First we will take an example of **Mixed Constraints** and after that we will take one example of **Unrestricted Constraints** in duality.

Maximize z = x_{1} + 5x_{2}

subject to

3x_{1} + 4x_{2} ≤
6

x_{1} + 3x_{2} ≥ 2

x_{1}, x_{2} ≥ 0

Solution.

The above problem can be written as

Maximize z = x_{1} + 5x_{2}

3x_{1} + 4x_{2} ≤ 6

-x_{1} - 3x_{2} ≤ -2

Now, the dual model of the problem can be formulated as follows:

Minimize z = 6w_{1} - 2w_{2}

subject to

3w_{1} - w_{2} ≥
1

4w_{1} - 3w_{2} ≥ 5

w_{1}, w_{2} ≥ 0

Maximize z = 2x_{1} + 3x_{2 }+ x_{3}

subject to

4x_{1}+ 3x_{2} + x_{3} = 6

x_{1} + 2x_{2} + 5x_{3} = 4

x_{1}, x_{2}, x_{3} ≥
0

Solution.

Converting equalities to inequalities

Any linear equality can be written as a set of like-directioned linear inequalities by imposing one additional constraint. The equation may be replaced by two weak inequalities. For instance, x = 10 can be written as x ≤ 10 and x ≥ 10, which in turn can be written as x ≤ 10 and -x ≤ -10.

So the constraints can be written as

4x_{1} + 3x_{2} + x_{3} ≥
6

4x_{1} + 3x_{2} + x_{3} ≤
6

x_{1} + 2x_{2} + 5x_{3} ≥
4

x_{1} + 2x_{2} + 5x_{3} ≤
4

Further, the above constraints can be written as

-4x_{1}- 3x_{2} - x_{3} ≤
-6

4x_{1} + 3x_{2} + x_{3} ≤
6

-x_{1} -2x_{2} - 5x_{3} ≤
-4

x_{1} + 2x_{2} + 5x_{3} ≤
4

Now, the *dual model* of the problem can be formulated as follows:

Minimize z = -6w_{1} + 6w_{2 }- 4w_{3 }+
4w_{4}

-4w_{1}+ 4w_{2} - w_{3} + w_{4} ≥
2

-3w_{1 }+ 3w_{2} - 2w_{3} + 2w_{4} ≥
3

-w_{1}+ w_{2} - 5w_{3} + 5w_{4} ≥
1

w_{1}, w_{2}, w_{3}, w_{4} ≥
0.

Simplifying the problem

Let y_{1} = w_{2} - w_{1} , y_{2} =
w_{4} - w_{3}

Minimize z = 6y_{1} + 4y_{2}

subject to

4y_{1} + y_{2} ≥ 2

3y_{1} + 2y_{2} ≥ 3

y_{1} + 5y_{2} ≥ 1

y_{1}, y_{2 } are unrestricted.

The above problem can be directly solved by using the following rules:

Primal | Dual |
---|---|

i^{th} relation an equality |
i^{th} variable unrestricted in sign |

j^{th} variable unrestricted in sign |
j^{th} relation an equality |

Minimize z = 6w_{1} + 4w_{2}

subject to

4w_{1} + w_{2} ≥ 2

3w_{1} + 2w_{2} ≥ 3

w_{1} + 5w_{2} ≥ 1

w_{1}, w_{2 } are unrestricted.