sec14_Notes

# sec14_Notes - 14 The revised simplex method Consider once...

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

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

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 14 The revised simplex method Consider once again the standard equality-form linear program maximize c T x = z subject to Ax = b x ≥ , Corresponding to any basis B is a unique tableau. Theorem 12.2 gave a formula for that tableau, which we can rewrite slightly as follows: z- X j ∈ N ( c j- y T A j ) x j = y T b x B + X j ∈ N A- 1 B A j x j = ˆ x B , where the vector y solves the equation A T B y = c B , and the vector ˆ x B (whose components are the current values of the basic variables) is A- 1 B b . Recall that N is a list of the nonbasic indices, the vector A j is the j th column of the matrix A ; as usual, A B is the basis matrix, and analogously, the vector c B has entries c i as the index i runs through the basis B . Using this notation, consider how an iteration of the simplex method proceeds. We begin by choosing an entering variable x k (where k ∈ N ) with positive reduced cost c k- y T A k . To do this, we must first calculate the vector y . Step 1: Solve the equation A T B y = c B for the vector y . Step 2: Choose an entering index k ∈ N such that c k > y T A k . The next step is the ratio test, for which we need the column vector whose components are the coefficients of the entering variable x k in the body of the tableau. We can see from the formula for the tableau that this vector, which we call d , is just A- 1 B A k . Step 3: Solve the equation A B d = A k for the vector d . Step 4: Calculate t ← min n ˆ x i d i : i ∈ B, d i > o , and choose a leaving index r from among those indices i attaining this minimum. 68 Since the ratio test at the next iteration will involve the new values of the basic variables, it is helpful to calculate these values. Step 5: Update the current values of the variables, ˆ x k ← t, ˆ x B ← ˆ x B- td, and then the basis: Replace r by k in the ordered list B. We write B ← B ∪ { k } \ { r } ....
View Full Document

{[ snackBarMessage ]}

### Page1 / 7

sec14_Notes - 14 The revised simplex method Consider once...

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

View Full Document
Ask a homework question - tutors are online