Question

Lecture Exercise #12 Critical Path Method Determine the early start, early finish, late start, late finish times for each activity Identify the critical path . Activity А Time (days) 6 7 3 Predecessor(s) None None B с A 2 A E 4 B F 6 B 10 C. E D. F H 7

Answer 1

EXERCISE - 12:

Solution:

Activity	Immediate Predecessors	Duration
A	-	6
B	-	7
C	A	3
D	A	2
E	B	4
F	B	6
G	C,E	10
H	D,F	7

Edge and it's preceded and succeeded node

Edge	Node1 → Node2
A	1→2
B	1→3
C	2→4
D	2→5
E	3→4
F	3→5
G	4→6
H	5→6

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

Forward Pass Method
E1=0

E2=E1+t1,2 [t1,2=A=6]=0+6=6

E3=E1+t1,3 [t1,3=B=7]=0+7=7

E4=Max{Ei+ti,4}[i=2,3]

=Max{E2+t2,4;E3+t3,4}

=Max{6+3;7+4}

=Max{9;11}

=11

E5=Max{Ei+ti,5}[i=2,3]

=Max{E2+t2,5;E3+t3,5}

=Max{6+2;7+6}

=Max{8;13}

=13

E6=Max{Ei+ti,6}[i=4,5]

=Max{E4+t4,6;E5+t5,6}

=Max{11+10;13+7}

=Max{21;20}

=21


Backward Pass Method
L6=E6=21

L5=L6-t5,6 [t5,6=H=7]=21-7=14

L4=L6-t4,6 [t4,6=G=10]=21-10=11

L3=Min{Lj-t3,j}[j=5,4]

=Min{L5-t3,5;L4-t3,4}

=Min{14-6;11-4}

=Min{8;7}

=7

L2=Min{Lj-t2,j}[j=5,4]

=Min{L5-t2,5;L4-t2,4}

=Min{14-2;11-3}

=Min{12;8}

=8

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

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

=Min{7-7;8-6}

=Min{0;2}

=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 and critical activities are B,E,G

The total project time is 21

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

For each non-critical activity, the total float, free float and independent float calculations are shown in Table

Activity (i,j) (1)	Duration (tij) (2)	Earliest time Start (Ei) (3)	(Ej) (4)	(Li) (5)	Latest time Finish (Lj) (6)	Earliest time Finish (Ei+tij) (7)=(3)+(2)	Latest time Start (Lj-tij) (8)=(6)-(2)	Total Float (Lj-tij)-Ei (9)=(8)-(3)	Free Float (Ej-Ei)-tij (10)=((4)-(3))-(2)	Independent Float (Ej-Li)-tij (11)=((4)-(5))-(2)
1-2	6	0	6	0	8	6	2	2	0	0
2-4	3	6	11	8	11	9	8	2	2	0
2-5	2	6	13	8	14	8	12	6	5	3
3-5	6	7	13	7	14	13	8	1	0	0
5-6	7	13	21	14	21	20	14	1	1	0