Labelling algorithms for the elementary shortest path problem with resource constraints considering EU drivers’ rules

Abstract

This paper describes how drivers’ rules according to EU social legislation can be formally modelled using the resource concept and how ‘legal’ vehicle routes and schedules can be computed by exact and heuristic labelling algorithms for solving the elementary shortest path problem with resource constraints.

Appendix: Resource extension functions

Note The subscripts indicating the vertex are suppressed, as the update of the resource variables is also performed on arcs, not only at the end vertex of each arc.

τ^drive,daily is set to zero,

• if, at the head vertex, at least one rest period of at least 660 min is taken, or

• if there has already been a split daily rest in the current daily driving time interval and one of the break or rest periods taken at the head vertex is at least 540 min long, or

• if it is still possible to take a reduced daily rest period in the current calendar week and one of the break or rest periods taken at the head vertex is at least 540 min long.

Otherwise, the daily driving time is the driving time since its last reset, which may have occurred on a previous arc or which occurs on the current arc

• if one of the break or rest periods taken along the arc, before reaching the head vertex, is at least 660 min long, or

• if there has already been a split daily rest in the current daily driving time interval and one of the break or rest periods taken along the arc is at least 540 min long, or

• if it is still possible to take a reduced daily rest period in the current calendar week and one of the break or rest periods taken along the arc is at least 540 min long.

$\tau^{drive,cur\_calendar\_week}$ is set to zero if the weekly or fortnightly driving time limit is reached. Otherwise, it is set to the total driving time since its last reset, which may have occurred on a previous arc or which occurs on the current arc if the weekly or fortnightly driving time limit is reached.

$\tau^{drive,cur\_calendar\_fortnight}$ is set to zero if the fortnightly driving time limit is reached. Otherwise, it is set to the total driving time since its last reset, which may have occurred on a previous arc or which occurs on the current arc if the fortnightly driving time limit is reached.

If either the weekly or the fortnightly driving time limit is reached when travelling along an arc, a break or rest period is taken until the end of the current calendar week.

The update of $\tau^{since\_last\_daily\_rest}$ is equivalent to that of τ^drive,daily, except that not only the driving time is counted, but also the waiting, break, and rest time and the active and passive service time.

Every time a daily or weekly rest ends, the value of $\tau^{max\_ext\_of\_last\_daily\_rest}$ is set to the maximum amount of time by which the last daily rest can be extended such that no time windows of subsequent vertices are violated. Before each vertex, $\tau^{max\_ext\_of\_last\_daily\_rest}$ is set to$\max\left\{\tau^{max\_ext\_of\_last\_daily\_rest}-\tau^{cur\_extension\_of\_last\_daily\_rest},0\right\}.$

The update of $\tau^{since\_last\_weekly\_rest}$ is equivalent to that of $\tau^{since\_last\_daily\_rest}.$

The update of $\tau^{max\_ext\_of\_last\_weekly\_rest}$ is equivalent to that of $\tau^{max\_ext\_of\_last\_daily\_rest}.$

$\tau^{since\_start\_of\_calendar\_week}$ is increased by the driving, waiting, break, and rest time and by the active and passive service time, and it is reset when it exceeds 168 h.

$n^{ext\_ddt}$ is increased by one each time the decision to extend the daily driving time to 10 h is taken until $n^{ext\_ddt}=2$ . It is reset to zero whenever $\tau^{since\_start\_of\_calendar\_week}$ is reset.

$\beta^{split\_break}$ is set to one if a split break of 15 min is taken. It is reset to zero after the next break or rest period of at least 30 min.

$\beta^{split\_rest}$ is set to one if a split daily rest of 180 min is taken. It is reset to zero after the next rest period of at least 540 min.

$\beta^{ddt\_extended}$ is set to one if the daily driving time is extended to 10 h. It is reset to zero after the next daily rest period.

$n^{red\_daily\_rests\_since\_last\_weekly\_rest}$ is increased by one each time the decision to take a reduced daily rest periods is taken until $n^{red\_daily\_rests\_since\_last\_weekly\_rest}=3$ . It is reset to zero whenever $\tau^{since\_last\_weekly\_rest}$ is reset.

For the update of XAT, LAT, and Duration, the reader is referred to [19].