A novel decision-making method using R-Norm concept and VIKOR approach under picture fuzzy environment
Introduction
Fuzzy set (FS) theory introduced by Zadeh (1965) has been proved to be an effective tool in measuring the uncertainty associated with the vague terms like ‘much more’, ‘slightly’ etc. This property of FSs helps the decision makers in Multi-Criteria Decision-Making (MCDM) problems to express their preferences of relevant alternatives in view of certain prefixed criteria. The virtue of FSs in dealing with vagueness involved in decision making problems inspired the researchers to make the concept more practical for real world problems. Introduction of Type-2 FSs by Zadeh (1975), interval-valued FSs by Zadeh (1975), Grattan-Guiness (1975), Jahn (1975), rough FSs by Dubois and Parde (1990), probability hesitant fuzzy sets by Zhu and Xu (2018), Singh and Lalotra (2019), neutrosophic set by Smarandache (2006) etc. indicate the efforts taken in the direction. Amongst various extensions of FSs, the Intuitionistic Fuzzy Sets (IFSs) proposed by Atanassov Atanassov (1986) gained much popularity with authors. In addition to the existing membership degree (μ) and non-membership degree (ν), Attansov Atanassov (1986) added one more component in the structure of FSs called ‘Intuitionistic Index’ or ‘Hesitancy Degree’ (π) satisfying . This can be better understood through an example on voting (Atanassov, 1986, Atanassov, 1999). In an election, some people vote for and some people vote against the government. The people who support the government constitute the membership degree and who do not vote for the government denote the non-membership degree. But in practical, there is a section of people which remains in dilemma to vote for whom. This ‘dilemma’ factor was introduced by Atanassov, 1986, Atanassov, 1999 as ‘hesitancy index’ in the existing structure of FSs. The addition of third component, that is, intuitionistic index made the FSs more realistic for practical situations. Further, the introduction of intuitionistic fuzzy entropy by Burillo and Bustince (2001) caught the attention of researchers world wide. This development caused the authors to propose the information measures based on IFSs from their viewpoint and applied them in distinct fields like image processing, pattern recognition, decision making problems etc. Joshi (2019); Joshi, Kumar, 2017a, Joshi, Kumar, 2018a, Joshi, Kumar, 2018b, Joshi, Kumar, 2018c, Joshi, Kumar, 2018e; Singh and Sharma (2019); Vlachos and Sergiadis (2007); Xu, Yager, 2006, Xu, Yager, 2008; Yager, 1979, Yager, 1988, Yager, 2004; Zeng, Yu, Yu, Chen, and Wu (2009); Zhao and Xu (2016). However, IFSs introduced by Atanasov Atanassov (1986) can handle the problems based on intuitionism in a better way than FSs but in practice, there are situations where the use of IFSs does not give accurate output. This can be better understood again with the help of an example on voting. During voting, the whole population can be divided into four categories: favor, abstain, against and refusal. The incapability of IFSs to handle this fourth factor, that is, ‘refusal’, limits its scope of applications particularly in voting and their use does not produce desired results. The solution to this problem was provided by Cuong and Kreinovich (2012) in the form of Picture Fuzzy Sets (PFSs) characterized by membership degree (μ ∈ [0, 1]), neutral membership degree (δ ∈ [0, 1]) and non-membership degree (ν ∈ [0, 1]) with only restriction given by . Moreover, the fourth parameter denoted by represents the degree of refusal. PFSs can express the decision makers’ viewpoints more accurately than that of IFSs including yes, abstain, no and refusal. Recently, several authors applied PFSs in decision making, clustering analysis problems, for example, Zhang, Wang, and Hu (2017) proposed some operational laws and aggregation operators on PFSs, Wei (2016) proposed cross-entropy for PFSs and applied it in decision making, Wang, Zhang, Wang, and Li (2018) introduced picture fuzzy normalized projection based VIKOR method and applied it in risk evaluation for construction project, Wei (2018a) suggested similarity measures for PFSs and Nie, Wang, and Li (2017) proposed a voting method based on 2-tuple linguistic picture preference relation etc. However, many authors have done so much work on PFSs, but, to the best of my knowledge, no research has yet been conducted on picture fuzzy entropy from probabilistic point of view like that of Hung and Yang (2006) for IFSs. This study is a sincere effort in this direction. In this study, a new framework is proposed as a criteria for picture fuzzy entropy based on probabilistic viewpoint. Another notable extension of FS studied by Zadeh (1965) was