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.