In this project you will use linear programming to maximize profit. This will help you understand business problems, translate the business requirements into mathematical terms, and apply linear algebra to problem$-$solving.

Problem:

A northern hardware company is studying a plan to open a new distribution center in the southeast. The company plans to rent a warehouse and an adjacent office and distribute its main products to the local dealers. The company has decided to initially start with four of its main products: pressure washers, go karts, generators, and water pumps. You will help the company maximize profits by using linear programming to find the best solution.

The table below describes how much each of the products will cost the company $($including transportation costs$)$:

Item

Cost $($in Dollars$)$

Pressure Washer

$330$

Go$-$Kart

$370$

Generator

$410$

$($Case of $5$ Water Pumps$)$

$635$

The company has set aside a purchasing monthly budget of $$170,000$ for the new location. The selling prices $($per unit$)$ for each item are given in the table below:

Item

Selling Price $($in Dollars$)$

Pressure Washer

$499\mathrm{.}99$

Go$-$Kart

$729\mathrm{.}99$

Generator

$700\mathrm{.}99$

Water Pump

$269\mathrm{.}99$

Other than the budget, another of the company$$s concern is the available space in the warehouse. The warehouse has $82$ shelves, and each shelf is $30$ ft long and $5$ ft wide. Pressure washers and generators each are stored on $5$ ft by $5$ ft pallets whereas each Go Kart is stored on an $8$ ft by $5$ ft pallet. Furthermore, a $5$ ft by $5$ ft pallet is used to store four cases of water pumps

For promoting its brand products, the company$$s marketing department has decided to allocate at least $30\%$ of its inventory to pressure washers and Go Karts and sell at least twice as many generators as water pumps.

Perform a monthly analysis using a linear programming model to maximize the company's net profit.

Question $1$

Write the mathematical formulation of the problem.

That is$,$ write down the:

Objective function

Budget constraint

Space constraint

Any other constraints for the business requirements

Non$-$negativity constraints