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 )
  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.
Any bug, improvement, feedback then Submit Here





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)




 
Copyright © 2020. All rights reserved. Terms, Privacy
 
This site is protected by reCAPTCHA and the Google Privacy Policy and Terms of Service apply.
 





Adblocker detected!

Dear user,

We've detected that you are using AdBlock Plus or some other adblocking software which is preventing the page from fully loading.

We don't have any banner, Flash, animation, obnoxious sound, or popup ad. We do not implement these annoying types of ads!

We need money to operate the site, and almost all of it comes from our online advertising.

Please add atozmath.com to your ad blocking whitelist or disable your adblocking software.

After unblocking website please refresh the page and click on find button again.

Thanks for your support

×
  .