I have $12$ unknown $a_i, b_i, c_i, i=1,\ldots,4$, that should satisfy equations
$$
\sum_{i=1}^4n_ia_i=a,\quad\sum_{i=1}^4n_ib_i=b,\quad\sum_{i=1}^4n_ic_i=c,
$$
where $n_i,\,i=1,\ldots,4$ and $a,b,c$ are positive and known constants.
Some other restrictions are $a_4=b_3=b_4=c_1=0$.
I am looking, if at all possible, for positive values of the unknown with the requirement that the four quantities $$ a_i+b_i+c_i,\quad i=1,\ldots,4 $$ have the same value or minimize as much as possible the differences. I have not a formal requirement for this, I am aware that this is somehow vague and can be defined in many different acceptable ways.
The problem arise from a distribution of work hours in three activities ($a, b, c$) of a project, that span over different and partly overlapping time periods ($i=1,\ldots,4$), and the tentative to optimize their distribution.
