Using Probabilistic Intelligence to Influence
Course of Action Planning and Optimization
by
Dennis E. Okon, Jr.
Submitted to the Department of Electrical Engineering and Computer
Science in partial fulfillment of the requirements for the degrees of
Bachelor of Science in Computer Science and Engineering
and
Master of Engineering in Electrical Engineering and Computer Science
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
May 17, 2000 -
L750ic
20C0
Copyright 2000, Dennis E. Okon, Jr. All rights reserved.
The author hereby grants to M.I.T. permission to reproduce and distribute
publicly paper and electronic copies of this thesis document in whole or part.
Author
Department of Electrical Engineering and Computer Science
May 17, 2000
Certified by
Howard Shrobe
Profetsor, Department of Electrical Engineering and Computer Science
Thesis Supervisor
Accepted by
eArthur C. Smith
Chairman, Department Committee on Gra uate Theses
MASSACHUSETTS INSTITUTE
OF TECHNOLOGY
ENG8
JUL 2 7 2000
LIBRARIES
Using Probabilistic Intelligence to Influence
Course of Action Planning and Optimization
by
Dennis E. Okon, Jr.
Submitted to the Department of Electrical Engineering and
Computer Science on May 17, 2000, in partial fulfillment of
the requirements for the degrees of
Bachelor of Science in Computer Science and Engineering
and
Master of Engineering in Electrical Engineering and Computer Science
Abstract
Several methods based on probabilistic data meant for biasing a genetic search
algorithm are presented. Each of the methods is tested against standard genetic algorithm
test problems and their performance is compared to neutral tests where the influence is
ignored. The intended end use is to influence a GA searching in a predictive mode useful
in intelligent game playing. More specifically, the GA is meant to run underneath the
FOX system and help predict enemy Courses of Action.
probabilistic information comes from military intelligence.
For this application, the
Thesis Supervisor: Howard Shrobe
Title: Professor, Department of Electrical Engineering and Computer Science
2
Acknowledgment
I would like to thank several people for making this work possible and for guiding my
thoughts and efforts while working on this project.
I would first like to thank my advisor, Professor Howard Shrobe, for having faith in me
and taking the risk of advising an off-campus thesis.
Secondly, I owe an immense amount of gratitude to Charles River Analytics,
particularly Greg Zacharias, for funding my entire year of work and allowing me full access
and support to their FOX system. I would especially like to give Harald Ruda and Janet
Burge recognition for working with me on FOX and helping me to understand the system
and the technology behind it.
I also want to acknowledge DARPA's CPoF project for exposing me to the "real
world" side of this work, especially for John Schmitt's electronic tactical decision games
(eTDGs), all of the Graybeards' expertise, and the mock combat experience at the JRTC
(Joint Readiness Training Center) in Ft. Polk, LA.
3
Table of Contents
1. Introduction ......................................................................................................... 15
1. 1
Motivation ............................................................................................ 15
1.2
Idea ....................................................................................................... 15
1.3
Incorporation ........................................................................................ 16
1.4
Paper Layout ......................................................................................... 18
2. FO X .....................................................................................................................19
2.1
Background ........................................................................................... 19
2.2
Abstraction ........................................................................................... 19
3. Goals and Design ................................................................................................. 24
3.1
Goals ..................................................................................................... 24
3.2
Design ...................................................................................................24
3.2. 1
Criteria ...................................................................................24
3.2.2
Design ....................................................................................25
4. Implementation and Testing ................................................................................28
4.1 Im plementation .....................................................................................28
4.2
4.3
4. 1. 1
Initial Skew ............................................................................ 28
4.1.2
Fitness Skew ........................................................................... 29
4.1.3
Crossover Skew ...................................................................... 29
Algorithm Test Cases ............................................................................ 31
4.2.1
Dejong's F1 ............................................................................ 35
4.2.2
Dejong's F2 ............................................................................ 39
4.2.3
Dejong's F3 ............................................................................ 43
4.2.4
4.2.5
Dejong's F4 ............................................................................ 47
F5: Inverted Shekel's Foxholes .............................................. 51
4.2.6
Schaffer's F6 ........................................................................... 56
Conclusion for Test Cases ..................................................................... 60
5. Interfacing with FOX and Scenarios .................................................................... 63
5.1
Interfacing with FOX ............................................................................ 63
5.2
FO X Scenario .......................................................................................64
4
5.2.1
The Battle of johnsonburg ..................................................... 64
5.2.2
eCOA without Influence ....................................................... 68
5.2.3
eCOA with Influence .............................................................69
6. Conclusion ...........................................................................................................72
7. References ............................................................................................................73
Appendix A : Data ......................................................................................................75
A .1. F 1 Data ................................................................................................. 75
A .1. 1. Stage 1 ....................................................................................75
A .1. 2. Stage 2 ....................................................................................78
A .2. F2 Data .................................................................................................81
A .2.1. Stage 1....................................................................................81
A .2.2. Stage 2 ....................................................................................84
A .3. F3 Data .................................................................................................87
A -3. 1. Stage 1 ....................................................................................87
A -3.2. Stage 2 ....................................................................................90
A .4. F4 Data .................................................................................................93
A-4. 1. Stage 1 ....................................................................................93
A-4.2. Stage 2 ....................................................................................
A -5. F5 Data .................................................................................................99
A -5.1. Stage 1 ....................................................................................
A .5.2. Stage 2 .................................................................................... 102
A .6. F6 Data .................................................................................................105
A .6. 1. Stage I .................................................................................... 105
A .6-2. Stage 2 .................................................................................... 108
5
List of Figures
Figure 1.3-1: FOX Framework........................................................................................17
Figure 2.2-1: Traditional COA Sketch..........................................................................
20
Figure 2.2-2: COA Sketch Overlaid with AAs and LDTs ............................................
21
Figure 2.2-3: Abstract Battlefield Grid ..........................................................................
22
Figure 2.2-4: Abstract Battlefield and Units.................................................................
22
Figure 4.1-1: Probabilities of Keeping A for Initial Skew...............................................
28
Figure 4.1-2: Fitness Skew (goal = 1)............................................................................
29
Figure 4.1-3: "Stickiness" Measure for Crossover Skew ..............................................
30
Figure 4.2-1: Fl Function Plot......................................................................................
36
Figure 4.2-2: F1 Confidence Plot...................................................................................
36
Figure 4.2-3: Fl Skewed Function Plot .......................................................................
36
Figure 4.2-4: Fl Initial Skew vs. Crossovers ................................................................
37
Figure 4.2-5: F1 Initial Skew vs. M utation ...................................................................
37
Figure 4.2-6: F1 Fitness Skew vs. Crossovers.................................................................
37
Figure 4.2-7: F1 Fitness Skew vs. M utation...................................................................
37
Figure 4.2-8: Fl Crossover Skew vs. Crossovers ............................................................
37
Figure 4.2-9: F1 Crossover Skew vs. Mutation............................................................
37
Figure 4.2-10: F1 Optim al Test .....................................................................................
38
Figure 4.2-11: F1 Non-O ptim al Test............................................................................
38
Figure 4.2-12: F2 Function Plot......................................................................................
40
Figure 4.2-13: F2 Confidence Plot .................................................................................
40
Figure 4.2-14: F2 Skewed Function Plot .......................................................................
40
Figure 4.2-15: F2 Initial Skew vs. Crossovers ................................................................
41
6
Figure 4.2-16: F2 Initial Skew vs. Mutation .................................................................
41
Figure 4.2-17: F2 Fitness Skew vs. Crossovers...............................................................41
Figure 4.2-18: F2 Fitness Skew vs. Mutation.................................................................
41
Figure 4.2-19: F2 Crossover Skew vs. Crossovers..........................................................
41
Figure 4.2-20: F2 Crossover Skew vs. Mutation..........................................................
41
Figure 4.2-21: F2 Optimal Test .....................................................................................
42
Figure 4.2-22: F2 Non-Optimal Test.............................................................................
42
Figure 4.2-23: F3 Function Plot....................................................................................
44
Figure 4.2-24: F3 Confidence Plot.................................................................................
44
Figure 4.2-25: F3 Skewed Function Plot .....................................................................
44
Figure 4.2-26: F3 Initial Skew vs. Crossovers ...............................................................
45
Figure 4.2-27: F3 Initial Skew vs. Mutation .................................................................
45
Figure 4.2-28: F3 Fitness Skew vs. Crossovers...............................................................45
Figure 4.2-29: F3 Fitness Skew vs. Mutation.................................................................45
Figure 4.2-30: F3 Crossover Skew vs. Crossovers..........................................................45
Figure 4.2-3 1: F3 Crossover Skew vs. Mutation..........................................................
45
Figure 4.2-32: F3 Optimal Test .....................................................................................
46
Figure 4.2-33: F3 Non-Optimal Test.............................................................................46
Figure 4.2-34: F4 Function Plot....................................................................................
48
Figure 4.2-35: F4 Confidence Plot.................................................................................48
Figure 4.2-36: F4 Skewed Function Plot ........................................................................
48
Figure 4.2-37: F4 Initial Skew vs. Crossovers ...............................................................
49
Figure 4.2-38: F4 Initial Skew vs. Mutation .................................................................
49
Figure 4.2-39: F4 Fitness Skew vs. Crossovers...............................................................
49
7
Figure 4.2-40:
Fitness Skew vs. M utation.................................................................
49
Figure 4.2-41:
Crossover Skew vs. Crossovers..........................................................
49
Figure 4.2-42:
Crossover Skew vs. M utation.............................................................49
Figure 4.2-43:
Optimal Test .....................................................................................
50
Figure 4.2-44:
Non-Optim al Test.............................................................................
50
Figure 4.2-45:
Function Plot....................................................................................
53
Figure 4.2-46:
Confidence Plot .................................................................................
53
Figure 4.2-47:
Skewed Function Plot .......................................................................
53
Figure 4.2-48:
Initial Skew vs. Crossovers .................................................................
54
Figure 4.2-49:
Initial Skew vs. M utation .................................................................
54
Figure 4.2-50:
Fitness Skew vs. Crossovers...............................................................
54
Figure 4.2-5 1:
Fitness Skew vs. M utation.................................................................
54
Figure 4.2-52:
Crossover Skew vs. Crossovers..........................................................
54
Figure 4.2-53:
Crossover Skew vs. M utation.............................................................54
Figure 4.2-54:
Optimal Test .....................................................................................
55
Figure 4.2-55:
N on-O ptimal Test.............................................................................
55
Figure 4.2-56:
Function Plot......................................................................................58
Figure 4.2-57:
Confidence Plot.................................................................................
58
Figure 4.2-58:
Skewed Function Plot .......................................................................
58
Figure 4.2-59:
Initial Skew vs. Crossovers .................................................................
59
Figure 4.2-60:
Initial Skew vs. M utation .................................................................
59
Figure 4.2-61:
Fitness Skew vs. Crossovers...............................................................
59
Figure 4.2-62:
Fitness Skew vs. M utation.................................................................
59
Figure 4.2-63:
Crossover Skew vs. Crossovers..........................................................
59
8
Figure 4.2-64: F6 Crossover Skew vs. Mutation..........................................................
59
Figure 4.2-65: F6 Optimal Test .....................................................................................
60
Figure 4.2-66: F6 Non-Optimal Test.............................................................................60
Figure 5.2-1: The Battle of Johnsonburg .....................................................................
66
Figure 5.2-2: A FOX Interpretation of The Battle of Johnsonburg...............................67
Figure 5.2-3: eCOA without Influence..........................................................................68
Figure 5.2-4: eCOA with Influence ...................................................................................
9
70
List of Tables
Table 3.2-1: Likelihood Function O utput Definitions .................................................
27
Table 3.2-2: Confidence M easure Defintions...............................................................
27
Table 4.2-1: Trial Param eter Sets.................................................................................
33
Table 4.3-1: Sum mary of Perform ance Effects...............................................................
61
Table 5.2-1: Position Belief Data....................................................................................69
Table 5.2-2: Com position Belief Data ..........................................................................
69
Table A-1: F1 Optimal: Initial Skew vs. Expected # Crossovers..................................75
Table A-2: F1 Optimal: Fitness Skew vs. Expected # Crossovers ...............................
75
Table A-3: F1 Optimal: Crossover Skew vs. Expected # Crossovers...........................
76
Table A-4: F1 Optimal: Initial Skew vs. Mutation Rate ...............................................
76
Table A-5: F1 Optimal: Fitness Skew vs. Mutation Rate.............................................
77
Table A-6: F1 Optimal: Crossover Skew vs. Mutation Rate ........................................
77
Table A-7: F1 Optimal: Initial Skew vs. Fitness Skew .................................................
78
Table A-8: F1 Optimal: Initial Skew vs. Crossover Skew.............................................
78
Table A-9: F1 Optimal: Fitness Skew vs. Crossover Skew .............................................
79
Table A-10: F1 Non-Optimal: Initial Skew vs. Fitness Skew........................................79
Table A-11: Fl Non-Optimal: Initial Skew vs. Crossover Skew ...................................
80
Table A-12: F1 Non-Optimal: Fitness Skew vs. Crossover Skew..................................80
Table A-13: F2 Optimal: Initial Skew vs. Expected # Crossovers...............................81
Table A-14: F2 Optimal: Fitness Skew vs. Expected # Crossovers ..............................
81
Table A-15: F2 Optimal: Crossover Skew vs. Expected # Crossovers..........................82
Table A-16: F2 Optimal: Initial Skew vs. Mutation Rate ............................................
Table A-17: F2 Optimal: Fitness Skew vs. Mutation Rate...........................................83
10
82
Table A-18: F2 Optimal: Crossover Skew vs. Mutation Rate ......................................
83
Table A-19: F2 Optimal: Initial Skew vs. Fitness Skew ...............................................
84
Table A-20: F2 Optimal: Initial Skew vs. Crossover Skew...........................................84
Table A-2 1: F2 Optimal: Fitness Skew vs. Crossover Skew..........................................85
Table A-22: F2 Non-Optimal: Initial Skew vs. Fitness Skew........................................85
Table A-23: F2 Non-Optimal: Initial Skew vs. Crossover Skew ...................................
86
Table A -24: F2 Non-Optimal: Fitness Skew vs. Crossover Skew.................................86
Table A-25: F3 Optimal: Initial Skew vs. Expected # Crossovers...............................87
Table A-26: F3 Optimal: Fitness Skew vs. Expected # Crossovers ..............................
87
Table A-27: F3 Optimal: Crossover Skew vs. Expected # Crossovers..........................88
Table A-28: F3 Optimal: Initial Skew vs. Mutation Rate ............................................
88
Table A-29: F3 Optimal: Fitness Skew vs. Mutation Rate...........................................89
Table A-30: F3 Optimal: Crossover Skew vs. Mutation Rate ......................................
89
Table A-3 1: F3 Optimal: Initial Skew vs. Fitness Skew ...............................................
90
Table A-32: F3 Optimal: Initial Skew vs. Crossover Skew...........................................90
Table A-33: F3 Optimal: Fitness Skew vs. Crossover Skew..........................................91
Table A-34: F3 Non-Optimal: Initial Skew vs. Fitness Skew........................................91
Table A-35: F3 Non-Optimal: Initial Skew vs. Crossover Skew ...................................
92
Table A-36: F3 Non-Optimal: Fitness Skew vs. Crossover Skew..................................92
Table A-37: F4 Optimal: Initial Skew vs. Expected # Crossovers...............................93
Table A-38: F4 Optimal: Fitness Skew vs. Expected # Crossovers ..............................
93
Table A-39: F4 Optimal: Crossover Skew vs. Expected # Crossovers..........................94
Table A-40: F4 Optimal: Initial Skew vs. Mutation Rate ............................................
Table A-41: F4 Optimal: Fitness Skew vs. Mutation Rate...........................................95
11
94
Table A-42: F4 Optimal: Crossover Skew vs. Mutation Rate ......................................
95
Table A-43: F4 Optimal: Initial Skew vs. Fitness Skew ...............................................
96
Table A-44: F4 Optimal: Initial Skew vs. Crossover Skew...........................................96
Table A-45: F4 Optimal: Fitness Skew vs. Crossover Skew ..........................................
97
Table A-46: F4 Non-Optimal: Initial Skew vs. Fitness Skew........................................97
Table A-47: F4 Non-Optimal: Initial Skew vs. Crossover Skew ...................................
98
Table A-48: F4 Non-Optimal: Fitness Skew vs. Crossover Skew..................................98
Table A-49: F5 Optimal: Initial Skew vs. Expected # Crossovers...............................99
Table A-50: F5 Optimal: Fitness Skew vs. Expected # Crossovers ..............................
99
Table A-51: F5 Optimal: Crossover Skew vs. Expected # Crossovers............................100
Table A-52: F5 Optimal: Initial Skew vs. Mutation Rate ...............................................
100
Table A-53: F5 Optimal: Fitness Skew vs. Mutation Rate..............................................101
Table A-54: F5 Optimal: Crossover Skew vs. Mutation Rate .........................................
101
Table A-55: F5 Optimal: Initial Skew vs. Fitness Skew ..................................................
102
Table A-56: F5 Optimal: Initial Skew vs. Crossover Skew..............................................102
Table A-57: F5 Optimal: Fitness Skew vs. Crossover Skew............................................103
Table A-58: F5 Non-Optimal: Initial Skew vs. Fitness Skew..........................................103
Table A-59: F5 Non-Optimal: Initial Skew vs. Crossover Skew .....................................
104
Table A-60: F5 Non-Optimal: Fitness Skew vs. Crossover Skew....................................104
Table A-6 1: F6 Optimal: Initial Skew vs. Expected # Crossovers..................................105
Table A-62: F6 Optimal: Fitness Skew vs. Expected # Crossovers ................................
105
Table A-63: F6 Optimal: Crossover Skew vs. Expected # Crossovers............................106
Table A-64: F6 Optimal: Initial Skew vs. Mutation Rate ...............................................
106
Table A-65: F6 Optimal: Fitness Skew vs. Mutation Rate..............................................107
12
Table A-66: F6 Optimal: Crossover Skew vs. Mutation Rate .........................................
107
Table A-67: F6 Optimal: Initial Skew vs. Fitness Skew ..................................................
108
Table A-68: F6 Optimal: Initial Skew vs. Crossover Skew..............................................108
Table A-69: F6 Optimal: Fitness Skew vs. Crossover Skew ............................................
109
Table A-70: F6 Non-Optimal: Initial Skew vs. Fitness Skew..........................................109
Table A-7 1: F6 Non-Optimal: Initial Skew vs. Crossover Skew .....................................
110
Table A-72: F6 Non-Optimal: Fitness Skew vs. Crossover Skew....................................110
13
Glossary of Terms
AA
Avenue of Approach
BLUFOR
Friendly forces
COA
Course of Action
eCOA
Enemy Course of Action
fCOA
Friendly Course of Action
FEBA
Forward Edge of Battle Area
GA
Genetic Algorithm
LDT
Line of Defensible Terrain
LOA
Limit Of Advance
MB
Maneuver Box
OPFOR
Enemy (Opposing) forces
TAA
Tactical Assembly Area
14
1. Introduction
1.1 Motivation
"Ifyou know your enemy and know yourself, you need not
fear the result of a hundred battles. If you know yourself but
not the enemy, for every victory gained you will also suffer a
defeat. If you know neither the enemy nor yourself, you will
succumb in every battle."
- Sun Tzu [23]
Although Sun Tzu wrote this over 2500 years ago, it is still a guiding principle for
military planning. "Knowing yourself' is often a logistics problem dependent on having a
good flow of information and the ability to look at yourself objectively. However,
"knowing your enemy" is a potentially harder problem, because no matter how good your
information, there is still an element of guesswork and prediction involved.
Throughout the life of Computer Science and Al research, predicting what the enemy
will do has been a basic problem. For instance, game playing strategies like Mini-Max
Search attempt to find the enemy's best moves and counter them. How successful would
Deep Blue have been if it had not considered what Gary Kasporav might do?
Unfortunately, fighting wars on any level, from a single Fire Team of riflemen
traversing a swamp to coordinating WWII over the Pacific and in Europe, is far from the
"simple" deterministic games of chess and checkers. However, while this fact might stop
us from fully automating our Armed Forces by replacing men with computers anytime in
the near future, we should not be discouraged from developing tools to aid commanders
in their command and control decision-making.
1.2 Idea
One such tool being developed is most commonly referred to as FOX or FOX-GA.
Originally, this tool was conceived of and developed at UIUC by Maj. J. L. Schlabach and
Caroline Hayes [7]. The intent of FOX is to generate Courses of Action (COAs) for
friendly units given a situation (i.e. mission goals, available resources, terrain information,
and enemy COAs). The work by Schlabach and Hayes, and more recently in case-studies
[5], showed the feasibility of abstracting the war simulation into an abstract, chessboardlike simulation and using genetic algorithms to search the COA-space defined in this
15
abstraction. Charles River Analytics used this work as a starting point to add
functionality, capabilities, power, and flexibility as well as make the system more robust
[9][12]. Throughout all the additions and modifications, the basic goal of FOX remained
constant: to suggest friendly COAs to a commander. However, that is not the only thing
FOX could possibly do; this thesis explores an alternative use for FOX: examining the
potential impact of enemy COAs on friendly plans.
This is motivated, in part, by Sun Tzu's quote, where we find that FOX could help a
commander "know [his] enemy" by predicting enemy COAs. If we pretend FOX is
working for the enemy and run it with the current situation (or at least the situation we
believe the enemy sees), its output may give us an idea for what the enemy will do and
then we can plan accordingly. Using FOX to predict enemy COAs rather than using a
human expert allows for a non-biased approach to predicting enemy actions. For
instance, a human expert might not see a very powerful plan the enemy might be cooking
up because it is not in his doctrinal training - it is simply something he would not think of
because it is never done. However, FOX does not inherently have this bias against nontraditional plans and might find them helpful. Another usefulness of this enemy
prediction is that FOX can play the enemy commander for training exercises.
Unfortunately, this straightforward rotation between BLUFOR and OPFOR is not
quite as powerful as it could be since it ignores (or rather simply does not use) any
intelligence that may exist about the enemy. The above idea needs to be modified slightly
to reflect this observation: FOX should output the best enemy COAs that are viable
based on the current intelligence about the enemy. In this way, FOX should be able to
predict likely, robust enemy COAs and the commander will better know his enemy.
1.3 Incorporation
In order for this idea to be useful in FOX we need a way to incorporate knowledge
that will govern any intelligence about the enemy. By looking at FOX's basic framework
(Figure 1.3-1 adapted from [12]) we see that there are only a couple of places this
knowledge can go:
1. The Abstract Wargamer's Rules
2. The Fitness Function's Criteria
3. The GA Search Engine itself
16
Abstract
Terrain
BLUFOR
data
fCOAs
OPFOR
COA
interactions
data
GA
Parameters
Figure 1.3-1: FOX Framework
The Wargamer's function is to simulate the engagement, so its rule-base should hold
knowledge about determining an outcome for a specific engagement; for example, it
should know about the fighting styles (e.g., Russian, English, American, etc.), momentum
of the battle, and cultural effects (e.g., religious holidays) and how these affect a given
battle. However, this is not where uncertain situational intelligence about the enemy is
really useful.
The Fitness Function's job is to evaluate the outcome of the Wargamer and assign a
score to the BLUFOR COA. A low score (close to 0.0) means the BLUFOR's mission
was not accomplished, many casualties were suffered, the OPFOR overran the BLUFOR,
etc. A high score (close to 1.0) means that BLUFOR was successful, very few casualties
were taken, the BLUFOR overran the OPFOR, etc. The knowledge needed by the
Fitness Function answers the basic question: "What is good and what is bad?" Again,
situational intelligence about the enemy does not belong here.
Lastly, the GA Search Engine could easily take into account probabilistic, situationaldependent intelligence simply by modifying how it searches COA-space (e.g., how it
17
chooses which individuals survive and breed). This looks like a perfect fit. The only
question now is in the details: How can the intelligence change the search? And, what
are the effects of changing the search?
