Decision-Making with Neutrosophic Set: Theory and Applications in Knowledge Management 1536194190, 9781536194197

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COMPUTATIONAL MATHEMATICS AND ANALYSIS

DECISION-MAKING WITH NEUTROSOPHIC SET THEORY AND APPLICATIONS IN KNOWLEDGE MANAGEMENT

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COMPUTATIONAL MATHEMATICS AND ANALYSIS

DECISION-MAKING WITH NEUTROSOPHIC SET THEORY AND APPLICATIONS IN KNOWLEDGE MANAGEMENT

HARISH GARG EDITOR

Copyright © 2021 by Nova Science Publishers, Inc. All rights reserved. No part of this book may be reproduced, stored in a retrieval system or transmitted in any form or by any means: electronic, electrostatic, magnetic, tape, mechanical photocopying, recording or otherwise without the written permission of the Publisher. We have partnered with Copyright Clearance Center to make it easy for you to obtain permissions to reuse content from this publication. Simply navigate to this publication’s page on Nova’s website and locate the “Get Permission” button below the title description. This button is linked directly to the title’s permission page on copyright.com. Alternatively, you can visit copyright.com and search by title, ISBN, or ISSN. For further questions about using the service on copyright.com, please contact: Copyright Clearance Center Phone: +1-(978) 750-8400 Fax: +1-(978) 750-4470 E-mail: [email protected]. NOTICE TO THE READER The Publisher has taken reasonable care in the preparation of this book, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of information contained in this book. The Publisher shall not be liable for any special, consequential, or exemplary damages resulting, in whole or in part, from the readers’ use of, or reliance upon, this material. Any parts of this book based on government reports are so indicated and copyright is claimed for those parts to the extent applicable to compilations of such works. Independent verification should be sought for any data, advice or recommendations contained in this book. In addition, no responsibility is assumed by the Publisher for any injury and/or damage to persons or property arising from any methods, products, instructions, ideas or otherwise contained in this publication. This publication is designed to provide accurate and authoritative information with regard to the subject matter covered herein. It is sold with the clear understanding that the Publisher is not engaged in rendering legal or any other professional services. If legal or any other expert assistance is required, the services of a competent person should be sought. FROM A DECLARATION OF PARTICIPANTS JOINTLY ADOPTED BY A COMMITTEE OF THE AMERICAN BAR ASSOCIATION AND A COMMITTEE OF PUBLISHERS. Additional color graphics may be available in the e-book version of this book.

Library of Congress Cataloging-in-Publication Data ISBN: H%RRN

Published by Nova Science Publishers, Inc. † New York

CONTENTS Preface

vii

Section I: Mathematical Aspects of Neutrosophic Set

1

Chapter 1

Neutrosophic Set Theory and Engineering Applications: A Study K. Bhargavi and B. Sathish Babu

3

Chapter 2

A New Type of Quasi Open Functions in Neutrosophic Topological Environment M. Parimala, C. Ozel, F. Smarandache and M. Karthika

27

Accordance with Neutrosophic Logic? A Multimoora Approach for Countries Worldwide Willem K. M. Brauers

37

Evaluation of Online Education Software under Neutrosophic Environment Fatma Kutlu Gündoğdu and Serhat Aydın

69

Chapter 3

Chapter 4

Chapter 5

A New Attribute Sampling Plan for Assuring Weibull Distributed Lifetime using Neutrosophic Statistical Interval Method P. Jeyadurga and S. Balamurali

Section II: Decision Making Problems with Neutrosophic Set Chapter 6

On Some Propositions of Boundary in Interval Valued Neutrosophic Bitopological Space Bhimraj Basumatary

91 111 113

vi Chapter 7

Chapter 8

Chapter 9

Chapter 10

Contents An Expected Value-Based Novel Similarity Measure for Multi-Attribute Decision-Making Problems with Single-Valued Trapezoidal Neutrosophic Numbers Palash Dutta and Gourangajit Borah TrNN-ARAS Strategy for Multi-Attribute Group Decision-Making (MAGDM) in Trapezoidal Neutrosophic Number Environment with Unknown Weight Rama Mallick and Surapati Pramanik An Application of Reduced Interval Neutrosophic Soft Matrix in Medical Diagnosis Somen Debnath Interval-Valued Neutrosophic N Soft Set and Intertemporal Interval-Valued Neutrosophic N Soft Set to Assess the Resilience of the Workers Amidst Covid-19 V. Chinnadurai and A. Bobin

Section III. Extension of the Neutrosophic Set Chapter 11

Chapter 12

Chapter 13

Chapter 14

2-Additive Choquet Cosine Similarity Measures for Simplified Neutrosophic Sets and Applications to Medical Diagnosis Ezgi Türkarslan, Murat Olgun, Mehmet Ünver and Şeyhmus Yardimci

133

163

195

219 257 259

Multi-Attribute Group Decision-Making Based on Uncertain Linguistic Neutrosophic Sets and Power Hamy Mean Operator Yuan Xu, Xiaopu Shang and Jun Wang

283

An n-Dimensional Neutrosophic Linguistic Approach to Poverty Analysis with an Empirical Study D. Ajay, J. Aldring and S. Nivetha

331

Multi-Granulation Single-Valued Neutrosophic Hesitant Fuzzy Rough Sets Tahir Mahmood and Zeeshan Ali

353

About the Editor

377

Contributors

379

Index

387

PREFACE With the complexity of the socio-economic environment, today's decision-making is one of the most notable ventures, whose mission is to decide the best alternative under the numerous known or unknown criteria. In cognition of things, people may not possess a precise or sufficient level of knowledge of the problem domain and hence they usually have some sort of uncertainties in their preferences over the objects. This will make the performance of the cognitive in terms of three-ways model namely acceptation, rejection, indeterminacy which is falls under the neutrosophic set theory, an extension of the fuzzy set theory. Now in days, many new extensions of the ordinary neutrosophic set are proposed and they are expected to be competitive with the other extensions in the future. In this book, a new extension of the fuzzy sets, entitled as Neutrosophic sets, is introduced by eminent researchers with several applications. In this set, the performance of the cognitive in terms of fuzzy environment is considered with the help of degrees of acceptation, rejection, indeterminacy. This book consists of three parts. The first part involves five chapters presenting the important mathematical aspects of the neutrosophic sets and its extensions. The second part contains five chapters presenting contribution on the information measures and the aggregation operators of neutrosophic sets and its extension which include neutrosophic fuzzy decision-making methods and different applications to the real-life problems. Finally, the last part contains four chapters on the theory of the interval-neutrosophic sets and their applications to the decision-making process. The first chapter in the first part of the book is to present a basic introduction to the neutrosophic sets and also bring out the difference between the classical seta and neutrosophic set. The application of the neutrosophic set over the different engineering disciplines namely civil, aerospace and mechanical engineering are discussed. The second chapter is to define the new type of the quasi open and closed functions in neutrosophic topological space. The fundamental properties and its characterizations are discussed in it. The third chapter considered the credit rating of the agencies different from the point of view of neutrosophic theory and hence discusses the scenario with the

viii

Harish Garg

help of the MULTIMOORA approach. In the fourth chapter, authors propose the neutrosophic MULTIMOORA (Multiobjective Optimization by Ratio Analysis plus Full Multiplicative Form) method to evaluate online learning tools concerning some critical factors which have an essential influence on student satisfaction. Comparative and sensitivity analyses are also performed to show the validity of the methodology. The fifth chapter introduces an acceptance sampling for assuring Weibull distributed lifetime of the products using neutrosophic interval method. The probabilities corresponding to nonfailure, failure and indeterminate case are obtained under Weibull distribution. The first chapter in the second part of the book is to present a concept on interval valued neutrosophic bitopological space by defining an interval valued neutrosophic interior, closure and relation between them. Also, author studied an interval valued neutrosophic boundary and some of their propositions in interval valued bitopological space. The second chapter presents a novel similarity measure with the single-valued trapezoidal neutrosophic numbers (SVTNNs). The proposed measure involves calculating the salient feature of expected value of a SVTNN, using the 𝛼 − cut method. Later on, based on this measure, a decision making algorithm is introduced to solve the multi-attribute decision making algorithms. The third chapter extends the ARAS (Additive Ratio Assessment) strategy to the trapezoidal neutrosophic number environment. Based on it, it presents an approach for multi-attribute group decisionmaking problems under the trapezoidal neutrosophic number environment. In addition, entropy measures are utilized to assess to compute the experts’ importance degrees. The fourth chapter introduces the notion of interval neutrosophic soft matrix (ivn-soft matrix) and defined some basic algebraic operations on them. Based on these new matrices, an algorithm has been developed to solve the decision making problems such as medical diagnosis problems etc. In the fifth chapter, the notions of an interval-valued neutrosophic N soft set (IVNNSS) and the quasi-hyperbolic discounting intertemporal interval-valued neutrosophic N soft set (QHDIIVNNSS) have been proposed. The approach is applied to determine the resilience of the workers in an organization amidst the coronavirus (COVID-19) pandemic. The last part of the book deals on the theory of the extension of neutrosophic sets such as uncertain linguistic neutrosophic set, neutrosophic hesitant fuzzy rough set and their applications to the solve the decision-making problems. In this part, the first chapter deals with cosine similarity measures for the simplified neutrosophic sets in which rating of each objects are treated as neutrosophic numbers. For it, six new measures are defined by considering 2-additive Choquet integral model. The advantages of their measures is to reduce the computational effort due to the help of 2-additivity. The utility of the measures is tested on to the medical diagnosis problems. The second chapter in this part deals with uncertain linguistic neutrosophic sets (ULNSs), an extension of the neutrosophic set, to solve the decision making problems. In this chapter, the definition, basic operational rules, comparison method and distance measure of ULNSs are presented and discussed.

Preface

ix

Also, a series of aggregation operators for ULNSs based on power average operator and Hamy mean operator are stated. Based on the proposed operators, it presents an approach for multiple attribute group decision-making problems under the ULNS environment. The third chapter explains how to measure the poverty with the help of neutrosophic set features. For it, an attempt to analyze poverty among the target group of people using nnumber of linguistic variables approach in neutrosophic environment is made in the chapter. In it, they generate neutrosophic membership functions for the poverty indicators which help to get the defuzzified values for fuzzy linguistic variables. Case study related to categorize the households according to their poverty level is discussed in the chapter. The last chapter of this book explores the theory of Single-valued neutrosophic hesitant fuzzy rough set (SVNHFRS) and its operational laws justified with the help of an example. Additionally, two kinds of multi-granulation SVNHFRS (MGSVNHFS), called optimistic MGSVNHFRS (OMGSVNHFRS) and pessimistic MGSVNHFRS (PMGSVNHFRS) are presented. The relation between MGSVNHFRS, OMGSVNHFRS, and PMGSVNHFRS are also discussed. We hope that this book will provide a useful resource of ideas, techniques, and methods for the research on the theory and applications of Neutrosophic sets. We are grateful to the referees for their valuable and highly appreciated works contributed to select the high quality of chapters published in this book. We would like to also thank the NOVA Publisher and his team for his supportive role throughout the process of editing this book. Dr. Harish Garg Patiala, India

SECTION I: MATHEMATICAL ASPECTS OF NEUTROSOPHIC SET

In: Decision-Making with Neutrosophic Set Editor: Harish Garg

ISBN: 978-1-53619-419-7 © 2021 Nova Science Publishers, Inc.

Chapter 1

NEUTROSOPHIC SET THEORY AND ENGINEERING APPLICATIONS: A STUDY K. Bhargavi1,* and B. Sathish Babu2 1

Department of CSE, Siddaganga Institute of Technology, Tumakuru, Karnataka, India 2 Department of CSE, R. V. College of Engineering, Bengaluru, Karnataka, India

ABSTRACT A typical meaning of uncertainty is an error which persists in planning, design, construction, and testing phases of engineering. The degree of uncertainty encountered varies depending on the discipline of engineering. Managing uncertainty is one of the important prerequisites for proper formulation of design to arrive at optimal solutions for complex computational problems. Neutrosophic sets are a potential framework for handling the uncertainties in the computing environment of engineering. This paper provides a basic introduction to neutrosophic sets and also brings out the differences between classical set and neutrosophic set. The neutrosophic set is applied to three engineering disciplines; Civil engineering, Aerospace engineering, and Mechanical engineering. Because the neutrosophic set is capable enough to handle commonly occurring uncertainty in the mentioned engineering disciplines which includes improper specification of boundary conditions, drastic fluctuation in the execution pattern, infected information channels, poor fabrication of hardware parts, varying sensitivity coefficient of materials, transducer errors, voltage measurement errors, and so on. The properties of neutrosophic sets and its usefulness in handling the uncertainties with respect to mentioned engineering discipline problems are discussed. The interval value analysis reveals that the efficiency achieved by neutrosophic set enabled engineering solutions are better than the classical solutions.

*

Corresponding Author’s Email: [email protected].

4

K. Bhargavi and B. Sathish Babu

Keywords: neutrosophic sets, uncertainty, engineering, efficiency, civil engineering, aerospace engineering, mechanical engineering

1. INTRODUCTION The neutrosophic set theory is a branch of philosophy invented by Florentin Smarandache, in the year 1998, which concentrates more towards neutralities and is used to model the problems involving high degrees of uncertainties. The study regarding neutrosophy is considered as an extension of dialectics and some of its derivatives are neutrosophic set, neutrosophic statistics, neutrosophic calculus, neutrosophic probability, and others. The neutrosophic set is the end product obtained after the generalization of the three traditional sets which includes classical set, rough set, and fuzzy set [1, 2, 3]. The main advantages of neutrosophic set theory are ability to handle the uncertainty in decision making is high, imposes a high degree of control over the decision making process, high degree of indeterminacy is handled using indeterminacy component, able to manage the nonlinear complex systems, rapid operation over uncertain parameters, high precision aggregation operations, effective in managing the noise or disturbance during operation, and so on. The existing approaches applied to handle uncertainty in engineering discipline includes rule based, fuzzy logic, crisp set, soft set theory, and so on. But these mathematical framework are subjected to limitations in terms of lower speed of operation, restriction on input parameters count, poor feedback incorporation, inability of scale for larger problems, improper tuning operations, wastage of resources, and so on. The neutrosophic set is capable enough to handle commonly occurring uncertainty precisely in various engineering disciplines like improper specification of boundary conditions, drastic fluctuation in the execution pattern, infected information channels, poor fabrication of hardware parts, varying sensitivity coefficient of materials, transducer errors, voltage measurement errors, and so on. This motivated towards the application of neutrosophic set in engineering disciplines to handle uncertainty. For any neutrosophic set of an element x is comprised of truth membership function 𝑇(𝑥), falsehood membership function F(x), and inderminacy membership function 𝐼(𝑥), i.e., 𝑁𝑆(𝑥) =< 𝑇(𝑥), 𝐹(𝑥), 𝐼(𝑥) > where the value of x ranges between ]0− and 1+ [. For example consider a neutrosophic set 𝑁𝑆(𝑥) =< 𝑇(𝑥) = 0.8, 𝐹(𝑥) = 0.2, 𝐼(𝑥) = 0.1 > representing whether the system is adaptive are not. The set is interpreted as follows 80% of the system is adaptable, 20% the system is inderminate about the adaptability, and 10% of the system is inadaptable.

Neutrosophic Set Theory and Engineering Applications

5

1.1. Types of Neutrosophic Set There are different types of neutrosophic set, the most popular forms of neutrosophic sets used for handling real-time uncertainties are as follows [4, 5, 6].

Interval Valued Neutrosophic Set 𝐼𝑉𝑁𝑆(𝑥) =< 𝑇(𝑥), 𝐹(𝑥), 𝐼(𝑥) >, where 𝑇(𝑥), 𝐹(𝑥), 𝐼(𝑥) ⊆ [0,1] Example: 𝐼𝑉𝑁𝑆(𝑥) =< 𝑇(𝑥) = 0.1, 𝐹(𝑥) = 0.8, 𝐼(𝑥) = 0.1 > indicates that the falsehood of the data is above average. Fuzzy Neutrosophic Set 𝐹𝑁𝑆(𝑥) =< 𝑇(𝑥), 𝐹(𝑥), 𝐼(𝑥) > , where 𝑇(𝑥), 𝐹(𝑥), 𝐼(𝑥) → [0,1]and 0− ≤ 𝑇(𝑥), 𝐹(𝑥), 𝐼(𝑥) ≤ 3+ . Example: 𝐹𝑁𝑆(𝑥) =< 𝑇(𝑥) = 0.5, 𝐹(𝑥) = 0.1, 𝐼(𝑥) = 0.8 > indicates that the majority portion of the data is in inderminate state. Intuitionistic Neutrosophic Set 𝐼𝑁𝑆(𝑥) =< 𝑇(𝑥), 𝐹(𝑥), 𝐼(𝑥) >, where min(𝑇(𝑥), 𝐹(𝑥)) ≤ 0.5), min(𝑇(𝑥), 𝐼(𝑥)) ≤ 0.5), and min(𝐹(𝑥), 𝐼(𝑥)) ≤ 0.5), 0 ≤ 𝑇(𝑥) + 𝐹(𝑥) + 𝐼(𝑥) ) indicates that the data is bounded within the interval of 0 and 2. Single Valued Neutrosophic Set 𝑆𝑉𝑁𝑆(𝑥) =< 𝑇(𝑥), 𝐹(𝑥), 𝐼(𝑥) >, where ∑𝑖=𝑛 𝑖=1

𝑥

, 𝑥 ∈ 𝑋 and X is a

discrete value; multi valued neutrosophic set 𝑀𝑉𝑁𝑆(𝑥) =< 𝑇(𝑥), 𝐹(𝑥), 𝐼(𝑥) >, where 𝑇(𝑥), 𝐹(𝑥), 𝐼(𝑥) → [0,1] and 𝑇(𝑥) ≤ 0𝐹(𝑥) ≤ 1, 𝐼(𝑥) ≤ 3 Example: 𝑆𝑉𝑁𝑆(𝑥) =< 𝑇(𝑥) = 0.2, 𝐹(𝑥) = 0.4, 𝐼(𝑥) = 0.5 > indicates that the data is bounded within the interval of 1 and 3 in which the rate of inderminacy is almost same as the falsehood.

Bipolar Neutrosophic Set 𝐵𝑁𝑆(𝑥) =< 𝑇 + (𝑥), 𝐹 + (𝑥), 𝐼 + (𝑥), 𝑇 − (𝑥), 𝐹 − (𝑥), 𝐼 − (𝑥), where 𝑇 + (𝑥), 𝐹 + (𝑥), 𝐼 + (𝑥) → [0,1] and 𝑇 − (𝑥), 𝐹 − (𝑥), 𝐼 − (𝑥) → [−1,0]. Example:

𝐵𝑁𝑆(𝑥) =< 𝑇 + (𝑥) = 0.1, 𝐹 + (𝑥) = 0.8, 𝐼 + (𝑥) = 0.8, 𝑇 − (𝑥) = 0.9, 𝐹 − (𝑥) =

0.8, 𝐼− (𝑥) = 0.1 > indicates that the upper and lower boundary of falsehood of the data is

identical whereas for truth and inderminacy of the data the boundaries values vary

6

K. Bhargavi and B. Sathish Babu

widely. The neutrosophic set differs from conventional classical set in many ways, some of them are list below. Table 1. Classical set versus neutrosophic set Classical set An element either belongs to the set or does not belong to the set using either true or false. Does not allow the elements to be partially in the set. Implements two valued logic. Classification of the elements happens through sharp boundary. Unable to handle the imprecision involved in the data. The law of excluded middle is satisfied. The flexibility is high in selection of union, intersection, and negotiation operations. Unable to handle the vagueness involved in the data. Commonly used in the design of digital systems. Relies on upper approximation and lower approximation to arrive at optimal solutions.

Neutrosophic set An element belongs to the set or not does not belongs to the set using truth, inderminacy, and falsehood membership grades. Allows the elements to be partially in the set. Implements three valued logic. Classification of elements happens through smooth boundary. Able to handle the imprecision involved in the data. The law of excluded middle is not satisfied The flexibility is poor in selection of union, intersection, and negotiation operations. Able to handle the vagueness involved in the data. Commonly used in the design of fuzzy systems. Relies on upper approximation, middle approximation, and lower approximation to arrive at optimal solutions.

1.2. Neutrosophic Set Operations The basic operations that could be carried out over the neutrosophic set are described as follows [7, 8].

Containment The neutrosophic set P is contained inside another neutrosophic set Q 𝐼𝑉𝑁𝑆(𝑃) ⊆ 𝐼𝑉𝑁𝑆(𝑄) if and only if inferior and superior functions of the truth and false membership functions are satisfied. < inf(𝑇(𝑃)) < inf(𝑇(𝑄)) and sup(𝑇(𝑃)) < sup(𝑇(𝑄)) > < inf(𝐹(𝑃)) ≥ inf(𝐹(𝑄)) and sup(𝑇(𝑃)) ≥ sup(𝑇(𝑄)) >

Complement Complement of a neutrosophic set yields a single valued neutrosophic set. < 𝑇 𝑐 (𝑃) = 1+ − 𝑇(𝑃)) > < 𝐼 𝑐 (𝑃) = 1+ − 𝐼(𝑃)) >

Neutrosophic Set Theory and Engineering Applications

7

< 𝐹 𝑐 (𝑃) = 1+ − 𝐹(𝑃)) >

Union Union of two neutrosophic sets P and Q produces another neutrosophic set R. < 𝑇(𝑅) = 𝑇(𝑃) + 𝑇(𝑄) − 𝑇(𝑃) ∗ 𝑇(𝑄) > < 𝐼(𝑅) = 𝐼(𝑃) + 𝐼(𝑄) − 𝐼(𝑃) ∗ 𝐼(𝑄) > < 𝐹(𝑅) = 𝐹(𝑃) + 𝐹(𝑄) − 𝐹(𝑃) ∗ 𝐹(𝑄) >

Intersection Intersection of two neutrosophic sets P and Q produces another neutrosophic set R. < 𝑇(𝑅) = 𝑇(𝑃) ∗ 𝑇(𝑄) > < 𝐼(𝑅) = 𝐼(𝑃) ∗ 𝐼(𝑄) > < 𝐹(𝑅) = 𝐹(𝑃) ∗ 𝐹(𝑄) >

Product Product of two neutrosophic sets P and Q provides extension for the existing intuitionistic relations between the neutrosophic sets. < 𝑇(𝑅) = 𝑇(𝑃) ∗ 𝑇(𝑄) > < 𝐼(𝑅) = 𝐼(𝑃) + 𝐼(𝑄) − 𝐼(𝑃) ∗ 𝐼(𝑄) > < 𝐹(𝑅) = 𝐹(𝑃) + 𝐹(𝑄) − 𝐹(𝑃) ∗ 𝐹(𝑄) >

Addition Addition of two neutrosophics sets yields a reduced form of single valued neutrosophic set. < 𝑇(𝑅) = 𝑇(𝑃) + 𝑇(𝑄) − 𝑇(𝑃) ∗ 𝑇(𝑄) > < 𝐼(𝑅) = 𝐼(𝑃) ∗ 𝐼(𝑄) > < 𝐹(𝑅) = 𝐹(𝑃) ∗ 𝐹(𝑄) >

Subtraction Subtraction of two neutrosophics sets yields a reduced form of single valued neutrosophic set. < 𝑇(𝑅) = 𝑇(𝑃) + 𝑇(𝑄) − 𝑇(𝑃) ∗ 𝑇(𝑄) > < 𝐼(𝑅) = 𝐼(𝑃) ∗ 𝐼(𝑄) > < 𝐹(𝑅) = 𝐹(𝑃) ∗ 𝐹(𝑄) >

8

K. Bhargavi and B. Sathish Babu

Equality The neutrosophic set P is equal to the neutrosophic set Q if they are subset of each other. < 𝑇(𝑃) ⊂ 𝑇(𝑄) < 𝐼(𝑃) ⊂ 𝐼(𝑄) > < 𝐹(𝑃) ⊂ 𝐹(𝑄) >

Inclusion The neutrosophic set P is included in the neutrosophic set Q if the inferior and superior functions of the truth, inderminacy, and falsehood membership functions are related to each other. < inf(𝑇(𝑃)) , sup(𝑇(𝑄)) < sup(𝑇(𝑄)) < inf(𝑇(𝑃)) < inf(𝐼(𝑄)) , sup(𝐼(𝑃)) > sup(𝐼(𝑄)) , inf(𝑇(𝑃)) >

Division The division of two neutrosophic sets P and Q provides a powerful structure to arrive at single valued number as answers. 𝑇(𝑃) > 𝑇(𝑄) 𝐼(𝑃) − 𝐼(𝑄) < 𝐼(𝑅) = > 1 − 𝐼(𝑄) 𝐹(𝑃) − 𝐹(𝑄) < 𝐹(𝑅) = > 1 − 𝐹(𝑄) < 𝑇(𝑅) =

The application of Neutrsophic set towards the engineering domains to handle the uncertainty is a novel approach which mainly tries to handle the uncertainty using truth, falsehood, and inderminacy functions. Here the properties of the neutrosophic set and its usefulness in handling the uncertainties in the engineering applications is discussed in specific with much more clarity. Whereas in the literature traditional sets are applied to engineering domains which fail to handle uncertainties [9, 10]. The objectives of the proposed chapter are listed below.   

Introduction towards neutrosophic soft set, Highlighting the benefits of the neutrosophic soft set over the traditional set. Identifying the sources causing uncertainty in engineering disciplines like aerospace, mechanical, and civil engineering.

Neutrosophic Set Theory and Engineering Applications  

9

Application of neutrosophic soft set theory to handle uncertainty in the engineering disciplines like aerospace, mechanical, and civil engineering. Efficiency analysis of the neutrosphic set theory in handling uncertainty in engineering disciplines like aerospace, mechanical, and civil engineering.

2. TERMINOLOGIES This section provides definition for some of the mathematical terminologies used in the paper which is shown in Table 2. Table 2. Terminologies used Terminology Neutrosophic set Uncertainty Indeterminacy Falsehood Aerospace Engineering Civil Engineering Mechanical Engineering S-Norm T-Norm Crisp set Membership function Accuracy Error rate

Meaning It is component of Neutrosophy which focuses on the study related to origin and scope of the neutralities. It refers to the epistemic situation which involves a lot of unclear information. It is one of the mathematical concepts referred to represent uncertainty. It is a state which represent untrue situation. It is a branch of Engineering which deals with the development of aircraft and spacecrafts. It is a branch of Engineering which deals with the design, development, and maintenance of structural components. It is a branch of Engineering which deals design, development, and maintenance of mechanical systems. Maximum operator Triangular Norm Set consists of elements which either belongs to set or not. It is generalization of the indicator function for classical sets. The degree to which result of measurement adheres to standard value. Degree to which errors are encountered during data transmission.

3. NEUTROSPOPHIC SET APPLICATIONS IN CIVIL ENGINEERING The civil engineering discipline mainly deals with the planning, design, construction, and maintenance of the physical bodies like buildings, road, dam, railways, airport, and so on [11, 12]. Uncertainties exist in every phase of the civil engineering, some of the sources of uncertainties are design of faulty construction model, use of probability based safety measures, improper specification of boundary conditions, estimated quality of the materials is far from reality, imprecise numerical computation, poor construction maintenance, drastic fluctuation in the execution pattern, and so on [13, 14, 15]. This uncertainty in civil engineering motivated towards the application of neutrosophic set theory. A high level view of the neutrosophic based civil engineering approach is given

10

K. Bhargavi and B. Sathish Babu

in Figure 1. The uncertainty existing in the every stage of the civil engineering beginning from planning to maintenance is handled using the truth, false, and inderminacy components of the neutrosophic logic. The properties of the neutrosophic set which are useful in handling civil engineering uncertainties are discussed below.

Property-1 Single valued triangular function of neutrosophic sets handles inderminacy of the data.

Figure 1. A high level view of the neutrosophic based civil engineering.

Applicability The property-1 of neutrosophic sets is applied in the civil engineering for addressing the issues like outbreak of power, material optimization procedure, formation of waste disposal policies, establishing synchronization between the design and process, and so on. Description Consider the real-time parameters of civil engineering process with truth, false, and inderminacy components 𝐶𝐸𝑃 = {𝑐𝑒𝑝1 , 𝑐𝑒𝑝2 , 𝑐𝑒𝑝3 , . . 𝑐𝑒𝑝𝑛 ; 𝑇(𝑐𝑒𝑝𝑖 ), 𝐹(𝑐𝑒𝑝𝑖 ), 𝐼(𝑐𝑒𝑝𝑖 )} , the weights assigned for truth, indeterminacy, and falsehood of 𝑐𝑒𝑝𝑖 𝑠 are 𝑊𝑡 ∈ [𝜑𝑡 , Ω𝑡 ], 𝑊𝑖 ∈ [𝜑𝑖 , Ω𝑖 ], and 𝑊𝑓 ∈ [𝜑𝑓 , Ω𝑓 ]], and where 𝑥 ∈ 𝑐𝑒𝑝𝑖 and i ranges between 1 to n.

Neutrosophic Set Theory and Engineering Applications

11

𝑇(𝑥) = (𝑥 − 𝑐𝑒𝑝1 )𝑊𝑡 , 𝑖𝑓 (𝑐𝑒𝑝1 < 𝑥 < 𝑐𝑒𝑝𝑛 ); 𝑊𝑡, if (𝑐𝑒𝑝𝑛 < 𝑥 < 𝑐𝑒𝑝𝑛+1 ) (𝑐𝑒𝑝𝑛 − 𝑐𝑒𝑝1 ) (𝑐𝑒𝑝𝑛 − 𝑥)𝑊𝑡 , 𝑖𝑓 (𝑐𝑒𝑝𝑛+1 < 𝑥 < 𝑐𝑒𝑝𝑛 ); 𝜑𝑡 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 (𝑐𝑒𝑝 𝑛 − 𝑐𝑒𝑝3 ) { [𝑐𝑒𝑝𝑛 − 𝑥 + 𝑊𝑖(𝑥 − 𝑐𝑒𝑝1 )] , 𝑖𝑓 (𝑐𝑒𝑝1 < 𝑥 < 𝑐𝑒𝑝𝑛 ); 𝑊𝑖 𝑖𝑓 (𝑐𝑒𝑝𝑛−1 < 𝑥 < 𝑐𝑒𝑝𝑛 ) (𝑐𝑒𝑝2 − 𝑐𝑒𝑝1 ) 𝐼(𝑥) = [𝑥 − 𝑐𝑒𝑝𝑛−1 + 𝑊𝑖(𝑐𝑒𝑝𝑛 − 𝑥)] , 𝑖𝑓 (𝑐𝑒𝑝𝑛 < 𝑥 < 𝑐𝑒𝑝𝑛+1 ); Ω𝑡 , 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 (𝑐𝑒𝑝4 − 𝑐𝑒𝑝3 ) { [𝑐𝑒𝑝𝑛 − 𝑥 + Ω𝑓 (𝑥 − 𝑐𝑒𝑝1 )] , 𝑖𝑓 (𝑐𝑒𝑝1 < 𝑥 < 𝑐𝑒𝑝𝑛−1 ); 𝑊𝑓 𝑖𝑓 (𝑐𝑒𝑝1 < 𝑥 < 𝑐𝑒𝑝𝑛 ) (𝑐𝑒𝑝𝑛 − 𝑐𝑒𝑝1 ) 𝐹(𝑥) = [𝑥 − 𝑐𝑒𝑝𝑛−1 + Ω𝑓 (𝑐𝑒𝑝𝑛 − 𝑥)] , 𝑖𝑓 (𝑐𝑒𝑝𝑛−1 < 𝑥 < 𝑐𝑒𝑝𝑛 ); Ω𝑓 , 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 (𝑐𝑒𝑝𝑛 − 𝑐𝑒𝑝𝑛−1 ) {

Usefulness The triangular function of neutrosophic set efficiently handles the indeterminate components of the planning, design, construction, and maintenance phases of the civil engineering. Property-2 The aggregated value of collection of trapezoidal neutrosophic numbers is also a trapezoidal neutrosophic number. Applicability The property-2 of neutrosophic sets is applied in the civil engineering for addressing the issues like maintaining sustainability of the structural elements, analyzing the productivity drop, recycling of the construction materials, estimating the vitality factor of the manufacturing plants, and so on. Description The aggregated neutrosophic set ANS is calculated over the real-time parameters of the civil engineering process. 𝐴𝑁𝑆(𝑐𝑒𝑝1 , 𝑐𝑒𝑝2 , 𝑐𝑒𝑝3 , . . 𝑐𝑒𝑝𝑛 ) = 𝑊1 𝑐𝑒𝑝1 ⊕ 𝑊2 𝑐𝑒𝑝2 ⊕ 𝑊3 𝑐𝑒𝑝3 ⊕. . 𝑊𝑛 𝑐𝑒𝑝𝑛

=⊕𝑖=𝑛 𝑖=1 (𝑊𝑖 𝑐𝑒𝑝𝑖 ) 𝐴𝑁𝑆(𝑐𝑒𝑝1 , 𝑐𝑒𝑝2 , 𝑐𝑒𝑝3 , 𝑐𝑒𝑝4 ) = 𝑖=𝑛

(𝑐𝑒𝑝𝑛 )𝑊𝑛 >

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K. Bhargavi and B. Sathish Babu

Usefulness The aggregated neutrosophic set ANS exhibits high stability towards dynamically changing parameters which is useful in handling the falsehood component while arriving at optimal multimodal solutions. Property-3 The S-norm and T-norm operation of neutrosophic set produces flexibility. Applicability The property-3 of neutrosophic sets is applied in the civil engineering for addressing the issues like identification of poor connection between the designs of the structure, reduction of disruption in construction activities, proper estimation of quality of the material, proper handling of the congestion during heavy traffic, forming defacto rules for construction materials, identification of ground water levels, and so on. Description The S-norm and T-norm is calculated over the real time parameters of the civil engineering process. √𝑆(𝑇(𝑐𝑒𝑝1 ), 𝑇(𝑐𝑒𝑝2 ), 𝑇(𝑐𝑒𝑝3 ), . . 𝑇(𝑐𝑒𝑝𝑛 ) 𝑆 − 𝑛𝑜𝑟𝑚(𝑐𝑒𝑝1 , 𝑐𝑒𝑝2 , 𝑐𝑒𝑝3 , . . 𝑐𝑒𝑝𝑛 ) = { 1 − √𝑆(𝐼(𝑐𝑒𝑝1 ), 𝐼(𝑐𝑒𝑝2 ), 𝐼(𝑐𝑒𝑝3 ), . . 𝐼(𝑐𝑒𝑝𝑛 ) 1 − √𝑆(𝐹(𝑐𝑒𝑝1 ), 𝐹(𝑐𝑒𝑝2 ), 𝐹(𝑐𝑒𝑝3 ), . . 𝐹(𝑐𝑒𝑝𝑛 ) 𝑇 − 𝑛𝑜𝑟𝑚(𝑐𝑒𝑝1 , 𝑐𝑒𝑝2 , 𝑐𝑒𝑝3 , . . 𝑐𝑒𝑝𝑛 ) √𝑆(𝑇(1 − 𝑐𝑒𝑝1 ), 𝑇(1 − 𝑐𝑒𝑝1 ), 𝑇(1 − 𝑐𝑒𝑝2 ), 𝑇(1 − 𝑐𝑒𝑝3 ), . . 𝑇(1 − 𝑐𝑒𝑝𝑛 ) ={

√𝑆(𝐼(𝑐𝑒𝑝1 ), 𝐼(𝑐𝑒𝑝2 ), 𝐼(𝑐𝑒𝑝3 ), . . 𝐼(𝑐𝑒𝑝𝑛 ) √𝑆(𝐹(𝑐𝑒𝑝1 ), 𝐹(𝑐𝑒𝑝2 ), 𝐹(𝑐𝑒𝑝3 ), . . 𝐹(𝑐𝑒𝑝𝑛 )

Usefulness The T-norm and T-conorm operations of the neutrosophic sets provide good flexibility in assessing the structure of the materials. It is very useful in precise measurement and sustainable assessment of the distance between the construction fields. Illustration of Interval Value Based Analysis of Efficiency Achieved by the Neutrosophic Based Civil Engineering Let us analyze the efficiency of the neutrosophic sets in handling the uncertainties involved in the civil engineering problems. The analysis begins by gathering the parameters of each stages of the civil engineering i.e., planning i.e., 𝑃 = {𝑝1 (80), 𝑝2 (75), 𝑝3 (55), 𝑝4 (45)} , design 𝐷 = {𝑑1 (77), 𝑑2 (80), 𝑑3 (22), 𝑑4 (32)} , construction 𝐶 = {𝑐1 (64), 𝑐2 (34), 𝑐3 (67), 𝑐4 (21)} , and maintenance 𝑀 =

Neutrosophic Set Theory and Engineering Applications

13

{𝑚1 (41), 𝑚2 (19), 𝑚3 (50), 𝑚4 (97)}. The efficiency achieved by the civil engineering parameters is varied between low (l), medium (m) and high (h). Create a matrix for interval value based civil engineering IVCE consisting of truth, false, and inderminacy values i.e., 𝐼𝑉𝐶𝐸(𝑃, 𝐷, 𝐶, 𝑀) =< 𝑇(𝑥), 𝐹(𝑥), 𝐼(𝑥) >. [0.2,0.6,0.2] [0.1,0.9,0.0] [0.6,0.0,0.4] [0.6,0.1,0.3]

[0.4,0.3,0.3] [0.3,0.1,0.6] [0.2,0.3,0.5] [0.3,0.3,0.4]

[0.3,0.5,0.2] [0.2,0.5,0.3] [0.6,0.1,0.3] [0.6,0.1,0.3]

Calculate the threshold for 𝐼𝑉𝐶𝐸(𝑃, 𝐷, 𝑀, 𝐶) = 𝐼𝑉𝐶𝐸(𝑃, 𝐷, 𝐶, 𝑀) + {[0.2,0.5,3.0], [0.2,0.4,0.4], [0.3,0.2,0.5], [0.1,0.9,0.0]} . Which is computed by adding the IVCE values computed for each phase of the civil engineering. Calculate the mean of 𝐼𝑉𝐶𝐸(𝑃, 𝐷, 𝑀, 𝐶)𝑚 i.e., 𝐼𝑉𝐶𝐸(𝑃, 𝐷, 𝐶, 𝑀)𝑚 =< [0.2,0.6,0.2] [0.5,0.5,0.0] [0.3,0.5,0.2] [0.1,0.5,0.4] 4

,

4

,

4

,

4

>.

Apply summation operation over each of the mean values obtained. [0.2] [0.1] [0.6] [0.6]

[0.3] [0.3] [0.5] [0.2]

[0.2] [0.2] [0.6] [0.3]

The efficiency achieved by the neutrosophic set in planning phase is max(0.2,0.3,0.2) = 0.3 which is the maximum among the obtained three values 0.2, 0.3, and 0.2, efficiency towards design phase is max(0.1,0.3,0.2) = 0.3 which is the maximum among the obtained three values 0.1, 0.3, and 0.2, efficiency towards construction phase is max(0.6,0.5,0.6) = 0.6 which is the maximum among the obtained three values 0.6, 0.5, and 0.6, and efficiency towards maintenance phase is max(0.6,0.2,0.3) = 0.6 which is the maximum among the obtained three values 0.6, 0.2, and 0.3. Hence an inference is drawn that efficiency achieved by the neutrosophy logic is above average i.e., 0.6 for construction and maintenance phases of civil engineering. Whereas, the efficiency towards planning and design is below average i.e., 0.2.

4. NEUTROSPOPHIC SET APPLICATIONS IN AEROSCAPE ENGINEERING The aerospace engineering discipline consists of three important phases i.e., conceptual design, preliminary design, and detailed design [16]. Uncertainties exist in every phase of the aerospace engineering, some of the sources of uncertainties are

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K. Bhargavi and B. Sathish Babu

spacecraft modeling errors, noise involved in observation, aircraft trajectory planning errors, infected information channels, multiple information sources, shock wave layer separation, geometric irregularities in transonic region, statistical manufacturing errors, increase in manufacturing cycle time, explosion in the demand and supply of aircraft components, attacks on aircraft tools, exhaustion in acoustic levels, weak satellite communication signals, inaccurate vehicle navigation system, increasing operational cost, speed constrained aero elasticity, imprecise measurement of takeoff and landing distances, poor fabrication of hardware parts, and so on [17, 18, 19, 20]. A high level view of the neutrosophic based aerospace engineering approach is given in Figure 2. The properties of the neutrosophic set which are useful in handling aerospace engineering uncertainties are discussed below.

Property-1 Neutrosophic based fuzzy t-norms, and a fuzzy t-conorms intersections and conjunctions operation allows optimization of imprecise topological models.

Figure 2. A high level view of the neutrosophic based aerospace engineering.

Applicability The property-1 of neutrosophic sets is applied in the aerospace engineering for addressing the issues like combustion control during transmission of acoustic signals, handling the thermal efficiency, aero elastic coupling of crucial artifacts components, and so on. Description Consider the real-time parameters of aerospace engineering process with truth, false, and inderminacy components 𝐴𝐸𝑃 = {𝑎𝑒𝑝1 , 𝑎𝑒𝑝2 , 𝑎𝑒𝑝3 , . . 𝑎𝑒𝑝𝑛 ; 𝑇(𝑥), 𝐹(𝑥), 𝐼(𝑥)} , where 𝑥 ∈ 𝑎𝑒𝑝𝑖 and i ranges between 1 to n.

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𝑇(𝑥)Λ𝑇(𝑦) = min{𝑇(𝑥), 𝑇(𝑦)} = 𝑇(𝑥) ∗ 𝑇(𝑦) 𝑇(𝑥)Λ𝑇(𝑦) = max{𝑇(𝑥) + 𝑇(𝑦) − 1,0} 𝐹(𝑥)Λ𝐹(𝑦) = min{𝐹(𝑥), 𝐹(𝑦)} = 𝐹(𝑥) ∗ 𝐹(𝑦) 𝐹(𝑥)Λ𝐹(𝑦) = max{𝐹(𝑥) + 𝐹(𝑦), 𝐹(𝑦)} 𝐼(𝑥)Λ𝐼(𝑦) = min{𝐼(𝑥), 𝐼(𝑦)} = 𝐼(𝑥) ∗ 𝐼(𝑦) 𝐼(𝑥)Λ𝐼(𝑦) = max{𝐼(𝑥) + 𝐼(𝑦), 𝐼(𝑥)}

Usefulness The neutrosophic based fuzzy t-norms, and a fuzzy t-conorms intersections and conjunctions operation efficiently handles the imprecise components of conceptual design and preliminary design phases of the aerospace engineering. Property-2 Neutrosophic nonstandard inequalities between the neutrosophic elements provide a single real number outputs with perfect uncertainty proof neutrosophic ordering between them. Applicability The property-2 of neutrosophic sets is applied in the aerospace engineering for addressing the issues like passive control of engine parts, orbit transfer control, positioning of space shuttle, developing architecture for reusable sensors, and so on. Description Consider the real-time parameters of aerospace engineering process with truth, false, and inderminacy components 𝐴𝐸𝑃 = {𝑎𝑒𝑝1 , 𝑎𝑒𝑝2 , 𝑎𝑒𝑝3 , . . 𝑎𝑒𝑝𝑛 ; 𝑇(𝑥), 𝐹(𝑥), 𝐼(𝑥)} , where 𝑥 ∈ 𝑎𝑒𝑝𝑖 and i ranges between 1 to 4. − 0 + 0+ −+ −0 + ),( ),( ),( ),( ),( ) ∈ 𝐴𝐸𝑃 ≅ |𝑇(𝑥)| ≥ 0 𝑇(𝑥) 𝑇(𝑥) 𝑇(𝑥) 𝑇(𝑥) 𝑇(𝑥) 𝑇(𝑥) − 0 + 0+ −+ −0 + ( ),( ),( ),( ),( ),( ) ∈ 𝐴𝐸𝑃 ≅ |𝐹(𝑥)| < 0 𝐹(𝑥) 𝐹(𝑥) 𝐹(𝑥) 𝐹(𝑥) 𝐹(𝑥) 𝐹(𝑥) − 0 + 0+ −+ −0 + ( ),( ),( ),( ),( ),( ) ∈ 𝐴𝐸𝑃 ≅ |𝐼(𝑥)| → 0 𝐼(𝑥) 𝐼(𝑥) 𝐼(𝑥) 𝐼(𝑥) 𝐼(𝑥) 𝐼(𝑥) (

Usefulness The nonstrandard inequalities between the neutrosophic elements with proper neutrosophic ordering efficiently handles the unsteadiness, and boundary layer separation

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factor among the aerospace elements during detailed design phase of the aerospace engineering.

Property-3 The division of two neutrosophic sets provides quality measure to validate and invalidate the scientific interpretation of complex operations. Applicability The property-3 of neutrosophic sets is applied in the aerospace engineering for addressing the issues like manufacturing cycle time reduction, structural damage analysis, aircraft noise control, turbulent air flow management, sustainable delivery of aircraft components, inspecting the malfunctioning of the critical components, rebuilding of reusable artifacts, and so on. Description Consider the real-time parameters of aerospace engineering process with truth, false, and inderminacy components 𝐴𝐸𝑃 = {𝑎𝑒𝑝1 , 𝑎𝑒𝑝2 , 𝑎𝑒𝑝3 , . . 𝑎𝑒𝑝𝑛 ; 𝑇(𝑥), 𝐹(𝑥), 𝐼(𝑥)} , where 𝑥, 𝑦 ∈ 𝑎𝑒𝑝𝑖 , and 𝜀1 and 𝜀2 are infinitesimals. 𝑇(𝑥) (T(x) − 𝜀1 , 𝑇(𝑥)) T(x) − 𝜀1 T(x) = = , 𝑇(𝑦) (T(y) − 𝜀2 , 𝑇(𝑦)) 𝑇(𝑦) 𝑇(𝑥) − 𝜀2 where

𝜀1 > 0, 𝜀2 > 0,

T(x)−𝜀1 𝑇(𝑦)




𝑇(𝑥)

𝑇(𝑦)−𝜀2 𝑇(𝑦)

𝐹(𝑥) (F(x) − 𝜀1 , 𝐹(𝑥) ∗ 𝐹(𝑦) F(x) − 𝜀1 F(x) ∗ F(y) = = , 𝐹(𝑦) (F(y) − 𝜀2 , 𝐹(𝑦)) 𝐹(𝑦) 𝐹(𝑥) − 𝜀2 where

𝜀1 > 0, 𝜀2 > 0,

F(x)−𝜀1 𝐹(𝑦)

==

𝐹(𝑥)

, and

𝐹(𝑦)

F(x)


0, 𝜀2 > 0,

I(x)−𝜀1 𝐼(𝑦)

≥

𝐼(𝑥)

, and

𝐼(𝑦)

I(x) 𝐼(𝑦)−𝜀2

. [0.1,0.4,0.5] [0.2,0.6,0.2] [0.8,0.0,0.2] [0.4,0.3,0.3]

[0.2,0.8,0.0] [0.5,0.0,0.5] [0.2,0.2,0.6] [0.1,0.8,0.1]

[0.1,0.5,0.4] [0.2,0.6,0.2] [0.4,0.1,0.5] [0.3,0.1,0.6]

Calculate the threshold for 𝐼𝑉𝐴𝐸(𝐶𝐷, 𝑃𝐷, 𝐷𝐷). 𝐼𝑉𝐴𝐸(𝐶𝐷, 𝑃𝐷, 𝐷𝐷) = 𝐼𝑉𝐴𝐸(𝐶𝐷, 𝑃𝐷, 𝐷𝐷) = +𝐼{[0.5,0.5,0.0], [0.2,0.2,0.6], [0.2,0.2,0.6], [0.1,0.0,3.0]}. Which is computed by adding the IVCE values computed for each phase of the aerospace engineering. Calculate the mean of 𝐼𝑉𝐴𝐸(𝐶𝐷, 𝑃𝐷, 𝐷𝐷)𝑚 . 𝐼𝑉𝐶𝐸(𝐶𝐷, 𝑃𝐷, 𝐷𝐷)𝑚 =

[0.4,0.5,0.1] [0.2,0.5,0.3] [0.3,0.4,0.3] [0.1,0.6,0.3] 4

,

4

,

4

,

4

>.

Apply summation operation over each of the mean values obtained. [0.5] [0.5] [0.1]

[0.3] [0.5] [0.1]

[0.2] [0.2] [0.2]

The efficiency achieved by the neutrosophic set in conceptual design phase is max(0.5,0.3,0.2) = 0.5 which is the maximum of the obtained values 0.5, 0.3, 0.2, efficiency towards preliminary phase is max(0.5,0.5,0.2) = 0.5 which is the maximum of the obtained values 0.5,0.5, and 0.2, and efficiency towards detailed design phase is max(0.1,0.1,0.2) = 0.25 which is the maximum of the obtained values 0.1, 0.1, and 0.2. Hence an inference is drawn that efficiency achieved by the neutrosophy logic is average i.e., 0.5 for conceptual design and preliminary design phases of aerospace engineering. Whereas, the efficiency towards detailed design is below average i.e., 0.2.

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5. NEUTROSPOPHIC SET APPLICATIONS IN MECHANICAL ENGINEERING The mechanical engineering discipline consists of five important phases i.e., planning, system level design, system level prototyping, testing, and pilot manufacturing [21, 22, 23, 24, 25]. Uncertainties exist in every phase of the mechanical engineering, some of the sources of uncertainties are unavoidable flaws in the structures, imprecision in the instrumental design, lack of information during modeling, probability based decision making, unbeatable design constraints, mechanical resistance of structural materials, sophisticated strength measurement models, varying sensitivity coefficient of materials, sample centering from measurement, equipment failures, transducer errors, voltage measurement errors, and so on [26, 27, 28, 29, 30]. A high level view of the neutrosophic based mechanical engineering approach is given in Figure 3.

Figure 3. A high level view of the neutrosophic based mechanical engineering.

Property-1 Neutrosophic based fuzzy t-norms, and a fuzzy t-conorms unions and disjunctions operation allows optimization of imprecise topological models. Applicability The property-1 of neutrosophic sets is applied in the mechanical engineering for addressing the issues like magnetic suspension in the vehicles, handling friction during movement of components, tilling of load carrying devices, additive manufacturing of powerful batteries, predicting the lifetime of mechanical devices, and so on.

Neutrosophic Set Theory and Engineering Applications

19

Description Consider the real-time parameters of mechanical engineering process with truth, false, and inderminacy components 𝑀𝐸𝑃 = {𝑚𝑒𝑝1 , 𝑚𝑒𝑝2 , 𝑚𝑒𝑝3 , . . 𝑚𝑒𝑝𝑛 ; 𝑇(𝑥), 𝐹(𝑥), 𝐼(𝑥)}, where 𝑥, 𝑦 ∈ 𝑚𝑒𝑝𝑖 . 𝑇(𝑥)⋁𝑇(𝑦) = max{𝑇(𝑥), 𝑇(𝑦)} = 𝑇(𝑥) + 𝑇(𝑦) − 𝑇(𝑥) ∗ 𝑇(𝑦) 𝑇(𝑥)⋁𝑇(𝑦) = min{𝑇(𝑥) + 𝑇(𝑦), 1} 𝐹(𝑥)⋁𝐹(𝑦) = max{𝐹(𝑥), 𝐹(𝑦)} = 𝐹(𝑥) + 𝐹(𝑦) − 𝐹(𝑥) ∗ 𝐹(𝑦) 𝐹(𝑥)⋁𝐹(𝑦) = min{𝐹(𝑥) + 𝐹(𝑦), 𝐹(𝑥)} 𝐼(𝑥)⋁𝐼(𝑦) = max{𝐼(𝑥), 𝐼(𝑦)} = 𝐼(𝑥) + 𝐼(𝑦) − 𝐼(𝑥) ∗ 𝐼(𝑦) 𝐼(𝑥)⋁𝐼(𝑦) = min{𝐼(𝑥) + 𝐼(𝑦), 𝐼(𝑦)}

Usefulness The neutrosophic based fuzzy t-norms, and a fuzzy t-conorms unions and disjunctions operation efficiently handles the imprecise components of prototyping and testing phases of the mechanical engineering. Property-2 For any two neutrosophic sets the De Morgan’s law holds good which increases the tolerance level of the state of the art practicing relating to engineering. Applicability The property-2 of neutrosophic sets is applied in the mechanical engineering for addressing the issues like development of scalable design, articulation between body and component, identification of loaded paths, redesigning of thermal components, implementation of moving parts of the machines, testing of engines, recognizing structural flaws, and so on. Description Consider the real-time parameters of mechanical engineering process with truth, false, and inderminacy components 𝑀𝐸𝑃 = {𝑚𝑒𝑝1 , 𝑚𝑒𝑝2 , 𝑚𝑒𝑝3 , . . 𝑚𝑒𝑝𝑛 ; 𝑇(𝑥), 𝐹(𝑥), 𝐼(𝑥)}, where 𝑥 ∈ 𝑚𝑒𝑝𝑖 and i ranges from 1 to n. 𝑐 𝑐 𝑐 𝑐 (⋃𝑖=4 𝑖=1 𝑇(𝑥)) = ∏𝑖∈𝐼 𝑇(𝑥) and (∏𝑖∈𝐼 𝑇(𝑥) ) = ⋃𝑖∈𝐼 𝑇(𝑥) 𝑐 𝑐 𝑐 𝑐 (⋃𝑖=4 𝑖=1 𝐹(𝑥)) = ∏𝑖∈𝐼 𝐹(𝑥) and (∏𝑖∈𝐼 𝐹(𝑥) ) = ⋃𝑖∈𝐼 𝐹(𝑥) 𝑐 𝑐 𝑐 𝑐 (⋃𝑖=4 𝑖=1 𝐼(𝑥)) = ∏𝑖∈𝐼 𝐼(𝑥) and (∏𝑖∈𝐼 𝐼(𝑥) ) = ⋃𝑖∈𝐼 𝐼(𝑥)

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Usefulness The De Morgan’s law of neutrosophic sets helps in handling the uncertainties involved in the design, evaluation, and controlling applications of the mechanical engineering. Property-3 The nonstandard addition and subtraction of two neutrosophic sets yields real neutrosophic numbers as output which is fault tolerant to adhoc neutrosophic engineering problems. Applicability The property-3 of neutrosophic sets is applied in the mechanical engineering for addressing the issues like plastic deformation, analysis of the stress on materials, identification of chances of failure early during design, noise vibration analysis, calculation of distortion in engines, pistons durability enhancement, thermal modeling, oil flow analysis, and so on. Description Consider the real-time parameters of mechanical engineering process with truth, false, and inderminacy components 𝑀𝐸𝑃 = {𝑚𝑒𝑝1 , 𝑚𝑒𝑝2 , 𝑚𝑒𝑝3 , … 𝑚𝑒𝑝𝑛 ; 𝑇(𝑥), 𝐹(𝑥), 𝐼(𝑥)} , where 𝑥, 𝑦 ∈ 𝑚𝑒𝑝𝑖 and 𝜀1 and 𝜀2 are infinitesimals 𝑇(𝑥) + 𝑇(𝑦) = (𝑇(𝑥) − 𝜀1 , 𝑇(𝑥)) + (𝑇(𝑦) − 𝜀2 , 𝑇(𝑦)) 𝑇(𝑥) − 𝑇(𝑦) = (𝑇(𝑥) − 𝜀1 ) − (𝑇(𝑦) − 𝜀2 ) = 𝑇(𝑥) − 𝑇(𝑦) − 𝜀1 + 𝜀2 𝐹(𝑥) + 𝐹(𝑦) = (𝐹(𝑥) − 𝜀1 , 𝐹(𝑥)) + (𝐹(𝑦) − 𝜀2 , 𝐹(𝑦)) 𝐹(𝑥) − 𝐹(𝑦) = (𝐹(𝑥) − 𝜀1 ) − (𝐹(𝑦) − 𝜀2 ) = 𝐹(𝑥) − 𝐹(𝑦) − 𝜀1 + 𝜀2 𝐼(𝑥) + 𝐼(𝑦) = (𝐼(𝑥) − 𝜀1 , 𝐼(𝑥)) + (𝐼(𝑦) − 𝜀2 , 𝐼(𝑦)) 𝐼(𝑥) − 𝐼(𝑦) = (𝐼(𝑥) − 𝜀1 ) − (𝐼(𝑦) − 𝜀2 ) = 𝐼(𝑥) − 𝐼(𝑦) − 𝜀1 + 𝜀2

Usefulness The addition and subtraction properties of neutrosophic sets provide good sustainable services by efficiently handling the inderminacy factors in the real-time engineering parameters. Illustration of Interval Value Based Analysis of Efficiency Achieved by the Neutrosophic Based Mechanical Engineering Let us analyze the efficiency of the neutrosophic sets in handling the uncertainties involved in the mechanical engineering problems. The analysis begins by gathering the

21

Neutrosophic Set Theory and Engineering Applications

parameters of each stages of the mechanical engineering i.e., planning i.e., 𝑃 = {𝑝1 (80), 𝑝2 (75), 𝑝3 (55), 𝑝4 (45)} , system design 𝑆𝐷 = {𝑠𝑑1 (77), 𝑠𝑑2 (80), 𝑠𝑑3 (22), 𝑠𝑑4 (32)} , system prototyping 𝑆𝑃 = {𝑠𝑝1 (64), 𝑠𝑝2 (34), 𝑠𝑝3 (67), 𝑠𝑝4 (21)}, and testing 𝑇 = {𝑡1 (41), 𝑡2 (19), 𝑡3 (50), 𝑡4 (97)}, and pilot manufacturing 𝑃𝑀 = {𝑝𝑚1 (41), 𝑝𝑚2 (19), 𝑝𝑚3 (50), 𝑝𝑚4 (97 . The efficiency achieved by the mechanical engineering parameters is varied between low (l), medium (m) and high (h). Create a matrix for interval value based civil engineering IVME consisting of truth, false, and inderminacy values i.e., 𝐼𝑉𝑀𝐸(𝑃, 𝑆𝐷, 𝑆𝑃, 𝑇, 𝑃𝑀) =< 𝑇(𝑥), 𝐹(𝑥), 𝐼(𝑥) >. [0.1,0.9,0.0] [0.4,0.4,0.2] [0.8,0.1,0.1] [0.2,0.3,0.5] [0.3,0.2,0.5]

[0.4,0.5,0.1] [0.5,0.2,0.3] [0.2,0.2,0.6] [0.6,0.1,0.3] [0.6,0.2,0.3]

[0.3,0.3,0.4] [0.6,0.2,0.2] [0.4,0.3,0.3] [0.2,0.5,0.3] [0.1,0.4,0.3]

Calculate the threshold for 𝐼𝑉𝐶𝐸(𝑃, 𝑆𝐷, 𝑆𝑃, 𝑇, 𝑃𝑀). 𝐼𝑉𝐶𝐸(𝑃, 𝑆𝐷, 𝑆𝑃, 𝑇, 𝑃𝑀)+= {[0.5,0.5,0.0], [0.2,0.6,0.2], [0.3,0.2,0.5], [0.1,0.8,0.1]} . Which is computed by adding the IVCE values computed for each phase of the mechnical engineering. Calculate

the

mean 𝑚

𝐼𝑉𝐶𝐸(𝑉𝐶𝐸(𝑃, 𝑆𝐷, 𝑆𝑃, 𝑇, 𝑃𝑀))

𝐼𝑉𝐶𝐸(𝑉𝐶𝐸(𝑃, 𝑆𝐷, 𝑆𝑃, 𝑇, 𝑃𝑀))

of

,

4

,

4

,

i.e.,

=
.

