Pearson Edexcel International Advanced Level Mathematics: Critical path analysis and scheduling (YMA01) – Revision Notes

These notes condense critical path analysis for Pearson Edexcel International Advanced Level Mathematics (YMA01), Specification Issue 3, April 2019: unit D1, section 4, topics 4.1 to 4.4. The specification states that D1 is an optional unit for IAS Mathematics and Further Mathematics and for IAL Mathematics and Further Mathematics. For full explanations, start with the study guide, then test yourself with the practice questions. Related links: course hub, checklist and all free 10-minute diagnostics.

Key terms

Term	Meaning
Activity	A job with a duration, drawn as an arc
Event	A node where activities finish and others start
Precedence table	Lists each activity’s immediate predecessors only
Dummy	A dashed arc of duration 0
eᵢ	Earliest time event i can occur
lᵢ	Latest time event i can occur without delaying the project
Critical activity	An activity with total float 0
Critical path	A path of critical activities from start to end
Total float	F(i, j) = lⱼ − eᵢ − duration (i, j)

Formulae at a glance

Quantity	Rule
Earliest start of (i, j)	eᵢ
Earliest finish	eᵢ + duration
Latest finish	lⱼ
Latest start	lⱼ − duration
Forward pass at an event	largest of (eᵢ + duration) over arriving arcs
Backward pass at an event	smallest of (lⱼ − duration) over leaving arcs
Lower bound on workers	total of durations ÷ project time, rounded up

Method in steps

Drawing a network (4.1)

1. Start all activities with no predecessor at event 1.
2. Give each other activity a start event reached by exactly its predecessors.
3. Add a dummy wherever one event would add an unwanted dependence, or where two activities would share both end events.
4. Finish all activities with no successor at one end event.
5. Number events so every arc points to a larger number.

Reading a table from a network (4.2)

1. Go to the start event of the activity.
2. Look back along the arcs arriving there.
3. Follow dummies back until you reach real activities.
4. Write only the activities that end immediately before.

Critical path (4.3)

1. Forward pass from e = 0; take maximums.
2. Backward pass from the end event; take minimums.
3. Mark an activity critical if eᵢ = lᵢ, eⱼ = lⱼ and eⱼ − eᵢ equals its duration.
4. Write the minimum project time and the critical path.

Float, Gantt, scheduling (4.4)

1. Total float = lⱼ − eᵢ − duration for every activity.
2. Draw one row per activity from its earliest start, float shown after.
3. Divide total duration by project time and round up for the minimum workers.
4. Build a schedule using float and check every precedence.

Worked reminder

Table: G (4 days, none), H (3 days, none), J (5 days, G and H), K (6 days, G), L (2 days, J and K).

G ends at its own event, where K starts. J needs H as well, so a dummy joins the end of G to J’s start event.

Arc	From	To	Duration
G	1	2	4
H	1	3	3
dummy	2	3	0
K	2	4	6
J	3	5	5
dummy	4	5	0
L	5	6	2

Forward:  e1 = 0, e2 = 4, e3 = max(3, 4) = 4, e4 = 4 + 6 = 10,
          e5 = max(4 + 5, 10) = 10, e6 = 12
Backward: l6 = 12, l5 = 10, l4 = 10, l3 = 10 - 5 = 5,
          l2 = min(5, 10 - 6) = 4, l1 = min(4 - 4, 5 - 3) = 0

Minimum project time 12. Critical path: G, K, L. Floats: H = 5 − 0 − 3 = 2 and J = 10 − 4 − 5 = 1. Total of durations is 20, and 20/12 = 1.67, so at least 2 workers.

Gantt chart and schedule for the same project

One column is one day: # is critical, = is non-critical, - is float.

day  0    5    10
G    ####........
H    ===--.......
K    ....######..
J    ....=====-..
L    ..........##

Worker 1 takes the critical activities: G (0–4), K (4–10), L (10–12). Worker 2 takes H (0–3) and J (4–9). J starts at day 4, after G and H have both finished, and L starts at day 10, after J and K. So 2 workers meet the 12-day project time, which matches the lower bound.

