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On optimization methods for branching multistage water resource systems
Author(s) -
Burt Oscar R.
Publication year - 1970
Publication title -
water resources research
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.863
H-Index - 217
eISSN - 1944-7973
pISSN - 0043-1397
DOI - 10.1029/wr006i001p00345
Subject(s) - mathematical optimization , bellman equation , transformation (genetics) , nonlinear system , dynamic programming , mathematics , markov process , state variable , constraint (computer aided design) , nonlinear programming , linear programming , function (biology) , computer science , biochemistry , chemistry , physics , statistics , geometry , quantum mechanics , gene , thermodynamics , evolutionary biology , biology
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 within 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 Markovian‐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.