Project planning: CPM and PERT

In short

CPM (Critical Path Method) and PERT (Programme Evaluation and Review Technique) are network-based planning techniques that break a project into activities, link them by dependency, and compute the shortest possible completion time and the slack available on every activity. CPM assumes a single deterministic duration per activity and is used for repetitive construction work; PERT uses three time estimates and gives the probability of finishing by a target date, so it suits research-type or first-of-a-kind work. For a civil engineer they decide when materials, formwork, labour gangs and payments are mobilised, and they are the technical basis of delay claims and extension-of-time disputes. The topic carries guaranteed marks in GATE Civil (floats, cost slope, PERT probability) and university construction-management papers, and it is asked in planning-engineer interviews at L&T, Tata Projects, Afcons, Shapoorji Pallonji, NCC, NHAI/DMRC consultancies and PMC firms, where Primavera P6 and MS Project are the working tools.

Explained

From bar charts to networks

• Bar (Gantt) chart: activities as horizontal bars against a time scale. Simple and good for communication, but it does not show *why* an activity starts when it does, cannot show interdependency, and does not reveal float — so the effect of a delay cannot be traced.
• Milestone chart: a bar chart with key events marked; an intermediate step towards networks.
• Network: a graphical model of activities and their logical sequence. It exposes the critical path (the longest path, which fixes project duration) and the float elsewhere.
• Origins: CPM developed at DuPont (1957) for plant maintenance; PERT developed for the US Navy Polaris programme (1958). Modern software merges both — deterministic CPM logic with optional probabilistic analysis.
• Three planning questions a network answers: minimum project duration, which activities must not slip, and where resources can be shifted without hurting the end date.

Network rules, AoA and AoN

• Activity: a task consuming time and resources (e.g. "cast slab"). Event (node in AoA): an instant marking the start/finish of activities; consumes no time.
• Activity-on-Arrow (AoA / i–j / arrow diagram): activities are arrows, events are circles. Needs dummy activities (dashed, zero duration) for two purposes — *logical* dummies to show a dependency correctly, and *identity (grammatical)* dummies to prevent two activities sharing the same start and end event numbers.
• Activity-on-Node (AoN / precedence diagram): activities are boxes, arrows are only dependencies. No dummies needed; this is what P6 and MS Project use.
• Rules: one start and one end event; no looping (a cycle, which implies an activity precedes itself); no dangling activity (an arrow ending nowhere except at the final event); each activity has a unique identity; arrows show logic, not length or scale (unless a time-scaled/squared network is drawn).
• Fulkerson's rule for event numbering: delete all arrows leaving the start event, number the new "start" events next, and repeat — so every arrow runs from a lower to a higher number (i < j).
• Terminology: *predecessor / successor*, *concurrent* activities, *burst* event (many activities start), *merge* event (many activities end).
• WBS first: planning starts with a Work Breakdown Structure (project → sub-project → work package → activity), then logic, then durations from quantities ÷ productivity.

CPM time analysis: passes, floats and the critical path

1. Forward pass (earliest times), starting at zero: ES of an activity = largest EF of its predecessors; EF = ES + D. In AoA, earliest event time Ej = max(Ei + Dij). The final value is the project duration.
2. Backward pass (latest times), starting with LF of the last activity = project duration: LF = smallest LS of its successors; LS = LF − D. In AoA, Li = min(Lj − Dij).
3. Event slack = Li − Ei. A critical event has zero slack; a critical activity has zero total float and lies on a chain of critical events.
4. Critical path: the continuous chain from start to finish with zero total float; it is the longest path in time, and it fixes the project duration. There may be more than one critical path; a path with small float is sub-critical and must be watched.
5. Total float TF = LS − ES = LF − EF. The delay an activity can absorb without delaying the project (but it eats into successors' float).
6. Free float FF = min(ES of successors) − EF (in AoA, FF = TF − slack of head event). Delay available without disturbing the early start of any following activity.
7. Independent float IF = ES of successor − LF of predecessor − D, taken as zero if negative (in AoA, IF = FF − slack of tail event). Float available even if predecessors finish as late as possible and successors start as early as possible.
8. Interfering float = TF − FF — the part of the float that, if used, pushes successors later.
9. Always TF ≥ FF ≥ IF. Float on a path is shared: once one activity consumes it, the others on that path lose it.
10. Advantages of CPM: pinpoints controlling activities, quantifies the effect of a delay, supports resource and cash planning, and gives a defensible record for claims. Limitations: needs reliable durations and complete logic; a network is only as good as its assumptions; it ignores resource limits unless levelling is done; and it is activity-based, so continuous repetitive work (pipelines, housing blocks) is often better shown by Line of Balance (LOB).

