Project Planning and Scheduling (CPM/PERT) - Cheat Sheet
Your 30-minute exam-ready reference for network scheduling, critical paths, float calculations, and PERT probability. Every formula, definition, and trap is here.
Network Fundamentals & Critical Path Method (CPM)
Formulas
EF = ES + d
Meaning
EF = Early Finish, ES = Early Start, d = duration
Watch out
Duration is always in the SAME UNITS as ES/EF (days, weeks, hours). Do NOT mix units.
When to use
Forward pass to calculate when each activity finishes earliest
ES (successor) = MAX(EF of all predecessors)
Meaning
An activity cannot start until ALL predecessors finish
Watch out
Take the MAXIMUM (latest) finish time, not the average or minimum.
When to use
Forward pass; when an activity has multiple incoming edges
LS = LF - d
Meaning
LS = Late Start, LF = Late Finish, d = duration
Watch out
Work BACKWARDS from the project end. Project LF = Project EF for the final activities.
When to use
Backward pass to calculate the latest an activity can start without delaying project
LF (predecessor) = MIN(LS of all successors)
Meaning
An activity must finish before ALL successors can start late
Watch out
Take the MINIMUM (earliest) late start, not the average. This is opposite of the forward pass logic.
When to use
Backward pass; when an activity has multiple outgoing edges
TF = LS - ES = LF - EF
Meaning
TF = Total Float (slack); how much an activity can slip without delaying the project
Watch out
Both formulas must give the SAME answer. If they don't, you made an arithmetic error. TF = 0 means CRITICAL.
When to use
After forward and backward pass to identify which activities have flexibility
FF = EF (successor) - EF (activity) - 0
Meaning
FF = Free Float; how much an activity can slip without delaying any SUCCESSOR (not the project)
Watch out
Free float ≤ Total float always. FF answers the question: 'Can this activity slip without hurting others?'
When to use
Identify activities with zero free float (must keep on schedule even if non-critical)
Important facts
- The CRITICAL PATH is the LONGEST path, not the shortest.
- All activities on the critical path have Total Float = 0; they are 'critical'.
- Non-critical activities have positive float and can be delayed without delaying the project.
- Forward pass = compute ES and EF. Backward pass = compute LS and LF.
- Project duration = EF of the final activity (or max EF if multiple final activities).
- To shorten project duration, ONLY crash activities on the critical path.
- Crashing the wrong activity (non-critical) does NOT shorten the project.
- Multiple critical paths can exist; all have the same length = project duration.
- AOA networks use nodes (circles/ovals) for events, arrows for activities. AON networks use nodes for activities, arrows for dependencies.
- Total float applies to the project; free float applies to individual successors.
Key definitions
Critical Path
Example
If paths are 15 days, 12 days, and 14 days, the critical path is 15 days.
The LONGEST path through the network; its length = minimum project duration; all activities on it have TF = 0.
Activity
Example
Foundation work (7 days), steel erection (5 days).
A task with a duration and logical predecessors/successors; consumes time and/or resources.
Event
Example
Start of project (day 0), end of excavation.
A point in time marking the start or finish of one or more activities; has zero duration.
Dummy Activity
Example
Used when activity X must finish before activity Z starts, but Y is independent of both.
A zero-duration logical link showing dependency only, used in AOA (Activity-on-Arrow) networks to clarify precedence.
Early Times
Example
Activity starts on day 5 (ES), finishes day 10 (EF) if predecessor finishes by day 5.
ES and EF computed in forward pass; earliest an activity can start/finish assuming all predecessors finish earliest.
Late Times
Example
Activity must finish by day 20 (LF) to keep the project on schedule.
LS and LF computed in backward pass; latest an activity can start/finish without delaying the project.
Float (Slack)
Example
An activity with TF = 3 days can slip 3 days without delaying the project.
Spare time an activity has; can be delayed by this amount without affecting project completion.
