CELE Construction Management & Methods — Project Planning and Scheduling (CPM/PERT)Cheat Sheet
Project Planning and Scheduling (CPM/PERT) cheat sheet — the reference card you wish you had on exam day. Condensed from the full study notes, this is the high-yield core of Project Planning and Scheduling (CPM/PERT) for CELE Construction Management & Methods. Download, print, revise.
Exam context
On the CELE 2026, the Construction Management & Methods subtest carries a "Core" weight in Professional Regulation Commission (PRC) — Board of Civil Engineering's pattern. Project Planning and Scheduling (CPM/PERT) lands at position 2nd out of 5 in the standard review order. Target score is 70% weighted average, no sub-test below 50%, and roughly a meaningful share of items come from Construction Management & Methods on a typical CELE paper.
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.
Sections
Formulas
Formula
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
Formula
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
Formula
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
Formula
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
Formula
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
Formula
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)
Section Title
Network Fundamentals & Critical Path Method (CPM)
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
Term
Critical Path
Example
If paths are 15 days, 12 days, and 14 days, the critical path is 15 days.
Definition
The LONGEST path through the network; its length = minimum project duration; all activities on it have TF = 0.
Term
Activity
Example
Foundation work (7 days), steel erection (5 days).
Definition
A task with a duration and logical predecessors/successors; consumes time and/or resources.
Term
Event
Example
Start of project (day 0), end of excavation.
Definition
A point in time marking the start or finish of one or more activities; has zero duration.
Term
Dummy Activity
Example
Used when activity X must finish before activity Z starts, but Y is independent of both.
Definition
A zero-duration logical link showing dependency only, used in AOA (Activity-on-Arrow) networks to clarify precedence.
Term
Early Times
Example
Activity starts on day 5 (ES), finishes day 10 (EF) if predecessor finishes by day 5.
Definition
ES and EF computed in forward pass; earliest an activity can start/finish assuming all predecessors finish earliest.
Term
Late Times
Example
Activity must finish by day 20 (LF) to keep the project on schedule.
Definition
LS and LF computed in backward pass; latest an activity can start/finish without delaying the project.
Term
Float (Slack)
Example
An activity with TF = 3 days can slip 3 days without delaying the project.
Definition
Spare time an activity has; can be delayed by this amount without affecting project completion.
Term
Crashing
Example
Pay overtime or hire more workers to finish concrete in 3 days instead of 5.
Definition
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
Formulas
Formula
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
Formula
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
Formula
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
Formula
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
Section Title
PERT (Program Evaluation and Review Technique) — Probabilistic Scheduling
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
Term
Optimistic Time (a)
Example
Foundation concrete: 3 days (if no rain, no equipment breakdown, extra labor).
Definition
Shortest reasonable time to complete an activity under ideal conditions; accounts for minimal delays.
Term
Most Likely Time (m)
Example
Foundation concrete: 5 days (typical conditions, normal crew, typical weather).
Definition
Best estimate of activity duration based on experience and normal conditions; the mode of distribution.
Term
Pessimistic Time (b)
Example
Foundation concrete: 11 days (heavy rain, equipment delays, labor shortage).
Definition
Longest reasonable time to complete an activity accounting for foreseeable problems; the worst case.
Term
Expected Duration (t_e)
Example
Foundation: t_e = (3 + 4×5 + 11)/6 = 6 days.
Definition
The weighted average of a, m, b; represents the statistically expected duration for an uncertain activity.
Term
Variance (σ²)
Example
If σ² = 4, then σ = 2 days; wide spread from a to b means high variance.
Definition
Measure of activity duration uncertainty; larger variance = more uncertainty in the estimate.
Term
Standard Deviation (σ)
Example
σ_project = 2 days means project duration has ±2-day variability (±1 standard deviation).
Definition
Square root of variance; expresses uncertainty in the same units as duration (days, weeks).
Term
Beta Distribution
Example
PERT assumes activity duration follows a beta curve, not a normal distribution (though project duration approaches normal).
Definition
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
Formulas
Formula
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
Formula
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
Section Title
Crashing & Time-Cost Trade-offs
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
Term
Normal Duration & Cost
Example
Foundation takes 7 days at normal cost of PHP 50,000 with one crew.
Definition
Standard schedule and cost assuming regular workforce, normal hours, routine methods; baseline plan.
