Problems:

A jeweler and her apprentice make silver pins and necklaces by hand. Each week they have $80$ hours of labor and $36$ ounces of silver available. It requires $8$ hours of labor and $2$ ounces of silver to make a pin and $10$ hours of labor and $6$ ounces of silver to make a necklace. Each pin also contains a small gem of some kind. The demand for pins is no more than six per week. A pin earns the jeweler $$400$ in profit, and a necklace earns $$100.$ The jeweler wants to know how many of each item to make each week to maximize profit.

a$.$ Formulate an integer programming model for this problem.