Three different classes of multiple points location-allocation problems in the Euclidean plane are considered under a discrete optimization criterion which minimizes the maximum cost based on certain interpoint distances. Each of these classes of geometric optimization problems is studied with three different distance metrics (Euclidean, Rectilinear, Infinity) as well as for feasible solution sets in the plane which are both discrete and infinite. All of these problems are shown to be polynomial-time reducible to each other and furthermore D^{p} complete.