Term
Crash Duration & Cost
Example
Same foundation in 5 days at crash cost of PHP 65,000 (overtime + extra crew).
Definition
Minimum possible time and associated cost by adding resources (overtime, extra labor, premium materials, expedited equipment).
Term
Crash Cost Slope
Example
Slope = (65k - 50k) / (7 - 5) = PHP 7,500/day to crash foundation 2 days.
Definition
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)
Section Title
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
Note
Start with ES = 0 for the first activity. Propagate forward to the end.
Equation
EF = ES + d (forward pass)
Conditions
For each activity, compute early finish from early start and duration
Note
Start with LF = EF of the last activity. Work backward to the start.
Equation
LS = LF - d (backward pass)
Conditions
For each activity, compute late start from late finish and duration
Note
If unequal, arithmetic error in forward or backward pass.
Equation
TF = LS - ES (float check)
Conditions
Verify with TF = LF - EF; both must be equal
Section Title
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
Rows
Values
- Single (deterministic), best-guess value
- Three estimates: optimistic (a), most likely (m), pessimistic (b)
Property
Duration Estimate
Values
- Well-defined, routine, predictable projects (building construction, manufacturing)
- High-uncertainty, R&D, first-time projects (space, new technology, research)
Property
Appropriate Use
Values
- d = single estimate
- t_e = (a + 4m + b) / 6
Property
Duration Formula
Values
- Deterministic; single project duration
- Probabilistic; project duration is a distribution with variance σ_project²
Property
Probability Analysis
Values
- Time management and critical path identification
- Uncertainty quantification and risk estimation
Property
Focus
Values
- Not used; no uncertainty measure
- Central to the method; σ² = ((b-a)/6)² for each activity
Property
Variance Concept
Columns
- Aspect
- CPM (Critical Path Method)
- PERT (Program Evaluation and Review Technique)
Table Title
CPM vs. PERT
Rows
Values
- How much an activity can slip without delaying the PROJECT
- How much an activity can slip without delaying any SUCCESSOR
Property
Definition
Values
- TF = LS - ES = LF - EF
- FF = EF(successor) - EF(activity)
Property
Formula
Values
- Project-level; affects overall completion
- Activity-level; affects only direct successors
Property
Scope
Values
- Typically larger; includes buffer for downstream activities
- Always ≤ TF; subset of total float
Property
Magnitude
Values
- TF = 0 (must be on critical path)
- FF = 0 (may or may not be critical)
Property
Critical Activity
Values
- Project scheduling and baseline planning
- Risk mitigation; identify 'hidden critical' activities with FF = 0
Property
When to Use
Columns
- Property
- Total Float (TF)
- Free Float (FF)
Table Title
Total Float vs. Free Float
Rows
Values
- Start → End (left to right in network)
- End → Start (right to left in network)
Property
Direction
Values
- Project start; set ES = 0 for first activity
- Project end; set LF = EF of last activity
Property
Starting Point
Values
- EF = ES + d for each activity
- LS = LF - d for each activity
Property
Computation
Values
- ES (next) = MAX(EF of predecessors)
- LF (predecessor) = MIN(LS of successors)
Property
At Merge Points
Values
- Early Start (ES) and Early Finish (EF) times
- Late Start (LS) and Late Finish (LF) times
Property
Output
Values
- Determine earliest possible project completion
- Determine latest allowable times to avoid delays
Property
Purpose
Columns
- Aspect
- Forward Pass
- Backward Pass
Table Title
Forward Pass vs. Backward Pass
Rows
Values
- Arrow (edge) between two event nodes
- Node (box) with arrows showing dependencies
Property
Activity Location
Values
- Node (circle/oval) represents start/end point
- Not explicitly shown; implicit between nodes
Property
Event
Values
- Often needed to clarify dependencies; zero duration
- Rarely needed; logic shown directly on arrows
Property
Dummy Activity
Values
- More complex; dummy activities add confusion
- Simpler; clearer representation of logic
Property
Ease of Use
Values
- Less common; historical method
- Standard; used in most project management software
Property
Modern Practice
Values
- Nodes labeled by number (e.g., 1, 2, 3); arrows by activity name
- Nodes labeled by activity name; arrows show precedence
Property
Notation
Columns
- Feature
- AOA (Activity-on-Arrow)
- AON (Activity-on-Node)
Table Title
Activity-on-Arrow (AOA) vs. Activity-on-Node (AON)
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