Question

Bryant's Pizza, Inc. is a producer of frozen pizza products. The company makes a profit of $1.00 for each regular pizza it produces and $1.50 for each deluxe pizza produced. Each 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 uses 1 pound of dough mix and 4 ounces of topping mix. Each 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 pizzas. How many regular and deluxe pizzas should the company make in order to maximize profits?

a)Write mathmatical formulation for linear programming model.

b) Show feasible region on graph

c) Solve the problem and write optimal solution

Regular pizza =

Deluxe pizza =

Profit =

d) If you can get 1lb of either dough or topping mix, which one will you choose and why?

Answer 1

To begin, we'll assign variables to represent the number of regular pizzas (x) and deluxe pizzas (y) produced. Now, we need to write a system of linear inequalities representing each constraint and a function representing the income for the pizza place.

First of all, we write the income function:

This is because the pizza place makes $1 for each regular pizza and $1.50 for each deluxe pizza.

Now, we need a system of inequalities to represent the constraints:

These represent the minimum number of each pizza that the pizza place expects to sell. Now we need inequalities to represent the production constraints caused by availability of materials.

The amount of dough used cannot exceed 150 pounds, and each pizza uses 1 pounds. Therefore,

Finally, the amount of toppings cannot exceed 50 pounds. Regular pizzas use 1/4 pound (4 ounces) of toppings and deluxe pizzas use 1/2 pound (8 ounces). Hence,

Next, we graph the system of inequalities, as shown below:

The optimal income will be found at one of the vertices of the polygonal region defined by the system of inequalities. The coordinates of the vertices are (50,25),(50,75),(100,50) and (125,25). We then evaluate each of these points in the profit function we wrote earlier and choose the one that gives the largest result.

f(50,25)=50+1.25(25)=87.5

f(50,75)=50+1.5(75)=162.5
f(100,50)=100+1.5(50)=175
f(125,25)=125+1.5(25)=162.5

So the maximum net income is $175 and will be attained if the pizza place produces 100 regular pizzas and 50 deluxe pizzas.

so,

Optimal solition is

Regular pizza=100

Deluxe pizza is 50

Profit=175

d)

The two inequalities are binding the solution is intersection of two inequalities

More of topping is bought because deluxe pizza requires more topping and profit is also more