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# final_s - ESE304 Introduction to Optimization(Final Exam...

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ESE304 - Introduction to Optimization (Final Exam) Fall Semester, 2010 M. Carchidi –––––––––––––––––––––––––––––––––––– Problem #1 (10 points) - Chapter 12 Determine the maximum and minimum values of f ( x 1 , x 2 , x 3 , . . . , x n ) = n X j =1 a j x j subject to the one non-linear constraint n X j =1 x 2 j 1 Hint : You may assume that the maximum and minimum values occur when the constraint is binding. Be sure to clearly justify whether your solution is a maximum or a minimum. –––––––––––––––––––––––––––––––––––– Problem #2 (20 points) - Chapter 9 State University must purchase 1 , 100 computers from three vendors. Ven- dor 1 charges \$500 per computer plus a fi xed delivery charge of \$5 , 000 . Vendor 2 charges \$350 per computer plus a fi xed delivery charge of \$4 , 000 and vendor 3 charges \$250 per computer plus a fi xed delivery charge of \$6 , 000 . Vendor 1 will sell the university at most 500 computers; vendor 2 at most 900 ; and vendor 3 at most 400 . Formulate an integer programming problem to minimize the cost of purchasing the needed computers. You need not solve the problem. ––––––––––––––––––––––––––––––––––––

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–––––––––––––––––––––––––––––––––––– Problem #3 (20 points) - Chapter 12 Consider the nonlinear function f ( x 1 , x 2 , x 3 ) = a ( x 2 1 + x 2 2 + x 2 3 ) + 2 x 1 x 2 + 2 x 2 x 3 , which has a critical point at ( x 1 , x 2 , x 3 ) = (0 , 0 , 0) . Determine the range of values for a for which this critical point is a: ( a ) local minimum point ( b ) local maximum point ( c ) saddle point. –––––––––––––––––––––––––––––––––––– Problem #4 (20 points) - Chapter 12 a.) (15 points) Determine the minimum and maximum values of f ( x, y ) = xy subject to the constraints 2 x 2 + 3 y 5 and 2 x + 3 y 1 . b.) (5 points) Determine the minimum and maximum values of f ( x, y ) = xy subject to the constraints 2 x 2 + 3 y 5 and 2 x + 3 y 1 along with the added restrictions that x and y are both integers . –––––––––––––––––––––––––––––––––––– 2
–––––––––––––––––––––––––––––––––––– Problem #5 (30 points) - Chapter 9 Determine the optimal solution to the following MIP problem. max z = 45 x 1 + 100 x 2 (Objective Function) s.t. 10 x 1 + 20 x 2 124 (Constraint #1) 20 x 1 + 50 x 2 292 (Constraint #2) x 2 = integer (Integer Restrictions) x 1 , x 2 0 (Sign Restrictions) You may start with the optimal tableau for the LP relaxation problem, Optimal Tableau For The LP Relaxation Problem Row z x 1 x 2 s 1 s 2 rhs BVs 0 [1] 0 0 5 / 2 1 602 z 1 0 [1] 0 1 / 2 1 / 5 18 / 5 x 1 2 0 0 [1] 1 / 5 1 / 10 22 / 5 x 2 for the LP relaxation of the above MIP.

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final_s - ESE304 Introduction to Optimization(Final Exam...

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