The application of Genetic Algorithms to the Berth Allocation Problem

The Application of Genetic
Algorithms to the Berth
Allocation Problem
Foo Hee Meng
18 August, 2000

Quick Summary
• The Berth Allocation Problem (BAP)
– Allocation of berthing space to container
vessels (ships) in a transshipment port
• Objective
– Minimize container handling effort
• Solution Method
– Genetic Algorithm (GA)
• Focus of Thesis
– devise good way to apply GA to BAP
– what are the problems, how to deal with them.

Presentation Outline
• Introduction
– The Berth Allocation Problem (BAP)
– Focus of thesis
• “Traditional” GA applied to the BAP
• Our solution - The BAP-GGA
• System Implementation
• Experimentation
• Conclusion

Vessels
Yard Cranes
Quay Crane
Prime
Mover
Section 1
Section
2
Container Yard Area
Wharf Area
The Port

The Berth Allocation Problem
Section 1
Section 2
Section 3
Planning Window
15 3
20
10
5
12
25
3045
20
(say 1 week)
Transshipment

Container Handling Effort
Section 1
Section 2
Section 3
Trans. Cost = sum of all {cross section distance moved for each container}
Crosssectiondistance
15 3
20
10
5
12
25
3045
20

Container Handling Effort
Section 1
Section 2
Section 3 15 3
20
10
5
12
300
300
Trans. Cost = (300  10) + (300  5) + (300  20) + (600  3) + (300  12) + (300  15)
= 20400

Why is this Problem Important?
• Large potential savings in container
handling costs
– worldwide traffic doubled between 1990 - 1998
to 175mil. TEU (World Bank)
– PSA handled 16 mil. TEU in 1999 (ST)
• Recurring costs
– Container traffic seasonal
• Limited land area
Twenty-foot Equivalent Units
Small percentage savings
mean large dollar savings
Plan for 1 week
use for next few weeks

Previous Work
• Leong and Hum 1992
– Modeled problem as a network flow problem.
– Did not consider cross-sectional distances.
• Important - PPT, Europe (Pallotino)
• Loh 1996
– Modeled BAP as a generalization of the QAP
– QAP = Quadratic Assignment Problem
– QAP factors in distances of assignment locations
– QAP is NP-Hard
– Proposed a Genetic Algorithm (GA) solution

Focus of Loh’s Work
• Establish basic Problem Model
– QAP  BAP
• Demonstrated feasibility of GA solution
– eval. QAP benchmarks using BAP
• Proof of concept system
– tested with 2 small, hand-crafted data sets
• (50 vessels, 3 sections)
Start of R&D effort by RAS Group

Our Adopted Solution Approach
Vessels Sections
1. Section Assignment
(Partitioning)
Allocation of vessels to
sections.
2. Wharf mark assignment
(Packing)
Assign wharf locations to
vessels within allocated
sections.

Partitioning?
Section 1
Section 2
Section 3
Planning Window
15 3
20
10
5
12
25
3045
20

Partitioning!
3
5
13
11
1
2
4
6
7
8
9
10
12
15 3
20
10
5
12
25
3045
20
Each vessel rep. by node
Transshipment flow rep. by directed edge
 min-cut multi-way partitioning problem

Assumptions
• Any vessel can be assigned to any section
– no bias
• All vessels are berthed upon arrival
– no delay
• Vessels can berth along the whole length of
the section
– no berthing between sections

The BAP Partitioning Problem
• Allocation of sections to vessels to minimize transshipment cost
• Do not consider which wharf location to allocate each vessel to.
• Hence, the need to incorporate capacity constraint

The Berth Capacity Constraint
• At any instance of time within the planning period, the total berth usage of all
vessels assigned to each section cannot exceed the length of that section.
t
0    L t S


The RAS BAP Road Map
[Loh96]
Date: 1996
Partitioning
Packing
Exptn.
System
Devt.
Delay
BAP-GA
1.0
Heuristic
Packer
[LeKu96]
Port
Survey
[Foo96]
Data
Gen.
[FoOn96]
Expt. &
Analysis
[FoOn96]
BAPS
GUI
[FYP97]
BAPS 1.0 Date: Aug 97
BAP-GA
2.0
BAP-GP
[Ong00]
BAP-BB
[Xu99]
DSA
[Quek98]
BAP-Pack
[Chen99]
Expt. &
Analysis
[FoOn98]
BAPS
GUI 2.0
[Ong00]
Interactive
BAPS
[Sim99]
Interactive
BAPS 2.0
[Nisha00]
Date: May 2000BAPS 2.0
BAP-MP
[Li99]

The BAP Partitioners
BAP-GA
1.0
BAP-GA
2.0
BAP-GP
[Ong00]
BAP-BB
[Xu99]

BAP-GA
1.0
BAP-GA
2.0
BAP-GP
[Ong00]
BAP-BB
[Xu99]
GAApproach
Objectives:
• Evaluate Loh’s
engine to realistic
data sets.
Lessons:
• Traditional GA -
do not solve BAP
Partitioning well.
• Not scalable.
Branch & Bound
(BAP-BB)
Optimal
Solutions
Very Slow
Modified
Fidducia-
Mattheyses
(BAP-GP)
Locally
Optimal Solutions
Very Fast
GA
Objectives:
• Address drawbacks
of [Loh96].
• Answer question:
why traditional GA
not suitable?

Modified
[Loh96]
BAP-GA
2.0
BAP-GP
[Ong00]
BAP-BB
[Xu99]
GAApproach
Objectives:
• 1st cut solver.
• Initial results.
Lessons:
• Traditional GA -
do not solve BAP
Partitioning well.
• Not scalable.
Branch & Bound
(BAP-BB)
Optimal
Solutions
Very Slow
Modified
Fidducia-
Mattheyses
(BAP-GP)
Good Solutions
Very Fast
GA
Better
Solutions
Fast

Focus of Thesis
• Devise a suitable GA for solving the BAP
Partitioning Problem
– Good (in terms of quality) solutions
– reasonable planning time (say 1 night)
• Answer question why “traditional” GA
methods fail

Thesis Contributions
• Devised good GA for BAP Partitioning
– < 19% improvement for 40% data sets over GP
– Locate feasible allocations where GP could not
– < 6 hrs on P2-300Mhz
• Identified reasons for “traditional” GA failure
– unsuitable encoding
– problem with busy schedules
• Conducted Extensive Experimentation
– ~60,000 individual expt. runs on various GA param. combi.

