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Learning Outcomes • Mahasiswa dapat menyelesaikan PL dengan menggunakan metode Karmarkar
Outline Materi: • Model Karmarkar (Interior point method) • Algoritma Karmarkar • Contoh penyelesaian &diskusi • Studi kasus
Interior Point Method Starts at feasible point Moves through interior of feasible region Always improves objective function Theoretical interest x2 40 20 x1 30 40
Karmarkar’s Method The LP form of Karmarkar’s method minimize z = CX subject to AX = 0 1X = 1 (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
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
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)
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
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.
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
Interior Point Method -contd Where ,
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
Interior Point Method - contd Step 4: compute the new solution Then let and return to Step 2.
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.
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.
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