Resolution of the berth allocation problem through a heuristic model based on

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R
esolu
tion of the
berth

allocation
problem
through
a

heuristic

model

based on
genetic

algorit
h
ms


Vanina Macowski Durski Silva
*
,
Antônio G. Novaes
a
,
Antônio Sérgio Coelho
b


Department of Production Engineering

Federal University of Santa Catarina, 88040
-
97
0, Florianópolis, SC, Brazil

*
Corresponding author
:

t
el.:

+
55 48 8412 4873
, fax:

+55 48
3721 7011
e
-
mail address: vaninadurski@gmail.com

a
novaes@deps.ufsc.br

;
b
coelho
@deps.ufsc.br


ABSTRACT


This article is
characterized by approaching one
of
the problems encountered in port systems
known as the Berth Allocation Problem. As the operational activities require a high degree of
time to be carried out and, in most cases they are done manually, it becomes necessar
y to use
an optimization tool. To obtain a good solution with a small amount of computational effort, a
heuristic model, based on concepts of Genetic Algorithms

(GA)
, is proposed, allowing this
concept to be learnt. Prepared generically, with some minor ad
justments in the data, the
method can be applied to solve the problem in any port, as the ports use a similar management
system. Finally, a numerical experiment examines and evaluates the results, noting their
effectiveness in the aid the improvement and r
efinement of the management system.


Keywords: genetic algorithm, transportation, berth allocation problem, scheduling, port management.



1 Introduction

The new global organization based on the creation of global markets requires the
establishment of effi
cient logistic systems that are capable of allow the efficient flow of
production to foreign markets. The ports, as a means of transport, should be considered as an
important link in the integration of the domestic and global markets, and their modernizati
on
is one of the main activities to be prepared with the domestic costs reducing plan, as well as
increasing exports.


It is perceived, therefore, that there is a gap still to be explored regarding research and
methods for one of the
most

important operati
onal problems found in the port system: the
berth allocation problem

(BAP)
.

One of the first studies on this subject in Brazil was
written by

Silva and Coelho (2007),
they researched how to determine an allocation plan for the vessels that dock at the port

berths, so that each vessel is placed in a berth for a period of time that is required for the
loading and unloading of the cargo.

The berthing plan is to combine the best of the best possible berth to serve each vessel in
order to respect the restriction
s that are imposed, resulting in a lower berthing cost. The
attention to these aspects makes the berthing preparation plan at
the port an expensive
operation and

requires the use of an optimization technique to prepare this plan. Therefore,
the general pur
pose of this article is to propose a heuristic computational model to help the
decision making of a container port complex on the best way to provide the best berth
-
vessel
allocation with the aim of reducing operational costs.


2

There
are
many papers about t
he berth allocation problem:
Brown et al. (1994, 1997)
,
Lim
(1998),

Moon (2000)
, Imai et al. (2003), Kim and Moon (2003)
,

Guan and Cheung (2004) and
Mulato and Oliveira (2006)

and others.

Brown et al. (1994, 1997) addressed the problem of berth allocation
in marine ports. The
authors identified the optimal set of vessel
-
to
-
berth attributes that maximizes the sum of
benefits of vessels while in port. The planning of berths in marine ports has some important
differences from the planning of berths in commerci
al ports. According to Imai et al. (2001),
in marine ports the change of berths occurs because of the ports own services, in other words,
a new vessel arrives and is assigned to a berth where another vessel is already berthed. This
treatment is unlikely to

occur in commercial ports. Thus, the exchange of berths and other
factors less relevant to commercial ports are considered in Brown et al. (1994, 1997) making
the issue inappropriate for commercial ports.

In the studies done by Lim (1998), the problem was

transformed into a restricted version
of the two
-
dimensional packing problem. Tong et al. (1999) in Dai et al. (2004) solved the
problem of assigning berths by using the optimization approach used by a colony of ants, but
the focus of the work
wa
s to mini
mize the length of the required pier.


