Bipolar Fuzzy Sets In Switchboard Automata And Optimisation Problems .

CHAPTER 1INTRODUCTION1.1Background and motivation of researchHANIMengineering, management and decision-making (Zimmermann, 2001), classification,ANcontrol, forecasting and function approximation (Dubois & Prade,1980; Wang &UTU et al., 2004, 2005).Mendel, 1992; Su & Stepanenko, 1994; Castro, 1995;KWangNUTFuzzy sets are the type of a set in set theory thatis imprecise and has no boundariesNAfuzzy set theory could be considered as aA(Klir & Yuan, 1995). Subsequently,KATgeneralisation of eitherclassical set theory or of a classical dual logic (Zimmermann,SUPR1999) sincetheE fuzzy set only deals with the degree of memberships or in otherPwords, it only allows elements to be partially in a set, in which crisp sets could notIn 1965, Zadeh proposed the fuzzy set theory (Zadeh, 1965). The idea of fuzzy setsATTPseemed to receive much attention from the researchers in a variety of fields such asaddress and handle. Since then, fuzzy sets have become an effective tool in order tomanage uncertainties and vagueness for a better understanding to the real-lifeproblems.In fuzzy set theory, the membership of an element lies between 0 and 1.However, in reality, fuzzy set may not forever be true since there may be somehesitation due to the degree of non-membership degree of an element in a fuzzy set isequal to 1 minus the membership degree. With this intention, fuzzy set theory hasbeen extended to type-2 fuzzy sets (Zadeh, 1975), intuitionistic fuzzy sets(Atanassov, 1986), fuzzy multisets (Yager, 1996), neutroshopic sets (Smandarache,1999), nonstationary fuzzy sets (Ozen et al., 2004), hesitant fuzzy sets (Torra, 2010),

2Pythagorean fuzzy sets (Yager, 2014) and bipolar fuzzy sets (Zhang, 1994).As a matter of fact, bipolarity tends to occur in our real life. Due to thisproblem, Zhang (1994) introduced bipolar fuzzy sets as a generalisation of fuzzy setsin the interval of [-1, 1]. Subsequently, Reza et al. (2018) proposed triangular bipolarfuzzy number (TBFN) to overcome the problem that cannot be handled by triangularintuitionistic fuzzy number (TIFN) proposed by Burillo et al. (1994). In a bipolarfuzzy set, the membership degree 0 of an element means that the element isirrelevant to the corresponding property. The membership degree (0, 1] of anelement somewhat satisfies the property and the membership degree [-1, 0) of anelement indicates that the element somewhat satisfies the implicit counter-property.The positive information represents what is granted to be possible, while negativeinformation is considered to be impossible or forbidden, or surely false (Jun &Kavikumar, 2011; Kavikumar et al., 2012; Benferhat et al., 2004; Dubois & Prade,2004; Isabelle, 2009)HANIfuzzy sets theory. By using triangular bipolar fuzzy number (TBFN) (Reza etMal.,AN2018), a unified approach to polarity and fuzziness could be formalisedindecisionUTU composition,analysis, bipolar clustering and coordination, bipolar fuzzyKaggregation,NU provides bipolar cognitivelinguistic description and mathematical computation,TNAmodelling and multivalent decisionanalysis, captured die bipolar or doubled sidedAKand side effect) nature of human perception andAT(negative and positive,oreffectSUPcognition RE(Kavikumar et al., 2012). For this reason, by using bipolar relations, thePcommon interests and the conflicts between two countries can be naturallyATTPIn particular, triangular fuzzy number is one of the extensions of bipolarcharacterised as bipolar fuzzy, which has both positive and negative poles (Zhang,1999). This is because the logical values in the two classical logical models, whichare Boolean logic and fuzzy logic, are unipolar models in nature and they lie in thepositive interval [0, 1] (Zhang, 1998).Notably, optimisation problem is an intrinsic part of life and human activity.By using optimisation method, the complex systems could be simplified to allow afuller exploitation of the advantages inherent to complex systems. Optimisationproblem works better than traditional “guess-and-check” methods and also it canreduce the cost of building and testing which is relatively less expensive. Intensiveresearches have been made in various fields such as project scheduled and cost

