Todor STOILOVProfessor Académie des Sciences Bulgares
The hierarchy is a term, which has a relative meaning. The hierarchy has predefined
domains of application as [Mesarovich and all., 1973] :
the levels of description of the system (strata) ;
the allocation of rights and obligations between the executive elements (echelon) ;
the allocation of problems, functions, tasks among the hierarchical levels (layers). The strata hierarchy is applied for the description and modeling of complex systems, for which it is not possible to define only a unique model for their operation.
The echelons represent the hierarchical levels, which contain the executive tools and elements in the complex system. The layers hierarchy concerns the allocation of tasks and control problems in the complex systems. This type of hierarchy is the main domain for formalization of decision-making and control processes. Particularly the layer hierarchy is formalized as hierarchically ordered
optimization problems. This hierarchy contains a sequence of sub-problems, which have to be solved independently. The solutions of each sub-problem influence the lower level problem by changing their constraints or local goals functions.
The hierarchical (layer hierarchy) decision/control processes received an increasing
interest in mathematical programming within the area of multilevel programming. In this hierarchical approach each decision-maker controls some variables. The higher-level decision-maker (the coordinator) optimizes the objective of the higher level of the hierarchical system under the constraints that the lower level managers (subsystems) define with their control variables, calculated after the optimization of local objectives and constrains. Sequentially, the coordinator solves its optimization problem and the corresponding optimal solutions are used to define the values of a set of parameters in the lower level optimization sub-problems. Then the subsystems solve their local optimization sub-problems and their inputs influence parameters of the upper level coordination problem. Hence the operations of lower levels are influenced by the upper level subsystems through
their chosen control variables, which determine the feasible choices of lower level
subsystems. The main drawback of such hierarchical operations and problem solutions is the iterative manner of problem solutions. The lower level problem can be solved after receiving the upper level solutions. But the last are influence from the lower level solutions. Thus an iterative sequence of calculations takes place till converging towards local or global solution of the hierarchically interconnected optimization problems. This iterative nature of problem solution
prevents the application of the hierarchical approach for control or decision scenarios where real time requirements are compensatory for the normal rules of operation. A particular solution for constraining the iterative manner of operation in hierarchical systems was derived under the notation method of “non-iterative coordination” in multilevel system. The presentation will provide the backgrounds of this method. Particular case for bi-level hierarchy is under consideration, which is the main hierarchical structure, applied in practical cases. Two general coordination strategies will be given : goal coordination and predictive coordination. Following the description of the non-iterative coordination related results and practical applications will be presented. They concern :
Derivation of analytical solutions for solving linear-quadratic mathematical programming problems. These relations allow to accelerate the solution of non-linear mathematical programming problem in comparison with the utilization of numerical iterative methods of solution ;
Providing analytical relations for the evaluation of inverse matrices with high scales. It has been derived relations, which applied computations for low-level inverse matrices but the result concerns high dimension matrix.
Application of the non-iterative analytical relations for controlling arterial transport section. A bi-level optimization problem is defined and solved, which increase the content of the optimal solutions both with the green light duration and the time cycle of the traffic lights.
Application of bi-level formalization for the definition and solution of a portfolio optimization problem. The classical portfolio problem has been extended in a way that additionally to the optimal weights of securities it has been evaluated in a common optimization problem the ratio between the risk and the portfolio return.
The presentation tries to convince that the hierarchical approach can be successfully
applied for real time application for control and decision-making.