Genetic algorithms (GA) are increasingly used to solve inherently intractable problems
such NP
-
hard problems like machine scheduling problems.
A special
aplication

is
from
Nishimura et al. (2001) where is proposed a m
odel for dynamic BAP that considers the arrival
of vessels after the start of the elaboration of the mooring plan. For such case,
GAs
are
applied where two different types of representation for chromosomes are utilized

with
satisfactory results.

Despite th
e fact that there are many ways to resolve the BAP, the nearest
literature to this article’s line of study is the series of articles of Imai et al. (2001) and
Nishimura et al. (2001), which will be used as basis for this research development.

The article
d
iscusses applying GAs to the berth allocation problem and
is organized as
follows: in section 2 the formulation of the model is shown, in section 3 the procedure for
solving the
GA

is given, in section 4 the numerical experiments and the final section
cont
ains

the conclusion of the study and suggestions for future research is given.

2
De
velopment of the Model

The objective of this study is to minimize the length of stay of vessels in berths/ports to
reduce operating costs. To develop a model that comes clos
e to the maximum of the real
problems are considered extremely important factors which have great impact on the
operational costs, such as the cost of the vessel staying in port (Silva and Coelho, 2008).

In practice it is known that a port receives vessels

of various sizes, therefore, different
rates are charged for the stay. For the development of this article the charges for berthing and
handling of loads shall be considered, which were judged to be the greatest impact in forming
the operational cost of d
ock side berthing.

2.1
Restri
ctions

Amongst the various influential restrictions in decision making for the berthing of a
vessel, are:
t
ime

restriction
,
draft restriction
,

length

restri
ction
,

vessel
s scheduling
restri
ction
.


2.2
Decision V
ariables

(
when an
d where to
a
l
loca
te each
vessel
)

The docking plan must be capable of allocating vessels to berths in the best possible way,
i.e., determine 'what' berth should moor each vessel as well as the 'moment' of berthing,
analyzing if the waiting time for each ves
sel, accounting costs, in order to optimize the
operation of the port system.



3

2.3
Formulation of the problem
:

Heuristic
Al
location
A
lgorithm

Initially a simplified algorithm is proposed that is able to consider a list of vessels (each
containing informati
on about the
a
rrival

moment
, draft, cargo, length and differentiated
tariffs), and a list of berths (release time, interval, depth, length, productivity rates). Then, it is
verified whether the restrictions are met, and subsequently allocates a vessel in e
ach given
berth.

To calculate the total allocation cost

(CA)
, consider:


Su
b
je
ct to
:




Where
:

CA: is the objective function of the allocation cost;

D
i
: draft of vessel
i
;

E
i
: vessel
i

waiting time;

D
j
: depth of berth
j
;

A
ij
: vessel
i

service time in the berth
j
;

TAtrac
j
: wharfage duty charged by berth
j

to vessel
i
;

CP
i
: Cost of vessel
i

stopped;

C
i
: load of vessel
i
;

Mcheg
i

: arrival moment of v
essel
i
;

TMov
j
: movement duty charged by berth
j

to vessel
i
;

Mlib
j
: release moment of berth
j
;

P: number of executed periods of time;

L
i
: length of vessel
i
;

x
ij
: binary variable type 0
-
1, where value 1 is considered
in the event of vessel
i
has been se
rviced in berth
j

and
value 0 is assumed in the event of the contrary;

N = {n
1
, n
2
, ..., n
k
}, represents the entirety of vessels;

B = {b
1
, b
2
, ..., b
m
}, represents the entirety of berths.

The function (1) minimizes the total cost of the allocation
,
which

is the sum of the
following plots:

cost of the waiting time and
cost of service time,
cost of cargo handling

and

rate of use
of
the

berth

length
, per period.

