Civil, Environmental and Architectural Engineering ...



University of ColoradoDepartment of Civil, Environmental and Architectural EngineeringCVEN 5393 Water Resources Development and ManagementHomework #5Due Feb 25, 2013Topics: Linear Programming Formulation, Graphical solutions and Lagrangian FunctionsProblem 1Two types of crops can be grown in particular irrigation area each year. Each unit quantity of crop A can be sold for a price PA and requires WA units of water, LA units of land, FA units of fertilizer, and HA units of labor. Similarly, crop B can be sold at a unit price of PB and requires WB, LB, FB and HB units of water, land, fertilizer, and labor, respectively, per unit of crop. Assume that the available quantities of water, land, fertilizer, and labor are known and equal W, L, F, and H, respectively.(a) Structure a linear programming model for estimating the quantities of each of the two crops that should be produced in order to maximize total income.(b) Solve the problem graphically, using the following data:Clearly show the feasible region on your graph, the optimal solution and the optimal objective function value. Problem 2An industry must have a water supply of at least 4x106 liters/day of a quality such that total dissolved solids (TDS) is kept below 100 mg/l. The water can be obtained from two sources: (1) purchase from the city system at $100 per million liters, and (2) pump from a nearby stream at $50 per million liters. The concentration of TDS in the city source is 50 mg/l. TDS in the stream is 200 mg/l. Water from the two sources is completely mixed before it is used. The city can supply up to 3.5x106 l/day, and water rights permit pumping up to 2x106 l/day from the stream.Formulate a linear program to optimize the amount of water used from each source. Define your decision variables and the meaning of the objective function and constraints.Use the graphical method to determine the optimal solution.Clearly show the feasible region on your graph, the optimal solution and the optimal objective function value. Problem 3An aqueduct constructed to supply water to industrial users has an excess capacity in the months of June, July, and August of 14,000 acft, 18,000 acft, and 6,000 acft, respectively. It is proposed to develop not more than 10,000 acres of new land by utilizing the excess aqueduct capacity for irrigation water deliveries. Two crops, hay and grain, are to be grown. Their monthly water requirements and expected net returns are given in the following table:Monthly Water Requirement (acft/acre)JuneJulyAugustReturn, $/acreHay211100Grain120120Formulate a linear program to optimize the irrigation development. Clearly define all the variables used and give their units.Use the graphical method to determine the optimal solution.Clearly show the feasible region on your graph, the optimal solution and the optimal objective function value.Problem 4A reservoir is designed to provide hydropower and water for irrigation. The turbine releases may also be used for irrigation as shown in Figure (a) below. At least one unit of water must be kept in the river each month at point A. The hydropower turbines have a capacity of 4 units of water per month (flows are constant during any single month), and any other releases must bypass the turbines. The size of farmed area is very large relative to the amount of irrigation water available, so there is no upper limit on usable irrigation water. The reservoir has a capacity of 10 units, and initial storage is 5 units of water. The ending storage must be equal to or greater than the beginning storage. The benefits per unit of water, and the estimated average inflows to the reservoir are given in Table below.(a)(b)Schematic of a Hydropower and Irrigation Supply ProblemHydropower and Irrigation Problem DataMonthInflow UnitsHydropower Benefits ($/unit)Irrigation Benefits ($/unit)121.61.0221.71.2331.81.9441.92.0532.02.2622.02.2Develop a Linear Programming problem for maximizing the economic benefits of reservoir operation for the reservoir configurations in Figures (a) and (b) above, separately.(Assume that the hydropower and irrigation water are separate quantities) Problem 5The storage volume for a proposed reservoir is given by the equation:S = 10 A hWhere A is the area of the reservoir site in acres and h is the height of the dam in feet. The cost of the land is $1000 per acre, and the cost of the dam is $5000 per foot of height. The total budget for the project is $200,000.Determine the optimal size for the reservoir area and dam height within the given project budget.b. If the project budget were increased by $1000, what corresponding increase in storage could be obtained?Solve using Lagrange Multipliers ................
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