Answer to Question #272108 in Operations Research for Prathibha Rose

Question #272108

Use big š‘€ method to solve Minimize š‘ = 6š‘„1 + 3š‘„2 + 4š‘„3 š‘†š‘¢š‘š‘—š‘’š‘š‘” š‘”š‘œ



š‘„1 ≄ 30; š‘„2 ≄ 50; š‘„3 ≄ 20; š‘„1 + š‘„2 + š‘„3 = 120

1
Expert's answer
2021-11-29T16:56:36-0500

Minimize "\ud835\udc4d = 6\ud835\udc65_1 + 3\ud835\udc65_2 + 4\ud835\udc65_3"

š‘†š‘¢š‘š‘—š‘’š‘š‘” š‘”š‘œ

"\ud835\udc651 \u2265 30"

"\ud835\udc652 \u2265 50"

"\ud835\udc653 \u2265 20"

"\ud835\udc65_1 + \ud835\udc65_2 + \ud835\udc65_3 = 120"


After introducing surplus,artificial variables

MaxĀ Z=6x1+3x2+4x3+0S1+0S2+0S3-MA1-MA2-MA3-MA4

subject to x1-S1+A1=30

x2-S2+A2=50

x3-S3+A3=20

x1+x2+x3+A4=120

andĀ x1,x2,x3,S1,S2,S3,A1,A2,A3,A4≄0



Negative minimumĀ Zj-CjĀ isĀ -2M-6Ā and its column index isĀ 1. So,Ā the entering variable isĀ x1.

Minimum ratio isĀ 30Ā and its row index isĀ 1. So,Ā the leaving basis variable isĀ A1.

∓ The pivot element is 1.

EnteringĀ =x1, DepartingĀ =A1, Key ElementĀ =1



Negative minimumĀ Zj-CjĀ isĀ -2M-4Ā and its column index isĀ 3. So,Ā the entering variable isĀ x3.

Minimum ratio isĀ 20Ā and its row index isĀ 3. So,Ā the leaving basis variable isĀ A3.

∓ The pivot element is 1.

EnteringĀ =x3, DepartingĀ =A3, Key ElementĀ =1



Negative minimumĀ Zj-CjĀ isĀ -2M-3Ā and its column index isĀ 2. So,Ā the entering variable isĀ x2.

Minimum ratio isĀ 50Ā and its row index isĀ 2. So,Ā the leaving basis variable isĀ A2.

∓ The pivot element is 1.

EnteringĀ =x2, DepartingĀ =A2, Key ElementĀ =1



Negative minimumĀ Zj-CjĀ isĀ -M-6Ā and its column index isĀ 4. So,Ā the entering variable isĀ S1.

Minimum ratio isĀ 20Ā and its row index isĀ 4. So,Ā the leaving basis variable isĀ A4.

∓ The pivot element is 1.

EnteringĀ =S1, DepartingĀ =A4, Key ElementĀ =1



Since allĀ Zj-Cj≄0

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

x1=50,x2=50,x3=20

MaxĀ Z=530


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