Introduction to Operations Research Revised Simplex Method

Linear programming 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? Maximize 5x1 + 3x2 + 4x3 Subject to: 2x1 + x2 + x3 ≤ 20 3x1 + x2 + 2x3 ≤ 30 x1, x2, x3 ≥ 0

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

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

Revised simplex method

Revised Simplex method Initial Tableau

Revised Simplex Method 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. Maximize Z = 3x1 + 5x2 Subject to: x1 ≤ 4 2x2 ≤ 12 3x1 + 2x2 ≤ 18 x1, x2 ≥ 0

Revised Simplex method Initial Tableau

Revised Simplex method

Revised simplex method 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

Initial Basis BxB =b

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

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.

Wyndor example Maximize Z = 3x1 + 5x2 Subject to: x1 ≤ 4 2x2 ≤ 12 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