Now how are the problems in practice related to this problem template or, more concretely, for which problem variant do we need an algorithm in order to solve a practical problem? Let us have a look at some meaningful examples of rulesets (and their interplay with the planning horizon and the optimization goal). In all these cases we expect the driver to be completely rested (with respect to all relevant types of breaks) at the beginning of the planning horizon.
• Suppose a dispatcher needs to plan trips for single days within the European Union. According to the Regulation (EC) No. 561/2006 of the EU (see section 2.1.1.1), a driver (of a vehicle with a total mass exceeding 3.5t) must no longer drive after an accumulated driving time of limitD = 4.5h. The accumulated driving time is reset after a break of at least break=0.75h. It is allowed to split the break in two. The first split break must be at least break1st =0.25h long and the second split break at least break2nd=0.5h long. In practice, it may be okay to not consider a break split. In this case, an algorithm for the ruleset {drive until driven } is sufficient to find a feasible schedule. But a shorter schedule may be found by an algorithm for the ruleset {drive until driven, first-second-split }. However, this is only half the truth because we have not yet taken two relevant provisions of the regulation into account. And it remains in the responsibility of the dispatcher to check these as well. First, there is a limit on the maximum driving time between two daily rests. This means the dispatcher has to check beforehand whether the sum of driving times∑ni=1drivei exceeds that limit or
not. And second, there is a limit on the maximum travel time between two daily rests. Here, the dispatcher has two options: Either he restricts the length of the planning horizon to the limit value a priori. Or he calls an algorithm with the objective to find a schedule with minimum duration. In this case, he checks whether the minimum duration meets the limit a posteriori.
26 Chapter 2. Fundamentals and Preliminaries
• According to the most recent hours-of-service regulation of the United States (see section 2.1.2.1), a driver must no longer drive after an accumulated travel time of limitT =8h. The accumulated travel time is reset after a break of at least break = 0.5h. So for a planning horizon of a single day in the US, the ruleset {drive until traveled } is sufficient.
But as before, two more checks remain in the responsibility of the dispatcher. Again, this is because there is a limit on the maximum driving time and the maximum travel time between two daily rests. The only difference is that ac- cording to US law, the driver is still allowed to perform service after the max- imum travel time is reached. To exploit this, the travel time check could be adjusted a little. Let us assume that we have an algorithm for the minimum duration variant, that the driving time to the last customer is greater than zero, and that a time window at the last customer is open when the planning ho- rizon closes (the last two assumptions are without loss of generality). In this case, the dispatcher could create a modified problem instance in which the last customer is always open and requests no service. Then there is a feasible solu- tion for the original instance if there is one for the modified instance that meets the travel time limit.
• According to an older hours-of-service regulation of the United States (Federal Motor Carrier Safety Administration, 2008, meanwhile extended by the provi- sion on rest breaks as mentioned above), a driver must no longer drive after an accumulated driving time of limitD = 11h or an accumulated travel time of limitT = 14h. The accumulated times are reset after a break of at least break = 10h. So for a planning horizon of one week in the US, the ruleset {drive until driven, drive until traveled } used to be sufficient. We mention this problem variant because it was studied in the literature before the “rest break provision” entered into force in 2013. Again, there remains an additional a pri- ori check for the dispatcher. He has to make sure that no more than 60 or 70 hours of on-duty time are assigned to the driver within the planning horizon. • In case of multi-manning, two drivers take turns. One driver may take a
lunch break while the other is driving. According to the Regulation (EC) No. 561/2006 of the European Union (see again section 2.1.1.1), both drivers must no longer drive after an accumulated driving time of limitD = 18h and not even work after an accumulated travel time of limitT =21h. The accumulated times are reset after a break of at least break=9h. So for a planning horizon of roughly a week in the EU, an algorithm for the ruleset {drive until driven, work until traveled } suffices in case of a doubly-manned vehicle.
However, it is up to the dispatcher to take care that the maximum driving time in two consecutive weeks is not exceeded and that the planning horizon is no longer than 6·24 hours such that enough time for a weekly rest remains. • Let us stick to a planning horizon of at most 6·24 hours and the provisions
of the EU regulation but suppose that there is only a single driver on board. Then we have to distinguish two kinds of breaks which we call “short” and “long” in the following. At least, we need an algorithm for the two rulesets
{drive until driven}shortand{drive until driven, work until traveled}long, where we set the two minimum break periods to be breakshort =0.75h and breaklong=11h,
and set the limit values as follows: limitDshort = 4.5h, limitDlong = 9h, and
2.4. Basic Definitions and Notation 27 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 0 1 2 3 4 Planning horizon W aiting tim e
FIGURE2.1: Example of a waiting time function W. Function in red and dashed. Time windows in the background.
Since both the short and the long break are allowed to be split, it would be even better to have an algorithm for the two rulesets {drive until driven, first-second- split}shortand{drive until driven, work until traveled, first-second-split}long. In this case, we have to set break1st
short =0.25h and break2ndshort =0.5h as well as break1stlong=
3h and break2nd
long = 9h in addition. The check of the maximum bi-weekly dri-
ving time can be conducted beforehand.