PERT: three time estimates and probability

• Used when durations are uncertain. Three estimates per activity: optimistic to, most likely tm, pessimistic tp.
• The activity duration is assumed to follow a beta distribution (unimodal, finite range, not necessarily symmetric), giving the weighted mean te = (to + 4tm + tp)/6 and standard deviation σ = (tp − to)/6 (taking the range as about 6σ).
• The network is then analysed exactly like CPM using te, giving the expected project duration Te along the critical path.
• By the central limit theorem, the sum of many independent activity durations tends to a normal distribution, so the project duration is treated as normal with mean Te and variance = sum of the variances of the critical activities only.
• Probability of meeting a scheduled date Ts: compute Z = (Ts − Te)/σ_path and read the normal table. Z = 0 gives 50%; the planner usually targets a probability of about 85–95%, which means adding a time contingency to Te.
• PERT is event-oriented (TE, TL and slack at events) and originally non-repetitive, probabilistic, time-only; CPM is activity-oriented, deterministic, and handles time–cost trade-off.
• Limitations of PERT: the three estimates are subjective; only the single critical path's variance is used, so merge-event bias makes the computed probability optimistic when several near-critical paths merge; Monte Carlo simulation in modern software avoids this.

Precedence networks, leads and lags

• Four relationships between activities i and j: FS (finish–start, the default), SS (start–start), FF (finish–finish) and SF (start–finish, rare).
• A lead/lag is a time offset: "SS + 5 d" means j may start 5 days after i starts. Used for overlapping (fast-tracked) work — e.g. start block-work 7 days after slab casting starts on a long floor.
• Ladder (staircase) networks model long, overlapping continuous activities such as excavation → bedding → pipe-laying → backfilling in a trench.
• Constraints such as "start no earlier than" (concrete curing, monsoon windows, permit dates) are imposed as date constraints; over-constraining destroys the network's logic and hides real float.
• Negative float appears when a constraint or an imposed finish date is earlier than the logic allows — a signal to re-plan, not a result to report.

Time–cost trade-off and crashing

• Direct cost (labour, material, plant) rises as duration is shortened; indirect cost (site establishment, supervision, overheads, rent, finance) rises with duration. Total cost shows a minimum at the optimum duration.
• Normal time/cost = most economical duration; crash time/cost = shortest technically possible duration with its higher cost. The relation is assumed linear between them.
• Cost slope = (Crash cost − Normal cost)/(Normal time − Crash time) — the extra rupees per day saved.
• Crashing procedure: (1) find the critical path; (2) crash the critical activity with the lowest cost slope, one step at a time; (3) after every step re-run the passes, because a new path may become critical (then parallel critical paths must be crashed together, comparing the *combined* slope); (4) stop when the cheapest available slope exceeds the indirect cost per day, or when all critical activities are at crash time (crash duration of the project).
• Crashing a non-critical activity, or crashing beyond the point where slope > indirect saving, only wastes money.
• Crashing vs fast-tracking: crashing adds resources to the same logic; fast-tracking overlaps activities (changes the logic) and adds rework risk.

Resources, monitoring, updating and Indian practice

• Resource allocation (loading) converts the schedule into a histogram of labour, plant and cash per day; the uneven peaks are then treated by:
  • Resource smoothing — project duration is fixed; non-critical activities are shifted within their float to even out demand.
  • Resource levelling — resource ceiling is fixed (e.g. only 2 batching plants); the project duration may be extended.
• Updating: the network is revised periodically with actual progress, revised durations for the balance work and changed logic; the critical path commonly shifts during execution.
• Monitoring: S-curve of planned vs actual cumulative progress/cost; earned-value measures (planned value, earned value, actual cost) to read schedule and cost variance; delay analysis (time-impact or as-planned-versus-as-built) for extension-of-time claims.
• Indian standards and practice: IS 15883 "Construction Project Management — Guidelines" is the Indian series (Part 1 general, with later parts covering time, cost, quality, safety and risk management — part numbering and years have been added to over time, so the current list should be checked on the BIS catalogue). CPWD Works Manual and MoRTH/NHAI, DMRC, Railways and metro contracts require the contractor to submit and update a CPM programme (bar chart plus network) as a contract document; EPC and FIDIC-type conditions make the approved programme the reference for delay and EoT assessment. Global practice follows PMBOK (PMI); schedules are prepared in Primavera P6, MS Project or Asta Powerproject, increasingly linked to a BIM model for 4D sequencing.
• Critical Chain (CCPM) is a later development: buffers (project buffer, feeding buffers) are managed instead of activity-level safety time, and resource dependency is treated as logic.

