This document proposes a new computational methodology for finding appropriate solutions to fuzzy equations of the form A=B, where A and B are known fuzzy numbers and X is an unknown fuzzy number. It introduces definitions of positive, negative, and half-positive/half-negative fuzzy numbers. Propositions and theorems are presented to show that the solution to such an equation can be a positive, negative, or half-positive/half-negative fuzzy number depending on the properties of A and B. Examples are provided to illustrate solving equations where A and B are positive, negative, or half-positive/half-negative fuzzy numbers. The methodology aims to overcome limitations in solving equations where X is an unknown fuzzy number.
COMPARISON OF DIFFERENT APPROXIMATIONS OF FUZZY NUMBERSijfls
The notions of interval approximations of fuzzy numbers and trapezoidal approximations of fuzzy numbers have been discussed. Comparisons have been made between the close-interval approximation, valueambiguity
interval approximation and distinct approximation with the corresponding crisp and trapezoidal fuzzy numbers. A numerical example is included to justify the above mentioned notions.
Research Inventy : International Journal of Engineering and Scienceresearchinventy
Research Inventy : International Journal of Engineering and Science is published by the group of young academic and industrial researchers with 12 Issues per year. It is an online as well as print version open access journal that provides rapid publication (monthly) of articles in all areas of the subject such as: civil, mechanical, chemical, electronic and computer engineering as well as production and information technology. The Journal welcomes the submission of manuscripts that meet the general criteria of significance and scientific excellence. Papers will be published by rapid process within 20 days after acceptance and peer review process takes only 7 days. All articles published in Research Inventy will be peer-reviewed.
Dear students get fully solved assignments
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COMPARISON OF DIFFERENT APPROXIMATIONS OF FUZZY NUMBERSijfls
The notions of interval approximations of fuzzy numbers and trapezoidal approximations of fuzzy numbers have been discussed. Comparisons have been made between the close-interval approximation, valueambiguity
interval approximation and distinct approximation with the corresponding crisp and trapezoidal fuzzy numbers. A numerical example is included to justify the above mentioned notions.
Research Inventy : International Journal of Engineering and Scienceresearchinventy
Research Inventy : International Journal of Engineering and Science is published by the group of young academic and industrial researchers with 12 Issues per year. It is an online as well as print version open access journal that provides rapid publication (monthly) of articles in all areas of the subject such as: civil, mechanical, chemical, electronic and computer engineering as well as production and information technology. The Journal welcomes the submission of manuscripts that meet the general criteria of significance and scientific excellence. Papers will be published by rapid process within 20 days after acceptance and peer review process takes only 7 days. All articles published in Research Inventy will be peer-reviewed.
Dear students get fully solved assignments
Send your semester & Specialization name to our mail id :
“ help.mbaassignments@gmail.com ”
or
Call us at : 08263069601
Using Alpha-cuts and Constraint Exploration Approach on Quadratic Programming...TELKOMNIKA JOURNAL
In this paper, we propose a computational procedure to find the optimal solution of quadratic programming
problems by using fuzzy -cuts and constraint exploration approach. We solve the problems in
the original form without using any additional information such as Lagrange’s multiplier, slack, surplus and
artificial variable. In order to find the optimal solution, we divide the calculation in two stages. In the first
stage, we determine the unconstrained minimization of the quadratic programming problem (QPP) and check
its feasibility. By unconstrained minimization we identify the violated constraints and focus our searching in
these constraints. In the second stage, we explored the feasible region along side the violated constraints
until the optimal point is achieved. A numerical example is included in this paper to illustrate the capability of
-cuts and constraint exploration to find the optimal solution of QPP.
In this paper, we investigate transportation problem in which supplies and demands are intuitionistic fuzzy numbers. Intuitionistic Fuzzy Vogel’s Approximation Method is proposed to find an initial basic feasible solution. Intuitionistic Fuzzy Modified Distribution Method is proposed to find the optimal solution in terms of triangular intuitionistic fuzzy numbers. The solution procedure is illustrated with suitable numerical example.
Hypersoft set is an extension of the soft set where there is more than one set of attributes occur and it is very much helpful in multi-criteria group decision making problem. In a hypersoft set, the function F is a multi-argument function. In this paper, we have used the notion of Fuzzy Hypersoft Set (FHSS), which is a combination of fuzzy set and hypersoft set. In earlier research works the concept of Fuzzy Soft Set (FSS) was introduced and it was applied successfully in various fields. The FHSS theory gives more flexibility as compared to FSS to tackle the parameterized problems of uncertainty. To overcome the issue where FSS failed to explain uncertainty and incompleteness there is a dire need for another environment which is known as FHSS. It works well when there is more complexity involved in the parametric data i.e the data that involves vague concepts. This work includes some basic set-theoretic operations on FHSSs and for the reliability and the authenticity of these operations, we have shown its application with the help of a suitable example. This example shows that how FHSS theory plays its role to solve real decision-making problems.
Open Data Semantic Web Community Barn RaisingBoris Mann
Open source is now a term that is increasingly understood, or at least talked about. One of the next battles in keeping the web open is keeping data open - from the content we "generate" for many social sites to the closed silos of government and businesses.
Just like open source, open data is a mix of technology, licensing, and attitude. Here in Vancouver, we've started experimenting with what I like to call a Semantic Web Community "Barn Raising" effort. Lots of people are interested in open data and related technology that supports it, like the semantic web, RDF, etc. But it's very hard to just have nice thought experiments about all this, we learn best by doing.
