Question

Find the Max and Mini values of the objective function Z=3x+4y on the region bounded by

2x+y>5

x+5y>16

2x+y<14

-x+4y<20

Needs to be solved with the simplex method

Step 1

First find the minimum value: so, the problem is,

**Min Z = 3 x _{1} + 4 x_{2}**

** **

Subject to

**2x _{1 }+ x_{2 }> 5**

**x _{1 }+ 5x_{2} > 16**

**2x _{1 }+ x_{2 }< 14**

**-x _{1 }+ 4x_{2 }< 20**

** **

And **x _{1}, x_{2} **

Now, **Max Z = -3 x _{1} – 4 x_{2}**

_{ }

The problem is converted to canonical form by adding slack, surplus and artificial variable:

**Max Z = -3 x _{1} - 4 x_{2 }+ 0 S_{1} + 0 S_{2} + 0 S_{3} +0 S_{4} -MA_{1}-MA_{2}**

** **

Subject to

**2x _{1 }+ x_{2 }- S_{1 } + A_{1 }= 5**

**x _{1 }+ 5x_{2} - S_{2 } + A_{2 }=16**

**2x _{1 }+ x_{2 + }S_{3 }=14**

**-x _{1 }+ 4x_{2 } + S_{4 }=20**

** **

And **x _{1}, x_{2}, S_{1,} S_{2,} S_{3,} S_{4,} A_{1,} A_{2 }**

** **

** **

**Iteration 1 **

**.**

Step 2

Pivot element is 5 and leaving basis variable is A_{2 }and entering variable is x_{2}.

Iteration table 2:

Step 3

Pivot element is** 1.8, entering = x1** and leaving basis is **A1**

** **

Iteration ...

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