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Water Resource System Optimization by Geometric Programming

W. L. Meier, C. S. Shih, D. J. Wray

Water resources planners and systems analysts are continually confronted with many complex optimization problems. Two major factors contribute to this problem. Firstly, mathematical modeling and system description capabilities in water resources have increased considerably. Secondly, planning and design analyses have become more complicated because of a desire by planners to represent more completely the problem to be solved.

In a recent appraisal of Federal Water Resources Research activities, research concerned with water resource planning methodology was labeled as the "most promising area of research" [22].

In an effort to meet the Congressional decree [54] for "optimum" water resources planning, the literature is filled with illustrations of the applicability and usefulness of optimization methods in water resources analyses [38]. Linear programming [53], dynamic programming [41,40], and nonlinear programming [19] have been shown to be useful in water resource optimization problems. These techniques have been and are being coupled with mathematical modeling and system simulation to produce practically useful planning algorithms.

The purpose of this report is to describe a new optimization technique which can be extremely useful in solving water resources optimization problems. This new and potentially powerful technique is called geometric programming. It is one of a class of mathematical programming techniques. Mathematical programming discussed extensively elsewhere [56,23] refers to a class of optimization problems dealing with the allocation of limited resources. Problems are described with an objective function which measures system effectiveness and constraints which describe limitations on the solution. If either the objective function or constraints are nonlinear, a nonlinear programming problem results.

A variety of algorithms are available for solving nonlinear programming problems. However, when the constraints are nonlinear, the objective function is higher than second degree or the problem of high dimensionality; most of the available techniques are not computationally efficient. Little previous research has been aimed at making geometric programming of use to water resource analysts. The purpose of this research was to explore ways of making geometric programming more useful to water resource systems analysts. Specifically, this research has as its objectives the development of a computer algorithm for solving geometric programming problems and the specific application of this algorithm in the solution of a water renovation and treatment process.

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