## A Linear Programming Formulation. You run a stand a your local farmers’ market. You sell two items: milk and butter.

### mathematics

##### Description

1 Formulations

1-1. [10 marks] A Linear Programming Formulation. You run a stand a your local farmers’ market. You sell two items: milk and butter. Each month, you can purchase milk from a local farm for a given cost, and you must meet your customers’ demands for milk and butter. These prices and demands for the following 4 months are summarized in the following table:

At the end of each month, you churn any leftover milk (which would not stay fresh until next month) into butter, which you can store in a freezer indefinitely—assume that one litre of milk yields χ (m→b) = 0.03125 litres of butter. Since you do not own a freezer, you must rent freezer space to store your butter at a cost of ψ (b) = \$0.02 per litre. You start at the beginning of month 1 with 6 litres of butter in storage, and you should end at the end of month 4 (when the farmers’ market shuts down for the winter) with no milk and no butter left. Your goal is to meet your customers’ needs as cheaply as possible. Formulate this problem as a linear program.

[10 marks] A Mixed-Integer Linear Programming Formulation. You are trying to choose your courses for next semester. There are a total of twelve courses you are interested in; for simplicity, they are simply numbered 1 – 12. Each course occurs in a particular time slot, has a particular credit value, and is from a particular subject, as summarized in the following table:

Your goal is to earn as many credits as possible next semester. Of course, you cannot take two courses whose time slots overlap, and you cannot take more than 5 courses next semester. Finally, in order to satisfy your depth requirements, you must take at least 3 MATH classes or at least 2 ENGL classes. Formulate this problem as a mixed-integer linear program (MILP).