For your class fundraiser youve Redded to sell cookies You a

For your class fundraiser you\'ve Redded to sell cookies. You are going to make three kinds of cookies: plain, iced, and chocolate chip. One dozen of the plain cookies requires .6 pounds of cookie mix. One dozen of the iced cookies requires 1 pound of cookie mix and .4 pounds of icing mix. One dozen of the chocolate chip cookies requires .8 pounds of cookie mix and .3 pounds of chocolate chips. You have 120 pounds of cookie mix, 32 pounds of icing mix. and 25 pounds of chocolate chips available. You plan to sell the plain cookies for $6.00 a dozen and it costs $4.50 a dozen to make those cookies. The iced cookies sell for $7.00 a dozen and cost $5.00 a dozen to make. The chocolate chip cookies sell for $10.00 a dozen and cost $7.00 to make. How many of each type of cookie should you make in order to make as much money as possible, and what is your maximum profit? Define you variables. Write the inequalities describing all the constraints. Find the coordinates of the vertices of the feasible polyhedron. Give the optimal solution and the profit.

Solution

Sol)

Let X1,X2,X3 be the decision variables

1) x1= Cookie mix

x2=iced mix

x3=cholocate mix

2)

Entering =X3, Departing =S3, Key Element = 3

R3 (new) =R3 (old) ÷3=R3 (old) 13

R1 (new) =R1 (old) 8R3 (new)

R2(new) =R2 (old)

Entering =X2, Departing =S1, Key Element = 1

R1 (new) =R1(old)

R2(new) =R2 (old)

R3(new) =R3 (old)

Entering =S3, Departing =X3, Key Element = 13

R3 (new) =R3 (old) ÷13=R3 (old) 3

R1(new) =R1 (old) +83R3(new)

R2(new) =R2 (old)

Since all CjZj0,

Optimum Solution is arrived with value of variables as :
X1=0

X2=120

X3=0

Maximise Z=840

MAX Z= 6 X1+ 7 X2+ 10 X3

Subject to constraints

6 X1+ 1 X2+ 8 X3 120 4 X1+ 0 X2+ 0 X3 32 0 X1+ 0 X2+ 3 X3 25

and X1,X2,X30