Use simplex method to maximize ๐=3๐ฅ+5๐ฆ+4๐ง
subject to the conditions
2๐ฅ+3๐ฆโค18
2๐ฅ+5๐ฆโค10
3๐ฅ+2๐ฆ+4๐งโค15
and ๐ฅ,๐ฆ,๐ง โฅ0.
Solution:
Assuming given variables x,y,z as x1, x2, x3 respectively.
Max Z=3x1+5x2+4x3
subject to
2x1+3x2โค18
2x1+5x2โค10
3x1+2x2+4x3โค15 and
x1,x2,x3โฅ0;
The problem is converted to canonical form by adding slack, surplus, and artificial variables as appropriate
1. As the constraint-1 is of type 'โค' we should add slack variable S1
2. As the constraint-2 is of type 'โค' we should add slack variable S2
3. As the constraint-3 is of type 'โค' 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,S3โฅ0
Since all Zj-Cjโฅ0
Hence, optimal solution is arrived with value of variables as :
x1=0,x2=2,x3=2.75
Max Z=21
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