T
he equation (2) represents the restrictions in the
length of the berth
j

which must be greater t
han the length of the vessel
i
, the restriction (3)
indicates that the draft of the vessel
i

must be less than the depth of the berth
j
, the restriction
(4) indicates the time the vessel berthed and the service of the vessel
i

and this should be more
than
the release moment from berth
j
, i.e. a vessel cannot be berth and serviced in a given
berth before this is released from servicing the previous vessel.

Because of the fact of greatly variable costs in a sequence of allocation to the other, it is
necessar
y to evaluate the various possibilities for the sequences that may exist
. To be

not
exhaustive, it is proposed to use a genetic algorithm to perform the analysis of possible
allocation sequences.




4

3 Genetic Algorithm for the Problem’s Solution

GAs are lik
e the heuristic methods
where

the optimality of the answers cannot be
determined. They work in the principle of evolving a population of trial solutions over many
iterations to adapt them to the fitness landscape expressed in the objective function.

As sho
wn in the previous section, the
BAP

is a non
-
linear programming problem that is
difficult to solve. To facilitate the solution of the process, a heuristic based on GAs is
proposed by Silva and Coelho (2008).

3.1
Condi
tions of the
P
roposed
H
eur
i
stic
s

In ad
dition to the factors that were carried out, this paper presents a difference to the
Nishimura et al. (2001)

model.

The authors work with the division of the problem in
n

sub
-
problems (SUB
s
) of

berth
allocation in terms of the time factor, as in

Fig.
1
.











Fig.
1

-

Berthing Schedule (NISHIMURA et al., 2001).

After the discharge of all the berths, the first sub
-
problem is solved by
GA
. Inheriting the
solution of the first SUB (i.e. when

all the berths are free again), the next SUB is solved. The
process is repeated until all SUB
s

have been resolved, in the meantime the final solution can
be affected by intermediate solutions. In the
proposed
model this does not occur, since it is
conside
red as a single issue, containing all the vessels for berthing.

Another difference with the work of Nishimura et al. (2001) is the way to characterize a
chromosome.
The cited authors used

a sequence of rand
om berthing of vessels and then

analyze
d

the fitne
ss of this allocation. In the proposed study, the genes are analyzed one by
one, choosing the best individual mooring to another vessel and then select another vessel for
the best berthing position. So, a sequence will be formed for berthing, in other word
s a
chromosome.

3.2
Genetic
A
lgorit
h
m

The GA as a heuristic method
does

not guarantee the optimality of the responses
but
search for the adequate solution of the problem.

This algorithm can be understood better by
following the steps below:


Step 1.

Gener
ate population of individuals.

Step 2.

Calculate value of objective function.

Step 3.

Select ancestors.

Step 4.

Make the crossover between ancestors.

Step 5.

Perform the mutation.

Step 6.

Calculate value of objective function.



5

Step 7.

If the value of the n
ew objective function is better than the
previous value of the objective function, go to Step 8.
Otherwise, go to Step 9.

Step 8.

Replace the individuals of the population by the descendants.

Step
9.

If the objective function is satisfactory, END. Otherwis
e, go
to Step 3.


T
o calculate the objective function

it´s used

the diagram below
:















Diag
.

1

-

Presentation of the calculus of the objective function

3.2.1 Representa
tion and genetic operator
s

In the proposed method,
a sequence of vessels, which can carry with themselves the
values of the variables (depth, length, cargo) is considered as a chromosome:


Where
N
n
k

represent
s the
k
th
vessel

from

the list
,
and each
vessel

has a

posi
tion in this l
ist

(
for example
:
vessel

N3 o
c
cup
ies position
1
in the list
).

To sort the individuals of the population during the search process a measurement is used
for the performance of fitness measured by (1). Being a problem of minimizing the costs
when the lower t
he value of the objective function, the better the adaptation of the individual,
and like this, the method always chooses the best performing individuals causing them to
reproduce.

Several rules exist in the literature to make the crossover between individ
uals and
therefore, for the problem of this research it was decided to differentiate and adopt as a rule,
the average of the positions occupied by individuals in the chromosomes
(Silva and Coelho,
2008).


