Answer to Question #235888 in Operations Research 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

Similarly, going further, we get the third iteration:

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