• No real data
• GA Evaluation Problem
– Many parameters to tune
– Long running times
• No real data
– Generate Realistic Data
• GA Evaluation Problem
– Adopt Successive Refinement Strategy
Difficulties Faced

• Introduction
• Experimentation
• Conclusion

Genetic Algorithms
• A class of stochastic, population based, search
algorithms that mimic the process of natural
selection in the natural world.
– One class of Evolutionary Algorithms (EA)
• Genetic Programming, Simulated Evolution
• Some history
– John Holland at the Univ. of Michigan 1970s
• Formal model of natural selection and genetics
– Ken. DeJong - late 70s
• Experimental work to verify Holland’s theory
– David Goldberg (at U. Mich.) 1980s
• first applied GA to solve a real-world problem

Applying GA to Optimization
Problem
Chromosomes
Encoding
Solution
Decoding
Genetic
Algorithm
Fittest
Chromosome

Generation
The GA Process
Randomly
Generate
Initial Population of
chromosomes
Select
chromosomes
for
Mating
based on
Fitness
Select
Survivors
based on
Fitness
Perform
Mutation
on
chromosomes
Put
Children
back into
Population
Recombine
chromosomes
to
Produce
Children
(Mating)
Evaluate
chromosomes
in
Population
based on
Fitness
N
Y
Stop?
Done

Applying GA To Problem
• Define Encoding scheme
– solution instance chromosome
• Define Fitness Function
– measure of individual’s (associated to chromosome) survivability
• Define Selection scheme
– ie. How to choose who should mate
• Define Crossover scheme
– recombination procedure (mating)
• Define Mutation scheme
– variation procedure
• Define Replacement scheme
– ie. Who should live and who should die
• “Roulette wheel” or proportional selection
– fitter have higher probability of getting to mate.
• Steady State Replacement
– replace fixed number of least fit in population with children.

Encoding The BAP
3
5
13
11
1
2
4
6
7
8
9
10
12
15 3
20
10 5
12
25
3045
20
2 3 4 5 6 7 8 9 131211101Vessel Number :
Section Number :
BAP Chromosome
Genes

Section 1
Section 2
Section 3
Encoding The BAP
3
5
13
11
1
2
4
6
7
8
9
10
12
15 3
20
10 5
12
25
3045
20
2 3 4 5 6 7 8 9 131211101
1 1 1 2 2 2 2 2 33331
Vessel Number :
Section Number :
BAP Chromosome
Object Membership Encoding

Performing Crossover
3 1 2 1 3 3 1 2 2 2113
1 1 1 2 2 2 2 2 33331
Cross-point Cross-point

Performing Crossover
1 1 1 1 3 3 1 2 2 3333
2 Point Crossover

Performing Mutation
1 1 1 1 3 3 1 2 2 33332
Mutated Gene
Flip Mutator

Applying GA To Problem
• Define Encoding scheme
– solution instance chromosome
• Define Fitness Function
– measure of individual’s (associated to chromosome) survivability
• Define Selection scheme
– ie. How to choose who should mate
• Define Crossover scheme
– recombination procedure (mating)
• Define Mutation scheme
– variation procedure
• Define Replacement scheme
– ie. Who should live and who should die

Calculating Fitness
• Objective Function
= Transshipment Cost + (   Overshoot Cost)
• BAP Objective is to Minimize
• But GA Objective is to Maximize Fitness
– Need to convert Min problem to Max problem
Primary Objective • Relaxation of Berth
Capacity Constraint
• Penalize infeasible
allocations.
• Discriminate among
infeasible allocations
• >> Trans. Cost
• Differentiate
feasible and
infeasible obj.
values

Calculating Overshoot
Planning Window
10 5
315
12
20
300
300

Calculating Overshoot
Planning Window
• Overshoot Cost = Average Overshoot + Max. Overshoot


• Object Membership Encoding
• Crossover schemes
– 2 point
• Mutation schemes
– flip
• Selection - Roulette Wheel
• popsize = 50, numgen = 50, 500
BAP-GA 1.0 (Prelim. Study)
• Crossover schemes
– 1 point, 2 point, Uniform
• Mutation schemes
– flip, Neighborhood Search (NS)
• Selection - Roulette Wheel
• popsize = 50, numgen = 50, 500
Neighborhood Search:
1-move, best neighbor
Best GA

Literature Survey
• Revisit of GA fundamentals
– Introductions: [Gold89], [Mich92], [Reev93]
– Theory: [Holl75]
• GAApplications to similar problems
– Jones and Beltramo [JoBe91]
• Initial application of GA on clustering/partitioning
– Bui and Moon ([BuMo94] and [BuMo94a])
• multiway partitioning and ratio-cut partitioning
– Kochhar and Heragu [KoHe98]
• GA approach to Multi-Floor Facility Layout (MFFLP)
– Falkenauer [Falk97]
• Grouping GA (GGA) for grouping problems

Key Survey Findings
• Better understanding of how GA works
– [Holl75], [Gold89], [Mich92], [Falk97]
• Which are the more important GA operators
– Encoding scheme
– Crossover scheme
– Mutation scheme
– [Reev93], [Falk97], [Kers97]
– The Focus of our study is on these 3 operators

“Traditional”
GA
Domain Specific
– reordering of chromosome [BuMo94]
– multi-point crossovers [BuMo94]
– swap mutator [JoBe91]
• Object Membership/Neighborhood Search Hybrid
– Limited NS [BuMo94] [BuMo94a]
– NS post-process [BuMo94] [BuMo94a]
• Order Based Encoding
– PMX, OX, CX crossovers [JoBe91]
– Greedy decoder [JoBe91]
– First Fit decoder [KoHe98]
• Grouping GA (GGA) [Falk97]
Key GAApproaches Studied
(Classified According to Encoding Scheme)

