On optimization methods for branching multistage water resource systems

Abstract

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...

A Fuzzy-Interval Dynamic Optimization Model for Regional Water Resources Allocation under Uncertainty

“…In the process of water resources optimization, it is essential to give a full consideration to dynamic characteristics, and thus provide the best scheme for water distribution at different stages of the planning period [28]. Dynamic programming cannot only solve the optimization problem of multi-stage decision process in water resources allocation [29], but also obtain the optimal strategy of the whole process and the optimal sub-strategy of each stage [30][31][32][33][34].…”