midterm - NAME: PAGE 2 OF 8 Question 1. [5 marks] Convert...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

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

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: NAME: PAGE 2 OF 8 Question 1. [5 marks] Convert the following linear program into standard equality form. min 3 x 1- 2 x 2 + x 3 subject to 2 x 1- x 2 2 3 x 2- 5 x 3 = 6- x 2 +3 x 3 - 3 x 1 ,x 3 Solution: max 3 x 1 + 2 x + 2- 2 x- 2- x 3 subject to- 2 x 1- x + 2 + x- 2- s 1 = 2 3 x + 2- 3 x- 2- 5 x 3 = 6- x + 2 + x- 2 + 3 x 3 + s 2 =- 3 x 1 , x + 2 , x- 2 , x 3 , s 1 , s 2 . NAME: PAGE 3 OF 8 Question 2. [10 marks] Consider the linear program (P) max x 1 + x 2 subject to x 1 + 3 x 2- 6 x 3 + x 4 = 5 + x 2- 3 x 3- x 4 = 0 x 1- x 2 + 6 x 3 + 5 x 4 = 5 x 1 , x 2 , x 3 , x 4 , . (a) Write the dual (D) of (P). Solution: min 5 y 1 + 5 y 3 subject to y 1 + y 3 1 3 y 1 + y 2- y 3 1- 6 y 1- 3 y 2 + 6 y 3 y 1- y 2 + 5 y 3 (b) State the complementary slackness conditions for (P) and (D). Solution: For feasible solutions x and y of (P) and (D), respectively, x and y are optimal if and only if each of the following conditions hold: x 1 = 0 or y 1 + y 3 = 1 x 2 = 0 or 3 y 1 + y 2- y 3 = 1 x 3 = 0 or- 6 y 1- 3 y 2 + 6 y 3 = x 4 = 0 or y 1- y 2 + 5 y 3 = 0 . (c) Is the feasible solution x = (2 , 3 , 1 , 0) T of (P) optimal? Solution: x is optimal if and only if there exists a feasible solution y for (D) satisfying the com- plementary slackness conditions with x . So x is optimal if and only if the following linear system...
View Full Document

Page1 / 7

midterm - NAME: PAGE 2 OF 8 Question 1. [5 marks] Convert...

This preview shows document pages 1 - 3. Sign up to view the full document.

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