Lecture 20110211

# Lecture 20110211 - IMSE2008 Operational Research Techniques...

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IMSE2008 Operational Research Techniques Lecture 02/11 Geometry of Linear Programs implex Method Simplex Method Miao Song Dept of Industrial & Manufacturing Systems Engineering

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eview of Lecture 01/28 Review of Lecture 01/28 How to linearize piecewise linear objective / constraints? Min piecewise-linear convex function (minimax) ax piecewise- ear concave function ( aximin Max piecewise linear concave function (maximin) Max(linear functions) linear function in(linear functions) linear function Min(linear functions) Geometric method to solve 2-dimentional LP 2/11/2011 2
eometric Method to Solve LP Geometric Method to Solve LP y 5 Identify the feasible region Shift the isoquant line so that 4 There is still at least one feasible point on the line Moving further would make all points on the line feasible 3 infeasible 2 x 1 3x + 5y = 16 2/11/2011 3 0 1 2 3 4 5 6 3x + 5y = 0 3x + 5y = 8 3x + 5y = 11

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genda Agenda Geometry of Linear Programming Geometric Properties of Feasible Set and p Optimal Solutions implex Method Simplex Method 2/11/2011 4
xtreme Points Extreme Points An extreme point or corner point of the feasible region is a point that is not the midpoint of two other points of the feasible region What are the corner points of this feasible region? 2/11/2011 5

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Which Shapes Are Convex? Which Are LP Feasible Regions? hich Are the Extreme Points? Which Are the Extreme Points? 2/11/2011 6
olving for the Extreme Point (2D) Solving for the Extreme Point (2D) y 5 In two dimensions, an extreme point lies at 4 the intersection of two lines 3 x + 2y = 6 What are the extreme 2 2x + 3y = 10 points in 3D? x 1 2/11/2011 7 0 1 2 3 4 5 6

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olving for the Extreme Point (3D) Solving for the Extreme Point (3D) In three dimensions, an extreme point lies at the intersection of three constraints z (planes) x y What is the red extreme oint? 0 x 2 y 2 point? 0 y 0 z 2 +z 3 2/11/2011 8 x – y + z 3
Can an extreme point in 2D be an intersection of more than two lines?

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## This note was uploaded on 12/09/2011 for the course IMSE 0301 taught by Professor Song during the Spring '11 term at HKU.

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Lecture 20110211 - IMSE2008 Operational Research Techniques...

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