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Mathematical Modeling and Applications

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IE 456 MATHEMATICAL MODELING AND APPLICATIONS Homework 1 1. [15 points] Consider the following linear programming problem: Min z = 3x12x23x3 Subject to x1+2x2 +x3  14 x1+2x2+4x3  12 x1 x2 +x3 = 2 x3  3 x1; x2 unrestricted (a) [6 points] Reformulate the problem so it is in standard format. (b) [6 points] Reformulate the problem so it is in canonical format. (c) [3 points] Convert the problem into a maximization problem. 2. [20 points] A lathe is used to reduce the diameter of a steel shaft whose length is 36 inches (in.) from 14 in. to 12 in. The speed x1 (in revolutions per minute), the depth feed x2 (in inches per minute), and the length feed x3 (in inches per minute) must be determined. The duration of the cut is given …show more content…

(1: tanks, 2: planes, 3: missile batteries). (a) [12 points] Max x1+ 3x2+ 2x3 (22) s.t. 600; 000x1+2; 000; 000x2+800; 000x3  900; 000; 000 (23) x1  200 (24) x2  200 (25) x2  300 (26) x2 4x3  0 (27) x2+ 2x3  0 (28) x1; x2; x3  0: (29) (22) gives the total utility obtained from the military logistics items to maximize. (23) is the budget constraint that restricts the money to spend on the items. (24) { (26) give the limits on the quantity of tanks and planes to acquire. (27) and (28) ensure the ratio of the tanks and missile batteries are within the given limits. Finally, (29) ensure non-negativity of the quantity of items to buy. (b) [3 points] (23) can be rewritten as: 0:6x1 + 2x2 + 0:8x3 

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