Models in I.E. Lectures 22-23
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Transcript of 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: DistancesShortest Auto Travel Routes
Example: TimesRouting messages on the internet
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
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)
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
Shortest path:More examples
Goal: have use of a car for 4 years at minimum cost
PurchaseCost
1st yearmaint.
1 yearResale
2nd yearmaint
2 yearResale
3rd yearmaint
3rd yearResale
NewCar
15000 1000 11000 1000 9000 1500 8000
UsedCar
5000 2000 4000 3000 3000 3500 2500
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)
Auto use: shortest path
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?
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
How we use optimization models
Real problem
Math Problem(Optimization
Model)
Solution to Math Problem
Data
Algorithm
How we use optimization models
Real problem
Math Problem(Optimization
Model)
Solution to Math Problem
Data
Algorithm
ConceptualModel
To use a model successfullyWe need TWO things
• The model must fit the real problem
• We must be able to solve the model
Realism orGenerality
Solvability orTractability
To use a model successfullyWe need TWO things
• The model must fit the real problem
• We must be able to solve the model
TENSION
Spectrum of Optimization Models
LessGeneralEasier to solveCan solve larger casesand/or can solve casesmore quickly
More general
Applies tomore problemsbut harder to solve, especiallyto solve large cases
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
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
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.
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
Spectrum of Optimization Models
Networks Networks+ LP Convex QP IP NLP
Shortest PathMin Span Tree Max Flow Assignment Transportation Min Cost Flow
portfoliooptimization
chemicalprocesses
materialsdesign
blending
planning
logistics
scheduling
production/distributionflow of materials
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.