Project Management, Monitoring and Control: formula sheet

Key formulas

Triple constraint

Project success = Scope/Quality delivered within Time and within Cost

Changing one constraint usually affects the other two. Use it to explain trade-offs.

Five-phase life cycle

Initiation → Planning → Execution → Monitoring and Control → Closure

Monitoring and control overlaps execution. Some books use fewer or more phases, so follow the model the question names.

Characteristics of a project

Defined objective + Start and end date + Limited budget + Uniqueness + Temporary team + Uncertainty

Use this list for 'features of a project' questions.

Hierarchy of a WBS

Project → Deliverables / Phases → Sub-deliverables → Work packages

Each lower level must add up to the level above it. No gaps and no overlaps.

Activity finish time

Finish date = Start date + Duration

Use the same time unit (days or weeks) throughout. Mind whether weekends count.

Start of a dependent activity

Earliest start = latest of the finish times of all its predecessors

An activity with two predecessors waits for the later one.

Project duration (simple sequence)

Total time = Σ durations of activities in sequence; for parallel paths, take the longest path

Activities done in parallel do not add up.

Gantt vs milestone chart

Gantt = activities against time (bars); Milestone = key events against dates (symbols)

Use this one-line contrast in comparison questions.

Earliest event time (forward pass)

E(j) = maximum of [E(i) + duration(i, j)] over all activities ending at event j

Start with E(1) = 0. Take the largest value where several arrows merge.

Latest event time (backward pass)

L(i) = minimum of [L(j) − duration(i, j)] over all activities starting at event i

Start with L(last) = E(last). Take the smallest value where several arrows leave.

Earliest start and finish

EST = E(i); EFT = EST + duration

For activity (i, j).

Latest finish and start

LFT = L(j); LST = LFT − duration

For activity (i, j).

Total float

Total float = L(j) − E(i) − duration = LST − EST = LFT − EFT

Zero total float means the activity is critical.

Free float

Free float = E(j) − E(i) − duration

Delay possible without affecting the earliest start of succeeding activities. Free float ≤ total float.

Independent float

Independent float = E(j) − L(i) − duration

If the result is negative, take it as zero. Independent float ≤ free float.

Critical path

Critical activity: E(i) = L(i), E(j) = L(j) and L(j) − E(i) = duration

Project duration equals the sum of durations on the critical path.

Expected time

te = (a + 4m + b) ÷ 6

a = optimistic, m = most likely, b = pessimistic.

Activity variance

σ² = ((b − a) ÷ 6)²

Depends only on a and b, not on m.

Activity standard deviation

σ = (b − a) ÷ 6

Square root of the variance.

Project expected time

Te = Σ te of critical path activities

Critical path is the longest path using expected times.

Project variance

σp² = Σ σ² of critical path activities

Add variances, never standard deviations. Use critical path only.

Z value

Z = (Ts − Te) ÷ σp

Ts = scheduled or target time. Look up Z in the normal table for P(project finishes by Ts).

Reading the Z

Z = 0 gives probability 0.5

Ts above Te gives Z positive and probability above 50%; Ts below Te gives probability below 50%.

Cost slope

Cost slope = (Crash cost − Normal cost) ÷ (Normal time − Crash time)

Assumes cost rises in a straight line between normal and crash points. It is the extra cost per unit of time saved.

Maximum crashing available

Maximum crash = Normal time − Crash time

You cannot shorten an activity beyond this limit.

Cost of crashing an activity

Extra cost = Cost slope × Time units crashed

Add this to the normal project cost to get the new direct cost.

Total project cost

Total cost = Direct cost + Indirect cost

Direct cost rises as you crash. Indirect cost usually falls with duration. Choose the duration with minimum total.

Net saving from a crash step

Net saving = Indirect cost saved − Crashing cost

Keep crashing only while this is positive.

Cost variance (CV)

CV = EV − AC

Negative means over budget; positive means under budget.

Schedule variance (SV)

SV = EV − PV

Negative means behind schedule; positive means ahead.

Cost performance index (CPI)

CPI = EV ÷ AC

Below 1 means you get less than ₹1 of work for each ₹1 spent.

Schedule performance index (SPI)

SPI = EV ÷ PV

Below 1 means progress is slower than planned.

Estimate at completion (EAC)

EAC = BAC ÷ CPI

BAC is budget at completion. Assumes the current cost efficiency continues.

Float (slack)

Total float = LST − EST (or LFT − EFT)

Float is the room to delay a non-critical activity without delaying the project. Critical activities have zero float.

Levelling versus smoothing

Levelling: may delay any activity, so end date may change. Smoothing: float only, end date fixed.

Both aim at even resource use. Levelling is used when resources are limited.

Quick revision

• A project is temporary, unique and has a defined start, end and objective.
• The three main constraints are time, cost and scope or quality.
• A WBS breaks the deliverable into smaller work packages that can be assigned and measured.
• Forward pass gives earliest times; backward pass gives latest times.
• Total float = latest start − earliest start (or latest finish − earliest finish).
• Critical activities have zero total float; the critical path is the longest path.
• PERT expected time = (O + 4M + P) ÷ 6.
• PERT variance of an activity = ((P − O) ÷ 6)².
• Project standard deviation = square root of the sum of critical-path variances.
• Z = (target time − expected project time) ÷ project standard deviation; read probability from the normal table.
• Crash cost per unit time = (crash cost − normal cost) ÷ (normal time − crash time).
• Crash only critical activities, cheapest first, and recheck for new critical paths.
• Control means comparing actual with plan, finding the variance and taking corrective action.

Common mistakes

• Treating any repeated activity as a project. Fix: Check for a fixed end date and a unique result. Daily production is routine operations.
• Placing monitoring and control only after execution. Fix: Say that monitoring and control runs through planning and execution, comparing actual with plan.
• Writing a WBS as a list of tasks in time order Fix: Organise a WBS by deliverables and levels, not by sequence. Sequence belongs to scheduling.
• Leaving out scope, such as testing or training, from the WBS Fix: Check each branch against the project objective. Include management, testing, approvals and handover.
• Taking the minimum in the forward pass or the maximum in the backward pass. Fix: Remember: forward pass takes the largest (the event waits for the slowest arrow); backward pass takes the smallest (the tightest deadline).
• Missing or wrongly placing dummy activities. Fix: For each activity, re-read its predecessors in the diagram after drawing. Add a dummy only when needed to show a dependence or to avoid two arrows with the same start and end events.
• Adding variances of all activities in the network. Fix: Mark the critical path first and sum variances for those activities alone.
• Adding standard deviations instead of variances. Fix: Add variances, then take the square root of the total.
• Crashing an activity that is not on the critical path. Fix: Choose the lowest slope only among critical activities. Non-critical crashing gives no time saving.
• Dividing by the wrong time difference in the slope. Fix: Always use normal time minus crash time in the denominator, and crash cost minus normal cost in the numerator.

Exam tips

• Definition plus features plus a distinction is the usual pattern. Prepare all three.
• In MCQs, check whether the situation is one-time and unique. If so, it is a project.
• Write phase answers as bold headings with an output line. It reads quickly and earns step marks.
• Always mention time, cost and quality. Examiners look for these three words.
• Revise this topic with planning, networks (CPM and PERT) and monitoring, since numerical questions build on it.
• For compare questions, such as Gantt chart vs milestone chart, write points side by side: what it shows, detail level, dependencies, use, limitation.
• In a WBS question, draw the hierarchy as an indented list. Label the levels and name the work packages.
• In numerical questions, show the table of start and finish times. Step marks come from it even if the final figure is wrong.