Critical path, Total float, Free float, Independent float

A	-	2
B	-	4
C	-	3
D	A	1
E	B	6
F	C	5
G	D,E	7
H	F,G	2

Solution:

Activity	Immediate Predecessors	Duration
A	-	2
B	-	4
C	-	3
D	A	1
E	B	6
F	C	5
G	D,E	7
H	F,G	2

Edge and it's preceded and succeeded node

Edge	Node1 `->` Node2
A	1`->`2
B	1`->`3
C	1`->`4
D	2`->`5
E	3`->`5
F	4`->`6
G	5`->`6
H	6`->`7

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

		2
	A(2)		D(1) E(6)	5
1	B(4)	3		G(7)
	C(3)			G(7)
		4	F(5)	6	H(2)	7

(Note: Same diagram as above. Move node position by dragging node.
If you are not able to select node then just click here for )

Forward Pass Method

`E_1=0`

`E_2=E_1 + t_(1,2)``=0 + 2``=2`

`E_3=E_1 + t_(1,3)``=0 + 4``=4`

`E_4=E_1 + t_(1,4)``=0 + 3``=3`

`E_5=Max{E_i + t_(i,5)} [i=2, 3]`

`=Max{E_2 + t_(2,5); E_3 + t_(3,5)}`

`=Max{2 + 1; 4 + 6}`

`=Max{3; 10}`

`=10`

`E_6=Max{E_i + t_(i,6)} [i=4, 5]`

`=Max{E_4 + t_(4,6); E_5 + t_(5,6)}`

`=Max{3 + 5; 10 + 7}`

`=Max{8; 17}`

`=17`

`E_7=E_6 + t_(6,7)``=17 + 2``=19`


Backward Pass Method

`L_7=E_7=19`

`L_6=L_7 - t_(6,7)``=19 - 2``=17`

`L_5=L_6 - t_(5,6)``=17 - 7``=10`

`L_4=L_6 - t_(4,6)``=17 - 5``=12`

`L_3=L_5 - t_(3,5)``=10 - 6``=4`

`L_2=L_5 - t_(2,5)``=10 - 1``=9`

`L_1=text{Min}{L_j - t_(1,j)} [j=2, 3, 4]`

`=text{Min}{L_2 - t_(1,2); L_3 - t_(1,3); L_4 - t_(1,4)}`

`=text{Min}{9 - 2; 4 - 4; 12 - 3}`

`=text{Min}{7; 0; 9}`

`=0`


(b) The critical path in the network diagram has been shown. This has been done by red lines by joining all those events where E-values and L-values are equal.
The critical path of the project is : `1-3-5-6-7` and critical activities are `B,E,G,H`

The total project time is 19
The network diagram for the project, along with E-values and L-values, is

		2		`E_(5)=10` `L_(5)=10`
	A(2)	`E_(2)=2` `L_(2)=9`	D(1) E(6)	5
1	B(4)	3		G(7)
`E_(1)=0` `L_(1)=0`	C(3)	`E_(3)=4` `L_(3)=4`	`E_(6)=17` `L_(6)=17`	G(7)		`E_(7)=19` `L_(7)=19`
	`E_(4)=3` `L_(4)=12`	4	F(5)	6	H(2)	7

(Note: Same diagram as above. Move node position by dragging node.
If you are not able to select node then just click here for )

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

Activity `(i,j)` `(1)`	Duration `t_(ij)` `(2)`	Earliest Start `E_i` `(3)`	`E_j` `(4)`	`L_i` `(5)`	Latest Finish `L_j` `(6)`	Earliest Finish `=E_i+t_(ij)` `(7)=(3)+(2)`	Latest Start `=L_j-t_(ij)` `(8)=(6)-(2)`	Total Float `=L_j-t_(ij)-E_i` `(9)=(8)-(3)`	Free Float `=E_j-E_i-t_(ij)` `(10)=(4)-(3)-(2)`	Independent Float `=E_j-L_i-t_(ij)` `(11)=(4)-(5)-(2)`	Critical Activity? `(12)`
1-2	2	0	2	0	9	`0+2=2`	`9-2=7`	`7-0=7`	`2-0-2=0`	`2-0-2=0`	No
1-3	4	0	4	0	4	`0+4=4`	`4-4=0`	`0-0=0`	`4-0-4=0`	`4-0-4=0`	Yes
1-4	3	0	3	0	12	`0+3=3`	`12-3=9`	`9-0=9`	`3-0-3=0`	`3-0-3=0`	No
2-5	1	2	10	9	10	`2+1=3`	`10-1=9`	`9-2=7`	`10-2-1=7`	`10-9-1=0`	No
3-5	6	4	10	4	10	`4+6=10`	`10-6=4`	`4-4=0`	`10-4-6=0`	`10-4-6=0`	Yes
4-6	5	3	17	12	17	`3+5=8`	`17-5=12`	`12-3=9`	`17-3-5=9`	`17-12-5=0`	No
5-6	7	10	17	10	17	`10+7=17`	`17-7=10`	`10-10=0`	`17-10-7=0`	`17-10-7=0`	Yes
6-7	2	17	19	17	19	`17+2=19`	`19-2=17`	`17-17=0`	`19-17-2=0`	`19-17-2=0`	Yes

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