Consider two chromosomes:


This is the average of the positions occupied by each individual in the two chromosomes
:

n1: occupy´s the positions 4 and 3, so the average is 3.5;


n2: occupy´s the positions 2 and 1, so the average is 1.5;


n3: occupy´s the positions 1 and 4, so the a
verage is 2.5

n4: occupy´s the positions 3 and 2, so the average is 2.5;


6

Then, list them in ascending order, and when there is tie, randomly picks one of them to
fill the position:


As there was tie between

the average of the ind
ividuals

n3

and
n4
,
it´s
chose
n

to insert in
the list, in the second position, the vessel
n3

and
following the vessel

n4
.

In this paper was not
compared this method with others encountered in the literature.

After doing the crossover, the mutation takes sp
ace.

4 Numeric Validation

4.1

Par
ameters for the search of the solution

To implement the proposed algorithm to solve the PAB, a software was developed in
Delphi® version 7, which was tested and the variables analyzed.

To use the

software

it is
needed to
i
nser
t the files that contain
informa
tion on the
vessel
s and the
berth
s

(
length
,
draft
,
dutie
s,
arrival
moment, etc.)
and

fill other
informations like

the

o
c
cu
pation period as well as
the mutation rate to be

consider
e
d.
Three different ways can be

used as s
top criterion
:
maximum
n
umber of
iteration
s achieved
;

maximum time of computing achieved and
,
val
ue of

epsilon
1

achieved
.

It was considered

three options of resolution methods to be chosen
:

M
e
t
h
od 1 (
based on the earliest release date of the
vessels)
,
M
e
t
h
od

2

(
based on lowest cost of
vessel allocation
) and
M
ethod
3

(
based on the relationship between Method 2 and the length
of time the vessels remains in the
port)
.

Method 1 s
tarts with a value for the Earliest Departure Date of the vessels as being
infinite
. Next, determine the value of Departure Date that each vessel would have, if the berth
and the berthing time varied. The Departure Date is found by the maximum value between the
berth release time plus the considered occupation time and the time of arriva
l of the vessel
plus its service time. If the value found for the Departure Date is less than the value for the
Earliest Departure Date, this should be replaced by the first one and the procedure be
repeated. Subsequently, calculate the value of the object
ive function using equation (1).


Method 2
i
nitially
accepts

a value for the minimum cost of allocation as being infinite.

Then, check the best cost of allocation in each of the berths for all vessels, using equation (1).
If the value of the Cost is less t
han the value of the Minimum Cost, it should be replaced by
the first one.

Method 3
is similar to the previous method, but differentiates itself from the
moment that the value of the cost is found and multiplied by the total time that the vessel
stays in t
he port (from arrival to departure). What is

realized is

that this method potentiates the
time by trying to find the balance between the length of stay of a vessel in the port and cost
allocation. Because this study has a heuristic nature this method was a
greed to analyze the
behavior.

4.2
Simulated and real experiments

The proposed method was applied and analysed for some simulated case
s
.
In every case it
was considered as comparing criterion
,

a popula
tion of
3
.
000 indiv
i
d
ual
s
and a mutation rate



1
Epsilon

is the adopted parameter to evaluate if the devition obtained between the fitness
of the

worst and the
best

cromossom
e is aceptable
.
The deviation is found by the following:

Deviation = (Worst Cromo Fitness


Best Cromo Fitness
) / Best Cromo Fitness


In this manner,
if
Deviation > Epsilon,
the

algorit
h
m

continues
.


7

of

2%,
wi
th an occupation

interval

of

2 ho
ur
s
, an occupation period of
6 ho
ur
s

and a
maximum
number of
iterations

of

500
.
000.

The tests were carried out fo
r the fixed processing
times of

3, 5 and 10 minutes, also varying the 3 different methods, this totaled 540 ex
amined
cases.

In order to evaluate the proposed system in terms of quality of results, the method was
applied to the case of
Itajaí

P
ort (Brazil), which currently does not have a tool for the
optimization of allocation of the transaction and, uses Excel® s
oftware.

