1. Penyelesaian PL Dengan Karmarkar Pertemuan 9 :
Mata kuliah : K0164/ Pemrograman Matematika
Tahun : 2008
Penyelesaian PL Dengan Karmarkar Pertemuan 9 :

2. Learning Outcomes
Mahasiswa dapat menyelesaikan PL dengan menggunakan metode Karmarkar

3. Outline Materi: Model Karmarkar (Interior point method)
Algoritma Karmarkar
Contoh penyelesaian &diskusi
Studi kasus

4. Interior Point Method x2 40 20 x1 30 Starts at feasible point
Moves through interior of feasible region
Always improves objective function
Theoretical interest

5. Karmarkar’s Method The LP form of Karmarkar’s method minimize z = CX
subject to
AX = 0
1X = (1)
X ≥ 0
This LP must also satisfy
satisfies AX = 0 (2)
Optimal z-value = 0 (3)
Where X= (x1, x2, …., xn)T , A is an m x n matrix

6. Karmarkar’s Method Suppose the LP is in the form (1) - (3)
Step 1: k = 0, start with the solution point and compute
Step 2: stop if CXk < ε, else go to Step 3
Step 3:
Define
Compute
Where

7. How to Transform any LP to the Karmarkar’s Form
Step 1: set up the dual form of the LP
Step 2: apply the dual optimal condition to form the combined feasible region “=“ form
Step 3: Convert the combined feasible region to the homogeneous form: AX = 0
Add the “ sum of all variables ≤ M” constraint
Convert this constraint to “=“ form
Introduce new dummy variable d2 = 1 to the system to convert the system to AX = 0 and 1X = M + 1
Step 4: convert the system to the form (1)-(3)
Introduce the set of new variables xj = (M +1)xj’ to convert the system to the form AX’ = 0 and 1X’ = 1
Introduce new dummy variable d3’ to ensure (2) and (3)

8. Examples Example 1: convert the following LP to the Karmarkar’s LP
Maximize z = 3x1 + x2
Subject to 2x1 – x2 ≤ 2
x1 + 2x2 ≤ 5
x1, x2 ≥ 0
Example 2: Perform one iteration of Karmarkar’s method for the following LP
Minimize z = 2x1 + 2x2 – 3x3
s.t x1 – 2x2 + 3x3 = 0
x1 + x2 + x3 = 1
x1, x2, x3 ≥ 0

9. Computational Method Interior Point Methods
Barrier or interior-point methods, by contrast, visit points within the interior of the feasible region. These methods derive from techniques for nonlinear programming that were developed and popularized in the 1960s by Fiacco and McCormick, but their application to linear programming dates back only to Karmarkar's innovative analysis in 1984.

10. Interior Point Method
Step 1: Choose any feasible interior point solution,
and set solution index t=0.
Step 2: If any component of is 0, or if recent steps have made no significant change in the solution value, stop. Current point is either optimal or very nearly so.
Step 3: Construct the next move direction

11. Interior Point Method -contd
Where ,

12. Interior Point Method - contd
Step 4: If there is no limit on feasible moves in the direction (all components are nonnegative), stop ; the given model is unbounded. Otherwise, construct the step size

13. Interior Point Method - contd
Step 4: compute the new solution
Then let and return to Step 2.

14. A Simple Example
The Marriott Tub Company manufactures two lines of bathtubs, called Model A and model B. Every tub requires a certain amount of steel and zinc; the company has available a total of 25,000 pounds of steel and 6,000 pounds of zinc. Each model A bathtub requires a total of 125 pounds of steel and 20 pounds of zinc, and each yields a profit of $90. Each model B bathtub can be sold for a profit of $70; it in turn requires 100 pounds of steel and 30 pounds of zinc. Find the best production mix of the bathtubs.

15. The Formulation Maximize Subject to
Where x and y are the numbers of model A and model B bathtubs that the company will make, respectively.

16. Solving by Interior Point Methodx1 x2 x3 x4
Obj
Initial
100
2500
1000
16000
1st
129.4
86.4
178.8
818.8
17698
2nd
152.8
58.7
3.3
1184
17861
3 rd
157
53.6
5.7
1250
17882
4 th
189
13.60
8.3
1810
17962
5 th
199.6
0.4
6.5
1995
17992
6 th
199.7
0.3
0.1
1994
17994
7 th
199.9
1999
17998
8 th
200
2000
18000

17. Terima kasih,
Semoga berhasil