Question: A jeweler and her apprentice make silver pins and necklaces by hand. Each week they have 80 ours 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 pin is no more than 6 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.

(b). Solve this problem by using the computer. Compare this solution to the solution without integer restrictions and indicate whether the rounded-down solution would have been optimal.