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It is really hard for me to describe my optimization goal. If anyone finds a better title, please just replace it.

Here is an example of my optimization problem. We have 3 people: A,B,C, and we have 5 apples, 5 bananas, and 3 pears. The constraint is that each person should get at least 4 fruits. If we index people with i and fruits with j, and $x_{ij}$ denotes the number of $j$th fruit allocated for the $i$th person. The constraint should be $$\sum_{j=1}^3 x_{ij} >= 4, ~~i=1,2,3$$ Our goal is to try our best to allocate the same type of fruit to just one person. So the solution can be A: 4 apples, B: 4 bananas, C: 3 pears + 1 apple. It can also be: A: 4 bananas, B: 4 apples, C: 3 pears + 1 banana, so many alternative optimal solution in this case. Let's say the metric is $$\sum\limits_{i=1}^3 |N_i - 1|,$$ where $N_i$ stands for the number of types for $i$th person. But how to mathematically model $N_i$? Thanks in advance.

MIMIGA
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  • If the first person gets 1 type of item, the second gets 1 type of item, and the third gets 3 types of items, is it better or worse than if they get 1, 2, and 2 types respectively? –  Dec 07 '17 at 23:03
  • @Rahul Just updated the post and added a metric, basically we consider your two cases equivalent. – MIMIGA Dec 07 '17 at 23:50
  • $N_i = |{j:x_{ij}\ne0}| = \displaystyle\sum_{j=1}^3\begin{cases}1&\text{if $x_{ij}\ne0$,}\0&\text{otherwise}\end{cases}$. This is sometimes called the $L^0$ norm, $|\mathbf x_i|0$, if $\mathbf x_i=\begin{bmatrix}x{i1}\x_{i2}\x_{i3}\end{bmatrix}$. –  Dec 08 '17 at 07:09

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