<s>
The	O
vehicle	B-Algorithm
routing	I-Algorithm
problem	I-Algorithm
(	O
VRP	O
)	O
is	O
a	O
combinatorial	O
optimization	O
and	O
integer	B-Algorithm
programming	I-Algorithm
problem	I-Algorithm
which	O
asks	O
"	O
What	O
is	O
the	O
optimal	O
set	O
of	O
routes	O
for	O
a	O
fleet	O
of	O
vehicles	O
to	O
traverse	O
in	O
order	O
to	O
deliver	O
to	O
a	O
given	O
set	O
of	O
customers	O
?	O
"	O
</s>
<s>
It	O
generalises	O
the	O
travelling	B-Algorithm
salesman	I-Algorithm
problem	I-Algorithm
(	O
TSP	O
)	O
.	O
</s>
<s>
In	O
1964	O
,	O
Clarke	O
and	O
Wright	O
improved	O
on	O
Dantzig	O
and	O
Ramser	O
's	O
approach	O
using	O
an	O
effective	O
greedy	B-Algorithm
algorithm	I-Algorithm
called	O
the	O
savings	O
algorithm	O
.	O
</s>
<s>
Therefore	O
,	O
commercial	O
solvers	O
tend	O
to	O
use	O
heuristics	B-Algorithm
due	O
to	O
the	O
size	O
and	O
frequency	O
of	O
real	O
world	O
VRPs	O
they	O
need	O
to	O
solve	O
.	O
</s>
<s>
Several	O
variations	O
and	O
specializations	O
of	O
the	O
vehicle	B-Algorithm
routing	I-Algorithm
problem	I-Algorithm
exist	O
:	O
</s>
<s>
Vehicle	B-Algorithm
Routing	I-Algorithm
Problem	I-Algorithm
with	O
Profits	O
(	O
VRPP	O
)	O
:	O
A	O
maximization	O
problem	O
where	O
it	O
is	O
not	O
mandatory	O
to	O
visit	O
all	O
customers	O
.	O
</s>
<s>
Vehicle	B-Algorithm
Routing	I-Algorithm
Problem	I-Algorithm
with	O
Pickup	O
and	O
Delivery	O
(	O
VRPPD	O
)	O
:	O
A	O
number	O
of	O
goods	O
need	O
to	O
be	O
moved	O
from	O
certain	O
pickup	O
locations	O
to	O
other	O
delivery	O
locations	O
.	O
</s>
<s>
Vehicle	B-Algorithm
Routing	I-Algorithm
Problem	I-Algorithm
with	O
LIFO	B-Application
:	O
Similar	O
to	O
the	O
VRPPD	O
,	O
except	O
an	O
additional	O
restriction	O
is	O
placed	O
on	O
the	O
loading	O
of	O
the	O
vehicles	O
:	O
at	O
any	O
delivery	O
location	O
,	O
the	O
item	O
being	O
delivered	O
must	O
be	O
the	O
item	O
most	O
recently	O
picked	O
up	O
.	O
</s>
<s>
Vehicle	B-Algorithm
Routing	I-Algorithm
Problem	I-Algorithm
with	O
Time	O
Windows	O
(	O
VRPTW	B-Algorithm
)	O
:	O
The	O
delivery	O
locations	O
have	O
time	O
windows	O
within	O
which	O
the	O
deliveries	O
(	O
or	O
visits	O
)	O
must	O
be	O
made	O
.	O
</s>
<s>
Capacitated	O
Vehicle	B-Algorithm
Routing	I-Algorithm
Problem	I-Algorithm
:	O
CVRP	O
or	O
CVRPTW	O
.	O
</s>
<s>
Vehicle	B-Algorithm
Routing	I-Algorithm
Problem	I-Algorithm
with	O
Multiple	O
Trips	O
(	O
VRPMT	O
)	O
:	O
The	O
vehicles	O
can	O
do	O
more	O
than	O
one	O
route	O
.	O
</s>
<s>
Open	O
Vehicle	B-Algorithm
Routing	I-Algorithm
Problem	I-Algorithm
(	O
OVRP	O
)	O
:	O
Vehicles	O
are	O
not	O
required	O
to	O
return	O
to	O
the	O
depot	O
.	O
</s>
<s>
Multi-Depot	O
Vehicle	B-Algorithm
Routing	I-Algorithm
Problem	I-Algorithm
(	O
MDVRP	O
)	O
:	O
Multiple	O
depots	O
exist	O
from	O
which	O
vehicles	O
can	O
start	O
and	O
end	O
.	O
</s>
<s>
Vehicle	B-Algorithm
Routing	I-Algorithm
Problem	I-Algorithm
with	O
Transfers	O
(	O
VRPWT	O
)	O
:	O
Goods	O
can	O
be	O
transfered	O
between	O
vehicles	O
at	O
specially	O
designated	O
transfer	O
hubs	O
.	O
</s>
<s>
Although	O
VRP	O
is	O
related	O
to	O
the	O
Job	B-Algorithm
Shop	I-Algorithm
Scheduling	I-Algorithm
Problem	O
,	O
the	O
two	O
problems	O
are	O
typically	O
solved	O
using	O
different	O
techniques	O
.	O
</s>
<s>
There	O
are	O
many	O
methods	O
to	O
solve	O
vehicle	B-Algorithm
routing	I-Algorithm
problems	I-Algorithm
manually	O
.	O
</s>
<s>
Due	O
to	O
the	O
difficulty	O
of	O
solving	O
to	O
optimality	O
large-scale	O
instances	O
of	O
vehicle	B-Algorithm
routing	I-Algorithm
problems	I-Algorithm
,	O
a	O
significant	O
research	O
effort	O
has	O
been	O
dedicated	O
to	O
metaheuristics	B-Algorithm
such	O
as	O
Genetic	B-Algorithm
algorithms	I-Algorithm
,	O
Tabu	B-Algorithm
search	I-Algorithm
,	O
Simulated	B-Algorithm
annealing	I-Algorithm
and	O
Adaptive	O
Large	O
Neighborhood	O
Search	O
(	O
ALNS	O
)	O
.	O
</s>
<s>
Some	O
of	O
the	O
most	O
recent	O
and	O
efficient	O
metaheuristics	B-Algorithm
for	O
vehicle	B-Algorithm
routing	I-Algorithm
problems	I-Algorithm
reach	O
solutions	O
within	O
0.5	O
%	O
or	O
1%	O
of	O
the	O
optimum	O
for	O
problem	O
instances	O
counting	O
hundreds	O
or	O
thousands	O
of	O
delivery	O
points	O
.	O
</s>
<s>
As	O
such	O
,	O
the	O
application	O
of	O
metaheuristic	B-Algorithm
techniques	O
is	O
often	O
preferred	O
for	O
large-scale	O
applications	O
with	O
complicating	O
constraints	O
and	O
decision	O
sets	O
.	O
</s>
<s>
Most	O
solutions	O
to	O
the	O
vehicle	B-Algorithm
routing	I-Algorithm
problems	I-Algorithm
can	O
be	O
classified	O
as	O
one	O
of	O
the	O
following	O
approaches	O
:	O
</s>