The Order Based Encoding
• [JoBe91] - clustering/partitioning,
[KoHe98] - Multi-floor facility layout
• Initially used to solve the TSP
• To solve other problems - need decoder
5 3 10 8 1 11 6 4 12 97132
5 3 10 8 1 11 6 4 12 97132
Decoding
Procedure
Section 1
Section 2
Section 3
3
5
13
11
1
2
4
6
7
8
9
10
12
15 3
20
10 5
12
25
3045
20

Unsuitability of Order Based GA
• The encoding scheme is very bad
• “Incompleteness” of decoder
– difficult (when solving practical problems) to
guarantee completeness of decoder
• Very high redundancy of encoding
– Number of possible permutations is n!
– Number of possible Object Membership
configurations is only mn
– Level of redundancy is (n! / mn)
• 100 vessels, 4 sections - (100! / 4100) ~ (10157/1060)
– Most of the time GA is doing nothing useful!

– Redundant and incomplete encoding
– Context insensitivity of crossover
– Problems handling busy schedules
– GA/NS Contribution
– Problems with NS Methods dealing with busy periods
Unsuitability of Traditional GA

Unsuitability of Obj. Mem. GA
(Context Insensitivity of Crossover)
Section 1
Section 2
Section 3
3
5
11
13
10
2
4
6
7
8
9
1
12 15 3
20
10 5
12
25
3045
20
2 3 4 5 6 7 8 9 131211101
1 1 1 2 2 2 2 2 33313
Vessel Number :
Section Number :
BAP Chromosome
Reason:
Traditional
crossover works
object-wise - not
group wise

Inversion or Reordering
• [BuMo94] - multiway partitioning
[BuMo94a] - ratio-cut partitioning
• Cluster groups of genes together so that
crossover has less chance of breaking up
good subgroups
• Use WDFSR preprocess - Weighted Depth
First Search Reordering
• However, still has high prob. of breaking up
good subgroups

• Random initialization - population infeasible
• GA waste time getting to feasible region

numgen
Unsuitability of Obj. Mem. GA
(Problem dealing with Busy Schedules)

Limitations of NS Methods
• BAPS 1.0 Partitioner, [Ong00] - BAP-GP
• NS Methods - Local Adjustment
– cumulatively improve solution by performing small adjustments
• Choice of neighborhood
– 1-move? 2-move? How to choose?
• The Problem maneuvering busy periods
– Overflow bucket [Ong00]

• Unsuitable encoding
– Use Grouping GA (GGA) encoding
• Difficulty locating feasible allocations
– Use greedy initializer
– Use timezone mutator
How we deal with Traditional GA
Limitations
• Difficulty locating feasible allocations

Falkenauer’s Insights on Encoding
• Good solutions of Grouping problems
usually composed of subgroups of objects
with high affinity to one another.
– Meaning of affinity - diff. for diff. prob.
– For BAP Partitioning - large transshipment
connections + “knapsack”-type goodness-of-fit
• GA encoding and recombination should
facilitate the creation and exchange of
subgroups with high affinity.

The GGA Encoding
Section 1
Section 2
Section 3
3
5
13
11
1
2
4
6
7
8
9
10
12 15 3
20
10
5
12
25
3045
20
1 2 3
1
2
3
4
5
6
7
8
10
11
12
13
9
Group
Part
Object
Part
2 Crossover schemes
• Inject crossover (Falkenauer)
• 1 point crossover (our own)

Inject Crossover
1 2 3
1
2
3
4
5
6
7
8
10
11
12
13
9
1 2 3
1
5
6
10
2
4
7
11
3
8
9
13
12

Inject Crossover
1
1
2
3
4
2
5
6
7
8
9
3
10
11
12
13
1
1
5
6
10
2
2
4
7
11
12
3
3
8
9
13

Inject Crossover
1
1
2
3
4
2
5
6
7
8
9
3
10
11
12
13
2
2
4
7
11
12
Recombination
of subgroups

Inject Crossover
1
1
2
3
4
2
5
6
7
8
9
3
10
11
12
13
2
2
4
7
11
12
Duplicates

Inject Crossover
1
1
2
3
4
2
5
6
7
8
9
3
10
11
12
13
2
4
11
12 • Removal of Duplicates
- use heuristic procedure
that makes use of affinity
of objects to choose
between 2 groups
• Mutation - use another
heuristic procedure

1-point Crossover
1 2 3
1
2
3
4
5
6
7
8
10
11
12
13
9
1 2 3
1
5
6
10
2
4
7
11
3
8
9
13
12

1-point Crossover
1
1
2
3
4
2
2
4
7
11
12
3
3
8
9
13
Duplicates

1-point Crossover
1
1
2
4
2
7
11
12
3
3
8
9
13
5 6 10Missing

1-point Crossover
1
1
2
4
2
7
11
12
3
3
8
9
135
6
10
• Removal of Duplicates
- use heuristic procedure
• Add Missing - use same
heuristic procedure

Affinity - The Greedy Heuristic
• Primary Heuristic - allocate to section to
cause least increase in transshipment cost.
• Secondary Heuristic - allocate to section
with largest available capacity
3
5
13
11
1 2
4
6
7
8
9
10
12 15 3
20
10 5
12
25
3045
20
3

Resolving Busy Schedules
• Random initialization - infeasible allocations
– Greedy Initializer
• Problem with Neighborhood Search
– Timezone Reallocator (mutator)
• Generate a random permutation of all the vessels
• Apply Greedy Heuristic on the list
• Higher density time periods are more likely to be cause of
infeasibility.
• However, cannot discount lower density time periods.
• k-move heuristics have problems dealing with them
• Instead of performing local adjustment - empty and reallocate

Time Zones
• Chop up planning window into non-interfering time periods called
time-zones
z1 z2 z4 z5 z6 z7 z8 z9z3

Zone Densities
•Make use of proportional selection to choose time-zone to have vessels
emptied and reallocated (greedy)
z1 z2 z4 z5 z6 z7 z8 z9z3
Timezone Mutator
13 15 16% 12 11 9 7 610

