Answer to Question #237077 in Operations Research for opr

Question #237077

Use the simplex method to obtain the optimal solution of the following linear programming model

š‘€š‘Žš‘„š‘–š‘šš‘–š‘§š‘’ š‘ = 35š‘„1 + 50š‘„2

š‘ š‘¢š‘š‘—š‘’š‘š‘” š‘”š‘œ

3š‘„1 + š‘„2 ā‰¤ 30

š‘„1 + 2š‘„2 ā‰¤ 15

4š‘„1 + 4š‘„2 ā‰¤ 40 š‘„1,

š‘„2 ā‰„ 0


1
Expert's answer
2021-09-27T16:07:29-0400

Problem is

MaxĀ  "Z=35x_1+50x_2" subject to:

"3x_1+x_2\u226430"

"x_1+2x_2\u226415"

"4x_1+4x_2\u226440"

andĀ "x_1,x_2\u22650" ;


The problem is converted to canonical form by adding slack, surplus and artificial variables as appropiate


1. As the constraint-1 is of type 'ā‰¤' we should add slack variableĀ "S_1"


2. As the constraint-2 is of type 'ā‰¤' we should add slack variableĀ "S_2"


3. As the constraint-3 is of type 'ā‰¤' we should add slack variableĀ "S_3"


After introducing slack variables

"Max \\ Z=35x_1+50x_2+0S_1+0S_2+0S_3"

subject to:

"3x_1+x_2+S_1=30\\\\\nx_1+2x_2+S_2=15\\\\\n4x_1+4x_2+S_3=40\\\\"

andĀ "x_1,x_2,S_1,S_2,S_3\u22650"



Negative minimumĀ Zj-CjĀ isĀ -50Ā and its column index isĀ 2. So,Ā the entering variable isĀ "x_2" .


Minimum ratio isĀ 7.5Ā and its row index isĀ 2. So,Ā the leaving basis variable isĀ "S_2" .


āˆ“Ā The pivot element isĀ 2.


EnteringĀ ="x_2" , DepartingĀ ="S_2" , Key ElementĀ =2




Negative minimumĀ Zj-CjĀ isĀ -10Ā and its column index isĀ 1. So,Ā the entering variable isĀ "x_1" .


Minimum ratio isĀ 5Ā and its row index isĀ 3. So,Ā the leaving basis variable isĀ "S_3" .


āˆ“Ā The pivot element isĀ 2.

EnteringĀ ="x_1" , DepartingĀ ="S_3" , Key ElementĀ =2




Since allĀ Zj-Cjā‰„0


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

"x_1" =5,"x_2" =5


MaxĀ Z=425


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