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 $6$ hours of labor and $3$ ounces of silver to make a

pin and $10$ hours of labor and $5$ 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 $$200$ in

profit, and a necklace earns $$300.$ 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.

b$)$ Solve this model by using the Solver Tool in Microsoft Excel. Compare this solution with the

solution without integer restrictions and indicate the difference in the profit and units produced.