Problem $1.$ In this question, we will explore an alternative formulation for the CVRP that we have seen in class.

$($a$)$ Suppose you have a mechanism of enumerating all possible feasible routes for a CVRP $($a feasible route is a sequence of nodes that starts and ends with the depot, visits a number of customers, and respects vehicle capacities$).$ Let's call this set $R.$ Define a binary variable $x_r$ which is equal to $1$ if we choose route rinR, and $0$ otherwise. Using only this variable, provide an equivalent formulation to the CVRP that was given in class. You are free to define any parameters you want, but make sure to clearly and completely define them $($what they represent, how to compute them, etc.$).$ Briefly explain the objective and constraints and dsicuss why this is equivalent to the CVRP$.$

$($b$)$ How would you incorporate the constraint that you have a maximum of $N$ vehicles into the formulation you developed in part $($a$)?$

$($c$)$ What is a huge disadvantage of the model $($and the particular definition of variable given to you in $(a)$

$($d$)$ Can we use the previous model developed in part $($a$)$ if you also have a problem that involves time windows? If so$,$ explain what you would need to modify. If not, explain why not.