A project manager has the following AON precedence network for a small project.

Time estimates in weeks are given as follows:

$\backslash$

table

$[$

$[$

Task

$,$

Optimistic,Most Likelv,Pessimistic

$]$

$,$

$[$

$1$

$,$

$4$

$,$

$6$

$,$

$1$

$4$

$]$

$,$

$[$

$2$

$,$

$1$

$,$

$3$

$,$

$1$

$1$

$]$

$,$

$[$

$3$

$,$

$2$

$,$

$9$

$,$

$1$

$6$

$]$

$,$

$[$

$4$

$,$

$1$

$,$

$3$

$,$

$8$

$]$

$]$

a

$.$

$$Find the expected duration of the project and its variance according to the PERT model.

b

$.$

$$Based on the PERT model, what is the probability that the project will be completed within

$1$

$3$

$$weeks$?$

c

$.$

$$What due date has a

$9$

$0$

$$percent chance

$($

or greater

$)$

$$of being met?

d

$.$

$$If the project manager uses the PERT model and wants to give a customer a due date that has at least a

$6$

$0$

$$percent chance of occurring, what is the earliest date that the project manager can promise the customer that the project will be completed?

e

$.$

$$A major criticism of the classic PERT model is that it relies only on the longest expected paths in the precedence network. One possible solution that has been suggested is to use both the longest path and the second

$-$

longest path to calculate the expected project duration. If this approach was used in this problem, what is the probability that the project will be completed within

$1$

$3$

$$weeks$?$