ISM 206 Lecture 2

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ISM 206Lecture 2

• Intro to Linear Programming

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Announcements

• Scribe Schedule on website

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Next Four Lectures Linear Programming

• Properties of LPs
• The Simplex Method
• Sensitivity and Duality
• Alternative Methods for solving

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Outline

• Typical Linear Programming Problems
• Standard Form
• Converting Problems into standard form
• Geometry of LP
• Extreme points, linear independence and bases
• Optimality Conditions
• The simplex method
• Graphically
• Analytically

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Product Mix Problem

• How much beer and ale to produce from three
  scarce resources
• 480 pounds of corn
• 160 ounces of hops
• 1190 pounds of malt
• A barrel of ale consumes 5 pounds of corn, 4
  ounces of hops, 35 pounds of malt
• A barrel of beer consumes 15 pounds of corn, 4
  ounces of hops and 20 pounds of malt
• Profits are 13 per barrel of ale, 23 for beer

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Transportation Problem

• A firm produces computers in Singapore and
  Hoboken.
• Distribution Centers are in Oakland, Hong Kong
  and Istanbul
• Supply, demand and costs summary

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Other LP examples

• Blending problem
• Diet problem
• Assignment problem

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Key Elements of LPs

• Proportionality
• Additivity
• Divisibility
• Building a Linear Model
• Identify activities
• Identify items
• Identify input-output coefficients
• Write the constraints
• Identify coefficients of objective function

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Geometry of LP

• Consider the plot of solutions to a LP

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Types of LP descriptions
To deal with different types of objectives and
constraints we convert each linear program to
standard form.
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Standard Form(according to Hillier and Lieberman)
Concise version
A is an m by n matrix n variables, m constraints
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Converting into Standard Form

• Slack/surplus variables
• Replacing free variables
• Minimization vs maximization

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Standard Form to Augmented Form
A is an m by n matrix n variables, m constraints
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Questions and Break
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Claim We only need to worry about corner points
(basic feasible solutions)

• Proof Assume there is a better interior point
• This is a convex combination of 2 extreme points
• Easy to show one must be at least as good

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Basic Feasible Solutions

• We have an equation Axb with more columns than
  rows
• How do we normally solve this?
• A basic solution corresponds to one that uses
  only linearly independent columns of A
• A basic feasible solution is also feasible

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Solutions, Extreme points and bases

• Linear independence of vectors
• Basis of a matrix
• A basic solution of an LP
• Basic Feasible solution (Corner Point Feasible)
• The vector x is an extreme point of the solution
  space iff it is a bfs of Axb, xgt0
• Key fact
• If a LP has an optimal solution, then it has an
  optimal extreme point solution

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Note on Rank of a matrix

• Rank of a matrix no. linearly independent cols
  (and rows)
• rankltminm,n
• A has full rank if rank(A)m
• If A is of full rank then there is at least one
  basis B of A
• B is set of linearly independent columns of A
• We will generally assume that A is of full rank

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Simplex Method

• Checks the corner points
• Gets better solution at each iteration
• 1. Find a starting solution
• 2. Test for optimality
• If optimal then stop
• 3. Perform one iteration to new CPF (BFS)
  solution. Back to 2.