3performance (Ghanbari et al., 2017), business (Mazlum & Güneri, 2017), projectcash flow (Mohagheghi et al., 2016), decision makers’ risk attitude (Oladeinde &Oladeinde, 2014) and other optimisation problems as stated in Chapter 2 in Section2.3.1 and Section 2.3.2, by using triangular fuzzy number and triangular intuitionisticfuzzy number.However, the problems still arise since the bipolarity is not considered toexist in real life. As a consequence, data become insufficient and indeterministic.Under those circumstances, TBFN could become a good model to real worldapplications in the bipolarity environment. In this research, the emphasis is made onthe optimisation problems so that it could enhance the idea of bipolarity in the realfield that is limited to critical path problem and reliability system of an automobile.Since the range of TBFN lies in the interval of [-1, 1], the problems can bedetermined successfully by using its positive membership value (acceptance area)and non-membership values (rejection area) of triangular bipolar fuzzy numbers.HANprovides mechanisms for the formulation and solution of general problems that can IMAbe applied to real-world problems in the future. There are a variety ofNconventionalUTautomata, such as Deterministic Finite State Automata (DFA),Non-DeterministicUKNFinite State Automata (NFA), Probabilistic (stochastic)U Automata (PA) and FuzzyTNyears, DFA have been applied in manyFinite State Automata (FFSA). OvertheAAKapplications (Doostfatemeh& Kremer, 2005). Therefore, the automata theory is ofATSscientists and also mathematicians.Ugreat use to engineers,PRPEHowever, there are still lacking in literature studies the automata theory inATTPOn the other hand, automata theory is the general system theory, whichbipolar environment. For that reason, the properties of bipolar fuzzy finite stateautomata (BFFSA) and transformation semigroups inspired by Jun & Kavikumar(2011) could be discussed. Meanwhile, due to the fact that mathematical ideas ofBFFSA are still in the early stage, the algebraic properties and their variousapplications in the real world phenomenon seem to be disputing. Thus, byincorporating the idea of bipolar set theory and switchboard automata, the propertiesof BGFSA could be studied and its topological properties could also be discussed. Itis hoped that BFFSA will become well known in numerous researches as the intervalof bipolar fuzzy sets have been extended from -1 to 1.

41.2Problem statementsIn many real-life problems, some of the information around the behaviour of thesystems encounters uncertainties and vagueness. Sometimes the evaluation of nonmembership values do not fulfill our satisfaction since it is indeterministic in natureand fuzzy set theory is not suited to deal with such problems. In this research, theelongation of the bipolar fuzzy set, which is known as triangular bipolar fuzzynumber is applied to solve the critical path problem for project network and is verywell known in operational research. Since bipolar fuzzy number is a powerful tool tohandle the satisfaction degree to counter-property, it can overcome the problems oftriangular intuitionistic fuzz number. In probability assumption, the system is fuzzilycharacterised in the context of probability measures. Since the data are imprecise andinaccurate, the prediction becomes very difficult to be adapted in the systems. Therejection area cannot simply be defined by using fuzzy sets and the binary sets. InHANsystem of an automobile. The idea that lies behind this explanation is connected with IMAaboutbipolar information, which is positive information and negative informationNUTthe given set. Positive information represents what is concededtobepossible, whileUKthe negative information represents what is consideredUN to be impossible. Hence,TNthrough the mathematical calculation,the rejection evidence (negative area) andAAKacceptable evidence (positivearea) of the reliability system of an automobile couldATSUbe configuredPsuccessfully.RE applications especially in mathematics and sciences face problemsPManyATTPthis research, the triangular bipolar fuzzy number is applied to solve the reliabilitybecause most of the information is not dealt with bipolarity. Since the bipolar fuzzysets are the extension of fuzzy sets and the interval are enlarged to [-1, 1], bipolarinformation is crucial for many applications and domains. In this research, theconcept of decomposition theorem of fuzzy automata and transformationssemigroups in bipolar environment are studied inspired by Jun & Kavikumar (2011).Next, the study of algebraic automata is extended as a truth structure in bipolarsetting.A switchboard automaton is a special kind of finite automata. Thecharacterisation of algebraic structures and its classification of finite state automatawith switchboard, that described the connection of switching and commutative were