Summary of BAP-GGA Features
– Grouping GA (GGA) encoding and crossover
• Busy schedules
– Greedy Initializer
– Timezone mutator

BAPS
The BAP Architecture
BAP
PackageData Sources
BAP
Solver
BAP-XX
Packer
BAP-XX
Packer
BAP-XX
PackerBAP-XX
Partitioner
BAP-XX
Partitioner
BAP-GA
Partitioner
Analysis
Module
Display
Module
Operations
System
(Interactive)

Structure of BAP-GGA.
GA Genetic Algorithm
GA Population
BAP
GGA
Chromosome
BAP
GGA
Chromosome
BAP
GGA
Chromosome
...
Chromosome Handling Functions
• Crossover - inject, 1-point
• Mutator - timezone, flip
Standard GA Processes
• Roulette wheel selector, Steady-state replacement
BAP
Fitness Function
GALib
Component
Self
implemented

• Introduction
• Experimentation
– Data Generation
– Experimentation System
– Evaluating the various GA studied
– Result Analysis
• Conclusion

Data Generation
• Obtain Domain Information
– Port Survey [Foo96]
– publicly available sources (PortView, WWW)
– domain data for data generation
• Design and build data generation system
– BAP Expt. Framework [Foo99]
– Flexible, comprehensive
• Generated 100 realistic data sets
– The BENCHMARK data set collection

BENCHMARK Data Set Collection
Port Structures
5 (Brani)1 2 3 4
60
16696 240 240 240
208120 302 302 302
250144 362 362 362
292422 422 422
0.25
0.4
0.5
0.6
0.7
A
B
D*
• 5 data sets in each
box.
• Num. in box is num.
of vessels
Port Section
Layouts
* Average Berthing Demand
600m
600m
600m
600m
600m
600m
600m
600m
600m
600m1,800m
1,200m
1,200m
600m
480m
800m
200m

BAP Experimentation System
BAP
Experimental
Framework
Result
Analysis
Subsystem
Data
Generation
Subsystem
Expt. Run Subsys.
BAPS
BAP
Package
BAP
Solver
BAP-XX
Packer
BAP-XX
Packer
BAP-XX
PackerBAP-XX
Partitioner
BAP-XX
Partitioner
BAP-XX
Partitioner

– WDFS Reordering, multi-point crossovers [BuMo94]
– swap mutator [JoBe91]
– Various hybridization schemes
– PMX, OX, CX crossovers [JoBe91]
– Greedy decoder [JoBe91], First Fit decoder [KoHe98]
• Grouping GA (GGA) [Falk97]
– Random, Greedy initialization
– Inject, 1-point crossover
– flip, timezone mutator
GA Combinations Evaluated
110 Various GA
Combinations!

Strategy for GA Evaluation Problem
• Focus on GA elements that give biggest payoff
– Encoding, Crossover, Mutation
• Quick prototyping of GA solvers
– Use library of GA components (GALib)
• “Production-line” experimentation system
– BAP Experimental Framework
– Automate expts, clone
• Successive Refinement
– Progressively evaluate larger data sets
– 4 stage evaluation process

Other Data Set Collections
• BENCHSUB collection
– 15 representative data sets from BENCHMARK
• SMALLDATA collection
– 16 data sets (26-42 vessels, 2-3 sections) - opt.
– Created by Xu in [Xu99]
• TESTDATA collection
– 21 data sets (6-15 vessels, 2-3 sections) - opt.
– handcrafted

Stage 0
PMX O
X
CX
First Fit
Greedy
Crossover
Decoder
Crossover
Mutator
Reordering
1pt 2pt 3pt 4pt 5pt
No Reorder
WDFS Reorder
flip
swap
Uniform
1pt
random
Inject
flip
timezone
Crossover
Mutator
greedy Initializer
BAP-GGA
Object
Membership
Order
Based

Stage 3
PMX O
X
CX
First Fit
Greedy
Crossover
Decoder
Crossover
Mutator
Reordering
1pt 2pt 3pt 4pt 5pt
No Reorder
WDFS Reorder
flip
swap
Uniform
1pt
random
Inject
flip
timezone
Crossover
Mutator
greedy Initializer
BAP-GGA
Object
Membership
Order
Based
Data sets:
- BENCHSUB collection
- 15 data sets from
BENCHMARK.
Popsize/numgen:
- 50/100, 50/400,
100/200
Limited Experimention
(still open)

The Elimination of Order Based
Ratio of Optimal
oga21 oga22 oga23 oga24 oga25 oga26
d011 1.97 1.97 1.97 1.97 1.97 1.97
d013 1.97 1.97 1.97 1.97 1.97 1.97
d015 1.97 1.97 1.97 1.97 1.97 1.97
d017 1.97 1.97 1.97 1.97 1.97 1.97
d021 1.08 1.08 1.08 1.08 1.08 1.08
d023 1.08 1.08 1.08 1.08 1.08 1.08
d025 1.08 1.08 1.08 1.08 1.08 1.08
d027 1.08 1.08 1.08 1.08 1.08 1.08
d031 1.13 1.13 1.13 1.13 1.13 1.13
d033 1.13 1.13 1.13 1.13 1.13 1.13
d012 inf inf inf inf inf inf
d022 1.00 1.00 1.00 1.00 1.00 1.00
d024 1.00 1.00 1.00 1.00 1.00 1.00
d026 1.00 1.00 1.00 1.00 1.00 1.00
d028 1.00 1.00 1.00 1.00 1.00 1.00
d041 1.00 1.00 1.00 1.00 1.00 1.00
Ratio of Optimal
gr001 ff001 gr010 ff010
d011 1.97 1.97 1.97 1.97
d013 1.97 1.97 1.97 1.97
d015 1.97 1.97 1.97 1.97
d017 1.97 1.97 1.97 1.97
d021 1.08 1.08 1.08 1.08
d023 1.15 1.08 1.08 1.08
d025 1.08 1.08 1.08 1.08
d027 1.08 1.15 1.08 1.08
d031 1.32 1.51 1.13 1.51
d033 1.13 1.51 1.13 1.51
d012 inf inf inf inf
d022 1.00 1.00 1.00 1.00
d024 1.00 1.00 1.00 1.00
d026 1.00 1.00 1.00 1.00
d028 1.00 1.00 1.00 1.00
d041 1.01 1.00 1.01 1.00
Incompleteness of Decoder
Random
Decoding

