WorkSHEET 8.1
Networks
Name: ___________________________
/50
3
1
(a)
Depots
(b)
Travel times
(c)
A directed network
The network shown in the figure above shows
the time, in minutes, taken to travel between
depots.
(a) What do the nodes in the network
represent?
(b) What do the arcs in the network
represent?
(c) What type of network is pictured?
2
What is the shortest time taken to travel from
A to F?
By inspection, the shortest path from A to F is
55 min.
4
6
3
The shortest path is 87.
What is the shortest path from P to Q?
7
4
Give the minimal spanning tree for this
network.
Maths Quest Maths A Year 12 for Queensland
Page 1 of 9
5
What is the length of the minimal spanning
tree?
6
A
A
B
C
D
E
F
B
13
C
25
20
D
30
E
The length of the spanning tree is 86.
4
6
F
10
30
10
15
The table above shows the cost, in thousands of
dollars, of networking a number of offices at a The minimum cost is $68 000.
large complex.
Draw a network to represent this information.
7
Referring to the information in question 6, by
identifying the minimal spanning tree calculate
the minimum cost of connecting these offices.
5
The minimum cost is $68 000.
Begin at A and write in each node the shortest
distance from A.
8
Calculate the length of the shortest path from
A to B.
The minimum distance from A to B is 60.
Maths Quest Maths A Year 12 for Queensland
Page 2 of 9
6
9
P
P
Q
R
S
T
U
Q
30
R
10
15
S
T
5
20
8
5
10
5
U
10
30
Six towns are connected by roads and the
distance between them, in kilometres, is shown
in the table.
Draw a network for this system of roads.
10
Use the information in question 9 to find the
shortest distance from P to U.
Recognise that the shortest path to Q is
P – R – T – Q.
The shortest distance from P to U is 38 km.
Maths Quest Maths A Year 12 for Queensland
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4
WorkSHEET 9.1
Critical path analysis and queuing
Name: ______________________
/50
11
Consider the network shown in the figure
below. It represents the data of an activity
table, with activities A, B, … J and associated
times (in hours).
(a)
5
Activity
Immediate
A
predecesso r

B

C
A
D
B
E
B
F
E
(a) Determine the immediate predecessors for
each activity.
G
C, D
H
C, D
(b)Perform a forward scan.
J
G
(b)
Enter latest completion time for each
pathway.
Activity A : 5 hours
Activity B: 4 hours
Activity C, D: either 5 + 6 (via A, C) or 4 + 8
(via B, D), so reject 11, use 12 hours.
Activity E: 4 + 5, (via B, E) = 9 hours
Activity G: 12 + 3 (via B, D, G) = 15 hours
Activity F, H, J: either 9 + 6 (via E, F), or
12 + 7 (via C, H) or 15 + 5 (via G, J), so
reject 15, 19 and use 20 hours.
Maths Quest Maths A Year 12 for Queensland
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12
Consider the network in question 1.
(a) Determine the earliest completion time for
the entire project.
(a)
The earliest completion time (from the
results in question 1) is 20 hours.
5
(b)
(b)Perform a backward scan.
Enter 20 in the right-hand side of the
finish node.
Path J: 20 – 5 = 15
Path H: either 20 – 7 (via H) or 15 – 3
(via G).
Reject 13, use 12.
Path F: 20 – 6 = 14
Path E: either 14 – 5 (via E) or 12 – 8
(via D)
Reject 9, use 4.
Path C: 12 – 6 = 6
Complete by entering a 0 in the
right-hand side of the start node.
13
In the network in questions 1 and 2:
(a) determine the critical path
(a)
Examine the diagram in question 2. Find
nodes where the number in the left-hand
side of the node equals the number in the
right-hand side of the node. This leads to
the pathway:
B – D – G – J.
(b)
Subtract activity time from next righthand side of the node and prior left-hand
side of the node numbers for non-critical
activities.
Float(H) = 20 – 12 – 7 = 1 hour
Float(F) = 20 – 9 – 6 = 5 hours
Float(E) = 14 – 4 – 5 = 5 hours
Float(C) = 12 – 5 – 6 = 1 hour
Float(A) = 6 – 0 – 5 = 1 hour
(b)determine the float times for any
non-critical activities.
