Develop the linear programming model, you do not need to solve the problem. Then, discuss the pros and cons of using this mathematical model in making a managerial decision. Analyze the production problem at Better Fitness, Inc., and prepare a report for BFI$$s president presenting your findings and recommendations. Include $($but do not limit your discussion to$)$ a consideration of the following items:

What is the recommended number of BodyPlus $100$ and BodyPlus $200$ machines to produce?

How does the requirement that the number of units of the BodyPlus $200$ produced be at least $25\%$ of the total production affect profits?

Where should efforts be expended in order to increase profits?

Include a copy of your linear programming model and graphical solution in an appendix to your report.

$$Better Fitness, Inc. $($BFI$),$ manufactures exercise equipment at its plant in Freeport, Long Island. It recently designed two universal weight machines for the home exercise market. Both machines use BFI$-$patented technology that provides the user with an extremely wide range of motion capability for each type of exercise performed. Until now, such capabilities have been available only on expensive weight machines used primarily by physical therapists.

At a recent trade show, demonstrations of the machines resulted in significant dealer interest. In fact, the number of orders that BFI received at the trade show far exceeded its manufacturing capabilities for the current production period. As a result, management decided to begin production of the two machines. The two machines, which BFI named the BodyPlus $100$ and the BodyPlus $200,$ require different amounts of resources to produce.

The BodyPlus $100$ consists of a frame unit, a press station, and a pec$-$dec station. Each frame produced uses $4$ hours of machining and welding time and $2$ hours of painting and finishing time. Each press station requires $2$ hours of machining and welding time and $1$ hour of painting and finishing time, and each pec$-$dec station uses $2$ hours of machining and welding time and $2$ hours of painting and finishing time. In addition, $2$ hours are spent assembling, testing, and packaging each BodyPlus $100.$ The raw material costs are $$450$ for each frame, $$300$ for each press station, and $$250$ for each pec$-$dec station; packaging costs are estimated to be $$50$ per unit.

The BodyPlus $200$ consists of a frame unit, a press station, a pec$-$dec station, and a leg$-$press station. Each frame produced uses $5$ hours of machining and welding time and $4$ hours of painting and finishing time. Each press station requires $3$ hours of machining and welding time and $2$ hours of painting and finishing time, each pec$-$dec station uses $2$ hours of machining and welding time and $2$ hours of painting and finishing time, and each leg$-$press station requires $2$ hours of machining and welding time and $2$ hours of painting and finishing time. In addition, $2$ hours are spent assembling, testing, and packaging each BodyPlus $200.$ The raw material costs are $$650$ for each frame, $$400$ for each press station, $$250$ for each pec$-$dec station, and $$200$ for each leg$-$press station; packaging costs are estimated to be $$75$ per unit.

For the next production period, management estimates that $600$ hours of machining and welding time, $450$ hours of painting and finishing time, and $140$ hours of assembly, testing, and packaging time will be available. Current labor costs are $$20$ per hour for machining and welding time, $$15$ per hour for painting and finishing time, and $$12$ per hour for assembly, testing, and packaging time. The market in which the two machines must compete suggests a retail price of $$2400$ for the BodyPlus $100$ and $$3500$ for the BodyPlus $200,$ although some flexibility may be available to BFI because of the unique capabilities of the new machines. Authorized BFI dealers can purchase machines for $70\%$ of the suggested retail price.

BFI$$s president believes that the unique capabilities of the BodyPlus $200$ can help position BFI as one of the leaders in high$-$end exercise equipment. Consequently, he has stated that the number of units of the BodyPlus $200$ produced must be at least $25\%$ of the total production$.$