COMPUTATION WITH 2-TUPLE LINGUISTIC VARIABLES AND ITS APPLICATION IN MATRIX GAMES

Abstract

Making a decision entails selecting from a set of options based on a preliminary analy sis, which frequently includes human intervention and uncertainty. Besides, grasping the meaning communicated by information in a qualitative setting is necessary before making a further analysis. One of the most challenging issues is to deal with statements, human thoughts preferences, feelings, and so on because of the inherent character of natural lan guage. Over the years, much work has been taken to account for the ambiguity and impre cision of linguistic information by using the theory of fuzzy set and fuzzy linguistic-based approach. Several computational methods have been created to deal with uncertainty, par ticularly when it is not of a probabilistic character. Specifically, in 2000, a new model known as the “2-tuple linguistic representation model” arose, which improved numerous linguistic processes for handling complex decision-making issues. It facilitates a contin uous representation of the linguistic terms, and henceforth, research concerned with the 2-tuple model is profuse and worthwhile considering in deep. The introspection of the distinguished literature in the 2-tuple model enabled us to re alize that some limitations still persist in the existing uncertain 2-tuple models. This mo tivated us to improve the existing uncertain models to make their implementation more flexible and consistent in decision-making processes. Therefore, in our study, we have ad dressed the constraints and challenges associated with the existing 2-tuple model and have worked towards its development to enhance its applicability. Further, we have supported our study by applying the 2-tuple model in the domain of matrix games and decision anal ysis. This has enabled us to contribute to the researchers worldwide who are working in this field and are also looking for exploration. The thesis entitled “Computation with 2-tuple linguistic variable and its application in matrix games” comprises of six chapters followed by the summary and future scope. The bibliography and the list of publications are provided at the end of the thesis. The introductory Chapter 1 presents a short overview of computing with words 2- xiii tuple based linguistic model as well as its elementary application anticipated in distinct decision-making models followed by its extension. Thus, the current chapter creates a background and motivates this thesis’s work. The chapter is based on a review paper, “A systematic review of developments in the 2-tuple linguistic model and its applications in decision analysis,” published in Soft Computing, Springer (2020). The Chapter 2 entitled, “Group operations and properties for 2-tuple linguistic vari ables with its application” establish the basis for a theory of 2-tuple linguistic groups un der the given binary operation in a classical impression. In literature, the concept of fuzzy algebra has been a subject of research for many years and has made significant progress. Nevertheless, the abstract theory of linguistic groups is pristine and yet to be explored. The use of fuzzy linguistic concepts to represent practical situations with qualitative data has shown to be a powerful approach. Several computational techniques have been intro duced to alleviate the computation between linguistic terms. Among these computational techniques, the proposal of a 2-tuple linguistic model is a useful tool by easing out the computations and avoiding information loss when applied in some practical decision making situations. In the study of linguistic information, the aggregation of 2-tuple lin guistic labels is a crucial problem. Several computing models existing in the literature are well-suited to deal with this problem. However, it is noted that the existing opera tional laws are not satisfying the closure property. Moreover, to the best of knowledge, no theory has been developed to support the concept of linguistic groups. For this reason, the foundation of the theory of 2-tuple linguistic groups under a crisp binary operation is a milestone in this direction, overcoming the constraints of the existing operational laws which operate without information loss. The chapter has given a formal methodology to claim that the 2-tuple linguistic term set forms an algebraic structure group. Further, a similarity relation between the linguistic groups is obtained, and some properties of the operational laws, group isomorphic and homomorphic relation, have been discussed in detail. Lastly, the physical meaning of the abstract concept so developed has been showcased in bipolar graphs and matrix games. The chapter is based on a research paper entitled, “Group operations and isomorphic relation with the 2-tuple linguistic variables”, published in Soft Computing, springer 24, 18287–18300 (2020) and “Group isomorphic properties with some novel operational laws for 2-tuple linguistic variables and its appli cation in linguistic matrix games” Communicated in IEEE Transactions on Systems, Man, and Cybernetics: Systems. A qualitative decision making problems with linguistic term set where all plausible lin xiv guistic descriptors provided by experts have symmetric and uniform distribution has been investigated by several scholars. Obviously, it might not be suitable in practical life de cision problems since the experts may prefer linguistic labels distributed non-uniformly and non-symmetrically. Numerous studies have been developed on theoretical and prac tical applications to handle an unbalanced linguistic context. However, the current unbal anced linguistic computational models are complex and computationally more expensive. Therefore, in Chapter 3 entitled, “Methodology for unbalanced linguistic terms” we pro pose a newly constructed methodology to handle a set of unbalanced linguistic terms and further develop a novel 2-tuple linguistic technique for the unbalanced linguistic set. The new 2-tuple unbalanced linguistic model is computationally less complicated and can avoid information loss. Finally, numerical illustrations present the concrete steps of the developed approach and manifest the practicality and flexibility of this model by eluci dating a comparative analysis with existing models. The chapter is based on a research paper titled, “A New 2-Tuple Linguistic Approach for Unbalanced Linguistic Term Sets”, published in IEEE Transactions on Fuzzy Systems 29 (8) 2158–2168 (2021). Chapter 4 entitled, “Matrix games with probabilistic multiplicative unbalanced linguis tic information” proposes a novel concept of the probabilistic multiplicative unbalanced linguistic term set considering the probabilities as well as non-uniformity of distinct lin guistic labels. Further, based on the proposed concept a unified mechanism to solve a two-person linguistic matrix game having probabilistic multiplicative unbalanced linguis tic information is suggested. The proposed approach can be perceived as a convenient technique for multiple criteria decision-making (MCDM) problems. Numerical illustra tions are presented to discuss the significance of the proposed methodology. The chapter is based on a research paper titled, “Probabilistic multiplicative unbalanced linguistic term set and its application in matrix games”, communicated in International journal of ma chine learning and cybernetics, Springer. In Chapter 5 entitled, “Matrix games with interval-valued 2-tuple linguistic informa tion” a 2-player non-cooperative zero-sum interval-valued 2-tuple fuzzy linguistic (IVTFL) matrix game is proposed, and interval-valued linguistic linear programming (IVLLP) methodology is suggested to solve such class of games. A hypothetical example is used to demonstrate the suggested method’s applicability in the practical world. The chapter is based on a research paper titled, “Methodology for Interval-Valued Matrix Games with 2-Tuple Fuzzy Linguistic Information”, published in In: Sergeyev Y., Kvasov D. (eds) Numerical Computations: Theory and Algorithms. NUMTA 2019. Lecture Notes in xv Computer Science, Springer, Cham. , 11974, (2020). https://doi.org/10.1007/978-3-030- 40616-5_12. Chapter 6 entitled, “Interval norm approach for solving two player zero sum matrix games with interval payoffs” present a new approach that gives a unique outlook for solv ing a two-player zero-sum interval-valued matrix game (ZSIMG) based on the interval matrix norm framework. The methodology presented in this chapter helps obtain an ap proximated interval game value for the corresponding ZSIMG without undergoing the existing process of solving traditional interval linear mathematical models. The chapter is based on the research paper titled, “Interval norm approach for solving two-player zero sum matrix games with interval payoffs” Submitted in Computational optimization and application, Springer. After chapter 6, we present the summary of the research work carried out in this thesis. In addition, the future scope of the thesis has been discussed briefly.