Recently, I gave a talk at DrupalCon Washington DC talking about the semantic web. I said that RDFa is "food for robots", so what better data to experiment with than something related to food?
The project we're tackling is to collectively source restaurant information. The information will be stored in part on Freebase, as well as various front end mashups, iPhone apps, and other tools. Eventually, we hope to create an economic incentive so that some restaurants and/or restaurant directory websites will expose their data semantically.
The talk will be a mix of community, policy, hand waving, and technology, as well as an open call to help with this project in your own city. You'll leave with some food for thought on how open data and the semantic web are evolving, and what you can do with it today.
Using Alpha-cuts and Constraint Exploration Approach on Quadratic Programming...TELKOMNIKA JOURNAL
In this paper, we propose a computational procedure to find the optimal solution of quadratic programming
problems by using fuzzy -cuts and constraint exploration approach. We solve the problems in
the original form without using any additional information such as Lagrange’s multiplier, slack, surplus and
artificial variable. In order to find the optimal solution, we divide the calculation in two stages. In the first
stage, we determine the unconstrained minimization of the quadratic programming problem (QPP) and check
its feasibility. By unconstrained minimization we identify the violated constraints and focus our searching in
these constraints. In the second stage, we explored the feasible region along side the violated constraints
until the optimal point is achieved. A numerical example is included in this paper to illustrate the capability of
-cuts and constraint exploration to find the optimal solution of QPP.
In this paper, we investigate transportation problem in which supplies and demands are intuitionistic fuzzy numbers. Intuitionistic Fuzzy Vogel’s Approximation Method is proposed to find an initial basic feasible solution. Intuitionistic Fuzzy Modified Distribution Method is proposed to find the optimal solution in terms of triangular intuitionistic fuzzy numbers. The solution procedure is illustrated with suitable numerical example.
Hypersoft set is an extension of the soft set where there is more than one set of attributes occur and it is very much helpful in multi-criteria group decision making problem. In a hypersoft set, the function F is a multi-argument function. In this paper, we have used the notion of Fuzzy Hypersoft Set (FHSS), which is a combination of fuzzy set and hypersoft set. In earlier research works the concept of Fuzzy Soft Set (FSS) was introduced and it was applied successfully in various fields. The FHSS theory gives more flexibility as compared to FSS to tackle the parameterized problems of uncertainty. To overcome the issue where FSS failed to explain uncertainty and incompleteness there is a dire need for another environment which is known as FHSS. It works well when there is more complexity involved in the parametric data i.e the data that involves vague concepts. This work includes some basic set-theoretic operations on FHSSs and for the reliability and the authenticity of these operations, we have shown its application with the help of a suitable example. This example shows that how FHSS theory plays its role to solve real decision-making problems.
Open Data Semantic Web Community Barn RaisingBoris Mann
Open source is now a term that is increasingly understood, or at least talked about. One of the next battles in keeping the web open is keeping data open - from the content we "generate" for many social sites to the closed silos of government and businesses.
Just like open source, open data is a mix of technology, licensing, and attitude. Here in Vancouver, we've started experimenting with what I like to call a Semantic Web Community "Barn Raising" effort. Lots of people are interested in open data and related technology that supports it, like the semantic web, RDF, etc. But it's very hard to just have nice thought experiments about all this, we learn best by doing.
Recently, I gave a talk at DrupalCon Washington DC talking about the semantic web. I said that RDFa is "food for robots", so what better data to experiment with than something related to food?
The project we're tackling is to collectively source restaurant information. The information will be stored in part on Freebase, as well as various front end mashups, iPhone apps, and other tools. Eventually, we hope to create an economic incentive so that some restaurants and/or restaurant directory websites will expose their data semantically.
The talk will be a mix of community, policy, hand waving, and technology, as well as an open call to help with this project in your own city. You'll leave with some food for thought on how open data and the semantic web are evolving, and what you can do with it today.
Linking Open, Big Data Using Semantic Web Technologies - An IntroductionRonald Ashri
The Physics Department of the University of Cagliari and the Linkalab Group invited me to talk about the Semantic Web and Linked Data - this is simply an introduction to the technologies involved.
Evolving the Web into a Giant Global DatabaseMarko Rodriguez
The Web as we know it today will not be the Web as we know it tomorrow. The Web of today is oriented towards the universal accessibility of files (e.g. web pages, images). The Web of today can be thought of as a large-scale, distributed file system. The Web of tomorrow will encode any datum (e.g. strings, integers, dates). The Web of tomorrow can be thought of as a large-scale, distributed database. This talk will discuss the the future Web with special focus on the supporting standards and application visions.
A Semantics-based User Interface Model for Content Annotation, Authoring and ...Ali Khalili
A Semantics-based User Interface Model for Content Annotation, Authoring and Exploration: My PhD defense slides
full version of thesis:
http://svn.aksw.org/papers/2014/Thesis_Ali/public.pdf
The Minimum Hamming Distances of the Irreducible Cyclic Codes of Length inventionjournals
Let be a finite field with elements and where are positive integers and are distinct odd primes and 1. In this paper, we study the irreducible factorization of over and all primitive idempotents in the ring Moreover, we obtain the dimensions and the minimum Hamming distances of all irreducible cyclic codes of length over
The aim of this paper is to investigate different definitions of soft points in the existing literature on soft set theory and its extensions in different directions. Then limitations of these definitions are illustrated with the help of examples. Moreover, the definition of soft point in the setup of fuzzy soft set, intervalvalued fuzzy soft set, hesitant fuzzy soft set and intuitionistic soft set are also discussed. We also suggest an approach to unify the definitions of soft point which is more applicable than the existing notions.
