# lecture3 - Quotes of the day 15.053 ! February 8, 2011 You...

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1 15.053 February 8, 2011 ! The Geometry of Linear Programs the geometry of LPs illustrated 2 Quotes of the day You don't understand anything until you learn it more than one way. Marvin Minsky One finds limits by pushing them. Herbert Simon Overview ! Views of linear programming Geometry Algebra Economic interpretations 3 What does the feasible region of an LP look like? 4 Three 2-dimensional examples

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Some 3-dimensional LPs 5 mathworld.wolfram.com/ ConvexPolyhedron.html 6 Goal of this Lecture: understand the geometrical nature of 2 and 3 dimensional LPs ! What properties does the feasible region have? ! What properties does an optimal solution have? ! How can one find the optimal solution: the l geometric method z The simplex method 7 A Two Variable Linear Program (a variant of the DTC example) x, y ! 0 2x + 3y " 10 x + 2y " 6 x + y " 5 y " 3 x " 4 z = 3x + 5y objective (1) (2) (3) (4) (5) (6) Constraints 8 Finding an optimal solution ! Introduce yourself to your partner ! Try to find an optimal solution to the linear program, without looking ahead.
9 Inequalities x y An inequality with two variables determines a unique half-plane x + 2y " 6 1 2 3 4 5 6 1 2 3 4 5 10 1 2 3 4 5 6 1 2 3 4 5 Graph the Constraints : 2x+ 3y " 10 (1) x ! 0 , y ! 0. (6) x y 2x + 3y = 10 Graphing the Feasible Region 11 1 2 3 4 5 6 1 2 3 4 5 Add the Constraint : x + 2y " 6 (2) x y x + 2y = 6 12 1 2 3 4 5 6 1 2 3 4 5 Add the Constraint : x + y " 5 x y x + y = 5 A constraint is called redundant if deleting the constraint does not increase the size of the feasible region. l x + y = 5 z is redundant

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13 1 2 3 4 5 6 1 2 3 4 5 Add the Constraints : x " 4; y " 3 x y We have now graphed the feasible region. 14 x y 1 2 3 4 1 2 3 The geometrical method for optimizing 3x + 5y Graph points such that 3x + 5y = p for various values of p.
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## This note was uploaded on 01/12/2012 for the course MANAGEMENT 15.053 taught by Professor Jamesorlin during the Spring '11 term at MIT.

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lecture3 - Quotes of the day 15.053 ! February 8, 2011 You...

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