Question

If someone could help me with how to solve this that would be amazing.

Linear Programming is a set of linear equations that help to create a "feasible" region.

This is commonly used in manufacturing and mass production of products.

Let's say you are starting your own cookie business and you have $1,000 to spend.

What ingredients will you need?
How much of each will you buy to start with?
Is there a maximum amount that you can purchase?

Answer 1

Fist, what ingredients will we need? Suppose we have n ingredients. Let the quantity of each of the n ingredients be specified by \[x_1, x_2, ..., x_n\].

Now, what are the constraints on these n ingredients? These constraints will be m linear inequalities, where constraint i is: \[f_i(x_1, ..., x_n) <= b_i\]

These m constraints will define a feasible region--a polyhedron as defined by the intersection of the m half-spaces. We can add as many constraints as needed. To give two examples, consider the following:

We have $1000 to spend. What is the cost of each ingredient? Let the cost of ingredient i be \[c_i\]. Then, the total cost constraint is:

\[c_{1}x_1 + c_{2}x_2 + ... c_{n}x_n <= 1000\]

Another constraint arises from the observation that negative quantities of any ingredient doesn't make sense. So:

\[x_1 >= 0, x_2 >= 0, ..., x_n >= 0\]