1.4 Paper Layout
The rest of this paper explores possible answers to those two questions. To begin
dealing with this, Section 2 more fully describes the FOX system; Section 3 describes the
design and implementation details; Section 4 demonstrates and analyzes the performance
of the newly developed GA on classic GA search problems; Section 5 describes the
interface to the existing FOX system and its performance on domain specific problems;
and Section 6 concludes the work.
18
2. FOX
2.1 Background
FOX is a planning support tool for assisting military intelligence and maneuver
battlestaff in rapidly generating and assessing battlefield courses of actions (COAs). The
first prototype of FOX (called FOX-GA) was designed and created by Maj. J. L.
Schlabach and Caroline Hayes at the University of Illinois, Urbana-Champaign (UIUC)
with Carolyn Fiebig and Robert Winkler. [5] [7] Under contract to the Army Research
Labroatory, Charles River Analytics is transitioning the work done by Hayes et al. from a
research oriented application into a usable product. [12] This transitioned version of FOX
will be used as part of the Command Post of the Future (CPoF), a program to research
technology for battlespace decision aids under the Department of Defense's (DoD)
Defense Advanced Research Projects Agency (DARPA). Other possible uses for FOX
exist in similar programs, such as the US Army Communications-Electronics Command's
(CECOM) Command Post 21 (CPXXI).
2.2 Abstraction
FOX's efficiency in generating large numbers of potential COAs stems from its highlevel (abstract) representation of the battlespace and forces (see Figure 2.2-1 through
Figure 2.2-4). Wargaming at an abstract level enables a rapid search through COA-space
for generally desirable, high-level COAs. In FOX's current configuration, these candidate
COAs are then presented to human analysts for a more in-depth analysis and detailed
planning effort. However, FOX's output can also be fed into a more detailed COA
analysis system, for instance Logica Carnegie Group's CADET system. [9]
FOX employs a high-level representation of battlefield engagements in which two
units (one BLUFOR and one OPFOR) attack and/or defend against one another. The
details of the battle are abstracted as follows:
*
The most important information about the terrain becomes encapsulated into a
generic maneuver box (MB), formally represented as a 2-dimensional grid
consisting of N parallel avenues of approach (AAs) crossed perpendicularly by M
lines of defensible terrain (LDTs). An LDT is a string of roughly adjacent choke
points cutting across all the AAs providing a naturally strong defensive position.
(see Figure 2.2-2)
19
"
Additional terrain information is encapsulated as "go" and "no-go" regions
between AAs. For instance, a mountain range or unfordable river would be
considered "no-go" and movement in that part of the battlefield would be
restricted.
* Offensive forces are modeled as moving from tactical assembly areas (TAAs)
behind the forward edge of the battle area (FEBA) toward an envisioned limit of
advance (LOA) beyond the furthest LDT. The units' objectives are a variable
mixture of two criteria:
1. Capturing as much territory and seizing as many objectives as possible.
2. Attriting the enemy as much as possible and being attrited as little as possible.
*
Defensive forces are modeled similarly, but their objectives are to hold territory
and objectives while warding off the attacking units.
Similarly, subordinate unit movements and the outcome of battles are abstracted
using a relatively simple, deterministic rule-base and attrition equations.
It E3
A
a ndi
P_ BLUE
FROZI
VC
EM
<= Fa
K otw
xIA
PAO"
TO"
PLGREE-4
X
cr~
WM
r
CW
AI-:
Pt OAMCE
[a_
PLAIW1 (LD)
Xxomim
PL srse
Figure 2.2-1: Traditional COA Sketch
20
PL PL AID
NXE FW4
Figure 2.2-1 illustrates a traditional COA sketch. It uses standard symbology to
display the basic terrain features, BLUFOR units' compositions, positions, and high-level
tasks, OPFOR units' compositions, positions, and objectives.
AA
LDT
MI
No-Go
CT)=Objective
Figure 2.2-2: COA Sketch Overlaid with AAs and LDTs
Figure 2.2-2 adds avenues of approach (AAs) and lines of defensible terrain (LDTs) to
the sketch according to the existing terrain, obstacles, and control measures already in
place. These lines all need to be added by an analyst or some other program since FOX
does not perform this function.
21
...........
...........
..................
....................
-..................
....
....
....
.....
...
I...............
..........................................
......................
.........
...........
....... I.I.I
.......
e.....
......
1
......
.....
.....
...............................................
..................................
........................................
........--
.......... ...........
Figure 2.2-3: Abstract Battlefield Grid
.4
<0*>
N/
............
I............
I.................
.............................
..........................
..............
..............
.............
..........
I....
........
....................
Y
<5>
101
-.1.1
...........
I.I.I.................
...........
...
............................
....................
..............
...........
X
W
19
<*.-. .......................
E)-.--...................
................
..............
.........................
...........
........
.........
.............
SLAM
Figure 2.2-4: Abstract Battlefield and Units
Using the AAs and LDTs to define a grid over the battlefield, the terrain, obstacles,
and control measures are abstracted away, leaving us with the representation shown in
Figure 2.2-3. The "no-go" regions marked are abstracted from the existence of the
mountains in the original sketch. Figure 2.2-4 adds the units in their abstracted positions;
this is the representation used by FOX. After FOX is finished, a COA can be translated
22
back into the sketch format by reversing this process, which again, an analyst or a
program other than FOX must do.
These simplifications and abstractions exist for two reasons: First, due to FOX's use of
a GA to search COA space, wargaming simulations are called thousands of times and
therefore need to be as fast as possible; abstraction obviously helps speed up the
simulation. Second, commanders tend to wargame on approximately this same level (two
echelons below where they are commanding) with a significant degree of fidelity and
usefulness.
23
3. Goals and Design
3.1 Goals
The main goal of this project is to augment FOX, in its rotated configuration, to take
advantage of intelligence about the enemy. This has the obvious advantage of
strengthening FOX's usefulness as a tool for commanders; the "smarter" FOX is, the more
likely commanders will find it useful and the more likely FOX will help commanders make
quality decisions.
This goal can be broken down into several subgoals, addressing several levels of this
problem:
1. How do we represent the intelligence?
2. How can we use the intelligence?
3. How can we test these methods?
4. What are the advantages/disadvantages of these methods?
This thesis will not address the entire scope of this problem - it is far too large for a
single project. Instead, this thesis will focus on designing and implementing a couple of
methods to combine intelligence into FOX's COA search, empirically analyzing the
performance of these methods on problems traditionally solved using GAs, and
extrapolating these results to modify FOX. This should both produce some preliminary
results as well as provide a starting point for future research on intelligence
representations, additional influence methods, effective combinations of these methods,
and testing the usefulness in various domains.
3.2 Design
3.2.1
Criteria
In order to accomplish the above goals, the design of this system should satisfy several
constraints:
1. When no intelligence is available the system should act just like a normal system
that does not try to use intelligence. - This is important because on a system
level, a lack of information should not mean anything. However, on a higher
level a lack of information may translate into useful intelligence. For instance, if
24
no reports are coming in from the field, it may be because the unit was destroyed
or captured.
2. The intelligence should be handled as an uncertain quantity. - Intelligence on a
battlefield is, at best, noisy and uncertain. For instance, units can easily be
misreported: e.g., it is not always easy to differentiate between friendly and enemy
forces; events and information can be reported several times: e.g., two reports of 6
casualties each does not necessarily mean 12 casualties; and reports may conflict:
e.g., one unit reports there are no enemies in sight while another reports it is
under attack.
3. Likely solutions (those that fit the intelligence best) should be considered even if
they are not as good as less likely solutions. - This is basically why we are
including the intelligence. Just because a solution is not the best, does not mean
the enemy is not using it.
4. Unlikely solutions that are extremely good should not be completely discounted.
- Commanders often want to know what the worst case is, even if it is unlikely.
3.2.2
Design
The design for this "influenced GA" is basically just a GA search engine with
extensions to control the influence measure, distribution, and use. The GA basis should
include control and feedback about:
*
*
Population Size: limited only by available memory
Fitness: from a single, user-defined function
*
Generations: unlimited
*
Minimization or maximization searches
Elitism: Retain a user-defined percentage (0% - 100%) of a population between
generations according to several schemes: Best', Roulette2 , or Random3 .
*
Best: chooses the top X% of the population.
Roulette: chooses a random X% of the population with each individual weighted according
to their
fitness.
1
2
' Random: chooses a random X% uniformly from the population.
25
*
*
*
*
Breeding: Choose individuals from a user-defined percentage (0% - 100%) of the
population according to several schemes: Best with Best, Best with Roulette, Best
with Random, Roulette with Roulette, Roulette with Random, or Random with
Random. Only one child should be produced per breeding.
Crossovers: User-defined expected number of crossovers per breeding - actual
crossover computation is done by making a random decision at each bit of the
gene while breeding.
Mutation: User-defined probability to mutate any bit of the gene during breeding.
Control to evolve for a certain number of generations, until some fitness threshold
is reached (with a maximum number of generations), or until a user-defined
percentage of the population has reached some fitness threshold (with a maximum
number of generations specified).
*
Ability to insert a phenotype into the current population at any time.
*
Feedback about any individual in the current population.
*
Statistics about the current population: average fitness, maximum fitness,
minimum fitness, rank of a fitness, and convergence measure.
*
Statistics about the amount of work done: number of calls to the fitness function
and number of generations evolved.
In addition to the basic functionality above, the Influenced GA engine needed to be
designed to accommodate for the supply and use of uncertain information. Therefore, the
following additions were made:
*
*
*
User-defined value for a fitness "goal"; i.e. the best-expected fitness.
A user-defined function to calculate the likelihood of an individual; this is best
thought of as a heuristic based measurement of the desirability of the individual.
Table 3.2-1 describes the meaning of its output. In the context of COAs, this
function should calculate how well an individual COA fits the observed
intelligence.
User access to set parameters controlling the confidence level placed on each type
of influence. Table 3.2-2 describes the meanings of various levels of confidence.
Note that all of these additions fit within the criteria mentioned in Section 3.2.1.
26
Likelihood
1.0
0.0
1.0
I
General Meaning
Meaning in COA Context
Very likely.
Fits all intelligence perfectly.
Likely, to a certain degree.
Probable, to some degree,
given current intelligence.
No information leading to a
likely or unlikely conclusion.
No intelligence available to
make a conclusion.
Unlikely, to a certain degree.
Not probable, to some degree,
given current intelligence.
Very unlikely.
Impossible given all known
intelligence.
Table 3.2-1: Likelihood Function Output Definitions
Confidence
1.0
Meaning
Absolute confidence: Fully use the
value of the likelihood function.
Some confidence: Only use a
proportion of the likelihood
function's value.
0.0
Unconfident: Do not use the value of
the likelihood function at all. i.e., act
like an uninfluenced GA.
Table 3.2-2: Confidence Measure Defintions
27
4. Implementation and Testing
4.1 Implementation
The above design for a GA engine extended to deal with uncertain information was
implemented in C+ + (specifically, Microsoft's Visual C+ + 6.0) and was implemented to
be fully extensible via C+ +'s class inheritance functionality. Using this engine, three
types of influence were implemented:
1. Influencing the initial population distribution: a.k.a. "Initial Skew"
2. Influencing the fitness of individuals: a.k.a. "Fitness Skew"
3. Influencing the crossover probabilities between parents during breeding: a.k.a.
"Crossover Skew"
4.1.1 Initial Skew
Normally, a GA generates its initial population by randomly generating individuals
uniformly over the search space. However, if the information available discounts some
solutions and supports others, it seems to make sense to generate the initial population
from a non-uniform distribution.
In order to take advantage of this idea, the following scheme was used to generate the
initial population:
1. Randomly create an individual, A, using a uniform distribution over the search
space.
2. Add A to the population with a probability:
P(addingA) = 1+C(L(A)-1)
C
Confidence measure of using Initial Skew and L(A)
Likeliness measure of A.
P (A)
C__0
.
O=0.50 . 6
0=1
.4
0.2
-1
-0.5
0.5
1
L (A)
L)
Figure 4.11 -: Probabilities of Keeping A for Initial Skew
28
3. Repeat until the population is full.
4.1.2 Fitness Skew
Another way to influence the GA's search is to modify the fitness of individuals based
on their likelihood. However, we do not want to modify the fitness function directly since
that would break any abstraction we have between the fitness function and the GA
procedures. Also, preserving the actual fitness is probably useful since that is the actual
measure of how good or bad the individual is. Instead, we will modify the fitness and use
the new, "skewed" fitness for elitism and breeding selections.
Fs (A) = F(A) + (C . L(A)Xgoal
-
F(A))
Fs (A) - Skewed fitness of A, F(A) = Fitness of A, C - Confidence measure of using
Fitness Skew, L(A) = Likeliness measure of A, and goal is the best-expected fitness in the
search space.
FS (A)
1
FS (T )
1
C=0
L(A)=1
0.75
(A)=1
0.5
0.25
-0.25
L (A)=0
0.75
L (A) =0.5
0.5
C=0 .5
SL(A)=0.5
0.25
0.2
0.4
0.6
0.8
(A)
1
0.4
-0.25
-0.5
-0.5
-0.75
-0.75
-1
0.6
0.8
1
-1
C=1
FS (A)
1
L (A)=0
1
0.75
0.5
L(A)=1
L(A)=0.5
0.25
0.2
0
0.6
0.8
1
k
-0.25
-0.5
-0.75
-1
Figure 4.1-2: Fitness Skew (goal = 1)
4.1.3
Crossover Skew
The last implemented influence allows a more likely individual to pass more genes on
to its children than its "mate." Unlike the previous two techniques, this has less to do
29
with the actual value of L(A), but rather uses a comparison of the likelinesses of the two
mating individuals. The scheme is as follows (using the same definitions of L(A) and C
as above):
1. For a breeding pair of individuals, A and B, calculate: r =). C(L(A)- L(B))+1
2
C=0
C=0 . 5
r
r
1
1
0.8
0 .8
0.4
0 .2
0
0.4
0 .2
0
1
0.8
1
0.8
06
L (B)
0.6
0.4
.
1 ()0.8
0.8
.6
.64
6
0
.
0.2
0.2
L.I
.6
0.4
.
0.2
L()
0. 2
L (A)
r
1
R.
0.8
0.6
0.4
0.2
0
0.8
0.8
60.6
0.
0.4
0.4
0.2
L (B)
0.2
L
t
0
Figure 4.1-3: "Stickiness" Measure for Crossover Skew
2. Let E be the expected number of crossovers and Igene be the length (in bits) of
the gene.
3. The probability that A donates the first bit of the gene is r; that it is from B is
1-r.
4. For any bit of the gene, the probability of crossing over is:
2(1- r)
P(crossover) =
2r E
genel
30
E
genel
if on A
if on B
This means that in the normal case:
P(crossover)=
C = 0 and L(A) = L(B) z
-2
and
E . Therefore, the expected number of crossovers is in fact E:
genel
Igenel
Igenel
E
1=1
Igenel
P(crossover)= I
However, in an influenced case: C > 0,
(Ir -+
r=
and P(crossover)
L(A) increases, and L(B) decreases a
E
C)
=E
if onA
. Thus, as C increases towards
'gene
(1+ C) E
if onB
geneI
1.0, the probability of crossover decreases if on A and increases if on B. This means that
A becomes "sticky" and contributes more genes than B, which becomes "slippery." The
reverse holds true if L(B)> L(A). The expected number of crossovers decreases slightly
as r varies from
, eventually dropping sharply to 0 for r = 0.0 and 1.0. As would be
expected, the expected number of bits donated by A rises (and bits from B decrease)
slowly as r increases from 1/2 eventually rising sharply to 100% when r = 1.0. Conversely,
the expected number of bits donated by B rises (and bits from A decrease) slowly as r
decreases from 1/2 eventually rising sharply to 100% when r = 0.0.
4.2 Algorithm Test Cases
The following tests were based on measuring the performance of the Influenced GA
presented here on "standard" GA problems. [1] [221 The main reason for doing this
testing is to gain knowledge about how the modifications affect GA searches. By using
standard problems with known landscapes and characteristics, we can gain some
understanding about how the modifications react under controlled conditions and then
use this information in more uncontrolled situations, like searching COA-space. For each
test case, the following settings were used:
*
Population Size = 100
*
Roulette breeding using 100% of the population
*
*
The best 10% of the last generation was retained (elitism)
100 trials were run at each combination of parameters
*
A maximum of 200 generations was allowed for each trial
31
Two types of tests were performed:
1. Optimal: Performance was measured as the number of generations taken until at
least one individual found the global optimum (with a maximum score of 200
generations).
2. Non-Optimal: Performance was measured as the number of generations taken
until at least 10% of the population was in approximately the most fit 0.05% of the
search space. Estimates for the fitness threshold that delimitated this top 0.05%
area were found using empirical evidence of measuring the fitness of several
hundred thousand randomly chosen individuals. Here too, the maximum score
was 200 generations.
Data was collected in two stages:
1.
Effects of Skews on Mutation and Crossover Rates: Performance was measured
using the first eight parameter sets in Table 4.2-1 to determine what, if any, effects
each Skew independently exerted on the choice of Mutation and Crossover rates.
This was done only for the Optimal testing criteria.
2. Skew Interaction: Performance was measured using the last three parameter sets
in Table 4.2-1 to determine what, if any, effects Skews exert on each other. The
Mutation and Crossover rates were chosen using the data from the first stage and
held constant for each problem (F1 - F6). This was done for both testing criteria.
32
Trial
Mutation
Rate
Expected #
Crossovers
Initial
Skew
Fitness
Skew
Crossover
Skew
Baseline M
0.0-0.05
3.0
0.0
0.0
0.0
Initial M
0.0 - 0.05
3.0
0.0- 1.0
0.0
0.0
Fitness M
0.0
0.05
3.0
0.0
0.0-1.0
0.0
Crossover M
0.0
0.05
0.0 15
3.0
0.0
0.0
0.0-11.0
1.0- 10.0
0.0
0.0
0.0
Initial C
0.015
1.0 - 10.0
0.0 - 1.0
0.0
0.0
Fitness C
0.015
1.0- 10.0
0.0
0.0- 1.0
0.0
Crossover C
0.015
1.0- 10.0
0.0
0.0
0.0 - 1.0
Initial v.
varied
varied
0.0- 1.0
0.0-1.0
0.0
Initial v.
Crossover
varied
varied
0.0- 1.0
0.0
0.0- 1.0
Fitness v.
varied
varied
0.0
0.0-1.0
0.0-1.0
Baseline C
-
Fitness
Crossover
Table 4.2-1: Trial Parameter Sets
For all problems, the genes were binary strings using reflected Gray-codes [3] [13] to
encode the data. This encoding was chosen to provide continuity between elements in
the search space and avoid so-called "Hamming cliffs." For instance, the binary
representations of 7 ("0111") and 8 ("1000") have a Hamming distance of 4, but are
likely to have fitness values very close in a somewhat smooth search space.
The following six sections (pages 35 - 60) present tests and analysis over standard GA
search problems.
These six were used because each problem displays different
characteristics of search spaces. Once we know how the Skews affect searches over these
characteristics, we can infer how they will affect searches on a more general problem if we
know something about what the characteristics of that search space are.
The information is organized as follows:
1. The Problem Space - presents the function and domain that define the search
space.
33
2. The Confidence Measure - presents a function that will be used to measure the
likelihood that a point is the global optimum. For the case of these analytic
functions, the confidence measures are created by applying heuristics that
emphasize the optimal part of the search space. These are necessarily linked to
the fitness functions so that performance can be easily measured and compared.
3. Stage 1 Test Results - six sets of data are presented and analyzed to see what
effect the confidence level of the Skews has on the ideal choice for Mutation and
Crossover Rates. Each data point is the average number of generations it took for
the Optimal Test criteria to be met (see above) for a particular pair of values (the
confidence in a Skew and either Mutation Rate or Expected # of Crossovers).
The fewer number of generations it took to find the optimum, the less effort was
needed and the better the search performed. From these results, we can
determine the ideal Mutation and Crossover Rates for each problem space.
4. Stage 2 Test Results - two sets of data are presented and analyzed to see what
affect the Skews had on each other for each test criteria, Optimal and NonOptimal. Each data point represents the average number of generations it took for
the respective criteria to be met (see above) for a particular set of confidence
levels in the three Skews. Due to the shear amount of data and computation
power needed to test all possible combinations, the problem's complexity was
reduced by keeping the confidence of at least one of the Skews 0.0. For each set
of data, dark areas represent a low number of generations (good performance) and
light areas represent high values (poor performance). The bar shown next to each
graph with the maximum and minimum numbers of generations displayed shows
the actual range of the data. For the same reasons mentioned above, this means
that dark regions correlate with good performance and light regions correlate with
poor performance.
34
4.2.1
4.2.1.1
DeJong's F1
The Problem Space
The function is:
3
f(x
1,x2 ,x3 )=
This function should be minimized over -5.12
(4-1)
x2
x,
5.12; the minimum occurs at
f(0,0,0)= 0. [1][22]
The Gray-Code representation for this domain uses 30 bits to represent three signed 9
bit numbers. (Therefore the actual domain searched over is -5.12
x, 5.11 with a
resolution of 0.01.) A plot of f(x,x 2 ,0) is shown in Figure 4.2-1.
4.2.1.2
The Confidence Measure
For the confidence measure of a point's likelihood of being the global solution, the
following function was used:
M2
c(xI, x2 , x3 )= 2e
-1
where m = maxix,1
i=1.3
(4-2)
This formula promotes the minimization of the maximum coordinate magnitude.
This heuristic seems a rather obvious choice simply by looking at equation (4-1). The
1
factor of -I was added to "spread out" the distribution to better cover the domain. A
10
plot of c(xI , x 2 ,0) is shown in Figure 4.2-2. A plot of equation (4-1) skewed by equation
(4-2) is shown in Figure 4.2-3 - notice how much steeper the skewed function becomes.
35
FI(xx2.0)
1
C1(x1 ,X
,0)
2
go-
0.80.6-
80
-
70-
60
0.4-
u
50
0.20
F-
-
4030
4
-0.2-4
-
-0.4
20-
0.8
10-
0
5
.1-
5
-5
5
X1-5
X
-5
Figure 4.2- 1: F1I Function Plot
.
0 X2
Figure 4.2,-: F1 Confidence Plot
Fs1(xjx 2 0)
90-
I
so70
-
60
so-
-
4030
-
20
-
1055
-5
-
Figure 4.2-3: F1 Skewed Function Plot
4.2.1.3 Stage 1 Test Results
For the Stage 1 Optimal test, the results are shown in Figure 4.2-4 - Figure 4.2-9 and
the actual data is in Appendix A.
36
Generations,7
40
Generations
r0.8
-*
20 -
20-
0
0.6
0.4
E xpected #
Crossovers
3
of
0.6
0,
Initial Skew
2
Mutation Rate
0
2
0.02
Fitness Skew
0.01
0
0
Figure 4.2-4: F1 Initial Skew vs. Crossovers Figure 4.2-5: F1 Initial Skew vs. Mutation
Generations
Generations
1000
40-0a
00
2010
0.4
0
0.05
Fitness Skew
3
1
Mutation
0
Rate
Fitness
40.4
0.03
Figure 4.2-6: F1 Fitness Skew vs.
Crossovers
Skew
0.2
0.0
01
0 0
Figure 4.2-7: F1 Fitness Skew vs. Mutation
Generationc+,
Generations
150-
5
40
30
20 -
O
0.2
5
Expected # of
Crossovers
0.6
5
80.
---
10--
0.6
0-
10
8
7
6
# of
Crossovers
E xpected
/ 04
Crossover Skew
112
3
2
0.05
0.04
.
0.03Mutation Rate
0
Figure 4.2-8: F1 Crossover Skew vs.
Crossovers
Crossover Skew
0.01
Figure 4.2-9: F1 Crossover Skew vs.
Mutation
37
As seen in the above figures, none of the Skews have much effect on performance
regardless of changes in the Mutation and Crossover Rates. Likewise, Crossover Rate has
little effect on performance, but the best value seems to be
at about 7. Mutation Rate has
Se
a dramatic effect, and the best value appears to be 0.005.
