university of pannonia · expert system for health promotion author: balázs gaál supervisor: dr....

University of PannoniaFaculty of Information Technology

Multi-Level Genetic Algorithms andExpert System for Health Promotion

Author:Balázs Gaál

Supervisor:Dr. György Kozmann

Dissertation presented to the Information Science & Technology PhD Schoolof University of Pannonia in partial fulllment of the requirements for the degree of

Doctor of Philosophy

December 1, 2009

Multi-Level Genetic Algorithms and Expert System

for Health Promotion

(Többszint¶ genetikus algoritmus és táplálkozás- és életmód-tanácsadó szakért®i rendszer)

Értekezés doktori (PhD) fokozat elnyerése érdekében

Írta:Gaál Balázs

Készült a Pannon Egyetem Informatikai Tudományok Doktori Iskolája keretében

Témavezet®: Dr. Kozmann György

Elfogadásra javaslom (igen / nem)(aláírás)

A jelölt a doktori szigorlaton ..........%-ot ért el

Veszprém ........................................a Szigorlati Bizottság elnöke

Az értekezést bírálóként elfogadásra javaslom:

Bíráló neve: .................................................. (igen / nem)(aláírás)

Bíráló neve: .................................................. (igen / nem)(aláírás)

A jelölt az értekezés nyilvános vitáján ..........%-ot ért el

Veszprém, ........................................a Bíráló Bizottság elnöke

A doktori (PhD) oklevél min®sítése .................................

..............................Az EDT elnöke

Tartalmi kivonat

A Többszint¶ genetikus algoritmus és táplálkozás- és életmód-tanácsadó szak-

ért®i rendszer cím¶ disszertáció tartalmát képez® kutatási célok, vizsgálatok és ered-ményeik bemutatása - ahogy az a dolgozat címéb®l kiolvasható - kett® egyenl® fontosságúrészre osztható. E két cím, két szakasz alatt bemutatott munka azonban szorosan össze-függ® és egymásba kapcsolódó.

A tézis els® jól körülhatárolható kutatási témájához a szerz® által kidolgozott új ge-netikus algoritmus (Genetic Algorithm (GA)) - amely a többszint¶ genetikus algoritmusnevet kapta (Multi-Level Genetic Algorithm (muleGA)) - és a hozzá kapcsolódó kísér-leti vizsgálatok bemutatása és eredményeinek értékelése tartozik. Ezen új algoritmuskifejlesztését a többszint¶, sokdimenziós és nagy keresési térrel rendelkez® optimalizá-lási feladatok megoldására jól alkalmazható módszerek iránti igény gerjesztette. (Ilyenoptimalizálási feladatként kerül megfogalmazásra a táplálkozási- és életmód tanácsadásprobléma is - lásd kés®bb). A többszint¶ optimalizálási problémának létezik egy ésszer¶felbontása részproblémákra, és optimális megoldásának megtalálásához felhasználható arészproblémák, vagy akár azok továbbontása során létrejött rész-részproblémák (és ígytovább) egyenkénti optimumának megkeresése. Az oszd-meg-és-uralkodj típusú dekom-pozíciós technológia tradicionális, azonban az evolúciós algoritmusok területén eddig nemkerült kell®en kiaknázásra. A muleGA kidolgozása egy lépés ebbe az irányba.

A dolgozat második jól körülhatárolható része egy olyan szakért®i rendszer mester-séges intelligencia moduljának kidolgozását mutatja be, amely személyhez szabott diet-etikai és életviteli tanácsokkal látja el a felhasználót: egyedi étrend- és zikai aktivitástervet készít a páciens igényeit gyelembe véve. E tevékenység anamnézis és szintézis al-feladatokra bontható. Az el®bbi a felhasználói igények felmérését, a személyes optimumértékeinek feltárását, utóbbi ezen értékek alapján végzett javaslat el®állítását foglalja ma-gában. A disszertációban bemutatott munka a szintézis alfeladatra fókuszál. E feladatinformatikai szemszögb®l megfogalmazva egy kombinatorikus optimalizálási probléma,amely nyomokban hasonlóságot mutat olyan ismert feladatokkal, mint az órarendterve-zés és ütemezés problémák (timetabling and scheduling) és a hátizsák probléma (knap-sack problem), lényeges tulajdonsága, hogy a megoldás egészére és részmegoldásaira isvonatkoznak mennyiségi és összeférhet®ségi megkötések. El®bbiek a javaslat tartalmiszempontok szerinti értékeléséhez, utóbbiak a megoldást el®állító részkomponensek tet-sz®leges kombinációjára felírt, és azok harmóniájára vonatkozó követelmények megtartá-sához szükségesek.

Abstract

The aim of the thesis related research is twofold, with a connection existing between thetwo topics.

In the rst part of the dissertation, a novel Genetic Algorithm (GA), the Multi-LevelGenetic Algorithm (muleGA) is presented, which has been developed by the author forthe solution of high-dimensional, multi-objective and multi-level optimization problems.Multi-Level Optimization Problems (MLOPs) are such problems, that have a reasonabledecomposition into subproblems which may also have separate and rational subproblems(and so on). This divide-and-conquer type decomposition is traditional in solving largescaled design optimization problems, but its potential has not been exploited in theeld of Evolutionary Computation (EC), and there is a lack of truly competitive GAbased methods for solving MLOPs. Furthermore, GAs lose their performance advantageover random search as the dimension of the search space increases. The introduction ofmuleGA is a step to overcome these limitations of GAs.

The second part of the thesis deals with the methods developed for a novel expertsystem, which attempts to generate an output that matches the quality of dietary menuplans and physical activity plans produced by human experts. A mathematical formaliza-tion of the goals of personalized planning is given, which reveals the task of personalizedplanning is a hierarchical, multi-objective MLOP whose subproblems have resemblacesto such traditional optimization problems like the Knapsack Problems (KPs) and thescheduling and timetabling problems. A muleGA based method is introduced for thesolution of this MLOP problem, as well as a rule-based assessment technique, which isresponsible for maintaining the harmony of the components of the plans. If conguredproperly, the novel methods succeed in creating quantitively and qualitatively adequatepersonalized dietary plans and workout timetables. Proof on the eectiveness of thenewly developed methods is given through empirical testing and statistical analysis.

Zusammenfassung

Mehrstuger genetischer Algorithmus und fachkundiges Ernährungs- und Le-

bensstilberatersystem. Wie aus dem Titel der Arbeit ersichtlich, hat der Autor denInhalt der Dissertation in zwei gleichwertige Teile gegliedert. Diese zwei Teile sind jedocheng miteinander verbunden.

Zum ersten Forschungsthema gehört der vom Autor ausgearbeitete genetische Algo-rithmus (GA), ein sogenannter mehrstuger genetischer Algorithmus (muleGA), und dieVorstellung der zu diesem Algorithmus gehörenden Untersuchungen und die Bewertungderen Ergebnisse. Die Anforderung auf eine gut verwendbare Methode für die Lösung dermehrstugen, mehrdimensionalen Optimierungsaufgaben für groÿe Suchräume, war dieGrundlage für die Entwicklung dieses neuen Algorithmus. (Als solche Optimierungsauf-gabe wird auch die Nahrungs- und Lebensstilberatung formuliert - siehe weiter unten.)Unter einer mehrstugen Optimierungsaufgabe versteht sich ein solches Problem, beidem eine sinnvolle Aufteilung in Teilprobleme existiert, und zum Finden der optimalenLösung dieses Problems die Einzeloptima der Teilprobleme, oder sogar weitere Aufteilungin Teilteilprobleme usw. verwendet werden können. Die Technologie divide et impera(teile und herrsche) gilt als traditionell, aber bisher wurde diese Technologie auf demGebiet der evolutionären Algorithmen nicht gebührend ausgenutzt. Die Ausarbeitungvon muleGA bedeutet einen Schritt in diese Richtung.

Der zweite Teil der Arbeit stellt die Ausarbeitung des Moduls einer künstlichen Intelli-genz eines fachkundigen Systems vor. Damit kann der Benutzer individuelle Ernährungs-und Lebensstillratschläge bekommen. Das Modul erstellt einen persönlichen Nahrungs-und Aktivitätsplan unter Berücksichtigung der Ansprüche des Patienten. Diese Tätigkeitkann in zwei Phasen - Anamnese und Synthese - geteilt werden. Erstere bedeutet dieErmittlung der Benutzeransprüche und die Erschlieÿung der persönlichen Optimalwerte,das Zweite die Erstellung eines Vorschlags, aufgrund der Ergebnisse der ersten Phase. Inder vorliegenden Arbeit hat sich der Autor eher auf die Synthese konzentriert. Diese Auf-gabe gilt aus informationstechnischer Sicht als kombinatorisches Optimierungsproblem,welches Ähnlichkeiten mit bekannten Aufgaben, wie etwa die Probleme der Zeitplaner-stellung, Aufgabenplanung und der sog. Rucksackproblem hat. Eine wesentliche Eigen-schaft dieses Problems ist es, dass sich quantitative und Kompatibilitätsbedingungensowohl auf die Gesamtlösung, als auch auf die Teillösungen beziehen. Die Ersteren sindzur Bewertung des Vorschlags vom inhaltlichen Standpunkt nötig, während die Letzterezur Einhaltung der Erfordernisse im Bezug auf die Harmonie der beliebigen Kombinationder zur Lösung benötigten Teilkomponenten gebraucht werden.

AcknowledgmentsFor who makes you dierent from anyone else? What do you have that you did not receive? And if you did receiveit, why do you boast as though you did not?

1 Corinthians 4:7

I thank God for guiding me in life and towards the research of Evolutionary Com-putation through which I am profoundly edied by the imperfection and fragility ofthe phenomenon of convergence occurring in the simulated population of individuals viaadaptation, natural selection, recombination and mutation. While inside the oce, wor-king on improving the performance of Genetic Algorithms on problems with large andhigh-dimensional search spaces, I was becoming increasingly fascinated by the utmostperfection of the outside: Nature from the subatomic to the cosmic and Life evolved inincomprehensible dimensions. I am deeply grateful for my journey as a research studentthrough which serendipitously I became a more perceptive observer of our world.

Hereby, I would like to express my gratitude to those who taught me, guided meand in many ways helped me in acquiring knowledge and experience I am honored torender into this thesis. I wish to convey my deepest thanks to the two people who werethoroughly supportive of my work, whose indispensable help and sincere assistance Iwarmly acknowledge and to whom I am greatly indebted. To my supervisor GyörgyKozmann and to the supervisor of my Master's Thesis István Vassányi. What I havelearnt from them is far more beyond the scope of the dissertation and research in general.

I am fortunate that after a few years of initial research and development and promisingresults, funds became available to create a pre-commercial level version of the expertsystem infrastructure, which would be built around the concepts and algorithms I made.This initiated a development, which at the time of writing is still ongoing, and of whichmany exceptional software developers are or have been part of. Here, I would like to saymany thanks to Ádám Endr®di and Balázs Pintér, the two most outstanding softwaredevelopers I had the honor to work together with, for their multi-year contribution tothe project, which is highly acknowledged. I would also like to thank Gábor Bata, BalázsVégs®, Attila Czigány, Zsuzsanna Szente for their eorts on establishing and improvingthe software infrastructure.

I express my thanks to the directors of the university organizations involved in theproject, namely Katalin Hangos head of Department of Electrical Engineering and Infor-mation Systems, University of Pannonia, István Szabolcs head of Department of Dieteticsand Nutrition Sciences, Semmelweis University, and Mária Figler head of Institute of Hu-man Nutrition and Dietetics, University of Pécs. I also thank each and every present and

former personnel of these organizations with whom I had great time working together.I would like to give special thanks to Erzsébet Mák, Anna Medve, Márta Nemes, Szil-via Szabó, Gábor Balázs, Tibor Dulai, Csaba Fazekas, Kristóf Haraszti, Zoltán Juhász,Emil Mógor, Dániel Muhi, István Takács and Zsolt Tarjányi for their kind support andcooperation, and to every co-author of my papers and everyone who has worked on ad-ministering and maintaining the development infrastructure and on implementing toolsfor the project, especially to Kornél Fülöp, Viktor Hercinger, Attila Keszi and AndrásKirály. My thanks go to the Information Science & Technology PhD School and the Fa-culty of Information Technology of the University of Pannonia for supporting my research.

I would like to say thank you to my friends Brigitta Éva Farkas, Zsóa Kerekes, EszterVárosi, János Balogh, László Embersits, László Farkas, Tamás Harczos, Tamás Kiezerand András Kutrovics. I deeply and sincerely thank my friends Evelin Berta, GáborCsullag, Péter Király, Attila Kutas, Tibor L®cze, Gábor Sárdi, Szabolcs Schmidt andBalázs Szabados for all of their generous and heartfelt help and encouragement. Here, Iwish to express my love and gratitude towards my whole family for all of their support.

I thank my parents Edit and János and my brother Zoltán for everything they havedone and given for me, and I gratefully and sincerely dedicate this thesis to them.

Contents

1 Introduction and Overview 11.1 Purpose and Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Goals and Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Document Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.4 How to Read this Document . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2 Evolutionary Algorithms and Global Optimization 102.1 Single-Objective Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2 Multi-Objective Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2.1 Pareto Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.3 Constrained Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.4 Multi-Level Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.4.1 Multi-Level Hierarchy Optimization . . . . . . . . . . . . . . . . . . . . . 152.4.2 Multidisciplinary Collaborative Optimization . . . . . . . . . . . . . . . . 182.4.3 Multi-Level Granularity Optimization . . . . . . . . . . . . . . . . . . . . 18

2.5 Global Optimization Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.5.1 Enumerative Search . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.5.2 Deterministic Algorithms for Knapsack Problems . . . . . . . . . . . . . . 212.5.3 Mathematical Programming methods for Knapsack Problems . . . . . . . 212.5.4 Stochastic Algorithms for Knapsack Problems . . . . . . . . . . . . . . . . 212.5.5 No Free Lunch (NFL) Theorems for Optimization . . . . . . . . . . . . . 23

2.6 Evolutionary Computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.6.1 Genetic Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.6.2 Evolution Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.6.3 Evolutionary Programming . . . . . . . . . . . . . . . . . . . . . . . . . . 252.6.4 Genetic Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.7 The class of Genetic Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.7.1 Base Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.7.2 Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.7.3 Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.7.4 Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.7.5 Feasibility of Osprings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.7.6 Formal denition of GAs . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.8 Multi-Objective Genetic Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . 322.8.1 A-priori Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.8.2 Progressive Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.8.3 A-posteriori Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

i

2.9 Constraint Handling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362.10 Co-Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362.11 Parallel Genetic Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372.12 Multi-Level and Hierarchical Genetic Algorithms . . . . . . . . . . . . . . . . . . 372.13 Hybrid methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382.14 Conclusion on Evolutionary Algorithms and Global Optimization . . . . . . . . . 38

3 Multi-Level Genetic Algorithm (muleGA) 403.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.2 Multi-Level Evolutionary Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.2.1 Conceptual Overview of Multi-Level Evolutionary Algorithm (MLEA) . . 413.2.2 Formal Denition of the Multi-Level Evolutionary Algorithm (MLEA) . . 423.2.3 Conclusion on Multi-Level Evolutionary Algorithms (MLEAs) . . . . . . . 53

3.3 Formal Denitions of Knapsack Problems . . . . . . . . . . . . . . . . . . . . . . 543.3.1 0-1 Knapsack Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543.3.2 Multi-Objective 0-1 Knapsack Problem . . . . . . . . . . . . . . . . . . . 543.3.3 Multi-Level Multi-Objective 0-1 Knapsack Problem . . . . . . . . . . . . 553.3.4 Knapsack Problems with Description Logic . . . . . . . . . . . . . . . . . 57

3.4 The Multi-Level Genetic Algorithm (muleGA) . . . . . . . . . . . . . . . . . . . . 583.4.1 Concepts of muleGA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583.4.2 Algorithmic Details of muleGA . . . . . . . . . . . . . . . . . . . . . . . . 623.4.3 Scheduling Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643.4.4 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

3.5 Algorithm Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 663.5.1 Goal of the Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 663.5.2 Test Problems and Measures of Performance . . . . . . . . . . . . . . . . 673.5.3 Experiment Design and Test Infrastructure . . . . . . . . . . . . . . . . . 673.5.4 Knapsack Problem Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . 683.5.5 Numerical Multi-Objective Problem (MOP) Tests . . . . . . . . . . . . . 73

3.6 Conclusion on the Multi-Level Genetic Algorithm . . . . . . . . . . . . . . . . . . 83

4 Expert System Design for Health Promotion 844.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

4.1.1 Health Promotion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 844.1.2 Impact of Diet and Nutrition . . . . . . . . . . . . . . . . . . . . . . . . . 854.1.3 Nutrition Counseling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 864.1.4 Food Composition Databases . . . . . . . . . . . . . . . . . . . . . . . . . 874.1.5 The Process of Dietary Menu Planning . . . . . . . . . . . . . . . . . . . . 87

4.2 Nutrition Counseling Expert Systems: The State of the Art . . . . . . . . . . . . 884.2.1 CAMP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 904.2.2 PRISM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 904.2.3 CAMPER . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 904.2.4 MIKAS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 914.2.5 DIETPAL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 914.2.6 Knowledge Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 914.2.7 Evolutionary Algorithm Based Methods . . . . . . . . . . . . . . . . . . . 914.2.8 On the Application of Expert Systems for Lifestyle and Nutrition Counseling 92

4.3 Objectives of Dietary Menu Planning . . . . . . . . . . . . . . . . . . . . . . . . . 934.3.1 Concepts and Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

4.3.2 Attributes of a Dietary Menu Plan . . . . . . . . . . . . . . . . . . . . . . 954.3.3 Overview of Dietary Menu Planning Objectives . . . . . . . . . . . . . . . 964.3.4 Analogous Objectives of Lifestyle Counseling and Physical Activity Planning 96

4.4 Formal Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 974.4.1 Dietary Menu Planning Problems . . . . . . . . . . . . . . . . . . . . . . . 984.4.2 Related Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

4.5 Expert System Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1074.5.1 Algorithm Choice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1074.5.2 Infrastructure of Menugene . . . . . . . . . . . . . . . . . . . . . . . . . . 1074.5.3 Employing Multi-Level Genetic Algorithms for solving the Hierarchical

Multi-Objective Menu Planning Problem with Harmony Rules . . . . . . 1084.5.4 Ontology-Based Knowledge Representation . . . . . . . . . . . . . . . . . 1084.5.5 Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1114.5.6 Rule-Based Guide of the Evolutionary Search Procedure . . . . . . . . . . 113

4.6 Conclusion on Expert System Design for Health Promotion . . . . . . . . . . . . 117

5 Conclusions 1195.1 Summary of Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1195.2 Summary of Scientic Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1225.3 Applications and Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

A Details on Evolutionary Algorithms and Multi-Objective Optimization 125A.1 Details on Global Optimization Methods . . . . . . . . . . . . . . . . . . . . . . . 125A.2 Details on Genetic Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129A.3 Details on the Formalization of Evolutionary Algorithms . . . . . . . . . . . . . . 132A.4 Details on Multi-Objective Genetic Algorithms . . . . . . . . . . . . . . . . . . . 135A.5 Details on Constraint Handling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141A.6 Details on Co-Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144A.7 Multi-Objective Numerical Test Functions . . . . . . . . . . . . . . . . . . . . . . 148A.8 MOP Numeric Test Functions with side constraints . . . . . . . . . . . . . . . . . 152

B Test Congurations and Results 157B.1 Random Knapsack Problem Congurations . . . . . . . . . . . . . . . . . . . . . 157B.2 Results of muleGA runs on Multi-Objective Test Problems . . . . . . . . . . . . 159

List of Figures

1.1 Goals and contributions of the thesis and their relations are shown in the gures.The order depicted by the black headed arrows on subgure 1.1(a) represents howthe ongoing research work necessitated contributions. The order illustrated bythe white headed arrows on subgure 1.1(b) shows how the results are used andbuilt on top of the other. Contributions of the primary goal (expert system) arehighlighted with dotted box, while those of the secondary goal (muleGA) withbroken-line ellipse. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1 The solutions (some of them marked with uppercase alphabetic characters) of atwo-dimension minimization problem are shown in the gure. Solutions A, B andD are non-dominated (with all those other ones in color red), thus belonging tothe Pareto front (PF∗), which is highlighted with a red line. Solutions C and E(and all of those colored blue) are dominated. . . . . . . . . . . . . . . . . . . . 13

2.2 The hierarchy of a three-level Multi-Level Optimization Problem (MLOP) P(·,·)(3,·) is

shown in the gure. The input problem mapping functions (f) are presented witharrows pointing towards the parent problems. Two special input problem mappingfunctions are depicted, one which maps decision variables to a lower level (dottedline), and one which maps to the top level from the bottom level (broken line). . 17

2.3 Classication of Global Optimization Methods . . . . . . . . . . . . . . . . . . . 19

2.4 Loose estimate on the application of various global optimization methods for solv-ing the Multi-Objective Knapsack Problem (MOKP). Results of the number ofco-occurrences of multi-objective knapsack problem and the respective algorithmnames indexed by Google. The percentage values show the share of the specicalgorithm from the number the search for the single string: multi-objective knap-sack problem results, which is 20, 300 at the 22th of October, 2009. . . . . . . . 20

2.5 A population with four individuals, with basic GA terminology highlighted. Thegenotype (binary string encoding) and phenotype (an unsigned integer representedat locus 1-2, and a signed integer represented in two's complement form at lo-cus 3-5) of the individuals are shown, as well as the calculated tness values((signed integer)(unsigned integer)), for each individual. . . . . . . . . . . . . . . 27

2.6 The most commonly used crossover operators for binary representations . . . . . 29

3.1 Mapping input problems a and b to subproblems A and B of solution 2. . . . . . 41

iv

3.2 Outer and inner algorithmic cycles of Multi-Level Evolutionary Algorithms (MLEAs)with the inner cycle evolving population P(1,·)

(2,1) from the population schedule P(c).

The outer cycle is at c = 3, with the rst population in P(3) being P(1,·)(2,1) (also

written interchangeably as P[2]). In case input problem mapping occurs, the f(2,1)(l,u)

functions map the information from the input problem's individual space to thesubspace of the individual space of population P(1,·)

(2,1). The dotted lines show op-tional events. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.3 An example for Multi-Level Evolutionary Algorithm (MLEA) populations andtheir relationships. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.4 All possible individuals of the parent problem renderable through input problemmappings are shown, considering each input problem population is made up ofthose two individuals highlighted on the gure. Note that the numbered circlesand boxes denote the genotypes of the individuals and that the f-mappings areequal to the identical function, thus appearing the same genotypes in the parentproblem's and subproblems' genotype spaces. . . . . . . . . . . . . . . . . . . . . 51

3.5 An example of KPs arranged in a hierarchical structure, forming a 3-level KP. Theaim of this multi-level KP is to maximize the price vectors of the ve rst level KPs(and therefore the price vectors of all the upper level ones), while satisfying theweight constraints of each KP. The dotted blue lines represent the assignmentsfrom all those rst level knapsacks to the top-level one, which are assigned toa second-level knapsack. The broken red line highlights a special case, when aknapsack is assigned to more upper level knapsacks. From Denition 10, it followsthat the content of this knapsack is duplicated, and half of this duplicated contentis assigned to one of the second-level knapsacks, the other half is assigned to theother second-level knapsack. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.6 The notation of muleGA concepts - sols denoted with circles, attrs denoted withsquares standing on one vertex. Lines denote connections. . . . . . . . . . . . . . 60

3.7 The concepts of taxonomy and combination vals are shown in this gure. Theassignments of vals to sols are inferred from problem specic knowledge-basesand taxonomies encoding the non-quantiable information about the sols. . . . . 61

3.8 The results of the rst test conguration with number of boxes ranging from23 · 2i,(i = 1, . . . , 5) (x-coord), weight and prot dimensions ranging from 2j ,(j =1, . . . , 10) (y-coord) and dierence between the results of muleGA and GA given inpercentage, calculated by subtracting the latter from the former (z-coord). Notethat the coloring is only for separating the congurations and increasing visibility. 70

3.9 Percentage dierences in function of decision space dimensions (number of boxes)and objective space dimensions (number of weight and prot dimensions) resultedfrom the rst test conguration. . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

3.10 The results of the second test conguration with number of boxes ranging from23 · 2i,(i = 1, . . . , 5) (x-coord), weight and prot dimensions ranging from 2j ,(j =1, . . . , 10) (y-coord) and dierences between the results of muleGA and GA givenin percentage, calculated by subtracting the latter from the former (z-coord). . . 71

3.11 The results of the third test conguration with number of boxes ranging from23 · 2i,(i = 1, . . . , 5) (x-coord), weight and prot dimensions ranging from 2j ,(j =1, . . . , 10) (y-coord) and dierence between the results of muleGA and GA givenin percentage, calculated by subtracting the latter from the former (z-coord). . . 71

3.12 Enumerated Pareto fronts of the Binh1 and Poloni problems . . . . . . . . . . . 75

3.13 Results of the binh1 based multi-level MOP related comparisons. The uppergure depicts the distance of PFknown to PFtrue, while the lower gure showsthe tness of the best individual (low-is-best). Each gure is normalized withrespect to the number of dimensions, which are indicated by the x-coord values(2i+1, i = 1, . . . , 4). The dark-green bar represents the results of the random inputproblem mapping tests, the green bar of the semi-order input problem mappingtests, the light-green bar of the full-order input problem mapping tests, while theyellow bar of the single-level Non-dominated Sorting Genetic Algorithm II (NSGA-II) tests. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

3.14 Results of the Poloni's problem based multi-level MOP related comparisons. Theconstruction of the gure is equivalent to that of Figure 3.13. . . . . . . . . . . . 79

3.15 Distance of PFknown to PFtrue . . . . . . . . . . . . . . . . . . . . . . . . . . . . 803.16 The minimal (best) tness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 803.17 Distance of PFknown to PFtrue . . . . . . . . . . . . . . . . . . . . . . . . . . . . 813.18 The minimal (best) tness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 813.19 Distance of PFknown to PFtrue . . . . . . . . . . . . . . . . . . . . . . . . . . . . 813.20 The minimal (best) tness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 813.21 Distance of PFknown to PFtrue . . . . . . . . . . . . . . . . . . . . . . . . . . . . 823.22 The minimal (best) tness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 823.23 Dierences in distances of PFknown to PFtrue in full-order input problem mapping

muleGA to NSGA-II. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 823.24 Dierences of minimal (best) tnesses of full-order input problem mapping muleGA

to NSGA-II. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

4.1 Contribution of diet and nutrition to the burden of disease in Europe. . . . . . . 854.2 Diet as cause of reductions in disease. Subgure 4.2(a) shows the Hungarian mor-

tality statistics for men with cardiovascular diseases giving 51.3% and neoplasiaallocating 23.9% of the pie-chart. Subgure 4.2(b) highlights the preventive eectresides in diet and nutrition, through which 25.6% of the male population couldextend the healthy and productive era of their lifetime. . . . . . . . . . . . . . . 86

4.3 Schematic infrastructure of a comprehensive expert system, which integrates intodaily life with its decision support logic readily accessible whenever and whereverneeded. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

4.4 Nutritional Structure (NS) with Nutritional Levels (NLs), Dietary Menu Plans(DMPs) and Nutrition Hierarchy Objects (NHOs). . . . . . . . . . . . . . . . . . 95

4.5 Hierarchical Multi-Objective Dietary Menu Planning Problem with Harmony (HMO-DMPP-H) in Multi-Level GA structure . . . . . . . . . . . . . . . . . . . . . . . 109

4.6 The multi-level GA scheduling strategies (top-down, bottom-up, credit propa-gation, mutation based) in function of algorithmic levels. The numbers in thetop-down and bottom-up columns show the order in which the various levels areevolved. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

4.7 The crossover operator is shown in the gure in function of the nutritional level.Legend: M-Monday, Tu-Tuesday, W-Wednesday, Th-Thursday, F-Friday, Sa-Saturday,Su-Sunday BF-Breakfast, MS-Morning Snack, L-Lunch, AS-Afternoon Snack, D-Dinner S-Soup, G-Garnish, T-topping, Dr-drink, De-Dessert, cp-Crossover Point 110

4.8 A sample rule cache . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1124.9 The occurrences (ordered by relative frequency, upper gure) of 120 of the 150

possible values of alleles in the best solutions of 15.000 runs. The lower gureshows the goodness of the best solution of which the respective allele was part of. 114

4.10 The relative occurrence of a particular solution (A) in function of the strictnessof two rules (penalizing alleles A and B). . . . . . . . . . . . . . . . . . . . . . . . 115

4.11 Occurrences of the 15 possible alleles in a solution for a DMP in 1.000 runs infunction of the strictness of the rule imposed on the 15th allele. . . . . . . . . . . 116

4.12 Statistical analysis of the distribution of the potential alleles in the best solutionsand the mean occurrence of the alleles (A,B) on which the rules were imposed. . 116

A.1 Binary encoded chromosomes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129A.2 Real-value encoded chromosomes . . . . . . . . . . . . . . . . . . . . . . . . . . . 129A.3 Permutation encoded chromosomes . . . . . . . . . . . . . . . . . . . . . . . . . . 130A.4 Value encoded chromosomes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130A.5 Tree encoded chromosomes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131A.6 The population transformation T deterministically maps the parent population P

(of size µ) to the ospring population P ′ (of size µ′). . . . . . . . . . . . . . . . . 133A.7 The Random Population Transformation (RPT) R maps the random event ω to

the population transformation T , which maps parent populations of size µ (whichis independent of ω) to ospring populations of some xed size µ′ ∈ Z+ (whichmay depend on ω). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

A.8 The evolutionary operator X maps the exogenous parameter(s) Θ to the RPTR. The underlying sample space of R is Ω. Each of the possible populationtransformations acts on populations of size µ. The ospring population size µ′ ∈Z+ may depend on Θ as well as on the random event ω ∈ Ω. . . . . . . . . . . . 134

A.9 Non-dominated Sorting Genetic Algorithm (NSGA) ow chart . . . . . . . . . . 136A.10 Categorization of co-evolutionary algorithm properties . . . . . . . . . . . . . . . 147

B.1 Distance of PFknown to PFtrue . . . . . . . . . . . . . . . . . . . . . . . . . . . 160B.2 Distance of PFknown to PFtrue . . . . . . . . . . . . . . . . . . . . . . . . . . . 160B.3 Distance of P to PFtrue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160B.4 Distance of P to PFtrue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160B.5 Distance of best sol to PFtrue . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160B.6 Distance of best sol to PFtrue . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160B.7 Number of pareto sols in P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161B.8 Number of pareto sols in P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161B.9 The minimal (best) tness values . . . . . . . . . . . . . . . . . . . . . . . . . . . 161B.10 The minimal (best) tness values . . . . . . . . . . . . . . . . . . . . . . . . . . . 161B.11 The maximal (worst) tness values . . . . . . . . . . . . . . . . . . . . . . . . . . 161B.12 The maximal (worst) tness values . . . . . . . . . . . . . . . . . . . . . . . . . . 161B.13 The average tness values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162B.14 The average tness values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162B.15 The diversity of P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162B.16 The diversity of P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162B.17 The variance of the sols in P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162B.18 The variance of the sols in P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162B.19 Distance of PFknown to PFtrue . . . . . . . . . . . . . . . . . . . . . . . . . . . 163B.20 Distance of P to PFtrue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163B.21 Distance of best sol to PFtrue . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163B.22 Number of pareto sols in P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163B.23 The minimal (best) tness values . . . . . . . . . . . . . . . . . . . . . . . . . . . 163B.24 The maximal (worst) tness values . . . . . . . . . . . . . . . . . . . . . . . . . . 163

B.25 The average tness values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164B.26 The diversity of P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164B.27 The variance of the sols in P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164B.28 Distance of PFknown to PFtrue . . . . . . . . . . . . . . . . . . . . . . . . . . . 164B.29 Distance of P to PFtrue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164B.30 Distance of best sol to PFtrue . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165B.31 Number of pareto sols in P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165B.32 The minimal (best) tness values . . . . . . . . . . . . . . . . . . . . . . . . . . . 165B.33 The maximal (worst) tness values . . . . . . . . . . . . . . . . . . . . . . . . . . 165B.34 The average tness values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165B.35 The diversity of P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165B.36 The variance of the sols in P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166B.37 Distance of PFknown to PFtrue . . . . . . . . . . . . . . . . . . . . . . . . . . . 166B.38 Distance of P to PFtrue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166B.39 Distance of best sol to PFtrue . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166B.40 Number of pareto sols in P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166B.41 The minimal (best) tness values . . . . . . . . . . . . . . . . . . . . . . . . . . . 167B.42 The maximal (worst) tness values . . . . . . . . . . . . . . . . . . . . . . . . . . 167B.43 The average tness values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167B.44 The diversity of P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167B.45 The variance of the sols in P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

List of Tables

4.1 Real-world counterparts of Nutrition Hierarchy Objects (NHOs) from correspond-ing NLs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

4.2 Real-world counterparts of Activity Hierarchy Objects (AHOs) from correspondingActivity Levels (ALs). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

4.3 Domain of relation according to the type of Simple Dietary Menu Planning Problem(S-DMPP), where O is the set of objects ([O] = m) and S is the set of slots ([S] = n). 99

4.4 Meaning of the variables used to formalize the Hierarchical Multi-Objective Di-etary Menu Planning Problem (HMO-DMPP). . . . . . . . . . . . . . . . . . . . 100

4.5 Meaning of the expressions used to formalize the HMO-DMPP. . . . . . . . . . . 1014.6 Example ontologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

B.1 The parameters of the test problems . . . . . . . . . . . . . . . . . . . . . . . . . 158

ix

Nomenclature

D Decoding function

X Evolutionary operator

Θ Exogenous parameter

Ω Feasible region

Ts Fitness scaling function

f Input problem mapping operator

I Individual space

f Objective function

PF∗ Pareto front

P∗ Pareto-optimal set

X Exogenous parameter space

P Payload matrix

PFcurrent Current Pareto front

PFknown Known Pareto front

PFtrue True Pareto front

P Population

µ Population size

T Population transformation operator

ω Random event

R Random population transformation operator

Ω Sample space

s Solution of a problem, Individual of a population

Ωs Search space

ι Termination condition

x

Acronyms

AHO Activity Hierarchy Object

AI Articial Intelligence

AL Activity Level

AS Activity Structure

ATP Activity Timetable Plan

CPU Central Processing Unit

EA Evolutionary Algorithm

EC Evolutionary Computation

EP Evolutionary Programming

ES Evolution Strategy

FCDB Food Composition Database

CO Collaborative Optimization

DL Description Logic

DMP Dietary Menu Plan

FOL First Order Logic

FOP First Order Programming

GA Genetic Algorithm

GP Genetic Programming

GPGPU General-purpose computing on graphics processing units

GPU graphics processing unit

G-MOEA Guided Multi-Objective Evolutionary Algorithm

HBGA Human-based Genetic Algorithm

IEC Interactive Evolutionary Computation

IT Information Technology

KP Knapsack Problem

LP Linear Programming

xi

MDO Multi-Disciplinary Optimization

MicroGA Micro-Genetic Algorithm for Multi-Objective Optimization

MILP Mixed Integer Linear Programming

MLEA Multi-Level Evolutionary Algorithm

MLOP Multi-Level Optimization Problem

MOGA Multi-Objective Genetic Algorithm

MOKP Multi-Objective Knapsack Problem

MOSCKP Multi-Objective Single-Constraint Knapsack Problem

MOMCKP Multi-Objective Multiple-Constraint Knapsack Problem

MOMGA Multi-Objective Messy Genetic Algorithm

MOMGA-II Multi-Objective Messy Genetic Algorithm

MOP Multi-Objective Problem

MOPSO Multi-Objective Particle Swarm Optimization

MOTS Multi-Objective Tabu Search

MPP Massive parallel processing

MOSA Multi-Objective Simulated Annealing

muleGA Multi-Level Genetic Algorithm

NBB Nutritional Building Block

NFL No Free Lunch

NHO Nutrition Hierarchy Object

NL Nutritional Level

NLP Nonlinear programming

NS Nutritional Structure

NPGA Niched Pareto Genetic Algorithm

NPGA2 Niched Pareto Genetic Algorithm 2

NSGA Non-dominated Sorting Genetic Algorithm

NSGA-II Non-dominated Sorting Genetic Algorithm II

PAES Pareto Archived Evolution Strategy

PESA Pareto Enveloped-based Selection Algorithm

RPT Random Population Transformation

PSO Particle Swarm Optimization

SA Simulated Annealing

SPEA Strength Pareto Evolutionary Algorithm

SPEA2 Strength Pareto Evolutionary Algorithm 2

S-DMPP Simple Dietary Menu Planning Problem

MO-DMPP Multi-Objective Dietary Menu Planning Problem

HMO-DMPP Hierarchical Multi-Objective Dietary Menu Planning Problem

HMO-DMPP-H Hierarchical Multi-Objective Dietary Menu Planning Problem with Harmony

TSP Travelling Salesman Problem

TS Tabu Search

VEGA Vector Evaluated Genetic Algorithm

Chapter 1

Introduction and Overview

In three sections, this chapter introduces the two main research topics of the disserta-tion and an overview is given on the contributions of the work and organization of thedocument. Only inner references to chapters and sections and no bibliography citationsare used in this chapter, as it is intended to be thoroughly self-contained, introducingresearch questions and summarizing results in short, while ushering the reader into theconcerned areas of the thesis.

1.1 Purpose and Scope

The title of the dissertation: Multi-Level Genetic Algorithms and Expert System forHealth Promotion captures the main emphases of the contained twofold research work.

The rst research topic of the thesis focuses on the specication and evaluation of anew Genetic Algorithm (GA) named Multi-Level Genetic Algorithm (muleGA) (detailedin Chapter 3). The need for muleGA is induced by the demands for truly powerfuland competitive algorithmical methods for solving Multi-Level Optimization Problems(MLOPs), either single-objective or multi-objective. As the usage of the epithet multi-level is ambiguous in the optimization literature, an exact denition of what is understoodin this thesis to qualify as a Multi-Level Optimization Problem (MLOP) is given inSection 2.4.1. In a few words, to the class of MLOPs belong those problems, on which areasonable decomposition to separate levels of subproblems can be carried out (top-downdecomposition), and those problems, which are constructed by coupling subproblems toform a main optimization problem (bottom-up composition). The subproblems of theMLOPs form a hierarchical structure, and they are at least partially coupled. Optimaof the MLOP are sought by solving these subproblems quasi independently, then usingtheir results for conguring their parent problems. While the divide-and-conquer typeof decomposition technique has its traditions in optimization, its potential has not beenthoroughly exploited in the eld of Evolutionary Computation (EC). The introductionof the novel muleGA is a step in this direction. As EC methods tend to lose theirstrength as the complexity and dimension of the optimization problem increases, theexploitation of the knowledge residing in the hierarchy of the underlying structure, or

2 Introduction and Overview

even the simple decomposition of the search space through dimension reduction canregain the performance of Evolutionary Algorithms (EAs). The eligibleness of muleGA isassessed through such hierarchical and Multi-Objective Problems (MOPs) and conclusionis drawn on how, when, and how much muleGA may improve optimization performance.

The second research topic of the thesis (detailed in Chapter 4) deals with the for-malization, specication, methodology and design of a novel Expert System for HealthPromotion. Denitions of health promotion are given in Section 4.1.1. In short, this um-brella term encompasses the concepts, practices, tools, science and art of enabling peopleto change their lifestyle and increase control over their health, and moving them towarda state of optimal health. The expert system presented in this thesis is developed tocontribute to health promotion by providing advices and personalized plans for the enduser regarding diet, nutrition and lifestyle. The goal of the expert system is to match theoutput of the human expert in quality, considering dietary menu plans, and as an optionalfeature, physical activity plans. These personalized consultancies in real-life practice areprovided by allied health professionals (dietitians, nutritionists, athletic trainers, exercisephysiologist and physiotherapists). The main focus is put on building the skeleton of theframework by mathematically formalizing the requirements of the personalized plans andarchitecting methods with which the encoding of expert knowledge is adequate and thedesired level of output quality is attainable. There is connection between the rst andsecond research work, as muleGA forms the basis of the expert system. In particular,the output of the expert system is the result of an optimization process, which computesthe optima of the mathematically formalized personalized planning problem (presentedin Section 4.4). As the problem formalization highlights the diet and lifestyle planningproblem is actually a MLOP, the need for muleGA is practically triggered by the expec-tations on the quality of the expert system generated personalized dietary and physicalactivity plans.

The goals and contributions of the thesis are presented in the following section andare introduced in the order in which the research topics induce one another, startingwith the primal aim of the research and following with those it necessitates. This diersfrom how the document is organized (presented in Section 1.3), as it is constructed inaccordance how the developed methods conduce one another. This bottom-up, built ontop of the other, conducing approach, which starts with the generic method (muleGA),then follows with the expert system, has been used here as well to outline the purposeand scope of the dissertation.

1.2 Goals and Contributions

The primary and secondary goals of the thesis are denoted with Gpri and Gsec. Note thatthe two goals are considered to be of equal importance, the order only expresses that thesolution of Gsec is a necessity of Gpri. The rst contribution to Gpri is denoted with C

pri1 ,

the ith with Cprii . Using this notation, the goals and contributions of the thesis and their

relations are presented in the following.

1.2 Goals and Contributions 3

Primary Goal (Gpri) Design and create a novel expert system framework and algo-rithm for health promotion, which, if congured properly with valid and suceexpert knowledge, can match the output of the domain experts in creating person-alized dietary menu plans. By learning from the limitations of previous approaches,provide better algorithmic solutions to the problem, so that requirements on nutri-tional adequacy and harmony are satised in each component and subcomponentof the computer generated plan.

The research presented in this thesis is ignited by the aims Gpri represents. Onemay notice the optional support of physical activity planning is not included in thedenition of the primary goal. Verily, it is not, because of the following reasons: theacts of personalized counseling, either with nutritional or physical concepts in mind,comprise very similar procedures and objectives. Although the acts and objectives arealike, which signals one algorithm may work for both tasks, the nutrition domain is larger,thus requiring a richer algorithmical toolbox. This comprehensive set of algorithmicaltools however may well support personalized physical activity planning out of the box.Focusing solely on the nutritional domain saves the repetitive discussion of concepts,notations and algorithms which are generally the same for both specialties. For this end,the domain of nutrition counseling and dietary menu planning, as one, if not the mostimportant area of health promotion is addressed through the construction of the expertsystem. Nonetheless the analogousness of the concepts and objectives of physical activityplanning with dietary menu planning is presented in the thesis (Section 4.3.4).

The ins and outs of the denition of Gpri are claried, so comes the Gpri related rstcontribution, which formalizes the dietary menu planning task belonging to the nutritiondomain of health promotion.

Formal denition of the Dietary Menu Planning Problem (Cpri1 ) is given start-ing from the simplest representation of the problem, then iteratively includingall necessary details to nally formulate the Hierarchical Multi-Objective DietaryMenu Planning Problem with Harmony (HMO-DMPP-H), which is an unprece-dentedly accurate mathematical rendition of the objectives of personalized dietarymenu planning (detailed in Section 4.4.1.5).

Altogether four representations of the Dietary Menu Planning Problem are given,starting with the simplest Simple Dietary Menu Planning Problem (S-DMPP), throughthe intermediate Multi-Objective Dietary Menu Planning Problem (MO-DMPP) andHierarchical Multi-Objective Dietary Menu Planning Problem (HMO-DMPP) until themost accurate Hierarchical Multi-Objective Dietary Menu Planning Problem with Harmony(HMO-DMPP-H) (all dened in Section 4.4). As the names of the representations re-veal, a single objective version of the problem is formalized rst, then it is extended tohandle multiple objectives, and after that, the hierarchical composition of the multi-levelproblem is given. The nal extension to the formalization is the concept of harmony,(see 4.4.1.4), whose function is to describe how harmonizing the components of the planare. It is introduced in HMO-DMPP-H. The formalizations of Cpri1 reveal HMO-DMPP


and HMO-DMPP-H are special Multi-Level Optimization Problems (MLOPs), mostlyresembling to a hierarchical structure of interconnected Knapsack Problems (KPs). Thismakes algorithms for KPs primal candidates for solving the optimization tasks of dietarymenu planning. Therefore overview of global optimization methods from the perspectiveof their viability in solving knapsack type optimization problems is given (Section 2). Thesummary of global optimization methods highlights GAs are the de facto techniques forsolving KPs, but there are several other algorithms which provide good results. The mainchallenges for any particular algorithm are how it copes with the high-dimensionality,underlying hierarchical structure and harmony related requisites of the dietary menuplanning problem. In general, all of the methods which provide satisfactory solutionsfor KPs with small search spaces, signicantly lose their potential as the size of theproblem increases, thus some kind of decomposition and parallelization is necessary. AGA based technology is chosen as a base (causes behind algorithm choice presented inSection 4.5.1), around which a skeleton needs to be constructed that handles the highdimensionality and hierarchical nature of the problem. This unveils the secondary goalof the thesis, which is the following.

Secondary Goal (Gsec) Create a novel Genetic Algorithm (GA) which rst, overcomesthe limitations of previous GAs regarding ineectiveness in solving high-dimensionalMOPs, and second, supports the handling of large optimization problems, whichare specially made up of a hierarchy of subcomponents also forming optimizationproblems.

