A tech startup was founded in $2021.$ The following table shows an estimate of active numbers ofcon$-$ tractors the company needs for any given year $($which includes new hires as well as those stayingon due to a longer$-$term contract$)$ over the next $5$ years:

The company can choose to sign different length contracts with each contractor it hires and wants tokeep more contractors than needed during some years of the planning horizon. The startup estimatesthat the cost of recruiting and maintaining the contractors is a function of their length of stay withthe company. The following table summarizes these estimates:

The startup is interested in the recruiting schedule to minimize the total cost.

$($Hint: Think about the number of contractors the startup needs to hire for different horizons $-1,2,3,$

$4,5$ years $-$ in each year of the planning horizon.$)$

$($a$)(15$ points$)$ Write the problem as an optimization problem. You can ignore the fact that the numbers of engineers cannot take fractional values.

$($b$)(5$ points$)$ Is this a linear or non$-$linear optimization problem? Please briefly explain.

$($c$)(20$ points$)$ Determine the optimal solution by using Excel's solver or Python's packages. If you use Excel, be sure to submit the Excel sheet and solver window. If you use Python, be sure to submit your fully commented code and output.

Please see the image for further details