Models in I.E. Lectures 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

Example: Times Routing 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

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 Conceptual Model

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

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

Spectrum of Optimization Models Less General Easier to solve Can solve larger cases and/or can solve cases more quickly More general Applies to more problems but harder to solve, especially to 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 Path Min Span Tree Max Flow Assignment Transportation Min Cost Flow portfolio optimization chemical processes materials design blending planning logistics scheduling production/distribution flow 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.