COMPARISON OF DIFFERENT APPROXIMATIONS OF FUZZY NUMBERSWireilla
ABSTRACT
The notions of interval approximations of fuzzy numbers and trapezoidal approximations of fuzzy numbers have been discussed. Comparisons have been made between the close-interval approximation, valueambiguity interval approximation and distinct approximation with the corresponding crisp and trapezoidal fuzzy numbers. A numerical example is included to justify the above mentioned notions.
A NEW APPROACH FOR RANKING SHADOWED FUZZY NUMBERS AND ITS APPLICATIONijcsit
In many decision situations, decision-makers face a kind of complex problems. In these decision-making
problems, different types of fuzzy numbers are defined and, have multiple types of membership functions.
So, we need a standard form to formulate uncertain numbers in the problem. Shadowed fuzzy numbers are
considered granule numbers which approximate different types and different forms of fuzzy numbers. In
this paper, a new ranking approach for shadowed fuzzy numbers is developed using value, ambiguity and
fuzziness for shadowed fuzzy numbers. The new ranking method has been compared with other existing
approaches through numerical examples. Also, the new method is applied to a hybrid multi-attribute
decision making problem in which the evaluations of alternatives are expressed with different types of
uncertain numbers. The comparative study for the results of different examples illustrates the reliability of
the new method.
In many decision situations, decision-makers face a kind of complex problems. In these decision-making problems, different types of fuzzy numbers are defined and, have multiple types of membership functions. So, we need a standard form to formulate uncertain numbers in the problem. Shadowed fuzzy numbers are considered granule numbers which approximate different types and different forms of fuzzy numbers. In this paper, a new ranking approach for shadowed fuzzy numbers is developed using value, ambiguity and fuzziness for shadowed fuzzy numbers. The new ranking method has been compared with other existing approaches through numerical examples. Also, the new method is applied to a hybrid multi-attribute decision making problem in which the evaluations of alternatives are expressed with different types of uncertain numbers. The comparative study for the results of different examples illustrates the reliability of the new method.
Zadeh conceptualized the theory of fuzzy set to provide a tool for the basis of the theory of possibility. Atanassov extended this theory with the introduction of intuitionistic fuzzy set. Smarandache introduced the concept of refined intuitionistic fuzzy set by further subdivision of membership and non-membership value. The meagerness regarding the allocation of a single membership and non-membership value to any object under consideration is addressed with this novel refinement. In this study, this novel idea is utilized to characterize the essential elements e.g. subset, equal set, null set, and complement set, for refined intuitionistic fuzzy set. Moreover, their basic set theoretic operations like union, intersection, extended intersection, restricted union, restricted intersection, and restricted difference, are conceptualized. Furthermore, some basic laws are also discussed with the help of an illustrative example in each case for vivid understanding.
A New Hendecagonal Fuzzy Number For Optimization Problemsijtsrd
A new fuzzy number called Hendecagonal fuzzy number and its membership function is introduced, which is used to represent the uncertainty with eleven points. The fuzzy numbers with ten ordinates exists in literature. The aim of this paper is to define Hendecagonal fuzzy number and its arithmetic operations. Also a direct approach is proposed to solve fuzzy assignment problem (FAP) and fuzzy travelling salesman (FTSP) in which the cost and distance are represented by Hendecagonal fuzzy numbers. Numerical example shows the effectiveness of the proposed method and the Hendecagonal fuzzy number M. Revathi | Dr. M. Valliathal | R. Saravanan | Dr. K. Rathi"A New Hendecagonal Fuzzy Number For Optimization Problems" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-1 | Issue-5 , August 2017, URL: http://www.ijtsrd.com/papers/ijtsrd2258.pdf http://www.ijtsrd.com/mathemetics/applied-mathamatics/2258/a-new-hendecagonal-fuzzy-number-for-optimization-problems/m-revathi
Numerical Solution of Nth - Order Fuzzy Initial Value Problems by Fourth Orde...IOSR Journals
In this paper, a numerical method for Nth - order fuzzy initial value problems (FIVP) based on
Seikkala derivative of fuzzy process is studied. The fourth order Runge-Kutta method based on Centroidal Mean
(RKCeM4) is used to find the numerical solution and the convergence and stability of the method is proved. This
method is illustrated by solving second and third order FIVPs. The results show that the proposed method suits
well to find the numerical solution of Nth – order FIVPs.
Some Properties of Determinant of Trapezoidal Fuzzy Number MatricesIJMERJOURNAL
ABSTRACT: The fuzzy set theory has been applied in many fields such as management, engineering, matrices and so on. In this paper, some elementary operations on proposed trapezoidal fuzzy numbers (TrFNs) are defined. We also defined some operations on trapezoidal fuzzy matrices (TrFMs). The notion of Determinant of trapezoidal fuzzy matrices are introduced and discussed. Some of their relevant properties have also been verified.
A New Approach for Ranking Shadowed Fuzzy Numbers and its Application IJCSITJournal2
n many decision situations, decision-makers face a kind of complex problems. In these decision-making
problems, different types of fuzzy numbers are defined and, have multiple types of membership functions.
