Problems:
Linear and Integer Programming Model
1. Consider the following linear programming model:
Maximize Z=5 x1 + 4x2
Subject to
3x1+4x2≤10
X1, x2≥0 and integer
2. Demonstrate the graphical solution of this model.
3. A tailor makes wool tweed sport coats and wool slacks. He is able to get a shipment of 150 square yards of wool cloth from Scotland each month to make coats and slacks, and he has 200 hours of his own labor to make them each month. A coat requires 3 square yards of wool and 10 hours to make, and a pair of slacks requires 5 square yards of wool and 4 hours to make. The tailor earns $50 in profit from each coat he makes and $40 from each pair of slacks. He wants to know how many coats and pairs of slacks to produce to maximize profit.
a. Formulate and integer linear programming model for this problem.
b. Determine the integer solution to this problem by using the computer. Compare this solution with the solution without integer restrictions and indicate whether the rounded down solution would have been optimal.
4. 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.
b. Solve this model by using the computer. Compare this solution without integer restrictions and indicate whether the rounded-down solution would have been optimal.
5. A glassblower makes glass decanters and glass trays on a weekly basis. Each item requires 1 pound of glass, and the glassblower has 15 pounds of glass each week. A glass decanter requires 4 hours of labor, a glass tray requires 1 hour of labor, and the glassblower works 25 hours a week. The profit from the decanter is $50, and the profit from the tray is $10. The glassblower wants to determine the total number of decanters (x1) and trays (x2) that he needs to produce in order to maximize his profit.
a. Formulate and integer programming model for this problem.
b. Solve this model by using the computer.