<s>
Arc	B-Protocol
routing	I-Protocol
problems	I-Protocol
(	O
ARP	O
)	O
are	O
a	O
category	O
of	O
general	O
routing	O
problems	O
(	O
GRP	O
)	O
,	O
which	O
also	O
includes	O
node	O
routing	O
problems	O
(	O
NRP	O
)	O
.	O
</s>
<s>
The	O
objective	O
of	O
arc	B-Protocol
routing	I-Protocol
problems	I-Protocol
involves	O
minimizing	O
the	O
total	O
distance	O
and	O
time	O
,	O
which	O
often	O
involves	O
minimizing	O
deadheading	O
time	O
,	O
the	O
time	O
it	O
takes	O
to	O
reach	O
a	O
destination	O
.	O
</s>
<s>
Arc	B-Protocol
routing	I-Protocol
problems	I-Protocol
can	O
be	O
applied	O
to	O
garbage	O
collection	O
,	O
school	O
bus	O
route	O
planning	O
,	O
package	O
and	O
newspaper	O
delivery	O
,	O
deicing	O
and	O
snow	O
removal	O
with	O
winter	O
service	O
vehicles	O
that	O
sprinkle	O
salt	O
on	O
the	O
road	O
,	O
mail	O
delivery	O
,	O
network	O
maintenance	O
,	O
street	O
sweeping	O
,	O
police	O
and	O
security	O
guard	O
patrolling	O
,	O
and	O
snow	O
ploughing	O
.	O
</s>
<s>
Arc	B-Protocol
routings	I-Protocol
problems	O
are	O
NP	O
hard	O
,	O
as	O
opposed	O
to	O
route	O
inspection	O
problems	O
that	O
can	O
be	O
solved	O
in	O
polynomial-time	O
.	O
</s>
<s>
For	O
a	O
real-world	O
example	O
of	O
arc	B-Protocol
routing	I-Protocol
problem	I-Protocol
solving	O
,	O
Cristina	O
R	O
.	O
Delgado	O
Serna	O
&	O
Joaquín	O
Pacheco	O
Bonrostro	O
applied	O
approximation	O
algorithms	O
to	O
find	O
the	O
best	O
school	O
bus	O
routes	O
in	O
the	O
Spanish	O
province	O
of	O
Burgos	O
secondary	O
school	O
system	O
.	O
</s>
<s>
There	O
are	O
generalizations	O
of	O
arc	B-Protocol
routing	I-Protocol
problems	I-Protocol
that	O
introduce	O
multiple	O
mailmen	O
,	O
for	O
example	O
the	O
k	O
Chinese	O
Postman	O
Problem	O
(	O
KCPP	O
)	O
.	O
</s>
<s>
Arc	B-Protocol
routing	I-Protocol
problems	I-Protocol
have	O
applications	O
in	O
school	O
bus	O
planning	O
,	O
garbage	O
and	O
waste	O
and	O
refuse	O
collection	O
in	O
cities	O
,	O
mail	O
and	O
package	O
delivery	O
by	O
mailmen	O
and	O
postal	O
services	O
,	O
winter	O
gritting	O
and	O
laying	O
down	O
salt	O
to	O
keep	O
roads	O
safe	O
in	O
the	O
winter	O
,	O
snow	O
plowing	O
and	O
removal	O
,	O
meter	O
reading	O
including	O
remote	O
radio	O
frequency	O
identification	O
meter	O
reading	O
technology	O
,	O
street	O
maintenance	O
and	O
sweeping	O
,	O
police	O
patrol	O
car	O
route	O
planning	O
,	O
and	O
more	O
.	O
</s>
<s>
Arc	B-Protocol
routing	I-Protocol
problems	I-Protocol
impact	O
strategic	O
,	O
tactical	O
,	O
and	O
operational	O
planning	O
decisions	O
.	O
</s>
<s>
The	O
decision	O
of	O
the	O
vehicle	O
fleet	O
size	O
and	O
vehicle	O
types	O
with	O
varying	O
specifications	O
relate	O
to	O
the	O
tactical	O
aspect	O
of	O
arc	B-Protocol
routing	I-Protocol
problems	I-Protocol
in	O
operations	O
research	O
.	O
</s>
<s>
Routing	O
and	O
scheduling	O
decisions	O
are	O
operational	O
planning	O
decisions	O
in	O
arc	B-Protocol
routing	I-Protocol
problems	I-Protocol
.	O
</s>
<s>
Vehicle	B-Algorithm
routing	I-Algorithm
decisions	O