The large optimization problems made up of a hierarchy of subcomponents are theMulti-Level Optimization Problems (MLOPs), which are dened in Section 2.4.1. Therst contribution of the thesis to Gsec is the creation of the Multi-Level Genetic Algorithm(muleGA) (detailed in Section 3.4) and is denoted with Csec1 .

Novel Genetic Algorithm (Csec1 ) named muleGA has been developed to overcome thelimitations of traditional GAs on problems with large search spaces and hierarchicalinner structure.

The formalization and specication of muleGA requisites the denition of what doesthe algorithmic class named Multi-Level Evolutionary Algorithm (MLEA) stand for andhow muleGA is related to this class of algorithms. There is no prior work formalizingMLEAs, presumably because these kind of algorithms are still fairly new. Those fewbelonging to this class are referred and cited in the thesis (Section 2.12). The secondcontribution of Gsec denoted by Csec2 is a framework for the unied formal description ofmulti-level type GAs.

Formalization (Csec2 ) of the Multi-Level Evolutionary Algorithm (MLEA) has beengiven, which provides a generic abstract framework to dene and describe newtype of GAs with multiple populations arranged in multi-level hierarchy.

1.2 Goals and Contributions 5

Utilizing the formalization, the operational sequence of muleGA and in general everyMLEA is laid down and put on paper (Section 3.2 and 3.4). However, the proof of thepudding is in the eating, hence empirical studies are carried out examining the behaviorof muleGA. These result in the third and last contribution (Csec3 ) to Gsec.

Empirical analyses (Csec3 ) prove the viability and superiority of muleGA over tradi-tional GA based methods in solving hierarchically structured Multi-Level Opti-mization Problems (MLOPs).

The contributions to Gsec are complete with Csec1 , Csec2 and Csec3 making the secondarygoal (Gsec) reached. However, Gsec is induced by the results of Csec1 belonging to the pri-mary goal (Gpri), which still requires contributions besides Csec1 to be reached. Naturally,the contributions of Gsec are utilized to reach Gpri through the application of muleGA forthe optimization tasks of the expert system. The concepts of the HMO-DMPP are con-verted to the abstract representation required by muleGA (presented in Section 4.5.3),and through this association, a new method is fabricated for solving the dietary menuproblem in such details previously unavailable.

New method (Cpri2 ) has been produced for the solution of the dietary menu planningproblem formalized in HMO-DMPP. The abstract concepts of muleGA have beenutilized for solving the problem ensuring each subcomponent is optimized by acorresponding GA responsible for seeking nutritional adequate combinations.

The HMO-DMPP formalization is able to accurately describe the nutritional require-ments (constraints and optima) of any Dietary Menu Plan (DMP). With the muleGAbased method, a near-optimal solution can be sought. Regardless of the statistically ver-ied and conrmed performance of the muleGA (see Section 3.5), the optimality of thefound solution is not guaranteed. Note that looking for the true optima of these multi-level, high-dimensional, complex combinatorial optimization problems is generally onlyavailable by the brute-force evaluation of the feasible region of the search space, whichis computationally expensive. The muleGA based method provides solutions with goodquality and a nicely approximated Pareto front more eectively. The fact muleGA onlyproduces an approximate optima is not that bad news at all, as for the actual dietaryeld, the dierence between a truly optimal solution and a near optimal one is smallerthan those errors generated by the unavoidable inaccuracies of the Food CompositionDatabases (FCDBs) (detailed in 4.1.4). The numerical constraints of HMO-DMPP aresatised, as good as they are, however a DMP is just as much about the harmony of itscomponents. For this reason, rule-based assessment is introduced assessing each of thesolutions of muleGA regardless of their status in the hierarchy. These are not productionrules specifying symbol substitutions, but rather a compilation of attributes accompaniedwith a percentage value referring to the value of the list (see Section 4.5.5). The valueover 100% favors, under 100% hinders the compilation. If the compilation matches asolution, then the rule belongs to the matching rules of the particular solution. How-ever, if there is more than one rule for a solution, only the more specic ones are red


considering they are commensurable. Non-commensurable rules assess dierent type ofattributes, thus their eect is considered independently of each other. When a rule isred, it modies the tness of the solution, thus adjusting the goodness of the individualand altering the convergence of the population as whole.

Rule-based assessment (Cpri3 ) has been introduced to the tness function of the muleGAbased dietary menu planning method. With expert knowledge available in the formof simple rules, the dietary menu plans are assessable according to the harmonyof their components. The eectiveness of the rule-based guide of the evolutionarysearch has been shown through empirical tests.

The rule-based assessment of the DMPs works as intended, but the great multitude ofrules needed for the formalization of the expert knowledge renders the technique usefulfor only testing purposes. No human expert is able to rate each and every possibleDMP according to the harmony of its components. Furthermore, the set of rules isnot set in stone, as their values might change in dierent circumstances (like personalpreferences or medical considerations). This calls for a more abstract representation ofharmony related rules. There is a vast amount of tools for knowledge representation, soone is chosen for the task. As OWL-DL is seemingly becoming the de facto tool for thedescription of nutrition and dietary related knowledge, it is the pick for the job. Theknowledge required by an expert system is more detailed and covers more aspects ofthe domain than for example presented in ontologies comprising semantic informationfor a simple semantic web application. Therefore, rather than creating a single ontologyrepresenting the concepts of the nutrition domain, a structure of a bunch of ontologies iscreated. The position of an ontology is assigned by its domain (what is described by theontology) and the aspect of the description and categorization of the domain (detailedin Section 4.5.4).

Novel dietary ontology (Cpri4 ) architecture has been developed based on innovativeorganization and classication philosophy. It utilizes the possibilities of OWL-DLand provides a straightforward way for dietitians to record expert knowledge.

Of course, the ontology based knowledge is pre-processed and converted to the simpleform used in the tness function of muleGA. Besides this application, the knowledgestored in the ontology can be used to categorize the computer generated DMP frompretty much any aspects, OWL-DL reasoners are available for this task. The applicationof OWL-DL also makes it possible to have taxonomy classes identied by OWL sentences,rather than by the manually constructed lists of their elements.

Through Cpri1 , Cpri2 , Cpri3 and C

pri4 the primary goal (Gpri) of the thesis is reached.

These contributions and the contributions (Csec1 , Csec2 , Csec3 ) of the secondary goal (Gsec)are discussed in detail in the thesis, although not in this inducing order (shown in Figure1.1(a)). The structure in which this document is organized is presented in the followingsection.

1.3 Document Organization 7

(a) Inducing order (b) Conducing order

Figure 1.1: Goals and contributions of the thesis and their relations are shown in the gures.The order depicted by the black headed arrows on subgure 1.1(a) represents howthe ongoing research work necessitated contributions. The order illustrated by thewhite headed arrows on subgure 1.1(b) shows how the results are used and built ontop of the other. Contributions of the primary goal (expert system) are highlightedwith dotted box, while those of the secondary goal (muleGA) with broken-line ellipse.

1.3 Document Organization

The organization of the dissertation is presented here along with the short description ofchapters. Also, tips are given on how to read the document considering reader interest.The thesis encompasses research work from two relatively distinct area (evolutionarycomputation and lifestyle and nutrition counseling), and the reader may not regard bothof the domains with the same importance.

The document is composed of fourteen main sections of which ve are initiatives, veare numbered chapters, two are appendices and two are bibliographies.

• Contents

• List of Figures

• List of Tables

• Acronyms

• Acknowledgments

• Introduction and Overview (Chapter 1)

• Evolutionary Algorithms and Global Optimization (Chapter 2)

• Multi-Level Genetic Algorithm (muleGA) (Chapter 3)

• Expert System Design for Health Promotion (Chapter 4)

• Conclusions (Chapter 5)


• Multi-Objective Optimization (Appendix A)

• Test Congurations and Results (Appendix B)

• Thesis Related Publications

• Bibliography

The contents of the initials (Contents, List of Figures, List of Tables, Acronyms andAcknowledgments) are apparent and no further introduction is given on them. The shortintroduction of the chapters and appendices are given in the following.

Introduction and Overview (Chapter 1) introduces the two main research topicsof the dissertation and an overview is given on the contributions of the work andorganization of the document in three sections. Intended to be thoroughly self-contained, only inner references to chapters and sections and no bibliography cita-tions are used.

Evolutionary Algorithms and Global Optimization (Chapter 2) aims to intro-duce the reader to the concepts of global optimization and Pareto terminology(through Sections 2.1, 2.2, 2.3). Utilizing these concepts, the domain of Multi-Level Optimization Problems (MLOPs) is dened (2.4.1), which is related to theultimate optimization problem of the thesis formulated in HMO-DMPP-H (4.4.1.5).

Multi-Level Genetic Algorithm (muleGA) (Chapter 3) presents a novel GA (inSection 3.4), the muleGA for solving MLOPs. A novel Multi-Level EvolutionaryAlgorithm (MLEA) framework is also presented (in Section 3.2), which describesthe structure of those EAs which evolve multiple populations arranged in a hi-erarchical structure. The muleGA (a new MLEA) is tested on multi-level KPs(Section 3.5.4) and numerical MOPs (Section 3.5.5), and proofs are given on theapplicability of the new algorithm in solving MLOPs.

Expert System Design for Health Promotion (Chapter 4) deals with the researchand development of a novel expert system. The personalized dietary menu planning(Section 4.3) and the physical activity timetable planning (Section 4.3.4) problemsare discussed and a mathematical formalization is given which adequately modelsthe goals of these problems 4.4. A new method utilizing muleGA is proposed forsolving the health promotion related optimization problems which employs a rule-based assessment technique integrated into the tness function of muleGA (Section4.5.5).

Conclusions (Chapter 5) summarizes the contributions and scientic results of thedissertation, and briey discusses the possible applications and future work.

Multi-Objective Optimization (Appendix A) contains complementary material forChapter 2. Those global optimization and evolutionary algorithm related conceptsare presented here, whose discussions would have broken the train of thought and

1.4 How to Read this Document 9

continuity in Chapter 2, but their introduction is important to make the disserta-tion more self-contained.

Test Congurations and Results (Appendix B) details the congurations and re-sults of those muleGA tests presented in Chapter 3 which would have increased thelength of the chapter excessively.

1.4 How to Read this Document

The thesis focuses on the development of an expert system for health promotion, thereforeit belongs to a multidisciplinary research area. This entails that readers of the dissertationmight have dierent qualications and interests, therefore, they may choose to read onlyspecic sections of this document. In the following, few possible approaches for readingthe thesis are presented to the reader. It is assumed here that the reader has alreadyread the document until this point.

The comprehensive approach is advised to the readers who would like to understandthe author's research work thoroughly. This approach involves the reading of Chap-ters 2, 3, 4 and 5 with following the references to the corresponding Appendices toread the details not presented in the core chapters. Readers familiar with the con-cepts of evolutionary computation and global optimization can safely jump Chapter2 and Appendix A.

The evolutionary computation approach is advised to those readers who are inter-ested in the EA related methods and results only, and do not intend to get involvedwith health informatics. This approach involves the reading of Chapters 2, 3 and 5with following the references to Appendix A. Readers familiar with the concepts ofevolutionary computation and global optimization can safely jump Chapter 2 andAppendix A.

The health informatics approach is advised to those readers who are only interestedin the expert system infrastructure and those results which are applicable for healthpromotion. This approach involves the reading of Chapters 4 and 5.

The quick-read approach is advised for those readers who are looking for a shortsummary of the author's research. This summary is available at the conclusion ofthe thesis (in Section 5), in which all the pieces of the research work is presentedbriey.

There are forward and backward references throughout the document to allow thereader navigate conveniently between the chapters. The electronic version of the disser-tation available at http://www.gaalbalazs.hu/phd/thesis.html provides in-documenthyperlinks which ease navigation in the document.

Chapter 2

Evolutionary Algorithms and GlobalOptimization

This chapter serves as an introductory overview of the concepts of the thesis relatedresearch. No contributions are presented here, bar the denition of Multi-Level Opti-mization Problem (MLOP) (in Denition 7). The content of the chapter is framed to sup-port the discussion of the details and results of the author's research work (presented inSections 3 and 4). Both research areas of the thesis are about optimization, as the Multi-Level Genetic Algorithm (muleGA) (detailed in Section 3.4) is a new stochastic methodfor nding the global optimal solution of a Multi-Level Optimization Problem (MLOP),while the Dietary Menu Planning Problem (Section 4.4) belongs to the eld of combina-torial optimization and is being a MLOP (Section 2.4.1). Therefore, the main conceptsof optimization are presented in this chapter.

First, the objectives of global optimization and the concepts of Pareto optimality areintroduced (in Sections 2.1, 2.2 and 2.3). These concepts of global optima and the Paretoterminology are utilized through the whole dissertation.

Second, introduction is given on constrained (Section 2.3), multi-level optimizationtechniques (Section 2.4). This is not l'art pour l'art, as the real-life problem of dietarymenu planning formalized in Section 4.4 is both constrained and multi-level.

Third, an overview is given on global optimization methods (Section 2.5), with theaim of summarizing their performances on combinatorial optimization problems such asthe Knapsack Problems (KPs) which resemble to the dietary menu planning problem.This is carried out to help choosing the algorithm for solving the dietary menu planningproblem. It is no secret that a GA based method has been chosen for the task, more tothat, muleGA is actually built with this purpose in mind. The overview summarizes thepossible solution methods so they can be referred when the algorithm choice is explainedand underlined (in Section 4.5.1).

Fourth, Evolutionary Computation (EC) methods are introduced in short, as GAsbelong to this class of algorithms. The aim behind this is to present the main principlesof evolutionary models. Through this principles are the concepts of the Multi-LevelEvolutionary Algorithm (MLEA) (presented in Section 3.2) apprehensible.

2.1 Single-Objective Optimization 11

Fifth, an overview on the class of Genetic Algorithms (GAs) is given (in Section2.7). The area of GAs forms the most signicant part of the author's research work asboth the problem independent muleGA, and the muleGA based method of the nutritioncounseling expert system belong to this eld. Thus, concepts, parameters, types andattributes of GAs are presented in short.

Sixth, the most important GA related concepts and properties are introduced inSections 2.8, 2.9, 2.10, 2.11, 2.12, 2.13. In these sections, Multi-Objective GeneticAlgorithms (MOGAs), constraint handling techniques, co-evolution methods, parallelGAs, multi-level and hierarchical GAs and hybrid methods are presented respectively.The objective behind this is to briey introduce the complete eld of GAs. Let's call itthe space of GAs, in which every attribute of the GAs form a dimension (some discrete,some continuous, e.g.: type of tness function, selection method and crossover rate). Weare not going to dene operators on this space, nor explicitly specify each of its dimen-sions. It is going to be used as an imaginary concept, a multi-dimensional space, in whichevery GA implementation has its unique position. By using this phrasing, the aim ofSection 2.7 is to briey introduce the dimensions of the imaginary GA space, so thatany new GA, and thus muleGA can be positioned properly in it by identifying all of itscoordinates (attributes).

2.1 Single-Objective Optimization

During the global optimization process, the minimum or maximum of a search space issought. The following denition formally summarizes the single-objective global opti-mization problem [1].

Denition 1 (Global Minimum) Given a function f : Ωs ⊆ Rn → R, Ωs 6= 0, for~x ∈ Ωs the value f∗ , f(~x∗) > −∞ is called a global minimum, if and only if

∀~x ∈ Ωs : f(~x∗) ≤ f(~x) (2.1)

Then, ~x∗ is the global minimum solution(s), f is the objective function, and the set Ωs isthe feasible region. The problem of determining the global minimum solution(s) is calledthe global optimization problem.

It is important to note that the domain of f (Ωs, the search space) can be multi-dimensional even in single-objective optimization. The search space (also called decisionspace or criteria space) denoted Ωs is often a proper subset of the Euclidean space Rn,usually specied by a set of constraints (more on constrained optimization in Section2.3). The dimensionality of the search space is equal to the number of decision variables(comprised by ~x) the optimization problem has. The vector of the decision variablesis written as ~x = [x1, x2, . . . , xn]T , where the n decision variables are denoted withx1, x2, . . . , xn. The decision variables are those numerical quantities for which valuesare to be chosen in an optimization problem. There is a branch of optimization calleddiscrete optimization [2], where the Ωs (or any of its subspaces, and thus any of its decision

12 Evolutionary Algorithms and Global Optimization

variables) is restricted to discrete values such as integers. A notable branch of discreteoptimization is combinatorial optimization closely intertwining in research literature withinteger programming [3]. Generally, combinatorial optimization problems are formalizedwith integer programming models. These integer programming models are not furtherdetailed here, but are presented in Section 3.3.1, where an integer programming modelof the well-known combinatorial optimization problem, the Knapsack Problem (KP) isgiven. Regarding combinatorial optimization, the word combinatorial refers to the factthat only a nite number of alternative feasible solutions of the problem exist, those,which have the proper combination of values represented by their decision variables. Itcan be concluded that, considering single-objective optimization, the only necessity isthe singular dimensionality of the objective space, which is the codomain (target) of theobjective function f , and there are no assumptions on the dimensionality and structure ofthe search space. Naturally, multi-objective optimization (to be presented in the followingsection) deals with problems where the objective space has plural dimensionality.

2.2 Multi-Objective Optimization

It is simple to formalize the concept of optimality in the case of single-objective optimiza-tion (see Denition 1). However, if the objective space is multi-dimensional, supplemen-tary concepts and notations are needed to compare and measure multi-objective solutionsagainst each other. These supplements are the Pareto concepts which are presented inthe following.

2.2.1 Pareto Terminology

The denitions of Pareto dominance, Pareto optimality, Pareto optimal set and Paretofront are given in the following, along with the denition of Multi-Objective Problem(MOP) global minimum. These Pareto concepts presented here are from [4], and areused through the whole dissertation.

Denition 2 (Pareto Dominance) A vector ~u = (u1, . . . , uk) is said to dominate an-other vector ~v = (v1, . . . , vk) (denoted by ~u ~v) if and only if u is partially less than v,i.e., ∀i ∈ (1, . . . , k) ui ≤ vi ∧ ∃i ∈ (1, . . . , k) : ui < vi

Solutions (vectors) of a two-dimensional optimization problem are shown in Figure2.1, with solutions dominated by no other solutions highlighted in red. These solutionsare optimal in the Pareto sense. The denition of Pareto Optimality is given in thefollowing.

Denition 3 (Pareto Optimality) A solution x ∈ Ωs is said to be Pareto optimal withrespect to Ωs if and only if there is no x′ ∈ Ωs for which ~v = ~f(x′) = (f1(x′), . . . , fk(x

′))dominates ~u = ~f(x) = (f1(x), . . . , fk(x)). Then ~f is the multi-dimensional objectivefunction and f1, . . . , fk are the k objective functions for each dimension. The phrase

2.2 Multi-Objective Optimization 13

Figure 2.1: The solutions (some of them marked with uppercase alphabetic characters) of a two-dimension minimization problem are shown in the gure. Solutions A, B and D arenon-dominated (with all those other ones in color red), thus belonging to the Paretofront (PF∗), which is highlighted with a red line. Solutions C and E (and all ofthose colored blue) are dominated.

Pareto optimal is taken to mean with respect to the entire decision variable space unlessotherwise specied.

The decision variables of each Pareto optimal solution specify a point in the searchspace. The set of those points that belong to Pareto optimal solutions is the Paretooptimal set.

Denition 4 (Pareto Optimal Set) For a given MOP ~f(x), the Pareto optimal set(P∗) is dened as:

P∗ := x ∈ Ωs | ¬∃x′ ∈ Ωs ; ~f(x′) ~f(x) (2.2)

Denition 5 (Pareto Front) For a given MOP ~f(x) and Pareto optimal set P∗, thePareto front (PF∗) is dened as:

PF∗ := ~u = ~f(x) = (f1(x), . . . , fk(x))|x ∈ P∗) (2.3)

Denition 6 (MOP Global Minimum) Given a function ~f : Ωs ⊆ Rn → Rk,Ωs 6=0,k ≥ 2, for ~x ∈ Ωs the set PF∗ , ~f(~x∗i ) > (−∞, . . . ,−∞) is called the global minimumif and only if

∀~x ∈ Ωs : ~f(~x∗i ) ~f(~xi) (2.4)

The Pareto front (the global minima) of a two-dimensional minimization problem isshown in Figure 2.1. The gure depicts that A is better than B considering the rstobjective (f1) and worse considering the second one (f2). Both A and B dominates C,


as f1(A) < f1(C), f2(A) < f2(C), f1(B) < f1(C) and f2(B) < f2(C). Note that notevery solution of the Pareto front dominates C (D is better than C considering f1, butworse considering f2), but none of the solutions of the Pareto front are dominated.

In this thesis, whenever the concept of optimality comes up during a discussion re-garding a multi-objective problem, these Pareto concepts are used, whether or not theterm Pareto is explicitly stated.

2.3 Constrained Optimization

Most of the real-world optimization problems have restrictions imposed by various char-acteristics of the environment or the nite number and amount of available resources,necessitated by physical limitations, time restrictions and other requisites. A certainsolution of a problem is only acceptable if it satises all of the restrictions. The re-strictions are called constraints, which describe the dependences of decision variables toconstants or parameters of the optimization problem, and are expressed in the form ofmathematical inequalities or equalities:

gi(~x) ≥ 0 i = 1, . . . , n (2.5)

hi(~x) = 0 i = 1, . . . , p (2.6)

The number of equality constraints (p) must be less than the number of decisionvariables (n), otherwise the problem is said to be overconstrained (in case p ≥ n), asthere are no degrees of freedom left for optimizing. The degree of freedom is equalto n − p. The constraints are either explicit (given in algebraic form) or implicit (~xmust be known to compute gi(~x)). Those solutions which satisfy the constraints arecalled feasible, while those not are called infeasible. The problem of nding a feasiblesolution in the search space without the aim of nding the best feasible one is called aconstraint satisfaction problem. If the best solution or solutions are sought, then it iscalled constrained optimization. More details on constrained optimization are given inSection 2.9 and in Appendix A.5 from the perspective of Evolutionary Computation (EC)related constraint handling techniques.

The Knapsack Problem (KP) is a typical constrained optimization problem (and itis going to be dened in Section 3.3.1), which is briey introduced here. The goal ofthe problem is to ll a knapsack with boxes (each of the boxes has a weight and a protvalue), while simultaneously maintaining the weight limit of the KP and maximizingthe prot of the knapsack's content. There are many versions of the KP, of which thepopular variants are dened in Section 3.3.

2.4 Multi-Level Optimization

In multi-level optimization, the main optimization problem is decomposed into subprob-lems through some type of decomposition method, which ensures that the solution of the

2.4 Multi-Level Optimization 15

(smaller and thus more easily solvable) subproblems in some way can contribute to thesolution of the original problem. There is a kind of vagueness in this denition (sometype, some way) waiting for clarication. The reason of this vague paraphrasing is thatin the literature, the term multi-level optimization is used for at least three types ofmethods, which are presented in the following, under the names of multi-level hierarchyoptimization, multi-disciplinary optimization and multi-level granularity optimization.Also dened in the following is the class of Multi-Level Optimization Problems (MLOPs),which is one of the main focus areas of the dissertation.

2.4.1 Multi-Level Hierarchy Optimization

This type of multi-level optimization deals with problems on which a reasonable de-composition to separate levels of subproblems can be carried out (top-down approach,decoupled optimization) and with problems which are built up of various subproblemsforming a hierarchical structure (bottom-up approach, coupled subproblems). In what-ever way the hierarchy of subproblems is designed, they are solved quasi-independently,and then their results are used to congure their parent problems. This is basically adivide-and-conquer approach, which consists of three steps: rst, breaking the probleminto subproblems (considering the top-down approach), which are smaller instances of thesame type of problem, second, recursively solving these subproblems, and third, appropri-ately combining their results. This type of multi-level optimization is particularly popularfor system design optimization, and generally deals with single-objective problems. Acomprehensive overview of multi-level design optimization methods (Optimization byLinear Decomposition, Concurrent SubSpace Optimization, Collaborative Optimization,Bi-Level Integrated System Synthesis, Analytical Target Cascading and Quasi-separableSubsystem Decomposition) is given in [5]. A decomposition based optimization method(not particularly for design optimization) is the Dantzig-Wolfe decomposition [6], whichis a technique used to solve such linear programs which have special structure (constraintsthat can be divided to a set of general constraints and to a set of constraints which havespecial block angular structure). The algorithm works by iterating between solving themaster problem and solving the subproblems. In each iteration, new objective functionscan be calculated for the subproblems so they can oer subsolutions that improve theoverall goodness of the master problem. Both the design optimization methods and theDantzig-Wolfe decomposition are eective for such problem structures, in which the sub-problems are loosely coupled and rather independent of each other. If the subproblemsare weakly coupled, they can be solved concurrently. However, a sequential approach ispreferred for solving strongly coupled subproblems, making the latest available boundarydata obtainable [5].

The hierarchical and multi-level structure of the dietary menu planning problem con-sidered in this thesis (HMO-DMPP-H dened in Section 4.4.1.5) has similarities to thoseproblems for which the above methods were developed for, but there are a few cleardierences. First, its subproblems are multi-objective combinatorial optimization prob-lems, which have great resemblance to Multi-Objective Multiple-Constraint KnapsackProblems (MOMCKPs). Second, most of its constraints are coupling, so they contain all


or a signicant number of the decision variables, therefore no independent subproblemscan be formed. Third, the objectives of the subproblems are explicitly dened consideringexpert knowledge and dietary guidances, so they cannot be altered to enhance the resultsof their parent problems. Fourth, because the harmony of all components of the planshould be considered, no predictions can be made on the expected goodness of an un-elaborated solution. Despite these dierences, this problem still has similarities to othermulti-level optimization problems, hence it is discussed here, together with the otherdecomposable problems. In the following, an exact denition is given for the Multi-LevelOptimization Problem (MLOP) to clearly formalize the problem class this thesis focuseson. So, this novel MLOP is similar to those problems on which the traditional divide-and-conquer approaches can be eectively used (like the bi-objective multi-dimensionalknapsack problem [7]). The main dierence of the MLOPs considered here are that eachof their subproblems (from now on called input problems) is a separate optimizationproblem in itself (not just a smaller instance of the same type of problem). Therefore,the denition of the MLOP is given here.

Denition 7 (Multi-Level Optimization Problem) is dened as follows.

• Let P(h,v)(l,u) denote a global optimization problem (with search space Ωs(h,v)

(l,u) ), whichis positioned in a hierarchy of problems in the following way: the problem is on thelth level, it is the uth input problem of an hth level problem, which is the vth inputproblem of its parent problem. The quadruple (l, u, h, v) unambiguously determines

the position of P(h,v)(l,u) in the hierarchy.

• Let f(h,v)(l,u) : Ωs(h,v)

(l,u) → Ωs

(l,u)→(h,v) denote the input problem mapping function which

maps the decision variables of P(h,v)(l,u) (~x

(h,v)(l,u) ) to the specic subspace of its parent

problem. Note that the domain and co-domain of the mapping function (f) do notneed to have the same dimensionality, nor should f be a surjective function.

• The parent problems' search space is the Cartesian product of an input problemindependent subspace (Ωs

i (h,v)) and those subspaces determined by the input problem

mapping functions. Ωs

(h,v) = Ωs

i (h,v) × Πr(P(h,v))

u=1 Ωs

(l,u)→(h,v), where r(P(h,v)) is thenumber of input problems P(h,v) has. Note that P(h,v) is the parent problem of

P(h,v)(l,u) . In this context, it does not matter whether P(h,v) has any parent problems

or not (if not, it is interchangeably denoted with P(·,·)(h,·) or Ph), hence missing the

parameters of its parent from the notation. Also note that those decision variablesof Ωs

(h,v) which not belong to Ωs

i (h,v) are only congurable through the input problemmapping operators, so only a change in the input problem decision variables caninduce changes in the parent problem's specic decision variables.

• Let Pq be a q-level problem. Pq is called the main problem of an Multi-Level Opti-

mization Problem (MLOP) if there is a set of input problems P1(h1,v1)(l1,u1) ,. . . ,Pn

(hn,vn)(ln,un)

2.4 Multi-Level Optimization 17

Figure 2.2: The hierarchy of a three-level Multi-Level Optimization Problem (MLOP) P(·,·)(3,·) is

shown in the gure. The input problem mapping functions (f) are presented witharrows pointing towards the parent problems. Two special input problem mappingfunctions are depicted, one which maps decision variables to a lower level (dottedline), and one which maps to the top level from the bottom level (broken line).

(n ≥ 1), and there is at least one input problem (Pi(hi,vi)(li,ui)

, 1 ≤ i ≤ n) such that

hi = q. Then f(hi,vi)(li,ui)

: Ωs(hi,vi)(li,ui)

→ Ωs

(li,ui)→(hi,vi)and Ωs

(li,ui)→(hi,vi)⊆ Ωs

q. Then thehierarchy of these problems form the MLOP.

Note that from the denition, it follows that a hierarchical structure of optimizationproblems classies as a MLOP, if there is at least one problem (the main problem), towhich decision variables of at least one input problem are mapped. A 3-level MLOPis shown in Figure 2.2. P(·,·)

(3,·) is the main problem of the MLOP. Its decision variables

(occupying points in the search space Ωs(·,·)(3,·)) are at least partly dependent on the decision

variables of the three input problems P(3,·)(2,1), P

(3,·)(2,2), P

(3,·)(1,1). Note that P(3,·)

(1,1) could havebeen represented as a second level problem. The assignment of problems to levels is notan exact task, however the rule of thumb is to have the equal type and equally complexproblems at the same levels. Input problem mapping operators usually map from thebottom-up, but can map top-down (f(1,1,2,2)

(2,1,3,·) ). In case of the latter, the parent's andthe input problem's full identier (the (l, u, v, h) quadruples) are used in the notation forunambiguity.

Now, that the characteristics of the MLOPs are presented, it is again stated that inthis dissertation, those traditional global optimization problems which are solved more ef-fectively through the hierarchical, divide-and-conquer type decomposition of their searchspace are considered under the term multi-level. The acronym MLOP is going to be notedwhenever the extended version of the multi-level problem (in which each subproblem con-sidered as an autonomous optimization problem) is discussed (Denition 7). Note thatboth the Multi-Level Multi-Objective 0-1 Knapsack Problem (dened in Section 3.3.3)and the Dietary Menu Planning Problem (dened in Section 4.4.1.5) are MLOPs.


In the following, two types of problems are presented, which are also referred as multi-level in the literature. In this thesis, they are referred as Multidisciplinary Collaborativeand Multi-Level Granularity optimization problems, respectively.

2.4.2 Multidisciplinary Collaborative Optimization

Multi-Disciplinary Optimization (MDO) deals with solving design problems incorporat-ing many disciplines. Collaborative Optimization (CO) (also called MultidisciplinaryCollaborative Optimization) is a technique for solving MDO problems [8]. In CO, themain optimization problem is decoupled to subspaces (subsystem-level) coordinated by asystem-level optimization procedure (thus sometimes referred as a multi-level approach).The subsystem-level problems represent disciplines which are optimized simultaneously,with interactions between the subproblems. Although the subproblems in a MLOP arenot restricted to the same discipline, in this thesis no MDO applications are considered.

2.4.3 Multi-Level Granularity Optimization

There are optimization methods also termed as multi-level, which restrict the detail of agiven problem description continuously from the nest level through the ne level, untilthe granularity of the problem description reaches the coarse and coarsest descriptionlevels. The number of levels is problem dependent. After the problem is solved on thecoarsest representation level, the solution is prolongated to a level with ner granularity.The dierences in detail of description can be caused by at least the following threereasons: dierent mathematical formulations, dierent discretization limits for the sameformulation and dierent approximate empirical models on each level of the multi-levelmodel. In these kind of multi-level optimization problems, where there is dierencesin detail between the various levels describing the same problem, the cost of evaluatingthe models is directly related to the models' accuracies. Because of this, the searchprocess is ecient if detailed models only used if necessary. Twenty search methodsfor this kind of multi-level optimization has been tested in [9] with a GA based comingout on top. Although the denition of the MLOP does not restrict the subproblemsfrom describing the same problem with dierent granularity, in this thesis, this type ofmulti-level decomposition is not considered.

2.5 Global Optimization Methods

We are looking for global optimization techniques for solving such problems similar to thesubproblems of the Dietary Menu Planning Problem (dened in Section 4.4.1.5), whichare quite similar to Multi-Objective Knapsack Problems (MOKPs) (or more preciselyto Multi-Objective Multiple-Constraint Knapsack Problems (MOMCKPs) , dened inSection 3.3.2). Well, generally, we would be looking for a method which can solve thewhole hierarchy of optimization problems making up the MLOP, which is a multi-levelmulti-objective KP with combinatorial harmony (to be dened in Section 3.3.3), a newtype of optimization problem, which, to the extent of the author's knowledge, is discussed

2.5 Global Optimization Methods 19

Enumerative Deterministic Stochastic

Greedy

Hill-Climbing

Branch and Bound

Depth-First

Breadth-First

Best-First

Calculus-Based

Random Search

Simulated Annealing

Monte Carlo

Tabu Search

Evolutionary Computation

Mathematical Programming

Global optimization and search methods

Figure 2.3: Classication of Global Optimization Methods

here for the rst time. As the problem itself has not been studied elsewhere, it is a goodinitial step to analyze algorithmic methods for the solution of the suproblems of theMLOP.

Therefore, this brief overview focuses on the applicability of various global optimiza-tion algorithms for solving MOKPs. The methodology of these algorithms are not pre-sented here, although a short introduction to them is given in the corresponding Detailson Global Optimization Methods section in Appendix A.1.

Global optimization algorithms are generally classied into two categories, determin-istic and stochastic methods. Here, enumerative algorithms will form a third category, asin [10, 11]. While enumerative search is deterministic, it employs no heuristics. Therefore,it is rmly dierent from all the other deterministic algorithms and handled separately.The classication of global optimization algorithms is shown in Figure 2.3 (based on[11]).

There is no statistics available that precisely measures the popularity of the variousglobal optimization methods for solving MOKPs. To have some clue about the popu-larity of various methods, results for co-occurrences of strings multi-objective knapsackproblem and (name of the algorithms) are queried from Google, and are shown in Fig-ure 2.4. GAs come out on top by far, appearing in 26.6% of those pages multi-objectiveknapsack problem appears. Well, these results are a long shot from being a scienticproof, but there is no smoke without re, GAs tend to be the de facto tools for solv-ing MOKPs, at least for those who publish in such way, which is available to Google


Figure 2.4: Loose estimate on the application of various global optimization methods for solvingthe Multi-Objective Knapsack Problem (MOKP). Results of the number of co-occurrences of multi-objective knapsack problem and the respective algorithm namesindexed by Google. The percentage values show the share of the specic algorithmfrom the number the search for the single string: multi-objective knapsack problemresults, which is 20, 300 at the 22th of October, 2009.

for indexing. Note that the results returned by Google are not equal to the amount ofpapers published using a specic method, but they may serve as indicator referring tothe citeability of the methods in connection to the MOKP. The naming convention usedin the research eld of KPs is ambiguous, as many type of KPs are referred as MOKPs(bi-objective KPs, Multi-Objective Single-Constraint Knapsack Problems (MOSCKPs),MOMCKPs). This may also bias the statistics presented in Figure 2.4.

Also note that this brief overview of global optimization methods is not a survey andit does not try to be comprehensive. Its sole purpose is to introduce the basic attributesof popular global optimization techniques (with special focus put on MOKP solvers), andto serve as a base for the explanation of the algorithm choice (presented in Section 4.5.1)for solving the MLOP.

2.5.1 Enumerative Search

The enumerative search is possibly the most simple algorithm for nding the globaloptimum. It employs no domain knowledge about the problem, just simply evaluates eachand every possible solution in the search space. In a nite search space, this can be donein nite time, however, most of the real-world problems are computationally hard, and thewhole search space cannot be evaluated in reasonable time. Enumerative Search is alsoreferred as brute-force and exhaustive search. The good thing about this search methodis that if there is available time to exhaustively enumerate the search space, nding


PFtrue is guaranteed. With the mainstream availability of General-purpose computing ongraphics processing units (GPGPU) [12], and thus of Massive parallel processing (MPP),enumerative search techniques may gain popularity, at least through the hybridizationof heuristic search methods for the brute-force exploitation of the promising areas of thesearch space.

2.5.2 Deterministic Algorithms for Knapsack Problems

Deterministic Algorithms are algorithms which work predictably. For any particularinput, a deterministic algorithm always produces the same output. They use domainspecic knowledge to narrow the search space. A short introduction to those deter-ministic algorithms (greedy algorithms, hill climbing algorithms and branch and boundalgorithms) which are applicable in solving KPs is given Appendix A.1.1.

There are eective deterministic algorithms for KPs, although most of them are forthe single-objective case. Almost all of the multi-objective exact algorithms are forthe bi-objective case (branch and bound [13]), however, there is a novel exact dynamicprogramming method for the tri-objective KP [14]. Furthermore, the exact algorithms arefor the MOSCKP, bar for two methods, which solve the three objective three constraintproblem [15, 16]. The latter being a multicriteria branch and bound algorithm whichbelongs to the eld of mathematical programming.

2.5.3 Mathematical Programming methods for Knapsack Problems

Mathematical Programming methods for solving MOKP include integer linear programsolvers [17], linear program based heuristics [18] and dynamic programming techniques.Almost all of the mathematical programming based approaches deal with the bi-objectiveand single-constraint case. The rst dynamic programming method for the tri-objectivesingle-constraint KP case is by Bazgan et al [14]. Brief details of Mathematical Pro-gramming related models and methods (dynamic programming, linear programming)are presented in Appendix A.1.3.

The fact that exact and mathematical programming methods are only available forthe singe-, bi- and tri-objective KPs turns our attention to the stochastic methods.

2.5.4 Stochastic Algorithms for Knapsack Problems

There are a lot of stochastic methods which have been successfully applied to solvevariants of MOKPs, and as it is shown in Figure 2.4, GAs are the most popular ones.Some of the algorithms applied for solving MOKPs (Simulated Annealing (SA), ParticleSwarm Optimization (PSO), Tabu Search (TS)) are briey discussed in Appendix A.1.2.Note that the termmulti-objective is not always noted in front of the algorithm names, butnaturally, multi-objective versions of these methods are used for solving MOKPs, whichare referred as Multi-Objective Genetic Algorithm (MOGA), Multi-Objective SimulatedAnnealing (MOSA) [19], Multi-Objective Particle Swarm Optimization (MOPSO) [20].MOGAs are the de facto tools, so we discuss how the others compare to them.


Although tabu search showed good results in the Google query, Multi-Objective TabuSearch (MOTS) [21] algorithms available in the literature are only for the MOSCKP [22],bi-objective [23], or single-objective multi-constraint KP versions [24]. Also, to the extentof the author's knowledge, no comparison of MOTS and MOGA solvers are available forany MOKP variant. In the following, results of MOSA, MOPSO and MOGA basedmethods are introduced briey.

2.5.4.1 Multi-Objective Simulated Annealing (MOSA) for Multi-Objective

Knapsack Problems (MOKPs)

The advantage of Multi-Objective Simulated Annealing (MOSA) over MOGAs is also itsdisadvantage, as MOSA only has one search agent, therefore it only nds one solution ata time instead of a set of solutions (as population based methods). However, because ofthis, MOSA does not need a large amount of memory to maintain all the information ofa whole population. Also, with those one search agent, MOSA can nd a small group ofPareto solutions in a short time. This phenomenon is also shown for the MOKP in [25],where it is concluded, that the SA based approach has better solution time than the (fairlyold) GA based counterpart [26], although the latter provides better solution quality. It isshowed in a recent study [27] that MOSA has great performance on small multi-objectiveproblems, however, when the problem size and epistasis (interaction between genes)becomes large, MOSA is outperformed by MOGAs. We can draw the conclusion thatMOSA techniques are still in an early phase of development considering the elaboratenessof various MOGA approaches. While there can be advantages in applying MOSA forsolving MOKP, many of these can be matched by ne-tuned MOGAs. The single searchagent nature of SA is the biggest drawback of the method, when the solution of MOKPsand multi-level MOKPs are considered.

2.5.4.2 Multi-Objective Particle Swarm Optimization (MOPSO) for Multi-

Objective Knapsack Problems (MOKPs)

The techniques for PSO based multi-objective optimizations have notably evolved inrecent years, and as it is summarized in [20], variants of the MOPSO include aggregat-ing approaches, Pareto-based approaches and sub-population approaches, techniques andconcepts that are very similar to MOGA (to be discussed in Section 2.8). This is notsurprising at all, as there are many similarities between GA and PSO concepts, hencePSO is considered an EA (also by its authors) [28]. As it was the case with MOSA,the main advantage of MOPSO can become its main disadvantage considering multi-objective optimization. As particles follow a leader to scan the search space, PSO hasfast convergence, however, in a multi-objective search space this may result in a prema-ture convergence to a local front. It was shown in [29] that MOPSO can better NSGA-IIin calculating a small number of Pareto-solutions of a bi-objective optimization problem.However, when larger number of Pareto-solutions were required, this tendency disap-peared, or even inverted. Nevertheless, MOPSO variants have been successfully applied


to solve MOKPs [30, 31], but general superiority (if there is any) of MOPSO over GAsin solving high-dimensional MOKPs has not been proven.

2.5.4.3 Multi-Objective Genetic Algorithms (MOGAs) for Multi-Objective

Knapsack Problems (MOKPs)

The most popular MOMCKP solvers are undoubtedly the MOGAs. Popular MOGAmethods, namely NSGA-II (introduced in Section 2.8.3.5) and Strength Pareto Evo-lutionary Algorithm 2 (SPEA2) (introduced in Section 2.8.3.6) often form the base ofcomparison to other MOKP solvers [16]. The most inuential contributions of the eldof MOGA based MOMCKP solving are [32, 33].

The evolvedness of MOGAs for solving MOKP makes them the default choice asstarting point in the quest for the MLOP solver. Therefore, discussion of the concepts ofGAs, and in general EAs is given in the following section (in Section 2.6). Before that,the concepts of No Free Lunch (NFL) theorems are presented in the following, whichunderline that the eectiveness of MOGAs may not come as a free lunch (but maybe asa free appetizer, at least).

2.5.5 No Free Lunch (NFL) Theorems for Optimization

In the work of Wolpert and Macready [34], it is shown that, considering general-purposeblack-box optimization algorithms such as GAs or Simulated Annealing, which exploitlimited knowledge about the optimization problem on which they are run (both forstatic and time-dependent optimization problems), average performance of any pair ofalgorithms across all possible problems is identical. If the performance of some algorithma1 is superior to that of another algorithm a2 over some set of optimization problems,then the reverse must be true over the set of all other optimization problems. Even if oneof the algorithms is random, this is still true. Over the set of all optimization problems,any arbitrary algorithm a1 performs worse than randomly just as readily as it performsbetter than randomly. The NFL theorems of Wolpert and Macready [34] mean that ifan algorithm does particularly well on average for one class of problems, then it mustdo worse on average for the remaining problems, thus comparison of algorithms on afew sample problems with particular parameter settings are of limited utility consideringthe general behavior of those algorithms, however, for a narrow range of problems, thecomparison can be accurate and informative.

In real-life optimization problems, the superiorness of a particular algorithm is clearlyperceivable in several scenarios (which naturally does not contradict with the NFL theo-rems). It is suggested in [35], that restricted black box optimization scenarios (opposedto the unrestricted scenario of the NFL theorems) should be used to compare algorithmperformance and to show that a specic algorithm performs better over a subset of theentire set of problems than another. Droste et al [35] conclude: Perhaps not a free lunch,but at least a free appetizer is achievable.


2.6 Evolutionary Computation

In the eld of Evolutionary Computation (EC) Darwinian principles of evolutionary mod-els are used for solving computationally dicult problems. An Evolutionary Algorithm(EA) is an algorithm designed with the EC concepts in mind. EAs perform well on vir-tually any kind of optimization problem, and succeed in nding near-optimal solutionsutilizing that no assumptions are taken about the search space. EAs build on the evo-lution of individuals which represent potential solutions. The individuals making up apopulation are stochastically evolved, keeping Darwin's notion of survival of the ttestin mind, hopefully driving the population to represent more powerful candidate solutions.

EAs earned success in a diverse eld of applications, including engineering, biology,economics, genetics, operations research, robotics, art, physics and medicine. EAs havealso been used in articial life research to validate theories about biological evolution andnatural selection. While the generality of the EC concepts makes an EA a good choicefor near-optimal optimization tasks, the No Free Lunch theorem still holds, meaning thata general EA is anticipated to be outperformed by a highly domain specic algorithm,or even an other EA which employs domain specic knowledge.

Three dierent interpretations of the EC principles, namely Evolutionary Programming(EP) by Fogel [36], Genetic Algorithm (GA) by Holland [37] and Evolution Strategy (ES)by Rechenberg and Schwefel [38, 39], were developed independently of each other duringthe late 60's and early 70's. The fourth type of EAs, Genetic Programming (GP) wasintroduced [40] in the early 90's.

2.6.1 Genetic Algorithm

GAs are the most widespread type of EAs, and will be discussed thoroughly in thefollowing section (Section 2.7), they are introduced here briey to provide a base ofcomparison with EP, ES and GP.

