Question #283934

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?



1
Expert's answer
2022-01-10T17:11:10-0500

profit:

z=120x1+100x2+270x3z=120x_1+100x_2+270x_3

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

for Islamic books:

2x1+2x2+3x32502x_1+2x_2+3x_3\le 250

for Science books:

2x1+4x2+4x33002x_1+4x_2+4x_3\le 300

for Geography books:

2x1+x2+5x32702x_1+x_2+5x_3\le 270


solution by Simplex method:


After introducing slack variables:

Max Z=120x1+100x2+270x3+0S1+0S2+0S3Z=120x_1+100x_2+270x_3+0S_1+0S_2+0S_3

subject to

2x1+2x2+3x3+S1=2502x_1+2x_2+3x_3+S_1=250

2x1+4x2+4x3+S2=3002x_1+4x_2+4x_3+S_2=300

2x1+x2+5x3+S3=2702x_1+x_2+5x_3+S_3=270

x1,x2,x3,S1,S2,S30x_1,x_2,x_3,S_1,S_2,S_3≥0


Negative minimum Zj-Cj 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 Zj-Cj 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 Zj-Cj 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 Zj-Cj≥0

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

x1=82, x2=16, x3=18

Max Z=16300


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