Lecture04 - IE 505 MATHEMATICAL PROGRAMMING Bahar Yetis...

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Unformatted text preview: IE 505 MATHEMATICAL PROGRAMMING Bahar Yetis Kara Lecture #4 Date: 23 th September 2008 1 Example • Solve 2 , 8 2 6 . . 3 m ax 2 1 2 1 2 1 2 1        x x x x x x t s x x 3 (0,6) (6,0) 6 2 1   x x (0,4) 8 2 2 1    x x 2 1 3 m ax x x  3 3 2 1   x x x 1 x 2 3 14 , 3 4 * 2 * 1   x x GEOMETRIC SOLUTION 4 GEOMETRIC SOLUTION Only suitable for solving very small problems (up to 2 variables) Provides a general insight 5 GEOMETRIC SOLUTION Consider • Among all points in F (feasible region) find a point with minimal value. • Points with the same objective function value satisfy the equation 6 . . m in   x b Ax t s x c T x c T x c z T  GEOMETRIC SOLUTION Since z is to be minimized, then the plane must be moved parallel to itself in the direction – c as much as possible. 7 x c z T  Feasible region c Optimal solution 8 Remember Two Crude Petroleum Example 9 Draw and See GEOMETRIC SOLUTION • For a maximization problem, must be moved parallel to itself in the direction c as much as possible. • 4 possible cases depending on F and Ω . 10 x c z T  One Possibility 11 (0,6) (6,0) 6 2 1   x x (0,4) 8 2 2 1    x x 3 3 2 1   x x x 1 x 2 3 14 , 3 4 * 2 * 1   x x Example • Solve 12 , 2 4 1 2 2 2 8 2 7 . . 1 0 5 m in 2 1 2 1 2 1 2 1       x x x x x x t s x x Example • Solve 13 , 2 4 1 2 2 2 8 2 7 . . 1 0 5 m in 2 1 2 1 2 1 2 1       x x x x x x t s x x (3.6, 1.4) GEOMETRIC SOLUTION: Case 1 • Unique Finite Optimal Solution: F is nonempty and Ω is a singleton....
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This note was uploaded on 04/15/2010 for the course INDUSTRIAL ie513 taught by Professor Zeynephuygur during the Spring '10 term at Bilkent University.

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Lecture04 - IE 505 MATHEMATICAL PROGRAMMING Bahar Yetis...

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