Home > Operation Research calculators > Project scheduling with uncertain activity times (Optimistic, Most likely, Pessimistic) example

7. Project scheduling : Activity, Predecessors, to, tm, tp 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

6. Critical path, Total float, Free float, Independent float : Activity i-j, Name of Activity, Duration
(Previous method)
8. Project scheduling : Activity i-j, to, tm, tp
(Next method)

1. Example-1



1. Project scheduling with uncertain activity times (Optimistic, Most likely, Pessimistic)
A-117
B-147
C-228
DA111
EB2514
FC258
GD,E3615


Solution:
Expected time of each activity,
Activity`t_o``t_m``t_p``t_e=(t_o + 4*t_m + t_p)/6``sigma^2=((t_p - t_o)/6)^2`
A11721
B14741
C22831
D11110
E251464
F25851
G361574



The earliest and latest expected time for each activity is calculated by considering the expected time `t_e`

ActivityImmediate PredecessorsDuration
A-2
B-4
C-3
DA1
EB6
FC5
GD,E7


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



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



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`


Backward Pass Method
`L_6=E_6=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` and critical activities are `B,E,G`

The total project time is 17
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`
 G(7) `G : 5->6`
`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`
`E_(6)=17`
`L_(6)=17`



This material is intended as a summary. Use your textbook for detail explanation.
Any bug, improvement, feedback then Submit Here





6. Critical path, Total float, Free float, Independent float : Activity i-j, Name of Activity, Duration
(Previous method)
8. Project scheduling : Activity i-j, to, tm, tp
(Next method)




 
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