Crashing
Example
Pay overtime or hire more workers to finish concrete in 3 days instead of 5.
Reducing activity duration by adding resources (labor, equipment, money); trades cost for time.
Diagrams to know
- Precedence diagram (network diagram) with activities, durations, and dependencies
- Gantt chart showing ES, EF, LS, LF, float, and critical path visually
- Forward and backward pass timeline with early and late times labeled
- Activity-on-Arrow (AOA) network with events and dummy activities
- Activity-on-Node (AON) network with dependencies shown as arrows between activity boxes
PERT (Program Evaluation and Review Technique) — Probabilistic Scheduling
Formulas
t_e = (a + 4m + b) / 6
Meaning
t_e = expected duration; a = optimistic, m = most likely, b = pessimistic
Watch out
The most likely (m) gets WEIGHT 4, not weight 1. Divide by 6, not 3. Common error: (a + m + b)/3 is WRONG.
When to use
When activity durations are uncertain; replace single estimates with expected values
σ² = ((b - a) / 6)²
Meaning
σ² = variance of activity duration; measures uncertainty spread
Watch out
The formula is ((b - a) / 6)², NOT (b - a)² / 36 (same thing, but easy to mess up the order). Use (b - a) in numerator.
When to use
Calculate variance for each activity; sum variances along critical path for project variance
σ_project = √(Σσ² along critical path)
Meaning
σ_project = standard deviation of project duration; square root of sum of variances
Watch out
Only sum variances of activities ON THE CRITICAL PATH, not all activities.
When to use
Determine project variability; use with normal distribution to estimate probability of meeting target dates
Z = (T_target - t_e_project) / σ_project
Meaning
Z = standard normal variable; T_target = desired finish date; used to find probability
Watch out
If Z is negative, probability of meeting the date is less than 50%. If Z > 3, probability approaches 100%.
When to use
Convert target date into Z-score; look up probability in standard normal table
Important facts
- PERT uses three estimates (a, m, b) to handle uncertainty; CPM uses single (deterministic) estimates.
- The expected duration formula weights the most likely time by 4 (out of 6), not equally.
- Activity variance depends ONLY on (b - a), not on m; wider range = higher uncertainty.
- Project duration variance = sum of variances along THE CRITICAL PATH only; non-critical activities don't contribute.
- By the Central Limit Theorem, project duration approaches a normal distribution even if individual activities are beta-distributed.
- A Z-score of 0 means 50% probability (target = expected). Z = 1 ≈ 84%, Z = 2 ≈ 98%, Z = 3 ≈ 99.9%.
- Negative Z-score = probability < 50%; the target date is earlier than expected.
- If critical path changes during risk analysis, variances must be recalculated for the new critical path.
- PERT is useful for R&D, new construction types, and high-uncertainty projects; less useful for well-defined, routine work.
Key definitions
Optimistic Time (a)
Example
Foundation concrete: 3 days (if no rain, no equipment breakdown, extra labor).
Shortest reasonable time to complete an activity under ideal conditions; accounts for minimal delays.
Most Likely Time (m)
Example
Foundation concrete: 5 days (typical conditions, normal crew, typical weather).
Best estimate of activity duration based on experience and normal conditions; the mode of distribution.
Pessimistic Time (b)
Example
Foundation concrete: 11 days (heavy rain, equipment delays, labor shortage).
Longest reasonable time to complete an activity accounting for foreseeable problems; the worst case.
Expected Duration (t_e)
Example
Foundation: t_e = (3 + 4×5 + 11)/6 = 6 days.
The weighted average of a, m, b; represents the statistically expected duration for an uncertain activity.
Variance (σ²)
Example
If σ² = 4, then σ = 2 days; wide spread from a to b means high variance.
Measure of activity duration uncertainty; larger variance = more uncertainty in the estimate.
Standard Deviation (σ)
Example
σ_project = 2 days means project duration has ±2-day variability (±1 standard deviation).
