Chapter I – Fuzzy Relational Equations – Basic Concepts
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Data Envelopment Analysis with Fuzzy Parameters: an Interactive Approach
International Journal of Operations Research and Information Systems, 2(3), 39-53, July-September 2011 39 Data Envelopment Analysis with Fuzzy Parameters: An Interactive Approach Adel Hatami-Marbini, Universite Catholique de Louvain, Belgium Saber Saati, Islamic Azad University, Iran Madjid Tavana, La Salle University, USA ABSTRACT Data envelopment analysis (DEA) is a methodology for measuring the relative efficiencies of a set of deci- sion making units (DMUs) that use multiple inputs to produce multiple outputs. In the conventional DEA, all the data assume the form of specific numerical values. However, the observed values of the input and output data in real-life problems are sometimes imprecise or vague. Previous methods have not considered the preferences of the decision makers (DMs) in the evaluation process. This paper proposes an interactive evaluation process for measuring the relative efficiencies of a set of DMUs in fuzzy DEA with consideration of the DMs’ preferences. The authors construct a linear programming (LP) model with fuzzy parameters and calculate the fuzzy efficiency of the DMUs for different α levels. Then, the DM identifies his or her most preferred fuzzy goal for each DMU under consideration. A modified Yager index is used to develop a ranking order of the DMUs. This study allows the DMs to use their preferences or value judgments when evaluating the performance of the DMUs. Keywords: Data Envelopment Analysis, Efficiency Evaluation, Fuzzy Mathematical Programming, Interactive Solution, Preference Modeling INTRODUCTION efficiency. A DMU is considered efficient when no other DMU can produce more outputs using The changing economic conditions have chal- an equal or lesser amount of inputs. -
Fuzzy Integer Linear Programming with Fuzzy Decision Variables
Applied Mathematical Sciences, Vol. 4, 2010, no. 70, 3493 - 3502 Fuzzy Integer Linear Programming with Fuzzy Decision Variables C. Sudhagar1 Department of Mathematics and Applied Sciences Middle East College of Information Technology, Muscat, Oman [email protected] K. Ganesan Department of Mathematics, SRM University, Chennai, India gansan [email protected] Abstract In this paper a new method for dealing with Fuzzy Integer Linear Programming Problems (FILPP) has been proposed. FILPP with fuzzy variables model was taken for solution. This solution method is based on the fuzzy ranking method. The proposed method can serve deci- sion makers by providing the reasonable range of values for the fuzzy variable, which is comparatively better than the currently available solu- tions. Numerical examples demonstrate the effectiveness and accuracy of the proposed method. Mathematics Subject Classification: 65K05, 90C10, 90C70, 90C90 Keywords: Fuzzy numbers, Ranking, Fuzzy integer linear programming 1 Introduction Linear Programming Problems(LPP) have an outstanding relevance in the field of Decision making, Artificial intelligence, Control theory, Management sciences, Job placement interventions etc. In many practical applications the available information in the system under consideration are not precise. In such situations, it is more appropriate to use the fuzzy LPP. 1Corresponding author 3494 C. Sudhagar and K. Ganesan The concept of fuzzy linear programming problems was first introduced by Tanaka et al.,[13, 12] After his work, several kinds of fuzzy linear programming problems have appeared in the literature and different methods have been proposed to solve such problems. Numerous methods for comparison of fuzzy numbers have been suggested in the literature. In Campos Verdegay paper[2] linear programming problems with fuzzy constraints and fuzzy coefficients in both matrix and right hand side of the constraint set are considered. -
Simple Laws About Nonprominent Properties of Binary Relations