The Elimination of Order Based
oga01 oga02 oga03 oga04 oga05 oga06 oga11 oga12 oga13 oga14 oga15 oga16 oga21 oga22 oga23 oga24 oga25 oga26
d011 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0
d012 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0
d013 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
d014 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
d015 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0
d016 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0
d017 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
d018 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
d021 30 31 28 30 31 28 139 110 109 139 110 109 22 26 24 22 26 24
d022 39 39 42 39 39 42 86 39 65 86 39 65 11 9 10 11 9 10
d023 48 33 17 48 33 17 129 61 130 129 61 130 12 21 25 12 21 25
d024 14 20 23 14 20 23 51 138 97 51 138 97 17 13 4 17 13 4
d025 24 36 16 24 36 16 169 174 191 169 174 191 11 24 1 11 24 1
d026 28 28 38 28 28 38 86 147 63 86 147 63 4 17 18 4 17 18
d027 42 52 27 42 52 27 107 137 73 107 137 73 19 13 10 19 13 10
d028 49 38 18 49 38 18 63 55 77 63 55 77 10 10 8 10 10 8
d031 58 71 63 58 71 63 148 121 207 148 121 207 13 34 7 13 34 7
d032 43 37 28 43 37 28 175 141 181 175 141 181 26 37 19 26 37 19
d033 39 44 46 39 44 46 177 127 183 177 127 183 26 25 43 26 25 43
d034 41 44 44 41 44 44 188 163 160 188 163 160 21 25 19 21 25 19
Points of Stability
100 gen. 400 gen. 100 gen.
Redundant Encoding

Stage 3
PMX O
X
CX
First Fit
Greedy
Crossover
Decoder
Crossover
Mutator
Reordering
1pt 2pt 3pt 4pt 5pt
No Reorder
WDFS Reorder
flip
swap
Uniform
1pt
random
Inject
flip
timezone
Crossover
Mutator
greedy Initializer
BAP-GGA
Object
Membership
+
NS
Postprocess
Order
Based
Best BAP-GGA
• GGA encoding
• Inject crossover
• timezone mutator
• greedy and random init.
Limited Experimention
(still open)

BAP-GGA Compared with B&B
Objective Cost Ratio of Optimal (bb001)
ABD gga45 gga46 gga47 gga48 optimal gga45 gga46 gga47 gga48
d212 0.55 1.61E+05 1.61E+05 1.61E+05 1.61E+05 1.61E+05 1.00 1.00 1.00 1.00
d213 0.57 1.62E+05 1.62E+05 1.69E+05 1.62E+05 1.62E+05 1.00 1.00 1.04 1.00
d215 0.56 1.45E+05 1.45E+05 1.45E+05 1.45E+05 1.45E+05 1.00 1.00 1.00 1.00
d217 0.68 2.02E+05 1.69E+05 1.69E+05 1.69E+05 1.69E+05 1.19 1.00 1.00 1.00
d341 0.35 4.86E+04 4.86E+04 4.86E+04 4.86E+04 4.86E+04 1.00 1.00 1.00 1.00
d342 0.40 6.45E+04 6.45E+04 6.45E+04 6.45E+04 6.45E+04 1.00 1.00 1.00 1.00
d343 0.37 5.67E+04 5.67E+04 5.67E+04 5.67E+04 5.67E+04 1.00 1.00 1.00 1.00
d344 0.39 6.39E+04 6.39E+04 6.39E+04 6.39E+04 6.39E+04 1.00 1.00 1.00 1.00
d345 0.38 9.69E+04 9.69E+04 9.69E+04 9.69E+04 9.69E+04 1.00 1.00 1.00 1.00
d346 0.47 2.23E+05 1.97E+05 1.97E+05 1.97E+05 1.97E+05 1.13 1.00 1.00 1.00
d347 0.49 1.25E+05 1.10E+05 1.10E+05 1.10E+05 1.10E+05 1.13 1.00 1.00 1.00
d348 0.49 2.37E+05 2.48E+05 2.01E+05 2.35E+05 2.01E+05 1.18 1.23 1.00 1.17
d349 0.47 9.39E+04 9.39E+04 9.39E+04 9.39E+04 9.39E+04 1.00 1.00 1.00 1.00
d350 0.47 1.25E+05 1.25E+05 1.25E+05 1.25E+05 1.25E+05 1.00 1.00 1.00 1.00
d351 0.55 3.95E+05 4.45E+05 3.41E+05 3.22E+05 3.22E+05 1.23 1.38 1.06 1.00
d352 0.59 3.50E+05 3.14E+05 2.85E+05 2.73E+05 2.73E+05 1.28 1.15 1.04 1.00
gga45 - random Init, 1-pt Crossover
gga46 - greedy Init, 1-pt Crossover
gga47 - random Init, inject Crossover
gga48 - greedy Init, inject Crossover
SMALLDATA
collection (26-42 vessels)

BAP-GA
2.0
BAP-GP
[Ong00]
BAP-BB
[Xu99]
Branch & Bound
(BAP-BB)
Optimal
Solutions
Very Slow (> 1 hour)
Modified
Fidducia-
Mattheyses
(BAP-GP)
Good Solutions
(~ 87 % Best)
Very Fast
(< 1 min per run)
GA
Better
Solutions
(~ 95% Best)
Fast
(~ 1 min per run)

