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)

Problem Definition

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

A refresher on the queueing approach to traffic flows q max q k j Traffic flow k 1 k 2 v2v2 v1v1 qq max vfvf Traffic flow Speed Speed-flow diagramSpeed-density diagram Flow-density diagram Density

Queueing framework Queueing QueueService Station (1/k j ) T: Congestion parameter

Travel Time Distribution: Mean P periods of equal length Δp with a different travel speed associated with each time period p (1 < p < 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

Travel Time Distribution: Variance I TT  k *  p (Previous slide)  Var(TT)   p 2 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

Travel Time Distribution: Variance III Speed v t0t0 v avg vv Time zones k A B Area A + Area B = 0   k =  v

Travel Time Distribution: Variance IV  k    v (and  ~ f(v, k avg,  p))  Var(  k)   2 Var(  v) Var(v) ?

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.

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

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

Travel Time Distribution: Overview TT ~ Lognormal distribution E(W) and Var(W) see e.g. approximations Whitt for GI/G/K queues

Data generation: Routing problem Traffic generation Finding solutions for the Stochastic Dynamic Routing Problem Solutions Heuristics Tabu Search Ant Colony Optimization

Objective Functions I Results for F 1 (S): Significant and consistent improvements in travel times observed (>15% gains) Different routes

Objective Functions II Objective Function F 2 (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

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 !

Questions? ?