4.2.1.4 Stage 2 Test Results
n
For the Stage 2 tests, the results are shown in Figure 4.2-10 and Figure 4.2-11; the
actual data is in Appendix A.
42
A Crossover Skew
Crossover Skew
10
12
31. 2
2.0
Initial Skew
Initial Skew
Fitness Skew
Figure 4.2-10: F1 Optimal Test
WPFitness
Skew
Figure 4.2-11: F1 Non-Optimal Test
In the Optimal test results (Figure 4.2-10), there is a slight enhancement in
performance for Fitness Skew when accompanied by a significant amount of Initial Skew
or Crossover Skew. For the Non-Optimal test (Figure 4.2-11), the performance is best for
large values of Crossover Skew, but the effect of Initial Skew and Fitness Skew does not
appear significant.
38
4.2.2
4.2.2.1
DeJong's F2
The Problem Space
The function is:
f~xi~x2)=00(Xf
+(I- _X,)2(43
+
2
This function should be minimized over - 2.048
f(1,1)= 0. [1][22]
x,
2.048 ; the minimum occurs at
The Gray-Code representation for this domain uses 24 bits to represent two signed 11
bit numbers. (Therefore the actual domain searched over is - 2.048 x, 2.047 with a
resolution of 0.001.) A plot of f(xl,x 2 ) is shown in Figure 4.2-12.
4.2.2.2 The Confidence Measure
For the confidence measure of a point's likelihood of being the global solution, the
following function was used:
c(xI,x 2 )=2e-
2
-1
where m= xi -x 2
(4-4)
This formula promotes the minimization of x1 - x 2 , which is the dominant term in
equation (4-3). A plot of c(x1 , x 2 ) is shown in Figure 4.2-13. A plot of equation (4-3)
skewed by equation (4-4) is shown in Figure 4.2-14 - notice how much steeper the
skewed function is than the original.
39
C2(x
F2(x1 ,x2)
,x2)
0.8 0.60.4
0.2 0-
7000
5000
-0.2-0.4-
-0.6
-0.8
-210
Figure 4.2-12: F2 Function Plot
Figure 4.2-13: F2 Confidence Plot
Figure 4.2-14: F2 Skewed Function Plot
4.2.2.3 Stage 1 Test Results
For the Stage 1 Optimal test, the results are shown in Figure 4.2-15 - Figure 4.2-16
and the actual data is in Appendix A.
40
Immmmommow-
-- -
Generations
Generations
150
--
150-
-
0.84
50-
108
s
Initial Skew
3
Expected # of
Crossovers
0
2
Figure 4.2-15: F2 Initial Skew vs.
Crossovers
Figure 4.2-16: F2 Initial Skew vs. Mutation
100-
0
-0.03
0.02
0.02'
/.2
02
0 0
Generations
Generations
150
-
100-
rosv.
50
0.6
-
0
1
9
a
0
Skew
-Fitness
0.2
5
F
Expected #of
Crossovers
0.8
0.04
Fitness
Mutation Rate
Skew
00
2
Figure 4.2-18: F2 Fitness Skew vs.
Figure 4.2-17: F2 Fitness Skew vs.
Crossovers
Mutation
Generations
150-
150.]
0.8
50-
0.05
0.2-
0.02
Mutation Rate
0.01
0.5
50-0.04
0.4
0.030
0.2
0.0
0 0
Figure 4.2-20: F2 Crossover Skew vs.
Mutation
Figure 4.2-19: F2 Crossover Skew vs.
Crossovers
41
As seen in the above figures, none of the Skews have much effect on performance
regardless of changes in the Mutation and Crossover Rates - although Fitness Skew
appears to have a slightly negative effect, degrading performance. Neither Crossover nor
Mutation Rate has much effect either, but the optimal values appear to be at about 6 and
0.02 (respectively for Crossovers and Mutation Rate).
4.2.2.4 Stage 2 Test Results
For the Stage 2 tests, the results are shown in Figure 4.2-21 and Figure 4.2-22; the
actual data is in Appendix A.
A Crossover Skew
tCrossover Skew
2,5
Initial Skew
Initial Skew
Fitness Skew
Fitness Skew
Figure 4.2-22: F2 Non-Optimal Test
Figure 4.2-21: F2 Optimal Test
In the Optimal test results (Figure 4.2-21), there is no discernable trends that
indictates the Skews help or hinder performance when varied two at a time. However, for
the Non-Optimal test (Figure 4.2-22), the performance is better for increasing Crossover
Skew, especially when accompanied by either Initial Skew or Fitness Skew in significant
strength.
42
4.2.3 DeJong's F3
4.2.3.1
The Problem Space
The function is:
5
|_xi J
Ax1, V, x5)=
(4-5)
This function can be minimized or maximized over -5.12
occurs within -5.12
5
x,
x, <-5 with f(x, ?
,x 5
x,
5.12; the minimum
)=-30; the maximum occurs within
5 5.12 with f(xj,? , x 5 )= 25. [1][22]
The Gray-Code representation for this domain uses 50 bits to represent five signed 9
bit numbers. (Therefore the actual domain searched over is -5.12
x, 5.11 with a
resolution of 0.01.) A plot of f(xI, x 2,0,0,0) is shown in Figure 4.2-23.
4.2.3.2 The Confidence Measure
For the confidence measure of a point's likelihood of being the global solution, the
following function was used:
c(x,? ,x
5
)=1-
2
where m = max(x,)
i=axx5
1+ e'
(4-6)
This formula promotes the minimization of the maximum coordinate. This heuristic
seems a rather obvious choice simply by looking at equation (4-5). A plot of
c(xI, x 2 ,0,0,0) is shown in Figure 4.2-24. A plot of equation (4-5) skewed by equation (46) is shown in Figure 4.2-25 - notice the "buckling" and "stretching" as compared to
Figure 4.2-23.
43
F3(x1.x2,0,0,0)
c3(xo, X2,0,0,0)
40-
0.8
0.6
30-
0.4
0.2gC
20-0.2-0.4-
Figure 4.2,24:- F3 Confidence Plot
Figure 4.2-23: F3 Function Plot
Figure 4.2-25: F3 Skewed Function Plot
4.2.3.3 Stage 1 Test Results
For the Stage 1 Optimal test, the results are shown in Figure 4.2-26 - Figure 4.2-31
and the actual data is in Appendix A.
44
Generations
Generations
50-
60-
A
403020--
40
-
20- 10
0.6
10
0.05
Initial
of
Expected #
4
3
Crossovers
2
2
1
02
0..02
0.2
ne
tial Skew
0,2
-
002
0.04
Skew
Mutation Rate
0.01
0
Figure 4.2-26: F3 Initial Skew vs.
Crossovers
Figure 4.2-27: F3 Initial Skew vs. Mutation
0.03
00
Generations
Generations
100-
40--
m
3020-
60
-
4020-
100.4
Fitness
.
0.6
0.04
Skew
040 Fitness
Skew
0.2
of
Expected #
Crossovers
3
2
Mutation Rate
0
Figure 4.2-28: F3 Fitness Skew vs.
Crossovers
0.01
Figure 4.2,29: F3 Fitness Skew vs.
Mutation
Generations
Generations
40
00
3020
50
0
10
.2
0.
of
Expected #
Crossovers
3
0.
Crossover Skew
0.6
0..04
Mutation Rate
2
0
Crossover
0.03
Skew
0.2
0.01
Figure 4.2-31: F3 Crossover Skew vs.
Mutation
Figure 4.2-30: F3 Crossover Skew vs.
Crossovers
45
As seen in the above figures, Initial Skew does not effect performance significantly,
and both Fitness and Crossover Skews have a negative effect. Crossover Rate has some
effect on performance, with the best value at approximately 5. Mutation Rate has a
dramatic effect between 0.0 and 0.005; its best value appears to be around 0.010
4.2.3.4 Stage 2 Test Results
For the Stage 2 tests, the results are shown in Figure 4.2-32 and Figure 4.2-33; the
actual data is in Appendix A.
A Crossover Skew
Crossover Skew
27,9
'0
2.5
Initial :
Initial Skew
1.0
Fitness Skew
0i tn.ss
F
k ew
Figure 4.2-33: F3 Non-Optimal Test
Figure 4.2-32: F3 Optimal Test
In the Optimal test results (Figure 4.2-32), there are no discernable trends that indict
the Skews help or hinder performance when varied two at a time. However, for the NonOptimal test (Figure 4.2-33), the performance is clearly better for significant Crossover
Skew, especially when accompanied by Initial Skew, and to a lesser degree, Fitness Skew.
46
4.2.4 DeJong's F4
4.2.4.1
The Problem Space
The function is:
Axi,
, O) N(,1)
+
ix4
(4-7)
N(m, d) = A random number chosen from a Gaussian distribution with mean = m
and standard deviation = d. This function should be minimized over -1.28
x, 1.28 ;
the expected minimum occurs at f(O, ,0) = 0. However, with the noise, values less
than 0 are possible as well as the minimum not being at (0,? ,0). [22]
The Gray-Code representation for this domain uses 240 bits to represent 30 signed 7
bit numbers. (Therefore the actual domain searched over is -1.28
xi 1.27 with a
resolution of 0.01.) A plot of f(xI, x 2 ,O,? ,0) is shown in Figure 4.2-34.
4.2.4.2
The Confidence Measure
For the confidence measure of a point's likelihood of being the global solution, the
following function was used:
c(xI,? , x 3 0 )= 2e-"' 2 -l
where m = maxlx,1
i=1..30
(4-8)
This formula promotes the minimization of the maximum coordinate magnitude.
This heuristic seems a rather obvious choice simply by looking at equation (4-7), which is
clearly dominated by the largest coordinate magnitude. A plot of p(xI , x 2 ,0,? ,0) is
shown in Figure 4.2-35. A plot of equation (4-7) skewed by equation (4-8) is shown in
Figure 4.2-36 - notice the much more defined bowl-shape as compared to Figure 4.2-34.
47
o4(x1X0... 0)
F4(xlx 2 0.---)
0.5
10
0,
5"
0
-0.5
X1
.0.
Wr
05 0
0.5
Figure 4.2-35: F4 Confidence Plot
Figure 4.2-34: F4 Function Plot
F,4(x,,x2,0,15,
5.
0,
X1
0.5
0
0.5
X2
Figure 4.2-36: F4 Skewed Function Plot
4.2.4.3 Stage 1 Test Results
For the Stage 1 Optimal test, the results are shown in Figure 4.2-37 - Figure 4.2-42
and the actual data is in Appendix A.
48
Generations
Generations
1500
so-
100
40
S0.8
10
0.4
0.05aw.0
Initia Sko
0.2
of
3
Expected
Crossovers
0.03
2
0.01
Mutation Rate
2
0
Figure 4.2-37: F4 Initial Skew vs.
Crossovers
Figure 4.2-38: F4 Initial Skew vs. Mutation
0
,~
'
0.02
Generations
0.6
0.2
Generatin
60
10
50-a
0-.
40
20
0/0
.6
10
SFitness
of
Skew
4
0.4
Fitness
Skew
*00
5
4
Expected #
Crossovers
3
Mutation Rate
0
2
Figure 4.2-39: F4 Fitness Skew vs.
Crossovers
Ooi
Figure 4.2-40: F4 Fitness Skew vs.
Mutation
Generations
Generations
100
150--
60
40
20
0.
10
Crossover
Crossover Skew
of
Expected #
Crossovers
2
Mutation
0
Rate
o.01
Skew
0.2
0.0/
4
o
0 0
Figure 4.2-42: F4 Crossover Skew vs.
Mutation
Figure 4.2-41: F4 Crossover Skew vs.
Crossovers
49
As seen in the above figures, none of the Skews have much effect on performance
regardless of changes in the Mutation and Crossover Rates. Crossover Rate has minimal
effect, and most of that is between 1 and 3, with the best value at approximately 8.
Mutation, on the other hand, has a dramatic effect, with the best performance at 0.005.
4.2.4.4 Stage 2 Test Results
For the Stage 2 tests, the results are shown in Figure 4.2-43 and Figure 4.2-44; the
actual data is in Appendix A.
I
I
Crossover Skew
Crossover Skew
ntlk
Initial Skew
Initial Skew
Fitness Skew
Fitness Skew
Figure 4.2-44: F4 Non-Optimal Test
Figure 4.2-43: F4 Optimal Test
In the Optimal test results (Figure 4.2-43), there is significant increase in performance
for high values of Crossover Skew and Fitness Skew; Initial Skew seems to have a negative
effect though. Likewise, for the Non-Optimal test (Figure 4.2-44), the performance is
clearly better for significant Crossover Skew, especially when accompanied by Fitness
Skew. Initial Skew seems to hinder performance here also.
50
4.2.5
4.2.5.1
F5: Inverted Shekel's Foxholes
The Problem Space
The function is:
f(xIx 2 )= 500-
(4-9)*
0.002+
1
i=1
ai is the ith coordinate of the jt ordered pair generated by {- 32,-16,0,16,32}2.
This
65.536; the maximum occurs at
function should be maximized over -65.536 s x,
f(- 32,-32)~ 499.01996. [1]
The Gray-Code representation for this domain uses 34 bits to represent two signed 16
bit numbers. (Therefore the actual domain searched over is - 65.536 x, < 65.535 with
a resolution of 0.001.) A plot of f(x
, x2 )
is shown in Figure 4.2-45.
4.2.5.2 The Confidence Measure
For the confidence measure of a point's likelihood of being the global solution, the
following function was used:
c(x 1, x 2 )= 2e 50
1
where m = min
j=L.25
(xx,
i=L.2
-a
)
(4-10)
By carefully analyzing equation (4-9), the following logic rationalizes the above
heuristic:
* There is some discrepancy among sources about the exact equation: [221 lists the following equation.
However, the character of the search problem is basically identical, the differences between maxima are just
more pronounced than in (4-9).
211
f(x,,x2 )=0.002+
22
j=1
51
j +
i=1
(X
x - a,
1.
1
(4-9) is maximized when
25
is minimized.
2
0.002+L
2
j1
)
(x -a.
6
:=1
1
2.
0.002+
is minimized when
2
1
2
1=1
is maximized
2
j +
+ 1j(x, - a,
Xx -a)6
i=1
(Since this must be > 0, there's no chance of (4-9) being greater than 500.)
3. A heuristic for maximizing
is to minimize the largest
2
j= +
i=1
(x, - a .
denominator in the sum.
4. The heuristic for finding the largest denominator is to find the maximum element
in the sum, namely: max((x, - a
5. However, this can be simplified to: ma x, - a
6.
Thus, we want to minimize overj (from #3) the max,
-
ad.
Equation (4-10) promotes this minimization. A plot of c(x1 , x 2 ) is shown in Figure
4.2-46. A plot of equation (4-9) skewed by equation (4-10) is shown in Figure 4.2-47.
52
c5(x
F5(xjex2)
0.8 -0.6-
500
40
000
X2)
6
0.-
-100--
0.2
-3000
-000-
-
20
0-
40
60
-300--02
-2200
-04
Figure 4.2-46:
F5 Confidence Plot
500
XI
20
-2
0
-53.0
-06
4
----
0~~
20
X
400
*5o0j
-400-
3 -s
s
.60
Figure 4.2-47: F5 Skewed Function Plot
4.2.5.3 Stage 1 Test Results
For the Stage 1 Optimal test, the results are shown in Figure 4.2-48 - Figure 4.2-53
and the actual data is in Appendix A.
53
Generations
Generations
r
30
20
40
''7
1o-
0
2008
0.6
/
0.6
0
10
0.
nitial Skew
0.00
0.2
Expected #
Crossovers
3
of
.
001
Mutation Rate
0
2
0 0
Figure 4.2-49: F5 Initial Skew vs. Mutation
Figure 4.2-48: F5 Initial Skew vs.
Crossovers
Generations
Generations
30-
40-
20-
010
0.a
20-
10-
0 6
OA Fitness
8
0
Skew
0.04
04 Fitness
0,2
3
Expected # of
Crossovers
Mutation Rate
0.01
2
Figure 4.2-50: F5 Fitness Skew vs.
Crossovers
Skew
O2
0.02
0
Figure 4.2-51: F5 Fitness Skew vs.
Mutation
/
Generations
Generationsin
100-
30-6
20
50
0.6
0.6
10s
04 Crossover Skew
Crossover Skew
6
of
E xpected #
Crossovers
4
Mutaio3
0.
3
2
Mutation Rate
12
0 02
--
0,2
0.01
0
0
Figure 4.2-53: F5 Crossover Skew vs.
Mutation
Figure 4.2-52: F5 Crossover Skew vs.
Crossovers
54
As seen in the above figures, none of the Skews have much effect on performance
regardless of changes in the Mutation and Crossover Rates. Crossover Rate has minimal
effect, and most of that is between 1 and 3, with the best value at approximately 8.
Mutation, on the other hand, has a dramatic effect, with the best performance at 0.025.
4.2.5.4 Stage 2 Test Results
For the Stage 2 tests, the results are shown in Figure 4.2-54 and Figure 4.2-55; the
actual data is in Appendix A.
to
4 Crossover Skew
4 Crossover Skew
1.0
2.7
16.2
Initial Sk
Initial S
::10
-A10t
Fitness Skew
Fitness Skew
Figure 4.2-55: F5 Non-Optimal Test
Figure 4.2-54: F5 Optimal Test
In the Optimal test results (Figure 4.2-54), there is little significant effect; Crossover
Skew with Fitness Skew might have a slight performance gain and Initial Skew might
have a slightly negative effect, but the differences do not look very significant. On the
other hand, for the Non-Optimal test (Figure 4.2-55), the performance is clearly better
for significant Crossover Skew and Fitness Skew; Initial Skew seems to hinder
performance.
55
4.2.6 Schaffer's F6
4.2.6.1
The Problem Space
The function is:
si2I 2
2
sin2 x + x 2
f(XI,IX2=
2
2
(4-11) *
22 2
2
1 +-
1
1000
This function should be maximized over -10.24
x
10.24; the maximum occurs at
f(0,0)=1. [1]
The Gray-Code representation for this domain uses 22 bits to represent two signed 11
bit numbers. (Therefore the actual domain searched over is -10.24
xi 10.23 with a
resolution of 0.01.) A plot of f(xI, x 2 ) is shown in Figure 4.2-56.
4.2.6.2 The Confidence Measure
For the confidence measure of a point's likelihood of being the global solution, the
following function was used:
c(xI,x
2
)=2e0 -15 where m = ma4xI,
x 2 1)
(4-12)
By carefully analyzing equation (4-11), the following logic rationalizes the above
heuristic:
* There is some discrepancy among sources about the exact equation: [22] lists the following equation.
However, the character of the search problem is basically identical, the differences between maxima are just
more pronounced than in (4-12).
sin 2 X2
xI+x
x2X
x2
1
2
2
2
2
56
+ (xI +x2
1000
sin2 vx2 +x
1.
(4-11) is maximized when
2
-
2
2
is minimized.
1+ X' +
1000
(K
2.
0ssin 2
1±
Jx
x 2 +x 2
2
1000
1 and
2
+
+
1000
>1,
so the fraction is minimized when
is minimized.
3. A heuristic to minimize 1+
magnitude: max(x
1
,|x
2
x2 +x2
1+
2
1000
is to minimize the largest coordinate
j).
Equation (4-12) promotes the minimization of this quantity. A plot of c(x, ,x 2 ) is
shown in Figure 4.2-5 7. A plot of equation (4-11) skewed by equation (4-12) is shown in
Figure 4.2-58 - notice the much smooth middle (near the maximum) and the much lower
troughs as compared to Figure 4.2-56.
57
C6(x1x2
F6(x,,x2)
-0.5
-1
10
100
10
-5
Fg
X
-5
-10
10
2-57
x2
.5
-10
-10
Figure 4.2-56: F6 Function Plot
-5
X2
.10
Figure 4.2-57: F6 Confidence Plot
F,6(xlpx2
0.1
1
0
10
-5
-10
X2
.10
Figure 4.2-58: F6 Skewed Function Plot
4.2.6.3 Stage 1 Test Results
For the Stage 1 Optimal test, the results are shown in Figure 4.2-59 - Figure 4.2-64
and the actual data is in Appendix A.
58
-. W'Pm
Generations
Generations
10100-
50
0.8
ISOi
50-
0.6
00.05
-0..2
~
Expected #aof
Crossovers
0.04
Mutation Rate
-,
nitial Skew
0.01
2
00
Figure 4.2-59: F6 Initial Skew vs.
Crossovers
Generations
0.4
0.
0.02
Figure 4.2-60: F6 Initial Skew vs. Mutation
I
Generations
10
56
0.6
0-
00
0.4
Fitness Skew
.
0.2
of
Expected #
Crossovers
OS
-----
0.03
Mutation Rate
3
2
Skew
0.01
0
Figure 4.2-61: F6 Fitness Skew vs.
Crossovers
Fitness
4.0.2
-~o
Figure 4.2-62: F6 Fitness Skew vs.
Mutation
Generations
Generations
150
100
-
500-
50
0.6
10
0.4
Crossover Skew
0
0.05
*
-
Crossover
0.2
Expected # of
Crossovers
Mutation Rate
3
2
0
0.02
O0
Skew
0.2
Figure 4.2-64: F6 Crossover Skew vs.
Mutation
Figure 4.2-63: F6 Crossover Skew vs.
Crossovers
59
As seen in the above figures, neither Initial Skew nor Crossover Skew has much
effect on performance regardless of changes in the Mutation and Crossover Rates.
However, Fitness Skew dramatically helps performance. Additionally, Crossover Rate has
little effect on performance, as does Mutation Rate. The best values for these parameters
seem to be approximately 0.03 for Mutation and 4 for Crossovers.
4.2.6.4 Stage 2 Test Results
For the Stage 2 tests, the results are shown in Figure 4.2-65 and Figure 4.2-66; the
actual data is in Appendix A.
Crossover Skew
I Crossover Skew
10
i
21.'
5
2,0
Initial Skew
Initial Skew
Fitness Skew
Figure 4.2-65: F6 Optimal Test
Fitness Skew
Figure 4.2-66: F6 Non-Optimal Test
In the Optimal test results (Figure 4.2-65), there is a clear performance boost for large
Initial Skew values, a slight improvement for increasing Crossover Skew values, and no
significant effect for Fitness Skew. On the other hand, for the Non-Optimal test (Figure
4.2-66), the performance is most improved for large values of Crossover Skew and
improvements along the other two axes is minimal.
4.3 Conclusion for Test Cases
From the performance of the various influence methods on the six test problems
above, several conclusions can be drawn. First, a summary of the performance results is
shown in Table 4.3-1. A "+" means performance was improved with greater influence; a
- means performance was degraded with greater influence, and a blank means there was
no significant effect on performance from the influence.
60
Problem
Description of Space
Performance on
Performance on Non-
Optimal criteria
optimal criteria
Initial Skew
F1
Fitness
Skew
Crossover! Initial Skew
Skew
Fitness
Skew
Gradient always points to global
+
optimum; no other local optima
F2
Crossover
Skew
Not quite as "easy" as F1, but there
are still no local optima besides the
global optimum
F3
F4
Staircase, but underlying gradient
always points to global optimum
Random roughness and high
dimensionality, but underlying
gradient always points to expected
optimum
F5
Many local optima with large changes
in fitness between them
F6
Global optimum is surrounded by a
ring of global minima; Rings of local
optima are easy to find and difficult to
get out of
+
+
+
-
+
+
-
+
+
+
Table 4.3-1: Summary of Performance Effects
It is clear from the summary that for Non-Optimal searches, such as those needed for
predicting eCOAs, Crossover Skew is definitely beneficial. However, not much can be
said for the Fitness Skew; although, it did help in F4 and F5. Initial Skew was also
relatively inert, but it clearly hindered the performance of F4 and F5 in the Non-Optimal
case. There are many potential reasons that the various Skews had the effects they did
(e.g., Initial Skew may lead to too much convergence too quickly), but actually exploring
these reasons and finding an explanation is outside the scope of this thesis and is best left
for further research.
