Parametric Linear Programming
Parametric Linear Programming-1
Systematic Changes in cj
n
•
Objective function
Z   c j x j is replaced by
j 1
n
Z      c j   j x j
j 1
•
Find the optimal solution as a function of θ
Z*(θ)
0
θ1
θ2
θ
Parametric Linear Programming-2
Example: Wyndor Glass Problem
• Z(θ) = (3 + 2θ) x1+(5 - θ) x2
Parametric Linear Programming-3
Example: Wyndor Glass Problem
Range
of θ
0 ≤ θ ≤ 9/7
Basic
Var.
Z
x1
x2
x3
x4
x5
RHS
Z(θ)
1
0
0
0
(9-7θ)/6
(3+2θ)/3
36-2θ
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
Parametric Linear Programming-4
Example: Wyndor Glass Problem
Range
of θ
9/7 ≤ θ ≤ 5
Basic
Var.
Z
x1
x2
x3
x4
x5
RHS
Z(θ)
1
0
0
(-9+7θ)/2
0
(5-θ)/2
27+5θ
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
Parametric Linear Programming-5
Example: Wyndor Glass Problem
Range
of θ
θ≥5
Basic
Var.
Z
x1
x2
x3
x4
x5
RHS
Z(θ)
1
0
-5+θ
3+2θ
0
0
12+8θ
x4
0
0
2
0
1
0
12
x5
0
0
2
-3
0
1
6
x1
0
1
0
1
0
0
4
Parametric Linear Programming-6
Procedure Summary for Systematic
Changes in cj
1. Solve the problem with θ = 0 by the simplex method.
2. Use the sensitivity analysis procedure to introduce the
Δcj = αjθ 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-7
Systematic Changes in bi
n
•
Constraints
a x
j
 bi for i  1,2,
a x
j
 bi  i for i  1,2,
j 1
n
j 1
•
ij
ij
,m
are replaced by
,m
Find the optimal solution as a function of θ
Z*(θ)
0
θ1
θ2
θ
Parametric Linear Programming-8
Example: Wyndor Glass Problem
• y1
+ 3y3 ≥ 3 + 2θ
2y2 + 2y3 ≥ 5 - θ
Parametric Linear Programming-9
Example: Wyndor Glass Problem
Range
of θ
0 ≤ θ ≤ 9/7
Basic
Var.
Z
y1
y2
y3
y4
y5
RHS
Z(θ)
1
2
0
0
2
6
-36+2θ
y3
0
1/3
0
1
-1/3
0
(3+2θ)/3
y2
0
-1/3
1
0
1/3
-1/2
(9-7θ)/6
Parametric Linear Programming-10
Example: Wyndor Glass Problem
Range
of θ
9/7 ≤ θ ≤ 5
Basic
Var.
Z
y1
y2
y3
y4
y5
RHS
Z(θ)
1
0
6
0
4
3
-27-5θ
y3
0
0
1
1
0
-1/2
(5-θ)/2
y1
0
1
-3
0
-1
3/2
(-9+7θ)/2
Parametric Linear Programming-11
Example: Wyndor Glass Problem
Range
of θ
θ≥5
Basic
Var.
Z
y1
y2
y3
y4
y5
RHS
Z(θ)
1
0
12
6
4
0
-12-8θ
y5
0
0
-2
-2
0
1
-5+θ
y1
0
1
0
3
-1
0
3+2θ
Parametric Linear Programming-12
Procedure Summary for Systematic
Changes in bi
1. Solve the problem with θ = 0 by the simplex method.
2. Use the sensitivity analysis procedure to introduce the
Δbi = αiθ 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.
Parametric Linear Programming-13