Apply summation operation over each of the mean values obtained. [0.6] [0.5] [0.6] [0.6] [0.4]

[0.5] [0.6] [0.8] [0.4] [0.4]

[0.7] [0.8] [0.9] [0.2] [0.5]

The efficiency achieved by the neutrosophic set in planning phase is max(0.6,0.5,0.7) = 0.7 which is maximum of obtained values 0.6, 0.5, and 0.7, efficiency towards system design phase is max(0.5,0.6,0.8) = 0.8 which is maximum of obtained values 0.5, 0.6, and 0.8, efficiency towards system prototyping phase is max(0.6,0.8,0.9) = 0.9 which is maximum of obtained values 0.6, 0.8, and 0.9, efficiency towards testing phase is max(0.6,0.4,0.2) = 0.6 which is maximum of obtained values 0.6, 0.4, and 0.2, and efficiency towards testing phase is max(0.4,0.4,0.5) = 0.5 which is the maximum of obtained values 0.4, 0.4, and 0.5. Hence an inference is drawn that efficiency achieved by the neutrosophy logic is high

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i.e., 07, 0.8, and 0.9 for planning, system design, and system prototype phases of mechanical engineering. Whereas, the efficiency towards pilot manufacturing is average i.e., 0.5.

CONCLUSION The paper discusses the application of neutrosophic set theory to handle the uncertainty issues in the three major engineering disciplines i.e., Civil engineering, aerospace engineering, and mechanical engineering. For each of the mentioned engineering disciplines the sources of uncertainty are mentioned and the properties of the neutrosophic sets useful in handling those uncertainties are also presented. Finally interval value analysis is performed to evaluate the efficiency achieved by neutrosophic sets in handling the uncertainties in the engineering process. The efficiency achieved in handling uncertainties is nearer to optimal. Further detailed analysis and performance modeling of the neutrosophic set theory in handling the uncertainties of other engineering disciplines like computer science, biochemical, electronics, electrical, and instrumentation technology can be carried out. The efficiency of the neutrosophic set theory in addressing uncertainty challenges in such engineering discipline can be quantified towards the performance metrics like accuracy, error rate, throughput, response time, and so on.

REFERENCES [1] [2]

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Peng, Xindong, Jingguo Dai. A bibliometric analysis of neutrosophic set: Two decades review from 1998 to 2017. Artificial Intelligence Review, Vol. 53, No. 1 (2020), pp. 199-255. Jana, Chiranjibe, Madhumangal Pal, Faruk Karaaslan, Wang, JQ. Trapezoidal neutrosophic aggregation operators and their application to the multi-attribute decision-making process.” Scientia Iranica. Transaction E, Industrial Engineering, Vol. 27, No. 3, (2020), pp. 1655-1673. Hanafy, IM; Salama, AA; Mahfouz, KM. Neutrosophic classical events and its probability. In International Journal of Mathematics and Computer Applications Research (IJMCAR), Vol. 3, No. 1, (2013), pp. 171-178. Liang, Jiye, Kwai-Sang Chin, Chuangyin Dang, Richard CM Yam. A new method for measuring uncertainty and fuzziness in rough set theory. International Journal of General Systems, Vol. 31, No. 4, 2002, pp. 331-342. Wang, Chong, Hermann G. Matthies. “Epistemic uncertainty-based reliability analysis for engineering system with hybrid evidence and fuzzy variables.” Computer Methods in Applied Mechanics and Engineering, Vol. 355, No. 9, (2019), pp. 438-455. Bilal M Ayyub; Chao, RU. Uncertainty modeling in civil engineering with structural and reliability applications. In Uncertainty modeling and analysis in civil engineering, (1997), pp. 1-8. de Borst, R; Lambertus J. Sluys, Muhlhaus, HB; Jerzy Pamin. Fundamental issues in finite element analyses of localization of deformation. International Journal for Computer-Aided Engineering, Vol. 10, No. 2, (1993), pp. 99-121. Jun Ye. Trapezoidal neutrosophic set and its application to multiple attribute decision-making. In Neural Computing and Applications, Vol. 26, No. 5, (2015), pp. 1157-1166. Abdel-Basset, Mohamed, Mumtaz Ali, Asmaa Atef. Uncertainty assessments of linear time-cost tradeoffs using neutrosophic set. Computers & Industrial Engineering, Vol. 141, No. 9, (2020), pp. 106-286. Hariri-Ardebili, Mohammad Amin, Bruno Sudret. Polynomial chaos expansion for uncertainty quantification of dam engineering problems, Engineering Structures, Vol. 203, No. 22, pp. 109-131. Upadhya, AR; Dayananda, GN; Kamalakannan, GM; Ramaswamy Setty, J; Christopher Daniel, J. Autoclaves for aerospace applications- Issues and challenges. In International Journal of Aerospace Engineering, Vol. 11, No. 7, (2011), pp. 111. Kalyan Mondal, Surapati Pramanik. Neutrosophic tangent similarity measure and its application to multiple attribute decision making. Neutrosophic sets and systems, Vol. 9, No. 9, (2015), pp. 80-87.

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[18] Said Broumi, Florentin Smarandache, Mohamed Talea, Assia Bakali. An introduction to bipolar single valued neutrosophic graph theory. In Applied Mechanics and Materials, vol. 841, No. 9, (2016), pp. 184-191. [19] Mukhopadhaya, Jayant, Brian T. Whitehead, John F. Quindlen, Juan J. Alonso, Andrew W. Cary. Multi-fidelity modeling of probabilistic aerodynamic databases for use in aerospace engineering., International Journal for Uncertainty Quantification, Vol. 10, No. 5, (2020), pp. 1-20 [20] Dasari, Siva Krishna, Abbas Cheddad, Petter Andersson. Predictive modelling to support sensitivity analysis for robust design in aerospace engineering., Structural and Multidisciplinary Optimization, Vol. 10, No. 2, (2020), pp. 1-16. [21] Kenneth W Chase, William H. Greenwood. Design issues in mechanical tolerance analysis. Manufacturing Review, Vol. 1, No. 1, (1988), pp. 50-59. [22] Paul D Arendt, Daniel W. Apley, Wei Chen. Quantification of model uncertainty: Calibration, model discrepancy, and identifiability. In Journal of Mechanical Design, Vol. 134, No. 10, (2012), pp. 1-10. [23] Said Broumi, Florentin Smarandache. Correlation coefficient of interval neutrosophic set. In Applied Mechanics and Materials, Vol. 436, No. 10, (2013), pp. 511-517. [24] Bhargavi, K; Sathish Babu, B. Uncertainty Aware Resource Provisioning Framework for Cloud Using Expected 3-SARSA Learning Agent: NSS and FNSS Based Approach. Cybernetics and Information Technologies, Vol. 19, No. 3, (2019), pp. 94-117. [25] Saqlain, Muhammad, Naveed Jafar, Sana Moin, Muhammad Saeed, Said Broumi. Single and Multi-valued Neutrosophic Hypersoft set and Tangent Similarity Measure of Single valued Neutrosophic Hypersoft Sets, Neutrosophic Sets and Systems, Vol. 32, No. 1, (2020), pp. 20-39. [26] Khalid, Huda E. Neutrosophic Geometric Programming (NGP) with (max-product) Operator, An Innovative Model, Neutrosophic Sets and Systems, Vol. 32, No. 1 (2020), pp. 1-16. [27] Iryna, Shchur, Yu Zhong, Wen Jiang, Xinyang Deng, Jie Geng. Single-Valued Neutrosophic Set Correlation Coefficient and Its Application in Fault Diagnosis, Symmetry, Vol. 12, No. 8, 2020, pp. 137-149. [28] Kumar, Anil, Gandhi, CP; Yuqing Zhou, Hesheng Tang, Jiawei Xiang. Fault diagnosis of rolling element bearing based on symmetric cross entropy of neutrosophic sets, Measurement, Vol. 152, No. 7, (2020), pp. 109-118. [29] Khalid, Huda E. Neutrosophic Geometric Programming (NGP) Problems Subject to (\/,.) Operator; the Minimum Solution, Neutrosophic Sets and Systems, Vol. 32, No. 1, (2020), pp. 1-20.

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[30] Son, Nguyen Thi Kim, Nguyen Phuong Dong, Mohamed Abdel-Basset, Gunasekaran Manogaran, Hoang Viet Long. On the stabilizability for a class of linear time-invariant systems under uncertainty, Circuits, Systems, and Signal Processing, Vol. 39, No. 2, 2020, pp. 919-960.

In: Decision-Making with Neutrosophic Set Editor: Harish Garg

ISBN: 978-1-53619-419-7 c 2021 Nova Science Publishers, Inc.

Chapter 2

A N EW T YPE OF Q UASI O PEN F UNCTIONS IN N EUTROSOPHIC T OPOLOGICAL E NVIRONMENT M. Parimala1,∗, C. Ozel2†, F. Smarandache3‡ and M. Karthika1,§ Department of Mathematics, Bannari Amman Institute of Technology, Sathyamangalam, Tamil Nadu, India 2 Department of Mathematics, King Abdulaziz University, Jeddah, KSA 3 Mathematics & Science Department, University of New Mexico, Gallup, NM, US 1

Abstract Neutrosophic topological space is a generalization of intuitionistic topological space. Each neutrosophic set in the neutrosophic topological space has three component namely, membership, indeterminacy and non-membership grade. These neutrosophic sets are called neutrosophic open sets. The objective of this chapter is to introduce neutrosophic quasi αψ open and αψ closed functions and investigate the properties of neutrosophic quasi αψ open and αψ closed functions whose image and pre-image are the subsets of neutrosophic αψ open and αψ closed sets. This chapter motivates to investigate the neutrosophic quasi αψ homeomorphism, neutrosophic quasi supra αψ open and αψ closed functions.

Keywords: neutrosophic αψ-closed set, neutrosophic αψ-open set and neutrosophic quasi αψ-f unction AMS(2010) Subject classification: 54A05, 54D10, 54F65, 54G05

1.

INTRODUCTION

Atanassov [1] defined the notion of intuitionistic fuzzy sets, which is a generalized form of Zadeh’s [2] fuzzy set. D. Coker [3] introduced intuitionistic topological space. Intuitionistic ∗

Corresponding Author’s Email: Corresponding Author’s Email: ‡ Corresponding Author’s Email: § Corresponding Author’s Email: †

[email protected]. [email protected]. [email protected]. [email protected].

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M. Parimala, C. Ozel, F. Smarandache et al.

fuzzy sets in intuitionistic fuzzy topological space are open sets. The basic operations and properties of them are studied in his work. Later, researchers introduced and investigated many notions as a generalization of intuitionistic fuzzy topological space. F. Smarandache [4, 5] introduced and studied neutrosophic sets (NS) and neutrosophic logic. Later, A. A. Salama [6] introduced and studied neutrosophic topology. This approach leads to many investigations in this area. Since then more research have been identified in the field of neutrosophic topology [7, 8, 9, 10, 11, 12, 13, 14, 15], neutrosophic ideals [13, 16, 17], etc. Parimala et al. [14] investigated the application of neutrosophic set in decision making. Neutrosophic minimal structure space is a space which satisfying first axiom of neutrosophic topological space. This new notion introduced by Karthika et al. [8] Application of neutrosophic minimal structure space in decision making studied in their work. Parimala et al. [12] introduced neutrosophic αψ closed set in neutrosophic topological space. This notion is a generalization of neutrosophic closed set. Later, Parimala et al. [11] investigated the connectedness between neutrosophic αψ open set and studied the properties of neutrosophic αψ connected space.

Motivation and Objective The notion of neutrosophic αψ closed set motivates to propose this novel neutrosophic quasi αψ open and αψ closed function. This novel notion is also a generalization of neutrosophic quasi open and closed function. The objective of our work is to introduce neutrosophic quasi αψ open and αψ closed functions whose image and pre-image are the subset of neutrosophic αψ open and αψ closed set in neutrosophic topological space.

Limitations Neutrosophic quasi open function is neutrosophic quasi αψ open function. Similarly, neutrosophic quasi closed function is neutrosophic quasi αψ closed function but converse may not be true. In this chapter, the range of an element is restricted to unit interval. The chapter is organized as follows: The definitions which are required for the sequel is presented in chapter 2. The proposed concept of neutrosophic quasi αψ open function is defined and its properties are studied in chapter 3. In chapter 4, neutrosophic quasi αψ closed function is defined and its properties are studied. Future work and Conclusion part of our work is presented in chapter 5. Through out this chapter, spaces means neutrosophic topological spaces on which no separation axioms are assumed unless otherwise mentioned and f : (X, τ ) → (Y, σ) denotes a neutrosophic function f of a space (X, τ ) into a space (Y, σ). Let A be a subset of a space X. The neutrosophic closure and the neutrosophic interior of A are denoted by ncl(A) and nint(A), respectively.

2.

BASIC D EFINITIONS

This section contains the collection of some existing definitions in [3, 8, 10, 11, 12, 13, 14, 15, 16, 18, 19] which are helpful for this work.

A New Type of Quasi Open Functions in Neutrosophic Topological Environment

29

Definition 2.1. Let X and I be non-empty sets and the interval [0, 1], respectively. A NS A is defined by A = {he, µA (e), σA (e), νA(e)i : e ∈ X} where the map of membership µA , indeterminacy σA and non-membership νA respectively defined from non-empty set X to I , ∀e ∈ X to the set A and with condition that the sum of µA (e), σA(e), νA (e) should not exceed 3 and less than zero, ∀e ∈ X. Definition 2.2. Let the two NSs be of the form A = {he, µA (e), σA(e), νA (e)i : e ∈ X} and B = {he, µB (e), σB (e), νB (e)i : e ∈ X}. Then (i) A is a subset of B if and only if membership of A is less than or equal to membership of B, indeterminacy and non-membership of A are respectively, greater than or equal to indeterminacy and non-membership of B. (ii) A = {he, νA (e), σA(e), µA (e)i : e ∈ X}; (iii) Union of two NS’s A and B is set of all maximum of membership of A and B, minimum of indeterminacy function of A and B and non-membership function of A and B, for each element in X. (iv) Intersection of two NS’s A and B is set of all minimum of membership of A and B, maximum of indeterminacy function of A and B and non-membership function of A and B, for each element in X. Definition 2.3. A neutrosophic topology (NT) on X is a collection τ of NS’s in X holds the following properties (i) Null set he, (0, 1, 1)i and universal set he, (1, 0, 0)i are in τ , each e ∈ X. (ii) Union of any NS’s of τ is in τ . (iii) Intersection of A, B ∈ τ is in τ ; Note: Every NS in τ is a neutrosphic open sets (NOS) and its complements are neutrosophic closed sets (NCS). Definition 2.4. Let A be a NS in NTS (X, τ ). Then neutrosophic interior of the given NS A is maximum of all NCS contained in A. Neutrosophic closure of the given NS A is minimum of all NOS contains A. Neutrosophic interior of A is denoted by nint(A) and neutrosophic closure of A is denoted by ncl(A). Definition 2.5. (X, τ ) is called

Let (X, τ ) be a neutrosophic topological space and a subset A of

1. a neutrosophic pre-open set, if A is a subset of neutrosophic interior of neutrosophic closure of A. 2. a neutrosophic semi-open set, if A is a subset of neutrosophic closure of neutrosophic interior of A.

30

M. Parimala, C. Ozel, F. Smarandache et al. 3. a neutrosophic α-open set, if A is a subset of neutrosophic interior of neutrosophic closure of neutrosophic interior of A.

Definition 2.6. Let (X, τ ) be a neutrosophic topological space and a subset A of (X, τ ) is called 1. a neutrosophic pre-closed set if the complement is a neutrosophic pre-open set. 2. a neutrosophic semi-closed set if the complement is a neutrosophic semi-open set. 3. a neutrosophic α-closed set if complement is a neutrosophic α-open set. Definition 2.7. A subset A of a neutrosophic topological space (X, τ ) is called 1. a neutrosophic semi-generalized closed (briefly, nsg-closed) set if intersection of all neutrosophic semi closed sets which contains A is a subset of U whenever A is a subset of U and U is semi-open in (X, τ ). 2. a neutrosophic ψ-closed set if scl(A) ⊆ U whenever A ⊆ U and U is N sg-open in (X, τ ). 3. a neutrosophic αψ-closed (briefly, nαψ-closed) set if ψcl(A) ⊆ U whenever A ⊆ U and U is N α-open in (X, τ ). Definition 2.8. Let A be a NS in neutrosophic topological space (X, τ ). Then 1. nαψ interior of A is the minimum of nαψOS in X contained in A and it is denoted by nαψint(A). 2. nαψ-closure of A is the maximum of nαψCS in X which contains A. and it is denoted by nαψcl(A).

3.

O N N EUTROSOPHIC QUASI αψ-O PEN F UNCTIONS

In this section, we introduce the definition of neutrosophic quasi αψ-open,nαψneighborhood of a point x . Definition 3.1. A neutrosophic function f : X → Y is said to be neutrosophic quasi αψ-open (briefly, nq-αψ-open), if the image of every neutrosophic αψ-open set in X is neutrosophic open in Y . It is evident that, the concept of neutrosophic quasi αψ-openness and neutrosophic αψ-continuity coinside if the function is bijective. Theorem 3.1. A neutrosophic function f : X → Y is nq-αψ-open if and only if for every subset U of X, f (nαψint(U )) ⊂ nint(f (U )) . Proof. Let f be a nq-αψ-open function. Now, we have nint(U ) ⊂ U and nαψint(U ) is a nαψ-open set. Hence, we obtain that f (nαψint(U )) ⊂ f (U ). As f (nαψint(U )) is open,f (nαψint(U )) ⊂ nint(f (U )).

A New Type of Quasi Open Functions in Neutrosophic Topological Environment

31

Conversely, assume that U is a nαψ-open set in X. Then f (U ) = f (nαψint(U )) ⊂ nint(f (U )) but nint(f (U )) ⊂ f (U ). Consequently, f (U ) = nint(f (U )) and hence f is nq-αψ- open. Theorem 3.2. If a neutrosophic function f : X → Y is nq-αψ-open, then nαψint(f −1 (G)) ⊂ f −1 (nint(G)) for every subset G of Y . Proof. Let G be any arbitrary subset of Y . Then, nαψint(f −1 (G)) is a nαψ-open set in X and f is nq-αψ-open, then f (nαψint(f −1 (G))) ⊂ nint(f (f −1 (G))) ⊂ nint(G). Thus, nαψint(f −1 (G)) ⊂ f −1 (nint(G)). Definition 3.2. A subset A is said to be a nαψ-neighborhood of a point x of X if there exists a nαψ-open set U such that x ∈ U ⊂ A. Theorem 3.3. For a neutrosophic function f : X → Y , the following are equivalent (i) f is nq-αψ-open; (ii) for each subset U of X, f (nαψint(U )) ⊂ nint(f (U )); (iii) for each x ∈ X and each nαψ-neighborhood U of x in X, there exists a neighborhood V of f (x) in Y such that V ⊂ f (U ). Proof. (i) ⇒ (ii) It follows from Theorem 3.1. (ii) ⇒ (iii) Let x ∈ X and U be an arbitrary nαψ-neighborhood of x ∈ X. Then, there exists a nαψ-open set V in X such that x ∈ V ⊂ U . Then by (ii), we have f (V ) = f (nαψint(V )) ⊂ nint(f (V )) and hence f (V ) is open in Y such that f (x) ∈ f (V ) ⊂ f (U ). (iii) ⇒ (i) Let U be an arbitrary nαψ-open set in X. Then for each y ∈ f (U ), by (iii) there exists a neighborhood Vy of y in Y such that Vy ⊂ f (U ). As Vy is a neighborhood of y, there exists an open set Wy in Y such that y ∈ Wy ⊂ Vy . Thus S f (U ) = {Wy : y ∈ f (U )} which is an open set in Y . This implies that f is nq-αψ-open function. Theorem 3.4. A neutrosophic function f : X → Y is nq-αψ-open if and only if for any subset B of Y and for any nαψ-closed set F of X containing f −1 (B), there exists a neutrosophic closed set G of Y containing B such that f −1 (G) ⊂ F . Proof. Suppose f is nq-αψ-open. Let B ⊂ Y and F be a nαψ-closed set of X containing f −1 (B). Now, put G = Y − f (X − F ). It is clear that f −1 (B) ⊂ F ⇒ B ⊂ G. Since f is nq-αψ- open, we obtain G as a neutrosophic closed set of Y . Moreover, we have f −1 (G) ⊂ F . Conversely, let U be a nαψ-open set of X and put B = Y − f (U ). Then X − U is a nαψ-closed set in X containing f −1 (B). By hypothesis, there exists a neutrosophic closed set F of Y such that B ⊂ F and f −1 (F ) ⊂ X − U . Hence, we obtain f (U ) ⊂ Y − F . On the other hand, it follows that B ⊂ F , Y − F ⊂ Y − B = f (U ). Thus we obtain

32

M. Parimala, C. Ozel, F. Smarandache et al.

f (U ) = Y − F which is neutrosophic open and hence f is a nq-αψ-open function. Theorem 3.5. A neutrosophic function f : X → Y is nq-αψ-open if and only if f −1 (cl(B)) ⊂ nαψcl(f −1(B)) for every subset B of Y . Proof. Suppose that f is nq-αψ-open. For any subset B of Y , f −1 (B) ⊂ αψcl(f −1 (B)). Therefore, by Theorem 3.3 there exists a neutrosophic closed set F in Y such that B ⊂ F and (f −1 (F )) ⊂ nαψcl(f −1 (B)). Therefore, we obtain f −1 (ncl(B)) ⊂ (f −1 (F )) ⊂ nαψcl(f −1 (B)). Conversely, let B ⊂ Y and F be a nαψ-closed set of X containing f −1 (B). Put W = nclY (B), then we have B ⊂ W and W is neutrosophic closed and f −1 (W ) ⊂ nαψcl(f −1 (B)) ⊂ F . Then by Theorem 3.4., f is nq-αψ-open. Theorem 3.6. Two neutrosophic function f : X → Y and g : Y → Z and g ◦ f : X → Z is nq-αψ-open. If g is continuous injective function, then f is nq-αψ-open. Proof. Let U be a nαψ-open set in X, then (g ◦ f )(U ) is open in Z, since g ◦ f is nq-αψ-open. Again g is an injective continuous function, f (U ) = g −1 (g ◦ f (U )) is open in Y . This shows that f is nq-αψ-open

4.

O N N EUTROSOPHIC QUASI αψ-C LOSED F UNCTIONS

Definition 4.1. A neutrosophic function f : X → Y is said to be neutrosophic quasi αψ-closed (briefly, nq-αψ-closed), if the image of every neutrosophic αψ-closed set in X is neutrosophic closed in Y . Theorem 4.1. Every nq-αψ-closed function is neutrosophic closed as well as neutrosophic αψ-closed. Proof. It is obvious. The converse of the above theorem need not be true by the following example. Example 4.1. Let a neutrosophic function f : X → Y . Let X = {p, q, r} and τN 1 = {0, A, B, C, D, 1} ia a neutrosophic topology on X, Where D p q r p q r p q r E A = x, ( , , ), ( , , ), ( , , ) , 0.4 0.3 0.2 0.3 0.4 0.5 0.2 0.3 0.2 D p q r p q r p q r E B = x, ( , , ), ( , , ), ( , , ) , 0.2 0.4 0.6 0.3 0.2 0.1 0.5 0.4 0.3 D p q r p q r p q r E C = x, ( , , ), ( , , ), ( , , ) , 0.4 0.4 0.6 0.3 0.2 0.1 0.2 0.3 0.2 D p q r p q r p q r E D = x, ( , , ), ( , , ), ( , , ) 0.2 0.3 0.2 0.3 0.4 0.5 0.5 0.4 0.3

A New Type of Quasi Open Functions in Neutrosophic Topological Environment

33

and Let Y = {p, q, r} and τN 2 = {0, E, F, G, H, 1} is a neutrosophic topology on Y , Where D p q r p q r p q r E E = y, ( , , ), ( , , ), ( , , ) , 0.1 0.2 0.3 0.3 0.2 0.3 0.5 0.6 0.4 D p q r p q r E p q r , ), ( , , ), ( , , ) , F = y, ( , 0.4 0.3 0.2 0.3 0.4 0.5 0.2 0.3 0.2 D p q r p q r p q r E , ), ( , , ), ( , , ) , G = y, ( , 0.4 0.3 0.3 0.3 0.2 0.3 0.2 0.3 0.2 D p q r p q r p q r E H = y, ( , , ), ( , , ), ( , , ) . 0.1 0.2 0.2 0.3 0.5 0.5 0.5 0.6 0.4 Here f (p) = p, f (q) = q, f (r) = r. Then clearly f is nαψ-closed as well as neutrosophic closed but not nq-αψ-closed. Lemma 4.1. If a neutrosophic function is nq-αψ-closed, then f −1 (nint(B)) ⊂ nαψint(f −1 (B)) for every subset B of Y . Proof. Let B be any arbitrary subset of Y . Then, nαψint(f −1 (G)) is a nαψ-closed set in X and f is nq-αψ-closed, then f (nαψ- nint(f −1 (B))) ⊂ nint(f (f −1 (B))) ⊂ nint(B). Thus, f (nαψint(f −1 (B))) ⊂ f −1 (nint(B)). Theorem 4.2. A neutrosophic function f : X → Y is nq-αψ-closed if and only if for any subset B of Y and for any nαψ-open set G of X containing f −1 (B), there exists an open set U of Y containing B such that f −1 (U ) ⊂ G. Proof. This proof is similar to that of theorem 3.4. Definition 4.2. A neutrosophic function f : X → Y is called nαψ ∗ -closed if the image of every nαψ-closed subset of X is nαψ-closed in Y . Theorem 4.3. If f : X → Y and g : Y → Z be any nq-αψ-closed functions, then g ◦ f : X → Z is a nq-αψ-closed function. Proof. It is obvious. Theorem 4.4. Let f : X → Y and g : Y → Z be any two neutrosophic functions, then (i) If f is nαψ-closed and g is nq-αψ-closed, then g ◦ f is neutrosophic closed; (ii) If f is nq-αψ-closed and g is nq-αψ-closed, then g ◦ f is nαψ ∗ -closed; (iii) If f is nαψ ∗ -closed and g is nq-αψ-closed, then g ◦ f is nq-αψ- closed. Proof. It is obvious. Theorem 4.5. Let f : X → Y and g : Y → Z be any two neutrosophic functions such that g ◦ f : X → Z is nq-αψ-closed.

34

M. Parimala, C. Ozel, F. Smarandache et al. (i) If f is nαψ-irresolute surjective, then g is is neutrosophic closed;

(ii) If g is nαψ-continuous injective, then f is nαψ ∗ -closed. Proof. (i) Suppose that F is an arbitrary neutrosophic closed set in Y . As f is nαψirresolute, f −1 (F ) is nαψ-closed in X. Since g ◦ f is nq-αψ-closed and f is surjective, (g ◦ f )(f −1 (F )) = g(F ), which is closed in Z. This implies that g is a neutrosophic closed function. (ii) Suppose F is any nαψ-closed set in X. Since g ◦ f is nq-αψ-closed, (g ◦ f )(F ) is neutrosophic closed in Z. Again g is a nαψ-continuous injective function, g −1 (g ◦ f (F )) = f (F ), which is nαψ-closed in Y . This shows that f is nαψ ∗ closed. Theorem 4.6. Let X and Y be neutrosophic topological spaces. Then the function f : X → Y is a nq-αψ-closed if and only if f (X) is neutrosophic closed in Y and f (V ) − f (X − V ) is open in f (X) whenever V is nαψ-open in X. Proof. Necessity: Suppose f : X → Y is a nq-αψ-closed function. Since X is nαψclosed, f (X) is neutrosophic closed in Y and f (V )−f (X−V ) = f (V )∩f (X)−f (X−V ) is neutrosophic open in f (X) when V is nαψ-open in X. Sufficiency: Suppose f (X) is neutrosophic closed in Y , f (V ) − f (X − V ) is neutrosophic open in f (X) when V is nαψ-open in X and let C be neutrosophic closed in X. Then f (C) = f (X) − (f (C − X) − f (C)) is neutrosophic closed in f (X) and hence neutrosophic closed in Y . Corollary 4.1. Let X and Y be two neutrosophic topological spaces. Then a surjective function f : X → Y is nq-αψ-closed if and only if f (V ) − f (X − V ) is open in Y whenever U is nαψ-open in X. Proof. It is obvious. Theorem 4.7. Let X and Y be neutrosophic topological spaces and let f : X → Y be nαψ-continuous and nq-αψ-closed surjective function. Then the topology on Y is {f (V ) − f (X − V ) : V is nαψ-open in X}. Proof. Let W be open in Y . Let f −1 (W ) is nαψ-open in X, and −1 −1 f (f (W )) − f (X − f (W )) = W . Hence all open sets of Y are of the form f (V ) − f (X − V ), V is nαψ-open in X. On the other hand, all sets of the form f (V ) − f (X − V ) where V is nαψ-open in X, are neutrosophic open in Y from corollary 4.1. Definition 4.3. A neutrosophic topological space (X, τ ) is said to be nαψ-normal if for any pair of disjoint nαψ-closed subsets F1 and F2 of X, there exists disjoint open sets U and V such that F1 ⊂ U and F2 ⊂ V . Theorem 4.8. Let X and Y be a neutrosophic topological spaces with X is nαψ-

A New Type of Quasi Open Functions in Neutrosophic Topological Environment

35

normal. If f : X → Y is nαψ-continuous and nq-αψ-closed surjective function. Then Y is normal. Proof. Let K and M be disjoint neutrosophic closed subsets of Y . Then f −1 (K), f −1 (M ) are disjoint nαψ-closed subsets of X. Since X is nαψ-normal, there exists disjoint neutrosophic open sets V and W such that f −1 (K) ⊂ V , f −1 (M ) ⊂ W . Then K ⊂ f (V ) − f (X − V ) and M ⊂ f (W ) − f (X − W ), further by corollary 4.11, f (V ) − f (X − V ) and f (W ) − f (X − W ) are neutrosophic open sets in Y and clearly (f (V )−f (X −V ))∩(f (W )−f (X −W )) = nφ. This shows that Y is normal.

F UTURE WORK AND C ONCLUSION Neutrosophic open set in neutrosophic topological spaces gained attention and leads to many generalization of open sets. Our proposed work is investigating the neutrosophic quasi αψ open and αψ closed function and it is one of the generalizations of neutrosophic quasi open and closed function. Neutrosophic quasi αψ open and αψ closed function motivates to investigate neutrosophic quasi αψ homeomorphism, neutrosophic quasi supra αψ open and αψ closed function and this work will be our future work.

R EFERENCES [1] Atanassov K. T., Intuitionstic fuzzy sets, Fuzzy sets and systems, 20, (1986), 87-96. [2] Zadeh L. A., Fuzzy Sets, Information and Control, 18, (1965), 338-353. [3] Coker D., An introduction to fuzzy topological spaces, Fuzzy sets and systems, 88, (1997), 81-89. [4] Smarandache F., Neutrosophy and Neutrosophic Logic, First International Conference on Neutrosophy, Neutrosophic Logic Set, Probability and Statistics; University of New Mexico, Gallup, NM, USA, (2002). [5] Smarandache F., A Unifying Field in Logics: Neutrosophic Logic. Neutrosophy, Neutrosophic Set, Neutrosophic Probability, American Research Press, Rehoboth, NM, USA, (1999). [6] Salama A. A., Alblowi S. A., Neutrosophic Set and Neutrosophic Topological Spaces, IOSR J. Math., 3, (2012), 31-35. [7] Karthika M., Parimala M., Jafari S., Smarandache F., Alshumrani Mohammad, Ozel Cenap, Udhayakumar R., Neutrosophic complex αψ connectedness in neutrosophic complex topological spaces, Neutrosophic sets and systems, 29, (2019), 158-164. [8] Karthika M., Parimala M. and Smarandache F., An introduction to neutrosophic minimal structure spaces, Neutrosophic Sets and Systems, 36, (2020), 378-388.

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[9] Parimala M., Jeevitha R., Jafari S., Smarandache F. and Udhayakumar R., Neutrosophic αψ-Homeomorphism in Neutrosophic Topological Spaces, Information, 9(103), (2018), 1-7. [10] Parimala M., Karthika M., Dhavaseelan R., Jafari S., On neutrosophic supra precontinuous functions in neutrosophic topological spaces, New Trends in Neutrosophic Theory and Applications, 2, (2018), 371-383. [11] Parimala M., Karthika M., Jafari S., Smarandache F., El-Atik A.A., Neutrosophic αψconnectedness, Journal of Intelligent and Fuzzy Systems, 38(1), (2020), 853-857. [12] Parimala M., Smarandache F., Jafari S. and Udhayakumar R., On Neutrosophic αψ -Closed Sets, Information, 9, (2018), 103, 1-7. [13] Parimala M., Karthika M., Jafari S., Smarandache F., Udhayakumar R., Neutrosophic nano ideal topological structures, Neutrosophic sets and systems, 24, (2019), 70-77. [14] Parimala M., Karthika M., Jafari S., Smarandache F., Udhayakumar R., DecisionMaking via Neutrosophic Support Soft Topological Spaces. Symmetry, 10(2018), 110. doi:10.3390/sym10060217. [15] Salama A. A., Smarandache Florentin and Kromov Valeri, Neutrosophic Closed Set and Neutrosophic Continuous Functions, Neutrosophic Sets and Systems, 4, (2014), 4-8. [16] Parimala M., Jafari S., and Murali S., Nano Ideal Generalized Closed Sets in Nano Ideal Topological Spaces, Annales Univ. Sci. Budapest., 60, (2017), 3-11. [17] Parimala M., Jeevitha R. and Selvakumar A., A New Type of Weakly Closed Set in Ideal Topological Spaces, International Journal of Mathematics and its Applications, 5 (4-C), (2017), 301-312. [18] Arokiarani I., Dhavaseelan R., Jafari S., Parimala M., On some new notions and functions in neutrosophic topological spaces, Neutrosophic Sets System, 16, (2017), 16-19. [19] Parimala M. and Perumal R., Weaker form of open sets in nano ideal topological spaces, Global Journal of Pure and Applied Mathematics, 12 (1), (2016), 302-305. [20] Kuratowski K., Topology Vol. II (transl.), Academic Press, New York, (1966).

In: Decision-Making with Neutrosophic Set Editor: Harish Garg

ISBN: 978-1-53619-419-7 © 2021 Nova Science Publishers, Inc.

Chapter 3

ACCORDANCE WITH NEUTROSOPHIC LOGIC? A MULTIMOORA APPROACH FOR COUNTRIES WORLDWIDE Willem K. M. Brauers* University of Antwerp, Department of Economics, Prinsstraat, Antwerpen, Belgium

Smarandache (1998) defined a Neutrosophic Set as composed of three components: Truth, Indeterminacy and Falsehood (T, I, F - concept), which is T% true, I% indeterminate and F% false, or more general a more refined concept: (T1, T2,…..; I1, I2,……; F1, F2,…….). After Webster’s Dictionary “indeterminate data” mean: “having inexact limits; indefinite; indistinct; vague”. Following references illustrate a new trend in Neutrosophic Theory: Şahin, Olgun et al. (2017); Şahin, Ecemiş et al. (2017). Other studies make the connection with the decision-making process: Broumi et al. (2019); Hassan et al. (2018); Şahin, Ecemis et al. (2017); Uluçay & Şahin (2020) and finally Ulucay, Şahin and Olgun (2018). From now on the link will be made between neutrosophic sets and multiple objectives or criteria decision analysis (MODA): Uluçay, Sahin and Hassan. present a neutrosophic expert set for MODA (2018). Are Credit Rating Agencies Opinions false from the point of view of Neutrosopic Theory? It could be as internal analysts from Credit Rating Agencies (CRA) are not followed by their management which has rather subjective opinions. Considering the Credit Rating of Firms, the CRA made significant mistakes during the Recession 2007-2009 and their judgment is considered to be too much American *

Corresponding Author’s Email: [email protected].

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Willem K. M. Brauers

orientated. Consequently, in Europe, many efforts were made to come to a new Agency, but all efforts failed. Nevertheless, the European Union set rules of conduct for the CRA regarding the monopoly situation of the big three (Credit Rating Agencies. Regulation (Ec) N°1060-2009). Perhaps the United Nations must do the same on world level.

1. THE CREDIT RATING OF FIRMS The three main rating offices: Standard & Poor’s, Moody’s and Fitch were founded in the period 1909-1924. Beside corporate bonds also government bonds were considered. Ipso facto the country involved was taken into consideration. In the early 1970s their policy changed: an “Issuer Pays Model”, whereby the entity issuing the bonds pays the rating firm, replaced “The Investor Pays model” (Deutsche Bank, 2011; White, 2010). What can be called, the Big Three represents 96.4% of the market share (SEC, 2016), a Quasi-Monopoly. Their ratings influence the credit rating of the shares and bonds of private companies. The credit rating of a firm is of great importance in terms of revealing the firms’ general credit worthiness (Ashbaugh-Skaife et al. 2006). There is also an influence on the government bonds and finally on the countries themselves. Not very much is known about the way of operating of the CRA’s. The agencies like Standard & Poor’s, Moody’s and Fitch are taken up with a complex judgmental process (Nickell et al. 2000). However, quantitative approaches would make it clearer and bring more certainty in this complex process (Bernard, 2006, p.294). In this quantitative direction Afonso et al. (2011) show that four variables have an impact on sovereign ratings: 1) level of GDP per capita 2) real GDP growth 3) public debt 4) government balance. For the long term they mention government effectiveness, level of external debt and external reserves of the government. Nevertheless, also a dummy, reflecting past sovereign defaults and fiscal variables, seem to be important. More details exist for Standard & Poor’s. Indeed, its director for Europe, Mrs. Myriam Fernández de Heredia (2012), affirmed that an analyst performs desk research and interviews per country. His conclusions are transmitted to the management, who determine the rating of a firm by simple majority voting. The question remains: how do the analysts work? Do they work after a standardized system or to they work by hazard? How the management can be specialist about cattle, engineering and construction, the pharmaceutical industry, banks and administration? Does it mean that the CRA are false from a Neutrosophic point of view? Reality is more complex than that.

A Multimoora Approach for Countries Worldwide

39

Companies Do they Work Scientifically? Companies work rather by guesswork, trial and error instead of by a scientific conduct. Some examples can illustrate this statement. 



Sometimes firms ask for lowering the size of a necessary sample in order to decrease the costs or they will not look after the standard deviations of a sample. Of course, taking into consideration the standard deviations with a fuzzy analysis would disfavor the position of a contractor (Brauers et al. 2018). Churchman et al. (1957) state that Operational Research provides managers of an organization with a scientific basis for solving their problems, with as major phases: 1. 2. 3. 4.



formulating the problem constructing a mathematical model deriving a solution from the model implementation.

Originally the researcher has to consult all stakeholders concerned, inclusive the principal, before he draws his final conclusions. The misinterpretation consists of the fact that the researcher returns to the principal with his report and the principal asks for some changes, position sometimes repeated several times. In the application of Operational Research, the analyst after taking into consideration all stakeholders is asked by the principal to change still his final conclusion. In Belgium interviewing the readers of a newspaper are satisfied with a standard deviation of 12% or a possible spread of 24%. In this way the publishers are satisfied as the result is better than a possible answer from the newspaper itself (CIM, 2013-2014; CIM, 2014).

Consequently, it is true from the Neutrosphic point of view that company’s behavior does not respect scientific laws and regulations. Concerning the credit rating of firms, the CRA made significant mistakes during the Recession 2007-2009 and their judgment is considered to be too much American orientated. Consequently, in Europe efforts were made to come to a new Agency, but all efforts failed (Berger, 2012; Bertelsmann Stiftung, 2012 and the European Commission). Nevertheless, the European Union set rules of conduct for the CRA regarding the monopoly situation of the big three CRA (Credit Rating Agencies, 2009; ESMA, 2011; Regulating Credit Rating Agencies, 2013).

40

Willem K. M. Brauers

True or false the monopoly situation of the CRA towards the judgment of companies has to be accepted under these circumstances. On world level, one could think of rules of conduct by the United Nations for the CRA after the example of the European Commission. The credit rating of countries is another story, however less important for the CRA: 11% of their activity (European Commission, 2016). Table 1. Composition of the revenue of the Credit Rating Agencies Corporate bond ratings Structured finance products Sovereign ratings Total

69.8% 18.9% 11.3% 100%

The research of underlying study concerns what can be done for the estimation of the credit rating of countries. Therefore, this study could ask if a more extensive quantitative approach, based on statistics and forecasts, is not preferable for the countries, instead of the solutions given by the CRA. As it will become clear in what is following, the approach towards firms and this one towards countries must be different.

2. CHOICE OF OBJECTIVES (CRITERIA) CHARACTERIZING THE ECONOMIES OF THE COUNTRIES The Choice of Criteria characterizing the Economy of a Country in the Present and in the Future is restricted to the available data. Criteria become Objectives when they are Maximized (+) or Minimized (-): 



Government: 1. Government Budget Deficit to GDP 2. Government Debt 3. Government consolidated Gross Debt 4. Current account deficit (B. of P.) 5. Government bond yields GDP 6. GDP in constant prices 7. GDP in PPP 8. GDP per capita (constant prices) 9. GDP per capita in PPP 10. GDP per capita compared to the other EU countries

41

A Multimoora Approach for Countries Worldwide

 





 

11. GDP growth rate 12. GDP per capita growth rate Inflation (13) Employment 14. Employment Rate 15. Unemployment Rate 16. Tertiary education Population Pyramid 17. Median Age 18. Proportion of population aged 0-14 19. Proportion of population aged 15-64 20. Proportion of population aged 65 and over Health 21. Number of physicians per 1000 inhabitants 22. Dwellings completed (per capita) in m2 Justice 23. Registered criminal offences (misdemeanors per 100,000 inhabitants) Pollution 24. Average pollutant emissions per km2 in kg.

At that moment it is possible to construct a Decision Matrix with the objectives as columns and the countries as rows. Such a Decision Matrix will have the form presented in Table 2. Table 2. Decision Matrix composition Obj. 1

Obj. 2

……………

Obj. i

Alternative 1

x11

x21

……………

xi1

Alternative 2

x12

x22

……………

xi2

xn2

…….

……..

……………

………

………..

x1j

x2j

……………

xij

xnj

……

………

……………

…….

………

x1m

x2m

……………

xim

xnm

………….. Alternative J ……….. Alternative M

Obj. n ………………

xn1

All the objectives have different denominations. Consequently, there is a question of uniformness, composed of normalization and importance. Therefore, a method must be chosen to respond to this necessity. It will be treated under next heading.

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3. A CHOICE OF A METHOD FOR THE MULTI-OBJECTIVE OPTIMIZATION OF THE RATING OF COUNTRIES 3.1. Neutrosophic False Based on weights would mean that the 24 criteria obtain a weight, with the 24 obtained and desired weights adding up to 1. Each weight represents a measure of normalization of different units (like money, units, %, meters, calories, etc.) and of importance (e.g., pollution is two times more important than value added). A choice of a weight for 24 criteria exceeds the possibilities of the human brain. After the psychologist Miller (1965), a scale from one to seven is considered as a limit of our capacity in that field. Moreover, if for instance 20 different stakeholders are involved the weights are a consensus of twenty different opinions.

3.2. Neutrosphic True Preference will be given to dimensionless measurements and namely as applied in the MULTIMOORA Method (Brauers & Zavadskas, 2010a). This method is chosen given its robustness (Brauers & Zavadskas, 2010b) and its possibility to summarize three methods, being all possible methods with dimensionless measures and with the possibility that they control each other. First, MOORA was developed, composed of two methods: Ratio Analysis and Reference Point Method (Brauers & Zavadskas, 2006). Later the Full Multiplicative Method was added (Brauers & Zavadskas, 2010a). Application in Neutrosophic Theory is stressed by Nabeeh et al. (2019).

3.3. Multi-Objective Optimization by Ratio Analysis (MOORA) 3.3.1. The First Part of MOORA: The Ratio Analysis The method starts with a matrix of responses of different alternatives on different objectives: ( xij )

with: xij as the response of alternative j on objective i i=1, 2,…, n as the objectives j=1, 2,…, m as the alternatives

43

A Multimoora Approach for Countries Worldwide

MOORA prefers a ratio system in which the response of each alternative on an objective is compared to a denominator, which is representative for all alternatives concerning that objective. Simple averages are taken by column when the Decision Matrix is read vertically: =

/∑

(1)

Table 3. An example with 2 objectives and 2 solutions

Solution 1 (chemical) Solution 2 (retail)

MAX. Employment in person years 20 80 100

MAX. Value Added in million € 70 30 100

Sum on basis of 1 0.9 1.10

Ranking 2 1

When the percentages are compared, they become Dimensionless Measurements. Already De Jong (1967) showed interest in dimensionless measurements. It is also the case with Miller and Star (1969). However, simple averages are inconsistent as they may change the sign and even lead to no sense results. Consequently, for the denominator the square root of the sum of squares of each alternative per objective is chosen. Brauers & Zavadskas (2006) proved that this is the most robust choice:

xij*=

𝒙𝒊𝒋 𝟐 √(∑𝒎 𝒋=𝟏 𝒙𝒊𝒋 )

(2)

with: xij = response of alternative j on objective i. j = 1, 2, ..., m; m the number of alternatives. i = 1, 2 ,…, n; n the number of objectives. xij* = a number representing the response of alternative j on objective i. Once the xij* from different objectives are compared they become dimensionless numbers, which are called from now on RATIOS, also a given name for this Method. For optimization these responses are added in case of maximization and subtracted in case of minimization:

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Willem K. M. Brauers

i= g i =n ∑ y j* = ∑ xij* xij* i =1 i = g +1

(3)

with: i = 1,2, …, g as the objectives to be maximized. i = g+1, g+2, …, n as the objectives to be minimized. yj* = the assessment of alternative j with respect to all objectives. An ordinal ranking in a descending order of the yj.* shows the final preference.

3.3.2. The Second Part of MOORA with the Reference Point For the second part of MOORA the Reference Point Theory is chosen. Which Reference Point? The following choices are possible: 1. Maximal Objective Reference Point: the point realizable, in the example of Table 2 a Point with as coordinates: (80; 70). 2. Utopian Objective Reference Point is farther away than the Maximal Objective Reference Point. 3. Aspiration Objective Reference Point is more nearby than the Maximal Objective Reference Point. The first choice is accepted here. For the Reference Point Methods, the Minkowski Metric brings the most general synthesis (Minkowski, 1896; 1911; Pogorelov, 1978):

Min .M j =

{

i=n ∑ (ri i =1

-

x* ij

)α }1 / α

(4)

Moving farther away than the Rectangular or the Euclidean positions namely until infinity, the Minkowski metric becomes the Tchebycheff Min-Max Metric (Karlin & Studden, 1966; Tchebycheff, 1947):

Min max ri ( j ) (i)

- xij*

(5)

ri = the ith co-ordinate of the reference point ││means the absolute value: if a coordinate of xij* is larger than the corresponding coordinate of ri , for instance with the Aspiration Objective Reference Point. Given the composition of equation (4) the results are ranked in an ascending order.

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3.4. MULTIMOORA As MOORA is based on dimensionless measures why not going a bit further by adding the remaining form which uses dimensionless measures, namely the Full Multiplicative Form? The following form for multi-objectives is called from now on a Full-Multiplicative Form:

n U j = ∏ xij i =1

(6)

with: j = 1,2, ..., m; m the number of alternatives. i = 1,2, …, n; n being the number of objectives. xij = response of alternative j on objective i. Uj = overall utility of alternative j. The overall utilities (Uj), obtained by multiplication of different units of measurement, become dimensionless. How is it possible to combine a minimization problem with the maximization of the other objectives? Therefore, the objectives to be minimized are denominators in the formula:

U 'j =

Aj Bj

(7)

with:

i A j = ∏ xgi g =1 j = 1,2, ..., m; m the number of alternatives. i = the number of objectives to be maximized.

Bj =

n ∏ xkj k =i +1

n-i = the number of objectives to be minimized. with: Uj’ : the utility of alternative j with objectives to be maximized and objectives to be minimized.

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Willem K. M. Brauers

In the Full Multiplicative Form, a problem may arise for zero and negative values making the results senseless. In order to escape of this nonsense solution, 0.1 replaces zero and for instance -2 becomes 0.02, etc.

3.5. The Theory of Ordinal Dominance The three methods of MULTIMOORA represent all possible methods with dimensionless measures in multi-objective optimization and one cannot argue that one method is better than or is of more importance than another one. However, three different rankings are a result from MULTIMOORA. How is it possible to come to an overall ranking? How to make a synthesis between the results of the three approaches: Ratio System, Reference Point Method, which uses the ratios obtained in the ratio system as coordinates, and the Full Multiplicative Form? Therefore Brauers; Zavadskas (2011) developed the following Theory of Dominance.

3.5.1. Axioms on Ordinal and Cardinal Scales 1. A deduction of an Ordinal Scale (a ranking), from cardinal data is always possible (Arrow, 1974). 2. An Ordinal Scale can never produce a series of cardinal numbers (Arrow). 3. An Ordinal Scale of a certain kind, a ranking, can be translated in an ordinal scale of another kind. All these axioms are also covered by Neutrosophic Logic. In application of axiom 3 the ordinal scale of the three methods of MULTIMOORA is translated in another one based on Dominance, being Dominated, Transitivity and Equability.

3.5.2. Dominance, being Dominated, Transitiveness and Equability Dominance Absolute Dominance means that an alternative, solution or project is dominating in ranking all other alternatives, solutions or projects which are all being dominated. This absolute dominance shows as rankings for MULTIMOORA: (first-first-first). General Dominance in two of the three methods with a P b P c Pd (P preferred to) is for instance of the form: (d-a-a) is generally dominating (c-b-b). (a-d-a) is generally dominating (b-c-b).

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(a-a-d) is generally dominating (b-b-c) and further on transitiveness plays fully. Transitiveness If a dominates b and b dominates c than also a will dominate c. Overall Dominance of One Alternative on Another For instance (a-a-a) is overall dominating (b-b-b) which is overall being dominated by (a-a-a). Equability Absolute Equability has the form: for instance (e-e-e) for 2 alternatives. Partial Equability of 2 on 3 exists e.g., (5-e-7) and (6-e-3). The same rules apply for the three methods of MULTIMOORA with no significance coefficients proposed as the three methods are considered to have the same importance.

4. INDETERMINACY TOWARDS NEUTROSOPHIC PHILOSOPHY In order to make the approach more determined one must concentrate on the Liquidity and the Solvency position of a Country.

4.1. The Liquidity of a Country being its Capacity to pay Debts on Time Due The composition of the Public Debt and the Budget Deficit of a country influence its liquidity position. Their financing depends of:

4.1.1. No Public Debt in Other Currencies If there is no public debt in other currencies it means no problems with currency rates. For instance in Belgium the public debt is only labeled in Euros. 2. Difference between External and Internal Public Debt For the Euro Countries it makes a difference if the Debt in € belongs to foreigners or to the own residents. For the Internal Public Debt, the government can more easily enforce a moratorium. If necessary, an urgent appeal can be made to the own public of the country to subscribe to government bonds on free will. Households could be willing to finance immediately the Public Debt if the interest rate is sufficiently high. It all depends of the savings capacity of the households, which for instance in the US is very low.