Now that we have some understanding of how the various influences presented above
affect searches over various types of spaces, we can more aptly choose the parameters to
use when applying this technology to other spaces. Specifically for FOX, we have learned
from experience that the COA-space, as constrained by FOX's wargamer and measured
by its fitness function, tends to have the following characteristics:
*
*
There are at least a couple relatively equal optima separated by spaces of
significantly worse COAs.
Most optimal solutions have a significantly broad area around them containing
slight variations that also score quite well.
61
*
*
Most of the COAs in the space are not good.
There are some step-like characteristics present in the space due to some hard
mathematical boundaries the wargamer relies on. For instance, a battle that is
determined by a very small difference in the opposing forces' strengths would have
two distinct outcomes that are based on very small mathematical differences in
the formulae used.
From all of this information, we can hypothesize that the normal COA-spaces used by
FOX have characteristics most similar to F5 and F6 but with some local characteristics
similar to F3. Currently, FOX's warganer is deterministic, but it may become based on a
Monte Carlo type of simulation instead; this would introduce randomness into the space
similar to F4.
Using the above equivalencies and the results of the associated test problems, we can
choose the confidence we have in each of the Skews. For the COA problem below, we
will use a confidence of 1.0 for Crossover Skew and 0.5 for both Fitness and Initial Skews.
62
5. Interfacing with FOX and Scenarios
5.1 Interfacing with FOX
The first step, of course, was to modify FOX's current search method, a traditional
GA with niches, to use the technology. This was done by using the abstractions already
present in FOX and simply replacing the current class with the one used above and
wrapping it to have a similar interface.
The only problem with doing this is that there is no automatic way to feed the new
GA any collected intelligence. To remedy this situation for the purposes here, a
somewhat generic set of assumptions was used to replace any run-time intelligence; the
actual knowledge needs to be modified in the code per scenario and described in detail
below. In order to get the full potential of incorporating intelligence into the GA, FOX
will have to make a couple of changes:
1. Modifying the input to include the intelligence or add the intelligence as a
separate input source.
2.
Pass this information into some knowledge-based application that can translate it
from the data collected into a form usable by the GA.
The first point is obviously necessary, but should not be too difficult once a standard
for communication is developed between FOX and whatever is providing it input.
Currently FOX is using canned scenarios created manually in a FOX-specific format, so
the inclusion of any needed information is simply a matter of making manual changes.
With the ongoing integration efforts in CPoF, a standardization that could potentially
support such uncertain and partial intelligence is emerging.
The second point is also just as necessary, but much more difficult to implement. The
difficulty stems from the implications of pieces of knowledge that are beyond the scope of
a simple likeliness measure. For instance, a human expert may know something particular
about his enemy's behavior: e.g., this particular Russian commander likes attacking from
the North. The problem of interpreting the data is not easy and is probably best handled
by some knowledge-based system developed outside of FOX.
63
5.2 FOX Scenario
The scenario used to demonstrate FOX integrated with this Influenced GA is from
CPoF's eTDG3 (electronic Tactical Decision Games) run by John Schmitt on September
14, 1999. This scenario was chosen for a couple of reasons:
1. It is of the complexity and design that FOX can currently handle. Later scenarios
involved more complicated tasks (e.g., evacuation), force structures (e.g.,
guerillas), and terrain (e.g., crowded cities).
2. A significant amount of information is available about it from both being
personally involved in the eTDG as well as having access to the after action
reports.
3. A large amount of the information from the eTDG is from experts who were
forced to explain their thinking aloud due to the environment: telephones and a
collaborative, web-enabled whiteboard.
5.2.1
The Battle of Johnsonburg
"The Battle of Johnsonburg" was the name of the scenario for eTDG3. [15] The basic
situation was as follows (see also Figure 5.2-1): [14]
You are the commander of an ad hoc brigade-size task force consisting of a
tank-heavy battalion (2d), 2 motorized infantry battalions (1st and 5th), a
reinforced light-armored reconnaissance (LAR) company (C), and 2 hostnation mechanized battalions (3d and 4th). Host-nation forces are not up
to U.S. standards, but are usually capable of most basic missions. You are
supported by 2 battalions of direct-support artillery. The terrain is rolling
and thickly wooded in places. The wooded areas are impassable to all but
infantry. Enemy forces are principally mechanized and motorized. Friendly
forces, advancing north, and enemy forces, advancing south, have clashed
along the trace of the Vopeist River. The Vopeist is a shallow, slowmoving river some 200-400 meters wide. Upon contact, in an effort to
seize the initiative, both forces started shifting east, trying to turn the
other's flank and establish a bridgehead on the far side of the river.
Unopposed crossings of the Vopeist at the various ford sites, although
time-consuming, are generally not difficult for vehicles or infantry. Assault
crossings are another story. The lesser streams feeding into the Vopeist are
not obstacles to movement.
Your task force is sent east along Rte. 85 with urgent instructions to secure
a bridgehead in the vicinity Johnsonburg-Ryerton-Hayesville in order to
facilitate the continued advance north of the division, or at least to deny
those crossing sites to the enemy. Combat intelligence indicates an enemy
mechanized regiment closing on Hayesville from the west along Rte. 81.
64
[Event-0] Charlie Company (LAR) races ahead and clashes with a
reinforced enemy company of tanks and mech at Roth Bridge at 1930.
[SitRep-0] After a heated engagement, Charlie repulses the enemy, which
withdraws to positions north of the river on Holcomb Hill. [Event-1] At
1945, 4th Battalion likewise clashes with enemy armored reconnaissance
at O'Neal Bridge, where a standoff develops. Both forces witdraw to their
respective sides of the river and continue to observe and enage at long
range. [SitRep-1] By 2100, 4th Battalion reports it holds positions in depth
south of O'Neal Bridge. Aout this same time, 1st Battalion arrives at Roth
Bridge. [Event-21 With heavy supporting arms and Charlie supporting by
fire, Furious First launches a hasty attack north across Roth Bridge at
2130. In the darkness, a close, confused engagement develops on the
wooded slopes of Holcomb Hill; the use of supporting arms becomes
problematic. The 1st Battalion commander reports he has only a rough
idea of current friendly-enemy dispositions and that the operation has
devolved into a series of intense small-unit actions.
[Event-3] By 2330, 2d Battalion has arrived south of Roth Bridge on Rte.
8, and 3d and 5th Battalions are moving up in trace of Slammin' Second.
Meanwhile reconnaissance reports indicate continued vehicular traffic
between Ryerton and Roth Bridge. Your logistics trains have already
replenished 4th Battalion in its positions and will have your units in the
vicinity Johnsonburg-Roth Bridge replenished by 0300. By 0030, 1st
Battalion commander reports the sounds of significant mechanized activity
near the bridge. He estimates the enemy at one or possibly two battalions.
He reports that the situation has stalemated, with friendly and enemy
forces interspersed in the thick woods of Holcomb Hill. [SitRep-2] He
believes that he holds a tenuous bridgehead north of the river but says that
the situation will not be sorted out until at least dawn. Around 0130 you
receive reports of concentrations of unidentified enemy forces west and
southeast of Hayesville, as well as movement south on Rte. 8 toward Roth
B. [SitRep-3]. It is now about 0200. 3d and 5th Battalions have been
refueled; 2d Battalion is about to start. A calm seems to have settled over
the battlefield. What will be your next move?
65
T2
F~~Z;
==P
[15]
Figure 5.2-1: The Battle of Johnsonburg
Later reports also provided the following information:
[SitRep-41: 2235: SA-8 Gecko Battery detected vic RYERTON
[Event-6]: At about 0230 you receive reports of enemy mechanized forces
moving southeast near Ryerton.
[Event-7]: At 0400 you receive a report that Kapler Bridge, some 12 km
east, was "dropped" by aviation at 0200.
Interpreting this into something FOX can handle means taking some liberties with the
abstraction into AAs and LDTs, ignoring some information that FOX cannot currently
handle (like airsupport), changing some information into something FOX can currently
handle (e.g., changing the Motorized Infantry and the Light-Armored Recon. units into
Infantry Battalions), giving BLUFOR a COA, and ignoring OPFOR for the moment; we
have a situation that looks like this:
66
U0F
R
L
R
...............
- - .-..
........................
......
Figure 5.2-2: A FOX Interpretation of The Battle of Johnsonburg
FOX also needs to know the Order of Battle for the OPFOR. We will use the
following:
1 Armored Battalion
2 Mechanized Infantry Battalions
1 Infantry Battalion (rather than a Light-Armored Reconnaissance)
And the intelligence about the enemy described above becomes:
The Infantry Battalion is probably small and on AA3.
One of the Mechanized Infantry Battalions is probably large and on AA2.
The other Mechanized Infantry Battalion is probably on AA1 or 2.
The Armored Battalion is probably on AA1 or 2.
This intelligence becomes the likeliness measure, so any eCOAs that agree with this
will be considered more likely, any that disagree will be considered less likely. Now FOX,
with its influenced GA, can figure out what the enemy might do.
67
5.2.2 eCOA without Influence
Without using the influence from the intelligence in the GA, FOX decides OPFOR's
best COA is to have an all out attack (holding no reserves) with the following force
composition (see Figure 5.2-3):
The Armored Battalion, consisting of 2 Mechanized Infantry Companies and 2
Armored Companies, leads the attack on AA3 to seize O'Neal Bridge.
The Infantry Battalion, consisting of 2 Mechanized Infantry Companies, 2 Infantry
Companies, and a single Armored Company, follows and supports the Armored
Battalion on AA3.
A Mechanized Battalion, consisting of 2 Armored Companies and a single
Mechanized Infantry Company, leads the attack on AA1 to seize McGee Heights.
The other Mechanized Battalion, consisting of 3 Mechanized Infantry Companies,
leads the attack on AA2 to seize Roth Bridge.
...............
....
........
..........
...
.......
.............
...........
.....................
.... .........
.....
..
.........
. ...........
......
.
..............
....
.......................
............
. ...........
.....
...........
7..................
<*L
...................
......................
............
...............
. ...............
................
............
1................
..., -...............
60
L
..........
F ......
............
................
............
.........................
. .............
....
....
...........
...........
......
.........
.............
......
........
I.....................
......
......
...
...................
Figure 5.2-3: eCOA without Influence
This plays out, according to FOX's wargaming engine, such that the enemy with a
fairly substantial amount of damage takes all three objectives. The BLUFOR forces
become reactionary, lose any momentum they had, and are overwhelmed. The fitness
score is 0.62 for the enemy.
This would be a fine solution to the problem, except it does not agree very much with
our intelligence about sizes and positions of the enemy units.
68
5.2.3 eCOA with Influence
The first step to using the intelligence is to create a measure of likeliness from it:
positivebelief = 0.0
negative belief = 0.0
for each battalion B
bl = lookup belief for position of B
if bl > 0 then increase(positive belief, bl)
if bl < 0 then increase (negative belief, -bl)
b2 = lookup belief for size of B
if b2 > 0 then increase(positive belief, b2)
if b2 < 0 then increase (negative belief, -b2)
next battalion
increase(double belief, double change)
{
}
belief = belief + change*(1 - belief)
The increase function is inspired by MYCIN's method for combining uncertain
quantities of belief in its rule-base. The data used in the above pseudo-code is as follows:
Battalion
AA1
AA2
AA3
Armored
0.4
0.4
-0.5
Infantry
-0.9
-0.5
0.9
Mech. Infantry 1
-0.8
0.9
-0.8
Mech. Infantry 2
0.4
0.4
-0.5
Table 5.2-1: Position Belief Data
# Companies
Battalion
2
3
4
5
Armored
0
0
0
0
Infantry
0.9
0.0
-0.5
-0.9
Mech. Infantry 1
-0.9
-0.5
0.5
0.9
Mech. Infantry 2
0
0
0
0
Table 5.2-2: Composition Belief Data
69
Using the influence from the intelligence in the GA with confidence values of 1.0 for
Crossover Skew and 0.5 for both Initial and Fitness Skews, FOX decides OPFOR's best
COA is to have an all out attack (holding no reserves) with the following force
composition (see Figure 5.2-4):
The Armored Battalion, consisting of 2 Mechanized Infantry Companies and 2
Armored Companies, leads the attack on AA1 to seize McGee Heights.
A Mechanized Battalion, consisting of a single Armored Company and 4
Mechanized Infantry Companies, leads the attack on AA2 to seize Roth Bridge.
The other Mechanized Battalion, consisting of 2 Mechanized Infantry Companies
and 2 Armored Companies, joins the first Mechanized Battalion.
The Infantry Battalion, consisting of 2 Infantry Companies, leads the attack on
AA3 to seize O'Neal Bridge.
L
Figure 5.2-4: eCOA with Influence
70
This plays out, according to FOX's wargaming engine, such that the enemy with
almost no damage takes McGee Heights, with moderate damage takes Roth Bridge, and
takes O'Neal Bridge after a significant fight and after substantial loses. BLUFOR loses
because its Reserve units are caught too far from the fight at Roth Bridge and McGee
Heights, where the Infantry forces are simply overrun by the Armored Battalion. By the
time the Reserves get there, they are too late, and are too far from then the battle for
O'Neal Bridge to turn a close fight. The score for the OPFOR is 0.61.
This COA scores slightly worse than the COA proposed by the uninfluenced search,
except it is significantly more probable since it agrees much more with our intelligence
about sizes and positions of the enemy units.
71
6. Conclusion
Several methods of influencing a GA which incorporate uncertain data as a heuristic
for searching the problem space have been presented. The three methods, Initial Skew,
Fitness Skew, and Crossover Skew, were applied to a traditional GA that uses fixedlength strings of bits to encode the problem space. All three were tested for effects on
Mutation and Crossover Rate selection. In general, the selection of Mutation and
Crossover Rates were independent of any amount of the influence measures used while
Crossover Skew most dramatically affected performance in the test cases using the NonOptimal criteria. The Non-Optimal cases are interesting since for the intended
application of this thesis, incorporation with FOX, it is most relevant to look for several
good solutions and not necessarily the global optimum.
The reasons behind the success, failure, or indifference exhibited by the various
influence combinations are still unanswered. Reasons for failure may range from Initial
Skew's propensity for promoting too much convergence too quickly to Fitness Skew's
effect of flattening out high and low ranges of fitness. Success might be due to Fitness
Skew's ability to ignore unimportant minima and maxima or maybe Crossover Skew
works because it works more like an intelligent form of Mutation than the equal
contribution of information between individuals. (Think of Crossover Skew as changing
one individual by using another individual as a reference for mutation.) Other possible
extensions to this research that may shed some light on the qualities of influence methods
include applying them to other gene representations, such as strings of integers or floats,
or using these influence techniques in combination with nicheing or hybrid search
technology. It could be very useful to keep the general idea of the Crossover Skew in
mind when developing operators to work on problem specific representations by allowing
one individual to contribute more than the other.
Incorporation with FOX was shown to be both possible and potentially useful. Due to
the additional information requirements, the integration could not be done perfectly, but
with minimal changes, FOX could use these devices to become more powerful, apparently
more intelligent, and better suited to solving COA problems with realistic solutions. This
allows commanders and training instructors a chance to better know their enemies.
72
7. References
[1]
A. A. Adewuya, "New methods in genetic search with real-valued chromosomes,"
MIT Theses Online (http://theses.mit.edu/), 1998.
[2]
H. Argo, E. J. Brennan, M. W. Collins, K. Gipson, C. Lindstrom, and S. W.
MacKinnon, "Level 1 Model for Battle Management Language (BML-1)," TEMO
Simulation Laboratory, March 23, 1999.
[3]
G. S. Bhat and C. D. Savage, "Balanced Gray Codes," North Carolina State
University, 1996.
[41
K. A. Dejong and W. M. Spears, "An Analysis of the Interacting Roles of
Population Size and Crossover in Genetic Algorithms," Proceedings of the First Int'l
Conference on Parrallel Problem Solving from Nature. October 1990. Dortmund,
Germany.
[5]
C. B. Fiebig-Brodie and C. C. Hayes, "Evaluating The utility of Decision Support
Tools to Assist in Army Tasks," Advanced Displays and Interactive Displays
Consortium Proceedings of the 4'" Annual FedLab Symposium. April 2000. College
Park, MD.
[6]
P. Giguere and D. Goldberg, "Population Sizing for Optimum Sampling with
Genetic Algorithms: A Case Study of the Onemax Problem," Proceedingsof the Third
Annual Genetic ProgrammingConference. July 1998. Madison, WI.
[7]
C. C. Hayes and J. L. Schlabach, "FOX-GA: A Planning Support Tool for Assisting
Military Planners in a Dynamic and Uncertain Environment," Proceedings of the
1998 Artifical Intelligence PlanningSymposium. June 1998. Pittsburgh, PA.
[8]
P. S. Ngan, M. L. Wong, W. Lam, K. S. Lueng, and J. C. Y. Cheng, "Medical data
mining using evolutionary computation," Artificial Intelligence in Medicine 16. 1999.
[9]
D. E. Okon, J. E. Burge, H. Ruda, and G. L. Zacharias, "FOX Installation and User
Guide," Charles River Analytics, R9835 1. September 29, 1999.
[10] M. Pelikan, D. E. Goldberg, and E. Cantu-Paz, "BOA: The Bayesian Optimization
Algorithm," Illinois Genetic Algorithms Laboratory, IlliGAL Report No. 99003.
January 1999.
[11] V. W. Porto, M. Hadrt, D. Fogel, K. Kreutz-Delgado, and L. J. Fogel, "Evolving
Tactics using Levels of Intelligence in Computer-Generated Forces," SPIE
Conference on Enabling Technology for Simulation Science III. April 1999. Orlando,
FL.
[12] G. J. Rinkus, J. E. Burge, and H. Ruda, "FOX COA Development and Planning
Tool Enhancement and Analysis," Charles River Analytics, R98371. May 27, 1999.
[13] C. D. Savage and P. Winkler, "Monotone Gray Codes and the Middle Levels
Problem," J. Comp Theory (A) 70:2. May 1995.
73
[14]
J. Schmitt,
"eTDG3: The Battle of Johnsonburg - Situa'tion Narrative," September
30, 1999.
[15]
J. Schmitt
et. al., "eTDG3: The Battle of Johnsonburg," September 14, 1999.
[16] A. Schultz, "Adapting the Evaluation Space to Improve Global Learning,"
Proceedings of the Fourth International Conference on Genetic Algorithms. July 1991.
San Diego, CA.
[17] A. C. Schultz and J. J. Grefenstette, "Improving Tactical Plans with Genetic
Algorithms," Proceedings of the IEEE Conference on Tools for Artificial Intelligence.
November 1990. Herndon, VA.
[18] W. M. Spears, "Recombination Parameters," The Handbook of Evolutionary
Computation, T. Baeck, D. Fogel and Z. Michalewicz (editors), IOP Publishing and
Oxford University Press. 1997.
[19] W. M. Spears and V. Anand, "A Study of Crossover Operators in Genetic
Programming," Proceedings of the Sixth International Symposium on Methodologies for
Intelligent Systems. 1991.
[20] W. M. Spears and K. A. Dejong, "An Analysis of Multi-Point Crossover,"
Proceedings of the Foundations of Genetic Algorithms Workshop. July, 1990.
Bloomington, Indiana.
[211 W. M. Spears and K. A. Dejong, "On the Virtues of Parameterized Uniform
Crossover," 4 th International Conference on Genetic Algorithms. July 1991. La Jolla,
CA.
[221 P.
D.
Surry,
http://www.quadstone.co.uk/rpl2/glossary/node14.html
nodel5.html, July 1995.
[23] Sun Tzu, "The Art of War." Ed.
J. Clavell;
74
and
Delacorte Press, New York, NY. 1983.
Appendix A: Data
All data shows the average measured number of generations taken to solve the
problem. A maximum number of 200 generations was used, and at least 100 trials were
run to compute the average. All Stage 1 data was run using the Optimal Test criteria,
Stage 2 data was run for both the Optimal and Non-Optimal Test criteria.
A.1. F1 Data
A.1.1. Stage 1
Initial Skew
Crossovers
1
0
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
48.62
48.83
60.99
56.49
48.96
61.45
50.29
54.67
54.50
45.22
39.57
42.08
40.12
45.49
43.42
50.22
47.25
46.36
46.14
41.02
45.68
40.73
44.56
42.76
44.63
42.86
39.40
39.27
43.90
45.62
38.23
43.74
36.12
42.95
42.08
44.19
41.96
44.63
39.07
42.68
40.91
39.16
43.69
44.72
44.96
51.06
43.63
39.05
41.67
39.92
43.62
41.30
45.52
40.49
40.19
42.48
40.12
42.67
42.00
40.31
48.34
44.24
45.32
41.48
44.01
44.86
43.23
42.82
43.02
42.84
42.62
44.84
41.34
48.45
44.89
43.38
43.53
47.55
35.67
40.35
45.75
43.03
43.09
45.98
44.71
46.37
41.86
36.58
39.99
0.1
0.2
56.52
48.79
2
48.43
49.68
3
47.69
45.90
4
43.04
43.76
5
40.30
48.00
6
45.73
43.12
7
42.37
41.76
8
9
10
50.26
47.88
44.98
44.64
38.89
38.79
47.28
Table A-1: F1 Optimal: Initial Skew vs. Expected # Crossovers
Fitness Skew
0
Crossovers
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1
56.52
55.58
55.49
54.99
50.81
49.45
59.11
55.72
61.38
49.21
50.60
2
48.43
49.02
45.55
45.98
48.47
46.59
47.95
51.47
49.66
51.31
50.08
3
47.69
40.36
44.85
47.13
47.57
47.66
37.98
42.30
51.06
47.26
44.74
4
43.04
42.21
45.54
44.00
45.31
46.94
48.45
45.36
48.75
45.41
49.03
5
40.30
41.01
49.36
41.54
46.35
44.29
44.05
47.12
42.61
44.28
49.22
4.r...7 3
35.84
42.45
43.72
42.86
34.98
39.77
45.72
49.82
48.16
53.28
7
42.37
38.97
43.58
44.19
50.79
41.43
44.27
45.62
45.41
43.36
44.73
8
9
10
50.26
43.56
44.62
49.01
41.86
43.43
43.32
41.34
45.87
41.67
47.37
44.64
44.96
45.11
44.49
46.80
44.54
45.14
45.37
48.35
44.60
48.27
38.79
42.50
39.12
39.29
45.63
38.86
43.98
45.94
51.05
50.18
44.66
.........
.. ......... ......
