Critical path, Total float, Free float, Independent float

1-2	4
2-3	6
2-4	2
3-4	0
3-6	2
4-5	7
5-6	4
6-7	8
7-8	3

Solution:
The given problem is

Activity	Duration
1-2	4
2-3	6
2-4	2
3-4	0
3-6	2
4-5	7
5-6	4
6-7	8
7-8	3

Edge and it's preceded and succeeded node

Edge	Node1 `->` Node2
A0	1`->`2
A1	2`->`3
A2	2`->`4
d	3`->`4
A4	3`->`6
A5	4`->`5
A6	5`->`6
A7	6`->`7
A8	7`->`8

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

			A1(6)	3	A4(2)	6	A7(8)	7	A8(3)	8
1	A0(4)	2		d(0)		A6(4)
			A2(2)	4	A5(7)	5

(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 + 4``=4`

`E_3=E_2 + t_(2,3)``=4 + 6``=10`

`E_4=Max{E_i + t_(i,4)} [i=2, 3]`

`=Max{E_2 + t_(2,4); E_3 + t_(3,4)}`

`=Max{4 + 2; 10 + 0}`

`=Max{6; 10}`

`=10`

`E_5=E_4 + t_(4,5)``=10 + 7``=17`

`E_6=Max{E_i + t_(i,6)} [i=3, 5]`

`=Max{E_3 + t_(3,6); E_5 + t_(5,6)}`

`=Max{10 + 2; 17 + 4}`

`=Max{12; 21}`

`=21`

`E_7=E_6 + t_(6,7)``=21 + 8``=29`

`E_8=E_7 + t_(7,8)``=29 + 3``=32`


Backward Pass Method

`L_8=E_8=32`

`L_7=L_8 - t_(7,8)``=32 - 3``=29`

`L_6=L_7 - t_(6,7)``=29 - 8``=21`

`L_5=L_6 - t_(5,6)``=21 - 4``=17`

`L_4=L_5 - t_(4,5)``=17 - 7``=10`

`L_3=text{Min}{L_j - t_(3,j)} [j=4, 6]`

`=text{Min}{L_4 - t_(3,4); L_6 - t_(3,6)}`

`=text{Min}{10 - 0; 21 - 2}`

`=text{Min}{10; 19}`

`=10`

`L_2=text{Min}{L_j - t_(2,j)} [j=3, 4]`

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

`=text{Min}{10 - 6; 10 - 2}`

`=text{Min}{4; 8}`

`=4`

`L_1=L_2 - t_(1,2)``=4 - 4``=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 : `1-2-3-4-5-6-7-8` and critical activities are `A0,A1,d,A5,A6,A7,A8`

The total project time is 32
The network diagram for the project, along with E-values and L-values, is

			A1(6)	3	A4(2)	6	A7(8)	7	A8(3)	8
1	A0(4)	2	`E_(3)=10` `L_(3)=10`	d(0)	`E_(4)=10` `L_(4)=10`	A6(4)	`E_(6)=21` `L_(6)=21`	`E_(7)=29` `L_(7)=29`		`E_(8)=32` `L_(8)=32`
`E_(1)=0` `L_(1)=0`		`E_(2)=4` `L_(2)=4`	A2(2)	4	A5(7)	5	`E_(5)=17` `L_(5)=17`

(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	4	0	4	0	4	`0+4=4`	`4-4=0`	`0-0=0`	`4-0-4=0`	`4-0-4=0`	Yes
2-3	6	4	10	4	10	`4+6=10`	`10-6=4`	`4-4=0`	`10-4-6=0`	`10-4-6=0`	Yes
2-4	2	4	10	4	10	`4+2=6`	`10-2=8`	`8-4=4`	`10-4-2=4`	`10-4-2=4`	No
3-4	0	10	10	10	10	`10+0=10`	`10-0=10`	`10-10=0`	`10-10-0=0`	`10-10-0=0`	Yes
3-6	2	10	21	10	21	`10+2=12`	`21-2=19`	`19-10=9`	`21-10-2=9`	`21-10-2=9`	No
4-5	7	10	17	10	17	`10+7=17`	`17-7=10`	`10-10=0`	`17-10-7=0`	`17-10-7=0`	Yes
5-6	4	17	21	17	21	`17+4=21`	`21-4=17`	`17-17=0`	`21-17-4=0`	`21-17-4=0`	Yes
6-7	8	21	29	21	29	`21+8=29`	`29-8=21`	`21-21=0`	`29-21-8=0`	`29-21-8=0`	Yes
7-8	3	29	32	29	32	`29+3=32`	`32-3=29`	`29-29=0`	`32-29-3=0`	`32-29-3=0`	Yes

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