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3. Project scheduling with uncertain activity times (Optimistic, Most likely, Pessimistic) example
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1. Example1
1. Project scheduling with uncertain activity times (Optimistic, Most likely, Pessimistic)
12  A  1  1  7  13  B  1  4  7  14  C  2  2  8  25  D  1  1  1  35  E  2  5  14  46  F  2  5  8  56  G  3  6  15 
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`  12  1  1  7  2  1  13  1  4  7  4  1  14  2  2  8  3  1  25  1  1  1  1  0  35  2  5  14  6  4  46  2  5  8  5  1  56  3  6  15  7  4 
The earliest and latest expected time for each activity is calculated by considering the expected time `t_e`
The given problem is
Activity  Activity  Duration  12  A  2  13  B  4  14  C  3  25  D  1  35  E  6  46  F  5  56  G  7 
Edge and it's preceded and succeeded node
Edge  Node1 `>` Node2  A  1`>`2  B  1`>`3  C  1`>`4  D  2`>`5  E  3`>`5  F  4`>`6  G  5`>`6 
The network diagram for the project, along with activity time, is
         D(1) `D : 2>5`  E(6) `E : 3>5` 
                    
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 Evalues and Lvalues are equal. The critical path of the project is : `1356` and critical activities are `B,E,G`
The total project time is 17 The network diagram for the project, along with Evalues and Lvalues, is
         D(1) `D : 2>5`  E(6) `E : 3>5` 
                    
This material is intended as a summary. Use your textbook for detail explanation. Any bug, improvement, feedback then







