Project scheduling with uncertain activity times (Optimistic, Most likely, Pessimistic)

A	-	1	1	7
B	-	1	4	7
C	-	2	2	8
D	A	1	1	1
E	B	2	5	14
F	C	2	5	8
G	D,E	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`
A	1	1	7	2	1
B	1	4	7	4	1
C	2	2	8	3	1
D	1	1	1	1	0
E	2	5	14	6	4
F	2	5	8	5	1
G	3	6	15	7	4

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

Activity	Immediate Predecessors	Duration
A	-	2
B	-	4
C	-	3
D	A	1
E	B	6
F	C	5
G	D,E	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

		2
	A(2)		D(1) E(6)	5
1	B(4)	3		G(7)
	C(3)			G(7)
		4	F(5)	6

(Note: Same diagram as above. Move node position by dragging node.
If you are not able to select node then just click here for )

Forward Pass Method

`E_1=0`

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

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

`E_4=E_1 + t_(1,4)``=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)``=17 - 7``=10`

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

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

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

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

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

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

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

`=0`


(b) The critical path in the network diagram has been shown. This has been done by red 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_(5)=10` `L_(5)=10`
	A(2)	`E_(2)=2` `L_(2)=9`	D(1) E(6)	5
1	B(4)	3		G(7)
`E_(1)=0` `L_(1)=0`	C(3)	`E_(3)=4` `L_(3)=4`		G(7)
	`E_(4)=3` `L_(4)=12`	4	F(5)	6	`E_(6)=17` `L_(6)=17`

(Note: Same diagram as above. Move node position by dragging node.
If you are not able to select node then just click here for )

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