DOCSLIB.ORG

Exploring Fractional Order Calculus As an Artificial Neural Network Augmentation

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Exploring Fractional Order Calculus As an Artificial Neural Network Augmentation Exploring Fractional Order Calculus as an Artificial Neural Network Augmentation by Samuel Alan Gardner A project document submitted in partial fulfillment of the requirements for the degree of Master of Science in Computer Science MONTANA STATE UNIVERSITY Bozeman, Montana April, 2009 c Copyright by Samuel Alan Gardner 2009 All Rights Reserved ii APPROVAL of a project document submitted by Samuel Alan Gardner This project document has been read by each member of the project committee and has been found to be satisfactory regarding content, English usage, format, citations, bibliographic style, and consistency, and is ready for submission to the Division of Graduate Education. Dr. John Paxton Approved for the Department of Computer Science Dr. John Paxton Approved for the Division of Graduate Education Dr. Carl A. Fox iii STATEMENT OF PERMISSION TO USE In presenting this project document in partial fulfullment of the requirements for a master's degree at Montana State University, I agree that the Library shall make it available to borrowers under rules of the Library. If I have indicated my intention to copyright this project document by including a copyright notice page, copying is allowable only for scholarly purposes, consistent with \fair use" as prescribed in the U.S. Copyright Law. Requests for permission for extended quotation from or reproduction of this project document in whole or in parts may be granted only by the copyright holder. Samuel Alan Gardner April, 2009 iv TABLE OF CONTENTS 1. INTRODUCTION .................................................................................... 1 2. BACKGROUND....................................................................................... 4 Artificial Neural Networks .........................................................................4 Neurons................................................................................................4 Topologies ............................................................................................5 Learning Algorithms..............................................................................8 Backpropagation.......................................................................................8 Evolutionary Algorithms ........................................................................... 10 Genetic Algorithms ................................................................................... 12 GNARL ................................................................................................... 15 Initialization ......................................................................................... 16 Selection............................................................................................... 16 Mutation .............................................................................................. 17 Parametric Mutation ......................................................................... 17 Structural Mutation........................................................................... 18 Fractional Order Calculus.......................................................................... 18 3. SYSTEM DESCRIPTION......................................................................... 22 Neural Network Implementation ................................................................ 22 Discrete Differintegral Computation ........................................................... 22 Fractional Calculus Neuron Augmentation.................................................. 25 Learning Algorithm................................................................................... 26 Initialization ......................................................................................... 27 Selection............................................................................................... 27 Crossover.............................................................................................. 27 Mutation .............................................................................................. 28 Parametric Mutation ......................................................................... 28 Structural Mutation........................................................................... 29 Statistic Collection.................................................................................... 30 4. EXPERIMENTS AND RESULTS.............................................................. 31 Artificial Life Simulation ........................................................................... 31 Common Experimental Parameters ............................................................ 33 Baseline.................................................................................................... 36 Non-Resetting........................................................................................... 40 Hidden Layer Size Comparison................................................................... 41 Fixed Q-Value Comparison........................................................................ 44 v GA Evolution ........................................................................................... 46 Sensed Walls............................................................................................. 48 Track ....................................................................................................... 50 Intercept .................................................................................................. 