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<br />When it is decided to automatically provide an estimate of the best size <br />or the "best" anything in a mathematical sense, a certain number of requirements <br />immediately become apparent. The first is that "best" must be precisely and <br />uniquely defined by an indicator or index that integrates all of the desired <br />performance characteristics of the system that is being analyzed, This index <br />is normally termed the objective function. In addition, the capability to adjust <br />automatically the size of each component within a feasible range and evaluate <br />the resulting change in performance of the system must be devised. Then a <br />search procedure that is as nearly foolproof as possible must be developed. <br />Objective Function,- The plan formulation strategy previously described in- <br />cluded initially determining an economically optimum system (unconstrained <br />maximum net benefits) as a starting point for determining a performance standard <br />for subsequent analyses, The unconstrained economic optimum can be charac- <br />terized by an index of the system performance (objective function) that consists <br />of the sum of the total annual system cost and the total value of the system's <br />expected annual flood damages, If we label this the total social cost of flooding, <br />then the objective is to find the combination of component sizes of the system <br />that results in the minimum total value of system social cost of flooding, Obviously, <br />the system that results in minimum total'social cost as previously defined is <br />exactly the system that will result in the maximum value of system net benefits. <br />The second sizing phase in plan formulation was to determine the component <br />sizes that would accomplish the performance standard (degree of protection) <br />most efficiently and economically, The objective function that was adopted <br />from among several that were tested for determining the system that will maximize <br />system net benefits while satisfying performance standards,if they exist, is <br /> <br />( . k ) [ k (DE V )4 ] <br />Z = ~ C, + ~ ADj ~ - . + CNST <br />,-I J-I J~I A Q, J <br /> <br />. , . . . . . . . , . , . . (I) <br /> <br />in which Z = system performance index (magnitude of objective function); Cj <br />= equivalent annual cost of system component i; AD j = expected annual damage <br />at location j; n = number of system components to be optimized; k = number <br />of damage locations (damage centers); DEV = (Q, - Q,) if the result is positive, <br />otherwise DEV = 0; Qz = flow (stage) for target degree of protection at damage <br />location j; Q, = target flow (stage) for target degree of protection at damage <br />location j; and A, CNST = normalizing constants and weights, usually 0.1 <br />and 1.0, respectively. The function is comprised of two parts; the total annual <br />social cost of flooding and a multiplier that penalizes the function whenever <br />the operation of the components results in performance that is not within a <br />certain tolerance of the desired system performance target, The penalty is merely <br />a devise for forcing the performance target to be met. When the flow, Q z' <br />is equal to or less than the target flows, Q t' for a given system, then for <br />a constant, CNST, of 1,0 the value of the objective function is the sum of <br />the total annual system cost and expected annual flood damage, The initial <br />"unconstrained" sizing problem is therefore solved by setting CNST to 1.0 <br />and Q, to a very high value. Providing a value of 0.1 for the normalizing constant, <br />A, in effect says that when performance Q z is within 10% of the target, Q t' <br />the weight between the social cost of flooding and the hydrologic performance <br />is equal, For deviations larger than 10% the components are penalized at the <br /> <br />, <br /> <br />f <br />