The following
informati
on was considered for this test:

each day
from N
ovember 20 to December 31, 2007
was analyzed where

63 vessels were scheduled to berth at the port. Currently the port operates
24 hours a day
,

7 days a week, with 3 berths.

It
is worth mentioning that

Itajaí

Port has a 740 meter long pier and does not operate using
fixed
-
size berths. In average there would be 3 berths x 246 meters each, but in practice this
length varies with the length of the vessel that is berthed. For example
, if the port receives a
vessel that is 260 meters long, it will use the first 246 meters of the first berth and another 14
meters belonging to the second berth.


To test the developed model

with actual data from
Itajaí

were

discarded

vessels with the
grea
test length, because the software works with individual berths.

Whereas, the arrival of
these longer vessels is not very likely, the result of the mooring plan will not be very different
to the plan drawn up by the employees of the port in question.

4.3
Re
sult
s obt
ained

Due to the number of simulations carried out some tests were prioritized with respect to
the performance obtained by the proposed tool
:
a
n
alysis of the number of
itera
tion
s
versus

epsilon

variation
;

an
alysis of the number of iterations
versu
s
the
maximum

processing t
ime
;

an
alysis of the computation time difference
versus
the problem
complexi
ty and

an
alysis of the
percentage of
difference
s of values

of the objective function
.

With regard to the number of iterations v
ersus

the
epsilon

variation
, the

three methods
have presented

a decreasing curve, in other words, as the epsilon value increases, the number
of iterations is considerably re
duced.

In relation to the analysis of the number of iterations
versus the maximum processing time required, it

was found that during 75.50% of tests, there
was divergence in the number of iterations between methods 1 and 2, i.e. method 1 required
more iterations than Method 2 compared to the same period of time. This behavior is also
repeated when compared to meth
ods 1 and 3.

In this analysis it can also be seen the extent to which there is variation in the number of
vessels for the same amount of berths (e.g. 15 berths, servicing 10, 70 and 100 vessels),
increases the difference in the number of iterations betwee
n methods 1 and 2 and methods 1
and 3.
It is notorious that the
di
f
feren
ce between

m
e
t
hod
s 1
and

2
,

and

2
and

3
is
around 30%
and that method 1 presents twice the iterations as compared to method 3.

The three methods
behave in most part of the cases as for

the number of iterations carried out by fixed time of
analysis.
As the complexity of the problem increases
,
the difference of behaviour between the
methods also increases
.


With the time convergence analysis of the results, it´s possible do see that by in
creasing
the complexity of the problem, all three methods tend to generate the optimal solution us
ing
the same computational time.
See
Fig.
2
.








8

Fig.
2

-

Difference in the convergence tim
e of between methods 1 and 2.


The values obtained in the objective function (cost allocation) which in most cases was
the difference between methods 2 and 1 take on the behavior of a growing curve, i.e., as the
number of vessels increases, it increases th
e value the percentage difference between the
methods, i.e. the value of the difference from method 2 to method 1 grows considerably as the
number of vessels increases.
Fig
.
3

shows this reasoning:












Fig
.
3

-

D
ifference in the best solution found with the methods 1 and 2.


In this case, when the problem has 100 vessels, the method 2 provides results
in

the
objective function that are 5.61% greater than method 1. As this research att
empts to obtain a
reduction in costs, the greater the value of the solution, the poorer result is.

It should be mentioned that tests carried out in the case of
Itajaí

Port were satisfactory
.

I
t
was possible to allocate all planned vessels and in some cases

there was disagreement as to
the length of stay of vessels in port (in real cases), probably due to exogenous variables (such
as a delay in arrival), which were not considered in the proposed software.


Thus, an analysis was made of the difference in all
ocation obtained from
Itajaí

Port and
proposed by the PAB, as shown in
Fig.
4
.








