Geometric and algorithmic developments for a hierarchical planning problem

This paper presents a new model for multiobjective planning in hierarchical systems that explicitly takes into consideration the order in which decisions are made. Interactions and conflicts that normally exist among the levels are introduced by specifying jointly controlled feasible regions and interdependent objective functions. At each level in the system, planners attempt to maximize net benefits in light of all higher-level decisions, and thus may influence but not control the behavior of others. The resultant formulation leads to the multilevel programming problem. The geometry of an all linear case is first examined wherein it is shown that the optimal solution must lie at a vertex of the original polyhedral constraint region. Next, a set of first order optimality conditions is derived for the general case and used as the basis of an algorithm for the linear problem. A number of examples are given to highlight the results.