linear programming fair share between 2

Question

I am trying to formulate a linear programming problem.

Say that a guy dies and you have to split what this person owns between his two sons in a fair way.

The person owned 14 objects, each with a certain value, in total worth 183000$.

The question is "What is the partition that minimizes the difference between the values of the two parts? Formulate a LP that solves this problem."

My problem is that I don't really know where to start. Wondered if someone could give a hint

Answer 1

Let's define $28$ variables, $a_i$ and $b_i$ for $1 \leq i \leq 14$. If $a_i = 1$, the first person gets item $i$, if $b_i = 1$, the second one gets it.

To turn it into a integer linear programming problem, we need a few steps:

First we have to make sure that for every $i$, the $i$-th item is received by exactly one of the persons:

For all $i$, add the conditions $a_i, b_i \geq 0$ and $a_i + b_i = 1$.

Now let $v_i$ be the value of item $i$. Then the value the first person gets is $$\sum_{i=1}^{14} a_i v_i$$ and...

I think you can continue from here ;)