Answer to Question #283934 in Operations Research for Sam

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=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\u22650"


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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