Case Study # 1: Tech Covers Inc Production Planning (Total marks: 25)

"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. 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)

Case Study # 2: Optimizing Cargo Distribution for XYZ Airline (Total marks: 25)

XYZ Airline has a cargo plane with three compartments for storing cargo: front, center, and back. These compartments have the following capacity limits on weight and space:

compartment	Weight capacity(lbs)	Space capacity(cubic ft)
front	4000	1000
center	5000	1500
back	3000	800

Furthermore, the weight of the cargo in each compartment must be the same proportion of that compartment's weight capacity to maintain the balance of the airplane. The following four cargoes have been offered for shipment on an upcoming flight, with their respective weights, space requirements, and profits:

cargo	Weight(lbs)	Space(cubic ft)	Profit($)
A	3000	500	500
B	2000	700	400
C	4000	1000	800
D	1000	300	200

Any portion of these cargoes can be accepted. The objective is to determine how much (if any) of each cargo should be accepted and how to distribute each among the compartments to maximize the total profit for the flight.

a) Formulate this problem as a linear program. You must clearly define your decision variables, objective function, and label constraints.