Parametric Linear Programming-1

1
Parametric Linear Programming
2
Systematic Changes in cj

• Objective function is
  replaced by
• Find the optimal solution as a function of ?

3
Example Wyndor Glass Problem

• Z(?) (3 2?) x1(5 - ?) x2

4
Example Wyndor Glass Problem
Range of ? Basic Var. Z x1 x2 x3 x4 x5 RHS
Z(?) 1 0 0 0 (9-7?)/6 (32?)/3 36-2?
x3 0 0 0 1 1/3 -1/3 2
x2 0 0 1 0 1/2 0 6
x1 0 1 0 0 -1/3 1/3 2
0 ? 9/7
5
Example Wyndor Glass Problem
Range of ? Basic Var. Z x1 x2 x3 x4 x5 RHS
Z(?) 1 0 0 (-97?)/2 0 (5-?)/2 275?
x4 0 0 0 3 1 -1 6
x2 0 0 1 -3/2 0 1/2 3
x1 0 1 0 1 0 0 4
9/7 ? 5
6
Example Wyndor Glass Problem
Range of ? Basic Var. Z x1 x2 x3 x4 x5 RHS
Z(?) 1 0 -5? 32? 0 0 128?
x4 0 0 2 0 1 0 12
x5 0 0 2 -3 0 1 6
x1 0 1 0 1 0 0 4
? 5
7
Procedure Summary for Systematic Changes in cj

• 1. Solve the problem with ? 0 by the simplex
  method.
• Use the sensitivity analysis procedure to
  introduce the ?cj aj? changes into Eq.(0).
• Increase ? until one of the nonbasic variables
  has its coefficient in Eq.(0) go negative (or
  until ? has been increased as far as desired).
• 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.

8
Systematic Changes in bi

• Constraints
  are replaced by
• Find the optimal solution as a function of ?

9
Example Wyndor Glass Problem

• y1 3y3 3 2?
• 2y2 2y3 5 - ?

10
Example Wyndor Glass Problem
Range of ? Basic Var. Z y1 y2 y3 y4 y5 RHS
Z(?) 1 2 0 0 2 6 -362?
y3 0 1/3 0 1 -1/3 0 (32?)/3
y2 0 -1/3 1 0 1/3 -1/2 (9-7?)/6
0 ? 9/7
11
Example Wyndor Glass Problem
Range of ? Basic Var. Z y1 y2 y3 y4 y5 RHS
Z(?) 1 0 6 0 4 3 -27-5?
y3 0 0 1 1 0 -1/2 (5-?)/2
y1 0 1 -3 0 -1 3/2 (-97?)/2
9/7 ? 5
12
Example Wyndor Glass Problem
Range of ? Basic Var. Z y1 y2 y3 y4 y5 RHS
Z(?) 1 0 12 6 4 0 -12-8?
y5 0 0 -2 -2 0 1 -5?
y1 0 1 0 3 -1 0 32?
? 5
13
Procedure Summary for Systematic Changes in bi

• 1. Solve the problem with ? 0 by the simplex
  method.
• Use the sensitivity analysis procedure to
  introduce the ?bi ai? changes to the right side
  column.
• Increase ? until one of the basic variables has
  its value in the right side column go negative
  (or until ? has been increased as far as
  desired).
• 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.