Simple Laws about Nonprominent Properties of Binary Relations Jochen Burghardt jochen.burghardt alumni.tu-berlin.de Nov 2018 Abstract We checked each binary relation on a 5-element set for a given set of properties, including usual ones like asymmetry and less known ones like Euclideanness. Using a poor man's Quine-McCluskey algorithm, we computed prime implicants of non-occurring property combinations, like \not irreflexive, but asymmetric". We considered the laws obtained this way, and manually proved them true for binary relations on arbitrary sets, thus contributing to the encyclopedic knowledge about less known properties. Keywords: Binary relation; Quine-McCluskey algorithm; Hypotheses generation arXiv:1806.05036v2 [math.LO] 20 Nov 2018 Contents 1 Introduction 4 2 Definitions 8 3 Reported law suggestions 10 4 Formal proofs of property laws 21 4.1 Co-reflexivity . 21 4.2 Reflexivity . 23 4.3 Irreflexivity . 24 4.4 Asymmetry . 24 4.5 Symmetry . 25 4.6 Quasi-transitivity . 26 4.7 Anti-transitivity . 28 4.8 Incomparability-transitivity . 28 4.9 Euclideanness . 33 4.10 Density . 38 4.11 Connex and semi-connex relations . 39 4.12 Seriality . 40 4.13 Uniqueness . 42 4.14 Semi-order property 1 . 43 4.15 Semi-order property 2 . 45 5 Examples 48 6 Implementation issues 62 6.1 Improved relation enumeration . 62 6.2 Quine-McCluskey implementation . 64 6.3 On finding \nice" laws . 66 7 References 69 List of Figures 1 Source code for transitivity check . .5 2 Source code to search for right Euclidean non-transitive relations . .5 3 Timing vs. universe cardinality . -
Connes on the Role of Hyperreals in Mathematics
Found Sci DOI 10.1007/s10699-012-9316-5 Tools, Objects, and Chimeras: Connes on the Role of Hyperreals in Mathematics Vladimir Kanovei · Mikhail G. Katz · Thomas Mormann © Springer Science+Business Media Dordrecht 2012 Abstract We examine some of Connes’ criticisms of Robinson’s infinitesimals starting in 1995. Connes sought to exploit the Solovay model S as ammunition against non-standard analysis, but the model tends to boomerang, undercutting Connes’ own earlier work in func- tional analysis. Connes described the hyperreals as both a “virtual theory” and a “chimera”, yet acknowledged that his argument relies on the transfer principle. We analyze Connes’ “dart-throwing” thought experiment, but reach an opposite conclusion. In S, all definable sets of reals are Lebesgue measurable, suggesting that Connes views a theory as being “vir- tual” if it is not definable in a suitable model of ZFC. If so, Connes’ claim that a theory of the hyperreals is “virtual” is refuted by the existence of a definable model of the hyperreal field due to Kanovei and Shelah. Free ultrafilters aren’t definable, yet Connes exploited such ultrafilters both in his own earlier work on the classification of factors in the 1970s and 80s, and in Noncommutative Geometry, raising the question whether the latter may not be vulnera- ble to Connes’ criticism of virtuality. We analyze the philosophical underpinnings of Connes’ argument based on Gödel’s incompleteness theorem, and detect an apparent circularity in Connes’ logic. We document the reliance on non-constructive foundational material, and specifically on the Dixmier trace − (featured on the front cover of Connes’ magnum opus) V. -
The Orthogonality Between Complex Fuzzy Sets and Its Application to Signal Detection
Article The Orthogonality between Complex Fuzzy Sets and Its Application to Signal Detection Bo Hu 1 ID , Lvqing Bi 2 and Songsong Dai 3,* ID 1 School of Mechanical and Electrical Engineering, Guizhou Normal University, Guiyang 550025, China; [email protected] 2 School of Electronics and Communication Engineering, Yulin Normal University, Yulin 537000, China; [email protected] 3 School of Information Science and Engineering, Xiamen University, Xiamen 361005, China * Correspondence: [email protected]; Tel.: +86-592-258-0135 Academic Editor: Hsien-Chung Wu Received: 25 July 2017; Accepted: 25 August 2017; Published: 31 August 2017 Abstract: A complex fuzzy set is a set whose membership values are vectors in the unit circle in the complex plane. This paper establishes the orthogonality relation of complex fuzzy sets. Two complex fuzzy sets are said to be orthogonal if their membership vectors are perpendicular. We present the basic properties of orthogonality of complex fuzzy sets and various results on orthogonality with respect to complex fuzzy complement, complex fuzzy union, complex fuzzy intersection, and complex