First Advisor

Abdul Qayum

Term of Graduation

Summer 1972

Date of Publication


Document Type


Degree Name

Doctor of Philosophy (Ph.D.) in Systems Science


Systems Science




Mathematical optimization, Programming (Mathematics)



Physical Description

1 online resource (vi, 114 pages)


In this thesis, the problem considered is that of linear static optimization of a large system which is composed of a finite number of subsystems, each characterized by its own constraint matrix and objective function.

The total system is itself constrained by resource availabilities and other factors, and its objective function is the mathematical linear sum of the objective function of the subsystems. The total system constraints couple together all the subsystems. The total system is first reformulated as a two-level problem by decoupling the total system constraints utilizing an arbitrary partition of the total system resources and other factors according to the number of subsystems. At the upper level we have a so-called central problem having as an objective function the sum of the optima of the subsystems achieved for any given partition and constrained by the total available resources and other factors which can be partitioned. At the lower level we have the subproblems which are small-dimension linear programming problems parameterized on the right-hand side of the part of the constraints which resulted from the decoupling of the total system constraints.

The resources vector of each subproblem's set of constraints contains the system common resources allocated to it by the central problem. Different allocations of these resources to each subsystem create multiparametric optimization problems for which we have solution methods. The subsystem solutions become functions of the central system allocation policies. Therefore, the major concern for optimization of the whole system is the discovery of the optimum allocation policy.

The method that we introduce finds the optimum allocation policy in a finite number of different allocation iterations.

The major steps in the development are the discovery that the minimal (in case of multiple solutions) shadow prices of the subsystems are equal at optimality to the central system shadow prices, and that a coordination of the subsystems for the purpose of achieving optimality of the total system can be organized by utilizing the concave relationships governing the subsystem shadow prices versus the resources allocated to these subsystems.

The method offers significant computational and conceptual advantages over present decomposition techniques, since it disposes with the solution of a central problem and the subproblems at each iteration and substitutes instead a simple coordination operation and subsystem parametric optimization at each iteration after the first, where a full solution of the subsystems takes place.


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