Question

Activity	Predecessor	Optimistic	Most Likely	Pessimistic
A		4	7	10
B		2	9	10
C	A,B	2	5	8
D	C	16	19	28
E	C	6	9	24
F	E	1	7	13
G	C	4	10	28
H	D,F	2	5	14
I	G,F	5	8	17
J	H,I	2	5	8

1. Determine the expected completion time for each activity.
2. Construct the AON network for the activities of this contract and determine the critical path for the project.

Answer 1

Expected time of each activity,

Activity	to	tm	tp	te=(to+4⋅tm+tp)/6	σ2=(tp-to/6)2
A	4	7	10	7	1
B	2	9	10	8	1.78
C	2	5	8	5	1
D	16	19	28	20	4
E	6	9	24	11	9
F	1	7	13	7	4
G	4	10	28	12	16
H	2	5	14	6	4
I	5	18	17	15.67	4
J	2	5	8	5	1

The earliest and latest expected time for each activity is calculated by considering the expected time te

Activity	Immediate Predecessors	Duration
A	-	7
B	-	8
C	A,B	5
D	C	20
E	C	11
F	E	7
G	C	12
H	D,F	6
I	G,F	15.67
J	H,I	5

Edge and it's preceded and succeeded node

Edge	Node1 → Node2
A	1→2
B	1→3
d	2→3
C	3→4
D	4→5
E	4→6
G	4→8
H	5→9
F	6→7
d	7→5
d	7→8
I	8→9
J	9→10

The network diagram for the project, along with activity time, is

Forward Pass Method

E1=0

E2=E1+t1,2 [t1,2=A=7]=0+7=7

E3=Max{Ei+ti,3}[i=1,2]

=Max{E1+t1,3;E2+t2,3}

=Max{0+8;7+0}

=Max{8;7}

=8

E4=E3+t3,4 [t3,4=C=5]=8+5=13

E5=Max{Ei+ti,5}[i=4,7]

=Max{E4+t4,5;E7+t7,5}

=Max{13+20;31+0}

=Max{33;31}

=33

E6=E4+t4,6 [t4,6=E=11]=13+11=24

E7=E6+t6,7 [t6,7=F=7]=24+7=31

E8=Max{Ei+ti,8}[i=4,7]

=Max{E4+t4,8;E7+t7,8}

=Max{13+12;31+0}

=Max{25;31}

=31

E9=Max{Ei+ti,9}[i=5,8]

=Max{E5+t5,9;E8+t8,9}

=Max{33+6;31+15.67}

=Max{39;46.67}

=46.67

E10=E9+t9,10 [t9,10=J=5]=46.67+5=51.67

Backward Pass Method

L10=E10=51.67

L9=L10-t9,10 [t9,10=J=5]=51.67-5=46.67

L8=L9-t8,9 [t8,9=I=15.67]=46.67-15.67=31

L7=Min{Lj-t7,j}[j=8,5]

=Min{L8-t7,8;L5-t7,5}

=Min{31-0;40.67-0}

=Min{31;40.67}

=31

L6=L7-t6,7 [t6,7=F=7]=31-7=24

L5=L9-t5,9 [t5,9=H=6]=46.67-6=40.67

L4=Min{Lj-t4,j}[j=8,6,5]

=Min{L8-t4,8;L6-t4,6;L5-t4,5}

=Min{31-12;24-11;40.67-20}

=Min{19;13;20.67}

=13

L3=L4-t3,4 [t3,4=C=5]=13-5=8

L2=L3-t2,3 [t2,3=d=0]=8-0=8

L1=Min{Lj-t1,j}[j=3,2]

=Min{L3-t1,3;L2-t1,2}

=Min{8-8;8-7}

=Min{0;1}

=0

The critical path in the network diagram has been shown. This has been done by double lines by joining all those events where E-values and L-values are equal.
The critical path of the project is : 1-3-4-6-7-8-9-10 and critical activities are B,C,E,F,d,I,J

The total project time is 51.67

The network diagram for the project, along with E-values and L-values, is

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