Critical path analysis flashcards

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Precedence (dependence) table
A table listing, for each activity, the activities that must be completed immediately before it can start.
Activity on arc
A network in which each activity is a directed arc (labelled with its letter and duration) and each node is an event.
Event
An instant in the project when every activity entering that node is complete; the activities leaving it can then start.
Dummy activity
An arc of duration 0, drawn dashed, used only to show a dependence correctly or to keep activities uniquely identified by their start and end events.
Early event time
The earliest time at which an event can occur, found by the forward pass.
Late event time
The latest time at which an event can occur without delaying the whole project, found by the backward pass.
Critical activity
An activity for which any increase in duration increases the project duration; its total float is zero.
Critical path
A path from the start to the finish made up entirely of critical activities (and possibly dummies); its length is the minimum project duration.
Total float
The amount of time an activity can be delayed or extended without delaying the completion of the project.
Cascade (Gantt) chart
A chart drawn to a time scale: critical activities in one row, every other activity on its own row starting at its earliest start, with its float shown after it.
Forward pass rule
Early time of an event = the LARGEST of (early time at the start of each incoming arc + its duration). Start with 0 at the source. (No — learn it)
Backward pass rule
Late time of an event = the SMALLEST of (late time at the end of each outgoing arc − its duration). Start at the sink with its early time. (No — learn it)
Total float of activity from event i to event j, duration d
late time of j − d − early time of i (latest finish − duration − earliest start). (No — learn it)
Lower bound for the number of workers
Sum of all activity durations ÷ critical path length, rounded UP. (No — learn it)
Lower bound for the completion time with k workers
Sum of all durations ÷ k, rounded up (and never less than the critical path length). (No — derive it)
Conditions for a critical activity from i to j
early time of i = late time of i, early time of j = late time of j, and late time of j − early time of i = duration. (No — learn it)
Event box on an activity network
A two-part box at each event: early event time on the left, late event time on the right.
How a dummy is drawn
A dashed arrow with no letter and duration 0; its direction shows which event must wait for which.
Float on a cascade chart
A dotted or dashed box straight after the activity's bar, ending at the activity's latest finish time.
Scheduling diagram
One row per worker against a time scale, each activity a labelled bar; idle time left blank.
Why is a dummy needed when D depends on A and B but C depends on A only?
C and D cannot leave the same event. C leaves the end of A; D leaves the end of B, with a dummy from the end of A to the end of B.
Two activities have the same predecessors and the same successors. What is needed?
A uniqueness dummy: end one of them at an extra event and join that event to the other's end with a dummy.
Why does the forward pass take the maximum?
An event cannot happen until EVERY activity entering it has finished, so it waits for the latest of them.
Is an activity with equal early and late times at both ends always critical?
No. It must also fill the gap: late time at end − early time at start must equal its duration.
When must an activity be happening at time t?
When latest start ≤ t < earliest finish: it has started even if left as late as possible and not finished even if started as early as possible.
Why is the worker lower bound not always achieved?
Precedences can force workers to wait, so more workers (or more time) may be needed. A schedule is needed to show the bound works.
Scheduling rule when a worker becomes free
Give them the available activity (all predecessors finished) with the smallest latest finish time; if none is available, they wait.
P (6) and Q (4) both leave the source, event 0. P ends at event 1, Q at event 2, and a dummy joins 1 to 2. Early time of event 2?
max(0 + 4, 6 + 0) = 6.
Activity of duration 6 from an event with early time 9 to an event with late time 23. Total float?
23 − 6 − 9 = 8.
Durations total 50 hours; critical path 18 hours. Lower bound for workers?
50 ÷ 18 = 2.78, so 3 workers.
Durations total 33 hours and only 2 workers are available. Lower bound for the completion time?
33 ÷ 2 = 16.5, so at least 17 hours (if this is more than the critical path length).
A (5) then C (4) then D (2) is the critical path; B (3) runs from the start to the start of D. Float of B?
Latest finish of B = 11 − 2 = 9; float = 9 − 3 − 0 = 6.
An activity with total float 3 is delayed by 4 days. Effect on the project?
The project is delayed by 4 − 3 = 1 day, and the activity becomes critical.
Checking the backward pass
The late time at the source must come back to 0. Any other value means an arithmetic slip.
Pointing a dummy
Everything leaving the arrowhead's event waits for everything entering the tail's event. Check that this matches the table, both ways.
Rounding the worker bound
Always round UP: 2.25 workers means 3. Two workers cannot fit the work into the critical path time.
Giving a critical path
List the activities in order from start to finish and check they add up to the project duration.

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