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# Lecture04 - IE 505 MATHEMATICAL PROGRAMMING Bahar Yetis...

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IE 505 MATHEMATICAL PROGRAMMING Bahar Yetis Kara Lecture #4 Date: 23 th September 2008 1

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Example Solve 2 0 , 8 2 6 . . 3 max 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 max x x 3 3 2 1 x x x 1 x 2 3 14 , 3 4 * 2 * 1 x x

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GEOMETRIC SOLUTION 4
GEOMETRIC SOLUTION Only suitable for solving very small problems (up to 2 variables) Provides a general insight 5

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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 0 . . min 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

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8 Remember
Two Crude Petroleum Example 9 Draw and See

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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

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Example Solve 12 0 , 24 12 2 28 2 7 . . 10 5 min 2 1 2 1 2 1 2 1 x x x x x x t s x x
Example Solve 13 0 , 24 12 2 28 2 7 . . 10 5 min 2 1 2 1 2 1 2 1 x x x x x x t s x x (3.6, 1.4)

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GEOMETRIC SOLUTION: Case 1 Unique Finite Optimal Solution: F is nonempty and Ω is a singleton. 14 Feasible region Feasible region c c
Example Solve 15 0 , 8 2 6 . .

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