HW _2 solutions

HW _2 solutions - NamaMaJZCQ/M Date: 3; flgéfi CEE...

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Unformatted text preview: NamaMaJZCQ/M Date: 3; flgéfi CEE 3000A: CIVIL ENGINEERING SYSTEMS Homework #2 Fall 2008 1. You have been asked to estimate the optimal size of a water treatment plant for a large community. Knowing that the size of such a plant will have both economic benefits and costs attached to it, you have decided to adopt an economist’s perspective on this problem, that is, you have decided to determine the size of the plant where net benefits are maximized and where marginal benefits equal marginal costs. From historical data on water treatment plants similar to the one being contemplated, you know that the total benefits and costs for the treatment plant can be estimated mathematically as shown below. If “x” is the size of the water treatment plant in million gallons per day (MGD), what is the optimal size of the plant? (5 points) Benefits = 1.73 + 6.84 (x) Costs = 1.24 + 1.67 (x) + 0.75 (x)2 m I -' . \. (a) The owner of a warehouse decides to fence in an area of SI2 square feet behind the warehouse. She plans to use the wall of the building as one of the four Sides that will enclose the rectangular area. If she would like to use the least amount of fencingJF necessary for the other three sides, how many feet of fence will be needed? (5 points) if , 1" .w t (:7: r x 7) i *dwmm;flfl (b) The same warehouse owner decides to fence in a rectangular patio behind her house. She has 150 feet of fence to use. One side of the rectangle is the back of the house. What should be the dimensions of the rectangular region if she wants to make the patio area enclosed as large as Qossible? (5 points) if" 11‘: 5": -fl \___ f. / s a. - / \ .. n {F '\ ‘1»: < . j .. .. f. a 2 3. A state transportation agency in the Snowbelt is looking for a new material that can be used to melt ice on the highways during winter months. From research, the agency knows that a certain combination of components will provide the best ice-meiting capability while minimizing the environmental effects on surrounding lands (that is, seeping into ground water). The limitations on the amount of these components in the inal ice—melting compound are: No more than 15.5% salt content No more than 40% silicate content No more than 20% calcium hydrate content The rest of the compound can be any base material that causes the combined components to stick together. The agency has three different locations from which it can obtain material having some proportion of the desired components. The concept is to obtain enough material from each source, at the lowest possible cost, that can then be combined into the desired ice—melting compound. The table below shows the relevant information for each source of material. E25 10% No constraint No constraint Source 1 Cestfi/ton) ‘% Salt per ton % silicate per ton % calcium h drateper ton $11 I constraint No more than 500 tons a) Ifagency engineers need 2,000 tons per month ofthe ice—melting compound, how many tons of material should they purchase from each source if they are trying to minimize total costs? Hand in the results obtained from the use of the Li) template and the Solver b) Undertake a sensitivity analysis of your results. in particular, you are interested in answering the following question. How wouid the optimal cost change if you were able to obtain 550 tons from location 1 instead of the current 500 tons? Show how you calcuiated this answer by referencing your sensitivity anaiysis form. fl}, 5‘ , , (7' y r,“ I ; H a , . ,0 f. __ . ;' t) . f; 2‘ t if] . c) You are the supplier o’f material from Source 1. Your daughter is a graduate of Georgia Tech and she had told you that you could make a lot more money by increasing your cost of the material purchased by the transportation agency. However, you are worried that if you increase the cost, the agency wili simply purchase material from elsewhere. Using your sensitivity report, calculate how much more money supplier 1 could make if he simply raised his costs without losing any of the sales he currently makes to the state transportation agency. (10 points) .r -:2 f fjr‘ Left Hand Side Coefficients x1 x2 x3 RHS Values Objective Function 1 Constraints Total Supply Source 1 Supply Salt Constraint Silicate Constraint 0.35 Calcium Hydrate Constaint —> User Entry _ Teal —> Calculated Optimal Decision Variable Values x1 x2 x3 1375 Optimized objective function value = 239375 Value RHS Limit Constraints Total Supply Source1 Supply Salt Constraint Silicate Constraint Calcium Hydrate Constaint Microsoft Excel 11.0 Sensitivity Report Worksheet: [LP Template.xls]Sheet1 Report Created: SIM/2008 4:11:59 PM Jr;mmwwmuaaumwzzzuA-mI:{Ami:‘mmzynazzfl-Anwmmz-nwman '(.\‘>r.c~}r.w>y\=':\w a MA .2;.s;.z;...v. Amman”. 'nw‘ “Fina: “*fisaucea ,, W. w, um nmmnvnxvmqm: hS‘rflk“2€~fii\‘[E‘-‘-’$‘I-CVu!‘a1!i’\\71’17’3/(74>31Vkflrnmv99m‘Jt‘flrflm4:7.“\fi\lhh}f’vfi';yi.>§n"‘fi?¥é&%§$1:\$wéI'AWKW'$\?¢\\£?JL5.w%(fltfi‘ofi‘fi‘fifliffififiiflh'flfifi :n-.~>u=u.»~c:v-- "‘me 'Kl‘iéfiéis :. .. _____ ., ., m. . H Mange“: : fits“: , ’ " 57143 7.7500. Constreints . _.$._Q$2§;Tj9t§!§uppiy Value ’ 800 4. Solve the following LP graphicaily. . .graph paper attached (5 points) max 2xl +9:2 SI. xl +x2 28 x, “2x2 £4 wxl +2x2 310 W X‘rfc’QYlfiQO 10 I a :14 .6?- .7 —n _. it, Zve.‘ 2(2M44;;~ :Jfl‘rwafi D. 1.2.2.)?1“ 4: 4‘7) 343 KC 5235 ; 239% iéfler a 52-7 a 2 ...
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HW _2 solutions - NamaMaJZCQ/M Date: 3; flgéfi CEE...

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