Square root of variance; expresses uncertainty in the same units as duration (days, weeks).
Beta Distribution
Example
PERT assumes activity duration follows a beta curve, not a normal distribution (though project duration approaches normal).
The assumed probability distribution for PERT durations; unimodal and skewed, with mean = t_e and variance = ((b-a)/6)².
Diagrams to know
- Beta distribution curve (skewed, unimodal) for a single activity duration
- Normal distribution curve for project duration (bell curve, centered at t_e_project)
- Three-point estimate diagram showing a, m, b on a timeline with t_e marked
- Risk vs. Duration graph showing probability of completion on different dates
Crashing & Time-Cost Trade-offs
Formulas
Crash cost per unit time = (Crash cost - Normal cost) / (Normal duration - Crash duration)
Meaning
Rate of cost increase per day (or hour) of time reduction; used to prioritize which activities to crash
Watch out
Do NOT crash activities that are not on the critical path; it wastes money without shortening the project. Recalculate critical path after each crash.
When to use
Identify the cheapest way to reduce project duration; crash lowest-cost-per-day activities first
Crash limit = Normal duration - Crash duration (max reducible time per activity)
Meaning
The maximum amount any single activity can be compressed; beyond this, no further reduction is possible
Watch out
Do not try to crash an activity beyond its crash limit; it's physically/logically impossible.
When to use
Determine how much time is practically reducible for each activity before crashing
Important facts
- Crashing only critical-path activities reduces project duration; crashing non-critical activities wastes money.
- Always crash the activity with the LOWEST crash cost slope first to minimize total project cost.
- After crashing an activity, the critical path MAY SHIFT; recheck which activities are critical before the next crash.
- A non-critical activity may become critical after the original critical path is shortened.
- Some activities have a crash limit of zero (cannot be crashed further, even with infinite cost).
- Total project cost = normal costs + sum of crash costs for crashed activities.
- Crash only the amount necessary to meet the deadline; further crashing increases cost without benefit.
- The trade-off curve (time vs. total cost) is typically U-shaped or J-shaped; shortening too much increases direct and indirect costs.
Key definitions
Normal Duration & Cost
Example
Foundation takes 7 days at normal cost of PHP 50,000 with one crew.
Standard schedule and cost assuming regular workforce, normal hours, routine methods; baseline plan.
Crash Duration & Cost
Example
Same foundation in 5 days at crash cost of PHP 65,000 (overtime + extra crew).
Minimum possible time and associated cost by adding resources (overtime, extra labor, premium materials, expedited equipment).
Crash Cost Slope
Example
Slope = (65k - 50k) / (7 - 5) = PHP 7,500/day to crash foundation 2 days.
Incremental cost per unit time reduction; directly tells you cost-effectiveness of shortening an activity.
Diagrams to know
- Time-cost trade-off curve (project duration vs. total cost)
- Crash slope diagram showing cost increase per day for each activity
- Normal and crash schedule comparison (Gantt chart side-by-side)
Network Analysis — Forward & Backward Pass Procedure
Important facts
- FORWARD PASS: Start from project start (ES = 0 for first activity). Compute EF = ES + d. Set ES (next) = max(EF of predecessors).
- BACKWARD PASS: Start from project end (LF = EF of last activity). Compute LS = LF - d. Set LF (predecessor) = min(LS of successors).
- Critical activities satisfy: ES = LS (equivalently, EF = LF or TF = 0).
- Non-critical activities have ES < LS, indicating they can start late and still be on time.
- An activity with TF = 0 is critical ONLY if it is on the longest path through the network.
- Parallel activities (no dependency) have independent float; they do not compete for resources in basic CPM.
- A 'merge point' (multiple predecessors) determines ES by the LATEST predecessor finish.
- A 'burst point' (multiple successors) determines LF by the EARLIEST successor start (backward pass).
- Float conservation: The sum of activity floats on non-critical paths equals the difference between critical-path length and that path's length.
