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exam_2_s

# exam_2_s - ESE304 Introduction to Optimization(Exam#2 Fall...

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ESE304 - Introduction to Optimization (Exam #2) Fall Semester, 2010 M. Carchidi –––––––––––––––––––––––––––––––––––– Problem #1 (25 points) Consider the following LP max z =11 x 1 +14 x 2 +18 x 3 (Objective Function) s.t. 2 x 1 +5 x 2 +4 x 3 44 (Constraint #1) 3 x 1 x 2 x 3 50 (Constraint #2) 6 x 1 x 2 +9 x 3 86 (Constraint #3) x 1 ,x 2 3 0 (Sign Restrictions) with optimal tableau given as follows. Optimal Tableau Row z x 1 x 2 x 3 s 1 s 2 s 3 rhs BVs 0 [1] 0 0 0 1 1 1 180 z 1 0 [1] 0 0 11 25 9 8 x 1 2 0 0 [1] 0 3 6 2 4 x 2 3 0 0 0 [1] 9 20 7 2 x 3 Suppose that an additional constraint given by x 1 + x 2 + x 3 b 4 is introduced into the problem. a.) (10 points) Determine the smallest value of b 4 so that the variables x 1 , x 2 and x 3 remain basic and their optimal values do not change. b.) (15 points) Suppose the value of b 4 is one ( 1 )lessthanthevaluedetermined in part (a). Determine the new optimal solution to this problem by starting with the above optimal tableau and using the Dual Simplex Method.

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–––––––––––––––––––––––––––––––––––– Problem #2 (25 points) Consider the following LP max z =3 x 1 +3 x 2 +4 x 3 (Objective Function) s.t. 4 x 1 +2 x 2 x 3 46 (Constraint #1) 2 x 1 x 2 x 3 36 (Constraint #2) 3 x 1 x 2 +5 x 3 50 (Constraint #3) x 1 ,x 2 3 0 (Sign Restrictions) with the following optimal tableau. A Portion of the Optimal Tableau Row z x 1 x 2 x 3 s 1 s 2 s 3 rhs BV 0 z 1 +1 / 3 2 / 3 +1 / 3 x 1 2 2 / 3 11 / 3 +10 / 3 x 2 3 +1 / 3 +10 / 3 8 / 3 x 3 a.) (12 points) Fill in the rest of the optimal tableau using only the matrix formulas discussed in Chapter 6. Note that if you f ll this in by using the Simplex Method, you will not receive any credit for part (a), but you will need the optimal tableau to do part (b) below. b.) (13 points) I ftheva lue so f b 1 =46 , b 2 =36 and b 3 =50 are changed to b 1 =46+ b 1 , b 2 =36+ b 2 and b 3 =50+ b 3 , determine the set of inequalities that b 1 , b 2 and b 3 should satisfy if x 1 , x 2 and x 3 are to remain as the basic variable. 2
–––––––––––––––––––––––––––––––––––– Problem #3 (25 points) Consider the following LP problem min w = b 1 y 1 + b 2 y 2 + b 3 y 3 + ··· + b n y n (Objective Function) st a 1 y 1 + a 2 y 2 + a 3 y 3 + + a n y n c (Constraint #1) y 1 ,y 2 3 ,...,y n 0 (Sign Restrictions) where all parameters: b 1 , b 2 , b 3 , ..., b n , a 1 , a 2 , a 3 , ..., a n and c are greater than zero. a.) (20 points) Use the Dual of this problem to determine a simple formula for w min in terms of b 1 , b 2 , b 3 , ..., b n , a 1 , a 2 , a 3 , ..., a n and c . Hint :Y ou r f nal answer should involve the minimum function de f ned as min { β 1 , β 2 , β 3 ,..., β n } = the smallest of { β 1 , β 2 , β 3 β n } .

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exam_2_s - ESE304 Introduction to Optimization(Exam#2 Fall...

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