final_s

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

Info iconThis preview shows pages 1–5. Sign up to view the full content.

View Full Document Right Arrow Icon
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 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 amax imumoram in imum . 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 f xed delivery charge of $5 , 000 . Vendor 2 charges $350 per computer plus a f xed delivery charge of $4 , 000 and vendor 3 charges $250 per computer plus a f 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.
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
–––––––––––––––––––––––––––––––––––– Problem #3 (20 points) - Chapter 12 Consider the nonlinear function f ( x 1 ,x 2 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 2 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 y 1 . b.) (5 points) Determine the minimum and maximum values of f ( x, y xy subject to the constraints 2 x 2 y 5 and 2 x y 1 along with the added restrictions that x and y are both integers . 2
Background image of page 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. 3
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
–––––––––––––––––––––––––––––––––––– Solution to Problem #1 Since f ( x 1 ,x 2 3 , ..., x n )= n X j =1 a j x j has no critical points, we may assume that the maximum and minimum values of f occur when the constraint is binding. With this, we may now use the method of Lagrange multipliers and construct L = n X j =1 a j x j + μ 1 n X j =1 x 2 j so that L x k = a k 2 μ x k =0 for k =1 , 2 , 3 ,...,m ,and L ∂μ n X j =1 x 2 j .
Background image of page 4
Image of page 5
This is the end of the preview. Sign up to access the rest of the document.

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.

Page1 / 14

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

This preview shows document pages 1 - 5. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online