introduced by Smarandache (2006) in the form of Neutrosophic Set (NS). Extending the idea of FS proposed by Zadeh (1965), Smarandache (2006) incorporated one more component called ‘Indeterminacy ’ in the existing framework of FSs. The new structure thus contained three components, namely, membership degree (T), indeterminacy degree (I) and non-membership degree (F) defined on a non-standard unit interval . The main difference between IFS studied by Atanassov (1986) and NS set introduced by Smarandache (2006) lies with sum of the three constituent components of two structures. In IFS, the sum of three components, that is, membership degree, indeterminacy degree and non-membership degree is exactly one, whereas, in NS set, the sum of three constituent components may be exactly one or may exceed or less than one. Thus, the NS set proposed by Smarandache (2006) is a more general concept than IFS studied by Atanassov (1986). The two sets, that is, IFS and NS set contain only three components called membership degree, indeterminacy degree and non-membership degree but do not contain refusal degree which is an added feature of PFSs proposed by Cuong and Kreinovich (2012). Also, the need of added feature was established by Cuong and Kreinovich (2012) through an example on voting as mentioned earlier. However, both the concepts, that is, IFS and NS set may be extended to PFS, but in this study, we will restrict ourselves only to IFS studied by Atanassov (1986).
Multi-Criteria Decision-Making (MCDM) is a process which includes a set of possible alternatives and a predefined set of criteria. The aim is to find an alternative which best satisfies the given criteria. Scholars from across the world proposed many theories and approaches to solve MCDM problems. For example, Technique for Order Preference by Similarity to Ideal Solutions (TOPSIS) method proposed by Hwang and Yoon (1981), VIKOR (Vlsekriterijumska Optimizacija i Kompromisno Resenje) method introduced by Opricovic (1998), PROMETHEE (Preference Ranking Organization Method for Enrichment Evaluations) introduced by Brans and Mareschel (1984), ELECTRE (elimination et choice translating reality) proposed by Benayoun, Roy, and Sussman (1966), and TODIM (TOmada de Decisao Interativa e Multicrit’erio) studied by Gomes and Lima (1991) etc. Highlighting the drawbacks of TOPSIS, PROMOTHEE and ELECTRE methods, Opricovic and Tzeng (2007) proposed an extended VIKOR method. Among the various MCDM methods reported by many authors in the literature, VIKOR method proposed by Opricovic and Tzeng (2007) gained much popularity with researchers due to its feature of providing a compromise solution. This study will employ the VIKOR method to decide the best alternative.
The weight-age assigned to different criteria plays a decisive role in the solution of a MCDM problem. Only proper assignment of criteria weights leads to the selection of most suitable alternative. Estimating the importance of criteria weights in MCDM problem, Chen and Li (2010b) categorized them into two parts: subjective evaluation and objective evaluation. Subjective evaluation of criteria weights is concerned with the preferences expressed by decision makers. Weighted least square method introduced by Chu, Kalaba, and Spingarn (1979), Analytical Hierarchy Process (AHP) method proposed by Saaty (1980) and Delphi method proposed by Hwang and Lin (1987) are some examples of subjective weights category. In objective evaluation of criteria weights, the criteria weights are determined by solving mathematical programming models. Multi-objective programming method proposed by Choo and Wedley (1985), Principle element analysis suggested by Fan (1996) and entropy method belong to the category of objective evaluation. In objective weights category, the entropy method is one of the most trusted and prevalent methods used for determining the criteria weights. Each method of criteria weight evaluation has its own benefits and drawbacks. However, subjective weights if available are very much beneficial, but it may not be possible every time to obtain reliable subjective weights. This may be due to incomplete information available about alternatives or time pressure or limited expertise about problem domain on the part of decision-maker etc. In such situations, objective evaluation of criteria weights become helpful. In this communication, we will be using the proposed picture fuzzy R-norm entropy to determine the criteria weights. The prime aims of this contribution are: (1). to propose a new axiomatic structure of entropy for PFSs, (2). to propose a new R-norm picture fuzzy information measure, (3). to use the information so obtained in election forecast through opinion polls and investment of funds.
