Coping with Variability in Dynamic Routing Problems

1
Coping with Variability in Dynamic Routing
Problems

• Tom Van Woensel (TU/e)
• Christophe Lecluyse (UA), Herbert Peremans (UA),
  Laoucine Kerbache (HEC) and Nico Vandaele (UA)

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Problem Definition
3
Previous work

• Deterministic Dynamic Routing Problems
• Inherent stochastic nature of the routing problem
  due to travel times
• Average travel times modeled using queueing
  models
• Heuristics used
• Ant Colony Optimization
• Tabu Search
• Significant gains in travel time observed
• Did not include variability of the travel times

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A refresher on the queueing approach to traffic
flows
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Queueing framework
Queueing
T Congestion parameter
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Travel Time Distribution Mean

• P periods of equal length ?p with a different
  travel speed associated with each time period p
  (1 lt p lt P)
• TT ? k ?p
• Decision variable is number of time zones k
• Depends upon the speeds in each time zone and the
  distance to be crossed

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Travel Time Distribution Variance I

• TT ? k ?p (Previous slide)
• ? Var(TT) ? ?p2 Var(k)
• Variance of TT is dependent on the variance of k,
  which depends on changes in speeds
• i.e. Var(k) is a function of Var(v)
• Relationship between (changes in k) as a result
  of (changes in v) needs to be determined ?k ?
  ?v

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Travel Time Distribution Variance III
Area A Area B 0 ? ?k ? ?v
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Travel Time Distribution Variance IV

• ?k ? ? ?v (and ? f(v, kavg, ?p))
• ? Var(?k) ? ?2 Var(?v)
• Var(v) ?

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Travel Time Distribution Variance V

• What is Var(1/W)?
• Not a physical meaning in queueing theory
• Distribution is unknown but
• Assume that W follows a lognormal distribution
  (with parameters ? and ?)
• Then it can be proven that (1/W) also follows a
  lognormal distribution with (parameters -? and ?)
• See Papoulis (1991), Probability, Random
  Variables and Stochastic Processes, McGraw-Hill
  for general results.

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Travel Time Distribution Variance VI

• With (1/W) following a Lognormal distribution,
  the moments of its distribution can be related to
  the moments of the distribution for W as follows

W LN
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Travel Time Distribution

• If W LN ? 1/W LN
• ? v LN
• ? TT LN
• Assumption is acceptable
• Production management often W LN
• E.g. Vandaele (1996) Simulation Empirics
• Traffic Theory often TT LN
• Empirical research e.g. Taniguchi et al. (2001)
  in City Logistics

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Travel Time Distribution Overview

• TT Lognormal distribution

E(W) and Var(W) see e.g. approximations Whitt for
GI/G/K queues
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Finding solutions for the Stochastic Dynamic
Routing Problem
Data generation Routing problem Traffic
generation
Heuristics
Ant Colony Optimization
Tabu Search
Solutions
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Objective Functions I

• Results for F1(S)
• Significant and consistent improvements in travel
  times observed (gt15 gains)
• Different routes

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Objective Functions II

• Objective Function F2(S)
• No complete results available yet
• Preliminary insights
• Not necessarily minimal in Total Travel Time
• Variability in Travel Times is reduced
• Recourse Less re-planning is needed
• Robust solutions

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Conclusions

• Travel Time Variability in Routing Problems
• Travel Times
• Lognormal distribution
• Expected Travel Times and Variance of the Travel
  Times via a Queueing approach
• Stochastic Routing Problems
• Time Windows !

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Questions?

• ?