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Models in I.E. Lectures 22-23

Models in I.E. Lectures 22-23. Introduction to Optimization Models: Shortest Paths. Shortest Paths : Outline. Shortest Path Examples: Distances Times Definitions More Examples Costs Reliability Optimization Models. Example: Distances Shortest Auto Travel Routes.

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Models in I.E. Lectures 22-23

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  1. Models in I.E.Lectures 22-23 Introduction to Optimization Models: Shortest Paths

  2. Shortest Paths: Outline • Shortest Path Examples: • Distances • Times • Definitions • More Examples • Costs • Reliability • Optimization Models

  3. Example: DistancesShortest Auto Travel Routes

  4. Example: TimesRouting messages on the internet

  5. 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 >= 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

  6. 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)

  7. 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 >= 0 • the instances are not too huge

  8. Shortest path:More examples

  9. Goal: have use of a car for 4 years at minimum cost

  10. 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)

  11. Auto use: shortest path

  12. 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?

  13. Reliable Paths • Reliability of a path = product of Pij for edges ij on path • Maximizing a product instead of minimizing a sum -- doesn’t seem to fit shortest path model • Method (trick used more than once): • set Cij = - log Pij

  14. How we use optimization models Data Math Problem (Optimization Model) Real problem Algorithm Solution to Math Problem

  15. How we use optimization models Data Real problem Math Problem (Optimization Model) Conceptual Model Algorithm Solution to Math Problem

  16. The model must fit the real problem We must be able to solve the model To use a model successfullyWe need TWO things Realism or Generality Solvability or Tractability

  17. The model must fit the real problem We must be able to solve the model To use a model successfullyWe need TWO things T E N S I O N

  18. 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

  19. 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

  20. 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

  21. 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.

  22. 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

  23. 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

  24. 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.

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