CELE Construction Management & Methods — Project Planning and Scheduling (CPM/PERT)Revision Notes

Exam Tips

• When given a list of activities with durations and predecessors, always enumerate ALL paths and their total durations before declaring the critical path.
• In small networks (≤8 activities), enumerate paths by inspection; in larger ones, use the forward/backward pass.
• If two paths have equal duration, BOTH are critical — state this explicitly in board exam answers.
• The minimum project duration equals the length of the critical path — this is a definition, not a derived result.

Key Points

• A project is broken into discrete activities, each with a defined duration and a set of predecessor activities that must be completed before it can start.
• Two network representations exist: Activity-on-Arrow (AOA / Arrow Diagram Method, ADM) and Activity-on-Node (AON / Precedence Diagram Method, PDM). PDM is the modern industry standard and the one most commonly tested.
• In PDM, each node (box) represents an activity. Arrows show dependency (finish-to-start unless otherwise stated).
• The project network has a unique Start node and a unique Finish node.
• The critical path is the LONGEST continuous path from Start to Finish — it sets the minimum project duration.
• Activities on the critical path have ZERO total float — any delay in them directly delays the project.
• Non-critical activities have positive total float, meaning they can be delayed within limits without affecting the project finish date.
• A project may have more than one critical path (multiple paths of equal length).

Definitions

Section Title

1. Network Fundamentals — Activities, Precedence, and the CPM Network

Common Mistakes

• Choosing the SHORTEST path as the critical path — it is always the LONGEST.
• Forgetting to include all paths when enumerating; missing a hidden longer path.
• Confusing Activity-on-Arrow with Activity-on-Node notation when reading network diagrams.
• Assuming there is only one critical path — there can be multiple paths of equal (maximum) duration.

Formulas

Exam Tips

• Always label each node with a four-value box: [ES | EF] on top and [LS | LF] on bottom (to be filled in backward pass).
• Work strictly left to right. Do not jump ahead to compute backward pass times until the entire forward pass is complete.
• Double-check by confirming: the project duration from the forward pass should equal the length of the path you identify as critical.
• Board problems often give you a partially completed network and ask you to find one missing ES or EF — apply EF = ES + d directly.

Key Points

• The forward pass computes the EARLIEST time each activity can start (Early Start, ES) and finish (Early Finish, EF).
• It moves from left to right (from Start toward Finish) through the network.
• For the very first activity (or any activity with no predecessors), ES = 0.
• EF = ES + duration (d). This is the single most important CPM formula.
• When an activity has MULTIPLE predecessors, its ES = the MAXIMUM EF among all its predecessors. (You must wait for ALL predecessors to finish before you can start.)
• The EF of the last activity (or the maximum EF of all activities ending the project) = project early finish = minimum project duration.
• All early times are computed using the MAXIMUM rule at merge points (nodes with multiple incoming arrows).

Definitions

Section Title

2. Forward Pass — Early Start and Early Finish

Common Mistakes

• At a merge point, taking the MINIMUM instead of the MAXIMUM EF of predecessors — this violates the logic that ALL predecessors must finish first.
• Starting the first activity at ES = 1 instead of ES = 0 — the convention is ES = 0 for the project start.
• Forgetting to update ES of ALL successors when computing EF of a given activity.
• Carrying forward only one predecessor's EF at a merge point and ignoring the others.

Formulas

Exam Tips

• Remember: Forward pass → MAX at merges; Backward pass → MIN at bursts. This asymmetry is a classic exam trap.
• After completing both passes, verify: for critical activities, ES = LS and EF = LF. If this does not hold, recheck your arithmetic.
• In board problems with 4–6 activities, always show both passes in a table with columns: Activity, d, ES, EF, LS, LF, TF — this prevents errors and earns partial marks.
• The LF of the project's terminal node must equal its EF from the forward pass — this is the anchor for the backward pass.

Key Points

• The backward pass computes the LATEST time each activity can start (Late Start, LS) and finish (Late Finish, LF) WITHOUT delaying the project.
• It moves from right to left (from Finish toward Start) through the network.
• For the very last activity (or any activity with no successors), LF = project duration (the EF from the forward pass).
• LS = LF − d. Paired with EF = ES + d for the forward pass.
• When an activity has MULTIPLE successors, its LF = the MINIMUM LS among all its successors. (The activity must finish in time for the earliest-starting successor.)
• All late times are computed using the MINIMUM rule at burst points (nodes with multiple outgoing arrows).
• Critical activities satisfy: ES = LS and EF = LF (equivalently, TF = 0).

Definitions

Section Title

3. Backward Pass — Late Start and Late Finish

Common Mistakes

• At a burst point, taking the MAXIMUM instead of the MINIMUM LS of successors — using the maximum would violate the constraint of not delaying any successor.
• Setting LF of the last activity equal to something other than the project duration from the forward pass.
• Confusing the forward-pass rule (MAX at merges) with the backward-pass rule (MIN at bursts).
• Not completing the full backward pass before computing float values.

Formulas

Exam Tips

• Board exam questions on float usually give you four values (ES, EF, LS, LF) and ask for TF — apply TF = LS - ES directly.
• If asked which activity is NOT on the critical path, identify the one with TF > 0.
• Free float is tested less often than total float, but know that FF = min(ES_successors) - EF.
• A common board question: 'By how many days can Activity X be delayed without affecting the project?' — this is asking for TF.