So, we need a standard form to formulate uncertain numbers in the problem. Shadowed fuzzy numbers are
considered granule numbers which approximate different types and different forms of fuzzy numbers. In
this paper, a new ranking approach for shadowed fuzzy numbers is developed using value, ambiguity and
fuzziness for shadowed fuzzy numbers. The new ranking method has been compared with other existing
approaches through numerical examples. Also, the new method is applied to a hybrid multi-attribute
decision making problem in which the evaluations of alternatives are expressed with different types of
uncertain numbers. The comparative study for the results of different examples illustrates the reliability of
the new method.
Sentient Arithmetic and Godel's Incompleteness TheoremsKannan Nambiar
For me, there is only one logic that we rational human beings are able to accept and appreciate, and that is the mathematical logic of ZF theory. But in the last century we found that ZF theory is not in a position to provide all that we want, and went in search of a new mode of thinking and got one which we called meta mathematics. My question is: if we can put the unambiguous logic of ZF theory on paper, why can't we do the same with meta mathematics. This paper is my feeble attempt in that direction.
GDG Cloud Southlake #33: Boule & Rebala: Effective AppSec in SDLC using Deplo...James Anderson
Effective Application Security in Software Delivery lifecycle using Deployment Firewall and DBOM
The modern software delivery process (or the CI/CD process) includes many tools, distributed teams, open-source code, and cloud platforms. Constant focus on speed to release software to market, along with the traditional slow and manual security checks has caused gaps in continuous security as an important piece in the software supply chain. Today organizations feel more susceptible to external and internal cyber threats due to the vast attack surface in their applications supply chain and the lack of end-to-end governance and risk management.
The software team must secure its software delivery process to avoid vulnerability and security breaches. This needs to be achieved with existing tool chains and without extensive rework of the delivery processes. This talk will present strategies and techniques for providing visibility into the true risk of the existing vulnerabilities, preventing the introduction of security issues in the software, resolving vulnerabilities in production environments quickly, and capturing the deployment bill of materials (DBOM).
Speakers:
Bob Boule
Robert Boule is a technology enthusiast with PASSION for technology and making things work along with a knack for helping others understand how things work. He comes with around 20 years of solution engineering experience in application security, software continuous delivery, and SaaS platforms. He is known for his dynamic presentations in CI/CD and application security integrated in software delivery lifecycle.
Gopinath Rebala
Gopinath Rebala is the CTO of OpsMx, where he has overall responsibility for the machine learning and data processing architectures for Secure Software Delivery. Gopi also has a strong connection with our customers, leading design and architecture for strategic implementations. Gopi is a frequent speaker and well-known leader in continuous delivery and integrating security into software delivery.
zkStudyClub - Reef: Fast Succinct Non-Interactive Zero-Knowledge Regex ProofsAlex Pruden
This paper presents Reef, a system for generating publicly verifiable succinct non-interactive zero-knowledge proofs that a committed document matches or does not match a regular expression. We describe applications such as proving the strength of passwords, the provenance of email despite redactions, the validity of oblivious DNS queries, and the existence of mutations in DNA. Reef supports the Perl Compatible Regular Expression syntax, including wildcards, alternation, ranges, capture groups, Kleene star, negations, and lookarounds. Reef introduces a new type of automata, Skipping Alternating Finite Automata (SAFA), that skips irrelevant parts of a document when producing proofs without undermining soundness, and instantiates SAFA with a lookup argument. Our experimental evaluation confirms that Reef can generate proofs for documents with 32M characters; the proofs are small and cheap to verify (under a second).
Paper: https://eprint.iacr.org/2023/1886
Essentials of Automations: The Art of Triggers and Actions in FMESafe Software
In this second installment of our Essentials of Automations webinar series, we’ll explore the landscape of triggers and actions, guiding you through the nuances of authoring and adapting workspaces for seamless automations. Gain an understanding of the full spectrum of triggers and actions available in FME, empowering you to enhance your workspaces for efficient automation.
We’ll kick things off by showcasing the most commonly used event-based triggers, introducing you to various automation workflows like manual triggers, schedules, directory watchers, and more. Plus, see how these elements play out in real scenarios.
Whether you’re tweaking your current setup or building from the ground up, this session will arm you with the tools and insights needed to transform your FME usage into a powerhouse of productivity. Join us to discover effective strategies that simplify complex processes, enhancing your productivity and transforming your data management practices with FME. Let’s turn complexity into clarity and make your workspaces work wonders!
Climate Impact of Software Testing at Nordic Testing DaysKari Kakkonen
My slides at Nordic Testing Days 6.6.2024
Climate impact / sustainability of software testing discussed on the talk. ICT and testing must carry their part of global responsibility to help with the climat warming. We can minimize the carbon footprint but we can also have a carbon handprint, a positive impact on the climate. Quality characteristics can be added with sustainability, and then measured continuously. Test environments can be used less, and in smaller scale and on demand. Test techniques can be used in optimizing or minimizing number of tests. Test automation can be used to speed up testing.
Observability Concepts EVERY Developer Should Know -- DeveloperWeek Europe.pdfPaige Cruz
Monitoring and observability aren’t traditionally found in software curriculums and many of us cobble this knowledge together from whatever vendor or ecosystem we were first introduced to and whatever is a part of your current company’s observability stack.