The early type of GAs used binary representations for encoding individuals [41], andwhile many of the present day GAs still use binary encodings, other type of representa-tions evolved [42]. Standard GAs use stochastic selection which prefers individuals withbetter tness values, and the chance of selecting an individual for reproduction is propor-tionate to the respective tness of that individual. There are two typical form of creatingand handling populations during the iterations of the GA, generational dynamics andsteady-state dynamics.

In the generational GA, parents are selected, then recombined and mutated to createthe ospring population of the same size as the parent population. So the individualsare totally replaced during each iteration. In the steady-state GA, only one ospring isgenerated at a time, and it replaces the worst individual in the population.

2.6.2 Evolution Strategy

What mostly distinguishes ESs from GAs are the real-valued representation they use, andthe dynamics they employ to form create and handle populations during the evolution.

2.6 Evolutionary Computation 25

Real-valued representation means that each individual is an array of real numbers. Withmost of the implementations, there is only one genetic operator, namely mutation, whichis employed by the algorithm. The most common mutation operator, working on thecommon real-valued representation, modies each gene by some random value, whichmean value is zero and its standard deviation is adapted during the run.

With ESs, the population of the new iteration is created in two steps [43]. First, anospring population is created, which does not have to be the same size of the parentpopulation. Second, the parent population of the new generation is created by selectingindividuals from the ospring population and the previous parent population.

There are two dierent type of dynamics, called plus strategy, and comma strategy.In a (µ+λ) ES, there are λ osprings competing with each other and with the µ parentsfor surviving into the next population. In a (µ, λ) ES parents cannot make it into thenext population, only individuals of the ospring population are competing for survival.

2.6.3 Evolutionary Programming

As with ESs, the main operator of Evolutionary Programming (EP) is mutation. In EPthe emphasis is on the behavioral linkage between parents and their osprings. Thereis no material exchange made between the individuals in the population. Osprings aregenerated for each parent employing Gaussian mutation. Considering the tnesses ofthe individuals in the parent and the newly created ospring population, probabilisticselection is made to form the new parent population, quite similarly [44] to a (µ+µ) ES.

2.6.4 Genetic Programming

Genetic Programming (GP) is a methodology to evolve a population of individuals rep-resenting computer programs [45]. The tness of an individual is determined accordingto how well the program (represented by the individual) performs in a given compu-tational task. GP can be considered as a method for automatic programming, as itevolves programs that produce desired outputs for particular inputs. GP is far morecomputationally intensive than GAs, ESs and EP, because of the computation neededto evaluate the individuals and the huge population-size recommended, compared to theother techniques.

Most of the implementations represent programs in a tree-like structure. Crossoverand mutation operators are applied during evolution. In case of a tree representation,crossover simply interchanges tree-nodes among two individuals, while mutation can re-place a whole node, or the information represented in a node. As computer hardwareevolved, results produced by GP methods reached the level of human-made programs.Human-competitive results are from application domains including electronic circuit con-struction, Robot soccer-player creation, quantum-algorithms programming [46, 47].


2.7 The class of Genetic Algorithms

A Genetic Algorithm (GA) is a global stochastic search algorithm used for the solutionof dicult problems by the application of the principles of evolutionary biology and com-puter science. GAs use techniques such as inheritance, mutation, natural selection andrecombination derived from biology. In GAs, populations of abstract representations ofcandidate solutions evolve toward better solutions. The evolution starts with a popula-tion containing random individuals and happens in generations, in which stochasticallyselected individuals are modied (via recombination and mutation) to form the popula-tion of the next iteration. The genes of the chromosomes contain the information whereeach gene represents a property.

In the following, the base concepts of GAs are presented along with their most com-mon operators and applications. The introductory material is neither exhaustive norcomprehensive. Its main purpose is to lay the ground for the base of assessment of thenew methods presented in this thesis.

2.7.1 Base Concepts

Whenever a GA is congured to solve a particular problem, each individual in the pop-ulation of the algorithm represents a candidate solution of the problem at hand. Thechromosome, which can be considered as the blueprint of the individual, stores the in-formation about the solution. How the data about the solution is encoded depends onthe genetic representation of the problem. The chromosome is made up of genes, whichcan take several values, called alleles. One type of encoding is the binary representation,in which the state of each gene is either 0 or 1, making the chromosome equivalent to abitstring. The allele aects the expression of a particular trait. The locus of a gene is itsposition in the chromosome. The genotype is the specic allele makeup of the individual.The phenotype is the observable characteristic of the individual, which is the set of alltraits. These concepts are shown in Figure 2.5.

During the course of the evolution, individuals are selected for reproduction and forsurvival. These are the two kinds of selection mechanism the GA is working with. Theformer (selection for reproduction) selects individuals according to their goodness (whichis represented by their tness values), and in most of the cases, creates new individualsby applying the recombination and mutation operators on the selected chromosomes.Recombination creates osprings by exchanging the genetic information of the parents,while mutation randomly changes the alleles of the newly created osprings. The latter(selection for survival) selects the individuals for the next generation from the set ofnewly created osprings and from the current generation.

2.7.2 Representation

The encoding of the problem is the mapping of the phenotype to the genotype, whiledecoding is the inverse operator, which calculates the object parameters (parameters ofphenotype) from the genotype. The genotype encodes the genetic information of the

2.7 The class of Genetic Algorithms 27

Genotype space Phenotype space

0 0 1 0

1 1 1 1popu

latio

n

chromosome

gene

allele

locus 5. 4. 3. 2. 1.

genotype

Encoding

Decoding

-4 2

3

0

1

3

-2

2 individual 21 = 2

phenotype -42 = 16

trait 33 = 9

fitness(-2,0) = -20 = 11 1 0 00

0

1

1 0 0 10

Figure 2.5: A population with four individuals, with basic GA terminology highlighted. Thegenotype (binary string encoding) and phenotype (an unsigned integer representedat locus 1-2, and a signed integer represented in two's complement form at lo-cus 3-5) of the individuals are shown, as well as the calculated tness values((signed integer)(unsigned integer)), for each individual.

individual, it is the representation of the problem. Crossover and mutation operators acton the genotype. In case every object parameter in the phenotype has a unique statein the genotype, then the representation is called direct, otherwise indirect. Traditionalrepresentations are the following: binary representation, real-coded representation, per-mutation encoding, value representation, tree representation. These are presented insome detail in Appendix A.2.1.

Usually binary encoding (Appendix A.2.1.1) is used for the representation of 0-1KPs (dened in Section 3.3.1), where each bit of the binary string represents whether aspecic item (identied by the locus of the bit) is in the knapsack or not. For bounded andunbounded KPs (where there are limited or unlimited copies of the same item available,respectively) integer encoding is used, meaning that each allele is an unsigned integervalue that represents the number of a particular item in the knapsack.

2.7.3 Evaluation

Evaluation of the individuals is done by calculating an objective-score for each individual.In case of single-objective optimization, the objective-score is a real number, in case ofmulti-objective optimization, the objective score is a vector of real numbers (as thereis an objective score for each objective). The tness of the individual, which is a realnumber, is calculated from the objective score(s) with a possibility of taking the otherindividuals of the population into account. The objective score (or considering multi-objective optimization, each objective score) is problem specic and therefore it shouldnot be modied with the aim of enhancing the convergence of the evolution process.However, the mapping of the objective score(s) to a tness value makes it available to


adjust the goodness of an individual for selection, as selection only takes the tness valueas parameter. Considering the multi-objective case, which will be detailed in Section 2.8,the objective-score to tness mapping is an Rn → R mapping, that takes the n-vector ofobjective-scores and represents it in a single tness value, thus hiding the multi-objectivenature of the problem from the selection operator. During the following part of thesection, only the single-objective case is considered.

The type of the objective-score to tness mapping is either scaling or ranking. Incase of scaling, the tness is a linear or non-linear function of the objective-score, whilein case of ranking, the population is sorted according to the objective-score, and eachindividual is assigned a tness value according to its position or rank in the population(two individuals with the same objective-score will get the same rank, but not the sameposition). A more detailed introduction to scaling and ranking is given in in [48].

Note that in many cases, the objective-score to tness value mapping is the identityfunction (f(x) = x), with the tness value being equal to the objective-score.

2.7.4 Operators

2.7.4.1 Selection

As it was mentioned earlier, in the general EA, two kinds of selection happen: selectionfor reproduction and selection for survival. The rst selects the individuals from thepopulation for ospring creation, while the second selects the individuals of the newpopulation. This section presents the selection techniques which select individuals forreproduction.

Selection for survival is a GA variant specic algorithm, and thus, it is usually de-tailed together with the evolution process of that particular GA. The default is to selecteach newly created ospring to survive into the next generation, and have all the individ-uals of the current population unselected. Algorithms which work this way are termedgenerational GAs, while those that not replace the whole population in each iteration arecalled steady-state GAs. In the following, the concepts of normalized tness and relativetness from [48] are presented here.

Normalized tness of a parent individual is the fraction of the population's total num-ber of ospring the given parent is assigned to produce in the next selection step.The value of the normalized tness is in the interval [0, 1], and is often seen as aprobability value, which is true for some implementations of selection algorithms.

Relative tness of an individual is the tness of the individual normalized by theaverage tness of the population. The relative tness value gives a direct indicationon how much the individual is better or worse than the current average individuals.

The most widely used selection algorithms are the following: roulette wheel selection,tournament selection, truncation selection, linear ranking selection, exponential rankingselection. These are presented in details in Appendix A.2.2. A more detailed descriptionof selection schemes is presented in [49].


2.7.4.2 Crossover

Crossover is a special form of recombination, used in GAs to create two osprings fromtwo parents by exchanging the genetic information stored in the parent chromosomes.Crossover is responsible for exploring the search space by making big jumps in it throughthe exchange of the alleles of parent individuals. There are many variants of crossover(three of the most popular are shown in Figure 2.6). Which one is preferred depends onthe encoding and on the problem.

In binary encoding, the most common crossover operators are single point crossover,two point crossover, n-point crossover, uniform crossover and arithmetic crossover.

(a) One point crossover (b) Two point crossover (c) Uniform crossover

Figure 2.6: The most commonly used crossover operators for binary representations

In single point crossover, shown in Figure 2.6(a), one crossover point is randomlyselected (in this example it points to the 4th gene) and the two osprings are created byinterchanging the bits of the parents after the crossover point. Analogously, in two pointcrossover (Figure 2.6(b)), the genes of the chromosomes are interchanged before and afterthe crossover points. N-point crossover works by interchanging genetic information be-tween odd intervals (determined by crossover points 2k− 1 and 2k, where k = 1, . . . , n/2) and keeping them between even intervals. In uniform crossover, shown in Figure 2.6(c)bits are randomly copied from the parents. Arithmetic crossovers perform arithmetic op-erations on parents to form the new osprings, for example, with AND or OR operators.In shue crossover, chromosomes are shued before single-point crossover occurs, andthey are consequently deshued after [50]. In reduced surrogate crossover, one of thepreviously presented crossover operators are applied, but only to the non-identical bitsof the chromosomes. This guarantees that the osprings created using reduced surrogatecrossover will dier from their parents.

2.7.4.3 Mutation

The mutation operator in GAs is responsible for maintaining the genetic diversity ofthe population. Mutation is the only operator that can introduce new alleles in thepopulation. Considering the binary representation, the most commonly used mutation


operator is the bit inversion, which randomly selects bits from the bitstring and invertsthem. Besides that only mutation can introduce new alleles, it also helps avoiding localminima, by preventing the individuals from becoming too similar. Mutation is alsoresponsible for exploiting promising areas of the search space, which is done by performingsmall alterations on some or all of the alleles.

2.7.5 Feasibility of Osprings

The crossover and mutation operators may render the ospring individuals into infeasiblesolutions of the problem they encode. The phenomena of infeasible osprings are frequentin the eld of constrained optimization and also in problems which use permutationencoding (Appendix A.2.1.3).

Basically, there are two ways to have feasible ospring: rst, by not rendering theminfeasible, which can be achieved by employing feasibility-preserving operators [51], orsecond, by applying repair algorithms. Popular repair schemes are the Lamarckian andBaldwinian methods, which are frequently applied in EAs solving KPs. Both are greedyrepair methods. Lamarckian repair modies the genetic information of the individual,while Baldwinian repair is executed on the y at each tness evaluation, keeping thegenetic information of the individual intact. The comparison of the two methods aregiven in [52]. More on the feasibility of solution regarding constraint handling is givenin Appendix A.5.

2.7.6 Formal denition of GAs

An abstract formalization of the EA concepts (representation, evaluation of individuals,operators) presented in the former sections are given in the following. A frameworkdeveloped by Merkle and Lamont in 1997 [53], and by Bäck in 1996 [1], is presented herein Denition 8 and in Algorithm 1 to formalize and mathematically describe evolutionaryalgorithms and GAs (as GAs belong to the class of EAs). The denition of the formalframework is presented in [11]. The formalism gives us a more precise specication onwhat EAs are and how they work.

Before giving the formalization of EAs (see Denition 8), Merkle and Lamont denethe following concepts: decoding function, tness function, population transformation,random population transformation, evolutionary operator, recombination operator, mu-tation operator, selection operator. The denitions of these concepts are given in Ap-pendix A.3 to increase the lucidity of Denition 8, which is quite compact and details of itare hardly apprehensible without the complementary denitions. A more simple outlineof the EA is given in Algorithm 2 to make the discussion of EAs more apprehensible.

Denition 8 (Evolutionary Algorithm) Let I be a non-empty set (the individualspace), µ(i)i∈N a sequence in Z+ (the parent population sizes), µ′(i)i∈N a sequencein Z+ (the ospring population sizes), Φ : I → R a tness function, ι :

⋃∞i=1 (Iµ)i →

true, false (the termination criterion), χ ∈ true, false, r a sequence r(i) ofrecombination operators r(i) : X(i)

r → T(

Ω(i)r , T

(Iµ

(i), Iµ

′(i)))

, m a sequence m(i) of


mutation operators m(i) : X(i)m → T

(Ω

(i)m , T

(Iµ′(i), Iµ

′(i)))

, s a sequence s(i) of se-

lection operators s(i) : X(i)s × T (I,R) → T

(Ω

(i)s , T

((Iµ′(i)+χµ(i)

), Iµ

′(i)))

, θ(i)r ∈ X(i)

r

(the recombination parameters), θ(i)m ∈ X(i)

m (the mutation parameters), θ(i)s ∈ X(i)

s (theselection parameters), where T (S1,S2) denotes the set of mappings from the set S1 to S2,sample spaces for random events associated with the evolutionary operators (see AppendixA.3 and [53] for details) are denoted with Ω. Then the algorithm shown in Algorithm 1is called an Evolutionary Algorithm.

For the sake of clarity and readability, a simplied, and thus very shallow outline ofEA is presented in Algorithm 2. This simple description hides the attributes of the evolu-tionary operators (selection, recombination, mutation) behind function calls (for examplelike recombine-individuals for recombination). Only input and output parameters ofthese functions are shown in this simplied description.

Algorithm 1 Evolutionary Algorithm Outline

χ ∈ true, false is a design parameter which is decided and set at compile-timet := 0;initialize P(0) := s1(0), . . . , sµ(0) ∈ Iµ(0);while (ι(P(0), . . . ,P(t)) 6= true) do

recombine: P ′(t) := r(t)

Θ(t)r

(P(t));

mutate: P ′′(t) := m(t)

Θ(t)m

(P ′(t));if χ then

P(t+ 1) := s(t)

(Θ(t)s,Φ)

(P ′′(t));else

P(t+ 1) := s(t)

(Θ(t)s,Φ)

(P ′′(t)) ∪ P(t));

end if

t := t+ 1;end while

Algorithm 2 Evolutionary Algorithm Outline (simple form)

t := 0;initialize-population(P(0))while (terminal-condition 6= true) do

P ′(t) := recombine-individuals(P(t))P ′′(t) := mutate-individuals(P ′(t))P(t+ 1) := select-individuals-for-survival(P(t),P ′′(t))t := t+ 1;

end while


2.8 Multi-Objective Genetic Algorithms

Until this point, only references were given to those type of GAs which can handle prob-lems having multiple objectives. These GAs are the Multi-Objective Genetic Algorithms(MOGAs), which are intended to nd the global optima of a given multi-objective prob-lem. Considering a minimization problem, the MOGA's goal is to produce solutionswhich belong to the global minimum dened in Denition 6. Because the problemsconsidered in this thesis are multi-objective, the MOGA based solution approaches arebriey introduced here. The classication of MOGA techniques from Veldhuizen andLamont [54] is employed here, in which MOGAs are grouped according to when thosecompromise between the multiple objectives is made, which picks the nal solution ofthe Multi-Objective Problem (MOP) out of the non-dominated solutions. This nalcompromised choice is made by the domain expert, and according to when it is made,either before, during or after the optimization process, the technique is called A Priori,Progressive or A Posteriori, respectively.

2.8.1 A-priori Techniques

A-priori techniques are the simplest way of handling multiple objectives in a GA, andthey are applied in scenarios, in which the signicance of each objective is known inadvance. In these cases, only the approximation of a small region of the PF∗ is needed.That region which contains those solutions of the problem that are good consideringthe most signicant objectives (or in case each objective has the same signicance, thenare equally good in all objectives). A-priori techniques order weights or goals to eachobjective before the search process starts. By using these, the goodness of any givensolution is averaged by some means to be represented as single-dimensional objectivevalue. It is discussed in [55] that aggregating a-priori techniques can outperform Pareto-dominance based MOGAs in high-dimensional problems. So, the aggregating approach issimple and computationally eective, and it also can perform better than more complexmethods in special circumstances. In the muleGA evaluation experiments, an aggregatingtness function is used for testing muleGA's performance on high-dimensional and multi-level KPs (Section 3.5.4). The popular a-priori methods are presented in the following.

2.8.1.1 Linear Fitness Combination

f(~x) =k∑i=1

ωifi(~x) (2.7)

Where fi is the objective function for the ith dimension. The relative importanceof the k objective functions are represented through the weighting coecients ωi ≥ 0.Usually it is assumed that the sum of the coecients is one.

k∑i=1

ωi = 1 (2.8)

2.8 Multi-Objective Genetic Algorithms 33

2.8.1.2 Nonlinear Fitness Combination Techniques

Nonlinear tness combination techniques include multiplicative techniques, target vectorapproaches and the minimax technique.

Multiplicative techniques The individual objective functions are combined throughmultiplication.

f(~x) =

k∏i=1

fi(~x) (2.9)

Target vector approaches Performance goals are assigned to each objective and thevalues of the solutions are evaluated considering their distance from their goals.Dierent metrics can be used.

f(~x) =∥∥∥[~f(~x)− ~g

]W−1

∥∥∥α

(2.10)

Where ~g is the target vector, W is the weighting matrix, and usually α = 2 makingthe distance Euclidean (the norm of an arbitrary vector ~v is denoted with ‖v‖).

Minimax technique The maximum dierence between the goals and the objectives isminimized.

f(~x) = maxi=1...k

|fi(~x)− gi|wi

(2.11)

2.8.2 Progressive Techniques

In the case of progressive techniques, the domain expert or decision maker is involvedin an interactive search process. During the evolutionary search, both a priori and aposteriori methods can be used. Because of the human interaction progressive techniquesresemble to Interactive Evolutionary Computation (IEC) in which human evaluation isemployed [56], and to Human-based Genetic Algorithms (HBGAs) in which all primaryevolutionary operators are outsourced to humans [57].

2.8.3 A-posteriori Techniques

A-posteriori techniques try to approximate the full Pareto front and nd a set of Paretosolutions. When the search has nished, the decision maker can evaluate the results andpick the nal solutions from the known Pareto solutions Ptrue. The muleGA tests onnumerical MOPs are carried out with Pareto-based techniques. Test conguration andresults are presented in Section 3.5.5. In the following, the most popular a-posterioriapproaches are briey introduced.

The majority of A-posteriori techniques presented here employ Pareto-based t-ness assessment to nd the Pareto optimal set (P∗) and the Pareto front (PF∗). The


only exception is the Vector Evaluated Genetic Algorithm (VEGA) which uses an ag-gregation selection technique. Details of NSGA-II, SPEA2 and Micro-Genetic Algo-rithm for Multi-Objective Optimization (MicroGA) are presented in Appendix A.4.1(as they are implemented in MOGALib, a multi-objective GA library, which has beendeveloped in connection with this thesis and on which the muleGA implementation isbuilt on. MOGALib is an extension to GALib, a popular GA library, and is available athttp://mogalib.uni-pannon.hu).

2.8.3.1 Vector Evaluated Genetic Algorithm (VEGA)

The main idea of the algorithm developed by Schäer [58, 59] is that k subpopulations

of sizeµ

kare selected with dierent objectives. The individuals of the subpopulations

are mixed and then crossover and mutation operators are applied on them in the usualway to form the succeeding population.

2.8.3.2 Goldberg's Pareto Ranking Algorithm

Goldberg suggested a GA [41] that assesses the individuals of a population using Paretodominance theory. He proposed the following iterative tness assignment. Individualsthat are Pareto non-dominated by the rest of the population are sought, and the highestrank is assigned to them. These individuals are then removed from the competition, andanother set of non-dominated individuals is sought in the remainder of the population.The individuals of this set are assigned with the second highest rank. This iterationends when all of the individuals get ranked. A niching technique called sharing was alsoproposed by Goldberg to prevent convergence to a single non-dominated vector. Sharingshould be performed by dividing tness values of individuals by a quantity proportionalto the number of neighboring individuals [41].

2.8.3.3 Multi-Objective Genetic Algorithm (MOGA)

The method proposed by Fonseca and Fleming [60] and used in the MOGA employsa Pareto-based tness assignment technique based on Goldberg's Pareto ranking. Therank of an individual is calculated as follows.

rank(xi, t) = 1 + pi(t) (2.12)

The rank of xi at generation t depends on pi(t) which denotes that how many indi-viduals dominate xi.

2.8.3.4 Non-dominated Sorting Genetic Algorithm (NSGA)

As dened by Srinivas and Deb in [61], only the way NSGA carries out selection variesfrom simple genetic algorithm. In NSGA, before the regular selection is performed usinga dummy tness value, individuals are ranked based on non-domination. The samedummy tness value is assigned to the individuals of the rst non-dominated front, and

2.8 Multi-Objective Genetic Algorithms 35

then sharing is performed on these tnesses, according to the density of the individuals onthe non-dominated front. These individuals are temporarily ignored then, and the secondnon-dominated front is sought and assessed in the same way, ensuring that the individualsof the second front will have smaller dummy tness values than the minimal shareddummy tness of the previous front. The process continues until the whole populationis classied into several fronts. Then, the population is reproduced, using the dummytness values into which multiple objectives were reduced to. The details of the algorithmare presented in Appendix A.4.1.1.

2.8.3.5 Non-dominated Sorting Genetic Algorithm II (NSGA-II)

Non-dominated Sorting Genetic Algorithm II (NSGA-II) from Deb, Agrawal, Pratap andMeyarivan [62, 63] is a modied version of NSGA [61]. NSGA-II improves the complexityof non-dominated sort from O(M ·N3) to O(M ·N2) (whereM is the number of objectivesand N is the size of the population) and in contrary to NSGA, it incorporates elitismwhich speeds up performance signicantly, and NSGA-II also eliminates the need tospecify the sharing parameter a priori. The pseudo-code of the fast non-dominated sortingalgorithm of the NSGA-II with time complexity of O(M ·N2) is presented in Algorithm7, in Appendix A.4.1.2. The pseudo-code of the crowding distance assignment algorithmis presented in Algorithm 8, while the pseudo-code of NSGA-II is presented in Algorithm9, also in Appendix A.4.1.2.

2.8.3.6 Strength Pareto Evolutionary Algorithm 2 (SPEA2)

The Strength Pareto Evolutionary Algorithm 2 (SPEA2) from Giannakoglou et al [64] isan improved version of Strength Pareto Evolutionary Algorithm (SPEA), and accordingto the authors, it performs better on all the test problems they used, than its predecessor[64]. The SPEA2 has an enhanced tness assignment strategy compared to SPEA, and itemploys new techniques for archive truncation and density-based selection. The pseudo-code of the SPEA2 algorithm is presented in Algorithm 10 and the details of the algorithmis introduced in Appendix A.4.1.3.

2.8.3.7 Micro-Genetic Algorithm for Multi-Objective Optimization (MicroGA)

The Micro-Genetic Algorithm diers from the previous algorithms as it maintains morethan one population. The pseudo-code of the Micro-Genetic Algorithm is shown inAlgorithm 11 in Appendix A.4.1.4.

2.8.3.8 Other A-posteriori algorithms

Other A-posteriori algorithms frequently cited and mentioned in the literature are theNiched Pareto Genetic Algorithm (NPGA) [65], Niched Pareto Genetic Algorithm 2(NPGA2) [66], Pareto Archived Evolution Strategy (PAES) [67], Strength Pareto Evo-lutionary Algorithm (SPEA) [68], Pareto Enveloped-based Selection Algorithm (PESA)


[69], Multi-Objective Messy Genetic Algorithm (MOMGA) [10] Multi-Objective MessyGenetic Algorithm (MOMGA-II) [70], NSGA-II with controlled elitism [71].

2.9 Constraint Handling

Another important attribute of GA implementations is if and how they handle con-straints. Basically, GAs are unconstrained optimization techniques, so for solving con-strained optimization problems (Section 2.3), constraint handling methods need to beincorporated into their tness function for rewarding feasible solutions and decreasingbreeding possibility of infeasible individuals. In the corresponding Details on ConstraintHandling section in Appendix A.5, an insight is given on the state of the art in constraint-handling techniques for EC, based on the comprehensive surveys of [72, 73, 74]. The mostcommon approach of incorporating constraints into EAs is done by the means of penaltyfunctions (detailed in Appendix A.5.1, with all the other methods for constraint-handlingare introduced in Appendix A.5.2).

Penalty functions transform a constrained optimization problem into an unconstrainedone by penalizing an infeasible individual according to the amount of constraint violation.Guidelines for engineering good penalty functions are also given in Appendix A.5.1.

2.10 Co-Evolution

There are so-called co-evolutionary GAs which evolve individuals whose tness is depen-dent on other individuals (residing in the same or in separate populations). The biologicalmeaning of co-evolution is the mutual evolutionary inuence involving two species, whichare exerting selective pressures on, and thereby biasing the evolution of each other. Inthe context of evolutionary computation, the co-evolution of individuals stands for suchan evolutionary process, where the tness of an individual is a subjective function of itsinteractions with other individuals. The relation of the individuals, according to whichthe tness is computed, is called symbiosis. Many variants of co-evolution methods exist,however, two main categories can be specied.

In the rst category of co-evolutionary implementations, pairs or groups of individualstake part in a tournament, and are assigned relative competitive tness values accordingto how they performed in the tournament. This way, the tness of an individual is aectedby other individuals from the same population, thus making the GA co-evolutionary [75].

To the second category belong the systems with multiple populations, where a prob-lem is broken into components, and the components are assigned to these populations.The tness values of the individuals of these populations are calculated by randomlychoosing complementary individuals from the other populations and evaluating how thenewly formed group of individuals solve the problem [76].

Details of co-evolution related concepts, notations and categorization of co-evolutionaryalgorithms are presented in the corresponding Details on Co-Evolution section in Ap-pendix A.6.

2.11 Parallel Genetic Algorithms 37

2.11 Parallel Genetic Algorithms

Another attribute of GA implementations is how the steps of the evolutionary iterationsare performed. The default way is the sequential execution of the evolutionary steps,which has been presented in Algorithm 2.

The parallel implementations of GAs can give substantial gains regarding performanceand they are relatively easy to implement [77]. The idea of parallel GAs uses the divide-and-conquer approach, which can be utilized many dierent ways.

Master-Slave model uses a single population, and only parallelizes the tness calcu-lations. The entire population is considered for selection and crossover in eachiteration, so the master-slave model is attributed as being global.

Island model GAs which are also termed multiple-population, multiple-deme or dis-tributed GAs are based on several isolated subpopulations. Controlled migrationoccurs between these subpopulations to exchange individuals.

Diusion model GAs are also called ne-grained GAs as each processor holds one toa few individuals, and the processing nodes are arranged in a spatial conguration.Recombination and mutation can only occur in neighboring nodes. Because of thislocality, good solutions can only spread the whole population by diusing throughthe means of recombination of neighboring nodes.

2.12 Multi-Level and Hierarchical Genetic Algorithms

The terms multi-level and hierarchical are used as attributes for GAs. A coherent namingconvention has not yet evolved for GAs utilizing hierarchical structures. Algorithms withquite diering mechanisms were given the attributes multi-level or hierarchical. In thefollowing, a short summary is given on the various type of GAs bearing the name orattribute multi-level or hierarchical.

The term hierarchical can refer to an optimization problem which has hierarchicalstructure, to spatial positions of populations of parallel GAs or to the structure of al-leles in the genomes. Examples for the latter, namely for hierarchical structures in thechromosome, are presented in [78] and [79]. In [79], genetic information is stored in asemi-hierarchical structure (thus the name structured genetic algorithm) where higherlevel genes determine the activation state of their corresponding lower level counterparts.This way, chromosomes store redundant information in the form of deactivated genes.The same mechanism is exploited in [78] (named Hierarchical Genetic Algorithm), wherethe states of the control genes dene the status of corresponding parameter genes.

Another algorithm also entitled as Hierarchical GA [80] works quite dierently. Itidenties and exploits the hierarchical structure of optimization problems with crossoverand mutation operators respecting and not disrupting the already found subsets of vari-ables which are near-optimal in some context.


Examples of GAs with multiple populations managed in hierarchical structure are[81], [82] and [83]. A similarly hierarchical algorithm with parallel populations is intro-duced in [77].

Most if not any of the multi-level termed GAs are multi-population algorithms. Twomain groups can be formed for multi-level termed GAs. In the rst group belong thoseimplementations which beside the traditional population(s) of the GA, also maintain ameta-level population with individuals encoding algorithmic parameters. The meta-levelGA is used to evaluate a large set of algorithmic settings for the base GAs [84, 85, 86].These type of algorithms are mostly called Meta-Level GAs.

Into the second group GAs termed multi-level belong those implementations whichuse multiple populations to solve parts (subproblems) of the overall problem with eachpopulation, and then use another population [87] or any other mechanism to merge thesubresults [83]. In most of the cases, the multi-level implementations are restrained to twolevels [87, 88, 89]. More recently, implementations with three or more levels have arisen[90]. Interestingly, two of these independently created algorithms were developed for com-puterized dietary menu planning [91, 92]. Recent publications deal with the automaticand adaptive division of the objective vectors into subcomponents [93]. Decompositionof the problem and co-evolving subcomponents in a single level of subpopulations also apopular research topic [94, 95], and in some way similar to hierarchical approaches withtwo levels.

2.13 Hybrid methods

Another attribute of GA implementations is whether they use other techniques duringthe evolutionary process. If so, they called hybrid GAs. The simple and straightforwardalgorithmic structure of GAs give the option of convenient hybridization. Many kindsof algorithms were partnered with GAs with success [96, 97, 98]. A relatively new evo-lutionary method which belongs to the class of hybrid GAs is the Memetic Algorithm.Unlike GAs, which evolve genes, Memetic Algorithms evolve memes, which can adaptthemselves in contrary to genes, which only evolve through the evolutionary operators ofthe GA. Basically, it is an extension of GAs with local search techniques [99], separateindividual learning and improvement procedures [100]. Anyway, there is no strict borderbetween GAs and Memetic Algorithms. The concept of self-adapting individuals is notat all new. For example, repair algorithms of crossover operators for Travelling SalesmanProblems (TSPs) also adapt and correct individuals.

2.14 Conclusion on Evolutionary Algorithms and Global Op-timization

In this introductory chapter, Evolutionary Algorithm (EA) and global optimization re-lated concepts and techniques were discussed, with the aim of laying the ground mainly

2.14 Conclusion on Evolutionary Algorithms and Global Optimization 39

for the discussion of muleGA, which is a contribution of this thesis (Csec1 ), and it is goingto be presented in the next chapter.

First, the various types of optimization problems were presented (single-objective,multi-objective, constrained and multi-level optimization problems in Sections 2.1, 2.2,2.3 and 2.4, respectively). As the sole contribution of the chapter, the denition of theMulti-Level Optimization Problem (MLOP) was given (in Denition 7), which is a spe-cial type of optimization problem this thesis focuses on. Next, such global optimizationalgorithms were discussed and compared (in Section 2.5) which perform well on Multi-Objective Knapsack Problems (MOKPs) (a problem which resembles to the subproblemsof the dietary menu planning problem, to be specied in Section 4.4.1.5). It was concludedin the introduction of methods for KPs that there are no algorithms which better GAsconsidering their performances on MOKPs. Therefore, a brief summary on EvolutionaryComputation (EC) was given (in Section 2.6) and the class of Genetic Algorithms (GAs)was presented (in Section 2.7). In a widespread overview on the many type of GAs,all their major concepts, attributes and implementations were briey introduced. Thesewere the following: evolutionary operators, individual encoding and evaluation, multipleobjectives and constraints handling, co-evolution and parallelism, multi-level, hierarchi-cal and hybrid implementations. Those concepts whose discussion would have brokenthe train of thought are presented in the corresponding appendix chapter: Details onEvolutionary Algorithms and Multi-Objective Optimization (Appendix A).

This introduction showed that there is a plethora of ways for extending GAs. All ofthese extensions and all the GA variants exist because of a reason. Regardless of themany methods developed so far, there are still shortcomings of GA based approachesin solving high-dimensional multi-objective problems. Furthermore, there has been verylittle research made on the multi-level extension of GAs, although the number of contri-butions has increased recently. One such contribution is presented in the next Chapter,for whose discussion, all the necessary concepts have been presented in this introductorypart of the dissertation. The goals behind the development of muleGA are to improveperformance of GA based approaches in solving high-dimensional multi-objective prob-lems, and to provide means for solving MLOPs. Employing the concepts presented inthis introduction, new methods and contributions are presented in the following Chapter.

Chapter 3

Multi-Level Genetic Algorithm(muleGA)

3.1 Introduction

A novel GA, the Multi-Level Genetic Algorithm (muleGA) is presented in this chapter,which has been developed by the author to provide an eective way for solving Multi-Level Optimization Problems (MLOPs) (see Denition 7) such as the dietary menu plan-ning problem (in Section 4.4.1.5), and other high-dimensional, multi-objective problems.

First, a conceptual overview is given on MLEAs (in Section 3.2.1), and a novel formalframework is presented for the uniform description and handling of Multi-Level Evolu-tionary Algorithms (MLEAs) (in Section 3.2.2). Then, the details of MLEAs and thefeatures of input problem mapping operators are discussed, along with the limitations ofthe framework.

Second, the formal denition of the multi-level multi-objective 0-1 KP is given byiteratively extending the simple 0-1 KP (Section 3.3.1). After the denition of the 0-1 KP,the formalism is extended to handle multiple objectives and constraints (Section 3.3.2).This multi-objective 0-1 KP is then extended to multiple levels (Section 3.3.3). Finally, aspecial multi-level KP is presented, which extends the underlying general Mixed IntegerLinear Programming (MILP) model by supporting relations between decision variablesdened with Description Logic (DL).

Third, the new MLEA, the muleGA is presented (in Section 3.4). The concepts(Section 3.4.1) and algorithmic details (Section 3.4.2) of muleGA are discussed, alongwith multi-level evolution scheduling algorithms (Section 3.4.3). The applicability of theconcepts of muleGA is demonstrated through a KP related example (in Example 3).

Fourth, the muleGA is tested on a Multi-Level Multi-Objective Knapsack Problem(MOKP), and on hierarchical, multi-level versions of well knownMulti-Objective Problems(MOPs) (Section 3.5). The results of muleGA are compared to those of traditional GAs,and through statistical analysis, the advantages of muleGA are shown (in Section 3.5).

3.2 Multi-Level Evolutionary Algorithms 41

3.2 Multi-Level Evolutionary Algorithms

The various types of GAs which have multi-level attributes were presented in Section2.12. In the context of this thesis, the term multi-level will be used to name algorithmswith multiple populations, which are connected in a divide-and-conquer manner. TheseMulti-Level Evolutionary Algorithms (MLEAs) presented here are for solving problemswith large search spaces, which can be solved by combining the solutions of problemswith smaller search spaces, through the divide-and-conquer decomposition of the searchspace into subspaces, with special attention given to the class of MLOPs (Denition 7).

In the following, a conceptual overview is given on MLEAs, then a formal denitionis presented, which is going to be used for the denition of a new type of multi-level GA,the Multi-Level Genetic Algorithm (muleGA).

3.2.1 Conceptual Overview of MLEA

The name MLEAs is proposed by the author for such EAs in which multiple populationsare evolving various input problems. The solutions of these input problems are used tocongure the parameters of subproblems of a given upper-level problem, named parentproblem. The mapping from input problems a and b to subproblems A and B is presentedon Figure 3.1.

Figure 3.1: Mapping input problems a andb to subproblems A and B ofsolution 2.

The gure shows a 2-level evolutionary al-gorithm with 3 populations (P(·,·)

(2,·), P(2,·)(1,1) and

P(2,·)(1,2)). Note that the dot symbol (·) denotes

that there are no u and/or h and/or v values

for a particular P(h,v)(l,u) . Also note the similar-

ity of the notation of the multi-level popula-tions with that used for the MLOPs (in Sec-tion 2.4.1). The populations are identied inthe multi-level hierarchy with the (l, u, h, v)quadruples. If there is no value for parame-ter h, that means that the population has noparent. Those (u, h, v) parameters without val-ues may be omitted from the representation(P(·,·)

(l,·) = Pl). Population P2 has four individu-als (1,2,3,4), which are congured through thetwo optimization problems residing in the base-level. The MLOP presented in Figure 3.1 is a two-level optimization problem, becausesolutions of P2 (for example individual 2) use subsolutions, also called lower-level solu-tions (A and B) which are calculated through mapping a and b from the input problemspaces to the subproblem spaces. If individual a or b had also used solutions of inputproblems generated by lower-level GAs, the algorithm would have been called a 3-levelEA.

42 Multi-Level Genetic Algorithm (muleGA)

In an MLEA, there are no constraints imposed on the GAs arranged in the multi-levelhierarchy. The GA types, evolutionary operators, mappings from input problem spacesto subproblem spaces, sequences of evolutionary steps are free of choice. Here, we recallthe imaginary space of GA attributes, the dimensions of which were roughly identiedin the introduction to EC (in Chapter 2). Analogously, we can picture a similar spacefor the attributes of any MLEA. In this MLEA space, there are numerous subspacesdescribing the attributes of the GAs residing in the multi-level hierarchy. Besides theparameters of the GAs, there are other attributes needed to fully specify an MLEA (tobe presented in Section 3.2.2).

The only necessities of a MLEA are the autonomous populations, among which thereis at least one parent population which gets loaded and updated with information comingthrough the mappings of individuals of input problems to its subproblems. This is quitesimilar to the concept of the MLOP (Denition 7). Here, for each subproblem there isan EA which tries to optimize it. In the following, the MLEA concepts are preciselyformalized.

3.2.2 Formal Denition of the Multi-Level Evolutionary Algorithm(MLEA)

The framework developed by Merkle and Lamont in 1997 [53], and by Bäck in 1996 [1],and presented in Section 2.7.6 is extended here in Denition 9 and Algorithm 3 for theformalization of the Multi-Level Evolutionary Algorithm (MLEA). Denitions of evolu-tionary operators are given in Appendix A.3. The objective of using and extending theEA framework by Merkle and Lamont for introducing the novel Multi-Level EvolutionaryAlgorithm (MLEA) is to present in details how traditional EAs are integrated and forma hierarchical structure, and to show which operators of EAs need to be extended andwhich stay intact. The presentation of these concepts can only be thorough if the mostcomprehensive EA formalism is used. This is why the work of Merkle and Lamont formsthe base of the denition of MLEAs.

The class of MLEAs describe such EAs which are built to eectively solve MLOPs(see Denition 7), as they evolve a population for each subproblem of a given MLOP,but naturally they can be used for solving other type of problems. Note that in thedenition of the MLOP it is stated that the input problem dependent decision variablesof any parent problem can only get congured through input problem mapping functions.However, the MLEAs (to be dened in the following) evolve the subproblems of a MLOP,and the decision variables of those problems are congured by GAs, hence they can changeduring the evolution process (regardless of being input problem dependent or not). Ifthere are such upper-level GAs in the hierarchy which alter the decision variables of theirsubproblems, then a method for maintaining the consistency in the MLOP hierarchy isneeded. Therefore, the concept of inverse input problem mapping function is discussedafter the denition of MLEA in Section 3.2.2.1.

The following denition is an extension of the denition of EAs (Denition 8). Thenotation introduced in Denition 7 (with the (l, u, v, h) quadruple serving as an identierfor each problem) is going to be used here as well for indexing the populations of the


MLEA hierarchy. Basically, the following denition is the mixture of the EA formalismand the MLOP denition. Each MLEA has an inner cycle, which is responsible forthe evolution of a single population, and an outer cycle, which schedules the multi-leveloptimization process. Because the MLEA comprises more EAs, in the inner cycle, EAsare going to be identied by the (l, u, h, v) quadruples. The inner cycle iterator is i, andbecause of the outer cycle, a new iterator j is introduced.

Denition 9 (Multi-Level Evolutionary Algorithm) The Multi-Level EvolutionaryAlgorithm with q levels is dened as follows.

• Let Iq be a non-empty set (the individual space for a level q problem Pq).

• Let I(q−1,1), . . . , I(q−1,r(Pq)) be r(Pq) pieces of non-empty sets (the individual spacesfor all the level (q − 1) input problems (P(q−1,1), . . . ,P(q−1,r(Pq))), where r(Pq) is

the number of input problems the qth level problem Pq has or was divided into).

• Let P(h,v)(l,u) be a level l problem, the uth input problem of a level h problem, which is

the vth input problem of its parent. Then I(h,v)(l,u) is a non-empty set, the individual

space of this problem. The notation P(h,v)(l,u) is interchangeably used with Pa, where a

refers to the quadruple (l, u, v, h). The same is true for every other symbol indexed

with(h,v)(l,u) .

• Let f(h,v)(l,u) : I

(h,v)(l,u) → I(l,u)→(h,v) be a function for mapping individuals from an

input population's individual space to one of its parent population's individual sub-spaces. The parent population's individual space is the Cartesian product of aninput problem independent subspace and its input problem dependent subspaces.

I(h,v) = Ii(h,v) ×Πr(P(h,v))

u=1 I(l,u)→(h,v)

• Let µ(i,j)(h,v)(l,u) i,j∈N be sequences in Z+ (the parent population sizes of P

(h,v)(l,u) , index

variables are i for the so-called inner cycle and j for the outer cycle).

• µ′(i,j)(h,v)(l,u) i,j∈N sequences in Z+ (the ospring population sizes of P

(h,v)(l,u) )

• Let P(j)j∈N denote the sequence of schedulings of the q-level problem's populations.

• Φ(j)(h,v)(l,u) : I

(h,v)(l,u) → Rj∈N a sequence of tness functions for P

(h,v)(l,u)

•ι(j)

(h,v)(l,u) :

⋃∞i=1

((Iµ

(h,v)(l,u) )i

)(j)→ true, false

j∈N the sequence of termination

criteria for P(h,v)(l,u)

• χ(h,v)(l,u) ∈ true, false, χr(h,v)

(l,u) ∈ true, false, χm(h,v)(l,u) ∈ true, false,

χs(h,v)(l,u) ∈ true, false, χP(h,v)

(l,u) ∈ true, false (compile-time decidable pa-

rameters for specifying the algorithm variant)


• r(h,v)(l,u) sequences r(i,j)(h,v)

(l,u) of recombination operators (one sequence for each inner

cycle),

r(i,j)(h,v)

(l,u) : X(i,j)r

(h,v)

(l,u) → T(

Ω(i,j)r

(h,v)

(l,u) , T(Iµ

(i,j) (h,v)

(l,u) , Iµ′(i,j) (h,v)

(l,u)

))• m(h,v)

(l,u) sequences m(i,j)(h,v)(l,u) of mutation operators,

m(i,j)(h,v)

(l,u) : X(i,j)m

(h,v)

(l,u) → T(

Ω(i,j)m

(h,v)

(l,u) , T(Iµ

(i,j) (h,v)

(l,u) , Iµ′(i,j) (h,v)

(l,u)

))• s(h,v)

(l,u) a sequence s(i,j)(h,v)(l,u) of selection operators,

s(i,j)(h,v)

(l,u) : X(i,j)s

(h,v)

(l,u) → T(

Ω(i,j)s

(h,v)

(l,u) , T(Iµ

(i,j) (h,v)

(l,u) , Iµ′(i,j) (h,v)

(l,u)

))• θ(i,j)

r ∈ X(i,j)r (the recombination parameters)

• θ(i,j)m ∈ X(i,j)

m (the mutation parameters)

• θ(i,j)s ∈ X(i,j)

s (the selection parameters)

Then the algorithm shown in Algorithm 3 is called a Multi-Level Evolutionary Algorithm.


