Question $1(10$ points$)$

Commander Gordon wants to move the Batcave closer to downtown Gotham City to enhance

collaboration with Batman amidst rising crime rates. He consults Bruce, who decides to apply the

facility and planning principles he learned at Gotham State University.

Q$1-1.$ When Thomas Wayne served on the Gotham City planning committee, he oversaw a

comprehensive redesign of the downtown area. The streets were organized into a strict grid

pattern with right$-$angle intersections along with avenues and streets. In this case, which

distance norm is most appropriate? Pick one of them: $1-$norm, $2-$norm, and Infinite norm.

Q$1-2.$ Using the chosen norm in Q$1-1,$ formulate an optimization model to identify the optimal

location for the new Batcave, aiming to minimize the farthest distance to any of the three existing

police stations at xy coordinates $(2,6),(4,10),$ and $(7,3)$ in Gotham City, respectively.

Q$1-3.$ By introducing a variable $z,$ do you think we can reformulate the optimization problem

developed in Q$1-2,$ eliminating the max operator in the objective function? If yes, what constraints

should be added?

Q$1-4.$ One of the constraints in Q$1-3$ is $|x-2|+|y-6|2.$ Bruce remembered that he could

replace this constraint with a set of linear constraints. What are they?