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Home > Operation Research calculators > Critical path, Total float, Free float, Independent float example
6. Critical path, Total float, Free float, Independent float : Activity i-j, Name of Activity, Duration example ( Enter your problem )
( Enter your problem )
  1. Example-1
 		Other related methods
  1. Network diagram : Activity, Predecessors
  2. Network diagram : Activity i-j
  3. Network diagram : Activity i-j, Name of Activity
  4. Critical path, Total float, Free float, Independent float : Activity, Predecessors, Duration
  5. Critical path, Total float, Free float, Independent float : Activity i-j, Duration
  6. Critical path, Total float, Free float, Independent float : Activity i-j, Name of Activity, Duration
  7. Project scheduling : Activity, Predecessors, to, tm, tp
  8. Project scheduling : Activity i-j, to, tm, tp
  9. Project scheduling : Activity i-j, Name of Activity, to, tm, tp
  10. Project crashing : Activity, Predecessors, Normal Time & Cost, Crash Time & Cost and Indirect Cost
  11. Project crashing : Activity i-j, Normal Time & Cost, Crash Time & Cost and Indirect Cost
  12. Project crashing : Activity i-j, Name of Activity, Normal Time & Cost, Crash Time & Cost and Indirect Cost
  13. Project crashing : Activity, Predecessors, Normal Time & Cost, Crash Time & Cost and varying Indirect Cost
  14. Project crashing : Activity i-j, Normal Time & Cost, Crash Time & Cost and varying Indirect Cost
  15. Project crashing : Activity i-j, Name of Activity, Normal Time & Cost, Crash Time & Cost and varying Indirect Cost
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


1. Critical path, Total float, Free float, Independent float
1-2A4
2-3B6
2-4C2
3-4d0
3-6D2
4-5E7
5-6F4
6-7G8
7-8H3


Solution:
The given problem is
ActivityActivityDuration
1-2A4
2-3B6
2-4C2
3-4d0
3-6D2
4-5E7
5-6F4
6-7G8
7-8H3


Edge and it's preceded and succeeded node
EdgeNode1 `->` Node2
A1`->`2
B2`->`3
C2`->`4
d3`->`4
D3`->`6
E4`->`5
F5`->`6
G6`->`7
H7`->`8



The network diagram for the project, along with activity time, is
 B(6) `B : 2->3`
3
 D(2) `D : 3->6`
6
 G(8) `G : 6->7`
7
 H(3) `H : 7->8`
8
1
 A(4) `A : 1->2`
2
 d(0) `d : 3->4`
 F(4) `F : 5->6`
 C(2) `C : 2->4`
4
 E(7) `E : 4->5`
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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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)


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