Critical path, Total float, Free float, Independent float : Activity i-j, Name of Activity, Duration Example-1

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6. Critical path, Total float, Free float, Independent float : Activity i-j, Name of Activity, Duration example ( Enter your problem )

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Example-1

Other related methods

5. Critical path, Total float, Free float, Independent float : Activity i-j, Duration
(Previous method)

7. Project scheduling : Activity, Predecessors, to, tm, tp
(Next method)

1. Example-1

Critical path, Total float, Free float, Independent float
1-2 A 4
2-3 B 6
2-4 C 2
3-4 d 0
3-6 D 2
4-5 E 7
5-6 F 4
6-7 G 8
7-8 H 3

Solution:
The given problem is

Activity	Activity	Duration
1-2	A	4
2-3	B	6
2-4	C	2
3-4	d	0
3-6	D	2
4-5	E	7
5-6	F	4
6-7	G	8
7-8	H	3

Edge and it's preceded and succeeded node

Edge	Node1 `->` Node2
A	1`->`2
B	2`->`3
C	2`->`4
d	3`->`4
D	3`->`6
E	4`->`5
F	5`->`6
G	6`->`7
H	7`->`8

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

B(6) `B : 2->3`

D(2) `D : 3->6`

G(8) `G : 6->7`

H(3) `H : 7->8`

A(4) `A : 1->2`

d(0) `d : 3->4`

F(4) `F : 5->6`

C(2) `C : 2->4`

E(7) `E : 4->5`

Forward Pass Method
`E_1=0`

`E_2=E_1 + t_(1,2)` [`t_(1,2) = A = 4`]`=0 + 4``=4`

`E_3=E_2 + t_(2,3)` [`t_(2,3) = B = 6`]`=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)` [`t_(4,5) = E = 7`]`=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)` [`t_(6,7) = G = 8`]`=21 + 8``=29`

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

Backward Pass Method
`L_8=E_8=32`

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

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

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

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

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

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

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

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

`=10`

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

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

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

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

`=4`

`L_1=L_2 - t_(1,2)` [`t_(1,2) = A = 4`]`=4 - 4``=0`

(b) 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 : `1-2-3-4-5-6-7-8` and critical activities are `A,B,d,E,F,G,H`

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

B(6) `B : 2->3`

3 `E_(3)=10`
`L_(3)=10`

D(2) `D : 3->6`

6 `E_(6)=21`
`L_(6)=21`

G(8) `G : 6->7`

7 `E_(7)=29`
`L_(7)=29`

H(3) `H : 7->8`

8 `E_(8)=32`
`L_(8)=32`

1 `E_(1)=0`
`L_(1)=0`

A(4) `A : 1->2`

2 `E_(2)=4`
`L_(2)=4`

`E_(3)=10`
`L_(3)=10`

d(0) `d : 3->4`

`E_(4)=10`
`L_(4)=10`

F(4) `F : 5->6`

`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`

C(2) `C : 2->4`

4 `E_(4)=10`
`L_(4)=10`

E(7) `E : 4->5`

5 `E_(5)=17`
`L_(5)=17`

`E_(5)=17`
`L_(5)=17`

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

Activity `(i,j)` `(1)`	Duration `(t_(ij))` `(2)`	Earliest time Start `(E_i)` `(3)`	`(E_j)` `(4)`	`(L_i)` `(5)`	Latest time Finish `(L_j)` `(6)`	Earliest time Finish `(E_i+t_(ij))` `(7)=(3)+(2)`	Latest time 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)`
2-4	2 `t_(2,4)=2`	4 `E_2=4`	10 `E_4=10`	4 `L_2=4`	10 `L_4=10`	6 `6=4+2` `(E_i+t_(ij))`	8 `8=10-2` `(L_j-t_(ij))`	4 `4=8-4` `(L_j-t_(ij))-E_i`	4 `4=(10-4)-2` `(E_j-E_i)-t_(ij)`	4 `4=(10-4)-2` `(E_j-L_i)-t_(ij)`
3-6	2 `t_(3,6)=2`	10 `E_3=10`	21 `E_6=21`	10 `L_3=10`	21 `L_6=21`	12 `12=10+2` `(E_i+t_(ij))`	19 `19=21-2` `(L_j-t_(ij))`	9 `9=19-10` `(L_j-t_(ij))-E_i`	9 `9=(21-10)-2` `(E_j-E_i)-t_(ij)`	9 `9=(21-10)-2` `(E_j-L_i)-t_(ij)`

This material is intended as a summary. Use your textbook for detail explanation.
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