for	O
the	O
location	O
of	O
a	O
depot	O
depend	O
on	O
the	O
cost	O
of	O
transporting	O
materials	O
over	O
a	O
geographical	O
region	O
.	O
</s>
<s>
al	O
applied	O
vehicle	B-Algorithm
routing	I-Algorithm
to	O
the	O
dial	O
a	O
ride	O
problem	O
.	O
</s>
<s>
Finding	O
an	O
efficient	O
solution	O
with	O
large	O
amounts	O
data	O
to	O
the	O
Chinese	O
Postman	O
Problem	O
(	O
CPP	O
)	O
,	O
the	O
Windy	O
Postman	O
Problem	O
(	O
WPP	O
)	O
,	O
the	O
Rural	O
Postman	O
Problem	O
(	O
RPP	O
)	O
,	O
the	O
k-Chinese	O
postman	O
problem	O
(	O
KCPP	O
)	O
,	O
the	O
mixed	O
Chinese	O
postman	O
problem	O
(	O
MCPP	O
)	O
,	O
the	O
Directed	O
Chinese	O
Postman	O
Problem	O
(	O
DCPP	O
)	O
,	O
the	O
Downhill	O
Plowing	O
Problem	O
(	O
DPP	O
)	O
,	O
the	O
Plowing	O
with	O
Precedence	O
Problem	O
(	O
PPP	O
)	O
,	O
the	O
Windy	O
Rural	O
Postman	O
Problem	O
(	O
WRPP	O
)	O
and	O
the	O
Windy	O
General	O
Routing	O
Problem	O
(	O
WGRP	O
)	O
requires	O
using	O
thoughtful	O
mathematical	O
concepts	O
,	O
including	O
heuristic	B-Algorithm
optimization	I-Algorithm
methods	I-Algorithm
,	O
branch-and-bound	B-Algorithm
methods	I-Algorithm
,	O
integer	B-Algorithm
linear	I-Algorithm
programming	I-Algorithm
,	O
and	O
applications	O
of	O
traveling	B-Algorithm
salesman	I-Algorithm
problem	I-Algorithm
algorithms	O
such	O
as	O
the	O
Held	B-Algorithm
–	I-Algorithm
Karp	I-Algorithm
algorithm	I-Algorithm
makes	O
an	O
improvement	O
from	O
to	O
.	O
</s>
<s>
In	O
addition	O
to	O
these	O
algorithms	O
,	O
these	O
classes	O
of	O
problems	O
can	O
also	O
be	O
solved	O
with	O
the	O
cutting	B-Algorithm
plane	I-Algorithm
algorithm	I-Algorithm
,	O
convex	O
optimization	O
,	O
convex	O
hulls	O
,	O
Lagrange	O
multipliers	O
and	O
other	O
dynamic	B-Algorithm
programming	I-Algorithm
methods	I-Algorithm
.	O
</s>
<s>
In	O
cases	O
where	O
it	O
is	O
not	O
feasible	O
to	O
run	O
the	O
Held	B-Algorithm
–	I-Algorithm
Karp	I-Algorithm
algorithm	I-Algorithm
because	O
of	O
its	O
high	O
computational	O
complexity	O
,	O
algorithms	O
like	O
this	O
can	O
be	O
used	O
to	O
approximate	O
the	O
solution	O
in	O
a	O
reasonable	O
amount	O
of	O
time	O
.	O
</s>
<s>
The	O
earliest	O
documented	O
reference	O
to	O
the	O
area	O
of	O
arc	B-Protocol
routing	I-Protocol
problems	I-Protocol
is	O
the	O
classic	O
bridges	O
of	O
Königsberg	O
challenge	O
,	O
which	O
Euler	O
proved	O
to	O
be	O
impossible	O
.	O
</s>
<s>
Arc	B-Protocol
routing	I-Protocol
problems	I-Protocol
(	O
ARPs	O
)	O
differ	O
in	O
their	O
goal	O
and	O
heuristics	B-Algorithm
.	O
</s>
<s>
The	O
undirected	O
capacitated	O
arc	B-Protocol
routing	I-Protocol
problem	I-Protocol
consists	O
of	O
demands	O
placed	O
on	O
the	O
edges	O
,	O
and	O
each	O
edge	O
must	O
meet	O
the	O
demand	O
.	O
</s>
<s>
The	O
heuristics	B-Algorithm
described	O
in	O
this	O
article	O
ignore	O
any	O
such	O
problems	O
that	O
arise	O
due	O
to	O
application	O
constraints	O
.	O
</s>
<s>
In	O
2000	O
,	O
Dror	O
published	O
a	O
book	O
describing	O
different	O
arc	B-Protocol
routing	I-Protocol