Compared with BAP-GA 1.0
BAP-GGA is best (<= 1.0)
BAP-GGA is worse than BAP-GA 1.0 (feasible) < 1.2
BAP-GGA did not find feasible soln (BAP-GA 1.0 did not)
BAP-GGA found feasible soln (BAP-GA 1.0 did not)
BAP-GGA worse than BAP-GA 1.0 (feasible) > 1.2
Port Structure
51 2 3 4
60
16696 240 240 240
208120 302 302 302
250144 362 362 362
292422 422 422
0.25
0.4
0.5
0.6
0.7
A
B
D
BAP-GGA did not find feasible soln (BAP-GA 1.0 found)
BAP-GA 1.0 Hybrid GA
- ~ 5 hours per run (10 runs)
BAP-GGA
- < 10 mins per run (10 runs)
72
21
1
0
6
0
Best Ratio
0.74

Compared with BAP-GP (800 runs)
BAP-GGA is worse than BAP-GP-800 (feasible) < 1.2
BAP-GGA did not find feasible soln (BAP-GP-800 did not)
BAP-GGA found feasible soln (BAP-GP-800 did not)
BAP-GGA worse than BAP-GP-800 (feasible) > 1.2
Port Structure
51 2 3 4
60
16696 240 240 240
208120 302 302 302
250144 362 362 362
292422 422 422
0.25
0.4
0.5
0.6
0.7
A
B
D
BAP-GGA did not find feasible soln (BAP-GP-800 found)
BAP-GGA
- low mutation rate (0.05)
38
53
0
0
7
2
Best Ratio
0.81
• Large solution space
- 5 sections
• All sections are small
• GA has not stabilized.

Longer BAP-GGA Runs
Port Structure
51 2 3 4
240
302
362
422
0.25
0.4
0.5
0.6
0.7
A
B
D
BAP-GGA
- double num of gen. (1000)
Best Ratio
0.96

Increasing Population Size
Port Structure
51 2 3 4
240
302
362
422
0.25
0.4
0.5
0.6
0.7
A
B
D
BAP-GGA
- double popsize (100)
Best Ratio
0.95
Larger solution space
requires larger GA
coverage

Compared with BAP-GP (800 runs)
Port Structure
51 2 3 4
60
16696 240 240 240
208120 302 302 302
250144 362 362 362
292422 422 422
0.25
0.4
0.5
0.6
0.7
A
B
D
BAP-GGA
38
53
0
0
7
2
Best Ratio
0.81
• Exisitence of very
small section (200m)
• Multi-start (800)NS
could not resolve.

Increasing Mutation Rate (Timezone)
Port Structure
51 2 3 4
250
292
0.25
0.4
0.5
0.6
0.7
A
B
D
BAP-GGA
- mutation rate (0.5)
Best Ratio
0.91

Summary of Analysis
• BAP-GGA gives good results but in longer
time (compared to BAP-GP).
– How long depends on size of solution space.
– < 19% better results possible
– finds feasible solutions where NS type methods have diff.
– < 20mins per run on slow P2-300Mhz
– < 6hrs total planning time
• BAP-GGA performs better than traditional GA
– Strength of GGA encoding, Inject crossover
• BAP-GGA able to resolve busy schedules
– Strength of Timezone mutator

Contributions
• Identified key problems of Traditional GA approaches
to solve the BAP Partitioning Problem
– unsuitable encoding
– difficulty handling busy schedules
• Devised GA (BAP-GGA) using techniques that address
these problems
– GGA encoding
– Greedy Initialization
– Timezone based reallocation

BAP-GA
2.0
BAP-GP
[Ong00]
BAP-BB
[Xu99]
Branch & Bound
(BAP-BB)
Optimal
Solutions
Very Slow
Modified
Fidducia-
Mattheyses
(BAP-GP)
Good Solutions
Very Fast
GA
Better
Solutions
Fast
Each addresses a different planning need

Short Term Extensions
• Other GA Operators
– Selection scheme
• rank-based [Whit89], stochastic univ. sel. [Hanc91]
– Replacement scheme
• replace-bad (not worst) [Reev97], replace-old [Mich91]
– what is a good value for ?
• Incorporation of fast FM heuristic [Ong00]
– how to incorporate?

Long Term Extensions
• Incorporate Yard Capacity to model.
– Container handling rate of Quay and Yard cranes.
– Storage capacity of Yard
• Study effects of various port structures.
• Study of solutions generated by various
combinations of Partitioning and Packing
solvers

Premature Convergence

Dealing with Premature Convergence
• Standard way - Increase Population Size
– costs - memory, running time
– increase cautiously
• Adaptive mutation
– drastically increase rate of mutation
• 0.05 to 1.0
– change mutation scheme (and inc. mut. rate)
• time-zone to flip

Schema
0
1
1
1
0
1
1
1
0
1
1
1
H1 = 1** H2 = *0* H3 = **1

The Berth Allocation Problem
Section 1
Section 2
Section 3

The Quadratic Assignment Problem
• Facility Layout Problem
• Given a floor area of a factory evenly partitioned into n equal blocks and n departments,
we want to assign all departments to floor blocks so as to minimize the total material
handling cost.
• NP-Hard

A Special Case of the BAP
• n vessels assigned to n sections to minimize container handling effort.
• Each vessel must be assigned a unique section.
• Same problem as QAP.

The General BAP
• n vessels assigned to m sections (m < n) to minimize transshipment cost.
• Vessels must be assigned a sections that have space to accommodate them.
• Generalization of QAP.
• NP-Hard

Planning Window
Applying GA To The BAP
15 3
20
10
5
12
25
3045
20
300
300

Key Ideas Behind GA
• All organisms endowed with different traits (attributes)
– eg. number of fingers, webbed feet, long neck
– determined at birth by their genes inherited from parents
• Genes encapsulated into structures called chromosomes
• Some traits are better suited for survival
– eg. long neck for reaching leaves on trees
• Fitness
– ie. how well each individual is able to survive in the environment
• All individuals compete for survival and mating
– strong will flourish, weak will die off
– genes from stronger individuals get passed on to next generation
• Mutation introduces variation to the genes
– new traits also appear

Key GAAbstractions
• Organism = chromosome = organism
– the chromosome is the organism
• Chromosome = string of chars
– each string element represents a gene
• Fitness = fitness function
– maps chromosome to a real number
– usually dependant on values of string elements
• Who lives and dies based on fitness value
– replacement scheme
• Selection for mating also based on fitness value
– ie. fitter organisms get to mate more
• Mutation acts to alter string elements

BAPS 1.0 Experimentation
• Focus - to perform evaluation of Loh’s proposed
GA combinations
• Constructed 100 realistic data sets
– BENCHMARK data set collection
• Best GA:
– Uniform crossover
– NS/flip mutator
• Long running times - days
– unbounded NS
• Compared not favorably with BAP-GP
– FM based partitioner (fast)

An Appeal
• Please support Open Source software efforts
and the Free Software Movement
• This research has benefited much from the
fact that there is a freely available,
powerful, stable OS from which to run
expts on cheap PCs.

