A TV company produces Smart TVs$.$ The company wants to plan production and workforce levels for

the next $6$ months. The table below shows the number of workdays and Smart TV demand $($in thousands$)$

for each month. Note that if the demand is exceeded, the leftovers can be carried to the next month. If the

demand is not satisfied, the items are backordered.

The annual inventory holding cost per Smart TV is $$120.$ If the company cannot satisfy a demand on

time, a backorder cost of $$30$ is incurred. Additionally, past data shows that $25$ workers can produce $60$

thousand Smart TVs in $20$ days. The company currently has $35$ workers and $3000$ Smart TVs in stock. The

company incurs a hiring cost of $$800$ and firing cost of $$1200$ per worker. The company cannot hire or fire

more than $10\%$ of its workforce in any month $($i$.$e$.,$ if there are $30$ workers at the end of a month, they can

hire or fire a maximum of $3$ people starting next month$).$ The payroll is $$60$ per worker per day.

Suppose that the company cannot change the number of workers they have for the first $3$ months. The

company also aims to have at least $25$ workers and $10$ Smart TVs in stock at the end of the $6$ month planning

horizon. Write a Linear Programming $($LP$)$ model that identifies the workforce and production plan, aiming

to minimize the total cost.