Final07.old - HACETTEPE UNIVERSITY DEPT. OF ELECTRICAL AND...

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HACETTEPE UNIVERSITY DEPT. OF ELECTRICAL AND ELECTRONICS ENGINEERING ELE 704 Optimization Midterm Examination, 17 April, 2007 Name : Hardworker ID # : # 1 Question 1 2 3 4 5 Total Mark 20 40 10 20 20 110 Q1. (20pts) A cardboard box for packing some stu/is to be manufactured as shown in Fig. 1. Find the dimensions of such a box that maximizes the volume for a given amount of cardboard, equal to 24 m 2 . y x z front (a) (4pts) Formulate the problem in a standard optimization problem form. maximize f ( x;y;z ) = xyz subject to xy + xz + yz = 12 x;y;z & 0 Since the inequality constraints are rather general and will be shown that they can be embedded to the problem, you can ignore them. (b) r f ( x ;y ;z ) + r h ( x ;y ;z ) = 0 2 4 yz xz xy 3 5 + 2 4 y + z x + z x + y 3 5 = 2 4 0 0 0 3 5 yz + ( y + z ) = 0 (1) xz + ( x + z ) = 0 (2) xy + ( x + y ) = 0 (3) xy + xz + yz = 12 1
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(c) (4pts) Find x, y, and z. ( xy + yz + xz ) + 2 ( x + y + z ) = 0 but using the constraint we can write ( x + y + z ) = 6 (4) Obviously, 6 = 0 and & < 0 since length can not be negative. If we let x = 0 , from Eqn. (2) y = 0 and if y = 0 from Eqn. (3) z = 0 which results in x = y = z = 0 which is not possible. Hence x , y and z are non-zero. Multiply Eqn. (1) by x and the second by y and subtract the two to obtain ( x y ) z = 0 Multiply Eqn. (2) eqn. by y and the third by z and subtract the two to obtain ( y z ) x = 0 Since x , y and
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Final07.old - HACETTEPE UNIVERSITY DEPT. OF ELECTRICAL AND...

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