Parametric Linear Programming-1 Parametric Linear Programming

Parametric Linear Programming-2 Systematic Changes in c j Objective function is replaced by Find the optimal solution as a function of θ Z*(θ) 0 θ1θ1 θ2θ2 θ

Parametric Linear Programming-3 Example: Wyndor Glass Problem Z(θ) = (3 + 2θ) x 1 +(5 - θ) x 2

Parametric Linear Programming-4 Example: Wyndor Glass Problem 0 ≤ θ ≤ 9/7 Range of θ Basic Var. Zx1x1 x2x2 x3x3 x4x4 x5x5 RHS Z(θ)1000(9-7θ)/6(3+2θ)/336-2θ x3x /3-1/32 x2x /206 x1x /31/32

Parametric Linear Programming-5 Example: Wyndor Glass Problem Range of θ Basic Var. Zx1x1 x2x2 x3x3 x4x4 x5x5 RHS Z(θ)100(-9+7θ)/20(5-θ)/227+5θ x4x x2x /201/23 x1x /7 ≤ θ ≤ 5

Parametric Linear Programming-6 Example: Wyndor Glass Problem Range of θ Basic Var. Zx1x1 x2x2 x3x3 x4x4 x5x5 RHS Z(θ)10-5+θ3+2θ0012+8θ x4x x5x x1x θ ≥ 5

Parametric Linear Programming-7 Procedure Summary for Systematic Changes in c j 1.Solve the problem with θ = 0 by the simplex method. 2.Use the sensitivity analysis procedure to introduce the Δc j = α j θ changes into Eq.(0). 3.Increase θ until one of the nonbasic variables has its coefficient in Eq.(0) go negative (or until θ has been increased as far as desired). 4.Use this variable as the entering basic variable for an iteration of the simplex method to find the new optimal solution. Return to Step 3.

Parametric Linear Programming-8 Systematic Changes in b i Constraints are replaced by Find the optimal solution as a function of θ Z*(θ) 0 θ1θ1 θ2θ2 θ

Parametric Linear Programming-9 Example: Wyndor Glass Problem y 1 + 3y 3 ≥ 3 + 2θ 2y 2 + 2y 3 ≥ 5 - θ

Parametric Linear Programming-10 Example: Wyndor Glass Problem 0 ≤ θ ≤ 9/7 Range of θ Basic Var. Zy1y1 y2y2 y3y3 y4y4 y5y5 RHS Z(θ) θ y3y3 01/301-1/30(3+2θ)/3 y2y2 0-1/3101/3-1/2(9-7θ)/6

Parametric Linear Programming-11 Example: Wyndor Glass Problem 9/7 ≤ θ ≤ 5 Range of θ Basic Var. Zy1y1 y2y2 y3y3 y4y4 y5y5 RHS Z(θ) θ y3y /2(5-θ)/2 y1y /2(-9+7θ)/2

Parametric Linear Programming-12 Range of θ Basic Var. Zy1y1 y2y2 y3y3 y4y4 y5y5 RHS Z(θ) θ y5y θ y1y θ Example: Wyndor Glass Problem θ ≥ 5

Parametric Linear Programming-13 Procedure Summary for Systematic Changes in b i 1.Solve the problem with θ = 0 by the simplex method. 2.Use the sensitivity analysis procedure to introduce the Δb i = α i θ changes to the right side column. 3.Increase θ until one of the basic variables has its value in the right side column go negative (or until θ has been increased as far as desired). 4.Use this variable as the leaving basic variable for an iteration of the dual simplex method to find the new optimal solution. Return to Step 3.