Variable in Objective Function that not compares in inequalities

I am new to linear programming and I have learned that the objective function has the form:

$$c^{T}x = c_{1}x_{1}+\cdots+c_{n}x_{n}.$$

And the equalities/inequalities that populate a Linear Programming Problem simply limit the domain of the objective function.

My question is, what happens to the problem if one or more of the variables that are in the equalities/inequalities are not in the objective function?

If the constraints literally have nothing to do with the objective function, then they should not be listed along with the problem as they have no effect. It's like asking "what's the best way to drive to work provided that Mars is red?" Mars being red literally has no effect on how you drive to work, so you don't even need to consider it.

That being said, you may need to see if there is an indirect relationship between the variables in your objective and the constraints.

Edit: For a quick, simple example consider the problem

\begin{align} &\min_x c^Tx\\ \text{s.t.}&\\ &b < 4 \end{align}

Here, $b$ has nothing to do with the problem and this condition can be ignored. If however you have something like

\begin{align} &\min_x c^Tx\\ \text{s.t.}&\\ &b < 4\\ &x_1 < b \end{align}

Then you would obviously need to make the necessary substitution for $b$.

• Can you show me some examples of problems with these indirect relationship and problems where is possible to simple cancel the unnecessary variables ? – Qwerto Feb 16 '18 at 13:41
• I edited my answer to add an example, does that help? – dt688 Feb 16 '18 at 17:56
• thanks so much :) – Qwerto Feb 17 '18 at 13:28