fuzzy inference methods. Finally, an example application of signal detection demonstrates the utility of the orthogonality of complex fuzzy sets. Keywords: orthogonality; complex fuzzy sets; complex fuzzy operations; complex fuzzy inference 1. Introduction Complex fuzzy sets [1] are an important extension of fuzzy set theory. In recent years, complex fuzzy sets have been successfully used in complex fuzzy inference systems for various applications, such as time series prediction [2–8], function approximation [9–11], and image restoration [12,13]. A complex fuzzy set A on a universe of discourse U is a mapping from U to the unit disc in the complex plane. -
Trivial Meet and Join Within the Lattice of Monotone Triangles John Engbers Marquette University, [email protected]
Marquette University e-Publications@Marquette Mathematics, Statistics and Computer Science Mathematics, Statistics and Computer Science, Faculty Research and Publications Department of 1-1-2014 Trivial Meet and Join within the Lattice of Monotone Triangles John Engbers Marquette University, [email protected] Adam Hammett Bethel College - Mishawaka Published version. Electronic Journal of Combinatorics, Vol. 21, No. 3 (2014). Permalink. © 2014 The Authors. Used with permission. Trivial Meet and Join within the Lattice of Monotone Triangles John Engbers Department of Mathematics, Statistics and Computer Science Marquette University Milwaukee, WI, U.S.A. [email protected] Adam Hammett Department of Mathematical Sciences Bethel College Mishawaka, IN, U.S.A. [email protected] Submitted: Jan 22, 2014; Accepted: Jul 10, 2014; Published: Jul 21, 2014 Mathematics Subject Classifications: 05A05, 05A16, 06A20 Abstract The lattice of monotone triangles (Mn; 6) ordered by entry-wise comparisons is studied. Let τmin denote the unique minimal element in this lattice, and τmax the unique maximum. The number of r-tuples of monotone triangles (τ1; : : : ; τr) with minimal infimum τmin (maximal supremum τmax, resp.) is shown to asymptotically r−1 approach rjMnj as n ! 1. Thus, with high probability this event implies that one of the τi is τmin (τmax, resp.). Higher-order error terms are also discussed. Keywords: monotone triangles; permutations; square ice; alternating sign matri- ces; meet; join 1 Introduction and statement -
A Clone-Based Representation of the Fuzzy Tolerance Or Equivalence Relations a Strict Order Relation Is Compatible With
Available online at www.sciencedirect.com ScienceDirect Fuzzy Sets and Systems 296 (2016) 35–50 www.elsevier.com/locate/fss A clone-based representation of the fuzzy tolerance or equivalence relations a strict order relation is compatible with ∗ Bernard De Baets a, , Lemnaouar Zedam b, Azzedine Kheniche b a KERMIT, Department of Mathematical Modelling, Statistics and Bioinformatics, Ghent University, Coupure links 653, B-9000 Gent, Belgium b Laboratory of Pure and Applied Mathematics, Department of Mathematics, Med Boudiaf University – Msila, P.O. Box 166 Ichbilia, Msila 28000, Algeria Received 17 May 2015; received in revised form 14 August 2015; accepted 16 September 2015 Available online 30 September 2015 Abstract We show that although there exists no non-trivial (fuzzy) tolerance relation a partial order relation is compatible with (in the sense of Belohlávek),ˇ the situation is quite different when considering its strict part. More specifically, we provide a representation of all fuzzy tolerance (and, in particular, all fuzzy equivalence) relations a strict order relation is compatible with. To that end, we introduce the notion of clone relation associated with a partially ordered set and discuss its basic properties. The mentioned representation is intimately connected with this clone relation. © 2015 Elsevier B.V. All rights reserved. Keywords: Clone relation; Compatibility; Equivalence relation; Fuzzy relation; Order relation; Tolerance relation 1. Introduction Order relations and equivalence relations are basic mathematical concepts that are fundamental to numerous math- ematical and computational disciplines. Not surprisingly, these notions have been generalized to the setting of fuzzy sets in the early days of fuzzy set theory [18], and have been the subject of many studies since, with new ones still appearing at a regular pace. -