51 Hide and Seek........................................................................................... 54 Structural Evolution.................................................................................. 57 Feed-Forward ........................................................................................ 58 Recurrent ............................................................................................. 60 Feed-Forward GA.................................................................................. 61 Summary of Results .................................................................................. 63 5. FUTURE WORK ..................................................................................... 65 More Experimental Data ........................................................................... 65 Structural Evolution.................................................................................. 66 Backpropagation....................................................................................... 66 Simulation Complexity .............................................................................. 67 6. CONCLUSIONS ....................................................................................... 69 REFERENCES.............................................................................................. 72 APPENDICES .............................................................................................. 75 Appendix A: Software ............................................................................. 76 Appendix B: Code Listings ...................................................................... 99 vi LIST OF TABLES Table Page 1 Link Restriction Options.................................................................... 26 2 Mutable Parameter Types.................................................................. 28 3 Sensor Cell Type Values .................................................................... 33 4 Base Experimental Evolution Parameters............................................ 34 5 Base Experimental Simulation Parameters .......................................... 34 6 Base Experimental Neural Network Parameters................................... 35 7 Baseline { 95% Confidence Intervals for Maximum Fitness................... 37 8 Baseline { Simulation Data From Peak Fitness Individuals................... 39 9 Non-Resetting { 95% Confidence Intervals for Maximum Fitness.......... 41 10 Hidden Layer Size { NN 95% Confidence Intervals for Maximum Fitness 43 11 Hidden Layer Size { FNN 95% Confidence Intervals for Maximum Fitness 43 12 Fixed Q Value { 95% Confidence Intervals for Maximum Fitness.......... 45 13 GA { 95% Confidence Intervals for Maximum Fitness.......................... 48 14 Sensed Walls { 95% Confidence Intervals for Maximum Fitness............ 49 15 Track { 95% Confidence Intervals for Maximum Fitness....................... 50 16 Intercept { 95% Confidence Intervals for Maximum Fitness.................. 53 17 Hide and Seek { 95% Confidence Intervals for Maximum Fitness.......... 56 18 Feed-Forward Structural { 95% Confidence Intervals for Maximum Fitness.............................................................................................. 59 19 Recurrent Structural { 95% Confidence Intervals for Maximum Fitness. 61 20 Feed-Forward Structural GA { 95% Confidence Intervals for Maximum Fitness.............................................................................................. 63 vii LIST OF FIGURES Figure Page 1 Anatomy of a Neuron ........................................................................4 2 A Feed-forward, Multi-layer Neural Network .......................................6 3 An Example of a Fully-connected, Feed-forward, Multi-layer Neural Network............................................................................................7 4 Roulette Selection Illustration with Population Size = 10..................... 14 5 Single Point Crossover ......................................................................
Load more

Recommended publications
  • List of Reviews by Gianni Pagnini in Mathematical Reviews MR2145045
    List of reviews by Gianni Pagnini in Mathematical Reviews MR2145045 Saxena R.K., Ram J., Suthar D.L., On two-dimensional Saigo–Maeda frac- tional calculus involving two-dimensional H-transforms. Acta Cienc. Indica Math., 30(4), pp. 813–822 (2004) MR2220224 Nishimoto K., N-fractional calculus of a logarithmic function and generali- zed hypergeometric functions. J. Fract. Calc., 29(1), pp. 1–8 (2006) MR2224671 Saxena R.K., Ram J., Chandak S., On two-dimensional generalized Saigo fractional calculus associated with two-dimensional generalized H-transfroms. J. Indian Acad. Math., 27(1), pp. 167–180 (2005) MR2266353 Lin S.-D., Tu S.-T., Srivastava H.M., Some families of multiple infinite sums and associated fractional differintegral formulas for power and composite functions. J. Fract. Calc., 30, pp. 45–58 (2006) MR2286840 Biacino L., Derivatives of fractional orders of continuos functions. Ricerche di Matematica, LIII(2), pp. 231–254 (2004) MR2330471 Chaurasia V.B.L., Srivastava A., A unified approach to fractional calculus pertaining to H-functions. Soochow J. of Math., 33(2), pp. 211–221 (2007) MR2332241 Chaurasia V.B.L., Patni R., Shekhawat A.S., Applications of fractional deri- vatives of certain special functions. Soochow J. of Math., 33(2), pp. 325–334 (2007) MR2355703 Agrawal R., Bansal S.K., A study of unified integral operators involving a general multivariable polynomial and a product of two H¯ −functions. J. Rajasthan Acad. Phy. Sci., 6(3), pp. 289–300 (2007) MR2390179 Benchohra M., Hamani S., Ntouyas S.K., Boundary value problems for dif- ferential equations with fractional order. Surv.
