Problem 1 (70 points) : Suppose that you are working in a team that is trying to put together a hydropower development plan for a region in Southeast Asia. Your main responsibility is to decide the optimal capacities for two dams on the same river, at sites 1 and 2. For the sake of this problem, let's assume that the annual hydroelectric generating capacity of a reservoir is roughly proportional to the annual flow rate through it times the reservoir height: E(kwh/yr) QC). H(m), where a = 68748.5. The annual revenue from hydropower per unit of hydroelectric generating capacity is $0.02. [You can refer to Chapter 11.7 from Loucks textbook for more information about how to calculate hydropower energy] The annual flow of the river at sites 1 and 2 is 100 () and 150 (mat), respectively. The cost of a dam depends primarily on the volume of concrete utilized in its construction (which means only at the beginning of the project). Assume that concrete costs $65 per m3 to buy and costs as much again to install. The volume of concrete required to make a dam is roughly proportional to the height of the dam raised to a power. That exponent depends on shape of the valley, which is 2.5 for site 1 and 3.0 for site 2. The constant of proportionality is 8 for site 1 and 5 for site 2. The maximum possible height for reservoir 1 is 100 (m) and for reservoir 2 is 75 (m). The discount rate used in projects of this type is 10% per year, and the dams are being considered for a lifetime of 50 years. 0 1 2 50 A A A Max Profit (present value) = -C + [P|A, 0.1, 50] The financing for this project comes from the World Bank as a one-time loan in the beginning of the project. Therefore, the cost of construction cannot exceed this loan. The maximum loan is $70M. Part 1-1) Given the problem statement, formulate an optimization problem that you are facing. Determine the decision variables, constraints and the objective function (write your objective function in terms of present value) Part 1-2) Solve the optimization problem using the KKT conditions method assuming that the loan from the World Bank is $40 million. (After you set up the KKT conditions, try to simplify the problem by testing if the dam height constraint for the can be binding or not. The answer to this test will simplify the problem largely) Problem 1 (70 points) : Suppose that you are working in a team that is trying to put together a hydropower development plan for a region in Southeast Asia. Your main responsibility is to decide the optimal capacities for two dams on the same river, at sites 1 and 2. For the sake of this problem, let's assume that the annual hydroelectric generating capacity of a reservoir is roughly proportional to the annual flow rate through it times the reservoir height: E(kwh/yr) QC). H(m), where a = 68748.5. The annual revenue from hydropower per unit of hydroelectric generating capacity is $0.02. [You can refer to Chapter 11.7 from Loucks textbook for more information about how to calculate hydropower energy] The annual flow of the river at sites 1 and 2 is 100 () and 150 (mat), respectively. The cost of a dam depends primarily on the volume of concrete utilized in its construction (which means only at the beginning of the project). Assume that concrete costs $65 per m3 to buy and costs as much again to install. The volume of concrete required to make a dam is roughly proportional to the height of the dam raised to a power. That exponent depends on shape of the valley, which is 2.5 for site 1 and 3.0 for site 2. The constant of proportionality is 8 for site 1 and 5 for site 2. The maximum possible height for reservoir 1 is 100 (m) and for reservoir 2 is 75 (m). The discount rate used in projects of this type is 10% per year, and the dams are being considered for a lifetime of 50 years. 0 1 2 50 A A A Max Profit (present value) = -C + [P|A, 0.1, 50] The financing for this project comes from the World Bank as a one-time loan in the beginning of the project. Therefore, the cost of construction cannot exceed this loan. The maximum loan is $70M. Part 1-1) Given the problem statement, formulate an optimization problem that you are facing. Determine the decision variables, constraints and the objective function (write your objective function in terms of present value) Part 1-2) Solve the optimization problem using the KKT conditions method assuming that the loan from the World Bank is $40 million. (After you set up the KKT conditions, try to simplify the problem by testing if the dam height constraint for the can be binding or not. The answer to this test will simplify the problem largely)