Answer to Question #178730 in Operations Research for Malama Mulenga

Question #178730

Use Simplex method to

Maximize Ζ = 4x + 10x2

Subject 2X+X2 ≤ 50

2X + 5X2 ≤ 100

2X + 3X2 ≤ 100

X, X2 ≥ 0


1
Expert's answer
2021-04-08T13:49:03-0400

"Max Z=4x_1+10x_2\\\\subject\\ to\\\\2x_1+x_2\u226450\\\\2x_1+5x_2\u2264100\\\\2x_1+3x_2\u2264100\\\\and\\ \\ x_1,x_2\u22650;"


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


1. As the constraint-1 is of type '≤' we should add slack variable S1


2. As the constraint-2 is of type '≤' we should add slack variable S2


3. As the constraint-3 is of type '≤' we should add slack variable S3


After introducing slack variables

"Max Z=4x_1+10x_2+0S_1+0S_2+0S_3\\\\subject\\ to\\\\2x_1+x_2+S_1=50\\\\2x_1+5x_2+S_2=100\\\\2x_1+3x_2+S_3=100\\\\and \\ x1,x2,S1,S2,S3\u22650"






Negative minimum"Z_j-C_j"  is -10 and its column index is 2. So, the entering variable is "x_2" .


Minimum ratio is 20 and its row index is 2. So, the leaving basis variable is "S_2" .


∴ The pivot element is 5.


Entering ="x_2" , Departing ="S_2" , Key Element =5




Since all "Z_j-C_j\u22650"


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

"x_1=0,x_2=20"

Max Z = 200

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