Algorithm 3 Multi-Level Evolutionary Algorithm Outline

χra,χma ,χ

sa, χ

Pa ∈ true, false are design parameters which are decided and set at compile-time

c := 0; The outer cycle counter t := 0; The inner cycle counter P(c) = create-population-schedule(Pq, c) Creates an array containing populations from a

subset of the set of all populations of Pq . The order of the populations in the array is algorithm specic

for ∀(l, u, h, v) quadruple such that ∃P(h,v)(l,u) do

initialize P(h,v)(l,u) (0) := s1

(h,v)(l,u) (0), . . . , sµ

(h,v)(l,u) (0) ∈ I(h,v)

(l,u)

µ(0);

end forInitializes all populations for P[a] = P(0), . . . ,P(last) doin each iteration P[a] refers to a concrete population P(h,v)

(l,u)also speciable with the (l,u,v,h) quadruple

initic(P[a]) initiates the inner cycle. Executes input-problem-mapping(Pa)while (ι(c)(P[a](0, c), . . . ,P[a](t, c)) 6= true) doif χra then pre-recombination-input-problem-mapping(Pa) end if

recombineml: recombination with optional evolution on sublevels 1

P[a]′(t, c) := ra

(t,c)

Θ(t,c)ra

(P[a](t, c))⊕ sublevel-evolutions(P[a](t, c));

end of recombineml if χma then pre-mutation-input-problem-mapping(Pa) end if

mutateml: recombination with optional evolution on sublevels

P[a]′′(t, c) := ma

(t,c)

Θ(t,c)m

(P[a]′(t, c))⊕ sublevel-evolutions(P[a]

′(t, c));

end pf mutateml;if χsa then pre-selection-input-problem-mapping(Pa) end if

if χa then

P[a](t+ 1) := sa(t,c)

(Θ(t,c)sa ,Φ)

(P[a]′′(t, c));

else

P[a](t+ 1) := sa(t,c)

(Θ(t,c)sa ,Φ)

(P[a]′′(t, c)) ∪ P[a](t, c));

end if

t := t+ 1;if χP

a then P[a] := adjust-population-schedule(P[a],t,Pq,c) end if

end while

c := c+ 1;P(c) := create-population-schedule(Pq, c)

end for

1Optionally, multi-level evolutionary operators (recombineml and mutateml) can re evolution onsublevels of the hierarchy. This functionality is represented through the method sublevel-evolutions().The symbol ⊕ is employed here to emphasize that the mappings of ra and sublevel-evolutions() canbe performed in any sequence. In which sequence they are performed are intrinsic parameters of themulti-level evolutionary operators.


outer cycle

inner cycle

population queues

if

then

if

then

inner cycles

inner cycles

if then

if

then adjust

create

initic( )

Figure 3.2: Outer and inner algorithmic cycles of Multi-Level Evolutionary Algorithms (MLEAs)with the inner cycle evolving population P(1,·)

(2,1) from the population schedule P(c). The

outer cycle is at c = 3, with the rst population in P(3) being P(1,·)(2,1) (also written

interchangeably as P[2]). In case input problem mapping occurs, the f(2,1)(l,u) functionsmap the information from the input problem's individual space to the subspace ofthe individual space of population P(1,·)

(2,1). The dotted lines show optional events.

Algorithm 3 is an extension of Algorithm 1. The Multi-Level Evolutionary Algorithmevolves multiple populations, while the EA only evolves one. In this formal denition,the populations are represented with P(h,v)

(l,u) or P[a], where there is an equivalence betweena and the quadruple (l, u, h, v). The form P[a] is used for increasing readability, while

the form P(h,v)(l,u) explicitly shows the position of the population and the problem in the

multi-level hierarchy. The owchart of the algorithm is shown in Figure 3.2, while anexample of the populations arranged in a multi-level hierarchy is shown in Figure 3.3.

A simplied outline of the MLEA is presented in Algorithm 4. The dierence innotation to Algorithm 3 is that in the simplied version, standard evolutionary operatorsare represented as function calls along with their input and output parameters. The


Figure 3.3: An example for Multi-Level Evolutionary Algorithm (MLEA) populations and theirrelationships.

two algorithms are equivalent, so the shallow simplied form is given only to enhancereadability.

3.2.2.1 Algorithmic Details of MLEA

The MLEA runs all traditional EA methods in the inner cycle, while the outer cycleis responsible for selecting a population for evolution. The traditional variables andoperators in the inner cycle are basically just extended with the outer cycle counter(denoted with j in Denition 9 and with c in Algorithm 3) and with the multi-level

hierarchy position identier (denoted with the sub and superscripts X(h,v)(l,u) or with the

subscript Xa, where X can represent any operator or variable, or with X[a] in case Xrepresents a population.)

New functions introduced to the inner cycle are the pre-recombination, pre-mutationand pre-selection functions for input problem mapping to the subproblems of parent prob-lems, the schedule adjuster and the dierences in the recombine and mutate operators,which are denoted with the ml superscript and with the sublevel-evolution function.The detailed inner mechanisms of these methods are not exposed because they are mainlyimplementation dependent. Nonetheless, input and output parameters and basic func-tionality of these methods are presented. The recombineml and mutateml operators,besides the same functionalities they have in EAs, are allowed to re evolution (run agiven number of evolutionary steps) in subpopulations. For the sake of simplicity and toavoid overcomplexity, this feature was not formalized in Denition 9 and Algorithm 3,and details of it have been hidden under the function calls (sublevel-evolution()).

An example mutation-based scheduling algorithm which utilizes the possibility of sub-level evolution will be presented later (in Section 3.4.3.1). Note that evolutionary op-


Algorithm 4 Multi-Level Evolutionary Algorithm Outline (simple form)

χra,χma ,χ

sa, χ

Pa ∈ true, false are design parameters which are decided and set at compile-time

c := 0; The outer cycle counter t := 0; The inner cycle counter P(c) = create-population-schedule(Pq, c)

for ∀(l, u, h, v) quadruple such that ∃P(h,v)(l,u) do

initialize-population(P(h,v)(l,u) )

end forInitializes all populations for P[a] = get-population-from-population-schedule() do

initic(P[a]) initiates the inner cycle. Executes input-problem-mapping(Pa)while (terminal-condition(c) 6= true) doif χra then pre-recombination-input-problem-mapping(Pa) end if

P ′(t, c) := recombine-individuals-and-perform-sublevel-evolutions(P(t, c))if χma then pre-mutation-input-problem-mapping(Pa) end if

P ′′(t, c) := mutate-individuals-and-perform-sublevel-evolutions(P ′(t, c))if χsa then pre-selection-input-problem-mapping(Pa) end if

P(t+ 1, c) := select-individuals-for-survival(P(t, c),P ′′(t, c))t := t+ 1;if χP

a then P[a] := adjust-population-schedule(P[a],t,Pq,c) end if

end while

c := c+ 1;P(c) := create-population-schedule(Pq, c)

end for


erators can modify all of the decision variables of an individual (both those which areindependent of the input problems and those which get congured by input problemmappings). This diers from the concepts of MLOP (dened in Section 2.4.1), which ne-cessitates that input problem dependent decision variables are only congurable throughinput problem mapping, so that the relating decision variables in the parent problem andthe input problem are in accordance. So, if a MLOP is handled by a MLEA, to keepthe integrity of the MLOP, either inverse input problem mapping operators, or specialrecombination and mutation operators (which do not aect the decision variables of thesubproblems) are needed. The concept of inverse input problem mapping is not explicitlypresented in the MLEA framework, but it is discussed in Section 3.2.2.4.

The inner cycle schedule adjuster adjust-population-schedule(P[a],t,Pq,c) is in-troduced to allow the formalization to support algorithms which use outer cycle and innercycle runtime parameters to on-the-y calculate population scheduling. As the function isparametrized with and thus aware of detailed runtime information (P[a],t,Pq,c), the innercycle population scheduling is as informed as its outer cycle counterpart. Formal detailsof the method are not given, however, an example credit propagation based schedulingalgorithm which utilizes this possibility will be presented later (in Section 3.4.3.4).

3.2.2.2 Input problem mapping

The most important part of MLEAs is the mapping of information from one populationto another. As it was declared earlier, each MLEA has at least one input problem tosubproblem mapping operator. For every MLEA, there are at least two populationsP(h1,v1)

(l1,u1) (let this be the parent population) and P(h2,v2)(l2,u2) (let this be the subpopulation in

which the input problem is evolved), for which, the following holds.

∃f(h2,v2)(l2,u2) : I

(h2,v2)(l2,u2) → I(l2,u2)→(h2,v2), (3.1)

such that: I(h1,v1)(l1,u1) = I(l2,u2)→(h2,v2) ×X, (3.2)

dim I(h1,v1)(l1,u1) = dim I(l2,u2)→(h2,v2) + dimX, (3.3)

0 ≤ dimX < dim I(l2,u2)→(h2,v2), (3.4)

where X is an arbitrary space, and also a subspace of I(h1,v1)(l1,u1) . The populations are

usually structured in a hierarchical tree fashion, therefore most of the time h2 = l1 andv2 = u1, but this is not necessary.

Example 1 (input problem mapping demonstration) As an example for demon-

strating the input problem mapping, let the populations P(3,·)(2,1), P

(3,·)(2,2),P

(3,·)(1,1) on Figure

3.3 which are input problem populations for P(·,·)(3,·) have the following individual spaces:

I(3,·)(2,1), I

(3,·)(2,2) and I

(3,·)(1,1) are the individual spaces of the input problem populations with

the dimensions dim I(3,·)(2,1) = 8, dim I

(3,·)(2,2) = 7 and dim I

(3,·)(1,1) = 5. The parent population's


individual space is I(·,·)(3,·) with dimension dim I

(·,·)(3,·) = 12. It is necessary for the sum of

the dimensions of the subproblem spaces (I(2,1)→(3,·), I(2,2)→(3,·) and I(1,1)→(3,·)) to alsobe equal to 12 (dim I(2,1)→(3,·) + dim I(2,2)→(3,·) + dim I(1,1)→(3,·) = 12). To ease under-standing, the individual spaces will be rather simple in this example: interval of real orinteger values for each dimension.

I(3,·)(2,1) = N3 × R3 × [0, 1]2 (3.5a)

I(3,·)(2,2) = N2 × R4 × Z+ (3.5b)

I(3,·)(1,1) = (R+ ∪ 0)

5, (3.5c)

I(·,·)(3,·) =

I(2,1)→(3,·)︷︸︸︷N3 × R2 ×

I(2,2)→(3,·)︷︸︸︷N3 × R2 ×

I(1,1)→(3,·)︷︸︸︷[0, 1]2 (3.5d)

From the individual spaces, the following input problem mapping function parametersresult.

f(2,1)→(3,·) : N3 × R3 × [0, 1]2 → N3 × R2 (3.6a)

f(2,2)→(3,·) : N2 × R4 × Z+ → N3 × R2 (3.6b)

f(1,1)→(3,·) : (R+ ∪ 0)5 → [0, 1]2 (3.6c)

The individuals are denoted with s, with their identier in subscript position sid. Ifonly the value of one of the many dimensions of sid is considered, then the dimensionunder consideration is written in superscript position sdimid . The individuals sa ∈ I(3,·)

(2,1),

sb ∈ I(3,·)(2,2), sc ∈ I

(3,·)(1,1) and sx ∈ I

(·,·)(3,·) have the following dimensions sa = s1

a, . . . , s8a,

sb = s1b , . . . , s

7b, sc = s1

c , . . . , s5c and sx = s1

x, . . . , s12x .

For the mapping function f which maps from Xn → Y d, its subfunctions mappingfrom Xn → Y are denoted with f1, . . . , fd, so f = f1, . . . , fd. Possible input problemmapping subfunctions for f(2,1)→(3,·)(sa) are the following then.

f1(2,1)→(3,·)(sa) = s1

a + s2a ∈ N (3.7a)

f2(2,1)→(3,·)(sa) = s1

a + s3a + bs5

ac ∈ N (3.7b)

f3(2,1)→(3,·)(sa) = s1

a · s2a · s3

a ∈ N (3.7c)

f4(2,1)→(3,·)(sa) = s4

a · s7a ∈ R (3.7d)

f5(2,1)→(3,·)(sa) = s6

a · s8a ∈ R (3.7e)

Innite number of input problem to subproblem mapping functions can be givenfor any two parent and subproblem populations. The simplest mapping function is theidentity function id : f(x) = x. This requires the input problem individual space and

the subproblem individual space to be the same (I(h2,v2)(l2,u2) = I(l2,u2)→(h2,v2)).


Figure 3.4: All possible individuals of the parent problem renderable through input problemmappings are shown, considering each input problem population is made up of thosetwo individuals highlighted on the gure. Note that the numbered circles and boxesdenote the genotypes of the individuals and that the f-mappings are equal to theidentical function, thus appearing the same genotypes in the parent problem's andsubproblems' genotype spaces.

The input-problem-mapping operators from Algorithm 3 and 4 are responsible formapping information from input problem spaces to subproblem spaces. It is importantto emphasize, when input problem mapping occurs, individuals are either altered orcreated (considering they have not been initialized before) in the target population. Theinput problem mapping functions are responsible for compiling the modied individualsin the parent population spaces. Which input problem solutions are mapped and whichsubproblem solutions are combined to frame the newly altered and rendered individualdepends on the implementation of the mapping functions. Variants of mapping functionsfor muleGA test problems and their characteristics are presented in Section 3.5.2. Figure3.4 shows all possible individuals compilable for the parent-population through inputproblem mappings. Note that just as the input problem GAs can alter their individualsallocating points in the input problem search spaces, parent GAs can also evolve theirsolutions and alter them to reach new portions of the search space, which may involvethe alteration of the positions in the subproblem search space, in case the recombinationand mutation operators of the parent problem are not keeping the subproblems intact.

3.2.2.3 MLEA Parallelism

As with multi-population co-evolutionary algorithms and parallel GAs, there is potentialreward in evolving some or all of the populations of the MLEA parallel. The MLEAsimpose no more constraints and restrictions on parallel population evolution than theother parallel EAs. Although the formalism presented in Algorithm 3 depicts a sequentialalgorithm ow, sequential population evolution is not a necessity for MLEAs. Note thatfor the simple case for a given pair (a, c) | ∀a : P[a] ∈ P(c) ∧ χra = χma = χsa = χP

a =

false, all populations of P(c) could evolve simultaneously, as there is no inter-problem


or inter-population communication in the inner cycle.

3.2.2.4 Implicit concepts of the framework and unsupported functions

The traditional EA functions are represented in the formalism as operators which mapfrom parameter spaces to random operator spaces, as they were taken from the formal-ization of Merkle and Lamont [53], while the new MLEA concepts are introduced asfunctions with return values and adjustable parameters. This creates a mixed formaldenition. EA operators are diverse in functionality and have many variants, thereforeit was necessary to introduce the formal denition from Merkle et al in Section 2.7.6because it properly and comprehensively describes EAs.

However, adding the time and random event dependencies of Merkle's frameworkto the new MLEA concepts would sorely increase complexity without any real benets.MLEAs are based on a newborn concept, and have only been introduced recently. Theouter cycle has a fairly simple algorithmic ow, which run traditional EAs in subiter-ations. However, in these subiterations (namely in the so-called inner cycles) any typeof EAs can evolve, from the simplest one to the evolved and complex versions, hencethe detailed description of the inner cycle. There are only a few known MLEAs, andtheir main algorithmic properties and functional behaviors are describable by the newlyintroduced MLEA framework, so, at the moment, the random parameter space based de-scription of the outer cycle is not necessitated. Therefore, the new MLEA framework is abit shallow (yet quite complex) in comparison to the depths of Merkle's EA framework.In the following, a short summary is given on what concepts are only implicitly (withoutexact formalization) contained by the framework, and what concepts and possibilitiesare unhandled.

The MLEA evolutionary operators recombineml and mutateml derive the randomoperators from the EA framework. The new concepts, according to which therecombination and mutation operators can re evolutionary steps in other popu-lations in the hierarchy, were not presented in the framework explicitly as randomoperators, but were hidden behind function calls. The parameter space to randomoperator space formalization of these enhanced recombination and mutation oper-ators would reach a level of complexity, which could render the framework useless,or impossible to use in practice. The mappings related to these enhanced opera-tors would include spaces produced by the Cartesian product of all populations inthe multi-level structure. Considering that the iteration of the populations is notsynchronized, the domains, codomains and ranges of these operators are hardlyformalizable with the concepts of Merkle's EA framework [53].

The input problem to subproblem mappings were dened to be time and itera-tion invariant functions. Future MLEA implementations may employ time variantmappings with dependence on random exogenous parameters. For the understand-ing and discussion of present day MLEAs, time and environment invariant handlingof mappings is suce.


The mapping from input problems to subproblems is one-way considering presentday MLEA implementations. However, new information can appear in the sub-problem space when recombination and mutation is performed on the individualsrepresenting the whole problem, so the evolutionary operators may alter the searchregion covered in the subproblem space. The new information which may arise thisway could also be useful for solving the input problems. For mapping the newlygenerated knowledge back to the input problems, subproblem to input problemmappings would be needed. If the function f(h,v)

(l,u) is surjective, then the subproblemto input problem mapping is a straightforward task, as the inverse function doesthe job. For every other case, a specic g(h,v)

(l,u) function would be needed for every

problem P(h,v)(l,u) for inverse mapping subproblem solutions to input problem solu-

tions. The inverse input problem mappings would always get executed right beforeinput problem evolution starts. Note that the recombination and mutation opera-tors of muleGA (Section 3.4) do not alter the decision variables in the subproblemspaces, therefore no inverse input problem mapping is necessitated. This is themain reason inverse input problem mapping has not been formalized explicitly inDenition 9.

3.2.3 Conclusion on Multi-Level Evolutionary Algorithms (MLEAs)

The aim of the framework for MLEAs is to provide means for the discussion and uniformhandling of such EAs which evolve multiple populations in a hierarchical structure. Thereare only a few such algorithms [87, 88, 89, 90], and two of them are for computerizeddietary menu planning [91, 92] (with the rst being the author's work).

From the denition of MLEAs, it can be seen that they share many similaritieswith island model parallel GAs (Section 2.11) and multi-population co-evolutionary GAs(Section 2.10). It is possible to evolve MLEA populations in parallel, just as the islandmodel parallel GAs evolves its populations, and the migrations between the parallel GApopulations resemble to input problem mappings. No explicit co-evolution happens inMLEAs, as each population is autonomous and closed in the sense that its tness functionis uninuenced by other individuals and populations. However, if we take any individualin a subpopulation, the chance of its survival may depend on how well the informationit provides to the parent problem performs with other subsolutions. It is not necessaryfor all MLEAs to have such meta-level co-evolution behavior, but the muleGA, to bepresented in Section 3.4, has it (and it is briey discussed in Section 3.4.2.6).

Despite the similarities to other type of EAs, MLEAs form a distinct class, intowhich special hierarchical, multi-population EAs belong. There can be many dierencesbetween MLEA implementations. The denition and formalization of MLEA providesgreat freedom, yet it strictly species the necessities an algorithm must fulll to becamea MLEA.


3.3 Formal Denitions of Knapsack Problems

3.3.1 0-1 Knapsack Problem

The 0-1 Knapsack Problem is a special Knapsack Problem (KP) with the followingdenition [33]. There are n items, a Knapsack with a size of c and one set of variablesand two sets of constants related to the items, namely: decision variables x1, . . . , xn,positive weights w1, . . . , wn, and prots p1, . . . , pn, where, ∀i ∈ [1, n], xi ∈ 0, 1 (xi iseither zero or one), wi ∈ R+ ∪ 0 represents the weight, pi ∈ R+ ∪ 0 represents theprot (also called prize and price) of the ith item. The single objective 0-1 KP is formallystated as follows.

maximize p =n∑i=1

pixi (3.8a)

subject to w =n∑i=1

wixi ≤ c (3.8b)

where xi ∈ 0, 1 (3.8c)

The problem dened in Equations (3.8) is single-objective and single-constrainedbecause the weights and prices are elements of R+ ∪ 0. Many variants of the single-objective KP, namely: unbounded, multiple-choice, multi-constrained knapsack problemsand the change-making problem are presented in [101]. References for the nested [102],nonlinear [103] and inverse-parametric [104] KPs are also cited.

3.3.2 Multi-Objective 0-1 Knapsack Problem

Multi-objective KPs have multi-dimensional price vectors and either single- or multi-dimensional weight vectors. Many interpretations could be formed for the real-life coun-terparts of the multi-dimensional prize p and weight w sums. In most of the cases, theMulti-Objective Knapsack Problem (MOKP) with m objectives is interpreted as a prob-lem with m single-dimension knapsacks with capacities c1, . . . , cm. Every selected objectmust be simultaneously placed into all m knapsacks. The weight and prize of an objectusually diers for every knapsack [52, 105].

In the scope of this thesis, the interpretation of the MOKP is given as a problemwith one multi-dimensional knapsack, rather than many single-dimensional knapsacks.Note that while the interpretation formed for the problem diers, the MOKP underconsideration is generally the same (the dimensions of the price and weight vectors neednot to be the same in the following denition in contrast to the traditional approaches).The denition of the 0-1 MOKP withmp dimensional prices andmw dimensional weightsis the following.

maximize ~p =

n∑i=1

~pixi (3.9a)

3.3 Formal Denitions of Knapsack Problems 55

subject to ~w =n∑i=1

~wixi ≤ ~c (3.9b)

where xi ∈ 0, 1, ~c = c1, . . . , cmw (3.9c)

~p = p1, . . . ,pmp, ~w = w1, . . . ,wmw (3.9d)

There are two versions of MOKPs, the Multi-Objective Single-Constraint KnapsackProblem (MOSCKP) and the Multi-Objective Multiple-Constraint Knapsack Problem(MOMCKP). Considering the former, mw = 1, in case of the latter, mw > 1. Inthis thesis, under the term MOKP, the MOMCKP is considered. In the literature,multi-constraint is sometimes referred as multi-dimensional. Basically, the three mainparameters which dene a KP are the following: the dimension of the prot vector(~p), the dimension of the weight vector ~w and the number of knapsacks (n). Otherimportant parameters are the type of constraints (tight, loose, etc) and the type of thedecision variables (boolean, integer, etc).

3.3.3 Multi-Level Multi-Objective 0-1 Knapsack Problem

The occurrence of the term Multi-Level Knapsack Problem in the literature is innitesi-mal as there is only one incidence of the term Multi-Level Knapsack Problems in [106].The same can be said about hierarchical knapsack problems. Basically, there is no ac-tive research involving any kind of knapsack related problems which have multi-level orhierarchical attributes by any means. This is far from being surprising, as the follow-ing, considerable new type of knapsack problem is signicantly more complex than theNP-hard KP.

Denition 10 (Multi-Level Multi-Objective 0-1 Knapsack Problem) Let ~pall de-note the price vector, which is the sum of the ~pj price vectors of the v rst level knapsacks,each knapsack with nj possible boxes. The z upper level knapsacks are denoted with ~w∗k.The value of Kk(i) denes which boxes from which knapsacks should also get packed intothe upper level knapsack k for weight constraint checking with ~c ∗k . Then, the following isa Multi-Level Multi-Objective 0-1 Knapsack Problem.

maximize ~pall =v∑j=1

~pj =v∑j=1

nj∑i=1

~p(i,j)x(i,j) (3.10a)

subject to ~wj =

nj∑i=1

~w(i,j)x(i,j) ≤ ~cj , (3.10b)

~w∗k =

|Kk|∑i=1

~w[Kk(i)]x[Kk(i)] ≤ ~c ∗k , (3.10c)

where x(i,j) ∈ 0, 1, ~cj = c(1,j), . . . , c(mw,j), ~c∗k = c∗(1,k), . . . , c

∗(mw,k) (3.10d)

~pj = p(1,j), . . . ,p(mp,j), ~wj = w(1,j), . . . ,w(mw,j), [Kk(i)] ∈ N2 (3.10e)


Figure 3.5: An example of KPs arranged in a hierarchical structure, forming a 3-level KP. Theaim of this multi-level KP is to maximize the price vectors of the ve rst levelKPs (and therefore the price vectors of all the upper level ones), while satisfying theweight constraints of each KP. The dotted blue lines represent the assignments fromall those rst level knapsacks to the top-level one, which are assigned to a second-levelknapsack. The broken red line highlights a special case, when a knapsack is assignedto more upper level knapsacks. From Denition 10, it follows that the content of thisknapsack is duplicated, and half of this duplicated content is assigned to one of thesecond-level knapsacks, the other half is assigned to the other second-level knapsack.

j = 1, . . . , v, k = 1, . . . , z, nj ∈ N+ (3.10f)

Note that the relation of the rst level knapsacks to the upper level ones are denedby theKk assignments. The denition allows a broad variety of problem congurations toexist, as each box of the rst level knapsacks can be assigned to any upper level knapsack.In the context of this thesis, Multi-Level Multi-Objective 0-1 Knapsack Problems will besimplied by imposing constraints on the possible Kk assignments, so not all the degreesof freedom provided by the denition is going to be used. Either all of the possible boxesof an arbitrary rst level knapsack is assigned to an upper level knapsack or none. Upperlevel knapsacks need to get assigned with boxes from at least two rst level knapsacks.An example arrangement of knapsacks into a 3-level hierarchical structure is shown inFigure 3.5.

Example 2 (Multi-Level Multi-Objective 0-1 Knapsack Problem concepts) Thenot-so-real concepts of Example 3 are further evolved here to illustrate the properties ofthe Multi-Level Multi-Objective 0-1 Knapsack Problems. The levels of the multi-levelconguration are the following: the level of burglars, the level of car trunks and thelevel of the helicopter (an example for a general 3-level KP is shown in Figure 3.5). Thescenario is the following: There are v burglars simultaneously robbing v houses. When-ever the burglars nished loading their knapsacks, they supposed to meet their partnerburglars at the getaway car. There is a getaway car for each pair of burglars, so thenumber of the cars is v

2 (as each burglar has a pair, number v is even). At the time

3.3 Formal Denitions of Knapsack Problems 57

two burglars meet at their getaway car, they are supposed to load the pilferage intothe trunk of the car. Each burglar's knapsack conguration is adequate, as they couldnot have left the houses with knapsacks violating weight constraints. However, as thetrunk has its own weight constraints, it is possible that two valid knapsack congurationsmay infringe them. In case all of the goods in the v knapsacks successfully t in the v

2trunks, the burglars drive the cars to the helicopter. Into the helicopter they load allof the goods residing in the trunks, and again, they check for weight constraints, thistime the helicopter's. In case each stolen object successfully loads into the helicopterwithout violating any weight constraints, the burglars carried out a valid congurationof the Multi-Level Multi-Objective 0-1 Knapsack Problem. In case the conguration isnon-dominated in the Pareto sense, then it is a Pareto-optimal conguration.

3.3.4 Knapsack Problems with Description Logic

The KPs presented above belong to the class of Mixed Integer Linear Programming(MILP) problems. There are, however, such necessities of the dietary menu planningproblems which are not expressible through the MILP type formalizations of the KPs.Some of the relationships of Dietary Menu Plan (DMP) decision variables are only for-malizable through description logic. More on these requirements are presented in Section4.4.1.4.

It is important to clarify here that for solving the DMPs, we are looking for such analgorithm which handles the KPs arranged in a hierarchical structure, forming a MLOP,and also provides some mechanism to handle the interconnected decision variables of theproblems, whose connection is only expressible through rst-order predicate logic. MILPmodels do not support reasoning about objects, classes of objects. Generally, there aresuch programs which are impossible to formulate or convert to a MILP, because MILPschemas are not Turing-complete, unlike First Order Programming (FOP), which is. Arecent work introducing FOP focuses on the integration of First Order Logic (FOL) andMILP under the name First-Order Logic Mixed Integer Linear Programming model ispresented in [107].

There are two scenarios the DMP solver should handle. In the rst scenario, onedecision variable (the hypothesis variable) infers the values of other variables (the con-clusion variables) according to some relation expressible through description logic. Here,this is named taxonomy relation, because this type of information is going to be used forrepresenting the implicit information about any given object (knapsack). In the secondscenario, a combination of decision variables infers the values of other variables. This iscalled the combination relation because the conclusion variables are set according to thecombination of the hypothesis variables. This type of information is going to be used bythe solver to assess the compilation of the DMP components. The real-world examplesfor the taxonomy and combination relation are given in Example 3 in Section 3.4.

So, the solver we are looking for should perform well in solving MOKPs, it shouldhandle the hierarchical structure of KPs, which form a MLOP, and it should supportthe handling of information about the boxes (objects) represented with description logic.Such an algorithm is proposed in the following section.


3.4 The Multi-Level Genetic Algorithm (muleGA)

The Multi-Level Genetic Algorithm (muleGA) is a novel MLEA for solving hierarchicallydecomposable problems (top-down decomposition), and problems which are composed ofsubproblems forming a hierarchical structure (bottom-up problem composition). For suchproblems, muleGA provides means to pseudo-simultaneously evolve subproblems throughthe evolution of input problem populations (which are also called subpopulations). Then,by using the solutions of these subproblems, the nal candidate solutions are evolvedon the uppermost level. Several variants of the algorithm will be presented in thissection, which dier mainly in their population scheduling methods and there are minordierences in their inner cycles. Employing the notations of the MLEA framework, thefollowing necessities and restrictions identify the muleGA.

Input problem to subproblem mapping is restricted, as there should be exactly onemapping function f(h,v)

(l,u) : I(h,v)(l,u) → I(l,u)→(h,v) for every input population P(h,v)

(l,u) (or

input problem P(h,v)(l,u) ). Input problems are only mapped to their one and only

parent problem's subproblem space. This renders the population hierarchy to be atree. All of the mapping functions are identity functions, so I(h,v)

(l,u) = I(l,u)→(h,v).

Inner cycle algorithms are GAs of any type, with any kind of tness functions.

Individual encoding is strictly muleGA specic and detailed in the following section(Section 3.4.1).

So, the input problem mapping, the type of the inner cycle algorithms and the in-dividual encoding are those MLEA concepts which are specical to the muleGA. Otherparameters remain to be free of choice.

The name of muleGA is derived from the rst syllables of the words multi and level,referring to the multi-level hierarchical structure the subpopulations are arranged in.In addition, mule also refers to that dierent type of GAs can be used for solving thedierent subproblems, which will actually breed the solution of the nal problem. Innature, horses and donkeys are dierent species with dierent numbers of chromosomes,and one type of hybrid they ospring is the mule. An evolutionary solution evolvedfrom the recombination of diering chromosomes. Note that the word mule not at allrefers to infertility and thus to the imperfectness of the muleGA (most female mules andall male mules are infertile). Actually, the test results prove muleGA provides bettersolutions than traditional thoroughbred GAs (test results are presented in Section 3.5).

3.4.1 Concepts of muleGA

The muleGA is intended to solve MLOPs in a divide-and-conquer manner, with MOKPlike subproblems that have special relations between decision variables expressed by de-scription logic. Therefore, individual space of muleGA is dened with the main designconsiderations being the representation of multi-level hierarchy and the handling of the

3.4 The Multi-Level Genetic Algorithm (muleGA) 59

taxonomy and combination relations of the decision variables. For the former, the con-cept of sol and attr is dened, while for the latter, the concept of val is introduced inthe following.

sol is the atomic component for muleGA. It is the abbreviation for solution, and con-ceptually it is an entity for any theoretical or real-world object. Two type of solsare dened: atomic and parametric. Parametric extends the atomic type with thepossibility of attributes the sol can be attributed with.

attr is the abbreviation for attribute. The function of attrs is to represent sols alongwith their actual quantity. The range of sols an attr can represent, and theirpossible amount is given for each attr. Three main types of attrs exist, single,pool and ga with the following meaning. Single type attr can only represent onesol. Pool type attributes can represent sols picked from a pool (set of sols). Thega type attr uses a GA to pick a sol from a given set of sols. The fourth type ofattr is the composite type, which acts as a wrapper for two or more attrs. Thesol represented by a composite attr is one of the sols represented by the wrappedattrs.

val is the abbreviation for value. Values can be attributed to sols to note a particularproperty, function or behavior. The vals represent the information derived fromthe sols' taxonomy and combination relations. Considering taxonomy relations ofsols, the vals represent sets or groups into which various sols belong to or can beclassied into, according to a corresponding theoretical or real-world point of view.Basically, they carry the taxological categorization of each sol. Considering thecombination relations of any given sol, its vals express the information derivablethrough the sols represented by the attrs of that particular sol. These two typeof vals are named taxonomy val and combination val, respectively.

The abstract concepts (shown in Figure 3.6) give a generalized base for the individualspace. With these concepts, the design and workow of the algorithm can be detailedindependently from real-world optimization problems. For not to hamper intelligibilitywith generability, some real-world counterparts of the concepts are given in the followingexample.

Example 3 (The muleGA concepts) The 0-1 Knapsack Problem, which is formallydened in Section 3.3.1, is taken as example to demonstrate the meaning of the concepts.

In this not-so-real-world example, a burglar has broken into a house, from whichhe is prepared to steal electronic devices and fruits. He has one knapsack to ll. Theknapsack has two constraints: volume and weight, while the burglar has two preferences:maximize the price of the stolen goods and maximize the human digestible energy amountof the stolen goods. At the time of writing, on the wake of the 21th century, electronicdevices are still more valuable on the black market than everyday fruits, while foodsstill contain more nourishment, although tendencies are leaping for both statements to


atomic sol without attrs

parametric sol with 3 attrs

composit-attr pool-attr

single-attr ga-attr pool

population

sol with two attrsactual, minimaloptimal/defaultmaximal valuesstored for eachconnection

atomic or parametric type sol(without highlighting attrs)

Figure 3.6: The notation of muleGA concepts - sols denoted with circles, attrs denoted withsquares standing on one vertex. Lines denote connections.

reach inadequacy (more on the increasing prices of foods and decreasing nutrient valuesin Section 4.1).

As the burglar is using muleGA for solving his multi-objective 0-1 Knapsack Problem,the counterparts of muleGA concepts are the following real-world objects and entities.Every concrete object which the burglar can put in his knapsack is a sol. Each of thesesols has four attributes: volume, weight, price and energy content. As attrs are denedto have the function of representing sols, the sols for the real-world concepts (volume,weight, price, energy content) should exist. These latter sols are atomic type, while theformer ones (electronic goods and fruits) are of parametric type. So the burglar has solsfor the goods he considers stealing. These sols have single type attrs which representatomic type sols along with their amount. Note that there should be a measure for eachamount, and as the muleGA concepts do not include any information on them, for eachamount, the measure given by common sense is used (weight: kilogram, volume: cubiccentimeter, price: euros, energy content: kilojoule). Thus, the measures are implicitlycontained by the problem denition (through the amounts represented in the attributes),so the decision maker (the burglar) needs to stick to the chosen measures for the wholeoptimization problem, because there are no conversions carried out by the muleGA.

Instances for the uppermost sols contain solutions for the optimization problem. Notethat these uppermost sols are evolved in a population, and are of parametric type, andtheir attrs represent goods with a given amount. The question is: how many attrs atop level sol should have, considering this problem? It depends on the type of the attrs.If single type attrs are used, then each attr can only represent one sol with a givenamount. The amount should be one if the knapsack contains the particular sol, andzero otherwise. This makes a sol with many attrs and the concept barely diers fromthe binary representations used for solving 0-1 Knapsack Problems. The other way is to


Figure 3.7: The concepts of taxonomy and combination vals are shown in this gure. Theassignments of vals to sols are inferred from problem specic knowledge-bases andtaxonomies encoding the non-quantiable information about the sols.

utilize another implicitly available parameter also known by common sense. It is knownby the burglar that at most three pieces of electronic devices or four pieces of fruitst in the knapsack, no more (derived from his experiences). The burglar can createtwo theoretical pools (sets of sols), one containing the electronic devices and the othercontaining the fruits. Now the rst three attrs can represent sols from the rst pool,while the other four from the second pool.

There has been no mention of vals yet. In the muleGA context, vals are used torepresent the unquantiable information about a sol renderable through taxonomy andcombination relations. However, none of the standard knapsack problems are dened tohandle unquantiable information. The concept of vals will be useful for the nutritioncounseling problem detailed in Chapter 4. To provide a brief introduction, the set offruits and the set of electric devices would make the vals in this example. Althoughthis information could be stored in a special attr representing a single sol named fruit,if the upper structure is considered as a fruit, or zero, if it is not. This qualiable butunquantiable information about sol is given with vals. Other taxonomy vals for thisproblem would be cool stus referring for the most wanted electronic equipments, androtten fruits referring to fruits with easily sensible visual aws. While the burglar wantsto maximize the energy amount, he still needs to parameter the muleGA to avoid rottentomatoes. This is actually a description logic type extension of the burglar's previouslydened 0-1 Knapsack Problem, the goals of which were to optimize volume, weight, priceand energy content. An example for a combination val is presented in the following.The combination vals provide non-quantiable information on the combination of theknapsack's content (so on the combination of the goods). For example, the best pilferageaward would go to those combination of goods, which only contain cool stus and alsocontain more than one fruit. The two types of vals and their assignments to sols arepresented in Figure 3.7.


3.4.2 Algorithmic Details of muleGA

3.4.2.1 Assessing individuals with the tness function

As it was dened in Denition 9, for each problem P(h,v)(l,u) in the multi-level hierarchy,

there is a tness function Φ(j)(h,v)(l,u) for every outer cycle j. The tnesses of muleGA are in

possession of every tiny bit of information about the individual which is being assessed.In muleGA, the individual s is a sol which is either atomic or parametric. In case itis atomic, it has no attrs, but it may have vals assigned to it. The valist(s) : S →Vmaxval(s) function returns the list of vals the individual s is associated with (maxval(s)is the maximum number of vals, the individual may be associated with).

In case the individual s is of parametric type, besides the valist(s) function, thetness function can calculate with the alleles of s, which are attrs. The functionattrs(s) : S → Amaxattr(s) returns the attrs of sol s, while the function sols(s) :

S → (S× R)maxsol(s) returns the set of sols with their respective amount, representedby the attrs of sol s. This way, the tness implementation can recursively traverse thetree of sols and attrs from its root sol s.

3.4.2.2 Input Problem Mapping in muleGA

It was stated in the introductory section of muleGA that the input problem mappingis the identity function. From the point of view of MLEA concepts, this is the correctphrasing. Considering the implementation of muleGA, there are no real decision variablemappings happening in it, as the various level GAs are using the same decision variablesin dierent level problems through the sols() and attrs() methods with which thetraversal of the whole multi-level hierarchy is attainable (this is equivalent to using theidentity function for mapping and inverse mapping the decision variables). Hence, anytime a decision variable of a problem is changed, the corresponding decision variables ofall connecting problems are changed. Therefore, there are no subproblem synchronizationissues in muleGA.

3.4.2.3 Recombination

During recombination, attrs of parent individuals s1p and s2p are mixed in a given style(one-point crossover, uniform crossover, etc.) to create the ospring individuals s1o ands2o. If the GA of the parent individuals is not generational and/or uses elitism, it ispossible then that the parents survive the iteration. This means that the attrs cannotbe just simply moved to the osprings, they have to get copied. Copying an attr meanscopying every bit of information it represents, i.e. the whole underlying structure ofthe multi-level structure. Note that attrs can contain single type sols, pools and evenGAs, which also can contain underlying structures. This fact makes the multi-levelrepresentation and evolution memory expensive and intensive. For this end, a copy-on-write type handling of attrs with managed memory references is necessary (more on theimplemented copy-on-write technique on Section 3.4.4). The recombination operator of


muleGA only creates changes by exchanging the attrs of two parent sols, so the wholelower-level structure belonging to the attrs are exchanged without inner modications.

3.4.2.4 Mutation

Mutation is highly problem dependent. There is no muleGA specic necessity for themutation method to mutate individual s. The option of evolving lower level populations(by running sublevel evolutions) and then using their results through input problem tosubproblem mappings is only an extension to traditional mutation methods. BecausemuleGA does not impose any requirements against the mutation operator, any typeof mutation can be implemented. Changes in the individual can be made by runningsublevel-evolutions or by running any other algorithms which alter the decision variablesof the problem (the sols represented by the attrs). Note that because in muleGA thedecision variables of the subproblems are actually the same decision variables the parentproblem uses (the input problem mapping function is the identity function), any changein the decision variables is promptly eectuated in the whole hierarchy.

3.4.2.5 Selection

The muleGA does not impose any requirements on the algorithms which perform selectionfor recombination and selection for survival. As these functions run in the inner cycle,any type of selection operators can be used which are compatible with the inner cycleGA. However, extensions to traditional selection methods may be carried out, which forexample could include the acquisition of any kind of running time information from themulti-level GA hierarchy to raise or ease selective pressure on the current population.

3.4.2.6 Meta-Level Co-Evolution

There is no explicit co-evolution in muleGA, however, the survival of an individual (if itis not an individual of the main problem) is not entirely self-dependent because of thefollowing. We take two arbitrary sols (s1, s2) from dierent attrs belonging to the same(upper-level) sol (sparent). Then s1 and s2 survived in their corresponding populationsand were selected to be represented by the attrs of sparent. This parent individual isgoing to be assessed according to the whole hierarchical structure it represents (sols,vals, amounts). If sparent is not selected in the parent population for reproduction, thenits attrs (and therefore the sols represented by the attrs) are not going to make intothe next generation of the parent population. Hence, in the case s1 and s2 have not beenrepresented by any other attrs, they are going to die out, because those populationsthey reside in are not going to be evolved further and going to be deleted. This isthe phenomenon of meta-level co-evolution, in which lower-level individuals can die outin spite of their good tness values in their populations. Note that whole lower-levelpopulations are deleted when a sol represented by an attr does not make into the nextiteration in the parent population.


3.4.3 Scheduling Algorithms

The scheduling algorithm of muleGA determines the order of the evolution of the popu-lations belonging to the multi-level GA hierarchy. The MLEA framework (Denition 9,Algorithm 3, Figure 3.2) provides three interaction points to dene or adjust the evolu-tion schedule. First, the population schedule at outer cycle iteration c is created in theouter cycle by the method create-population-schedule(Pq, c) and stored in variableP(c). This P(c) is used then for each inner cycle iteration. Second, in the inner cycle,there are two interaction points to alter the evolution determined by P(c), the rst beingthe possibility to re evolution processes on lower-level populations by the recombineml

and mutateml operators. While this method does not alter P(c), it res such evolution-ary steps which were not scheduled to run in P(c). Third, the second interaction pointin the inner cycle is the adjust-population-schedule(P[a],t,Pq,c) method, which,depending on the implementation, can rearrange the schedule or even create a new one.

Each variant of muleGA is, among other factors, determined by the scheduling meth-ods it uses. For each variant, it is the behavior of the evolution schedule alterationmethods which should be precisely dened. Four scheduling alternatives are presentedin the following.

3.4.3.1 Mutation-based scheduling

The muleGA variant which uses mutation-based scheduling works as follows. Consideringa q-level problem Pq, the outer cycle P(c) is initialized with the uppermost population

P(·,·)(q,·). So, the inner cycle is intended to evolve P(·,·)

(q,·) only, according to P(c). All of the

evolution red in other populations of the multi-level hierarchy is red by the mutateml

operator. The mutateml operator is only allowed to re evolution in the subtree of theactual iteration's population. For the mutation-based scheduling algorithm, this is nar-rowed down to input problem populations. Using the introduced muleGA terminology,this means that whenever a sol from P(h,v)

(l,u) gets mutated in the inner cycle, the mutationoperator can re evolutionary operators in its input problem populations, which are main-tained by the sol's ga-type attrs. The sublevel evolutions are not partial in the sense thatmutateml either res one or a few complete inner cycles in some P(l,u)

(l′ ,u′ )input problem

populations, or initiates no evolution in populations of the subtree. So, if sublevel evolu-tion is executed, then at least a full inner cycle is run. Mutation of pool-type attributesmean randomly changing the represented sol with another sol from the pool. Single-typeattrs are not aected by mutations as they can only represent one sol. Note that be-cause a full inner cycle is run in the input problem populations, the mutateml operatorsof those inner cycles can also re evolutions in lower levels, so the mutation of populationscan recursively travel the tree of the multi-level hierarchy. Also note that for a muleGAwith mutation-based scheduling χr(h,v)

(l,u) = false, χm(h,v)(l,u) = false, χs(h,v)

(l,u) = true, for

every population P(h,v)(l,u) , meaning that there is only pre-selection-input-problem-mapping

carried out, right after the mutateml operation. As only mutateml can re evolution onlower levels, the input problem populations are unchanged before mutateml, so there is


no reason for carrying out input problem to subproblem mappings anywhere else in theinner cycle.

3.4.3.2 Top-down scheduling

In muleGA, the multi-level population hierarchy is restricted to be a tree, rather thanan arbitrary graph. The root of the tree is the uppermost population, considering aq-level problem Pq, this is P(·,·)

(q,·). The top-down scheduling method uses the breadth-rst

traversal of the population tree to form P(c). The breadth-rst tree traversal starts at theroot of the tree, and populations are appended to the end of P(c). The queue acts as aFIFO, as P(·,·)

(q,·) is always the rst population in P(c). There is no inner cycle adjustment

of the population schedule, nor recombineml or mutateml red evolutionary steps. Thereis no input problem mapping carried out in the inner cycle. Input problem mappings arerun during the initialization of each inner cycle with the method initic(Pa).