Zerohack Zer0pwn Youranonnews Yevgeniy Anikin Yes Men
Zerohack Zer0Pwn YourAnonNews Yevgeniy Anikin Yes Men YamaTough Xtreme x-Leader xenu xen0nymous www.oem.com.mx www.nytimes.com/pages/world/asia/index.html www.informador.com.mx www.futuregov.asia www.cronica.com.mx www.asiapacificsecuritymagazine.com Worm Wolfy Withdrawal* WillyFoReal Wikileaks IRC 88.80.16.13/9999 IRC Channel WikiLeaks WiiSpellWhy whitekidney Wells Fargo weed WallRoad w0rmware Vulnerability Vladislav Khorokhorin Visa Inc. Virus Virgin Islands "Viewpointe Archive Services, LLC" Versability Verizon Venezuela Vegas Vatican City USB US Trust US Bankcorp Uruguay Uran0n unusedcrayon United Kingdom UnicormCr3w unfittoprint unelected.org UndisclosedAnon Ukraine UGNazi ua_musti_1905 U.S. Bankcorp TYLER Turkey trosec113 Trojan Horse Trojan Trivette TriCk Tribalzer0 Transnistria transaction Traitor traffic court Tradecraft Trade Secrets "Total System Services, Inc." Topiary Top Secret Tom Stracener TibitXimer Thumb Drive Thomson Reuters TheWikiBoat thepeoplescause the_infecti0n The Unknowns The UnderTaker The Syrian electronic army The Jokerhack Thailand ThaCosmo th3j35t3r testeux1 TEST Telecomix TehWongZ Teddy Bigglesworth TeaMp0isoN TeamHav0k Team Ghost Shell Team Digi7al tdl4 taxes TARP tango down Tampa Tammy Shapiro Taiwan Tabu T0x1c t0wN T.A.R.P. Syrian Electronic Army syndiv Symantec Corporation Switzerland Swingers Club SWIFT Sweden Swan SwaggSec Swagg Security "SunGard Data Systems, Inc." Stuxnet Stringer Streamroller Stole* Sterlok SteelAnne st0rm SQLi Spyware Spying Spydevilz Spy Camera Sposed Spook Spoofing Splendide -
Decision Analysis in the UK Energy Supply Chain Risk Management: Tools Development and Application
Decision Analysis in the UK Energy Supply Chain Risk Management: Tools Development and Application by Amin Vafadarnikjoo Registration number: 100166891 Thesis submitted to The University of East Anglia for the Degree of Doctor of Philosophy (PhD) August 2020 Norwich Business School “This copy of the thesis has been supplied on condition that anyone who consults it is understood to recognise that its copyright rests with the author and that use of any information derived therefrom must be in accordance with current UK Copyright Law. In addition, any quotation or extract must include full attribution.” 1 Abstract Large infrastructures like electricity supply networks are widely presumed to be crucial for the functioning of societies as they create conditions for essential economic activities. There has always been a continuing concern and complexity around risks in the field of energy security and particularly power grids within energy supply chain. Drawing on this complexity and a need for useful tools, this research contributes to developing and utilising proper decision-making tools (i.e. methods and models) to deal with the risk identification and mitigation in the UK energy supply chain as a compound networked system. This thesis is comprised of four study phases (Figure I.A.). It is aimed at developing decision-making tools for risk identification, risk interdependency analysis, risk prioritisation, and long-term risk mitigation strategy recommendations. The application of the tools has focused on the UK power supply chain. The five new tools which are introduced and applied in this thesis are: (1) Proposed Expert Selection Model (ESM) and its application under hesitant fuzzy environment (i.e. -
Fuzzy Relational Maps and Neutrosophic Relational Maps
University of New Mexico UNM Digital Repository Faculty and Staff Publications Mathematics 2004 FUZZY RELATIONAL MAPS AND NEUTROSOPHIC RELATIONAL MAPS Florentin Smarandache University of New Mexico, [email protected] W.B. Vasantha Kandasamy [email protected] Follow this and additional works at: https://digitalrepository.unm.edu/math_fsp Part of the Algebraic Geometry Commons, Analysis Commons, and the Set Theory Commons Recommended Citation W.B. Vasantha Kandasamy & F. Smarandache. FUZZY RELATIONAL MAPS AND NEUTROSOPHIC RELATIONAL MAPS. Church Rock: Hexis, 2004. This Book is brought to you for free and open access by the Mathematics at UNM Digital Repository. It has been accepted for inclusion in Faculty and Staff Publications by