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Coping with Variability in Dynamic Routing Problems

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

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Coping with Variability in Dynamic Routing Problems

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  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)

  2. 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

  4. Speed vf Speed-density diagram Speed-flow diagram v2 v1 Density Traffic flow k1 kj k2 q qmax q qmax Flow-density diagram Traffic flow A refresher on the queueing approach to traffic flows

  5. Queue Service Station (1/kj) Queueing framework Queueing T: Congestion parameter

  6. 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

  7. 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

  8. Speed v A vavg v B t0 Time zones k Travel Time Distribution: Variance III Area A + Area B = 0k =  v

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

  10. 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.

  11. 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

  12. 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

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

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

  15. Objective Functions I • Results for F1(S): • Significant and consistent improvements in travel times observed (>15% gains) • Different routes

  16. 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

  17. 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 !

  18. Questions? ?

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