6.4 Critical Path and Schedule Calculations
Key Takeaways
- The forward pass calculates early start (ES) and early finish (EF) for each activity from left to right through the network
- The backward pass calculates late start (LS) and late finish (LF) for each activity from right to left through the network
- Total float = LS - ES = LF - EF — the amount of time an activity can be delayed without affecting the project end date
- A critical path activity has zero total float and any delay will directly delay the project completion date
- Leads reduce the gap between predecessor and successor (negative lag), while lags add waiting time between activities
Critical Path and Schedule Calculations
Critical path analysis and schedule calculations are among the most quantitative and heavily tested topics on the CAPM exam. You must understand forward pass, backward pass, and float calculations.
Network Diagram Basics
A project schedule network diagram shows the logical relationships (dependencies) between activities. Each activity is represented by a node with the following information:
┌────────────────────────┐
│ ES │ Duration │ EF │
├──────┼──────────┼──────┤
│ │ Activity │ │
├──────┼──────────┼──────┤
│ LS │ Float │ LF │
└────────────────────────┘
| Term | Abbreviation | Definition |
|---|---|---|
| Early Start | ES | Earliest time an activity can start |
| Early Finish | EF | Earliest time an activity can finish |
| Late Start | LS | Latest time an activity can start without delaying the project |
| Late Finish | LF | Latest time an activity can finish without delaying the project |
| Duration | D | Time needed to complete the activity |
Forward Pass (Calculate ES and EF)
The forward pass moves left to right through the network, calculating the earliest each activity can start and finish.
Rules:
- EF = ES + Duration
- First activity: ES = 0 (or 1, depending on convention)
- When an activity has multiple predecessors: ES = MAX of all predecessor EFs
- When using finish-to-start: ES of successor = EF of predecessor
Example Forward Pass:
Activity A (Duration 3): ES = 0, EF = 0 + 3 = 3 Activity B (Duration 5, follows A): ES = 3, EF = 3 + 5 = 8 Activity C (Duration 4, follows A): ES = 3, EF = 3 + 4 = 7 Activity D (Duration 2, follows B and C): ES = MAX(8, 7) = 8, EF = 8 + 2 = 10
Project Duration = 10 (EF of the last activity)
Backward Pass (Calculate LF and LS)
The backward pass moves right to left through the network, calculating the latest each activity can start and finish without delaying the project.
Rules:
- LS = LF - Duration
- Last activity: LF = Project Duration
- When an activity has multiple successors: LF = MIN of all successor LSs
Example Backward Pass (continuing from above):
Activity D: LF = 10, LS = 10 - 2 = 8 Activity B: LF = 8 (LS of D), LS = 8 - 5 = 3 Activity C: LF = 8 (LS of D), LS = 8 - 4 = 4 Activity A: LF = MIN(3, 4) = 3, LS = 3 - 3 = 0
Calculating Float
Total Float = LS - ES = LF - EF
| Activity | ES | EF | LS | LF | Total Float | On Critical Path? |
|---|---|---|---|---|---|---|
| A | 0 | 3 | 0 | 3 | 0 | Yes |
| B | 3 | 8 | 3 | 8 | 0 | Yes |
| C | 3 | 7 | 4 | 8 | 1 | No |
| D | 8 | 10 | 8 | 10 | 0 | Yes |
Critical Path: A → B → D (all activities with zero float)
Leads and Lags
| Concept | Definition | Effect |
|---|---|---|
| Lead | Allows the successor to start before the predecessor finishes | Compresses schedule (negative lag) |
| Lag | Requires waiting time after the predecessor before the successor can start | Extends schedule |
Examples:
- Lead: Activity B can start 2 days before Activity A finishes → FS - 2 days
- Lag: After pouring concrete (A), wait 3 days for curing before building walls (B) → FS + 3 days
Schedule Compression Calculations
Crashing
When crashing, select the activity on the critical path with the lowest crash cost per unit time:
| Activity | Normal Duration | Crash Duration | Normal Cost | Crash Cost | Crash Cost/Day |
|---|---|---|---|---|---|
| A | 10 days | 8 days | $5,000 | $7,000 | $1,000/day |
| B | 8 days | 6 days | $4,000 | $7,000 | $1,500/day |
| C | 6 days | 5 days | $3,000 | $4,500 | $1,500/day |
If A and B are on the critical path, crash Activity A first ($1,000/day is lower than $1,500/day).
Exam Tip: Always crash the critical path activity with the lowest crash cost per time unit first. Never crash non-critical activities (it won't shorten the project). After crashing, recalculate to see if the critical path has shifted.
During a forward pass, when an activity has multiple predecessors, the Early Start (ES) is calculated as:
An activity has ES = 5, EF = 12, LS = 8, LF = 15. What is the total float?
Allowing a successor activity to start 3 days before its predecessor finishes is an example of: