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Home > Operation Research calculators > Critical path, Total float, Free float, Independent float example
4. Critical path, Total float, Free float, Independent float : Activity, Predecessors, 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
3. Network diagram : Activity i-j, Name of Activity
(Previous method)		5. Critical path, Total float, Free float, Independent float : Activity i-j, Duration
(Next method)

1. Example-1


1. Critical path, Total float, Free float, Independent float
A-2
B-4
C-3
DA1
EB6
FC5
GD,E7
HF,G2


Solution:
ActivityImmediate PredecessorsDuration
A-2
B-4
C-3
DA1
EB6
FC5
GD,E7
HF,G2


Edge and it's preceded and succeeded node
EdgeNode1 `->` Node2
A1`->`2
B1`->`3
C1`->`4
D2`->`5
E3`->`5
F4`->`6
G5`->`6
H6`->`7



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



Forward Pass Method
`E_1=0`

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

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

`E_4=E_1 + t_(1,4)` [`t_(1,4) = C = 3`]`=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)` [`t_(6,7) = H = 2`]`=17 + 2``=19`


Backward Pass Method
`L_7=E_7=19`

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

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

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

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

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

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

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

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

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

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



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)`
1-2 2 `t_(1,2)=2` 0 `E_1=0` 2 `E_2=2` 0 `L_1=0` 9 `L_2=9` 2 `2=0+2`
`(E_i+t_(ij))`		 7 `7=9-2`
`(L_j-t_(ij))`		 7 `7=7-0`
`(L_j-t_(ij))-E_i`		 0 `0=(2-0)-2`
`(E_j-E_i)-t_(ij)`		 0 `0=(2-0)-2`
`(E_j-L_i)-t_(ij)`
1-4 3 `t_(1,4)=3` 0 `E_1=0` 3 `E_4=3` 0 `L_1=0` 12 `L_4=12` 3 `3=0+3`
`(E_i+t_(ij))`		 9 `9=12-3`
`(L_j-t_(ij))`		 9 `9=9-0`
`(L_j-t_(ij))-E_i`		 0 `0=(3-0)-3`
`(E_j-E_i)-t_(ij)`		 0 `0=(3-0)-3`
`(E_j-L_i)-t_(ij)`
2-5 1 `t_(2,5)=1` 2 `E_2=2` 10 `E_5=10` 9 `L_2=9` 10 `L_5=10` 3 `3=2+1`
`(E_i+t_(ij))`		 9 `9=10-1`
`(L_j-t_(ij))`		 7 `7=9-2`
`(L_j-t_(ij))-E_i`		 7 `7=(10-2)-1`
`(E_j-E_i)-t_(ij)`		 0 `0=(10-9)-1`
`(E_j-L_i)-t_(ij)`
4-6 5 `t_(4,6)=5` 3 `E_4=3` 17 `E_6=17` 12 `L_4=12` 17 `L_6=17` 8 `8=3+5`
`(E_i+t_(ij))`		 12 `12=17-5`
`(L_j-t_(ij))`		 9 `9=12-3`
`(L_j-t_(ij))-E_i`		 9 `9=(17-3)-5`
`(E_j-E_i)-t_(ij)`		 0 `0=(17-12)-5`
`(E_j-L_i)-t_(ij)`


This material is intended as a summary. Use your textbook for detail explanation.
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3. Network diagram : Activity i-j, Name of Activity
(Previous method)		5. Critical path, Total float, Free float, Independent float : Activity i-j, Duration
(Next method)


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