0
Table A-2: F1 Optimal: Fitness Skew vs. Expected # Crossovers
75
Crossovers
0
Crossover Skew
0.4
0.5
0.6
0.1
0.2
0.3
0.7
0.8
0.9
1
1
56.52
55.56
55.94
45.66
54.00
52.57
49.77
55.61
49.70
46.47
55.19
2
48.43
49.88
45.49
44.21
48.33
47.75
50.87
42.63
50.70
44.98
48.60
3
47.69
42.53
42.59
47.33
43.62
42.67
42.12
44.31
45.30
43.04
40.01
4
43.04
44.65
44.31
41.59
42.63
39.81
44.11
51.98
40.79
40.70
47.30
5
40.30
42.28
42.51
41.89
41.47
38.94
40.44
38.98
42.24
42.29
40.55
6
7
45.73
42.19
40.19
38.55
35.55
40.42
34.12
39.36
36.09
42.22
43.58
42.37
46.50
39.88
41.74
39.39
47.15
40.22
36.21
42.14
30.55
44.53
8
50.26
41.82
41.36
38.77
43.87
40.13
40.66
38.07
38.10
36.65
38.54
9
44.64
38.21
40.12
45.67
37.94
42.97
39.40
43.71
40.10
44.64
44.51
10
38.79
41.38
37.92
46.09
35.43
43.10
40.39
37.43
45.70
43.26
42.32
Table A-3: F1 Optimal: Crossover Skew vs. Expected # Crossovers
Initial Skew
0
0.1
0.2
0
94.66
81.29
82.25
160.57
50.76
101.37
0.005
34.96
42.13
35.62
42.72
46.38
51.26
0.01
0.015
0.02
0.025
0.03
0.035
41.03
44.36
41.84
39.54
42.61
47.69
45.90
46.14
41.02
45.68
46.89
48.41
47.95
43.27
53.30
62.26
64.61
004
72.60
65.78
68.20
0.045
0.05
73.47
75.57
63.94
88.09
89.93
91.04
Mutation
0.3
0.4
0.5
0.6
0.7
0.8
0.9
107.21
105.68
106.19
48.66
47.19
36.56
42.29
41.35
40.04
43.46
40.95
39.64
42.59
41.13
39.39
40.73
44.56
42.76
44.63
42.86
39.40
43.53
4542
40.71
44.15
44.64
43.83
43.86
41.93
53.85
43.83
47.26
51.58
48.43
47.39
51.10
-8.40
1
50.95
50.54
45.22
46.05
49.22
49.01
48.73
54.59
57.72
55.96
56.81
55.92
55.73
56.05
53.44
59.98
54.22
51.68
52.22
55.99
56.73
56.95
59.34
61.03
57.30
65.26
64.06
65.39
60.94
69.81
79.33
68.29
63.74
71.68
69.95
81.07
65.72
60.86
70.3
78.05
84.13
7823
7832
79.51
8193
8902
Table A-4: F1 Optimal: Initial Skew vs. Mutation Rate
76
Mutation
0
0.1
0.2
0.3
0.4
Fitness Skew
0.5
0.6
0.7
0.8
0.9
1
0
0.005
94.66
97.70
112.29
71.43
91.45
68.89
104.24
84.72
87.25
83.05
96.95
34.96
39.40
48.82
44.81
50.68
49.14
38.25
46.23
47.97
44 73
48.35
0.01
0.015
0.02
0.025
0.03
0.035
41.03
42.01
41.96
47.19
44.88
42.17
51.36
41.92
42.81
41.95
35.37
47.69
40.36
44.85
47 13
47.57
47.66
37.98
42.30
51.06
47.26
44.74
46.89
46 45
47 48
52.96
49.74
48.08
46.01
44.45
46.70
50.50
49.73
50.95
48.59
52.12
55.38
54.38
47.97
52.11
54.99
53.47
56.76
55.70
48.73
48.23
53.05
53.76
61.45
58.14
61.29
54.08
55.75
58.49
61.49
53.30
59.75
59.71
49.75
64.93
62.34
69.33
59.62
61.80
59.43
67.83
0.04
0.045
0.05
72.60
58.91
67.50
59.46
66.04
70.71
63.08
63.34
59.74
69.07
73.41
73.47
80.87
78.06
63.73
68.27
82.47
58.68
68.92
76.25
80.35
83.24
88 09
81.06
83.42
74.6
69.98
81.2
64.98
84.75
68.37
73.1
79.95
Table A-5: F1 Optimal: Fitness Skew vs. Mutation Rate
Mutation
0
0.005
0.01
0.015
0
94.66
0.1
0.2
0.3
Crossover Skew
0.4
0.5
0.6
0.7
0.8
0.9
1
146.84
147.99
118.06
133.96
160.73
36.76
43.71
47.88
40.41
42.27
40.21
41.85
35.99
41.29
43.98
41.24
37.17
42.67
42.12
44.31
45.30
43.04
40.01
44.43
45.59
38.02
41.61
47.70
43.07
41.41
40.90
46.97
47.84
44.93
51.75
46.01
4663
48.57
100.77
89.11
100.14
147.29
34.96
38.59
34.31
44.67
39.18
41.03
41.30
42.01
42.63
39.36
47.69
42.53
42.59
47.33
43.62
0.02
46.89
42.69
41.72
42.34
0.025
0.03
0.035
0.04
0.045
0.05
50.95
48.59
44.43
47.22
124.21
48.73
53.53
52.99
55.02
47.83
51.70
48.82
52.14
43.36
46.29
53.30
57.72
60.59
45.40
49.32
54.00
43.80
51.55
52.04
52.32
54.19
72.60
64.51
64.30
54.98
67.77
51.81
64.23
54.16
56.08
60.96
63.72
73.47
69.42
72.35
65.75
79.37
53.08
57.92
64.10
72.23
67.92
64.18
88.09
77.99
62.45
75.56
74.03
74.91
81.52
70.46
87.18
74.8
80.63
Table A-6: F1 Optimal: Crossover Skew vs. Mutation Rate
77
A.1.2. Stage 2
Initial Skew
0.4
0.5
0.6
0
0.1
0.2
0.3
0.7
0.8
0.9
1
Skew
..................................................................
t ..................................
..................................
.............................................................
....................................
38.51
38.85
41.56
38.93
39.49
35.83
35.87
37.56
40.37
37.44
40.87
0
40.4
37.94
39.23
39.12
41.94
39.16
38.64
40.39
38.15
0.1
39.52
40.32
40.77
39.08
40.81
35.56
38.65
3
9.21
38.34
42.24
40.47
41.32
0.2
38.09
39.14
39.12
38.37
40.81
37.05
36.67
43.79
40.86
41.71
0.3
35.81
35.21
42.89
40.59
41.11
46.65
39.56
4 0.78
39.32
37.56
42.32
36.54
0.4
36.95
38.49
41.51
36.92
4261
45.26
4
1.85
40.79
39.06
39.02
37.34
40.03
0.5
43.74
37.73
36.64
40.84
40.95
42.42
43.05
34.9
40.12
35.31
34.77
0.6
42.59
41.44
39.05
40.92
41.62
38.18
38.72
38.7
39.39
38.76
0.7
40.13
39.24
38.58
42.58
40.17
42.81
42.18
41.63
34.54
38.04
36.09
41.75
0.8
41.91
42.17
41.8
39.68
45.36
39.86
43.71
41.98
41.41
0.9
40.56
38.52
44.84
41.17
41.69
46.01
42.41
4
3.62
45.28
47.19
39.71
44.05
41.37
1.0
Fitness
Table A-7: F1 Optimal: Initial Skew vs. Fitness Skew
Crossover
Skew
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Initial Skew
0.5
0.6
0
0.1
0.2
0.3
0.4
38.51
38.85
41.56
38.93
39.49
35.83
43.59
39.26
38.14
36.48
35.99
41.55
37.37
39.63
35.75
43.24
37.65
35.73
38.26
36.32
36.6
34.77
36.62
41.27
0.7
0.8
0.9
1
35.87
37.56
40.37
37.44
40.87
38.93
39.11
37.5
35.57
37.09
39.12
38.84
36.42
36.64
36.86
35.12
38.41
39.03
33.78
36.02
37.08
40.08
38.6
40.28
40.68
41.24
38.55
37.84
33.01
31.55
36.85
37.86
40.63
36.95
37.69
36.72
39.81
34.42
36.42
40.92
37.55
33.21
36.65
41.33
42.38
41.18
37.12
38.12
34.56
42.7
39.44
41.09
35.27
39.85
39.37
38.41
37.54
31.22
40.77
41.51
39.3
37.84
42.1
33.72
38.02
37.37
37.99
36.9
43.07
37.79
42.78
34.53
40.33
35.56
38.08
38.33
38.32
36.91
38.81
39.39
38.46
39.06
35.56
38.97
37.81
36.94
40.15
43.91
41.62
39.73
45.08
36.96
38.7
43.68
42.99
38.67
32.78
Table A-8: F1 Optimal: Initial Skew vs. Crossover Skew
78
Fitness Skew
Crossover
Skew
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
38.51
40.4
40.77
39.14
42.89
38.49
43.74
42.59
39.24
41.91
44.84
43.59
38.77
41.17
39.26
38.95
35.6
42.35
37.95
42.58
41.56
43.1
37.37
36.98
41.3
37.89
37.47
41.05
38.38
40.52
39.05
41.05
44.71
38.26
43.24
36.94
43.05
36.25
40.11
41.86
39.04
41.26
42.65
41.54
37.08
40.67
41.14
41.23
38.76
37.39
39.17
45.02
35.49
44.08
49.21
37.86
37.15
37.31
44.62
38.97
44.45
38.07
37.96
40.23
38.71
40.05
36.65
44.7
38.35
45.14
35.95
42.46
39.46
39.99
37.48
41.38
41.19
39.85
39.7
37.01
39.46
39.79
37.43
36.84
40.57
44.8
36.71
41.42
38.02
40.07
40.31
40.66
38.8
35.75
38.02
41.71
47.08
39.8
41.83
38.33
46.97
40.81
42.63
42.4
44.99
43.25
38.92
40.08
39.81
38.02
40.15
37.63
39.16
43.87
41.03
38.67
44.86
44.03
42.35
42.36
40.34
Table A-9: F1 Optimal: Fitness Skew vs. Crossover Skew
0
Fitness
Skew
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0.1
0.2
0.3
0.4
Initial Skew
0.5
0.6
0.7
0.8
0.9
1
7.18
9.25
8.86
9.12
7.87
7.66
10.16
9.28
7.43
5.83
7.56
9.28
7.99
8.78
8.02
8.31
8.68
6.5
7.16
7.52
9.02
6.38
7.8
5.54
10.43
8 99
9.29
9.91
6.23
6.33
8.38
10.27
5.36
7.73
8.71
8.74
9.51
6
10.2
7.55
6
9.8
7.69
6.08
7.66
8.6
7.59
11.22
8.17
6.47
9.76
8.21
8.84
8.42
5.7
7.27
6.79
8
9.92
8.11
9.69
9.45
6.76
6.94
6.83
7.28
8.34
7.68
7.2
10.38
8.27
7.17
9.42
9.33
7.85
6.11
8.55
9.88
9.71
9.36
10.16
7.16
8.36
9.84
9.27
6.65
7.63
7.36
9.34
8.49
7.67
8.42
9.99
7.48
8.42
6.29
8.27
6.84
8.62
9.98
8.25
6.46
9.19
9.87
9
10.2
7.46
8.96
9.83
6.51
9.28
9.34
8.8
8.38
8.65
10.42
9.52
7.33
9.56
7.89
9.15
Table A-10: F1 Non-Optimal: Initial Skew vs. Fitness Skew
79
Crossover
Skew
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0
0.1
0.2
0.3
Initial Skew
0.5
0.6
0.4
0.7
0.9
0.8
7.43
5.83
6.53
6
6.44
6.44
7.52
7.11
8.19
7.4
7.03
7.68
3.27
9.32
5.82
6.62
7.18
7.42
8.53
6.5
5.9
6.04
6.62
6.46
8.29
5.68
6.26
4.78
4.54
8.09
8.72
4.73
3.87
3.69
7
8.71
5.43
4
7.46
7.16
4.49
5.36
7.38
5.55
6.38
6.27
6.84
6.26
5.42
7.08
5.15
5.54
8.68
7.77
6.88
8.21
6.35
9.44
4.98
9.66
7.84
8.8
8.27
8.38
6
7.64
3.57
7.56
7.18
9.25
8.86
9.12
7.87
7.66
10.16
9.28
9.34
7.75
6.38
9.4
6.36
9.9
7.64
8.26
7.24
8.6
6.02
7.87
7.6
5.83
6.58
8.22
7.35
8.02
8.54
4.9
8.77
6.12
6.57
7.63
8.39
6.82
2.62
4.34
5.38
7
7.18
2
7.17
8.86
8.92
7.28
6.07
2
7.73
4.93
4.87
6.86
8.2
6.92
7.56
Table A-11: F1 Non-Optimal: Initial Skew vs. Crossover Skew
Fitness Skew
Crossover
Skew
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
7.18
9.28
7.8
7.73
7.66
7.27
8.34
9.88
9.34
9.98
9.28
9.34
6.5
5.56
9.45
7.27
6.4
7.76
9.8
9.52
8.73
9.57
8.26
5.32
8.71
9.58
6.72
7.01
3.68
8.74
9.87
9.82
9.2
6.58
4.58
5.98
3.72
7.23
9.05
10.6
6
8.39
9.12
9.53
8.77
7.79
5.6
5.7
11.36
6.59
9.34
9.56
9.26
8.4
6.56
6.82
8.5
7.85
8.54
7.89
9.56
7.56
8.65
9.34
9.82
9.56
7.18
7.31
8.41
8.23
8.18
9.16
2
8.1
8.98
9.12
9.69
8.92
9.02
4.16
5.95
8.6
9.28
3.14
6.86
6.88
9.1
8.15
2
6.64
6.29
9.81
8.29
7.94
2.41
4.56
7.5
9.42
7
4.87
5.86
7.46
11.06
7.94
5.59
2
9.87
7.16
6.22
5.44
8.2
5.56
5
10.72
5.43
8
5.58
8.6
9.79
12
3.33
Table A-12: F1 Non-Optimal: Fitness Skew vs. Crossover Skew
80
A.2. F2 Data
A.2.1. Stage 1
Crossovers
Initial Skew
0.4
0.5
0.6
0
0.1
0.2
0.3
0.7
0.8
0.9
1
164.30
175.33
138.18
127.79
188.12
186.80
128.32
172.77
158.28
145.57
157.59
2
156.96
137.45
147.66
118.15
141.54
175.77
156.65
147.29
176.82
165.14
173.70
1
3
122.64
141.29
125.02
158.98
141.48
126.18
131.07
142.95
159.00
143.04
153.92
4
123.01
132.92
136.65
151.35
127.27
124.50
139.58
148.30
148.89
11475
61.10
5
160.80
107.15
126.89
148.74
147.32
11873
107.08
149.35
149.92
141.87
136.81
6
143.02
145.54
131.49
110.45
152.79
108.01
145.58
161.99
130.73
119.24
112.39
7
119.90
118.88
10693
137.95
150.17
120.79
138.27
112.91
121.88
139.03
11827
8
130.12
117.76
152.92
150.69
150.93
120.68
97.47
112.64
124.56
112.92
148.81
9
138.55
159.18
158.20
86.94
119.30
155.94
102.89
113.33
125.69
93.55
131.92
10
108.36
92.73
138.45
150.86
124.19
126.04
145.04
132.69
132.13
143.57
127.76
Table A-13: F2 Optimal: Initial Skew vs. Expected # Crossovers
Fitness Skew
0.4
0.5
0.6
Crossovers
1
0
0.1
0.2
0.3
0.7
0.8
0.9
1
164.30
18315
152.40
173.18
142.89
173.79
164.85
198.98
187.82
183.12
17755
2
156.96
148.63
123.76
171.14
179.58
171.82
157.32
152.64
185.94
183.84
193.10
3
I
122.64
115.17
138.37
130.32
148.04
136.17
137.90
177.85
141.33
138.97
180.63
4
123.01
134.39
107.91
148.48
151.77
130.15
181.16
121.64
163.08
150.14
192.94
5
160.80
146.90
141.64
161.64
151.76
155.20
115.64
151.15
145.73
162.17
184.03
6
143.02
160.21
136.00
87.82
138.01
151.27
147.85
103.04
104.62
123.39
176.38
119.90
118.74
145.01
142.03
107.25
116.30
151.81
144.32
119.14
136.37
193.27
130.12
143.51
106.67
129.16
142.79
135.68
14180
133.73
85.13
116.23
174.86
9
138.55
127.84
100.91
111.48
115.30
115.56
147.54
155.47
112.79
133.67
149.34
10
108.36
118.86
138.36
109.18
101.61
140.68
126.45
108.82
109.24
120.17
195.76
7
8
j
Table A-14: F2 Optimal: Fitness Skew vs. Expected # Crossovers
81
Crossovers
Crossover Skew
0.4
0.5
0.6
0
0.1
0.2
0.3
0.7
0.8
0.9
1
164.30
165.54
172.62
165.33
185.25
192.09
174.03
179.01
173.55
15657
178.42
2
156.96
172.31
168.33
153.93
142.90
173.84
172.90
167.45
163.27
156.77
172.97
1
3
122.64
138.12
147.78
126.27
109.54
118.28
142.17
134.74
151.50
158.45
165.70
4
123.01
164.78
127.97
135.59
143.51
119.89
150.31
135.73
141.32
120.47
97.43
5
160.80
112.94
115.27
137.24
120.42
129.49
135.45
142.67
108.68
110.74
113.13
6
143.02
118.38
116.87
116.91
131.95
132.19
94.11
82.92
87.90
149.59
119.95
7
119.90
131.60
125.12
102.27
125.39
143.68
108.44
118.35
150.27
91.48
90.56
8
130.12
121.75
153.80
104.62
119.53
118.99
137.48
126.24
163.46
133.02
90.60
9
138.55
105.38
110.83
136.19
103.35
128.61
138.98
96.14
118.31
121.39
117.16
10
108.36
152.09
149.53
12269
12741
11284
102.02
15623
104.40
104.03
83.91
Table A-15: F2 Optimal: Crossover Skew vs. Expected # Crossovers
Initial Skew
Mutation
0
0.005
0.01
0.015
0.()2
0.025
0.03
0.035
0.04
0.045
0.05
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
187.94
159.18
199.00
147.45
155.80
199.00
199.00
183.20
199.00
172.75
184.04
167.69
124.63
165.14
122.32
164.75
161.15
158.44
179.54
186.56
172.77
173.65
125.79
172.18
128.87
135.07
168.95
173.88
146.09
121.18
59.29
144.99
129.64
122.64
141.29
125.02
158.98
141.48
126.18
131.07
142.95
159.00
143.04
153.92
142.19
159.62
157.01
140.39
79.95
180.95
136.50
119.62
153.96
134.32
140.78
130.51
170.68
170.16
168.05
159.42
135.84
164.44
126.15
125.38
144.99
151.81
148.69
148.02
145.81
155.81
146.77
121.61
138.26
146.84
157.27
147.02
128.86
140.59
154.47
166.72
158.55
152.36
171.71
174.91
154.57
100.20
163.33
126.35
169.42
163.77
149.61
182.69
165.14
165.37
103.96
137.23
146.20
141.98
148.94
172.53
173.10
168.43
18266
174.18
174.83
160.51
10.94
16396
142.97
167.39
187.02
186.27
146.47
133.29
177.63
166.64
161.72
161.29
149.49
173.51
167.99
Table A-16: F2 Optimal: Initial Skew vs. Mutation Rate
82
1
Fitness Skew
Mutation
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
0.05
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
187.94
199.00
181.51
199.00
147.29
170.10
155.93
160.54
10876
17335
17425
167.69
122.66
187.72
189.22
139.72
183.70
148.96
169.21
168.57
179.70
199.00
1
125.79
157.14
163.11
158.76
179.17
146.24
160.10
160.00
184.02
170.39
182.80
122.64
115.17
138.37
130.32
148.04
136.17
137.90
177.85
141.33
138.97
180.63
142.19
130.59
158.14
164.62
158.57
184.90
171.12
108.11
156.01
175.78
99.12
130.51
142.63
127.66
142.14
156.96
117.31
142.42
124.94
151.27
168.42
184.69
148.69
123.93
158.74
147.86
135.99
182.96
126.16
162.57
16313
174.53
179.92
140.59
168.37
164.27
174.84
103.07
179.25
152.77
170.84
126.78
165.53
186.40
j169.42
157.49
144.07
181.38
143.56
172.81
134.50
134.73
176.70
166.71
192.50
172.53
144.43
180.56
150.60
176.43
154.28
176.83
178.08
183.66
167.19
198.41
187.02
151.64
180.36
174.71
140.67
166.47
160.36
179.38
180.95
191.69
182.82
Table A-17: F2 Optimal: Fitness Skew vs. Mutation Rate
Mutation
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
0.05
0
187.94
Crossover Skew
0.4
0.5
0.6
0.1
0.2
0.3
125.29
146.60
165.94
149.59
101.79
199.00
0.7
0.8
0.9
144.10
199.00
199.00
199.00
1
167.69
177.28
166.33
157.34
148.90
163.59
196.88
198.10
184.49
164.09
174.99
125.79
169.88
182.16
116.40
159.02
99.55
136.70
166.07
117.94
131.08
168.92
122.64
138.12
147.78
126.27
109.54
118.28
142.17
134.74
151.50
158.45
165.70
142.19
141.28
136.61
127.86
104.71
125.51
146.37
162.13
138.19
120.61
131.49
130.51
172.76
118.32
164.94
152.17
139.80
152.68
166.81
158.14
113.91
148.94
148.69
170.00
115.51
156.90
140.54
146.99
121.41
158.74
133.35
143.62
146.92
140.59
173.85
138.62
134.61
148.13
152.87
80.79
145.97
151.15
137.10
146.60
169.42
148.87
185.22
158.09
168.84
132.74
171.57
121.69
145.92
136.86
140.13
172.53
174.21
169.35
160.54
154.29
115.54
148.81
131.55
145.51
143.70
140.32
187.02
191.85
129.5
173.46
180.16
14865
180.41
161.59
165.38
118.91
155.85
Table A-18: F2 Optimal: Crossover Skew vs. Mutation Rate
83
A.2.2. Stage 2
Fitness
Skew
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0
Initial Skew
0.4
0.5
0.6
0.1
0.2
153.92
148.7
139.21
139.89
131.45
122.74
110.76
126.11
134.92
152.76
128.21
147.09
141.67
132.07
124.43
143.03
142.01
150.59
155.36
122.98
130.55
141
140.16
142.03
142.48
129.1
126.76
139.6
123.7
136.36
105.08
163.29
148.65
134.19
130.12
133.24
150.28
145.19
135.18
153.42
134.86
135.81
148.33
141.64
150.29
136.96
147.52
131.24
159.68
123.29
142.43
141.42
167.59
139.64
146.47
149.97
142.49
0.3
0.7
0.8
0.9
1
129.32
165.83
126.96
126.65
160.45
140.45
146.96
144.24
143.52
122.74
144.24
133.39
118.5
148.3
118.12
119.28
125.93
111.76
142.55
131.08
144.1
112.26
161.01
153.96
129.31
125.07
104.35
142.69
154.2
139.59
158.24
136.57
157.99
140.92
139.51
129.07
155.66
122.55
146.12
154.63
147.83
133.33
169.55
139.43
135.31
104.31
130.72
127.82
139.53
151.29
130.66
153.92
115.7
179.36
194.96
188.79
169.1
180.87
179.18
170.81
167.03
173.37
191.53
139.26
1
Table A-19: F2 Optimal: Initial Skew vs. Fitness Skew
Crossover
Skew
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0
0.1
Initial Skew
0.5
0.6
0.2
0.3
0.4
139.89
131.45
122.74
0.7
0.8
0.9
129.07
129.32
165.83
126.96
126.65
153.92
148.7
139.21
133.25
136.8
121.84
150.49
108.51
155.38
138.07
142.55
123.29
131.85
141.55
113.99
145.56
110.1
146.65
136.49
156.09
132.35
123.82
121.88
96.27
130.14
128.27
142.98
149.06
149.27
118.09
108.24
148.73
131.77
134.02
151.8
135.3
121.48
142.03
137.97
146.12
126.71
122.4
134.03
135.65
120.1
147.67
126.35
132.02
127.73
106.6
138.49
137.87
145.62
144.6
110.12
132.68
133.02
124.9
142
145.44
135.44
128.7
112.07
107.5
128.9
105.31
129.59
117.32
108.55
125.51
143.2
134.32
145.1
138.26
136.52
116.2
94.82
110.05
123.25
128.18
108.8
125
118.08
138.08
145.89
105.77
14194
141.15
106.98
127.6
145.46
117.41
142.83
133.45
128.13
145.52
140.99
120.59
135.76
136.45
136.94
122.35
135.63
132.76
146.22
148.91
142.96
116.15
136.29
115.93
149.87
139.69
.21
Table A-20: F2 Optimal: Initial Skew vs. Crossover Skew
84
Crossover
Skew
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0
0.1
0.2
0.3
Fitness Skew
0.4
0.5
0.6
0.7
0.8
0.9
1
153.92
110.76
124.43
140.16
105.08
135.18
136.96
141.42
155.66
169.55
179.36
133.25
142.16
98.07
145.58
150.1
145.81
138.77
148.63
145.65
166.73
169.08
113.99
122.14
133.14
127.51
137.6
127.81
160.94
138.22
116.28
148.22
187.64
128.27
135.3
117.6
156.97
121.06
163.43
125.71
140.32
126.6
129.9
184.76
121.48
132.33
153.33
136.87
138.39
145.08
105.89
112.97
124.47
124.88
175.07
132.02
117.38
131.86
145.29
116.7
142.69
141.38
150.39
113.77
123.46
166.1
85.21
122.05
119.03
128.62
151.19
153.33
151.35
120.36
127.37
132.61
170.07
108.55
134.93
140.18
125.03
150.85
116.25
153.37
143.99
145.69
134.06
167.52
128.18
101.57
139.84
133.87
147.5
135.37
138.03
115.79
158
116.37
194.53
145.46
152.96
133.34
143.07
143.67
132.91
139.46
156.7
119.07
165.6
188.28
122.35
132.62
85.12
123.73
148.19
137.88
140.65
112.71
153.51
149.87
186.45
Table A-21: F2 Optimal: Fitness Skew vs. Crossover Skew
Fitness
Skew
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Initial Skew
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
................................