    [Show full text]
  • A Novel Approach to Fractional Calculus -.:: Natural Sciences
    Progr. Fract. Differ. Appl. 4, No. 4, 463-478 (2018) 463 Progress in Fractional Differentiation and Applications An International Journal http://dx.doi.org/10.18576/pfda/040402 A Novel Approach to Fractional Calculus: Utilizing Fractional Integrals and Derivatives of the Dirac Delta Function Evan Camrud1,2 1 Department of Mathematics, Concordia College, Moorhead, MN, USA 2 Department of Mathematics, Iowa State University, Ames, IA, USA Received: 8 Jan. 2018, Revised: 28 Feb. 2018, Accepted: 2 Mar. 2018 Published online: 1 Oct. 2018 Abstract: While the definition of a fractional integral may be codified by Riemann and Liouville, an agreed-upon fractional derivative has eluded discovery for many years. This is likely a result of integral definitions including numerous constants of integration in their results. An elimination of constants of integration opens the door to an operator that reconciles all known fractional derivatives and shows surprising results in areas unobserved before, including the appearance of the Riemann Zeta function and fractional Laplace and Fourier transforms. A new class of functions, known as Zero Functions and closely related to the Dirac delta function, are necessary for one to perform elementary operations of functions without using constants. The operator also allows for a generalization of the Volterra integral equation, and provides a method of solving for Riemann’s complimentary function introduced during his research on fractional derivatives. Keywords: Fractional calculus, fractional differential equations, integral transforms, operations with distributions, special functions. 1 Introduction The concept of derivatives of non-integer order, commonly known as fractional derivatives, first appeared in a letter between L’Hopital and Leibniz in which the question of a half-order derivative was posed [1].
    [Show full text]
  • A Public Transport Bus Assignment Problem: Parallel Metaheuristics Assessment
    David Fernandes Semedo Licenciado em Engenharia Informática A Public Transport Bus Assignment Problem: Parallel Metaheuristics Assessment Dissertação para obtenção do Grau de Mestre em Engenharia Informática Orientador: Prof. Doutor Pedro Barahona, Prof. Catedrático, Universidade Nova de Lisboa Co-orientador: Prof. Doutor Pedro Medeiros, Prof. Associado, Universidade Nova de Lisboa Júri: Presidente: Prof. Doutor José Augusto Legatheaux Martins Arguente: Prof. Doutor Salvador Luís Bettencourt Pinto de Abreu Vogal: Prof. Doutor Pedro Manuel Corrêa Calvente de Barahona Setembro, 2015 A Public Transport Bus Assignment Problem: Parallel Metaheuristics Assess- ment Copyright c David Fernandes Semedo, Faculdade de Ciências e Tecnologia, Universidade Nova de Lisboa A Faculdade de Ciências e Tecnologia e a Universidade Nova de Lisboa têm o direito, perpétuo e sem limites geográficos, de arquivar e publicar esta dissertação através de exemplares impressos reproduzidos em papel ou de forma digital, ou por qualquer outro meio conhecido ou que venha a ser inventado, e de a divulgar através de repositórios científicos e de admitir a sua cópia e distribuição com objectivos educacionais ou de investigação, não comerciais, desde que seja dado crédito ao autor e editor. Aos meus pais. ACKNOWLEDGEMENTS This research was partly supported by project “RtP - Restrict to Plan”, funded by FEDER (Fundo Europeu de Desenvolvimento Regional), through programme COMPETE - POFC (Operacional Factores de Competitividade) with reference 34091. First and foremost, I would like to express my genuine gratitude to my advisor profes- sor Pedro Barahona for the continuous support of my thesis, for his guidance, patience and expertise. His general optimism, enthusiasm and availability to discuss and support new ideas helped and encouraged me to push my work always a little further.