While the dev and ops silo continues to crumble….many organizations still relegate monitoring & observability as the purview of ops, infra and SRE teams. This is a mistake - achieving a highly observable system requires collaboration up and down the stack.
I, a former op, would like to extend an invitation to all application developers to join the observability party will share these foundational concepts to build on:
GraphSummit Singapore | The Future of Agility: Supercharging Digital Transfor...Neo4j
Leonard Jayamohan, Partner & Generative AI Lead, Deloitte
This keynote will reveal how Deloitte leverages Neo4j’s graph power for groundbreaking digital twin solutions, achieving a staggering 100x performance boost. Discover the essential role knowledge graphs play in successful generative AI implementations. Plus, get an exclusive look at an innovative Neo4j + Generative AI solution Deloitte is developing in-house.
LF Energy Webinar: Electrical Grid Modelling and Simulation Through PowSyBl -...DanBrown980551
Do you want to learn how to model and simulate an electrical network from scratch in under an hour?
Then welcome to this PowSyBl workshop, hosted by Rte, the French Transmission System Operator (TSO)!
During the webinar, you will discover the PowSyBl ecosystem as well as handle and study an electrical network through an interactive Python notebook.
PowSyBl is an open source project hosted by LF Energy, which offers a comprehensive set of features for electrical grid modelling and simulation. Among other advanced features, PowSyBl provides:
- A fully editable and extendable library for grid component modelling;
- Visualization tools to display your network;
- Grid simulation tools, such as power flows, security analyses (with or without remedial actions) and sensitivity analyses;
The framework is mostly written in Java, with a Python binding so that Python developers can access PowSyBl functionalities as well.
What you will learn during the webinar:
- For beginners: discover PowSyBl's functionalities through a quick general presentation and the notebook, without needing any expert coding skills;
- For advanced developers: master the skills to efficiently apply PowSyBl functionalities to your real-world scenarios.
Generative AI Deep Dive: Advancing from Proof of Concept to ProductionAggregage
Join Maher Hanafi, VP of Engineering at Betterworks, in this new session where he'll share a practical framework to transform Gen AI prototypes into impactful products! He'll delve into the complexities of data collection and management, model selection and optimization, and ensuring security, scalability, and responsible use.
Securing your Kubernetes cluster_ a step-by-step guide to success !KatiaHIMEUR1
Today, after several years of existence, an extremely active community and an ultra-dynamic ecosystem, Kubernetes has established itself as the de facto standard in container orchestration. Thanks to a wide range of managed services, it has never been so easy to set up a ready-to-use Kubernetes cluster.
However, this ease of use means that the subject of security in Kubernetes is often left for later, or even neglected. This exposes companies to significant risks.
In this talk, I'll show you step-by-step how to secure your Kubernetes cluster for greater peace of mind and reliability.
GraphSummit Singapore | The Art of the Possible with Graph - Q2 2024Neo4j
Neha Bajwa, Vice President of Product Marketing, Neo4j
Join us as we explore breakthrough innovations enabled by interconnected data and AI. Discover firsthand how organizations use relationships in data to uncover contextual insights and solve our most pressing challenges – from optimizing supply chains, detecting fraud, and improving customer experiences to accelerating drug discoveries.
GridMate - End to end testing is a critical piece to ensure quality and avoid...ThomasParaiso2
End to end testing is a critical piece to ensure quality and avoid regressions. In this session, we share our journey building an E2E testing pipeline for GridMate components (LWC and Aura) using Cypress, JSForce, FakerJS…
DevOps and Testing slides at DASA ConnectKari Kakkonen
My and Rik Marselis slides at 30.5.2024 DASA Connect conference. We discuss about what is testing, then what is agile testing and finally what is Testing in DevOps. Finally we had lovely workshop with the participants trying to find out different ways to think about quality and testing in different parts of the DevOps infinity loop.
A new computational methodology to find appropriate
1. Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.2, No.1, 2011
A New Computational Methodology to Find Appropriate
Solutions of Fuzzy Equations
Shapla Shirin * Goutam Kumar Saha
Department of Mathematics, University of Dhaka, PO box 1000, Dhaka, Bangladesh
* E-mail of the corresponding author: shapla@univdhaka.edu
Abstract
In this paper, a new computational methodology to get an appropriate solution of a fuzzy equation of the
form , where , are known continuous triangular fuzzy numbers and is an unknown fuzzy
number, are presented. In support of that some propositions with proofs and theorems are presented. A
different approach of the definition of ‘positive fuzzy number’ and ‘negative fuzzy number’ have been
focused. Also, the concept of ‘half-positive and half-negative fuzzy number’ has been introduced. The
solution of the fuzzy equation can be ‘positive fuzzy number’ or ‘negative fuzzy number’ or ‘half positive
or half negative fuzzy number’ which is computed by using the methodology focused in the proposed
propositions.
Keywords: Fuzzy number, Fuzzy equation, Positive fuzzy number, Negative fuzzy number, half positive
and half negative fuzzy number, of a fuzzy number.
1. Introduction
In most cases in our life, the data obtained for decision making are only approximately known. The concept
of fuzzy set theory to meet those problems have been introduced [11]. The fuzziness of a property lies in
the lack of well defined boundaries [i.e., ill-defined boundaries] of the set of objects, to which this property
applies. Therefore, the membership grade is essential to define the fuzzy set theory.
