Problem $1$: You are taking a trip to Busch Gardens and are a roller coaster enthusiast. There

are $8$ roller coasters at Busch Gardens: Pantheon $($P$),$ Griffon $($G$),$ Alpengeist $($A$),$ Tempesto $($T$),$

Lock Ness Mosnter $($L$),$ Apollo's Chariot $($C$),$ InvadR $($I$),$ and Verbolten $($V$).$ In order to expedite

pedestrian flow, they have implemented several one way paths which make the distances between

coasters non equidistant. Specifically, the table below gives the distance between each two coasters:

Thus, for example, if you are walking from Pantheon to Tempesto, you are traveling $1\mathrm{.}6$ miles, but

walking from Tempesto to Pantheon is a travel distance of $1\mathrm{.}3$ miles.

You want to ride each roller coaster while walking as little as possible.

a$.$ Formulate a concrete model whose solution will give you a tour that visits all $8$ roller coasters

exactly once. Hint: Do not write all the subtour elimination constraints, you can write one or

two then move on$.$ Also, if you're having trouble getting started draw out the network for the

problem.

b$.$ Convert your concrete model above to a parameterized model.

c$.$ Suppose after solving your model, the solver returns the following solution.

The following edges should be selected: $(P,C),(G,P),(A,I),(T,A),$

$(L,G),(C,L),(I,V),(V,T)$

$($a$)$ What are the values of your variables associated with this solution?

$($b$)$ What is the total distance traveled by this solution?

$($c$)$ Is this solution optimal for your TSP problem?

$($d$)$ If the solution is not optimal, write a constraint you could add to your model to remove

this solution from your feasible region.