Question

Bryant's Pizza, Inc. is a producer of frozen pizza products. The company makes a profit of $1.00 for cach regular pizza it produces and $1.50 for deluxe pizza producedEach pizza includes a combination of dough mix and topping mix Currently the firm has 150 pounds of dough mix and 50 pounds of topping mix Each regular pizza usesI pound of dough mix and 4 ounces of topping mixEnch deluxe pizza uses 1 pound of dough mix and 8 ounces of topping mix Based on past demand Bryant can sell at least 50 regular pizzas and at least 25 deluxe plzzas, How many regular and deluxe pizzas should the company make in order to maximize profits?

a) Write mathematical formulation for linear programming model.

b) Show the feasible region on the graph.

c) Solve the problem and write optimal solution

Regular Pizza =

Deluxe Pizza =

Profit =

d) which constraints are binding?

e) if you can get 1 lb. of either dough or topping mix, which one will you chose and why?

Answer 1

(A)

• Let the Profit be P.
• No. of Regular Pizza produced be X and no. of Delux Pizza produced to be Y.
• Given, the profit from Regular and Delux Pizza is $1.00 and $1.50 respectively.
• Therefore, the total profit:
• So, we have to maximize P in order to get the solution.
• Constraints:

1. Both the Pizza requires 1 lb of dough mix and the total available dough mix is 150 lb, i.e.,
2. A Regular Pizza requires 4 ounces(0.25 lb) of topping mix and a Delux Pizza requires 8 ounces(0.5 lb) of topping and the total available topping mix is 50 lb, i.e.,
3. At least 50 Regular and 25 Delux Pizza are sold. i.e., and

(B)

• To obtain a feasible region, the region satisfying all the constraints is found.
• 

(C)

• To obtain the optimal solution, all the corners of the feasible region are found out by solving the pair of lines forming the vertices.
• The vertices are (50,75), (100,50), (125,25), (50,25).
• 
• Substituting the values in the equation to be maximized:

• X	Y	P = (1)X + (1.5)Y
  50	75	162.5
  100	50	175
  125	25	162.5
  50	25	87.5

• So, the Maximum Profit = $175 which is obtained when they produce 100 Regular and 50 Delux Pizza.

(D)

• A constraint is binding if the LHS and RHS are equal at the optimal solution.

1. At optimal solution: 100 + 50 = 150. Therefore, binding.
2. At the optimal solution: 25 + 25 = 50. Therefore, binding.
3. At the optimal solution: 100>50. Therefore, non-binding.
4. At the optimal solution: 50>25. Therefore, non-binding.

(E)

• When we consider 1 lb extra dough mix:
• The constraint related to dough mix becomes:
• Corresponding graph and solution:
• 

• X	Y	P = (1)X + (1.5)Y
  50	75	162.5
  102	49	175.5
  126	25	163.5
  50	25	87.5

• When we consider 1 lb extra topping mix:
• The constraint related to dough mix becomes:
• Corresponding graph and solution:
• 

• X	Y	P = (1)X + (1.5)Y
  50	77	165.5
  96	54	177
  125	25	162.5
  50	25	87.5

• Comparing both the cases, we get a profit of $175.5 for 1 lb extra of dough mix while $177 profit for 1 lb extra of topping mix. Hence, a 1 lb extra topping mix is desirable.