The notion of fuzzy numbers has been introduced from the idea of real numbers [4] as a fuzzy subset of the
real line. There are arithmetic operations, which are similar to those of the set of real numbers, such that +,
–, . , /, on fuzzy numbers [6 8]. Fuzzy numbers allow us to make the mathematical model of linguistic
variable or fuzzy environment, and are also used to describe the data with vagueness and imprecision.
The definition of ‘positive fuzzy number’ and ‘negative fuzzy number’ have been introduced [5, 9]. The
shortcoming of the definitions [5] has been focused [10] and the concept of ‘nonnegative fuzzy numbers’
has been introduced [10] as well. None has introduced the notion of ‘half-positive and half-negative fuzzy
number’. In this paper, a different approach of the definitions of ‘positive fuzzy number’ and ‘negative
fuzzy number’ have been focused; and a new notion of ‘half-positive and half-negative fuzzy number’ has
been introduced. There are another notion in the fuzzy set theory is the concept of the solution of fuzzy
equations [8] of the form and , which have been discussed in [1 3, 8]. It is easy to
solve the fuzzy equation of the form , where , are known fuzzy numbers and is an
unknown fuzzy number [8], but there are some limitations to solve the fuzzy equation of the form ,
where is an unknown fuzzy number. Our main objective is to introduce a new computational
methodology to overcome the limitations to get a solution, if it exists, of the fuzzy equation of the form
where and are known continuous triangular fuzzy numbers. Here it is noted that the core of
a known continuous triangular fuzzy number is a singleton set.
2. Preliminaries
In this section, some definitions [1 11] have been reviewed which are important to us for representing
1
2. Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.2, No.1, 2011
our main objective in the later sections. Let be the set of all fuzzy numbers and means that is a
fuzzy number whose membership function is .
2.1 Definition : The of a fuzzy set is denoted by and is defined by
, .
2.2 Definition : The strong of a fuzzy set is denoted by and is
defined by , .
2.3 Definition : The support of a fuzzy set is denoted by and is defined by
.
2.4 Definition : A fuzzy set is normal if there exist , s.t .
2.5 Definition : A fuzzy number is a fuzzy set, whose membership function is denoted by ,
which satisfies the conditions as under :
(a) is normal fuzzy set;
(b) is a closed interval ;
(c) support of , i.e., is a bounded set in the classical sense.
That is, a fuzzy number satisfies the condition of normality and convexity.
2.6 Definition [5] : A fuzzy number is called positive (negative), denoted by ( ), if its
membership function satisfies .
2.7 Definition [10] : A fuzzy number is called positive, denoted by , if its membership function
satisfies .
2.8 Definition [10] : A fuzzy number is called nonnegative, denoted by , if its membership
function satisfies .
3. Existence of a Solution of a Fuzzy Equation
Consider the fuzzy equation , where , are known fuzzy numbers and is an unknown fuzzy
number. If , and are
of , and , respectively, then the fuzzy equation has a solution if and only if the
equation
(A)
has a solution and satisfies the following conditions [8] :
Condition 1: . (B)
Condition 2 : If then (C)
4. New Proposed Definitions
Here we have introduced some definitions which will help us to solve the fuzzy equation of the form
, where , are known continuous fuzzy numbers and is an unknown fuzzy number. The
definitions are as follows and will be used in the next section.
4.1 Definition : A triangular fuzzy number is called negative, denoted by , if
there exist where ), , such that
, and .
4.2 Example : is a negative fuzzy number which is defined by
2
3. Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.2, No.1, 2011
( )= ,
where , and
.
4.3 Definition : A triangular fuzzy number is called positive, denoted by , if there
exist where ), , such that
, and .
4.4 Example : is a positive fuzzy number which is defined by
( )= ,
where , and
.
4.5 Definition [Half positive and half negative] : A triangular fuzzy number is called ‘half-positive and
half-negative’, denoted by , if there exist where
), , such that
, and .
4.6 Example : is a half-positive and half-negative fuzzy number which is defined by
( )=
where and .
Figure 1 represents the fuzzy numbers which are given in examples 4.2, 4.4, and 4.6.
5. Problems, Discussions, and Results
In this section, we have proposed some propositions with their proofs, which will help us to solve the fuzzy
equation without any difficulties and within a reasonable time. We have also established related
theorems. In support of that some problems and their solutions have also been investigated.
5.1 Proposition : If are known fuzzy numbers and is any unknown fuzzy number, then the
solution of the fuzzy equation is a positive fuzzy number.
Proof : Given that and the fuzzy equation . Then, and
, where and .
Now, via representation, we have, = .
Then, and such that , and
. That is, ( )] ( )]
is true if each is positive. Hence, the solution of the fuzzy equation
is a ‘positive fuzzy number’.
5.2 Problem : Suppose that and are two triangular negative continuous fuzzy numbers, where
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= ; = .
Solve the fuzzy equation for the unknown fuzzy number
Solution : Given the fuzzy equation ,
where and are known negative fuzzy numbers and the unknown fuzzy number. Here,
and . Now, we solve the following equation for
the unknown ,
i.e., (2)
Since , , we choose three cases for unknown fuzzy number :
.
Case (i) : Consider . Then, , where .
Therefore,
.
So, . Since satisfies (A), (B)
and (C) , it is a solution of equation (2) and hence, is the solution of the fuzzy equation (1)
whose membership function is as follows :
.
