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# hmk_4_s - ESE304 - Introduction to Optimization (Homework...

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ESE304 - Introduction to Optimization (Homework #4) Fall Semester, 2010 M. Carchidi –––––––––––––––––––––––––––––––––––– Problem #1 (20 points) ToyWorld makes soldiers ( x 1 ), trains ( x 2 ) and dolls ( x 3 )andth eLPfo r maximizing ToyWorld’s monthly pro f t (in dollars) is as follows. max z =\$20 x 1 +\$43 x 2 + \$43 x 3 (Pro f t Objective Function) s.t. 2 x 1 +4 x 2 +5 x 3 250 (Hours) (Carpentry Constraint) x 1 +2 x 2 +3 x 3 140 (Hours) (Sanding Constraint) 3 x 1 +7 x 2 x 3 320 (Hours) (Painting Constraint) x 1 ,x 2 3 0 (Sign Restrictions) We are given that the optimal solution results in ToyWorld making some of all three types of toys: soldiers, trains and dolls, and 245 123 375 1 = 11 15 2 45 1 12 0 . a.) (5 points) Using only matrix algebra (no simplex method) determine the optimal tableau, and from this determine how many of each (soldiers, trains and dolls) should be made and ToyWorld’s monthly pro f t. b.) (5 points) Determine the shadow prices for each of the three constraints c.) (5 points) ToyWorld wants to consider also making cars ( x 4 ), and has determined that each car requires 3 carpentry hours, 2 sanding hours and 3 painting hours. Determine how much pro f tmustbemadepercarforitto become worthwhile to make any cars. d.) (5 points) Set the pro f t per car equal to that determined in part (c) plus one dollar and then determine which of the soldiers, trains or dolls will no longer be pro f table to make.

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–––––––––––––––––––––––––––––––––––– Problem #2 (15 points) ToyWorld makes soldiers ( x 1 ), trains ( x 2 ) and dolls ( x 3 )andth eLPfo r maximizing ToyWorld’s monthly pro f t (in dollars) is as follows. max z =\$20 x 1 +\$30 x 2 + \$40 x 3 (Pro f t Objective Function) s.t. 2 x 1 +4 x 2 +5 x 3 250 (Hours) (Carpentry Constraint) x 1 +2 x 2 +3 x 3 140 (Hours) (Sanding Constraint) 3 x 1 +7 x 2 x 3 320 (Hours) (Painting Constraint) x 1 ,x 2 3 0 (Sign Restrictions) With the given pro f ts per toy, this time the optimal solution results in ToyWorld making only soldiers and dolls (not trains), and 250 131 350 1 = 1 5 50 5 30 2 45 1 By how much must the pro f t per train increase so that it becomes pro f table to start making trains? Problem #3 (15 points) Construct the Dual of the following LP problem. min w = 20 y 1 +30 y 2 +40 y 3 +50 y 4 (Objective Function) s.t. 2 y 1 y 2 y 3 + y 4 250 (Constraint #1) y 1 y 2 y 3 y 4 =140 (Constraint #2) 3 y 1 y 2 y 3 y 4 320 (Constraint #3) y 1 is urs ,y 2 3 0 and y 4 0 (Sign Restrictions) You f nal answer should have only 3 variables ( x 1 , x 2 and x 3 )and4con - straints. 2
–––––––––––––––––––––––––––––––––––– Problem #4 (20 points) Consider the following primal LP problem.

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## This note was uploaded on 03/02/2011 for the course ESE 304 taught by Professor Michaela.carchidi during the Winter '11 term at UPenn.

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hmk_4_s - ESE304 - Introduction to Optimization (Homework...

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