Apple Inc., must determine how many new Apple Watch to produce each of the next four quarters of $2023.$ The demand for each of the next four quarters is as follows: first quarter, $6000$ Watches; second quarter, $5500$ Watches; third quarter, $7500$ Watches, fourth quarter, $6500$ Watches. Apple must meet demand on time. At the beginning of the first quarter, Apple Inc., has an inventory of $450$ Watches. At the beginning of each quarter, Apple Inc., must decide how many Watches should be produced during that quarter. For simplicity, we assume that Watches manufactured during a quarter can be used to meet demand for that quarter.

During each quarter, Apple Inc., produces up to $3500$ Watches with regular$-$time labor at a total cost of $$200$ per Watch. By having employees work overtime during a quarter, Apple Inc., can produce additional Watch with overtime labor at a cost of $$300$ per Watch. At the end of each quarter $($after production has occurred and the current quarter$$s demand has been satisfied$),$ a carrying or holding cost of $$10$ per Watch is incurred.

Formulate the linear programming to determine a production schedule to minimize the sum of production and inventory costs during the next four quarters.

$($Attach Excel Solution showing model display and screen shot of solver$)$