from source to sink. It involves calculating earliest and latest event times via forwards and backwards passes, then computing total float. The critical path is the longest path through the network and determines project completion time.

1. What is Critical Path Analysis?

Critical path analysis (CPA) uses an activity network to determine: the minimum duration of the project; the critical activities (must start and finish at earliest possible times); the critical path (source to sink through critical activities); and the total float (how much non-critical activities can be delayed).

Critical activities have total float zero. Any delay to them increases the project duration.

The critical path is the path from source node to sink node that includes all critical activities. There may be more than one critical path.

Total float is the time an activity can be delayed without affecting the minimum project duration.

2. Earliest Event Times (Forwards Pass)

The earliest event time at a node is the earliest time any activity can leave that node. It determines the minimum project duration.

Method: Start at source node with 0. For each node, consider all arcs leading into it. For each arc: early_end = early_start + duration. Take the maximum of these values.

Formula: EET(j) = max(EET(i) + duration(i,j)) over all activities (i,j) leading to node j.

This is called a forwards pass. Remember: dummies have duration 0.

3. Latest Event Times (Backwards Pass)

The latest event time at a node is the latest time we can leave that node without increasing the minimum project duration.

Method: Start at sink node — set it equal to the earliest event time (minimum project duration). Work backwards. For each arc: late_start = late_end - duration. Take the minimum of these values.

Formula: LET(i) = min(LET(j) - duration(i,j)) over all activities (i,j) leaving node i.

This is called a backwards pass. Earliest times must be completed before latest times.

4. Total Float

Total float of activity (i,j): F(i,j) = LET(j) - EET(i) - duration(i,j).

In words: latest finish time minus earliest start time minus duration. Equivalently: how long the start can be delayed without affecting project duration.

Critical activities have total float zero. Non-critical activities have positive float.

Total float is a single value per activity — do not confuse with other float types.

5. Finding the Critical Path and Critical Activities

6. Worked Example

7. Examiner Tips

Critical Paths

Summary

1. What is Critical Path Analysis?

Critical activities have total float zero. Any delay to them increases the project duration.

The critical path is the path from source node to sink node that includes all critical activities. There may be more than one critical path.

Total float is the time an activity can be delayed without affecting the minimum project duration.

2. Earliest Event Times (Forwards Pass)

The earliest event time at a node is the earliest time any activity can leave that node. It determines the minimum project duration.

Method: Start at source node with 0. For each node, consider all arcs leading into it. For each arc: early_end = early_start + duration. Take the maximum of these values.

Formula: EET(j) = max(EET(i) + duration(i,j)) over all activities (i,j) leading to node j.

This is called a forwards pass. Remember: dummies have duration 0.

3. Latest Event Times (Backwards Pass)

The latest event time at a node is the latest time we can leave that node without increasing the minimum project duration.

Formula: LET(i) = min(LET(j) - duration(i,j)) over all activities (i,j) leaving node i.

This is called a backwards pass. Earliest times must be completed before latest times.

4. Total Float

Total float of activity (i,j): F(i,j) = LET(j) - EET(i) - duration(i,j).

In words: latest finish time minus earliest start time minus duration. Equivalently: how long the start can be delayed without affecting project duration.

Critical activities have total float zero. Non-critical activities have positive float.

Total float is a single value per activity — do not confuse with other float types.

5. Finding the Critical Path and Critical Activities

Critical events: nodes where EET = LET. Critical activities connect critical events and have zero float.

Caution: A non-critical activity can lie between two critical events. Always verify float is zero.

Critical path: sequence of activities from source to sink, e.g. A-C-G-I. Write as a path with hyphens.

Critical activities: list of activities, e.g. A, C, G, I. Write as a comma-separated list.

Minimum project duration: the EET (and LET) at the sink node.

6. Worked Example

Given early and late times at each node, identify nodes where EET = LET. Possible critical activities: A, C, G, I.

Check total float: A: 5-0-5=0; C: 12-5-7=0; G: 18-12-6=0; I: 23-18-5=0. All zero.

Critical path: A-C-G-I. Critical activities: A, C, G, I. Minimum project duration: 23 days.

7. Examiner Tips

Distinguish critical path (sequence, e.g. A-B-D-F) from critical activities (list, e.g. A, B, D, F).

Include dummies in forwards and backwards passes — they have duration 0.

If all activities have the same unspecified duration, the critical path is the path with the fewest edges from source to sink.