We study a two-level system having N local systems in the lower level subordinate to a central system in the higher one, such that both central and local systems have decision-making units. The central system is a coordinating agency and the local ones are semi-autonomous operating devisions. The basic principle of planning for this organization is that the central system allocates resources so as to optimize its own objective, while the local ones optimize their own objectives using the given resources.
A local objective function, fn, is a function of the lower level decision variable vector x=(x1,・・・, xN) and the higher level one a=(a1,・・・, aN), where an is a resource vector allocated to the local system n. Since the functions ■ are mutually independent, the lower level composes a multi-objective system, in which the lower level decision-makers minimize a vector objective function f =(f1,・・・,fN) with respect to x in cooperation with each other. Thus, the lower level generates a set of noninferior (i.e. Pareto optimal) solutions ■(a) being parametric with respect to a.
The central decision-maker, then, chooses the optimal resource allocation a⁰ and the best noninferior solution ■⁰ corresponding to a⁰ from among a set of ■(a).
The above problem becomes a decentralized two-level optimization, when the local system contains only its own variables (xn, an).
Several theorems and iterative algorithms for the formulated problems are obtained by use of mathematical programming techniques.