We use cookies to improve your experience on our site and to show you relevant advertising. By browsing this website, you agree to our use of cookies. Learn more   

AtoZmath.com
Support us
(New) All problem can be solved using search box
I want to sell my website www.AtoZmath.com with complete code
Home What's new College Algebra Games Feedback About us
Algebra Matrix & Vector Numerical Methods Statistical Methods Operation Research Word Problems Calculus Geometry Pre-Algebra


Home > Operation Research calculators > PERT and CPM 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


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.
Any bug, improvement, feedback then Submit Here


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)


Share this solution or page with your friends.


Home What's new College Algebra Games Feedback About us
 
Copyright © 2023. All rights reserved. Terms, Privacy
 
 

.