Set S, which is an non-empty finite subset of $\{ (i,j) : i, j \in N \land i \neq j \}$, is given. E.g. $S=\{(1,3), (2,3), (1,4), (2,4), (3,1), (3,4)\}$ . For each element $(i,j)$, we have weight $w_{ij}=c(i)>0$, where $c$ is some benefit function, and a binary decision variable $x_{ij}$. The optimization problem is defined as follows:
$$\text{maximize} \sum_{(i,j)\in S} w_{ij}x_{ij} $$ $$\text{s.t.} x_{ij}+x_{ik} \le 1 $$ $$x_{ij}+x_{jk} \le 1 $$
Note this problem is not the Maximum Weighted Matching problem as edges that share the same end point are allowed.