Question $1.$ Tech Covers Inc. Production Planning

"Tech Covers Inc." is a company that produces covers for three types of devices: laptops, tablets,

and smartphones. The company's production facilities allow them to produce $8000$ laptop covers,

$5000$ tablet covers, or $7000$ smartphone covers in one day only if the entire production is

devoted to one type of cover $($in other words, producing $8000$ laptop covers, or $5000$ tablet

covers, or $7000$ smartphone covers takes one full working day$).$ The production schedule is one

week $(5$ working days$),$ and the week's production must be stored before distribution. Storing $1000$

laptop covers $($packaging included$)$ takes up $60$ cubic feet of space, while storing $1000$ tablet covers

$($packaging included$)$ takes up $30$ cubic feet of space and storing $1000$ smartphone covers $($packaging

included$)$ takes up $40$ cubic feet of space. The total storage space available is $8000$ cubic feet.

Due to agreements with major electronics retailers, Tech Covers Inc. has to deliver at least

$6000$ laptop covers and $4000$ tablet covers per week to strengthen the product's market presence.

The marketing department estimates that the weekly demand for laptop covers, tablet covers, and

smartphone covers does not exceed $12000,10000,$ and $15000$ units, respectively. Therefore, the

company does not want to produce more than these amounts for laptop, tablet, and smartphone

covers. Finally, the net profit per each laptop cover, tablet cover, and smartphone cover is $$5,$$$8,$

and $$10,$ respectively.

$($a$)$ Write a Linear Programming formulation for the problem. You must label each constraint.

$($Hint: define decision variables as the number of items of each produced over the week$).$

$($b$)$ Solve your developed model using Excel and obtain the optimal weekly production plan of this

company.