Shortcomings of linear programming in optimizing river basin allocation

Abstract

Numerous computer models for river basin planning and management have been developed and used extensively since the mid‐1970s. Early developments have relied on the use of network flow algorithms (NFA), due mainly to higher execution speed than the standard Simplex solvers. However, subsequent efforts to include proper modeling of hydraulic and hydrologic constraints introduced iterative schemes into the NFA‐based models, which diminished the initial advantages in execution speed and which also caused concerns over the accuracy of the convergence schemes. Hence full‐blown commercial linear programming (LP) solvers were introduced as a replacement to the iterative solution strategy of the NFA approach. This paper demonstrates one possible failure to solve a simple allocation problem using the NFA‐based model and shows how this problem can be solved using the standard LP approach. It then identifies cases when even a full‐blown LP approach cannot properly model two critical aspects of river basin management, one related to reservoirs with multiple outflows and the other one related to modeling of hydrologic channel routing. For NFA‐based models the failures are the result of the inability to include relationships between flows on different model components directly into the search process. For the models based on LP solvers, the failures are caused by the fact that integrated reservoir outflow capacity between the starting and the ending storage levels is assumed over the entire length of the assumed time step, while the actual outflow can only take place during the portion of the time step when the storage level is above the invert of the outlet structure.