Models in I.E. Lecture 21

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Models in I.E.Lecture 21

• Introduction to Optimization Models Shortest
  Paths

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Shortest Paths Outline

• Shortest Path Examples
• Distances
• Times
• Definitions
• More Examples
• Costs
• Reliability
• Optimization Models

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Example DistancesShortest Auto Travel Routes
180
a
c
20
200
100
t
S
40
100
150
70
b
d
80
distances are in miles
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Example DistancesShortest Auto Travel Routes
180
a
c
20
200
100
t
S
40
100
150
70
b
d
80
Optimal solution to this case has length 270
miles. Note it does not use edge sa
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Example DistancesShortest Auto Travel Routes
180
a
c
20
200
100
t
S
40
100
150
70
b
d
80
Algorithm actually finds a tree giving
shortest paths from s to every node in graph
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Example TimesRouting packets on the internet
a
1.0
.2
c
.6
S
t
1.1
.8
.1
.5
.5
d
b
.3
Costs are in milliseconds
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Shortest Path Definitions

• Graph G (V,E)
• V vertex set, contains special vertices s and t
• E edge set
• Costs Cij on edges (i,j) in E
• Cij gt 0 The model we are studying
• no cycles with negative total cost
• arbitrary costs (rarely used too hard to solve)
• Cost of a path sum of edge costs
• Objective find min cost path from s to t

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Shortest Path

• Shortest Path is a particular kind of math
  problem, as is finding the roots of a quadratic
  polynomial or maximizing a differentiable
  function in one variable.
• Shortest Path is an Optimization Problem. It has
• A set of possible solutions (paths from s to t)
• An objective function (minimize the sum of edge
  costs)

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Shortest path as an optimization problem

• Shortest path has something else, which makes it
  useful...
• An algorithm that correctly and quickly solves
  cases of the shortest path problem, provided that
• the instances satisfy Cij gt 0
• the instances are not too huge

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Shortest pathMore examples
Shortest path is a math problem. It
doesnt matter if the edge costs are distances,
times, money, etc. It only matters that the goal
is to minimize the sum of costs on the path.
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Goal have use of a car for 4 years at minimum
cost
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Auto use example

• Vertices of graph need not represent physical
  locations
• V 0,1,2,3,4
• time 0, 1,...,4 in years
• Seek least expensive path from 0 to 4
• Edge cost from i to j cost of buying a car at
  time i, using it, and selling it at time j
• for each edge, pick cheapest alternative (new or
  used)

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Auto use shortest path
4
0
1
3
2
Our s is 0 our t is 4.
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Auto use shortest path
4
0
1
3
2
Buy at 1, sell at 2
Buy at 0 sell at 1
Buy at time 0, keep 2 years, sell at time 2
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calculating edge costs
Keep new car 1 year 15000 1000 - 11000
5000 Keep used car 1 year 5000 2000 -
4000 3000 Keep new car 2 years 15000
2000 - 9000 8000 Keep used car 2 years 5000
5000 - 3000 7000 edge cost is
the cheaper of the two alternatives
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Auto use shortest path
4
0
1
3
2
3000
3000
7000
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Auto use shortest path
4
0
1
3
2
3000
3000
etc.
7000
Note in this case, edges are actually
directed. You cant get from 2 to 1 at cost
3000. The shortest path model permits directed
edges.
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Example Reliability

• Send a packet on a network from s to t
• Transmission fails if any arc on path fails
• Arc ij successfully transmits a packet with
  probability Pij. Probabilities are independent.
• Problem what path on the network has the
  highest probability of successful transmission
  from s to t?

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Reliable Paths

• Reliability of a path product of Pij for edges
  ij on path
• Maximizing a product instead of minimizing a sum
  -- doesnt seem to fit shortest path model
• Method (trick used more than once)
• set Cij - log Pij

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How we use optimization models
Data
Math Problem (Optimization Model)
Real problem
Algorithm
Solution to Math Problem
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How we use optimization models
Data
Real problem
Math Problem (Optimization Model)
Conceptual Model
Algorithm
Solution to Math Problem
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To use a model successfullyWe need TWO things

• The model must fit the real problem

• We must be able to solve the model

Realism or Generality
Solvability or Tractability
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To use a model successfullyWe need TWO things
T E N S I O N

• The model must fit the real problem

• We must be able to solve the model

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Spectrum of Optimization Models
Less General
More general Applies to more problems but harder
to solve, especially to solve large cases
Easier to solve Can solve larger cases and/or
can solve cases more quickly
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Modeling

• Modeling is almost always a tradeoff between
  realism and solvability
• Good modelers know
• computational limits of different models
• how to make a model fit a wider range of real
  problems
• how to make a real problem fit into a model
• Advanced modelers know
• how to solve a wider range of models
• how to extend the range of cases that can be
  solved with software tools

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How to make a model fit a wider range of real
problems

• I. Mathematical agility
• example taking logs to convert max product to
  min sum
• example robot cleanup, minimax assignment
• II. Conceptual agility
• example Shortest path model for automobile use
  . Realizing that nodes on a graph need not
  represent physical locations or objects.
• example Shortest path model for stocking paper
  rolls at a cardboard box manufacturer

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How to make a real problem fit into a model

• JUDGEMENT (how to teach???)
• Cutting corners
• Approximating
• if your data are inexact....
• Aggregating
• Simplifying
• Example in automobile problem, we could decide
  to sell and purchase at any time, not just at
  start of year. But a continuous time decision
  model is more complex.

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modeling

• When you have a choice between two models, both
  of which capture the same information about the
  problem, use the model that is easier to solve

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Spectrum of Optimization Models
Networks Networks LP Convex QP IP
NLP
portfolio optimization
Shortest Path Min Span Tree Max Flow
Assignment Transportation Min Cost Flow
logistics scheduling
blending planning
chemical processes materials design
production/distribution flow of materials
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Preparation for Next Class

• We will concentrate on LP (linear programming)
  formulation
• Read the problems posted before class. We will
  not have time to read them during lecture.