3.4.3.3 Bottom-up scheduling

The bottom-up scheduling method is basically the same as its top-down scheduling sister.The only dierence is the insertion of the populations to P(c), as in bottom-up scheduling,populations are inserted into the front of the population queue P(c), so the root populationP(·,·)

(q,·) is always the last population of P(c). There is no inner cycle input problem mapping,

as it is carried out only during the initialization of each inner cycle with the methodinitic(Pa).

3.4.3.4 Credit-propagation scheduling

The idea of credit-propagation scheduling is that each P(h,v)(l,u) is given some credit by its

evolver (the outer cycle or parent GA's inner cycle), and let it decide what to do with it.

Then P(h,v)(l,u) can use the acquired credit to evolve itself, or it can pass some or all of the

credit to its attrs. The population P(h,v)(l,u) (or more correctly the attr which maintains

the GA of P(h,v)(l,u) ) can even return the credit or part of it. Highly problem dependent

decision factors should be implemented for tailoring proper credit distribution algorithms.Having them can increase the eciency of the multi-level evolution process, as the sensiblemanaging of credits can eliminate the randomness of mutation-based scheduling and theiterative traversal and evolution of the whole multi-level population structure occurringwith top-down and bottom-up scheduling.

3.4.4 Implementation

The muleGA framework (MGLib) is implemented in C++. It is built on top of GALib[108] and GSLib [109] (in-house development, generic data-structure management frame-work with AVL-tree, copy-on-write and reference counting support). The concepts ofmuleGA (sol, attr, etc.) are implemented as C++ classes (CSol, CAttr, etc.). Basically,


the CSol class comprises pointers to CAttrs which manage and evolve GAPopulations.In these populations the individuals are CSols. For each muleGA related class, copy-on-write is supported by GSLib. Unless copy-on-write, the muleGA implementation wouldbe overly memory expensive, as copying an attr would mean copying its entire sub-tree. Copy on write is done by reference counting objects. Once a copy of an objectis requested, only a virtual copy is made, which is only a reference to the original one.The actual copy is carried out when a modication is requested on an object, which hasmore than one referrer. Then the modication is performed on a real copy of the objectwhich was produced by copying the original. The modication requester will now havereference to the new and modied object, while the original version of the object staysintact. Because of the well-known properties of the crossover operators (they create newsolutions by combining the alleles of existing individuals), many individuals of a popula-tion share common alleles. Through the utilization of copy-on-write, memory utilizationis more economical and moderate.

3.5 Algorithm Evaluation

3.5.1 Goal of the Experiments

The goal of the following investigations is to measure the absolute and relative perfor-mance of muleGA. Absolute performance will be expressed by testing the algorithm onsearch spaces already evaluated by exhaustive search, while relative performance is calcu-lated in comparison with traditional GAs. Emphasis is going to be put on the discoveryand study of the assets and shortcomings of the multi-level evolutionary process. For thisend, those parameters are going to be examined closely which determine the muleGA.The inner cycle GAs and the single-level GA of the analyzed congurations are going tobe parameterized with standard values. The eects of conventional GA parameters andGA variants to the multi-level evolutionary process is only briey discussed.

As problem solving with muleGA involves the decomposition of the search space intosubspaces, it is expected that the algorithm performs exceptionally well on problemswhose subproblems are easily solvable with GAs and where the subproblems are notlinked, or only loosely linked. The linkage of a variable is said to be order k if it neverdepends on more than k−1 other variables [80]. However, in most real-life problems, thereare some linkages between all of the variables, although with diering signicances. Whenthere are many conicting objectives with a high order linkage, GAs lose performance.It was shown in [110] that MOGAs are outperformed by traditional non-Pareto GAswith aggregated objective functions if the number of objectives is equal to or greaterthan four. For problems with more than ten objectives, a purely random search mayperform better than the traditional MOGA approaches [111]. A detailed study of theoptimization of many conicting objectives is presented in [55]. It concludes the numberof objectives dramatically changes the behavior of MOGAs, thus the performance of analgorithm conguration for a small number of objectives cannot be generalized to largernumber of objectives, and good proximity to the trade-o surface with high diversity

3.5 Algorithm Evaluation 67

is hard to achieve. Although there are variants of MOGAs to overcome these problemsproviding good results for search spaces with 50 objectives.

The muleGA supports a hierarchical decomposition to unlimited levels, which can in-crease solution performance on high-dimensional problems. This study attempts to com-pare the performances of muleGA to traditional GAs (both with aggregating and Pareto-based techniques) on search spaces ranging from being bi-objective to high-dimensional.The variants of the Knapsack Problem (KP) are used for testing, as KPs are popular testproblems for analyzing the performances of EAs, and they share many similarities withthe real-world dietary menu planning problems presented in Section 4.4.1. Traditionalnumeric test functions for MOPs are also going to get utilized in testing to demonstratethe capabilities of muleGA. Note that these numeric functions are test beds for tradi-tional MOGAs, which perform well on few-dimensional search spaces, usually two to fourdimensional ones. This is why the traditional numeric functions implemented in MOGALib

and presented in Section A.7 are suitable for this dimension range.

3.5.2 Test Problems and Measures of Performance

The measure of the performance of the algorithms is traditionally expressed in the dis-tance of the found solution(s) from the optimal one(s) [112]. Other metrics are eciency(CPU time, computational eort to obtain solutions, number of iterations, use of spa-tial and temporal resources) and eectiveness (accuracy and convergence, robustness,scalability). Test function characteristics are the following [11]: continuous vs. discon-tinuous vs. discrete, dierentiable vs. non-dierentiable, convex vs. concave, unimodalvs. multi-modal, numerical vs. alphanumeric, quadratic vs. non-quadratic, type ofconstraints (equalities, inequalities, linear, nonlinear, etc.), low vs. high dimensionality(genotype, phenotype), deceptive vs. non-deceptive, biased vs. nonbiased portions ofPFtrue. Concluding from the number of characteristic features a numerical test problemmay have, a high number of possible test congurations are needed to test the perfor-mance of any new algorithm. From the No Free Lunch (NFL) theorems it derives thatno algorithm is expected to perform equally well on every test problem. As a matter offact, if every possible conguration of test problems is utilized, each algorithm comes outwith an average performance.

The test congurations which are going to be utilized here bear such parametersthrough which the muleGA's particular characteristics can be expressed. The mainpurpose is to determine the problem characteristics with which the muleGA is expectedto perform better than conventional GAs and MOGAs.

3.5.3 Experiment Design and Test Infrastructure

Two types of tests were assembled to assess muleGA performance. The rst type oftests compare muleGA to traditional GAs with aggregating tness functions in solvingMulti-Level Multi-Objective 0-1 Knapsack Problems, the second type of tests deal withnumerical MOP test functions and compare performances of muleGAs to MOGAs. Thesecond type of tests were run on exhaustively evaluated search spaces, so the goodness of


every outcome is expressed in light of the true optima. The results of the test runs wererecorded in a PostgreSQL database, statistical hypothesis testing and visualization wererun as batch processes using Matlab and GNUPlot. Results of the comparisons were alsorecorded in the database and gures were saved as image les.

3.5.4 Knapsack Problem Tests

Single-Level Knapsack Problems were generated randomly using the congurations de-tailed in Appendix B.1. Using these single-level problems, three Multi-Level KP con-gurations with second and third level extensions were constructed with the followingparameters.

First knapsack conguration has one level-3 knapsack, two level-2 knapsacks andfour base-level knapsacks. Each of the level-2 knapsacks constrains the contentsof two base-level knapsacks, the level-3 knapsack constrains the contents of thelevel-2 knapsacks. The weight constraints of the level-2 knapsacks equal the sumof the weight constraints of their corresponding base-level knapsacks. The level-3knapsack's weight constraints equal the sum of the level-2 knapsacks.

Second knapsack conguration is generally the same as above, with the only dier-ence being that 25% of the level 2 and 3 weight constraints were randomly decreasedto 80%.

Third knapsack conguration has no modications to the weight constraints of therst conguration, but each level-2 knapsack constrains four base-level knapsacks,and there are eight base-level knapsacks in total.

The GAs compared in the experiments are congured as follows. The single-levelGA runs an aggregating tness function which rewards the individuals according to howmuch prot they represent. Whenever a weight constraint of any knapsack (base-level,level-2 or level-3) is breached, the individual's tness is zeroed. The KPs are encodedas binary strings, the ith bit of a string encoding a KP (boxes-to-knapsack assignment)tells if the ith box is in the knapsack. Each individual of the single-level GAs encodesall of the various level knapsack problems by recording all of the assignments in a singlegenome. Uniform mutation is used in each GA, and mutation is done on a gene byrandomly ipping its value (FlipMutation).

The muleGA evolves the base-level knapsacks in separate populations, then collectstheir best individuals through input problem mapping and creates the top-level individ-uals from the rst n best combinations of the input problem solutions, where n is thesize of the top-level population. At each level, aggregating tness function is used. Thebase-level tness functions only reward a single knapsack and only penalty the weightconstraint of the corresponding knapsack. The top-level tness function rewards thelevel-2 and level-3 knapsacks and penalizes breaching their constraints.

Three pairs of test congurations were assembled, each comparing the results of aGA with its muleGA counterpart. The tests were congured as follows.


First Test Conguration has the single-level GA congured with the following pa-rameters. Population size is 40, number of generations is 10 (the rate of convergencedecreased signicantly after ten iterations), chance of crossover is 90%, chance ofmutation is 10%. The muleGA populations are sized 40 for each base-level GA andfor the top-level GA as well. Base-level populations are evolved for 10 iterationseach, top-level population is only evolved for 1 iteration (just collecting and oncerecombining and mutating its individuals). The two GAs are compared with eachother on the rst knapsack conguration. Note that each base-level GA in themuleGA conguration runs for the same number of iterations the single-level GA isrun. This, however, does not mean the muleGA has more iterations to run, becausethe single-level GA works with individuals encoding the whole level-3 problem, sothe length of its chromosomes equal the length of the sum of the chromosomesof the base-level GAs. The computation resources the two congurations use aregenerally the same, so the comparison is not biased.

Second Test Conguration has the same GA congurations as the rst one, but herethe GAs evolve the problems dened in the second knapsack conguration. Asthe weight constraints of the upper-level knapsacks were decreased, such inputproblem mapping method was utilized for this conguration which does not allowthe mapping of an input problem combination if it breaches some constraints ofthe parent problem. If it was necessary, the introduced input problem mappingfunction iteratively changed bits from one to zero in the mapped subproblems untilthe newly created individual satised all of the constraints.

Third Test Conguration evolves the third knapsack conguration problems with theGAs having the same parameters as in the rst test conguration, only the numberof base-level populations dier in the muleGA, as there are eight base-level GAsevolving the corresponding base-level knapsacks.

Each test conguration was run a hundred times and the results were analyzed withstatistical hypothesis testing. The signicance level of each test was chosen to be 5%uniformly. Normal distribution of the results were tested with Lilliefors-tests [113]. If theresults were of normal distribution, then two-sample T-tests were run to check whetherthe results of the single-level GA and the muleGA have equal means and variances.Two-sample Kolmogorov-Smirnov tests [114] and paired, two-sided Wilcoxon signed ranktests [115] were both run to compare the results. The latter two tests do not need thedistributions of the results to be normal.

Figures 3.8, 3.9(a), 3.9(b), 3.10, 3.11 highlight the dierences between the results ofthe single-level GA and muleGA. As it is shown in Table B.1 (in Appendix B), there areve congurations for the number of knapsacks (21 · 8 = 16, . . . , 25 · 8 = 256), and tencongurations for the number of weight and prot dimensions (21 = 2, . . . , 210 = 1024).The combinations of the settings give a total number of 50 distinct congurations. The3D gures 3.8, 3.10, 3.11 show the dierences of the results (the sum of the prot in thetop-level knapsack) in function of the number of knapsacks (decision space parameter)and the number of dimensions (objective space parameter).


12

34

56

78

910

1 2 3 4 5

−0.05

0

0.05

0.1

0.15

0.2

Figure 3.8: The results of the rst test conguration with number of boxes ranging from 23·2i,(i =1, . . . , 5) (x-coord), weight and prot dimensions ranging from 2j ,(j = 1, . . . , 10) (y-coord) and dierence between the results of muleGA and GA given in percentage,calculated by subtracting the latter from the former (z-coord). Note that the coloringis only for separating the congurations and increasing visibility.

1 2 3 4 50

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

(a) Decision space dimension congurations

1 2 3 4 5 6 7 8 9 100

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

(b) Objective space dimension congurations

Figure 3.9: Percentage dierences in function of decision space dimensions (number of boxes)and objective space dimensions (number of weight and prot dimensions) resultedfrom the rst test conguration.


12

34

56 7

89

101

23

45

−0.05

0

0.05

0.1

0.15

0.2

Figure 3.10: The results of the second test conguration with number of boxes ranging from23 · 2i,(i = 1, . . . , 5) (x-coord), weight and prot dimensions ranging from 2j ,(j =1, . . . , 10) (y-coord) and dierences between the results of muleGA and GA givenin percentage, calculated by subtracting the latter from the former (z-coord).

12

34

56

78

910

12

34

5

−0.05

0

0.05

0.1

0.15

0.2

Figure 3.11: The results of the third test conguration with number of boxes ranging from23 · 2i,(i = 1, . . . , 5) (x-coord), weight and prot dimensions ranging from 2j ,(j =1, . . . , 10) (y-coord) and dierence between the results of muleGA and GA given inpercentage, calculated by subtracting the latter from the former (z-coord).


It can be concluded from the KP tests that the muleGA is capable of nding signif-icantly better solutions for the KPs in congurations were the number of boxes in thebase-level knapsacks is low. Figure 3.9(a) highlights muleGA runs result in some 17%better knapsack prot in 16 box base-knapsack congurations. This means single-levelGA is reaching at most 83% of the maximum possible prot in solving the 64(4 · 16)box 3-level KP. No exhaustive search was run on the KP test congurations, but froman iterative exploration of the promising regions of the search space it can be concludedthat for the 4 · 16 box case, muleGA converges to the true optima while the single-levelGA stucks just above 80%.

As the number of the boxes increases, the potential of muleGA decreases. The phe-nomenon is the same in all test congurations, including the third one where there areeight base-level GAs. The total number of knapsacks does not have any inuence onthe quality of the solution in contrary to the number of knapsacks in the base-level GAswhich do have. The perceivable performance loss in function of increasing the number ofknapsacks is attributable to the base-level GAs, which lose performance as the dimensionof their decision spaces increase. Therefore, they are not being able to bring the top-levelsolution closer to the true optimum.

The statistical tests showed that the results of muleGA are signicantly better thanthe results of the single-level GA in all congurations where base-level box numbers wereunder 128. Only a handful of cases in high objective space dimensions produced insigni-cant dierences in muleGA and GA results in 128 box base-level knapsack congurations.In the 256 box base-level congurations, the muleGA and GA produce results which arestatistically equivalent, regardless of the objective space dimensions.

As the gures show, the eect of the number of objective space dimensions on theresults is minute. Only a mild decline is depicted by the results of the experiments asthe number of dimension increase. Note that aggregating tness functions were used fortesting, thus the goodness of each individual was summed in a one dimensional protvalue which may explain why the results are insensitive to the objective space dimensionchanges.

The tests on the multi-level KPs underline the eectiveness of using sub-level GAs fornding subsolutions in subspaces of multi-level KPs. These subsolutions can congurehigher level GAs, in which extra constraints can be introduced, just as they were inthe second test conguration, as 1

4 of the weight limits were decreased to 80%, still themuleGA showed signicantly better results than the single-level GA. Note that for KPslike in the rst and third test congurations and unlike the second test conguration,there is no point running the top-level GA for many iterations, since on the top-level thedimension of the decision space is higher. However, for problems which introduce newconstraints in upper-levels, like the problem in the second test conguration, iteration ofupper-level GAs are necessary to have them evolve solutions which satisfy the upper-levelconstraints as well as the base-level ones. Note that the aggregating tness functions forsolving KPs are not the most eective GA based solution methods. Lamarckian andBaldwinian GAs [52] would nd better solutions in the single-level implementation, buttheir advancements would also appear in the multi-level implementation too.


Evolving only the base-level GAs is equivalent to the divide-and-conquer decomposi-tion of the problem and solving the subproblems with GAs. The divide-and-conquer typeof decomposition of decision space and objective space and then solving the subproblemswith GAs have been tested and presented in [116] for multi-dimensional numerical testproblems. Solving the subproblems with GAs and then combining their results werefound signicantly more eective than solving the whole problem with a single GA.

3.5.5 Numerical MOP Tests

The eectiveness of the divide-and-conquer decomposition of multi-dimensional numeri-cal test problems is presented in [116]. The paper describes tests which employ scalabletest problems [117, 118] with adjustable decision space and objective space dimensions.The KPs are conveniently scalable, because it is easy to adjust the dimensions of thecriterion space and objective space, as the number of boxes and the dimension of theweight constraints are easily congurable. However, nding Ptrue and PFtrue for KPsis a computationally expensive process as the whole feasible region of the search spaceneeds to be evaluated. This makes KPs less practical for such testing purposes wherethe comparison of the results with Ptrue and PFtrue is needed. Some of the numericaltest problems are constructed in such a way that Ptrue is known right away, as it can beformulated mathematically. The scalable test problems of Deb et al [118] are problemsof this type and became a standard in testing MOGAs in higher dimension (three ormore) problem solving. However these scalable problems are single-level problems, sotheir decomposition would be a top-down problem decomposition without introducingsubproblem specic constraints for each subproblem. To test muleGA in its intendedfunction, namely in solving multi-level problems, multi-level numerical test problems areformulated here (with a bottom-up problem composition approach) by combining tradi-tional numerical test problems presented in Appendix A.7. Combining single-objectiveor bi-objective MOPs to a higher dimension problem is the easiest way of creating multi-level test problems [117]. However, besides combining the traditional MOPs, new objec-tives are also introduced for the upper-level GA, just as it has been done in the secondconguration of the KP tests.

The traditional MOPs chosen for the experiments are the Binh1 [119, 120] and Poloni[121] problems, which are two-dimensional both in decision space and objective space.The enumerated Pareto fronts of the problems are shown in Figure 3.12.

3.5.5.1 Multi-Level Problem Formulation

The bi-objective MOPs (binh1 and poloni) are combined to form 4, 8, 16 and 32 di-mension problems, both in criteria space and objective space. The Ptrue sets of the newproblems are the combination of the base-problems Ptrue sets, and the same holds truefor the PFtrue sets.


Pbinh1.4Dtrue = Pbinh1

true ×Pbinh1true (3.11)

Pbinh1.8Dtrue = Pbinh1.4D

true ×Pbinh1.4Dtrue (3.12)





Ppoloni.4Dtrue = Ppoloni

true ×Ppolonitrue (3.15)

Ppoloni.8Dtrue = Ppoloni.4D

true ×Ppoloni.4Dtrue (3.16)





So, the newly formed problems have 4, 8, 16 and 32 dimensional decision spaces andobjective spaces, respectively. A traditional MOGA would have diculties in convergingto the Pareto fronts of these newly formed problems, because of the following: in the high-dimensional objective space, there can be many individuals assigned with rank 1, whileonly standing out according to one objective (and therefore being non-dominated) andhaving worse than average results according to the other objectives. As the dimension ofthe objective space increases, the chance of converging to PFtrue decreases, because thewhole population of the GA is spread with rst ranked solutions (PFknown). Becausemany individuals get assessed with rst rank while actually being mediocre, selectivepressure is lost. This selective pressure can be reintroduced by adding new levels tothe GA. If each base-level GA evolves the traditional 2-dimensional MOPs and theirsolutions are mapped to the subproblems of the high-dimensional problem, it is possibleto simultaneously converge to the Pareto fronts of the subproblems as well as to thePareto front of the parent problem. The results in [116] show that the divide-and-conquer decomposition signicantly increases the quality of the solutions. If the upper-level problem is simply the composition of the subproblems without any new constraintsand objectives, then the running of the upper-level GA can have diculties in increasingthe quality of the already good solutions found by the sublevel solvers (because of thelesser selection pressure and the larger dimensionality of the upper-level GA). A Multi-Level Optimization Problem (MLOP), however, has some specic objective assigned foreach level, therefore a simple solution composition at the upper-level is not suce.

For the sake of testing, the upper-level (and as being a two-level problem, the top-level) GA's individuals are going to get assessed according to the following measures.For each individual, two parameters are calculated: rank and inner distance. The rankis calculated in the high-dimensional (4 to 32) search space, and is being the individ-ual's Pareto rank. The distance parameter is the sum of the Euclidean distances of thesubsolutions, which is calculated according to the following. Let x1 and y1 denote thedecision-variables of the rst subproblem, and xn and yn denote the objective-variablesof the nth subproblem. Then, the distance is calculated as follows.


pareto front50

00

(a) Binh1

pareto front

-70-55

0

-5

(b) Poloni

Figure 3.12: Enumerated Pareto fronts of the Binh1 and Poloni problems

inner-distance =

n∑i=2

√(x1 − xi)2 + (y1 − yi)2 (3.19)

Note that the distance parameter dened in Equation (3.19) is an intrinsic attributeof each top-level individual, as it reects the distances of the individual's subsolutions toeach other.

Once the distance attribute is calculated for each individual, it is multiplied by theindividual's rank. This results in a tness value in which both the individuals' rank (andthus some implication on the relative position of the individuals to the others and thePareto frontier) and its inner-distance is represented. The introduction of the inner-distance was necessary to make the problem multi-level in the MLOP sense. Withoutintroducing extra constraints to the upper-level GA, the solving of the problem wouldbe a simple decomposition of the search space. However, note that for problems whichdo not have multi-level attributes, the upper-level GA may still play a role in nding thematching subproblems, which form the best compilation for the top-level problem.

3.5.5.2 Test Congurations

The true Pareto fronts of the numeric MOPs have been enumerated with high granu-larity and the coordinates of the discretized Pareto front have been stored in relational-database. The distances to the true Pareto front (PFtrue) were calculated using theresults of the enumerative reference measurements. Four type of GAs were compared toeach other in solving the multi-level MOPs formerly presented. The four type of GAsare the following.


muleGA with random individual mapping runs n base-level GAs, where n is thenumber of base-problems, and one top-level GA. Each GA is an NSGA-II, withthe top-level one congured to use the tness function which multiplies the rankswith the inner-distances. The mapping of the input problem solutions is carried outrandomly, which means the individuals of the top-level GA are formed by randomlyselecting subsolutions from the subproblems and combining them together. Onlysubsolutions which ranks are 1 allowed for selection.

muleGA with semi-order input problem mapping only diers from the randomindividual mapping version in the mapping function, as semi-ordering means thereis some order in which the subsolutions get mapped to form the top-level individual.The order is introduced to increase the chance of the top-level individuals to containsuch subsolutions which distance is minimal. Semi-ordering means that consideringthe n base-problems, the closest individual to the actually selected subsolution inthe 1st base-problem is selected from the second base-problem. Only the rstobjectives are minimized, so using the notation introduced in Equation (3.19), if asubsolution is chosen from the rst base-problem with x1 and y1 decision variables,then such a solution is chosen from the second base-problem for which |x1 − x2| isminimal. If n > 2 then the remaining subsolutions are chosen randomly.

muleGA with full-order input problem mapping is generally the same as the semi-order input problem mapping version with the dierence that the minimum of|x1 − xi| is sought for each i = 2, . . . , n.

NSGA-II which solves the 2 · n dimensional top-level problem and employs the tnessfunction which incorporates the distance measure.

Because the multi-level problem has been composed of the same 2 dimensional MOPs,the sublevel GAs are providing solutions for the same problems. Therefore, a fth-version of the test conguration could have been introduced which would have reducedthe multi-level problem into a simple divide-and-conquer decomposition. Irrelevant of thedimension of the top-level problem, if the base-problems are the same bi-objective MOPs,then a single GA would be enough to evolve a single MOP and then its Pareto solutionscould be used by combining them with themselves to form the higher-dimensional top-level problem. This way, the distance between the subsolutions would be always zero,and assuming the base-level GA nds PFtrue, the 4, 8, 16 and 32 dimensional top-levelproblems could be solved by a single bi-objective MOGA. Note that this is only true ifthe base-level problems are the same, and this is why this straightforward approach isnot presented here.

In the tests, bottom-up scheduling was used, so rst the base-level GAs were evolved,then the top-level GA. One outer cycle iteration has been run in each test, so inputproblem mapping occurred once per test. If n denotes the number of base-problems, theneach iteration of the top-level GA was counted as n total iterations, while the sum of thesingle iterations of every base-level GA was also counted as being a total of n iterations.This calculation was based on the assumption that the top-level GA uses n-times more


computational resources than one base-level GA. This derives from the fact that thetime complexity of the non-dominated sort in NSGA-II, which is the most computationalexpensive process of the MOGA is O(M ·N2), where N is the size of the population andM is the number of objectives. Note that input problem mapping only happens once inmany iterations, and its time-complexity, which is highly implementation dependent, isin the worst-case O(n · N), where n is the number of base-level problems and N is thesize of the base-level populations. In each conguration of the MOP tests, the size ofthe top-level GA's population was chosen to be n-times the size of the base-level GAspopulation size, which means the assumption that one top-level iteration equals to n base-level iterations in computational cost is very generous to the fourth test congurationwhich only evolves a single-level GA encoding the top-level problem. The results of thetests, however, show that the muleGA approaches beat the traditional singe-level GA,even with the iteration handicap.

3.5.5.3 Test Runs and Measurements

The 4, 16 and 32 dimensional MOPs were solved 400 times, the 8 dimensional MOP wassolved 600 times with each GA conguration. The results were tested with the statisticalhypothesis tests presented in the previous KP tests, namely Lillifors-test, two-sampleT-test, Komlogorov-Smirnov test and Wilcoxon-test, all with 5% signicance level. Theresults of the test runs were measured at every icth inner cycle iteration, and the followinginformation were written into database.

Information on the iteration numbers of the base-level GAs and top-level GA. Infor-mation from the top-level GA includes: the distance of PFknown to PFtrue, the distanceof P to PFtrue, the distance of the best-individual from PFtrue (the minimal distance ofthe individuals in P to PFtrue), the size of PFknown, the size of P, the minimal, averageand maximal tness values, the diversity of P and the variance of the objective scores.

3.5.5.4 Test Results

The results of the runs on the binh1 problem are shown in Figure 3.13. Both the base-level GAs and the top-level GA were congured to use tournament selection. In caseof 4 dimensional top-level MOP, all of the runs ended with same results regardless ofthe GA conguration. This coincides with the results of former studies, which impliedthat traditional MOGAs achieve good results for up to 4 dimensional problems. Withincreasing decision space and objective space dimensions NSGA-II loses ground to themuleGAs both in Pareto distance and best (minimal) tness.

It is interesting though that NSGA-II does not fall even further behind when thenumber of dimensions is increased from 16 to 32 and that the results are generally thesame for both measures in both the 16 and 32 dimensional congurations. Note thatthe results are normalized, as both best tness and sum of distances to the Pareto frontincrease together with the dimension of the problem, therefore the results are divided bythe number of dimensions. The characteristic of the 16 and 32 dimensional search spaces


could be similar in the sense that a relatively decent solution can still be found by theNSGA-II in both congurations despite of the dimension increase.

In the Poloni tests, uniform selection was used in the base-level GAs as the tour-nament selection method of GALib does not support negative objective functions, onefeature Poloni's problem has (see Figure 3.12(b)). Note that using tournament selec-tion would have been still possible if the Poloni-problem was inverted, and the aim ofthe optimization was changed from minimization to maximization. However, the lossof selective pressure in the base-level GAs creates another handicap for the muleGAimplementations, which creates an interesting competition.

Results of the tests shown in Figure 3.14 depict that muleGA still outperforms theNSGA-II in high-dimensional congurations, with the full-order variant being signi-cantly better in each result of each conguration, not counting the 4-dimensional resultswhich are equally good. The results show that even the multi-level architecture can behandicapped by the loss of selective pressure, and in the 8-dimensional case, it loses outto the traditional MOGA both in best tness and Pareto distance. Note that this isthe only problem conguration, and the only type (the random input problem mappingtype) of muleGA which results' not matched or bettered the NSGA-II results. Also notethat the loss of selective pressure in the base-level GAs restrain the convergence to thePareto front, and by employing the random type input problem mapping, only averagerandom individuals form the initial population of the top-level GA. Therefore, the top-level GA has half the iterations to catch up the NSGA-II which is an impossible missionconsidering the top-level GA this muleGA tests run is the very same NSGA-II which runsin the single-level test conguration. The fact that in the 16 or more dimensional casesthe single-level GA is unable to match even the random input problem mapping typemuleGA underlines how badly the Pareto GAs work in high-dimensional search spaces.The test results detailed in Appendix B.2 show that even for an 8-dimensional problem,generally the whole top-level GA population is lled with Pareto individuals. It is easierto stand out in something if there are more things to stand out in. That is why thechance of adding a random individual to a population so that it is not dominated by anyindividual from that population increases together with the dimension of the objectivespace.

The results of the test runs are shown in more details on the Figures 3.15 to 3.22,which detail the results of the 16-dimensional bin1-based MOP with boxplots [122]. Pairsof gures show the distance of PFknown to PFtrue and minimal (best) tness values fora particular conguration.

Figures 3.15 and 3.16 show the results of the random input problem mapping typemuleGA. The checkpoints for the measurements were each 40th iteration. The numberof iterations was calculated summing the base-level GAs iteration numbers with thetop-level GA iteration number multiplied with the number of subproblems. Consideringthe test conguration whose results the gures represent, 320 total iterations have beencounted, therefore each base-level GA has evolved for 20 base-level iterations and sohas the top-level GA for another 20 top-level iterations. Although input problem tosubproblem mapping would have only been carried out after the 160th iteration, after


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which the top-level GA was going to evolve the top-level individuals, for the sake ofacquiring statistical information, input problem mappings had been carried out at eachcheckpoint to be able to record information on the state of the top-level problem. Fromthe 0th to 160th iteration, the base-level GAs work on what they do best, approachingPFtrue. Note that the base-level GAs do not worry about the inner-distances of thetop-level individuals. Inner-distance is only managed by the semi-order and full-orderinput-mapping operators and by the tness function of the top-level GA. The top-levelGA takes over after the 160th iteration and reaches its rst checkpoint at the 220th

iteration. In its rst 40 iterations, the top-level GA manages to create individuals withbetter inner-distances altough at the expense of the distance from PFtrue. This clearlydepicts the dierent functions of the various level GAs of a muleGA structure. In a multi-level problem, there are various constraints for each level and each GA tries to nd theoptimal solutions of its problem at its level. Most of the times, the various level objectivesare contradictive, thus the need for an outer cycle which repeatedly gives focus to eachGAs in the multi-level structure. Considering these numerical MOPs, the applicationof inverse mapping (mapping information from the subproblem to the input problems)would also enhance the co-operation between the participants of the multi-populationevolution.

Results of the semi-order input problem mapping muleGA variant are shown onFigures 3.17 and 3.18. Note that the scaling of the y-coordinates dier from the previousgures, and the semi-order variant nds signicantly better solutions than the randomvariant. Looking into the data recording the distances from PFtrue, the same eectobserved as in the previous tests. After the top-level GA takes over, a signicant increasein the distance is observed. The same cannot be told about the minimal (best) tnesses,as there is not much of an improvement performed by the top-level GA. The semi-orderinput problem mapping operator done so well regarding the distances that the top-levelGA hardly nds better solutions.

The same tendencies are perceivable on the Figures 3.19 and 3.20 showing the resultsof the full-order input problem mapping muleGA variant. There is a setback in the


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goodness of distance to PFtrue and even less progression is generated by the top-levelGA considering the minimal (best) tness values.

The Figures 3.21 and 3.22 draw a dierent picture showing the results of the single-level MOGA, the NSGA-II: undiminished progression to PFtrue and to better tnessvalues, no setbacks, no lurching. The convergence of NSGA-II is convincing even ifdealing with a 16-dimensional multi-level MOP. However, one should look at the scal-ing of the y-coordinates of the gures to discover how far really NSGA-II is from themuleGA solvers regarding both distance to PFtrue and minimal (best) tness values.The full-order type muleGA is better in both measures (distance and best tness) ineach presented iteration than the best of the single-level NSGA-II.

The distances of the full-order input problem mapping muleGA variant and the single-level NSGA-II are shown in Figures 3.23 and 3.24. Note that the single-level GA contin-uously gets closer to the results of the muleGA although most of this happens becausethere is not much room left for muleGA to improve after its rst couple iterations.


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3.6 Conclusion on the Multi-Level Genetic Algorithm 83

3.5.5.5 Conclusion of the MOP tests

The test results showed that muleGA properly congured can produce ve-times bettersolutions for multi-level MOPs, using the same or less computational resources thanNSGA-II. Both multi-level decomposition of the problem and input problem mappingcontributes to the results. The base-level GA is responsible for calculating valuablesolutions for the upper-level problem, while the input problem mapping operators areresponsible for selecting the bests from the better solutions.

3.6 Conclusion on the Multi-Level Genetic Algorithm

A novel evolutionary algorithm framework has been introduced (in Section 3.2) which isintended to form an uniform basis for describing Multi-Level Evolutionary Algorithms(MLEAs). The MLEA framework extends the EA formalism by Merkle and Lamont [53]by introducing the handling of multiple populations and an outer cycle algorithm, whichis responsible for the scheduling of the evolution of the multi-level hierarchy of EAs. Thefunctionality of MLEAs has been dened by specifying the input and output parametersand describing the inner mechanisms of those methods (function calls) every MLEA canor must implement. The concept of mapping information from one EA to an other (bothresiding in the multi-level hierarchical structure) has been discussed. A detailed examplewas also presented about input problem mapping.

A new MLEA, the Multi-Level Genetic Algorithm (muleGA) has been dened (inSection 3.4) with the aim of providing an algorithm which betters the previous approachesin solving such MLOPs which have MOKP like subproblems and relationships betweendecision variables speciable by description logic. All the MLEA and GA specicalparameters of muleGA were given, including four outer cycle scheduling algorithms.

Extensive testing of muleGA was carried out on enhancements of traditional GA andMOGA test problems. The tests (presented in Section 3.5) resulted in conrming thesuperiority of muleGA over traditional GAs and MOGAs in solving high-dimensionaland multi-level problems. A new type of KP, the Multi-Level MOKP (which is basicallyequivalent to the Dietary Menu Plan (DMP) from the point of view of quantitative con-tent) was introduced for testing the muleGA. Because GAs rapidly lose performance insolving MOKP when the number of the boxes (decision variables) increases, the applica-tion of problem decomposition muleGA supports is particularly reasonable for KP likeproblems with high number of decision variables (like the DMPs).

The proven viability of muleGA for solving Multi-Level MOKPs makes it an idealchoice for solving the Dietary Menu Plans (DMPs) (which are specied in the followingchapter of the dissertation).

Chapter 4

Expert System Design for HealthPromotion

4.1 Introduction

The chapter introduces the scope, domain, goals, system architecture and results of thedietary menu planning systemMenugene. In this introductory section, a brief descriptionis given on health promotion and the importance of nutrition, on the nutrition counselingprocess and on its methodical and technological background. Section 4.2 presents thestate-of-art in nutrition counseling expert systems by introducing the main features anddesign philosophies of the systems developed so far. In Section 4.3, the objectives ofthe dietary menu planning problems are exhibited through introducing concepts whichsupport the formalization of the problem. The formalization of the dietary menu plan-ning problem is carried out in Section 4.4, where the similarities to Knapsack Problemsare also discussed along with the analogousness of DMP related concepts to physical ac-tivity timetable planning. The architecture of the expert-system is discussed in Section4.5, where the application of Multi-Level Genetic Algorithms (muleGAs) to the dietarymenu planning problem is presented, as well as how the nutrition related information isdescribed in a special ontology and how the expert rules guide the evolutionary searchprocess. Overall results of the testing of the system are also presented here.

4.1.1 Health Promotion

Health Promotion is best known to be the science and art of helping people change theirlifestyle to move toward an optimal health [123]. Other denitions are the following.Health promotion is the process of enabling people to increase control over, and to im-prove, their health" dened in [124]. In [125] health promotion is dened as the set ofthree mechanism, namely self-care" ; "mutual aid, or the actions people take to helpeach other cope"; and "healthy environments".

Health Promotion can be performed in many ways (like persuasions through one-to-one advices, displays and exhibitions, mass campaigns, etc) and in many locations

4.1 Introduction 85

(health care facilities, schools, workplaces, etc). It is now widely believed that healthpromotion strategies only achieve signicant behavioral changes if people are addresseddirectly with personalized health action plans. The workplace is one of the settings whichreceive special attention for promoting health. A survey of workplace health promotionprograms concludes that strong evidence exists on the positive eects of worksite healthpromotion programs, which have inuence on dietary intake and physical activity [126].The Internet is an ubiquitous medium and is available at the workplaces, schools andtherefore it could and should be utilized for providing information services and healthrelated personalized advices for the masses. The proposed expert system Menugene aimsto provide the technological background for these type of health promoting services andmainly focuses on the dietary domain. Thus, in the following, the impact of diet andnutrition on the individual's health status is discussed.

4.1.2 Impact of Diet and Nutrition

The link between health, diet, nutrition, food and food supplies is a documented scienticevidence [127, 128]. The idea of food safety and the seriousness of foodborne diseasesspread through contaminated food are notorious, mainly because the relatively shortincubation period of foodborne illnesses. On the other hand, it is not in the generalpublic's mind that the quality of one's diet may play a role of same importance (or evenof more importance) in developing diseases, although during longer time periods.

Figure 4.1: Contribution of diet and nutritionto the burden of disease in Europe.

The share of disease attributed to poordiet and that avoidable through better dietis distinguished and measured separately in[128]. The contribution of diet as cause tothe burden of disease is shown in Figure4.1, while the eect of diet on the preven-tion of diseases is presented on Figure 4.2.The quality of diet pays great contributionto the prevalent lifestyle diseases, such ascardiovascular diseases [129, 130], neoplasia[131] and diabetes [132] and provides pos-sibilities for preventive medicine, primary,secondary and ternary prevention [133, 134].Right after smoking, overweight and obesitycame out close second as cause of most pre-ventable deaths [135]. The everyday atten-tion is still not maintained by the majority of the western population, however recentstudies claim the public is aware of the importance of the situation [136] as poor diet andlack of physical activity is perceived more likely as priorities for prevention than before.The responsibility is on the individual to maintain a consistent and healthy lifestyle. Theeects of poor diet and lack of physical activity not only aects the life of the individualbut also the population. This means work years lost due to chronic diseases developedin consequence to irresponsible lifestyles.

86 Expert System Design for Health Promotion

(a) Mortality statistics (b) Preventive eect

Figure 4.2: Diet as cause of reductions in disease. Subgure 4.2(a) shows the Hungarian mortalitystatistics for men with cardiovascular diseases giving 51.3% and neoplasia allocating23.9% of the pie-chart. Subgure 4.2(b) highlights the preventive eect resides indiet and nutrition, through which 25.6% of the male population could extend thehealthy and productive era of their lifetime.

Besides the signicant eects on health and economy, diet also has a remarkableimpact on the environment. As western diets are rich in animal-based fats, their produc-tion requires several orders of magnitude of energy (land and water resources) more thanwould be needed by producing the same calories from cereals and vegetables [137, 138].The transportation of goods across continents and local transportation to customers(measured in food miles) [139] and the high usage of fossil energy in food productionmeans there is space for improvement, optimization and rationalization from an econom-ical point of view. It has been shown that a healthier diet and the return to traditionalfarming could help reduce fossil energy fuel consumption in US food system by 50 percent[140]. The food production accounts for around 20 percent of the total energy use in theUnited States.

4.1.3 Nutrition Counseling

Many denitions of Nutrition Counseling exist. Generally, the denitions bear the samemeaning, and only dier in textual composition. In the following, two denitions arecited.

• Nutrition Counseling is an ongoing process in which a health professional, usuallya registered dietitian, works with an individual to assess his or her usual dietaryintake and identify areas where change is needed. The nutrition counselor providesinformation, educational materials, support, and follow-up to help the individualmake and maintain the needed dietary changes [141].

• Nutrition Counseling is a process by which a health professional with special train-ing in nutrition helps people make healthy food choices and form healthy eating

4.1 Introduction 87

habits. In cancer treatment, the goal of nutritional counseling is to help patientsstay healthy during and after treatment and to stay strong enough to ght infec-tions and the recurrence of disease. Also called dietary counseling [142].

Inducting from the denitions, the tasks performed by dietitian experts during theircontinuous contributions either fall into the category of assessment or planning.

• Nutrition Assessment is an in-depth evaluation of both objective and subjectivedata related to an individual's food and nutrient intake, lifestyle and medical his-tory. The data for a nutritional assessment falls into four categories: anthropometric(human body related physical parameters, e.g.: weight, body mass index), bio-chemical (e.g.: blood tests, urine tests), clinical (e.g.: longitudinal patient records,medical history), and dietary (e.g.: diet history, food frequency analysis) [143].

• Dietary Planning activities, amongst others, include individual dietary planning,creating dietary guidances, institutional food planning, military food and nutritionplanning, food fortication [127].

4.1.4 Food Composition Databases

One cornerstone of the nutritional counseling process is the Food Composition Database(FCDB) involved. As one main purpose of the Dietary Menu Planning Process is tocreate Dietary Menu Plans (DMPs) which contain the right amount of nutrients, theamount of nutrients in foods needs to be known.

In the FCDB, information is collected over the nutritional composition of foods. Thedetails of FCDBs can vary from compiling only basic information on nutrition content(energy, and main nutrients, such as carbohydrate, protein, fat, etc.) to comprehensivelistings of ingredients. The USDA National Nutrient Database for Standard Reference[144] records the amounts of around two hundred ingredients per food.

4.1.5 The Process of Dietary Menu Planning

For a qualied dietitian, it takes about three hours to plan a personalized, weekly dietarymenu for an individual. When the necessary information is gathered about the patientduring the nutrition assessment the dietary planning starts.

The task of the dietitian is to prepare a Dietary Menu Plan (DMP), which basicallyrecords the parameters of each dish served in the meals for a given period of time. Inpractice, the DMP is at least for 3 days and usually not for more than 10 days. TheDMP and if there is any, also its subcomponents (daily plan, a single meal, etc.) has tosatisfy the requirements previously set in the nutritional assessment.

4.1.5.1 Recipe Database

The recipe database is connected with the nutrient database (also called food composi-tion database) and records the type and portion of food items comprising dishes. There


are a lot of commercial software tools which are bundled with a detailed recipe database.These software applications are using proprietary code, and the data storage is mostlikely unique for each implementation. There are, however, standardized tools for man-aging recipes, in the form of XML schemes (RecipeML, CookML). These open standardsallow the conversion of data among software tools which use the same food compositiondatabase.

4.1.5.2 Dietary Menu Planning

The creation of a dietary menu plan is carried out by lling menu patterns with dishesselected from the recipe database. The menu pattern denes what kind of items themenu contains. For example, a typical menu pattern for a lunch is <soup, garnish,

topping, drink, dessert>. The duration of the Dietary Menu Plan (DMP) can beanything from a simple meal to several weeks. The number of slots in the menu patternsis in linear proportion to the duration of the DMP.

4.1.5.3 Assessment of the Dietary Menu Plan

The assessment of a DMP has at least two aspects. Firstly, we must consider the quantityof nutrients. There are well dened constraints for the intake of nutrient componentssuch as carbohydrate, fat or protein which can be computed for anyone, given their age,gender, body mass, type of work, age and diseases. Optimal and extreme values can bespecied for each nutrient component [127, 145]. So as for quantity, the task of planninga meal can be formulated as a constraint satisfaction and optimization problem.

Secondly, the harmony of the DMP's components should be considered. Plans satis-fying nutritional constraints should also be appetizing. The dishes of a meal should gotogether. By common sense, some dishes or nutrients do not appeal in the way others do.This common sense of taste and cuisine should be incorporated in any nutritional coun-selor designed for practical use. There could also be conicting numerical constraints orharmony rules. A study found that even menus made by professionals may fail to satisfyall of the nutrient constraints [146].

4.2 Nutrition Counseling Expert Systems: The State of theArt

The rst referenced mathematical solution of the nutrition and diet related problem datesback to the 1940s. In fact, it was one of the rst applications of the simplex method.Dantzig describes the problem in his book Linear programming and extension (1963)[147] as follows:

One of the rst applications of the simplex algorithm was to the determination of anadequate diet that was of least cost. In the fall of 1947, Jack Laderman of theMathematical Tables Project of the National Bureau of Standards undertook, as atest of the newly proposed simplex method, the rst large-scale computation in this

4.2 Nutrition Counseling Expert Systems: The State of the Art 89

eld. It was a system with nine equations in seventy-seven unknowns. Using hand-operated desk calculators, approximately 120 man-days were required to obtain asolution. ... The particular problem solved was one which had been studied earlierby George Stigler (who later became a Nobel Laureate) who proposed a solution basedon the substitution of certain foods by others which gave more nutrition per dollar.He then examined a "handful" of the possible 510 ways to combine the selectedfoods. He did not claim the solution to be the cheapest but gave his reasons forbelieving that the cost per annum could not be reduced by more than a few dollars.Indeed, it turned out that Stigler's solution (expressed in 1945 dollars) was only 24cents higher than the true minimum per year $39.69.

