![](bg7.png)
ber of columns of the RMP within component CG at
time-limit. Columns ”cost (e)” and ”gap%” report
the final solution provided by our algorithm and its
percentage gap against the lower bound. Entries with
(-) indicates that the algorithm was not able to pro-
duce a lower bound for the instance. Moreover, in
columns 4 and 7, we mark with an asterisk (*) the
solutions of the MIPs that are proven optimal.
From the examination of Table 2, we observe that
our algorithm without time-limit was able to produce
a lower bound for all instances but 4 (for instances
10.03.2010, 11.03.2010, 05.05.2010 and 04.05.2010
the DPE algorithm ran out of memory). Both variants
of our algorithm (CG-STD and CG-IS) always find a
feasible solution. CG-STD performs best in 6 over 14
instances while CG-IS does it in 9 over 14 instances.
The solution of instance 05.05.2010 is the same for
both algorithms. We remark that CG-IS performs
best for the 6 more difficult instances (08.03.2010,
10.03.2010, 11.03.2010, 04.05.2010, 05.05.2010 and
14.06.2010). The average cost of the solutions pro-
vided by CG-STD is 30’394.16 e and the average gap
is 6.18%, while the average cost of the solution pro-
vided by CG-IS is 29’930.16 e and the average gap
is 4.77% (the average gap computation does not con-
sider the instances where the lower bound is not avail-
able). CG-STD at time-limit generates always more
columns than CG-IS. The generated columns using
CG-STD is in average 47’863.50, while their aver-
age using CG-IS is 16’898.21. As consequence, the
solver S can solve easier MIPs generated using CG-IS
than those generated using CG-STD. In fact, S proved
the optimality of 5 MIPs, which were all produced by
CG-IS.
Our algorithm allows to deal with real-world in-
stances providing results within 30 minutes of com-
putation and a feasible solution is always ensured by
the initialization provided by ANT.
The devised incremental search (CG-IS) is effec-
tive and improves the CG-STD solutions by 1.52%.
Indeed, the number of columns within the RMP at
time-limit is dramatically reduced (-55%) without af-
fecting their quality. As consequence, the produced
MIP can be solved more easily by the general purpose
solver S.
7 INDUSTRIAL ASPECTS
The efficient definition of a distribution plan is
a highly relevant activity for logistic companies
(Golden et al., 2008; Ceselli et al., 2009). Our indus-
trial partner currently performs this task manually. Its
route planning office, composed by 5 planners, every
morning define a 24 hours plan to serve the customers
using a given fleet of vehicles. We want to evaluate
the potential impact of the introduction of our opti-
mization algorithm into the existing infrastructure of
our industrial partner. Our evaluation considers the
following aspects (1) time required to plan, (2) distri-
bution cost and (3) quality of the plan. Our analysis is
based on a comparison between two handmade plans
and the solutions provided by our algorithm.
Our industrial partner gave us two hand-
made plans concerning instance 06.08.2010 and
23.08.2010. We don’t have access to the plans of the
other 12 instances. The manual definition of a plan
requires 5 employees for 4 hours, which are 20 man-
hours. Our algorithm is executed with a time-limit of
30 minutes.
In Table 3 we report the cost comparison between
the handmade plans and the solutions of our algo-
rithm. We observe that our algorithm saves respec-
tively 598 and 590 e. Since our industrial partner
reported that its average operating costs are about
30’000 e/day and the percentage gain achieved by
our algorithm is about 3%, we estimate a potential
saving of 1’050 e/day that are 315’000 e/year (our
partner operates 6 days a week). We remark that al-
gorithm solutions don’t necessarily require the use
of less vehicles than handmade ones. Indeed, for
the instance 06.08.2010 the algorithm solution re-
quires 2 vehicles less than the handmade one (21 in-
stead of 23), while for the instance 23.08.2010 it re-
quires 3 more vehicles (30 instead of 27). Beside cost
minimization, planners implicitly try to minimize the
number of vehicles used. To this extent they decide to
violate some constraints.
Table 3: Costs comparison between handmade plans and
solutions of our algorithm.
Planners Algorithm
20 man/hours 30 min. execution
Instance vehicles cost e vehicles cost e
06.08.2010 23 16723 21 16125
23.08.2010 27 24163 30 23573
To evaluate qualitatively handmade solutions we
validate them using the constraints encoded in our
model. Tables 4 and 5 report the detailed description
of each constraint violation discovered in the hand-
made plans. Table 4 reports the 19 violations related
to instance 06.08.2010, while Table 5 reports the 28
violations releted to instance 23.08.2010. We observe
that most of them represent minor violations but in
the plan related to instance 06.08.2010 we report a
”capacity” violation of 3’686.15 kg, which represents
the 14% of the total capacity of the vehicle, and is thus
HYBRID COLUMN GENERATION-BASED APPROACH FOR VRP WITH SIMULTANEOUS DISTRIBUTION,
COLLECTION, PICKUP-AND-DELIVERY AND REAL-WORLD SIDE CONSTRAINTS
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