a$)$ Below is the new product development project consisting of $10$ activities. Use the predecessors of each activity and duration and find critical path and the project duration of the network.

b$)$ From your critical path method, you naturally find the slack of each activity. What is the slack of activity B and activity $\backslash ($ A $\backslash )?$ What are the differences between these two activities, and what understanding does it give you?

C$)$ Since we realized that while the product development to launch takes more than a year, our prediction of activity duration can be fluctuated. Therefore, we have to use probabilistic activity times. Below, you can find the optimistic duration, most likely duration, and worst duration. Use the formula:

$-$ Expected Activity Time: $\backslash (\backslash$frac$\{$a$+4$ m$+$b$\}\{6\}\backslash )$

$-$ Activity Variance: $\backslash (\backslash$left$(\backslash$frac$\{$b$-$a$\}\{6\}\backslash$right$)^\{2\}\backslash )$

PERT Steps

$-$ STEP $1$: Find the Expected Time and Activity Variance for each activity.

$-$ STEP $2$: Use the expected time of the activities found in step $1$ and the critical path method to find the critical path and expected project duration.

$-$ STEP $3$: Find the project variance by summing up the variances of the critical activities.

$-$ STEP $4$: Your client requests that the project be completed within $52$ weeks. What is the probability that the project can be completed within $52$ weeks?

$($You found the project duration from Step $2$ and variance from Step $3.$ Don't forget to square root the variance to plug in the standard deviation into the formula $\backslash ($ Z$=\backslash$frac$\{$x$-$x$\}\{$S D$\}\backslash ))$