CMA Final · Strategic Cost Management

Network Analysis - PERT, CPM: formula sheet

Full chapter guide

Key formulas

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. Where several activities merge into an event, take the largest value.
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 event) = E(last event). Where several activities leave an event, take the smallest value.
Earliest start and finish of an activity: ES = E(tail event); EF = ES + duration; In activity-on-node tables, ES = largest EF among all predecessors.
Latest finish and start of an activity: LF = L(head event); LS = LF − duration; In activity-on-node tables, LF = smallest LS among all successors.
Total float of an activity: Total float = LS − ES = LF − EF = L(j) − E(i) − duration(i, j); Total float is zero for every critical activity.
Critical path rule: Critical activity: E(i) = L(i), E(j) = L(j) and L(j) − E(i) = duration(i, j); The critical path is the chain of critical activities from start to end. Project duration = E of the last event.
Event slack: Slack = L − E; L = latest event time, E = earliest event time. Zero for events on the critical path.
Total float: TF = L(j) − E(i) − D = LST − EST = LFT − EFT; Activity i–j with duration D. i is the tail event and j the head event.
Free float: FF = E(j) − E(i) − D = TF − slack of head event j; Never more than total float.
Independent float: IF = E(j) − L(i) − D = FF − slack of tail event i; If the result is negative, take it as zero.
Interfering float: Interfering float = TF − FF = slack of head event; Part of total float that is shared with the following activities.
Activity times: EFT = EST + D; LST = LFT − D; EST of an activity = E of its tail event. LFT = L of its head event.
Critical activity: TF = 0 (with FF = IF = 0); All floats of a critical activity are zero.
Expected time of an activity: te = (to + 4tm + tp) ÷ 6; Use te as the duration when finding the critical path.
Variance of an activity: σ² = ((tp − to) ÷ 6)²; Only to and tp enter the variance. tm does not.
Standard deviation of an activity: σ = (tp − to) ÷ 6; Do not add standard deviations across activities. Add variances.
Expected project duration: TE = Σ te of critical-path activities; If two paths tie as critical, treat the question as asking for one path unless it says otherwise.
Project variance and standard deviation: σ²(project) = Σ σ² of critical-path activities; σ(project) = √σ²(project); Assumes activity times are independent.
Standard normal variable: Z = (T − TE) ÷ σ(project); T is the target or scheduled completion time. Probability of finishing by T = Φ(Z) from the normal table.
Time for a given probability: T = TE + Z × σ(project); Use for questions like: in how many days is completion 95% likely? Z is about 1.645 for 95% (one-sided).
Cost slope: Cost slope = (Crash cost − Normal cost) ÷ (Normal time − Crash time); Extra cost per day saved. Assumes cost rises in a straight line between normal and crash points.
Maximum crashing available: Maximum days crashable = Normal time − Crash time; You cannot crash an activity by more than this.
Total project cost: Total cost = Total direct cost + Total indirect cost; Direct cost = normal costs plus crashing cost added. Indirect cost = rate per day × project duration.
Crashing decision rule: Crash one more day if cost of saving that day < indirect cost saved per day; If equal, total cost does not change. Stop when the cost is higher.
Crashing cost of a step: Crashing cost = Days crashed × Cost slope; For parallel critical paths, add the slopes of the activities crashed together.
Total float: Total float = LST − EST = LFT − EFT; Float is the room available to shift an activity without delaying project completion. Critical activities have zero total float.
Free float: Free float = Earliest start of succeeding activity − EFT of this activity; Use free float to shift an activity without affecting even the next activity's earliest start.
Resource demand per period: Demand in a period = Σ resource units of all activities scheduled in that period; Build this row from a time-scaled bar chart and compare it with the cap or the average.
Average resource requirement: Average = Total resource-days ÷ Project duration; Smoothing aims to bring daily demand close to this figure.
Updated completion time: Revised completion = Elapsed time + Longest remaining path from the status date; Recompute with remaining durations only for incomplete activities.

Quick revision

Critical path is the longest path through the network and fixes the project duration.
Activities on the critical path have zero total float.
Forward pass gives earliest times; backward pass gives latest times.
Total float = latest finish − earliest start − duration.
Free float = earliest time of head event − earliest time of tail event − duration. This equals the earliest time of the head event − earliest finish of the activity.
Independent float = earliest time of head event − latest time of tail event − duration. A negative value is taken as zero. It assumes the tail event occurs at its latest time (the predecessor finishes as late as possible) and the head event at its earliest time (the successor starts as early as possible), so using it does not affect any other activity. Independent float is less than or equal to free float, which is less than or equal to total float (IF ≤ FF ≤ TF).
PERT expected time te = (to + 4tm + tp) ÷ 6.
PERT variance of an activity = ((tp − to) ÷ 6)².
Project variance is the sum of variances of critical-path activities only; standard deviation = √variance.
Z = (target time − expected project time) ÷ project standard deviation.
Cost slope = (crash cost − normal cost) ÷ (normal time − crash time).
Crash the critical activity with the lowest slope first, and recheck whether other paths become critical.

Common mistakes

Taking the smaller value at a merge event in the forward pass Fix: In the forward pass always take the maximum, because the event cannot occur until all incoming activities are done. In the backward pass always take the minimum.
Marking a path critical just because its events have E = L Fix: Also check that the activity duration equals L(j) − E(i). If not, the activity has float and is not critical.
Using the same event time for both ends, such as L(j) − L(i) − D for total float. Fix: Total float always uses L of the head and E of the tail: L(j) − E(i) − D. Write it down before substituting.
Taking the maximum in the backward pass instead of the minimum. Fix: Forward pass takes the maximum of the incoming values. Backward pass takes the minimum of the outgoing values.
Adding variances of all activities in the network instead of only critical-path activities. Fix: Mark the critical path first. Add variances only for those activities.
Adding standard deviations instead of variances. Fix: Add variances, then take the square root of the total once.
Crashing an activity that is not on the critical path Fix: Choose only from critical activities. Non-critical activities save no project time.
Crashing beyond the gap to the next path without recalculating Fix: Limit each step to the gap to the next longest path. After that, crash all critical paths together.
Shifting a critical activity to reduce a peak. Fix: Mark critical activities first and treat them as fixed. Only non-critical activities may move, and only within float.
Treating smoothing and levelling as the same thing. Fix: Smoothing holds the project duration. Levelling holds the resource cap and may extend duration. Say which one the question asks for.

Exam tips

Draw a small box at each event with E on the left and L on the right. It keeps the two passes visible and earns method marks even if one number slips.
Always finish with a one-line answer: the critical path in letters and the duration with its unit.
Check every merge and burst point twice. Almost all lost marks come from taking the wrong maximum or minimum there.
Questions often continue into float, crashing or PERT probability. A correct critical path is the base for all of them, so do not rush it.
In MCQs, you often only need the project duration. Do the forward pass first and read the answer before attempting the backward pass.
Show the forward and backward pass values on the network or in a table. Marks are given for event times even if a float is wrong.
Always write the formula with the event labels (for example L5 − E3 − D) before substituting. It prevents head-tail mix-ups.
Tabulate EST, EFT, LST, LFT, TF, FF and IF for each activity in one table. It makes checking easy and fits case-based questions.