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Decision Maths
Critical Path Analysis
Problem
Every morning you have three pieces of bread (A,B and C) under the grill.
The grill will take two pieces of bread at a time and will take 30 seconds to toast one side of the bread.
How long will it take to toast all three pieces of bread?
Solution
You could follow the schedule below.
Or you could follow a more efficient plan.
Time
Toast one side of A & B 30s
Toast other side of A & B 30s
Toast one side of C 30s
Toast the other side of C 30s
Total Time 120s
Time
Toast one side of A and B 30s
Toast one side of C and other of A 30s
Toast other side of B and C 30s
Total Time 90s
Critical Path Analysis
This was a trivial example of how time can be saved by careful planning.
We can apply similar thinking to larger problems involving construction and maintenance.
Critical Path Analysis enables us to plan and monitor complex projects, so that they are approached and carried out as efficiently as possible.
Critical Path Analysis
Imagine you have a project to do which involves doing lots of different activities, some of which cannot be started until others have been completed.
For example the stages involved in recording and promoting a compact disc are shown in the table below
Critical Path Analysis
Activity Duration Preceding Activities
A Tape the performance 10
B Design the cover 9
C Book adverts in press 3
D Tape to CD 2 A
E Produce Cover 2 B
F Packing 1 D,E
G Promotion Copies to radio 1 D,E
H Dispatch to shops 3 F
I Played on radio 2 G
J Adverts in press 1 C,H,I
Critical Path Analysis
Given the conditions previously shown, what is the shortest possible time the project can be completed in.
We will assume that tasks can be carried out simultaneously whenever the conditions allow it.
The first stage is construct a Precedence Network.
Precedence Network
Draw a “start node” From the start node
add the activities that can be done immediately.
1
DurationPrec
Act DurationPrec
Act
A 10 F 1 D,E
B 9 G 1 D,E
C 3 H 3 F
D 2 A I 2 G
E 2 B J 1 C,H,I
A(10)
B(9)
C(3)
Precedence Network
Activity D needs A to have been done immediately before it.
Add a node to A. Have D coming from
that node.
2
DurationPrec
Act DurationPrec
Act
A 10 F 1 D,E
B 9 G 1 D,E
C 3 H 3 F
D 2 A I 2 G
E 2 B J 1 C,H,I
1
A(10)
B(9)
C(3)
D(2)
Precedence Network
We can do a similar thing with activity E
1
2
3
DurationPrec
Act DurationPrec
Act
A 10 F 1 D,E
B 9 G 1 D,E
C 3 H 3 F
D 2 A I 2 G
E 2 B J 1 C,H,I
A(10)
B(9)
C(3)
D(2)
E(2)
Precedence Network
Activities F and G have both of D and E as their immediately preceding activities.
Bring D and E into a single node.
F and G emerge from that node.
4
DurationPrec
Act DurationPrec
Act
A 10 F 1 D,E
B 9 G 1 D,E
C 3 H 3 F
D 2 A I 2 G
E 2 B J 1 C,H,I
1
2
3
A(10)
B(9)
C(3)
D(2)
E(2)
F(1)
G(1)
Precedence Network
Add nodes to the end of activities F and G.
Activity H must follow F. Activity I must follow G.
1
2
3
4
5
6
DurationPrec
Act DurationPrec
Act
A 10 F 1 D,E
B 9 G 1 D,E
C 3 H 3 F
D 2 A I 2 G
E 2 B J 1 C,H,I
A(10)
B(9)
C(3)
D(2)
E(2)
F(1)
G(1)I(2)
H(3)
Precedence Network
J must follow C,H and I, so bring all of those in to one node.
J can now be added and the network is complete.
1
2
3
4
5
6
7
DurationPrec
Act DurationPrec
Act
A 10 F 1 D,E
B 9 G 1 D,E
C 3 H 3 F
D 2 A I 2 G
E 2 B J 1 C,H,I
A(10)
B(9)
C(3)
D(2)
E(2)
F(1)
G(1)I(2)
H(3)
8J(1)
Critical Path Analysis More Complicated Examples The example we
have just looked through was a reasonably easy one.
Consider the example to the right, it will give us more to think about.
