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Unformatted text preview: AMS 341 (Spring, 2009) Exam 1  Solution notes Estie Arkin Mean 77.6, median 81, high 99 (2 of them!), low 34. 1. (20 points) Consider the following LP: max z = 2 x 1 x 2 + x 3 s . t . x 1 + 2 x 2 x 3 ≥ 1 3 x 1 2 x 2 + x 3 ≤ 20 x 1 , x 2 ≥ x 3 unrestricted (a). Rewrite the LP in standard form. max z = 2 x 1 x 2 + x ′ 3 x ′′ 3 s . t . x 1 + 2 x 2 x ′ 3 + x ′′ 3 e 1 = 1 3 x 1 2 x 2 + x ′ 3 x ′′ 3 + s 2 = 20 x 1 , x 2 , x ′ 3 , x ′′ 3 , e 1 , s 2 ≥ Common mistake: keeping x 3 unrestricted. (b). What is the dual of the given LP? (You can either state the dual of the original problem or the dual of its standard form.) min z = y 1 + 20 y 2 s . t . y 1 + 3 y 2 ≥ 2 2 y 1 2 y 2 ≥  1 y 1 + y 2 = 1 y 1 ≤ y 2 ≥ 2. (20 points) Each day Eastinghouse produces capacitors during three shifts: 8am4pm, 4pm midnight, midnight8am. The hourly salary paid to the employees of each shift, the number of capacitors produced by a worker during the shift, the price charged for each capacitor made during each shift, and the number of defects in each capacitor produced during a shift are given in the table below. Each of the company’s 25 workers can be assigned to one of the three shifts. Because of machinary limitations, no more than 10 workers can be assigned to each shift. Each day, at most 250 capacitors can be sold, and the average number of defects per capacitor for the day’s production cannot exceed three. Formulate an LP to maximize Eastinghouse’s daily profit (sales revenue minus labour cost). (Your formulation does NOT have to be put into standard form. Do NOT solve, just formulate!) (a). Define the variables you are using in the formulation. x i number of workers assigned to shift i (84am is shift 1 etc)....
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 Spring '08
 Arkin,E
 Optimization, auto loans, Common mistake, objective function

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