Problem Statement:

You are hired as a project manager for a software development company launching a new mobile application. Your task is to plan, schedule, and optimize the project timeline while ensuring the project goals are met.

Part $1$: CPM $($Critical Path Method$)$

The project involves the following activities:

$1.$ Construct the project network diagram and identify the critical path.

$2.$ Calculate the project completion time and total float for each activity.

$3.$ Which activities need to be monitored closely to avoid project delays?

Part $2$: PERT $($Program Evaluation and Review Technique$)$

Given the uncertainties in development, the duration of each task may vary. Use the three$-$time estimate $($Optimistic$,$ Most Likely, Pessimistic$)$ for the Backend development $($Activity C$)$:

$-$ Optimistic $($O$)$: $4$ days

$-$ Most Likely $($M$)$: $6$ days

$-$ Pessimistic $($P$)$: $10$ days

$1.$ Calculate the expected time and variance for Activity C$.$

$2.$ Find the probability that the project will be completed within $\backslash (\backslash$mathbf$\{18\}\backslash )$ days.

Part $3$: Goal Programming

The project must meet multiple objectives:

$-$ Goal $1$: Complete the project within $20$ days.

$-$ Goal $2$: Keep the cost below $\backslash$$$50,000.$

$-$ Goal $3$: Achieve a user satisfaction score of $\backslash (90\backslash \%\backslash ).$

However, there are trade$-$offs between these goals $($e$.$g$.,$ reducing time increases cost$).$ Formulate this as a goal programming problem and describe how you would balance these competing objectives.