...............................
..................................
................
..........
..........................
1............................
.......................
;,
0
0.1
23.51
16.17
17.49
16.7
15 91
16.15
0.2
11.93
19.62
14.07
15.25
12.89
17.01
17.57
17.49
16.57
16.31
15.39
16.81
15.25
22.87
14.98
17.22
14.3
11.85
16.58
19.05
19.12
13.24
18.21
18.45
19.79
13.14
16.42
19.88
14.76
16.14
21.42
16.25
21.65
14.78
15.72
9.95
16.54
12.8
13.65
17.2
17.32
16.68
15.01
16.14
19.63
12.33
15.54
16.1
14.21
16.45
19.24
19.36
16.73
15.83
13.83
14.27
14.64
17.65
13.59
17.91
15.77
18.38
18.23
18.42
1547
12.02
12.62
16.97
15.26
16.03
12.16
16.59
20.67
16.74
20.49
15.49
16.77
17.45
15.32
18.7
13.21
14.41
19.65
22 12
11.78
14.59
21.54
14.98
20.82
17.71
16.09
21.7
13.32
15.74
11.68
25.24
16.03
20.32
17.46
13.21
16.34
14.58
17.28
10.87
22.37
27.62
21.05
21.4
15.05
14.8
16.57
17.56
21.61
13.62
12.09
Table A-22: F2 Non-Optimal: Initial Skew vs. Fitness Skew
85
Initial Skew
Crossover
Skew
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0
0.1
0.2
0.3
0.4
0.5
0.6
23.51
16.17
11.93
19.62
14.07
17.49
16.57
22.21
16.75
22.71
16.9
18.8
11.33
16.27
12.66
15.12
16.76
14.5
19.42
9.26
13.22
18.57
13.3
11.9
13.07
19.31
1017
9.1
12.11
14.26
12.81
22
11.74
11.65
17.25
12.66
19.46
17.85
13.29
16.6
14.9
8.86
11.94
10.34
12.34
18.46
13.78
9.53
14.16
3
9.73
20.45
15.4
10.12
6.85
10.06
12.2
15.14
1141
9.94
16
11.53
0.9
12.44
18.03
15.2
14.25
1.0
21.86
16.82
15.32
10.84
0.7
16.31
0.8
0.9
15.39
1
16.81
15.25
12.99
13.66
11.69
13.88
12.47
13.82
13.31
8.46
1084
15.93
12.59
14.83
9.56
12.77
8.86
9.42
17.07
14.22
12.04
17.26
16.11
10.76
12.66
7.96
8.1
1412
12.62
8.8
5.
1399
12.23
13.67
11.3
7.52
11.56
14.07
9.6
11.28
13.27
12.43
11.81
15.07
19.86
4.56
13.04
8.12
9.72
Table A-23: F2 Non-Optimal: Initial Skew vs. Crossover Skew
Fitness Skew
Crossover
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
23.51
17.49
15.91
19 88
13.65
16.45
15.77
16.59
19.65
15 74
22.37
Skew
0
0.1
0.2
22.21
17.83
18.16
22.78
12.52
14.09
16.12
16.52
18.33
17-11
24.71
1 15.12
17.64
14.84
16.27
1289
17.82
15
15.28
16.21
16.27
22.71
0.3
0.4
0.5
0.6
0.7
11.9
15.33
17.15
10.66
16.3
13.31
15.49
8.3
10.17
12.54
24.68
22
12.44
21.76
13.62
8.57
12.39
11.53
9.6
8.52
8.51
10.9
13.29
5.15
8.56
12.27
9.66
15.19
11.46
13.08
12
16.98
7.39
18.46
12.82
15.92
14.81
13.63
14.16
18.36
22.1
5.6
12.46
13.91
20.45
14.27
12.55
7.07
4.49
19.98
11.81
7.91
8.68
10.4
22.88
0.8
15.14
2.54
10.85
10.4
13.49
16.14
8.77
10.09
9.24
10.44
18.23
0.9
12.44
7.58
17.93
18.16
15.28
23.82
10.62
22.38
7.98
8.86
27.88
1.0
21.86
13.69
22.88
12.36
13.69
19.7
9.4
21
3.3
11.39
8.31
Table A -24: F2 Non-Optimal: Fitness Skew vs. Crossover Skew
86
A.3. F3 Data
A.3.1. Stage I
0
0.1
Initial Skew
0.4
0.5
0.6
0.2
0.3
0.7
0.8
0.9
1
1
45.44
45.69
47.49
53.43
49.57
45.54
37.98
54.58
36.28
40.93
43.20
2
4040
37.26
30.36
34.14
32.83
28.28
35.70
38.58
32.17
32.92
26.78
3
29.84
31.70
32.02
32.93
35.91
28.50
22.84
32.31
31.24
27.24
26.31
4
34.17
26.71
29.29
24.27
26.28
35.25
38.39
29.82
31.32
28.22
26.53
5
26.75
27.58
23.68
32.60
27.40
33.81
27.51
30.47
24.46
29.77
23.20
6
22.30
24.38
26.99
28.05
23.47
30.93
31.87
30.87
26.20
28.39
21.25
7
28.76
25.64
29.04
36.54
30.07
29.14
29.10
33.36
32.75
25.11
23.37
8
30.29
30.84
30.02
28.92
31.14
27.74
24.27
22.40
26.25
28.07
30.64
9
30.27
28.17
25.57
26.06
33.89
27.97
26.30
22.68
30.88
22.85
19.31
10
24.08
29.17
34.32
28.39
27.73
27.86
28.17
30.00
26.09
30.00
23.53
Crossovers
Table A-25: F3 Optimal: Initial Skew vs. Expected # Crossovers
Fitness Skew
Crossovers
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
47.35
43.05
53.95
45.93
46.19
50.22
59 78
1
45.44
40.55
39.28
46.06
2
40.40
34.87
38.82
37.98
39.72
498
8
37.06
49.29
44.39
40.94
49.59
3
29.84
30.11
31.03
33.90
34.17
36.04
35.68
41.08
42.02
40.05
48.23
4
34.17
25.26
27.89
37.56
32.82
38.68
37.63
38.85
35.72
41.07
45.00
5
26.75
20.73
35.01
29.72
32.81
30.66
33.30
34.70
38.80
36.23
46.97
6
22.30
24.62
29.02
30.92
36.92
34.18
35.36
40.43
36.34
41.68
44.24
7
28.76
28.99
29.25
32.85
32.62
31.85
33.40
34.45
37.64
39.48
48.28
8
30.29
27.08
32.03
32.24
28.26
33.40
34.12
38.98
40 19
36.54
4119
9
30.27
26.99
27.04
35.44
29.68
34.98
34.54
38.16
36.26
43.89
46.22
10
24.08
27.01
35.09
32.07
33.18
38.33
40.12
34.77
38.82
38.44
42.07
Table A-26: F3 Optimal: Fitness Skew vs. Expected # Crossovers
87
Crossovers
0
Crossover Skew
0.4
0.5
0.6
0.1
0.2
0.3
0.7
0.8
0.9
1
1
45.44
43.43
53.70
49.90
49.69
50.14
46.62
56.62
52.84
51.33
45.07
2
3
4
5
6
7
40.40
38.90
45.22
41.62
34.68
41.66
48.87
48.48
41.50
40.21
47.93
29.84
31.35
29.00
27.95
34.15
35.52
38.30
35.12
39.75
36.06
37.81
34.17
32.53
29.60
35.35
28.24
37.11
38.98
33.55
42.34
38.42
36.33
26.75
22.97
37.37
27.63
37.03
35.79
30.16
33.63
39.04
39.22
39.16
22.30
24.61
29.76
3282
29.13
31.97
35.90
31.49
36.92
41.47
40.16
28.76
26.00
36.84
33.65
34.07
35.20
34.07
40.62
35.19
38.65
38.69
8
30.29
27.77
31.42
31.25
31.97
34.46
32.62
33.29
32.31
34.65
38.82
9
| 30.27
30.75
25.71
30.48
34.30
38.26
35.37
31.11
37.88
35.48
36.67
24.08
30.60
26.84
31.60
33.81
31.70
38.20
35.18
31.83
32.31
43.28
10
Table A-27: F3 Optimal: Crossover Skew vs. Expected # Crossovers
Mutation
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
0.05
0
Initial Skew
0.4
0.5
0.6
0.1
0.2
0.3
0.7
0.8
0.9
1
70.56
54.60
51.93
79.28
65.77
84.25
88.15
34.64
54.90
34.93
39.63
30.52
36.73
37.64
34.34
34.74
24.38
32.93
26.31
31.23
33.73
20.68
26.32
33.83
34.05
33.55
34.21
35.60
39.37
24.26
37.19
27.61
32.01
29.84
31.70
32.02
32.93
35.91
28.50
22.84
32.31
31.24
27.24
2631
32.76
29.79
32.56
34.34
35.92
30.53
34.49
26.73
28.96
26.41
21.62
36.86
32.88
36.63
30.96
37.90
37.65
38.75
32.53
34.58
27.50
31.61
37.86
35.43
35.20
33.88
35.95
38.28
34.03
34.86
32.39
34.05
35.65
42.07
44.64
36.14
41.78
36.88
38.27
43.40
36.13
42.13
37.01
33.91
48.46
46.43
51.37
47.86
48.33
45.38
4792
37.23
49.29
46.43
49.51
50.89
52.47
48.67
53.33
52.61
53.06
39.39
41.15
53.56
53.17
53.96
56.67
53.24
59.8
54.92
53.35
56.95
53.58
55.69
58.69
56.83
49.8
Table A-28: F3 Optimal: Initial Skew vs. Mutation Rate
88
Fitness Skew
Mutation
0
0.005
0.1
0.2
0.3
0.4
0.5
0.8
0.9
1
70.56
0
73.82
73.02
83.48
104.76
100.92
59.72
110.62
109.46
6175
78.50
30.52
33.64
32.26
35.36
32.91
35.48
36.53
36.95
36.65
35.15
52.47
0.01
26.32
28.17
33.85
35.11
36.41
35.35
38.25
37.68
42.15
41.27
44.38
0.015
0.02
0.025
0.03
0.035
0.04
0.045
0.05
29.84
30.11
31.03
33.90
34.17
36.04
35.68
41.08
42.02
40.05
48.23
32.76
28.88
36.32
31.74
33.99
35.87
44.58
43.68
39.29
43.47
49.64
36.86
32.74
34.12
38.48
37.53
44.37
34.43
46.49
45.97
43.06
54.24
37.86
31.28
3591
36.22
39.78
40.27
38.68
49.11
50.61
42.89
51.34
42.07
44.00
41.29
43.22
45.42
50.65
45.44
48.85
42.85
49.79
58.95
48.46
45.29
42.91
46.22
52.53
47.38
57.06
48.12
50.92
57.85
62.98
50.89
44.22
46.27
49.27
43.86
56.62
51.45
62.86
54.96
58.24
66.25
56.67
45.37
55.07
56.1
59.8
53.47
55.12
60.67
69.19
65.7
65.81
0.6
0.7
Table A-29: F3 Optimal: Fitness Skew vs. Mutation Rate
Crossover Skew
Mutation
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
0.05
0
0.1
0.2
0.3
70.56
64.83
90.90
97.64
109.69
30.52
46.35
30.15
42.59
53.75
60.62
26.32
31.52
31.72
31.30
42.13
44.98
29.84
31.35
29.00
27.95
34.15
35.52
38.30
32.76
30.82
40.13
34.88
29.33
32.21
44.25
36.86
37.46
39.25
38.43
46.49
43.21
45.22
37.86
30.43
38.74
31.30
37.54
36.03
42.07
40.46
32.59
43.39
36.71
41.24
48.46
35.74
44.43
44.78
45.82
51.83
50.89
45.70
40.60
47.90
60.43
56.67
58.82
58.93
42.79
66.1
0.4
0.5
0.6
0.7
0.8
0.9
1
180.12
144.22
160.36
136.70
166.24
32.58
54.60
47.86
65.78
63.43
43.27
43.08
37.93
42.85
35.17
35.12
39.75
36.06
37.81
36.40
3945
42.73
36.38
39.59
43.70
33.26
38.93
49.10
37.02
34.64
46.85
42.52
33.30
41.85
43.94
51.24
34.32
41.29
52.39
46.48
50.22
50.15
60.48
55.14
49.01
71.61
51.17
62.39
65.96
56.23
58.47
56.98
61.84
64.93
93.17
Table A-30: F3 Optimal: Crossover Skew vs. Mutation Rate
89
A.3.2. Stage 2
Fitness
Skew
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0
Initial Skew
0.4
0.5
0.6
0.1
0.2
0.3
28.07
24.88
29.26
30.84
22.91
27.23
26.2
28.48
23.52
35.57
32.96
26.63
27.31
29.57
28.28
26.62
29.09
31.16
30.39
33.98
29.9
24.3
25.72
33.43
33.84
35.81
38.61
36.56
36.97
34.58
36.66
39.64
31.82
33.68
31.7
39.33
33.2
31.9
40.15
39.89
38.29
0.7
0.8
0.9
1
26.45
24.85
28.68
21.24
22.05
30.13
23.91
28.96
27.8
24.22
22.97
34.22
32.99
28.4
23.94
28.72
27.19
28.27
27.95
30.9
28.53
28.36
31.78
25.96
33.38
35.12
29.57
31.63
34.19
30.5
24.88
28.26
35.21
30.32
30.59
26.04
32.22
32.27
37.89
36.84
38.05
38.65
39
24.97
28.77
40.38
34.59
34.27
35.12
34.1
32.95
33.47
38.35
33.61
38.78
36.35
37.53
29.64
35.37
40 23
40.09
31.2
38.22
34.98
42.01
35.53
38.76
35.1
34.61
40.04
40.05
44.34
50.82
42.57
51.27
44.39
43.82
44.98
45.31
41.79
Table A-3 1: F3 Optimal: Initial Skew vs. Fitness Skew
Crossover
Skew
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Initial Skew
0.5
0.6
0
0.1
0.2
0.3
28.07
24.88
29.26
30.84
22.91
27.23
29.29
22.05
24.76
24.35
27.27
29.8
26.85
27.91
31.71
32.69
32.49
30.99
29.17
35.67
28.07
31.76
32.08
25.38
33.69
38.93
33.8
35.88
37.58
34.83
44.21
34.87
34.38
32.65
36.01
30.72
41.99
43.08
33.61
46.83
43.83
38.24
37.96
42.78
37.83
35.74
37.87
39.43
34.64
37.35
34.33
32.44
41.94
42.11
40.56
40.19
45.26
0.4
0.7
0.8
0.9
1
26.45
24.85
28.68
21.24
22.05
20.94
31.25
24.53
25.89
25.8
25.47
29.14
34.72
28.04
30.19
22.56
29.15
31.13
37.2
26.77
28.85
33.44
24.28
27.08
35.9
38.41
39.71
36.08
27.59
34.42
37.99
40.11
34.98
23.27
23.65
33.04
29.74
33.58
33.18
22.18
36.97
39.23
36.99
33.02
29.85
25.58
30.19
32.24
32.55
41.53
27.83
22.66
39.24
43.02
38.18
35.85
38.8
17.96
29.05
39.71
26.97
41.56
39.51
31.96
Table A-32: F3 Optimal: Initial Skew vs. Crossover Skew
90
Crossover
Skew
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0
0.1
0.2
0.3
Fitness Skew
0.4
0.5
0.6
0.7
0.8
0.9
1
28.07
26.2
26.63
29.09
29.9
33.84
36.97
31.82
33.2
35.37
40.04
29.29
30.95
33.25
35.16
29.36
31.53
31.27
37.7
37.31
42.06
52.31
29.8
29.86
34.78
37.91
32.4
36.26
37.19
37.29
35.52
37.16
44.45
32.49
25.89
29.83
31.73
37.34
41.78
36.99
34.42
32.37
32.82
45.91
31.76
33.26
36.18
37.83
38.2
41.05
35.04
39.48
40.49
42.45
44.9
33.8
37.97
36.48
35.43
39.1
38.68
38.36
41.16
44.89
41.3
50.73
34.87
36.49
40.23
33.66
39.62
40.98
37.73
40.74
42.02
41.35
48.2
43.08
28.95
37.32
36.21
38.09
39.51
41.45
39.14
46.62
45.94
52.93
37.96
35.03
36.46
39.51
43.17
37.65
37.01
39.21
39.31
44.05
54.18
39.43
28.14
32.78
40.31
37.79
38.12
42.71
47.24
39.37
48.01
63.88
41.94
32.66
37.91
42.16
40.37
39.36
38.11
39.66
48.41
40.91
50.1
Table A-33: F3 Optimal: Fitness Skew vs. Crossover Skew
Initial Skew
0
Fitness
Skew
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0.1
0.2
0.3
0.4
0.5
5.5
5.86
6.2
6.37
6.59
5
6.28
5.34
6
6.25
7
6
7.24
6.35
5
6.36
4.24
6.75
4.4
6.28
5.76
6.04
5.38
6.52
6.64
8.19
5
3.75
6.84
6.12
7.12
6.66
7.33
6.64
7.74
5.03
7.11
6.47
5.84
6.96
5.66
6.61
5.97
5.93
5.69
5.71
6.17
7.28
6.44
8.11
5.99
8.17
6.01
6.35
6.45
8.29
5.8
9.03
7.96
4.88
0.6
0.7
0.9
0.8
1
5.3
6.75
5.95
4
5.12
6.71
5.07
4.49
5.2
2.46
6.01
6.64
6.27
5.28
5
3.1
6.1
4.58
5.22
4.7
3.29
5.41
5.79
5.64
505
584
3.97
5.21
4.88
6.69
4.58
4.17
4.03
4.76
6.16
5.66
5.74
4.2
6.05
5.1
5.57
4.68
3.13
5.76
5.59
5.64
4.95
4.15
7.23
4.8
5.28
4.87
5.38
3.53
5.45
5.97
4.2
7.39
2.51
2.65
2.96
............
. . .......
Table A-34: F3 Non-Optimal: Initial Skew vs. Fitness Skew
91
Initial Skew
Crossover
Skew
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0
0.1
0.2
0.3
0.4
0.5
0.6
5.5
5.86
6.2
6.37
6.59
5
5.3
6.62
5.11
5.53
4.74
6.41
6.36
5.43
4.39
6.66
4.08
5 28
5
5.42
4.66
5.23
6.29
5.82
5.11
5.56
5.63
2.9
4.82
6.19
4 28
8.26
3.41
6.42
5.64
6.91
6.4
4.86
5.47
6.27
8.38
3.68
2.38
6.92
4.76
6.35
4.7
5.84
4.58
7.56
0.7
0.8
0.9
1
2.96
6.75
5.95
3.94
6.27
5.1
5.38
2.59
4.43
5.68
4.54
4.9
2.6
5.34
4.44
5
2.68
3.17
5.58
5.22
4.62
5.42
3.8
3.01
2.9
4.5
4.72
5.6
2.96
2.22
5.77
6.24
3.51
3.62
5.67
3.79
2.94
5.36
4.66
3.12
3.62
3.94
2.82
2.58
5.98
2.38
3.25
3.12
4.71
6
3.36
3.66
5
7.45
4.6
2.6
3.82
3.41
5.9
3.24
2.56
5.73
5.08
7.06
5.41
6.34
6.22
2.78
4.42
2.2
4
Table A-35: F3 Non-Optimal: Initial Skew vs. Crossover Skew
Fitness Skem
Crossover
Skew
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
..........................................................................................................
...................................
..................................
..................................
..................................
........
4.24
5.38
3.75
7.33
5.84
5.69
5.99
8.29
6
5.98
5.7
5.7
5.82
7.85
6.37
6.2
6.04
5.13
7.57
5.7
3.66
5.41
7.02
6.13
5.96
7.1
5.8
5.37
7.56
9.19
4.18
6.65
5.51
5.02
7.37
4.52
5.56
5.72
6.64
4.7
6.57
5.34
5.22
5.98
5.75
7.41
5.9
4.28
5.16
6.61
5.65
7.05
8.16
6.12
7.99
6.37
6.65
10.19
6.91
6.6
2
2
7.6
4.36
8.03
7.33
6.88
7.17
8.74
6.27
6.19
4.88
5.17
7.04
7.51
2
9.38
5.31
8.61
8.11
S6.92
7.71
7.17
4.04
8.24
6.03
6.8
7.48
9.61
8.04
4.98
4.7
6.27
6.37
5.48
2.18
4.51
6.43
8.59
8.92
9.16
4.55
4.58
6.04
5.27
6.1
5.96
6.27
6.57
6.36
8.47
5.5
6.28
6.62
5.76
5.43
7.33
5.42
.2
[able A-36: F3 Non-Optimal: Fitness Skew vs. Crossover Skew
92
8.56
A.4. F4 Data
A.4.1. Stage 1
Initial Skew
0.4
0.5
0.6
Crossovers
1
0
0.1
0.2
0.3
0.7
0.8
0.9
113.61
106.09
110.37
102.72
110.24
101.15
107.59
106.92
106.18
109.03
104.79
2
89.10
90.64
95.43
91.49
86.95
85.93
89.70
86.32
84.72
83.66
89.45
3
78.43
78.70
79.44
73.39
78.74
76.48
80.78
76.70
79.54
75.98
83.45
4
74.58
77.13
75.62
74.71
75.92
71.16
72.39
75.14
75.25
73.90
72.00
5
6
74.49
70.86
68.82
76.50
71.95
73.31
70.65
69.15
70.55
76.05
64.56
69.41
72.89
72.08
70.27
71.45
65.32
71.49
69.62
69.53
71.47
77.59
7
74.08
72.16
70.33
67.90
71.45
73.78
70.46
71.19
69.86
72.32
69.56
8
66.84
72.21
72.93
68.31
71.23
70.09
68.15
68.15
71.90
72.47
67.06
9
70.82
69.99
69.50
69.27
69.35
68.96
66.69
65.56
69.95
69.85
75.15
10
66.63
73.18
69.44
71.48
69.54
71.10
65.92
66.62
71.43
72.91
71.98
Table A-37: F4 Optimal: Initial Skew vs. Expected # Crossovers
Fitness Skew
Crossovers
0 I...........
0.1
0.2
0.3
0.4
0.5
0.6
1
0.7 ..................................
0.8 ............................
0.9 ......................
....................
..........
......
...................................
...................................
...................................
:................................
-:..................................
o..................................