Running the GA
•Wide coverage of solution space
•Not susceptible to local optima

Section 2
Section 1
Section 3
Encoding The BAP
3
5
13
11
1
2
4
6
7
8
9
10
12
15 3
20
10 5
12
25
3045
20
2 3 4 5 6 7 8 9 131211101
3 1 2 1 3 3 1 2 2 2113
Vessel Number :
Section Number :
BAP Chromosome

Strategy for GA Evaluation Problem
• Focus on GA elements that give biggest payoff
– Which are they?
• Quick prototyping of GA solvers
– Use library of GA components (GALib)
• “Production-line” experimentation system
– BAP Experimental Framework
– Automate expts, clone
• Successive Refinement
– 4 stage evaluation process

Calculating Fitness
= Transshipment Cost + (  Overshoot Cost)
– relaxation of Berth Capacity Constraint
– deal with infeasible allocations
–  is a very large number
•  >> Transshipment Cost
• differentiate feasible and infeasible allocations
• BAP Objective is to Minimize
• But GA Objective is to Maximize Fitness
– Need to convert Min problem to Max problem

BAPS 1.0 Conclusions
• Results
– GA without NS perform badly
– Results not very good compared with BAP-GP
• Questions
– Why does GA (without NS) perform badly?
– What is NS contribution to search?

The Fitness Landscape
• Height of each point
- determined by fitness function
• How points located next to
one another - determined by the
neighborhood function
• Encoding, crossover and mutation
scheme determine the landscape

Data Generation Parameters
• Port Structure
– section lengths, distances between them
• Berthing Demand
– how busy the port is
• Planning window - 84 hrs (1/2 week)
• Mix of vessels - 70% SML, 30% MED class
• Num. containers moved ~ 50% of capacity

Port Structures
• Port Structure 1
• Port Structure 5 (Brani)
600m
section 1
600m
section 2
600m
section 1
600m
section 2
600m
section 4
600m
section 5
600m
section 3
1200m
section 1
1200m
section 3
600m
section 2
600m
section 1
600m
section 3
1800m
section 2
600m
section 1
480m
section 2
800m
section 3
200m
section 4

Total Berth Demand
The Average Berth Demand
Planning Window
Total
Berthing
Length
• ABD = Total Berth Demand / (Total Berthing Length × Planning Window)
Density

Port Survey
• Objective - to obtain primary data figures
and gather domain related info.
• All info. from publicly available sources
– PortView, WWW
• Documented in [Foo96] (same title)
– RAS Internal document
• Key primary data:
– vessel classes (eg. panamax, post-panamax)
• vessel lengths, durations of stay, capacities
– port structures

Order Based Expt. Grid
PMX OX CX
First Fit
Greedy
Crossover
Decoder

Object Membership Expt. Grid
Crossover
Mutator
Reordering
1pt 2pt 3pt 4pt 5pt
No Reorder
WDFS Reorder
flip
swap
Uniform

Hybrid GA Expt. Grid
POST
LOCAL
Hybridization Scheme
FLIPN
LOCAL
1STEP
LOCAL
RANDOM
LOCAL

GGA Expt. Grid
Initializer
1pt
random
Inject
flip
timezone
Crossover
Mutator
greedy

Choice of Hybridization Scheme
Objective Cost Ratio of Optimal
RND 1STEP FLIPN POST optimal RND 1STEP FLIPN POST
d011 3.30E+04 3.30E+04 3.30E+04 3.30E+04 3.30E+04 1.00 1.00 1.00 1.00
d013 3.30E+04 3.30E+04 3.30E+04 3.30E+04 3.30E+04 1.00 1.00 1.00 1.00
d015 3.30E+04 3.30E+04 3.30E+04 3.30E+04 3.30E+04 1.00 1.00 1.00 1.00
d017 3.30E+04 3.30E+04 3.30E+04 3.30E+04 3.30E+04 1.00 1.00 1.00 1.00
d021 1.89E+05 1.89E+05 1.89E+05 1.82E+05 1.59E+05 1.19 1.19 1.19 1.15
d023 2.31E+05 2.31E+05 1.87E+05 1.87E+05 1.59E+05 1.46 1.46 1.18 1.18
d025 1.59E+05 1.59E+05 1.67E+05 1.67E+05 1.59E+05 1.00 1.00 1.05 1.05
d027 1.99E+05 1.99E+05 2.32E+05 1.83E+05 1.59E+05 1.26 1.26 1.46 1.15
d031 2.10E+08 2.10E+08 1.65E+05 1.65E+05 1.39E+05 1512.73 1512.73 1.19 1.19
d033 2.69E+05 2.69E+05 1.10E+08 2.69E+05 1.39E+05 1.94 1.94 793.45 1.94
d012 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 1.00 1.00 1.00 1.00
d014 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 1.00 1.00 1.00 1.00
d016 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 1.00 1.00 1.00 1.00
d018 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 1.00 1.00 1.00 1.00
d022 3.15E+04 3.15E+04 8.00E+04 6.30E+04 0.00E+00 inf inf inf inf
d024 0.00E+00 0.00E+00 1.26E+05 6.75E+04 0.00E+00 1.00 1.00 inf inf
d026 9.10E+04 9.10E+04 9.00E+04 0.00E+00 0.00E+00 inf inf inf 1.00
d028 5.60E+04 5.60E+04 1.43E+05 0.00E+00 0.00E+00 inf inf inf 1.00
d032 2.10E+08 2.10E+08 0.00E+00 0.00E+00 0.00E+00 inf inf 1.00 1.00
d034 2.21E+05 2.21E+05 1.10E+08 9.00E+04 0.00E+00 inf inf inf inf
d041 1.44E+13 1.44E+13 1.44E+13 1.44E+13 1.44E+13 1.00 1.00 1.00 1.00

