$.$ A tech startup was founded in $2021.$ The following table shows an estimate of active numbers ofcontractors 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:

Year $20222023202420252026$

# of software engineers $1101307016550$

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:

Length of employment contract $($in years$)12345$

Total cost for all of the years per contractor $110140170230250$

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.$)$