5established by Sato & Kuroki (2002). Subsequently, Kavikumar et al., (2012)introduced bipolar fuzzy finite switchboard state automata (BFFSSA) andinvestigated some of algebraic properties. However, the algebraic approach is stilllacking. Hence, it is necessary to study the algebraic properties of the general fuzzyswitchboard automata in bipolar fuzzy environment based on the idea ofDoostfatemeh & Kermer (2005). Moreover, on that point there are some possibilitiesof topological concepts that are available for bipolar general fuzzy switchboardautomata.1.2Research objectivesThe objectives of this research are as follows:(i)To investigate the critical path problem and reliability system of anautomobile by using the triangular fuzzy number in the bipolar fuzzy context.HANItransformation semigroup in bipolar fuzzy finite state automata.MA(iii) To investigate the algebraic and topological properties of bipolargeneralNUTfuzzy switchboard automata.KUNUTN1.3Scope of the researchAAKATSUBipolar fuzzyPsetsare the yardstick for this research. Triangular bipolar fuzzyREP(TBFN), which is the extension of bipolar fuzzy sets theory limited to benumber(ii)ATTPTo establish decomposition theorem and introduce the properties ofapplied in optimisation problems which cover critical path problems and reliabilitysystem of an automobile as motivation studies in bipolarity environment. The scopeof the research is limited to the decomposition theorem and transformationsemigroups of bipolar fuzzy finite state automata. Meanwhile, in understanding thebipolar general switchboard automata, the properties of switching and commutative,strong subsystems and its topological properties are studied.1.4Significance of studySome of the significances of study that can be referred to this research are given asfollows:

61.The investigation is expected to be useful for the development of bipolarfuzzy number and automata theory in real life problems.2.Critical path problems could be determined by using triangular bipolar fuzzynumbers successfully.3.In a reliability system of an automobile, the triangular bipolar fuzzy numberscould be applied efficiently.4.The new concept of the idea of mathematics, which is known as bipolargeneral fuzzy switchboard automata could be seen through its algebraicproperties and topological properties applied in computer and sciencedisciplines. Therefore, the research is of some significance and worthy ofeffort.5.The concept of switching and commutative in bipolar general switchboardautomata allows the flow of information in the system to be more efficient inbipolar environment. Thus, it has important use in many fields such asHANIMA1.5Organisation of thesisNUTUKNThe thesis consists of eight chapters including Uthe Introduction chapter. The otherTNfollows.remaining seven chapters are organisedasAAK review on the development of automata theory isIn Chapter 2, theliteratureATSSome applications in the real-life situations that relate to theUpresented andPdiscussed.REPbipolar fuzzy sets are reviewed. The literature of optimisation problems that areATTPartificial intelligence, optimal control and also production theory.focused on critical path problem and a reliability system of an automobile are alsodiscussed.In Chapter 3, the discussion on the critical path problem by using thetriangular bipolar fuzzy numbers (TBFN) is introduced. In addition, the algorithm isalso proposed and the illustrative example for a project network is presented forbetter understanding in order to find the critical path problem in terms of a bipolarenvironment by using α-cut technique and Euclidean ranking technique for orderingthe critical path problem inspired by Elizabeth & Sujatha (2015). Throughout thischapter, the new term so-called bipolar fuzzy critical path (BFCP) is introduced.In Chapter 4, the triangular bipolar fuzzy number (TBFN) is further used tosolve the reliability system of an automobile and could be represented as a system

7and the cause and sub-cause of the system that could be represented as subsystems.Some of the properties of triangular bipolar fuzzy numbers in this optimisationproblem are discussed. The series, parallel, series-parallel and parallel-series systemof the reliability circuit, which adhered approximation triangular bipolar fuzzynumber, is then focused. Theorems and calculations of the fault-tree to start anautomobile that were proposed by Shaw & Roy (2012) and Mahapatra & Roy (2013)are discussed. An illustrative example is given in this chapter. As a result, thereliability system of an automobile can be modelled by using triangular bipolar fuzzynumber in order to find the rejection area (negative area) and the acceptance(positive area) through the given illustrative example.In Chapter 5, the decomposition theorem and transformation semigroups ofbipolar fuzzy finite state automata are discussed. Some of the algebraic properties areinvestigated. The theorems and proving are also discussed. In this chapter, thedecomposition property can be extended to bipolar submachines since every fuzzyHANsemigroups of bipolar fuzzy finite state automata are introduced by considering the IMAstate membership as bipolar fuzzy sets. Inspired by Jun & Kavikumar(2011), theNUbipolar fuzzyTconcepts of decomposition of bipolar fuzzy finite state automataandUKNtransformation semigroups have been generalisedUas a truth structure of the transitionTNthe interval [-1, 1].in bipolar studies of algebraic automatainAAKAIn Chapter 6, theidea of bipolar general fuzzy switchboard automata isTSUdiscussed.RForPthepurpose of the study of algebraic automata, the concept of productEand Pcovering, which was inspired by Sato & Kuroki (2002), Horry (2016),ATTPfinite state automaton can be decomposed to primary submachines. TransformationKavikumar et al., (2012) and Doostfatemeh & Kermer (2005) is presented. In orderto show whether the bipolar general fuzzy state automata are switching andcommutative, the examples are presented for better understanding. In addition, analgorithm for constructing bipolar general fuzzy switchboard automata is presented.The properties of BGFSA in terms of switching and commutative are studied.Subsequently, the notion of asynchronous bipolar general fuzzy switchboardautomata is introduced and its onto-switching homomorphic image is studied.In Chapter 7, the discussion on the subsystem of bipolar general fuzzyswitchboard automata is presented. Next, the concept of switchboard subsystem,strong switchboard subsystem and homomorphism of bipolar general fuzzyswitchboard automata is introduced, and the idea of a switchboard is incorporated in

