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# Quantitative methods: Linear programming

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Consider the following integer linear programming problem.
Max Z = 3x1 + 2x2
Subject to: 3x1 + 5x2 <= 30
4x1 + 2x2 <= 28
x1 <= 8
x1, x2 >= 0 and integer
The solution to the linear programming relaxation is: x1 = 5.714, x2= 2.571.
What is the optimal solution to the integer linear programming problem? State the value of the objective function Z.

Max Z = 5x1 + 6x2
Subject to: 17x1 + 8x2 <= 136
3x1 + 4x2 <= 36
x1, x2 >= 0 and integer
What is the optimal solution?

3. The Metro Food Services Company delivers fresh sandwiches each morning to vending machines throughout the city. The company makes three kinds of sandwiches- ham and cheese, bologna, and chicken salad. A ham and cheese sandwich requires a worker 0.45 minutes to assemble, a bologna sandwich requires 0.41 minutes, and a chicken salad sandwich requires 0.50 minutes to make. The company has 960 available minutes (16 hours) each night for sandwich assembly. Vending machine capacity is available for 2000 sandwiches each day. The profit for a ham and cheese sandwich is \$0.35, the profit for a bologna sandwich is \$0.42, and the profit for a chicken salad sandwich is \$0.37. The company knows from past sales records that their customers buy as many or more of the ham and cheese sandwiches than the other two sandwiches combined, but customers need a variety of sandwiches available, so Metro stocks at least 200 of each. Metro Management wants to know how many of each sandwich it should stock to maximize profit.

Formulate a linear programming model for this problem. Clearly state your decision variables, objective function, and the constraints.
Solve the linear programming problem you formulated in part (a) using the computer (EXCEL or QM WINDOWS).
If Metro Food Services could hire another worker and increase its available assembly time by 8 hours (480 minutes), or increase its vending machine capacity by 100 sandwiches, which should it do? Why? How much additional profit would your decision result in?
What would the effect be on the optimal solution if the requirement that at least 200 sandwiches of each kind be stocked were eliminated? Compare the profit between the optimal solution and this solution, and indicate which solution you would recommend.
What would the effect be on the optimal solution if the profit for a ham and cheese sandwich were increased to \$0.40?
What would the effect be on the optimal solution if the profit for a ham and cheese sandwich were increased to \$0.45?

4. Mary Smith makes pottery by hand in her basement. She has 20 hours available each week to make bowls and vases. A bowl requires 3 hours of labor, and a vase requires 2 hours of labor. It requires 2 pounds of special clay to make a bowl and 5 pounds to produce a vase; she is able to acquire 35 pounds of clay per week. Mary sells her bowls for \$50 each and her vases for \$40 each. She wants to know how many of each item to make each week to maximize her revenue.

Formulate an integer programming model for this problem.
Solve this model using the computer.
Compare the solution in part (b) with the solution without integer restrictions and indicate if the rounded-down solution would have been optimal.

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