Develop a model of this decision problem in Excel, in a worksheet titled $"$NHT$".$

NHT$,$ a sports equipment manufacturer located in New Hampshire, sells high$-$quality camping tents. The demand for these tents is seasonal. To meet the camping season's higher summer demands, in the lowseason NHT produces more tents than are immediately needed, and carries inventory into the summer. NHT can produce at most $200$ tents per month using regular labor at a cost of $$80$ each; optionally, up to $60$ additional tents can be produced by utilizing overtime labor at a cost of $$120$ each. These tents are then sold for $$350$ each. NHT maintains a monthly safety stock of $20$ tents from October to March, and $35$ tents from April to September. Each tent held in inventory from one month to the next incurs a cost of $$10.$

Assume that you develop a production plan for $2025.$ In January $2025,$ NHT will have $20$ tents in inventory. The supposedly fairly accurate monthly demand estimates for $2025$ are shown below.

$\backslash$table$[[$Jan$,$Feb,Mar,Apr,May,Jun,July,Aug,Sep,Oct,Nov,Dec$],[25,30,70,80,150,350,500,450,350,200,30,20]]$

By standard company policy, NHT wants to meet all demands throughout the year.

a$)$ State an initial estimate of the regular production required level in each month, assuming that the overtime production level is set to $0.$ Use this estimate when you formulate and solve a linear programming model to determine how many tents should be produced each month, in order to maximize NHT$\text{'}$s total profit in $2025.$ Make sure that your solution is given in integers $($number of tents$).$

b$)$ What would happen if you applied the production and inventory management policy resulting in a$),$ but the $2025$ demands for tents fluctuate? You can assume that the monthly demands $($approximately$)$ follow independent discrete distributions, and suggest suitable distribution models. How would you set up your modified model calculations, and then analyze this situation? Could you propose a modified management policy? What are the expected consequences, in terms of profits? You don't have to carry out calculations, just answer the questions in a few sentences based on your experience gained in this course.