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hw-05-lagrange

# hw-05-lagrange - Homework 5 Langrange Multipliers 1(this...

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Homework 5: Langrange Multipliers 1. (this question has little to do with optimization; it's just about the electric power industry). Visit http://www.midwestiso.org/page/LMP+Contour+Map+%26+Data and look up the 5-minute real-time "LMP" (Locational Marginal Price) at the location DECO.MONROE3 I think that's the large coal-fired power plant in Monroe, MI. The LMP is essentially the Lagrange multiplier for a large NLP; it is also the wholesale price of electricity at that point. I believe the units are \$/MWh. Report the value you find, and the date/time you looked. Also, convert it to cents per kWh. The retail price for electricity (including distribution charges, taxes, etc.) is around 11 cents per kWh, just for comparison. Also, consider buying an 20-pack of AA batteries for \$10.86 (I got the price from walmart.com). Each AA battery contains, let's say, 2500 milli-Amp-hours, which when multiplied by 1.5 volts gives 3.75 Watt-hours. How much are you paying per kWh? Pretty easy homework question, right? --------------------------------------------------------- 2. Read the article "Economic Congestion Relief Across Multiple Regions Requires Tradable Physical Flow-Gate Rights" at http://people.emich.edu/aross15/e/electricity-dualization.pdf and the questions/responses from a previous year's student, included below in this homework assignment. Using Excel or Matlab/SciLab (I recommend Excel), duplicate the NLP shown at the bottom-left of page 3 of the document, using the values shown in Figure 1; you may check your work against the results shown in Figure 4. Also, write down any questions or thoughts you have about the article. --------------------------------------------------------- 3. "Transportation Problems" are a subclass of network flow problems. You have a set of source cities and a set of demand cities, the amount of supply in each source city, the amount of demand in each demand city, and the shipping costs for each (source,demand) city pair. You must exactly meet demand in each demand city, and each source city must ship out exactly what it produces (no more or less). a) Formulate and solve a transportation problem as follows: Source cities: San Francisco and Los Angeles (SF and LA) Demand cities: Las Vegas and Phoenix (LV and Ph) Costs: to: LV Ph from SF 50 300 LA 320 60 (costs are per item shipped). Supply: 5 in SF, 10 in LA Demand: 7 in LV, 8 in Ph Find the optimal shipping amounts and the total cost. When Solver says it has a solution, ask it for the Sensitivity report as well, but don't look at it yet.

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b) Add one unit of supply in SF and one unit of demand in Ph, re-solve, and report the new optimal decisions and total cost. Comment as appropriate. c) Go back to the sensitivity report you generated in part (a) and look at the Lagrange Multipliers, and comment on them in relation to part (b). d) In both (a) and (b), total supply and total demand matched each other perfectly. What would you do if supply exceeded demand? What if demand exceeded supply?
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