CEE111_Midterm_Exam_W10_Solution

CEE111_Midterm_Exam_W10_Solution - SOLUTION CEE 111 Midterm...

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SOLUTION CEE 111 Midterm Exam Winter 2010 February 17, 2010 Closed Book, Closed Notes Instructions: Answer all questions in the space provided. Show all work. Justify your answers . Problem 1. (20 total points) A city is growing in population and the public works department projects the need for an increased water supply. The current water supply is drawn from a reservoir on a local stream and is of good quality, but future sources—ones to be tapped next—have various problems. Water from a nearby aquifer is available in adequate supply, but its hardness level is too high unless it is either blended with a lower hardness source or chemically treated—chemical treatment can reduce the hardness level by 50%, but is also costly. The total pounds of hardness per million gallons is limited to 1,200 (lb/mg) in the final blended supply. Water from a distant stream is of sufficient quality, but a pipeline would have to be built to convey it, and the cost to pump the water to the water treatment plant is quite high. The costs to obtain water in dollars per million gallons ($/mgd), the supply limits in millions of gallons per day (mgd), and the hardness in pounds per million gallons (lb/mg) for each of the three sources—reservoir (source 1), aquifer (source 2), and distant stream (source 3)—are shown in Table 1. Table 1 Source Property Reservoir (Source 1) Aquifer (Source 2) Stream (Source 3) Untreated Treated Supply limit (mgd) 25 120 100 Hardness (lb/mgd) 200 2300 1150 70 Cost ($/mgd) 500 1,000 2,000 2,000
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Question 1.1 (10 points) Formulate the linear programming model that would provide a total of 150 million gallons per day to the city at minimum cost. Let 1 2 2 3 mgd to be drawn from Source 1 (Reservoir) mgd to be drawn from Source 2 (Aquifer) and left untreated mgd to be drawn from Source 2 (Aquifer) and treated mgd to be drawn from Source 3 (S U T x x x x tream) Constraints 1. Demand: City needs at least a total of 150 million gallons per day 12 2 3 150 UT xx x x  2. Supply: There is a limited supply available from each source 1 22 3 25 120 100 x xx x  3. Hardness: The total pounds of hardness per million gallons is limited to 1,200 (lb/mg) in the final blended supply 3 200 2300 1150 70 (1,200lb/mgd) (150 mgd) 180,000 x x  (Note: Here we can assume that constraint #1 will be satisfied at the equality, since the city will not buy more than it needs.) 4. Non-negativity constraints ,2 3 ,, 0 Objective Minimize 2 3 Cost 500 1,000 2,000 2,000 Z x x 
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Total Model Minimize 12 2 3 500 1,000 2,000 2,000 UT Z xx x x  Subject to: 2 3 1 22 3 3 ,2 3 150 25 120 100 200 2300 1150 70 180,000 ,, 0 xx x x x x x x   
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Question 1.2 (10 points) Find an initial feasible solution. Minimize 12 2 3 500 1,000 2,000 2,000 UT Z xx x x  Subject to: 2 3 1 22 3 3 ,2 3 150 25 120 100 200 2300 1150 70 180,000 ,, 0 xx x x x x x x    Convert to equation form for Simplex Tableau: Minimize 1 2 2 312345 500 1,000 2,000 2,000 0 0 0 0 0 100,000 Z x x S S S S S A Subject to: 2 31 223 34 3 5 ,2 312345 150 25 120 100 200 2300 1150 70 180,000 , ,,,,,,, 0 xx x xSA xS xxS x x S xx x xSSSSSA 
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This note was uploaded on 03/30/2012 for the course CIVIL ENGI CEE111 taught by Professor Recker during the Winter '12 term at UC Irvine.

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CEE111_Midterm_Exam_W10_Solution - SOLUTION CEE 111 Midterm...

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