(15 points] A regional health authority would like to locate an ambulance station so that the service is as close as possible to all towns to be serviced by this ambulance. There are n towns. All the towns are connected by a major hichway. Hence the location of each town i = 1....,n can be defined by a single number d; giving the distance along the highway from the first town. (a) (b) [4 points] The location of the ambulance station x is to be optimised. The objective (penalty to be minimised) is the average square of the distance between the station and all towns. (This penalises having towns far from the station more strongly than nearby towns). Write down the objective function f(z) and derive an expression for the optimal x in terms of the d; fori =1.... . n. [6 points] Some people on the board of the health authority disagree with the objective function and would like to minimise simply the average distance of the station from all towns. (i) Write down the objective function g(x) that gives the average (non-squared) distance of the station at x to all of the towns i = 1,....n. (ii) explain why the minimum of g(x) can net be found by solving g'(x) = 0, (iii) provide a plot of g(z) if n = 9 and the distances are: d = [0, 6, 13, 22, 39, 50, 69, 90, 108] (iv) find the ming(x) (e.g. by inspection from the plot) and discuss how you can be sure there is no better solution outside the range of the plot, (v} compare this solution (minimum of g) with the optimal solution to min f(zx). [5 points] The region decided to purchase a helicopter to more directly service the towns. Now they need to locate a helipad to best service towns. Assume that the aim is to minimise the average squared distance between the helipad at location (r,y) (distance east-west and north-south) with town locations given by coordinates (u;,v;). Since towns have different population sizes (p;). the optimal helipad location is to be based on minimising the population average of squared distances. That is. the average squared distance per person, or sum of population size times squared distance all divided by the total population. (For this exercise we take the distance to each person to be the distance to the town centre). (i) Formulate the objective as a function of both r and y then show that this is really the sum of two independent funections ( f{x)+ g(y)). so that the problem can be solved by minimising f(x) and g(y) independently. ({ii) Obtain the derivative of these funetions to find the optimal solution .y in terms of the data (u;, v; and p; parameters). (iii) Show that the problem is convex using the second derivative of the two functions. Hint: To compute the distance between two location (z,y) and (a,b) we can use Pythagoras' theorem: distance = /(r a)? + (y b)2