Diagrams to know
- Forward pass timeline (ES, EF marked on each activity)
- Backward pass timeline (LS, LF marked on each activity)
- Combined forward + backward diagram highlighting critical path (TF = 0 activities)
Reactions or equations
EF = ES + d (forward pass)
Note
Start with ES = 0 for the first activity. Propagate forward to the end.
Conditions
For each activity, compute early finish from early start and duration
LS = LF - d (backward pass)
Note
Start with LF = EF of the last activity. Work backward to the start.
Conditions
For each activity, compute late start from late finish and duration
TF = LS - ES (float check)
Note
If unequal, arithmetic error in forward or backward pass.
Conditions
Verify with TF = LF - EF; both must be equal
Common Exam Traps & Pitfalls
Important facts
- TRAP 1: Confusing LONGEST path with shortest. Critical path is ALWAYS the longest; length = minimum project duration.
- TRAP 2: Forgetting to do a full backward pass. Errors in LF/LS propagation lead to wrong float calculations.
- TRAP 3: Using the average EF or minimum LS instead of MAX(EF) and MIN(LS). Merge and burst points require extremes, not averages.
- TRAP 4: Applying PERT formula as (a + m + b)/3. The correct formula is (a + 4m + b)/6 with the '4' being critical.
- TRAP 5: Summing variance of all activities, not just critical-path activities. Only critical-path variances determine project variability.
- TRAP 6: Crashing non-critical activities. It wastes money and does NOT shorten the project.
- TRAP 7: Ignoring the fact that crashing can shift the critical path. Must recalculate after each crash decision.
- TRAP 8: Confusing Total Float with Free Float. Total = project delay; Free = successor delay.
- TRAP 9: Treating dummy activities as having duration. Dummy = 0 duration, logical link only.
- TRAP 10: Misinterpreting negative Z-score in PERT. Z < 0 means probability of meeting target < 50%.
Must remember
- CRITICAL PATH = LONGEST path through network; its length = minimum project duration. All activities on it have TF = 0.
- PERT formula is t_e = (a + 4m + b) / 6, NOT (a + m + b) / 3. The '4' weighting of 'm' is essential.
- Forward Pass: EF = ES + d, and ES(next) = MAX(EF of all predecessors). Backward Pass: LS = LF - d, and LF(predecessor) = MIN(LS of all successors).
- Total Float (TF) = LS - ES = LF - EF. If TF ≠ 0, the values are inconsistent (error in forward/backward pass).
- Only CRASH activities on the CRITICAL PATH to shorten project duration. Crashing non-critical activities wastes money.
- Activity Variance σ² = ((b - a) / 6)². Only sum variances of activities ON THE CRITICAL PATH for project variance.
- Z-score: Z = (T_target - t_e_project) / σ_project. Positive Z = probability > 50%; negative Z = probability < 50%.
- After crashing an activity, RECALCULATE the critical path. A new activity may become critical, and the old critical path may no longer be longest.
- Free Float (FF) ≤ Total Float (TF) always. FF indicates if an activity can slip without hurting successors; TF indicates if it can slip without hurting the project.
- Merge Point: Take MAX of predecessor EF values. Burst Point: Take MIN of successor LS values (backward pass). Never average or use median.
Last minute tips
- ALWAYS double-check the critical path is the LONGEST, not shortest. If your calculated project duration is less than the longest path you identified, stop and recalculate.
- In PERT, memorize that the weight on 'm' is 4 and total divisor is 6: t_e = (a + 4m + b) / 6. Practice this formula three times before the exam.
- When computing float, verify TF = LS - ES equals TF = LF - EF. If they differ, you have an arithmetic error somewhere in the forward or backward pass.
- Before crashing, identify the critical path. Check the crash cost slope for each critical activity. Crash the lowest-slope activity first. After each crash, recheck the critical path.
- In PERT probability questions, use a Z-table or normal distribution approximation. Memorize: Z = 0 → 50%, Z = 1 → 84%, Z = 2 → 98%, Z = 3 → 99.9%.