Activity Duration IPA
A 1
B 2
C 4 A
D 4 A,B
E 3 D
Complicated Example 1
Draw a “start node” From the start node add
the activities that can be done immediately.
Connect nodes to the ends of these activities.
Activity Duration IPA
A 1
B 2
C 4 A
D 4 A,B
E 3 D
1
A(1)
B(2)
2
3
Complicated Example 1
Activity C follows A. D must follow both A
and B. Where should D go?
Activity Duration IPA
A 1
B 2
C 4 A
D 4 A,B
E 3 D
1
A(1)
B(2)
2
3
C(4)
Complicated Example 1
You must add in a dummy activity. This will have a duration of 0.
Now to progress from node (3) you will have to have completed A and B.
Why does the dummy go from node (2) to (3)?
Activity C needs to follow A but has no need to follow B. If the dummy went the other way then it would be impossible to place activity C.
Activity Duration IPA
A 1
B 2
C 4 A
D 4 A,B
E 3 D
1
A(1)
B(2)
2
3
C(4)
(0)
Complicated Example 1
Activity D can now be attached to node (3).
Now place a node at the end of D.
Activity E can be assigned.
Activity Duration IPA
A 1
B 2
C 4 A
D 4 A,B
E 3 D
1
A(1)
B(2)
2
3
(0)
D(4)4
E(3)
C(4)
5
Complicated Example 2
This starts in a similar way and we can assign A and B to node 1.
Both C and D must follow A and B, Where does the dummy go?
The dummy is set from (3) to (2) because E comes from B and does not follow A.
Activity Duration IPA
A 1
B 2
C 4 A,B
D 4 A,B
E 3 B
1
A(1)
B(2)
2
3
(0)
Complicated Example 2
C and D can now be added to node (2).
E can be added to node (3).
They all join the final node (4).
Activity Duration IPA
A 1
B 2
C 4 A,B
D 4 A,B
E 3 B
1
A(1)
B(2)
2
3
(0)
D(4)
E(3)4
C(4)
Complicated Example 3
Activities A and B are simple.
C and E are also simple as they follow A and B respectively.
Two dummies are required to a new node (4).
D is attached and all meet at (5)
Activity Duration IPA
A 1
B 2
C 4 A
D 4 A,B
E 3 B
1
A(1)
B(2)
2
3
4 5
(0)
(0)
C(4)
D(4)
E(3)
Precedence Network
1 142
3
4 6
10
7
5
8
9 11
12
13
A(10)
B(25)
C(10)
D(16)
F(16)
H(18)
G(12)
E(9)
I(32)
J(32)
K(16)
L(21) S(14)
O(11)R(16)
U(50)
V(21)Q(48)
N(34)
M(31)
T(24)P(25)
Below is a precedence Network for a real life construction project. The times on the arc represent days. What is the quickest time that the project can be completed in?
170 days.
Precedence Network
1 142
3
4 6
10
7
5
8
9 11
12
13
A(10)
B(25)
C(10)
D(16)
F(16)
H(18)
G(12)
E(9)
I(32)
J(32)
K(16)
L(21) S(14)
O(11)R(16)
U(50)
V(21)Q(48)
N(34)
M(31)
T(24)P(25)
Due to a resource problem activity H is delayed by 10 days. What effect will this have on the whole project?
The entire project is delayed by 3 days.
Precedence Network
1 142
3
4 6
10
7
5
8
9 11
12
13
A(10)
B(25)
C(10)
D(16)
F(16)
H(18)
G(12)
E(9)
I(32)
J(32)
K(16)
L(21) S(14)
O(11)R(16)
U(50)
V(21)Q(48)
N(34)
M(31)
T(24)P(25)
The supervisor in charge of activity I wants to use overtime to reduce the completion time. How would you respond to this request?
No – there is plenty of spare time for the activity to be completed.
Precedence Network
1 142
3
4 6
10
7
5
8
9 11
12
13
A(10)
B(25)
C(10)
D(16)
F(16)
H(18)
G(12)
E(9)
I(32)
J(32)
K(16)
L(21) S(14)
O(11)R(16)
U(50)
V(21)Q(48)
N(34)
M(31)
T(24)P(25)
Money is available to reduce the time of activity B or Q. Where would you advise that the money is spent and why?
Activity B – It can reduce the overall completion time of the project.