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3. The Reserves of the Central Bank The gold reserves take a special place in these reserves. However, selling and buying of the gold stock is a function of a regulation between the central Banks worldwide. 4. The Money Machine In the US money creation is only a privilege of the Federal Reserve System and for the Euro Countries of the Central European Bank. The Central Banks of the other European Union countries are not entirely free, indirectly linked to the Euro. Money creation is a function of different diverging monetary theories. For the Euro countries, especially for Germany, the fear for inflation is a decisive factor.

4.2. The Solvency of a Country A traditional solvency ratio relates the own capital to the balance total. For a government solvency relates its monetary obligations to the government balance sheet together with taking into consideration inflation risks. A National Balance Sheet for a country is much broader and then solvency of the country relates its monetary obligations to the wealth of the whole nation (Prof. Goldsmith is the pioneer on national balance sheets, 1963, 1975). A National Balance Sheet for a country is composed of: 1. Natural resources. 2. Produced stock in households, enterprises and government. Some points of discussion may arise from intangibles and man-made non-reproducible tangible assets. In micro-economic bookkeeping intangibles are considered as phantoms which have to be withered away as soon as possible. The same attitude is taken for macro-economics (Brauers, 1990, p. 5). Man-made non-reproducible tangible assets such as museum and personal collections of arts, coins, stamps, etc. are also in part a hedge against inflation. 3. Net monetary means abroad. 4. Internal monetary reserves. 5. Human capital (Brauers, 1990, p. 6). However, for the Solvency of a Country the inclusion of human capital is not realistic. In addition, all points of the National Balance Sheet are not only a function of monetary considerations but are subjected to multiple objectives such as ecological conditions, environmental perspective and sustainable development including climate change, greenhouse gas emissions and other pollutions and clean technology. Also,

A Multimoora Approach for Countries Worldwide

49

personal appreciation plays a role. For instance, the Vatican art collection is subject to the admiration of a worldwide crowd of visitors and consequently cannot be used as a guaranty for the bad financial situation of the Vatican. Returning to the Solvency of a Country, unhappily, most of these additional data are not available in official statistics. Therefore, it is perhaps necessary that country monographs are needed.

5. POINTS STILL TO BE DISCUSSED Have still to be discussed: 1. 2. 3. 4.

The importance of an objective or criterion The choice of Stakeholders The choice of the Objectives (Criteria) The choice of the Solutions.

5.1. The Importance of Each Objective or Criterion In the traditional SAW methods (MacCrimmon, 1968; Hwang & Yoon, 1981) Importance and Normalization are mixed in what is called Weights, without knowing the respective part of importance and of normalization; i.e., “false after Neutrosphic Logic”. Indeed, it may lead to false conclusions. In MULTIMOORA normalization was discussed already, but not importance, now treated separately. It is true that a huge chemical plant will be very useful for an economy, however with an eventually bad influence on pollution. There are several ways to treat the importance problem.

5.1.1. Multiplication with a Coefficient of Importance (False after Neutrosphic Logic) As shown in Appendix A not only the ranks do not change but also total outcome will not change. The example on Lithuania concerns MOORA but the Multiplicative Form will present the same results. Moreover, after Miller and Starr (1969), for the Multiplicative Form, multiplying a column with 10, 102, 103, etc. will not change the results. 5.1.2. Adding a Number to an Objective (False after Neutrosphic Logic) Adding a number to an Objective would favor the lowest quoted solutions with possible deterioration of the results. Nevertheless, several tests have shown that even then the final ranking do not change.

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5.1.3. Multiplying an Objective with an Exponent As shown in Appendix B the increase will be exponential, meaning a too large favoring of an objective regarding towards the other objectives. 5.1.4. Dividing an Objective in Different Sub-Objectives (True after Neutrosphic Logic) Pollution is taken as an example. As shown in Appendix C total pollution is split into two parts: gas and liquid on the one side and solid pollution on the other. In Appendix C concerning Lithuania, Pollution gets a share in the number of objectives of 2/17 instead of 1/16. Nevertheless, case by case the outcome has to be verified.

5.2. All Stakeholders In principle, in Multi-Objective Optimization all stakeholders, interested in the issue under consideration, have to be involved; in fact, a very difficult task. In the following case the solution was less complicated. It concerned a study of 2002 about the future of the Facilities Sector in Lithuania (Brauers, Lepkova, 2003). The stakeholders were represented by a group of 15 delegates from the sector, the ministerial departments concerned with the issue and specialized professors in the field. In 2002 in Lithuania, no representative association of consumers was existing and given the only small firms of the sector, there was no representative workers’ association. It was assumed that the professors were knowing the aspirations of the workers and of the consumers.

5.3. The Choice of Objectives (Criteria) All the stakeholders concerned will make the choice of the objectives for the issue under consideration. Therefore, instead of a “face-to-face discussion” they will participate in an Ameliorated Nominal Group or/and a Delphi exercise (Brauers, 2008; 2004).

5.4. The Choice of Solutions A wide choice of possible solutions exists going from general economics to engineering, construction, regions, medicine, etc. It is essential that a certain form of competition exist between the several proposed solutions.

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6. FINAL CLASSIFICATION OF THE COUNTRIES BY MULTIMOORA AND ORDINAL DOMINANCE 6.1. Previous Studies For Europe are mentioned:  Brauers & Zavadskas, 2013.  Brauers & Lepkova, 2019 For the World:  Türe, H. et al., 2016.

6.2. Comparison with Standard & Poor’s Rates 2020 A comparison with the most important Credit Rating, viz. Standard & Poor’s 2020 rates (worldgovernmentbonds.com/world-credit-ratings/2020), given in following Table 4, measures the forecasting capacity of the MULTIMOORA data from 2012-2015. Table 4. The Standard & Poor’s classification AAA AA+ AA AAA+ A ABBB+ BBB BBBBB+ BB BBB+ B BCCC+ CCC CCCD

Excellent-Maximal Security High or Good Quality

Average Quality

Less than Average Quality

Speculative

More than Speculative

Important Risk Excessive Speculative Nearly Bankruptcy Bankruptcy

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MULTIMOORA proposes the following comparison with S&P’s (Table 5). Table 6 presents the final classification of MULTIMOORA on basis of Dominance Theory. The first ranking, called CORE, is composed of ten countries (Table 6) like the Group of 10 mightiest countries in the world economy (for S&P’s see: Sovereign Ratings List 2020). Table 5. Comparison with S&P’s

Core Semi-Core Semi-Periphery Periphery Rubbish

Quantity 10 10 10 Max. 30 Remainder

S&P’s (approximate) AAA Other A’s Triple B Simple B C and D

Table 6. The final classification of MULTIMOORA A. The CORE-Countries MULTIMOORA Countries 1 Singapore 2 Sweden 3 Norway 4 Switzerland 5 Finland 6 Netherlands 7 Denmark 8 United Arab Emirates 9 Germany 10 Austria B. The Semi-Core-Countries MULTIMOORA Countries 11 Belgium 12 Estonia 13 New Zealand 14 South Korea 15 Australia 16 Slovakia 17 Lithuania 18 Slovenia 19 Czech Rep. 20 Canada

S&P 2020 AAA AAA AAA AAA AA+ AAA AAA AA (2018) AAA AA+

S&P 2020 AA AA- (2018) AA (with a positive outlook) AA (2018) AAA (with a negative outlook) A+ A+ AAAAAAA

A Multimoora Approach for Countries Worldwide C. The Semi-Periphery Countries MULTIMOORA Countries 21 France 22 Hungary 23 Latvia 24 Malta 25 Poland 26 Israel 27 Cyprus 28 U.K. 29 Malaysia 30 Japan

S&P 2020 AA BBB A+ AAAABBBAA AA+ (with a positive outlook)

D. The Periphery Countries MULTIMOORA Countries 31 Chile 32 Usa 33 Italy 34 Romania 35 Spain 36 Ireland 37 Portugal 38 Bulgaria 39 Mauritius 40 Uruguay 41 Thailand 42 Turkey 43 Mexico 44 Georgia 45 Belarus 46 Greece 47 Croatia 48 Serbia 49 Peru 50 China 51 Indonesia 52 Kazakstan 53 South Africa 54 Philippines 55 Sri Lanka 56 Paraguay 57 Jamaica 58 Russia 59 Albania 60 Colombia

S&P 2020 A+ AA+ BBB (with a negative outlook) BBB- (with a negative outlook) A AaBBB BBB (with a positive outlook) nihil BBB (2018) BBB+ B+ BBB (with a negative outlook) BB- (2018) B (2018) BbBBbBB+ BBB+ A+ BBB (with a negative outlook) BBB-(2018) BBBBB+ BBB (2018) B (2018) BBBB+ (2018) BBB- (with a negative outlook)

E. Very Risky: 61) Namibia. 62) Bosnia and Herzegovina.63). Mongolia. 64) El Salvador. 65) Senegal. 66) Moldova. 67) Nicaragua. 68) Rwanda. 69) Uganda. 70) Ukraine. 71) Bangladesh. 72) Kyrgyz. 73) Brazil. 74) Malawi. 75) Mozambique.

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Willem K. M. Brauers

6.2. Missing Countries For Europe: Andorra, Iceland, Liechtenstein, Monaco, Montenegro, NorthMacedonia, San Marino, Vatican,  

     

Liechtenstein is a well-known tax-shelter country. It got from S&P a AAA rating (2018). Also, Monaco is a well-known tax-shelter country, but did not receive a Credit Rating from any CRA, perhaps because Monaco has to share its sovereignty with France. Andorra got a BBB rating from S&P (2018). Iceland got a A rating from S&P (2018). Montenegro got a B+ rating from S&P (2018). North-Macedonia got a BB- rating from S&P (2018). San Marino got a BBB- rating from Fitch (2018). No rating is found for Vatican.

Any other country despite a S&P’s notation but with insufficient official statistics like: Angola, Argentina, Azerbaijan, Burkina Faso, Bahrain, Bolivia, Bahamas, Botswana, Belize, Democratic Republic of Congo, Republic of the Congo, Cameroon, Colombia, Costa Rica, Cape Verde, Dominican Republic, Ecuador, Egypt, Fiji, Ghana, Guatemala, Hong Kong, Honduras, India, Jamaica, Jordan, Kenia, Cambodia, Kuwait, Lebanon, Morocco, Sri Lanka, Nigeria, Oman, Panama, Papua, Pakistan, Paraguay, Qatar, Rwanda, Saudi Arabia, Trinidad and Tobago, Taiwan, Uganda, Zambia. Any other country by error forgotten.

6.3. Luxemburg: Another Exception First, Luxemburg has to be excluded from any ranking of honesty, given a typical case of illegal (black) money transfer (Brauers, 1995 and 1998). A company named AMFA, specialized in bakery products, with as official address only a letter box in a small street in the city of Luxemburg, with a negative capital, meaning that it was in fact bankrupt, with no activity during several years and categorized as “very risky” by a rating office, would provide 75 million € (1995-rate) to build a European Football Stadium with 30,000 seats in Berchem-Antwerp. The plan was withdrawn given the two negative reports (Brauers, 1995 and 1998) and a unique anti-demonstration of the population.

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Secondly, the Minister of Finance declared to Michel (2014) that tax evasion systems are the normal way of acting” (profit transfers). The minister also maintained that Luxemburg has investments under management for 300 billion €, a world leader behind the US. Thirdly, the policy of letterboxes without any complete office behind is a well-known practice in Luxemburg (Obermayer, 2014). Beside “tax evasion” and “profit transfers” still other terms are used. Dharmapalla, 2019 speaks rather of “Profit shifting” or “cross-border tax avoidance” and “strategic transfer pricing”. Anyway, a difference has to be made with multinationals which really transfer their head office and an important part of their activity to tax friendly countries. Nevertheless, Luxemburg gets the highest ranking of S&P’s, namely AAA, whereas other dubious countries like Mauritius is only getting position 39 with no S&P’s rating and Panama with a BBB rating, classified here as having insufficient official statistics. After this study of falsification, Luxemburg has to be excluded from a fair ranking of countries, as being an example of bad conduct and a lesson for well-known other tax evasion countries like Monaco, Panama, Mauritius, etc. (more details are given in: Brauers, Lepkova, 2019). Aside its creditability, the real economic capability of Luxemburg is certainly weakened by the decline of the steel industry, before its main source of income (Brauers & Lepkova, 2019). The population seems not to be extremely wealthy, confirmed by the Human Development Index (HDI) of the United Nations. Indeed, the HDI is here a good indicator, not including Gross Domestic Product (GDP) with the total production on the territory, but rather Gross National Income of citizens and permanent residents in Purchasing Power Parity per capita (Human Development Report, 2018). After that Report, Luxemburg ranks last (21) of the Benelux Countries, after the Netherlands (10) and Belgium (17). Recently, the government has done a gesture towards the population by offering a continuous free ride on any public transport: trams busses and trains. Anyway, countries have to be condemned which sign individual favorable fiscal contracts with foreign companies. The European Union tries to fine some countries for these practices; but mostly case per case and unhappily not as a rule. If the Credit Rating Agencies are taken their position seriously, they would exclude countries like Luxemburg from their ranking. Therefore, after Neutrosophic Theory, their attitude is designated as being “false”.

6.4. Another Hot Issue: Ireland In this study Ireland is only classified as a Semi-Periphery Country, still as a function from the fact that Ireland was one of the countries, which suffered the most from

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Recession 2007-20009. It was accepted that Ireland restored much faster than expected and consequently would take its previous position as an extreme prosperous country. Anyway; for the period 2012-18 S&P changed the rate of Ireland seven times (Sovereigns Ratings List, 2018). Nevertheless, instead of GDP one has rather to look after National Income or after GNP. For Ireland, after recalculation of the 2014 statistics, one may conclude that national income only represents 62% of GDP or a difference of 38%, which is enormous (OECD, Economic Outlook, 2018). There is even something more, namely Tax Rulings: low taxation for companies practicing profit transfers from higher taxed countries, as was explained for Luxemburg. In August 2016 the European Commission ruled that the Irish Government light tax treatment towards Apple for an amount of 13bn € amounted to state aid, illegal under EU rules. It is a sum equivalent to the county’s entire healthcare budget or two thirds of its annual social welfare bill (Hodge, 2017).

6.5. The United Kingdom This study classifies the United Kingdom only as a Semi-Periphery Country corresponding to “less than Average Quality of S&P’s” whereas S&P’s gives “High or Good Quality”. However, one has not to forget that the UK was still ranking as AAA stable in 2010. Since then and till 2018 the rate changed 5 times (Sovereigns Ratings List, 2018). The older generation in the United Kingdom, but mainly in England, still lives in the mentality: “Britannia rules the waves” as before the First World War. However, after that war the United Kingdom lost much of its prestige. The English Pound as world currency was substituted by the American Dollar. After the Second World War it lost all its colonies and shortly after that war the public debt increased to 250% of GDP (Slater, 2018). An Oxfam study states that the UK only ranks 109th in the world for budget spending in education much worse than some developing countries and that the poor are often unable to cover living costs (Oxfam, the Guardian, 2017).

6.6. The United States For the US as a periphery country, compared to the AA+ rate of S&P, coming from an AAA rate, has much to do with: 

the huge Public Debt: 106% of GDP in 2014 (IMF, 2014);

A Multimoora Approach for Countries Worldwide     

57

since 2000 a government budget deficit financed by foreigners like Chinese, as Americans do not save very much; an international trade deficit every year: like 635 billion dollars in 2010 (Trade Statistics, 2011); the economy is moved by consumer spending for as much as 68% (2015) with limited savings (Fed Graph, 2015); redistribute less income through government actions than in European States (Journard et al., 2012); 2.3 million people incarcerated or one on every 100 adults (Sawyer, Wagner, 2020). Adding the dead penalty and the individual freedom of the possession of weapons it all characterizes a government as being weak in democratic authority.

7. ECONOMIC CAPABILITY PER COUNTRY: A METHOD OF FORECASTING? 7.1. The Necessity to Come to a Structural Credit Rating System for Countries Based on Continuity As the classifications of Moody’s and Fitch are very similar to these of Standard & Poor’s the outcome of the comparison with MULTIMOORA would be the same for the other Credit Rating Agencies. It seems necessary to come to a Structural Credit Rating System for countries based on continuity. From now on it will be called: Economic Capability. This kind of study would rely not on “Fingerspitzengefühl” but exclusively on desk research based on:  official statistics  other data which have undergone the test either of complete census or of a sampling with a scientific accepted standard deviation.  exceptionally in last resort: on simulation. Contrary to many other definitions, simulation is defined in this study in a rather broad sense. Gordon, Enzer and Rochberg (1970, 241) give the most complete description of simulation as mechanical, metaphorical, game or mathematical analogs. These authors conclude that simulations: “are used where experimentation with an actual system is too costly, is morally impossible, or involves the study of problems which are so complex that analytical solution appears impractical”.

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7.2. S&P’s and Forecasting It becomes False after Neutrosophic Logic when S&P’s starts with forecasting. A forecast (2013) only based on government debt and on yearly government expenses and due to the ageing of population, S&P’s comes, for 2050, to sovereign ratings of BBB for Belgium and even “junk” for the Netherlands, contrary to Italy with AAA (S&P, http://img.iex.nl. 2013). Gagnon (2011) speaks about an outlook over 25 years. Due to the consideration of many criteria of all kind, forecasting with MULTIMOORA, even more with Fuzzy MULTIMOORA, is certainly superior to other methods. In the research over the period 2012-2015, MULTIMOORA brings results comparable to S&P’s findings for 2020. Nevertheless, forecasting for a period over the traditional juridical term of 30 years remains senseless due to so many uncertainties like: migration flows, European integration and disintegration with uncertainties concerning the banking system, the policy of the European Central Bank and the Euro, the climate, etc. Nevertheless, very long time forecasting with conditional changes are possible. For instance, the year 2075 will be a climatically disaster for the world unless adequate measures are taken (on basis of: World Bank, 2011).

CONCLUSION Rating offices rate the solvability of a borrower. Three Rating Agencies of American origin have a quasi-monopoly for credit rating of companies and governments: Standard & Poor’s. Moody’s and Fitch. They follow an “Issuer Pays Model”. In that case the proverb could play: “bread binds”, but if the CRA would follow that proverb it would be the end of their business. Recently, there was much critique against the CRA’s: they did not foresee the bankruptcy of different huge American banks during the recession 2007-2009 and they were suspected of underestimating European companies. Therefore, in Europe several initiatives were taken to set up new CRA’s but of European origin and for European firms. They all failed. Only the European Commission adopted rules on Credit Rating Agencies. On basis of this experience, the monopoly situation of the three main credit rating agencies, is a situation to be accepted. However, after the example of the control of the European Commission the United Nations have to set up a control mechanism. In addition, the firms would have not to pay for the judgment of the CRA but a professional association representing the firm in question.

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However, the credit rating of countries is another story. Therefore, the research of underlying study concerns what can be done for the estimation of the credit rating of countries. Instead of qualitative judgment by rating offices, quantitative estimation of the economic rating of the States by Multi-Objective Optimization is preferred. Therefore 24 selected objectives will characterize each State. Next problem was the choice of an effective method of Multi-Objective Optimization. This method must use complete and not partial aggregation as an overall view of the countries is needed. It has also to avoid the use of weights, this last one being dual on normalization and importance. Therefore, methods based on dimensionless measures are preferred. MULTIMOORA responding to all these conditions was finally chosen. In addition, MULTIMOORA is composed of three approaches each controlling each other. In this way all possible methods based on dimensionless measures are included. Having the results of the three approaches, Ratio Analysis System, Reference Point Approach and Full Multiplicative Form, the problem remains how to come to a final and unique solution. For that purpose, the correlation of ranks is senseless. A Theory of Ordinal Dominance is rather preferred. All States were assigned to one of the groups in the rating system. It seems necessary, beside this regular and necessary re-estimation of credit ratings for countries, to come to a Structural Credit Rating System for countries based on continuity. From now on this will be called: Economic Capability. Such a system would not rely on “Fingerspitzengefühl” but exclusively on desk research based on:  official statistics  other data which have undergone the test either of complete census or of a sampling with a scientific accepted standard deviation.  exceptionally in last resort on simulation.

ACKNOWLEDGMENTS I thank Türe Hasan and Koçak Deniz both of Ankara Hacı Bayram Veli University, Department of Econometrics, Turkey and Doğan Seyyide of Karamanoğlu Mehmetbey University, Department of Econometrics, Turkey for providing me with information on countries worldwide beside Europe.

3 1.9 1 5.9

2

4

3

9

16

9 29

5.385

EEU

Secession sum of squares

square roots

3.689

1 13.61

3.61

9

3.689

1 13.61

3.61

9

5.9

1

1.9

3

0.743

0.557

EEU

Secession

0.271

0.515

0.813 0.271

0.515

0.813

Note: MULTIMOORA will react in the same way.

0.371

EMU

c - Objectives divided by their square roots and MOORA

4

EMU

Projects

b - Sum of squares and their square roots

Secession Totals

EMU EEU

0.536

0.370

0.759

22.4

144 501.9

68.89

289

37.3

12

8.3

17

a - Matrix of Responses of Alternatives on Objectives: (xij) Yearly 1. MIN 2. MIN 3. MIN 4. MIN

0.535

0.267

0.802

3.741657

4 14

1

9

6

2

1

3

5. Increase real wages in% MAX

0.632

0.562

0.534

71.2

2025 5069

1600

1444

123

45

40

38

6. MIN

0.489

0.622

0.612

11.25

30.25 126.6

49

47.33

19.38

5.5

7

6.88

7. MAX

0.048

0.706

0.706

10.34

0.25 106.8

53.29

53.29

15.1

0.5

7.3

7.3

8. MAX

Table A-1. A MOORA Simulation Lithuania (2006-2012)

0.585

0.574

0.573

14.87

75.69 221.1

72.76

72.69

25.756

8.70

8.53

8.53

9. MIN

0.816

0.408

0.408

4.899

16 24

4

4

8

4

2

2

10. MIN

APPENDIX A: MULTIPLICATION OF AN OBJECTIVE WITH A COEFFICIENT TWO

2 1 3

-2.0913 -2.5967

rank

sum -2.1523

3 1.9 1 5.9

2

4

3

9

16

9

29

5.385

EEU

Secession

∑ of square

square roots

3.689

13.61

1

3.61

9

3.689

13.61

1

3.61

9

5.9

1

1.9

3

0.743

0.557

EEU

Secession

0.271

0.515

0.813 0.271

0.515

0.813

Note: MULTIMOORA will react in the same way.

0.371

EMU

c - Objectives divided by their square roots and MOORA

4

EMU

Projects

b - Sum of squares and their square roots

Secession Totals

EEU

EMU

0.536

0.370

0.759

22.4

501.9

144

68.89

289

37.3

12

8.3

17

a - Matrix of Responses of Alternatives on Objectives: (xij) Yearly 1. MIN 2. MIN 3. MIN 4. MIN

0.535

0.267

0.802

7.483315

56

16

4

36

12

4

2

6

5. Δ real wages in % MAX

0.632

0.562

0.534

71.2

5069

2025

1600

1444

123

45

40

38

6. MIN

0.489

0.622

0.612

11.25

126.6

30.25

49

47.33

19.38

5.5

7

6.88

7. MAX

0.048

0.706

0.706

10.34

106.8

0.25

53.29

53.29

15.1

0.5

7.3

7.3

8. MAX

0.585

0.574

0.573

14.87

221.1

75.69

72.76

72.69

25.756

8.70

8.53

8.53

9 MIN

0.816

0.408

0.408

4.899

24

16

4

4

8

4

2

2

10. MIN

2 1 3

-2.0913 -2.5967

rank

sum -2.1523

Table A-2. A MOORA Simulation Lithuanian Sustainable Development (2006-2012) with increase in x-wages x 2 (n°5)

sum -2.044997 -2.257522 -2.727157

0.54213 0.243957 -

0.5421269 0.2439571 0

0.38834 0.000000 0.16516

0 0.808122036 0.505076272

0 0.0280911 0.0983189

0.5421269 0.8081220 0.6579037

0 0.37139068 0.18569534

0 0.000000 0.65790365

0.70627892

0.62217

0.01067 0.00000 0.13332

0.70627892 0.706279 0.04837527

0.61150 0.6221684 0.4888466

0 0.000269 0.0117007

0.5733346

0.5733346 0.5736036 0.5850353

14.870897

0 0 0.40824829

0.40824829

0.40824829 0.40824829 0.816496581

4.898979486

4 4 16 24

2 2 4 8

10. Other Pollution MIN.

EMU EEU Secession

10.3358599

11.250973

72.692676 72.7609 75.69 221.14358

8.53 8.53 8.70 25.756

9. CO2 ton/cap. MIN.

max.

53.29 53.29 0.25 106.83

8. Minus % Energy Consumpt. MAX (a) 7.3 7.3 0.5 15.1

47.3344 49 30.25 126.5844

6.88 7 5.5 19.38

7. Increase GDP (in %) MAX.

A - Matrix of Responses of Alternatives on Objectives: (xij) 2. Increase 3. Def. Public 4. Unemploy. 5. Increase 6. Shop time Yearly 1. Inflation Public Debt Budget (% (in % labor real wages (in weekly Numbers (in %) MIN. (% GDP) GDP) MIN. force) MIN. in% MAX. hours) MIN. MIN. EMU 2 3 3 17 38 9 EEU 4 1.9 1.9 8.3 40 1 Secession 3 1 1 12 45 4 Totals 9 5.9 5.9 37.3 123 14 B - Sum of squares and their square roots Projects EMU 4 9 9 289 81 1444 EEU 16 3.61 3.61 68.89 1 1600 Secession 9 1 1 144 16 2025 Sum of 29 13.61 13.61 501.89 98 5069 squares Square 5.38516481 3.68917335 3.6891733 22.4029016 9.899494937 71.19691 roots C- Objectives divided by their square roots and MOORA EMU 0.37139068 0.813190 0.8131903 0.75883028 0.90913729 0.533731 EEU 0.74278135 0.515021 0.5150205 0.370488 0.101015254 0.5618221 Secession 0.55708601 0.271063 0.2710634 0.5356449 0.404061018 0.6320499 D - Reference Point Theory with Ratios: co-ordinates of the reference point equal to the maximal objective values Ri 0.37139068 0.271063 0.2710634 0.37049 0.90913729 0.533731 E - Reference Point Theory: Deviations from the reference point

Table B-1. A MOORA Simulation Lithuanian Sustainable Development (2006-2012) z with increase in x-wages with exponent 2 (n°5)

APPENDIX B.

rank min. 1 3 2

rank 1 2 3

Al. Kau Kla Mari Pane Šiau Taur Tel Ut. Vil

MAX

1.04 0.938 1.123 1.014 0.95 0.986 1.029 1.052 1.15 0.978

MAX

migra-tion

0.331 0.378 -0.018 0.887 -0.655 -0.545 -1.131 -0.142 0.517 1.089

2

revenue

19.950 13.660 16.430 18.600 20.120 26.800 30.600 21.970 17.330 16.330

8.6 4.6 5.1 8 7.6 9 8.8 6.4 7 5.7

844 865 950 755 851 807 724 936 982 1061

22.2 22.5 19.5 19.7 22.8 21 21.4 21 24.4 19.6

88 80 84 71 89 86 79 73 93 86

4.79 3.49 3.72 5.65 4.69 4.22 5.79 4.41 5.29 3.29

1314 1189 1133 1252 1039 1143 1289 1084 943 1428

1927 4999.5 3917.9 2904.1 2870.8 3410.2 2306.3 4177.8 2078.4 6776.3

1138.3 1393.2 3294.1 737.5 1489 949.1 365.3 2144.2 1440 2457.3

820.3 916.4 1003.9 437.9 848.7 546 202.3 1166.3 900.2 1248

109.3 143.03 97.5 109.31 102.77 83.26 110.53 85.12 84.25 225.97

135 215 245 174 216 176 172 154 135 250

MIN

3

expenditure

MAX

4

unemploym.

MAX

5

earnings

MAX

6

floor-space

MAX

7

pre-schools

MAX

8

schools MAX

9

animal products

MAX

10

retail trade

MAX

11

investm.

MAX

12

construction

MIN

13

dwellings

MAX

14

criminal acts

1

physicians 25.7 43.5 32.9 23.5 26 23.9 17.5 18.3 22.7 44.1

MAX

15

463 1886 2694 437 539 722 386 7716 288 1296

434 1726 2565 391 512 613 353 7640 270 1233

MIN

29 160 129 46 27 109 33 76 18 63

MIN

16

total pollution MIN

16(2)

16(1)

pollution: gas + liquid

APPENDIX C. SHARE OF POLLUTION FOR LITHUANIAN COUNTIES 2002 solid pollution

64

Willem K. M. Brauers

REFERENCES Afonso, A., Gomes, P. & Rother, P. (2011). Short and Long-Run Determinants of Sovereign Debt Credit Ratings, International Journal of Finance and Economics, 16, 1-15. Arrow, K. J. (1974). General Economic Equilibrium: Purpose, Analytic Techniques, Collective Choice, American Economic Review, June, 253-272. Ashbaugh-Skaife, H., Collins, D. W. & LaFond, R. (2006). The Effects of Corporate Governance on Firms’ Credit Ratings, Journal of Accounting and Economics, 42, 203-243. Berger, R. (2012). Corriere della Sera, January 12, 2012; cited by “De Tijd” Flemish Financial Newspaper, January 12, 2012. Bernard, H. R. (2006). Research Methods in Anthropology, Qualitative and Quantitative Approaches, 4th. Edition, Alta Mira Press, USA. Bertelsmann Stiftung. (2012). Blueprint for Incra, An International Non-Profit Credit Rating Agency, On July 1, 2017 Retrieved from: http://www.bfna.org. Brauers, W. K. M. (2008). Group Decision Making with Multi-Objective Optimization, Foundations of Computing and Decision Sciences, vol. 33, N°2, 167-179., Brauers, W. K. M. (2004). Optimization Methods for a Stakeholder Society, A Revolution in Economic Thinking by Multi-Objective Optimization, Series: Nonconvex Optimization and its Applications Volume 73, Isbn 1-4020-7681-9, Kluwer Academic Publishers and Springer, Boston-Dordrecht-London, 342 p. Brauers, W. K. M. (1998). Internationaal Voetbalstadion te Antwerpen (Dutch For International Football Stadium in Antwerp). Consulting, Antwerp. Brauers, W. K. M. (1995). Onderzoek nopens de oprichting van een Eurostadion te Berchem-Antwerpen, Consulting, Antwerp. Brauers, W. K. M. (1990). Measurement of capital for production in macro-economics, University of Antwerp, Institute for Developing Countries, Antwerpen. Brauers, W. K. M., Baležentis, A. & Baležentis, T. (2018). Fuzzy Analysis a Possible Solution for the Standard Error in Sampling? Journal of Modern Technolgy and Engineering, Vol. 3, N°1, Baku University and Jomard Publishing. Brauers, W. K. M. & Lepkova, N. (2019). Is Credit Rating Reserved Territory for Credit Rating Agencies? A MULTIMOORA Approach for European Firms and Countries, Transformations in Business & Economics, VGTU, Vilnius. Brauers, W. K. M. & Lepkova, N. (2003). The Application of the Nominal Group Technique to the Business Outlook of the Facilities Sector of Lithuania over the Period 2003-2012”, International Journal of Strategic Property Management (Sjr Scopus 0.046) Issn 1648-715x), Vol. 7, N°1, 1-9.

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Goldsmith, R. W. (1975). A Note on the National Balance Sheet of Belgium 1850-1971, Workshop on Quantitative Economic History, Discussion Paper, 7403, Center for Economic Studies, Catholic University of Leuven. Goldsmith, R. W. & Lipsey R. E. (1963). Studies in the National Balance Sheet of United States, Vol. 1, Princeton University Press for the Bureau of Economic Research, Princeton. Gordon, T. J., Enzer, S. & Rochberg, R. (1070). An Experiment in Simulation Gaming for Social Policy Studies. Technological Forecasting, 1(3), 241-261. Hassan, N., Uluçay, V. & Şahin, M. (2018). Q-Neutrosophic Soft Expert Set and its Application in Decision Making. International Journal of Fuzzy System Applications (Ijfsa), 7(4), 37-61. Hodge, N. (2017). http://www.complianceweek.com/authors/neil-hodge, October 24, 2017. Human Development Report. (2018). Human Development Indices and Indicators, HDRO. United Nations Development Programme, United Nations. Hwang, C. L. & Yoon, K. (1981). Multiple Attribute Decision Making, Springer, Berlin. IMF, World Economic Outlook, Database, April 2014, 5: report for selected countries. Journard, I., Pisu, M. & Bloch, D. (2012). Tackling income inequality. The role of taxes and transfers, OECD. Karlin, S. & Studden, W. J. (1966). Tchebycheff Systems: with Applications in Analysis and Statistics, Interscience Publishers, New York. MacCrimon, K. R. (1968). Decision Making Among Multiple Attribute Alternatives. A Survey and Consolidated Approach, Santa Monica: Rand Corporation. Rm-4823Arpa. Michel, A. (2014). Le Luxembourg, plaque tournante de l’évasion fiscale, Le Monde, 511, 2014. Miller, D. W. & Starr, M. K. (1969). Executive Decisions and Operation Research, Prentice-Hall, Englewood Cliffs. Miller, G. A. (1965). The Magical Number Seven Plus or Minus Two: Some Limits on Our Capacity for Processing Information, Psychological Review, 63, 81–97. Minkowsky, H. (1896). Geometrie der Zahlen, Teubner, Leipzig. Minkowsky, H. (1911). Gesammelte Abhandlungen, Teubner, Leipzig. Nabeeh, N. A., Abtel-Monem, A. & Abdelmouty, A. (2019). A hybrid approach of Neutrosophic with MULTIMOORA in application of Personnel Selection, Neutrosophic Sets and Systems, vol. 30, pp. 1-21. Nickell, P., Perraudin, W. & Varotto, S. (2000). Stability of Rating Transitions, Journal of Banking & Finance, vol. 24, 203-227. Obermayer, B. (December 5, 2014). Day in a Fiscal Paradise, chasing Letter Box Leads in Luxemburg (translation), Süddeutsche Zeitung.

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Uluçay, V., Şahin, M. & Hassan, N. (2018). Generalized Neutrosophic Soft Expert Set for Multiple-Criteria Decision-Making. Symmetry, 10(10), 437. Ulucay, V., Şahin, M. & Olgun, N. (2018). Time-Neutrosophic Soft Expert Sets and its Decision Making Problem. Matematika, 34(2), 246-260. White, L. J. (2010). Markets: The Credit Rating Agencies, Journal of Economic Perspectives, 24(2), 211-226. World Bank, World Development Indicators Database. (2011). http://data.worldbank.org/data-catalog/world-development-indicators. Accessed December 17, 2011. Worldgovernmentbonds.com/world-credit-ratings/2020.

In: Decision-Making with Neutrosophic Set Editor: Harish Garg

ISBN: 978-1-53619-419-7 © 2021 Nova Science Publishers, Inc.

Chapter 4

EVALUATION OF ONLINE EDUCATION SOFTWARE UNDER NEUTROSOPHIC ENVIRONMENT Fatma Kutlu Gündoğdu* and Serhat Aydın Industrial Engineering Department, National Defence University, Turkish Air Force Academy, Istanbul, Turkey,

ABSTRACT Nowadays, the online learning system is developing in higher education. Several causes of this occur, including pandemic restrictions, flexible access to content and instruction at any place, and cost-effectiveness for education institutions. Online learning can also enhance the availability of learning experiences for scholars. For this reason, several options must be evaluated for online educators. The goal is to highlight which factors students find essential in guaranteeing quality learning outcomes in the online learning environment. Therefore, we detected evaluation criteria, including qualitative factors, and we used neutrosophic sets, which is a generalization of the classic set, to deal with inaccurate data with qualitative factors. For this aim, we proposed the neutrosophic MULTIMOORA (Multiobjective Optimization by Ratio Analysis plus Full Multiplicative Form) method to evaluate online learning software concerning some critical factors that have significant effects on student satisfaction. For the validity of the proposed method, we also present comparative and sensitivity analyses. Finally, we performed a comparative analysis with neutrosophic TOPSIS method.

Keywords: neutrosophic sets, MULTIMOORA method, group decision-making, online education

*

Corresponding Author’s Email: [email protected].

70

Fatma Kutlu Gündoğdu and Serhat Aydın

1. INTRODUCTION The concept of fuzzy sets was introduced by Zadeh in 1965. The ordinary fuzzy set defines one real degree between 0 and 1 that is 𝜇𝐴̃ (𝑢) to represent the grade of membership of fuzzy set A on universe X. For almost 50 years, fuzzy sets have been employed in many real-life problems to handle vagueness. Additionally, the fuzzy set theory has been extended to several fuzzy sets (Kutlu Gündoğdu and Kahraman, 2019). Some fuzzy sets that are related to our chapter are summarized as follows: In expert systems, the membership degree is supported by the evidence and the nonmembership degree against the evidence is taken into consideration. To deal with the imprecision in data which employs a membership degree and a non-membership degree, the intuitionistic fuzzy set (IFS) theory was developed by Atanassov in 1986 as one of the significant extensions of the ordinary fuzzy set (FS) theory. According to this theory, the sums of the membership function degrees must be less than or equal to 1. Pythagorean fuzzy sets (PyFS) were proposed by Yager (2013) based on the logic of intuitionistic fuzzy sets. Like the IFS, membership and non-membership degrees are defined, and the squared sum of these degrees may at most be 1.0. PFS provides a larger preference area to assign membership functions for decision-makers. Some generalized Pythagorean fuzzy aggregation operators based on Einstein operations were proposed by Garg (2016, 2017) for improving these sets. Different from the above-mentioned fuzzy sets, neutrosophic sets (NS) were proposed by Smarandache in 1998 based on a different philosophy. Neutrosophic set is a robust general formal framework that generalizes the concept of the classic set, fuzzy set, and extensions of the fuzzy set (Wang et al., 2010). They can be independently assigned; therefore, their sum may take any value between 0 and 3. Neutrosophic sets do both handle the indeterminacy of the system and decrease the indecisiveness of inconsistent information. The neutrosophic set generalizes the above-mentioned sets from a philosophical point of view. These sets are defined in three dimensions: a truthiness degree (𝑇𝐴̃ ), an indeterminacy degree (𝐼𝐴̃ ), and a falsity degree (𝐹𝐴̃ ). The indeterminacy degree is independent of truthiness and falsity values. Some applications of neutrosophic sets include decision-making, machine learning, pattern recognition, medical diagnosis, robotics, and education. There are several types of neutrosophic sets such as singlevalued neutrosophic sets, interval-valued neutrosophic sets, linguistic neutrosophic numbers, and neutrosophic soft sets (Ye, 2014). There are many research articles about neutrosophic sets in many research areas, such as; Garai et al., (2020) developed a ranking method based on possibility mean for multi-attribute decision-making with single valued neutrosophic numbers, Garg and Rani (2020) developed new generalized Bonferroni mean aggregation operators of complex intuitionistic fuzzy information based on Archimedean t-norm and t-conorm, Edalatpanah (2020) proposed data envelopment analysis based on triangular neutrosophic numbers. Edalatpanah (2020a) investigated the

Evaluation of Online Education Software …

71

System of Neutrosophic Linear Equations based on the embedding approach. Yang et al., (2020b) developed neutrosophic data envelopment with triangular single valued Neutrosophic fuzzy numbers. Garai et al. (2020) presented a new ranking methodology of SVN-numbers for solving multi-attribute decision-making problems. Garg (2020a) proposed some new kinds of operational laws named as neutrality addition and scalar multiplication for the pairs of single-valued neutrosophic numbers. Garg (2020b) proposed a multiple attribute decision making method based on immediate probabilities aggregation operators for single-valued and interval neutrosophic sets. Multi-criteria decision-making algorithms handle real-life problems to find the best solution from a set of alternatives concerning the criteria. Various types of MCDM methods are laid out in literature under three categories, such as value measurement methods, distance-based methods, and outranking methods. Weighted aggregated sum product assessment (WASPAS), simple additive weighting (SAW), and the weighted product models (WPM) are examples of value measurement methods. Distance-based methods arethe technique for order preference by similarity to ideal solution (TOPSIS), vise kriterijumska optimizacija i kompromisno resenje (VIKOR), and combinative distance-based assessment (CODAS). Theoutranking methods are the elimination et choix traduisant la realité (ELECTRE), the preference ranking organization method for enrichment of evaluations (PROMETHEE), and gained and lost dominance score (GLDS). A variety of ranking aggregation techniques can be used to integrate these three categories. One of the most popular ranking aggregation software in literature is MULTIMOORA (Hafezalkotob et al., 2019). Brauers and Zavadskas (2006) presented MOORA (Multi-Objective Optimization based on a Ratio Analysis) method. This method includes two stages that are Ratio System and Reference Point Approach. They also (2010) enhanced the MOORA method to MULTIMOORA (Multi-Objective Optimization based on a Ratio Analysis) adding the full Multiplicative form method and using dominance theory to obtain a final integrative ranking based on the results. Ratio System and Full Multiplicative Form belong to the category of value measurement techniques while Reference Point Approach falls in the category of distance-based models. The MULTIMOORA method has applied in various fields such as project management, mining, personnel management, construction, entertainment, logistics, and aviation (Zavadskas et al., 2015). There are also many extensions of MULTIMOORA based on the fuzzy set theory. Brauers et al. (2011) proposed a MULTIMOORA model to update EU member states by combining fuzzy triangular numbers. Liu et al. (2015) proposed a new hybrid decision approach using the fuzzy MULTIMOORA method to evaluate health-care waste treatment. Balezentiene et al. (2013) developed a fuzzy decision support methodology using the ordinary fuzzy MULTIMOORA method for sustainable energy crop selection. In another research, Baležentis et al. (2012a; 2012b) presented a decision-making method based on linguistic reasoning with an application on personnel selection. Deliktas and

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Fatma Kutlu Gündoğdu and Serhat Aydın

Ustun (2017) performed student selection by combining fuzzy MULTIMOORA and goal programming. Eghbali-Zarch et al. (2018) proposed a hybrid approach that includes fuzzy SWARA and MULTIMOORA methods applied to pharmacological therapy selection of type 2 diabetes. For the risk assessment based on FMEA, a hybrid method combining fuzzy AHP and MULTIMOORA methods was proposed by Fattahi and Khalilzadeh (2018). Cebi and Otay (2016) suggested a two-stage fuzzy approach for supplier selection, considering the lead time and quantity discounts. Further fuzzy extensions of the MULTIMOORA method are as follows: Stanujkic et al. (2015) presented the MULTIMOORA method based on interval-valued triangular fuzzy numbers by integrating the weighted averaging and geometric operators. Dorfeshan et al. (2018) developed the MULTIMOORA method based on interval type-2 fuzzy sets for selecting a project-critical path. Baležentis et al. (2014) integrated the MULTIMOORA method and intuitionistic fuzzy theory, and it was applied to performance management. Besides, interval-valued intuitionistic fuzzy MULTIMOORA was developed by Zhao et al. (2017) and Zavadskas et al. (2015) for group decision-making problems. Zeng et al. (2013) proposed the MULTIMOORA method under a hesitant fuzzy environment for group decision-making. Li (2014) also presented an extension of the MULTIMOORA method based on hesitant fuzzy sets. Dong (2018) presented the improved version of the hesitant fuzzy MULTIMOORA method and applied it to assess universities’ innovative ability. For robot assessment and selection, the hesitant fuzzy linguistic MULTIMOORA method was used by Liu et al. (2019). Lin et al. (2020) proposed a new MCDM model for site selection of car-sharing stations under the picture fuzzy environment. Liang et al. (2019) developed a power average based MULTIMOORA method based on Pythagorean fuzzy set for group decision-making. Li et al. (2020) presented the Pythagorean fuzzy MULTIMOORA method for passenger satisfaction based on a large group environment. Spherical fuzzy MULTIMOORA was proposed by Kutlu Gündoğdu (2019) and was employed for personnel selection. Moreover, this approach was extended to intervalvalued spherical fuzzy MULTIMOORA by Aydın and Kutlu Gündoğdu (2020) to evaluate Industry 4.0 technologies. Stanujkic et al. (2017) extended the MULTIMOORA method to the neutrosophic MULTIMOORA method. Zavadskas et al. (2017) developed neutrosophic MULTIMOORA for selecting residential house elements and material selection. Aydın (2018) used the neutrosophic MULTIMOORA technique for the selection of augmented reality goggles. Liang et al. (2018) improved neutrosophic MULTIMOORA method based on a novel weighted and geometric Heronian mean operators. Nabeeh et al. (2019) proposed a hybrid approach of the neutrosophic MULTIMOORA method for personnel selection. Tian et al. (2017) introduced an improved MULTIMOORA method by using neutrosophic linguistic information. To the best of our knowledge, there is no research based on online education software such as Microsoft Teams, Zoom, Blackboard concerning determined essential factors based on the neutrosophic MULTIMOORA method. Therefore, the main contribution of this

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chapter is handling the evolution of online education software problems by using neutrosophic MULTIMOORA for the first time in literature. Online education has become a more popular concept in recent years providingflexible learning in education (Park et al., 2009). The number of learners and universitiesthat have used online learning platforms has grown rapidly in 2020 due to restrictions caused by the COVID-19 pandemic. The merging of online teaching and learning into the stream of everyday practices at schools, and the increasingly important role of online education software in institutions has vital importance. Combining interactive technology and more active modes of learning requires student skills and responsibility for their learning, therefore students may give differing reactions to online education software. Hence, determining which factors are significant for student satisfaction in terms of online learning and which online learning software is better than others concerning these factors have a critical importance. (Gikandi, et al. (2011). Numerous studies have tried to identify which factors have definitive influence on students’ satisfaction during online education (Alptekin and Karsak, 2011; Begicevic et al. 2007; Bhuasiri et al., 2012; Zare et al., 2016). The online education software evaluation problem involves substantial vagueness and uncertainty; therefore, we need to utilize fuzzy logic rather than classical logic. Neutrosophic sets are one of the extensions of fuzzy sets, which introduce a new component called “indeterminacy”, and consider more information than fuzzy sets. In this chapter, the neutrosophic MULTIMOORA method is applied to determine the best online education software based on the essential factors that are defined to have a remarkable impact on students’ satisfaction, according to literature. Comparative and sensitivity analyses are performed to demonstrate the validity of the methodology as well. The rest of the paper is organized as follows. Section 2 summarizes some concepts and preliminaries of neutrosophic sets. Section 3 introduces the methodology, which is neutrosophic MULTIMOORA. This method is applied to the selection of the best online learning software concerning the criteria mentioned in Section 4. In Section 5, sensitivity analysis is performed to show the robustness of the proposed decision-making method. In Section 6, we have performed a comparative analysis of neutrosophic TOPSIS. Finally, some final remarks and future research suggestions are offered in Section 7.

2. NEUTROSOPHIC SETS Many systems have been developed to deal with approximate and uncertain reasoning since the introduction of fuzzy logic. Among the latest and most general proposals, the neutrosophic logic, introduced by Smarandache (1998), is a generalization of fuzzy logic. Fuzzy logic extends classical logic by assigning membership between 0 and 1 to variables. Smarandache represented a new term of ‘‘indeterminacy,” which

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carries more information than fuzzy logic. A neutrosophic set 𝐴 in a universal set 𝑋 is characterized by a truth-membership function 𝑇𝐴 (𝑥) , an indeterminacy-membership function 𝐼𝐴 (𝑥), a falsity-membership function 𝐹𝐴 (𝑥). The functions 𝑇𝐴 (𝑥), 𝐼𝐴 (𝑥), 𝐹𝐴 (𝑥) in 𝑋 are three subsets of the nonstandard interval ]¯0,1⁺[i.e., 𝑇𝐴 (𝑥) ⊆ ]¯0,1⁺[ , 𝐼𝐴 (𝑥) ⊆ ]¯0,1⁺[ , and 𝐹𝐴 (𝑥) ⊆ ]¯0,1⁺[ . There is no restriction on the sum of 𝑇𝐴 (𝑥), 𝐼𝐴 (𝑥), 𝐹𝐴 (𝑥)i.e., 0 ≤ 𝑠𝑢𝑝𝑇𝐴 (𝑥) + 𝑠𝑢𝑝𝐼𝐴 (𝑥) + 𝑠𝑢𝑝𝐹𝐴 (𝑥) ≤ 3⁺ (Ye, 2014). However, it is not easy to apply the neutrosophic set and set-theoretic operators in the real application. Therefore, Wang et al. (2010) proposed a single-valued neutrosophic set extending neutrosophic sets. For each point x in X, we have TA ( x), I A ( x), FA ( x)  0,1 , and 0  TA ( x), I A ( x), FA ( x)  3.

Preliminaries of the Single Valued Neutrosophic Set In this section, basic concepts of single valued neutrosophic sets are given as follows.

Definition 1 Let X be a space of objects, with a generic element in X denoted by x. The Neutrosophic set 𝐴 in 𝑋 is as follows: (Smarandache, 1998) A

 x, T ( x), I A

A

( x), FA ( x) x  X



(1)

where, 𝑇𝐴 (𝑥) is characterized by the truth-membership function, 𝐼𝐴 (𝑥) is the indeterminacy membership function and 𝐹𝐴 (𝑥) is the falsity-membership function. TA ( x) : X    0,1  , I A ( x) : X    0,1  , FA ( x) : X    0,1 

(2)

There is no restriction on the sum of TA ( x) , I A ( x) and FA ( x) , so 

0  sup TA ( x)  sup I A ( x)  sup FA ( x)  3 .

Definition 2 If the functions TA ( x) , I A ( x) and FA ( x) are singleton subintervals/ subsets in the real standard [0,1], that is TA ( x) : X  0,1 , I A ( x) : X  0,1 , FA ( x) : X  0,1 Then, a simplification of neutrosophic set A is denoted by (Ye, 2014);

A

 x, T ( x), I A

A

( x), FA ( x) x  X



(3)

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which is called a single valued neutrosophic set. It is a subclass of neutrosophic sets. 0  TA ( x), I A ( x), FA ( x)  3.

(4)

Definition 3 For a single valued neutrosophic set 𝐴 in 𝑋, the triple 〈𝑡𝐴 , 𝑖𝐴 , 𝑓𝐴 〉 is called the single valued neutrosophic number. Definition 4 Let 𝑥1 = 〈𝑡1 , 𝑖1 , 𝑓1 〉 and 𝑥2 = 〈𝑡2 , 𝑖2 , 𝑓2 〉 be two single-valued neutrosophic numbers and 𝜆 > 0; then the basic operators are as follows (Ye, 2014): x x .x 1

2



x



t t ,i  i

1

2

1



2

t

1

1

 t 2  t 1t 2, i 1i 2,

2

 i 1i 2,

f

1



f f 1

f

2



2

(5)

,

f f 1

2

,

(6)

 x1  1  (1  t1 ) , i1 , f1 ,

(7)

x1  t1 ,1  (1  i1 ) ,1  (1  f1 ) .

(8)

Definition 5 Let 𝑥 = 〈𝑡𝑥 , 𝑖𝑥 , 𝑓𝑥 〉 be a single valued neutrosophic number; then the score function 𝑠𝑥 of 𝑥 can be as follows (Ye, 2014):

sx  (1  tx  2ix  f x ) / 2

(9)

where 𝑠𝑥 ∈ [−1,1].

Definition 6 Let 𝑥1 = 〈𝑡1 , 𝑖1 , 𝑓1 , 〉 and 𝑥2 = 〈𝑡2 , 𝑖2 , 𝑓2 , 〉 are Single Valued Neutrosophic Numbers and the maximum distance between 𝑥1 and 𝑥2 is as follows (Ye, 2014):  t  t , x1 , x2   max , d max ( x1 , x2 )   1 2  f1  f 2 , x1 , x2   min .

(10)

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76

Definition 7 Let 𝐴𝑗 = 〈𝑡𝑗 , 𝑖𝑗, 𝑗𝑗 〉 be a collection of Single Valued Neutrosophic Sets and 𝑊 = (𝑤1 , 𝑤2 , … . , 𝑤𝑛 )𝑇 be an associated weighting vector. Then, the Single Valued Neutrosophic Weighted Average operator of 𝐴𝑗 is as follows (Ye, 2014): Single Valued Neutrosophic Weighted Average (𝐴1 , 𝐴2 , … . , 𝐴𝑛 ) 𝑛

𝑛

=∑ 𝑗=1

𝑤𝑗 𝐴𝑗 = (1 − ∏

𝑛

(1 − 𝑡𝑗 )𝑤𝑗 , ∏

𝑗=1

𝑗=1

𝑛

(𝑖𝑗 )𝑤𝑗 , ∏

𝑗=1

(𝑓𝑗 )𝑤𝑗 )

(11)

n

where: 𝑤𝑗 is the element 𝑗 of the weighting vector, 𝑤𝑗 ∈ [0,1] and

w j 1

j

 1.

Definition 8 Let 𝐴𝑗 = 〈𝑡𝑗 , 𝑖𝑗, 𝑗𝑗 〉 be a collection of Single-Valued Neutrosophic Sets and 𝑊 = (𝑤1 , 𝑤2 , … . , 𝑤𝑛 )𝑡 be an associated weighting vector. Then the Single-Valued Neutrosophic Weighted Geometric operator of 𝐴𝑗 is as follows (Ye, 2014): Single-Valued Neutrosophic Weighted Geometric Operator (𝐴1 , 𝐴2 , … . , 𝐴𝑛 ) 𝑛

=∏

𝑛

(𝐴𝑗 )𝑤𝑗 = (∏

𝑗=1

𝑛

(𝑡𝑗 )𝑤𝑗 , 1 − ∏

𝑗=1

𝑛

(1 − 𝑖𝑗 )𝑤𝑗 , 1 − ∏

𝑗=1

(1 − 𝑓𝑗 )𝑤𝑗 )

(12)

𝑗=1

n

Where: 𝑤𝑗 is the element 𝑗 of the weighting vector, 𝑤𝑗 ∈ [0,1] and

w j 1

j

 1.

3. NEUTROSOPHIC MULTIMOORA METHOD In this section, the steps of the Neutrosophic MULTIMOORA method (Stanujkic et al., 2017) are given.

3.1. Neutrosophic MOORA- Ratio Method Step 1. Calculate 𝑌𝑖+ and 𝑌𝑖− by using the Single Valued Neutrosophic Weighted Average Operator, as follows:

Evaluation of Online Education Software …   Yi   1   (1  t j ) w j ,  (i j ) w j ,  ( f j ) w j  jmax jmax  jmax 

 w w w Yi   1   (1  t j ) j ,  (i j ) j ,  ( f j ) j j  j  j   min min min

  

77

(13) (14)

where 𝑌𝑖+ and 𝑌𝑖− denote the importance of the alternative i obtained based on the benefit and cost criteria, respectively; 𝑌𝑖+ and 𝑌𝑖− are Single Valued Neutrosophic Numbers. Step 2. To get the overall importance of each alternative, we need to get defuzzified values of 𝑌𝑖+ and 𝑌𝑖− . Hence, we utilized Eq. (15-17).

s Yi

yi  s Yi  

(15)

yi  s Yi  

(16)

  (1  t  2i  f s Y    1,1 yi

yi

yi

)/2

(17)

i

Step 3. Compute The overall importance of each alternative as follows:

yi  yi  yi

(18)

Step 4. Rank the alternatives based on the value of 𝑦𝑖 in descending order and so the alternative with the highest value is the best alternative.

