Question

Problem 1   (TAY 44-45/161 adjusted)                           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 0.90 minutes to assemble, a bologna sandwich requires a worker 0.80 minutes, and a chicken salad sandwich requires a worker 1.00 minutes to make. The company has 6 workers available each night to assemble sandwiches (i.e. 2,880 minutes). Vending machine capacity is available for 3,000 sandwiches each day. The profit for a ham and cheese sandwich is 70¢ (i.e. $0.70), the profit for a bologna sandwich is 85¢, and the profit for a chicken salad sandwich is 75¢. 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 300 of each sandwich type. Metro management wants to know how many of each sandwich it should stock to maximize profit.

Formulate this decision question as a Linear Programming Model. Define the variables, write down the constraints and the objective function in mathematical (algebraic) terms (variables, inequalities, etc.).

Find the optimal solution using SOLVER.

Answer as many parts below as possible without re-running Solver. (do each part independently from the others) Use the SENSITIVITY REPORT whenever possible.

If Metro Food Services could hire another worker and increase its available assembly time by 480 minutes, or increase its vending machine capacity by 200 sandwiches, which should it do? How much additional profit would your decision result in?

What would the effect be on the optimal solution if the profit for a ham and cheese sandwich were increased to 80¢?   to 90¢?  

Answer 1

Maximize: Z = 0.70*x + 0.85*y + 0.75*z

Subject to:

0.90*x + 0.80*y + 1.0*z <= 2880 (6 workers available each night to assemble sandwiches)

x + y + z = 3000 ( Vending machine capacity)

-x + y + z <= 0 ( y+z< = x, customers buy as many or more of the ham and cheese(x) sandwiches than the other two sandwiches combined (y+z) )

x >= 300 , y >= 300, z >= 300 ( Minimum no. of sandwiches)

x, y, z >= 0 (non- negative constraint)

Solver:

Optimal solution:

No. of ham and cheese sandwiches, x= 1500

No. of ham and bologna sandwiches, y= 1200

No. of chicken sandwiches, z= 300

Maximum profit = $ 2295

If Metro Food Services could hire another worker and increase its available assembly time by 480 minutes, or increase its vending machine capacity by 200 sandwiches, :then it should increase the capacity of vending machine.

Effect on the optimal solution if the profit for a ham and cheese sandwich were increased to $0.80

Effect on the optimal solution if the profit for a ham and cheese sandwich were increased to $0.90