Q1.
Figure 1 below shows an activity diagram for a construction project. The time needed for each activity is given in days.
Figure 1
 
(a)     Find the earliest start time and the latest finish time for each activity and insert their values on Figure 1.
(4)
(b)     Find the critical paths and state the minimum time for completion of the project.
Critical paths are ____________________________________________________
__________________________________________________________________
Minimum completion time is ______________ days.
(3)
(c)     On Figure 2, draw a cascade diagram (Gantt chart) for the project, assuming that each activity starts as early as possible.
Figure 2
         
(5)
(d)     Activity J takes longer than expected so that its duration is x days, where x ≥ 3. Given that the minimum time for completion of the project is unchanged, find a further inequality relating to the maximum value of x.
(2)
(Total 14 marks)
Q2.
The diagram shows the activity network and the duration, in days, of each activity for a particular project. Some of the earliest start times and latest finish times are shown on the diagram.
 
(a)    Find the values of the constants x, y and z.
(3)
(b)    Find the critical paths.
(2)
(c)    Find the activity with the largest float and state the value of this float.
(2)
(d)    The number of workers required for each activity is shown in the table.
 
 	Activity	A	B	C	D	E	F	G	H	I	J	
 	Number of workers required	4	2	3	4	2	4	3	3	5	6	
Given that each activity starts as early as possible and assuming that there is no limit to the number of workers available, draw a resource histogram for the project on the figure below, indicating clearly which activities are taking place at any given time.

(5)
(e)     It is later discovered that there are only 9 workers available at any time. Use resource levelling to find the new earliest start time for activity J so that the project can be completed with the minimum extra time. State the minimum extra time required.
(2)
(Total 14 marks)

Mark schemes
Q1.
(a)
 
Forward pass
condone one slip (follow through)
M1
all correct
A1
Backward pass
condone one slip (follow through)
M1
all correct
A1
4
(b)     Critical paths
B E G K N
first path correct
M1
D F I L N
second path and no others
A1
Minimum completion time is 21 days
B1
3
(c)     Cascade diagram
may be in blocks or bars (see examples)
One of ‘their’ CPs correct
ft their CP
M1
B , D , E , F , G , I , K , L , N
these activities correct
A1
A , C , H , J , M
3 of these with correct start and duration
M1
3 correct with correct slack indicated
A1
all 5 correct with correct slack
A1
 
 	 	 	Slack	
 	A	0 − 7	7 − 8	
 	C	0 − 5	5 − 6	
 	H	10 − 12	12 − 14	
 	J	6 − 8	8 − 16	
 	M	15 − 18	18 − 19	
5
(d)     (Max value of x is) 10
considering Jlatest − Jearliest
M1
⇒ x ≤ 10
(condone x < 11 for SC2) 
NMS x ≤ 10 award M1 A1
A1 cao
2
[14]
Q2.
(a)     x = 4
B1
y = 12
B1
z = 13
B1
3
(b)     B  D  H  J    and  C  E  I  J
first correct path
M1
2nd correct and no others
A1
2
(c)     G
B1
Float = 3
B1
2
(d)     One of their CPs correct height
and correct durations
M1
B, D, H, J and C, E, I correct
and correct durations
A1
 
 	Number
of workers		
          Number of days
 
A starting at 0 and ending at 3
one correct with correct height
M1
F starting at 6 and ending at 11
two correct with correct height
A1
G starting at 13 and ending at 14
all correct with correct height
withhold first A1 earned if it is not clear which activities
take place at any given time
withhold another A1 if “holes” appear in histogram
A1
5
(e)     New earliest    J        22   days
assuming activities continuous
B1
Minimum extra time    5 days
assuming activities continuous
B1
2
[14]