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UFO

Niklaus EggenbergMatteo Salani and Prof. M. Bierlaire

Uncertainty Feature Optimization, an Implicit Paradigm for Problems with Noisy Data

Transport and Mobility Laboratory, EPFL, Switzerland

CTW2008

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Schedule

Index

Motivations

The UFO framework

Existing Methods seen as UFO

Example: Problem with Multiple Knapsack

Constraints (MKC)

Preliminary Simulation Results for MKC

Future Work and Conclusions

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Motivation

Motivations I

• Uncertainty should not be neglected

• Uncertainty is hard to characterize exactly

• Problems under uncertainty are hard to solve in

general

• Few guarantees on real solution

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Four Approaches

Motivations II

1. Neglect and solve deterministic problem

Not realistic (Herroelen 2005, Sahinidis 2004)

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Four Approaches

Motivations III

1. Neglect and solve deterministic problem

2. On-line Optimization

Data-driven

Not feasible for some problems (e.g. airline

schedules)

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Four Approaches

Motivations IV

1. Neglect and solve deterministic problem

2. On-line Optimization

3. Characterize the Uncertainty and solve robust or

stochastic problems

Need explicit Uncertainty characterization

Hard to characterize/model in general

Leads to difficult problems

Solutions tend to “simple” properties

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Examples from Airline Scheduling

Motivations V

o Increase plane’s idle time (Al-Fawzana & Haouari 2005)

o Decrease plane rotation length (Rosenberger et al. 2004)

o Departure de-peaking (Jiang 2006, Frank et al. 2005)

o More plane crossings (Bian et al. 2004, Klabjan et al. 2002)

o …

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Four Approaches

Motivations VI

1. Neglect and solve deterministic problem

2. On-line Scheduling

3. Characterize the Uncertainty

4. Model Uncertainty Implicitly => Uncertainty Features

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Uncertainty Feature Optimization

Objectives

I. Increase robustness/stability (e.g. idle time)

II. Increase recoverability (e.g. plane crossings)

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UF: Definition

UFO Framework I

Given a problem with Decision Variables x

UF: a function (x) measuring the “quality” of a solution x

OBJECTIVE: MAX (x)

s.t. x feasible solution to initial problem

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How to Derive an UF ?

UFO Framework II

Know WHAT changes, not HOW

UF is problem dependent use practitioner’s experience/intuition

If recovery strategy is known

seek UF improving recovery’s performanceRECOVERABILITY

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General Optimization Problem

UFO Framework III

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UFO: Multi-Objective Problem

UFO Framework IV

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UF and Optimality Budget

UFO Framework V

Uncertainty Feature

Original Optimum

Maximal Optimality Gap

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UFO with Budget Relaxation

UFO Framework VI

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UFO Properties

UFO Framework VI

I. Complexity not changed if (x) similar to f(x)

II. Implicit modeling of uncertainty

III. Differentiate solutions on optimal facet

IV. “Plug” tool for any existing method

V. Can use UF based on explicit uncertainty set

VI. Generalizes existing methods

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Stochastic Problem as an UFO

UFO Extension – Stochastic I

Given an Uncertainty Set U with a probability measure on it

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Stochastic problem as an UFO

UFO Extension – Stochastic II

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Robust problem as an UFO

UFO Extension – Robust I

Original LP Problem

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Robust problem as an UFO

UFO Extension – Robust II

Formulation of Bertsimas and Sim (2004)

21UFO Extension – Robust III

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Start with Feasibilty Problem

UFO Extension – Robust IV

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Define UF and budget

UFO Extension – Robust III

Where

and

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UFO formulation

UFO Extension – Robust IV

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Replace Elements in Constraint

UFO Extension – Robust IV

=

Which is equivalent to

=

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Retrieve Robust Formulation

UFO Extension – Robust V

Q.E.D.

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BONUS

UFO Extension – Robust VI

Gives methodology to compute maximal

values of to ensure a robust solution

exists.

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Multiple Knapsack Constraints

Application – MKC I

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MKC with Max Taken Object UFO

Application – MKC II

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Other derived UF

• Max Taken (MTk):

• Diversification (Div):

• Impact Ratio (IR):

• 2Sum:

Application – MKC III

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MKC Simulator

• Generation of problems

• Solve Models inc. Robust (combining possible)

• Simulation with user-defined parameters

Application – MKC IV

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Simulation Results

• Partial results on 8 of 24 classes• Classes according to

i. Cost-correlated A matrixii. Granularityiii. Number of Constraintsiv. Number of varying coefficients

MKC – Results I

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Simulation Results

• Simulations according toa. Exact variability matrix Âb. Variability matrix based on Ac. Random Variability Matrixd. High or Low variances

MKC – Results II

34MKC – Results III

21470

25.56

18.07

98.13

Nbr Unfeasible

% of unfeasible

Avg OptimalityGap (%)

Max OptimialityGap (%)

ROBUST Â

25763

30.67

11.66

96.89

ROBUST0.1*A

40680

48.43

9.25

96.4

IR rho = 10%

40841

48.62

9.23

96.4

_MTk + 2Sumrho = 10%

50212

59.78

7.94

96.89

Div rho = 10%

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1.00

4.10

33.55

761

50.73

9.59

90.01

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3.13

9.67

24.97

117

7.8

9.45

24.98

1186

79.07

6.57

89.22

35MKC – Results IV

2

<0.01

3.95

37.43

Nbr Unfeasible

% of unfeasible

Avg OptimalityGap (%)

Max OptimialityGap (%)

ROBUST Â _MTk rho = 10%

 0.2*A RANDOM

Used Model

Var. Matrix  0.2*A RANDOM

36Future Work

Future Work• Extended Tests on MKC

• Application of UFO to Airline Transportation

• Find an UF generator ?

37Conclusions

Conclusions• UFO allows to cope with uncertainty IMPLICITLY

• Use explicit uncertainty model is still possible

• UFO can be combined with any already existing method

• It is NOT an alien method !

38Conclusions

THANKS for your attention

Any Questions?