    [Show full text]
  • Fractional-Order Derivatives and Integrals: Introductory Overview and Recent Developments
    KYUNGPOOK Math. J. 60(2020), 73-116 https://doi.org/10.5666/KMJ.2020.60.1.73 pISSN 1225-6951 eISSN 0454-8124 ⃝c Kyungpook Mathematical Journal Fractional-Order Derivatives and Integrals: Introductory Overview and Recent Developments Hari Mohan Srivastava Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia V8W3R4, Canada and Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan, Republic of China and Department of Mathematics and Informatics, Azerbaijan University, 71 Jeyhun Ha- jibeyli Street, AZ1007 Baku, Azerbaijan e-mail : [email protected] Abstract. The subject of fractional calculus (that is, the calculus of integrals and deriva- tives of any arbitrary real or complex order) has gained considerable popularity and im- portance during the past over four decades, due mainly to its demonstrated applications in numerous seemingly diverse and widespread fields of mathematical, physical, engineer- ing and statistical sciences. Various operators of fractional-order derivatives as well as fractional-order integrals do indeed provide several potentially useful tools for solving dif- ferential and integral equations, and various other problems involving special functions of mathematical physics as well as their extensions and generalizations in one and more variables. The main object of this survey-cum-expository article is to present a brief ele- mentary and introductory overview of the theory of the integral and derivative operators of fractional calculus and their applications especially in developing solutions of certain interesting families of ordinary and partial fractional “differintegral" equations. This gen- eral talk will be presented as simply as possible keeping the likelihood of non-specialist audience in mind.
    [Show full text]
  • What Is... Fractional Calculus?
    What is... Fractional Calculus? Clark Butler August 6, 2009 Abstract Differentiation and integration of non-integer order have been of interest since Leibniz. We will approach the fractional calculus through the differintegral operator and derive the differintegrals of familiar functions from the standard calculus. We will also solve Abel's integral equation using fractional methods. The Gr¨unwald-Letnikov Definition A plethora of approaches exist for derivatives and integrals of arbitrary order. We will consider only a few. The first, and most intuitive definition given here was first proposed by Gr¨unwald in 1867, and later Letnikov in 1868. We begin with the definition of a derivative as a difference quotient, namely, d1f f(x) − f(x − h) = lim dx1 h!0 h It is an exercise in induction to demonstrate that, more generally, n dnf 1 X n = lim (−1)j f(x − jh) dxn h!0 hn j j=0 We will assume that all functions described here are sufficiently differen- tiable. Differentiation and integration are often regarded as inverse operations, so d−1 we wish now to attach a meaning to the symbol dx−1 , what might commonly be referred to as anti-differentiation. However, integration of a function is depen- dent on the lower limit of integration, which is why the two operations cannot be regarded as truly inverse. We will select a definitive lower limit of 0 for convenience, so that, d−nf Z x Z xn−1 Z x2 Z x1 −n ≡ dxn−1 dxn−2 ··· dx1 f(x0)dx0 dx 0 0 0 0 1 By instead evaluating this multiple intgral as the limit of a sum, we find n N−1 d−nf x X j + n − 1 x = lim f(x − j ) dx−n N!1 N j N j=0 in which the interval [0; x] has been partitioned into N equal subintervals.
