Figure $1.$ Delivery Routes

$1.$ Consider the network diagram given in Figure $1.$ Assume that the amount on each branch is the distance in miles between the respective nodes. What is the shortest route from the source node $($node $1)$ to nodes $2,3,$ and $4?$ Indicate the total distance for each route.

$2.$ Consider the network diagram given in Figure $1.$ Assume that the amount on each branch is the distance in miles between the respective nodes. What is the shortest route from the source node $($node $1)$ to nodes $5$ and $6?$Indicate the total distance for each route.

$3.$ Consider the network diagram given in Figure $1.$ Assume that the amount on each branch is the distance in miles between the respective nodes. Also assume that it is not possible to travel from a node with a higher number to a node with a lower number. Write the constraint associated with the second node $($node $2)$ for the $\backslash (0-1\backslash )$ integer linear programming formulation of the shortest route problem.

$4.$ Consider the network diagram given in Figure $1.$ Assume that the amount on each branch is the distance in miles between the respective nodes. Also assume that it is not possible to travel from a node with a higher number to a node with a lower number. Write the constraint associated with the fifth node $($node $5)$ for the $0-1$ integer linear programming formulation of the shortest route problem.

$5.$ Consider the network diagram given in Figure $1.$ Assume that the numbers on the branches indicate the length of cable $($in miles$)$ between each pair of the six nodes on a telecommunication network. What is the minimum number of miles of cable to be used to connect all six nodes?