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# lec4 - 15.053 z The Geometry of Linear Programs The simplex...

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15.053 February 15, 2007 z The Geometry of Linear Programs The simplex algorithm More properties of linear programs Pentagonal prism 1

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2 Overview of Lecture z Review of Geometry z The Simplex Algorithm z More on convexity z RHS Sensitivity Analysis
Quotes of the Day Geometry is not true, it is advantageous. Jules H. Poincare I've always been passionate about geometry and the study of three- dimensional forms. Erno Rubik 3

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4 x y 1 2 3 4 1 2 3 Review of Geometry A set S is convex if for every two points in the set, the line segment joining the points is also in the set; that is, p 1 p 2 Theorem. The feasible region of a linear program is convex. if p 1 , p 2 S, then so is(1- λ )p 1 + λ p 2 for λ∈ [0,1]
Corner Points z A corner point of the feasible region is a point that is not the midpoint of two other points of the feasible region. z All feasible LPs with non-negativity constraints have at least one corner point. If an LP is feasible, has non- negativity constraints, and has an optimal solution, then there is a corner point that is optimal. 5

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Solving for Corner Points z In two dimensions, a corner point is the intersection of two equality constraints. z In three dimensions, a corner point is the intersection of three constraints. (3 planes) x y 0 x 2 0 y 2 z 0 z 2 x - y + z 3 The red corner point is the intersection of three planes x = 2 z = The unique solution is x = 2, y = 1, z = 2. 2 x - y + z = 3 6
7 There are 3 Types of Linear Programs Those whose objective value is unbounded Those with no feasible solution. 1 2 3 4 1 2 3 1 23456 1 2 3 4 5 Isoprofit line Those with an optimal solution 1 2 3 4 1 2 3

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8 The Simplex Method in Two Dimensions 8 1 2 3 4 5 6 1 2 3 4 5 x y Start at any feasible corner point. Move to an adjacent corner point with better objective value. Move along an edge of the feasible region. Continue until no adjacent corner point has a better objective value. Max z = 3 x + 5 y 3 x + 5 y = 19
The Simplex Method Again Start at any feasible corner point. Move to an adjacent corner point with better x objective value. Move along an edge of the feasible region. 5 Continue until no adjacent corner point has a better objective value. 4 3 2 1 1 2 3 4 5 6 y 9

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The Simplex Method in 3 Dimensions Start at any feasible corner point. Move to an adjacent corner point with better objective value. Move along an edge of the feasible region. Continue until no adjacent corner point has a better objective value. Note: in two dimensions, the “edges” are the intersections of two constraints. The corner points are the intersection of three constraints.
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lec4 - 15.053 z The Geometry of Linear Programs The simplex...

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