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 in order to maximize profit. a. (2 pts) Formulate an integer programming model for this problem. b. (3 pts) Solve this model using the branch and bound method. Compare this solution with the solution without integer restrictions and indicate if the rounded-down solution would have been optimal.

Jeweler should make 6 pins and 3 necklaces to maximize the profit.

Explanation:

Demand for pins is 6 per week and profit is $400

Demand for necklaces is not known and profit is $100

Let x = pins and y = necklaces

The profit maximizing quantity equation will be :

Q = 400x + 100y

Constraints are labor hours and silver availability.

Labor hours available  = 80 hours

Silver availability = 36 ounces

The constraint equation will be :

For labor hours : 8x + 10y ≤ 80

For silver : 2x + 6y ≤ 36

Solving the equation we will get maximum profit which is $2,700 and the quantity it should make is 6 pins and 3 necklaces to maximize the profit.

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