Question $2,$ Blending Problem:

Context: Your food business has become famous for its Chili with Beef. Due to different demands, you are now producing three styles of chili: original, catering, and restaurant, each with different requirements for beef, onion, and miscellaneous ingredients. Your challenge is to maximize your weekly profit while adhering to resource proportions, costs, and production minimums by deciding how many lbs of each resource to use in each style of chili.

Resource Specifications and Costs:

$-$ Beef Cost: $\backslash$$$7$ per IV

$-$ Onion Cost: $\backslash$$$1$ per Ib

$-$ Misc. Ingredients Cost: $\backslash$$$2$ per ld

$-$ Weekly Budget for Resources: $\backslash (\backslash$$ $5000\backslash )$

Chili Specifications:

Objective: Determine the optimal production quantity $($in pounds$)$ for each chili style to maximize total profit, considering the costs of resources and the total budget available.

Constraints:

$1.$ Resource proportions of beef and onion for each chili style must adhere to the specified ranges. Note that there is no specific proportion requirements on miscellaneous ingredients.

$2.$ The total cost of resources must not exceed the weekly budget of $\backslash (\backslash$$ $5000\backslash ).$

$3.$ Each chili style must meet its minimum production requirement.

$4.$ The total profit needs to be maximized within these constraints.

Questions:

a$)$ Formulate a linear programming model for this problem. $(15$ pts$)$

b$)$ Solve the model by using the computer and describe the optimal solution. $(35$ pts $)$