Hierarchical optimization: A satisfactory solution

Hierarchical optimization or multi-level programming techniques are extensions of Stackelberg games for solving decentralized planning problems with multiple decision makers in a hierarchical organization. They become more important for contemporary decentralized organizations where each unit or department seeks its own interests. Traditional approaches include vertex enumeration algorithms and approaches based on Kuhn-Tucker conditions or penalty functions. These are not only technically inefficient, but also lead to a paradox that the follower's decision power dominates the leader's. In this study, concepts of memberships of optimalities and degrees of decision powers are proposed to solve the above problems efficiently. In the proposed hierarchy decision process, the leader first sets memberships of optimalities of his/her possible objective values and decisions, as well as his/her decision power; and then asks the follower for his/her optima calculated in isolation under given constraints. The follower's decision with the corresponding levels of optimalities and decision powers are submitted to and modified by the leader with considerations of overall benefit for the organization and distribution of decision power until a best preferred solution is reached.