3.2. Neutrosophic Moora-Reference Point Method Step 1. Each coordinate of the reference point 𝑟 ∗ = {𝑟1,∗ 𝑟2,∗ … . . , 𝑟𝑛,∗ } is a Single Valued Neutrosophic Number, 𝑟𝑗∗ = 〈𝑡𝑗∗ , 𝑖𝑗∗ , 𝑓𝑗∗ 〉, and 𝑟𝑗∗ can be calculated as follows:   max t , min i , min f , , j   , ij ij ij max  i i i  * rj     min tij , max iij , max fij , , j   min .  i i i 

(19)

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Fatma Kutlu Gündoğdu and Serhat Aydın

Step 2. Calculate the maximum distance from each alternative to the reference point as follows: * dimax , j  d max ( rij , rj ) w j

(20)

𝑚𝑎𝑥 Where: 𝑑𝑖,𝑗 denotes the maximum distance of the alternative 𝑖 obtained based on

the criteria 𝑗 determined by Eq. (10). Step 3. Determine the maximum distance of each alternative, as follows: dimax  max dimax ,j

(21)

j

Step 4. Rank the alternatives based on the value of the 𝑑𝑚𝑎𝑥 in descending order and so the alternative with the lowest value is the best alternative.

3.3. Neutrosophic MOORA-Full Multiplicative Form Step 1. Let 𝐴𝑖 = 〈𝑡𝐴𝑖 , 𝑖𝐴𝑖 , 𝑓𝐴𝑖 〉 and 𝐵𝑖 = 〈𝑡𝐵𝑖 , 𝑖𝐵𝑖 , 𝑓𝐵𝑖 〉 are Single Valued Neutrosophic Numbers

 w w w Ai    (t j ) j ,1   (1  i j ) j ,1   ( f j ) j jmax jmax  jmax

  

(22)

 w w w Bi    (t j ) j ,1   (1  i j ) j ,1   ( f j ) j j min j min  jmin

  

(23)

Step 2. Calculate the Score Function of 𝐴𝑖 and 𝐵𝑖 as follows: ai  s  Ai  bi  s  Bi 

(24)

Step 3. Calculate the overall utility for each alternative as follows: ui 

ai bi

(25)

Step 4. Rank the alternatives based on the value of the 𝑢 𝑖 in descending order and so, the alternative with the highest value is the best alternative.

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3.4. Dominance Theory The final ranking of the alternatives can be calculated based on dominance theory. Brauers and Zavadskas (2011) developed the dominance theory based on Dominance, Dominated, Transitivity, and Equability. The theory is employed to summarize the three ranks provided by respective parts of MULTIMOORA into one.

4. APPLICATION In the application section, we evaluate the distance education software to the determined criteria. We evaluate three different online education software used approximately all over the world in the COVID-19 pandemic. Then, we determine six criteria by conducting a literature review, including qualitative factors. 𝐶1 - Network Infrastructure (benefit), 𝐶2 - Ease of use (benefit), 𝐶3 - Exam Management System (benefit), 𝐶4 - Reporting (benefit), 𝐶5 - Privacy (benefit), 𝐶6 - Usage rate of internet (non-benefit). Weights of alternatives are determined by using classical Analytical Hierarchy Process (Saaty, 1980). The weights are 0.34, 0.10, 0.19, 0.12, 0.13, 0.09, respectively. Alternatives are evaluated by the three experts, whose weights are 0.3, 0.4, 0.3 respectively, using online education software for three months. First, three experts evaluated the alternatives according to the determined criteria by using any scale. Experts freely assign the neutrosophic numbers as shown in Table 1-3. Table 1. Decision matrix for online education software evaluation by Expert 1

𝐴1 𝐴2 𝐴5

𝑪𝟏 Benefit 〈0.7,0.3,0.2〉 〈0.4,0.5,0.3〉 〈0.4,0.4,0.5〉

𝑪𝟐 Benefit 〈0.8,0.2,0.1〉 〈0.7,0.4,0.5〉 〈0.6,0.4,0.5〉

𝑪𝟑 Benefit 〈0.5,0.5,0.6〉 〈0.6,0.5,0.8〉 〈0.7,0.5,0.6〉

𝑪𝟒 Benefit 〈0.5,0.7,0.6〉 〈0.4,0.6,0.1〉 〈0.2,0.4,0.3〉

𝑪𝟓 Benefit 〈0.2,0.8,0.4〉 〈0.7,0.5,0.2〉 〈0.7,0.2,0.5〉

𝑪𝟔 Non-benefit 〈0.6,0.4,0.5〉 〈0.8,0.1,0.2〉 〈0.3,0.4,0.5〉

Table 2. Decision matrix for online education software evaluation by Expert 2

𝐴1 𝐴2 𝐴5

𝑪𝟏 Benefit 〈0.6,0.4,0.4〉 〈0.5,0.6,0.4〉 〈0.4,0.4,0.5〉

𝑪𝟐 Benefit 〈0.7,0.3,0.4〉 〈0.2,0.5,0.4〉 〈0.5,0.5,0.4〉

𝑪𝟑 Benefit 〈0.4,0.5,0.4〉 〈0.7,0.5,0.2〉 〈0.7,0.4,0.2〉

𝑪𝟒 Benefit 〈0.3,0.4,0.6〉 〈0.4,0.5,0.2〉 〈0.3,0.4,0.3〉

𝑪𝟓 Benefit 〈0.4,0.5,0.4〉 〈0.6,0.5,0.2〉 〈0.6,0.1,0.4〉

𝑪𝟔 Non-benefit 〈0.7,0.2,0.1〉 〈0.7,0.2,0.2〉 〈0.5,0.2,0.1〉

Fatma Kutlu Gündoğdu and Serhat Aydın

80

Table 3. Decision matrix for online education software evaluation by Expert 3

𝐴1 𝐴2 𝐴5

𝑪𝟏 Benefit 〈0.6,0.4,0.2〉 〈0.3,0.4,0.6〉 〈0.5,0.2,0.3〉

𝑪𝟐 Benefit 〈0.8,0.1,0.1〉 〈0.5,0.3,0.5〉 〈0.5,0.4,0.6〉

𝑪𝟑 Benefit 〈0.6,0.4,0.4〉 〈0.6,0.4,0.2〉 〈0.5,0.1,0.2〉

𝑪𝟒 Benefit 〈0.5,0.1,0.3〉 〈0.4,0.5,0.2〉 〈0.4,0.3,0.4〉

𝑪𝟓 Benefit 〈0.4,0.5,0.5〉 〈0.6,0.4,0.3〉 〈0.5,0.5,0.1〉

𝑪𝟔 Non-benefit 〈0.4,0.6,0.2〉 〈0.7,0.2,0.2〉 〈0.4,0.2,0.3〉

Then, decision matrices are aggregated by using Eq. (11). The aggregated decision matrix is shown in Table 4. Table 4. The aggregated decision matrix for online education software evaluation

𝐴1 𝐴2 𝐴5

𝐴1 𝐴2 𝐴5

𝐶1 Benefit 〈0.598,0.402,0.298〉 〈0.374,0.529,0.454〉 〈0.402,0.356,0.460〉 𝐶4 Benefit 〈0.407,0.342,0.513〉 〈0.369,0.566,0.191〉 〈0.279,0.402,0.369〉

𝐶2 Benefit 〈0.735,0.215,0.191〉 〈0.471,0.430,0.501〉 〈0.499,0.469,0.529〉 𝐶5 Benefit 〈0.312,0.617,0.469〉 〈0.598,0.501,0.265〉 〈0.570,0.251,0.309〉

𝐶3 Benefit 〈0.471,0.469,0.495〉 〈0.598,0.501,0.356〉 〈0.606,0.309,0.327〉 𝐶6 Non-benefit 〈0.546,0.402,0.251〉 〈0.700,0.191,0.309〉 〈0.374,0.381,0.284〉

After getting aggregated matrix, the steps of Neutrosophic MULTIMOORA method are applied.

Neutrosophic MOORA- Ratio Method The ranking results of alternatives are obtained based on Neutrosophic MOORARatio Method by using Eq. (13) to Eq. (18) as shown in Table 5. Table 5. The ranking of alternatives based on Neutrosophic MOORA- Ratio Method

𝐴1 𝐴2 𝐴3

𝒀𝒊+

𝒀𝒊−

𝒔(𝒀𝒊+ )

𝒔(𝒀𝒊− )

〈0.499,0.441,0.397〉 〈0.441,0.546,0.396〉 〈0.441,0.383,0.436〉

〈0.075,0.914,0.872〉 〈0.113,0.849,0.890〉 〈0.045,0.909,0.883〉

0,110 −0,024 0,120

−0,812 −0,737 −0,827

𝒚𝒊 0,923 0,714 0,947

Rank 2 3 1

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Neutrosophic Moora-Reference Point Method The Neutrosophic Moora-Reference Point Method starts with the assigning the reference point (𝑟𝑗∗ ) by using Eq. (19). The reference points are shown in Table 6. Table 6. The reference points

𝑟𝑗∗

𝑪𝟏 〈0.598,0.356,0.298〉

𝑪𝟐 〈0.735,0.215,0.191〉

𝑪𝟑 〈0.606,0.309,0.327〉

𝑟𝑗∗

𝐶4 〈0.407,0.342,0.191〉

𝐶5 〈0.598,0.251,0.265〉

𝐶6 〈0.374,0.402,0.309〉

The ranking results of alternatives are obtained based on Neutrosophic MooraReference Point Method by using Eq. (19) to Eq. (21) as shown in Table 7. Table 7. The ranking results of alternatives are obtained based on Neutrosophic Moora-Reference Point Method * dimax , j  d max ( rij , rj ) w j

𝐴1 𝐴2 𝐴5

0.000 0.078 0.068

0.000 0.028 0.025

0.026 0.001 0.000

0.000 0.005 0.016

0.039 0.000 0.004

0.006 0.000 0.003

dimax  max dimax ,j j

Rank

0.039 0.078 0.068

1 3 2

Neutrosophic MOORA-Full Multiplicative Form The ranking results of alternatives are obtained based on Neutrosophic MOORA-Full Multiplicative Form by using Eq. (22) to Eq. (25) as shown in Table 8. Table 8. The ranking results of alternatives are obtained based on Neutrosophic MOORA-Full Multiplicative Form

𝐴1 𝐴2 𝐴3

𝑨𝒊 〈0.537,0.397,0.359〉 〈0.491,0.478,0.352〉 〈0.488,0.325,0.378〉

𝑩𝒊 〈0.942,0.050,0.028〉 〈0.965,0.021,0.036〉 〈0.907,0.046,0.033〉

𝒂𝒊 0,192 0,091 0,230

𝒃𝒊 0.907 0.944 0.891

𝑼𝒊 0,212 0,097 0,258

Rank 2 3 1

Dominance Theory The ranking of the alternatives obtained by three different MULTIMOOORA methods can be seen in Table 9. As shown in Table 9, the Neutrosophic MOORA- Ratio Method and Neutrosophic MOORA-Full Multiplicative Form results are the same. At the

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same time, the Neutrosophic Moora-Reference Point Method is different. Finally, we get the final ranking of alternatives according to dominance theory, as seen in Table 9. Table 9. The final ranking of alternatives according to the Neutrosophic MULTIMOORA method Neutrosophic Ratio Method 𝐴1 𝐴2 𝐴3

2 3 1

Neutrosophic Reference Point Method 1 3 2

Neutrosophic Full Multiplicative Form 2 3 1

Dominance Theory 2 3 1

The ranking of alternatives is 𝐴3 > 𝐴1 > 𝐴2 according to the Neutrosophic MULTIMOORA method.

5. SENSITIVITY ANALYSIS In this section, a sensitivity analysis was performed. Different weights were assigned to the criteria. They analyzed to observe how much it would influence the final rankings of alternatives. In the first case, we chanced the weights of criteria as follows: 0.5, 0.1, 0.1, 0.1, 0.1, 0.1 and the ranking of alternatives is gotten as follows 𝐴1 > 𝐴3 > 𝐴2 In the second case, we changed the weights of criteria as follows: 0.1, 0.5, 0.1, 0.1, 0.1, 0.1 and the ranking of alternatives is gotten as follows 𝐴1 > 𝐴3 > 𝐴2 . In the third case, 0.1, 0.1, 0.5, 0.1, 0.1, 0.1 0.1 are used respectively and the ranking of alternatives provided as follows 𝐴3 > 𝐴2 > 𝐴1 . In the fourth case, we modified the weights as follows: 0.1, 0.1, 0.1, 0.5, 0.1, 0.1 and the ranking of alternatives obtained as follows 𝐴2 > 𝐴3 > 𝐴1 . In the next case, 0.1, 0.1, 0.1, 0.1, 0.5, 0.1 are used, and the ranking of alternatives as follows 𝐴3 > 𝐴2 > 𝐴1 . In the last case, 0.1, 0.1, 0.1, 0.1, 0.1, 0.5 are used and the ranking of alternatives is gotten as follows 𝐴2 > 𝐴3 > 𝐴1 . The ranking of alternatives for all cases can be seen in Table 10. Table 10. The results of sensitivity analysis 𝑨𝟏 Case 1 Case 2 Case 3 Case 4 Case 5 Case 6

1 1 3 3 3 3

𝑨𝟐 𝑨𝟑 Ranking 3 2 3 2 1 2 1 2 2 1 1 2

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6. COMPARATIVE ANALYSIS In this section, we performed a comparative analysis of neutrosophic TOPSIS (NS-TOPSIS. Due to space constraints, the steps of the comparative analysis will not be explained in detail. Instead of this, the pseudo algorithm of NS-TOPSIS is given as follows (Şahin and Yiğider, 2014).

Algorithm 1. Pseudo Representation of NS-TOPSIS Input: 𝑛 : number of evaluation criteria (i  1,2,...m) ,

( j  1,2,...n)

, 𝑚 : number of alternatives

𝑠: number of evaluators (k  1,2,...s)

Output: Order of alternatives 1: for 𝑘 = 1: 𝑠 do 𝑘 𝑘 Experts are freely assigned the neutrosophic numbers as 𝑟̃ 𝑘 𝑖𝑗 = 〈[𝑡𝑖𝑗 ], [𝑖𝑖𝑗 ], [𝑓𝑖𝑗𝑘 ]〉 end for 2: Aggregate decision matrices to obtain consensus neutrosophic decision matrix 𝑅 = (𝑟̃𝑖𝑗 )𝑚×𝑛

𝑅 = 𝑟̃𝑖𝑗 =

s

 r

k k ij

k 1

s s s    1   (1  tijk )k ,  (iijk )k , ( f ijk )k  k 1 k 1 k 1  

where the weight vector of the evaluator is  j  0.3,0.4,0.3 . T

3: Calculate the weights of criteria based on the opinion of evaluators by using classical AHP wj  0.34, 0.10, 0.19, 0.12, 0.13, 0.09 ⟹

Based on Section 4

4: Construct weighted consensus NS decision matrix (𝑅∗ = (𝑟̃𝑖𝑗𝑤 )𝑚×𝑛 ) 𝑟̃𝑖𝑗𝑤 = ∑𝑛𝑗=1 𝑤𝑗 𝑟̃𝑖𝑗 , 𝑖 = 1,2, … , 𝑚 𝑛 ∗

𝑅 = where 𝑟̃𝑖𝑗𝑤 =

𝑟̃𝑖𝑗𝑤



n



j 1

n

n



= ∑ 𝑤𝑗 𝑟̃𝑖𝑗 = 1   (1  t j ) w ,  (i j ) w , ( f j ) w 

𝑗=1 𝑤 𝑤 〈[𝑡𝑖𝑗 ], [𝑖𝑖𝑗 ], [𝑓𝑖𝑗𝑤 ]〉.

j

j

j 1

j

j 1



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5: Calculate the neutrosophic positive ideal solutions ( r j* ) and negative ideal solutions ( rj )   max t w , min i w , min f w , , j  J ij ij ij benefit ,  i i i  * rj     min tijw , max iijw , max f ijw , , j  J cos t  i i i  where rj*   t *j , i*j , f j* 

  min t w , max i w , max f w , , j  J ij ij ij benefit ,  i i i   rj     max tijw , min iijw , min f ijw , , j  J cos t  i i i  where rj   t j , i j , f j 

6: Calculate distance measures from r j* and rj for 𝑖 = 1: 𝑚 do Si* 



1 n  * w  t j  tij 3 j 1 

  i 2

*

j

 iijw

  f 2

*

j



2  fijw  

end for 7: Calculate distance measures from rj for 𝑖 = 1: 𝑚 do Si 



1 n   w  t j  tij 3 j 1 

  i 2



j

 iijw

  f 2

 j

2  fijw   

end for 8: Compute the relative closeness ratio ( Ci ) for 𝑖 = 1: 𝑚 do Ci 

end for

Si Si  Si*

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9: Order the alternatives in descending order of 𝐶𝑖 ∗ The ranking of alternatives is 𝐴1 > 𝐴3 > 𝐴2 according to the Neutrosophic TOPSIS method. Table 11. The ranking based on neutrosophic TOPSIS method Alternatives

S i

Si*

Ci

𝐴1 𝐴2 𝐴3

0.122198 0.161814 0.127155

0.142513 0.093758 0.131467

0.538372 0.366856 0.508337

Ranking 1 3 2

CONCLUSION Nowadays, all countries in the world struggle with the epidemic disease COVID–19. It has been required to apply to online learning platforms for the continuity of educational activities. Online learning becomes a useful tool by linking digital content delivery with learning services. However, at this point, we face the problem of determining which online learning platform on the market universities should be suitable for our students. For this reason, it should be investigated which criteria should be taken into consideration in choosing the most suitable online learning platform for educational institutions in selecting the best platform from the alternatives. For this purpose, we applied one of the essential extensions of the decision-making approach, the neutrosophic MULTIMOORA method, to find the best online education software. First, we determined the evaluation criteria, which are the most used ones in literature. Then, we determined the weights of criteria by using the classical Analytical Hierarchy Process (Saaty, 1980). Then, we applied the Neutrosophic MOORA- Ratio Method, Neutrosophic Moora-Reference Point Method, Neutrosophic MOORA-Full Multiplicative Form into the problem, respectively. In the last step, dominance theory was applied to get the final rank, and the third alternative is selected as the best alternative according to the Neutrosophic MULTIMOORA method. Then we performed the sensitivity analysis to show the model’s robustness, and we got satisfactory results according to the sensitivity analysis. We also implemented comparative analysis with the NS-TOPSIS method, and we got similar ranking results. Therefore, the neutrosophic MULTIMOORA method can effectively handle the online education software evaluation problem and efficiency. Online learning is becoming a more critical issue day by day. It has gained much more importance during the Covid-19 pandemic, and we are sure that it will be used continuously after the Covid-19 pandemic. Therefore, online education software will be

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used in many fields of online learning. This chapter provides a model for determining the suitability of distance education software. For further studies, the application results can be compared with the other extensions of fuzzy sets, and the number of criteria can be increased for the evaluation process. Some types of linguistic fuzzy operators (Garg, 2019; Garg 2020c) can be implemented into the problem. Moreover, different fuzzy MCDM models can be used in online education software evaluation problems.

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In: Decision-Making with Neutrosophic Set Editor: Harish Garg

ISBN: 978-1-53619-419-7 © 2021 Nova Science Publishers, Inc.

Chapter 5

A NEW ATTRIBUTE SAMPLING PLAN FOR ASSURING WEIBULL DISTRIBUTED LIFETIME USING NEUTROSOPHIC STATISTICAL INTERVAL METHOD P. Jeyadurga and S. Balamurali Department of Computer Applications, Kalasalingam Academy of Research and Education, Krishnankoil, TN, India

ABSTRACT The existing acceptance sampling plans designed under classical statistics make acceptance/rejection decision using precise data obtained by inspecting a lot of products. Such sampling plans cannot be applied for lot sentencing if the data contain indeterminacy, vague, and imprecise observations. Hence, the researchers concentrate on designing the sampling plans under neutrosophic statistics in recent days, extending of classical statistics. Neutrosophic statistics is considered as an appropriate one for handling data under uncertain environment. This chapter introduces an acceptance sampling plan for assuring Weibull distributed lifetime of the products using the neutrosophic interval method. The probabilities corresponding to non-failure, failure and indeterminate case are obtained under Weibull distribution. The probabilities for acceptance and rejection of the lot and probability for indeterminate case under the proposed plan are then calculated at various combinations of plan parameters. The advantages of the proposed plan are discussed and an appropriate application of the proposed sampling plan is explained with the help of real-time data. In this study, it is observed that the proposed plan has good discriminating power since it provides minimum chance to accept the lot and high chance to reject the lot when product quality is poor. Compared to the existing single sampling plan, the proposed plan reduces the probability of acceptance of the lot by over 74% when product quality is poor and it also exhibits a better discriminatory power of the proposed plan.

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Keywords: attribute sampling plan, neutrosophic statistics, percentile life, Weibull distribution

1. INTRODUCTION In the production field, quality is the most important measure of excellence or a state without any defects, deficiencies and significant variations. In practice, consumers determine certain specifications on product quality and expect that the products must have standard quality levels. The responsibilities of the producer are to enhance the product’s quality and ensure that the lot of products adheres to the specified standards. Obviously, this assurance should be provided in advance of marketing the products for consumer use in order to avoid the acceptance of defective or failure products. It is to be mentioned that two inspections namely complete inspection (or 100% inspection) and sampling inspection can be employed for evaluation of quality and to provide quality assurance. Complete inspection can be avoided unless it is imperative to find out all defective products due to the consumption of inspection time and cost. Hence, sampling inspection is preferred for the following reasons: (i) It is the best alternative for destructive testing; (ii) It prohibits the inspection of entire lot of products and provides sufficient quality assurance by inspecting a fraction of total products; (iii) It reduces the inspection time and cost; (iv) It requires few inspectors, and it helps to avoid the incorrect classification of products due to inspector’s fatigue. Acceptance sampling is one form of sampling inspection implemented to make accept/reject decision on the concerned lot submitted for the inspection. Acceptance sampling facilitates the process of decision making on the submitted lot with minimum effort based on the quality of the random sampled products. However, acceptance sampling involves risks for both producer as well as consumer. It means that there may be a chance to reject the lot, which consists of good products or a chance to accept the lot of products that not possessing the expected quality. Then the probabilities for rejecting the lot with good products and accepting the lot with unsatisfactory quality level are termed as producer’s risk, denoted by (α) and consumer’s risk, denoted by (β) respectively. These aforementioned risks are occurred due to the reason of including only the random sampled results for lot sentencing and the random sample will not reflect the quality conditions of all products in the lot. On the other hand, acceptance sampling encourages the producer to tighten the quality control techniques by rejecting the entire lot when the sample items do not satisfy the quality specifications. In this way, the problem of consumers is to weed out the defective products is eliminated. Also, acceptance sampling safeguards consumers from choosing unsatisfactory products. Acceptance sampling plans state the sampling procedures along with the criteria/criterion for acceptance. The optimal sampling plans are determined by utilizing statistical as well as optimization techniques simultaneously. Dodge [1], who is

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called the father of acceptance sampling, initially introduced the theory of designing of sampling plans. The basic classifications of acceptance sampling plans are attribute sampling plans and variables sampling plans. Attribute sampling plans are executed where the measurements obtained from the inspection for quality assurance are of a discrete nature. In attribute inspection, the sampled item is classified as either defective or non-defective based on some of the non-measurable quality characteristics such as appearance, color, etc. Finally, counting the number of defective items is used for lot sentencing purpose under discrete distribution. In contrast, variables sampling plans are applicable where measurements (i.e., height, volume, etc.) are measured on a continuous scale and consequently the underlying probability distributions are continuous. It should be pointed out that for any sampling plans, the graph namely the operating characteristic (OC) curve depicts the performance of the sampling plans. That is, how the sampling plan discriminates poor quality lot among good quality lots can be shown by the OC curve. Further, the sampling plans can be classified on the basis of a number of samples considered for either attribute or variables inspection. For instance, a single sampling plan (SSP) utilizes just one random sample for making acceptance/rejection decision, double sampling plan (DSP) may utilize one more additional random sample to make the final decision if the results of the first sample will fall under the intermediate zone, etc. However, SSP involves only the easiest sampling procedure among the different sampling procedures. Different sampling plans exist in the literature to handle both attribute and variables inspection processes at any stage such as inspection of raw materials, in-process products and finished products. Also, conditions for the application of such sampling plans will vary concerning the situation. One of the most familiar methods frequently used in determining sampling plans is two points on the OC curve approach in which the producer and consumer risks are simultaneously considered along with their respective quality levels namely acceptable quality level and limiting quality level. The literature of acceptance sampling contains many works on the determination of attribute and variables sampling plans based on such an approach. For instance, Guenther [2] provided the procedures and examples to obtain the optimal SSP under binomial, hypergeometric and Poisson distributions. Sommers [3] determined the optimal parameters of variables DSP and matched SSP based on two points on the OC curve approach. Pearn and Wu [4] introduced the sampling plan for a low fraction of defectives based on process capability index Cpk using exact sampling distribution. Vijayaraghavan et al. [5] designed SSP for attribute inspection under gamma-Poisson distribution. Wu and Liu [6] developed a variables sampling plan based on yield index Spk for making a decision on the lot of products which possesses a meager fraction of nonconformities. As mentioned above, there are mainly two quality characteristics, such as attribute and variables quality characteristics. Besides, lifetime is the most important quality characteristics under both attribute and variables inspection. In order to inspect the

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lifetime of the products, the life test is conducted. The measurement of lifetime is directly used in some of the sampling plans designing and the designing of such sampling plans was discussed by several authors. For instance, Balasooriya [7] discussed the failurecensored sampling plans for two-parameter exponential distribution by considering m random samples each with size n. The first failure time of each of m samples is needed and it was proved that such sampling plan is effective, involves shorter test-time and saving of resources. Balasooriya and Balakrishnan [8] proposed reliability sampling plans based on progressively censored samples under 2-parameter log-normal distribution. Jun et al. [9] studied variables SSP and DSP for lot acceptance under sudden death testing where the products’ lifetime follows Weibull distribution. The product is classified as failure product or non-failure product based on the lifetime in attribute inspection. However, too much time has to be spent to obtain the exact lifetime of all the sampled products. Hence, time truncated life test is preferred; that is, the life test is performed for a pre-defined time. Acceptance sampling plans are designed for quality evaluation (or assurance) based on mean lifetime or percentile lifetimes of the products under attribute inspection. It should be mentioned that such sampling plans are designed under the assumption that the lifetime follows some lifetime distributions namely exponential, gamma, Weibull, etc. Several authors contributed their work on designing of sampling plans for lifetime assurance under different distributions on the basis of time truncated life test. In particular, some of the authors determined minimum sample size to ensure mean (or percentile or median) lifetime of the products under distinct lifetime distributions for different combinations of acceptance numbers, confidence levels and the ratios of the fixed experiment time to the specified mean (or percentile or median) lifetime. For example, Gupta and Groll [10] – gamma distribution, Kantam et al. [11] – log-logistic distribution, Tsai and Wu [12] – generalized Rayleigh distribution, Balakrishnan et al. [13] – generalized Birnbaum-Saunders distribution, Lio et al. [14] – Birnbaum–Saunders distribution, Gui and Aslam [15] – weighted exponential distribution, Hu and Gui [16] – Burr type X distribution. Two points on the OC curve approach has also been used to determine sampling plans for lifetime assurance. For instance, Aslam et al. [17] designed DSP and group sampling plan for assuring Birnbaum–Saunders distributed mean lifetime. Aslam et al. [18] proposed multiple dependent state repetitive group sampling plan for assuring percentile lifetime of the products under Burr type XII distribution. Among various lifetime distributions, Weibull distribution is the most applicable distribution to model failure time of the products. One can also see the designing of sampling plans under Weibull distribution. For example, Aslam et al. [19] proposed a repetitive group sampling plan for Weibull distributed percentile lifetime assurance. Balamurali et al. [20] designed a quick switching sampling system under Weibull distribution to ensure the mean lifetime of the products. In general, the optimal sampling plan is selected under the assumption that the sampling parameters involved in plan determination, such as the proportion defective

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which is the ratio between the observed defective products and inspected sample products is crisp and precise value. In practice, it is not always possible to have the sampling parameters are precise but it may be vague or imprecise. Sometimes, the classical set cannot be formed by collecting such values (or elements) because in classical set theory, the elements of the set can be said with clarity and there will be no uncertainty/ambiguity. Hence, Zadeh [21] introduced the concept of fuzzy set and fuzzy logic to handle the uncertain environment. A fuzzy set A defined on universal set X which can be characterized by the function namely membership function and it represents the grade of membership of an element i.e., μA(x) ∈ [0, 1]. Later, intuitionistic fuzzy sets were introduced by Atanassov [22] as an extension of fuzzy sets by including the grade of nonmembership (falsity-membership) along with the grade of membership (truthmembership) to handle the uncertain environment. Although the aforementioned sets are applicable for handling incomplete or imprecise information and solving decision-making problems in practice, they fail to handle the problems consist of indeterminate and inconsistent information. However, we can realize the indeterminate case in our routine life. For instance, in two players namely player A and player B game, there is 60% chance to win player A, (i.e., true case), 35% chance to lose player A, (i.e., false case), and 25% chance that the game to be a tie (i.e., indeterminate case). It is one of the real examples of facing the neutrosophy case. Suppose you are invited by your close friend to attend his family function. Then you may or may not accept your friend’s invitation. In neutrosophic terms, ‘You will accept invitation’ can be described as follows it is 50% true, 30% indeterminate and 40% false. In the production field, it is difficult to produce all the products with extreme quality due to some factors. Hence, we can see the products with good quality, moderate quality and poor quality. In this case, moderate quality is an example of indeterminacy and neutrosophy. In this circumstance, it is necessary to include the indeterminacy case along with true and false. For this purpose, Smarandache [23] introduced a new set named as a neutrosophic set which includes the indeterminate or inconsistent or neutrosophy situation of the problems. In neutrosophic set, the degree of truth, degree of falsity and degree of indeterminacy are considered simultaneously and they are independent of each other. Similarly, Wang et al. [24] pointed out “Neutrosophic set is a part of neutrosophy which studies the origin, nature, and scope of neutralities.” Further, to specify neutrosophic set technically, some set theoretic operations was defined on neutrosophic set and the resultant set was called as single valued neutrosophic (SVN) set by Wang et al. [24] and also, they investigated the distinct properties of SVN set. It is to be mentioned that SVN set has many real-life applications including science and engineering fields, decision-making problems, etc. To show the significance of neutrosophic set, El-Hefenawy et al. [25] made a review on the applications of neutrosophic set in various fields. Later, some algebraic operators namely subtraction and division of SVN numbers were defined by Smarandache [26]. Based on the basic operational laws such as “addition”, “multiplication”, “scalar multiplication”, Peng et al.

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[27] developed aggregation operators. Similarly, many authors developed different aggregation operators based on some operational laws to solve multi-attribute decision making (MADM) problems under neutrosophic environment. For example, Garg and Nancy [28] developed different weighted averaging and geometric aggregation operators based on logarithm operational laws for SVN numbers. They explained the applications of their work by solving the MADM problem involved in ‘Goods and Service Tax Mitra Scheme’ using SVN number and by the comparative study they proved that their work is more efficient than existing works. In addition, Garg and Nancy [29] provided novel algorithms with single valued and interval neutrosophic sets and Garg [30] introduced neutrality addition and scalar multiplication laws for SVN numbers to solve MADM problems. One can find the work based on neutrosophy in the literature on various fields including mathematical sciences. It denotes that the mathematical concepts namely probability, statistics, operations research (see Abdel-Basset [31]), have been discussed under neutrosophy environment. Smarandache [32] introduced neutrosophic probability in which the possibility of indeterminate case of an event was considered. Then he developed neutrosophic statistics for analyzing such events. Different probability distributions have been introduced in the literature to solve the problem with ambiguity or impreciseness. For example, Alhasan and Smarandache [33] studied neutrosophic Weibull distribution and some other distributions related to Weibull distribution such as neutrosophic inverse Weibull, neutrosophic Rayleigh, neutrosophic beta Weibull and neutrosophic Weibull with three, five and six parameters. They suggested that these distributions can be applied to deal with indeterminate or inaccurate problems involved in different domains, such as reliability, electrical engineering, quality control, etc. and the properties with some examples for these distributions were discussed. Recent days, one can see the applications of neutrosophic logic in many fields including acceptance sampling plans designing, see for instance, Refs. [34-39]. In general, there are two major assumptions considered in the determination of optimal sampling plans. The first assumption is that the parameters involved in sampling are crisp and the second one is such sampling parameters are having vague or imprecise values, particularly the case of expressing parameters only by linguistic variables. When designing the acceptance sampling plans using crisp values of sampling parameters, the authors can use classical statistics techniques. Obviously, the applications of classical statistics can be seen in various fields including production. In the production field, quality control techniques use statistical measures to quality control and quality evaluation (or assurance) purposes. Numerous authors have contributed their precious time and efforts for sampling plans designing with crisp parameters using classic statistical theory (see [2-20]). In addition, in the determination of optimal sampling plans in practice, the case of defining exact values for sampling parameters is not simple, and in most cases the estimation of such parameters is done on the basis of prior information of production process and pre-defined quality requirement. However, classical statistics

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only describes the possibilities of occurrence and non-occurrence of sampling parameters and provides the exact value. An indeterminacy of such sampling parameters is not considered by classical statistics. The sampling plans already exist under classical statistics cannot be applied where the sampling parameters have vague or imprecise values. Similarly, there is no discussion on what will happen if the sampling parameter that is, proportion defective is indeterminate when designing the sampling plan. Hence, considering such indeterminate case in sampling plan designing, we attempt to design a sampling plan using neutrosophic statistics introduced by Smarandache [32]. In classical statistics, crisp values are involved whereas neutrosophic statistics deals with set values which are ambiguous, vague, incomplete and even unknown. Neutrosophic probability also consists of three values in the set that describes the possibilities of true, false and indeterminacy cases. Hence, the sampling plans designed under neutrosophic probability attract the researcher attention. There are numerous reliability sampling plans that have been designed under neutrosophic statistical distributions, for example, Aslam and Arif [40] designed group sampling plan based on sudden death testing under neutrosophic Weibull distribution, Aslam et al. [41] proposed group sampling plan for attribute inspection based on time truncated life test under the neutrosophic Weibull distribution. However, those plans have been designed under neutrosophic Weibull distribution having neutrosophic shape parameter and neutrosophic scale parameter. In such plans designing, the indeterminate case of proportion defective or failure probability has not been considered by the authors. Hence, Aslam [42] designed SSP using neutrosophic statistical interval method by considering the indeterminate case of proportion defective items. From the literature survey, it is found that none of the authors designed the sampling plan for lifetime assurance by considering the indeterminate case of failure probability of the products under Weibull distribution. Therefore, in this chapter, we briefly discuss the designing of SSP using neutrosophic statistics to provide quality assurance for the products in terms of Weibull distributed percentile lifetime and indeterminacy case of failure probability is also considered in this designing. We provide the sampling procedure of the proposed plan under neutrosophic statistics. Tables are constructed to report the probabilities of lot acceptance, rejection and indeterminacy case under the proposed sampling plan using the neutrosophic binomial distribution for different specified parameters. It is expected that the proposed sampling plan will be more effective, flexible, adequate and suitable for uncertainty environment.

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2. DESIGNING OF SAMPLING PLAN UNDER WEIBULL DISTRIBUTION USING NEUTROSOPHIC STATISTICS When designing the sampling plan for providing lifetime assurance using neutrosophic statistics, the indeterminacy case of failure probability will be considered. In such designing, the product with lifetime greater than specified lifetime is classified as a good product whereas the product can be said to have poor quality if its lifetime is less than a specified lifetime. Suppose the true lifetime of the product is equal to the specified lifetime. Then the product is said to have indeterminacy quality level. In this situation, the existing sampling plans designed using classical statistics are not applicable. Now, we discuss the designing of SSP using neutrosophic statistics with the assumption on lifetime of the product is it follows the Weibull distribution. The cumulative distribution function of the Weibull distribution denoted by F and is given as follows.

(1) where δ and λ respectively denote the shape and scale parameters in which the scale parameter is unknown. The Weibull distributed cumulative lifetime of the product until the time t is denoted by F(t). Hence, the probability that the product to attain failure (i.e., failure probability) before the experiment time t0 is defined as

p  F t 0 

(2)

The qth percentile of the Weibull distribution is obtained as 1

  1      q    ln  1  q   

(3)

Then unknown scale parameter can be obtained as follows.



q 1

(4)

  1    ln     1  q   

The failure probability of a product before the experiment time t0 under the Weibull distribution is given as

A New Attribute Sampling Plan for Assuring Weibull …

  t0   p  1  exp         

99

(5)

It will be convenient if the experiment time t0 is expressed as a multiple of the specified qth percentile life θ0, i.e., t0 = aθ0 for a constant a (experiment termination ratio). While substituting the values of t0 and λ, equation (5) becomes as follows.

 p  1  exp   a  

   0    ln  1   1 q       q

(6)

Hence, the failure probability of a product before the experiment time t0 having Weibull distributed lifetime with known shape parameter is calculated by equation (6). In the above equation, θq/θ0 represents the ratio of true percentile life and specified percentile life. It is to be mentioned that while designing the sampling plan in order to assure lifetime of the products using classical statistics, the ratio θq/θ0 takes the value greater than or equal to 1 and it does not consider the case of θq/θ0 less than 1. Specifically, the failure probabilities obtained for θq/θ0 > 1 and θq/θ0 = 1 are used in sampling plan designing under classical binomial distribution. In addition, the existing sampling plans designed for assuring lifetime of the products using classical statistics lead to make a decision on the lot based on number of failed items before the experiment time. In which, the failure (i.e., success) and non-failure (i.e., failure) of the product are considered as an outcome of the trial. In particular, classical binomial distribution considers only success and failure cases. In such sampling plans designed under classical statistics, the lot will be accepted if number of failed products in the sample of size n is less than or equal to the acceptance number c, otherwise, the lot will be rejected. From the above discussion, it is observed that the existing sampling plans only consider the case of either acceptance or rejection and it does not include the situation of indeterminacy. However, when designing the sampling plan to prove the lifetime, three cases including indeterminacy case to be considered such as the ratio θq/θ0 > 1 (i.e., non-failure case), θq/θ0 < 1 (i.e., failure case) and θq/θ0 = 1 (i.e., indeterminacy case). Hence, we use neutrosophic statistics to design the proposed plan in particular, neutrosophic binomial distribution (see Smarandache [32]) is applied and also the proposed sampling plan will be suitable for uncertainty environment. Smarandache [32] defined an indeterminacy threshold (say, IT) where there is T number of trials whose outcome is indeterminate among n trials, i.e., T  {1, 2, …, n}. If IT < T, then the cases will belong to the indeterminate part, while cases with IT ≥ T will belong to the determinate part. In our work, the case where the true percentile lifetime of the product is lesser than the specified lifetime is success case, case where the

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true percentile lifetime of the product is greater than the specified lifetime is failure case and the case where the true percentile lifetime of the product is equal to the specified lifetime is indeterminacy. Then the probabilities for such cases can be expressed as neutrosophic interval probability (NIP) namely, p 

 pF , pI , pNF 

. Note that pF

denotes the probability that a particular product will fail, pI denotes the probability that a particular product will result indeterminacy case, pNF denotes the probability that a particular product will not fail. The total probability for the three cases may satisfy pF + pI + pNF ≥ 1. For specified values of a, q and δ, the probabilities pF, pI and pNF are calculated as follows.

 pF  1  exp   a  

   0    ln  1   where θq/θ0 < 1 1 q       q



or pF  1  exp   a rF





 1    , where rF = θq/θ0 ln  1  q  

  1    , here θq/θ0 = 1 pI  1  exp   a ln  1  q    

or pI  1  exp   a rI



pNF

  1  exp   a  





(8)

 1    , where rI = 1 ln  1 q 

   0    ln  1   where θq/θ0 > 1 1 q       q

or pNF  1  exp   a rNF



(7)



(9)

 1    , where rNF = θq/θ0 ln  1 q 

The sampling procedure of the proposed plan using neutrosophic statistics is described as follows. Step 1: Decide the values of θ0 (specified percentile life), t0 (pre-defined testing time) and indeterminacy threshold IT. The number of outcomes resulting in indeterminacy case (i.e., T) is compared with IT and such cases will add to indeterminate part if IT < T. Draw a random sample with size n from the submitted lot. Carry out the life test for pre-defined time t0 on the sampled items. Classification of sampled items as failure is determined in the following two ways.

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(i) The sampled item can be classified as a failure if its lifetime is less than t0 when θ0 = t0 . (ii) The sampled item can be classified as a failure if its lifetime is less than θ0 when θ0 > t0 . Denote the number of failure items by d. Clearly, the failure time of the item is exactly the true lifetime of the item (θq). It is obvious that the products, which do not meet failure after θ0 is classified as a non-failure item. Step 2: Accept the lot if d ≤ c when the cases belong to determinate part (i.e., IT ≥ T). Otherwise, reject the lot. The proposed sampling plan is characterized by the parameters n and c. The lot acceptance, intermediate and rejection probabilities are written in the form of neutrosophic statistical interval format such as (PA, PI, PR) for NIP

 pF , pI , pNF 

where

the probabilities PA, PI, PR are determined as follows. T nd n!  pI k  pNF n  d  k  pF d   d  0 d!n  d ! k 0  k 

(10)

PI 

n z n  z   n! z  pF k  pNF n  z  k   p  I   z T 1 z!n  z ! k 0  k 

(11)

PR 

T n  d  n! d   pI k  pNF n  d  k   p   F   k d ! n  d ! d  c 1 k 0  

(12)

c

PA   n

n

It is obvious that the lot will be accepted under the proposed plan if the number of failures is less than or equal to the acceptance number c when indeterminacy case T is less than or equal to indeterminacy threshold IT (i.e., determinate part, T ≤ IT). Similarly, the lot will be rejected if number of failures exceeds c. The indeterminate part is considered if T > IT. From above equations, it is clearly understood that the probabilities computed under neutrosophic statistics will be equal to the same calculated under classical statistics when T = 0 and there will be success and failure outcomes only (i.e., there is no indeterminacy case and pNF = 1–pF). It should be noted that the following combinatorial formula given by Smarandache [32] will be easier than the calculation of PR value. PR = (pF + pI + pNF)n – PA – PI

(13)

The lot acceptance, indeterminate and rejection probabilities (PA, PI, PR) are calculated for fixed values of percentile, δ, a, n, c, T, rF, rI, rNF and reported in Table 1. The following observations can be made from Table 1.

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An increment in acceptance number (c) or decrement in sample size (n) with other fixed values increases the probability of acceptance and vice-versa. An increment either in indeterminacy threshold or in experiment termination ratio increases the probability of acceptance.

From these observations, it can be concluded that the proposed plan will provide a minimum probability of acceptance when failure probability is high and provide a high chance to reject the lot. Table 1. The probabilities (PA, PI, PR) for assuring 50th percentile life when δ = 1, rF = 0.4, rI = 1, rNF = 2 n 5 10 n 5 10 n 5 10 15 n 5 10 15

c=0 (0.0018644, 0.4910007, 0.5071350) (0.0000007, 0.8647507, 0.1352486) -

a = 0.5, IT = 1 c=1 (0.0141869, 0.4384901, 0.5473230) (0.0000050, 0.8240212, 0.1759738) IT = 2 c=1 (0.0494703, 0.1424873, 0.8080423) (0.0000400, 0.5732549, 0.4267050) a = 1.0, IT = 1 c=1 (0.0233741, 0.4910007, 0.4856252) (0.0000183, 0.8647507, 0.1352310) -

c=0 (0.0075621, 0.1757142, 0.8167236) (0.0000057, 0.6418548, 0.3581395) -

IT = 2 c=1 (0.0771154, 0.1757142, 0.7471704) (0.0001360, 0.6418548, 0.3580092) -

c=0 (0.0008907, 0.4384901, 0.5606192) (0.0000001, 0.8240212, 0.1759786) c=0 (0.0038488, 0.1424873, 0.8536639) (0.0000013, 0.5732549, 0.4267438)

c=2 (0.0897313, 0.4384901, 0.4717786) (0.0000766, 0.8240212, 0.1759022) c=2 (0.2427634, 0.1424873, 0.6147492) (0.0005433, 0.5732549, 0.4262018) c=2 (0.1179205, 0.4910007, 0.3910788) (0.0002174, 0.8647507, 0.1350318) (0.0000002, 0.9700667, 0.0299332) c=2 (0.3066956, 0.1757142, 0.5175902) (0.0014440, 0.6418548, 0.3567011) (0.0000016, 0.8883589, 0.1116394)

3. APPLICATION OF THE PROPOSED PLAN Obviously, one can judge whether the submitted lot is to be accepted or rejected while executing the existing sampling plans designed using classical statistics since the counting of failed items under life test is only considered in lot sentencing. In the existing sampling procedure, the sampled item whose lifetime is exactly equal to the specified lifetime is also considered as the non-failure item. The lot will be accepted in this circumstance if all the sampled items work up to the specified lifetime. After accepting

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the lot, the minimum level of expectations such as the product should have long lifetime will be satisfied because the minimum lifetime (i.e., percentile ratio = 1) is also acceptable. It expresses that the existing sampling plans designed by using classical statistics overlook the indeterminacy cases while making a decision but the sampling plan designed using neutrosophic statistics take into account such indeterminacy cases. In the proposed plan, the chance will decrease for the lot to be accepted even a few sampled items have specified lifetime because such products will belong to the indeterminacy group. Hence, the lot accepted under the proposed plan will definitely satisfy the expectation on product’s lifetime since it classifies the sampled item as good if the lifetime is greater than the specified one. In order to describe the application of the proposed plan in manufacturing industries, the execution of the proposed plan is explained using the lifetime data of the products in this section. One of the essential factors involved in determining the service life of ball bearings is its fatigue life and hence, the fatigue life test on ball bearing products is constantly conducted by the manufacturers to obtain the information relating fatigue life to load and other factors. Suppose the quality personnel of ball bearing manufacturing industry would like to inspect the 50th percentile life of the ball bearing via time truncated life test experiment, it is to be mentioned that the lifetime of the ball bearing is represented in terms of million revolutions. The quality personnel specifies that 50th percentile life of the ball bearing is at least 33 million revolutions (i.e., θ0 = 33) and it is decided to carry out the life test for pre-defined time of t0 = 33. From this, the termination ratio is obtained as a = 1.0. Based on the lifetime, the sampled items are classified as failed item, non-failed item and indeterminacy case. That is, the sampled item will be called as a failed item if the product meets the failure before the specified lifetime θ0 (i.e., θq < θ0) or the lifetime is less than t0, in this case θq/θ0 < 1. Similarly, the sampled item is classified as a nonfailed item if the product will work beyond the specified lifetime θ0 (i.e., θq > θ0) or the lifetime is greater than experiment time t0, in this case θq/θ0 > 1. The indeterminacy case is considered when the lifetime is exactly equal to the experiment time, i.e., θq = θ0. For illustration purpose, we only consider the lifetime data of 10 ball bearings from 23 observations given by Lieblein and Zelen [43] and are as follows; 17.88, 28.92, 33.00, 84.12, 93.12, 98.64, 128.04, 105.84, 127.92, 105.12 It is found that the maximum likelihood estimate of the shape parameter of these data is 1.3034 ≈ 1.0. i.e., δ = 1. From the data, we observe that there are 2 failed ball bearings out of 10 ball bearings, 7 non-failed ball bearings out of 10 ball bearings, and 1 ball bearing out of 10 ball bearings belong to indeterminacy group. In this case, the lot acceptance, indeterminate and rejection probabilities are (PA, PI, PR) = (0.0002174, 0.8647507, 0.1350318). Hence, the chance to accept the lot of ball bearing product is 0.02%.

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4. COMPARATIVE STUDY The sampling plan, which correctly distinguishes the poor quality lot among good quality lots will be admirable and also such performance is one of the most expected characteristics (i.e., discriminating power) of any sampling plan. In addition, the sampling plan with such characteristic must consider all the possibilities of the lot (i.e., indeterminacy case also). Hence, the authors consider the proposed plan under neutrosophic statistics. In order to investigate whether the proposed plan possesses discriminating power higher than that of sampling plan designed under classical statistics or not, we compare the NIP (PA, PI, PR) of the proposed plan with a probability of acceptance of the lot under existing sampling plan. The probabilities are calculated for pre-defined values of δ = 1, rF = 0.4, rI = 1, rNF = 2, a = 0.5 and summarized in Table 2. Table 2. The probabilities (PA, PI, PR) obtained under neutrosophic statistics and OC value of SSP for assuring 50th percentile life when δ = 1, a = 0.5, rF = 0.4, rI = 1, rNF = 2 (IT, n, c) (1, 5, 0) (1, 10, 0) (1, 5, 1) (1, 10, 1) (1, 5, 2) (1, 10, 2)

(PA, PI, PR) (0.0008907, 0.4384901, 0.5606192) (0.0000001, 0.8240212, 0.1759786) (0.0141869, 0.4384901, 0.5473230) (0.0000050, 0.8240212, 0.1759738) (0.0897313, 0.4384901, 0.4717786) (0.0000766, 0.8240212, 0.1759022)

Pa(p) 0.01313900.00017230.10369400.00255220.35333850.0173126

From this table, it is clear that the existing plan only decides whether to accept the lot or to reject it, and it fails to provide the probabilities of indeterminacy case as well as rejection. On the other hand, the probabilities for acceptance, rejection and indeterminacy are obtained by the proposed plan. For instance, when n = 5, the possibility to accept the lot is 0.0897313, the possibility for indeterminacy (i.e., neither acceptance nor rejection) is 0.4384901. It represents that when the proposed plan is executed for inspection with n = 5, IT = 1 and c = 2, there is 44% (approximately) chance that the quality personnel get confusion about lot sentencing. Also, from the comparison of existing and proposed sampling plans, it can be observed that the chance to accept the lot under the proposed

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plan is minimum rather than existing sampling plan. It should be noted that when percentile ratio is 0.4, the product will have a high chance to fail (i.e., failure probability is high). In this situation, the existing plan provides more probability and it will affect the consumer. For example, the probability of acceptance of the lot under the proposed plan is 0.0897313 when n = 5, IT = 1 and c = 2 while the existing sampling plan yields 0.3533385 chance to accept the lot. That is, acceptance probability under the existing plan is 74% more than the proposed plan even when the quality is poor. From this, it is assured that the proposed plan designed under neutrosophic statistics will definitely protect the consumers at an unsatisfied quality level. There is a possibility to provide another chance for the producer by considering indeterminacy case under the proposed plan. Hence, the proposed sampling plan designed using neutrosophic statistics will be more effective and adequate under uncertainty environment.

CONCLUSION A new sampling plan under neutrosophic statistics to assure Weibull distributed percentile lifetime of the product is discussed in this chapter. In the proposed sampling plan designing, indeterminacy case is also considered with the cases of acceptance and rejection. The consideration of indeterminacy case will be a benefit for producer since it does not lead to immediate rejection. However, this feature cannot be seen in the sampling plan designed using classical statistics. The formulae to find the probabilities for determinate case, indeterminate case and acceptance of a lot of the product have been reported in tables for practical use. The proposed plan has been compared with the existing sampling plan designed under classical statistics in terms of discriminating power. From the comparative study, it has been confirmed that the proposed plan safeguards the consumer when the lifetime is lower than specified one whereas it yields additional chance when the original lifetime is equal to the specified lifetime. From this, it is concluded that the proposed sampling plan will be more effective and more suitable under uncertainty environment rather than the existing sampling plan designed under classical statistics. As a future study, some extensions of the proposed plan for different lifetime models should be designed and investigated under uncertainty environment. The designing of the proposed plan with economic aspect might also be of interest for future research.

ACKNOWLEDGMENTS The first author would like to thank Kalasalingam Academy of Research and Education for providing financial support through Post-doctoral fellowship. The authors

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would also like to thank the anonymous reviewers and the Editor for their precious comments and significant suggestions which led to improving the quality of this chapter.

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[29] Garg, H. & Nancy. (2020). Multiple attribute decision making based on immediate probabilities aggregation operators for single-valued and interval neutrosophic sets. Journal of Applied Mathematics and Computing, 63, 619–653. [30] Garg, H. (2020). Novel neutrality aggregation operator-based multi attribute group decision-making method for single-valued neutrosophic numbers. Soft Computing, 24, 10327–10349. [31] Abdel-Basset, M., Gunasekaran, M., Mohamed, M. & Smarandache, F. (2019). A novel method for solving the fully neutrosophic linear programming problems. Neural Computing and Applications, 31, 1595–1605. [32] Smarandache, F. (2014). Introduction to Neutrosophic Statistics. Sitech and Education Publishing: Craiova, Romania; Columbus, OH, USA. [33] Alhasan, K. F. H. & Smarandache, F. (2019). Neutrosophic of Weibull and neutrosophic family Weibull distribution. Neutrosophic Sets and Systems, 28, 191– 199. [34] Majumdar, P. (2015). Neutrosophic sets and its applications to decision making. Computational intelligence for big data analysis, Springer, Cham, pp. 97–115. [35] Aiwu, Z., Jianguo, D. & Hongjun, G. (2015). Interval valued neutrosophic sets and multi-attribute decision-making based on generalized weighted aggregation operator. Journal of Intelligent and Fuzzy Systems, 29(6), 2697–2706. [36] Abdel-Basset, M., Mohamed, M., Zhou, Y. & Hezam, I. (2017). Multi-criteria group decision making based on neutrosophic analytic hierarchy process. Journal of Intelligent and Fuzzy Systems, 33(6), 4055–4066. [37] Abdel-Basset, M., Zhou, Y., Mohamed, M. & Chang, V. (2018). A group decision making framework based on neutrosophic VIKOR approach for e-government website evaluation. Journal of Intelligent and Fuzzy Systems, 34(6), 4213–4224. [38] Broumi, S., Bakali, A., Talea, M., Smarandache, F. & Ali, M. (2017). Shortest path problem under bipolar neutrosphic setting. Applied Mechanics and Materials, 859, 59–66. [39] Abdel-Basset, M. & Mohamed, M. (2018). The role of single valued neutrosophic sets and rough sets in smart city: imperfect and incomplete information systems. Measurement, 124, 47–55. [40] Aslam, M. & Arif, O. (2018). Testing of grouped product for the Weibull distribution using neutrosophic statistics. Symmetry, 10(9), 1–10. [41] Aslam, M. & Raza, M. A. (2019). Design of new sampling plans for multiple manufacturing lines under uncertainty. International Journal of Fuzzy Systems, 21, 978–992. [42] Aslam, M. (2019). A new attribute sampling plan using neutrosophic statistical interval method. Complex and Intelligent Systems, 5, 365–370.

