Below is a maximal flow problem. Node $1$ is the origin, Node $5$ is the destination. Assume that there are capacity limits for each arc on the diagram $($they are just not shown$).$

Let Xij be the amount of traffic flow $($hundreds of cars per hour$)$ from node i to node j$.$

$1.$ Write down the objective function:

$2.$ Write down the following constraints:

a$.$ Constraint for Node $1.$

b$.$ Constraint for Node $3.$

c$.$ Constraint for Node $4.$