Key Points

• Float (also called slack) measures the scheduling flexibility of a non-critical activity.
• TOTAL FLOAT (TF): The maximum amount of time an activity can be delayed from its ES without delaying the PROJECT finish. TF = LS - ES = LF - EF.
• FREE FLOAT (FF): The maximum amount of time an activity can be delayed from its ES without delaying the ES of ANY of its IMMEDIATE SUCCESSORS. FF = min(ES of successors) - EF of the activity.
• TF ≥ FF always — total float is the more general measure.
• Critical activities have TF = 0 (and FF = 0).
• Non-critical activities have TF > 0 and may have FF = 0 (they can use their float only at the expense of sharing it with other activities on the same sub-path).
• If an activity uses all its total float, it becomes critical — this is called a SECONDARY critical path.
• Crashing (compressing) a non-critical activity does NOT shorten the project duration.

Definitions

Section Title

4. Float (Slack) — Total Float and Free Float

Common Mistakes

• Confusing Total Float and Free Float — TF is relative to the project; FF is relative to successors.
• Computing TF as EF - ES (this gives the duration, not the float).
• Stating that an activity with TF > 0 can be delayed by TF without affecting anything — it can affect successor float even if not the project finish.
• Forgetting that using float on one activity reduces the float available to other activities on the same path.

Formulas

Exam Tips

• Memorize the PERT formulas as a pair: t_e = (a+4m+b)/6 and σ = (b-a)/6. The σ² = [σ]² naturally.
• When asked for project standard deviation from variances, ALWAYS sum σ² first, THEN take √ — never sum σ directly.
• For probability problems, set up the Z-score clearly: Z = (T - T_E)/σ_project, then look up the Z-table. Board exams often provide the Z-table or specify the probability for given Z values.
• If the problem gives you t_e and asks you to back-calculate a or b, set up the t_e equation algebraically and solve — this is a common variation.
• A quick check: t_e should always be between a and b, and t_e ≥ m only if b is much larger than m (skewed distribution).

Key Points

• PERT (Program Evaluation and Review Technique) is used when activity durations are UNCERTAIN and cannot be fixed with confidence.
• Three time estimates are used for each activity: optimistic (a), most likely (m), and pessimistic (b).
• The expected (mean) duration t_e uses a BETA distribution approximation: t_e = (a + 4m + b) / 6.
• The most likely estimate m is WEIGHTED BY 4 — a critical formula detail tested every board cycle.
• The variance of each activity: σ² = ((b - a) / 6)².
• The standard deviation of each activity: σ = (b - a) / 6.
• For the CRITICAL PATH, the project expected duration = sum of t_e values of critical activities.
• Project variance = SUM of σ² values of critical path activities only (not all activities).
• Project standard deviation: σ_project = √(Σσ²_critical path).
• To find the probability of completing by a target date T: compute Z = (T - T_E) / σ_project, then use the standard normal distribution table.
• PERT uses the same forward/backward pass and float calculations as CPM, but with t_e replacing fixed durations.

Definitions

Section Title

5. PERT — Three-Point Estimates, Expected Duration, and Variance

Common Mistakes

• Forgetting the weight of 4 for m: writing t_e = (a + m + b) / 3 instead of (a + 4m + b) / 6.
• Dividing by 3 instead of 6 in the t_e formula.
• Including the variance of NON-critical activities in the project variance — only critical path variances are summed.
• Computing σ = (b-a)/6 but then squaring it incorrectly: σ² = (b-a)²/6 instead of the correct ((b-a)/6)² = (b-a)²/36.
• Using σ_project instead of σ²_project when summing — always sum VARIANCES (σ²), then take the square root at the end.

Formulas

Exam Tips

• Board problems on crashing usually ask: 'What is the minimum cost to reduce project duration by X days?' — apply the crash slope ranking procedure.
• Always state the critical path at each step of the crashing process, as it may change.
• If two paths become simultaneously critical, you must crash one activity on each path simultaneously — add both their crash slopes for the combined cost.
• The total crash cost = Σ(crash slope × days crashed) for each activity crashed.

Key Points

• Crashing means REDUCING the duration of an activity by adding resources (more workers, equipment, overtime) at an additional cost.
• The CRASH COST SLOPE (cost per day saved) = (Crash Cost - Normal Cost) / (Normal Duration - Crash Duration).
• Only crashing CRITICAL PATH activities shortens the project duration.
• Among critical path activities, crash the one with the LOWEST cost slope first (cheapest to crash).
• After each crash iteration, re-examine the network — the critical path may shift as you compress it.
• Continue crashing until: (a) the target duration is reached, (b) all critical activities are fully crashed, or (c) a NEW path becomes critical at equal length, requiring simultaneous crashing of multiple paths.
• The minimum project duration achievable is bounded by the fully crashed critical path.
• The optimal solution balances crashing cost against indirect cost savings (overhead) from shorter duration.

Definitions

Section Title

6. Crashing — Cost–Time Trade-offs

Common Mistakes

• Crashing a NON-critical activity and expecting to shorten the project — this only wastes money.
• Forgetting to re-evaluate the critical path after each crashing step — new paths may become critical.
• Computing crash slope with the formula inverted: (ND - CD)/(CC - NC) — this gives days saved per peso, not pesos per day.
• Crashing beyond the crash duration — each activity has a physical lower limit on duration regardless of resources added.