    [Show full text]
  • Solving the Maximum Weight Independent Set Problem Application to Indirect Hex-Mesh Generation
    Solving the Maximum Weight Independent Set Problem Application to Indirect Hex-Mesh Generation Dissertation presented by Kilian VERHETSEL for obtaining the Master’s degree in Computer Science and Engineering Supervisor(s) Jean-François REMACLE, Amaury JOHNEN Reader(s) Jeanne PELLERIN, Yves DEVILLE , Julien HENDRICKX Academic year 2016-2017 Acknowledgments I would like to thank my supervisor, Jean-François Remacle, for believing in me, my co-supervisor Amaury Johnen, for his encouragements during this project, as well as Jeanne Pellerin, for her patience and helpful advice. 3 4 Contents List of Notations 7 List of Figures 8 List of Tables 10 List of Algorithms 12 1 Introduction 17 1.1 Background on Indirect Mesh Generation . 17 1.2 Outline . 18 2 State of the Art 19 2.1 Exact Resolution . 19 2.1.1 Approaches Based on Graph Coloring . 19 2.1.2 Approaches Based on MaxSAT . 20 2.1.3 Approaches Based on Integer Programming . 21 2.1.4 Parallel Implementations . 21 2.2 Heuristic Approach . 22 3 Hybrid Approach to Combining Tetrahedra into Hexahedra 25 3.1 Computation of the Incompatibility Graph . 25 3.2 Exact Resolution for Small Graphs . 28 3.2.1 Complete Search . 28 3.2.2 Branch and Bound . 29 3.2.3 Upper Bounds Based on Clique Partitions . 31 3.2.4 Upper Bounds based on Linear Programming . 32 3.3 Construction of an Initial Solution . 35 3.3.1 Local Search Algorithms . 35 3.3.2 Local Search Formulation of the Maximum Weight Independent Set . 36 3.3.3 Strategies to Escape Local Optima .
    [Show full text]
  • Cognicrypt: Supporting Developers in Using Cryptography
    CogniCrypt: Supporting Developers in using Cryptography Stefan Krüger∗, Sarah Nadiy, Michael Reifz, Karim Aliy, Mira Meziniz, Eric Bodden∗, Florian Göpfertz, Felix Güntherz, Christian Weinertz, Daniel Demmlerz, Ram Kamathz ∗Paderborn University, {fistname.lastname}@uni-paderborn.de yUniversity of Alberta, {nadi, karim.ali}@ualberta.ca zTechnische Universität Darmstadt, {reif, mezini, guenther, weinert, demmler}@cs.tu-darmstadt.de, [email protected], [email protected] Abstract—Previous research suggests that developers often reside on the low level of cryptographic algorithms instead of struggle using low-level cryptographic APIs and, as a result, being more functionality-oriented APIs that provide convenient produce insecure code. When asked, developers desire, among methods such as encryptFile(). When it comes to tool other things, more tool support to help them use such APIs. In this paper, we present CogniCrypt, a tool that supports support, participants suggested tools like a CryptoDebugger, developers with the use of cryptographic APIs. CogniCrypt analysis tools that find misuses and provide code templates assists the developer in two ways. First, for a number of common or generate code for common functionality. These suggestions cryptographic tasks, CogniCrypt generates code that implements indicate that participants not only lack the domain knowledge, the respective task in a secure manner. Currently, CogniCrypt but also struggle with APIs themselves and how to use them. supports tasks such as data encryption, communication over secure channels, and long-term archiving. Second, CogniCrypt In this paper, we present CogniCrypt, an Eclipse plugin that continuously runs static analyses in the background to ensure enables easier use of cryptographic APIs. In previous work, a secure integration of the generated code into the developer’s we outlined an early vision for CogniCrypt [2].
    [Show full text]
  • Fractional Calculus
    faculty of mathematics and natural sciences Fractional Calculus Bachelor Project Mathematics October 2015 Student: D.E. Koning First supervisor: Dr. A.E. Sterk Second supervisor: Prof. dr. H.L. Trentelman Abstract This thesis introduces fractional derivatives and fractional integrals, shortly differintegrals. After a short introduction and some preliminaries the Gr¨unwald-Letnikov and Riemann-Liouville approaches for defining a differintegral will be explored. Then some basic properties of differintegrals, such as linearity, the Leibniz rule and composition, will be proved. Thereafter the definitions of the differintegrals will be applied to a few examples. Also fractional differential equations and one method for solving them will be discussed. The thesis ends with some examples of fractional differential equations and applications of differintegrals. CONTENTS Contents 1 Introduction4 2 Preliminaries5 2.1 The Gamma Function........................5 2.2 The Beta Function..........................5 2.3 Change the Order of Integration..................6 2.4 The Mittag-Leffler Function.....................6 3 Fractional Derivatives and Integrals7 3.1 The Gr¨unwald-Letnikov construction................7 3.2 The Riemann-Liouville construction................8 3.2.1 The Riemann-Liouville Fractional Integral.........9 3.2.2 The Riemann-Liouville Fractional Derivative.......9 4 Basic Properties of Fractional Derivatives 11 4.1 Linearity................................ 11 4.2 Zero Rule............................... 11 4.3 Product Rule & Leibniz's Rule................... 12 4.4 Composition............................. 12 4.4.1 Fractional integration of a fractional integral....... 12 4.4.2 Fractional differentiation of a fractional integral...... 13 4.4.3 Fractional integration and differentiation of a fractional derivative........................... 14 5 Examples 15 5.1 The Power Function......................... 15 5.2 The Exponential Function.....................
