Development and assessment of the SHARP
and RandSHARP algorithms for the arc
routing problem
Sergio González
a
,
Alejandra Pérez

Bonilla
a
, Angel A.Juan
a
, Daniel Riera
a
{sgonzalezmarti, aperezbon, ajuanp, drierat, jcastellav}@uoc.edu
a
Department of Computer Science, Multimedia and Telecommunication, IN3
–
Open University
of Catalonia, Barcelona, Spain.
http://dpcs.uoc.edu
Cyted

Harosa Int. Workshop
Valparaiso, Chile. November 12

13, 2012
1.
The Capacitated ARP (deterministic)
The
Capacitated
Arc
Routing
Problem
(CARP)
is
a
well

known
NP

hard
problem
Golden
et
al
.
(
1981
)
:
The
set
of
edges
constitute
an
undirected,
non

complete
graph
or
network
.
A
set
of
edges’
demands
must
be
supplied
by
a
fleet
of
(homogeneous)
vehicles
.
Resources
are
available
from
a
depot
.
Moving
a
vehicle
from
one
node
i
to
another
j
has
associated
(symmetric)
costs
c(i,
j)
>
0
.
An
edge
might
be
traversed
several
times
by
different
vehicles,
but
it
is
served
by
just
one
vehicle
.
Additional
constraints
must
be
considered
:
maximum
load
capacity
per
vehicle,
service
times,
etc
.
Depot
(resources)
Edges with
demands
Nodes
0
2
3
1
4
5
6
7
Goal
:
to
obtain
an
‘optimal’
solution
,
i
.
e
.
a
set
of
routes
satisfying
all
constraints
with
minimum
costs
2.
Variants of the
C
ARP
Different
ARP
variants
have
been
proposed
:
Directed
CARP
(DCARP)
:
A
version
of
the
CARP
where
the
Arcs
can
be
traversed
in
a
single
direction
(Maniezzo
&
Roffilli,
2008
)
.
Mixed
CARP
(MCARP)
:
A
version
of
the
CARP
with
mixed
graphs
(
Belenguer,
Benavent,
Lacomme,
&
Prins,
2006
)
.
Min

Max
k

CPP
:
CARP

like
problem
excluding
capacity
constraints
on
vehicles
(Ulusoy,
1985
)
.
Periodic
CARP
(PCARP)
:
CARP
extended
to
the
planning
of
P
days
(
Lacomme,
Prins,
Ramdane

Chérif,
2002
)
.
CARP
with
Stochastic
demands
(ARPSD)
:
CARP
where
the
demand
to
be
served
is
not
known
beforehand
(Fleury,
Lacomme,
&
Prins,
2005
)
.
Etcetera
.
0
2
3
1
4
5
6
7
3.
Practical Applications
Some
ARP
practical
applications
are
:
Refuse
collection
(Almeida
&
Mourão,
2000
)
.
Snow
removal
(Eglese,
1994
)
.
Inspection
of
distributed
systems
(Lee,
1989
)
.
Routing
of
electric
meter
readers
(Stern
&
Dror,
1979
)
.
School
bus
routing
(Ferland
&
Gudnette,
1990
)
.
0
2
3
1
4
5
6
7
4.
Mathematical Model
The
first
ILP
formulation
was
probably
the
one
presented
by
Golden
and
Wong
in
1981
.
(B
.
L
.
Golden
and
R
.
T
.
Wong
1981
)
.
A
formulation
using
undirected
variables
is
presented
by
Belenguer
and
Benavent
.
In
his
PhD
dissertation,
Letchford
gives
several
ILP
formulations
of
the
CARP,
and
derives
additional
valid
inequalities
and
separation
algorithms
for
the
problem
.
Other
mathematical
formulations
are
due
to
Welz
[
22
]
and
Eglese
[
23
],
all
of
them
for
the
undirected
case
.
5.
Solving Approaches (1/2)
Exact
Methods
(optimal
solutions
for
small

scale
problems)
:
Branch
and
Bound
techniques
(Kiuchi,
Hirabayashi,
Saruwatari,
&
Shinano,
1995
)
.
Tour
construction
algorithms
(Hirabayashi,
Nishida,
&
Saruwatari,
1992
)
.
Subtour
elimination
algorithms
(Saruwatari,
Hirabayashi,
&
Nishida,
1992
)
.
ILP
formulations
(Belenguer
&
Benavent,
1992
)
.
Heuristics
(fast,
‘good’
solutions)
:
Path
Scanning
algorithm,
Augment

