Answer to Question #238851 in Linear Algebra for lavanya

Question

Use simplex method to maximize 𝑓=3𝑥+5𝑦+4𝑧

subject to the conditions

2𝑥+3𝑦≤18

2𝑥+5𝑦≤10

3𝑥+2𝑦+4𝑧≤15

and 𝑥,𝑦,𝑧 ≥0.

Answer 1

Solution:

Assuming given variables x,y,z as x1, x2, x3 respectively.

Max Z=3x1+5x2+4x3

subject to

2x1+3x2≤18

2x1+5x2≤10

3x1+2x2+4x3≤15 and 

x1,x2,x3≥0;

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=3x1+5x2+4x3+0S1+0S2+0S3

subject to

2x1+3x2+S1=18

2x1+5x2+S2=10

3x1+2x2+4x3+S3=15 and 

x1,x2,x3,S1,S2,S3≥0

Since all Zj-Cj≥0

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

x1=0,x2=2,x3=2.75

Max Z=21