The city councils of three rural towns have agreed to build jointly a solid waste transfer...

The city councils of three rural towns have agreed to build jointly a solid waste transfer station at a location between the towns that will minimize the sum of distances to the transfer station from the towns. Map coordinates, in miles, are (x, y) = (10, 30), (40, 10), and (60, 50) for cities 1, 2, 3, respectively. Recall that

Assume the location is to be determined using the method of gradient search. The search space is limited to points inside a rectangle encompassing all three cities, and a random starting point is selected as (x, y) = (20, 20). Assume a step one-hundredth the length of the smallest side of the encompassing rectangle. Determine the optimal direction to move at the first step, and the associated improvement in the objective function.