Research in the eld of computer aided nutrition counseling has begun in the 1960s.In 1964, Balintfy developed a linear programming method for optimizing menus [148].Balintfy's computer code has been developed to plan menus by nding minimal costcombinations of menu items such that the daily dietary, gastronomic and productionrequirements can be satised for a sequence of days. While the menus met nutritionalconstraints, they did not satisfy the aspects of harmony. From an economical pointof view, according to the author, up to 30% percent of food cost saving was possiblewith the software, however a considerable amount of data processing had to precede theimplementation of the system. The code was written in FORTRAN and was runningon the IBM 1410 computer. In 1967 Eckstein used random search to satisfy nutritionalconstraints for meals matching a simple meal pattern. Each menu was composed ofseven components which were meat, starchy food, vegetable, salad, dessert, beverageand bread. Food items were randomly chosen for each component and were evaluated bycriteria including calories, cost, color and variety [149]. The program was iterated untilthe menu generated became satisfactory. A mixed-integer linear programming methodwas developed in 1993 [150] to select appropriate food items from a database. Themethod was improved to only allow foods in a diet plan which are part of the samerecipe [150]. Other mixed-integer linear programming approaches followed [151].

Later, articial intelligence methods were developed mostly using Case-Based Rea-soning (CBR), Rule-Based Reasoning (RBR) and the combination of these two with othertechniques [152, 153]. A hybrid CBR-RBR system, CAMPER [154] integrates the ad-vantages of the two independent implementations: the case-based menu planner, CAMP[155, 156, 157] and PRISM [158].

Although most of the tools that have been developed are entitled as nutrition coun-seling expert systems, they are primary focused on dietary menu planning. While froma computational intelligence point of view, algorithms which are capable of classifyingusers according to their nutritional assessment parameters and calculate their lifestyleand nutritional goals are non-trivial, planning dietary menus considering the personalizedgoals of a patient and general nutritional guidelines is a more complex problem.

Note that dietary planning activities involve the usually repetitive assessment of theplan, according to how it complies with the regulations, and how it assists the patientsreaching their goals, while the nutritional assessment is a straightforward and sequential


task. The more recent nutrition counseling expert systems are briey detailed in thefollowing.

4.2.1 CAMP

CAMP is a case-based reasoner, its case-base holds 84 daily menus obtained from rec-ognized nutrition sources, reviewed by experts for adequacy and modied to ensure thateach one conforms to the Reference Daily Intakes and aesthetic standards. However,there is not a menu which is good for all individuals, as they may vary in their tastesand nutrition needs. CAMP stores solutions for daily menus and also records their use-fulness which is computed according to the menus' nutrient vector, the types of mealsand number of snacks and foods included. CAMP operates by retrieving and adaptingdaily menus from its case base. CAMP's adaptation framework is based on the manualapproach of nutritional experts to design menus. Meal-level and food-level variations areperformed before nutrient-level adaptations. A detailed description of CAMP is given in[155, 157].

4.2.2 PRISM

The rule-based menu planner PRISM performs the same task as CAMP but in a dierentfashion. PRISM relies on menu and meal patterns and its approach to menu creationconsists of the following steps: generate, test, and repair. A daily menu is generatedby successively rening patterns of meals, dishes, foods, and lling general pattern slots.After a menu is generated to t both user specications and common sense, it is testedto see if it meets nutritional constraints. PRISM uses a backtracking process to repairsolutions, in which new foods, dishes or meals are substituted for those found to benutritionally lacking. [158].

4.2.3 CAMPER

CAMPER is an integration of the techniques employed by PRISM and CAMP. The CBRmodule was taken intact from CAMP and the RBR module was modeled on PRISM.The database of CAMPER is also more sophisticated than those of its predecessors,containing data for 608 food items, and describing each role a food can fulll. Accordingto the authors, CBR and RBR complement each other in CAMPER. CBR contributesan initial menu that meets design constraints by building on food combinations thatproved satisfactory in the past, and RBR allows the analysis of alternatives, so thatinnovation becomes possible. The ability to produce new cases for later use by a CBRmodule is signicant. This enables the system to improve its performance over timeCAMPER, taking advantage of CBR/RBR synergy, provides a capability which neitherCAMP nor PRISM provided. At rst, CAMPER obtains the menu planning criteria andselects the best daily menu from its case base. The selection of the menu is based onthe ease of its adaptation to meet the actual criteria. The selected menu is then adaptedusing strategies employed by human experts. Snippets, parts of other menus and domain

4.2 Nutrition Counseling Expert Systems: The State of the Art 91

specic adaptation rules are also employed. The user is allowed to execute What ifanalysis to add, delete or replace foods, making customization available to the user [154].

4.2.4 MIKAS

A more recent CBR approach is MIKAS, menu construction using incremental knowledgeacquisition system [159]. MIKAS allows the incremental development of its knowledge-base. Whenever the results are unsatisfactory, an expert will modify the system-produceddiet manually. MIKAS asks the experts for simple explanations for each of the manualactions they take and incorporates the explanations into its knowledge base [160].

4.2.5 DIETPAL

A web based system entitled Dietpal models the workow of dietitians. It has been builtin Malaysia recently for dietary menu generation and management [161]. According tothe authors, the main novelty of their tool is the use of the complete dietary-managementsystem currently adhered by dietitians in Malaysia. In addition, Dietpal is implementedas a Web-based application, therefore the outreach of the system for use by dietitiansand health professionals within the same hospital or at other locations is increased. Thesystem is also capable of storing and organizing dietary records and other health anddiet related information which allows dietitians to eectively evaluate and monitor thepatient's dietary changes throughout the period of consultations.

While more than a few expert systems have been developed recently for nutritioncounseling, a solution that at least tries to satisfy each and every aspect of an ideal menuplanning is still missing.

4.2.6 Knowledge Modeling

Collections of knowledge on dietetics, food and nutrition are mainly available in theform of textbooks and reference guidelines. Only a few attempts have been made toformalize expert knowledge by the means of expert systems [158, 154, 159] or ontologies[162, 163, 164, 165]. Recently, the emphasis is put on formalizing food and nutritionrelated information through ontologies. An overview of the ontology-based nutritionrelated knowledge modeling approaches is presented in [164].

4.2.7 Evolutionary Algorithm Based Methods

Recently, EA based methods turned up for dietary menu planning, utilizing the poten-tial of GAs for near-optimum search. The essence of these approaches [91, 92] is in thesimplicity of the EA concepts, namely the survival of the better. The dietary plan isconsidered as an individual of a population which is evolved by the GA. All of the prob-lem specic knowledge is implemented in the tness function, which in this applicationis responsible for assessing the dietary plans according to quantitative and qualitativeparameters. The methods developed by the author and presented in [91] are detailed in


Section 4.5. Do not mistake the dietary menu planning problem with planning menusfor Graphical User Interfaces which is also on the plate of the applications of GAs [166].

4.2.8 On the Application of Expert Systems for Lifestyle and NutritionCounseling

The expert systems provide means for dietitians and people to exploit the potential ofubiquitous computing resources in their everyday life by leaning on the instructions of theArticial Intelligence (AI) tools when dealing with lifestyle related questions. However,the lone Information Technology (IT) device with integrated knowledge and decisionsupport techniques for lifestyle and nutrition counseling will not save the situation. Itssole reason of existence is pedagogical, and the author's opinion is that overly excessiveutilization of these methods could result in counter eects such as addiction and obsessivecompulsive behavior. The user of such expert systems ought to learn from the system andlearn along with the system. An expert system like those mentioned before are for the userwho intends to use it, learns to use it and learns how to utilize it. The high-tech gadgets,the intercommunicating agents shown in Figure 4.3 might help exposing the importanceof the user's minute decisions regarding diet and physical activity. Regardless of theadvances in AI, these tools are for decision support and the user remains the decisionmaker.

Intelligentshopping cart recipe flowchart

medical expert

dietary logintelligent fridge

24/7 wearablesensors

ambientsensors

activity monitoringworkout planning

EXPERT SYSTEMmonitoringanalyzingreasoning

Figure 4.3: Schematic infrastructure of a comprehensive expert system, which integrates intodaily life with its decision support logic readily accessible whenever and whereverneeded.

4.3 Objectives of Dietary Menu Planning 93

4.3 Objectives of Dietary Menu Planning

To be able to handle the Dietary Menu Planning Problem from an algorithmic perspec-tive, new concepts and notations are introduced.

4.3.1 Concepts and Notations

For the discussion of the Dietary Menu Planning Problem, the following concepts andnotations are introduced: Nutritional Level (NL), Nutrition Hierarchy Object (NHO),Dietary Menu Plan (DMP) and Nutritional Structure (NS).

4.3.1.1 Nutritional Level (NL)

Nutritional Levels (NLs) are the levels of nutrients, foodstus, dishes, meals, daily menuplans, weekly menu plans. These are the traditional NLs dietary experts are workingwith. Each nutritional object involved in the Dietary Menu Planning Problem wouldbelong to one of these levels. In case of special need, further NLs can be introduced.Virtually unlimited number of upper levels could be introduced, after the level of weeklyplans, monthly plans and annual plans would follow, and so on. Introducing NLs down-wards could go as far as current physical and chemical understanding goes.

4.3.1.2 Nutrition Hierarchy Object (NHO)

A Nutrition Hierarchy Object (NHO) stands for any nutritional object which is part ofthe solution of the dietary menu planning problem, regardless of the object's NutritionalLevel (NL), function in the menu plan, its composition or integrity. The concept ofhierarchy is introduced later in this section. Real-world counterparts of NHOs are shownin Table 4.1.

NL real-world counterpart of the NHOWeek menu plan for a weekDay menu plan for a dayMeal menu plan for a single mealDish record of a recipe database

Foodstu record of a food composition databaseNutrient column of a food composition table

Table 4.1: Real-world counterparts of Nutrition Hierarchy Objects (NHOs) from correspondingNLs.

4.3.1.3 Dietary Menu Plan (DMP)

A Dietary Menu Plan (DMP) in practice, is basically a list of dishes and their servingtimes and portions for a given time period. For example, if the DMP is a meal plan (the


DMP Level is the level of meals), then the list of dishes it records forms a single meal.If the DMP is a daily plan, then its list of dishes forms a menu plan for a whole day.Considering the latter, subsets from these dishes form the meals of the day (breakfast,lunch, dinner, etc.), which are DMPs on the level of meals.

Considering an arbitrary DMP which is not a single meal, its subcomponents shouldalso be handled as DMPs. For human experts, this information is implicitly encoded inthe serving times of the dishes. If two dishes are served at the same day, then they arepart of the same daily plan, while if they are served in the same hour, they are part ofthe same meal.

In the context of this thesis, to be able to represent the DMP related informationexplicitly, a more complex structure is used for DMPs. Every DMP has a dened numberof slots for storing Nutrition Hierarchy Objects (NHOs) from a specic NL, along withserving times and portions. In this way, the DMPs form a hierarchical structure.

For example, a DMP is a meal plan, which has four slots for soup, garnish, topping anddrink. Four NHOs which represent dishes would be selected for the four slots respectivelyfrom a recipe database (see Section 4.1.5.1). Again, for example, a DMP for a daily menuplan with ve slots for breakfast, morning snack, lunch, afternoon snack and dinner,would represent meal-level DMPs in its slots, explicitly indicating the information thatsubcomponents of the daily menu plan are forming a whole in their own, namely, otherDMPs in the level of meals.

4.3.1.4 Nutritional Structure (NS)

The Nutritional Structure (NS) is the hierarchical, multi-level structure of NHO. Themenu pattern, and consequently the number of slots of a DMP are proportional to theduration of the DMP. Depending on the number of DMP slots, subcomponents of theDMP may form other DMPs which would belong to a lower NL. In this case, the DMPwith subsets of slots also representing DMPs from lower NLs is actually built up of thoselower level DMPs, which are consequently the building blocks of the upper level DMP.

4.3.1.5 Other concepts and notations

• DMP Slot: A slot in a DMP can be either left empty, or lled with a NHO.

• DMP Slot Level: The level of a slot denes that NHOs from which NL can be usedto ll the slot.

• DMP Level: denotes that for which Nutritional Level (NL) the DMP was compiled.The DMP Level can be any NL starting from the level of meals. In practice, theDMP Level is either the level of meals, daily plans or weekly plans.

• DMP Duration: the time period the DMP is for.

There is no generally accepted method for producing a good Dietary Menu Plan(DMP). Additionally, a DMP can only be evaluated as a whole after it is fully con-structed. However, subcomponents of the DMP can still be assessed during the planning

4.3 Objectives of Dietary Menu Planning 95

weekly plan level

daily plan level

meal level

dish level

foodstuff level

. . .. . . . . .

... ... ......

... ... ...

... ... ... ...

DMPNHO

DMPNHO

DMPNHO

NHO

NHO

NHOnutrition level

Figure 4.4: Nutritional Structure (NS) with Nutritional Levels (NLs), Dietary Menu Plans(DMPs) and Nutrition Hierarchy Objects (NHOs).

period, but the overall goodness of the DMP can only be calculated in the view of all ofits components. Note that the duration of a DMP is virtually unlimited, but it is usuallylimited to a single meal, a daily plan or a weekly plan.

4.3.2 Attributes of a Dietary Menu Plan

A DMP, and as well, any NHO may virtually have an unlimited number of attributes,which can be taken into consideration during its assessment. These attributes can bequantiable or non-quantiable. For example, the amount of protein in a NHO is quan-tiable and can be expressed with a numerical value, while the seasonality (which forexample can be either a month or a season in which the particular NHO is availablefor consumption) or the type of the NHO (which for example can be normal, vegan or


vegetarian) cannot be naturally expressed with a numerical value. Of course, numericalvalues can be assigned to the months of a year and to the NHO types, but still, these in-tegers do not cumulate or should not get cumulated when a NHO is created by combiningtwo NHOs (either in space- or in time-domain).

4.3.2.1 Assessment of the Objects of the Nutritional Hierarchy

A NHO is assessed by rating and examining its attributes and the attributes of theoverall content of its slots. A NHO represents itself and all those NHOs which it containsthrough its slots and it is assessed according to all the quantiable and non-quantiableattributes these NHOs have. This means that each NHO a DMP encompasses inuencesthe overall goodness of the DMP, so in the hierarchical NS, every NHO as a building-blockcontributes to the worth of upper-level NHOs it is built into.

4.3.3 Overview of Dietary Menu Planning Objectives

Concluding the aboves, solutions of the Dietary Menu Planning Problem can be consid-ered as hierarchical structures. The solution for the DMP is considered as a NutritionHierarchy Object (NHO), and it is made up of NHOs from lower levels of the hierarchy.The aim of the menu planning process is to have those NHOs selected for the solution,which both individually and collectively satisfy the expectations of the patient the DMPis addressed to.

Note that the DMPs form a multi-level structure, which perfectly matches the hierar-chical concepts of the MLOPs (Section 2.4.1). The similarities of DMPs to KPs remainsto be presented in Sections 4.4 and 4.4.2.

4.3.4 Analogous Objectives of Lifestyle Counseling and Physical Ac-tivity Planning

It was declared in the introductory section of the thesis that the dietary related conceptsof health promotion were going to be used through the dissertation for specifying theexpert system's logic and functionality. Also it was mentioned that the planning ofphysical activity timetables is basically the same task as planning dietary menus. In thefollowing, the similarities of the two domains are discussed in short.

The goal of the physical activity timetable planning is to provide the users withpersonalized plans recording such bodily activities that maintain or enhance their physicaltnesses and overall health. The main focus is on maintaining the calorie intake to calorieburn ratio, therefore such low intensity everyday activities like walking can also be partof the plan. Naturally, workout exercises for body building can be synthesized with thesame methods if the corresponding data on the exercises is available. Which are thesedata, and how they compare to the structure of the NLs, NHOs, DMPs and NSs? Thecorresponding activity related concepts are introduced in the following. Note that theterm level refers to some position in the hierarchy and not to the rate of exercise intensity.

4.4 Formal Problem Statement 97

Activity Level (AL) is analogous to the NL. The levels of eect, routine, exercise, dayand week are distinguished here. Each Activity Hierarchy Object (AHO) belongsto one of these levels. Naturally, further levels can be introduced.

Activity Hierarchy Object (AHO) is analogous to the NHO. The AHO stands forany activity related object which resides in the Activity Structure (AS). Real worldcounterparts of AHOs are shown in Table 4.2.

AL real-world counterpart of the AHOWeek collection of exercises for a weekDay collection of exercises for a day

Exercise collection of exercise routinesRoutine a single exercise routineEect amount of calorie burn, chemical increasement, time taken

Table 4.2: Real-world counterparts of Activity Hierarchy Objects (AHOs) from correspondingALs.

Activity Timetable Plan (ATP) is analogous to DMP. It is a list of routines per-formed at some point during the duration of the ATP. ATPs have slots to representlower-level AHOs, just as this was the case in the nutrition domain. An exerciseconsists of one or more routines, which have eects on the performer (energy burn,increasing chemicals such as dopamine, serotonin, etc.). The time it takes to per-form a particular routine is also represented as an eect on the performer.

Activity Structure (AS) is analogous to NS. It refers to the hierarchical, multi-levelstructure of AHOs.

The aim of physical activity planning is to create such an ATP that consists ofexercises which are matching the needs of the performer. Routines in an exercise programshould harmonize (e.g.: should not put too much strain on any specic muscle group,etc). Also, the sum of the eects of the exercises should be close to the personal optimagiven by physical activity guidelines [167].

From the analogous concepts of physical activity planning and dietary menu planningit can be seen that the two type of plans (DMP and ATP) are built up of quite similarcomponents. Therefore, the computer-based generation of these plans can be carried outwith the same approach. In the following, the nutritional domain is going to be used forthe discussion of the a solver developed for the DMP (and ATP).

4.4 Formal Problem Statement

In the following section, the Dietary Menu Planning Problem is going to be examinedfrom a computational intelligence point of view. Exact problem denitions will be given,and the similarities of the various DMPs to the KPs will be discussed.


4.4.1 Dietary Menu Planning Problems

Four DMPs are dened in the following. We start at the simplest, single-objective single-level version. Then an extension which handles multiple objectives is given. After that,the problem is extended to multiple levels. In the nal extension, the handling of knowl-edge and harmony is introduced.

4.4.1.1 Simple Dietary Menu Planning Problem (S-DMPP)

An assignment A := (ai,j)m×n of objects to slots is sought, where pi is the payload of theith object, cmin and cmax are the lower and upper constraints, and the following holds.

cmin ≤m∑i=1

n∑j=1

(pi · ai,j) ≤ cmax (4.1)

Equation (4.1) denes the satisability part of the S-DMPP problem. In many cases,lower and upper values of constraints imposed on the payload are quite wide, with anoptimal value for the payload existing somewhere between the lower and upper values.To represent this optimal value of the payload, the value copt is introduced, and theproblem is extended with the optimization part, which minimizes the deviation of thesum of the payloads from its optimal value, as dened in Equation (4.2). Note that forcopt : cmin ≤ copt ≤ cmax, but copt not necessary equal to (cmin + cmax)/2.

minimize

∣∣∣∣∣∣ copt −m∑i=1

n∑j=1

(pi · ai,j)

∣∣∣∣∣∣ (4.2)

For any specic S-DMPP, the payload for the ith object is its energy content given inkilojoules considering that the aim of the optimization is to optimize the energy contentof the DMP. The constraints cmin and cmax are lower and upper values for the energycontent of the DMP, with copt being the optimal value. Four type of S-DMPPs can bedened depending on how the assignments of objects to slots are restricted.

According to the relations of objects to slots, the following types of the problem aredened.

• (1,1)-S-DMPP: An object may belong to one slot, slots may hold one piece of anobject at most.

• (1,∞)-S-DMPP: An object may belong to one slot, slots may hold many pieces ofan object.

• (∞,1)-S-DMPP: An object may belong to many slots, slots may hold one piece ofan object at most.

• (∞,∞)-S-DMPP: An object may belong to many slots, slots may hold many piecesof an object


The domain of each S-DMPP is shown in Table 4.3. Note that for each S-DMPPversion, the maximal number of non-zero entries in the matrix is the number of slots.Although in some cases slots may hold many pieces of an object, but those pieces shouldbe instances of the same object.

problem-type domain of object-slot relation(1,1) [O × S](1,∞) [O × S × N+](∞,1) [O × Sn](∞,∞) [O × Sn × N+]

Table 4.3: Domain of relation according to the type of S-DMPP, where O is the set of objects([O] = m) and S is the set of slots ([S] = n).

4.4.1.2 Multi-Objective Dietary Menu Planning Problem (MO-DMPP)

In the dietary practice, there are more than one goal for every DMP. One of these goalsis the optimization of the energy amount of the compilation, however, other parameters(such as the amount of fat, carbohydrate and ber, to name only the most importantattributes) should be considered. If we count each and every aspect from which a DMPcan be assessed, the number of goals can surpass one thousand (only the number ofnutrition components is more than two hundred in a comprehensive FCDB, and yetthere are other goals).

An assignment A := (ai,j)m×n of objects to slots is sought, where P := (pi,k)m×o,and pi,k is the kth payload of the ith object, and the following holds.

cTmin ≤

m∑i=1

n∑j=1

([PT]i · ai,j) ≤ cTmax (4.3)

where cTmin and cT

max are column vectors, with each pair of their values in the samerow representing the minimal and maximal values of a constraint for a given payload.

Note that Equation (4.3) does not dierentiate the feasible solutions, as it eitherdeclares a solution feasible or infeasible. A solution is sucient (feasible) if it satises theconstraints, and insucient (infeasible) if not. To have more delity in the formalization,the column vector cT

opt should also be dened as well as the counterpart of Equation (4.2),which would minimize the deviation of the payloads from the optimum. For modesty'ssake, the assessment according to the distance from the optimum cT

opt is omitted fromthe formalism, however it is noted, that whenever an algorithmic solution is sought forthe problem, it is preferred to nd the best conguration among those which satisfy theconstraints. If there is no feasible solution, which may happen considering the variety ofrecommended dietary intakes [146], then still the best possible solution is sought amongstthe non-feasible ones.


4.4.1.3 Hierarchical Multi-Objective Dietary Menu Planning Problem (HMO-

DMPP)

Considering a DMP which describes a real-world menu plan (for example for a week),it encompasses sublevel DMPs (plans for a day, plans for meals, etc.) which form ahierarchical structure. This hierarchical connection of the subproblems cannot be rep-resented with the MO-DMPP model, as it does not support the handling of subprob-lems. Therefore, it is extended here to handle assignment matrices arranged in a multi-level and hierarchical structure. [Al+1] is an assignment of objects to slots consideringan (l + 1)-level problem. [Pl+1] is the payload matrix of this (l + 1)-level problem.Al,u := (al(i,j))m(l,u)×n(l,u)

assignment matrix of level l objects and slots for the uth sub-problem of Al+1 with the payload matrix [P(Al,u)].

The assignments of the subproblems need not be of the same dimension. Neither thenumber of slots, nor the number of objects should be equal for the subproblems. Thisalso holds for the payload vectors.

As the level lmatrices build up the [Al+1] assignment matrix, fl,u functions are neededwhich transform the number of r subproblems' assignment matrices height to match theheight of [Al+1]. Basically the fl,u functions, which are unique for each subproblem, arefor adding objects which were not present in the assignment matrix but were at least inone of the other subproblems' matrices. Note that these newly added objects would notbe assigned to any slots in the modied f(A) matrices.

[Al+1] = [fl,1([Al,1]) . . . fl,r([Al,r])] , 1 ≤ l ≤ q (4.4)

Variable Index Variable(s) Meaningm i number of objectsn j number of slotso k number of payloadsq h,l number of levelsr u,v number of subproblems

Table 4.4: Meaning of the variables used to formalize the HMO-DMPP.

The Hierarchical Multi-Objective Dietary Menu Planning Problem (HMO-DMPP) isformulated as follows, where q is the number of levels the problem has, with Aq beingthe uppermost assignment matrix, which has no parent problems, meaning that it is nota subproblem of any other problem. The meaning of the variables and expressions usedfor the formalization are presented in Table 4.4 and Table 4.5. Notice the similarities ofthe notations to those used for dening the MLOP in Denition 7 (in Section 2.4.1).

[Aq] =[fq−1,1([Aq−1,1]) . . . fq−1,r(Aq)([Aq−1,r(Aq)])

](4.5a)


cTmin(Aq) ≤

m(Aq)∑i=1

n(Aq)∑j=1

([P(Aq)T]i · ai,j) ≤ cT

max(Aq) (4.5b)

recursively-evaluate-problem([Aq]) (4.5c)

Algorithm 5 recursively-evaluate-problem([Ah,v])

for all l, u such that ∃Ah,vl,u do

evaluate cTmin(Ah,v

l,u ) ≤∑m(Ah,v

l,u )

i=1

∑n(Ah,vl,u )

j=1 ([P(Ah,vl,u )T]i · ai,j) ≤ cT

max(Ah,vl,u )

recursively-evaluate-problem([Ah,vl,u ])

end for

Expression Meaning

[Ah,vl,u ] The assignment matrix A represents a solution for a level l problem,

which is the uth subproblem of its level h parent, which is the vth

subproblem of its parent.[Ah

l,u] The assignment matrix A represents a solution for a level l problem,which is the uth subproblem of its level h parent. The same as theabove one, but without denoting whether the level h parent problemhas a parent or not.

[Al,u] The assignment matrix A represents a solution for a level l problem,which is the uth subproblem of its parent. The same as the above one,but without denoting the level of the parent.

[Al] The assignment matrix A represents a solution for a level l problem.The same as the above one, but without denoting whether the problemis a subproblem

m(Ah,vl,u ) the height (the number of rows) of the [Ah,v

l,u ] matrix, namely the num-ber of objects

n(Ah,vl,u ) the width (the number of columns) of the [Ah,v

l,u ] matrix, namely thenumber of slots

Table 4.5: Meaning of the expressions used to formalize the HMO-DMPP.

4.4.1.4 Introducing Knowledge and Harmony

From the sole point of view of ingredient quantication, the HMO-DMPP model for-malizes the problem. If an object is assigned to a slot, its payloads take account in thesummation and in the assessment of the assignment, which in turn is a DMP.


Taking the example of a meal plan and an assignment of a food object to a meal planslot, the USDA SR21 FCDB object Apple, raw, with skin with medium size (3" indiameter, 182 gram in weight) is assigned to the slot dessert. It is easily calculable fromthe data of the FCDB that, for example, how much energy (397kJ) and protein (0.47g)this object adds to the DMP.

This assignment holds implicit information for the human expert, therefore thisknowledge has to be expressed by some means in the model. What information is pre-sented in this assignment which has not been expressed explicitly? The Apple, raw,

with skin with medium size is an apple, is also a fruit, and is also a dessert made offruit. This is basically the taxonomic information of the component. More to that, thisassignment makes the DMP one which contains fruit, one which contains apple, and onewhich has fruit for dessert. Moreover, if this meal plan does not contain any meat, itbelongs to the class of vegetarian meals with fruit, which information is derived from thecombination of the meal's components. Here we recall the concepts of the taxonomy andcombination vals of muleGA (Section 3.4.1) which were introduced to support the han-dling of these non-quantiable and implicit type information. This knowledge descriptionof Menugene regarding object, object-slot, object-DMP and object-slot-DMP relations isway more detailed and will be dealt with in Section 4.5.4. For now, the function denotedwith ρ (standing for harmony) will be used to take an assignment matrix as parameterand transform it to a column vector with real values, with each row representing the valueof a harmony payload. Each of the harmony payloads represents one of the followingconcepts.

Taxonomy information is inferred from the assignment matrix of each DMP and rep-resents implicit and unquantiable knowledge about that DMP (handled in muleGAby taxonomy vals). In the harmony vector, taxonomy information is representedwith the values 0 or 1. At the moment, the handling of fuzzy membership valuesis not introduced in the representation of the non-quantiable information, but itremains an option for the future. The zero value means the DMP is not the memberof the taxonomy class, while one means it is a member.

Combination information for a DMP is inferred from the assignment matrices of thesublevel DMPs and represents the unquantiable information the combination ofthe sublevel DMPs bear (handled in muleGA by combination vals). In the harmonyvector, taxonomy information is represented with the values 0 or 1.

Combination harmony value is calculated similarly as the combination informationbut results in a numerical value rating the harmony of the combination of the com-ponents making up the DMP (represented in muleGA through objective values).In the harmony vector, taxonomy information is represented with real values.

For example, if the assignment matrix of an arbitrary DMP has 2 pieces of Apple,raw, with skin assigned to one slot, and 3 pieces of Pear, raw, with skin to anotherslot, and no other fruits and no meat at all, then the harmony vector will have the value5 for the harmony payload entitled fruits (taxonomy information). If the DMP is for


a lunch, then the harmony vector will have the value 1 for the harmony payload calledvegetarian lunch with fruit, and 0 for the harmony payload called dinner with fruit(combination information).

The example presented the most simple case, when the NHO Apple, raw, with

skin was selected to ll a specic slot in the DMP. In this case, the apple was a simpledish, which had only one ingredient, one apple. However, DMP slots usually assignedwith NHO representing dishes with more ingredients. Take the soup slot of a DMP formeal as an example, and assign chicken soup to it. The payload of this object can becalculated by summing the data of each component of the recipe provided by the FoodComposition Database (FCDB).

Besides taxonomy and combination information, the harmony vector has to expressthe harmony of the objects assigned to slots of the same DMP through the combinationharmony values. The harmony payloads for representing these kind of information wouldbe called harmony of the assignment, harmony of fruits in the assignment and so on.These harmony payloads will get zero value if there is no information for example on theharmony of fruits. Positive value will mean harmony, with a bigger value meaning moreharmonizing components, while negative values are for representing disharmony. For eachassignment matrix, the calculation of the harmony constraints (taxonomy information,combination information, combination harmony values) is done according to Equation(4.6).

hTmin(Ah,vl,u ) ≤ ρ(Ah,vl,u ) ≤ hT

max(Ah,vl,u ) (4.6)

To express harmony in the model, the calculation of the harmony vector accordingto Equation (4.6) should be added to the HMO-DMPP model, namely to Equation (4.5)and to Algorithm 5.

Note that the complex calculation of the harmony vector is hidden in the functionρ. The computational complexity of the calculation of ρ is exponential in time in thefunction of the number of the slots (n), because each subset of the objects associatedwith slots should get evaluated according to harmony. The harmony of each subsetwill be expressed through the harmony payloads. Not including the empty set, thenumber of subsets is 2n − 1, and this many checks are needed to calculate the valueof each harmony payload. This makes the proof of the decision problem 'whether theassignment's payloads are within the constraints' veriable in exponential time. For NPcomplexity the proof would have to be veriable in polynomial time by a deterministicTuring machine. Therefore, the introduction of harmony makes the decision problemharder than NP.

4.4.1.5 Hierarchical Multi-Objective Dietary Menu Planning Problem with

Harmony (HMO-DMPP-H)

The HMO-DMPP-H model extends the previous HMO-DMPP model with the conceptof the harmony function, which is represented by ρ. This function calculates the tax-onomy and combination information and the combination harmony values. The actual


calculation of the harmony payloads, which can involve description logic inferencing andstrenuous numerical calculations, is hidden behind the function ρ. The HierarchicalMulti-Objective Dietary Menu Planning Problem with Harmony (HMO-DMPP-H) isformulated as follows, where q is the number of levels the problem has, with Aq beingthe uppermost assignment matrix, which has no parent problems, meaning that it is nota subproblem of any other problem.

[Aq] =[fq−1,1([Aq−1,1]) . . . fq−1,r(Aq)([Aq−1,r(Aq)])

](4.7a)

cTmin(Aq) ≤

m(Aq)∑i=1

n(Aq)∑j=1

([P(Aq)T]i · ai,j) ≤ cT

max(Aq) (4.7b)

hTmin(Aq) ≤ ρ(Aq) ≤ hT

max(Aq) (4.7c)

recursively-evaluate-problem-with-harmony([Aq]) (4.7d)

Algorithm 6 recursively-evaluate-problem-with-harmony([Ah,v])

for all l, u such that ∃Ah,vl,u do

evaluate cTmin(Ah,v

l,u ) ≤∑m(Ah,v

l,u )

i=1

∑n(Ah,vl,u )

j=1 ([P(Ah,vl,u )T]i · ai,j) ≤ cT

max(Ah,vl,u )

evaluate hTmin(Ah,v

l,u ) ≤ ρ(Ah,vl,u ) ≤ hT

max(Ah,vl,u )

recursively-evaluate-problem-with-harmony([Ah,vl,u ])

end for

4.4.2 Related Problems

The Dietary Menu Planning Problems dened in Section 4.4.1 share similarities withthe Knapsack Problems (KPs) presented in Section 3.5.2. However, extensions of thosetraditional KPs is needed to make them really similar to the DMPs. In the following, twoextensions to the traditional MOKPs are given, which introduce the concept of harmony.

4.4.2.1 Similarities to Knapsack Problems (KPs)

The Simple Dietary Menu Planning Problem (S-DMPP) resembles to the single-objective0-1 KP. The transformation of S-DMPP to 0-1 KP is done by dening the parameters ofthe pi prot and wi weight of the KP to bear the attributes of the corresponding objectof the S-DMPP with pi payload. The boxes of the KP are dened with pi = wi values,or a small dierence can be introduced to represent that the actual nutritional content isa function of a probability distribution, where the variables pi < wi denote the minimumand maximum deviations with a given condence. Whether pi = wi or pi < wi, the aim


of the KP is to maximize the prot (satisfy the cmin constraint of the S-DMPP) withsuce weight (satisfy the cmax constraint of the S-DMPP).

Analogously, the Multi-Objective Dietary Menu Planning Problem (MO-DMPP) istransformable to a Multi-Objective Knapsack Problem (MOKP). The interpretation ofthe MOKP is the same as in the single-objective case, only with multi-dimensional protand weight constants. Extending the problems to multiple levels, the similarity remains,as HMO-DMPP is easily transformable to the Multi-Level Multi-Objective KnapsackProblem. The rst level objects-to-slots assignments of the HMO-DMPP are representedas the rst-level knapsacks and boxes of the Multi-Level MOKP. The upper-level objects-to-slots assignments get represented as the upper-level knapsacks.

While these dietary planning problems are not in full equivalence with the corre-sponding KPs, with a simple transformation, a valid solution of one can be convertedto a valid solution of the other and vice versa. From the point of view of quantity, thepreviously dened versions of KPs were suce, however the harmony related informationcannot be represented with them. To further demonstrate the similarities of the DietaryMenu Planning Problems to variants of Knapsack Problems, two new KPs are denedhere.

4.4.2.2 Multi-Level Multi-Objective 0-1 KP with Combinatorial Harmony

The combinatorial harmony means that each combinations of the boxes residing in oneknapsack contribute to the overall harmony of the knapsack. The overall harmony rep-resented in the harmony vector could violate harmony constraints set for the knapsack.In the multi-level conguration, there are harmony constraints for each knapsack.

Denition 11 (MLMO 0-1 KP with Combinatorial Harmony) Let ~pall denote theprice vector, which is the sum of the ~pj price vectors of the v rst level knapsacks, eachknapsack with nj possible boxes. The z upper-level knapsacks are denoted with ~w∗k. Thevalue of Kk(i) denes which boxes from which knapsacks should also get packed into theupper level knapsack k for weight constraint checking with ~c ∗k . The harmony vector for

the rst-level knapsack j and upper-level knapsack k is denoted with ~hj and ~hk, with

the minimum and maximum constraints being denoted with ~hminj , ~hmink and ~hmaxj , ~hmaxk

respectively. Then, the following is a Multi-Level Multi-Objective 0-1 Knapsack Problemwith Combinational Harmony.

maximize ~pall =v∑j=1

~pj =v∑j=1

nj∑i=1

~p(i,j)x(i,j) (4.8a)

subject to ~wj =

nj∑i=1

~w(i,j)x(i,j) ≤ ~cj , (4.8b)

~w∗k =

|Kk|∑i=1

~w[Kk(i)]x[Kk(i)] ≤ ~c ∗k , (4.8c)


~hminj ≤ ~hj ≤ ~hmaxj , ~hmink ≤ ~hk ≤ ~hmaxk (4.8d)

where x(i,j) ∈ 0, 1, ~cj = c(1,j), . . . , c(mw,j), ~c∗k = c∗(1,k), . . . , c

∗(mw,k) (4.8e)

~pj = p(1,j), . . . ,p(mp,j), ~wj = w(1,j), . . . ,w(mw,j), [Kk(i)] ∈ N2 (4.8f)

j = 1, . . . , v, k = 1, . . . , z, nj ∈ N+ (4.8g)

~hj = c-harmony-ks(j), 1 ≤ j ≤ v, ~hk = c-harmony-ulks(k), 1 ≤ k ≤ z (4.8h)

The denition is the extension of Denition 10 and Equations (3.10) with the con-straints on the harmony vectors and the methods c-harmony-ks() and c-harmony-ulks().The function c-harmony-ks() is responsible for computing the harmony of a knapsack,while c-harmony-ulks() is used for calculating the harmony of upper-level knapsacks.The distinction between assessing rst-level and upper-level knapsacks is made becauseof the roles the dierent type of knapsacks have in the problem. While in the rst-level knapsacks, there is no spatial and timewise dierentiation between the boxes, onupper-levels, the spatial properties of the boxes, namely that which box came from whichknapsack(s) (one rst-level and zero to more upper-level knapsacks) can aect harmony.Combinational harmony discards any time-related information (precedence, order of theboxes in which they were put into the knapsack), so the harmony of a combination isthe same regardless of the order the boxes were put into the knapsack. Generally, theharmony functions return the harmony vector for a specic knapsack. The size of thevector and what meaning each dimension carries is problem dependent. The harmonyvector would be the listing of all taxonomy and combination vals and combination har-mony values if the muleGA concepts of val dened in Section 3.4.1 were considered. Allof the possible combinations of the boxes in the knapsack can contribute to the overallharmony of the compilation, thus each of the combinations should get assessed. Thenumber of assessments is in factorial proportion to the number of boxes in the knapsack.This entails the harmony versions of the KP are not in NP, as their solutions cannot beveried in polynomial time.

4.4.2.3 Multi-Level Multi-Objective 0-1 KP with Permutational Harmony

The permutational harmony means that each permutation of the boxes residing in oneknapsack contributes to the overall harmony of the knapsack. Because there is a har-mony contribution for each permutation, the order in which the boxes were loaded intothe knapsacks matters. As there are only minor dierences between the combinatorialand permutational versions of the problems, Denition 11 is not echoed here, only thedierences are highlighted.

As well as in the combinational case, functions for accessing rst-level and upper-level knapsacks according to harmony are introduced. The function p-harmony-ks()

represents and encompasses the steps needed for accessing a rst-level knapsack according

4.5 Expert System Design 107

to permutational harmony, and the function p-harmony-ulks() does the same for theupper-level ones. The dierences between the two are the same as in the combinationalcase, namely, there is no spatial distinction between the boxes on the rst-level, whilethere is on the upper-levels. Both of these permutational functions dier from theircombinational counterparts, as the order of the boxes in which the knapsacks were loadedcontributes to the harmony of the knapsack. This renders the permutational versionof the problem to be the most computational intensive, as the permutations for eachcombination should get accessed.

4.5 Expert System Design

4.5.1 Algorithm Choice

Looking at the formalization of the HMO-DMPP-H and MLMO 0-1 KP with Combinato-rial Harmony, at rst glance, it would seem reasonable to try using some Integer ProgramSolver (ILOG CPLEX for example). Only at rst glance, because there are at least tworeasons traditional MILP solvers would not work. First, the relations between the deci-sion variables cannot be described by linear or nonlinear equations and inequations butonly with rst-order description logic or any Turing-complete alternative. Second, thehigh dimensionality of the decision space and objective space makes it impossible to solvethe problem with deterministic methods.

The suitability of GAs and particularly muleGA in solving KPs and multi-level KPshas been presented in Section 3.5. The KP related tests were carried out to underlinethe eectiveness of muleGA in creating multi-level DMPs, which from the point of quan-tiable objectives are identical to a corresponding multi-level KP. The necessity of non-quantiable objectives resulted in the introduction of two new KPs, the combinationaland permutational harmony variants. These new type of KPs as well as the multi-levelKPs are not actively researched, there are no references or indications to them. Becauseof their complexity, there would be not much point in examining them unless there werereal-world problems with the exact same characteristics. Considering the Dietary MenuPlanning Problem, this is exactly the case, as the similarities of the problem to the KPshave been presented in Section 4.4.2. The problems are generally equivalent from analgorithmic point of view, only the terminology diers as, regarding the KPs, there areknapsacks and boxes, and in the Dietary Menu Planning Problems there are DMPs andNHOs. Because this section deals with the design of the expert system for dietary menuplanning, the terminology of the latter will be used for proposing and testing the algo-rithm variant which deals with the Hierarchical Multi-Objective Dietary Menu PlanningProblem with Harmony (HMO-DMPP-H) or in other terms Multi-Level Multi-Objective0-1 Knapsack Problem with Combinatorial/Permutational Harmony.

4.5.2 Infrastructure of Menugene

The Menugene system is in development, and its nalization is the goal of a researchproject for human competitive lifestyle and dietary counseling system. Besides the di-


etary menu planning algorithm which is detailed in this thesis, graphical user interfaces,databases and administrative tools were implemented. The aim is to have all the toolsand services accessible from the http://www.menugene.com web-portal. At present, de-velopment work is ongoing for the following services: Menugene Dietary - a dietary menuplanning software for dietitians, which provides tools for maintaining and extending theknowledge base of Menugene. It also provides assessing and enhancing menu plans cre-ated by the human experts. Menugene Dietlog - a web 2.0 tool, which allows everydayusers to record and analyze their dietary intakes. Menugene Flowchart - a web 2.0 toolfor recipe exchange and analysis. Menugene Workout - a web 2.0 based virtual coachimplementation.

4.5.3 Employing Multi-Level Genetic Algorithms for solving the Hier-archical Multi-Objective Menu Planning Problem with HarmonyRules

For creating nutritionally adequate and harmonizing DMPs, Menugene employs muleGAfor the solution of the HMO-DMPP. In the following, the association of muleGA conceptsdened in Section 3.4.1 to DMP concepts presented in Section 4.4 is introduced. Theassociation is quite straightforward, as both of the conceptualizations share the samemulti-level, hierarchical and divide-and-conquer attributes.

Using the notation introduced in Section 4.4, let [Aq] denote the assignment matrixof the q-level HMO-DMPP-H, and let [Ah,v

l,u ] denote an arbitrary assignment matrix for

a level l problem, which is the uth subproblem of its level h parent which is the vth sub-problem of its parent. Then for each [Ah,v

l,u ], the muleGA solver runs a population P(h,v)(l,u) ,

which intends to nd a proper assignment dened in Section 4.4.1.5. The input prob-lem mapping operators simply use the DMPs created in the input problem populationsas building-blocks to congure the subsolutions of higher-level DMPs. The architectureof the multi-level DMP evolving algorithm is presented in Figure 4.5, the scheduling ofthe multi-level evolution process is detailed in Figure 4.6 and the evolutionary operatorsacting on DMPs are shown in Figure 4.7.

4.5.4 Ontology-Based Knowledge Representation

There are virtually unlimited ways to encode the nutrition and diet related knowledge,starting from natural (free-text) language to relational databases, from rst-order logicto semantic networks. The Web Ontology Language [168] OWL-DL was chosen for de-scribing knowledge in Menugene. OWL-DL is computationally complete, decidable andthere are semantic reasoners such as Pellet working with OWL-DL out of the box. Thereare readily available knowledge bases about the nutritional domain in the form of com-prehensive and evolved food coding systems [169, 170]. These knowledge bases provide awell organized taxonomical classication of food items, but a single taxonomy (assembledfrom a single point of view) is not sucient for providing human competitive results. Em-ploying OWL-DL for describing the information content from various databases makes


Figure 4.5: HMO-DMPP-H in Multi-Level GA structure


Figure 4.6: The multi-level GA scheduling strategies (top-down, bottom-up, credit propagation,mutation based) in function of algorithmic levels. The numbers in the top-down andbottom-up columns show the order in which the various levels are evolved.

Figure 4.7: The crossover operator is shown in the gure in function of the nutritional level. Leg-end: M-Monday, Tu-Tuesday, W-Wednesday, Th-Thursday, F-Friday, Sa-Saturday,Su-Sunday BF-Breakfast, MS-Morning Snack, L-Lunch, AS-Afternoon Snack, D-Dinner S-Soup, G-Garnish, T-topping, Dr-drink, De-Dessert, cp-Crossover Point


it possible to retrieve and use information in Menugene from a diverse set of sources. Forthis end, a general way of knowledge encoding is introduced.

Rather than using a single ontology which classies and organizes the nutrition re-lated concepts into categories, multiple ontologies are created. For each ontology, twoparameters are recorded. What are the entities the ontology categorizes? From whataspect does the ontology the categorization? Example ontologies are presented in Table4.6.

what is categorized from what aspectfoods food originfoods seasonal availabilitydishes heat transfer used for cookingvegetarian dishes heat transfer used for cookingmeals cultural aspectsmeals meal types

Table 4.6: Example ontologies

All nutrition related concept is categorized by dietitians. This categorization makesit possible for the expert system, for example, to only include menus in the breakfastplan which are allowed to be served for breakfasts. Also the categorization makes itpossible for Menugene to decide whether the menu plan it created is vegetarian, vegan,semi-vegetarian, pescatarian, raw vegan, macrobiotic, etc.

