Question

52) 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 each night for sandwich assembly. Vending machine capacity is available for 2,000 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 its customers buy as many ham and cheese sandwiches as the other two sandwiches combined, if not more so, but custom- ers need a variety of sandwiches available, so Metro stocks at least 200 of each. Metro manage- ment wants to know how many of each sandwich it should stock to maximize profit. Formulate a linear programming model for this problem. (USE EXCEL SOLVER) (SHOW WORK)

53. Solve the linear programming model formulated in Problem 52 for Metro Food Services Company by using the computer. (USE EXCEL SOLVER) (SHOW WORK)

a. If Metro Food Services could hire another worker and increase its available assembly time by 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?

b. What would the effect be on the optimal solution if the requirement that at least 200 sand- wiches of each kind be stocked was eliminated? Compare the profit between the optimal solution and this solution. Which solution would you recommend?

c. What would the effect be on the optimal solution if the profit for a ham and cheese sandwich was increased to $0.40? to $0.45?

Answer 1

Let the no. of ham and cheese, bologna and chicken
sandwich made and sold be : "x", "y", "2" respectively
Total profit = no. of each type of sandwiches
sold*profit per sandwich

= 0,.35x+0.0.42y+0.37z. We have to maximize this
fucntion, subject to the constraints:

1. Total available minutes for assembly time = 960
minutes.

Constraint function: 0.45x+0.41y+0.50z<=960

2. Capacity of vending machine is 2,000 sandwiches.
Constraint function: x+y+z<=2,000

3. Ham and cheese samdwich is as many as or greater
than boligna and chicken sandwiches.

Constraint function: x>=y+z

4. 200 of each type of sandwich is stocked.

constraint functions: x>=200, y>=200, z>=200

The above is modelled in the excel sheet and the table
is given below:

We will now use solver function in excel to solve the
above table, incorporating the constraints as mentioned earlier:

Thus, 1,000 of ham and cheese, 800 of bologna and
200 of chicken sandwiches will be made each day.

2a. profit by increasing assembly time: new available
time = 960+480 = 1440 hours. putting this new
constraint in solver, we get the following solution:
profit = 760, in case of 1440 hours available.

If capacity of vending machine is increased to become
= 2000+100 = 2100 sandwiches, then the solution will
be:

As the profits are higher in case of increasing vending
machine capacity, we should do this i.e. increase
vending machine capacity by 100 to make it to 2100. Additional profit = 798.5 - 760 = $38.5

b. If this constraint was removed than, the solution will
be:

Change in profit = 770-760 = $10. Thus profit will
increase by $10 when compared to optimal solution.
c. In case of change in profit of ham and cheese to
0.45, solution will be:

In case, profits were 0.40, optimal solution will be: