A project consists of five activities (A, B, C, D and E). The duration of each activity follows beta distribution. The three time estimates (in weeks) of each activity and immediate predecessor(s) are listed in the table. The expected time of the project completion is __________ weeks (in integer).

Activity Time estimates (in weeks) Immediate predecessor(s) Optimistic time Most likely time Pessimistic time A 4 5 6 None B 1 3 5 A C 1 2 3 A D 2 4 6 C E 3 4 5 B, D

Activity	Time estimates (in weeks)			Immediate predecessor(s)
	Optimistic time	Most likely time	Pessimistic time
A	4	5	6	None
B	1	3	5	A
C	1	2	3	A
D	2	4	6	C
E	3	4	5	B,D

The correct answer is 15.

Step 1: Understand the Expected Activity Duration Formula
In project scheduling using PERT (Program Evaluation and Review Technique), the activity durations follow a beta distribution. The expected time ($T_E$) for each activity is calculated using its three-time estimates: optimistic time ($t_o$), most likely time ($t_m$), and pessimistic time ($t_p$) by the formula:
$$T_E=\frac{t_o+4t_m+t_p}{6}$$

Step 2: Calculate the Expected Time ($T_E$) for Each Activity
Using the table data provided in the question and shown in the image, we compute the expected times for activities A, B, C, D, and E:

• Activity A (Predecessor: None):
  $T_E\left(A\right)=\frac{4+4\left(5\right)+6}{6}=\frac{30}{6}=5\text{weeks}$
  
• Activity B (Predecessor: A):
  $T_E\left(B\right)=\frac{1+4\left(3\right)+5}{6}=\frac{18}{6}=3\text{weeks}$
  
• Activity C (Predecessor: A):
  $T_E\left(C\right)=\frac{1+4\left(2\right)+3}{6}=\frac{12}{6}=2\text{weeks}$
  
• Activity D (Predecessor: C):
  $T_E\left(D\right)=\frac{2+4\left(4\right)+6}{6}=\frac{24}{6}=4\text{weeks}$
  
• Activity E (Predecessors: B, D):
  $T_E\left(E\right)=\frac{3+4\left(4\right)+5}{6}=\frac{24}{6}=4\text{weeks}$
  

Step 3: Analyze the Network Paths and Project Completion Time
By tracing the predecessors in the network diagram from the image, we can identify all possible paths from the start of the project to its completion:

1. Path 1 (A → B → E):
   $\text{Duration}=T_E\left(A\right)+T_E\left(B\right)+T_E\left(E\right)=5+3+4=12\text{weeks}$
   
2. Path 2 (A → C → D → E):
   $\text{Duration}=T_E\left(A\right)+T_E\left(C\right)+T_E\left(D\right)+T_E\left(E\right)=5+2+4+4=15\text{weeks}$
   

The expected project completion time is determined by the critical path, which is the path with the longest total duration. Comparing the two paths:
$\text{Project Completion Time}=max(12,15)=15\text{weeks}$
Thus, the expected time of the project completion is 15 weeks.