$($Class $20)$ You are tasked with determining vehicle routes in order to minimize the total

travel time $($or costs$)$ across all routes. All vehicles leave from a central depot d and there are

$5$ points that must be visited. Each point has a demand level and each vehicle has a capacity

of $400$ meaning that it cannot deliver more than $400$ units of the product along its route. The

travel times between all entities are given in the figure below.

$\backslash$table$[[$d$,,1,2,3,4,5],[$d$,0,30,25,15,20,15],[1,30,0,10,20,20,25],[2,25,10,0,30,10,25],[3,15,20,30,0,20,5],[4,20,20,10,20,0,12\mathrm{.}5],[5,15,25,25,5,12\mathrm{.}5,0]]$

Point $1$ has a demand of $120,$ Point $2$ has a demand of $200,$ Point $3$ has a demand of $160,$ Point

$4$ has a demand of $80,$ and Point $5$ has a demand of $240.$ Apply the cost$-$savings algorithm to

determine the set of vehicle routes that should be implemented, while respecting the capacity

of the vehicles. For full credit, at each step, provide the set of current routes, the time taken

to travel each of the current routes, and the total time required to travel all current routes.

You can provide arguments for why you no longer need to consider merging a particular

route with another one in order to save you the effort of examining all pairs of points during

the implementation of the cost$-$savings algorithm.