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Unformatted text preview: Notes on Simplex Algorithm CS 149 Staff October 18, 2007 Until now, we have represented the problems geometrically, and solved by finding a corner and moving around. Now we learn an algorithm to solve this without drawing a graph, and feasible regions. Once we have a standard form of LP, we can construct a simplex tableau, which looks like following. c T A b as in lecture slide 5 on Simplex. In this phase, we assume the initial tableau represents feasible solution. (We will later deal with the case that it doesn’t) So, in the lecture slide 8 on Simplex, we see a tableau, which corresponds to a bfs, x = 0 ,y = 0. This is a Simplex tableau representation of the following problem: Min- x- 2 y- x + y ≤ 2 x- y ≤ 3 x + y ≤ 5 x ≤ 4 y ≤ 3 x,y ≥ We can see that the five variables added in front are slack variables to make it standard form. In this case our bfs and the vectors are: x = 2 3 5 4 3 y 1 = 1- 1- 1- 1 1 y 2 = - 1 1- 1- 1 1 With the instructions given in the slide 7, we choose a column with negative c T , which can either be 6th or 7th. In this case we ’arbitrarily’ choose 6th column. And 1 the next procedure is to find λ = min ( 3 1 , 5 1 , 4 1 ) = 3. Since it was the second row that gave us minimum, we choose second row. Therefore 6th column goes into the basis, and second column goes out. We now follow the direction of y 1 for length 3....
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- Spring '09
- Optimization, Column, Simplex algorithm, negative cT