Merge
algorithm,
Construct

Strike
algorithm
(Golden,
DeArmon,
&
Baker,
1983
)
.
Node
duplication
heuristic
(Wøhlk,
2005
)
.
The
Cycle
Assignment
algorithm
(Benavent,
Campos,
Corberan,
&
Mota,
1990
)
.
Parallel
insert
algorithm
(Chapleau,
Ferland,
Lapalme,
&
Rousseau,
1984
)
.
6.
Solving Approaches (2/2)
Meta

heuristics
(pseudo

optimal
solutions)
:
Tabu
Search
algorithms
(Hertz,
Laporte,
&
Mittaz,
2000
;
Greistorfer,
2003
)
.
Simulated
Annealing
(Eglese,
1994
)
.
Genetic
Algorithms
(Lacomme,
Prins,
&
Ramdane

Chérif,
2001
)
.
Ant
Colony
Optimization
(Lacomme,
Prins,
&
Tanguy,
2004
)
.
Memetic
Algorithms
(Lacomme,
Prins,
&
Ramdane

Chérif,
2004
)
.
7. Our SHARP procedure for the CARP
Based
on
the
savings
heuristic
for
the
CVRP
(Clarke
&
Wright,
1964
)
.
Designed
to
provide
a
‘good’
starting
point
for
CARP
metaheuristics
.
SHARP
main
steps
:
Use
the
Floyd

Warshall
algorithm
to
compute
the
shortest
paths
for
all
pairs
of
nodes
in
the
network
.
This
allows
to
treat
the
graph
as
if
it
was
a
complete
one
.
Compute
the
savings
associated
with
demanding
edges
(real
or
virtual)
connecting
any
two
nodes
and
create
a
sorted
savings
list
.
Create
a
dummy
(initial)
solution
by
assigning
a
route
to
each
demanding
arc
.
Iterate
over
the
list
of
savings
and
look
at
each
node
in
the
selected
arc
to
see
which
routes
(if
any)
have
that
node
as
an
exterior
node,
attempting
to
merge
these
routes
if
possible
.
Reconstruct
the
final
solution
by
computing
the
shortest
path
between
the
edges
in
the
route
.
González,
S
.;
Juan,
A
.;
Riera,
D
.;
Castellà,
Q
.;
Muñoz,
R
.;
Pérez,
A
.
(
2012
)
:
”Development
and
Assessment
of
the
SHARP
and
RandSHARP
Algorithms
for
the
Arc
Routing
Problem”
.
AI
Communications,
Volume
25
,
pp
.
173

189
(indexed
in
ISI
SCI,
2011
IF
=
0
.
500
,
Q
4
)
.
ISSN
:
0921

7126
.
1
1
)
(
k
k
X
P
s
k
,...,
2
,
1
s
k
k
s
k
k
1
1
1
1
1
1
1
SHARP
the
first
edge
(the
one
with
the
most
savings)
is
the
one
selected
.
RandSHARP
introduces
randomness
in
this
process
by
using
a
quasi

geometric
statistical
distribution
edges
with
more
savings
will
be
more
likely
to
be
selected
at
each
step,
but
all
edges
in
the
list
are
potentially
eligible
.
Notice
:
Each
time
RandSHARP
is
run,
a
random
feasible
solution
is
obtained
.
By
construction,
chances
are
that
this
solution
outperforms
the
SHARP
one
hundreds
of
‘good’
solutions
can
be
obtained
after
some
seconds/minutes
.
Good results with
0.10 <
α
< 0.20
8. RandSHARP: a Multi

