## Optimum operation of a multi-reservoir water supply systemProject on Engineering-Economic Planning, Stanford University, 1967 - Water-supply engineering - 129 pages |

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allocation amount of water Appendix assumed assumptions component of grad computational constant constraints continuously differentiable convex convex function current release decreasing demand denoting derivatives DGJ(R dynamic programming eigenvalues equal equations exists fact feasible Gessford and Karlin Gessford-Karlin given GJ(R grad G hence Hessian matrix implies increment inflows initial storage vector Jacobian matrix jth period jth stage L(SJ lemma loss function marginal expectation marginal utility Masse's maximum minimize minimum multi-reservoir systems nondecreasing nonincreasing nonnegative orthant objective function occur optimization optimum decision optimum operating policy optimum policy optimum release piecewise linear points positive definite possible probability density function problem proof properties R J(R random variables release vector satisfies 2.24 shortage shown single reservoir solution stochastically independent storage level Theorem total amount total expected loss total release transfer capacities various reservoirs water in storage water resources systems Xj)AR zero transfer costs