Maths Quest Maths A Year 12 for Queensland
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4
14
Construct a network diagram from the
following activity table representing the
production of fruit at a cannery. Times are in
hours.
Since A, D, E have no immediate predecessors,
they can all start at once with three branches.
4
B follows A, and C follows B.
F follows D.
Activity
letter
A
B
C
D
E
F
G
H
J
K
15
Activity
Receive
from farm
Sort the fruit
Slice the
fruit
Produce
cans
Produce
syrup
Produce
labels
Store in
cans
Package in
boxes
Ship the
fruit
Remove
waste
Immediate
predecessors
–
Activity
time
1
A
B
2
1
G, K can now begin separately.
–
3
J follows H, K.
–
3.5
D
2.5
C,E,F
1.5
G
1
H,K
2
C,E,F
2
Perform a forward scan on the network of
question 4 to determine earliest completion
time for the whole project.
Three branches come together when C, E, F are
completed.
H follows K.
Enter times for activities A(1), B(2) and D(3).
Maximum time where 3 branches meet is
3 + 2.5 (reject 1 + 2 + 1 and 3.5).
Enter time for activity G(5.5 + 1.5).
Maximum time where 2 branches meet is
5.5 + 1.5 + 1 (reject 5.5 + 2).
Enter time for activity J = 8 + 2.
Earliest completion time = 10 hours.
Maths Quest Maths A Year 12 for Queensland
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5
16
Perform a backward scan on the network of
questions 4 and 5 to determine:
(a) the critical path
(a)
(b)float times for non-critical activities.
Enter 10 in the right-hand side of the last
node.
Enter 8 for next node before activity J.
Enter 7 for node before activity H.
Enter either 8 – 2 (path J, K) or
8 – 1 – 1.5 (path H, G) for node before
activities G, K.
Reject path J, K.
Enter 5.5 – 2.5 = 3 for node before
activity F.
Enter 5.5 – 1 = 4.5 for node before
activity C.
Enter 4.5 – 2 = 2.5 for node before
activity B.
8
Critical path is where numbers in both
sides of the nodes are the same:
D – F – G – H – J.
(b)
17
Float times for non-critical activities:
Float(K) = 8 – 5.5 – 2 = 0.5 hours
Float(C) = 5.5 – 3 – 1 = 1.5 hours
Float(B) = 4.5 – 1 – 2 = 1.5 hours
Float(A) = 2.5 – 0 – 1 = 1.5 hours
Float(E) = 5.5 – 0 – 3.5 = 2 hours
Complete a forward scan for the critical path
network shown below.
3
Earliest completion time = 30.
Determine the earliest completion time.
Maths Quest Maths A Year 12 for Queensland
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18
For the network in question 7, complete a
backward scan, and determine:
(a) the critical path
6
(b)the float times for non-critical activities.
(a) Critical path = A – B – L – M – N
(b)Float(D) = 30 – 19 – 10 = 1
Float(C) = 20 – 15 – 4 = 1
Float(H) = 30 – 11 – 10 = 9
Float(F) = 17 – 5 – 6 = 6
Float(G) = 21 – 11 – 4 = 6
Float(E) = 11 – 0 – 5 = 6
Float(K) = 21 – 5 – 8 = 8
Float(J) = 15 – 5 – 4 = 6
19
The project plan for a new computer software
program is shown in the figure below.
By forward scanning…
4
The earliest completion time = 147 days.
Time is measured in days.
Determine the earliest completion time.
Maths Quest Maths A Year 12 for Queensland
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20
For the network in question 9, determine:
By backward scanning…
6
(a) the critical path
(b) the float times for non-critical activities.
Maths Quest Maths A Year 12 for Queensland
(a)
the critical path is A – B – D – G – L – P
– R.
(b)
Float(C) = 25 – 6 – 11 = 8 days
Float(E) = 42 – 17 – 17 = 8 days
Float(F) = 90 – 41 – 23 = 26 days
Float(H) = 61 – 34 – 19 = 8 days
Float(J) = 93 – 34 – 21 = 38 days
Float(K) = 115 – 64 – 25 = 26 days
Float(M) = 117 – 55 – 24 = 38 days
Float(N) = 115 – 88 – 26 = 1 day
Float(Q) = 147 – 114 – 32 = 1 day
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