Answer to Question #176438 in Operations Research for Nomi

Question

The Cookie Monster Store at South Acres Mall makes three types of cookies—chocolate chip,

pecan chip, and pecan sandies. Three primary ingredients are chocolate chips, pecans, and sugar.

The store has 120 pounds of chocolate chips, 40 pounds of pecans, and 300 pounds of sugar. The

following linear programming model has been developed for determining the number of batches

of chocolate chip cookies pecan chip cookies and pecan sandies to make to maximize profit:

maximize Z = 10x1 + 12x2 + 7x3 (profit, $)

subject to:

x1 + 2x3 smaller than or equal to 40 (pecans, lb.)

10x1 + 5x2 smaller than or equal to 120 (chocolate chips, lb.)

20x1 + 15x2 + 10x3 smaller than or equal to 300(sugar, lb.)

x1, x2, x3 bigger than or equal to 0

Solve this model using the simplex method.

Answer 1

Standard form:

Max Z =  10x1 + 12x2 + 7x3 + 0s1 + 0s2 + 0s3

s.t.

20x1 + 15x2 + 10x3 + 1s1 + 0s2 + 0s3 = 300

10x1 + 5x2 + 0x3 + 0s1 + 1s2 + 0s3 = 120

1x1 + 0x2 + 2x3 + 0s1 + 0s2 + 1s3 = 40

x1, x2, x3, s1, s2, s3 $\geq$ 0

negative minimum Zj-Cj is -12 and its column index is 2. So, the entering variable is x2.

The minimum ratio is 20 and its row index is 1. So, the leaving basic variable is S1.

So, the pivot element is 15.

Entering =x2, Departing =S1, Key Element =15

Row operations: 

• R1(new)=R1(old) / 15
• R2(new)=R2(old) - 5 * R1(new)
• R3(new)=R3(old)

Since all Zj-Cj ≥ 0, the optimality condition is reached and the optimal solution is as follows:

x1 = 0

x2 = 20

x3 = 0

Max. Z = 240