Adaptive Large Neighbourhood Search for the Vehicle Routing Problem with Drivers Working Hours Asvin Goel MIT-Zaragoza International Logistics Program Zaragoza Logistics Center EU/ME 08, Troyes, 23 th October – 24 th October 2008

Drivers working hours New EC regulation 561/2006 entered into force in April 2007 Shippers and carriers can be made liable for infringements committed by the drivers Carriers must organise the work of drivers in such a way that drivers are able to comply with the respective regulations

Provisions of EC regulation 561/ After a driving period (i.e. the accumulated driving time between subsequent breaks and rest periods) of 4½ hours a driver shall take an uninterrupted break of not less than 45 minutes, unless she/he takes a rest period. 2.The daily driving time (i.e. the accumulated driving time between the end of one daily or weekly rest period and the beginning of the following daily or weekly rest period) shall not exceed 9 hours. A regular daily rest period is any period of rest of at least 11 hours.

Provisions of EC regulation 561/2006 (continued) 3.Within each period of 24 hours after the end of the previous daily rest period a driver shall have taken a new daily rest period. 4.The weekly driving time (i.e. the accumulated driving time during a week) shall not exceed 56 hours. 5.A weekly rest period shall start no later than 144 hours after the end of the previous weekly rest period.

Examples A time constrained example with alternative schedules:

Further provisions of EC regulation 561/2006 (not considered in the remainder) 6.The daily driving time may be extended to at most 10 hours not more than twice during the week. 7.The daily rest period may be reduced to 9 hours not more than 3 times during the week. 8.The break may be replaced by a break of at least 15 minutes followed by a break of at least 30 minutes. 9.The daily rest period may be taken in two periods, the first of which must be an uninterrupted period of at least 3 hours and the second an uninterrupted period of at least 9 hours.

Extended daily driving times and reduced daily rest periods In case of delays extended daily driving times and reduced daily rest periods can be used to repair otherwise infeasible schedules:

Splitting up regular breaks EC regulation 561/2006 explicitly states the following reason for replacing the old regulation: It has proved possible under the rules of Regulation (EEC) No 3820/85 to schedule daily driving periods and breaks to enable a driver to drive for too long without a full break, leading to reduced road safety and a deterioration in the drivers working conditions. For t ! 0 the goal of the new regulation is not achieved!!!

Naïve method for scheduling with drivers working hours Drive as long as possible Stop if destination is reached Add daily rest period if required Add break

Naïve method for scheduling with drivers working hours (continued) Let n be the first node in tour Repeat while n is not last node in tour Determine labels for successor node of n using the naïve method If there is enough slack between arrival and begin of time window add break or rest period Set n à successor node of n

Limitations of naïve method found by naïve method Not found by naïve method

Multi-label method for scheduling with drivers working hours Drive as long as possible Add label if destination is reached Add daily rest period if required Add break Generate new label with additional rest period Recursively call method to expand new label

Multi-label method for scheduling with drivers working hours (continued) Let n be the first node in tour Repeat while n is not last node in tour Determine all labels for successor node of n using the multi-label method Copy each label calculated by the multi-label method and add optional break and rest period Delete all dominated labels Set n à successor node of n

Naïve vs. Multi-label method found by naïve method found by multi- label method

Include provisions not considered thus far So far provision 3 has not been considered: Within each period of 24 hours after the end of the previous daily rest period a driver shall have taken a new daily rest period. Increase duration of rest periods in order to make sure that all daily rest periods are completed within the 24 hour limit (not always possible!!!)

Large Neighbourhood Search Initialisation: Find an initial solution s choose a stopping condition Repeat the following until the stopping condition is met: Choose a number k Until k customers are removed from their tour repeat: –Randomly choose a customer n to be removed –Remove customer n from its tour if naïve or multi-label method for scheduling drivers working hours finds a feasible solution Apply an insertion method to re-insert all removed customers If all customers are re-inserted and the modified solution s has lower costs than the current solution s set s := s

Concluding Remarks Drivers working hours have significant impact on total travel times Carriers can be made liable for infringements committed by the drivers Labelling methods can be used to generate tours complying with the new regulations Computational experiments have shown that LNS based on multi-label approach significantly outperforms LNS based on naïve method for scheduling drivers working hours Current and future research will study how to remove customers more effectively, in particular for PDP-DWH

Thank you! Merci! Asvin Goel MIT-Zaragoza International Logistics Program Zaragoza Logistics Center