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(fig. 4). The Precambrian crystalline rock is assumed to form the base of the active ground-water flow system, because it is virtually impermeable. In the upper aquifer, precipitation recharges ground water in topographically high areas and move- ment is toward discharge areas such as streams and lakes in nearby, topographically low areas. Recharge to the sandstone aquifer occurs mainly to the west of the Lower Fox River Valley, where the Maquoketa-Sin- nipee confining unit is absent and the sandstone aquifer is in good hydrologic connection to the upper aquifer. Ground-water movement in the sandstone aquifer prior to development was generally west to east. Since development, the direction of ground-water movement is towards the Lower Fox River Valley near the Central Brown County and the Fox Cities pumping centers. Water Use and Description of Wells The CBCWC and ECWRPC provided estimates of municipal water use for the year 2030. These estimates were needed for the purpose of comparison of opti- mized to non-optimized solutions. In the Central Brown County area, a 240 percent increase for the period 1990 to 2030 is projected, from 7.32 to 24.7 Mgal/d (million gallons per day). In the Fox Cities Heart-of-the-Valley communities, a 41 percent increase for the period 1990 to 2030 is projected, from 3.9 to 5.5 Mgal/d. For the Fox Cities Western Towns, a 110 percent increase for the period 1990 to 2030 is pro- jected, from 1.7 to 3.6 Mgal/d. Water use in the Fond du Lac area, in the southern portion of the model, was assumed to remain fixed at 1990 rates. Numerous high-capacity wells in the Central Brown County area and Fox Cities area have had a regional effect on water levels in the sandstone aquifer. Wells withdrawing water from the upper aquifer are typically shallow domestic wells with low pumping rates of about 5 to 10 gal/min (gallons per minute). Such wells typically affect water levels only locally in the upper aquifer and can therefore be ignored. Wells withdrawing water from the sandstone aquifer are typ- ically deep, high-capacity municipal, industrial, and commercial wells that pump about 500 to 1,000 gall mmi. Pumping rates for 1990 and 2030, along with descriptions of the high-capacity wells included in the ground-water model, have been compiled and are included in the Appendix. OPTIMIZATION MODELING Optimization modeling is a general class of prob- lems in which an objective function is either minimized or maximized subject to a series of constraints. The objective function and constraints are expressed as known mathematical functions of the variables of inter- est, termed decision variables. There are several classes of optimization models, depending in part on the form of the objective function and constraints. These include linear programming (linear objective function and con- straints with continuous decision variables), integer programming (linear objective function and constraints with integer decision variables), mixed integer pro- gramming (linear objective function and constraints with integer and continuous decision variables), and nonlinear programming (nonlinear objective function and decision variables). Most introductory texts on operations research describe the classes of optimization models (for example, Gue and Thomas, 1968). Ground-water optimization involves applying optimization modeling to problems of ground-water flow. A review of ground-water optimization tech- niques is given elsewhere (Gorelick, 1983). In most cases, linear programming has been applied to prob- lems of ground-water flow due to its ability to handle large numbers of decision variables and constraints and the relative speed of the solution technique. Specification of the objective function is a crucial step in optimization modeling. The objective function should represent the overall goal of the optimization. Typically the objective function is written using well pumping rates as the decision variables. Objective functions can range from a simple summation of pump- ing rates (for example, maximize total withdrawal) to a detailed function involving pumping rates and water levels (for example, minimize total cost). The constraints impose limits on the decision vari- ables and are very important in ensuring a realistic opti- mal solution. The constraints can vary from simple limits to more complex expressions. Examples of sim- ple limits include upper bounds on pumping rates, lower bounds on water levels, and upper bounds on drawdowns. Examples of more complex limits include upper bounds on horizontal gradients and upper and lower bounds on flow velocity or direction. Several approaches have been devised for repre- senting the ground-water flow system as a linear sys- tem, but the most common approach is to use a response matrix to represent the response of the aquifer OPTIMIZATION MODELING 7