Answer to Question #283005 in Operations Research for Khan

Question

Q2. A school organized a book fair and in this book fair a book seller is selling his books under the following rules:

There are three different packages available.

First package contains 2 Islamic books, 2 Science books and 2 Geography books, second package contains 2 Islamic books, 4 Science books and 1 Geography books and third package contains 3 Islamic books, 4 Science books and 5 Geography books. The book fair has a total of 250 Islamic books, 300 Science books, and 270 Geography books. First package makes a profit of Rs. 120, second package makes Rs.100 and third package makes Rs.270 per pack.

How many packs should be made to maximize book fair profits?

What will the profit be?

Answer 1

profit:

$z=120x_1+100x_2+270x_3$

where x₁, x₂, x₃ are numbers of different packages.

for Islamic books:

$2x_1+2x_2+3x_3\le 250$

for Science books:

$2x_1+4x_2+4x_3\le 300$

for Geography books:

$2x_1+x_2+5x_3\le 270$

solution by Simplex method:

After introducing slack variables:

Max $Z=120x_1+100x_2+270x_3+0S_1+0S_2+0S_3$

subject to

$2x_1+2x_2+3x_3+S_1=250$

$2x_1+4x_2+4x_3+S_2=300$

$2x_1+x_2+5x_3+S_3=270$

$x_1,x_2,x_3,S_1,S_2,S_3≥0$

Negative minimum Z_j-C_j is -270 and its column index is 3. So, the entering variable is x₃.

Minimum ratio is 54 and its row index is 3. So, the leaving basis variable is S₃.

∴ The pivot element is 5.

Entering =x₃, Departing =S₃, Key Element =5

Negative minimum Z_j-C_j is -46 and its column index is 2. So, the entering variable is x₂.

Minimum ratio is 26.25 and its row index is 2. So, the leaving basis variable is S₂.

∴ The pivot element is 3.2.

Entering =x₂, Departing =S₂, Key Element =3.2

Negative minimum Z_j-C_j is -6.25 and its column index is 1. So, the entering variable is x₁.

Minimum ratio is 82 and its row index is 1. So, the leaving basis variable is S₁.

∴ The pivot element is 0.625.

Entering =x₁, Departing =S₁, Key Element =0.625

Since all Z_j-C_j≥0

Hence, optimal solution is arrived with value of variables as :

x₁=82, x₂=16, x₃=18

Max Z=16300