The graphical representation of , and ������ are shown in Figure 2 where the graph of is shown by dashed
lines.
Case (ii) : Consider . Then, , where .
So, , and it does not satisfy the
equation (A) for . Therefore, is not a solution of (1).
Case (iii) : Suppose that Then, ,
where . Now, we have
, and it does not satisfy the
equation (A) for . So, for the case , is not a solution of (1).
5.3 Proposition : If are known fuzzy numbers and is any unknown fuzzy number, then the
solution of the fuzzy equation is a positive fuzzy number.
Proof : Given that and the fuzzy equation . Then, and
, where and .
Now, via representation, we have . Then, and
such that , and .
That is, is true only if each
is positive. Hence, the solution of the fuzzy equation is a ‘positive
fuzzy number’.
5.4 Problem : Suppose that are two triangular fuzzy numbers, where
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; .
Show that the solution of the fuzzy equation is a positive fuzzy number .
Solution : Given the fuzzy equation .
where and are known positive fuzzy numbers and the unknown fuzzy number. Here,
and . Now, we solve the following equation for the
unknown ,
.
Since , , we choose three cases for unknown fuzzy number :
.
Case (i) : Suppose that . Then, . Since satisfies
(A), (B) and (C) , it is a solution of equation (2) and hence, ������ is the solution of the fuzzy equation
(1) whose membership function is as follows :
.
The graphical representation of , and ������ are shown in Figure 3 where the graph of is shown by dashed
lines.
Case (ii) : Suppose that . Then, . Here, satisfies
the conditions (B) and (C). and does not satisfy the equation (A) for . So, for the case , is not
a solution of (1).
Case (iii) : Suppose that Then, . Here,
satisfies the conditions (B) and (C), but does not satisfy the equation (A) for . So, for the case
, is not a solution of (1).
5.5 Proposition : If and are known fuzzy numbers and is any unknown fuzzy number, then
the solution of the fuzzy equation is a negative fuzzy number.
Proof : Given that , and the fuzzy equation . Then,
and ,
where and . Now, via cut
representation, we have . Then, and such that
, either (i) and ;
or (ii) and .
That is, is verified only if
each is negative. Hence, the solution of the fuzzy equation is a
‘negative fuzzy number’.
5.6 Problem : Suppose that and > 0 are two triangular fuzzy numbers, where
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= ; = .
Then, show that the solution of the fuzzy equation is a negative fuzzy number.
Solution : Given that and the fuzzy equation . (1)
That is, (2)
We have = [ ] and = [ ]. Since , so we
choose three cases for unknown fuzzy number : .
Case (i) : Suppose that . Then, . Here,
satisfies the conditions (B) and (C), but does not satisfy the equation (A) for . So, for the case
, is not a solution of (1).
Case (ii) : Suppose that . Then, . Here,
satisfies the conditions (A), (B) and (C) .
Therefore, is a solution of (2) and hence
is the solution of the fuzzy equation . The membership function is as follows :
.
The graphical representation of , and ������ are shown in Figure 4 where the graph of is shown by dashed
lines.
Case (iii) : Suppose that . Then, .
Here, does not satisfy the equation (A) for . So, for the case , is not a solution of
(2).
5.7 Proposition : If and , a half positive and half negative, are known fuzzy numbers and is any
unknown fuzzy number, then the solution of the fuzzy equation is a half positive and half negative
fuzzy number.
Proof : Given that , is a half positive and half negative fuzzy number, and the fuzzy equation
, where is an unknown fuzzy number. Then, = ( )] and
, where and .
Now, via representation, we have .
Then, and such that
, and .
Which implies that and . Therefore, is the
solution of , that is, the
corresponding fuzzy number , which is a ‘half positive and half negative fuzzy number’, is the solution of
.
5.8 Problem : Suppose that and , a half positive and half negative, are two triangular fuzzy
numbers, where
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= ; = .
Prove that the solution of the fuzzy equation is a half positive and half negative fuzzy number.
Solution : Given the fuzzy equation (1)
That is, (2)
Case (i) : Suppose that . Then, . Here,
satisfies the conditions (B) and (C), but does not satisfy the equation (A) for . So, for the case
, is not a solution of (1).
Case (ii) : Suppose that . Then, . Here,
satisfies the conditions (B) and (C), but does not satisfy the equation (A) for . So, for the case
, is not a solution of (1) too.
Case (iii) : Suppose that . Then, . Here,
satisfies the conditions (A), (B) and (C) .
Therefore, is a solution of (2) and hence is a
solution of the fuzzy equation . The membership function is as follows :
.
So, for the case , is the solution of the fuzzy equation . The graphical representation
of , and ������ are shown in Figure 5 where the graph of is shown by dashed lines.
5.9 Proposition : If and , a half positive and half negative fuzzy number, are known fuzzy number
and is any unknown fuzzy number, then than the solution of the fuzzy equation is a half
positive and half negative fuzzy number.
Proof : The proof is similar to Proposition.5.7.
5.10 Problem : Let and , a half positive and half negative be two triangular fuzzy numbers, where
= ; = .
Then, the solution of the fuzzy equation is a ‘half positive and half negative fuzzy number’ ,
where
.
The graphical representation of , and ������ are shown in Figure 6, where the graph of is shown by dashed
lines.
