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Step 2: Dynamic decision of daily fleet size. In
(Phillipson and De Koff, 2020) the (exact) number
of parcels is known in advance and from that we can
compute the number of routes we need, given only
a capacity constraint on the number of parcels. How-
ever, now we don’t know the number of required vehi-
cles for two reasons. Firstly we don’t (exactly) know
the number of parcels per day and, secondly, we don’t
know the location of the parcels and thus we don’t
know the (time) lengths of the resulting routes. This
means that we have to decide on the number of routes,
somewhere before, during or after the assignment of
arriving parcels. We actually do all three: we start
with a certain number, add extra routes during the
sorting process and finally, we combine, where pos-
sible, routes into one route after sorting. This also
means that Step 2 and Step 3 are happening some-
what in parallel: the number of tours are not defined
beforehand but during the assignment. For the sake
of clarity, we will use the term tour for the first three
steps. The result of the third step will then be a num-
ber of tours with parcels assigned to it. In Step 4 the
tours will be assigned to routes, that can be executed
by vehicles.
In Step 2 we decide on the number of tours to start
with, and on the procedure and conditions to add an
extra tour to the problem. We looked at two possible
approaches for this step. For both approaches we as-
sume a (rough) estimation of the number of arriving
parcels, coming from a seasonal or weekly pattern.
We estimate the number of tours to distribute these
parcels and call this number N for a certain day. The
two approaches are now:
1. ‘Overall capacity based’ - Start with nN < N
(0% < n < 100%) tours. When the overall tour
load gets higher than a certain value c (0% < c <
100%) during the assignment phase, we add an
extra tour.
2. ‘Tour capacity based’ - Start with nN < N (0% <
n < 100%) tours. When the tour load of a specific
tour is higher than a certain value c (0% < c <
100%) when trying to assign a parcel to it, we add
an extra tour and assign this parcel to it.
We found in prior analyses that approach 2 performs
better than approach 1 and that approach 2 performed
best with parameters n = 75% and c = 99%. Note
that these parameters are case specific.
Step 3: Dynamic assignment of arriving parcels.
Next, three methods for the third step, the dynamic
assignment of arriving parcels, are proposed:
1. Based on minimal insertion costs – for all tours
we calculate the minimal cost of inserting the ar-
riving parcel destination to the tour. Details of this
approach are in the next section. We assume that
the assigned parcels are ordered in the routing of
the tour, so we can calculate the cost by trying to
insert the parcel between each pair of consecutive
parcels in the tour. The insertion that is cheapest
is selected.
2. Based on minimal insertion costs with penalty –
for all tours we calculate the minimal cost of in-
serting the arriving parcel destination to the tour.
Again, as we assume that the assigned parcels are
ordered in the routing of the tour, we can calcu-
late the cost by trying to insert the parcel between
each pair of consecutive parcels in the tour. The
cost is multiplied by a penalty factor, depending
on the load of the tour. The insertion that is cheap-
est will be selected. The calculation of the penalty
is explained in the next sub-section.
3. Based on fixed clusters – here the parcel is simply
assigned to the tour it belongs to, using the total
geographical separation of the initial stage, based
on customer or postal code of the customer.
When, by one of these methods, the assignment is
determined, the parcel is inserted on the right place
in the route of the selected vehicle, meeting the
assumption of ordering.
Step 4: Post-processing. In the post-processing step
we assign tours to vehicles. Due to the fact that we
do not know the exact number of parcels and the driv-
ing times, and due to the heuristic character of Step 2,
we can end with tours that are small and can be com-
bined (together) to one vehicle. Again we consider
two flavours.
1. Combining: Assume that the tours start and end
at the depot, such that any two tours can be com-
bined, as long as the time restriction, the maxi-
mum driving time of the driver, is met. The re-
striction on the number of parcels is not impor-
tant, due to the extra stop at the depot. It also
is not important which two tours are combined,
where we assume that there is no gain to obtain
by combining two (near) tours smartly. In this
post-processing step we just try to minimise the
number of vehicles, given the tours. We do so in
a greedy way, start with the longest tour and try
to add the longest tour as possible that fits the re-
quirements.
2. Integrating: Optimise further by integrating com-
bined tours, such that they do not have to go to
the depot in between. For this we use a greedy
heuristic as shown below:
(a) Sort the tours on available capacity;
Dynamic Assignment Vehicle Routing Problem with Generalised Capacity and Unknown Workload
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