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SECTION II: DECISION MAKING PROBLEMS WITH NEUTROSOPHIC SET

In: Decision-Making with Neutrosophic Set Editor: Harish Garg

ISBN: 978-1-53619-419-7 © 2021 Nova Science Publishers, Inc.

Chapter 6

ON SOME PROPOSITIONS OF BOUNDARY IN INTERVAL VALUED NEUTROSOPHIC BITOPOLOGICAL SPACE Bhimraj Basumatary Department of Mathematical Sciences, Bodoland University Kokrajhar, BTAD, India

ABSTRACT In this chapter, we introduce the idea of interval-valued neutrosophic bitopological space. We have defined definitions of interval-valued neutrosophic interior, closure, the relation between them. We studied the definition of interval-valued neutrosophic boundary and some of their propositions in interval-valued bitopological space.

Keywords: [ 𝜏𝑖 , 𝜏𝑗 ] -IVNB-interior, [ 𝜏𝑖 , 𝜏𝑗 ] -IVNB-closure, [ 𝜏𝑖 , 𝜏𝑗 ] -IVNB-boundary, interval valued neutrosophic bitopological space

1. INTRODUCTION After the discovery of fuzzy set theory by Zadeh [1], in 1965, many researchers conducted the idea on generalizations of fuzzy set theory. Atnasov [2] introduced the idea of the non-membership function and proposed intuitionistic fuzzy set theory. After the introduction of fuzzy sets, several types of research were conducted on the generalizations of the notions of fuzzy sets. The theory of fuzzy sets actually has been a 

Correspoonding Author’s Email: [email protected].

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generalization of the classical theory of sets in the sense that the theory of sets should have been a special case of the theory of fuzzy sets. After the generalization of fuzzy sets, many researchers have applied the generalization of fuzzy sets theory in many branches of science and technology. Garg [23] studied a modified score function for ranking order of interval-valued Pythagorean fuzzy sets. Based on Pythagorean fuzzy sets, a Pythagorean fuzzy technique for order of preference by similarity to ideal solution (TOPSIS) method by taking the preferences of the experts in the form of interval-valued Pythagorean fuzzy decision matrices was discussed. Also, different explorations of the theory of Pythagorean fuzzy sets can be seen in [24–30]. Yager [31] proposed the q-rung orthopair fuzzy sets, in which the sum of the qth powers of the membership and nonmembership degrees is restricted to one [32]. Peng and Liu [33] studied the systematic transformation for information measures for q-rung orthopair fuzzy sets. Pinar and Boran [34] applied a q-rung orthopair fuzzy multi-criteria group decision-making method for supplier selection based on a novel distance measure. Cuong [35] proposed the new theory of picture fuzzy sets and claimed that the framework of the intuitionistic fuzzy set could not be applied to some real-life scenarios such as voting and some other situations where opinions are of more than two types. Mahmood et al. [37] proposed the idea of a spherical fuzzy set and consequently a T-spherical fuzzy set (TSFS). The Einstein aggregation operators for T-spherical fuzzy sets are developed and incorporated in MADM problems in Garg et al. [36]. Later on, many researchers applied the generalization of fuzzy sets theory to different branches of science & technology. Chang [5] introduced fuzzy topology. Coker [6] defined the notion of intuitionistic fuzzy topological spaces. Many researchers have studied topology on neutrosophic sets, such as Lupianez [7–10] and Salama [11]. In 1963, Kely [12] defined the study of Bitopological spaces. Kandil et al.[13] discussed on fuzzy bitopological spaces. Lee et al. [14] discussed some properties of Intuitionistic Fuzzy Bitopological Spaces. Turksen [18] introduced the idea of interval-valued fuzzy set theory. Mondal and Samantha [8] gave the idea of interval-valued topology in a fuzzy sense. Atanassov & Gargov [20] generalized intuitionistic fuzzy sets in Interval-valued system. Warren [38] studied the boundary of a fuzzy set in fuzzy topology. Warren [38] studied some properties of the boundary of a fuzzy set and found that some properties are not the same as the properties of the crisp boundary of a set do. Later, many authors studied on properties of the boundary of a fuzzy set. Tang [39] has made heavy use of the notion of fuzzy boundary. Kharal [40] studied Frontier and Semifrontier in Intuitionistic Fuzzy Topological Spaces. Samarandache [3, 4] generalized the concepts of fuzzy set theory and intuitionistic fuzzy set theory and given the idea of the degree of indeterminacy as an independent component, and introduced neutrosophic set theory, in 1995. Later Salama and Alblowi [17] given the idea on neutrosophic topology. Muchahary & Basumatary [21] discussed

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neutrosophic bitopological space. Wang et al. [18] discussed an interval-valued neutrosophic set. Nanthini and Pushpalatha [22] given the idea on interval-valued neutrosophic topological space. Salama et. al [41] studied generalized neutrosophic topological space where they have discussed on properties of generalized closed sets. Many authors have studied on properties of the boundary of a fuzzy set by several methods (fuzzy set, intuitionistic fuzzy set, and neutrosophic set) but some of its properties are not the same as the properties of the crisp boundary of a set. In this chapter, we study the concept of interval-valued Netrosophic Bitopological Spaces. Next, we introduce the concepts of the interval-valued neutrosophic interior, interval-valued neutrosophic closure, and interval-valued neutrosophic boundary. Also, we discussed some propositions related to interval-valued neutrosophic interior, neutrosophic closure, and neutrosophic boundary.

2. BASIC OPERATIONS Definition 2.1. [18] A neutrosophic set (NS) A on the universe of discourse X is defined as 𝐴 = {< 𝑥, 𝜇𝐴 (𝑥), 𝜎𝐴 (𝑥), 𝛾𝐴 (𝑥) >∶ 𝑥𝜖 𝑋} where 𝜇𝐴 (𝑥), 𝜎𝐴 (𝑥), 𝛾𝐴 (𝑥) ∶ 𝑋 ⟶ [0− , 1+ ] and 0− ≤ 𝜇𝐴 (𝑥) + 𝜎𝐴 (𝑥) + 𝛾𝐴 (𝑥) ≤ 3+ , 𝜇𝐴 (𝑥) represents degrees of membership function, 𝜎𝐴 (𝑥) is the degree of indeterminacy and 𝛾𝐴 (𝑥) is the degree of non-membership function.

Definition 2.2. [18] An interval-valued neutrosophic set (IVNS) 𝐴 is said to be contained in IVNS 𝐵, if and only if inf 𝑇𝐴(𝑥) ≤ inf 𝑇𝐵(𝑥), sup 𝑇𝐴(𝑥) ≤ sup 𝑇𝐵(𝑥) inf (𝑥) ≥ inf 𝐼𝐵(𝑥), sup 𝐼𝐴(𝑥) ≥ sup 𝐼B (𝑥) inf 𝐹𝐴(𝑥) ≥ inf 𝐹𝐵(𝑥), sup 𝐹𝐴(𝑥) ≥ sup 𝐹𝐵(𝑥) for all 𝑥 in 𝑋.

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Definition 2.3. [18] Complement of IVNS 𝐴 is denoted by AC and is defined as 𝑇𝐴𝐶 (𝑥) = 𝐹 (𝑥); inf 𝐼𝐴𝐶 (𝑥) = 1 − sup (𝑥); sup 𝐼𝐴𝐶 (𝑥) = 1 – inf 𝐼𝐴(𝑥); 𝐹𝐴𝐶 (𝑥) = 𝑇𝐴 (𝑥) for all 𝑥 in 𝑋.

Definition 2.4. [18] The Intersection of two IVNS 𝐴 and 𝐵 is D = 𝐴 ⊓ 𝐵 and the truth-membership, indeterminacy – membership and false – membership of D are defined as Inf 𝑇D(𝑥) = min[inf 𝑇𝐴(𝑥),inf 𝑇𝐵(𝑥)] Sup𝑇D(𝑥) = min[sup 𝑇𝐴(𝑥),sup 𝑇𝐵(𝑥)] Inf𝐼D (𝑥) = max[inf 𝐼𝐴(𝑥),inf 𝐼𝐵(𝑥)] Sup𝑇D(𝑥) = max[sup 𝐼𝐴(𝑥),sup 𝐼𝐵(𝑥)] Inf𝐹D (𝑥) = max[inf 𝐹𝐴(𝑥),inf 𝐹𝐵(𝑥)] Sup𝑇D(𝑥) = max[sup 𝐹𝐴(𝑥),sup 𝐹𝐵(𝑥)] for all 𝑥 in 𝑋.

Definition 2.5. [18] Union of two IVNS 𝐴 and 𝐵 is the smallest IVNS containing both 𝐴 and 𝐵.

Definition 2.6. [22] Interval-Valued Neutrosophic topological spaces (IVNTS) Let 𝜏 be a collection of all interval-valued neutrosophic subsets on X. Then 𝜏 is called an Interval-Valued Neutrosophic topological spaces (IVNTS) in X if the following conditions hold i. 0𝑋 and 1𝑋 is belong to 𝜏. ii. Union of any number of IVN sets in 𝜏 is again belong to 𝜏. iii. Intersection of any two IVN set in 𝜏 is belong to 𝜏.

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Then the pair (X, 𝜏) is called Interval-Valued Neutrosophic topological spaces on X.

Definition 2.7. [7] Let (X, τ) be a neutrosophic topological space over X and A be any neutrosophic set on X. Then, the neutrosophic interior of A, denoted by Int(A) is the union of all neutrosophic open subsets of A. Int(A) is the biggest neutrosophic open set over X which containing A.

Definition 2.8. [7] Let (X,τ) be a neutrosophic topological space over X and A any neutrosophic set on X. Then, the neutrosophic closure of A, denoted by Cl(A) is the intersection of all neutrosophic closed supersets of A. Cl(A) is the smallest neutrosophic closed set over X which contains A.

3. MAIN RESULTS Definition 3.1. Interval Valued Neutrosophic Bitopological space (IVNBS) is a system (X, 𝜏𝑖 , 𝜏𝑗 ) consisting of a set X with two IVN topologies 𝜏𝑖 and 𝜏𝑗 on X. Here 𝑖 ≠ 𝑗 and 𝑖, 𝑗 ∈ {1, 2}.

Example 3.1. Let X={a, b} and 𝑎

𝑏

𝐴 = {< [0.2,0.3],[0.4,0.5],[0.4,0.6] >, < [0.5,0.6],[0.2,0.3],[0.4,0.6] >} 𝑎 [0.3,0.5],[0.6,0.7],[0.2,0.3]

>,
}.

Then 𝜏1 = {0𝑋 , 1𝑋 , 𝐴 } and 𝜏2 = {0𝑋 , 1𝑋 , 𝐵} then [X, 𝜏1 , 𝜏2 ] is IVNBS.

𝐵 = {
∶ 𝑥𝜖 𝑋} , neutrosophic [ 𝜏𝑖 , 𝜏𝑗 ]- IVNB-Interior of A is the union of all [ 𝜏𝑖 , 𝜏𝑗 ]-IVNB-open sets of X contained in A and is defined as follows [ 𝜏𝑖 -𝜏𝑗 ]-IVBN-Int(A) = {< 𝑥,⊔ 𝜏𝑖 ⊔𝜏𝑗 𝜇𝑖𝑗 , ⊓ 𝜏𝑖 ⊓𝜏𝑗 𝜎𝑖𝑗 , ⊓ 𝜏𝑖 ⊓𝜏𝑗 𝛾𝑖𝑗 >∶ 𝑥𝜖 𝑋}.

Example 3.2. Let X={a, b} 𝑎

𝑏

𝑎

and 𝐴 = {< [0.2,0.3],[0.4,0.5],[0.4,0.6] >, < [0.5,0.6],[0.2,0.3],[0.4,0.6] >} , 𝐵 = {< [0.3,0.5],[0.6,0.7],[0.2,0.3] >,
}, 𝐷 = {
,
}.

Then 𝜏1 = {0𝑋 , 1𝑋 , 𝐴 } and 𝜏2 = {0𝑋 , 1𝑋 , 𝐵} then [X, 𝜏1 , 𝜏2 ] is interval valued neutrosophic bitopological space Let 𝐷 = {
,
}

𝜏2 -IVNB-Int (D) =B and 𝜏1 -IVNB-Int (B) =𝐴 Hence [𝜏1 , 𝜏2 ]-IVNB-Int(D)= 𝐴

Theorem 3.1. Let (X, 𝜏𝑖 , 𝜏𝑗 ) be IVNBS then i

[ 𝜏𝑖 , 𝜏𝑗 ]-IVNB-Int (0𝑋 ) = 0𝑋 , [ 𝜏𝑖 , 𝜏𝑗 ]-IVNB-Int(1𝑋 ) = 1𝑋

ii

[ 𝜏𝑖 , 𝜏𝑗 ]-IVNB-Int(𝐴) ≤ A.

iii A is IVN-open set iff A= [ 𝜏𝑖 , 𝜏𝑗 ]-IVNB-Int(𝐴) iv [ 𝜏𝑖 , 𝜏𝑗 ]-IVNB-Int{ [ 𝜏𝑖 , 𝜏𝑗 ]-IVNB-Int(𝐴)} = 𝐴 v

A≤ B implies [ 𝜏𝑖 , 𝜏𝑗 ]-IVNB-Int(A) ≤ [𝜏𝑖 , 𝜏𝑗 ]-IVNB-Int(B)

vi [ 𝜏𝑖 , 𝜏𝑗 ]-IVNB-Int(A) ⊔ [ 𝜏𝑖 , 𝜏𝑗 ]-IVNB-Int(B) ≤ [ 𝜏𝑖 , 𝜏𝑗 ]-IVNB-Int(A⊔B) vii [ 𝜏𝑖 , 𝜏𝑗 ]-IVNB-Int(A ⊓ B) = [𝜏𝑖 , 𝜏𝑗 ]-IVNB-Int(A) ⊓ [ 𝜏𝑖 , 𝜏𝑗 ]-IVNB-Int(B).

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Proof: From the definition of [ 𝜏𝑖 , 𝜏𝑗 ]- IVNB-Interior we have clearly the result of i. and ii. iii. Let A be IVNB open set over X, then A is IVNB open set over X which contains A. So, A is the largest IVNB open set contained in A and A = [ 𝜏𝑖 , 𝜏𝑗 ]-IVNB-Int(𝐴). Conversely, let A = [ 𝜏𝑖 , 𝜏𝑗 ]-IVNB-Int(𝐴). Then A is IVNB open set. iv. Let [ 𝜏𝑖 , 𝜏𝑗 ]-IVNB-Int(𝐴) = B. Then [ 𝜏𝑖 , 𝜏𝑗 ]-IVNB-Int(𝐵) = B Hence [ 𝜏𝑖 , 𝜏𝑗 ]-IVNB-Int{ [ 𝜏𝑖 , 𝜏𝑗 ]-IVNB-Int(𝐴)} = 𝐴. v. Suppose that A ≤ B. As [ 𝜏𝑖 , 𝜏𝑗 ]-IVNB-Int(𝐴) ≤ A ≤ B. [ 𝜏𝑖 , 𝜏𝑗 ]-IVNB-Int(𝐴) is IVNB open subset of B. So clearly we have [ 𝜏𝑖 , 𝜏𝑗 ]-IVNB-Int(A) ≤ [𝜏𝑖 , 𝜏𝑗 ]-IVNBInt(B). It is clear that A ≤ A ⊔ B and B ≤ A ⊔ B. Thus, [ 𝜏𝑖 , 𝜏𝑗 ]-IVNB-Int(𝐴) ≤ [ 𝜏𝑖 , 𝜏𝑗 ]-IVNB-Int(A ⊔ B) and [ 𝜏𝑖 , 𝜏𝑗 ]-IVNB-Int(B) ≤ [ 𝜏𝑖 , 𝜏𝑗 ]-IVNB-Int(A⊔B). Thus, we have [ 𝜏𝑖 , 𝜏𝑗 ] − IVNB -Int(A) ⊔ [ 𝜏𝑖 , 𝜏𝑗 ] − IVNB -Int(B) ≤ [ 𝜏𝑖 , 𝜏𝑗 ] − IVNB -Int(A⊔B). vi. Prove is similar to the prove of iv and v.

Remark 3.1. [ 𝜏𝑖 , 𝜏𝑗 ]-IVNB-Int(A) ≠ [ 𝜏𝑗 , 𝜏𝑖 ]-IVNB-Int(A) when i ≠ 𝑗. For this we cite an example.

Example 3.3. From example 3.1, we have [ 𝜏𝑖 , 𝜏𝑗 ]-IVNB-Int(D)= 𝐴 𝜏1 -IVNB-Int(D)= 𝐴, [ 𝜏𝑗 , 𝜏𝑖 ]-IVNB-Int(D) = 0𝑋 . Hence [ 𝜏𝑖 , 𝜏𝑗 ]-IVNB-Int(A)≠ [ 𝜏𝑗 , 𝜏𝑖 ]- IVNB-Int(A).

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Definition 3.3. Let (X, 𝜏𝑖 , 𝜏𝑗 ) be an IVNBS. Then for a set 𝐴 = {< 𝑥, 𝜇𝑖𝑗, 𝜎𝑖𝑗 , 𝛾𝑖𝑗 >∶ 𝑥𝜖 𝑋} , neutrosophic [ 𝜏𝑖 , 𝜏𝑗 ]-IVNB-Closure of A is the intersection of all [ 𝜏𝑖 , 𝜏𝑗 ]-IVNB-Closed sets of X contained in A and is defined as follows [ 𝜏𝑖 , 𝜏𝑗 ]-IVNB-Cl(A) = {< 𝑥, ⊓ 𝜏𝑖 ⊓𝜏𝑗 𝜇𝑖𝑗 ,⊔ 𝜏𝑖 ⊔𝜏𝑗 𝜎𝑖𝑗 , ⊔ 𝜏𝑖 ⊔𝜏𝑗 𝛾𝑖𝑗 >∶ 𝑥𝜖 𝑋}.

Example 3.4. From example 3.1 𝑎

Let 𝑄 = {< [0.5,0.7],[0.3,0.4],[0.1,0.2] >,
}.

Now 𝜏2 -IVNB-Cl(Q)= 1𝑋 and 𝜏1 -IVNB-Cl(1𝑋 ) =1𝑋 Hence [ 𝜏𝑖 , 𝜏𝑗 ]-IVNB-Cl(Q)= 1𝑋 .

Theorem 3.2. If (X, 𝜏𝑖 , 𝜏𝑗 ) be IVNBS. Then i

[ 𝜏𝑖 , 𝜏𝑗 ]-IVNB-Cl(0𝑋 ) = 0𝑋 , [ 𝜏𝑖 , 𝜏𝑗 ]-IVNB-Cl(1𝑋 ) = 1𝑋

ii

A ≤ [ 𝜏𝑖 , 𝜏𝑗 ]-IVNB-Cl(𝐴).

iii A is IVNB closed set iff A=[ 𝜏𝑖 , 𝜏𝑗 ]-IVNB-Cl(𝐴) iv [ 𝜏𝑖 , 𝜏𝑗 ]-IVNB-Cl [[ 𝜏𝑖 , 𝜏𝑗 ]-IVNB-Cl(𝐴)] = 𝐴 v

A≤ B implies [ 𝜏𝑖 , 𝜏𝑗 ]-IVNB-Cl(A) ≤ [ 𝜏𝑖 , 𝜏𝑗 ]-IVNB-Cl(B).

vi [ 𝜏𝑖 , 𝜏𝑗 ]-IVNB-Cl(A⊔B) = [ 𝜏𝑖 , 𝜏𝑗 ]-IVNB-Cl(A)⊔[ 𝜏𝑖 , 𝜏𝑗 ]-IVNB-Cl(B) vii [ 𝜏𝑖 , 𝜏𝑗 ]-IVNB-Cl(A⨅B)≤[ 𝜏𝑖 , 𝜏𝑗 ]-IVNB-Cl(A) ⨅ [ 𝜏𝑖 , 𝜏𝑗 ]-IVNB-Cl(B). Proof: From the definition of [ 𝜏𝑖 , 𝜏𝑗 ]- IVNB-Closure, we have the results of i. and ii. iii. If A be IVNB closed set over X then A is itself an IVNB closed set over X which contains A. Therefore, A is the smallest IVNB closed set containing A and A = [ 𝜏𝑖 , 𝜏𝑗 ]-

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121

IVNB-Cl(𝐴). Conversely, suppose that A = [ 𝜏𝑖 , 𝜏𝑗 ]-IVNB-Cl(𝐴). As A is an IVNB closed set, so A is an IVNB closed set over X. iv. Since A in IVNB open set hence by iii, [ 𝜏𝑖 , 𝜏𝑗 ]-IVNB-Cl [[ 𝜏𝑖 , 𝜏𝑗 ]-IVNB-Cl(𝐴)] = 𝐴. v. Let A ≤ B. Then every IVNB closed superset of B will also contain A. This means that every IVNB closed superset of B is also an IVNB closed superset of A. Hence the intersection of IVNB closed supersets of A is contained in the intersection of IVNB closed supersets of B. Thus [ 𝜏𝑖 , 𝜏𝑗 ]-IVNB-Cl(A) ≤ [ 𝜏𝑖 , 𝜏𝑗 ]-IVNB-Cl(B). Following prove of i to v prove of vi and vii is straightforward.

Remark 3.2. [ 𝜏𝑖 , 𝜏𝑗 ]-IVNB-Cl(A)≠ [ 𝜏𝑗 , 𝜏𝑖 ]-IVNB-Cl(A) when i≠ 𝑗. For this we cite an example.

Example 3.5. Let X={a, b } and 𝑎

𝑏

𝐴 = {< [0.3,0.4],[0.6,0.7],[0.6,0.7] >, < [0.4,0.5],[0.7,0.8],[0.5,0.6] >} 𝑎 [0.4,0.5],[0.6,0.7],[0.5,0.6]

>,
∶ 𝑥𝜖 𝑋}. Then 𝐴𝐶 = {< 𝑥, 𝛾𝑖𝑗 , 1 − 𝜎𝑖𝑗 , 𝜇𝑖𝑗 >∶ 𝑥𝜖 𝑋} and [ 𝜏𝑖 , 𝜏𝑗 ]-IVNB -Int(A) = {< 𝑥,⊔ 𝜏𝑖 ⊔𝜏𝑗 𝜇𝑖𝑗 , ⊓ 𝜏𝑖 ⊓𝜏𝑗 𝜎𝑖𝑗 , ⊓ 𝜏𝑖 ⊓𝜏𝑗 𝛾𝑖𝑗 >∶ 𝑥𝜖 𝑋} Now [ 𝜏𝑖 , 𝜏𝑗 ] − IVNB − Cl(AC ) = {< 𝑥, ⊓ 𝜏𝑖 ⊓𝜏𝑗 𝛾𝑖𝑗 , 1 − ⊓ 𝜏𝑖 ⊓𝜏𝑗 𝜎𝑖𝑗 ,⊔ 𝜏𝑖 ⊔𝜏𝑗 𝜇𝑖𝑗 >∶ 𝑥𝜖 𝑋} So, [[ 𝜏𝑖 , 𝜏𝑗 ] − IVNB − Cl(AC )]C= {< 𝑥,⊔ 𝜏𝑖 ⊔𝜏𝑗 𝜇𝑖𝑗 , ⊓ 𝜏𝑖 ⊓𝜏𝑗 𝜎𝑖𝑗 , ⊓ 𝜏𝑖 ⊓𝜏𝑗 𝛾𝑖𝑗 >∶ 𝑥𝜖 𝑋} Hence [ 𝜏𝑖 , 𝜏𝑗 ]-IVNB -Int(A) = [ 𝜏𝑖 , 𝜏𝑗 ]-IVNB -Cl(AC)]C Proof of proposition iv is similar to proposition iii.

Definition 3.4. Let A be an IVN set in (X, 𝜏𝑖 , 𝜏𝑗 ), then [ 𝜏𝑖 , 𝜏𝑗 ]-IVNB-boundary of A is defined as [ 𝜏𝑖 , 𝜏𝑗 ]-IVNB-Bd(A) = [ 𝜏𝑖 , 𝜏𝑗 ]-IVNB-Cl(A) ⨅ [ 𝜏𝑖 , 𝜏𝑗 ]-IVNB-Cl(AC).

On Some Propositions of Boundary …

123

Proposition 3.1. Let A be an IVN set in (X, 𝜏𝑖 , 𝜏𝑗 ). Then [ 𝜏𝑖 , 𝜏𝑗 ]-IVNB-Bd(A) ⊔A ≤ [𝜏𝑖 , 𝜏𝑗 ]-IVNBCl(A).

Proof: From the definition [ 𝜏𝑖 , 𝜏𝑗 ]-IVNB-Bd(A)≤[𝜏𝑖 , , 𝜏𝑗 ]-IVNB-Cl(A) and A≤ [𝜏𝑖 , , 𝜏𝑗 ]IVNB-Cl(A) and hence [ 𝜏𝑖 , 𝜏𝑗 ]-IVNB-Bd(A)⊔A≤ [𝜏𝑖 , , 𝜏𝑗 ]-IVNB-Cl(A).

Remark 3.3. The converse part of the proposition is not true. For this we cite an example.

Example 3.6. From Example 3.1. Let 𝑄 = {
,
}

[𝜏𝑖 , , 𝜏𝑗 ]-IVNB-Cl(Q)=1X [𝜏𝑖 , , 𝜏𝑗 ]-IVNB-Bd(Q)=AC and hence [ 𝜏𝑖 , 𝜏𝑗 ]-IVNB-Bd(A)⊔A≤ [𝜏𝑖 , , 𝜏𝑗 ]-IVNB-Cl(A) but [𝜏𝑖 , 𝜏𝑗 ]-IVNB-Bd(A)⊔A≠ [𝜏𝑖 , , 𝜏𝑗 ]-IVNB-Cl(A).

Proposition 3.2. Let A and B be neutrosophic sets in (X, 𝜏𝑖 , 𝜏𝑗 ). Then i

[ 𝜏𝑖 , 𝜏𝑗 ]-IVNB-Bd(A) = [𝜏𝑖 , 𝜏𝑗 ]-IVNB-Bd(𝐴𝐶 ).

ii

If A be [ 𝜏𝑖 , 𝜏𝑗 ]-IVNB-closed set then [ 𝜏𝑖 , 𝜏𝑗 ]-IVNB-Bd(A) ≤A

iii If A be [ 𝜏𝑖 , 𝜏𝑗 ]-IVNB-open set then [ 𝜏𝑖 , 𝜏𝑗 ]-IVNB-Bd(A) ≤ AC Proof: [ 𝜏𝑖 , 𝜏𝑗 ]-IVNB-Bd(A) =[ 𝜏𝑖 , 𝜏𝑗 ]-IVNB-Cl(A) ⊓ [ 𝜏𝑖 , 𝜏𝑗 ]-IVNB-Cl(AC) = {< 𝑥, ⊓ 𝜏𝑖 ⊓𝜏𝑗 𝜇𝑖𝑗 ,⊔ 𝜏𝑖 ⊔𝜏𝑗 𝜎𝑖𝑗 , ⊔ 𝜏𝑖 ⊔𝜏𝑗 𝛾𝑖𝑗 >∶ 𝑥𝜖 𝑋}⨅{𝑥, ⊓ 𝜏𝑖 ⊓𝜏𝑗 𝛾𝑖𝑗 , 1 − ⊓ 𝜏𝑖 ⊓𝜏𝑗 𝜎 ,⊔ 𝜏𝑖 ⊔𝜏𝑗 𝜇𝑖𝑗 >∶ 𝑖𝑗

𝑥𝜖 𝑋}

124

Bhimraj Basumatary Also [ 𝜏𝑖 , 𝜏𝑗 ]-IVNB-Bd(𝐴𝐶 ) = [ 𝜏𝑖 , 𝜏𝑗 ]-IVNB-Cl(AC) ⊓ [ 𝜏𝑖 , 𝜏𝑗 ]-IVNB -Cl(A) {< 𝑥, ⊓ 𝜏𝑖 ⊓𝜏𝑗 𝛾𝑖𝑗 , 1 − ⊓ 𝜏𝑖 ⊓𝜏𝑗 𝜎𝑖𝑗 ,⊔ 𝜏𝑖 ⊔𝜏𝑗 𝜇𝑖𝑗 >∶ 𝑥𝜖 𝑋} ⊓

=

{< 𝑥, ⊓ 𝜏𝑖 ⊓𝜏𝑗 𝜇𝑖𝑗 ,⊔ 𝜏𝑖 ⊔𝜏𝑗 𝜎𝑖𝑗 , ⊔ 𝜏𝑖 ⊔𝜏𝑗 𝛾𝑖𝑗 >∶ 𝑥𝜖 𝑋}

Hence [ 𝜏𝑖 , 𝜏𝑗 ]-IVNB-Bd(A) = [ 𝜏𝑖 , 𝜏𝑗 ]-IVNB-Bd(𝐴𝐶 ). Let A be [ 𝜏𝑖 , 𝜏𝑗 ]-IVNB-closed set then [ 𝜏𝑖 , 𝜏𝑗 ]-IVNB-Cl(A) = A Now [ 𝜏𝑖 , 𝜏𝑗 ]-IVNB-Bd(A) = [ 𝜏𝑖 , 𝜏𝑗 ]-IVNB-Cl(A) ⊓ [ 𝜏𝑖 , 𝜏𝑗 ]-IVNB-Cl(AC) ≤ [ 𝜏𝑖 , 𝜏𝑗 ]-IVNB-Cl(A) = A

Hence [ 𝜏𝑖 , 𝜏𝑗 ]-IVNB-Bd(A) ≤ A.

Remark 3.4. The converse part of the proposition is not true. For this we cite an example.

Example 3.7. From the examples of 3.1 & 3.6 Let 𝑄 = {
,
}

and 𝑎 𝑏 >, < >} [0.3, 0.4], [0.7, 0.8], [0.7, 0.8] [0.3,0.5], [0.7, 0.8], [0.6, 0.7] [ 𝜏𝑖 , 𝜏𝑗 ]-IVNB-Cl(𝑄) = 1𝑋 and [ 𝜏𝑖 , 𝜏𝑗 ]-IVNB-Cl(𝑄 𝐶 ) = 𝐴𝐶 and [ 𝜏𝑖 , 𝜏𝑗 ]-IVNB -Bd(𝑄)= 𝐴𝐶 𝑄 𝐶 = {
𝐴4 > 𝐴1 > 𝐴3

𝐴2

𝐴3

𝐴1 > 𝐴4 > 𝐴3 > 𝐴2

𝐴1

𝐴2

Aal et al. [51] arithmetic and geometric ranking method Garai et al. [49] possibility mean and standard deviation based ranking method Nancy and Garg [52] novel divergence measure Garg and Nancy [37] logarithmic SVN-weighted averaging aggregation operator Garg and Nancy [37] logarithmic SVNweighted geometric aggregation operator Garg and Nancy [53] biparametric distance measure 𝒑 = 𝟏, 𝒕 = 𝟑

𝐴1 > 𝐴2 > 𝐴4 > 𝐴3

𝐴1

𝐴3

𝐴4 > 𝐴1 > 𝐴2 > 𝐴3

𝐴4

𝐴3

𝐴4 > 𝐴1 > 𝐴2 > 𝐴3

𝐴4

𝐴3

𝐴4 > 𝐴1 > 𝐴3 > 𝐴2

𝐴4

𝐴2

𝐴1 > 𝐴4 > 𝐴3 > 𝐴2

𝐴1

𝐴2

𝐴4 > 𝐴1 > 𝐴2 > 𝐴3

𝐴4

𝐴3

𝒑 = 𝟏, 𝒕 = 𝟓

𝐴4 > 𝐴1 > 𝐴2 > 𝐴3

𝐴4

𝐴3

𝒑 = 𝟓, 𝒕 = 𝟑

𝐴4 > 𝐴1 > 𝐴2 > 𝐴3

𝐴4

𝐴3

𝒑 = 𝟓, 𝒕 = 𝟓

𝐴1 > 𝐴4 > 𝐴2 > 𝐴3

𝐴1

𝐴3

𝐴4 > 𝐴2 > 𝐴1 > 𝐴3

𝐴4

𝐴3

Deli and Subas [18] ranking method Ye [50] score function method

Proposed Method

Our method is a maiden attempt to swot up SM based neutrosophic decision making using - cut technique. The evaluation procedure is very much different, useable and effective from the existing ones, which is its novelty and yet logical outcomes are achieved through the presented approach, which shows its feasibility.

An Expected Value-Based Novel Similarity Measure …

157

CONCLUSION The role of SM is very much crucial in solving MADM problems. In this work, a novel SM for SVTNNs is proposed. The reason behind such consideration of neutrosophic numbers is that, they can describe an ill-defined quantity very comprehensively, compared to those of fuzzy numbers and intuitionistic fuzzy numbers based on whom the concept has been generalized. Dealing with decision-making problems where the attribute values are expressed with the help of SVTNNs is not an easy task to accomplish because the ranking of alternatives with neutrosophic numbers is not similar to that of ranking methods for ordinary real numbers. Thus, as a maiden initiative we have devised a SM which involves the expected value between SVTNNs (evaluated through the  - cut technique) as a prominent feature. We have then validated the structural stability and applicability of the proposed method by conducting a comparative study with the existing methods, which suggests that the proposed method has better flexibility and desirable qualities of its own. Thereafter we have discussed the procedure to solve a MADM problem using the proposed SM approach. The results obtained by the proposed method are logical and at par with common human intuition. Comparison studies reveal the rationality, practicability, favourability and effectiveness of the proposed approach. Therefore, the developed method presents a new direction for dealing with decision-making problems under SVN-fuzzy environment. In the future direction, we will try to formulate the variance expression for SVTNNs, and also further extend it to the more general SVTNNs. However, it is a routine exercise to extend the proposed SM concept to generalized SVTNNs. Hopefully it will find applications in the field of information fusion system, game theory, optimization problems, risk analysis, expert system, medical diagnoses, pattern recognition, clustering analysis, complex decision-making problems, etc. Furthermore, the use of aggregation operators involving SVTNNs with new operational rules shall be investigated in near future.

CONFLICT OF INTEREST Authors declare that they have no conflict of interest

ETHICAL APPROVAL This article does not contain any studies with animals performed by any of the authors.

158

Palash Dutta and Gourangajit Borah

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In: Decision-Making with Neutrosophic Set Editor: Harish Garg

ISBN: 978-1-53619-419-7 © 2021 Nova Science Publishers, Inc.

Chapter 8

TRNN-ARAS STRATEGY FOR MULTI-ATTRIBUTE GROUP DECISION-MAKING (MAGDM) IN TRAPEZOIDAL NEUTROSOPHIC NUMBER ENVIRONMENT WITH UNKNOWN WEIGHT Rama Mallick1,* and Surapati Pramanik2,† 1

Department of Mathematics, Umeschandra College, Kolkata, West Bengal, India 2 Department of Mathematics, Nandalal Ghosh B. T. College, Panpur, West Bengal, India

ABSTRACT This main objective of the chapter is to extend the ARAS (Additive Ratio ASsessment) strategy to trapezoidal neutrosophic numbers environment. ARAS strategy can effectively evaluate and rank the feasible alternatives. This chapter first develops an ARAS strategy, which we call TrNN- ARAS strategy for multi-attribute group decisionmaking in trapezoidal neutrosophic numbers environment. An entropy measure for trapezoidal neutrosophic numbers is proposed to derive the weights of the criteria as well as weights of the decision makers. To show the applicability and effectiveness of the proposed strategy, we solve an illustrative problem of multi-attribute group decision making in trapezoidal neutrosophic numbers environment and present a comparison with the existing strategy in the literature.

Keywords: multi attribute group decision making, neutrosophic set, single valued neutrosophic set, trapezoidal fuzzy number, trapezoidal neutrosophic number, ARAS strategy  †

Corresponding Author’s Email: [email protected]. Corresponding Author’s Email: [email protected].

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1. INTRODUCTION Fuzzy Set (FS) (Zadeh, 1965) only deals with the membership to encounter uncertainty. Atanassov (1986) extended the FS to the Intuitionistic FS (IFS) by inducing non-membership function as an independent component to deal with uncertainty. In IFS, hesitation or indeterminacy is dependent on membership and non-membership functions. When indeterminacy is independent, FS and IFS fail to deal with the situation. A new set namely Neutrosophic Set (NS) (Smarandache, 1998) has been defined to deal with uncertainty, indeterminacy and inconsistency by introducing membership, indeterminacy and non-membership functions as independent components. NS is an extended form of a classical set, FS, and IFS. NS effectively represents real-world problems by considering truth, falsity, and indeterminacy factors of a decision- making situation (Garg & Nancy 2018; Pramanik, Mallick, & Dasgupta 2018). In today’s competitive world, decision making problems are generally complicated with multiple conflicting criteria involving uncertainty, indeterminacy and inconsistency. Multi- Criteria Decision Making (MCDM) is the most widely used field of operational research. MCDM is divided into Multi Objective Decision Making (MODM) and MultiAttribute Decision Making (MADM). When multiple decision-makers or experts are involved in decision making, it is termed as a Multi-Attribute Group Decision Making (MAGDM). In the MADM problem, we determine the rank of the alternatives with respect to a set of decision attributes. MADM is the most widely used decision strategies in science, government, and business. The MADM can improve the quality of a decision by making the DM process more rational, explicit, and efficient. The selection of the process can be made by comparing the different strategies to select the best alternative. Many MADM strategies have been developed in the literature such as:        

SAW- Simple Additive Weighting (MacCrimon, 1968), AHP- Analytic Hierarchy Methods (Saaty, 1977; 1994), TOPSIS- Technique for Order Preference by Similarity to Ideal Solution (Hwang & Yoon, 1981), PROMETHEE- Preference Ranking Organization Method for Enrichment Evaluation (Brans, Mareschal, & Vincke, 1984), GRA- Grey Relational Analysis (Deng, 1989; 2005), ELECTRE- ELimination Et Choice Translating Reality (Roy, 1990; 1996), TODIM- TOmada de Decisão Interativa e Multicritério (Gomes and Lima, 1992), COPRAS- Complex Proportional Assessment (Zavadskas & Kaklauskas, 1996; Zavadskas & Kaklauskas, 2007; Zavadskas, Kaklauskas, & Vilutienė, 2009),

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VIKOR- VlseKriterijumska Optimizacija I Kompromisno Resenje (Opricovic, 1998; Opricovic & Tzeng, 2002), MOORA (Brauers & Zavadskas, 2006), MULTIMOORA- Multiple Objective Optimization on the basis of Ratio Analysis plus Full Multiplicative Form (Brauers & Zavadskas, 2010), WASPAS-Weighted Aggregated Sum Product Assessment (Zavadskas, Turskis, Antucheviciene, & Zakarevicius, 2012), EDAS- Evaluation based on Distance from Average Solution (Ghorabaee, Zavadskas, Olfat, & Turskis, 2015).

To deal with neutrosophic MADM and MAGDM problems, several strategies have been proposed in the literature such as:                  

Correlation coefficient (Ye, 2013), GRA (Biswas, Pramanik, & Giri, 2014a; 2014b), Aggregation operator (Liu, Chu, Li, & Chen, 2014; Sodenkamp, Tavana, & Di Caprio, 2018). WASPAS (Zavadskas, Baušys, & Lazauskas, 2015; Morkunaite, Bausys, & Zavadskas, 2019), COPRAS (Bausys, Zavadskas, & Kaklauskas, 2015), TOPSIS (Biswas, Pramanik, & Giri, 2016b; 2019), AHP (Radwan, Senousy, & Riad, 2016: Abdel-Basset, Mohamed, Zhou, & Hezam, 2017; Abdel-Basset, Manogaran, Mohamed, & Chilamkurti, 2018), MULTIMOORA (Stanujkic, Zavadskas, Smarandache, Brauers, & Karabasevic, 2017), TODIM (Xu, Wei, &Wei, 2017), Similarity measure (Pramanik, Biswas, & Giri, 2017; Garg, H., & Nancy 2019; Mondal, Pramanik, & Giri 2018a; 2018b), Projection measure (Ye, 2017a), VIKOR (Abdel-Basset, Zhou, Mohamed, & Chang, 2018), Cross entropy (Pramanik, Dalapati, Alam, Smarandache, & Roy, 2018), DEcision MAking Trial and Evaluation Laboratory (DEMATEL) (Abdel-Basset, Manogaran, Gamal, & Smarandache, 2018) ANP (Zaied, Ismail, & Gamal, 2019), MULTIMOORA and SWARA (Zavadskas, Čereška, Matijošius, Rimkus, & Bausys, 2019). INVAR (INvestment Value Assessments along with Recommendation) (Zavadskas, Kaklauskas, Bausys, Naumcik, & Ubarte, 2020). Ranking of SVNS (Garai, Dalapati, Garg, & Roy, 2020).

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Recent theoretical developments and applications of MADM and MAGDM in neutrosophic environment scan be found in the research (Smarandache & Pramanik 2016; 2018; Broumi et al., 2018; Peng & Dai 2020; Pramanik 2020). Biswas, Pramanik, and Giri (2014c) defined trapezoidal fuzzy neutrosophic numbers and applied it to solve the MADM problem. Ye (2017b) proposed Trapezoidal Neutrosophic Number (TrNN) by extending Intuitionistic Fuzzy Number (IFNs) (Wang and Zhang 2009) and Intuitionistic Trapezoidal Fuzzy Numbers (ITFNs) (Wang and Zhang 2009) and SVNS (Wang et al., 2010). In the literature few researches in TrNN environments have been developed such as:          

Ranking of TrNN strategy (Biswas, Pramanik, & Giri, 2016b; Deli & Şubaş, 2017a; Suresh, Prakash, & Vengataasalam, 2020), Aggregation operator (Deli, & Şubaş, 2017b; Jana, Pal, Karaaslan, & Wang, 2020), TOPSIS (Biswas, Pramanik, & Giri, 2018a), VIKOR (Pramanik & Mallick, 2018), GRA (Pramanik & Mallick, 2019a; 2020a; Tan, Zhang, & Chen, 2019), Expected value-based strategy (Biswas, Pramanik, & Giri, 2018b), TODIM (Pramanik & Mallick, 2019b), MULTIMOORA (Mallick & Pramanik, 2020b), PROMETHEE II (Narayanamoorthy, Chithra, & Kang, 2020), Centroid based ranking strategy (Suresh, Prakash, & Vengataasalam, 2020)

1.1. Motivation of the Work Till date Additive Ratio ASsessment (ARAS) strategy for MAGDM is not proposed in TrNNs environment. To fill up the research gap, we extend the ARAS strategy in TrNNs environment.

1.2. Research Methodology  

The extended ARAS strategy is employed to solve the MAGDM problem involving robot selection in TrNNs environment. Entropy method is employed to determine objectively the completely unknown weights of the decision makers and the criteria.

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1.3. Research Contribution  

In this chapter, we first extend the ARAS strategy in the TrNNs environment for group decision making. An entropy measure for TrNNs is objectively developed to derive the weights of the criteria as well as weights of the decision makers.

The structure of the chapter is as follows: Literature review of the ARAS strategy is presented in Section 2. The basic definitions of FS, NS, TrNNs are presented in Section 3; In section 4, we propose an entropy measure to determine unknown weights of the decision makers as well as weights of the criteria; The extended TrNN-ARAS strategy is proposed in Section 5; In section 6, a robotic selection problem is solved to show the applicability of the proposed strategy; In Section 7, we compare the proposed strategy with existing VIKOR strategy. In section 8, we discuss the advantages of the proposed strategy. Finally, Section 9 presents the conclusion and some future research directions.

2. LITERATURE REVIEW In this section, we present an overview of the ARAS strategy. Zavadskas and Turskis (2010a) proposed the ARAS strategy. ARAS is the “ratio of the sum of normalized and weighted criteria scores.” They presented a real case study of microclimate evaluation in office rooms by analyzing the inside climate of the premises, where people work, and defining measures to be taken to improve their environment. The case study used the following attributes of climate evaluation: air turnover inside the premises, air flow rate, air humidity, illumination intensity, air temperature, and dew point. They determined the attribute weight by pairwise comparison strategy. Zavadskas, Turskis, and Vilutiene (2010) used the ARAS strategy to select the best foundation instalment alternative in redeveloping the building. The main attributes of the selection were: instalment duration, costs of instalment, the complexity of decisions, transferability and maintainability of installed foundation system, advantages, and disadvantages of decisions, past experience. These attributes were selected based on client interest and the goal and safety of future buildings and factors that influence the construction process’s efficiency. Turskis and Zavadskas (2010a) extended the ARAS strategy in a fuzzy environment by applying the Analytic Hierarchy Process (AHP) to evaluate the criteria. They also presented a case study to analyze the fuzzy MCDM. The study mainly focused on helping the stakeholders with the performance evaluation in an uncertain environment to select the best site for the logistic center among a set of available alternatives, where triangular fuzzy numbers described the attributes.

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Turskis, and Zavadskas (2010b) also developed the Grey Additive Ratio Assessment (ARAS-G) strategy by presenting the process potential supplier’s selection. The attributes of this selection are delivery price, financial position, standards and relevant certificates, production specifications, commercial strength, and the suppliers’ performances, which are expressed in terms of grey numbers (Zavadskas, Kaklauskas, Turskis, & Tamosaitiene, 2009). Bakshi and Sarkar (2011) used the AHP and ARAS strategy to select the best project. To illustrate the strategy, they used survey data on the expansion of optical fiber for a telecommunication sector. Balezentiene and Kusta (2012) used the ARAS strategy to investigate how to reduce greenhouse gas emissions. Turskis, Lazauskas, and Zavadskas (2012) combined fuzzy ARAS and AHP to present a case study dealing with multiple criteria assessments of construction site alternatives for non-hazardous waste incineration plant in Vilnius city and identified the best alternatives by evaluating seven alternative sites for construction of the waste incineration plant in Vilnius city. Stanujkic (2015) extended the ARAS strategy using linguistic variables for group decision making and presented a faculty website evaluation process. Büyüközkan and Göçer (2017) extended the ARAS strategy for Interval Valued Intuitionistic Fuzzy (IVIF) environment and presented a case study dealing with the digital supply chain. Koçak, Kazaz, and Ulubeyli (2018) applied the ARAS strategy in a real case study in subcontractor selection involving eleven different criteria and eight alternatives. Ghram and Frikha (2019) developed the multiple criteria Hierarchical ARAS (ARAS-H) strategy by adopting a bottom-up approach to analyze criteria in different levels of the hierarchy. Based on partial pre-orders obtained at the level of elementary criteria (the lowest level of the hierarchy), they constructed outranking relations at the hierarchy’s upper level. The alternatives are ranked in a decomposed manner i.e., at one level of the hierarchy and the intermediate ones. The developed ARAS-H strategy has been employed in a hierarchy of four levels to rank tourist destination brand’s websites. Liu and Cheng (2019) extended the ARAS strategy for MAGDM problems in probability multi-valued neutrosophic set environments by defining multi-valued neutrosophic normalized weighted Bonferroni distance. Maximizing deviation model has been extended to evaluate weight vector of attributes and entropy measure has been utilized to determine the weights of the decision-makers. Pinem, Handayani, & Huizen (2020) presented a comparison between the ELECTRE, SMART (Simple Multi-Attribute Rating Technique), and ARAS strategies in determining priority for the handling of areas affected by natural earthquake disasters.

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3. PRELIMINARIES Definition 2.1. Assume that L is a fuzzy set (Zadeh, 1965) in a universal set V  , which is defined by L  { v,  L (v)  v  V }

(1)

where  L (v) : V   [0,1] is known as the membership function of L, and the value of  L (v) is called the degree of membership. Definition 2.2. A fuzzy number L is called a Trapezoidal Fuzzy Number (TrFN) (Kumar et al., 2010) if its membership function is defined by  z  m11 m11  z  m12 m  m , 11  12 1, m12  z  m13  L (v )   m  z  14 , m13  z  m14  m14  m13  otherwise 0,

(2)

The TrFN L is denoted by L   m11 , m12 , m13 , m14  where m11 , m12 , m13 , m14 are real numbers and m11  m12  m13  m14 . Definition 2.3. Assume that V  is a universal set and v be an element in V  . Then an SVNS (Smarandache, 1998) M  in V  is defined by M   {v,  CM  (v), DM  (v), EM  (v)  v  V }

where

CM  (v) : V   [0,1], DM  (v) : V   [0,1]

and

(3)

(3)

EM  (v) : V   [0,1]

with

0  CM  (v)  DM  (v)  EM  (v)  3 for all v V  . The functions CM  (v), DM  (v) and EM  (v) are

the truth membership, the indeterminacy membership and the falsity membership function, respectively. Definition 2.4. Suppose that  be a Single Valued Trapezoidal Neutrosophic Number (SVTrNN) (Ye, 2017b). Its truth membership function, indeterminacy membership function, and falsity membership functions are defined by

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Rama Mallick and Surapati Pramanik  ( z   m11 )c , m11  z   m12   (m12  m11 ) c , m12  z   m13 c ( z )    (m14  z )c , m13  z   m14  (m  m ) 14 13  otherwise. 0,

(4)

(4)

 (m12  z )  ( z   m11 )d  , m11  z   m12  (m12  m11 )  d  , m12  z   m13 d  ( z )    z   m13  (m14  z )d  , m13  z   m14  m14  m13  otherwise. 0,

(5)

(5)

(6)

(6)

 m12  z   ( z   m11 )e , m11  z   m12  m12  m11  e , m12  z   m13 e ( z )    z   m13  (m14  z )e , m13  z   m14  m14  m13  otherwise. 0, 0  c ( z )  1, 0  d  ( z )  1 where ,

0  e ( z )  1

and

0  c ( z )  d  ( z )  e ( z )  3; m11 , m12 , m13 , m14  R .

Then the SVTrNN  is presented as   ( m11 , m12 , m13 , m14 ; c , d  , e ) . Definition

2.5.

Assume

that

m1  ( m11 , m12 m13 , m14  ; c1 , d1 , e1 )

m2  ( m21 , m22 , m23 , m24  ; c2 , d2 , e2 ) are any two TrNNs and

and

0 . Then we have the

following operations (Ye, 2017b). 1.m1  m2  ( m11  m21 , m12  m22 , m13  m23 , m14  m24  ; c1  c2  c1c2 , d1d 2 , e1e2 ) 2.m1  m2  ( m11 m21 , m12 m22 , m13 m23 , m14 m24 ; c1c2 , d1  d 2  d1d 2 , e1  e2  e1e2 )

3. m1  (  m11 ,  m12 ,  m13 ,  m14  ;1  (1  c1 ) ,(d1 ) ,(e1 ) ) 4.(m1 )  ( (m11 ) ,(m12 ) ,(m13 ) ,(m14 )  ;(c1 ) ,1  (1  d1 ) ,1  (1  e1 ) )

Definition 2.6. The complementary of a TrNN m1  ( m11 , m12 m13 , m14  ; c1 , d1 , e1 ) is defined by

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m1c  ( m11 , m12 m13 , m14  ;1  e1 , d1 ,1  c1 )

Definition 2.7. Suppose m1  ( m11 , m12 m13 , m14  ; c1 , d1 , e1 ) is a TrNN on R, and m11  m12  m13  m14 . Then the score function (Ye, 2017b) is defined as: Sc(m1 ) 

1 (m11  m12  m13  m14 )(2  c1  d1  e1 ) 12

(7)

(7)

Definition 2.8. Assume that m j  ( m j1 , m j2 , m j3 , m j4  ; cm j , dm j , em j )( j  1, 2,...., p) is a collection of TrNNs and w  ( w1 , w2 ,..., wp )t is the weight vector of m j ( j  1, 2,...., p) such p

that w j  [0,1],  w j  1 . Then the Trapezoidal Neutrosophic Number Weighted Arithmetic j 1

Averaging (TrNNWAA) operator (Ye, 2017b) is defined as: TrNNWAA(m1 , m2 ,...., m p )

(8)

p

  wj m j j 1

p p p p p p  p  w w w     w j m j1 ,  w j m j 2 ,  w j m j 3 ,  w j m j 4  ;1   (1  cmj ) j ,,  (d mj ) j  (e j ) j  j 1 j 1 j 1 j 1 j 1 j 1   j 1 

Definition 2.9. Let m1  ( m11 , m12 m13 , m14 ; c1 , d1 , e1 ) and m2  ( m21 , m22 , m23 , m24 ; c2 , d2 , e2 ) be any two TrNNs. Then the normalized Hamming distance (Biswas, Pramanik & Giri, 2018a) between m1 and m2 is defined as follows: d (m1 , m2 ) 

1  m11 (2  c1  d1  e1 )  m21 (2  c2  d 2  e2 )  m12 (2  c1  d1  e1 )  m22 (2  c2  d 2  e2 )    12   m13 (2  c1  d1  e1 )  m23 (2  c2  d2  e2 )  m14 (2  c1  d1  e1 )  m24 (2  c2  d 2  e2 ) 

(9)

4. ENTROPY MEASURE FOR TRNNS Entropy reflects the uncertainty of the attribute information. The greater value of entropy implies the greater uncertainty. The concept of entropy (Chen & Li, 2011) is extended in this chapter. The weight calculating process using entropy is described as follows:

172

Rama Mallick and Surapati Pramanik  t11 t12   t21 t22 Let a decision matrix be of the form: T     t y1 t y 2  

t1z   t2 z    t yz   

The value H s is the expression of deviation of the rating value, which is calculated using average rating values: H (t )  ( (t1 , t  ),  (t2 , t  ),....,  (t z , t  ))

Here

t

(10)

(10)

is the average rating value, which is calculated using the following equation

z z z 1 1 1 1 z 1 z 1 z 1 z  t     rs1 ,  rs 2 ,  rs 3 ,  rs 4  ;1   1  as1  z ,   bs1  z ,  cs1  z z z z z s 1 s 1 s 1  s 1  s 1 s 1 j 1

(11)

(11)

 (t1 , t  ) are calculated by using equation (9). The corresponding normalized distance vector is given by:    ts , t      , s  1, 2,...., z H t     max   ts , t   

(12)

The entropy measure of the s-th criterion C s for y available alternatives is obtained from     H   H s   1 s   es  t    ln ,   y  ln( y ) s 1  y   H H   r   r   r 1   r 1 z

(13)

The entropy measure satisfies the following properties: i. ii.

es (t , t )  0 if t is a crisp set; es (t11 , t12 )  1 if t11  0,0.5,0.5,0.5;0.5,0,0.5 and t12  0,0,0.5,0.5;0.5,0.5,0

iii.

es (t , t11 )  es (t , t12 ) if t11  t12

iv.

es (t , t11 )  es (t , t11c ) for all t T

TrNN-ARAS Strategy for Multi-Attribute Group …

173

Proof: i. If t is a crisp set, then t is denoted as

 x, x, x, x;1,0,0 ; x 0,1 in the form of

TrNN. It is clear that es (t , t )  0 . ii. t11  0,0.5,0.5,0.5;0.5,0,0.5 and t12  0,0,0.5,0.5;0.5,0.5,0 then     H    H 1 s s      0.998  1 es  t    ln   y  ln( y ) s 1  y   H H   r   r   r 1   r 1 z

Therefore es (t11 , t12 )  1 . If t11  t12 then      H 1  1 z  H s1 s  , es  t , t11    ln   ln( y ) s 1  y 1  y 1     H s   H s   r 1   r 1      H 2  1 z  H s2 s  , es  t , t12    ln   ln( y ) s 1  y 2  y 2     H s   H s   r 1   r 1

       H1  H 1  z  H 2  H 2  s s s s       ln  y ln  y  y y   s 1  s 1  1 1  2 2    H s   H s    H s   H s   r 1   r 1   r 1  r 1        H 1   H 2  z  z  H s1 H s2 s s          y ln y   y ln y   s 1  s 1  1 1  2 2  H H H H   s   s    s   s   r 1   r 1   r 1  r 1        H 1  2  H2 z  H 1 z  H s1 1 s s      ln  y ln  y s   y y   ln( y ) s 1  ln( y ) s 1  1 1  2 2   H s   H s    Hs   Hs  r 1   r 1  r 1  r 1  es (t , t11 )  es (t , t12 ) z

     

It is clear from above expression that if t11  t12 then es (t , t11 )  es (t , t12 ) t , t11 , t12  T Assume that t11  [r11 , r12 , r13 , r14 ]; a1 , b1 , c1  and t12  [r21 , r22 , r23 , r24 ]; a2 , b2 , c2  be two TrNNs.