4.5.5 Rules

Rules are allowed to be formulated in OWL-DL language. The rules formed as OWL-DL sentences can describe content or harmony related information. Example rules arethe following: do not have more than two main meals in a day, do not have starchgarnishes in successional main meals, if possible, do not have tomato soup with tomatodrink,reward meals with no meat or slim in meat. Many of the rules come from nationalcuisine, standard dietary guidances, but also the users of the system can record their ownpreferences.

The rules are used in assessing the result of the menu planning process, and also inthe tness function of the evolutionary search. Repetitive querying of the knowledge-basewould not be eective, so for making the rules rapidly accessible, they get pre-processed.After preprocessing, rules record a list of sols and vals and a goodness value expressedin percentages describing the opinion of the reasoner on the contents of the list. Bothtaxonomy and combination vals are allowed. Note that the sols and vals containedby the list comprises parts of a DMP or can refer to a complete DMP. Sample rulesare: (sol: carrot, val: afternoon snack, goodness: 150%),(sol: tomato-soup, sol: tomato-drink, 50%). The majority of the rules are not extreme in the sense that they do notunambiguously prohibit any particular pairing of sols and vals. The neutral goodness


value is 100%, which means according to the rule-base, the particular DMP which isbeing assessed is no worse and no better than its goodness calculated according to itsnumerical constraints. If the goodness value is less than 100%, for example 50%, thenthe DMP under evaluation is half as good as a DMP which has the same quantitativecontent while not breaching the harmony constraints.

The application of rules in the tness function is generally the on-the-y calculationof the ρ harmony function and synthetization of the harmony constraints dened inEquation (4.6) using the information stored in the rule-base. As it was described inSection 4.4.1.4, the calculation of the ρ function is computationally expensive, as eachsubset of the slots needs to get assessed according to harmony, in other words, matchingrules are searched for the objects residing in the actually assessed slots.

There are algorithms for eciently enumerating the subsets of a set [171]. A similaralgorithm is used byMenugene, which with ne-tunings, enhances the search for matchingrules. First of all, each rule is further processed to a form that only records sols, anda goodness value. So if there is a pre-processed rule, which, for example, records twosols and one val, then all the sols referred by the val is used to create new rules byrepeatedly writing them down replacing the val. For example, if r = s1, s2, v1, 50%and v1 = s3, s4, then the fully processed rules are going to be r1 = s1, s2, s3, 50%and r2 = s1, s2, s4, 50%. In case of combination vals, two or more sols replace the val.

Figure 4.8: A sample rule cache

If all of the vals were replaced in the pre-processed rules to sols, the rule cache is built,which stores the rules in a specic order tospare the traversal of the whole rule cachewhile looking for matching rules. A sample rulecache is shown in Figure 4.8. The cached andsimplied rules, which basically are made up ofa list of sols and a goodness value are storedin linked-lists stitched to each other. The rule-searching algorithm starts by traversing thoseindividual's sols which is under assessment. Itis assumed that the individual encodes its so-lutions ordered by their id. For example, ifa sol represents ve solutions through its at-tributes, s = s7, s8, s25, s97, s96, such a rulewith length of ve is searched, which recordsall of the sols' identiers and a corresponding goodness value. Considering the rulecache shown in Figure 4.8 is used, there is no such rule. Then the algorithm tries tond matching rules with the length of four, again with no success. It continues searchingfor rules with the length of three. There is one matching rule r1 = s7, s25, s96, 25%.It applies the 25% modication on the individual's tness, then continues to search forrules with length of two, with the condition, that they should not be part (subset) of anypreviously applied rule. Therefore, rsub = s7, s96, 75% is not red, because it is partof r1. Another rule with the length of two is found, namely r2 = s96, s97, 120% which


rewards the joint appearance of solutions s96 and s97. Because s96 already appearedin a rule longer than rone = s96, rone is not red. So only two matching rules canbe found in the example rule cache, r1 = s7, s25, s96, 25% and r2 = s96, s97, 120%,which means the tness function of the individual s = s7, s8, s25, s97, s96 is multipliedby 0.25 · 1.2 = 0.3.

4.5.6 Rule-Based Guide of the Evolutionary Search Procedure

Rules have to get expressed in the tness function of the GA both for providing sucientvariety in the DMPs by omitting repetitive components and for making it possible to as-sess DMPs according to the harmony of their components. By modifying the goodnessesof individuals containing particular alleles or pairs of alleles, rules are for adjusting theexpected occurrences of the alleles of the GA.

The method of modifying the tness values according to the indication of the match-ing rules is tested here to analyze whether the tness modication drives the evolutionarysearch by adjusting the expected occurrences of alleles recorded in the rule.

In test runs, the variety of DMPs were measured with reasonable constraints (regulardietary recommendations for women aged between 19-31 with mental occupation). Thevariety of the allele which represents one from the 150 possible alleles is shown in Figure4.9. The gure shows the occurrence (ordered by frequency) of each of the 120 possiblealleles which were present more than 15 times (0.1%) in the best solutions in 15.000 testruns. The most frequent allele in the best solutions of 15.000 runs was present 482 times(3.2%), the second 426 times (2.8%) and the 50th 102 times (0.7%).

Figure 4.9 also shows (lower part) the goodness of the best solution to which thecorresponding alleles belong. The goodness of an allele is computed by summing itsbest tness (i.e. the best tness value of all solutions the allele was part of) with theweighted best tness values of its 8 neighbors. The goodness of the ith allele is denedas: g[i] = f [i] +

∑4j=1 [(1− 0.2j) · (f [i− j] + f [i+ j])] , where f [i] is the tness of the

ith allele. The trend curve (lower part) shows that solutions containing frequent alleleshave a generally better goodness. The results show that allele values appearing in goodsolutions are used more often by the algorithm and the frequency of usage is about inverseproportional to the tness of the best solution generated by using the particular allele.

The eect of the rules were measured on the variety and mean occurrence of thealleles considering the solutions for a DMP for a meal. The results of the statisticalanalysis are shown in Figure 4.12. Two rules (rA and rB) where imposed on two alleles,respectively.

The strictness of the rules was decreased from 100% to 75%, 50% and nally to 25%,giving a total of 16 congurations. Rule rA penalized the solutions which containedan allele with the value A, while rB penalized solutions with allele B. The relativeoccurrences of A are shown in the function of the strictness of the rules in Figure 4.10.

Two-sample Kolmogorov-Smirnov goodness-of-t hypothesis tests with the signi-cance level of 5% were run on the two random samples created by recording the allelesrepresenting drinks in neighboring congurations, running 1.000 times each (using 10


Figure 4.9: The occurrences (ordered by relative frequency, upper gure) of 120 of the 150 pos-sible values of alleles in the best solutions of 15.000 runs. The lower gure shows thegoodness of the best solution of which the respective allele was part of.


Figure 4.10: The relative occurrence of a particular solution (A) in function of the strictness oftwo rules (penalizing alleles A and B).

random populations, running each one 100 times). Figure 4.12 lists those P values (de-noted with KS) for which the Kolmogorov-Smirnov test has shown signicant dierencein the distribution of the two independent samples. Signicant dierences were observedfor all of the test pairs with respect to increasing the strictness of rA (rows in Figure4.12), however the same was not true for all of the pairs with respect to increasing thestrictness of rB (columns in Figure 4.12.), so P values are not listed for such pairs. Theexplanation of this phenomenon is hidden in the dierences between the occurrences ofalleles A and B that were measured without penalties. These occurrences are 436 (43.6%)for A and 68 (6.8%) for B, out of the 1.000 possible. Since the number of instancesof A is comparable to the number of possible instances, rule rA not only changes themean occurrence of A, but signicantly changes the distribution of the alleles. In caseof penalizing the DMP with allele A by 75%, the occurrence count of A decreases by133 (30%) from 436 (100%) to 303 (70%). As Figure 4.11 shows, the 103 occurrences areshared somewhat proportional among the other alleles (A is the 15th, B is the 12th al-lele in Figure 4.11). Single sample Lilliefors hypothesis tests of composite normality wereperformed on the samples with an element size of 10, in 100 runs of the algorithm, with10 dierent starting populations, where the occurrences of A and B were counted.The distribution of the occurrences of A in function of the starting populations provednormal, except for one case. Again, due to the few occurrences of B in the test runs, itsdistribution could have not been determined. The samples of neighboring congurationswere paired, and if both had normal distribution with a signicance level of 5%, paired


Figure 4.11: Occurrences of the 15 possible alleles in a solution for a DMP in 1.000 runs infunction of the strictness of the rule imposed on the 15th allele.

Figure 4.12: Statistical analysis of the distribution of the potential alleles in the best solutionsand the mean occurrence of the alleles (A,B) on which the rules were imposed.

4.6 Conclusion on Expert System Design for Health Promotion 117

t-tests were run to check if there is a signicant dierence in the mean occurrences ofalleles A and B. The results of the paired t-tests are shown in Figure 4.12, denotedwith T. In case of signicant dierences, the corresponding P values are also listed. Thesamples which did not have normal distribution are marked with an asterisk (*).

Since the sample distribution was not known for more than half of the samples, theWilcoxon signed rank test of equality of medians with the signicance level of 5% wasemployed on each sample pair, to measure whether there is signicant dierence betweenthe mean occurrences. Results are shown in Figure 4.12 with the corresponding P values,and are denoted with Sr. There were only 3 situations where there was no signicantdierence between the mean occurrences. These cases are marked with a hyphen (-).

The results of the tests reveal that the application of the rules result in proper answersfrom the search process, as the GA based search is steered to those regions of the searchspace where the penalized combinations of sols are not presented. It has also beenrevealed that the strictness of the rules represented by the goodness values are suitableto correctly and proportionally reward or penalize a sol or a combination of sols.

4.6 Conclusion on Expert System Design for Health Pro-motion

The chapter presented a novel approach for an expert system for health promotion.After a brief overview on previous nutrition counseling expert systems (in Section 4.2)a comprehensive formalization of the dietary menu planning problem was presented (inSections 4.3 and 4.4), where also the analogous objectives of physical activity timetableplanning were discussed. Next, the similarity of the dietary menu planning problemsto KPs were introduced, that lead to the explanation of the algorithm choice. Prior tothis chapter, a novel MLEA, the muleGA was presented, which provides almost all thefunctionalities needed to solve multi-level DMPs out of the box. Only one extension wasnecessary for solving DMPs, and it was provided by the introduction of the rule-basedassessment technique. The eects of this new technique have been tested, and the resultsproved that it is possible for the novel muleGA based method to eectively solve a MLOPthat is built up by special MOKPs that necessitate the inner harmony of the content ofthe KPs.

The DMP tests proved that the muleGA based expert system logic is capable ofplanning nutritionally adequate diet plans, and that it provides means to eectively assessand promote harmonizing menu and food combinations. The goals and constraints ofthe DMP tests dened considering regular dietary recommendations. Since the expertsystem provides a nutritionally and harmonically adequate menu plan, it replicates theoutput of the expert. Therefore, the output of the system cannot be distinguished fromthe output of an expert. This is a necessity for satisfying the Subject Matter ExpertTuring test (also called the Feigenbaum test [172]).

Although no comprehensive head to head comparisons of human and computer cre-ated dietary menu plans were made, it can be concluded from the results and experiencesthat the DMP solver can outperform the human expert considering nutritional adequacy


related objectives and the time needed for the construction of the personalized plans.This makes the system truly applicable in the dietitian practice. The rule-base of thesystem is still under construction and it remains to be in development for some time. Therule-based assessment of the DMPs proved to be successful, however, it remains to beseen how well the (sometimes contradicting) goals of nutritional adequacy and harmonycan be managed trough the adjustments of the weights of the rules and quantity relatedobjectives.

In summary, a comprehensive mathematical formalization of the real-world dietarymenu planning problem has been given. Algorithmical tools were constructed for thesolution of this problem. At the moment there is no such approach foreseen by theauthor which could signicantly improve the performance of the DMP solver consideringthe quality of the DMPs. Therefore, any improvement of the expert system is consideredto result from the evolution of the knowledge base of Menugene.

Chapter 5

Conclusions

Herein, the contributions of the dissertation are summarized, and an outlook on futurework is given. The thesis presented a research work whose aim was to create an expertsystem engine for health promotion. This ultimate aim triggered many additional goalsand contributions. The steps in which the author's research work was carried out arepresented in the following.

5.1 Summary of Contributions

In the following, the contributions of the thesis are presented in a conducing order (asopposed to the inducing order used in Section 1.2).

First, the mathematical formalization of the dietary menu planning problem wasgiven with unprecedented detail (in Section 4.4). Because of the abstraction employed inthe formalism, similar combinatorial optimization problems (such as the physical activitytimetable planning problem) can also be described by the model. The formalization ofthe dietary menu planning problem uses the following abstract concepts: independent ofthe domain, every plan consists of empty slots, to which (elementary) objects could andshould be assigned. These objects may have an unlimited number of payloads, which candescribe quantiable and harmony related taxonomy information. Through the payloadsare described the properties of each plan (for example, a quantiable parameter is theamount of vitamin C in bell pepper, and a taxonomy information is that the bell pepperbelongs to the Solanaceae family). The number of slots depends on the size of the plan.Considering the dietary domain, a dietary menu plan for a week has about seven times theslots of a daily plan. For the proper solution of the planning problems, it was necessaryto support the handling of subsets of slots (subplans of the plan) as separate plans andto allow the denition of constraints and objectives for each of these subproblems aswell as for the main problem. It was also necessary to support the assessment of eachcombination of objects assigned to the slots of any particular plan. This necessity madethe problem harder than NP, because for NP complexity, the proof of the solution shouldhave to be veriable in polynomial time by a deterministic Turing machine. This doesnot hold for the dietary problem considered in this thesis, because the assessment of

120 Conclusions

each combination of the objects is of factorial complexity. A further necessity was tosupport the handling the subplans as objects. These non-elementary objects (subplans)summarize their whole underlying hierarchy of so called lower-level subplans, and arealso assignable with taxonomy and harmony related information. Considering all ofthese necessities, the formalization of the HMO-DMPP-H was given in Section 4.4.1.5.

Second, the comparison of the newly formalized dietary menu planning problem totraditional optimization problems was carried out. The similarities to the well knownMOKPs have been demonstrated (in Section 4.4.2), and therefore, popular KP solverswere discussed (in Section 2.5). Also, extensions were given to the traditional MOKPfor making it virtually equivalent to the dietary menu planning problem, thus providinga way for discussing a novel problem in a familiar context. The following new KPswere dened: Multi-Level Multi-Objective 0-1 KP (MLMO01KP), and the harmonyrelated versions, namely MLMO01KP with permutational harmony (MLMO01KP-PH)and with combinational harmony (MLMO01KP-CH). For solving the new KPs, a GAbased method was chosen, and because of the high-dimensional and multi-level natureof the problem, a novel GA, the Multi-Level Genetic Algorithm (muleGA) has beenintroduced (in Section 3.4). The abstract concepts of the muleGA are the following: solas solution, attrtext as attribute, val as value. In the main population of muleGA, top-level sols are evolved. The sols are composed of attrs, and through these attrs, theyrepresent lower-level sols, which are subsolutions of the main problem. The quantitativeinformation is represented by sols, while the non-quantiable taxonomy and harmonyrelated information is described with vals. For the assessment of each sol, its wholeunderlying structure can be evaluated, which consists the list of lower-level sols andvals. The muleGA concepts are easily utilizable to form any Multi-Level OptimizationProblem (MLOP).

Third, an abstract framework based on the work of Merkle and Lamont [53] for de-scribing the class of Evolutionary Algorithms (EAs) which evolve multiple populationsin a multi-level hierarchical structure has been introduced. The framework of Merkleand Lamont denes the concepts of population transformation and random populationtransformation, and the EA operators: mutation, selection, recombination, consideringtheir iteration number and random dependences. As being a comprehensive formalism,it provides a good base for specifying the class of Multi-Level Evolutionary Algorithms(MLEAs) (Section 3.2), because it clearly depicts how the hierarchical co-operation ofthe multiple populations in an MLEAs modify the inner structure of the traditionalEA, and also that which components remain unchanged. The methods that are inte-grated into the traditional EAs and specify each MLEA implementation are the following:create-population-schedule, adjust-population-schedule, sublevel-evolutions,input-problem-mapping, pre-recombination-, pre-mutation- and pre-selection-

input-problem-mapping, whose functions and input- and output parameters were alsodetailed (in Section 3.2.2). A MLEA implementation is specied through these methodsand the parameters of the EAs residing in the multi-level hierarchy. By utilizing theconcepts of the MLEA formalism, the position of muleGA among the class of MLEAswere given.

5.1 Summary of Contributions 121

Fourth, the advantages and benets of muleGA over traditional GA based methodsin solving 3-level MLMO01KPs (Section 3.3.3) were proven through empirical tests (theharmony related tests were conducted with dietary menu planning test problems - to bediscussed later). From the 50 test congurations, muleGA produced signicantly betterresults in 38 and worse in none of the congurations. The largest margin in favor ofmuleGA was measured to be 17% (which means the traditional GA approach reached atmost 83% in that specic problem conguration). The performance of muleGA showed adecrease in function of the number of base-level knapsacks. Its advantage over traditionalGAs vanished when the number of base-level knapsacks reached 256. This is the problemsize when the convergence in the base-level GAs disappears, and the search performancedrops to the level of random search. For overcoming this symptom, decomposition of thebase-level GAs would be needed. Besides the KP tests, experiences were carried out withmulti-level versions of well-known numerical Multi-Objective Problems (MOPs) (Binh1[119] and Poloni [121]). The numerical test problems made it possible to express thedistances of muleGA and NSGA-II [62] created solutions from the true Pareto fronts (inthe KP tests, only the relative performances of the muleGA and traditional GA basedmethods were available, because of the enormous sizes of the search spaces of the multi-level KPs, these search spaces could be enumerated in rational time). The test resultsshowed that the distances of the Pareto solutions of muleGA from the true Pareto frontcan be one-fth of the distances measured for the Pareto solutions of NSGA-II, whiletheir average goodness can be ve times better. During the comparison of muleGA andNSGA-II performances on multi-level multi-objective numeric test problems, besides theaverage distances of the Pareto solutions to the true Pareto fronts and the best tnessvalues, the following parameters were measured: average distance of the whole populationto the true Pareto front, the distance of the best individual to the true Pareto front, thenumber of Pareto solutions in the population, the average and worst tness values, thedivergence of the population and the variance of the objective values. Through thesemeasurements, the properties and good features of muleGA were presented (in Section3.5). It was also discussed how and on which type of problems muleGA can providecommanding results.

Fifth, through the utilization of the muleGA, a novel solver was presented for theHMO-DMPP. This is basically the application of the method elaborated in the sec-ond segment of the author's research work for the solution of the problem formalized inthe rst segment. The associations between the abstract muleGA concepts (sol, attr,val) and DMP parameters have been dened (in Section 4.5.3). This new dietary menuplanning problem solver proved capable to solve the multi-level dietary menu planningproblem. Extension of muleGA was only needed to support the harmony related assess-ment of the menu plans, which is presented in the following.

Sixth, a rule-based assessment technique has been integrated to the tness function ofthe muleGA solving the multi-level dietary menu planning problem. As every individual(sol) of the muleGA maintains the list of the sols and vals residing in its subhierarchy,the assessment of each individual can utilize the information available through theselists. Therefore, the harmony rules can refer to any subcomponent of a DMP. The

122 Conclusions

harmony rules only deal with the combination of the components of the DMP. A simplerule consists of a list of sols and vals and a penalty or reward value, which rates theharmony of the compilation represented by the list. For the assessment of an individual(solencoding a DMP), every combination of its subcomponents should be checked if thereis a corresponding harmony rule which penalizes or rewards that specic combination.This task is computationally expensive, therefore a rule-cache based approach has beenintroduced (Section 4.5.5). The rule-cache maintains a list of preprocessed harmony rules,only containing sols and one penalty value (vals are replaced to the sols they represent).These preprocessed rules are arranged in a hierarchy, and for each solution, only the mostcorresponding preprocessed rules are selected. It was proven with empirical tests that thesimple harmony rules adequately modify the convergence of the muleGA and properlyincrease or decrease the occurrences of the penalized or rewarded solutions, respectively.

Seventh, a hierarchical structure of ontologies has been created with which the lifestyleand nutrition related knowledge can be described in unprecedented details (Section 4.5.4).The ontology structure contributes to the rule-based assessment, because it extends thelimited expressivity of the simple harmony rules, and it provides a proper way to describethe taxonomy of the application domain. The novelty of this approach is that separateontologies are dened for describing a domain of expertise, and the position of eachontology in the hierarchical ontology structure is determined by its domain (which canbe a very narrow eld of a knowledge area) and the point of view from which the taxonomyof that domain is constructed.

In the following, the summary of the author's scientic results is presented, and anoutlook is given on the application of the results, and on the future work.

5.2 Summary of Scientic Results

Here, the thesis related new scientic results are summarized in brief. The correspondingpublications of the author are cited.

Thesis 1 Novel Multi-Level Evolutionary Algorithm Framework and Multi-Level Ge-netic Algorithm (muleGA). Thesis related publications are [T1, T2, T3, T6, T7,T8, T9, T10].

1.1 Formalization of the Multi-Level Evolutionary Algorithm has been given,which provides a generic abstract framework to dene and describe new typeof GAs with multiple populations arranged in multi-level hierarchy.

1.2 Novel Genetic Algorithm named Multi-Level Genetic Algorithm (muleGA)has been developed, and the cornerstones of the algorithm have been speciedaccording to the Multi-Level Evolutionary Algorithm formalism.

1.3 Proof of viability and superiority of muleGA over traditional GA basedmethods in solving hierarchically structured multi-level MOPs has been giventhrough extensive empirical comparison and evaluation.

5.3 Applications and Future Work 123

Thesis 2 Novel Nutrition Counseling Expert System Design has been presented. Thesisrelated publications are [T1, T2, T3, T5, T6, T7, T8, T9, T10, T11, T12, T13, T14].

2.1 Formal Denition of the Dietary Menu Planning Problem has been givenstarting from the simplest representation of the problem, then iterativelyincluding all necessary details to nally formulate the Hierarchical Multi-Objective Dietary Menu Planning Problem with Harmony (HMO-DMPP-H),which is an accurately rendition of the objectives.

2.2 Novel Dietary Ontology Architecture has been developed based on hier-archical organization and classication philosophy. It utilizes the possibilitiesof OWL-DL and provides a straightforward way for dietitians to record expertknowledge.

2.3 New Algorithm has been produced for the solution of the HMO-DMPP-Hutilizing the abstract concepts of muleGA, which ensures each subcomponentis optimized by a corresponding GA responsible for seeking nutritional ade-quate combinations.

2.4 Rule-based Assessment has been introduced to hybridize the muleGA forincluding expert knowledge dened in the dietary ontology for assessing plansaccording to harmony. The adequateness of the rule-based guidance of theevolutionary search has been shown through empirical tests.

5.3 Applications and Future Work

From the new methods and algorithms resulted from the author's research work, thenutrition and lifestyle counseling expert system has the greatest application potential.It makes the lifestyle and dietetics related expert knowledge available for the massesin an integrated, personalized and inexpensive way, by utilizing the existing internetand telecommunication infrastructure, and without the need for a human expert. Itssignicance lies in its applicability for exploiting the remarkable eects of nutrition andphysical activity on lifestyle diseases for primary, secondary and ternary prevention. Withadequate nutrition, the development of cardiovascular disease, neoplasm and diabetescan be prevented or prolonged. Therefore, the pedagogical utilization of the expertsystem can contribute in decreasing and solving very important health related sociallevel problems.

The primary application area of the novel Multi-Level Genetic Algorithm (muleGA)is the synthesis of personalized dietary and physical activity timetable plans as the en-gine of an expert system for health promotion. However, as the muleGA is a problemindependent algorithm, therefore it can be utilized for solving a diverse set of optimiza-tion problems, and it is expected to perform remarkably well on problems which belongor similar to the class of Multi-Level Optimization Problems (MLOPs). The potentialof muleGA and the technique of hierarchical decomposition has been presented by theauthor in parameter estimation of lumped-parameter cardiovascular models. [T1, T4].

124 Conclusions

The most interesting development area of muleGA is the utilization of General-purpose computing on graphics processing units (GPGPU) in the multi-level optimizationprocess for Massive parallel processing (MPP). Those graphics processing units (GPUs)that support MPP provide a previously unimaginable processing power for virtually any-one who can aord a mainstream personal computer. Numerical calculations on thesemodern GPUs can run more than a thousand times faster than on conventional CentralProcessing Units (CPUs). However, only massively parallelizable algorithms can exploitthe structure of the GPUs, hence, only these, usually simple algorithms benet the mostfrom the GPGPU technology and gain advantage over more complex and advanced algo-rithms which are usually highly sequential and only running on CPUs. GAs do not tendto have three orders of magnitude advantage over random search methods in real-worldapplications, therefore, a GPU based simple random search algorithm possibly outper-forms a more sophisticated and CPU based GA implementation. However, another goodthing about GAs is that they are partially or even fully massively parallelizable, thereforetheir performance can be increased by orders of magnitude by utilizing the novel GPGPUtechnology and programming philosophy.

Another research possibility for the future is the development of automatized problemdecomposition and multi-level hierarchy formation methods and new and intelligent inputproblem mapping operators, which transform information between the GAs residing inthe multi-level hierarchy. Also, the integration of local search algorithms into the muleGAremains a research possibility for the future.

The muleGA related software implementations and documentations are accessible atthe http://mulega.uni-pannon.hu website. The MOGALib multi-objective GA libraryon which muleGA is built on and which is maintained by the author is accessible at thehttp://mogalib.uni-pannon.hu website.

Appendix A

Details on Evolutionary Algorithmsand Multi-Objective Optimization

A.1 Details on Global Optimization Methods

A.1.1 Details on Deterministic Algorithms

A.1.1.1 Greedy Algorithms

Greedy algorithms [173] always make locally optimal choices at each stage, assuming thatthe locally optimal subsolution is part of the globally optimal one. Greedy algorithmswork well on problems which have optimal substructure, in which, the global optimalsolution contains optimal solution for subproblems. Because greedy algorithms neverreconsider their decisions, subproblems have to be non-overlapping, otherwise greedyalgorithms may fail to nd the global optimum, opposed to dynamic programming meth-ods, which, by reconsidering previously made decisions, work exhaustively on the dataavailable, thus nding the global optimum in problems with optimal substructure andoverlapping subproblems.

Greedy algorithms are irrevocable. Considering a search in a graph, once a decisionhas been made about which node to expand, it is impossible to reconsider that decisionas there is no tracking of past expanded nodes.

A.1.1.2 Hill Climbing Algorithms

The hill climbing algorithm [174] is a greedy local search algorithm which tries to nd theminimum or maximum of a function f(x), where x represents a discrete state, typically avertex in a graph. Considering a maximization problem, by examining the neighbors ofthe actual vertex xa simple hill climbing chooses the rst vertex xni where f(xa) < f(xni),while steepest ascent hill climbing examines all of the neighboring vertices (xn1 , . . . , xnk

)and chooses xnj such that f(xa) < f(xnj ) and f(xnj ) >= f(xni), where i = 1, . . . , k andk is the number of neighboring vertices. The same algorithm which works on continuoussearch space is called the gradient ascent or gradient descent algorithm.

126 Details on Evolutionary Algorithms and Multi-Objective Optimization

There are randomized stochastic versions of Hill Climbing algorithms, such as theRandom-restart hill climbing algorithm, which tries to overcome the problem of stuckingin local optimum by simply restarting the hill climbing process from random pointsof the search space. The comparison of deterministic and randomized versions of HillClimbing algorithms on multi-objective KPs is presented in [175], concluding that hillclimbing methods with proper heuristics can perform well on multi-objective KnapsackProblem (KP).

A.1.1.3 Branch and Bound algorithms

Branch and Bound algorithms try to nd the minimal or maximal value of a functionf(x). First the search space is split up into subregions, which process is called branching.Typically, branching is repeated recursively, and the subregions form a tree-structurecalled search-tree or branch-and-bound tree, where the nodes of the tree represent theregions.

Second, upper and lower bounds are assigned to the optimal solutions of the nodes(subregions). Considering a minimization problem, regions with lower bounds larger thanthe upper bound of a previously examined region can be safely omitted from followingexamination, which is called pruning.

If the upper and lower bounds assigned to a node are equal, then the node is solved,and it represents the optimal value of the corresponding subregion. Solved nodes canstill get pruned. The algorithm stops if all the nodes in the search-tree are either prunedor solved. Solved nodes will represent the global optimal solution for the search problem.

A.1.1.4 Other Deterministic Algorithms

The deterministic algorithms Depth-First Search, Breadth-First Search, Best-First Searchand Calculus-Based Search are detailed in the book Introduction to Algorithms [176].

A.1.2 Details on Stochastic Algorithms

A.1.2.1 Simulated Annealing

Simulated annealing [177] is a technique that can be used in various elds of optimization.The method tries to explore the search space and nd the global optimum, with respect toa cost function of the system to be optimized. The working space is searched via a seriesof random modications or jumps. If a jump is accepted, then the old system state isreplaced with the modied state. All jumps that decrease the cost function are accepted.Cost increasing jumps are accepted with the Boltzmann probability P = e−

∆CT , where

∆C is the cost increase, and T is the temperature parameter. During the search process,T is gradually decreased, but the best cooling schedule is application dependent.

A.1 Details on Global Optimization Methods 127

A.1.2.2 Swarm Intelligence

Swarm Intelligence deals with the behavior of a natural or articial system which is madeup of individuals which coordinate through self-organization and decentralized control[178, 179]. The focus is on the collective behaviour of the individuals, which is the result oftheir local interactions. Typical example studies in swarm intelligence deal with articialant colonies, schools of sh, ocks of birds, herds of land animals. Swarm Intelligenceis also used for solving optimization problems. The two main methods belonging to theclass of Swarm Intelligence techniques are Particle swarm optimization [180] and Antcolony optimization[181].

Ant Colony Optimization is a method for solving problems which can be reduced tonding good paths in a graph, through the modeling of the actions of an ant colony.When ants nd food, they lay down pheromones to mark paths on their way back totheir colony. If other ants nd this path by sensing the pheromone, they are likely tofollow on that path.

Particle swarm optimization is a population based direct search method, which modelsa swarm in a multi-dimensional search space. The particles of the swarm have a memoryrecording their own best position, and some or all of the other particles' best.

The computational eectiveness of PSO on single-objective continuous optimizationproblems has been presented, concluding that PSO needs less computation eort fornding such high quality solutions that are found by the GA. However, if the optimizationproblem is constrained and/or discrete, the advantage of PSO fades or disappears [182].

A.1.2.3 Tabu Search

Tabu Search (TS) is a meta-heuristic local search method, which in a short-term memory,keeps a record of the visited solutions along with their corresponding paths through whichthey were explored to avoid repeatedly visiting the same solutions [183]. Every potentialsolution is marked as taboo, once it has been determined. The algorithm repeatedlymoves from a current position to the best of the neighboring positions, while avoidingthose which have been already visited. Most Tabu Search based methods are for thesingle-objective KP. Multi-objective version of TS is called MOSA [184] can navigatehighly constrained search spaces successfully, because of their local search heuristics.

A.1.2.4 Other Stochastic Algorithms

Monte-Carlo methods use pure random search for nding solutions which are fully in-dependent of previously found ones. The search is based on random walks. The actualbest solution along with its decision variables are stored as a comparator [185].


A.1.3 Mathematical Programming

A.1.3.1 Dynamic programming

Dynamic programming is a method to solve problems with the properties of overlappingsubproblems, meaning that the problem is made up of subproblems which are reusedseveral times, and optimal substructures, meaning that the optimal nal solution is con-structed of optimal solutions to its subproblems [186]. Dynamic programming methodsmake use of memoization (also called caching) [187], a technique for the avoidance of thecalculation of results which were already calculated previously.

A.1.3.2 Linear programming

The denition of Linear Programming (LP) is the following [188].

Denition 12 For parameters n, d > 0, a linear programming problem in standard formconsists of nding a nonnegative vector x ∈ Rd that minimizes a linear function cTxsubject to n linear inequalities

∑dj=1 aijxj ≤ bi, i = 1, . . . , n. In compact form, the

denition is the following.

(LP) minimize cTx (A.1)

subject to Ax ≤ b, (A.2)

x ≥ 0, (A.3)

where c is a d-vector, b is an n-vector and A is an (n× d)-matrix.

Typical algorithms for solving LP problems are the Simplex Method [189] and Kar-markar's algorithm [190]. One of the rst applications of the simplex method was relatedto nutrition. The diet problem deals with nding the cheapest combination of foods thatwill satisfy all the daily nutritional requirements of a person [191].

A.1.3.3 Nonlinear programming

The denition of Nonlinear programming (NLP) is the following [192, 193].

Denition 13

minimize f(x) (A.4)

subject to ci(x) ≤ 0 (A.5)

i = 1, . . . ,m (A.6)

where it is assumed that, f : Rn → R and ci : Rn → R are twice continuouslydierentiable, and x is an n-vector of free (i.e., unbounded) variables. The problem isconvex if f is convex and each of the ci's is concave.

A.2 Details on Genetic Algorithms 129

Algorithms for solving NLP problems include approaches which use special formula-tions of linear programming problems, or involve branch and bound techniques, wherethe problem is divided into convex problems or to linear approximations.

A.2 Details on Genetic Algorithms

A.2.1 Types of individual representation

A.2.1.1 Binary representation

The rst works on GAs used binary encoding. It is the most commonly used representa-tion. In binary encoding, every chromosome is a bitstring, the alleles are either ones orzeros.

0 1 0 0 1

1 0 0 1 0

Chromosome A

Chromosome B

Figure A.1: Binary encoded chromosomes

Figure A.1 shows two binary encoded chromosomes. The phenotype of the individualdepends on the genotype-phenotype mapping. In case the bitstring represents a binarynumber, then the phenotype of chromosome A is 0 · 24 + 1 · 23 + 0 · 22 + 0 · 21 + 1 · 20 = 9and of chromosome B is 1 · 24 + 0 · 23 + 0 · 22 + 1 · 21 + 0 · 20 = 18.

A.2.1.2 Real-coded representation

In real-coded representation, each individual is encoded as a string of real-values. Incase the object parameters in the phenotype are identical to the alleles in the genotype,then the GA operates on the so-called natural representation, meaning that there is aone-to-one mapping between the gene values and the object parameters, the genotypeand phenotype are identical.

5.125Chromosome A

Chromosome B

8.917 28.758 79.778

3.701 6.327 8.812 1.432

Figure A.2: Real-value encoded chromosomes

Given the two chromosomes shown in Figure A.2, the phenotypes of the chromosomesconsidering the natural representation are the following: (5.125), (8.917), (28.758), (79.778)and (3.701), (6.327), (8.812), (1.432).


In case of a direct representation, where every allele is multiplied by itself to get theobject parameters of the phenotype (26.265625), (79.512889), (827.022564), (6364.529284)and (13.697401), (40.030929), (77.651344), (2.050624).

In case of an indirect representation, where for example pairs of genes are summed upto get the object parameters, the phenotypes of the individuals are (14.042), (108.536)and (10.028), (10.244).

A.2.1.3 Permutation encoding

Permutation encoding is used to code solutions for ordering problems, such as TSP or taskordering problem. The chromosomes are string of numbers which represent sequences.Can be eective for ordering problems, however, special operators are needed to keep theospring chromosomes consistent after reproduction and mutation.

1 3 2 5 4

2 3 5 4 1

Chromosome A

Chromosome B

Figure A.3: Permutation encoded chromosomes

Figure A.3 shows two chromosomes, both of them encoding a sequence of ve.

A.2.1.4 Value representation

Direct value encoding is for storing complicated problem specic value in the genes, suchas strings, pointers, arrays or arbitrary structures. The mapping of these alleles to objectparameters can be natural, direct or indirect.

(forward)Chromosome A

Chromosome B

(up) (left) (down)

(right) (right) (up) (up)

Figure A.4: Value encoded chromosomes

Figure A.4 shows two chromosomes encoding special values in the genes. For example,these chromosomes can encode the following four steps of an agent in a labyrinth game.

A.2.1.5 Tree representation

Tree encoding is mainly used for GP, for evolving programs or expressions. Every chro-mosome encodes objects in a tree structure. Functional languages such as Lisp use tree

A.2 Details on Genetic Algorithms 131

structures to represent expressions, making it straightforward to use tree representationfor encoding them in a chromosome.

a

+

/

3 b

Chromosome

Figure A.5: Tree encoded chromosomes

A sample tree representation of a mathematical expression for a+ (3/b) is shown inFigure A.5.

A.2.2 Details on selection operators

In the following, the most widely used selection algorithms are presented. It is assumedhere that the objective-score to tness value mapping has been already carried out foreach individual in the population. A detailed description of selection schemes is presentedin [49].

Roulette Wheel Selection The simplest selection method is the Roulette Wheel Se-lection. Each individual represents a segment in a roulette wheel, according to itstness value. The probability pi of the ith individual to be selected is proportionalto the tness of the individual and inverse proportional to the population's t-ness, which is the sum of the tnesses of the individuals making up the population.Basically, pi is the normalized tness of the ith individual.

pi =f(xi)∑µj=1 f(xj)

(A.7)

The method is also referred as tness proportionate selection. Although individualswith higher tnesses have better chance, even the weakest one can be selected forreproduction, which is an advantageous behavior, considering that weak solutionsmay still contain useful genetic information.

Tournament Selection Individuals are chosen randomly from the population for a so-called tournament, in which the individual with the best tness is selected as thewinner. The number of individuals chosen for the competition is determined bythe tournament size, which can vary between 2 and µ (where µ is the size of thepopulation). The winner of the tournament can either be removed from, or kept inthe population, if it is to allow or disallow to select an individual multiple times.Tournament selection has a time complexity of O(N), where N is the size of theospring population (selecting N individuals has O(N) time complexity). Theselection pressure is easily adjustable through the size of the tournament. The


intensity of selection in function of tournament size t is the following, as shown in[49].

Sel.Inttournament(t) ≈√

2 · (ln(t)− ln(√

4.14 · ln(t)) (A.8)

Truncation Selection The individuals are ordered according to their tness values,and with threshold T (0 < T ≤ 1), only the ttest dT · µe individuals are used forreproduction, and they have the same, 1

T probability to be engaged in reproduction.Truncation selection has a time complexity of O(NlogN) (where N is the size ofthe population), because the sorting of the individuals according to their tnessvalues is needed.

Linear Ranking Selection The individuals are sorted according to their tness andthe best is assigned the rank µ, while the worst gets rank 1. The selection probabil-ity is linearly assigned to each individual, according to its rank. The ith individual'sselection probability pi is the following:

pi =1

µ

(η− + (η+ − η−)

i− 1

µ− 1

); i ∈ 1, . . . , N (A.9)

where η+

µ and η−

µ are selection probabilities for the best and worst individuals to beselected, respectively. Considering xed population size, η+ = 2− η− and η− > 0must be satised. Note that all individuals, even those with the same tness valueswill get dierent rankings, and consequently, dierent selection probabilities.

As the linear ranking selection needs a sorted population, its time complexity isO(NlogN) (where N is the size of the population).

Exponential Ranking Selection only diers from linear ranking selection in the cal-culation of the selection probabilities of the ranked individuals as they are weightedexponentially. After rank N is assigned to the best individual and the rank 1 tothe worst individual, the calculation of the probabilities is the following.

pi =cN−i∑Nj=1 c

N−j; i ∈ 1, . . . , N (A.10)

The base of the exponent is the parameter c, where 0 < c < 1. As the exponen-tial ranking selection needs a sorted population, its time complexity is O(NlogN)(where N is the size of the population).

A.3 Details on the Formalization of Evolutionary Algorithms

The following denitions are from the framework developed by Merkle and Lamont in1997 [53]. They are presented here to support the discussion of the formal denition ofEvolutionary Algorithms (EAs) presented in Section 2.7.6.

A.3 Details on the Formalization of Evolutionary Algorithms 133

Denition 14 (Decoding function) Let I be a non-empty set (the individual space),and f : Rn → R (the objective function). If D : I → Rn is total, i.e. the domain of D isall of I, then D is called a decoding function.

The mapping D is not necessarily surjective, as the range of D determines the subsetof Rn actually available for exploration by the EA.

The tness of an individual s is an indication of the quality of the candidate solutionD(s) ∈ Rn. The tness function yields this indication, which is dened in the following.

Denition 15 (Fitness function) Let I be a non-empty set, D : I → Rn, f : Rn → R,and Ts : R → R (the tness scaling function). Then Φ , Ts f D is called a tnessfunction.

The objective function f is determined by the application, while the decoding functionD and the tness scaling function Ts are design issues.

Denition 16 (Population transformation) Let I be a non-empty set, and µ, µ′ ∈Z+ (the parent and ospring population sizes, respectively). A mapping T : Iµ → Iµ

′is

called a population transformation. If T (P) = P ′ then P is called a parent population andP ′ is called an ospring population. If µ = µ′ then they are called simply the populationsize.

Figure A.6: The population transformation T deterministically maps the parent population P(of size µ) to the ospring population P ′ (of size µ′).

The population transformation is shown in Figure A.6. Population transformationsin EAs often depend on outcomes of random experiments, which induces the introductionof the concept of random population transformation.

Denition 17 (Random population transformation) Let I be a non-empty set, µ ∈Z+, and Ω a set (the sample space). A random function R : Ω → T

(Iµ,⋃µ′∈Z+ Iµ

′)is

called a Random Population Transformation (RPT).

The random population transformation is shown in Figure A.7.

Denition 18 (Evolutionary operator) Let I be a non-empty set, µ ∈ Z+, X a set(the parameter space), and Ω a set. A mapping

X : X→ T

Ω, T

Iµ, ⋃µ′∈Z+

Iµ′

(A.11)


Figure A.7: The Random Population Transformation (RPT) R maps the random event ω to thepopulation transformation T , which maps parent populations of size µ (which isindependent of ω) to ospring populations of some xed size µ′ ∈ Z+ (which maydepend on ω).

is called an evolutionary operator. The set of evolutionary operators in the form ofEquation A.11 is denoted EVOP(I, µ,X,Ω). The evolutionary operator is presented inFigure A.8.

Figure A.8: The evolutionary operator X maps the exogenous parameter(s) Θ to the RPT R.The underlying sample space of R is Ω. Each of the possible population transfor-mations acts on populations of size µ. The ospring population size µ′ ∈ Z+ maydepend on Θ as well as on the random event ω ∈ Ω.

The RPTX(Θ) is denotedXΘ. The population transformationXΘ(ω) is also denotedwith XΘ except where confusion may arise. The ospring population [XΘ(ω)](P) isdenoted XΘ(P). If X has no parameters, i.e. X ∈ EVOP(I, µ, ,Ω), then the ospringpopulation is denoted X(P).

Denition 19 (Recombination operator) Let r ∈ EVOP(I, µ,X,Ω). If there existP ∈ I, Θ ∈ X and ω ∈ Ω such that at least one individual in the ospring populationrΘ(P) depends on more than one individual of P then r is called a recombination operator.

Denition 20 (Mutation operator) Let m ∈ EVOP(I, µ,X,Ω). If for every P ∈ Iµ,every Θ ∈ X, and every ω ∈ Ω, each individual in the ospring populationmΘ(P) dependson at most one individual of P then m is called a mutation operator.

Denition 21 (Selection operator) Let s ∈ EVOP(I, µ,X×T (I,R),Ω). If for everyP ∈ Iµ, every Θ ∈ X, and every tness function Φ : I → R, s satises s ∈ s(Ω,Φ)(P) ⇒s ∈ P, then s is called a selection operator.

A.4 Details on Multi-Objective Genetic Algorithms 135

A.4 Details on Multi-Objective Genetic Algorithms

A.4.1 Details on A-posteriori Techniques

NSGA-II, SPEA2 and MicroGA are A-posteriori MOGAs and are implemented in MOGALib,a multi-objective GA library, which has been developed in connection with this thesisand on which the muleGA implementation is built on. MOGALib is an extension to GALib,a popular GA library, and is available at http://mogalib.uni-pannon.hu.

A.4.1.1 Details of Non-dominated Sorting Genetic Algorithm (NSGA)

Each individual of NSGA gets ranked and then is given a dummy tness value, samevalue for individuals with the same rank. These dummy tness values are modied bysharing. The sharing function between two individuals in the same front is calculated asfollows:

Sh(dij) =

1−

(dij

σshare

)2, if dij < σshare;

0, otherwise.(A.12)

where dij is the phenotypic distance between two individuals i and j in the actualfront which is being assessed, and the maximum phenotypic distance between two indi-viduals to become the members of the same niche is dened in σsharing. A niche count iscalculated for each individual by summing the sharing function values for each individualin the front. The shared tness is then calculated by dividing the dummy tness valueby the niche count for each individual. The owchart of the algorithm from Srinivas andDeb [61] is shown in Figure A.9.


