PFA

PFA


1) (10 points) Explain why the optimum for linear systems lies at an extreme point. Describe two situations where a unique optimum will not exist. 2) (30 points) For the system: Min Z = 4x1 - 5x2 6x1 - 3x2 -12 4x1 -2x2 24 3x1 + 2x2 30 X2 6 a) What is the optimum solution? b) Indicate whether each constraint is binding or non-binding at the optimum. For the non-binding constraints, indicate how much slack or surplus is in the equation at the optimum. c) By how much could the “b value” of each of the binding constraints be increased or decreased before the optimum is bound by different binding constraints? (Note: it is not necessary to identify the new binding constraints). d) We have learned that alternate optima may occur when a constraint has the same slope as the objective function. If the constraint 4x1 -2x2 24 is changed to 4x1 -5x2 24, will this problem have alternate optima? Explain. 3) (30 points) Two coal-fired power plants owned by a single utility company provide electricity to two neighboring cities. Power plants 1 and 2 discharge 12,000 and 20,000 kg of SO2 per month, respectively. The pollutants are transported into the cities, causing air pollution problems and acid deposition. The following table gives the amount of SO2 pollution deposited in each city from each power plant: Amt Deposited on City 1 (g/km2 deposited/ kg SO2 emitted) Amt Deposited on City 2 (g/km2 deposited/ kg SO2 emitted) Power Plant 1 0.0075 0.0025 Power Plant 2 0.0025 0.0075 The EPA passes a new regulation stating that the maximum amount of SO2 deposition in each city is 50 grams/km2 per month. The cost of reducing emissions at both plants is $1000 per kilogram SO2. a. Construct a system of equations that will help the power utility determine the optimum emissions reductions for each power plant. What is the best decision if the objective is to minimize cost? b. Graph the system, identifying the feasible region and the binding constraints. c. City 1 determines that a 50 g/km2 per month standard is too high and still endangers their citizens. It decides to sue the EPA in order to make the regulation limit more stringent. By how much could the regulation limit for City 1 be decreased before one or both of the plants are forced to operate at zero emissions? d. Up to how much would the utility be willing to pay its lawyers every year to ensure that the proposed regulation in Part C does not occur? 4) (30 points) Ann Arbor City Council wants to create as many “green jobs” as possible within the city. After lobbying federal and state agencies, they have secured funding to assist with job creation. The federal government will provide up to $15 million. The state government will provide up to $5 million. A recent study found that for every $1 million invested in the renewable energy sector, 40 jobs are created. For every $1 million invested in green construction, 30 jobs are created. As expected from the government, there are very specific rules on how money can be spent. The cost share contract states that for any renewable energy investment, 25% of the project will be funded by the state and 75% of the project will be funded by the federal government. For green construction investment, 50% will come from state funds and 50% will come from federal funds. How should Ann Arbor spend the money in order to maximize job creation? a. Create a system of equations that describe the decision. What is the best use of the money to maximize job creation? b. If Ann Arbor could apply to the state of Michigan for additional funding beyond the amounts stated above, would they? How much money would they request to take full advantage of the Federal cost share opportunity? (In other words, how much money would Ann Arbor request before the state money is no longer binding -- if it is binding?) c. How many additional jobs would be provided with each additional $1 million of state funding?