Example $7\mathrm{.}9.[$A Maximum$-$Flow Problem $($Past Exam Question$)]$ A parent has five children and five household chores $($i$.$e$.,$ tasks$)$ to assign to them. Past experience has shown that forcing chores on a child is counterproductive $($i$.$e$.,$ not productive$).$ With this in mind, the children are asked to list their preferences among the five chores, as the following table shows:

$\backslash$table$[[$Child$,1,2,3,4,5],[$Preferred chore$($s$),3,4,$ or $5,1,1$ or $2,1,2$ or $5,2]]$

The parent wants as many chores as possible to finish while abiding by $($i$.$e$.,$ satisfying$)$ the children's preferences. Draw the corresponding maximum flow network graph, showing and explaining the nodes, arcs, and any relevant parameters. Then, formulate this problem as a maximum flow network model.