    [Show full text]
  • Amit Chouhan
    INVESTIGATION INTO FRACTIONAL DIFFERINTEGRAL OPERATORS, AND THEIR APPLICATION INTO VARIOUS DISCIPLINES A THESIS Submitted for the Award of Ph.D. degree of University of Kota, Kota (Mathematics-Faculty of Science) by AMIT CHOUHAN Under the supervision of Dr. Satish Saraswat (M.Sc., Ph.D. ) Lecturer Department of Mathematics Government College Kota, Kota – 324001(India) UNIVERSITY OF KOTA, KOTA (2013) Dr. Satish Saraswat Lecturer, (M.Sc., Ph.D.) Department of Mathematics Govt. College Kota, Kota -324001. CERTIFICATE I feel great pleasure in certifying that the thesis entitled “INVESTIGATION INTO FRACTIONAL DIFFERINTEGRAL OPERATORS, AND THEIR APPLICATION INTO VARIOUS DISCIPLINES”, embodies a record of the results of investigations carried out by Mr. Amit Chouhan under my guidance. I am satisfied with the analysis of data, interpretation of results and conclusions drawn. He has completed the residential requirement as per rules. I recommend the submission of thesis. Date : (Dr. Satish Saraswat) Research Supervisor DECLARATION I hereby declare that the (i) The thesis entitled “ INVESTIGATION INTO FRACTIONAL DIFFERINTEGRAL OPERATORS, AND THEIR APPLICATION INTO VARIOUS DISCIPLINES ” submitted by me is an original piece of research work, carried out under the supervision of Dr. Satish Saraswat. (ii) The above thesis has not been submitted to this university or any other university for any degree. Date: Signature of Candidate (Amit Chouhan) ACKNOWLEDGEMENTS I express my heartful gratitude to the “ ALMIGHTY GOD ” for his blessing to complete this piece of work. I wish to express my unfeigned indebtedness to my research supervisor Dr. Satish Saraswat , Department of Mathematics, Government College, Kota for his constant inspiration, supervision and able guidance in making this endeavor a success.
    [Show full text]
  • Analysis and Processing of Cryptographic Protocols
    Analysis and Processing of Cryptographic Protocols Submitted in partial fulfilment of the requirements of the degree of Bachelor of Science (Honours) of Rhodes University Bradley Cowie Grahamstown, South Africa November 2009 Abstract The field of Information Security and the sub-field of Cryptographic Protocols are both vast and continually evolving fields. The use of cryptographic protocols as a means to provide security to web servers and services at the transport layer, by providing both en- cryption and authentication to data transfer, has become increasingly popular. However, it is noted that it is rather difficult to perform legitimate analysis, intrusion detection and debugging on data that has passed through a cryptographic protocol as it is encrypted. The aim of this thesis is to design a framework, named Project Bellerophon, that is capa- ble of decrypting traffic that has been encrypted by an arbitrary cryptographic protocol. Once the plain-text has been retrieved further analysis may take place. To aid in this an in depth investigation of the TLS protocol was undertaken. This pro- duced a detailed document considering the message structures and the related fields con- tained within these messages which are involved in the TLS handshake process. Detailed examples explaining the processes that are involved in obtaining and generating the var- ious cryptographic components were explored. A systems design was proposed, considering the role of each of the components required in order to produce an accurate decryption of traffic encrypted by a cryptographic protocol. Investigations into the accuracy and the efficiency of Project Bellerophon to decrypt specific test data were conducted.