The

compliment

of

t11  [r11 , r12 , r13 , r14 ]; a1 , b1 , c1 

is

denoted

by

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Rama Mallick and Surapati Pramanik

t11c  [r11 , r12 , r13 , r14 ];1  c1 , b1 ,1  a1  and t13  [r31 , r32 , r33 , r34 ]; a3 , b3 , c3  is the average of the t11  [r11 , r12 , r13 , r14 ]; a1 , b1 , c1  and t12  [r21 , r22 , r23 , r24 ]; a2 , b2 , c2  .

Then, d (t11 , t13 ) 

1  r11 (2  a1  b1  c1 )  r31 (2  a3  b3  c3 )  r12 (2  a1  b1  c1 )  r32 (2  a3  b3  c3 )    12   r13 (2  a1  b1  c1 )  r33 (2  a3  b3  c3 )  r14 (2  a1  b1  c1 )  r32 (2  a3  b3  c3 ) 

d (t12 , t13 ) 

1  r21 (2  a2  b2  c2 )  r31 (2  a3  b3  c3 )  r22 (2  a2  b2  c2 )  r32 (2  a3  b3  c3 )    12   r23 (2  a2  b2  c2 )  r33 (2  a3  b3  c3 )  r24 (2  a2  b2  c2 )  r32 (2  a3  b3  c3 ) 

d (t11c , t13 )  

1  r11 (2  1  c1  b1  1  a1 )  r31 (2  a3  b3  c3 )  r12 (2  1  c1  b1  1  a1 )  r32 (2  a3  b3  c3 )    12   r13 (2  1  c1  b1  1  a1 )  r33 (2  a3  b3  c3 )  r14 (2  1  c1  b1  1  a1 )  r32 (2  a3  b3  c3 ) 

1  r11 (2  a1  b1  c1 )  r31 (2  a3  b3  c3 )  r12 (2  a1  b1  c1 )  r32 (2  a3  b3  c3 )    12   r13 (2  a1  b1  c1 )  r33 (2  a3  b3  c3 )  r14 (2  a1  b1  c1 )  r32 (2  a3  b3  c3 ) 

Since d (t11 , t13 )  d (t11c , t13 ) . 

 (ts , t  )    H 2 (t11c , t13 )   max  (ts , t ) 

Therefore, H1(t11 , t13 )  



s



and     H (t , t  )     1  1 11 12 ln  H1 (t11 , t12 )    e1s  t11c , t13  e1s  t11 , t13     y y   ln( y ) s 1    H r (t11 , t12 )   H r (t11 , t12 )    r 1   r 1 z

Therefore, es (t , t11 )  es (t , t11c ) for all t T .

4.1. Determination of the Unknown Weights of the Decision Makers and Weights of the Criteria Using the Proposed Entropy Measure Using equation (13), weight of the r-th decision maker is obtained as: y

kr 

e r 1

r

y

Using equation (13), weight of the s-th criterion is obtained as:

(14)

TrNN-ARAS Strategy for Multi-Attribute Group …

175

z

as 

e

s

s 1

(15)

z

We finally obtained the normalized weight of r-th decision maker r 

1  kr

(16)

y

 (1  k ) r

r 1

We finally obtained the normalized weight of the s-th criterion ws 

1  as

(17)

z

 (1  as ) s 1

Consequently, we get the normalized weight vectors   {1 ,  2 ,....,  y } where

0   r  1, r  1,2,....., y and w  (w1 , w2 ,...., wz ) 0  ws  1, s  1, 2,....., z.

5. EXTENDED ARAS STRATEGY FOR MAGDM IN TRNNS ENVIRONMENT Assume that there are y (  2 ) alternatives L  {L1 , L2 ,...., Ly } , z (  2 ) decision attributes E  {E1 , E2 ,...., Ez } and P (  2 ) decision makers in a group decision making problem. The rating value of the alternative Lr over the attribute Es is presented in the form TrNNs. The steps of the group decision making (see Figure 1) based on TrNNs are presented as follows: Step 1. Construct the decision matrices: Suppose that Q p  (nrsp ) y z is the p-th decision matrix where information about the alternative Lr is provided by the p-th decision maker with respect to the attribute p-th decision matrix is defined as follows:

Es .

The

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Rama Mallick and Surapati Pramanik

Q p  (nrsp ) y z

 n11p  p  n21    n yp1   

n1pz   n2pz    p n yz    

n12p n22p n

p y2

(18)

where p = 1, 2, …P. Step 2: Normalize individual decision-matrix: Assume that in the neutrosophic nrsp



([nrsp1 , n rsp 2 , nrsp3 , nrsp 4 ]; cn p

rs

p-th

decision

decision

, dn p , en p ) is the rating value of alternative rs

maker

rs

with

respect

to

attribute

matrix

(18),

Lr provided by the Es

such

that

0  cn p  1,0  d n p  1, 0  en p  1,0  cn p  cn p  cn p  3 rs

rs

rs

rs

rs

rs

To remove the effects derived from different physical dimensions, the decision matrix (nrsp ) y z is standardized. We employ the following technique (Biswas, Pramanik, & Giri 2018a) to obtain the standardized decision matrix X p  ( xrsp ) y z , in which the component xrsp of the entry xrsp  ([ xrsp1 , xrsp 2 , xrsp3 , xrsp 4 ]; cx p , d x p , ex p ) in the matrix X p is rs rs rs considered as: For benefit criterion xrsp  ([

nrsp1 nrsp 2 nrsp 3 nrsp 4 , , , ]; cx p , d x p , ex p ) rs rs rs ks ks ks ks

(19)

For cost criterion xrsp  ([

ks ks ks ks , , , ]; c p , d p , e p ) nrsp 4 nrsp 3 nrsp 2 nrsp1 xrs xrs xrs

(20)

Here ks  max{nrsp 4 r  1, 2,...., y} and ks  min{nrsp1 r  1, 2,...., y} for s =1, 2, …, z. Then we obtain the following standardized decision matrix:

X p  ( xrsp ) y z

 x11p  p  x21    x yp1   

x12p x22p x

p y2

x1pz   x2pz    p x yz    

(21)

TrNN-ARAS Strategy for Multi-Attribute Group …

177

Step 3: Determine the weights of the decision makers: Determination of the weights of the decision makers is presented in section 3. Using equation (16), we calculate the weight vector of the decision makers as:   {1 ,  2 ,....,  y } , y

where  r  [0,1],   r  1 . r 1

Step 4: Aggregate individual decision-making matrices into a group decision matrix: Assume that X p  ( xrsp ) y z is the TrNN decision matrix of the p-th decision maker and

  {1 ,  2 ,....,  y } is the weight vector of the decision makers. In the group decision making process, all the individual decisions need to be fused into a group opinion to construct an aggregated TrNN decision matrix. In order to do so, we employ the TrNNWAA operator (Ye, 2017b). X rs  TrNNWAA ( xrs(1) , xrs(2) ,...., xrs( y ) )   1 xrs(1)   2 xrs(2)  ....   y xrs( y ) y y y y y y  y  w w w     wp xrs( p1) ,  wp xrs( p 2) ,  wp xrs( p 3) ,  wp xrs( p 4)  ;1   (1  cx p , ) p ,  (d x p ) p ,  (ex p ) p  p 1 rs rs rs p 1 p 1 h 1 p 1 p 1 p 1  

  

(22) Then aggregated TrNN decision matrix is obtained as:

X   xrs  y z

 x11   x21    x y1  

x12 x22 xy 2

x1z   x2 z    x yz   

(23)

Here xrs = y y y y y y  y w w w  ( p1) ( p 2) ( p 3) ( p 4)    wp xrs ,  wp xrs ,  wp xrs ,  wp xrs  ;1   (1  cxrsp ,) p ,  (d xrsp ) p ,  (exrsp ) p  ,(r  1, 2,..., y; s  , 2,..., z) p 1 p 1 h 1 p 1 p 1 p 1    p 1 

is the rs-th element of the aggregated TrNN decision matrix. Step 5: Construct the improved group decision matrix using score values: Using equation (7), we calculate the score values of the elements of (23) and construct the improved group decision matrix as follows:

178

Rama Mallick and Surapati Pramanik  11 12    21  22 G    y1  y 2  

1z   2z 

(24)

   yz   

where rs  Sc( xrs ), r  1, 2,...., y; s  1, 2,...., z. Step 6: Determine the weights of the attributes: Determination of the weights of the attributes is shown in section 3. Using equation (17), we calculate the weight vector of the attributes as:

w  (w1 , w2 ,...., wz ) 0  ws  1, s  1, 2,....., z. Step 7: Construct the weighted improved group decision-making matrix: The weighted- values of all the entries of the matrix (24) are calculated as follows: ˆrs   rs ws ,

r  1, 2,.... y; s  1, 2,...., z

(25)

where  rs is the normalized evaluation value of the alternative Lr over the attribute Es; ws z

is the weight of the attribute Es and

w s 1

s

1 .

The weighted group decision matrix is obtained as:

ˆ  (ˆrs ) y z

 ˆ11 ˆ12   ˆ21 ˆ22    ˆ ˆ  y1 y 2  

ˆ1z   ˆ2 z    ˆyz   

(26)

(26)

Step 8: Determine optimal function values: The values of the optimality function denoted by S r is calculated as: z

Sr  rs , r  1, 2,...., y. s 1

The larger the value of S r reflects the better performance of the alternative Lr .

(27)

TrNN-ARAS Strategy for Multi-Attribute Group …

179

Step 9: Evaluate alternative utility degree: The degree of alternative utility is calculated by a comparison of the optimal function value with the ideal best one S b . The utility degree of the alternative  r is presented as:

r 

Sr ; r  1, 2,...., y. Sb

(28)

Here the ideal best S b  max S1 , S 2 ,..., S r  Step 10: Rank the alternatives: The relative priority of feasible alternatives is determined by ascending  r . The alternative with the highest value of  r is the best choice.

Figure 1. Flowchart of the extended TrNN-ARAS strategy for MAGDM.

6. NUMERICAL EXAMPLE The considered problem has been adapted from Ghorabaee (Ghorabaee, 2016). An auto company desires to select a suitable robot for its production process. The selection of robots for a particular application and manufacturing environment is a difficult task for decision makers. It has become more and more complicated due to the increase in advanced features, complexity, and facilities that are continuously being incorporated into the robots by different manufacturers. The decision makers need to identify and select the best-suited robot in order to achieve the desired output with respect to prescribed criteria. For this propose, the auto company forms a team of three decision makers for selecting the best robot. The members of this team are production mangers of the factory production line. Decision-makers have access to brochures and data of the alternatives (robots). There are four alternatives, and they are denoted by L1 , L2 , L3 , and L4 . Decision makers are denoted by D1 , D2 , D3 . The weight vectors of the decision makers and

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Rama Mallick and Surapati Pramanik

criteria are completely unknown. Seven criteria are considered by the decision-making team, which are stated as follows: 1. Inconsistency with infrastructure  E1  2. Supporting channel partner performance  E2  3. Programming flexibility  E3  4. Vender’s service contract  E4  5. Man-Machine interface  E5  6. Stability  E6  7. Compliance  E7  The objective of the decision-making is to identify the ideal Robot with unknown weight information. The first criterion  E1  one is the cost type criterion and the remaining six are benefit type criteria. Step-1. Construct the individual decision matrices: Based on the ratings of the alternatives provided by the decision makers with reference to the prescribed criteria, the following individual decision matrices are constructed. 1 Table 1. Decision Matrix Q1 by decision maker D1

   E1 E  2  E3   E4   E5 E  6  E7 

L1

0.25, 0.35, 0.40, 0.42;0.6, 0.2, 0.1 0.35, 0.43, 0.45, 0.53;0.7, 0.3, 0.2  0.42, 0.45, 0.54, 0.58;0.8, 0.2, 0.2  0.32, 0.34, 0.42, 0.45;0.6, 0.2, 0.1 0.25, 0.32, 0.41, 0.52;0.5, 0.2, 0.3 0.15, 0.22, 0.28, 0.34;0.7, 0.2, 0.3 ([0.43, 0.49, 0.62, 0.65];0.8, 0.2, 0.1)

L2

L3

L4



0.63, 0.65, 0.72, 0.75;0.8, 0.2, 0.4  0.45, 0.64, 0.71, 0.87;0.6, 0.2, 0.3 0.25, 0.33, 0.55, 0.63;0.6, 0.3, 0.3  0.62, 0.64, 0.77, 0.82;0.8, 0.3, 0.2  0.32, 0.44, 0.52, 0.63;0.7, 0.3, 0.4  ([0.42, 0.53, 0.62, 0.71];0.8, 0.4, 0.3)  ([0.41, 0.63, 0.74, 0.82];0.6, 0.4, 0.2) 0.25, 0.32, 0.55, 0.64 ;0.6, 0.4, 0.2  0.33, 0.42, 0.53, 0.64 ;0.7, 0.3, 0.4    (0.54, 0.61, 0.74, 0.81 ;0.8, 0.3, 0.2) ([0.41, 0.63, 0.74, 0.82];0.6, 0.4, 0.2) 0.45, 0.52, 0.71, 0.82 ;0.6, 0.2, 0.3  0.42, 0.58, 0.61, 0.74;0.8, 0.5, 0.3 0.32, 0.41, 0.52, 0.54;0.6, 0.4, 0.2  ([0.42, 0.53, 0.62, 0.71];0.8, 0.4, 0.3)  0.62, 0.73, 0.84, 0.91;0.6, 0.2, 0.1 0.62, 0.64, 0.77, 0.82;0.8, 0.1, 0.3 0.37, 0.44, 0.51, 0.72;0.7, 0.2, 0.3  0.37, 0.44, 0.51, 0.72;0.6, 0.3, 0.2  (0.54, 0.61, 0.74, 0.81;0.8, 0.4, 0.2) 0.25, 0.33, 0.55, 0.63;0.6, 0.2, 0.1 

(29) Table 2. Decision Matrix Q by decision maker D2 1 2

   E1 E  2  E3   E4   E5 E  6  E7 

L1

0.60, 0.62, 0.65, 0.65;0.5, 0.4, 0.2  0.45, 0.53, 0.55, 0.63;0.4, 0.2, 0.2  0.37, 0.43, 0.50, 0.54;0.6, 0.3, 0.2  0.32, 0.34, 0.42, 0.5;0.5, 0.2, 0.1 0.55, 0.62, 0.65, 0.72;0.8, 0.2, 0.3 0.43, 0.46, 0.53, 0.60;0.7, 0.3, 0.4  ([0.22, 0.43, 0.52, 0.61];0.6, 0.2, 0.3)

L2

L3

L4



0.23, 0.31, 0.43, 0.55;0.6, 0.3, 0.2  0.55, 0.57, 0.61, 0.77 ;0.8, 0.2, 0.3 0.15, 0.23, 0.25, 0.33;0.5, 0.2, 0.2   0.52, 0.64, 0.67, 0.72;0.7, 0.4, 0.3 0.42, 0.44, 0.48, 0.53;0.7, 0.3, 0.2  ([0.22, 0.43, 0.45, 0.51];0.6, 0.2, 0.3)  ([0.51, 0.53, 0.64, 0.70];0.6, 0.3, 0.2) 0.22, 0.32, 0.45, 0.54 ;0.6, 0.4, 0.1 0.33, 0.36, 0.43, 0.47 ;0.7, 0.1, 0.2    (0.64, 0.71, 0.74, 0.81 ;0.5, 0.2, 0.1) ([0.51, 0.65, 0.74, 0.80];0.5, 0.3, 0.2) 0.45, 0.52, 0.61, 0.64 ;0.6, 0.4, 0.2    0.52, 0.58, 0.63, 0.65 ;0.7, 0.3, 0.3 0.42, 0.51, 0 .54, 0.54 ;0.6, 0.4, 0.3 ([0.42, 0.45, 0.52, 0.61];0.7, 0.2, 0.3)        0.72, 0.73, 0.80, 0.85;0.8, 0.2, 0.4  0.72, 0.75, 0.80, 0.82;0.8, 0.2, 0.3 0.35, 0.40, 0.44, 0.52;0.6, 0.1, 0.2   0.47, 0.54, 0.61, 0.70;0.5, 0.1, 0.2  (0.24, 0.33, 0.40, 0.51;0.7, 0.2, 0.4) 0.15, 0.32, 0.35, 0.43;0.5, 0.2, 0.3 

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181

1 Table 3. Decision Matrix Q3 by decision maker D3    E1 E  2  E3   E4   E5 E  6  E7 

L1

0.35, 0.42, 0.53, 0.57;0.6, 0.2, 0.1 0.45, 0.53, 0.55, 0.60;0.7, 0.2, 0.3 0.52, 0.55, 0.60, 0.64;0.6, 0.3, 0.1 0.44, 0.54, 0.65, 0.70;0.8, 0.2, 0.4  0.78, 0.80, 0.80, 0.82;0.7, 0.2, 0.3 0.53, 0.55, 0.63, 0.65;0.6, 0.2, 0.4  ([0.22, 0.33, 0.42, 0.51];0.5, 0.2, 0.1)

L2

L3



L4

0.43, 0.51, 0.63, 0.65;0.8, 0.3, 0.2  0.75, 0.78, 0.82, 0.87;0.8, 0.3, 0.4  0.15, 0.26, 0.35, 0.40;0.5, 0.1, 0.3  0.72, 0.74, 0.77, 0.82;0.6, 0.1, 0.2  0.42, 0.48, 0.54, 0.60;0.6, 0.1, 0.2  ([0.32, 0.37, 0.44, 0.50];0.7, 0.2, 0.3)  ([0.51, 0.55, 0.64, 0.72];0.7, 0.3, 0.2) 0.15, 0.32, 0.45, 0.54 ;0.7, 0.3, 0.1 0.37, 0.47, 0.51, 0.60 ;0.8, 0.3, 0.1   (0.74, 0.76, 0.78, 0.81 ;0.8, 0.2, 0.2) ([0.51, 0.65, 0.77, 0.84];0.8, 0.4, 0.3) 0.12, 0.22, 0.27, 0.32 ;0.6, 0.1, 0.3  0.32, 0.35, 0.41, 0.54;0.6, 0.4, 0.1 0.22, 0.32, 0.42, 0.44;0.7, 0.4, 0.4  ([0.47, 0.55, 0.67, 0.73];0.8, 0.4, 0.2)  0.22, 0.25, 0.36, 0.41;0.5, 0.1, 0.2  0.42, 0.54, 0.67, 0.72;0.8, 0.2, 0.1 0.37, 0.44, 0.54, 0.62;0.6, 0.1, 0.2  0.41, 0.45, 0.55, 0.55;0.8, 0.3, 0.2  (0.64, 0.66, 0.78, 0.85;0.6, 0.3, 0.2) 0.75, 0.76, 0.80, 0.83;0.7, 0.2, 0.1 

(31) Step-2. Standardized the individual decision matrices: Using equation (20) and (21) 1 Table 4. Standardized decision matrix X 1 by decision maker D1

   E1 E  2  E3   E4   E5 E  6  E7 

L1

0.60, 0.62, 0.71,1;0.6, 0.2, 0.1 0.66, 0.81, 0.85,1;0.7, 0.3, 0.2  0.72, 0.78, 0.93,1;0.8, 0.2, 0.2  0.71, 0.76, 0.93,1; 0.6, 0.2, 0.1 0.48, 0.61, 0.79,1;0.5, 0.2, 0.3 0.44 0.65, 0.82,1;0.7, 0.2, 0.3 ([0.66, 0.75, 0.95,1]; 0.8, 0.2, 0.1)

L2

L3

L4



0.84, 0.88, 0.97,1;0.8, 0.2, 0.4  0.52, 0.63, 0.70,1 ;0.6, 0.2, 0.3 0.40, 0.45, 0.76,1;0.6, 0.3, 0.3  0.76, 0.78, 0.94,1;0.8, 0.3, 0.2  0.51, 0.70, 0.82,1;0.7, 0.3, 0.4  ([0.59, 0.75, 0.87,1];0.8, 0.4, 0.3)  ([0.5, 0.55, 0.90,1];0.6, 0.4, 0.2)  0.39, 0.45, 0.86,1 ;0.6, 0.4, 0.2   0.52, 0.62, 0.83, 1 ;0.7, 0.3, 0.4    (0.67, 0.75, 0.91,1 ;0.8, 0.3, 0.2) ([0.5, 0.77, 0.90,1]; 0.6, 0.4, 0.2)  0.55, 0.63, 0.86,1 ;0.6, 0.2, 0.3  0.57, 0.78, 0.82,1; 0.8, 0.5, 0.3 0.59, 0.76, 0.96,1;0.6, 0.4, 0.2  ([0.59, 0.75, 0.87,1];0.8, 0.4, 0.3)  0.68, 0.80, 0.92,1;0.6, 0.2, 0.1 0.76, 0.78, 0.93,1;0.8, 0.1, 0.3 0.51, 0.61, 0.71,1;0.7, 0.2, 0.3  0.51, 0.61, 0.71,1;0.6, 0.3, 0.2  (0.67, 0.75, 0.91,1; 0.8, 0.4, 0.2) 0.40, 0.52, 0.87,1;0.6, 0.2, 0.1 

(32) 1 Table 5. Standardized Matrix X 2 by decision maker D2

   E1 E  2  E3   E4   E5 E  6  E7 

L1

0.92, 0.92, 0.97,1;0.5, 0.4, 0.2  0.71, 0.84, 0.87,1;0.4, 0.2, 0.2  0.68, 0.80, 0.92,1; 0.6, 0.3, 0.2  0.64, 0.68, 0.84,1;0.5, 0.2, 0.1 0.76, 0.86, 0.90,1 ;0.8, 0.2, 0.3 0.72, 0.77, 0.88,1;0.7, 0.3, 0.4  ([0.42, 0.70, 0.85,1]; 0.6, 0.2, 0.3)

L2

L3

L4



0.42, 0.53, 0.74,1; 0.6, 0.3, 0.2  0.71, 0.90, 0.96,1;0.8, 0.2, 0.3 0.45, 0.6, 0.65,1;0.5, 0.2, 0.2   0.72, 0.89, 0.93,1;0.7, 0.4, 0.3 0.79, 0.83, 0.90,1;0.7, 0.3, 0.2  ([0.43, 0.84, 0.88,1];0.6, 0.2, 0.3)  ([0.74, 0.76, 0.91,1];0.6, 0.3, 0.2)  0.41, 0.59, 0.83,1 ; 0.6, 0.4, 0.1 0.70, 0.76, 0.91,1 ; 0.7, 0.1, 0.2    ( 0.79, 0.88, 0.91,1 ;0.5, 0.2, 0.1) ([0.64, 0.81, 0.92,1];0.5, 0.3, 0.2) 0.70, 0.81, 0.95,1 ;0.6, 0.4, 0.2   0.8, 0.89, 0.97,1; 0.7, 0.3, 0.3 0.78, 0.94,1,1;0.6, 0.4, 0.3 ([0.69, 0.74, 0.85,1]; 0.7, 0.2, 0.3)  0.85, 0.86, 0.94,1;0.8, 0.2, 0.4  0.88, 0.91, 0.98,1;0.8, 0.2, 0.3 0.67, 0.77, 0.85,1;0.6, 0.1, 0.2   0.67, 0.77, 0.87,1;0.5, 0.1, 0.2  (0.47, 0.65, 0.78,1; 0.7, 0.2, 0.4) 0.35, 0.74, 0.81,1 ; 0.5, 0.2, 0.3 

(33) Table 6. Standardized Matrix X by decision maker D2 1 2

   E1 E  2  E3   E4   E5 E  6  E7 

L1

0.61, 0.66, 0.83,1;0.6, 0.2, 0.1 0.75, 0.88, 0.92,1;0.7, 0.2, 0.3 0.81, 0.86, 0.94,1;0.6, 0.3, 0.1 0.63, 0.77, 0.93,1;0.8, 0.2, 0.4  0.95, 0.98, 0.98,1;0.7, 0.2, 0.3 0.82, 0.85, 0.97,1;0.6, 0.2, 0.4  ([0.43, 0.65, 0.82,1]; 0.5, 0.2, 0.1)

L2

L3

L4



0.66, 0.68, 0.84,1; 0.8, 0.3, 0.2  0.86, 0.91, 0.96,1;0.8, 0.3, 0.4  0.38, 0.43, 0.58,1;0.5, 0.1, 0.3  0.88, 0.90, 0.94,1;0.6, 0.1, 0.2  0.7, 0.8, 0.9,1; 0.6, 0.1, 0.2  ([0.64, 0.74, 0.88,1];0.7, 0.2, 0.3)  ([0.71, 0.76, 0.89,1];0.7, 0.3, 0.2) 0.28, 0.59, 0.83,1;0.7, 0.3, 0.1  0.62, 0.78, 0.85,1 ;0.8, 0.3, 0.1   (34) (0.91, 0.94, 0.96,1 ;0.8, 0.2, 0.2) ([0.61, 0.77, 0.92,1];0.8, 0.4, 0.3) 0.38, 0.69, 0.84,1; 0.6, 0.1, 0.3  0.59, 0.65, 0.76,1;0.6, 0.4, 0.1 0.5, 0.73, 0.95,1;0.7, 0.4, 0.4  ([0.64, 0.75, 0.92,1];0.8, 0.4, 0.2)  0.54, 0.61, 0.88,1;0.5, 0.1, 0.2  0.58, 0.75, 0.93,1;0.8, 0.2, 0.1 0.60, 0.71, 0.87,1;0.6, 0.1, 0.2   0.74, 0.82,1,1;0.8, 0.3, 0.2  (0.75, 0.78, 0.92,1;0.6, 0.3, 0.2) 0.90, 0.92, 0.96,1;0.7, 0.2, 0.1 

(34)

182

Rama Mallick and Surapati Pramanik Step-3: Determine the weights of the decision makers: Using equation (16), weights of the decision makers are obtained and shown in Table

7. Table 7. Weights of decision makers 1

2

2

0.2788

0.3640

0.3572

Step-4: Aggregate individual decision-making matrices into a group decision matrix: Using the TrNNWAA operator (22), we obtain group decision matrix (See Table 8). Table 8. Group decision matrix    E1 E  2  E3   E4   E5 E  6  E7 

L1

0.72, 0.74, 0.85,1;0.57, 0.25, 0.13 0.71, 0.84, 0.88,1;0.61, 0.22, 0.23 0.74, 0.81, 0.93,1;0.67, 0.26, 0.15 0.66, 0.73, 0.90,1;0.66, 0.2, 0.16  0.75, 0.83, 0.90,1;0.70, 0.2, 0.3 0.68, 0.76, 0.90,1;0.67, 0.23, 0.34  ([0.49, 0.70, 0.87,1];0.64, 0.2, 0.15)

L2

L3

L4



0.62, 0.68, 0.84,1;0.74, 0.27, 0.24  0.71, 0.83, 0.89,1;0.76, 0.23, 0.33 0.41, 0.49, 0.65,1;0.53, 0.17, 0.26   0.78, 0.86, 0.94,1;0.70, 0.22, 0.23 0.68, 0.78, 0.88,1;0.67, 0.20, 0.24  ([0.55, 0.77, 0.87,1];0.7, 0.24, 0.3)  ([0.66, 0.70, 0.90,1];0.63, 0.32, 0.2)  0.35, 0.55, 0.841,1 ;0.64, 0.36, 0.12   0.62, 0.73, 0.86,1 ;0.74, 0.20, 0.19    (35) ( 0.80, 0.86, 0.93,1 ;0.72, 0.22, 0.16) ([0.59, 0.78, 0.91,1];0.66, 0.36, 0.23)  0.54, 0.72, 0.88,1 ;0.6, 0.20, 0.25   0.66, 0.77, 0.85,1;0.70, 0.38, 0.20  0.63, 0.81, 0.97,1;0.63, 0.4, 0.30  ([0.64, 0.74, 0.88,1];0.77, 0.31, 0.26)  0.69, 0.75, 0.91,1;0.66, 0.16, 0.21 0.74, 0.82, 0.95,1;0.8, 0.16, 0.20  0.60, 0.70, 0.81,1;0.63, 0.12, 0.22   0.65, 0.74, 0.87,1;0.66, 0.20, 0.2 (0.63, 0.72, 0.86,1;0.70, 0.28, 0.23) 0.56, 0.74, 0.88,1;0.61, 0.2, 0.15 

(35) Step-5: Construct the improved group decision matrix using score values: Using equation (7), we calculate the score values of the elements of (35) and construct the improved group decision matrix (see Table 9).

Table 9. Improved group decision matrix    E1  E2   E3 E  4  E5 E  6 E  7

L1 0.6025 0.6192 0.6526 0.6293 0.6389 0.5742 0.5851

L2 0.5853 0.6714 0.5747 0.7015 0.5804 0.6421 0.6158

L3 0.6278 0.6193 0.4945 0.5676 0.5528 0.7108 0.5873

L4   0.4478  0.5776   0.6304  (36) 0.5619   0.5994  0.5946   0.6003 

Step 6: Calculate the weights of the criteria: Using equation (17), weights of the criteria are obtained and shown in Table 10.

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183

Table 10. Weights of the criteria w1

w2

w3

w4

w5

w6

w7

0.1330

0.1570

0.1504

0.1655

0.1801

0.0988

0.1552

Step 7. Construct the weighted-improved group decision matrix: Using equation (25), we construct the weighted -improved group decision matrix (see Table 11). Table 11. Weighted-improved group decision matrix    E1  E2   E3 E  4  E5 E  6 E  7

L1 0.0801 0.0972 0.0982 0.1041 0.1151 0.0567 0.0674

L2 0.0778 0.1054 0.0864 0.1161 0.1045 0.0635 0.0709

L3 0.0835 0.0972 0.0744 0.0939 0.0996 0.0702 0.0677

L4   0.0596  0.0907   0.0948  0.0930   0.1080  0.0588   0.0692 

Step 8: Evaluating optimality function: Using equation (27), the optimality function values can be obtained. S1  0.6188, S2  0.6247, S3  0.5865, S4  0.5739

So, S b = S2  0.6247 Step 9: Calculate the alternative utility degree: The degree of alternative utility can be calculated by a comparison of the optimal function value with the ideal best one S b . The utility degree of alternative  r is presented as: 1  0.2574, 2  0.2599, 3  0.2440, 4  0.2387

Step 10: Rank of the alternatives: Based on the utility degree of alternatives, the obtained ranking of alternatives (see Table 12) is presented as: L2  L1  L3  L4

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Therefore, after applying the proposed strategy and utilizing the seven attributes, we obtain the above ranking order of the alternatives (see Table 12). It implies that the 2 nd alternative i.e., L2 is the best alternative and 4th alternative i.e., L4 is the worst alternative. Table 12. Optimal function value, utility degree and ranking

L1

L2

L3

L4

Sr

0.6188

0.6247

0.5865,

0.5739

r

0.2574

0.2599

0.2440

0.2387

Ranking

2

1

3

4

7. COMPARATIVE ANALYSIS In the literature, VIKOR (Pramanik & Mallick, 2018) has been already proposed in the TrNN environment for group decision making. The ranking order of alternatives using proposed TrNN-ARAS and VIKOR strategy is shown in Table 13. In VIKOR strategy, we first determine the best and the worst attribute functions. Then we compute the values average and worst group scores for the alternative. Here we take value 0.5 as the decision-making mechanism coefficient. Then we rank the alternatives according to the traditional rule of VIKOR strategy. From the comparison Table 13, we see that the second alternative is the best choice for the VIKOR and the extended TrNN- ARAS strategy. But the ranking order differs. Table 13. Comparison of ranking of the alternatives obtained from ARAS, VIKOR strategies Strategy TrNN-ARAS strategy (Proposed)

Rank of the alternative

VIKOR Strategy (Pramanik & Mallick, 2018)

L2  L4  L1  L3

L2  L1  L3  L4

8. ADVANTAGES OF THE PROPOSED STRATEGY COMPARE TO VIKOR STRATEGY The extended TrNN-ARAS strategy is a simple strategy compared to VIKOR strategy (Pramanik & Mallick, 2018). In the VIKOR strategy, weights of the decision makers and the criteria are known. But in the extended TrNN strategy weights of the

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decision makers and the criteria are completely unknown. So, the proposed extended strategy is more comprehensive and more applicable.

CONCLUSION AND FUTURE RESEARCH DIRECTION In this chapter, we have proposed an extended ARAS strategy in TrNNs environment. The extended TrNN-ARAS strategy is capable of dealing with uncertain, inconsistent and indeterminate information. It is a flexible scientific strategy in the decision-making process. In the extended TrNN- ARAS strategy, the utility function directly determines the complex efficiency of a feasible alternative, which is directly proportional to the relative effect of values and weights of the attribute. The best alternative is chosen based on the utility function value. It is also convenient to determine and rank alternatives using this strategy. In this chapter, we have solved an MAGDM problem dealing with the robot selection problem in the context of the TrNN environment. Lastly, we compare the proposed ASAR strategy with the existing three MAGDM strategies in the literature. We hope that in the future, the proposed ARAS strategy will be widely applicable in MAGDM problems such as multiple target tracking (Fan, Hu, & Li, 2019; Fan, Xie, Pei, Hu, & Li, 2018), stock trending analysis (Jha et al., 2019), teacher selection (Pramanik & Mukhopadhyaya, 2011), pattern recognition (Ali, Deli & Smarandache, 2016), etc. From the practical point of view, the developed strategy allows for applications to a great variety of real-life decision- making situations. For examples, the developed TrNNARAS strategy could also interesting to carryout research on sustainable collaborative networks selection problems (Verdecho, Alfaro-Saiz, & Rodríguez-Rodríguez, 2010), collaborative supplier selection problems involving uncertain, inconsistent and incomplete data within a hierarchical structure. In the future, we will employ the proposed strategy to some real applications such as weaver selection (Dey, Pramanik, & Giri, 2015a; 2015b); school choice (Mondal, & Pramanik, 2015); Brick selection (Mondal & Pramanik, 2014), Radio Frequency Identification (RFID) technology selection (Wang, Wang & Dong, 2016) and supply chain management (Yazdani, Zarate, Coulibaly, &Zavadskas, 2017) problems. Future research could focus on more complex extensions involving interval neutrosophic hesitant fuzzy set (Liu & Shi, 2015), multivalued neutrosophic linguistic set (Li, Wang, Yang, & Li, 2018), multiple-valued neutrosophic uncertain linguistic set (Yang & Li, 2020), multi-valued interval neutrosophic linguistic soft set (Kamal, Abdullah, Abdullah, & Saqlain, 2020), neutrosophic linguistic neutrosophic trapezoid fuzzy linguistic set (Broumi & Smarandache, 2014). In the future, we will extend the proposed TrNN-ARAS strategy to interval TrNNs environment (Biswas, Pramanik, & Giri, 2018c; Mallick & Pramanik, 2019).

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In: Decision-Making with Neutrosophic Set Editor: Harish Garg

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Chapter 9

AN APPLICATION OF REDUCED INTERVAL NEUTROSOPHIC SOFT MATRIX IN MEDICAL DIAGNOSIS Somen Debnath* Department of Mathematics, Tripura University Suryamaninagar, Agartala, Tripura, India

ABSTRACT In 2014, Irfan Deli introduced the notion of interval neutrosophic soft sets (ivn-soft sets). The interval neutrosophic soft set is an extension of the neutrosophic soft set. In this paper, the notion of interval neutrosophic soft matrix (ivn-soft matrix) has been studied and defined different types of ivn-soft matrices with examples. There are so many indeterminacies existence in real life. The concept of ivn-soft matrix theory is one of the recent topics developed to deal with indeterminacy present in our surroundings with a more wide range of domain. In this work, we have tried to use the concept of ivn-soft matrices in a medical diagnosis problem with the help of a proposed algorithm. The main objective of this article is to study the decision-making problems by using ivn-soft matrices.

Keywords: soft set, neutrosophic set, neutrosophic soft set, interval neutrosophic soft matrix

*

Corresponding Author’s Email: [email protected].

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1. INTRODUCTION It is Prof. Lotfi Aliasker Zadeh [41], who first introduced the concept of fuzzy set in 1965. It is an extension of the classical set (or crisp set) to solve uncertain problems. It is considered as one of the most successful theoretical tool, which can tackle various kinds of uncertainties existence in the real world quite adequately. The fuzzy set is described with an aid of a fuzzy membership function  : X [0,1]. Many researchers and mathematicians used the fuzzy set theory in different areas, such as artifici-al intelligence, medicine, engineering, economics, computer science, control theory, etc. But due to more complexity exist in the real world, it is difficult to set the membership function in each particular case. For example, to define the attractiveness of a house, if one considered the membership degree as 0.8, then everyone may understand this in his/her own manner. So, there is a question about the uniqueness of the membership function.Later on, the fuzzy set theory has been more generalized by introducing the Lfuzzy set [14], vague set [13], intuitionistic fuzzy set [1], interval-valued fuzzy set [15], interval-valued intuitionistic fuzzy set [2], fuzzy rough set [17], etc. The difficulty of the membership function has been removed by the soft set by introducing parameter. Russian mathematician D. Molodtsov [25] introduced the novel concept of the soft set to handle uncertainty or vagueness in different ways by using the notion of parametrization in data. In soft set theory, to define an object, no need to introduce a membership function. For example, an attractiveness of a house can be described very easily via some parameters instead of considering membership function. The concept of soft set theory has been progressed very rapidly and it can be applied in different fields including game theory, social science, medical science, operation research, decision-making, pattern recognition, algebra, etc. In a soft set, we can take any parameter by using word, sentence, mapping, function, etc. Our considered parameters may not be always crisp, but maybe in fuzzy words. So, such type of vagueness demanded several kinds of extensions of soft set theory. Some of the extensions and recent works of soft set theory discussed in [3, 7, 11, 16, 18, 19, 20, 28]. Intuitionistic fuzzy set is a generalization of the fuzzy set. But, it can only handle the incomplete information by using both the membership and non-membership function. However, it fails to represent indeterminate and inconsistent information which exists in the real world. So, we need another effective tool which provides a general framework to represent incomplete, indeterminate and inconsistent information in an organized manner. It leads to the introduction of neutrosophic set theory, proposed by Smarandache in [35]. It is a generalization of the intuitionistic fuzzy set. Neutrosophic set based on triple membership functions, which are the truth-membership function, the indeterminacymembership function and the falsity-membership function and, by using these three functions we can handle the incomplete, indeterminate and inconsistent information elegantly. For scientific or engineering work, the concept of the single-valued

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neutrosophic set [38] is very useful. In 2005, Wang et al. [37] defined interval neutrosophic sets and it has been widely used in decision making. By combining neutrosophic set with the soft set, Maji [21, 22] introduced neutrosophic soft sets, where each parameter is a neutrosophic word or sentence or mapping or function and it is the main advantage of including soft set to describe uncertainty, incompleteness and indeterminacy in a more generalized form. Sahin et al. [34] defined the neutrosophic soft expert to deal with indeterminate and inconsistent information. Some other contributions related to neutrosophic sets, neutrosophic soft sets and their extensions given in [8, 9, 10, 12, 33, 39]. To store large data permanently in a computer, we need to reduce such type of data in a concise form so that it consumes less memory. For that purpose, we need to represent big data in a matrix form. Earlier, classical matrices are used in solving problems. But, classical matrices failed to represent various types of uncertainties (vagueness) involved in real-life. Mondal et al.[26] introduced soft matrices, by combining soft set and fuzzy matrix, a new concept known as fuzzy soft matrix [4, 5] has been introduced. Rajarajeswari et al. [31] described an interval fuzzy soft matrix. Yang et al. [40] initiated a matrix representation of fuzzy soft sets and their applications. Mitra Basu et al. [23] initiated the fuzzy soft matrix theory and its application. In [6], Chetia et al. and in [29, 30], Rajarajeswari et al. defined the intuitionistic fuzzy soft matrix theory and its application and in [32], Rajarajeswari et al. introduced the interval intuitionistic fuzzy soft matrix. In [27], Anjan et al. give an application of the interval intuitionistic fuzzy soft matrix in medical diagnosis. Basu et al. [24] introduced the neutrosophic soft matrix and its application in solving a group decision-making problem. In the neutrosophic soft matrix, each entry can be represented by a triplet. Each triplet comprised with truth-membership value, indeterminate-membership value and the falsity-membership value and for each membership value we assign a single real value, taken from the unit closed interval [0, 1]. But, sometimes it is difficult to represent indeterminacy by a single real value due to the more complexity involved to define indeterminacy in solving problems. So, there is a demand to extend the notion of the neutrosophic soft matrix. In this paper, we have introduced the notion of the interval neutrosophic soft matrix and the main motivation behind the introduction of interval neutrosophic soft matrix is to generalize the notion of neutrosophic soft matrix by widening the range of domain of membership functions so that we can handle the concept of indeterminacy in an effective manner. In an interval neutrosophic soft matrix, we assign an interval to each membership values and the interval contained in the unit closed interval [0, 1]. In interval intuitionistic fuzzy soft matrices, the membership interval and the non-membership interval are depend on each other. But, in interval neutrosophic soft matrices, the intervals are independent to one another. Then, we have defined different types of interval neutrosophic soft matrices with examples and perform some basic algebraic operations on them. Based on some new types of interval neutrosophic soft

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matrices, a new algorithm has been developed to solve interval neutrosophic soft set based real life decision-making problems. Finally, we apply this algorithm to solve medical diagnosis problem in an organized manner with the assistance of the soft medical expert. The present paper is organized as follows: Section-2 comprised with fuzzy set along with its generalization, soft set along with its generalization, generalization of fuzzy soft matrices, neutrosophic set and neutrosophic soft set. In section-3, firstly defined the interval neutrosophic soft matrices and theirkinds and some algebraic operations on them. In the next section (in section-4), an algorithm has been constructed which is based on parameterized interval neutrosophic soft sets in decision-making problem. In the last section (in section-5), with the help of a suitable example, it has been shown that the proposed algorithm has been applied successfully in medical diagnosis.

2.PRELIMINARIES In this section we recall some basic definitions which are relevant as far as the paper is concerned.

2.1. Definition [41] Let X be a non-empty set. Then a fuzzy set A is a set having the form

A

 x, 

A

 x  : x  X  ,

where the function  A : X   0,1 is called the membership

function and  A  x  is called the degree of membership of each element x  X .

2.2. Definition [1] Let X be a non-empty set. An intuitionistic fuzzy set (if-set) A in X is an object

A  {( x,  A ( x),  A ( x)) : x  X } , where the functions  A : X  [0,1] anddenote the degree of membership (namely  A ( x) ) and the degree of non-membership (namely

 A ( x ) ) of each element x  X to the set A respectively and 0   A ( x)   A ( x)  1 for each x  X .

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2.3. Definition [15] An interval fuzzy set (ivf-set) A over X is given by a function  A  x  where

 A : X  Int 0,1 , the set of all sub-intervals of unit interval i.e., for every x  X ,

 A  x  is an interval within  0,1 .

2.4. Definition [2] An interval intuitionistic fuzzy set (ivif-set) A over universe set X is defined as

A

 x, 

A

 x, A  x

: x X



,

where

 A  x  : X  Int 0,1

and

 A  x  : X  Int 0,1 (where Int 0,1 is the set of all closed intervals of  0,1 are functions such that the condition: x  X , 0  sup A  x   sup A  x   1 is satisfied.

2.5. Definition [35] Let X be a universe of discourse, with a generic element in X denoted by x, then the neutrosophic set (ns) is an object having the form

A

 x:

A

 x  , A  x  ,  A  x 



,x X ,

where the functions  , ,  : X    0,1  defined respectively the degree of membership (or Truth),the degree of indeterminacy, and the degree of nonmembership(or Falsehood) of the element x  X to the set 

A with the condition

0   A  x   A  x   A  x   3

From a philosophical point of view, the neutrosophic set takes the value from real standard or non-standard subsets of   0,1  .So, instead of   0,1  we need to take the interval [0,1] for technical applications, because   0,1  will be difficult to apply in the real applications such as in scientific and engineering problems.

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2.6. Definition [37] Let U be a space of points (objects), with a generic element in U denoted by u . An interval neutrosophic set (ivn-set) A in U is characterized by truth-membership function TA ,an indeterminacy-membership function I A and a falsity-membership function FA .

For each point u U ; TA , I A and FA  [0,1] . Thus, an ivn-sets over U can be represented by the set of the following form A

 TA u  , I A u  , FA u 



/ u : u U





Here, TA u  , I A u  , FA u  is called interval neutrosophic number for all u U and all interval neutrosophic numbers over U will be denoted by IVN (U ) .

2.7. Example Let U  u1 , u2  be the universe of discourse and A be an interval neutrosophic set in U . Then A can be expressed as follows:

A

 0.1, 0.6 , 0.4, 0.7, 0.3, 0.8 / u1, 0.1, 0.5, 0.3, 0.7 , 0.4, 0.7 / u2

2.8. Definition [25] Let U be an initial universe and E be a set of parameters. Let P U  denotes the power set of U and A  E . Then the pair  F , A  is called a soft set over U , where F is a mapping given by F : A  P U  . A soft set over U is a parameterized family of subsets of the universal set U where, e  E , F  e  may be considered as the set of e -approximate elements of the soft set

 F , e . 2.9. Definition [16] U Let U be an initial universe and E be a set of parameters. Let IVIFS be the set of

all interval intuitionistic fuzzy sets on U and A  E . Then the pair  F , A  is called an

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interval intuitionistic fuzzy soft set (ivif-soft set for short) over U , where F is a mapping U

given by F : A  IVIFS .

2.10. Definition [32]





Let U  c1, c2 , c3 ,..., cm be a universal set and E be the set of parameters given by





E  e1, e2 , e3 ,..., en . Let A  E and  F , A  be an interval intuitionistic fuzzy soft set

over U ,where F is a mapping given by F:A  I ,where I U denotes the collection of all interval intuitionistic fuzzy subsets of U .Then the interval intuitionistic fuzzy soft set can be expressed in a matrix form as U

Aˆ mn  aij    mn or Aˆ  a  i  1, 2,......, m; j  1, 2,......., n ij

  aij    

 



 

 



 

 

  jL ci ,  jU ci   jL ci ,  jU ci  ([0, 0][1,1])



if if

ej  A ej  A

  jL  ci  ,  jU  ci  denotes the membership of ci in the interval intuitionistic fuzzy

 

set F e j .

 jL  ci  ,  jU  ci  denotes the non-membership of ci in the interval intuitionistic

 

fuzzy set F e j with the condition  jU  ci    jU  ci   1 . Note that if  jU  ci  =  jL  ci  and  jU  ci  =  jL  ci  then the ivif-soft matrix reduces to an if-soft matrix.

2.11. Definition [21] Let U be an initial universe set and E be a set of parameters. Consider A  E .Let P U  denotes the set of all neutrosophic sets of U .The collection  F , A is termed to be

the soft neutrosophic set over U where, F is a mapping given by F : A  P U  .

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2.12. Example Let U be the set of houses under consideration and E be the set of parameters. Each parameter is a neutrosophic word or sentence involving neutrosophic words. Considered E ={beautiful, wooden, costly, very costly, moderate, green surroundings, in good repair, in bad repair, cheap, expensive}.In this case, to define a neutrosophic soft set means to point out beautiful houses, wooden houses, houses in the green surroundings and so on. Suppose that, there are five houses in the universe U given by, U = h1 , h2 , h3 , h4 , h5 and the set of parameters A  e1 , e 2 , e3 , e 4 ,where, e1 stands for the parameter ‘beautiful’, e 2 stands for the parameter ‘wooden’, e3 stands for the parameter ‘costly’ and the parameter

e4 stands for ‘moderate’. Suppose the neutrosophic soft set  F , A is given by

 

  

beautiful houses  h , 0.5, 0.6, 0.3 , h , 0.4, 0.5, 0.6 , h , 0.6, 0.2, 0.3 , h , 0.6, 0.3, 0.4 , h , 0.8, 0.2, 0.3 ,  5 1 2 3 4   wooden houses  h , 0.6, 0.3, 0.5 , h , 0.7, 0.5, 0.3 , h , 0.8, 0.1, 0.2 , h , 0.7, 0.2, 0.3 , h , 0.8, 0.3, 0.6 ,  5 1 2 3 4      F , A  cos tly houses  h1, 0.7, 0.4, 0.3 , h2 , 0.6, 0.7, 0.2 , h3 , 0.7, 0.3, 0.5 , h4 , 0.8, 0.2, 0.6 , h5 , 0.8, 0.3, 0.4 ,  ,   mod erate houses  h1, 0.8, 0.6, 0.4 , h2 , 0.7, 0.9, 0.5 , h3 , 0.7, 0.6, 0.4 , h4 , 0.7, 0.5, 0.6 , h5 , 0.6, 0.5, 0.7     







The tabular representation of the neutrosophic soft set  F , A is given by:

U h 1 h2 h  3 h4 h 5

beautiful

 0.5, 0.6, 0.3  0.4, 0.5, 0.6  0.6, 0.2, 0.3  0.6, 0.3, 0.4  0.8, 0.2, 0.3

wooden

 0.6, 0.3, 0.5  0.7, 0.5, 0.3  0.8, 0.1, 0.2  0.7, 0.2, 0.3  0.8, 0.3, 0.6

cos tly

 0.7, 0.4, 0.3  0.6, 0.7, 0.2   0.7, 0.3, 0.5  0.8, 0.2, 0.6  0.8, 0.3, 0.4

 0.8, 0.6, 0.4  0.7, 0.9, 0.6  0.7, 0.6, 0.4  0.7, 0.5, 0.6  0.6, 0.5, 0.7  mod erate

3. INTERVAL NEUTROSOPHIC SOFT MATRICES In this section we defined different types of interval neutrosophic soft matrices and perform some algebraic operations on them. We also proposed an algorithm in decisionmaking problem for medical diagnosis.

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3.1. Definition





Let U  c1, c2 , c3 ,..., cm be a universal set and E be the set of parameters given by





E  e1, e2 , e3 ,..., en . Let A  E and  F , A  be an interval neutrosophic soft set over U

U

,where F is a mapping given by F:A  I ,where I U denotes the collection of all interval neutrosophic subsets of U . Then the interval neutrosophic soft set can be expressed in a matrix form as follows: Aˆ mn   aij    mn or Aˆ  aij  i  1, 2,......, m; j  1, 2,......., n



  aij   

 

jL



 ci  ,  jU  ci  ,  jL  ci  ,  jU  ci  ,  jL  ci  , jU  ci 

 [0,0],[1,1],[1,1] 



if

ej  A

if

ej  A

  jL  ci  ,  jU  ci  represents the truth-membership value of ci in the interval

 

neutrosophic set F e j

 jL  ci  ,  jU  ci  represents the indeterminacy-membership

 

value of ci in the interval neutrosophic set F ej and  jL  ci  ,  jU  ci  represents the

 

falsity-membership value of ci in the interval neutrosophic set F ej with the condition

 jU  ci    jU  ci    jU  ci   3 . Note that if  jU  ci  =  jL  ci  ,  jU  ci  =  jL  ci  and  jU  ci    jL  ci  then the interval neutrosophic soft matrix (IVNSM) is reduces to neutrosophic soft matrix (NSM). Now we shall discuss some important types of interval neutrosophic soft matrices and operations on them as given below:

3.2. Definition 





Let A  [aij ] IVNSM mn , B  [bij ] IVNSM mn , Then A is an interval neutrosophic 





soft sub matrix of B , denoted by A  B if  AL   BL ,  AU   BU ;  AL   BL ,  AU   BU and

 AL  BL ,  AU  BU for all i , j.

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3.3. Definition An interval neutrosophic soft matrix of order m  n is called an interval neutrosophic soft null(zero) matrix if all its elements are

 0, 0 , 0, 0 , 1,1  and it is denoted by  .

3.4. Definition An interval neutrosophic soft matrix of order m  n is called an interval neutrosophic soft universal matrix if all its elements are

 1,1 , 0, 0 , 0, 0  and it is denoted by U .

3.5. Definition 







Let A  [aij ] IVNSM mn , B  [bij ] IVNSM mn , Then A is equal to B , denoted by   A  B if

 AL   BL ,  AU   BU ,  AL   BL ,  AU   BU and  AL  BL ,  AU  BU for

all i and j.

3.6. Definition 

Let A  [aij ] IVNSM mn where aij





[  jL (ci ),  jU (ci )],[ jL (ci ),  jU (ci )],  jL (ci ), jU (ci )   

  jL  ci  ,  jU  ci  represents the truth-membership value of



ci in the interval

 

neutrosophic set F e j

 jL  ci  ,  jU  ci  represents the indeterminacy-membership value of ci in the

 

interval neutrosophic set F ej

and  jL  ci  ,  jU  ci  represents the falsity-

 

membership value of ci in the interval neutrosophic set F ej

 jU  ci    jU  ci    jU  ci   3 . Then 



transpose matrix of A if A T = [a ji ] .

with the condition

 T A is called the interval neutrosophic soft

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3.7. Definition 







If A  [aij ] IVNSM mn , B  [bij ] IVNSM mn , then the sum of A and B denoted by   A B

and

defined

as

  A B

=

[cij ]mn

 [max(  AL ,  BL ),max(  AU ,  BU )],[min( AL ,  BL ),min( AU ,  BU )]   [min( ,  ),min( ,  )]  AL BL AU BU

=

,

 For all i and j.  

3.8. Example Let us consider the two IVNSMs as  ([0.5, 0.8],[0.1, 0.2], 0.4, 0.5) ([0.6, 0.8],[0.1, 0.2], 0.3, 0.5)  A  ([0.4, 0.6],[0.1, 0.2], 0.4, 0.8) ([0.7, 0.9],[0.1, 0.3], 0.3, 0.6)

22

 ([0.6, 0.8],[0.2, 0.4], 0.3, 0.5) ([0.7, 0.8], [0.1, 0.2], 0.4, 0.5)  B  ([0.5, 0.6],[0.3, 0.5], 0.6, 0.8) ([0.6, 0.9], [0.1, 0.2], 0.4, 0.6) 

22

Then sum of these two is   ([0.6, 0.8],[0.1, 0.2], 0.3, 0.5) ([0.7, 0.8], [0.1, 0.2], 0.3, 0.5)  A B    ([0.5, 0.6],[0.1, 0.2], 0.4, 0.8) ([0.6, 0.9], [0.1, 0.2], 0.3, 0.6) 

22

3.9. Definition 







If A  [aij ] IVNSM mn , B  [bij ] IVNSM mn , then the difference of A and B  

denoted by A  B and defined as   A  B = [cij ]mn

 [min(  AL ,  BL ),min(  AU ,  BU )],[max( AL ,  BL ),max( AU ,  BU )]  [max( ,  ),max( ,  )]  AL BL AU BU



,

  

206

Somen Debnath For all i and j.

