D1 June 2013 Q7

EdexcelOld spec17 marksCritical Path Analysis

Figure 5: activity network with events 1 to 12 and activities A to S — Figure 5

[The sum of the duration of all activities is 172 days]

A project is modelled by the activity network shown in Figure 5. The activities are represented by the arcs. The number in brackets on each arc gives the time, in days, to complete the activity. Each activity requires one worker. The project is to be completed in the shortest possible time.

(a) Complete Diagram 1 in the answer book to show the early event times and late event times. (4)

(b) Calculate the total float for activity M. You must make the numbers you use in your calculation clear. (2)

(i) A 2 day delay on the early start of activity P.

(ii) A 2 day delay on the early start of activity Q. (2)

(d) Calculate a lower bound for the number of workers needed to complete the project in the shortest possible time. (1)

Diagram 2 in the answer book shows a partly completed cascade chart for this project.

(e) Complete the cascade chart. (4)

(f) Use your cascade chart to determine a second lower bound on the number of workers needed to complete the project in the shortest possible time. You must make specific reference to times and activities. (2)

(g) State which of the two lower bounds found in (d) and (f) is better. Give a reason for your answer. (2)

Activity network with early and late event times: 1 0/0, 2 5/7, 3 7/7, 4 7/7, 5 15/18, 6 18/18, 7 26/31, 8 27/31, 9 28/28, 10 36/42, 11 42/42, 12 53/53 — Mark scheme (a)

Scheme	Marks
	M1 A1 M1 A1
	(4)

Scheme	Marks
Float on M = 42 – 26 – 8 = 8	M1 A1
	(2)

Mark scheme (c)

Scheme	Marks
(c)(i) 2 day delay on P – no effect on the project completion date (float on P is 4)	B1
(c)(ii) 2 day delay on Q – project finishes 2 days late (Q is a critical activity)	B1
	(2)

Notes

c1B1 CAO

c2B1 CAO

Scheme	Marks
(172/53 = 3.245, so) a minimum of 4 workers needed	B1
	(1)

Cascade chart: critical activities C, F, L, Q, S on the top row; other activities with floats shaded — Mark scheme (e)

Scheme	Marks
	M1 A1 (any 6 more) M1 A1 (all 11)
	(4)

Mark scheme (f)

Scheme	Marks
E.g. Activities H, I, J, K and L together with \(22 \lt \text{time} \lt 26\) stated. So 5 workers needed	M1 A1
	(2)

Notes

Examples for part (f):

Example 1: Activities H, I, J, K and L with \(22 \lt \text{time} \lt 26\) so 5 workers needed.

Example 2: At \(10 \lt \text{time} \lt 14\), F, D, E and H must be happening. Activity G must be happening \(7 \lt \text{time} \lt 18\) but its duration is 5 so it must also occur at some point in the interval \(10 \lt \text{time} \lt 14\) so 5 workers needed.

f1M1 Example 1: A statement with the correct number of workers (5) and the correct activities (H, I, J, K and L) with some mention of time, or
Example 2: A statement with the correct number of workers (5), the correct activities (F,D,E and H) with some mention of time and an indication that G must be happening with the other four activities at some point - give bod but e.g. ‘at time 11 F, D, E, G and H must be happening’ is M0). Scheduling the activities only scores M0.

f1A1 A correct, complete full statement with details of both time and activities. Candidates only need to give a time within the intervals stated.
Please note strict inequalities for the time. Allow e.g. on ‘day 23’ as equivalent to \(22 \lt \text{time} \lt 23\).

Mark scheme (g)

Scheme	Marks
The cascade gives a higher lower bound, so (f) is better.	M1 A1
	(2)
	(17 marks)

Notes

g1M1 Must have attempted both parts (d) and (f). Their higher lower bound chosen + attempt at a reason.
Allow for the M mark a reason which argues that e.g. the cascade chart gives a better lower bound (e.g. it takes into account exactly when activities must be taking place) or e.g. the calculation gives a better lower bound (e.g. as it takes into account the sum of all the activities) but without specifically answering the question of which of the two bounds is better. Give bod on an attempt at a reason.

g1A1 CAO plus a correct reason given. Acceptable reasons e.g. the cascade gives a larger value or the bound for the cascade shows that the project cannot be done with fewer workers, etc.