Problem $1.(11$pts$)$ A beer distributor needs to plan how to make deliveries from its warehouse $($Node $1-$ origin$/$start$)$ to a supermarket $($Node $7-$destination$/$terminal node$),$ as shown in the network below.

All the roads allow for bi$-$directional traffic $($e$.$g$.,$ you can go from Node $3$ to Node $4$ and from Node $4$ to Node $3).$ The numbers above each arc $($road$)$ denote length of road in miles $($in either direction$).$

The distributor is looking to find the shortest route from the warehouse to the supermarket.

Let $x_{ij}=1$ if you take the road from Node i to Node $j,x_{ij}=0$ if not.

Write down the following constraints:

a$.$ Constraint for Node $1($origin$/$start node$).$

b$.$ Constraint for Node $7($destination$/$terminal node$).$

c$.$ Constraint for Node $4.$

d$.$ Constraint for Node $5.$