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<br />., <br /> <br />. <br /> <br />." <br /> <br />. <br /> <br />~ 1;)1,. <br />;-.. <br /> <br />w::.- <br />I\,) <br />N <br />o <br /> <br />(ST)i = the maximum achievable salinity recluction achieved by lining the <br />entire length of the ith canal, Mgm/yr: ancl <br /> <br />n = the number of canals or ditches that csn be linsll to recluce salinity. <br /> <br />This curve of cost versus salt recluction a1wsys has the property of <br />convexity, a necessary condition for optimality (Wilde and Beight1er, 1967), <br />and can be consiclered what Er1enkotter ancl Scherer (1977) refer to as a "con- <br />tinuous project." In other words, a cost csn continually be assignecl for any <br />variable value of salt reduction. The functionsl relationship of this curve <br />remains the same throughout the entire optimization process. The curve has <br />the same basic shape ancl properties for canal lining in an incliviclual area as <br />well as for a basin-wide salinity control program. This property greatly <br />. simplifies the optimization process ancl the determination of the indiviclual <br />components of salinity control at any level of ~ontrol. <br /> <br />". '..~i'itilplifying' the' Optimization-- <br />As notecl earlier, the optimization process requires <br />.solved repeateclly for values of S ranging from zero to <br /> <br />. . <br /> <br />'::,,'" <br /> <br />that Equations 5-7 be <br />n , generating <br />I (S'l')i <br />i=l - <br />data from which the optimal function for canal lining is c1arivecl. The result- <br />ing canal lining cost-effectiveness function is characterizecl by increasing <br />marginal costs with scale but the nonlinearity is not great. These functional <br />features provide the opportunity to condense Equation 1 into a simple regres- <br />sion function. For example, the following expression has been found to pro- <br />duce good rasults: <br /> <br />~j, <br /> <br />C = <br />0. <br /> <br />Sl <br />ASl + B <br /> <br />(8) <br /> <br />For specific canal, ditch, or lateral, the only unknowns in Equation 8 are C <br />- c <br />(depenclent vsriable) sncl Sl (inclependent variable). A range of Sl values <br />within the intsrvsl from zero to the maximum value of ST can be generatecl from <br />Equation 1 when different lengths L are arbitrarily substitutecl into the <br />. equation.. Corresponding values of C are then calculstecl p1'oviding the x-y <br />c <br />data for a regression fitting. A linear regression can be used for curve <br />fitting if Equation 8 is transformecl to: <br /> <br />. . - ~, , <br /> <br />y=Ax+Bx <br /> <br />(9) <br /> <br />where y = liCe ancl x a 1/S~. <br /> <br />This function can also be compared in an optimizationsl context with <br />other similar strategies to formulate plans on a large scale. w~ile this <br />d.evelopment still requires some prior unclerstanding of operations research <br />methoclologies for those not so prepared, Equation 8 leads to a simple optimi- <br />zation solution basecl on the unique algebraic structure of the modified cost- <br />effectiveness functions. The complex optimization procedure is reducecl to a <br />facile series of arithmetical calculations. If necessary, most of this <br />procedure cou1c1 be done with hand-held calculators. <br /> <br />43 <br /> <br />,. <br />