113.61
102.83
105.06
106.28
104.12
101.51
101.68
104.50
103.21
97.65
100.65
2
89.10
86.13
81.97
85.45
77.93
76.80
79.75
79.70
84.10
78.10
81.09
3
78.43
79.40
74.80
75.45
73.76
77.59
72.81
73.78
74.67
73.33
75.77
4
74.58
69.33
72.68
73.86
71.16
70.40
71.97
72.35
75.75
67.84
71.42
5
74.49
68.76
71.48
72.98
69.33
70.43
70.53
67.93
69.06
68.15
72.42
6
69.41
74.31
70.74
67.87
67.78
67.58
71.08
68.89
70.48
71.89
72.01
7
74.08
66.41
68.43
67.69
68.01
65.92
68.79
71.07
68.77
68.95
68.47
8
9
10
66.84
67.19
70.85
65.92
72.43
69.88
67.16
67.31
67.36
69.43
71.26
70.82
71.49
71.18
65.34
65.82
64.36
71.95
68.05
69.51
72.05
71.57
66.63
67.75
67.16
69.33
66.09
67.20
66.66
69.81
67.85
69.96
68.92
Table A-38: F4 Optimal: Fitness Skew vs. Expected # Crossovers
93
Crossover Skew
Crossovers
1
2
3
4
5
6
7
8
9
10
0
0.1
0.2
0.3
0.4
0.6
0.7
0.8
0.9
1
113.61
107.70
111.24
93.58
91.45
100.35
96.89
96.75
92.40
96.31
90.86
89.10
84.60
80.62
80.94
75.78
77.04
75.76
76.84
74.01
77.61
74.20
78.43
74.67
71.89
71.82
68.95
68.31
69.00
66.33
67.36
67.07
66.66
74.58
74.40
69.80
65.90
65.45
66.04
63.58
65.56
65.41
62.86
64.51
64.13
0.5
74.49
74.38
69.96
67.66
64.71
62.54
62.45
63.35
61.82
63.16
69.41
67.26
69.19
65.49
64.51
63.16
66.16
63.16
59.25
63.15
59.45
74.08
72.39
66.18
64.57
62.86
62.87
61.46
60.14
62.41
60.44
63.61
66.84
65.50
62.04
66.08
63.20
60.88
61.70
59.80
59.53
59.96
61.51
70.82
65.96
65.40
64.57
62.62
65.54
61.24
57.07
55.89
59.57
58.56
66.63
66.56
66.79
63.53
61.44
59.25
59.99
58.88
59.14
60.18
58.87
Table A-39: F4 Optimal: Crossover Skew vs. Expected # Crossovers
Mutation
0
0.005
0.01
0.015
11 0
99.19
0.1
0.2
0.3
101.51
107.77
108.38
Initial Skew
0.4
0.5
0.6
116.15
0.7
0.8
0.9
93.47
101.14
91.66
109.99
112.79
92.81
1
57.55
55.42
57.46
59.26
59.23
58.48
57.79
55.68
58.81
58.29
60.11
63.42
65.74
68.29
65.50
63.16
64.53
61.88
61.55
60.58
63.96
64.55
78.43
78.70
79.44
73.39
78.74
76.48
80.78
76.70
79.54
75.98
83.45
0.02
100.21
107.96
99.37
97.51
94.05
94.35
100.69
96.85
95.72
103.03
100.87
0.025
0.03
0.035
0.04
0.045
0.05
126.70
127.32
134.11
133.58
130.22
128.47
131.52
127.00
122.15
129.52
131.50
160.22
154.07
152.85
159.35
154.51
144.19
155.00
158.46
161.68
159.70
160.47
180.29
174.75
182.72
167.33
180.78
181.68
177.66
179.84
174.43
183.78
174.74
188.21
187.84
190.89
189.35
193.12
190.25
189.94
188.05
193.42
189.80
187.65
196.66
198.09
196.05
195.37
193.64
196.81
192.50
197.39
192.96
194.98
195.33
197.55
198.98
197.72
197.54
198.3
197.25
198.56
197.78
197.83
197.82
198.52
Table A-40: F4 Optimal: Initial Skew vs. Mutation Rate
94
Fitness Skew
0.4
0.5
0.6
0
0.1
0.7
0.8
0.9
99.19
99.48
92.17
119.3 0
92.63
113.03
110.88
121.30
117.10
118.91
122.38
57.55
56.97
59.60
58.99
59.30
60.72
57.13
58.18
62.49
61.50
62.96
63.42
63.86
63.29
65.38
63.26
61.82
62.86
61.77
63.97
67.86
68.73
78.43
79.40
74.80
75.45
73.76
77.59
72.81
73.78
74.67
73.33
75.77
100.21
102.41
96.28
89.58
95.73
87.54
94.81
92.30
92.05
88.10
91.14
126.70
120.69
125.08
117.1 9
108.02
109.77
118.53
111.42
111.38
107.64
117.02
160.22
154.31
142.11
151.0 2
143.12
137.42
141.11
140.82
142.09
134.65
139.12
180.29
17381
170.96'
170.7 2
162.14
168.42
160.80
157.79
166.24
161.96
163.45
188.21
191.53
187.40
181.8 2
182.99
176.75
183.37
183.59
179.16
183.51
174.04
0.045
196.66
193.11
192.18
194.0 5
192.56
190.77
192.65
191.55
188.17
185.01
191.17
0.05
197.55
194.07
196.97
196.9 4
196.13
195.37
195.59
195.14
193.07
195.59
194.24
Mutation
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.2
0.3
1
Table A-41: F4 Optimal: Fitness Skew vs. Mutation Rate
Crossover Skew
0
0.1
99.19
91.62
57.55
0.3
0.4
0.5
0.6
0.7
0.8
0.9
97.91
114.24
112.18
108.86
134.11
121.49
121.39
146.75
124.55
60.10
55.27
61.08
56.69
54.86
54.09
59.11
60.4
57.88
57.66
63.42
61.21
61.90
59.04
60.59
56.89
57.96
59.84
56.00
59.09
54.73
78.43
74.67
71.89
71.82
68.95
68.31
69.00
66.33
67.36
67.07
66.66
0.02
100.21
89.40
92.16
85.16
87.39
87.26
84.07
82.02
74.41
78.34
81.10
0.025
0.03
0.035
0.04
0.045
0.05
126.70
128.19
112.55
109.69
107.15
107.81
111.57
108.40
97.82
107.09
112.33
160.22
157.22
161.82
148.18
151.86
137.46
139.14
131.74
137.13
133.25
131.36
180.29
173.64
180.15
177.38
170.02
160.84
162.57
166.60
162.86
156.22
155.86
188.21
190.14
180.72
185.06
184.30
186.38
177.21
183.28
182.41
173.86
188.12
196.66
195.48
191.56
192.88
193.01
189.58
193.50
189.20
188.49
188.57
188.66
197.55
198.88
198.53
195.64
193.84
194.83
195.97
195.76
198.71
197.11
190.89
Mutation
0
0.005
0.01
0.015
0.2
Table A-42: F4 Optimal: Crossover Skew vs. Mutation Rate
95
1
A.4.2. Stage 2
Fitness
Skew
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0
Initial Skew
0.4
0.5
0.6
0.1
0.2
50.84
51.64
50.97
48.9
49.07
48.46
5171
48.36
4933
50.13
50.16
51.47
52.65
50.4
51.92
51.42
48.66
48.8
50.35
52.39
51.83
52.46
5114
49.27
50.5
51.48
53.76
51.26
54.81
51.1
49.64
54.22
53.48
50.15
50.6
51.25
51.34
53.99
0.3
0.7
0.8
0.9
1
50.81
52.36
50.62
50.88
50.34
49.24
47.19
50.78
49.9
48.19
48.15
50.68
49.42
51.21
50.05
50.62
48.66
50.47
49.35
49.17
48.62
49.19
53.62
50.73
51
51.68
50.23
51.25
5262
51.92
5363
5191
54.13
53.3
51.66
49.46
52.03
49.55
53.12
54.9
53.28
48.41
51.13
52.33
49.75
49.94
52.1
52.45
50.76
54.33
51.41
55.84
52.09
50.85
52.17
51.98
52.01
53.47
51.27
52.03
54.86
55.22
54.43
52.55
52.31
54.91
53.94
53.49
54.12
52.92
52.93
54.28
53.29
54.96
55.82
53.81
52.08
55.17
52.51
52.12
52.01
52.36
54.92
Table A-43: F4 Optimal: Initial Skew vs. Fitness Skew
Crossover
Skew
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0
Initial Skew
0.5
0.6
0.1
0.2
50.84
51.64
50.97
48.9
49.07
48.46
0.3
0.4
0.7
0.8
0.9
1
50.81
52.36
50.62
50.88
50.34
51.45
50.36
50.11
49.18
48.24
47.52
47.23
47.67
49.3
49.2
49.36
49
48.53
49.39
47.52
51.03
48.04
45.36
46.46
47.95
49.24
48.77
48.02
49.24
48.14
46.54
44.91
47.53
46.84
46.32
50.83
48.22
45.09
48.79
46.43
50.19
47.46
50.24
47.69
45.41
43.48
47.83
48.01
48.41
49.46
48.13
46.73
47.2
43.98
46.09
48
47.53
46.36
44.63
46.04
49.58
47.96
49.64
44.71
46.7
43.58
45.81
46.24
45.78
44.5
46.52
49.47
48.61
46.55
46.61
45.61
47.82
46.16
46.95
45.62
48.58
44.59
46.74
46.16
48.3
46.88
48.8
45.99
45.25
44.09
48.91
48.2
45.92
50.26
48.44
51.1
46.83
49.02
4665
4866
43.71
48.49
45.8
4
50.9
51.53
50.04
44 68
47.08
48.23
46.86
47.27
45.45
46.64
48.57
Table A-44: F4 Optimal: Initial Skew vs. Crossover Skew
96
Fitness Skem
Crossover
Skew
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0
0.1
0.2
0.3
50.84
51.71
52.65
51.45
49.97
48.82
49
46.98
48.02
48.79
0.4
0.5
0.6
52.39
51.48
54.22
52.62
51.43
51.17
49.49
51.21
49.63
47.54
51.06
51.81
51.59
46.94
50.68
50.79
50.88
51.77
50.52
50.56
50.94
48.3
50.85
0.7
0.8
0.9
1
54.9
51.41
55.22
53.29
50.1
51.73
56.19
52.38
53.77
51.09
57.34
55.44
50.19
50.62
53.49
52.39
52.79
48.65
49.32
53.08
52.37
52.82
54.26
52.06
52.99
52.37
49.46
49.08
49.57
49.12
50.02
52
49.67
51.16
49.58
48.34
48.96
48.64
51.07
47.85
50.94
49.64
51.91
52.56
54.04
49.47
49.17
45.43
50.4
49.33
53.37
49.25
51.76
53.68
52.45
50.57
46.74
49.43
47.92
50.12
53.14
51.42
51.77
52.6
51.52
51.28
53.16
50.26
47.47
51.21
52.5
51.53
51.91
53.65
50.86
48.69
52.64
51.96
50.9
49.86
48.82
49.14
51.82
54.43
48.2
52.92
51.85
52.92
54.05
1
Table A-45: F4 Optimal: Fitness Skew vs. Crossover Skew
Initial Skew
Fitness
Skew
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
20.47
20.54
21.31
20.58
20.87
21.24
20.74
20.97
20.98
20.94
21.08
21.18
20.8
20.69
20.18
20.78
21.5
20.25
20.62
21.76
20.97
21.1
20.92
21.02
20.32
20.2
19.87
21.06
20.51
20.54
20.83
21.1
21.25
20.97
20.36
20.7
20.67
20.75
20.59
22.16
20.84
20.66
21.94
21.13
21.84
20.29
20.95
21.51
21.31
21.98
21.1
20.54
21.4
21.15
21.43
22.07
20.79
22.74
20.81
22.26
21.25
20.61
20.65
20.65
21.56
22.25
21.86
21.93
21.23
20.79
21.66
21.88
22.33
21.64
21.97
21.05
21.15
21.71
20.73
22.36
21.08
21.71
21.89
21.07
22.94
21.69
22.1
21.84
21.78
20.82
21.5
20.9
21.74
21.91
21.4
21.76
21.73
22.87
21.77
21.86
23.08
22.61
22.38
21.87
22.25
22.66
23.15
22.05
20.85
22.65
22.3
22.45
22.24
22.98
22.83
23.06
21.87
22.77
22.63
22.96
22.41
Table A-46: F4 Non-Optimal: Initial Skew vs. Fitness Skew
97
Crossover
Skew
0
0.1
0.2
0.3
0.4
0
Initial Skew
0.4
0.5
0.6
0.1
0.2
0.3
20.87
21.24
20.74
20.97
20.98
20.94
20.34
21.52
20.81
21.07
20
20.18
19.27
20.07
20.32
20.17
20.38
19.57
20
20.71
20.01
20.11
20.78
19.71
0.7
0.8
0.9
20.47
20.54
21.31
20.58
20.7
20.92
21.15
21.14
19.78
20.02
19.69
19.96
20.52
21.12
20.2
20.03
20.1
19.86
20.25
19.84
1
20.06
20.61
1963
19.69
19.33
19.77
20.02
21.32
20.06
20.27
19.97
0.5
20.53
19.91
19.82
19.17
19.86
21.01
20.78
20.13
19.15
19.79
19.88
0.6
20.21
19.99
19.57
19.78
19.59
19.96
20.38
20.35
19.6
20.29
20.09
0.7
19.75
21.02
19.55
20.13
19.63
19.99
20.09
19.18
19.94
19.96
18.71
0.8
20.25
19.07
19.91
19.03
19.72
19.88
19.48
19.82
19.67
19.44
20.29
0.9
19.94
19.88
19.78
20.38
19.46
20.04
20.19
19.6
20.03
19.56
19.14
1.0
19.83
19.04
20.04
19.5
20.43
18.57
20.01
20.08
19.6
19.6
19.44
Table A-47: F4 Non-Optimal: Initial Skew vs. Crossover Skew
Crossover
Skew
0
0
20.87
0.1
0.2
0.3
0.4
21.08
Fitness Skew
0.5
0.6
0.7
0.8
0.9
21.2186
21.71
21.78
21.86
22.3
21.36
22.42
21.63
22.17
1
0.1
0.2
20.34
20.93
20.46
20.54
20.84
19 27
21.01
20.35
22.23
20.59
20.73
21.83
20.41
22.85
22.6
21.7
0.3
0.4
20
199
21. 20
2059
20.5
21.08
21.27
21.19
21.01
21.99
21.99
20 06
20.12
1983
20.53
21.01
2038
21.41
22.22
21.52
23.42
20.65
0.5
0.6
0.7
0.8
20.53
20.33
19.82
20.12
19.68
20.79
21.78
21.58
21.31
21.54
21.77
2021
20.56
202
19. 20
20.28
19.91
21.33
22.02
21.74
22.53
2189
19.75
20.48
20 08
20.06
21.17
19.44
19.72
21.27
20.7
20.4
21.48
20.25
20.37
20.2
19.28
21.27
20.79
21.05
21.4
21.45
21.68
208
0.9
19.94
19.96
19.66
20.54
21.51
20.61
19.46
21.41
21.59
21.49
21.63
1.0
19.83
20.
19.57
33
20.88
20.89
20.78
20.32
21.77
21.99
21
2177
21.37
21.68
Table A-48: F4 Non-Optimal: Fitness Skew vs. Crossover Skew
98
A.5. F5 Data
A.5.1. Stage I
Crossovers
1 .
2
3
4
5
6
7
8
9
10
Initial Skew
0.4
0.5
0.6
0
0.1
0.2
0.3
0.7
0.8
0.9
1
36.09
31.51
30.78
31.57
31.16
33.33
33.03
30.95
26.51
29.73
29.49
26.27
24.77
29.80
25.66
26.85
26.54
2 3.19
27.68
25.83
24.39
24.41
26.99
25.71
25.33
24.62
26.37
26.65
2 1.43
24.14
24.56
21.44
18.21
26.94
24.84
19.17
23.94
24.81
18.64
26.98
23.91
24.71
22.41
25.22
25.85
20.70
24.06
24.28
21.03
25.45
22.46
20.45
21.23
25.25
22.81
20.69
21.24
21.82
24.41
23.12
21.53
2 1.28
23.48
20.03
21.57
21.15
22.20
21.81
20.82
18.73
24.12
18.97
20.79
19.99
19.26
20.45
22.08
23.16
20.64
23.09
22.89
23.51
21.74
20.43
18.58
19.79
22.74
19.49
24.49
22.50
19.30
22.52
22.05
17.91
22.83
22.45
22.95
20.50
21.94
23.43
23.98
21.81
22.57
23.23
23.69
23.36
20.71
20.47
22.03
24.05
Table A-49: F5 Optimal: Initial Skew vs. Expected # Crossovers
Crossovers
0
0.1
0.2
0.3
0.4
Fitness Skew
0.5
0.6
0.7
0.8
0.9
1
1
136.09
37.28
34.36
32.62
35.11
35.44
34.39
33.69
37.85
33.92
37.30
2
26.27
29.66
26.91
26.51
27.35
29.87
27.45
30.03
28.24
31.53
27.64
3
26.99
25.65
22.87
24.22
24.32
28.02
27.70
26.00
26.75
28.63
26.85
4
26.94
24.49
24.95
30.33
23.03
27.33
25.22
25.77
24.62
24.08
33.59
5
25.85
23.26
21.81
23.14
27.64
25.29
22.16
22.01
25.74
24.82
26.14
6
20.69
22.43
25.23
27.62
22.30
26.60
23.13
26.35
23.83
22.64
26.78
7
24.12
18.60
20.47
21.99
23.70
24.51
21.66
24.22
25.04
26.94
22.91
8
9
10
23.16
23.43
22.92
24.84
23.56
22.18
20.74
24.75
24.38
24.79
28.41
24.49
23.47
26.23
22.62
22.76
22.52
28.37
26.03
21.14
27.84
23.97
23.43
21.04
22.29
22.35
24.13
23.73
25.20
25.56
20.33
26.01
23.08
Table A-50: F5 Optimal: Fitness Skew vs. Expected # Crossovers
99
Crossover Skew
Crossovers
0
0.1
0.2
0.3
36.09
32.92
35.04
32.07
2
26.27
28.70
27.90
3
26.99
26.80
26.50
4
26.94
22.97
5
25.85
6
20.69
7
24.12
8
9
10
0.4
0.5
0.6
0.7
0.8
0.9
1
33.15
34.01
35.79
35.78
38.67
35.22
32.46
31.52
28.31
31.04
28.73
27.57
27.97
25.93
29.54
25.24
27.34
25.79
24.87
25.93
23.66
25.09
26.96
23.37
24.02
22.03
24.65
24.55
24.95
25.06
23.28
27.59
19.45
24.72
24.30
22.76
26.04
27.59
20.14
22.71
24.29
25.18
22.62
24.93
23.93
24.37
24.27
21.47
24.73
24.32
23.20
22.95
24.81
23.66
25.67
23.53
22.36
22.66
24.16
24.23
24.81
24.35
23.16
24.34
23.56
22.73
23.51
23.95
24.65
24.09
23.35
22.67
23.36
24.49
23.72
20.97
23.88
20.58
24.14
22.84
23.77
24.30
24.47
24.27
23.43
23.81
23.56
19.81
24.43
21.66
23.52
23.74
22.70
21.61
21.77
Table A-5 1: F5 Optimal: Crossover Skew vs. Expected # Crossovers
Mutation
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
0.05
0
Initial Skew
0.4
0.5
0.6
0.1
0.2
0.3
0.7
0.8
0.9
1
60.53
71.04
76.39
77.03
52.69
65.65
36.52
34.12
38.57
27.61
28.65
32.75
27.04
28.54
24.27
23.32
26.04
23.43
24.04
24.22
23.36
22.10
29.53
27.31
20.58
27.66
23.04
23.87
23.04
18.81
24.56
20.67
22.72
26.99
25.71
25.33
24.62
26.37
26.65
21.43
24.14
24.56
21.44
18.21
22.28
21.77
23.65
23.13
26.38
25.88
25.17
20.84
23.77
24.49
24.52
26.82
24.83
20.05
29.17
23.45
27.44
29.83
24.01
26.06
21.09
24.20
29.15
27.76
23.39
31.25
27.51
28.56
25.83
22.77
27.60
25.97
28.18
19.25
31.58
25.39
28.45
27.47
29.13
30.05
24.64
26.04
27.28
21.29
29.15
29.09
26.67
28.96
28.12
27.83
27.80
28.16
22.70
28.03
29.62
33.44
32.73
28.10
29.71
33.02
25.85
26.87
23.30
25.09
28.39
28.31
34.42
31.65
32
32.29
27.37
30.89
31.68
26.52
32.49
32.44
29.66
Table A-52: F5 Optimal: Initial Skew vs. Mutation Rate
100
Fitness Skew
Mutation
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
0.05
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
68.94
59.64
56.17
64.02
42.93
45.65
58.69
58.70
42.07
58.61
32.75
29.75
25.76
28.16
25.08
29.54
28.93
27.59
27.34
26.02
28.84
29.53
27.34
26.86
22.04
25.32
24.63
29.32
30.78
24.82
28.37
34.49
26.99
25.65
22.87
24.22
24.32
28.02
27.70
26.00
26.75
28.63
26.85
22.28
29.37
25.08
29.17
28.50
28.65
26.26
30.04
25.52
29.20
27.01
26.82
26.91
28.74
25.85
32.93
28.34
25.44
26.02
28.89
27.34
28.95
29.15
21.95
29.16
29.35
27.57
29.09
30.95
26.52
30.79
32.54
26.69
19.25
24.32
25.73
27.83
27.68
28.07
29.78
30.55
33.12
32.31
21.87
29.15
27.35
31.60
31.89
33.71
28.79
29.57
27.87
31.00
34.98
31.03
33.44
28.85
33.76
30.68
34.80
29.56
30.23
33.07
35.85
38.50
39.62
34.42
29.96
35.66
32.14
30.07
34.65
38.22
33.31
33.91
37.29
42.12
60.53
Table A-53: F5 Optimal: Fitness Skew vs. Mutation Rate
Crossover Skew
Mutation ...................................
0 .................................
0.1 ..
0.2
0.3
0.4
0.5
0.6
0.7 ..................................
0.8 4............................
0.9 ......................
1 a
.............................................................
....................................................................
........................................................
:...................................
p .........................
