1 / 14

Introduction to Operations Research

Revised Simplex Method. Introduction to Operations Research. How many basic variables are there in this problem? If we know that x 2 and x 3 are non-zero, how can we find the optimal solution directly?

osma
Télécharger la présentation

Introduction to Operations Research

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Revised Simplex Method Introduction to Operations Research

  2. How many basic variables are there in this problem? • If we know that x2 and x3 are non-zero, how can we find the optimal solution directly? • How can this help us use the simplex method more efficiently, assuming we start from the usual initial solution? • What if I told you that x3 was the only non-zero decision variable in a solution. How could you find that solution? Linear programming Maximize 5x1 + 3x2 + 4x3 Subject to: 2x1 + x2 + x3 ≤ 20 3x1 + x2 + 2x3 ≤ 30 x1, x2, x3 ≥ 0

  3. Linear Programming Description Maximize Z = c1x1 + … + cnxn Subject to. a11x1 + … + a1nxn ≤ b1 am1x1 + … + amnxn ≤ bm x1 ≤ 0, …, xn ≤ 0

  4. Revised Simplex Method • Re-write problem in matrix form Max Z = cx s.t. Ax ≤ b and x ≥ 0 c = [c1,c2,…,cn] contribution of each xi to Z

  5. Revised simplex method

  6. Revised Simplex method • Initial Tableau

  7. Put Wyndor Glass problem in Matrix form Fill in values for each Matrix or vector: c, A, x, b, I. xs Convince yourself that this really represents the initial tableau. Revised Simplex Method Maximize Z = 3x1 + 5x2 Subject to: x1 ≤ 4 2x2 ≤ 12 3x1 + 2x2 ≤ 18 x1, x2 ≥ 0

  8. Revised Simplex method • Initial Tableau

  9. Revised Simplex method

  10. Recall that in any basic solution m variables are basic (positive) and n variables are non-basic (equal to zero) • Ignore all the non-basic variables. • Let xB be the vector of basic variables • Let B be the Matrix of coefficients of basic variables. • Then it is true that BxB=b Revised simplex method

  11. Initial Basis BxB =b

  12. Revised Simplex method Consider the following constraint set, in augmented form. (x3 and x4 are slacks) x1 + 2x2 + x3 = 5 2 x1 + 3x2 + x4 = 8 If x1 and x3 are basic, then we know that x2 and x4 each equal 0, so we can rewrite the constraint set as x1 + x3 = 5 2 x1 = 8 In matrix form: xB = [x1, x3] and You can see that BxB = b

  13. Revised Simplex method • In any iteration it is true that BxB = b • We can solve for xB • B-1BxB = B-1b • xB = B-1b • So, if I tell you the basis, you can tell me the value of the basic variables.

  14. Wyndor example • The basic variables are x1, x3, x4. What are their values? • xB = B-1b Maximize Z = 3x1 + 5x2 Subject to: x1 ≤ 4 2x2 ≤ 12 3x1 + 2x2 ≤ 18 x1, x2 ≥ 0

More Related