secn6 - 6 The simplex method We next formalize the method...

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: 6 The simplex method We next formalize the method we developed in the previous section. We again consider a general linear program in standard equality form: maximize c T x subject to Ax = b x ≥ . As before, we introduce a new variable z to keep track of the objective value, so the system of equations defining the linear program is (6.1) z- c T x = Ax = b. We break down each iteration of the solution algorithm, which is called the simplex method , into a sequence of steps. Step 0: Initialization. We (somehow) find an initial feasible basis B , form a list N consisting of its complement, and find the corresponding tableau: (6.2) z- X j ∈ N ¯ c j x j = ¯ v x i + X j ∈ N ¯ a ij x j = ¯ b i ( i ∈ B ) . This system is equivalent to the original constraints (6.1), and from it, we can read off the corresponding basic feasible solution: basic variables x i = ¯ b i ( i ∈ B ) nonbasic variables x j = 0 ( j ∈ N ) objective value z = ¯ v. Step 1: Check optimality. If each reduced cost satisfies ¯ c j ≤ 0 (for all indices j ∈ N ), then we stop: the current solution is optimal . To see this, notice that any feasible solution satisfies z = ¯ v + X j ∈ N ¯ c j x j ≤ ¯ v, since ¯ c j ≤ 0 and x j ≥ 0 (for all j ∈ N ). Thus no feasible solution can have objective value larger than ¯ v . But the current solution attains this value, so it must be optimal. Step 2: Choose entering index. If we don’t stop in Step 1, we can find an entering index k ∈ N with corresponding reduced cost ¯ c k > 0. Consider now the effect of increasing the value of the corresponding variable x k (called the 36 entering variable ) away from its current value zero to a new value t ≥ 0. At the next iteration of the simplex method, if the value of x k is nonzero, the index k must have entered the basis: hence the term “entering index”....
View Full Document

This note was uploaded on 10/31/2010 for the course ORIE 5300 at Cornell.

Page1 / 4

secn6 - 6 The simplex method We next formalize the method...

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