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6. Conclusion
In this paper we have established a new methodology to overcome the discussed shortcomings or limitations
of the method [8] of the solutions of a fuzzy equation of the form , where , are known positive
or negative continuous fuzzy numbers and is an unknown fuzzy number. For this reason, different
approaches of the definitions of ‘positive fuzzy number’ and ‘negative fuzzy number’ have been introduced.
A new notion of ‘half positive and half negative fuzzy number’ has also been innovated. Some propositions
with their proofs and some related problems with their solutions have been discussed. The propositions will
help to assume the sign of unknown fuzzy number of the fuzzy equation for which we will be
able to get a solution of the fuzzy equation easily. After that, some related theorems are presented. There is
none who has discussed these notions yet. Without this notion it is very difficult to solve a fuzzy equation of
the form discussed above.
References
[1] Bhiwani, R. J., & Patre, B. M., (2009), “Solving First Order Fuzzy Equations : A Modal Interval
Approach”, IEEE Computer Society, Conference paper.
[2] Buckley, J. J., & Qu, Y., (1990), “Solving linear and quadratic fuzzy equations”, Fuzzy Sets and
Systems, Vol. 38, pp. 43 – 59.
[3] Buckley, J. J., Eslami, E. & Hayashi, Y. , (1997), “Solving fuzzy equation using neural nets”, Fuzzy
Sets and Systems, Vol. 86, No. 3, pp. 271 – 278.
[4] Dubois, D., & Prade H., (1978), “Operations on Fuzzy Numbers”, Internet. J. Systems Science, 9(6),
pp. 13 626.
[5] Dubois, D., & Prade H., (1980), “Fuzzy sets and systems: Theory and applications”, Academic Press,
New York, p. 40.
[6] Gaichetti, R. E. & Young, R. E., (1997), “A parametric representation of fuzzy numbers and their
arithmetic operators”, Fuzzy Sets and Systems, Vol. 91, No. 2, pp. 185 – 202.
[7] Kaufmann, A., & Gupta, M. M., (1985), “Introduction to Fuzzy Arithmetic Theory and Applications”,
Van Nostrand Reinhold Company Inc., pp. 1 43.
[8] Klir, G. J., & Yuan, B., (1997), “Fuzzy Sets and Fuzzy Logic Theory and Applications”, Prentice-
Hall of India Private Limited, New Delhi, pp. 1 117.
[9] Dehghan, M., Hashemi, B., & Ghattee, M., (2006), “Computational methods for solving fully
fuzzy linear systems, Applied Mathematics and Computation”, 176, pp. 328–343.
[10] Nasseri, H., (2008), “Fuzzy Numbers : Positive and Nonnegative” , International Mathematical
Forum, 3, No. 36, pp. 1777 – 1780.
[11] Zadeh, L. A., (1965), “Fuzzy Sets”, Information and Control, 8(3), pp. 338 353.
Shapla Shirin The author has born on 16th January, 1963, in Bangladesh. She obtained her M.Sc degree in
Pure Mathematics from the University of Dhaka in the year 1984. In 1996 she also received M. S. Degree
(in Fuzzy Set Theory) from La Trobe University, Melbourne, Australia. Her main topic of interest is Fuzzy
Set Theory and its applications. The author is an Associate Professor of Department of Mathematics,
University of Dhaka, Bangladesh. She is a member of Bangladesh Mathematical Society.
Goutam Kumar Saha The author has born on 14th October, 1985, in Bangladesh. He is a student of M.S.
(Applied Mathematics), Department of Mathematics, University of Dhaka, Bangladesh. His area of interest
is Fuzzy Set Theory.
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Membership function Membership function Membership function
1 1 1
0.8 0.8 0.8
0.6 0.6 0.6
0.4 0.4 0.4
0.2 0.2 0.2
x x x
-10 -8 -6 -4 -2 2 -1 1 2 3 4 5 6 -2 2 4 6
Figure 1 : Graphs of fuzzy numbers which are given in examples 4.2, 4.4, and 4.6.
x x
Membership function Membership function x
1 1
0.8 0.8
0.6 0.6
0.4 0.4
0.2 0.2
x
-10 -8 -6 -4 -2 2 x
-1 -0.5 0.5 1
Figure 2 : Graphs of fuzzy numbers , and the solution fuzzy number , respectively.
Membership function x x
1 Membership function x
0.8 1
0.6 0.8
0.4 0.6
0.2 0.4
x 0.2
2 4 6 8 10
x
-1 -0.5 0.5 1 1.5 2
Figure 3 : Graphs of fuzzy numbers , and the solution fuzzy number , respectively.
x Membership function x
1 x Membership function
0.8 1
0.8
0.6
0.6
0.4 0.4
0.2 0.2
x x
-20 -15 -10 -5 5 10 15 -3 -2.5 -2 -1.5 -1 -0.5
Figure 4 : Graphs of fuzzy numbers , and the solution fuzzy number , respectively.
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x x
1 x
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2 0.2
x x
-10 -7.5 -5 -2.5 2.5 5 -1 -0.5 0.5 1
Figure 5 : Graphs of fuzzy numbers , and the solution fuzzy number , respectively.
x
x 1
1
0.8
0.8
0.6 0.6
0.4 0.4
0.2 0.2
x x
-10 -8 -6 -4 -2 2 -0.6 -0.4 -0.2 0.2 0.4 0.6
Figure 6 : Graphs of fuzzy numbers , and the solution fuzzy number , respectively.
The above tables and figures have been discussed to the relevant sections of this paper.
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