3.10. Example Let us consider the two IVNSMs as follows:  ([0.5, 0.8],[0.1, 0.2], 0.4, 0.5) ([0.6, 0.8], [0.1, 0.2], 0.3, 0.5)  A  ([0.4, 0.6],[0.1, 0.2], 0.4, 0.8) ([0.7, 0.9], [0.1, 0.3], 0.3, 0.6) 

22

 ([0.6, 0.8],[0.2, 0.4], 0.3, 0.5) ([0.7, 0.8],[0.1, 0.2], 0.4, 0.5)  B  ([0.5, 0.6],[0.3, 0.5], 0.6, 0.8) ([0.6, 0.9],[0.1, 0.2], 0.4, 0.6) 

22

 

Then difference of these two is A  B =

([0.5, 0.8],[0.2, 0.4], 0.4, 0.5) ([0.6, 0.8],[0.1, 0.2], 0.4, 0.5)  ([0.4, 0.6],[0.3, 0.5], 0.6, 0.8 ) ([0.6, 0.9],[0.1, 0.3], 0.4, 0.6 )       22 

3.11. Definition 







If A  [aij ] IVNSM mn , B  [bij ] IVNSM mn , then the product of A and B denoted  

by A * B and defined as   A * B = [cij ]m p =    

[max min(  AL ,  BL ),max min(  AU ,  BU )],[max min( AL ,  BL ),min max( AU ,  BU )], j j j j j j j j [min max( AL ,  BL ),min max( AU ,  BU )] j j j j

For all i and j.

3.12. Example Let us consider the two IVIFSMs as

   

An Application of Reduced Interval Neutrosophic Soft Matrix …

207

  0.6, 0.70.3, 0.60.2, 0.3 A   0.4, 0.50.5, 0.70.3, 0.4

0.5, 0.60.7, 0.90.1, 0.2 0.3, 0.40.3, 0.60.4, 0.5 0.4, 0.50.6, 0.70.2, 0.3  0.3, 0.40.4, 0.50.4, 0.5 0.6, 0.70.4, 0.60.2, 0.3 0.7, 0.80.2, 0.40.1, 0.2     0.5, 0.60.3, 0.50.1, 0.2 0.6, 0.70.7, 0.80.2, 0.3 0.2, 0.30.6, 0.70.5, 0.6 0.3, 0.40.5, 0.60.4, 0.5  34

0.5,0.60.4,0.80.3,0.4  0.4,0.50.1,0.20.2,0.3 B 0.7,0.80.5,0.80.1,0.2 0.3,0.50.5,0.80.4,0.5        

0.4,0.50.5,0.7 0.3,0.4   0.7,0.80.5,0.60.1,0.2   0.6,0.70.4,0.50.2,0.3   0.1,0.20.4,0.7 0.5,0.6  42

 

Then product of these two is A * B =

 0.5, 0.60.5, 0.80.2, 0.3   0.6, 0.70.4, 0.50.2, 0.3  0.5, 0.60.5, 0.80.2, 0.3  





0.5, 0.60.5, 0.70.1, 0.2   0.1, 0.70.5, 0.60.2, 0.3  0.6, 0.70.5, 0.70.2, 0.3 32



Clearly, A * B  B * A

3.13. Definition Let

 A  [aij ]  IVNSM mn

where

aij  ([  jL (ci ),  jU (ci )],[ jL (ci ),  jU (ci )],[ jL (ci ), jU (ci )])

with

 c  jU (ci )   jU (ci )   jU (ci )  3 .Then A is called an interval neutrosophic soft complement 

matrix if Ac  [bij ]mn where aij  ([ jL (ci ),  jU (ci )],[ jL (ci ),  jU (ci )],[ jL (ci ),  jU (ci )]) with

 jU (ci )   jU (ci )   jU (ci )  3 3.14. Example Let

 ([0.5, 0.8],[0.1, 0.2], 0.4, 0.5) ([0.6, 0.8],[0.1, 0.2], 0.3, 0.5)  A  ([0.4, 0.6],[0.1, 0.2], 0.4, 0.8) ([0.7, 0.9],[0.1, 0.3], 0.3, 0.6)

be 22

neutrosophic soft matrix then its complement is

  Ac =  ([0.4, 0.8],[0.1, 0.2], 0.4, 0.6)

([0.4, 0.5],[0.1, 0.2], 0.5, 0.8 ) ([0.3, 0.5], [0.1, 0.2], 0.6, 0.8) 







([0.3, 0.6], [0.1, 0.3], 0.7, 0.9) 

22

an

interval

208

Somen Debnath

3.15. Definition 

Let A  [aij ] IVNSM mn , 

 B  [bij ] IVNSM n p









and A1L , A1U ; B1L , B1U be the reduced



matrices of A and B respectively then the product of these reduced matrices defined as









AL * BL  [c ]m p = ij

and AU * BU = [cij ]m p =





max min(  AL ,  BL ), max min( AL ,  BL ),min max( AL ,  BL ) j j j j j j

max min(  AU ,  BU ), max min( AU ,  BU ),min max( AU ,  BU ) j j j j j j





For all i,j.

4. DECISION MAKING PROBLEM BY USING THE INTERVAL NEUTROSOPHIC SOFT SETS FOR MEDICAL DIAGNOSIS Here we construct an algorithm for determining score in decision-making problem for medical diagnosis and the steps of the algorithm are discussed as follows.

Algorithm Step 1 Input the interval neutrosophic soft sets (N,D) and (N, D)cover the set of symptoms where D is the set of diseases under consideration. We also write the interval neutrosophic soft matrices of (N, D) and(N, D)c corresponding to the soft medical knowledge 1 and 2 respectively by reassessing the symptoms.

Step 2 Input the interval neutrosophic soft sets (N1, E) over the set of patients where E is the set of symptoms and write its relation matrix R. Step 3 Compute the product

1  R * 1

and

2  R * 2

and write the reduced matrices

1   X1L , X1U  ,  2   X 2 L , X 2U  , 1  1L , 1U  ,  2   2 L ,  2U  and R   R1L , R1U           

An Application of Reduced Interval Neutrosophic Soft Matrix …

209

Step 4 Compute the product of the reduced matrices as follows: X1L  RL * 1L , X1U  RU * 1U ; and X 2 L  RL *  2 L , X 2U  RU *  2U

Let us consider the non-disease matrices X 3 L , X 3U , X 4 L and X 4U corresponding to

X 1L , X 1U , X 2 L and X 2U respectively such that

 

X 3L  RL * 1L

c



, X 3U  RU * 1U



c



; and X 4 L  RL * 2 L



c



, X 4U  RU * 2U



c

Step 5 Compute the diagnosis score matrices as follows:      DSM  A   1     

     p , d ,   p , d    , max min   X  pi , d j  ,  X  pi , d j   ,min   X   X        4L  2U  i j  3U  i j     1L  

     DSM  B   2     

   p , d   ,min    p , d ,   p , d    , max min   X  pi , d j  ,  X  X      X       4U  i j   2L  i j  3L  i j     1U  

     p , d  ,  p , d    , max min   X  pi , d j  ,  X  pi , d j   ,max   X  i j X         4L  3U  i j     1L   2U       p , d  ,  p , d    min max   X  pi , d j  ,  X  pi , d j   ,max   X   X        4L  2U  i j  3U  i j     1L  

   p , d   ,max    p , d ,  p , d    , max min   X  pi , d j  ,  X  X     i j X        4U  i j   3L  i j     1U   2L     p , d   ,max    p , d  ,  p , d    min max   X  pi , d j  ,  X  X      X       4U  i j   2L  i j  3L  i j     1U  

                     

Step 6 Find the total score of A and B separately by adding all three membership values of each entry. Step 7 Then we obtain the absolute difference value between total score of A and B. Step 8 Among all the entries, the entry which has the maximum absolute difference value along the patient will go for under treatment corresponding to the disease. Here, we assume any value which lies in the interval [0.3, 0.5] as a risk value. If there is a tie or no such value exists then we stop here and repeat the process by reassessing the symptoms for the patients.

210

Somen Debnath

5. APPLICATION OF INTERVAL NEUTROSOPHIC SOFT MATRICES IN MEDICAL DIAGNOSIS Suppose, in a private hospital there are three patients with symptoms vi-sion problem, excessive thirst, frequent night time urination and fatigue, and let the possible diseases relating to the above symptoms be hypertension(high blood pressure) and diabetes mellitus(type 1 or type 2) and they are under observation by a medical team of three doctors. Each doctor gives equal preference to all the three patients. Now we consider the set of symptoms as E= {ei: i=1,2,3,4}, where e1, e2, e3 ande4 represent the symptoms vision problem, excessive thirst, frequent night time urination and fatigue respectively, the set of diseases denoted by D={d1,d2}, where d1 and d2 represent parameterized hypertension and diabetes mellitus respectively. We also denote the set of patients by P= {p1, p2, p3} and the set of doctors be denoted by T= {t1,t2, t3}. The interval neutrosophic soft set (N,D) is a parameterized family {N(d1), N(d2)} of all interval neutrosophic sets over the set E are determined with the help of medical documentation and construct their matrices, called symptom-disease matrices and they arearedenoted disease matrices and they denotedby by Γ1 and Γ2 . The interval neutrosophic soft set (N1, E) is another parameterized family of all interval neutrosophic sets and gives an approximate description of patient-symptom in the hospital and construct a matrix R called the patient- symptom matrix. Then Then by combining R separately by combining R separatelywith with 1 and  2 ,we get two matrices X1 and X2, called the patient-disease and the patient-non disease matrices respectively.

Step 1 Suppose,

N d1 

 

 e1 ,   e3 ,

0.5, 0.60.4, 0.80.3, 0.4  , e2 , 0.4, 0.50.1, 0.20.2, 0.3  ,  0.7, 0.80.5, 0.80.1, 0.2  , e4 , 0.3, 0.50.5, 0.80.4, 0.5  

 

 e1 ,   e3 ,

0.4, 0.50.5, 0.70.3, 0.4  , e2 , 0.7, 0.80.5, 0.60.1, 0.2  ,  0.6, 0.70.4, 0.50.2, 0.3  , e4 , 0.1, 0.20.4, 0.70.5, 0.6  

N d2 

d1

e1   e2  Then, 1   e3  e4 

0.5,0.60.4,0.80.3,0.4 0.4,0.50.1,0.20.2,0.3 0.7,0.80.5,0.80.1,0.2 0.3,0.50.5,0.80.4,0.5

d2

0.4,0.50.5,0.70.3,0.4   0.7,0.80.5,0.60.1,0.2   0.6,0.70.4,0.50.2,0.3  0.1,0.20.4,0.70.5,0.6 

An Application of Reduced Interval Neutrosophic Soft Matrix …    c    1    2    

0.3,0.40.4,0.80.5,0.6 0.2,0.30.1,0.20.4,0.5 0.1,0.20.5,0.80.7,0.8 0.4,0.50.5,0.80.3,0.5

0.3,0.40.5,0.7 0.4,0.5  0.1,0.20.5,0.6 0.7,0.8  0.2,0.30.4,0.50.6,0.7  0.5,0.60.4,0.7 0.1,0.2 

Step 2 Again,

 p1 , 0.6, 0.70.3, 0.60.2, 0.3  ,  p2 , 0.4, 0.50.5, 0.70.3, 0.4  ,  p3 , 0.5, 0.60.3, 0.50.1, 0.2  N  e    p1 , 0.5, 0.60.7, 0.90.1, 0.2  ,  p2 , 0.3, 0.40.4, 0.50.4, 0.5  ,  p3 , 0.6, 0.70.7, 0.80.2, 0.3  N  e    p1 , 0.3, 0.40.3, 0.60.4, 0.5  ,  p2 , 0.6, 0.7 0.4, 0.60.2, 0.3  ,  p3 , 0.2, 0.30.6, 0.7 0.5, 0.6  N  e    p1 , 0.4, 0.50.6, 0.70.2, 0.3  ,  p2 , 0.7, 0.80.2, 0.40.1, 0.2  ,  p3 , 0.3, 0.40.5, 0.60.4, 0.5   

N1 e1  1

2

1

3

1

4

So,  0.6, 0.70.3, 0.60.2, 0.3  R   0.4, 0.50.5, 0.70.3, 0.4  0.5, 0.60.3, 0.50.1, 0.2 

0.5, 0.60.7, 0.90.1, 0.2 0.3, 0.40.4, 0.50.4, 0.5 0.6, 0.70.7, 0.80.2, 0.3

0.3, 0.40.3, 0.60.4, 0.5 0.6, 0.70.4, 0.60.2, 0.3 0.2, 0.30.6, 0.70.5, 0.6

0.4, 0.50.6, 0.70.2, 0.3   0.7, 0.80.2, 0.40.1, 0.2  0.3, 0.40.5, 0.60.4, 0.5 

Step 3 Now we compute the following product:

 0.5, 0.60.5, 0.80.2, 0.3  X1  R * 1   0.6, 0.70.4, 0.50.2, 0.3  0.5, 0.60.5, 0.80.2, 0.3 

0.5, 0.60.5, 0.70.1, 0.2   0.1, 0.70.5, 0.60.2, 0.3  0.6, 0.70.5, 0.70.2, 0.3 

 0.4, 0.50.5, 0.80.3, 0.5  X 2  R *  2   0.4, 0.50.4, 0.50.3, 0.5  0.3, 0.40.5, 0.80.4, 0.5 

0.5, 0.60.5, 0.60.1, 0.2   0.5, 0.60.5, 0.60.1, 0.2  0.3, 0.40.5, 0.70.4, 0.5 

Step 4 Now we consider the following reduced matrices

 0.5, 0.5, 0.2 X1L   0.6, 0.4, 0.2   0.5, 0.5, 0.2

 0.6, 0.8, 0.3  0.1, 0.5, 0.2 , X1U   0.7, 0.5, 0.3   0.6, 0.5, 0.2   0.6, 0.8, 0.3 0.5, 0.5, 0.1 

0.6, 0.7, 0.2 

0.7, 0.6, 0.3 



0.7, 0.7, 0.3 

211

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Somen Debnath

  1L     

  1U     

0.5, 0.4, 0.3 0.4, 0.1, 0.2 0.7, 0.5, 0.1 0.3, 0.5, 0.4

0.6, 0.8, 0.4 0.5, 0.2, 0.3 0.8, 0.8, 0.2 0.5, 0.8, 0.5

0.4, 0.5, 0.3 

 0.3, 0.4, 0.5   0.7, 0.5, 0.1  ,   c   0.2, 0.1, 0.4 1L 0.6, 0.4, 0.2   0.1, 0.5, 0.7   0.4, 0.5, 0.3 0.1, 0.4, 0.5   0.5, 0.7, 0.4 

 0.4, 0.8, 0.6  0.8, 0.6, 0.2 c        0.3, 0.2, 0.5 1U 0.7, 0.5, 0.3   0.2, 0.8, 0.8   0.5, 0.8, 0.5 0.2, 0.7, 0.6   ,

 0.6, 0.3, 0.2 RL   0.4, 0.5, 0.3   0.5, 0.3, 0.1

 0.7, 0.6, 0.3 RU   0.5, 0.7, 0.4   0.6, 0.5, 0.2

  2L         2U     

0.3, 0.4, 0.5 0.2, 0.1, 0.4 0.1, 0.5, 0.7 0.4, 0.5, 0.3

0.4, 0.8, 0.6 0.3, 0.2, 0.5 0.2, 0.8, 0.8 0.5, 0.8, 0.5

 0.4, 0.5, 0.3 X 2 L   0.4, 0.4, 0.3   0.3, 0.5, 0.4

0.3, 0.5, 0.4 

0.1, 0.5, 0.7 



0.2, 0.4, 0.6 

0.5, 0.4, 0.1 

0.4, 0.7, 0.5 

  0.3, 0.5, 0.7  0.6, 0.7, 0.2  0.2, 0.6, 0.8

0.4, 0.6, 0.2 

0.5, 0.7, 0.1

0.3, 0.3, 0.4

0.3, 0.4, 0.4

0.6, 0.4, 0.2

0.7, 0.2, 0.1 

0.6, 0.7, 0.2

0.2, 0.6, 0.5

0.3, 0.5, 0.4 

0.6, 0.9, 0.2

0.4, 0.6, 0.5

0.5, 0.7, 0.3 

0.4, 0.5, 0.5

0.7, 0.6, 0.3

0.8, 0.4, 0.2 

0.7, 0.8, 0.3

0.3, 0.7, 0.6

0.4, 0.6, 0.5 

0.3, 0.5, 0.4 

   ,   c   2L 0.2, 0.4, 0.6     0.5, 0.4, 0.1   0.1, 0.5, 0.7 



0.5, 0.4, 0.3

0.7, 0.5, 0.1 

0.7, 0.5, 0.1

0.6, 0.4, 0.2 

0.3, 0.5, 0.4

 0.6, 0.8, 0.4  c  0.5, 0.2, 0.3 0.2, 0.6, 0.8  ,   2U    0.3, 0.5, 0.7   0.8, 0.8, 0.2  0.5, 0.8, 0.5  0.6, 0.7, 0.2    0.5, 0.8, 0.5  0.5, 0.5, 0.1 , X 2U   0.5, 0.5, 0.5   0.3, 0.5, 0.1   0.4, 0.8, 0.5

Obtain the non-disease relation matrices:

0.4, 0.5, 0.3 

0.4, 0.1, 0.2

0.4, 0.7, 0.5 

0.5, 0.5, 0.1 





0.1, 0.4, 0.5 

0.5, 0.7, 0.4 

0.8, 0.6, 0.2 



0.7, 0.5, 0.3 

0.2, 0.7, 0.6  0.6, 0.6, 0.2  0.6, 0.6, 0.2 



0.4, 0.7, 0.5 

An Application of Reduced Interval Neutrosophic Soft Matrix …

















X 3L  RL * 1L

X 3U  RU * 1U

X 4 L  RL *  2 L

X 4U  RU *  2U

c

c

c

c

 0.4, 0.5, 0.3   0.4, 0.4, 0.3   0.3, 0.5, 0.4

0.4, 0.5, 0.2 

 0.5, 0.7, 0.5   0.5, 0.7, 0.5   0.4, 0.7, 0.5

0.5, 0.7, 0.3 

 0.5, 0.5, 0.2   0.6, 0.4, 0.2   0.5, 0.5, 0.2  0.6, 0.7, 0.3   0.7, 0.7, 0.3   0.6, 0.7, 0.3

  0.3, 0.5, 0.4  0.5, 0.5, 0.1

  0.4, 0.6, 0.5 

0.6, 0.7, 0.2

0.5, 0.5, 0.1 

  0.6, 0.5, 0.2 

0.6, 0.5, 0.2

0.6, 0.7, 0.2 

  0.7, 0.6, 0.3 

0.7, 0.7, 0.3

Step 5 Now we calculate the diagnosis score matrices as follows:

 0.5, 0.8, 0.2 DSM1  A  = 0.6, 0.4, 0.2   0.5, 0.8, 0.2

0.5, 0.7, 0.1 

0.6, 0.7, 0.2 



0.6, 0.7, 0.2 

 0.6, 0.7, 0.3

0.6, 0.7, 0.2 

  0.6, 0.7, 0.3

0.7, 0.6, 0.3 

DSM 2  B   0.7, 0.5, 0.3

0.7, 0.6, 0.1 



Step 6 Now we calculate the total score matrices of A and B

1.5 1.3   Total score (A) = 1.2 1.5  1.5 1.5 

213

214

Somen Debnath

1.6 1.5    Total score (B) = 1.5 1.4  1.6 1.5  Step 7 d1

d2

P1  0.1 0.2  Absolute difference value between A and B = P2 0.3 0.1   P3  0.1 0.0  Step 8 From the above matrix it has been observed that the value along P2 and d1 is 0.3 which is maximum among the other entries. So P2 has the highest risk and he/she will go for under treatment. Value along P3 and d 2 is 0.0. So we are sure about that P3 is not suffering from d 2 . For the other cases, to get the better results, we need to reassess their symptoms with expert consultations and repeat all the steps.

CONCLUSION Here we have introduced the concept of ivn-soft matrices and perform some algebraic operations on them.In this work, by using the notion of reduced ivn-soft matrices, an algorithm has been developed for decision-making problem. Finally, with the help of an example it has been shown that, this newly constructed algorithm has been applied successfully in decision-making for medical diagnosis problem. In future, instead of using soft set, there is a scope of using hypersoft set,pilthogenic set[36] to extend the concept used in this paper and apply it in various problems in real world. The main objective of introducing interval neutrosophic soft matrix is that, it is the most generalized (till date) form of fuzzy soft matrix and its extensions. Earlier many researchers have shown the applications of fuzzy soft matrices and its extensions in medical diagnosis problem. In this paper also we have shown an application in medical diagnosis by using interval neutrosophic soft matrix. Because if it is successfully applied by using the most generalized form, then it is automatically applicable for the earlier existing notions of matrices and we always put effort to attain the most generalized form.

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215

ACKNOWLEDGMENTS Author is thankful to the reviewers for their valuable suggestions.

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[8] [9]

[10]

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[12]

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[15] Gorzalczany, M. “A method of inference in approximate reasoning based on interval-valued fuzzy sets”, Fuzzy Sets and Systems,1987, 21, 1-17. [16] Jiang, Y; Tang, Y; Chen, Q; Liu, H; Tung, J. “Interval-valued intuitionistic fuzzy soft sets and their properties”, Computers and Mathematics with Applications”, 2010, 60, 906-918. [17] Kumar, M; Yadav, N. “Fuzzy rough sets and its application in data mining field”, Advances in Computer Science and Information Technology,2015, 2, 237-240. [18] Maji, PK; Biswas, R; Roy, AR. “Soft set theory”, Computers and Mathematics with Applications, 2003, 45, 555-562. [19] Maji, PK; Biswas, R; Roy, AR. “Fuzzy soft sets” Journal of Fuzzy Mathematics, 2001, 9, 589-602. [20] Maji, PK; Biswas, R;Roy, AR. “Intuitionistic fuzzy soft sets”, Journal of Fuzzy Mathematics, 2004, 12, 669-683. [21] Maji, PK. “Neutrosophic soft set”, Annals of Fuzzy Mathematics and Informatics, 2013, 5, 157-168. [22] Maji, PK. “A neutrosophic soft set approach to a decision making problem”, Annals of Fuzzy Mathematics and Informatics,2012, 3, 313-319. [23] Mitra Basu, T; Mahapatra, NK; Mondal, SK. “Matrices in fuzzy soft set theory and their applications in decision making problems”, South Asian Journal of Mathematics,2012, 2, 126-143. [24] Mitra Basu, T; Mondal, SK. “Neutrosophic soft matrix and its application in solving group decision making problems from medical science”, Computer Communication and Collaboration, 2015,3,1-31. [25] Molodtsov, D. “Soft set theory-first results”, Computers and Mathematics with Application, 1999, 37, 19-31. [26] Mondal, S; Pal, M. “Soft matrices”, African Journal of Mathematics and Computer Science Research, 2011, 4, 379-388. [27] Mukherjee, A; Debnath, S; datta, M. “An application of interval-valued intuitionistic fuzzy soft matrices in medical diagnosis”, Annals of Fuzzy Mathematics and Informatics,2016, 12,573-584. [28] Peng, X; Garg, H. “Algorithms for interval-valued fuzzy soft sets in emergency decision making based on WDBA and CODAS with new information measure”, Computers and Industrial Engineering, 2018, 119, 439-452. [29] Rajarajeswari, P; Dhanalakshmi, P. “Intuitionistic fuzzy soft matrix theory and its application in decision making”, International Journal of Engineering Research and Technology,2013, 2, 1100-1111. [30] Rajarajeswari, P; Dhanalakshmi, P. “Intuitionistic fuzzy soft matrix theory and its application in medical diagnosis”, Annals of Fuzzy Mathematics and Informatics, 2014, 7, 765-772.

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[31] Rajarajeswari, P; Dhanalakshmi, P. “Interval-valued fuzzy soft matrix”, Annals of pure and Applied Mathematics,2014, 7, 61-72. [32] Rajarajeswari, P; Dhanalakshmi, P. “Interval-valued intuitionistic fuzzy soft matrix theory”, International Journal of Mathematical Archive,2014, 5, 152-161. [33] Rani, D; Garg, H. “Some modified results of the subtraction and division operations on interval neutrosophic sets”, Journal of Experimental and Theoretical Artificial Intelligence, 2019, 31, 677-698. [34] Sahin, M; Alkhazaleh, S;Ulucay, V. “Neutrosophic soft expert sets”, Applied Mathematics,2015, 6, 116-127. [35] Smarandache, F. “Neutrosophic set-a generalization of the intuitionistic fuzzy sets”, International Journal of Pure and Applied Mathematics, 2005, 24, 287–297. [36] Smarandache, F. “Extension of soft set to hypersoft set, and then to pilthogenic hypersoft set”, Neutrosophic Sets and Systems, 2018, 22, 168-170. [37] Wang, H; Smarandache, F; Zhang, YQ; Sunderraman, R. “Interval neutrosophic sets and logic theory and applications in computing Hexis” Neutrosophic book series, No 5, 2005. [38] Wang, H; Smarandache, F; Zhang, YQ; Sunderraman, R. “Single valued neutrosophic sets”, Review of the Air Force Academy, 2010, 1, 10-14. [39] Yang, HL; Bao, YL; Gu, ZL. “Generalized interval neutrosophic rough sets and its application in multi-attribute decision making”, Filomat, 2018, 32, 11-33. [40] Yang, Y; Chenli, J. “Fuzzy soft matrices and their applications part I”, Lecture notes in Computer Science, 2011, 7002, 618-627. [41] Zadeh, LA. “Fuzzy set”, Information and Control, 1965, 8, 338-353.

In: Decision-Making with Neutrosophic Set Editor: Harish Garg

ISBN: 978-1-53619-419-7 c 2021 Nova Science Publishers, Inc.

Chapter 10

I NTERVAL -VALUED N EUTROSOPHIC N S OFT S ET AND I NTERTEMPORAL I NTERVAL -VALUED N EUTROSOPHIC N S OFT S ET TO A SSESS THE R ESILIENCE OF THE W ORKERS A MIDST COVID-19 1

V. Chinnadurai1,∗ and A. Bobin1,† Department of Mathematics, Annamalai University, Tamilnadu, India

Abstract We interpret the notions of an interval-valued neutrosophic N soft set (IV N N SS) and the quasi-hyperbolic discounting intertemporal interval-valued neutrosophic N soft set (QHDIIV N N SS). We have used these concepts to determine the resilience of the workers in an organization amidst the coronavirus (COV ID−19) pandemic. We apply the stages of denial, anger, bargaining, depression, and acceptance (DABDA) mentioned in the Kubler-Ross model (KRM ) to identify the resilience of the workers during this pandemic. We present the essential perspectives of this manuscript as below. i) to assess the stages of the workers by using IV N N SS, score function (SF ), the rating scale distribution, and risk-level norms. ii) a longitudinal model to assess the resilience and agility of the workers by using QHDIIV N N SS during the COV ID − 19 lockdown sessions. iii) a comparative study to show the significance of the QHDIIV N N SS model.

Keywords and phrases: interval-valued neutrosophic N soft set, intertemporal intervalvalued neutrosophic N soft set, Quasi-Hyperbolic discounting function

1.

INTRODUCTION

Decision-makers (DM s) implemented the concept of fuzzy set (F S) [1] and intuitionistic F S (IF S) [2] to deal with multi-criteria decision making (M CDM ) problems. Later, ∗ †

Email: [email protected], [email protected] (Corresponding author). Email: [email protected]

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V. Chinnadurai and A. Bobin

Molodtsov [3] proposed the concept of soft set theory to interpret solutions for many decision-making problems. The combination of the fuzzy hybrid set and the soft set resulted in many advantages for solving M CDM problems. Garg and Arora [4] presented the power aggregation operators to solve M CDM problems using intuitionistic fuzzy soft set (IF SS). Garg and Arora [5] defined the correlation measures by using the technique for order of preference by similarity to ideal solution for IF SS. Athira et al. [6] defined concepts on entropy measures by using Pythagorean fuzzy soft set to estimate the degree of fuzziness. Fathima et al. [7] defined spherical fuzzy soft sets, a generalization of the soft set, and established their properties. Smarandache [8] presented the concept of neutrosophic set (N S) to deal with indeterminacy. N S is a combination of truth, indeterminate, and falsity membership values. During uncertainty, the membership value of indeterminacy plays a vital role in ranking the alternatives. F S and IF S theories could not handle the indeterminacy part, but N S could manage it. The domination of neutrosophic concepts in various fields started from thereon. Wang et al. [9] introduced the notion of a single-valued neutrosophic set (SV N S) with a restriction to the membership values. Later, Wang et al. [10] presented the notion of an interval-valued neutrosophic set (IV N S) to provide the membership values in intervals. We highlight some of the neutrosophic works which aid in solving M CDM problems below. Deli [11] presented the concept of interval-valued neutrosophic soft set (IV N SS) by combining soft set with IV N S. Chinnadurai et al. [12] determined a unique highest score for each object in an M CDM problem by including additional criteria from the parameter set. Chinnadurai and Bobin [13] presented a study under the IV N S environment for ranking the alternatives in M CDM by using prospect theory. Broumi et al. [14] introduced the concept of interval-valued neutrosophic graph theory to deal with M CDM problems. Surapati and Partha [15] proposed an approach for solving multi-level linear programming problems in a neutrosophic environment through a goal programming strategy. Romualdas et al. [16] presented the concept of a complex proportional assessment method to determine accurate results in the engineering field. Rani and Garg [17] defined the subtraction and division operators in IV N S. Ridvan [18] established an M CDM method based on SF in which the values for the alternatives are in SV N S and IV N S. Nancy and Garg [19] proposed an improved SF for ranking the SV N S and IV N S by including indeterminacy value between the truth and false values. Garg and Nancy [20] introduced operators to solve the M CDM problems under SV N S environment. Garg and Nancy [21] developed a method to solve an M CDM problem under the linguistic SV N S environment. Garg [22] defined operational laws such as neutrality addition and scalar multiplication for the SV N S pairs. Also, defined sine-trigonometric operations laws [23] to solve the M CDM problems under SV N Ss environment. Garg and Nancy [24] presented averaging and geometric aggregation operators for the collection of the SV N S and IV N S. Garg and Nancy [25] proposed the concept of distance measures to find the difference between the SV N Ss. Bhimraj and Broumi [26] presented an interval-valued triangular neutrosophic number to solve the linear programming problem. Ajay et al. [27] introduced a neutrosophic cubic fuzzy geometric Bonferroni mean operator and discussed its properties. Broumi et al. [28] presented a trapezoidal IV N S of Bellman’s algorithm to solve the neutrosophic shortest path problem.

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Zadeh [1] introduced the concept of F S to deal with uncertainty. The prospects inside the human brain for learning, understanding, and describing are naturally imprecise. Therefore, the judgment that develops from the human brain also becomes uncertain. In the late 80s, amid controversies, F S gained creditability in psychology [29]. Rosch [30], Hersh and Caramazza [31], Rubin [32] and Oden [33] conducted experimental research using F S theory. Horowitz and Malle [34] examined depression by using the fuzzy concept. Alliger et al. [35] applied the concept of the fuzzy approach in decision-making problems of personnel assessment and selection. Nandita et al. [36] detailed the aspects of mental health and presented the details using soft computing and neuro-fuzzy techniques. Wang et al. [37] identified the various psychological dysfunctions in construction designs. They developed a fuzzy mapping to determine the influence of psychological disorders in the time’s context, cost, and quality of construction. Sanpreet [38] designed an expert system to aid the psychiatrists in assessing the mental health of the individuals. Nuovo et al. [39] implemented a method to classify the mental retardation level. These manuscripts bring out the significance of the F S and other hybrid sets in analyzing personality assessment, diagnosis of disorders, and counseling rather than using traditional set theory. Although the usage of the F S and other hybrid theory is clear, it is not widespread. The psychiatrists are used to analyze scaled data with statistical techniques. They are always in the mindset to follow the traditional method of handling scale construction and classical test-theory. These conventional concepts have forced the psychiatrists to use scale construction rather than F Ss and other hybrid sets. Also, most of the psychological study deals with questionnaires to study human behavior. In this process, we can never ignore the prejudice of the subject when the subjects express their thought process using a questionnaire. That is the reason when the information received by a questionnaire are imprecise since raw values include hidden risks. Neutrosophic logic acts as a vital tool to deal with uncertainty. The reason for introducing the neutrosophic concept in the study of resilience is that much of the data received by the questionnaire is vague. Using neutrosophic information instead of raw data has the advantage of reducing vagueness. Hence, there is a need to define a novel set that is user friendly for psychiatrists to assess the psychological behaviors of human beings. Fatimah et al.[40] introduced the notion of N -soft set (N SS) with real-life illustrations. Later, Akram et al. coined the definition of fuzzy N SS (F N SS) [41], hesitant N SS (HN SS) [42] and intuitionistic N SS (IN SS) [43]. Riaz et al. [44] detailed on neutrosophic N SS (N N SS) along with their properties. The limitations of these theories are highlighted in section 3. In 1968, Phelps and Pollak [45] introduced the notion of the quasi-hyperbolic discounting function (QHDF ). In 1997, David [46] coined the definition of QHDF to capture the qualitative properties. Later, Peter and Botond [47] extended the concept proposed by David to deal with discounting issues . Liu et al.[48] proposed an intertemporal hesitant fuzzy soft set and showed the significance of the set with M CDM problems. Alcantud and Torrecillas [49] introduced the intertemporal framework to fill the gap in the fuzzy soft set theory. Although the temporal logic plays a significant role in considering the immediate effect from different parameters and sessions, it is still open for research in neutrosophic theory.

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On studying these manuscripts, we present the following research scopes in a nutshell. i) a change in the environment is common, which affects the psychosocial aspects of humans. So, it is vital to analyze the resilience of the workers during this pandemic. ii) scope for intensifying neutrosophic theory in the psychology field. iii) standardized psychological tool has limitations, a simple rating scale distribution cannot determine the risk level. Upon analysis, it is evident that psychiatrists are comfortable in using raw data and rating scale criteria. So, a novel set that could handle both indeterminacy and the traditional method of assessing human behaviors is needed. Finally, the treatment process has many sessions (longitudinal study) to diagnose socially unacceptable behaviors. Hence, implementing intertemporal data is being preferred by the psychiatrists. The principal objectives of this manuscript are to overcome the mentioned research gap. i) to define IV N N SS, by combining SF value of IV N SS with N SS. ii) to define QHDIIV N N SS, a combination of intertemporal IV N N SS (IIV N N SS) with QHDF . This set enables us to record the intertemporal information and assess the risk level associated in each session with the help of QHDF . We define these two novel sets to bridge the gap and aid the psychiatrist. We organize this manuscript as below. In section 2, we recall some of the existing definitions. In Section 3, we present the concept of IV N N SS. Section 4 provides the method, algorithm, and flowchart to assess the workers. Section 5 illustrates a case study to show the resilience of the workers by using IV N N SS. Section 6 introduces the definition of QHDIIV N N SS. Section 7 provides the method, algorithm, and flowchart to assess the workers on a longitudinal study. Section 8 illustrates a case study by using QHDIIV N N SS. Section 9 shows the superiority of QHDIIV N N SS, and section 10 ends with limitations, conclusion, and future study notes.

2.

P RELIMINARIES

In this section, we discuss some essential definitions required for understanding this manuscript. Throughout this manuscript, let U denote a universal set, u ∈ U , P a set of parameters, E ⊆ P and 2U the power set of U , unless otherwise specified. Definition 2.1. [9] A single-valued neutrosophic set (SV N S) is of the form N = {u, hTN (u), IN (u), FN (u)i}, where TN (u) : U → [0, 1], IN (u) : U → [0, 1] and FN (u) : U → [0, 1] be the membership values of truth, indeterminacy and falsity respectively with a condition 0 ≤ TN (u) + IN (u) + FN (u) ≤ 3. Definition 2.2. [10] An interval-valued neutrosophic set (IV N S) is of the form I =  ˜ u, TI (u), I˜I (u), FI˜(u) , where TI˜(u) : U → D[0, 1], I˜I (u) : U → D[0, 1] and FI˜(u) : U → D[0, 1] be the interval-valued membership of truth, indeterminacy and falsity respectively. Here D[0, 1] represents the set of all closed sub-intervals of[0,1]. The lower  ˜(u), I˜ (u) and F ˜(u) are denoted respectively by T − (u), T +(u) , and upper limits of T I I I I I     − II (u), II+(u) , and FI− (u), FI+ (u) , where 0 ≤ TI+ (u) + II+ (u) + FI+ (u) ≤ 3. Let I U denote the collection of all IV N Ss over U . Definition 2.3. [3] A pair (F , E) is called a soft set (SS) over U , F is a mapping given by F : E → 2U . Thus SS is a parameterized family of subsets of U .

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Example 2.4. Let U ={s1 , s2 , s3 } be a set of subjects with psychosocial disorders and E={p1 , p2 , p3 } be the set of parameters which stand for insomnia, coping skills and cognitive impairment respectively. A SS (F , E) is a collection of subsets of U , based on the description given in Table 1. Table 1. Representation of subjects with psychosocial disorders in SS form U s1 s2 s3

insomnia(p1 ) 0 1 1

coping skills(p2) 1 1 0

cognitive impairment(p3 ) 1 0 1

F (insomnia subjects) = {s2 , s3 } , F (coping skills subjects) = {s1 , s2 } , F (cognitive impairment subjects) = {s1 , s3 } . Definition 2.5. [11] An interval-valued neutrosophic soft set (IV N SS) over U is defined as a pair (F , E), where F : E → I U . For any parameter p ∈ E, F (p) is an (p) can be represented as an IV N S such that I˜ = F (p) = n IV D N SS. Clearly, FEo u, TI˜˜(u), I˜I˜(u), FI˜˜(u) , where TI˜˜(u), I˜I˜(u) and FI˜˜(u) be the interval-valued membership of truth, indeterminacy and falsity respectively. Let U = {u1 , u2 , ..., um} be the universal set. Let E = {p1 , p2, ..., pn} be set of parameters. An IV N SS can be expressed in matrix form as, u1

I ∗ = [vij ] =

u2 . .. um

p1 2 v11 6 6 v21 6 6 . 4 . .

vm1

p2 v12

... ...

v22 . .. vm2

... .. . ...

pn v1n 3

7 v2n 7 7, . 7 .. 5 vmn

 where [vij ] = TI˜∗ , I˜I ∗ , FI˜∗ ; i = 1, 2, ..., m and j = 1, 2, ..., n. I ∗ is an m × n intervalvalued neutrosophic soft matrix (IV N SM ). Example 2.6. Let U and E be as in Example 2.4. An IV N SS (F , E) describes the subset of clients with psychosocial disorders in terms of interval-valued membership of truth, indeterminacy, and falsity as given in Table 2. The IV N SM for the Example 2.4 is given below:

∗

I =

s1 s2 s3

p1 h[0.50, 0.60], [0.20, 0.30], [0.40.50]i 4h[0.65, 0.75], [0.40, 0.50], [0.35, 0.45]i h[0.80, 0.90], [0.55, 0.65], [0.10, 0.20]i 2

p2 h[0.70, 0.80], [0.50, 0.60], [0.50, 0.60]i h[0.30, 0.40], [0.05, 0.15], [0.35, 0.45]i h[0.20, 0.30], [0.30, 0.40], [0.10, 20]i

p3 3 h[0.85, 0.90], [0.90, 1.00], [0.15, 0.25]i h[0.30, 0.40], [0.40, 0.50], [0.20, 0.30]i 5 h[0.45, 0.55], [0.15, 0.25], [0.55, 0.65]i

.

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Table 2. Representation of subjects with psychosocial disorders in IV N SS form (F , E) U s1 s2 s3

insomnia(p1 ) h[0.50, 0.60], [0.20, 0.30], [0.40.50]i h[0.65, 0.75], [0.40, 0.50], [0.35, 0.45]i h[0.80, 0.90], [0.55, 0.65], [0.10, 0.20]i

coping skills(p2 ) h[0.70, 0.80], [0.50, 0.60], [0.50, 0.60]i h[0.30, 0.40], [0.05, 0.15], [0.35, 0.45]i h[0.20, 0.30], [0.30, 0.40], [0.10, 20]i

cognitive impairment(p3 ) h[0.85, 0.90], [0.90, 1.00], [0.15, 0.25]i h[0.30, 0.40], [0.40, 0.50], [0.20, 0.30]i h[0.45, 0.55], [0.15, 0.25], [0.55, 0.65]i

Definition 2.7. [40] Let G = {0, 1, ..., N − 1} be a set of ordered grades, where N ∈ {2, 3, ...}. Then, (F , E, N ) is a N -soft set (N SS) on U if F : E −→ 2U×G with the condition that for each p ∈ E there exists a unique (u, gp) ∈ U × G, such that (u, gp) ∈ F (p), u ∈ U , gp ∈ G. Definition 2.8. [47] In a T-horizon game, the quasi-hyperbolic discounting function (QHDF ) for the period t’s is given as u(qt ) + β

T −t X

δ i u(qt+i ),

(1)

i=1

with β, δ ∈ [0, 1] and represent the short-term and long-term discounting parameters. 

Definition 2.9. [13] Let I ∗ = [vij ] = TI˜∗ , I˜I ∗ , FI˜∗ . Then define the score function (SF ) for the IV N SM as [T ˜∗ + I˜I ∗ ] − [FI˜∗ ] S(I ∗) = [wij ] = I . (2) 2 ˜ Similarly, the SF for the IV N SS can be denoted as S(I). Example 2.10. The SF for the Example 2.6 is computed below. S(I)∗ =

3.

p1 s1 "0.35 s2 0.75 s3 1.30

p2 0.75 0.05 0.45

p3 1.62# 0.55 . 0.10

INTERVAL -VALUED N EUTROSOPHIC N -SOFT SET AND M ATRIX

The implementation of fuzzy and its hybrid sets with N SS is evident from the above research works. But, there are some limitations with these combinations. i) cannot accommodate the membership value of indeterminacy in F N SS and HN SS. ii) restricts the DMs from providing the membership values in intervals. iii) We would like to cite Example 2.5 in Akram et al. [43] manuscript. They determine the grading criteria based on the membership values and discard the non-membership values in IF S. Similarly, in Example 5.1, Riaz et al. [44] decide on the grading criteria based on the truth membership values in SV N S and discard the indeterminacy and falsity membership values. By discarding, the non-membership values in IF S and the indeterminacy and falsity membership

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values in SV N S may restrict in analyzing the psychological aspects of human beings. In this section, we define the notions of interval-valued neutrosophic N -soft set (IV N N SS) and interval-valued neutrosophic N -soft matrix (IV N N SM ) with suitable examples to overcome the said research gap. Definition 3.1. Let U be the universal set and P be a set of parameters, E ⊆ P. Let G = {1, 2, ..., N } be a set of rating scales, where N ≥ 2. Then the triple (ψ, J , N ) is said to be an interval-valued neutrosophic N soft set (IV N N SS), where J = (F , E, N ) ˜ is a N SS over U and ψ maps every parameter in E with a SF value of IV N SS, S(I) over F (p) which is clearly a subset of U × G and p ∈ E. That is, for each parameter p D ∈ E, there exists E a unique (u, gp) ∈ U × G such that (u, gp) ∈ F (p), u ∈ U D, gp ∈ G Eand ˜ ∈ ψ(p). IV N N SS can be denoted as I(N ˜ ) = ψ(p)(u) = gp, S(I) ˜ . (u, gp), S(I) Definition 3.2. Let U = {u1 , u2 , ..., um} be the universal set. Let P = {p1 , p2 , ..., pn} be set of parameters and G = {1, 2, ..., N } be a set of rating scale. Then IV N N SS (ψ, J , N ) can be expressed in matrix form as u1

I ∗ (N ) =

u2 . .. um

p1 ¸ ˙ 2 gp11 , w11

6˙ ¸ 6 gp , w21 21 6 6 . 4 .. ¸ ˙ gpm1 , wm1

p2 ˙ ¸ gp12 , w12

¸ ˙ gp22 , w22 . .. ¸ ˙ gpm2 , wm2

... ... ... .. . ...

˙

pn ¸ gp1n , w1n 3

¸7 gp2n , w2n 7 7 7 . 5 .. ¸ ˙ gpmn , wmn ˙

 such that I ∗ (N ) = gpj , wij , i = 1, 2, ..., m and j = 1, 2, ..., n. Then I ∗ (N ) is called an m × n interval-valued neutrosophic N soft matrix (IV N N SM ) of the IV N N SS (ψ, J , N ). Example 3.3. Consider a scenario where a psychiatrist observes the behavior of workers to understand their resilience conditions. Let the psychiatrist provide the values in IV N N SS, as in Example 2.6. Consider a 5 point rating scale (5-soft set) for positive and negative statements with the rating scale distribution, as in Tables 4 and 5, respectively. The values in Tables 4 and 5 are flexible based on the psychological aspects of human behavior. The positive statements denote socially acceptable behavior, whereas the negative statements denote socially deviant or problematic behavior. Here, let’s assume that the psychiatrist constructs positive statements for the parameters p1 and p2 and for p3 , negative statement. Then, we compute the I ∗ (N ) for Example 2.10, as below.

I ∗ (5) =

4.

p1 s1 "h3, 0.35i s2 h2, 0.75i s3 h1, 1.30i

p2 h2, 0.75i h4, 0.05i h3, 0.45i

p3 h5, 1.62i # h3, 0.55i . h2, 0.10i

TO A SSESS THE R ESILIENCE OF WORKERS A MIDST COV ID − 19 U SING IV N N SM

COV ID − 19 has placed the world in a lockdown situation. Recently, EMA Partners has surveyed over four hundred organizations in twenty-five different countries to understand

226

V. Chinnadurai and A. Bobin Table 3. Shows the rating scale details Positive statement 1 2 3 4 5

Negative statement 5 4 3 2 1

Table 4. Shows the rating scale distribution Positive statement 1 2 3 4 5

Negative statement 5 4 3 2 1

Score values wij ≥ 0.8 0.6 ≤ wij < 0.8 0.3 ≤ wij < 0.6 0.0 ≤ wij < 0.3 wij < 0.0

the impact of COV ID − 19. The report suggests that there has been a negative impact on their business and requires human resources who can handle uncertainty, strategic reasoning, and resilience skills to cope with this pandemic. For the progress of the organization, it is vital to understand the resilience of the workers amidst this pandemic. To deal with this, we construct the concept of IV N N SM , which supports the resilience assessment of the individuals. In this section, we put forward a method to assess the stage of workers amidst COV ID − 19 with an algorithm and flowchart. We explain the feasibility and validity of the application with real-life case studies in the following section. Consider a scenario where an organization approaches the psychiatrist and would wish to assess the resilience of the workers amidst the pandemic, COV ID − 19. Let us assume that the psychiatrist selects a partially standardized method like video conferencing or telephonic conversations to assess the pandemic. Let U = {s1 , s2 , ..., sm} be the set of subjects and E = {p1 , p2 , ..., pn} be the set of parameters to assess the conditions. Let the psychiatrist frames the following details, namely; positive and negative statements for parameters, rating scales with distribution criteria as in Tables 3 and 4. The psychiatrist frames the standard norms as the maximum value among the five stages determines the present ability of the worker. Also, if the workers secure a top score in denial, anger, bargaining, and depression (DABD) stages, then they are under risk and require immediate attention or treatment. The psychiatrist evaluates the workers by considering the parameters (for each question) and presents the results in the IV N SSs form. Now, we have to assess the resilience of the workers with the help of scores and norms.

Applications to Assess the Resilience of Workers

4.1.

227

Methodology to Assess the Resilience of the Workers

Construct the IV N SSs, I˜r , r = {1, 2, ..., k} for each positive or negative statement by observing or understanding the behavior of the worker based on the parameters in tabular form. Apply SF Definition 2.9 to the IV N SSs and tabulate the resultant values in matrix form as S(Ir∗ ). Now compare the entries and construct the Ir∗ (N ) by using Definition ∗ (N ) 3.2 and with the help of the framed rating scale distribution (Table 4). Construct the I+ ∗ ∗ ∗ values by summing the corresponding entries of I1 (N ), I2 (N ), ..., Ik (N ). Finally, tabulate the details and assess the stages of each worker. If the workers have a maximum value in DABD stages, then it requires risk and requires immediate attention or treatment.

4.2.

Algorithm to Assess the Resilience of Workers

The following steps facilitate the psychiatrist to assess the resilience of workers: Step 1: Identify the problem, select the workers and the parameters. Step 2: Frame the required details namely; positive and negative statements, rating scale with distribution, scoring keys and norms. Step 3: Construct IV N SSs, I˜ , r = {1, 2, ..., k} for each question by observing the ber

havior of the workers. Step 4: Determine SF in matrix form, S(Ir∗) by using Definition 2.9. Step 5: Construct IV N N SM , Ir∗ (N ) by comparing it with rating scale distribution details. ∗

Step 6: Determine the total value, I+ (N ) by summing the corresponding entries of I1∗ (N ), I2∗(N ), ..., Ik∗(N ) and tabulate the values. Step 7: Assess the ability of the workers by using the norms (maximum value among the five stages). Continue the remedy process until the worker reaches the acceptance stage.

4.3.

Flowchart of IVNN SM

In this subsection, we depict the flow of the problem on how to assess the resilience of workers. We have shown below a step-by-step process to understand the nature and the complexity of the problem.

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V. Chinnadurai and A. Bobin

Psychiatrist

defines the problem

selects the workers

selects the parameters

frames the required details

positive and negative statements for parameters

rating scales and distribution criteria

scoring and level norms

forms IV N SSs for each question

computes SF using Definition 2.9

constructs Ir∗ (N )r0 and determines the ∗ (N ) 0 values I+ r

acceptance

constructs and predicts the risk level by using the norms

psychiatrist to start the recovery treatment

no

yes

treatment may not be required

5.

C ASE STUDY U SING IV N N SM

Let us assume an organization approaches a psychiatrist to assess the resilience of the workers. Step 1: Suppose that U = {w1 , w2, w3 , w4, w5 , } be the set of workers and P = {d1 , d2 , d3, d4 , d5, } be the set of parameters where d1 = denial, d2 = anger, d3 = bargaining, d4 = depression and d5 = acceptance. Step 2: The psychiatrist frames six positive questions for each parameter. Let’s assume a 5 point rating scale (5-soft set) for the statements as in Table 3, the rating scale distribution as in Table 4, and the norms for each parameter. Step 3: The psychiatrist observes the behavior of each worker records the values in IV N SS (I˜1 , I˜2 , ..., I˜6) form for each question as in Tables 5, 6, 7, 8, 9, and 10.