8the bipolar general fuzzy switchboard automata is initiated. In this chapter, theproperties of switchboard subsystem are also discussed in order to show that everyswitchboard subsystem is a strong switchboard subsystem. Finally, the concept oftopology on bipolar general fuzzy switchboard automata in terms of thesecharacterisations is formulated, as an example of its application.Finally, Chapter 8 summarises the findings presented in the previous chaptersand some conclusions are drawn from them. The chapter is ended with somerecommendations for the future research. Summary of the above outline or map ofthe thesis is presented in Figure 1.1.HANIMANUTKUNUTNAAKATSUPPERATTP

9BIPOLAR FUZZY SETS IN SWITCHBOARDAUTOMATA AND OPTIMISATION PROBLEMSCHAPTER 1: IntroductionCHAPTER 2: Literature ReviewCHAPTER 3: Triangular Bipolar FuzzyNumber in a Critical Path ProblemOptimisationProblemsCHAPTER 4: Triangular Bipolar FuzzyNumber in a Reliability System of anAutomobileHANIMANUTAutomataCHAPTER 6: Bipolar General FuzzyKUNTheorySwitchboard AutomataUTNAAK 7: Subsystem of BipolarCHAPTERATSGeneral Fuzzy Switchboard AutomataUPPERCHAPTER 8: Conclusion and FutureCHAPTER 5: Decomposition Theoremof Bipolar Fuzzy Finite State Automataand Transformation SemigroupsATTPResearchFigure 1.1: Summary of the thesis

CHAPTER 2LITERATURE REVIEW2.1IntroductionHANIMfuzzy set theory is considered as the interest and the main theme of investigationANwithin this domain. The literatures that are reviewed in thisTsectionU show therelationship of the works and ideas of mathematics byKtheUprevious researchers inNUT work in the progress of theorder to accommodate justification to the consequentNAAresearch process. The discussionhelps to understand the nature of mathematicalKATessence and its relationto the research work in this report.SU definitions of bipolar fuzzy sets, applications in the real lifePRInthischapter,EPand some useful results are given in this preliminaries section, namely Section 2.2.Due to uncertainties and vagueness encountered in some problems that are modelledATTPby the real-life problems in the optimisation problems and automata theory, bipolarSubsequently, some relevant concepts and results are discussed are also discoveredin the same section. Next, the development automata theory is revised in Section 2.3.After that, the optimisation problems are reviewed on critical path problems andreliability systems by the previous researchers as motivation of study in bipolarenvironment are discussed in Section 2.4.1 and Section 2.4.2 respectively.

112.2PreliminariesIn this section, definition of bipolar fuzzy sets and its applications to the real-lifeproblems are discussed in order to give a good portrayal to the flow of the thesis.Some of the main concepts and results that are useful for this research are presented.Motivating examples of real-life problems of bipolar fuzzy setsExample 1:Bipolar fuzzy set “young”In Figure 2.1, a bipolar fuzzy set “young” is shown. It is manifested that 50 is anirrelevant age to the p

properties of bipolar general fuzzy switchboard automata are discussed in term of switching and commutative by proving the theorems that are related into these concepts. . 2.3 Automata theory 18 2.4 Optimisation problems 23 2.4.1 Critical path problem 23 . Deterministic finite automata FSM - Finite state machine FSA - Finite state automata .