60.53
101.08
121.50
87.96
81.54
106.11
110.52
113.81
0
105.52
98.70
108.55
32.75
25.06
26.74
26.40
26.06
28.44
24.27
23.76
29.13
0.005
34.09
34.58
29.53
29.12
26.76
26.41
28.25
27.70
28.31
27.19
0.01
27.24
26.11
27.90
26.99
26.80
26.50
27.34
25.24
25.79
24.87
25.93
23.66
0.015
25.09
26.96
22.28
26.39
26.39
27.79
26.67
26.28
26.90
0.02
27.67
27.21
27.92
26.32
26.82
28.23
27.51
26.68
27.37
25.99
29.14
27.08
0.025
25.12
27.16
26.62
29.15
29.20
27.88
27.55
28.59
26.72
26.72
27.27
0.03
24.52
22.65
25.13
0.035
19.25
27.63
28.87
27.73
31.51
26.99
26.69
27.26
30.21
28.47
25.49
0.04
0.045
0.05
29.15
28.94
31.42
28.71
27.33
28.21
30.81
26.96
27.95
27.40
25.39
33.44
31.68
31.96
31.74
27.12
31.62
29.60
28.69
27.45
28.67
29.49
34.42
31.32
30.45
31.83
30.47
31 91
33.24
31.51
33.27
32.1
30.3
Table A-54: F5 Optimal: Crossover Skew vs. Mutation Rate
101
A.5.2. Stage 2
Initial Skew
0
0.1
0.2
Skew
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
24.72
24.82
23.23
25.29
20.07
27.94
21.84
23.35
25.73
25.18
25.06
26.88
0.3
0.4
0.5
0.6
0.7
0.8
25.19
24.18
23.05
26.87
23.51
22.06
22.6
22.88
23.52
24.43
23.4
22.58
26.79
22.56
21.29
22.58
21.77
25.29
21.11
24.07
23.82
25.23
25.39
25.44
22.53
23.02
26.49
24.32
24.37
21.3
23.17
23.5
19.39
23.77
24.87
23.79
22.44
27.83
24.52
25.07
26.66
23.49
22.95
21.27
22.81
25.25
27.08
24.75
26.64
25.52
27.6
24.14
16.83
24.36
24.89
23.53
25.15
27.63
29.73
24.15
26.43
27.4
22.03
27.08
24.09
20.88
27.94
25.54
29.3
28.28
25.82
22.81
25.64
29.18
23.96
21.67
30.18
26.72
28.23
23.76
22.2
26.71
27.01
26.43
21.67
23.25
25.02
27.89
27.2
30.05
25.89
28.8
26.33
25.98
25.12
21.95
26.16
26.93
29.57
26.56
28.11
27.1
30.95
24.36
24.71
24.8
24.83
28.24
25.43
0.9
1
Fitness
0.9
1
Table A-55: F5 Optimal: Initial Skew vs. Fitness Skew
Initial Skew
Crossover
Skew
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
24.72
24.82
23.23
25.19
24.18
23.05
26.87
23.51
22.06
22.6
22.88
24.81
24.31
21.5
22.19
21.51
19.83
23.44
23.82
21.54
24.41
19.74
26.13
23.26
23.76
22.92
21.35
21.88
22.51
20.73
23.85
20.23
22.18
18.31
21.85
22.7
22.32
23.29
21.26
22.27
22.05
20.81
22.92
20.54
26.91
22.34
17.79
21.56
21.26
22.73
24.26
23.54
23.13
24.67
23.88
23.25
25.27
23.73
24.89
26.88
23.67
21.43
18.57
22.06
22.21
21.62
25.26
21.22
24.71
25.74
23.95
23.63
23.42
22.1
23.6
21.92
21.49
22.19
23.59
27.19
23.18
22.31
22.16
22.86
20.4
21.93
22.43
22.55
26
25.19
26.58
19.74
24.88
25.31
23.56
22.17
23.78
23.1
20.61
20.23
25.5
25.95
23.49
20.77
22.82
18.52
23.52
24.35
25.42
24.7
23.35
27.46
25.32
16.21
25.37
19.84
23.03
21.94
19.78
23.97
21.94
0
Table A-56: F5 Optimal: Initial Skew vs. Crossover Skew
102
Fitness Skew
Crossover
Skew
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
24.72
25.29
27.94
23.35
25.73
25.18
25.06
24.81
17.46
25.14
22.78
26.45
25.95
27.26
26.13
19.74
24.1
26.67
26.46
28.22
18.31
20.24
27.3
20.25
27.3
26.91
24.14
21.05
26.52
21.43
23.25
23.94
23.91
25.36
25.26
24.08
25.59
22.19
17.9
23.54
26
24.9
20.23
23.79
23.35
20.84
0.8
0.9
1
26.88
30.18
27.89
29.57
27.66
27.07
27.16
27.27
27.94
26.68
25.03
23.7
27.22
27.22
24.85
25.06
23.34
27.68
28.32
25.58
22.06
23.22
26.16
25.53
24.94
27.24
24.81
23.2
24.89
24.19
23.89
27.19
23.36
25.38
20.35
24.29
25.59
24.52
27.69
27.11
24.16
26.02
27.09
24.72
25.41
24.51
28.55
29.63
18.24
26.28
24.88
22.64
25.5
27.33
26.33
24.96
27.44
27.18
25.3
26.29
25.93
23.11
26.28
24.56
28.14
26.32
21.82
23.48
21.1
23.58
26.6
24.42
27.28
24.03
25.42
0.9
1
Table A-57: F5 Optimal: Fitness Skew vs. Crossover Skew
0
Fitness
Skew
0.1
0.2
0.3
Initial Skew
0.4
0.5
0.6
0.7
0.8
.............................................
..................................
..................................
..................................
4 ..................................
.................................
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
6.39
6.86
2.68
4.75
4.9
5.83
4.67
5.1
5.03
5.18
4.43
5.84
4.6
5.85
5.62
4.29
5.17
5.76
4.92
4.95
5.28
4.64
6.51
5.66
5.32
5.67
5.06
5.77
5.46
5.74
5.1
5.62
4.22
6.28
6.67
6.8
5.85
5.13
6.27
5.38
5.82
5.9
5.31
5.13
7.25
6.15
6.42
6.73
5.25
6.33
5.63
5.5
5.53
4.16
5.48
6.36
6.94
6.2
7.35
6.46
5.9
7.56
6.17
5.46
5.23
5.14
7.58
5.84
6.04
6.81
6.88
7.65
6.87
5.9
6.49
6.55
4.89
6.71
6.86
6.96
7.14
6.88
7.37
5.67
6.09
6.81
6.35
6.81
7.62
7.88
6.97
7.44
6.88
7.85
6.51
6.59
6.82
6.09
6.02
7.96
6.76
8.49
7.01
7.95
7.96
7.68
8
7.46
6.9
6.27
6.9
5.55
6.57
7.62
6.13
7.01
6.23
7.27
5.63
7.04
6.36
Table A-58: F5 Non-Op timal: Initial Skew vs. Fitness Skew
103
Initial Skew
Crossover
Skew
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
6.39
6.86
2.68
4.75
4.9
5.83
4.67
5.1
5.03
5.18
4.43
5.79
4.17
5.01
5.48
4.52
4.81
4.66
5.38
6.28
4.46
4.47
3.82
4.47
5.38
5.41
6.09
4.38
5.32
3.89
4.32
5.44
5.73
4.23
5.08
5.67
5.42
5.27
3.85
4.7
5.72
4.15
4.79
4.85
7.51
5.36
4.24
5.42
3.17
3.27
4.36
4.85
4.87
4.37
4.73
5.44
4.22
4.46
4.81
3.77
4.6
6.3
4.64
4.4
4.55
4.1
3.33
6
4.25
3.56
5.13
4.47
5.08
3.32
4.6
3.77
3.39
4.75
5.26
3.84
4.04
5.28
4.6
4.11
4.53
4.27
3.74
3.58
5.45
5.18
5.02
4.91
4.2
3.58
5.13
4.56
3.56
3.81
4.47
5.08
5 76
5.28
5.9
4.69
4.57
5.38
5.29
4.21
4.45
4.72
5.47
7.25
5.61
5.12
4.28
5.34
6.37
3.69
3.87
4.49
5
Table A-59: F5 Non-Optimal: Initial Skew vs. Crossover Skew
Fitness Skew
Crossover
Skew
0
0.1
0.2
0.3
0.4
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
6.39
5.84
6.51
6.28
7.25
6.36
7.58
6.71
7.62
7.96
6.9
5.79
6.89
5.8
6.18
6.42
5.67
7.1
7.09
7.64
7.22
8.2
3.82
5.29
6.43
7.5
7.07
5.76
6.53
7.07
6.09
7.91
6 75
4.23
4.18
5.12
6.03
6.6
654
647
6.47
7.17
734
8.76
7.51
5.31
5.83
5.06
4.37
5.76
6.03
6.18
8.08
6.31
6.31
0.5
0.6
5.44
3.33
7.35
56
5.97
6.72
5.87
6.26
6.34
7.36
8.31
6.98
6.06
3 78
5.5
6.58
4.82
7.27
6.35
7.4
7.52
631
0.7
0.8
0.9
4.75
3.54
6.66
5.4
7.29
7.3
6.33
8.43
5.68
6.82
9.42
5.45
5 17
5.95
7.62
5.14
5.86
6.42
7.34
6.09
6.54
7.17
08
5 03
4.12
6.39
5.24
7.28
6.76
5.86
5.95
8.39
5.64
1.0
5.47
5.19
4.72
6 12
6.92
7.31
7.05
6.92
4.18
7.36
5.69
5
Table A-60: F5 Non-Optimal: Fitness Skew vs. Crossover Skew
104
A.6. F6 Data
A.6.1. Stage 1
Initial Skew
0
Crossovers
1
2
3
4
5
6
7
8
9
10
0.1
0.2
0.3
0.4
0.5
163.67
147.12
0.6
0.7
0.8
116.72
199.00
161.92
148.30
162.80
159.78
181.49
86.45
146.80
141.14
147.18
178.84
98.20
141.90
163.20
110.63
146.43
129.86
154.56
108.74
174.69
167.29
166.81
180.36
168.95
149.12
131.79
153.57
159.60
136.00
143.32
146.86
124.50
142.09
140.97
85.64
142.07
95.52
199.00
128.66
117.06
158.92
190.42
127.98
184.60
153.60
157.20
171.54
181.33
154.27
178.86
155.06
186.44
179.85
156.25
155.97
184.87
182.22
172.98
143.63
149.87
160.82
105.90
169.23
161.72
180.95
138.90
146.27
166.22
193.00
146.07
160.54
168.53
163.26
102.40
157.16
177.42
149.99
149.58
191.52
198.76
185.68
133.85
92.31
160.14
136.07
154.37
110.20
145.83
157.89
181.13
199.00
137.28
177.55
170.36
134.10
172.77
189.84
164.14
179.13
118.96
199.00
155.10
156.51
153.60
158.93
169.25
174.41
|
0.9
1
Table A-61: F6 Optimal: Initial Skew vs. Expected # Crossovers
Crossovers
2
Fitness Skem
0.5
0.7 ...................................
1
0.8 ...................................
0.9 ..............................
.........................
..................................
0
0.1
0.2
0.3
0.4
163.67
195.64
147.61
115.71
167.78
119.29
114.98
130.59
91.89
93.55
41.65
141.14
138.44
129.87
102.32
115.21
118.58
73.73
64.17
88.62
73.07
27.19
3
174.69
136.32
156.16
119.65
137.22
104.38
139.18
94.50
98.11
57.07
27.29
4
146.86
133.08
136.71
136.92
132.98
165.49
169.76
126.01
113.03
80.74
30.62
5
190 42
104.02
115.58
157.04
114.61
115.77
120.82
11454
111.92
54.61
32.41
6
179.85
96.03
176.83
155.71
132.82
103.45
103.01
88.99
43.32
88.06
31.93
71
161.72
138.03
182.39
166.24
122.45
150.48
111.64
119.28
93.41
84.54
30.28
8
9
157.16
175.95
144.53
103.72
100.94
100.71
100.79
97.57
81.18
57.38
27.59
154.37
181.29
135.34
182 14
156.03
95.79
97.82
66.74
98.01
82.30
31.78
189.84
158.77
153.09
147.49
155.39
145.28
136.15
131.08
79.25
99.23
28.27
10
|
Table A-62: F6 Optimal: Fitness Skew vs. Expected # Crossovers
105
Crossover Skew
Crossovers
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
163.67
162.96
163.68
183.55
178.40
150.35
162.00
127.19
123.89
122.31
152.63
2
141.14
171.64
135.47
152.85
110.70
138.51
99.63
138.26
110.73
118.60
139.05
3
174.69
143.52
163.36
139.81
149.98
153.56
149.88
121.23
134.86
113.43
117.15
4
146.86
119.60
158.16
157.43
109.71
146.92
108.57
131.12
114.21
129.76
147.60
5
190.42
150.25
166.46
166.57
139.24
106.24
108.33
107.83
112.53
14919
159.11
6
179.85
1-714
151 1r
170.66
99.96
11955
126.84
148.91
141.84
113.51
141.48
7
161.72
165.32
149.08
163.02
166.75
151.29
104.11
168.87
91.49
107.78
99.09
8
9
10
157.16
166.71
147.20
136.38
137.23
96.81
176.63
113.58
137.97
131.92
163.86
154.37
167.60
131.04
116.22
151.89
152.08
129.82
156.04
107.08
141.39
162.90
189.84
147.23
149.27
147.68
172.14
173.89
130.86
150.66
153.30
134.13
148.14
.0
1
Table A-63: F6 Optimal: Crossover Skew vs. Expected # Crossovers
Initial Skew
Mutation
0
0.005
0.01
0.015
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
171.25
106.88
136.44
120.49
138.61
199.00
160.50
199.00
132.06
168.58
199.00
140.22
190.75
119.44
164.73
183.34
160.38
158.03
180.96
119.29
162.76
156.44
142.16
158.74
116.81
123.66
187.08
105.95
125.12
168.37
118.87
163.71
124.23
174.69
167.29
166.81
180.36
168.95
149.12
131.79
153.57
159.60
136.00
143.32
0.02
145.39
134.39
171.50
101.00
139.55
111.48
155.06
139.05
130.87
167.68
131.16
0.025
0.03
0.035
0.04
0.045
0.05
157.61
142.79
111.07
104.14
142.57
161 25
174.04
149.60
110.37
137.05
153.52
139.30
161.57
184.57
70.07
149.31
138.15
138.85
125.78
153.58
171.50
165.89
119.79
113.32
112.39
152.80
145.90
136.41
148.66
126.26
111.92
125.55
113.28
131.25
122.74
154.02
146.43
149.02
114.69
174.30
116.58
170.34
165.32
147.89
142.09
182.74
164.37
130.19
146.21
146.24
141.21
164.89
144.32
172.82
147.51
119.9
127.96
144.58
173.34
156.78
98.79
147.19
163.19
135.49
138.49
136.98
Table A-64: F6 Optimal: Initial Skew vs. Mutation Rate
106
1
Mutation
0
0.005
0.01
0.015
0.02
0.025
0.023
0.03
0
Fitness Skew
0.4
0.5
0.6
0.1
0.2
0.3
1.25
199.00
173.24
165.18
163.60
165.56
199.00
1 0.22
145.14
169.73
138.94
118.55
146.05
2.16
166.88
93.09
131.82
109.10
133.00
0.7
0.8
0.9
1
180.30
52.46
82.99
58.39
128.40
131.41
89.95
82.96
26.37
130.45
134.58
53.10
62.47
22.92
98.11
57.07
27.29
4.69
136.32
156.16
119.65
137.22
104.38
139.18
94.50
5.39
116.45
83.93
150.78
131.12
137.17
112.07
126.29
92.58
67.30
34.04
7.61
188.57
93.71
149.87
98.45
142.01
142.66
83.98
102.86
61.48
34.98
9.30
160.15
63.76
147.85
104.32
88.25
110.75
97.46
100.18
82.45
29.52
0.035
11 9.79
137.26
118.73
115.39
125.55
107.19
90.62
92.03
100.62
76.58
40.51
0.04
1.25
138.28
115.67
133.20
108.19
95.33
82.69
82.48
100.50
73.19
33.82
2.09
139.08
101.62
136.36
130.88
127.17
110.06
97.10
71.08
62.82
35.73
9.9
98.07
153.48
96.76
140.82
101.8
92.89
90.21
86.07
48.06
44.73
0.045
0.05
13d
Table A-65: F6 Optimal: Fitness Skew vs. Mutation Rate
Crossover Skew
Mutation
0
0.005
0.01
0.015
0 ..
0.1
0.2
0.3
0.4
0.5 ..................................
0.6 I..................................
0.7 ..................................
0.8 ...............................
0.9 .........................
1 .......
...
..................................
....................................................................
...................................
...................................
:...................................
171.25
179.63
197.29
133.48
163.80
156.30
146.47
140.22
146.17
169.46
175.85
169.47
126.33
155.34
142.16
180.51
139.81
140.29
126.37
140.72
174.69
143.52
163.36
139.81
149.98
153.56
0.02
145.39
122.41
164.63
144.79
131.22
0.025
0.03
0.035
0.04
0.045
157.61
145.96
142.87
132.43
139.30
169.84
157.35
117.92
119.79
112.74
166.87
131.25
142.46
148 92
142.09
123.21
134.36
0.05
119.9
153.11
106.69
149.1
167.88
4
115.00
167.02
146.57
80.80
104.43
87.28
143.08
112.35
108.36
134.78
124.88
112.86
149.88
121.23
134.86
113.43
117.15
163.62
159.87
127.71
140.87
126.87
131.54
126.09
125.35
108.54
126.34
120.08
92.09
150.89
125.44
157.62
134.21
103.32
108.81
160.15
142.55
139.64
140.56
127.34
131.64
102.78
129.86
121.54
111.30
151.04
110.85
158.36
134.78
107.15
152.01
122.94
130.41
146.73
167.86
136.16
121.34
104.76
113.11
127.59
117.74
139.88
108.83
113.05
97.05
100.79
142.46
131.85
Table A-66: F6 Optimal: Crossover Skew vs. Mutation Rate
107
A.6.2. Stage 2
Fitness
Skew
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0
Initial Skew
0.4
0.5
0.6
0.1
0.2
0.3
169.2
120.28
161.98
161.37
149.61
123.69
139.06
69.38
96.78
162.85
115.03
153.63
117.89
144.13
156.66
143.54
117.54
125.69
133.9
132.48
128.92
132.39
145.07
117.32
114.1
128.62
128.8
117.43
0.7
0.8
0.9
144.79
109.7
164.32
129.59
120.4
153.94
116.52
134.41
108.7
145.83
147.01
104.03
126.16
123.12
163.68
106.19
142.86
131.43
132.59
146.97
128.82
150.56
80.95
98.53
138.78
98.92
107.71
158.78
86.51
127.09
119.46
125.56
126.72
136.72
85.43
60.4
110.16
74.38
1
117.54
119.96
112.79
114.34
114.49
88.78
84.09
82.04
109.57
130.58
64.15
92.72
55.11
84.52
123.6
96.1
112.01
95.68
63.57
83.01
101.79
105.76
52.02
68.54
66.81
100.53
94.25
88.93
98.79
86.93
66.89
75.5
59.48
53.1
56.27
60.34
67.97
63.03
71.59
58.28
61.99
64.11
66.62
57.66
34.35
33.48
37.95
38.67
32.47
25.39
33.33
27.06
37.04
28.6
28.73
Table A-67: F6 Optimal: Initial Skew vs. Fitness Skew
Crossover
Skew
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0
0.1
0.2
0.3
0.4
Initial Skew
0.5
0.6
0.7
0.8
0.9
1
------------
169.2
120.28
119.04
81.27
161.98
161.37
107.19
180.78
161.9
104.13
91.16
138.25
125.48
97.82
93.33
149.61
123.69
182.66
166.38
110.38
97.34
97.02
187.53
150.12
113.34
139.48
126.94
92.13
111.53
109.61
107.53
108.92
107.82
78.88
53.09
100.41
117.83
113.13
125.09
144.79
109.7
164.32
129.59
120.4
133.44
149.37
135.4
117.38
157.08
68.4
136
159.29
144.47
165
151.69
161.34
121.91
102.61
97.39
100.95
158.52
115.84
118.32
112.7
141.51
131.27
174.6
156.77
136.66
118.72
94.4
131.67
89.71
131.98
136.24
97.01
97.81
127.09
113.77
157.54
82.1
84.14
89.77
56.51
69.59
81.03
114.46
158.82
104.03
149.61
117.31
151.91
86.28
87.33
111.13
118.01
125.43
100.28
88.39
126.89
68.63
112.27
119.7
141.32
132.42
85.11
85.72
109.58
77.43
98
76.49
90.66
85.65
157.94
98.25
98.72
142.26
96.19
67.09
82.74
130.36
Table A-68: F6 Optimal: Initial Skew vs. Crossover Skew
108
Fitness Skew
Crossover
Skew
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0
0.1
0.2
0.3
0.4
169.2
139.06
153.63
117.54
128.92
114.1
119.04
164.89
121.89
135.62
115.38
87.23
81.27
0.5
0.6
0.7
0.8
117.54
92.72
52.02
53.1
34.35
91.58
97.62
60.89
46.39
31.62
57.55
73.4
31.6
0.9
1
112.27
115.45
81.04
118.12
118.33
98.33
94.84
91.16
112.76
138.99
111.76
104.23
64.6
93.58
82.84
59.21
53.93
32.14
125.48
88.08
104.43
103.97
65.38
72.81
123.3
84.82
43.03
53.06
31.57
93.33
131.19
121.81
107.14
77.96
126.22
70.85
66.33
65.76
52.78
30.68
109.61
136.02
142.76
57.65
108.54
68.42
57.82
85.38
89.37
21.05
28.95
107.82
103.52
76.42
95.77
140.44
82
75.51
61.51
57.35
43.68
28.61
100.41
148.84
60.35
91.14
96.46
61.39
103.93
74.54
51.89
49.61
29.5
113.13
74.76
100.05
87.74
51.84
65.39
91.64
66.18
73.94
51.29
28.47
125.09
85.18
125.48
72.29
65.27
59.05
57.4
69.62
59.36
53.41
30.3
Table A-69: F6 Optimal: Fitness Skew vs. Crossover Skew
Initial Skew
Fitness
0
0.1
0.2
0.3
0.4
0.5
0.8
0.9
1
0.6
0.7
Skew
. . . . . . . . . . . . . . . . . ..................
4 ..................................
..................................
...................................
...................................
..........................
:...................................
14.09
10.29
8.44
10.77
7.74
7.93
9.42
9.75
8.11
6
7.24
0
7.81
10.73
5.97
9.71
8.78
10.21
8.12
8.18
7.87
6.62
4.66
0.1
8.48
9.69
9.53
8.59
9.62
6.96
10.3
7.46
8.32
7.82
5.38
0.2
6.57
7.76
8.39
10.25
8.66
8.44
7
7.76
8.97
10.26
9.44
0.3
8.95
8.61
8.64
7.09
10.4
8.89
7.88
8.16
7.57
8.85
5
0.4
8.02
9.41
9.7
7
6.43
6
6.97
8.09
4.33
9.45
6.75
0.5
7
11.53
7.02
7.86
9.94
9.34
7.34
6.3
7.95
7.18
5.54
0.6
9.16
8.68
10.23
7.19
9.04
6.75
8.73
6.1
9.52
7.62
6.56
0.7
6.68
5.32
11.04
9.77
8.1
4.86
5.02
7.96
8.27
6.97
5.28
0.8
6
5.39
5.97
8.38
6.76
7.39
6.94
8.06
7.58
6.78
8.96
0.9
6.17
8.32
6.28
10.93
8.09
13.76
4.75
9.54
7.06
10.06
4
1.0
Table A-70: F6 Non-Optimal: Initial Skew vs. Fitness Skew
109
Crossover
0
0.1
0.2
0.3
Initial Skew
0.4
0.5
0.6
0.7
0.8
0.9
1
Skew
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
14.09
10.29
8.44
10.77
7.74
7.93
9.42
9.75
8.11
10
8.46
9.59
8.16
9.25
5
5.92
6.28
7.79
7.45
9.17
8.98
7.54
6.78
7.8
6.4
7.46
6.46
7.99
6.24
7.99
8.89
5.59
S8.23
6.69
7.62
6.8
4.75
5
6.49
7.37
8.11
6
7.68
8.41
7.77
6
7.24
6.92
9.18
6.58
7.8
7.64
6.17
6.68
5.73
8.36
7.24
5.64
4.84
6
6.39
6.52
5.89
5.82
6.76
8
8.31
8.66
5.71
3.32
6.35
5.6
4.6
6.15
7.52
5.98
10.09
5.9
6.92
5.6
6.88
7.32
3.89
6.6
5
7.96
4.55
7.69
8.92
3.8
7.63
6.31
5.28
2
8.08
4.55
5.89
6.38
6
6.64
2
5.76
7.52
5.76
5.28
3.7
3.57
7.6
6.6
3.72
4.3
4
3.6
4
4.51
3.52
5.82
6.1
3
i4.66
Table A-71: F6 Non-Optimal: Initial Skew vs. Crossover Skew
Fitness Skew
Crossover
Skew
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0
..................
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
.........
...................................
I...................................
...................................
:...................................
A...................................
; ......................................................................
4 ..................................
..................................
..................................
14.09
7.81
8.48
6.57
8.95
8.02
7
9.16
6.68
6
6.17
10
10.34
6.49
6.19
4.66
7.56
7.6
6.89
9.27
7.76
6.25
7.79
7.78
9.98
6.78
7.06
9.71
4.18
6 71
7.61
8.77
5.8
7.46
4.17
7.52
9.54
7.39
8.4
6.51
8.89
7.91
4.58
4
8.23
8.2
9.76
4.17
7.43
9.54
7.7
4.48
6.2
9.85
6.52
7.37
6.73
7.08
5.53
6.62
6.88
6.34
5.46
10.65
9.38
4
8.36
5.7
8.96
7.75
4.84
5.3
8.45
5.39
7
8.17
3.64
10.09
7
6.12
6
4.48
5.15
5.03
7.64
7.9
9.86
7.67
7.69
6
4.44
6.26
4.47
6.38
8.46
6.08
7.72
4.16
10.25
5.23
4
8.88
5.12
6.88
5.22
7.79
5.38
5.51
11.44
8.6
6.54
6.69
9.74
6.94
8.39
5.73
4.24
5.8
2
4.66
Table A-72: F6 Non-Optimal: Fitness Skew vs. Crossover Skew
110