    [Show full text]
  • MULTIVALUED SUBSETS UNDER INFORMATION THEORY Indraneel Dabhade Clemson University, [email protected]
    Clemson University TigerPrints All Theses Theses 8-2011 MULTIVALUED SUBSETS UNDER INFORMATION THEORY Indraneel Dabhade Clemson University, [email protected] Follow this and additional works at: https://tigerprints.clemson.edu/all_theses Part of the Applied Mathematics Commons Recommended Citation Dabhade, Indraneel, "MULTIVALUED SUBSETS UNDER INFORMATION THEORY" (2011). All Theses. 1155. https://tigerprints.clemson.edu/all_theses/1155 This Thesis is brought to you for free and open access by the Theses at TigerPrints. It has been accepted for inclusion in All Theses by an authorized administrator of TigerPrints. For more information, please contact [email protected]. MULTIVALUED SUBSETS UNDER INFORMATION THEORY _______________________________________________________ A Thesis Presented to the Graduate School of Clemson University _______________________________________________________ In Partial Fulfillment of the Requirements for the Degree Master of Science Industrial Engineering _______________________________________________________ by Indraneel Chandrasen Dabhade August 2011 _______________________________________________________ Accepted by: Dr. Mary Beth Kurz, Committee Chair Dr. Anand Gramopadhye Dr. Scott Shappell i ABSTRACT In the fields of finance, engineering and varied sciences, Data Mining/ Machine Learning has held an eminent position in predictive analysis. Complex algorithms and adaptive decision models have contributed towards streamlining directed research as well as improve on the accuracies in forecasting. Researchers in the fields of mathematics and computer science have made significant contributions towards the development of this field. Classification based modeling, which holds a significant position amongst the different rule-based algorithms, is one of the most widely used decision making tools. The decision tree has a place of profound significance in classification-based modeling. A number of heuristics have been developed over the years to prune the decision making process.
    [Show full text]
  • The Maritime Pickup and Delivery Problem with Cost and Time Window Constraints: System Modeling and A* Based Solution
    The Maritime Pickup and Delivery Problem with Cost and Time Window Constraints: System Modeling and A* Based Solution CHRISTOPHER DAMBAKK SUPERVISOR Associate Professor Lei Jiao University of Agder, 2019 Faculty of Engineering and Science Department of ICT Abstract In the ship chartering business, more and more shipment orders are based on pickup and delivery in an on-demand manner rather than conventional scheduled routines. In this situation, it is nec- essary to estimate and compare the cost of shipments in order to determine the cheapest one for a certain order. For now, these cal- culations are based on static, empirical estimates and simplifications, and do not reflect the complexity of the real world. In this thesis, we study the Maritime Pickup and Delivery Problem with Cost and Time Window Constraints. We first formulate the problem mathe- matically, which is conjectured NP-hard. Thereafter, we propose an A* based prototype which finds the optimal solution with complexity O(bd). We compare the prototype with a dynamic programming ap- proach and simulation results show that both algorithms find global optimal and that A* finds the solution more efficiently, traversing fewer nodes and edges. iii Preface This thesis concludes the master's education in Communication and Information Technology (ICT), at the University of Agder, Nor- way. Several people have supported and contributed to the completion of this project. I want to thank in particular my supervisor, Associate Professor Lei Jiao. He has provided excellent guidance and refreshing perspectives when the tasks ahead were challenging. I would also like to thank Jayson Mackie, co-worker and friend, for proofreading my report.
    [Show full text]