Start approach
Juan,
A
.;
Faulin,
J
.;
Jorba,
J
.;
Riera,
D
.;
Masip
;
D
.;
Barrios,
B
.
(
2011
)
:
“On
the
Use
of
Monte
Carlo
Simulation,
Cache
and
Splitting
Techniques
to
Improve
the
Clarke
and
Wright
Savings
Heuristics”
.
Journal
of
the
Operational
Research
Society
,
Vol
.
62
,
pp
.
1085

1097
.
Average Gap w.r.t. BKS in MaxTime = 180s
Averages
Heuristics
BEST10
AVG10
Set
Instances
Nodes
Arcs
Density
PSH
SHARP
RandPSH
RandSHARP
RandPSH
RandSHARP
egl
24
109
144
2,65%
24,61%
6,20%
13,34%
1,32%
13,76%
1,75%
gdb
23
12
29
53,77%
7,09%
7,56%
0,46%
0,26%
0,72%
0,53%
kshs
6
8
15
55,95%
9,87%
9,87%
0,98%
0,62%
0,98%
0,62%
val
34
36
63
10,59%
14,98%
11,93%
2,74%
1,60%
4,00%
2,85%
Totals
87
Averages
47,79
73
22,94%
15,20%
9,05%
4,94%
1,10%
5,62%
1,78%
(*)
BKS
values
obtained
from
(
2010
)
(**)
Java
SE
1
.
6
.
0

Intel
Core
Quad
Q
9400
@
2
.
5
GHz
8
GB
RAM
9. Computational Experiments
RandSHARP
>>
RandPSH
>>
SHARP >> PSH
RandSHARP
>>
RandPSH
>>
SHARP >> PSH
9. Computational Experiments
We
have
presented
the
SHARP
procedure
solving
the
CARP
.
This
procedure
is
based
on
the
savings
heuristic
for
the
CVRP
.
We
have
introduced
the
RandSHARP
algorithm
for
the
CARP
.
This
algorithm
uses
biased
randomization
to
improve
the
SHARP
procedure
.
We
have
also
proposed
an
approach
for
solving
the
ARPSD
.
This
approach
combines
MCS
with
parallel
computing
and
the
RandSHARP
algorithm
.
The
basic
idea
is
to
consider
a
vehicle
capacity
lower
than
the
actual
VMC
when
constructing
CARP
solutions
.
This
way,
this
capacity
surplus
or
safety
stocks
can
be
used
when
necessary
to
significantly
reduce
the
expected
costs
due
to
expensive
route
failures
.
Our
approach
provides
the
decision

maker
with
a
set
of
alternative
solutions
with
different
properties
(number
of
routes,
fixed
and
expected
variable
costs,
reliability
indices,
etc
.
)
It
offers
flexibility
since
it
does
not
assume
any
particular
behavior
of
the
customers’
stochastic
demands
.
Therefore,
the
probabilistic
distributions
which
describe
demands
can
be
generic
.
The
randomized
algorithm
is
easily
parallelizable
,
which
allows
to
generate
pseudo

optimal
solutions
in
‘reasonable’
clock
times
.
18. Conclusions
http://dpcs.uoc.edu  http://ajuanp.wordpress.com
Development and assessment of the SHARP
and RandSHARP algorithms for the arc
routing problem
Sergio González
a
,
Alejandra Pérez

Bonilla
a
, Angel A.Juan
a
, Daniel Riera
a
{sgonzalezmarti, aperezbon, ajuanp, drierat, jcastellav}@uoc.edu
a
Department of Computer Science, Multimedia and Telecommunication, IN3
–
Open University
of Catalonia, Barcelona, Spain.
Cyted

Harosa Int. Workshop
Valparaiso, Chile. November 12

13, 2012
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