Key points

• Critical path - longest path through the network; it has zero total float and sets the project duration.
• Forward pass - earliest times; ES = max EF of predecessors, EF = ES + D.
• Backward pass - latest times; LF = min LS of successors, LS = LF − D.
• Total float - LS − ES = LF − EF; delay possible without extending the project.
• Free float - delay possible without disturbing the earliest start of any successor; TF ≥ FF ≥ IF.
• Independent float - float left when predecessors finish latest and successors start earliest; never negative.
• Event slack - Li − Ei; zero at every critical event.
• Dummy activity - zero time, zero resource arrow in AoA used for correct logic or unique activity identity.
• Fulkerson's rule - numbering scheme ensuring every arrow goes from a smaller to a larger event number.
• Looping and dangling - the two illegal network constructions.
• te = (to + 4tm + tp)/6 - PERT expected time from a beta distribution; σ = (tp − to)/6.
• Project variance - sum of variances of critical activities only; σ_path = √(Σσ²).
• Z = (Ts − Te)/σ_path - standard normal variate giving the probability of finishing by Ts.
• Merge-event bias - PERT probability is optimistic when several near-critical paths merge at an event.
• Cost slope - (crash cost − normal cost)/(normal time − crash time), in ₹ per day.
• Optimum duration - duration at minimum total cost; crash only while cost slope < indirect cost per day.
• Smoothing vs levelling - smoothing keeps the duration and uses float; levelling keeps the resource limit and may extend the duration.
• CPM vs PERT - deterministic, activity-oriented, repetitive work with cost trade-off versus probabilistic, event-oriented, novel work.

Formulas and examples

Example 1 — CPM: critical path and all three floats (AoN)

• Forward pass: E starts at max(EF of B = 6, EF of C = 9) = 9; G starts at max(16, 11) = 16 → project duration = 19 days.
• Backward pass: LF of A = min(LS of C = 4, LS of D = 9) = 4, so LS of A = 0.
• Critical path = A–C–E–G = 4 + 5 + 7 + 3 = 19 days.
• D and F share 5 days of float on the path A–D–F–G (4+3+4+3 = 14 days, i.e. 19 − 14 = 5). D's free float is 0 (F already starts at its earliest), so D's interfering float = 5 − 0 = 5 days: delaying D pushes F.
• B: TF = FF = IF = 3 days, because B has no predecessor activity and E is its only successor.

Example 2 — CPM on an AoA network: event times and slack

Activities (i–j, duration in weeks): 1–2 = 3, 1–3 = 5, 2–4 = 7, 3–4 = 4, 3–5 = 6, 4–6 = 5, 5–6 = 2.

• Forward: E1 = 0; E2 = 3; E3 = 5; E4 = max(3 + 7, 5 + 4) = 10; E5 = 5 + 6 = 11; E6 = max(10 + 5, 11 + 2) = 15 weeks.
• Backward: L6 = 15; L5 = 15 − 2 = 13; L4 = 15 − 5 = 10; L3 = min(10 − 4, 13 − 6) = 6; L2 = 10 − 7 = 3; L1 = min(3 − 3, 6 − 5) = 0.
• Event slack: 1 → 0, 2 → 0, 3 → 1, 4 → 0, 5 → 2, 6 → 0.
• Critical path 1–2–4–6 = 3 + 7 + 5 = 15 weeks. Activity 3–5 has TF = L5 − E3 − 6 = 13 − 5 − 6 = 2 weeks; FF = E5 − E3 − 6 = 11 − 5 − 6 = 0, so its float is entirely interfering.

Example 3 — PERT: expected duration and probability

Critical path activities P–Q–R–S (in series):

• Te = 6 + 6 + 8 + 4 = 24 days; Σσ² = 4 + 1 + 4 + 1 = 10 → σ_path = √10 = 3.16 days.
• Probability of finishing in 27 days: Z = (27 − 24)/3.16 = 0.95 → from the normal table, ≈ 83%.
• Probability of finishing in 21 days: Z = (21 − 24)/3.16 = −0.95 → ≈ 17%.
• Date for 90% confidence: Z = 1.28 → Ts = 24 + 1.28 × 3.16 = 28.0 → 28 days, i.e. a 4-day time contingency over Te.

Example 4 — Crashing to the optimum duration

Three activities in series; indirect cost ₹14,000 per day.

• Normal: duration 19 d, direct ₹1,90,000, indirect 19 × 14,000 = ₹2,66,000 → total ₹4,56,000.
• Step 1 — crash A by 2 d (slope ₹10,000 < ₹14,000): duration 17 d, direct ₹2,10,000, indirect ₹2,38,000 → ₹4,48,000.
• Step 2 — crash B by 2 d (slope ₹12,000 < ₹14,000): duration 15 d, direct ₹2,34,000, indirect ₹2,10,000 → ₹4,44,000.
• Step 3 — only C is left, slope ₹15,000 > ₹14,000, so crashing it raises cost: duration 14 d, direct ₹2,49,000, indirect ₹1,96,000 → ₹4,45,000.
• Optimum duration = 15 days at ₹4,44,000, a saving of ₹12,000; the crash (minimum) duration is 14 days but costs more.

