Modeling Integer LP. In the next semester, the school plans to replace the tutorials by self-organized student groups. Your task is to find the best possible partition of students into groups and to appoint a group leader for each group. To achieve this task, you are given a list of students students. Each student needs to be assigned to exactly one group, and each group needs to have a (student) leader. Every student can be a leader, but we can have only one per group.

Modeling Integer LP
In the next semester, the school plans to replace the tutorials by self-organized student groups. Your task is to find the best possible partition of students into groups and to appoint a group leader for each group. To achieve this task, you are given a list of students students. Each student needs to be assigned to exactly one group, and each group needs to have a (student) leader. Every student can be a leader, but we can have only one per group. A leader needs to be assigned to the group he is leading. The number of groups needs to be at least g and at most G. The quality of the groups depends on two criteria. First, the collaboration quality of a student i in a group with leader j is given by collaboration[i,j]. Furthermore, each student i has a certain leadership quality given by leadership[j]. You have to maximize the overall collaboration quality plus the sum of the leadership qualities of the leaders.
I have come up with the following and was wondering if I'm missing something or if someone can show me in the right direction:
I used the variable $x_{i,j}$ if a student i is assigned to the group with leader j ($x_{i,j}=1$) or not ($x_{i,j}=0$). The variable yj shall denote whether student j is a leader ($y_j$) or not ($=0$).
I assumed that there are n students and m leaders. Then I came up with the following LP:
max $\underset{i=1}{\overset{n}{\sum <!--\; \sum \; -->}}\underset{j=1}{\overset{m}{\sum <!--\; \sum \; -->}}{x}_{i,j}\cdot <!--\; \cdot \; -->{\text{collaboration}}_{i,j}+\underset{j=1}{\overset{m}{\sum <!--\; \sum \; -->}}{y}_j\cdot <!--\; \cdot \; -->{\text{leadership}}_j$s.t.
$\underset{j=1}{\overset{m}{\sum <!--\; \sum \; -->}}{x}_{i,j}=1\mathrm{\forall <!--\; \forall \; -->}i\in <!--\; \in \; -->\{1,\dots <!--\; \dots \; -->,n\}$
$g\le <!--\; \le \; -->\underset{j=1}{\overset{m}{\sum <!--\; \sum \; -->}}{y}_j\le <!--\; \le \; -->G$
$x_{i,j}\le <!--\; \le \; -->y_j\mathrm{\forall <!--\; \forall \; -->}i\in <!--\; \in \; -->\{1,\dots <!--\; \dots \; -->,n\}\mathrm{\forall <!--\; \forall \; -->}j\in <!--\; \in \; -->\{1,\dots <!--\; \dots \; -->,m\}$
$x_{i,j}\in <!--\; \in \; -->\{0,1\}\mathrm{\forall <!--\; \forall \; -->}i\in <!--\; \in \; -->\{1,\dots <!--\; \dots \; -->,n\}\mathrm{\forall <!--\; \forall \; -->}j\in <!--\; \in \; -->\{1,\dots <!--\; \dots \; -->,m\}$
$y_j\in <!--\; \in \; -->\{0,1\}\mathrm{\forall <!--\; \forall \; -->}j\in <!--\; \in \; -->\{1,\dots <!--\; \dots \; -->,m\}$
This is as far as I have gotten. I know I'm missing the connection between $x_{i,j}$ and $y_j$. Also I'm not sure about my function I want to maximize. Help would be appreciated.