Given an m by n binary matrix $P$ and a 'cost' vector $v_1$ I want to find the set of $P$ matrices where the product $Pv_1=v_2$ results in all the elements of $v_2$ being within a certain lower and upper bound (i.e. $\pm$ 20% of the sum of the elements of $v_1$/m).
This is for a fair allocation problem where 1000 items are being distributed into 20 groups (each item being allocated only once). The costs of the items are not normally distributed so the groups will necessarily be of different sizes to achieve a set of fair costs. I am currently using iterative methods but I was wondering if there was a neater way of approaching this.
Many thanks
Thanks to the edited tags on this question I now see it is a variation of the 'fair allocation problem', and there is a good resource here on the topic: http://recherche.noiraudes.net/resources/papers/comsoc-chapter12.pdf
Often it's just a case of finding out what your problem's name is :-)