Reading a table from a network

Take arcs X (1→2), Y (1→3), a dummy (2→3), Z (3→4) and W (2→4). Z starts at event 3, which is reached by Y and, through the dummy, by X, so Z has predecessors X and Y. W starts at event 2, reached by X only, so W has the single predecessor X. The dummy is why Z has two predecessors while W has one.

Presenting your working

The specification asks you to show clearly how an algorithm has been applied. A tidy layout makes your reasoning easy to follow and easy to check:

• Write the forward pass as a list, one line per event, showing the values being compared.
• Write the backward pass the same way, working from the end event.
• Show the event times in a table or beside each node, in the form eᵢ and lᵢ.
• Name the critical path in full and total its durations to confirm the project time.
• Draw dummies dashed, and label every arc with its letter and duration.

Must-know distinctions

Idea	Distinction
Event time and activity time	Event times belong to nodes. Start and finish times belong to arcs and come from eᵢ and lⱼ
Forward and backward pass	Forward uses the largest value, backward the smallest
Logic dummy and uniqueness dummy	One carries a dependence between events, the other separates two arcs with the same end events
Lower bound and schedule	The bound is only a minimum. A valid schedule must be shown
Critical activity and critical event	An event can have e = l while an arc into it is not critical

A project can have more than one critical path. Delaying a critical activity delays the project by the same amount. Delaying a non-critical activity by more than its float delays the project by the excess.

Float is shared along a chain. In the worked reminder, H has 2 days of float and J has 1, but they sit on the same route into event 5, so delaying H by 2 days leaves J with no float at all. Check every activity on a route before you promise that a delay is harmless.

Quick self-test

1. Write the formula for total float F(i, j).
2. An activity (3, 5) lasts 6 days. e₃ = 8 and l₅ = 17. Find its total float.
3. An activity (4, 6) lasts 4 days. e₄ = 9, l₄ = 12, e₆ = 14 and l₆ = 19. Find its earliest and latest start and finish.
4. In the worked reminder, what is the total float of H?
5. In the worked reminder, which activities are on the critical path?
6. Total duration is 46 days and the project takes 20 days. What is the lower bound on workers?
7. State two reasons for using a dummy.
8. A network has two activities with no successors. How are they drawn?
9. A critical activity is delayed by 3 days. What happens to the project?
10. A non-critical activity with float 5 is delayed by 7 days. What happens to the project?
11. In the worked reminder, what are L’s immediate predecessors?
12. Can a network have two critical paths?

Answers

1. F(i, j) = lⱼ − eᵢ − duration (i, j).
2. 17 − 8 − 6 = 3 days.
3. Earliest start 9, earliest finish 13, latest finish 19, latest start 15. The total float is 19 − 9 − 4 = 6.
4. 2 days.
5. G, K and L.
6. 46/20 = 2.3, so at least 3 workers.
7. To show a dependence correctly (logic), and to give two activities different pairs of end events (uniqueness).
8. Both end at the single final event.
9. The project finishes 3 days late.
10. The project finishes 2 days late (7 − 5).
11. J and K.
12. Yes, and every activity on either path has float 0.

Where marks are usually lost

• Writing every earlier activity in a precedence table instead of immediate predecessors only.
• Missing a dummy, so an activity depends on something it should not.
• Giving a dummy a duration, or counting it in a path total.
• Using the wrong operation in a pass: largest forward, smallest backward.
• Not showing each event’s working, so one slip hides the method used.
• Declaring a path critical without totalling its durations against the project time.
• Reading float off the wrong events: it is lⱼ − eᵢ − duration, with i the start event and j the end event.
• Stopping at the lower bound for workers and giving no schedule.
• Starting a Gantt bar before a predecessor has finished.

Official syllabus

Pearson Edexcel International Advanced Subsidiary/Advanced Level in Mathematics, Further Mathematics and Pure Mathematics, Specification Issue 3, April 2019 (first teaching September 2018), published by Pearson Education Limited. Unit D1, section 4, topics 4.1 to 4.4, and the D1 glossary definition of total float.