Limited multi-stage stochastic programming for managing water supply systems

Decision-making processes often involve uncertainty. A common approach for modeling uncertain scenario-based decision-making progressions is through multi-stage stochastic programming. The size of optimization problems derived from multi-stage stochastic programs is frequently too large to be addressed by a direct solution technique. This is due to the size of the optimization problems, which grows exponentially as the number of scenarios and stages increases. To cope up with this computational difficulty, solution schemes turn to decomposition methods for defining smaller and easier to solve equivalent sub-problems, or through using scenario-reduction techniques. In our study a new methodology is proposed, titled Limited Multi-stage Stochastic Programming (LMSP), in which the number of decision variables at each stage remains constant and thus the total number of decision variables increases only linearly as the number of scenarios and stages grows. The LMSP employs a decision-clustering framework, which utilizes the optimal decisions obtained by solving a set of deterministic optimization problems to identify decision nodes, which have similar decisions. These nodes are clustered into a preselected number of clusters, where decisions are made for each cluster instead of for each individual decision node. The methodology is demonstrated on a multi-stage water supply system operation problem, which is optimized for flow and salinity decisions. LMSP performance is compared to that of classical multi-stage stochastic programming (MSP) method.