Question #238851

Use simplex method to maximize ๐‘“=3๐‘ฅ+5๐‘ฆ+4๐‘ง

subject to the conditions

2๐‘ฅ+3๐‘ฆโ‰ค18

2๐‘ฅ+5๐‘ฆโ‰ค10

3๐‘ฅ+2๐‘ฆ+4๐‘งโ‰ค15

and ๐‘ฅ,๐‘ฆ,๐‘ง โ‰ฅ0.


1
Expert's answer
2021-09-23T17:12:52-0400

Solution:

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

Max Z=3x1+5x2+4x3

subject to

2x1+3x2โ‰ค18

2x1+5x2โ‰ค10

3x1+2x2+4x3โ‰ค15 and 

x1,x2,x3โ‰ฅ0;

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


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


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


3. As the constraint-3 is of type 'โ‰ค' 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,S3โ‰ฅ0




Since all Zj-Cjโ‰ฅ0

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

x1=0,x2=2,x3=2.75

Max Z=21


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