The federal ministry responsible for the development of water resources in Brazil once considered the feasibility of constructing a reservoir on the Rio Pardo, which flows through a relatively arid region of the country. The reservoir would be used to irrigate fields of maize, the chief irrigable crop of the region. A suitable site was identified, but the storage capacity of the proposed reservoir was not determined. Because the Rio Pardo feeds into a larger river, the Rio Parana, which flows through Paraguay and Argentina, it was determined that the minimum downstream release from the reservoir must be at least 0.2 x 107 m per month in order to satisfy international agreements on water use. The table below summarizes data from a preliminary study by a hydrologic consultant. Column 2 lists the quantity of water expected to flow into the reservoir each month from the Rio Pardo. In addition, water will be added to the reservoir directly through precipitation during the 6-month rainy season as shown in column 3. Evaporation from the reservoir's surface is expected to vary monthly according to temperature, humidity, wind, and insolation, and estimates of the monthly evaporation are shown in column 4. Finally, column 5 lists the monthly irrigation requirements to support 1 kin of maize, after direct rainfall on the fields has been considered. Because of high infiltration rates and plant uptake in the soil of the maize fields, no irrigation water is expected to flow back into the river. Month January February March April May June July August September October November December ALL WATER DATA IN UNITS OF 107 m3 Flow Into Rainfall Evaporation Net Irrigation The Into From Requirement for Reservoir Reservoir Reservoir 1km? 4.26 0.08 0.09 0.013 5.38 0.08 0.12 0.015 6.21 0.12 0.11 0.008 9.93 0.07 0.09 0.012 5.00 0.10 0.027 2.04 0.12 0.032 1.32 0.13 0.037 1.18 0.15 0.043 0.2 0.14 0.042 1.62 0.13 0.035 3.44 0.07 0.09 0.014 4.56 0.09 0.08 0.009 ---- ---- The cost of constructing the dam and laying irrigation pipe is expected to be proportional to the storage volume of the reservoir. Assuming the dam will be designed for a 30-year life, it has been estimated that this cost must be amortized (i.e. made into an equivalent uniform annual cost, EUAC, i.e. an annual worth of costs) at the rate of 1.40 Brazilian cruzeiros per year for each cubic meter of storage volume. Farming should produce increased income amounting to 6 million cruzeiros per year for each square kilometer of land irrigated. [Note: Pay very close attention to units to get your costs and benefits correct!] Consider that the same area of maize is irrigated every month year-round. a) Formulate and solve this situation as a linear programming problem to determine simultaneously the total storage volume of the reservoir and the area of land that can be irrigated if the yearly income minus the amortized yearly cost is to be maximized. Neglect all other potential water losses (such as seepage) in this problem. Be careful of your units! Determine the change in net income, area, and reservoir size if the minimum downstream release was increased to 0.4 x 107 m during the six dry season months. b) The federal ministry responsible for the development of water resources in Brazil once considered the feasibility of constructing a reservoir on the Rio Pardo, which flows through a relatively arid region of the country. The reservoir would be used to irrigate fields of maize, the chief irrigable crop of the region. A suitable site was identified, but the storage capacity of the proposed reservoir was not determined. Because the Rio Pardo feeds into a larger river, the Rio Parana, which flows through Paraguay and Argentina, it was determined that the minimum downstream release from the reservoir must be at least 0.2 x 107 m per month in order to satisfy international agreements on water use. The table below summarizes data from a preliminary study by a hydrologic consultant. Column 2 lists the quantity of water expected to flow into the reservoir each month from the Rio Pardo. In addition, water will be added to the reservoir directly through precipitation during the 6-month rainy season as shown in column 3. Evaporation from the reservoir's surface is expected to vary monthly according to temperature, humidity, wind, and insolation, and estimates of the monthly evaporation are shown in column 4. Finally, column 5 lists the monthly irrigation requirements to support 1 kin of maize, after direct rainfall on the fields has been considered. Because of high infiltration rates and plant uptake in the soil of the maize fields, no irrigation water is expected to flow back into the river. Month January February March April May June July August September October November December ALL WATER DATA IN UNITS OF 107 m3 Flow Into Rainfall Evaporation Net Irrigation The Into From Requirement for Reservoir Reservoir Reservoir 1km? 4.26 0.08 0.09 0.013 5.38 0.08 0.12 0.015 6.21 0.12 0.11 0.008 9.93 0.07 0.09 0.012 5.00 0.10 0.027 2.04 0.12 0.032 1.32 0.13 0.037 1.18 0.15 0.043 0.2 0.14 0.042 1.62 0.13 0.035 3.44 0.07 0.09 0.014 4.56 0.09 0.08 0.009 ---- ---- The cost of constructing the dam and laying irrigation pipe is expected to be proportional to the storage volume of the reservoir. Assuming the dam will be designed for a 30-year life, it has been estimated that this cost must be amortized (i.e. made into an equivalent uniform annual cost, EUAC, i.e. an annual worth of costs) at the rate of 1.40 Brazilian cruzeiros per year for each cubic meter of storage volume. Farming should produce increased income amounting to 6 million cruzeiros per year for each square kilometer of land irrigated. [Note: Pay very close attention to units to get your costs and benefits correct!] Consider that the same area of maize is irrigated every month year-round. a) Formulate and solve this situation as a linear programming problem to determine simultaneously the total storage volume of the reservoir and the area of land that can be irrigated if the yearly income minus the amortized yearly cost is to be maximized. Neglect all other potential water losses (such as seepage) in this problem. Be careful of your units! Determine the change in net income, area, and reservoir size if the minimum downstream release was increased to 0.4 x 107 m during the six dry season months. b)