18reviewoflinearprogramming

# 18reviewoflinearprogramming - 15.082 and 6.855J Review of...

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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.
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 x ij = Decision variable . Describes a decision to be made Objective Function 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. Constraints
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
7 Graphing 2-Dimensional LPs Example 1: 3 012 y 0 1 2 4 3 Feasible Region x Optimal Solution Maximize x + y x + 2 y 2 Subject to: x 3 y 4 x 0y 0 These LP animations were created by Keely Crowston.

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8 Graphing 2-Dimensional LPs Example 2: Feasible Region Multiple Optimal Solutions! 4 1 x 3 12 y 0 2 0 3 Minimize ** x - y -2 x + 2 y 4 Subject to: 1/3 x + y 4 x 3 x 0y 0
9 Graphing 2-Dimensional LPs Example 3: Feasible Region y x 30 10 20 0 10 20 40 0 30 40 Minimize x + 1/3 y x + y 20 Subject to: -2 x + 5 y 150 x 5 x 0y 0 Optimal Solution

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Do We Notice Anything From These 3 Examples? y y y x 0 1 01 2 2 3 4 3 x 30 10 20 0 10 20 40 x 0 1 2 3 4 012 3 40 30 0
11 A Fundamental Point y y y If an optimal solution exists, there is always a corner point optimal solution! x 0 1 01 2 2 3 4 3 x 30 10 20 0 10 20 40 x 0 1 2 3 4 012 3 40 30 0

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12 Graphing 2-Dimensional LPs Example 1: x 3 012 y 0 1 4 2 3 Feasible Region x 0y 0 x + 2 y 2 y 4 x 3 Subject to: Maximize x + y Optimal Solution Initial Corner pt.
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## This note was uploaded on 03/15/2010 for the course IE 505 taught by Professor Yok during the Spring '10 term at Galatasaray Üniversitesi.

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18reviewoflinearprogramming - 15.082 and 6.855J Review of...

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