{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

# S4 - C&O 350 Linear Optimization Fall 2009 Solutions to...

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

C&O 350 Linear Optimization – Fall 2009 – Solutions to Assignment 4 Question # Max. marks Part marks 1 8 2 6 2 8 6 2 3 8 4 4 4 8 2 2 2 2 5 8 4 4 6 (Bonus) (5) Total 40 + (5) 1. 8 marks = 2+6 Solution: (a) We start with: z + x 1 - x 2 + x 3 + x 4 + x 5 - x 6 = 0 x 1 + x 4 + 6 x 6 = 9 3 x 1 + x 2 - 4 x 3 + 2 x 6 = 2 x 1 + 2 x 3 + x 5 + 2 x 6 = 6 The tableau with basis { 1 , 2 , 3 } is: z - 9 4 x 4 + 5 2 x 5 - 29 x 6 = - 77 2 x 1 + x 4 + 6 x 6 = 9 x 2 - 5 x 4 + 2 x 5 - 24 x 6 = - 31 x 3 - 1 2 x 4 + 1 2 x 5 - 2 x 6 = - 3 2 This does not give us a feasible solution. The tableau with basis { 1 , 4 , 5 } is: z - 2 3 x 2 - 7 3 x 3 - 25 3 x 6 = - 43 3 - 1 3 x 2 + 4 3 x 3 + x 4 + 16 3 x 6 = 25 3 x 1 + 1 3 x 2 - 4 3 x 3 + 2 3 x 6 = 2 3 - 1 3 x 2 + 10 3 x 3 + x 5 + 4 3 x 6 = 16 3 This gives us the feasible solution x * = [ 2 3 , 0 , 0 , 25 3 , 16 3 , 0] T . We will use this tableau to start the simplex method in (b). (b) There are many possible solutions to this. We’ll present one that follows Dantzig’s rule (see page 78 of coursenotes). Starting with the tableau that corresponds to the basis { 1 , 4 , 5 } , we can choose between x 2 , x 3 or x 6 as our entering variable. Since max { c 2 , c 3 , c 6 } = max 2 3 , 7 3 , 25 3 = c 6 , so Dantzig’s rule chooses x 6 to enter. Next, we compute t = min 25 3 / 16 3 , 2 3 / 2 3 , 16 3 / 4 3 = min 25 16 , 1 , 4 = 1, so x 1 leaves. z + 25 2 x 1 + 7 2 x 2 - 19 x 3 = - 6 - 8 x 1 - 3 x 2 + 12 x 3 + x 4 = 3 3 2 x 1 + 1 2 x 2 - 2 x 3 + x 6 = 1 - 2 x 1 - x 2 + 6 x 3 + x 5 = 4 1

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

View Full Document
Now the only varible that has positive reduced cost is x 3 , so x 3 enters our basis. We compute min 3 12 , - , 4 6 = 1 4 , so x 4 leaves. The next tableau is z - 1 6 x 1 - 5 4 x 2 + 19 12 x 4 = - 5 4 - 2 3 x 1 - 1 4 x 2 + x 3 + 1 12 x 4 = 1 4 1 6 x 1 + 1 6 x 4 + x 6 = 3 2 2 x 1 + 1 2 x 2 - 1 2 x 4
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}