Comparison tables
CPM vs. PERT
| Aspect | CPM (Critical Path Method) | PERT (Program Evaluation and Review Technique) |
|---|---|---|
| Duration Estimate | Single (deterministic), best-guess value | Three estimates: optimistic (a), most likely (m), pessimistic (b) |
| Appropriate Use | Well-defined, routine, predictable projects (building construction, manufacturing) | High-uncertainty, R&D, first-time projects (space, new technology, research) |
| Duration Formula | d = single estimate | t_e = (a + 4m + b) / 6 |
| Probability Analysis | Deterministic; single project duration | Probabilistic; project duration is a distribution with variance σ_project² |
| Focus | Time management and critical path identification | Uncertainty quantification and risk estimation |
| Variance Concept | Not used; no uncertainty measure | Central to the method; σ² = ((b-a)/6)² for each activity |
Total Float vs. Free Float
| Property | Total Float (TF) | Free Float (FF) |
|---|---|---|
| Definition | How much an activity can slip without delaying the PROJECT | How much an activity can slip without delaying any SUCCESSOR |
| Formula | TF = LS - ES = LF - EF | FF = EF(successor) - EF(activity) |
| Scope | Project-level; affects overall completion | Activity-level; affects only direct successors |
| Magnitude | Typically larger; includes buffer for downstream activities | Always ≤ TF; subset of total float |
| Critical Activity | TF = 0 (must be on critical path) | FF = 0 (may or may not be critical) |
| When to Use | Project scheduling and baseline planning | Risk mitigation; identify 'hidden critical' activities with FF = 0 |
Forward Pass vs. Backward Pass
| Aspect | Forward Pass | Backward Pass |
|---|---|---|
| Direction | Start → End (left to right in network) | End → Start (right to left in network) |
| Starting Point | Project start; set ES = 0 for first activity | Project end; set LF = EF of last activity |
| Computation | EF = ES + d for each activity | LS = LF - d for each activity |
| At Merge Points | ES (next) = MAX(EF of predecessors) | LF (predecessor) = MIN(LS of successors) |
| Output | Early Start (ES) and Early Finish (EF) times | Late Start (LS) and Late Finish (LF) times |
| Purpose | Determine earliest possible project completion | Determine latest allowable times to avoid delays |
Activity-on-Arrow (AOA) vs. Activity-on-Node (AON)
| Feature | AOA (Activity-on-Arrow) | AON (Activity-on-Node) |
|---|---|---|
| Activity Location | Arrow (edge) between two event nodes | Node (box) with arrows showing dependencies |
| Event | Node (circle/oval) represents start/end point | Not explicitly shown; implicit between nodes |
| Dummy Activity | Often needed to clarify dependencies; zero duration | Rarely needed; logic shown directly on arrows |
| Ease of Use | More complex; dummy activities add confusion | Simpler; clearer representation of logic |
| Modern Practice | Less common; historical method | Standard; used in most project management software |
| Notation | Nodes labeled by number (e.g., 1, 2, 3); arrows by activity name | Nodes labeled by activity name; arrows show precedence |
Diagrams
CPM Forward & Backward Pass Decision Tree
Step-by-step logic for identifying the critical path using forward and backward passes. Shows when to use each computation and how to verify results.
CPM/PERT Key Formulas & Relationships
Comprehensive mind map showing all major CPM and PERT formulas, their relationships, and the logical flow from network analysis to risk quantification and cost optimization.
PERT Activity Duration Estimation & Project Risk Assessment
Complete flowchart for PERT estimation and risk assessment. Shows how to gather three-point estimates, compute expected duration and variance per activity, aggregate critical-path variance, and use Z-scores for probability analysis.
Typical Exam Question Distribution — CPM/PERT
Distribution of question types likely to appear on professional engineering exams. Critical path and float questions dominate; PERT calculations are second-most frequent.