Fig.
4

-

Analysis of the difference in allocation at the Port of Itajaí and the PAB

T
he grap
h contains

the 30 cases which

were analyzed and the difference obtained between
the real berthing time at the port and the time proposed by the PAB.
It´s possible to
see that
for most of the cases that were examined, a gain of up to 5 hours was obtained before the

9

allocation proposed
by the PAB, in other words, the PAB provided the berthing details by up
to 5 hours in most of dockings.


The Histogram 1 below shows an easier way to
interpret the allocations made. In

approximately 70% of cases
was
showed a difference in allocation
,

betwe
en real data and the
proposed method
,

of up to 5 hours;
which shows that the developed model is valid and can
provide satisfactory results
.


Histog
.

1
-

Frequency difference of the allocation in the Port of Itajaí and the PAB

Thi
s does not guarantee the efficiency of the proposed method compared to the system
in
Itajaí Port
. For the cases tested in the PAB, it was considered that the berths were made
available at the completion of the berthing plan, which in practice, in Itajaí, t
here were
probably other vessels occupying the berths and delaying the berthing of vessels used in this
analysis. However, the importance of the proposed software cannot be discarded, as the
suggested dockings were similar to those that occurred in practic
e, achieving the proposed
goal of reducing costs as well as the ease of preparing of the berthing plan.

5
Conclu
sions and Recommendations

A heuristic tool to support the decision
-
making on the best allocation of vessel
-
berths,
using
GAs
, was presented in t
his paper.
Although this study has been conducted on other
references,
it was not possible to compare
with the proposed model, since the
analyzed
data
were not the same
.

This method has shown to be effective and with a low degree of
implementation difficul
ty, and may be validated for use in container terminals.


As the behavior of simulations varies with the value of
epsilon
, it can be confirmed that
the results were not very stable but, in most cases, the number of iterations was reduced as the
value of
ep
silon

increased
.

F
or the number of iterations and maximum processing time, it was concluded that method
1 is more suited to carry out the processing, since it performs more iterations than methods 2
and 3. In relation to the results obtained for the value
of the objectiv
e function of the three
methods

was

expected

for

method 2 the best result for the allocation

cost
, because this was its
purpose: allocate
minimizing costs
, but it was seen that method 2 presented a greater
allocation cost than method 1 that
increased the complexity of the problem.

The

decision
of the best method
depend
s on the aimed result
:
lesser
mooring

time of the
vessel
s, or lesser
a
l
loca
tion costs
.
About the

analysis of convergence
time
of the results it is
concluded that by increasing t
he complexity of the problem, the methods tend to have similar
behavior reducing the difference between the computational time and results for the optimal
solution.

The proposed tool can have wide applications
in

t
he container port system management
,
facil
itating the work of logistics operators,
taht

conduct, in most cases, manually the berthing
plan. It also promotes the reduction of errors dur
ing this activity, as this is

computerized,
allowing for a reduction in operating costs.


10

To proceed with this rese
arch it is recommended to consider the existence of different
cargoes moving around the port (bulk, liquid
, etc.
), the availability of equipment at
the berths

to receive a certain vessel for handling the cargo, and distance to be traveled by the cargo
from

the vessel to the warehouse or even the method of transport that will be used later. It is
also recommended to change the proposed algorithm to allow more than one vessel
berthing
to optimize the available space, making a greater number of practical tests

in order to increase
the reliability of the system.

References

Brown G G, Lawphongpanich S, Thurman K

P
(1994)
Optimizing ship berthing
. Naval Research Logistics
,

41,
1
-
15.

B
rown

G

G, Cormican K J, L
awphongpanich

S, W
iddis

D B

(1997)
Optimizing submarine
berthing with a
persistence incentive.
Naval Research Logistics,

44.

Guan Y, Cheung R

K

(2004)
The berth allocation problem: models and solutions methods.
OR Spectrum
,

26
,

75
-
92.

Imai
A, Nishimura E, Papadimitriou S

(2001)

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