The authors present a method to solve a large class of multilevel programming problems which are applicable to certain hierarchical decision-making systems. These problems are characterized by a set of ordered levels where the objective function and the set of feasible solutions are determined on several levels. [For a general definition of such problems, see the authors in IEEE Trans. Autom. Control AC-27, 211- 214 (1982; Zbl 0487.90005)]. The method is described by taking the two- level linear programming problem. Geometric characterizations and algorithms are presented with some examples. The authors also record computational experience on one problem.