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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