Applying GA To The BAP
Planning Window
15 3
20
10
5
12
25
3045
20
300
300

Encoding The BAP
3
5
13
11
1
2
4
6
7
8
9
10
12
15 3
20
10
5
12
25
3045
20

Selecting Who Should Mate
• Standard way (Holland) -
known as “roulette wheel” or
proportional selection.
• Individuals in population have
probability of being selected
based on their fitness relative
to the population average.
• Fitter have higher prob. of
mating.
Fitness Roulette
1000
2000
5000
8000
1000
Individual 1
Individual 2
Individual 3
Individual 4
Individual 5

Fitness Roulette
1000
2000
5000
8000
1000
Individual 1
Individual 2
Individual 3
Individual 4
Individual 5
Fitness Roulette
6%
12%
29%
47%
6%
Individual 1
Individual 2
Individual 3
Individual 4
Individual 5
Proportional Selection

More Fit
Individuals
Less Fit
Individuals
Who Should Live and Who Should Die
Population of
Generation n
Individuals
From
Generation n
New
Individuals
Population of
Generation n+1
Number of individuals to kill = Number of children to produce
= Replacement Rate × Population Size
Steady State Replacement

Stage 1
PMX O
X
CX
First Fit
Greedy
Crossover
Decoder
Crossover
Mutator
Reordering
1pt 2pt 3pt 4pt 5pt
No Reorder
WDFS Reorder
flip
swap
Uniform
1pt
random
Inject
flip
timezone
Crossover
Mutator
greedy Initializer
BAP-GGA
Object
Membership
Order
Based
Data sets:
- TESTDATA collection
- 6-15 ves., 2-3 sect.
Popsize/numgen:
- 5/100, 5/400, 20/100

Stage 2
PMX O
X
CX
First Fit
Greedy
Crossover
Decoder
Crossover
Mutator
Reordering
1pt 2pt 3pt 4pt 5pt
No Reorder
WDFS Reorder
flip
swap
Uniform
1pt
random
Inject
flip
timezone
Crossover
Mutator
greedy Initializer
BAP-GGA
Object
Membership
Order
Based
Data sets:
- SMALLDATA
- 26-42 ves., 2-3 sect.
Popsize/numgen:
- 20/100, 20/400, 80/100

Elimination of Flip Mutator (BAP-GGA)
ggaX1 ggaX2 ggaX3 ggaX4 ggaX5 ggaX6 ggaX7 ggaX8
gga2X 0 1 6 9 7 9 11 11
gga3X 4 8 8 9 12 14 14 13
gga4X 5 10 12 11 14 11 14 15
ggaX1 ggaX2 ggaX3 ggaX4 ggaX5 ggaX6 ggaX7 ggaX8
gga2X 2.58 1.60 1.30 1.10 1.13 1.08 1.04 1.03
gga3X 1.38 1.19 1.20 1.08 1.02 1.01 1.01 1.02
gga4X 1.59 1.13 1.08 1.05 1.05 1.05 1.01 1.01
Optimal Count (of 16 data sets)
Average Ratios over Optimal
Flip Timezone

BAP-GGA : Inject over 1-pt Crossover
gga55 gga56 gga57 gga58 gga65 gga66 gga67 gga68 gga75 gga76 gga77 gga78
d105 1.70 1.02 1.02 1.03 1.16 1.02 1.02 1.03 1.19 1.01 1.01 1.00
d144 12.11 13.83 2.31 1.23 1.46 5.98 1.07 1.00 1.55 3.30 1.04 1.00
d164 2.48 1.38 1.39 1.16 1.70 1.16 1.05 0.98 1.64 1.23 1.10 0.91
d181 1.85 1.60 1.50 1.40 1.32 1.37 1.18 1.23 1.39 1.42 1.18 1.19
d126 1.56 1.56 1.63 1.52 1.25 1.33 1.17 1.17 1.35 1.37 1.25 1.22
d146 3.54 2.40 1.52 1.45 1.72 1.63 1.06 1.14 1.72 1.71 1.06 0.96
d166 1.71 1.51 1.42 1.36 1.36 1.25 1.18 1.05 1.42 1.33 1.14 1.08
d186 48.48 1.55 1.49 1.46 1.32 1.36 1.16 1.24 1.37 1.45 1.19 1.22
d131 1.55 1.55 1.62 1.55 1.27 1.34 1.16 1.21 1.34 1.39 1.28 1.23
d151 2.24 1.81 1.44 1.38 1.51 1.44 0.97 1.02 1.55 1.48 1.04 0.95
d171 1.63 1.63 1.43 1.51 1.35 1.39 1.24 1.18 1.41 1.44 1.27 1.20
d191 1760.62 622.52 1353.64 537.37 644.76 521.11 500.22 427.24 842.87 409.69 588.11 100.05
d140 987.41 220.48 851.97 130.80 627.67 123.79 560.61 87.09 659.37 24.05 526.61 1.30
d160 522.77 1.62 50.88 1.32 1.30 1.34 0.94 0.97 1.35 1.38 1.06 0.95
d200 126.74 42.26 78.57 35.74 60.65 31.81 42.94 29.84 63.53 33.40 48.94 14.48
Ratios over best BAP-GA 1.0
50/100 50/400 100/200
ggaX5, ggaX6 - 1-pt
ggaX7, ggaX8 - Inject

Calculating Transshipment Cost
Planning Window
10 5
315
12
20
300
300

Another View of the Port
Section 1
Section 2
Section 3
Time8am 10am 12pm 1pm
Cagney
Lacey
12
Transshipment

The application of Genetic Algorithms to the Berth Allocation Problem

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The application of Genetic Algorithms to the Berth Allocation Problem