Q1) A precedes C.
B precedes D & E.
C precedes F & G.
D precedes G.
E precedes H,I,J, & L.
F precedes J & L
H precedes K.
I is preceded by G.
J precedes K.
Draw the AoA and AoN network diagrams for the given precedence relations.
Solution:
AoA Network Diagram:
AoN Network Diagram:
Q2) Let us consider a restaurant arrangement and construction. The activities are as
follows:
a) Draw the AOA representation of this project. Indicate the duration of each activity
on the corresponding arc. Specify the node numbers such that an arc always goes from a
smaller node number to a larger node number. Keep the number of dummy activities at
minimum.
b) Draw the AON representation of this project. Indicate the duration of each activity
on the corresponding node.
Solution:
a) The AOA network:
8
6
B
1
A
4
D
F
2
E
3
M
Ǿ
C
I
J
G
7
Ǿ
11
2
H
9
5
L
K
N
10
b) AoN Network Diagram:
Q3) For the AOA type of project network given below with durations indicated on each activity:
5
3
5
7
8
1
10
10
2
6
9
5
2
4
Find the critical path and the critical activities.
Solution:
E1=0
E2=10
E3= 10+9=19
E4= max{1,19+2}=21
E5= max{5,10+8}=18
E6=max{18+7,10+10,19+4,21+5}=26
E7=max{18+3,21+4,26+8}=34
Critical path length is 34.
L7= 34
L6= 34-8=26
L5= min{34-3,26-7}=19
L4= min{34-4,26-5}=21
L3= min{26-4,21-2}=19
L2=min {19-8, 19-9, 26-10} =10
L1=min {10-10, 19-5, 21-1} =0
Critical Path = {1, 2, 3, 4, 6, 7}
Act Dur
(1,2)* 10
(1,4)
1
(1,5)
5
(2,3)*
9
(2,5)
8
(2,6)
10
ES
0
0
0
10
10
10
EF
10
1
5
19
18
20
LS
0
20
14
10
11
16
7
4
3
1
8
LF TS
10
0
21 20
19 14
19
0
19
1
26
6
4
(3,4)*
(3,6)
(4,6)*
(4,7)
(5,6)
(5,7)
(6,7)*
2
4
5
4
7
3
8
19
19
21
21
18
18
26
21
23
26
25
25
21
34
19
22
21
30
19
31
26
21
26
26
34
26
34
34
0
3
0
9
1
13
0
Q4)
Consider the following activity network with (optimistic, most likely, pessimistic) estimates for
activity duration are indicated on each arc. Assuming Beta distribution for each arc's activity
duration time and the independence of these random variables, find
a) expected critical path length
b) the probability that the project will be completed in (or before) 16 time units.
2
(2,5,7)
(2,3,5)
1
(1,3,4)
(4,7,12)
(1,2,2)
(3,4,5)
5
4
(1,1,2)
7
(1,3,5)
(3,6,9)
(2,4,5)
3
6
(3,5,7)
Obtain the standard deviation using the following relation: s=(b-a)/3.2.
SOLUTION
a)
* =>Critical activities
Activity
Mean
(1,2)*
4.833
(1,3)
3.833
(1,4)
2.833
(2,4)*
3.167
(2,5)
1.833
(2,7)
7.333
(3,6)
5
(4,6)*
6
(5,6)
1.167
Variance
2.441
0.879
0.879
0.879
0.097
6.250
1.563
3.516
0.097
(5,7)
(6,7)*
4
3
0.391
1.563
2
4.833
7.333
1.833
3.167
1
2.833
7
4
5
4
1.167
3
6
3.833
6
3
I
1
2
3
4
5
6
7
Ei
0
4.833
3.833
8
6.666
14
17
5
Li
0
4.833
9
8
12.833
14
17
Critical Path:{1,2,4,6,7}; CPL=17.
b)
P( z £
16 - 17
) s 2 =2.441+0.879+3.516+1.563=8.399
2
s
P(z≤-0.345)=0.3651
There is a 0.3651 probability that the project will be completed in 16 or less time units.