17_Review_of_LP - 15.082 and 6.855J Review of Linear...

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1 15.082 and 6.855J Review of Linear Programming
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2 Overview Describe LP and IP min cost flow as an LP Graphical solution Basic feasible solutions. Simplex Method Basic feasible solutions in matrix form Duality Note: this will cover lots of material. We will also have a recitation.
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3 The Minimum Cost Flow Problem u ij = capacity of arc (i,j). c ij = unit cost of shipping flow from node i to node j on (i,j). x ij = amount shipped on arc (i,j) Minimize (i,j) A c ij x ij j x ij - k x ki = b i for all i N . and 0 x ij u ij for all (i,j) A.
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4 Terminology j x ij - k x ki = b i for all i N . and 0 x ij u ij for all (i,j) A. x ij = Decision variable . Describes a decision to be made Minimize (i,j) A c ij x ij Objective Function Constraints
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5 Terminology j x ij - k x ki = b i for all i N . and 0 x ij u ij for all (i,j) A. Minimize (i,j) A c ij x ij Objective Function Constraints In a linear program, the objective function and the constraints are all linear. Typically, but not always, the variables are constrained to be non-negative. If variables are constrained to be integers, it is called an integer program.
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6 Production Planning : Given several products with varying production requirements and cost structures, determine how much of each product to produce in order to maximize profits. Scheduling : Given a staff of people, determine an optimal work schedule that maximizes worker preferences while adhering to scheduling rules. Portfolio Management : Determine bond portfolios that maximize expected return subject to constraints on risk levels and diversification. And an incredible number more. Some Applications of LPs + IPs
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7 Graphing 2-Dimensional LPs Example 1: x 3 0 1 2 y 0 1 2 4 3 Feasible Region x 0 y 0 x + 2 y 2 y 4 x 3 Subject to: Maximize x + y Optimal Solution These LP animations were created by Keely Crowston.
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8 Graphing 2-Dimensional LPs Example 2: Feasible Region x 0 y 0 -2 x + 2 y 4 x 3 Subject to: Minimize ** x - y Multiple Optimal Solutions! 4 1 x 3 1 2 y 0 2 0 3 1/3 x + y 4
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9 Graphing 2-Dimensional LPs Example 3: Feasible Region x 0 y 0 x + y 20 x 5 -2 x + 5 y 150 Subject to: Minimize x + 1/3 y Optimal Solution x 30 10 20 y 0 10 20 40 0 30 40
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y x 0 1 2 3 4 0 1 2 3 x 30 10 20 y 0 10 20 40 0 30 40 Do We Notice Anything From These 3 Examples? x y 0 1 2 3 4 0 1 2 3
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A Fundamental Point If an optimal solution exists, there is always a corner point optimal solution! y
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This note was uploaded on 05/05/2010 for the course EE 15.082 taught by Professor Orlin during the Spring '10 term at Visayas State University.

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17_Review_of_LP - 15.082 and 6.855J Review of Linear...

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