CPM versus PERT

Where it is used

• Construction and infrastructure contractors: L&T, Tata Projects, Afcons, Shapoorji Pallonji, NCC, Dilip Buildcon — the contract programme, monthly updates and look-ahead schedules are CPM networks in Primavera P6.
• Planning / billing engineer and PMC roles: preparing baseline schedules, S-curves, resource histograms and cash-flow forecasts for NHAI, DMRC, metro, airport, port and smart-city packages; delay analysis for extension-of-time and claim submissions.
• Public works departments and PSUs: CPWD, PWDs, NBCC, RITES, IRCON and Railways require a network programme with the tender/agreement, and progress is reviewed against it.
• Project finance and lenders' engineers: physical progress against the approved schedule triggers milestone payments, drawdowns and liquidated-damages assessment.
• Industrial shutdowns and maintenance: refinery and power-plant turnarounds (IOCL, BPCL, NTPC) use CPM for hour-by-hour sequencing — the original DuPont application.
• BIM-based 4D planning: linking the schedule to the model for sequence visualisation, clash-free sequencing and site logistics on large metro and high-rise projects.

Interview questions

• Q: Why is the critical path the longest path and not the shortest? A: Every path must be completed, so the project cannot finish before the longest chain of dependent activities is finished; that chain therefore has no spare time.
• Q: What is the difference between total float and free float? A: Total float is the delay possible without extending the project but it consumes successors' float; free float is the delay possible without disturbing the earliest start of any succeeding activity.
• Q: Can an activity have zero total float but not lie on the critical path? A: No — zero total float is exactly the definition of a critical activity; if several such chains exist there are multiple critical paths.
• Q: Can total float be negative, and what does it mean? A: Yes, when an imposed finish date or a date constraint is earlier than the network logic permits; it signals that the schedule is unachievable as planned and must be re-sequenced or crashed.
• Q: Why does PERT use a beta distribution for activity time? A: Activity durations have a finite minimum and maximum, a single most likely value and are usually skewed; the beta distribution fits that shape, whereas a normal distribution would be unbounded and symmetric.
• Q: Why is the project duration in PERT treated as normal even though activities are beta-distributed? A: By the central limit theorem, the sum of a reasonably large number of independent activity durations tends to a normal distribution.
• Q: What is a dummy activity and why is it needed? A: A zero-duration, zero-resource arrow used in activity-on-arrow networks to show a dependency correctly or to give two parallel activities distinct start–finish event pairs; precedence (AoN) networks do not need it.
• Q: Which activity would you crash first and when do you stop? A: The critical activity with the lowest cost slope, one step at a time, re-checking the critical path after each step; stop when the cheapest remaining slope exceeds the indirect cost per day or all critical activities are at crash time.
• Q: Difference between resource levelling and resource smoothing? A: Levelling fixes a resource ceiling and may extend the duration; smoothing keeps the duration fixed and shifts non-critical activities within their float to even out demand.
• Q: What is merge-event bias in PERT? A: When several near-critical paths merge, the event cannot occur until all are complete, so using only the single longest path's mean and variance understates the duration and overstates the probability of on-time completion.
• Q: How is CPM used in a delay claim? A: The approved baseline is updated with actual progress and the delay event is inserted; the shift in the critical path shows whether the delay was excusable and compensable, and how much extension of time is justified.
• Q: When would you prefer Line of Balance over CPM? A: For repetitive work such as multi-storey housing, pipelines or road layers, where the concern is maintaining a steady rate of output per crew rather than one-off activity sequence.

Common mistakes

• Taking the critical path as the path with the most activities, or as the shortest path, instead of the longest in time.
• Computing total float as LF − ES (which gives the activity's available time window, not its float) instead of LS − ES or LF − EF.
• Adding the variances of all activities instead of only the critical activities when finding σ_path, or adding standard deviations instead of variances.
• Using tm instead of te in the PERT network, or dividing the range by 3 instead of 6 for σ.
• Crashing a non-critical activity, or forgetting to redo the forward and backward passes after each crash step, so a newly critical parallel path is missed.
• Treating float as belonging to one activity when it is shared along a path — consuming it once leaves none for the activities that follow.
• Mixing up Z-values and probabilities (a negative Z gives a probability below 50%), and ignoring that an 85–95% probability date, not Te, is what should be promised to a client.

Related topics

Estimation and costing · Contracts, tenders and specifications · Equipment management and productivity · Quality management in construction · AutoCAD, STAAD.Pro and BIM basics · Infrastructure financing and smart cities