The subject contents of this contribution are managed as follows: The prime aim of introducing the manuscript, source of inspiration and work reported by earlier authors in the field are presented in Section 1. Section 2 contains some basic concepts and definitions related with the topic under study. Section 3 is utilized to propose a new framework for fuzzy entropy for PFSs. In Section 4, a new R-norm picture fuzzy information measure is proposed and validated. Section 5 is devoted to the application of proposed information measure in decision making. Two numerical examples are used to describe the practical utility of proposed measure in Section 6. Finally, the paper is concluded with ‘Concluding Remarks’ in Section 7.
In next Section, we give some definitions and concept necessary to understand the paper.
Section snippets
Preliminaries
Definition 2.1 Let be the finite universe of discourse. Then the FS on X is defined bywhere represents the membership function of and denotes the membership degree of hi ∈ X in .(Fuzzy Set (Zadeh, 1965))
Since FSs introduced by Zadeh (1965) have membership degree and non-membership degree only, that is, ‘yes’ or ‘no’ but not the hesitancy element. Atanassov (1986) extended the idea of FSs to IFSs by incorporating the third component in the
Background
Contrary to fuzzy measures, measures of fuzziness denote the degree of fuzziness of a FS. The entropy of a FS measures the fuzziness of a FS. Referring to the Shannon’s probability theory, Luca and Termini (1972) proposed a set of postulates for fuzzy set entropy given by Definition 3.1 A real function is called a fuzzy entropy if it satisfies the following properties: For all if and only if is a crisp set. The value of is maximum if and only if for all (Luca & Termini, 1972)
Literature review
Let be a set of complete probability distribution. For any Ω ∈ △n, Shannon defined an information measure given bywhich is popularly known as Shannon Entropy (Shannon, 1948). Afterwards, Renyi (1961) proved that this is not only the Shannon entropy but any measure satisfying a certain set of axioms can behave as an information measure. To justify his claim, Renyi (1961) put forward an information measure given by
Application of proposed picture fuzzy information measure in MCDM problem
As mentioned earlier, a MCDM problem consists of a set of alternatives and a set of prefixed criteria. The main aim is to chose an alternative which gives optimal output satisfying the given criteria. A MCDM problem can be represented by a matrix say (PFM)m × n with rows representing the m-alternatives and columns denoting n-criteria . The degrees to which an alternative satisfies the criteria are represented by using PFNs, that
Illustrativenumerical examples
Example 1 Consider an example of a election bound country where elections are going to take place in near future. Suppose there are five political parties say (Alt1, Alt2, Alt3, Alt4, Alt5) in the race. Besides local issues, there are six issues of national importance denoted by: 1. external security (Cr1) 2. internal security (Cr2), 3. inflation (Cr3), 4. foreign policy (Cr4), 5. stability (Cr5), 6. corruption (Cr6). A television channel conducted an opinion poll on a sample size of one thousand
Concluding remarks
In this paper, we have successfully introduced a new axiomatic structure of fuzzy entropy for picture fuzzy sets. Apart from this, a new R-norm picture fuzzy information measure is proposed and validated as an extension of well-known R-norm entropy studied by Boekee and Lubbe Boekee and Vander Lubbe (1980) from probability settings to picture fuzzy settings. Further, to show the practicality of proposed information measure in real world problems, its applications are given in the form of
CRediT authorship contribution statement
Rajesh Joshi: Conceptualization, Data curation, Formal analysis, Funding acquisition, Investigation, Methodology, Project administration, Resources, Software, Supervision, Validation, Visualization, Writing - original draft, Writing - review & editing, Visualization.
Acknowledgments
The author is thankful to anonymous reviewers for their precious suggestions to improve this manuscript and enhance my knowledge.
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