Optimization Of Branching Multistage Systems: A Reply to a Comment by D. P. Loucks

Meier, W. L.; Beightler, Charles S.

doi:10.1029/wr004i006p01385

Water Resources Research

1968

DOI: 10.1029/wr004i006p01385

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Optimization Of Branching Multistage Systems: A Reply to a Comment by D. P. Loucks

W. L. Meier¹,

Charles S. Beightler²

Abstract: The authors wish to express their appreciation to Loucks [1968] for his constructive criticism of their recent paper [Meier and Beightler, 1967]. Loucks' comment correctly interprets the authors' enthusiasm for dynamic programming and its usefulness in water resource analysis. The authors believe that dynamic programming OWRR GRANT B-024-TEX and FWPCA GRANT WP-01186-01. The authors wish to ex-press their appreciation to Robert W. Lawless for his assistance in preparation of this reply.

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On optimization methods for branching multistage water resource systems

Burt¹

1970

Water Resources Research

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The recent paper by Meier and Beightler [1967] and subsequent discussions [Loucks, 1968;Meier and Beightler, 1968] are interesting contributions to optimization methods in water resources development and management. However, it occurred to me that the positions taken in the exchange between Loucks and Meier and Beightler seemed to rule out the use of linear programing (and nonlinear programing) jointly with dynamic programing withi n the same optimization scheme. Presentation of such models for application to branching multistage systems is the primary objective of this letter.First it is noted that the class of problems considered by Meier and Beightler [1967] had the following properties: (1) additivity in the criterion function, (2) linearity in the constraints, and (3) essentially the so-called angular form in the constraint equations [Dantzig, 1963, p. 466]. The first property is very common in dynamic programing models, but other forms of criterion functions meet Bellman's definition of the Markovjan-type process [Bellman, 1961, p. 54]. It is very likely that the additive type will suffice in most water resource applications; but if it does not, a dynamic programing formulation may contain so many state variables that it becomes infeasible as an empirical model. The Markovian requirement can usually be met by introduction of a sufficient number of state variables. The second property is a direct consequence of an assumed linear transformation function for the state variable [Meier and Beightler, 1967, p. 648]. In problems such as water quality control, some of the transformation functions are apt to be nonlinear, but this causes no difficulty in dynamic programing. The third property is a direct consequence of the problem being amenable to a dynamic programing formulation. It ..was• surprising that the Dantzig-Wolfe decomposition principle [Dantzig, 1963, ch. 23] was not mentioned by either Loucks or Meier and Beightler since it is so applicable to problems with the three properties listed above. If we are willing to assume concavity of the relationships denoted by equation 4 [Meier and Beightler, 1967, p. 647], the problem can be approximated by linear programing, as was shown by Loucks. The complexity and dimensionality problems [Meier and Beightler, 1968, p. 1385] are not so overwhelming with the bounded variable method of handling separable concave programs [Dantzig, 1963, p. 484] used in conjunction with the decomposition principle [ Dantzig, 1963, p. 466]. The decomposition principle is especially important for problems comprised of many stages. In cases where the second property is violated (i.e., nonlinear constraints are required), the Dantzig-Wolfe decomposition principle is lost as an applicable method. Therefore we now turn to dynamic programing and view it as merely another method of decomposition.The recurrence relation of dynamic programing permits recursive optimization stage by stage, but there is nothing to prevent the use of linear or nonlinear programing in the optimization at each stag...

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On optimization methods for branching multistage water resource systems

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Optimization Methods for Branching Multistage Water Resource Systems—A Reply to a Comment by E. F. Brater

Meier¹,

Beightler²

1969

Water Resources Research

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We appreciate the interest shown in our paper [Meier and Beightler, 1967] by Brater in his comment. Brater raises two significant and valid points concerning the paper. First and foremost, there is a numerical error in the authors' paper, as indicated in a previous reply [Meier and Beightler, 1968] . The correct values of d• and d• in Equation 16of the authors' paper should be 7.11 and 2.11, respectively. Furthermore, the total maximum return is slightly greater than 355. These are typographical errors and were not occasioned through the use of the particular optimization method employed.The second point raised by Brater concerns the fact that optimization as an end-point to rational decision-making is not new. Engineers in the design process have always attempted to find the set of design variables that would best

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Optimization Of Branching Multistage Systems: A Reply to a Comment by D. P. Loucks

Cited by 2 publications

References 3 publications

On optimization methods for branching multistage water resource systems

On optimization methods for branching multistage water resource systems

Optimization Methods for Branching Multistage Water Resource Systems—A Reply to a Comment by E. F. Brater

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