Figure A.9: NSGA ow chart


A.4.1.2 Details of Non-dominated Sorting Genetic Algorithm II (NSGA-II)

Algorithm 7 fast-non-dominated-sort(P)

1: for all p ∈ P do

2: Sp = ∅3: np = 04: for all q ∈ P do5: if p ≺ q then if p dominates q

6: Sp = Sp ∪ q Add q to the set of solutions dominated by p 7: else if q ≺ p then

8: np = np + 1 Increment the domination counter of p 9: end if

10: end for

11: if np = 0 then p belongs to the rst front 12: prank = 113: F1 = F1 ∪ p14: end if

15: end for

16: i = 1 Initialize the front counter 17: while Fi 6= ∅ do18: Q = ∅ Used to store the members of the next front 19: for all p ∈ Fi do20: for all q ∈ Sp do21: nq = nq − 122: if nq = 0 then q belongs to the next front 23: qrank = i+ 124: Q = Q ∪ q25: end if

26: end for

27: end for

28: i = i+ 129: Fi = Q30: end while


Algorithm 8 crowding-distance-assignment(I)

1: l = |I| number of solutions in I 2: for all i do3: I[i]distance = 0 initialize distance 4: end for

5: for all m objective do6: I = sort(I,m) sort using each objective value 7: I[1]distance = I[l]distance =∞ so that boundary points are always selected 8: for i = from 2 to (l − 1) do for all other points 9: I[1]distance = I[1]distance + (I[i+ 1].m− I[i− 1].m)/(fmaxm − fminm )10: end for

11: end for

Algorithm 9 NSGA-II

1: Rt = Pt ∪Qt2: F = fast-non-dominated-sort(Rt)3: Pt+1 = ∅ és i = 14: repeat

5: crowding-distance-assignment(Fi)6: Pt+1 = Pt+1 ∪ Fi7: i = i+ 18: until |Pt+1|+ |Fi| ≤ N9: Sort(Fi,≺n)10: Pt+1 = Pt+1 ∪ Fi[1 : (N − |Pt+1|)]11: Qt+1 = make-new-pop(Pt+1)12: t = t+ 1


A.4.1.3 Details of Strength Pareto Evolutionary Algorithm 2 (SPEA2)

Algorithm 10 SPEA2

1: INPUTS: N - population size, N - archive size, T - maximum number of generations

OUTPUT: A - non-dominated set 2: Step 1: Initialization 3: generate initial population P0

4: create the empty archive (external set) P0 = 05: set t = 06: while true do7: Step 2: Fitness Assignment 8: calculate tness values of individuals in Pt9: calculate tness values of individuals in Pt10: Step 3: Environmental selection 11: copy all non-dominated individuals in Pt and Pt to Pt+1

12: if size of Pt+1 exceeds N then

13: reduce Pt+1 by means of the truncation operator14: else

15: ll Pt+1 with dominated individuals in Pt and Pt16: end if

17: Step 4: Termination 18: if t ≥ T or another stopping criterion is satised then19: set A to the set of decision vectors represented by the non-dominated individuals

in Pt+1

20: STOP21: end if

22: Step 5: Mating selection 23: perform binary tournament selection with replacement on Pt+1 in order to ll the

mating pool24: Step 6: Variation 25: apply recombination and mutation operators to the mating pool and set Pt+1 to

the resulting population26: t = t+ 127: end while


A.4.1.4 Details of Micro-Genetic Algorithm

Algorithm 11 Micro-GA

1: Generate starting population P of size N and store its contents in the populationmemory M Both portions of M will be lled with random solutions

2: i = 03: while i < Max do4: Get initial population for the micro-GA (Pi) from M5: while Nominal convergence is reached do6: Apply binary tournament selection based on nondominance7: Apply two-point crossover and uniform mutation to the selected individuals8: Apply elitism (retain only one non-dominated vector)9: Produce the next generation10: end while

11: Copy two non-dominated vectors from Pi to the external memory E12: if E is full when trying to insert individuals then13: use adaptive-grid to insert14: end if

15: Copy two non-dominated vectors from Pi to M16: if i mod replacement-cycle then17: apply second form of elitism18: end if

19: i = i+ 120: end while

The tness assignment in step 2 is done by assigning a strength value denoted withS(i) to each individual i in Pt and Pt. S(i) represents the number of solutions theindividual i dominates in the archive Pt and in the population Pt. The raw tness ofindividual i is R(i), and it is calculated according to the S values. The raw tness R(i)is the sum of those S(j) values where individual j from Pt or Pt dominates individual i.The tness is to be minimized in SPEA2, so the raw tness of an individual is determinedaccording to the strengths of its dominators from both the archive and the population, asopposed to the SPEA in which, only the individuals from the archive are considered. Rawtness assignment could fail in situations when most of the individuals do not dominateeach other. To discriminate individuals with identical raw tness values, a value calleddensity information is introduced. At any point, the density is a decreasing function ofthe distance to the kth nearest data point, an adaptation of the k-th nearest neighbormethod [194]. The tness value of the individual i is F (i) and is calculated by addingtogether the raw tness value of i, R(i), and the density information of i, D(i).

In step 3 of SPEA2, all non-dominated individuals are copied from Pt and Pt to Pt+1.In case the number of non-dominated individuals is less than the requested size of Pt+1,denoted with N , then the best dominated individuals are used to ll this gap. In case, thenumber of non-dominated individuals exceeds N , a truncation operator is used, which

A.5 Details on Constraint Handling 141

eliminates individuals from Pt+1 in an iterative fashion, until exactly N individuals arecontained by Pt+1. In each iteration, the truncation operator removes the individualwith the minimal distance to another individual. If there are several individuals withthe same minimum distance, the second smallest distances are taken into account, andso on, until an individual is chosen for removal.

A.5 Details on Constraint Handling

The classication of constraint-handling techniques presented in [72] is followed in thissection. The most common approach of incorporating constraints into EvolutionaryAlgorithm (EA) is done by the means of penalty functions, which are detailed in SectionA.5.1. All the other methods for constraint-handling are introduced in Section A.5.2.

A.5.1 Penalty Functions

The idea of penalty functions is to transform a constrained optimization problem intoan unconstrained one. This is done by adding (considering a minimization problem)or subtracting (considering a maximization problem) a certain value depending on theamount of constraint violation, to or from the objective function.

Generally there are two types of penalty functions, exterior and interior. The latteris dened to keep the solution from leaving the feasible region, as its value will tend toinnity as the constraint boundaries are approached. In case of interior penalty functions,the optimization should get started from a feasible solution. Exterior penalty functionsdo not require an initial feasible solution, and thus are the more commonly used methodsfor EA constraint-handling, as for most of the problems solved with EAs, nding an initialfeasible solution may well be an NP-hard complexity problem in itself.

Considering the following general non-linear programming problem in Equation A.13with n inequality constraints, and m equality constraints, the general formulation of theexterior penalty function is presented in Equation A.14.

minimize f(~x)gi(~x) ≤ 0 i = 1, . . . , nhj(~x) = 0 i = 1, . . . ,m

(A.13)

fe(~x) = f(~x)±

n∑i=1

ri ×Gi +m∑j=1

cj × Lj

(A.14)

where fe is the expanded objective function, ~x is the vector of solutions, Gi and Ljare functions of the constraints gi(~x) and hj(~x), respectively, and ri and cj are positiveconstants called penalty factors. The most common form of Gi and Lj are the followingwhere β and γ are normally 1 or 2.

Gi = max[0, (gi(~x))β]Lj = |(hj(~x))γ | (A.15)


The penalty should be kept as low as possible, however if the penalty is too low, lot ofsearch time will be spent to explore the infeasible region. Too high penalty will push thesearch into the feasible region, and will not allow it to move back towards the boundary.This is a problem in case of disjoint feasible regions, as the search cannot continue in another feasible region, once it has stuck in one, because of the high penalty.

The relationship between the feasible region of the search space and an infeasibleindividual should be dened according to the following guidelines [73].

• Regardless of the amount of an individual's constraint violation and of the informa-tion about how far it is from the feasible region, the individual might get penalized,just because it is not feasible.

• The infeasibility of an individual can be measured and presented in its penalty.

• The eort which would be needed to make an infeasible solution feasible can becalculated and presented in the penalty.

The guidelines for GAs with penalty functions [195] dene the cornerstone of thegood penalty function.

• Penalties calculated in function of the individual's distance from feasibility per-form better than penalties which are calculated according to how many constraintsviolation an individual has.

• Penalties solely depending on the number of violated constraints may fail to succeedin producing any solutions for problems with limited number of constraints andlimited number of feasible solutions.

• For a good penalty function, the two quantities of maximum completion cost andexpected completion cost are needed. The completion cost is a measure of thedistance to feasibility.

• The more accurate the penalty, the better solution is found, so penalties shouldbe close to the expected completion cost, however they should not fall below it fre-quently. If the penalty underestimates the completion cost, the search will possiblyfail to nd a solution.

The guidelines give a helping hand in constructing penalty functions, however, there isno generally accepted method to derive a tness function which uses accurate penalties.Also, the guidelines may be dicult to follow in some cases. Penalty functions canhandle both equality and inequality constraints, but usually equalities transformed toinequalities by introducing a small ε.

|hj(~x)| − ε ≤ 0 (A.16)

In the exhaustive survey by Coello [72], penalty functions are classied into thefollowing subcategories.

A.5 Details on Constraint Handling 143

Static Penalties are techniques which are not in any way dependent on the actualgeneration of the evolution process, and are constant during the entire algorithmicrun.

Dynamic Penalties are techniques depending on the actual generation number whencomputing the corresponding penalty factors. In most of the cases the penaltyfunction is dened in a way that it increases as the evolution process progresses.

Annealing Penalties are also dynamic penalties, which are based on the idea of sim-ulated annealing. In case the algorithm is stuck in a local optimum, the penaltycoecients are changed. Penalty is increased over time.

Adaptive Penalties are also dynamic penalties, which take feedback from the searchprocess. For example, in one implementation the penalty is decreased if all thebest individuals of the last k generations were feasible, or is increased if all of themwere infeasible [196]. Other algorithmic parameters can also aect the penalty, forexample the tness of the best solution found so far [197].

Co-evolutionary Penalties was proposed by Coello [198]. In Coello's approach, thepenalty is split into two values. One counts the number of constraint violation, theother sums the amount of the extent of the violations. The pairs of these penaltyfactors were encoded in chromosomes and were evaluated in a secondary popula-tion P2, while the individuals encoding the optimization problem were evolved inthe rst population P1 with each having a corresponding individual encoding thepenalty factors in P2.

Segregated Genetic Algorithm uses two penalty parameters for each constraint, whichaim to balance between heavy and moderate penalties by running two subpopula-tions of individuals instead of one [199]. Each individual is evaluated according totwo penalty functions, and two lists are made having the solutions ranked accordingto the two penalty functions. The best individuals of the two lists are chosen tobecome parents of the next generation.

Death Penalty is a technique which rejects infeasible solutions. It is computationallyecient, and the easiest way to handle constraints, as no calculation of the degreeof infeasibility is needed.

A.5.2 Other Constraint-Handling Techniques

Special representations and operators were developed by researchers to cope withproblems for which the generic binary representation scheme was not appropriate.Special operators are normally used to preserve the feasibility of solutions at alltimes. Some of the implementations assume a feasible initial point, others assumea feasible initial population.


Repair algorithms are used for problems, where it is relatively easy to make an in-feasible individual feasible, so in other words, repair it. It has been shown withempirical tests that repair algorithms increase EA both speed and performance onsome combinatorial optimization problems [200].

Separation of constraints and objectives techniques handle the objectives of theproblem separable from the constraint violations. Implementations that largelydier on the actual interpretation of the idea of separating objectives from con-straints are detailed and compared in [72].

Hybrid methods for constraint-handling usually employ two methods which are cou-pled, one of them is normally a numerical optimization approach. Detailed com-parison of hybrid methods are compared in [72].

A.6 Details on Co-Evolution

The denitional framework of [201] is utilized here to describe the symbiotic relations ofco-evolving individuals.

Denition 22 (Symbiosis) Symbiosis is a relationship between two (or more) indi-viduals such that the tness of one individual directly aects the tness of the otherindividual(s)

Denition 23 (Symbiotic connection) A symbiotic connection, A B (A aectsB) between two individuals A and B exists if and only if the tness of A has a directeect on the tness of B.

Denition 24 (Protagonization) A+ B (A protagonizes B) if and only if there ex-

ists a connection A B such that as the tness of A increases, the tness of B increases,and the tness of A decreases, the tness of B decreases.

Denition 25 (Antagonization) A− B (A antagonizes B) if and only if there exists

a connection A B such that as the tness of A increases, the tness of B decreases,and the tness of A decreases, the tness of B increases.

Denition 26 (Amensalism) Amensalism occurs between two individuals, Host and

Amensal, if and only if Host− Amensal and ¬(Amensal Host).

Denition 27 (Commensalism) Commensalism occurs between two individuals, Host

and Commensal, if and only if Host+ Commensal and ¬(Commensal Host).

Denition 28 (Competition) Competition occurs between two individuals, competi-

tors A and B, if and only if A− B and B

− A.

A.6 Details on Co-Evolution 145

Denition 29 (Predation) Competition occurs between two individuals, Predator and

Prey, if and only if Predator− Prey and Prey

+ Predator.

Denition 30 (Mutualism) Competition occurs between two individuals, symbionts A

and B, if and only if A+ B and B

+ A.

Denition 31 (Objective measure) A measurement of an individual is objective ifthe measure considers that individual independently from any other individuals, asidefrom scaling or normalization eects.

Denition 32 (Subjective measure) A measurement of an individual is subjective ifthe measure is not objective.

Denition 33 (Internal measure) A measurement of an individual is internal if themeasure inuences the course of evolution in some way.

Denition 34 (External measure) A measurement of an individual is external if themeasure cannot inuence the course of evolution in any way.

Denition 35 (Objective tness) The objective tness of individual xi at time t,foi (t), depends only on the genotype and phenotype of xi at t, and excludes the eects ofother individuals.

Denition 36 (Subjective tness) The subjective tness of individual xi at time t,f si (t), depends on the genotype and phenotype of xi at t, and on all individuals xj suchthat xj xi.

Denition 37 (Red Queen eect) The Red Queen eect is the phenomena, whenpopulations seem to be changing, but the internal subjective measure shows no progress.Internal subjective tness measurements provide no external information about the be-havior of the system.

A comprehensive hierarchical categorization of co-evolutionary algorithm propertieswas presented in [202] by Wiegand. The categorization shown in Figure A.10 formsthe basis of the introduction to co-evolutionary algorithms. Parts of the hierarchy notstrongly relevant to the context of this work are not discussed here.

A.6.1 Evaluation

A.6.1.1 Payo Quality

Competitive Co-Evolution involves a co-evolutionary algorithm, where the symbiosisof the individuals is antagonist. The individuals A and B involved in symbiosis are

competing with each other, A− B and B

− A.


Cooperative Co-Evolution involves individuals which are assessed according to howwell they perform together. In this mutualism the individuals A and B are collab-

orators, and in this collaboration A+ B and B

+ A.

Non-Competitive Co-Evolution problems are those in which there the same pairingof individuals are sometimes rewarded, and at other times depreciated. Thesesituations are referred as non-competitive.

A.6.1.2 Methods of Fitness Assignment

Implicit Fitness Assignment means that the tness value for each individual is sub-jectively calculated considering the state of the individuals in the co-evolutionarypopulation and/or populations. For example, sharing, which involves the modica-tion of the tness values of neighboring individuals depending on the crowdednessof the search space is an implicit tness assignment method.

Explicit Fitness Assignment means that the tness function is clearly stipulated bythe decision maker, and there is no inner mechanism which modies or scales it.

In multiple population congurations, it is possible to have some populations withimplicit and other populations with explicit tness assignment.

A.6.1.3 Methods of Interaction

The method of interaction tells in which congurations individuals (possible collabora-tors or competitors) are combined or paired in case of binary interactions. For binaryinteractions (a single interaction involves two individuals) the two extremes are completepairwise interaction (also called complete mixing) and when an individual is only involvedin one interaction. Interactions not executing complete mixing can choose competitorsor collaborators in many ways, ranging from uniformly random to tness biased selectionfor collaboration.

A.6.1.4 Update Timing

As co-evolutionary GAs employ more than one population, the clarication of the conceptof generation is needed. In traditional GAs, a generation is the complete iteration ofthe population (selection for reproduction, recombination, mutation, tness assignment,selection for survival).

A.6.2 Representation

The main attributes of problem representation in co-evolutionary GAs are problem de-composition, spatial topology and population structure.

Problem Decomposition deals with the division of the problem into smaller compo-nents, for which, how and when the problem is decomposed should be decided, and

A.6 Details on Co-Evolution 147

PayoffQuality

Evaluation

Representation

Methods ofFitnessAssignment

Methods ofInteraction

UpdateTiming

ProblemDecomposition

SpatialTopology

PopulationStructure

Cooperative

Competitive

Non-Competitive

Implicit

Explicit

Sample Size

Selective Bias

Credit Assignment

Implicit

Explicit

PartitioningMethods

DecompositionTemporality

Spatial Embedding

Non-Spatial Embedding

Single

Multiple

Static

Dinamic

Adaptive

Figure A.10: Categorization of co-evolutionary algorithm properties

also, whether these methods are static (unchanged during evolution) or dynamic(adaptive during evolution).

Spatial Topology denes whether the individuals are in a spatial structure. For ex-ample, the ne-grained parallel genetic algorithm assigns one individual to eachprocessing node, and individuals can only be engaged in evolutionary operationswith neighboring individuals.

Population Structure denes whether multiple populations or a single population isused for co-evolution.


A.7 Multi-Objective Numerical Test Functions

The following test functions are implemented in the multi-objective GA library MOGALib,a thesis related development, available at http://mogalib.uni-pannon.hu.

A.7.1 Binh(1) [119, 120]

F = (f1(x, y), f2(x, y)) (A.17a)

where

f1(x, y) = x2 + y2, (A.17b)

f2(x, y) = (x− 5)2 + (y − 5)2, (A.17c)

−5 ≤ x, y ≤ 10 (A.17d)

A.7.2 Binh(3) [203]

F = (f1(x, y), f2(x, y), f3(x, y)) (A.18a)

where

f1(x, y) = x− 106, (A.18b)

f2(x, y) = y − 2 ∗ 10−6, (A.18c)

f3(x, y) = xy − 2, (A.18d)

10−6 ≤ x, y ≤ 106 (A.18e)

A.7.3 Fonseca [204]

F = (f1(x, y), f2(x, y)) (A.19a)

where

f1(x, y) = 1− exp(−(x− 1)2 − (y + 1)2

), (A.19b)

f2(x, y) = 1− exp(−(x+ 1)2 − (y − 1)2

)(A.19c)

A.7.4 Fonseca(2) [205]

F = (f1(~x), f2(~x)) (A.20a)

where

f1(~x) = 1− exp

(−

n∑i=1

(xi −

1√n

)2), (A.20b)

f2(~x) = 1− exp

(−

n∑i=1

(xi +

1√n

)2), (A.20c)

−4 ≤ xi ≤ 4 (A.20d)

A.7 Multi-Objective Numerical Test Functions 149

A.7.5 Kursawe [206]

F = (f1(~x), f2(~x)) (A.21a)

where

f1(~x) =

n−1∑i=1

(−10e(−0.2)∗

√x2i+x2

i+1

), (A.21b)

f2(~x) =

n∑i=1

(|xi|0.8 + 5 sin(xi)

3)

(A.21c)

A.7.6 Laumanns [207]

F = (f1(x, y), f2(x, y)) (A.22a)

where

f1(x, y) = x2 + y2, (A.22b)

f2(x, y) = (x+ 2)2 + y2, (A.22c)

−50 ≤ x, y ≤ 50 (A.22d)

A.7.7 Lis [208]

F = (f1(x, y), f2(x, y)) (A.23a)

where

f1(x, y) = 8√x2 + y2, (A.23b)

f2(x, y) = 4√

(x− 0.5)2 + (y − 0.5)2, (A.23c)

−5 ≤ x, y ≤ 10 (A.23d)

A.7.8 Murata [209]

F = (f1(x, y), f2(x, y)) (A.24a)

where

f1(x, y) = 2√x, (A.24b)

f2(x, y) = x(1− y) + 5, (A.24c)

1 ≤ x ≤ 4, (A.24d)

1 ≤ y ≤ 2 (A.24e)


A.7.9 Poloni [121]

Maximize F = (f1(x, y), f2(x, y)) (A.25a)

where

f1(x, y) = −[1 + (A1 −B1)2 + (A2 −B2)2], (A.25b)

f2(x, y) = −[(x+ 3)2 + (y + 1)2], (A.25c)

−π ≤ x, y ≤ π, (A.25d)

A1 = 0.5 sin 1− 2 cos 1 + sin 2− 1.5 cos 2, (A.25e)

A2 = 1.5 sin 1− cos 1 + 2 sin 2− 0.5 cos 2, (A.25f)

B1 = 0.5 sinx− 2 cosx+ sin y − 1.5 cos y, (A.25g)

B2 = 1.5 sinx− cosx+ 2 sin y − 0.5 cos y (A.25h)

A.7.10 Quagliarella [210]

F = (f1(~x), f2(~x)) (A.26a)

where

f1(~x) =

√A1

n, (A.26b)

f2(~x) =

√A2

n, (A.26c)

A1 =

n∑i=1

[(xi)2 − 10 cos [2π(xi)] + 10], (A.26d)

A2 =

n∑i=1

[(xi − 1.5)2 − 10 cos [2π(xi − 1.5)] + 10], (A.26e)

−5.12 ≤ xi ≤ 5.12, n = 16 (A.26f)

A.7.11 Rendon [211]

F = (f1(x, y), f2(x, y)) (A.27a)

where

f1(x, y) =1

x2 + y2 + 1, (A.27b)

f2(x, y) = x2 + 3y2 + 1, (A.27c)

−3 ≤ x, y ≤ 3 (A.27d)

A.7.12 Rendon(2) [211]

F = (f1(x, y), f2(x, y)) (A.28a)

where

f1(x, y) = x+ y + 1, (A.28b)

f2(x, y) = x2 + 2y − 1, (A.28c)

−3 ≤ x, y ≤ 3 (A.28d)

A.7 Multi-Objective Numerical Test Functions 151

A.7.13 Shaer [212]

F = (f1(x), f2(x)) (A.29a)

where

f1(x, y) = x2, (A.29b)

f2(x, y) = (x− 2)2 (A.29c)

A.7.14 Shaer(2) [213, 214]

F = (f1(x), f2(x)) (A.30a)

where

f1(x, y) = −x, if x ≤ 1, (A.30b)

= −2 + x, if 1 < x ≤ 3, (A.30c)

= 4− x, if 3 < x ≤ 4, (A.30d)

= −4 + x, if x > 4, (A.30e)

f2(x, y) = (x− 5)2, (A.30f)

−5 ≤ x ≤ 10 (A.30g)

A.7.15 Vicini [215]

F = (f1(x, y), f2(x, y)) (A.31a)

where

f1(x, y) = −

(20∑i=1

Hi exp

[(x− xi)2 + (y − yi)2

2σ2i

])+ 3, (A.31b)

f2(x, y) = −

(20∑i=1

Hi exp

[(x− xi)2 + (y − yi)2

2σ2i

])+ 3, (A.31c)

0 ≤ Hi ≤ 1, (A.31d)

−10 ≤ x, xi, y, yi ≤ 10, (A.31e)

1.5 ≤ σi ≤ 2.5 (A.31f)

A.7.16 Viennet [216]

F = (f1(x, y), f2(x, y), f3(x, y)) (A.32a)

where

f1(x, y) = x2 + (y − 1)2, (A.32b)

f2(x, y) = x2 + (y + 1)2 + 1, (A.32c)

f3(x, y) = (x− 1)2 + y2 + 2, (A.32d)

−2 ≤ x, y ≤ 2 (A.32e)


A.7.17 Viennet(2) [216]

F = (f1(x, y), f2(x, y), f3(x, y)) (A.33a)

where

f1(x, y) =(x− 2)2

2+

(y + 1)2

13+ 3, (A.33b)

f2(x, y) =(x+ y − 3)2

36+

(−x+ y + 2)2

8− 17, (A.33c)

f3(x, y) =(x+ 2y − 1)2

175+

(2y − x)2

17− 13, (A.33d)

−4 ≤ x, y ≤ 4 (A.33e)

A.7.18 Viennet(3) [216]

F = (f1(x, y), f2(x, y), f3(x, y)) (A.34a)

where

f1(x, y) = 0.5 ∗ (x2 + y2) + sin(x2 + y2), (A.34b)

f2(x, y) =(3x− 2y + 4)2

8+

(x− y + 1)2

27+ 15, (A.34c)

f3(x, y) =1

(x2 + y2 + 1)− 1.1e(−x

2−y2), (A.34d)

−3 ≤ x, y ≤ 3 (A.34e)

A.8 MOP Numeric Test Functions with side constraints

A.8.1 Belegundu [217]

F = (f1(x, y), f2(x, y)) (A.35a)

where

f1(x, y) = −2x+ y, (A.35b)

f2(x, y) = 2x+ y, (A.35c)

0 ≤ x ≤ 5, (A.35d)

0 ≤ x ≤ 3, (A.35e)

0 ≥ −x+ y − 1, (A.35f)

0 ≥ x+ y − 7 (A.35g)

A.8 MOP Numeric Test Functions with side constraints 153

A.8.2 Binh(2) [218]

F = (f1(x, y), f2(x, y)) (A.36a)

where

f1(x, y) = 4x2 + 4y2, (A.36b)

f2(x, y) = (x− 5)2 + (y − 5)2, (A.36c)

0 ≤ x ≤ 5, (A.36d)

0 ≤ x ≤ 3, (A.36e)

0 ≥ (x− 5)2 + y2 − 25, (A.36f)

0 ≥ −(x− 8)2 − (y + 3)2 + 7.7 (A.36g)

A.8.3 Binh(4) [219]

F = (f1(x, y), f2(x, y), f3(x, y)) (A.37a)

where

f1(x, y) = 1.5− x(1− y), (A.37b)

f2(x, y) = 2.25− x(1− y2), (A.37c)

f3(x, y) = 2.625− x(1− y3), (A.37d)

−10 ≤ x, y ≤ 10, (A.37e)

0 ≥ −x2 − (y − 0.5)2 + 9, (A.37f)

0 ≥ (x− 1)2 + (y − 0.5)2 − 6.25 (A.37g)

A.8.4 Jimenez [220]


where

f1(x, y) = 5x+ 3y, (A.38b)

f2(x, y) = 2x+ 8y, (A.38c)

x, y ≥ 0, (A.38d)

0 ≥ x+ 4y − 100, (A.38e)

0 ≥ 3x+ 2y − 150, (A.38f)

0 ≥ 200− 5x− 3y, (A.38g)

0 ≥ 75− 2x− 8y (A.38h)


A.8.5 Kita [221]


where

f1(x, y) = −x2 + y, (A.39b)

f2(x, y) =1

2x+ y + 1, (A.39c)

x, y ≥ 0, (A.39d)

0 ≥ 1

6x+ y − 13

2, (A.39e)

0 ≥ 1

2x+ y − 15

2, (A.39f)

0 ≥ 5x+ y − 30 (A.39g)

A.8.6 Obayashi [222]


where

f1(x, y) = x, (A.40b)

f2(x, y) = y, (A.40c)

0 ≤ x, y ≤ 1, (A.40d)

x2 + y2 ≤ 1 (A.40e)

A.8.7 Osyczka [223]

F = (f1(x, y), f2(x, y)) (A.41a)

where

f1(x, y) = x+ y2, (A.41b)

f2(x, y) = x2 + y, (A.41c)

2 ≤ x ≤ 7, (A.41d)

5 ≤ y ≤ 10, (A.41e)

0 ≤ 12− x− y, (A.41f)

0 ≤ x2 + 10x− y2 + 16y − 80 (A.41g)

A.8 MOP Numeric Test Functions with side constraints 155

A.8.8 Osyczka(2) [223]

F = (f1(~x), f2(~x)) (A.42a)

where

f1(~x) = −(25(x1 − 2)2 + (x2 − 2)2 + (x3 − 1)2 + (A.42b)

+(x4 − 4)2 + (x5 − 1)2), (A.42c)

f2(~x) = x21 + x22 + x23 + x24 + x25 + x26, (A.42d)

0 ≤ x1, x2, x6 ≤ 10, (A.42e)

1 ≤ x3, x5 ≤ 5, (A.42f)

0 ≤ x4 ≤ 6, (A.42g)

0 ≤ x1 + x2 − 2, (A.42h)

0 ≤ 6− x1 − x2, (A.42i)

0 ≤ 2− x2 + x1, (A.42j)

0 ≤ 2− x1 + 3x2, (A.42k)

0 ≤ 4− (x3 − 3)2 − x4, (A.42l)

0 ≤ (x5 − 3)2 + x6 − 4 (A.42m)

A.8.9 Srinivas [213]

F = (f1(x, y), f2(x, y)) (A.43a)

where

f1(x, y) = (x− 2)2 + (y − 1)2 + 2, (A.43b)

f2(x, y) = 9x− (y − 1)2, (A.43c)

−20 ≤ x, y ≤ 20, (A.43d)

0 ≥ x2 + y2 − 225, (A.43e)

0 ≥ x− 3y + 10 (A.43f)

A.8.10 Tamaki [224]

Maximize F = (f1(x, y, z), f2(x, y, z), f3(x, y, z)) (A.44a)

where

f1(x, y, z) = x, (A.44b)

f2(x, y, z) = y, (A.44c)

f3(x, y, z) = z, (A.44d)

0 ≤ x, y, z, (A.44e)

1 ≥ x2 + y2 + z2 (A.44f)


A.8.11 Tanaka [225]


where

f1(x, y) = x, (A.45b)

f2(x, y) = y, (A.45c)

0 ≤ x, y ≤ π, (A.45d)

12 ≥

(x− 1

2

)2

+

(y − 1

2

)2

, (A.45e)

0 ≥ −(x2)− (y2) + 1 + 0.1 ∗ cos

(16 arctan

(x

y

))(A.45f)

A.8.12 Viennet(4) [216]

F = (f1(x, y), f2(x, y), f3(x, y)) (A.46a)

where

f1(x, y) =(x− 2)2

2+

(y + 1)2

13+ 3, (A.46b)

f2(x, y) =(x+ y − 3)2

175+

(2y − x)2

17− 13, (A.46c)

f3(x, y) =(3x− 2y + 4)2

8+

(x− y + 1)2

27− 15, (A.46d)

−4 ≤ x, y ≤ 4, (A.46e)

y < −4x+ 4, (A.46f)

x > −1, (A.46g)

y > x− 2 (A.46h)

Appendix B

Test Congurations and Results

B.1 Random Knapsack Problem Congurations

The Knapsack Problems (KPs) were generated randomly using the parameter intervalsshown in Table B.1. The number of boxes the problem has is shown of the titles of thesubtables. The meaning of the parameters are the following. For each randomly gener-ated box, Pdim is the dimension of the prot vector,Wdim is the dimension of the weightvector, Pstep is the granularity of the prots, Wstep is the granularity of the weights,the prot for each dimension is chosen randomly. Considering the random(min,max)function returns a number in [min,max], the prot and weight for each dimension iscalculated as follows.

pi = random(random(Pmin,Pmin′),random(Pmax′,Pmax)) (B.1)

wj = random(random(Wmin,Wmin′), random(Wmax′,Wmax)) (B.2)

The knapsacks are created with corresponding weight constraints calculated for eachdimension as follows.

random(random(Kmin,Kmin′),random(KWmax′,KWmax)). (B.3)

The parameters of the randomly generated problem congurations were chosen to besimilar to those of the typical dietary menu planning problems.

158 Test Congurations and Results

16 box Knapsack Problems

Pdim Wdim Pstep Wstep Pmin Pmax Pmin' Pmax' Wmin Wmax Wmin' Wmax' KWmin KWmax KWmin' KWmax'

2 2 10 1 10 1000 400 600 1 10 2 8 80 120 90 100

4 4 10 1 10 1000 400 600 1 10 2 8 80 120 90 100

8 8 10 1 10 1000 400 600 1 10 2 8 80 120 90 100

16 16 10 1 10 1000 400 600 1 10 2 8 80 120 90 100

32 32 10 1 10 1000 400 600 1 10 2 8 80 120 90 100

64 64 10 1 10 1000 400 600 1 10 2 8 80 120 90 100

128 128 10 1 10 1000 400 600 1 10 2 8 80 120 90 100

256 256 10 1 10 1000 400 600 1 10 2 8 80 120 90 100

512 512 10 1 10 1000 400 600 1 10 2 8 80 120 90 100

1024 1024 10 1 10 1000 400 600 1 10 2 8 80 120 90 100



2 2 10 1 10 1000 400 600 1 10 2 8 120 200 140 150

4 4 10 1 10 1000 400 600 1 10 2 8 120 200 140 150

8 8 10 1 10 1000 400 600 1 10 2 8 120 200 140 150

16 16 10 1 10 1000 400 600 1 10 2 8 120 200 140 150

32 32 10 1 10 1000 400 600 1 10 2 8 120 200 140 150

64 64 10 1 10 1000 400 600 1 10 2 8 120 200 140 150

128 128 10 1 10 1000 400 600 1 10 2 8 120 200 140 150

256 256 10 1 10 1000 400 600 1 10 2 8 120 200 140 150

512 512 10 1 10 1000 400 600 1 10 2 8 120 200 140 150

1024 1024 10 1 10 1000 400 600 1 10 2 8 120 200 140 150



2 2 10 1 10 1000 400 600 1 10 2 8 200 400 300 320

4 4 10 1 10 1000 400 600 1 10 2 8 200 400 300 320

8 8 10 1 10 1000 400 600 1 10 2 8 200 400 300 320

16 16 10 1 10 1000 400 600 1 10 2 8 200 400 300 320

32 32 10 1 10 1000 400 600 1 10 2 8 200 400 300 320

64 64 10 1 10 1000 400 600 1 10 2 8 200 400 300 320

128 128 10 1 10 1000 400 600 1 10 2 8 200 400 300 320

256 256 10 1 10 1000 400 600 1 10 2 8 200 400 300 320

512 512 10 1 10 1000 400 600 1 10 2 8 200 400 300 320

1024 1024 10 1 10 1000 400 600 1 10 2 8 200 400 300 320



2 2 10 1 10 1000 400 600 1 10 2 8 200 800 300 400

4 4 10 1 10 1000 400 600 1 10 2 8 200 800 300 400

8 8 10 1 10 1000 400 600 1 10 2 8 200 800 300 400

16 16 10 1 10 1000 400 600 1 10 2 8 200 800 300 400

32 32 10 1 10 1000 400 600 1 10 2 8 200 800 300 400

64 64 10 1 10 1000 400 600 1 10 2 8 200 800 300 400

128 128 10 1 10 1000 400 600 1 10 2 8 200 800 300 400

256 256 10 1 10 1000 400 600 1 10 2 8 200 800 300 400

512 512 10 1 10 1000 400 600 1 10 2 8 200 800 300 400

1024 1024 10 1 10 1000 400 600 1 10 2 8 200 800 300 400



2 2 10 1 10 1000 400 600 1 10 2 8 800 1200 900 1000

4 4 10 1 10 1000 400 600 1 10 2 8 800 1200 900 1000

8 8 10 1 10 1000 400 600 1 10 2 8 800 1200 900 1000

16 16 10 1 10 1000 400 600 1 10 2 8 800 1200 900 1000

32 32 10 1 10 1000 400 600 1 10 2 8 800 1200 900 1000

64 64 10 1 10 1000 400 600 1 10 2 8 800 1200 900 1000

128 128 10 1 10 1000 400 600 1 10 2 8 800 1200 900 1000

256 256 10 1 10 1000 400 600 1 10 2 8 800 1200 900 1000

512 512 10 1 10 1000 400 600 1 10 2 8 800 1200 900 1000

1024 1024 10 1 10 1000 400 600 1 10 2 8 800 1200 900 1000

Table B.1: The parameters of the test problems

B.2 Results of muleGA runs on Multi-Objective Test Problems 159

B.2 Results of muleGA runs on Multi-Objective Test Prob-lems

The gures show the results of the multi-level MOP related test results recorded repeat-edly at given inner cycles from the start of the run until the end. The number of theiteration the result was recorded is shown on the x-coordinates of the gures. FiguresB.1 to B.45 show the results of the 8-dimensional binh1-based MOP tests. For the ran-dom input problem mapping type of tests, the results are shown with and without themeasurements of the initial (random) iteration, thus there are two times as many guresfor this type of test than are for the others.

muleGA with random individual-mapping gures are B.1 to B.18.

muleGA with semi-order input problem mapping gures are B.19 to B.27.

muleGA with full-order input problem mapping gures are B.28 to B.36.

NSGA-II gures are B.37 to B.45.


0

50

100

150

200

250

300

350

400

450

500

0 20 40 60 80 100 120 140 160

Figure B.1: Distance of PFknown to PFtrue

0

5

10

15

20

25

30

35

20 40 60 80 100 120 140 160


0

100

200

300

400

500

0 20 40 60 80 100 120 140 160

Figure B.3: Distance of P to PFtrue

0

5

10

15

20

25

30

35

20 40 60 80 100 120 140 160


0

20

40

60

80

100

120

140

160

0 20 40 60 80 100 120 140 160

Figure B.5: Distance of best sol to PFtrue

0

0.5

1

1.5

2

20 40 60 80 100 120 140 160



90

100

110

120

130

140

150

160

0 20 40 60 80 100 120 140 160

Figure B.7: Number of pareto sols in P

157

157.5

158

158.5

159

159.5

160

20 40 60 80 100 120 140 160


0

50

100

150

200

0 20 40 60 80 100 120 140 160

Figure B.9: The minimal (best) tness values

0

10

20

30

40

50

20 40 60 80 100 120 140 160


500

1000

1500

2000

2500

3000

3500

0 20 40 60 80 100 120 140 160

Figure B.11: The maximal (worst) tness values

200

300

400

500

600

700

800

900

1000

1100

20 40 60 80 100 120 140 160



100

200

300

400

500

600

700

0 20 40 60 80 100 120 140 160

Figure B.13: The average tness values

60

80

100

120

140

160

180

200

220

20 40 60 80 100 120 140 160


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 20 40 60 80 100 120 140 160

Figure B.15: The diversity of P

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

20 40 60 80 100 120 140 160


0

0.5

1

1.5

2

x 105

0 20 40 60 80 100 120 140 160

Figure B.17: The variance of the sols in P

2000

4000

6000

8000

10000

12000

14000

16000

18000

20 40 60 80 100 120 140 160



0

5

10

15

20

25

30

35

20 40 60 80 100 120 140 160


0

5

10

15

20

25

30

35

20 40 60 80 100 120 140 160


0

0.5

1

1.5

2

2.5

20 40 60 80 100 120 140 160


145

150

155

160

20 40 60 80 100 120 140 160


0

5

10

15

20

25

20 40 60 80 100 120 140 160


200

300

400

500

600

700

20 40 60 80 100 120 140 160



50

60

70

80

90

100

110

120

20 40 60 80 100 120 140 160


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

20 40 60 80 100 120 140 160


1000

2000

3000

4000

5000

6000

7000

20 40 60 80 100 120 140 160


This concludes the test result gures ofmuleGA with semi-order input problem

mapping on the 8-dimensional binh1-basedMOP. The following gures show theresults of muleGA with full-order inputproblem mapping on the 8-dimensional

binh1-based MOP.

0

5

10

15

20

25

30

20 40 60 80 100 120 140 160


0

5

10

15

20

25

30

20 40 60 80 100 120 140 160



0

0.5

1

1.5

2

2.5

3

3.5

20 40 60 80 100 120 140 160


100

110

120

130

140

150

160

20 40 60 80 100 120 140 160


0

1

2

3

4

5

6

7

8

20 40 60 80 100 120 140 160


0

100

200

300

400

500

600

700

20 40 60 80 100 120 140 160


10

20

30

40

50

60

70

20 40 60 80 100 120 140 160


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

20 40 60 80 100 120 140 160



0

2000

4000

6000

8000

10000

20 40 60 80 100 120 140 160


This concludes the test result gures ofmuleGA with full-order input problem

mapping on the 8-dimensional binh1-basedMOP. The following gures show the

results of NSGA-II on the 8-dimensionalbinh1-based MOP.

0

50

100

150

200

250

20 40 60 80 100 120 140 160


0

50

100

150

200

250

20 40 60 80 100 120 140 160


0

10

20

30

40

50

20 40 60 80 100 120 140 160


159.5

159.6

159.7

159.8

159.9

160

160.1

160.2

160.3

160.4

160.5

20 40 60 80 100 120 140 160



0

20

40

60

80

100

120

20 40 60 80 100 120 140 160


200

300

400

500

600

700

800

900

1000

20 40 60 80 100 120 140 160


50

100

150

200

250

20 40 60 80 100 120 140 160


0.9

0.92

0.94

0.96

0.98

1

20 40 60 80 100 120 140 160


2000

4000

6000

8000

10000

12000

14000

16000

20 40 60 80 100 120 140 160


This concludes the 8-dimensionalbin1-based MOP test result gures.

Thesis Related Publications

[T1] B. Gaál. Model based decision support and articial intelligence in medicine. In IEEEHungary Section, Proceedings of the 2nd PhD Mini-Symposium, Veszprém, 2004, 2004.(rst prize).

[T2] B. Gaál. An evolutionary divide and conquer method for long-term dietary menu plan-ning. In medicine,IEEE Hungary Section, Proceedings of the 3rd PhD Mini-Symposium,Veszprém, 2005, 2005.

[T3] B. Gaál. Nutrition counseling expert system using hierarchical genetic algorithms. Inmedicine,IEEE Hungary Section, Proceedings of the 4th PhD Mini-Symposium, Veszprém,2006, 2006. (rst prize).

[T4] B. Gaál, Endr®di Á., K. Fülöp, Vassányi I. Király, A., and G. Kozmann. Evolutionaryalgorithms for cardiovascular decision support. In Proceedings of the 7th InternationalConference on Measurement May 20 - 23, 2009, Smolenice Castle., 2009.

[T5] B. Gaál, I. Vassányi, and G. Kozmann. Automated planning of weekly menus for personal-ized cardiovascular risk counselling. In Health Data in the Information Society, Proceedingsof MIE 2003, Int. Congress Medical Informatics Europe, St Malo, France, 4-7 May, 2003.

[T6] B. Gaál, I. Vassányi, and G. Kozmann. Automated planning of weekly dietary menus forpersonalized nutrition counselling. In M. H. Hamza, IASTED International Conference onArticial Intelligence and Applications, part of the 23rd Multi-Conference on Applied Infor-matics, Innsbruck, Austria, February 14-16, 2005, Articial Intelligence and Applications,pages 300305. IASTED/ACTA Press, 2005.

[T7] B. Gaál, I. Vassányi, and G. Kozmann. An evolutionary divide and conquer method forlong-term dietary menu planning. Lecture Notes in Articial Intelligence, 3581(5):419423,2005. IF=0.251 (2004).

[T8] B. Gaál, I. Vassányi, and G. Kozmann. A novel articial intelligence method for weeklydietary menu planning. Methods Inf Med, 44(5):655664, 2005. IF=1.338 (2004)Four non-self-citations (two journal and two bibliography citations)

Journal of the American Dietetic Association, 2007, IF=2.564

Current Bioinformatics, 2008, IF=1.255

An Indexed Bibliography of Genetic Algorithms in Biosciences, 2008

An Indexed Bibliography of Genetic Algorithms in Medicine, 2009 .

[T9] B. Gaál, I. Vassányi, and G. Kozmann. Expert system for lifestyle and nutrition counsel-ing. UNESCO-WABT Bio-Economy and Sustainable Technologies for Health Conference,Budapest, August, 2006.

THESIS RELATED PUBLICATIONS 169

[T10] B. Gaál, I. Vassányi, and G. Kozmann. Advanced Computational Intelligence Paradigms inHealthcare 2, volume 65 of Springer Studies in Computational Intelligence, chapter Appli-cation of Articial Intelligence for Weekly Dietary Menu Planning, pages 2749. Springer-Verlag Berlin, 2007.

[T11] B. Gaál, I. Vassányi, and G. Kozmann. Internet based health promotion and diseaseprevention. eVITA 2008 Conference and Exhibition and European launch event of theplanned AAL Joint Programme, Budapest, April, 2008.

[T12] B. Gaál, I. Vassányi, G. Kozmann, Zs. Szente, E. Mák, and Sz. István. Dietary ontol-ogy for nutrition counseling expert system. Abstract and Poster Presentation at the 10thInternational Protégé Conference July 15-18, 2007 Budapest, Hungary, 2007.

[T13] Balázs Gaál. Web-based expert system for nutrition counseling. Abstract and PosterPresentation at the 11th World Congress of Internet in Medicine October 13-20, 2006Toronto, Canada, 2006.

[T14] I. Vassányi, B. Gaál, K. Fülöp, G. Kozmann, and E. Mák. Personalized dietary coun-seling for tele-care using evolutionary programming and ontological reasoning. In MalinaJordanova (ed.) Global Telemedicine and eHealth Updates: Knowledge Resources, pages272276. Springer-Verlag, Berlin, Germany, 2009.

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