How to basically solve Integer programming problems?

Question

I learnt to solve the task below Linear Programmming is employed. I have tried studying texts to better understand what methods to employ out of the following:

$(A) Gomory's-cut$

$(B) Mixed-Gomory's-cut $

$(C) Branch-and-Bound $

But I am finding it difficult to comprehend any of the three and all the example I see are some how more complex and non related.Is there a way to simplify the problem below given the constraint is just $a,b,c,d,e,f\in \{4,2\} $

\begin{align} a+b+c+d+e+f &= 18, \tag{1} \\ b+d &= 4, \tag{2} \\ e+f &= 8, \tag{3} \end{align}

find $a,b,c,d,e,f$ ?

Answer 1

If all the variable takes value from $\{2,4\}$.

The constraints reduces to

$$a+c=6$$

$$b+d=4 \implies b=d=2$$

$$e+f=8 \implies e=f=4$$

Hence $(a,c) = (4,2)$ or $(a,c)=(2,4)$.

Answer 2

In your case, there is no need to advanced methods, unless you want to practice how to use them. If you only want to solve this specific problem you can do this:

$a+b+c+d+e+f=18$

is

$a+(b+d)+c+(e+f) =18$

so

$a+4+c+8=18$

so

$a+c=6$

since $a,b,c,d,e,f\in \{4,2\} $

you can easily determine integer values for $a,c$ from the last equation.