Tutorial_4

Tutorial_4 - 4 1 = X 2 2 = X complementary slackness...

This preview shows pages 1–2. Sign up to view the full content.

IELM202’05 Tutorial 4 1. Consider the linear program (P) Max Z= 2 1 5 3 X X + Subject to 4 1 X 12 2 2 X 18 2 3 2 1 + X X 0 , 2 1 X X Write down the dual of this LP According to the following table Primal model (MAX) Dual model (MIN) Constraint j is ≤ Variable yj ≥ 0 Constraint j is = Variable yj is unrestricted Constraint j is ≥ Variable yj ≤ 0 Variable xi ≥ 0 Constraint i is ≥ Variable xi is unrestricted Constraint i is = Variable xi ≤ 0 Constraint i is ≤ The Dual is Min W= 3 2 1 18 12 4 y y y + + St. 0 , 0 , 0 5 2 2 3 3 3 2 1 3 2 3 1 + + y y y y y y y 2. Max Z= 2 1 2 3 X X + s.t. 10 2 2 1 + X X 6 2 3 2 1 + - X X 6 2 1 + X X 0 , 0 2 1 X X (a) Construct a dual for this primal problem. (b) Show that , 4 1 = X 2 2 = X is an optimal to (P) by solving its dual using the Complementary Slackness Theorem.

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

View Full Document
IELM202’05 Tutorial 4 (a) the dual is Min W= 3 2 1 6 6 10 y y y + + s.t. 3 3 2 3 2 1 + - y y y 2 2 3 2 1 + + y y y 0 1 y 2 y 0, 3 y 0 (b) Check primal feasibility: 2*4+2=10<=10 -3*4+2*2=-8<=6 4+2=6<=6. Hence the give primal solution is feasible. Obtain dual solution using complementary slackness implies. Since
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: , 4 1 = X 2 2 = X , complementary slackness implies S 1 =10-10=0 S 2 =6-(-8)=14 S 3 =6-6=0 Since S i *y i =0 then we got the conclusion 2 = y (1) 3 3 2 3 2 1 1-+-= y y y e 2 2 3 2 1 2-+ + = y y y e * 1 1 = e x * 2 2 = e x Since X 1 =4 and X 2 =2 so 1 = e and 2 = e We got 3 3 2 3 2 1 =-+-y y y (2) And 2 2 3 2 1 =-+ + y y y (3) Solving equation (1), (2), and (3), we have 1 1 = y = 2 y 0, = 3 y 1 ( ) , , * 3 * 2 * 1 y y y =(1,0,1) W ( ) , , * 3 * 2 * 1 y y y = 3 2 1 6 6 10 y y y + + =16 Z( ) , * 2 * 1 x x =3*4+2*2=16 Z*=W* Based on Strong Duality Theorem: In a primal-dual LP pair, if either the Primal or the Dual has an optimal feasible bounded solution, then the two optimal objective values are equal...
View Full Document

{[ snackBarMessage ]}

Page1 / 2

Tutorial_4 - 4 1 = X 2 2 = X complementary slackness...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online