problems	I-Protocol
.	O
</s>
<s>
The	O
windy	O
postman	O
problem	O
is	O
an	O
arc	B-Protocol
routing	I-Protocol
problem	I-Protocol
(	O
ARP	O
)	O
that	O
contains	O
the	O
Mixed	O
Chinese	O
Postman	O
Problem	O
MCPP	O
as	O
a	O
special	O
case	O
.	O
</s>
<s>
Benavent	O
proposed	O
an	O
integer	B-Algorithm
linear	I-Algorithm
programming	I-Algorithm
formulation	O
and	O
different	O
heuristics	B-Algorithm
and	O
lower	O
bounds	O
for	O
the	O
WRPP	O
.	O
</s>
<s>
Benavent	O
et	O
al	O
published	O
an	O
evaluation	O
of	O
several	O
heuristic	B-Algorithm
methods	O
used	O
for	O
solving	O
the	O
WRPP	O
in	O
a	O
few	O
seconds	O
with	O
a	O
deviation	O
no	O
greater	O
than	O
1%	O
from	O
the	O
lower	O
bound	O
on	O
medium	O
sized	O
graphs	O
.	O
</s>
<s>
The	O
WRPP	O
is	O
an	O
important	O
Arc	B-Protocol
Routing	I-Protocol
Problem	I-Protocol
which	O
generalizes	O
many	O
of	O
the	O
single-vehicles	O
Arc	B-Protocol
Routing	I-Protocol
problems	I-Protocol
.	O
</s>
<s>
In	O
the	O
future	O
comparisons	O
with	O
other	O
meta-heuristic	O
methods	O
could	O
be	O
researched	O
,	O
including	O
Non-dominated	O
Sorting	O
Genetic	O
Algorithm	O
(	O
NSGA	O
-	O
)	O
,	O
multi-objective	O
particle	O
swarm	O
optimization	O
algorithm	O
(	O
MOPSO	O
)	O
and	O
multi-objective	O
Imperialist	O
Competitive	O
Algorithm	O
.	O
</s>
<s>
A	O
branch	B-Algorithm
and	I-Algorithm
cut	I-Algorithm
algorithm	O
was	O
published	O
by	O
Angel	O
Corberan	O
for	O
the	O
windy	O
postman	O
problem	O
.	O
</s>
<s>
The	O
algorithm	O
is	O
based	O
on	O
heuristic	B-Algorithm
and	O
exact	O
methods	O
for	O
manipulating	O
odd-cut	O
inequality	O
violations	O
.	O
</s>
<s>
Typically	O
,	O
this	O
is	O
assessed	O
as	O
an	O
arc	B-Protocol
routing	I-Protocol
problem	I-Protocol
with	O
a	O
graph	O
.	O
</s>
<s>
Another	O
aspect	O
that	O
must	O
be	O
considered	O
when	O
applying	O
arc	B-Protocol
routing	I-Protocol
to	O
snow	O
plowing	O
is	O
the	O
fact	O
that	O
on	O
steep	O
streets	O
it	O
is	O
either	O
difficult	O
or	O
impossible	O
to	O
plow	O
uphill	O
.	O
</s>
<s>
This	O
was	O
modeled	O
with	O
a	O
heuristic	B-Algorithm
algorithm	I-Algorithm
that	O
approximates	O
a	O
lower	O
bound	O
by	O
Dussault	O
,	O
Golden	O
and	O
Wasil	O
.	O
</s>
<s>
The	O
simulation	O
for	O
the	O
DPP	O
deadheaded	O
unplowed	O
street	O
about	O
5%	O
of	O
the	O
time	O
,	O
which	O
is	O
a	O
topic	O
for	O
future	O
graph	O
theory	O
and	O
arc	B-Protocol
routing	I-Protocol
research	O
.	O
</s>
<s>
Benevant	O
's	O
algorithm	O
approximated	O
the	O
solution	O
by	O
transforming	O
the	O
DRPP-TP	O
into	O
an	O
asymmetrical	O
traveling	B-Algorithm
salesman	I-Algorithm
problem	I-Algorithm
(	O
ATSP	O
)	O
.	O
</s>
<s>
Methods	O
of	O
solving	O
the	O
URPP	O
after	O
the	O
pre-processing	O
is	O
done	O
consist	O
of	O
the	O
cutting	B-Algorithm
plane	I-Algorithm
algorithm	I-Algorithm
and	O
the	O
branch	O
&	O
cut	O
methodology	O
.	O
</s>
<s>
This	O
is	O
a	O
list	O
of	O
computational	O
complexities	O
for	O
different	O
arc	B-Protocol
routing	I-Protocol
problems	I-Protocol
.	O
</s>
