This paper concerns with (p$_{1}$, p$_{2}$)-Plant Location problem in which entire set of candidate sites for facility locations is divided by two overlapping subsets, each of which with its own number of facilities to be established. We propose an algorithm which attempts to solve the Lagrangean dual of (p$_{1}$, p$_{2}$)-PLP by dividing sub-problem into twop-Plant Location Problems and solving them based on the convexity of the Lagrangean dual problem with respect to the number of facilities to be established. In doing so, Orthogonal Move procedure is proposed to provide easy-to-obtain lower bound to the Lagrangean Dual of (p$_{1}$, p$_{2}$)-PLP. Computational experience is reported.