Question

Ahmadi, Inc. manufactures laptop and desktop computers.  In the upcoming production period, Ahmadi needs to decide how many of each type of computers should be produced to maximize profit.  Each computer goes through two production processes.  Process I,involves assembling the circuit boards and process IIis the installation of the circuit boards into the casing.  Each laptop requires 24 minutes of process I time and 16 minutes of process II time. Each desktop requires 8 minutes of process I time and 32 minutes of process II time.  In the upcoming production period, 240 minutes are available in process I and 320 minutes in process II.  Each laptop costs $1,800 to produce and sells for $2,250.  Each desktop costs $600 to produce and sells for $1,000.

      Let your decision variables be:

                  X1 = Number of laptops to produce

                  X2 = Number of desktops to produce

1. Formulate an LP problem to maximize profit.  Write your problem formulation.
2. Graph the constraints and show the region of feasible solutions.
3. Determine the extreme points of the feasible region and find the profit at each extreme point.
4. Draw the isoprofit line and indicate the optimum point.
5. Are there any slacks at optimum?
6. If the selling price per desktop decreases to $700 per unit would there be any change in the optimum solution?  If yes, what would be the new optimum solution, and would there be any slack?
7. Assume the company does not want to produce more than 9 laptops in this production period.  Add this constraint to the originalproblem and show the region of feasible solutions.
8. Solve the problem as stated in part g and find the optimal solution.  Let the profit per Laptop to be $450 and per Desktop to be $100.
9. Let the selling price of Desktops to be $750 and solve the problem as stated in part g.

I just need help with part G, H, I.

Answer 1

Profit per Laptop = 2250 - 1800 = $ 450

Profit per Desktop = 1000 - 600 = $ 400

LP model and graphical representation is as follows:

g)

The additional constraint is: X1 <= 9

Adding this constraint to the original problem, the revised LP model and graph of feasible region is drawn as below

h)

The new profit per Laptop is given to be $ 450 and per Desktop to be $ 100

Solution of the problem for this new objective function is:

New optimal solution is:

Number of Laptops to be produced (X1) = 9

Number of Desktops to be produced (X2) = 3

New total profit = $ 4350

i)

Considering the selling price of Desktop to be $ 750, new profit per Desktop = 750-600 = $ 150

Solution of this new problem is following:

In the graph, we see that the objective function curve coincides with C1.

Therefore, there are multiple optimal solutions.

Two of the optimal solutions lie at the two extreme points as shown in the table, which are:

Solution 1: X1 = 9, X2 = 3

Solution 2: X1 = 8, X2 = 6

Total profit = $ 4500