an authorized administrator of UNM Digital Repository. For more information, please contact [email protected], [email protected], [email protected]. W. B. VASANTHA KANDASAMY FLORENTIN SMARANDACHE FUZZY RELATIONAL MAPS AND NEUTROSOPHIC RELATIONAL MAPS HEXIS Church Rock 2004 FUZZY RELATIONAL MAPS AND NEUTROSOPHIC RELATIONAL MAPS W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of Technology, Madras Chennai – 600036, India e-mail: [email protected] web: http://mat.iitm.ac.in/~wbv Florentin Smarandache Department of Mathematics University of New Mexico Gallup, NM 87301, USA e-mail: [email protected] HEXIS Church Rock 2004 1 This book can be ordered in a paper bound reprint from: Books on Demand ProQuest Information & Learning (University of Microfilm International) 300 N. Zeeb Road P.O. Box 1346, Ann Arbor MI 48106-1346, USA Tel.: 1-800-521-0600 (Customer Service) http://wwwlib.umi.com/bod/ and online from: Publishing Online, Co. -
On Comparability of Random Permutations
ON COMPARABILITY OF RANDOM PERMUTATIONS DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of the Ohio State University By Adam Hammett, B.S. ***** The Ohio State University 2007 Dissertation Committee: Approved by Dr. Boris Pittel, Advisor Dr. Gerald Edgar Advisor Dr. Akos Seress Graduate Program in Mathematics ABSTRACT Two permutations of [n] := {1, 2, . , n} are comparable in the Bruhat order if one can be obtained from the other by a sequence of transpositions decreasing the number of inversions. We show that the total number of pairs of permutations (π, σ) with π ≤ σ is of order (n!)2/n2 at most. Equivalently, if π, σ are chosen uniformly at random and independently of each other, then P (π ≤ σ) is of order n−2 at most. By a direct probabilistic argument we prove P (π ≤ σ) is of order (0.708)n at least, so that there is currently a wide qualitative gap between the upper and lower bounds. Next, emboldened by a connection with Ferrers diagrams and plane partitions implicit in Bressoud’s book [13], we return to the Bruhat order upper bound and show that for n-permutations π1, . , πr selected independently and uniformly at random, −r(r−1) P (π1 ≤ · · · ≤ πr) = O n , thus providing an extension of our result for pairs of permutations to chains of length r > 2. Turning to the related weak order “” – when only adjacent transpositions are ∗ admissible – we use a non-inversion set criterion to prove that Pn := P (π σ) is pn ∗ submultiplicative, thus showing existence of ρ = lim Pn . -
Proceedings of the Third International Workshop on Neural Networks and Fuzzy Logic
NASA Conference Publication 10111 Proceedings of the Third - _ International Workshop on Neural Networks and T_ Fuzzy Logic , k ('_ASA-CP-1OIII-Vol-2) PRQCEEOINGS N93-22206 i]F ThE TH[_O INTERNATIONAL WORKSHOP --THRU-- ON NEURAL NETWORKS AND FUZZY LOGIC, N93-22223 V_LU_E 2 (NASA) i83 p Unclas Volume II G3/63 0150400 a workshop held at ,nson Space Center Houston, Texas June 1 - 3, 1992 p_ _q_r NASA Conference Publication 10111 Proceedings of the Third International Workshop on Neural Networks and Fuzzy Logic Volume II Christopher J. Culbert, Editor NASA Lyndon B. Johnson Space Center Houston, Texas Proceedings of a workshop held at Lyndon B. Johnson Space Center Houston, Texas June 1 - 3, 1992 National Aeronautics and Space Administration January 1993 THIRD INTERNATIONAL WORKSHOP ON NEURAL NETWORKS AND FUZZY LOGIC Program Schedule Monday June 1, 1992 7:30-8:00 Registration 8:00-8:30 Robed T. Savely, Chief Scientist, Information Systems Directorate, NASA/Lyndon B. Johnson Space Center, Houston, TX. Welcoming Remarks. 8:30-9:30 Jon Erickson, Chief Scientist, Automation and Robotics Division, NASNLyndon B. Johnson Space Center, Houston, TX. Space Exploration Needs for Supervised Intelligent Systems. 9:30-9:45 Break Plenary Speakers 9:45-10:30 Piero P. Bonnisone, General Electric, Fuzzy Logic Controllers: A Knowledge-Based Systems Perspective. 10:30-11:15 Robed Farber, Los Alamos National Laboratory, Efficiently Modeling Neural Networks on Massively Parallel Computers. 11:15-1:00 Lunch pRLI_EOING P.._SE BLANK NOT RLMIED iii. w 1:00-1:30 Lawrence O. Hall and Steve G. Romaniuk, University of South Florida, Learning Fuzzy Information in a Hybrid Connectionist, Symbolic Model.