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Table 5. I˜1 shows the IV N SS data for the first question

w1 w2 w3 w4 w5

w1 w2 w3 w4 w5

d1 h[0.62, 0.65] , [0.45, 0.50] , [0.46, 0.50]i h[0.53, 0.55] , [0.35, 0.40] , [0.66, 0.70]i h[0.42, 0.50] , [0.31, 0.40] , [0.28, 0.30]i h[0.34, 0.40] , [0.21, 0.30] , [0.47, 0.50]i h[0.30, 0.35] , [0.35, 0.40] , [0.45, 0.50]i

d2 h[0.67, 0.70] , [0.25, 0.28] , [0.40, 0.44]i h[0.35, 0.42] , [0.35, 0.38] , [0.45, 0.54]i h[0.45, 0.54] , [0.45, 0.48] , [0.65, 0.68]i h[0.45, 0.60] , [0.35, 0.48] , [0.45, 0.46]i h[0.45, 0.55] , [0.33, 0.35] , [0.65, 0.70]i

d4 h[0.75, 0.81] , [0.35, 0.44] , [0.45, 0.48]i h[0.20, 0.25] , [0.45, 0.48] , [0.35, 0.38]i h[0.26, 0.31] , [0.25, 0.30] , [0.35, 0.40]i h[0.75, 0.81] , [0.35, 0.38] , [0.42, 0.48]i h[0.25, 0.28] , [0.25, 0.28] , [0.35, 0.42]i

d3 h[0.78, 0.80] , [0.32, 0.43] , [0.45, 0.51]i h[0.28, 0.30] , [0.28, 0.31] , [0.35, 0.41]i h[0.36, 0.40] , [0.22, 0.25] , [0.55, 0.61]i h[0.80, 0.85] , [0.32, 0.42] , [0.45, 0.48]i h[0.45, 0.55] , [0.42, 0.45] , [0.38, 0.40]i

d5 h[0.41, 0.50] , [0.41, 0.48] , [0.68, 0.72]i h[0.71, 0.85] , [0.31, 0.43] , [0.42, 0.49]i h[0.51, 0.56] , [0.33, 0.38] , [0.62, 0.68]i h[0.41, 0.45] , [0.45, 0.52] , [0.32, 0.43]i h[0.25, 0.28] , [0.25, 0.34] , [0.38, 0.48]i

Table 6. I˜2 shows the IV N SS data for the second question

w1 w2 w3 w4 w5

w1 w2 w3 w4 w5

d1 h[0.72, 0.70] , [0.41, 0.48] , [0.42, 0.52]i h[0.51, 0.57] , [0.31, 0.38] , [0.62, 0.67]i h[0.41, 0.52] , [0.35, 0.45] , [0.21, 0.27]i h[0.31, 0.42] , [0.25, 0.34] , [0.41, 0.47]i h[0.32, 0.42] , [0.31, 0.38] , [0.46, 0.52]i

d2 h[0.51, 0.60] , [0.26, 0.30] , [0.42, 0.47]i h[0.31, 0.40] , [0.34, 0.39] , [0.44, 0.56]i h[0.41, 0.50] , [0.44, 0.49] , [0.61, 0.67]i h[0.51, 0.61] , [0.32, 0.42] , [0.41, 0.45]i h[0.54, 0.62] , [0.31, 0.45] , [0.61, 0.65]i

d4 h[0.65, 0.70] , [0.30, 0.35] , [0.45, 0.50]i h[0.15, 0.20] , [0.40, 0.45] , [0.35, 0.40]i h[0.20, 0.30] , [0.20, 0.25] , [0.35, 0.45]i h[0.70, 0.80] , [0.30, 0.35] , [0.42, 0.50]i h[0.10, 0.20] , [0.20, 0.25] , [0.30, 0.35]i

d3 h[0.45, 0.54] , [0.30, 0.45] , [0.41, 0.51]i h[0.35, 0.47] , [0.25, 0.34] , [0.36, 0.41]i h[0.47, 0.55] , [0.44, 0.54] , [0.51, 0.65]i h[0.84, 0.87] , [0.35, 0.45] , [0.41, 0.48]i h[0.91, 0.94] , [0.45, 0.55] , [0.31, 0.35]i

d5 h[0.40, 0.50] , [0.41, 0.50] , [0.65, 0.70]i h[0.70, 0.80] , [0.31, 0.35] , [0.45, 0.50]i h[0.50, 0.60] , [0.33, 0.40] , [0.65, 0.70]i h[0.40, 0.45] , [0.45, 0.50] , [0.35, 0.50]i h[0.20, 0.30] , [0.25, 0.30] , [0.35, 0.45]i

Table 7. I˜3 shows the IV N SS data for the third question

w1 w2 w3 w4 w5

w1 w2 w3 w4 w5

d1 h[0.45, 0.50] , [0.70, 0.77] , [0.42, 0.45]i h[0.35, 0.40] , [0.60, 0.66] , [0.62, 0.65]i h[0.60, 0.65] , [0.56, 0.66] , [0.21, 0.25]i h[0.30, 0.35] , [0.45, 0.54] , [0.41, 0.45]i h[0.40, 0.45] , [0.71, 0.81] , [0.46, 0.52]i

d2 h[0.43, 0.55] , [0.20, 0.25] , [0.40, 0.50]i h[0.45, 0.50] , [0.30, 0.35] , [0.35, 0.40]i h[0.54, 0.64] , [0.40, 0.50] , [0.55, 0.60]i h[0.44, 0.54] , [0.30, 0.40] , [0.45, 0.50]i h[0.54, 0.64] , [0.25, 0.30] , [0.65, 0.70]i

d4 h[0.25, 0.35] , [0.35, 0.40] , [0.45, 0.50]i h[0.45, 0.55] , [0.35, 0.45] , [0.25, 0.35]i h[0.75, 0.85] , [0.25, 0.40] , [0.38, 0.40]i h[0.55, 0.65] , [0.35, 0.40] , [0.45, 0.50]i h[0.25, 0.30] , [0.25, 0.30] , [0.35, 0.50]i

d3 h[0.40, 0.45] , [0.30, 0.35] , [0.40, 0.45]i h[0.25, 0.30] , [0.20, 0.32] , [0.30, 0.45]i h[0.45, 0.50] , [0.40, 0.54] , [0.50, 0.55]i h[0.85, 0.90] , [0.32, 0.45] , [0.40, 0.45]i h[0.90, 0.95] , [0.42, 0.52] , [0.30, 0.35]i

d5 h[0.65, 0.75] , [0.45, 0.50] , [0.65, 0.70]i h[0.75, 0.85] , [0.35, 0.40] , [0.45, 0.50]i h[0.79, 0.82] , [0.15, 0.30] , [0.10, 0.12]i h[0.50, 0.55] , [0.45, 0.50] , [0.35, 0.40]i h[0.55, 0.60] , [0.25, 0.40] , [0.38, 0.40]i

230 w1 w2 w3 w4 w5

V. Chinnadurai and A. Bobin d4 h[0.65, 0.70] , [0.30, 0.35] , [0.45, 0.50]i h[0.15, 0.20] , [0.40, 0.45] , [0.35, 0.40]i h[0.20, 0.30] , [0.20, 0.25] , [0.35, 0.45]i h[0.70, 0.80] , [0.30, 0.35] , [0.42, 0.50]i h[0.10, 0.20] , [0.20, 0.25] , [0.30, 0.35]i

d5 h[0.45, 0.50] , [0.35, 0.45] , [0.65, 0.75]i h[0.65, 0.70] , [0.45, 0.50] , [0.45, 0.55]i h[0.55, 0.60] , [0.55, 0.60] , [0.65, 0.75]i h[0.45, 0.50] , [0.65, 0.70] , [0.35, 0.55]i h[0.25, 0.30] , [0.75, 0.80] , [0.35, 0.50]i

Table 8. I˜4 shows the IV N SS data for the fourth question

w1 w2 w3 w4 w5

w1 w2 w3 w4 w5

d1 h[0.15, 0.25] , [0.95, 1.00] , [0.40, 0.45]i h[0.25, 0.35] , [0.85, 0.90] , [0.45, 0.50]i h[0.35, 0.45] , [0.75, 0.80] , [0.50, 0.55]i h[0.45, 0.55] , [0.65, 0.70] , [0.55, 0.60]i h[0.55, 0.65] , [0.55, 0.60] , [0.60, 0.65]i

d2 h[0.55, 0.57] , [0.25, 0.38] , [0.40, 0.60]i h[0.45, 0.47] , [0.35, 0.48] , [0.45, 0.55]i h[0.35, 0.37] , [0.45, 0.58] , [0.65, 0.75]i h[0.65, 0.67] , [0.35, 0.48] , [0.45, 0.65]i h[0.75, 0.77] , [0.35, 0.55] , [0.65, 0.75]i

d4 h[0.65, 0.85] , [0.35, 0.48] , [0.45, 0.56]i h[0.35, 0.55] , [0.45, 0.58] , [0.35, 0.46]i h[0.15, 0.30] , [0.25, 0.38] , [0.38, 0.48]i h[0.75, 0.85] , [0.35, 0.45] , [0.45, 0.58]i h[0.38, 0.40] , [0.25, 0.35] , [0.35, 0.48]i

d3 h[0.55, 0.60] , [0.35, 0.45] , [0.40, 0.45]i h[0.35, 0.40] , [0.25, 0.35] , [0.30, 0.35]i h[0.45, 0.55] , [0.25, 0.30] , [0.50, 0.65]i h[0.40, 0.45] , [0.35, 0.45] , [0.40, 0.55]i h[0.25, 0.35] , [0.45, 0.55] , [0.30, 0.45]i

d5 h[0.44, 0.50] , [0.44, 0.50] , [0.65, 0.75]i h[0.73, 0.74] , [0.34, 0.45] , [0.45, 0.55]i h[0.53, 0.58] , [0.35, 0.40] , [0.65, 0.75]i h[0.24, 0.45] , [0.42, 0.45] , [0.35, 0.45]i h[0.28, 0.30] , [0.21, 0.25] , [0.38, 0.45]i

Table 9. I˜5 shows the IV N SS data for the fifth question

w1 w2 w3 w4 w5

w1 w2 w3 w4 w5

d1 h[0.41, 0.62] , [0.35, 0.45] , [0.45, 0.50]i h[0.51, 0.63] , [0.33, 0.55] , [0.55, 0.70]i h[0.43, 0.53] , [0.38, 0.48] , [0.20, 0.30]i h[0.61, 0.72] , [0.25, 0.35] , [0.35, 0.50]i h[0.31, 0.42] , [0.35, 0.45] , [0.40, 0.45]i

d2 h[0.55, 0.70] , [0.25, 0.36] , [0.45, 0.50]i h[0.45, 0.57] , [0.35, 0.46] , [0.34, 0.50]i h[0.55, 0.67] , [0.45, 0.56] , [0.54, 0.70]i h[0.60, 0.65] , [0.35, 0.66] , [0.43, 0.50]i h[0.50, 0.60] , [0.33, 0.46] , [0.54, 0.70]i

d4 h[0.55, 0.65] , [0.30, 0.35] , [0.45, 0.55]i h[0.45, 0.55] , [0.40, 0.45] , [0.35, 0.45]i h[0.67, 0.77] , [0.20, 0.30] , [0.38, 0.46]i h[0.78, 0.88] , [0.30, 0.35] , [0.45, 0.56]i h[0.90, 0.92] , [0.20, 0.25] , [0.35, 0.44]i

d3 h[0.72, 0.80] , [0.38, 0.40] , [0.45, 0.48]i h[0.22, 0.30] , [0.28, 0.30] , [0.35, 0.48]i h[0.32, 0.40] , [0.20, 0.30] , [0.55, 0.68]i h[0.70, 0.85] , [0.32, 0.40] , [0.45, 0.58]i h[0.24, 0.55] , [0.44, 0.50] , [0.35, 0.44]i

d5 h[0.45, 0.66] , [0.45, 0.55] , [0.52, 0.70]i h[0.75, 0.82] , [0.35, 0.55] , [0.42, 0.50]i h[0.55, 0.62] , [0.33, 0.45] , [0.62, 0.70]i h[0.20, 0.23] , [0.45, 0.52] , [0.32, 0.40]i h[0.20, 0.33] , [0.25, 0.32] , [0.42, 0.52]i

Step 4: Apply SF Definition 2.9, and represent the details in S(I1∗), S(I2∗), S(I3∗), S(I4∗), S(I5∗) and S(I6∗ ) form.

Applications to Assess the Resilience of Workers

231

Table 10. I˜6 shows the IV N SS data for the sixth question

w1 w2 w3 w4 w5

d1 h[0.85, 0.90] , [0.35, 0.50] , [0.45, 0.50]i h[0.55, 0.60] , [0.33, 0.40] , [0.65, 0.70]i h[0.75, 0.85] , [0.38, 0.40] , [0.25, 0.30]i h[0.65, 0.70] , [0.25, 0.30] , [0.40, 0.50]i h[0.80, 0.85] , [0.35, 0.40] , [0.45, 0.50]i

S(I1∗ ) =

w1 w2 w3 w4 w5

S(I2∗ ) =

w1 w2 w3 w4 w5

S(I3∗ ) =

w1 w2 w3 w4 w5

S(I4∗ ) =

w1 w2 w3 w4 w5

S(I5∗) =

S(I6∗ ) =

w1 w2 w3 w4 w5

w1 w2 w3 w4 w5

d2 h[0.65, 0.70] , [0.25, 0.30] , [0.40, 0.50]i h[0.80, 0.84] , [0.12, 0.15] , [0.10, 0.15]i h[0.25, 0.35] , [0.55, 0.65] , [0.65, 0.70]i h[0.45, 0.65] , [0.35, 0.50] , [0.45, 0.50]i h[0.38, 0.48] , [0.33, 0.40] , [0.25, 0.30]i

d3 h[0.75, 0.80] , [0.35, 0.40] , [0.45, 0.50]i h[0.25, 0.30] , [0.25, 0.30] , [0.35, 0.50]i h[0.81, 0.85] , [0.15, 0.30] , [0.10, 0.12]i h[0.75, 0.85] , [0.35, 0.40] , [0.45, 0.50]i h[0.20, 0.55] , [0.45, 0.50] , [0.35, 0.40]i

d1 20.630

d2 0.530 0.255 0.295 0.485 0.165

d3 0.685 0.205 0.035 0.730 0.545

d4 0.710 0.325 0.185 0.695 0.145

d5 0.2003 0.6957 0.2407 7 0.5405 0.130

d1 20.685

d2 0.390 0.220 0.280 0.500 0.330

d3 0.410 0.320 0.420 0.810 1.095

d4 0.525 0.225 0.075 0.615 0.050

d5 0.2303 0.6057 0.2407 7 0.4755 0.125

d1 20.775

d2 0.265 0.425 0.465 0.365 0.190

d3 0.325 0.160 0.420 0.835 1.070

d4 0.525 0.225 0.075 0.615 0.050

d5 0.1753 0.6507 0.4507 7 0.7005 0.625

d1 20.750

d2 0.375 0.375 0.175 0.525 0.510

d3 0.550 0.350 0.200 0.350 0.425

d4 0.660 0.560 0.110 0.685 0.275

d5 0.2403 0.6307 0.2307 7 0.3805 0.105

60.235 60.525 6 40.140 0.225

60.240 60.625 6 40.220 0.225

60.370 61.005 6 40.390 0.695

60.700 60.650 6 40.600 0.550 d1 20.440

60.385 60.660 6 40.540 0.340 d1 20.825

60.265 60.915 6 40.500 0.725

d2 0.455 0.495 0.495 0.665 0.325

d2 0.500 0.830 0.225 0.500 0.520

d3 0.685 0.135 −0.005 0.620 0.470

d3 0.675 0.125 0.945 0.700 0.475

d4 0.425 0.525 0.550 0.650 0.740

d4 0.200 0.600 0.735 0.500 0.125

d5 0.4453 0.7757 0.3157 7 0.3405 0.080 d5 0.5003 0.7007 0.9207 7 0.6255 0.510

Step 5: By using Definition 3.2, compare the score values with rating scale distribution (Table 4) and construct the following matrices I1∗ (5), I2∗ (5), I3∗ (5), I4∗ (5), I5∗ (5) and I6∗ (5).

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I1∗ (5) =

w1 w2 w3 w4 w5

I2∗ (5) =

w1 w2 w3 w4 w5

I3∗ (5) =

w1 w2 w3 w4 w5

I4∗ (5) =

w1 w2 w3 w4 w5

I5∗ (5) =

I6∗ (5) =

w1 w2 w3 w4 w5

w1 w2 w3 w4 w5

d1 2h2, 0.630i

d2 h3, 0.530i h4, 0.255i h4, 0.295i h3, 0.485i h4, 0.165i

d3 h2, 0.685i h4, 0.205i h4, 0.035i h2, 0.730i h3, 0.545i

d4 h2, 0.710i h3, 0.325i h4, 0.185i h2, 0.695i h4, 0.145i

d5 h4, 0.200i 3 h2, 0.695i 7 h4, 0.240i 7 7 h3, 0.540i 5 h4, 0.130i

d1 2h2, 0.685i

d2 h3, 0.390i h4, 0.220i h4, 0.280i h3, 0.500i h3, 0.330i

d3 h3, 0.410i h3, 0.320i h3, 0.420i h1, 0.810i h1, 1.095i

d4 h3, 0.525i h4, 0.225i h4, 0.075i h2, 0.615i h4, 0.050i

d5 h4, 0.230i 3 h2, 0.605i 7 h4, 0.240i 7 7 h3, 0.475i 5

d1 2h2, 0.775i

d2 h4, 0.265i h3, 0.425i h3, 0.465i h3, 0.365i h4, 0.190i

d3 h3, 0.325i h4, 0.160i h3, 0.420i h1, 0.835i h1, 1.070i

d4 h3, 0.525i h4, 0.225i h4, 0.075i h2, 0.615i h4, 0.050i

d5 h4, 0.175i 3 h2, 0.650i 7 h3, 0.450i 7 7 h2, 0.700i 5 h2, 0.625i

d1 2h2, 0.750i

d2 h3, 0.375i h3, 0.375i h4, 0.175i h3, 0.525i h3, 0.510i

d3 h3, 0.550i h3, 0.350i h4, 0.200i h3, 0.350i h3, 0.425i

d4 h2, 0.660i h3, 0.560i h4, 0.110i h2, 0.685i h4, 0.275i

d5 h4, 0.240i 3 h2, 0.630i 7 h4, 0.230i 7 7 h3, 0.380i 5 h4, 0.105i

6h4, 0.235i 6h3, 0.525i 6 4h4, 0.140i h4, 0.225i

6h4, 0.240i 6h2, 0.625i 6 4h4, 0.220i h4, 0.225i

6h3, 0.370i 6h1, 1.005i 6 4h3, 0.390i h2, 0.695i

6h2, 0.700i 6h2, 0.650i 6 4h2, 0.600i h3, 0.550i d1 2h3, 0.440i 6h3, 0.385i 6h2, 0.660i 6 4h3, 0.540i h3, 0.340i d1 2h1, 0.825i

6h4, 0.265i 6h1, 0.915i 6 4h3, 0.500i h2, 0.725i

d2 h3, 0.455i h3, 0.495i h3, 0.495i h2, 0.665i h3, 0.325i d2 h3, 0.500i h1, 0.830i h4, 0.225i h3, 0.500i h3, 0.520i

d3 h2, 0.685i h4, 0.135i h5, −0.005i h2, 0.620i h3, 0.470i d3 h2, 0.675i h4, 0.125i h1, 0.945i h2, 0.700i h3, 0.475i

d4 h3, 0.425i h3, 0.525i h3, 0.550i h2, 0.650i h2, 0.740i d4 h4, 0.200i h2, 0.600i h2, 0.735i h3, 0.500i h4, 0.125i

h4, 0.125i

d5 h3, 0.445i 3 h2, 0.775i 7 h3, 0.315i 7 7 h3, 0.340i 5 h4, 0.080i

d5 h3, 0.500i 3 h2, 0.700i 7 h1, 0.920i 7 7 h2, 0.625i 5 h3, 0.510i

∗ (5), by summing the corresponding entries of I ∗ (5), I ∗(5), ..., I ∗(5) Step 6: Determine I+ 1 2 6 values as in Table 11.

∗ Table 11. I+ (5) shows the consolidated data for all the questions

w1 w2 w3 w4 w5

d1 12 20 11 19 18

d2 19 18 22 17 20

d3 15 22 20 11 14

d4 17 19 21 13 22

d5 22 12 19 16 21

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Applications to Assess the Resilience of Workers

Analysis: From Table 11, we observe that worker, w1 is in the acceptance stage, with a maximum score value of 22. Since w1 is in the acceptance stage, the worker may not require the help of the psychiatrist. Similarly, the worker, w2 is in the bargaining stage with a maximum score of 22, w3 is in the anger stage with a maximum score of 22, w4 is in the denial stage, with a maximum score of 19, and w5 is in the depression stage, with a maximum score value of 22. The psychiatrist may continue the therapeutic process for the workers, w2 , w3 , w4 , and w5 who are yet to show resilience.

6.

INTERTEMPORAL INTERVAL -VALUED N EUTROSOPHIC N -SOFT SET

Definition 6.1. An intertemporal interval-valued neutrosophic N -soft set (IIV N N SS) is  l represented as a finite sequence of IV N N SS over U , and denoted by (ψ s , J s , N ) s=k 0 for a session k, l ∈ N such that (k ≤ k ≤ l).  l Definition 6.2. Let (ψ s, J s , N ) s=k for a session k, l ∈ N be an IIV N N SS, then the quasi-hyperbolic discounting intertemporal interval-valued neutrosophic N -soft set (QHDIIV N N SS) computed from  s s l 0 0 (ψ , J , N ) s=k at session k , (k ≤ k ≤ l) is defined as, 0

˜ ) 0 = ψ k (p)(u) = I(N k

*

gp ,



1 ˜ 0 +β S(I) k l − k0 + 1

0 l−k X

˜ 0 δ s .S(I) k +s

s=1

  +  ,

(3)

where δ ∈ [0, 1] and β ∈ [0, 1) are the long-term and short-term discounting parameters ˜ 0 and S(I) ˜ 0 are the SF s of IV N SS for the session k0 and k0 + s. respectively. S(I) k k +s Definition 6.3. Let U = {u1 , u2 , ..., um} be the universal set. Let P = {p1 , p2 , ..., pn} be set of parameters and G = {1, 2, ..., N } be a set of rating scale. Then the  l 0 0 QHDIIV N N SS computed from (ψ s, J s , N ) s=k at session k , (k ≤ k ≤ l) is defined as, u1

I ∗ (N )k0 = [oij ] =

u2 . .. um

p1 2 o11 6 6 o21 6 6 . 4 . .

om1

p2 o12

... ...

o22 .. . om2

... . .. ...

pn o1n 3

7 o2n 7 7, . 7 .. 5 omn

such that I ∗ (N )k0 = [oij ] =

*

gpij ,



1 (wij ) 0 + β 0 k l−k +1

0 l−k X

s=1

δ s .(wij )k0 +s

  +  ,

(4)

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i = 1, 2, ..., m and j = 1, 2, ..., n. Then I ∗ (N )k0 is called an m × n, quasi-hyperbolic discounting intertemporal interval-valued neutrosophic N -soft matrix (QHDIIV N N SM )  l of the QHDIIV N N SS (ψ s, J s , N ) s=k .

7.

A N A PPLICATION TO A SSESS THE R ESILIENCE OF WORKERS U SING QHDIIV N N SM

Consider a scenario where the psychiatrist has intended for a longitudinal study. The study comprises l counseling sessions to examine the resilience of the workers in an organization. 0 0 The therapist would like to assess the workers after k sessions, i.e., (k < k < l) to check the progression of the workers. This scheme of evaluation eventually helps the psychiatrist to understand the level of progress of each worker. Hence, the notion of QHDIIV N N SM plays a significant role in assessing the resilience of the workers. We present the method, an algorithm, and a flowchart to study the process involved in a longitudinal problem. Let U = {w1 , w2, ..., wm} be the set of workers and E = {d1 , d2 , ..., dn} be the set of parameters. Let the psychiatrist frames the following details, namely; positive and negative statements for parameters, rating scales with distribution criteria as in Tables 3 and 4. The psychiatrist frames the standard norms as the maximum value among the five stages determines the longitudinal ability of the worker. The psychiatrist evaluates the workers and 0 present the results in IV N SSs form, (I˜r )k0 , where r = {1, 2, ..., h} for each session, k . Now, we have to assess the resilience of the worker with the help of scores and norms.

7.1.

Methodology to Assess the Resilience of the Workers

Construct the IV N SSs, (I˜r )k0 , r = {1, 2, ..., h} for positive or negative statements by observing the behavior of the worker in each session. Apply SF Definition 2.9, to the IV N SSs and represent the resultant values as S(Ir∗)k0 . Compare the entries in each S(Ir∗)k0 and determine the Ir∗ (N )k0 by using Definition 6.3. The values of δ = 0.9, β = 0.5 can be used to compare the values with the rating scale distribution (Table 4). Determine ∗ (N ) 0 data by adding the corresponding entries of I ∗ (N ) 0 , I ∗ (N ) 0 , ..., I ∗(N ) 0 the I+ 1 2 h k k k k values. Now assess the risk level for each parameter by using the level norms. During the evaluation stage, if the worker shows the progress, then the worker has responded to the treatment. The therapist can continue the remedy process. If otherwise, then the psychiatrist should incite an alternate remedy process for the workers to facilitate them to reach the acceptance level.

7.2.

Flowchart of QHDIIVNN SM Method

In this subsection, we depict the flow of the problem to assess the resilience of workers. A structured process is shown below to understand the nature of the problem.

Applications to Assess the Resilience of Workers

235

Psychiatrist

defines the problem

selects the workers

selects the parameters

frames the required details

positive and negative statements for parameters

rating scales and distribution criteria

scoring and level norms

forms IV N SSs for each session (longitudinal study)

computes SF using Definition 2.9

constructs Ir∗ (N )r0 and determines the ∗ (N ) 0 values I+ r

progress

constructs to assess the stages by using the norms

change the remedy process for unsatisfactory workers from next session

no

yes

continue the remedy process

7.3.

Algorithm to Assess the Resilience among the Workers

The following steps provide an insight to assess the resilience among the workers. Step 1: Identify the problem, select the workers and the parameters. Step 2: Frame the required details namely; positive and negative statements, rating scale with distribution, scoring keys and risk level. Step 3: Construct (I˜r )k0 , where r = {1, 2, ..., h} for each question by observing the 0 behavior of the workers at k session. Step 4: Evaluate SF by using Definition 2.9. Step 5: Construct Ir∗ (N )k0 for each session by using Definition 6.3 and compare it with rating scale distribution details. ∗ (N ) 0 value by summing the corresponding entries of Step 6: Determine I+ k I1 (N )k0 , I2 (N )k0 , ..., Ih(N )k0 values. Step 7: Tabulate and assess the resilience by using risk level norms. Step 8: If in case the worker is yet to show the dominance towards the acceptance stage, then the psychiatrist should end the current remedy process and start an alternative treatment from the forthcoming session.

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8.

V. Chinnadurai and A. Bobin

C ASE STUDY U SING QHDIIV N N SM

In a longitudinal study, the psychiatrist requires multiple sessions to assess the resilience of the workers. Now, when there are varied sessions, it may be a challenge for the psychiatrist to assess the workers. To overcome this gap, we present a method to assess the workers with the data recorded in each session. Consider a scenario where the psychiatrist observes the behavior of the workers and records the information using IV N SS for every lockdown session during the pandemic. Let’s assume the workers are at the beginning of the fourth lockdown session, and the psychiatrist would like to assess the workers and determine the DABDA levels associated with them during the previous lockdown session. Step 1: Suppose that U = {w1 , w2, w3 , w4, w5 } be the set of workers and P = {d1 , d2 , d3, d4 , d5} be the set of parameters, where d1 = denial, d2 = anger, d3 = bargaining, d4 = depression and d5 = acceptance. Step 2: Consider the psychiatrist frames six positive questions for each parameter across the three lockdown sessions. Let the rating scale and distribution be as in Table 4. Step 3: Psychiatrist observes the behavior of each worker and provides the data in IV N SSs, (I˜1 )1 , (I˜2 )1 , (I˜3 )1 , (I˜4 )1 , (I˜5 )1 and (I˜6 )1 as in Tables 12, 13, 14, 15, 16 and 17 for the first lockdown session. Table 12. (I˜1 )1 shows the IV N SS data for the first question

w1 w2 w3 w4 w5

w1 w2 w3 w4 w5

d1 h[0.65, 0.75] , [0.35, 0.40] , [0.45, 0.50]i h[0.83, 0.88] , [0.35, 0.40] , [0.45, 0.50]i h[0.45, 0.50] , [0.38, 0.40] , [0.25, 0.30]i h[0.35, 0.55] , [0.40, 0.50] , [0.30, 0.40]i h[0.25, 0.45] , [0.35, 0.50] , [0.35, 0.40]i

d2 h[0.25, 0.55] , [0.45, 0.50] , [0.35, 0.40]i h[0.90, 0.92] , [0.35, 0.40] , [0.45, 0.50]i h[0.45, 0.50] , [0.45, 0.50] , [0.65, 0.70]i h[0.73, 0.83] , [0.35, 0.40] , [0.45, 0.50]i h[0.30, 0.55] , [0.45, 0.50] , [0.35, 0.40]i

d4 h[0.80, 0.85] , [0.35, 0.40] , [0.45, 0.50]i h[0.20, 0.55] , [0.45, 0.50] , [0.35, 0.40]i h[0.20, 0.30] , [0.25, 0.40] , [0.38, 0.40]i h[0.79, 0.83] , [0.34, 0.40] , [0.45, 0.50]i h[0.25, 0.30] , [0.25, 0.30] , [0.35, 0.50]i

d3 h[0.75, 0.80] , [0.35, 0.40] , [0.45, 0.50]i h[0.25, 0.30] , [0.25, 0.30] , [0.35, 0.50]i h[0.70, 0.80] , [0.35, 0.40] , [0.45, 0.50]i h[0.75, 0.85] , [0.35, 0.40] , [0.45, 0.50]i h[0.20, 0.55] , [0.45, 0.50] , [0.35, 0.40]i

d5 h[0.45, 0.65] , [0.35, 0.50] , [0.45, 0.50]i h[0.35, 0.40] , [0.35, 0.40] , [0.45, 0.50]i h[0.55, 0.60] , [0.33, 0.40] , [0.65, 0.70]i h[0.90, 0.95] , [0.35, 0.40] , [0.45, 0.50]i h[0.20, 0.30] , [0.25, 0.40] , [0.38, 0.40]i

Table 13. (I˜2 )1 shows the IV N SS data for the second question

w1 w2 w3 w4 w5

d1 h[0.75, 0.80] , [0.35, 0.40] , [0.45, 0.50]i h[0.41, 0.57] , [0.21, 0.38] , [0.52, 0.67]i h[0.31, 0.52] , [0.25, 0.45] , [0.11, 0.27]i h[0.71, 0.81] , [0.25, 0.45] , [0.31, 0.48]i h[0.22, 0.42] , [0.21, 0.38] , [0.36, 0.52]i

d2 h[0.80, 0.90] , [0.35, 0.55] , [0.21, 0.35]i h[0.80, 0.85] , [0.35, 0.40] , [0.45, 0.50]i h[0.10, 0.30] , [0.10, 0.25] , [0.25, 0.45]i h[0.76, 0.80] , [0.25, 0.45] , [0.31, 0.48]i h[0.71, 0.94] , [0.35, 0.55] , [0.21, 0.35]i

d3 h[0.35, 0.54] , [0.20, 0.45] , [0.31, 0.51]i h[0.60, 0.80] , [0.21, 0.35] , [0.35, 0.50]i h[0.37, 0.54] , [0.20, 0.45] , [0.31, 0.51]i h[0.74, 0.87] , [0.25, 0.45] , [0.31, 0.48]i h[0.81, 0.94] , [0.35, 0.55] , [0.21, 0.35]i

Applications to Assess the Resilience of Workers w1 w2 w3 w4 w5

d4 h[0.55, 0.70] , [0.20, 0.35] , [0.35, 0.50]i h[0.10, 0.20] , [0.30, 0.45] , [0.25, 0.40]i h[0.31, 0.50] , [0.35, 0.49] , [0.51, 0.67]i h[0.82, 0.87] , [0.25, 0.45] , [0.31, 0.48]i h[0.05, 0.20] , [0.10, 0.25] , [0.20, 0.35]i

237

d5 h[0.62, 0.70] , [0.31, 0.48] , [0.32, 0.52]i h[0.21, 0.40] , [0.24, 0.39] , [0.34, 0.56]i h[0.40, 0.60] , [0.23, 0.40] , [0.55, 0.70]i h[0.31, 0.80] , [0.20, 0.35] , [0.32, 0.50]i h[0.10, 0.30] , [0.15, 0.30] , [0.25, 0.45]i

Table 14. (I˜3 )1 shows the IV N SS data for the third question

w1 w2 w3 w4 w5

w1 w2 w3 w4 w5

d1 h[0.80, 0.85] , [0.35, 0.40] , [0.45, 0.50]i h[0.25, 0.40] , [0.50, 0.66] , [0.52, 0.65]i h[0.50, 0.65] , [0.48, 0.66] , [0.15, 0.25]i h[0.65, 0.75] , [0.22, 0.45] , [0.30, 0.45]i h[0.30, 0.45] , [0.61, 0.81] , [0.40, 0.52]i

d2 h[0.75, 0.85] , [0.32, 0.52] , [0.20, 0.35]i h[0.20, 0.30] , [0.10, 0.32] , [0.35, 0.45]i h[0.10, 0.30] , [0.10, 0.25] , [0.30, 0.45]i h[0.70, 0.80] , [0.22, 0.45] , [0.30, 0.45]i h[0.90, 0.95] , [0.32, 0.52] , [0.20, 0.35]i

d4 h[0.55, 0.70] , [0.20, 0.35] , [0.35, 0.50]i h[0.10, 0.20] , [0.30, 0.45] , [0.30, 0.40]i h[0.44, 0.64] , [0.30, 0.50] , [0.45, 0.60]i h[0.70, 0.75] , [0.22, 0.45] , [0.30, 0.45]i h[0.08, 0.20] , [0.10, 0.25] , [0.20, 0.35]i

d3 h[0.30, 0.45] , [0.20, 0.35] , [0.30, 0.45]i h[0.55, 0.70] , [0.35, 0.50] , [0.35, 0.55]i h[0.35, 0.45] , [0.20, 0.35] , [0.30, 0.45]i h[0.75, 0.90] , [0.22, 0.45] , [0.30, 0.45]i h[0.80, 0.95] , [0.32, 0.52] , [0.20, 0.35]i

d5 h[0.35, 0.50] , [0.60, 0.77] , [0.32, 0.45]i h[0.35, 0.50] , [0.20, 0.35] , [0.25, 0.40]i h[0.45, 0.60] , [0.45, 0.60] , [0.55, 0.75]i h[0.60, 0.80] , [0.20, 0.35] , [0.32, 0.50]i h[0.15, 0.30] , [0.65, 0.80] , [0.25, 0.50]i

Table 15. (I˜4 )1 shows the IV N SS data for the fourth question

w1 w2 w3 w4 w5

w1 w2 w3 w4 w5

d1 h[0.72, 0.85] , [0.35, 0.40] , [0.45, 0.50]i h[0.15, 0.35] , [0.75, 0.90] , [0.35, 0.50]i h[0.25, 0.45] , [0.65, 0.80] , [0.40, 0.55]i h[0.68, 0.78] , [0.38, 0.45] , [0.52, 0.58]i h[0.45, 0.65] , [0.45, 0.60] , [0.50, 0.65]i

d2 h[0.15, 0.30] , [0.35, 0.55] , [0.40, 0.45]i h[0.25, 0.40] , [0.20, 0.35] , [0.25, 0.35]i h[0.25, 0.30] , [0.32, 0.38] , [0.31, 0.48]i h[0.70, 0.80] , [0.38, 0.45] , [0.52, 0.58]i h[0.25, 0.35] , [0.35, 0.55] , [0.40, 0.45]i

d4 h[0.55, 0.85] , [0.40, 0.48] , [0.50, 0.56]i h[0.30, 0.55] , [0.50, 0.58] , [0.42, 0.46]i h[0.25, 0.37] , [0.48, 0.58] , [0.62, 0.75]i h[0.55, 0.55] , [0.60, 0.70] , [0.45, 0.60]i h[0.35, 0.40] , [0.30, 0.35] , [0.42, 0.48]i

d3 h[0.50, 0.60] , [0.30, 0.45] , [0.35, 0.45]i h[0.70, 0.74] , [0.25, 0.45] , [0.40, 0.55]i h[0.40, 0.60] , [0.30, 0.45] , [0.35, 0.45]i h[0.35, 0.45] , [0.25, 0.45] , [0.35, 0.55]i h[0.20, 0.35] , [0.35, 0.55] , [0.40, 0.45]i

d5 h[0.10, 0.20] , [0.85, 1.00] , [0.30, 0.45]i h[0.35, 0.47] , [0.40, 0.48] , [0.35, 0.55]i h[0.45, 0.58] , [0.30, 0.40] , [0.55, 0.75]i h[0.70, 0.85] , [0.38, 0.45] , [0.52, 0.58]i h[0.25, 0.30] , [0.10, 0.25] , [0.35, 0.45]i

238

V. Chinnadurai and A. Bobin Table 16. (I˜5 )1 shows the IV N SS data for the fifth question

w1 w2 w3 w4 w5

w1 w2 w3 w4 w5

d1 h[0.32, 0.66] , [0.40, 0.55] , [0.50, 0.70]i h[0.41, 0.63] , [0.23, 0.55] , [0.45, 0.70]i h[0.33, 0.53] , [0.28, 0.48] , [0.10, 0.30]i h[0.65, 0.75] , [0.25, 0.35] , [0.40, 0.56]i h[0.21, 0.42] , [0.25, 0.45] , [0.30, 0.45]i

d2 h[0.45, 0.55] , [0.40, 0.50] , [0.30, 0.44]i h[0.20, 0.30] , [0.20, 0.30] , [0.30, 0.48]i h[0.53, 0.77] , [0.10, 0.30] , [0.28, 0.46]i h[0.75, 0.85] , [0.25, 0.35] , [0.40, 0.56]i h[0.44, 0.55] , [0.40, 0.50] , [0.30, 0.44]i

d4 h[0.50, 0.65] , [0.25, 0.35] , [0.40, 0.55]i h[0.40, 0.55] , [0.35, 0.45] , [0.30, 0.45]i h[0.50, 0.67] , [0.40, 0.56] , [0.50, 0.70]i h[0.60, 0.72] , [0.28, 0.35] , [0.20, 0.50]i h[0.80, 0.92] , [0.10, 0.25] , [0.30, 0.44]i

d3 h[0.62, 0.80] , [0.28, 0.40] , [0.40, 0.48]i h[0.65, 0.82] , [0.25, 0.55] , [0.30, 0.50]i h[0.60, 0.80] , [0.28, 0.40] , [0.40, 0.48]i h[0.60, 0.85] , [0.35, 0.40] , [0.40, 0.58]i h[0.34, 0.55] , [0.40, 0.50] , [0.30, 0.44]i

d5 h[0.31, 0.62] , [0.25, 0.45] , [0.35, 0.50]i h[0.35, 0.57] , [0.25, 0.46] , [0.24, 0.50]i h[0.50, 0.62] , [0.30, 0.45] , [0.60, 0.70]i h[0.70, 0.88] , [0.25, 0.35] , [0.40, 0.56]i h[0.10, 0.33] , [0.23, 0.32] , [0.40, 0.52]i

Table 17. (I˜6 )1 shows the IV N SS data for the sixth question

w1 w2 w3 w4 w5

w1 w2 w3 w4 w5

d1 h[0.60, 0.75] , [0.40, 0.50] , [0.60, 0.70]i h[0.50, 0.60] , [0.35, 0.40] , [0.60, 0.70]i h[0.70, 0.85] , [0.24, 0.40] , [0.20, 0.30]i h[0.65, 0.75] , [0.30, 0.40] , [0.40, 0.50]i h[0.60, 0.85] , [0.30, 0.40] , [0.40, 0.50]i

d2 h[0.45, 0.55] , [0.40, 0.50] , [0.30, 0.40]i h[0.20, 0.30] , [0.20, 0.30] , [0.30, 0.50]i h[0.60, 0.85] , [0.15, 0.40] , [0.30, 0.40]i h[0.65, 0.80] , [0.30, 0.40] , [0.40, 0.50]i h[0.15, 0.30] , [0.20, 0.30] , [0.40, 0.50]i

d4 h[0.20, 0.35] , [0.30, 0.40] , [0.40, 0.50]i h[0.40, 0.55] , [0.20, 0.45] , [0.20, 0.35]i h[0.30, 0.35] , [0.50, 0.65] , [0.60, 0.70]i h[0.60, 0.70] , [0.20, 0.30] , [0.35, 0.50]i h[0.10, 0.30] , [0.20, 0.30] , [0.40, 0.50]i

d3 h[0.70, 0.80] , [0.30, 0.40] , [0.40, 0.50]i h[0.60, 0.85] , [0.25, 0.40] , [0.35, 0.50]i h[0.75, 0.80] , [0.30, 0.40] , [0.40, 0.50]i h[0.65, 0.85] , [0.30, 0.40] , [0.40, 0.50]i h[0.20, 0.55] , [0.40, 0.50] , [0.30, 0.40]i

d5 h[0.80, 0.90] , [0.30, 0.50] , [0.40, 0.50]i h[0.70, 0.84] , [0.10, 0.15] , [0.05, 0.15]i h[0.70, 0.82] , [0.10, 0.30] , [0.05, 0.12]i h[0.50, 0.65] , [0.30, 0.40] , [0.40, 0.50]i h[0.50, 0.60] , [0.35, 0.40] , [0.30, 0.40]i

Table 18. (I˜1 )2 shows the IV N SS data for the first question

w1 w2 w3 w4 w5

d1 h[0.52, 0.65] , [0.35, 0.50] , [0.36, 0.50]i h[0.43, 0.55] , [0.30, 0.40] , [0.56, 0.70]i h[0.32, 0.50] , [0.21, 0.40] , [0.18, 0.30]i h[0.31, 0.45] , [0.45, 0.52] , [0.22, 0.43]i h[0.20, 0.35] , [0.30, 0.40] , [0.35, 0.50]i

d2 h[0.67, 0.77] , [0.25, 0.38] , [0.40, 0.54]i h[0.35, 0.52] , [0.35, 0.48] , [0.45, 0.64]i h[0.26, 0.42] , [0.25, 0.40] , [0.35, 0.50]i h[0.45, 0.70] , [0.35, 0.58] , [0.45, 0.56]i h[0.45, 0.65] , [0.33, 0.45] , [0.65, 0.80]i

d3 h[0.68, 0.80] , [0.32, 0.53] , [0.45, 0.61]i h[0.61, 0.85] , [0.31, 0.50] , [0.42, 0.59]i h[0.26, 0.40] , [0.22, 0.35] , [0.55, 0.69]i h[0.75, 0.85] , [0.32, 0.53] , [0.45, 0.58]i h[0.40, 0.55] , [0.42, 0.55] , [0.38, 0.50]i

Applications to Assess the Resilience of Workers w1 w2 w3 w4 w5

d4 h[0.80, 0.81] , [0.35, 0.44] , [0.40, 0.48]i h[0.15, 0.25] , [0.40, 0.48] , [0.30, 0.38]i h[0.50, 0.54] , [0.40, 0.48] , [0.60, 0.68]i h[0.79, 0.81] , [0.30, 0.38] , [0.40, 0.48]i h[0.20, 0.28] , [0.20, 0.28] , [0.30, 0.42]i

239

d5 h[0.31, 0.50] , [0.31, 0.48] , [0.68, 0.82]i h[0.21, 0.30] , [0.28, 0.41] , [0.35, 0.42]i h[0.41, 0.56] , [0.33, 0.48] , [0.62, 0.78]i h[0.30, 0.40] , [0.21, 0.40] , [0.47, 0.60]i h[0.20, 0.28] , [0.25, 0.44] , [0.38, 0.59]i

Table 19. (I˜2 )2 shows the IV N SS data for the second question

w1 w2 w3 w4 w5

w1 w2 w3 w4 w5

d1 h[0.62, 0.75] , [0.31, 0.50] , [0.32, 0.60]i h[0.41, 0.60] , [0.21, 0.40] , [0.52, 0.70]i h[0.31, 0.55] , [0.25, 0.55] , [0.11, 0.30]i h[0.30, 0.50] , [0.35, 0.60] , [0.25, 0.45]i h[0.22, 0.45] , [0.21, 0.45] , [0.36, 0.50]i

d2 h[0.41, 0.65] , [0.16, 0.35] , [0.32, 0.50]i h[0.21, 0.45] , [0.24, 0.45] , [0.34, 0.60]i h[0.10, 0.35] , [0.10, 0.30] , [0.25, 0.55]i h[0.41, 0.65] , [0.22, 0.45] , [0.31, 0.65]i h[0.44, 0.72] , [0.21, 0.55] , [0.51, 0.60]i

d4 h[0.55, 0.60] , [0.20, 0.40] , [0.35, 0.60]i h[0.10, 0.30] , [0.30, 0.50] , [0.25, 0.50]i h[0.31, 0.40] , [0.35, 0.67] , [0.51, 0.65]i h[0.60, 0.70] , [0.20, 0.45] , [0.32, 0.40]i h[0.05, 0.15] , [0.10, 0.35] , [0.20, 0.30]i

d3 h[0.35, 0.45] , [0.20, 0.40] , [0.31, 0.41]i h[0.60, 0.70] , [0.21, 0.30] , [0.35, 0.45]i h[0.37, 0.45] , [0.33, 0.50] , [0.41, 0.55]i h[0.74, 0.80] , [0.25, 0.40] , [0.31, 0.45]i h[0.81, 0.90] , [0.35, 0.50] , [0.21, 0.45]i

d5 h[0.30, 0.40] , [0.31, 0.40] , [0.55, 0.80]i h[0.25, 0.37] , [0.15, 0.44] , [0.26, 0.51]i h[0.40, 0.50] , [0.23, 0.50] , [0.55, 0.60]i h[0.21, 0.32] , [0.15, 0.44] , [0.31, 0.50]i h[0.10, 0.20] , [0.15, 0.40] , [0.25, 0.55]i

Table 20. (I˜3 )2 shows the IV N SS data for the third question

w1 w2 w3 w4 w5

w1 w2 w3 w4 w5

d1 h[0.35, 0.55] , [0.60, 0.80] , [0.32, 0.50]i h[0.25, 0.45] , [0.50, 0.70] , [0.52, 0.70]i h[0.50, 0.70] , [0.48, 0.75] , [0.15, 0.30]i h[0.35, 0.45] , [0.55, 0.65] , [0.30, 0.60]i h[0.30, 0.50] , [0.61, 0.75] , [0.40, 0.50]i

d2 h[0.33, 0.60] , [0.10, 0.35] , [0.30, 0.55]i h[0.35, 0.65] , [0.20, 0.45] , [0.25, 0.45]i h[0.10, 0.35] , [0.10, 0.35] , [0.30, 0.55]i h[0.34, 0.64] , [0.25, 0.45] , [0.40, 0.65]i h[0.44, 0.70] , [0.15, 0.35] , [0.55, 0.75]i

d4 h[0.55, 0.65] , [0.20, 0.40] , [0.35, 0.45]i h[0.10, 0.35] , [0.30, 0.60] , [0.30, 0.35]i h[0.44, 0.70] , [0.30, 0.40] , [0.45, 0.55]i h[0.60, 0.75] , [0.20, 0.45] , [0.32, 0.45]i h[0.08, 0.25] , [0.10, 0.35] , [0.20, 0.40]i

d3 h[0.30, 0.55] , [0.20, 0.30] , [0.30, 0.50]i h[0.55, 0.65] , [0.35, 0.45] , [0.35, 0.48]i h[0.35, 0.55] , [0.30, 0.65] , [0.40, 0.52]i h[0.75, 0.82] , [0.22, 0.50] , [0.30, 0.42]i h[0.80, 0.85] , [0.32, 0.45] , [0.20, 0.37]i

d5 h[0.35, 0.45] , [0.25, 0.55] , [0.55, 0.65]i h[0.20, 0.35] , [0.10, 0.42] , [0.35, 0.50]i h[0.45, 0.55] , [0.45, 0.55] , [0.55, 0.65]i h[0.20, 0.45] , [0.40, 0.45] , [0.31, 0.55]i h[0.15, 0.35] , [0.65, 0.70] , [0.25, 0.45]i

240

V. Chinnadurai and A. Bobin Table 21. (I˜4 )2 shows the IV N SS data for the fourth question

w1 w2 w3 w4 w5

w1 w2 w3 w4 w5

d1 h[0.10, 0.30] , [0.85, 0.90] , [0.30, 0.40]i h[0.15, 0.40] , [0.75, 0.80] , [0.35, 0.45]i h[0.25, 0.50] , [0.65, 0.85] , [0.40, 0.60]i h[0.35, 0.55] , [0.32, 0.50] , [0.25, 0.50]i h[0.45, 0.60] , [0.45, 0.50] , [0.50, 0.60]i

d2 h[0.45, 0.70] , [0.30, 0.40] , [0.45, 0.65]i h[0.35, 0.50] , [0.40, 0.50] , [0.35, 0.70]i h[0.25, 0.40] , [0.32, 0.45] , [0.31, 0.50]i h[0.55, 0.80] , [0.41, 0.55] , [0.40, 0.60]i h[0.65, 0.72] , [0.42, 0.60] , [0.60, 0.80]i

d4 h[0.55, 0.80] , [0.40, 0.50] , [0.50, 0.64]i h[0.30, 0.50] , [0.50, 0.60] , [0.42, 0.50]i h[0.25, 0.30] , [0.48, 0.65] , [0.62, 0.80]i h[0.70, 0.80] , [0.38, 0.45] , [0.52, 0.60]i h[0.35, 0.45] , [0.30, 0.53] , [0.42, 0.50]i

d3 h[0.50, 0.68] , [0.30, 0.48] , [0.35, 0.58]i h[0.70, 0.78] , [0.25, 0.58] , [0.40, 0.48]i h[0.40, 0.58] , [0.30, 0.38] , [0.45, 0.62]i h[0.35, 0.48] , [0.25, 0.48] , [0.35, 0.52]i h[0.20, 0.38] , [0.35, 0.58] , [0.40, 0.52]i

d5 h[0.45, 0.60] , [0.40, 0.45] , [0.60, 0.65]i h[0.25, 0.45] , [0.20, 0.30] , [0.25, 0.40]i h[0.45, 0.55] , [0.30, 0.35] , [0.55, 0.70]i h[0.55, 0.65] , [0.60, 0.65] , [0.45, 0.55]i h[0.25, 0.35] , [0.10, 0.15] , [0.35, 0.40]i

Table 22. (I˜5 )2 shows the IV N SS data for the fifth question

w1 w2 w3 w4 w5

w1 w2 w3 w4 w5

d1 h[0.31, 0.60] , [0.25, 0.40] , [0.35, 0.55]i h[0.41, 0.65] , [0.23, 0.50] , [0.45, 0.60]i h[0.33, 0.55] , [0.28, 0.45] , [0.10, 0.35]i h[0.10, 0.25] , [0.35, 0.55] , [0.22, 0.45]i h[0.21, 0.45] , [0.25, 0.48] , [0.30, 0.50]i

d2 h[0.45, 0.65] , [0.10, 0.25] , [0.35, 0.40]i h[0.35, 0.55] , [0.25, 0.40] , [0.24, 0.30]i h[0.53, 0.75] , [0.10, 0.20] , [0.28, 0.45]i h[0.50, 0.60] , [0.30, 0.50] , [0.33, 0.55]i h[0.40, 0.55] , [0.23, 0.42] , [0.44, 0.49]i

d4 h[0.50, 0.55] , [0.25, 0.30] , [0.40, 0.50]i h[0.40, 0.45] , [0.35, 0.40] , [0.30, 0.40]i h[0.50, 0.55] , [0.40, 0.50] , [0.50, 0.60]i h[0.70, 0.75] , [0.25, 0.45] , [0.40, 0.55]i h[0.80, 0.85] , [0.10, 0.35] , [0.30, 0.45]i

d3 h[0.62, 0.70] , [0.28, 0.38] , [0.40, 0.50]i h[0.65, 0.80] , [0.25, 0.46] , [0.30, 0.45]i h[0.22, 0.30] , [0.23, 0.34] , [0.50, 0.65]i h[0.60, 0.65] , [0.35, 0.43] , [0.40, 0.50]i h[0.34, 0.45] , [0.40, 0.55] , [0.30, 0.35]i

d5 h[0.32, 0.65] , [0.40, 0.60] , [0.50, 0.65]i h[0.20, 0.35] , [0.20, 0.40] , [0.30, 0.45]i h[0.50, 0.60] , [0.30, 0.50] , [0.60, 0.65]i h[0.60, 0.70] , [0.28, 0.45] , [0.20, 0.60]i h[0.10, 0.40] , [0.23, 0.30] , [0.40, 0.50]i

Table 23. (I˜6 )2 shows the IV N SS data for the sixth question

w1 w2 w3 w4 w5

d1 h[0.80, 0.85] , [0.30, 0.40] , [0.40, 0.55]i h[0.50, 0.65] , [0.35, 0.45] , [0.60, 0.65]i h[0.70, 0.80] , [0.24, 0.35] , [0.20, 0.35]i h[0.40, 0.60] , [0.40, 0.45] , [0.30, 0.45]i h[0.60, 0.80] , [0.30, 0.50] , [0.40, 0.55]i

d2 h[0.60, 0.75] , [0.20, 0.35] , [0.35, 0.45]i h[0.70, 0.85] , [0.10, 0.25] , [0.05, 0.20]i h[0.60, 0.90] , [0.15, 0.25] , [0.30, 0.35]i h[0.30, 0.70] , [0.25, 0.55] , [0.40, 0.45]i h[0.20, 0.30] , [0.30, 0.45] , [0.20, 0.35]i

d3 h[0.70, 0.75] , [0.30, 0.50] , [0.40, 0.45]i h[0.60, 0.65] , [0.25, 0.45] , [0.35, 0.55]i h[0.75, 0.80] , [0.20, 0.35] , [0.05, 0.10]i h[0.65, 0.70] , [0.30, 0.35] , [0.40, 0.45]i h[0.20, 0.30] , [0.40, 0.45] , [0.30, 0.35]i

Applications to Assess the Resilience of Workers w1 w2 w3 w4 w5

d4 h[0.20, 0.30] , [0.30, 0.45] , [0.40, 0.45]i h[0.40, 0.50] , [0.20, 0.30] , [0.20, 0.30]i h[0.30, 0.40] , [0.50, 0.60] , [0.60, 0.65]i h[0.50, 0.60] , [0.30, 0.45] , [0.40, 0.60]i h[0.10, 0.20] , [0.20, 0.35] , [0.40, 0.55]i

241

d5 h[0.60, 0.70] , [0.40, 0.45] , [0.60, 0.65]i h[0.20, 0.40] , [0.20, 0.25] , [0.30, 0.45]i h[0.70, 0.80] , [0.10, 0.20] , [0.05, 0.10]i h[0.60, 0.75] , [0.20, 0.35] , [0.35, 0.40]i h[0.50, 0.65] , [0.35, 0.45] , [0.30, 0.35]i

Likewise, forms (I˜1 )2 , (I˜2 )2 , (I˜3 )2 , (I˜4 )2 , (I˜5 )2 and (I˜6 )2 for the second lockdown session as in Tables 18, 19, 20, 21, 22 and 23. Similarly, forms (I˜1 )3 , (I˜2 )3 , (I˜3 )3 , (I˜4 )3 , (I˜5 )3 and (I˜6 )3 for the third lockdown session as in Tables 24, 25, 26, 27, 28 and 29. Table 24. (I˜1 )3 shows the IV N SS data for the first question

w1 w2 w3 w4 w5

w1 w2 w3 w4 w5

d1 h[0.62, 0.65] , [0.45, 0.50] , [0.46, 0.50]i h[0.53, 0.55] , [0.35, 0.40] , [0.66, 0.70]i h[0.42, 0.50] , [0.31, 0.40] , [0.28, 0.30]i h[0.41, 0.45] , [0.45, 0.52] , [0.32, 0.43]i h[0.30, 0.35] , [0.35, 0.40] , [0.45, 0.50]i

d2 h[0.67, 0.70] , [0.25, 0.28] , [0.40, 0.44]i h[0.35, 0.42] , [0.35, 0.38] , [0.45, 0.54]i h[0.26, 0.31] , [0.25, 0.30] , [0.35, 0.40]i h[0.45, 0.60] , [0.35, 0.48] , [0.45, 0.46]i h[0.45, 0.55] , [0.33, 0.35] , [0.65, 0.70]i

d4 h[0.75, 0.81] , [0.35, 0.44] , [0.45, 0.48]i h[0.20, 0.25] , [0.45, 0.48] , [0.35, 0.38]i h[0.45, 0.54] , [0.45, 0.48] , [0.65, 0.68]i h[0.75, 0.81] , [0.35, 0.38] , [0.42, 0.48]i h[0.25, 0.28] , [0.25, 0.28] , [0.35, 0.42]i

d3 h[0.78, 0.80] , [0.32, 0.43] , [0.45, 0.51]i h[0.71, 0.85] , [0.31, 0.43] , [0.42, 0.49]i h[0.85, 0.90] , [0.30, 0.40] , [0.55, 0.61]i h[0.80, 0.85] , [0.32, 0.42] , [0.45, 0.48]i h[0.45, 0.55] , [0.42, 0.45] , [0.38, 0.40]i

d5 h[0.41, 0.50] , [0.41, 0.48] , [0.68, 0.72]i h[0.28, 0.30] , [0.28, 0.31] , [0.35, 0.41]i h[0.51, 0.56] , [0.33, 0.38] , [0.62, 0.68]i h[0.34, 0.40] , [0.21, 0.30] , [0.47, 0.50]i h[0.25, 0.28] , [0.25, 0.34] , [0.38, 0.48]i

Table 25. (I˜2 )3 shows the IV N SS data for the second question

w1 w2 w3 w4 w5

w1 w2 w3 w4 w5

d1 h[0.72, 0.70] , [0.41, 0.48] , [0.42, 0.52]i h[0.51, 0.57] , [0.31, 0.38] , [0.62, 0.67]i h[0.41, 0.52] , [0.35, 0.45] , [0.21, 0.27]i h[0.40, 0.45] , [0.45, 0.50] , [0.35, 0.50]i h[0.32, 0.42] , [0.31, 0.38] , [0.46, 0.52]i

d2 h[0.51, 0.60] , [0.26, 0.30] , [0.42, 0.47]i h[0.31, 0.40] , [0.34, 0.39] , [0.44, 0.56]i h[0.20, 0.30] , [0.20, 0.25] , [0.35, 0.45]i h[0.51, 0.61] , [0.32, 0.42] , [0.41, 0.45]i h[0.54, 0.62] , [0.31, 0.45] , [0.61, 0.65]i

d4 h[0.65, 0.70] , [0.30, 0.35] , [0.45, 0.50]i h[0.15, 0.20] , [0.40, 0.45] , [0.35, 0.40]i h[0.41, 0.50] , [0.44, 0.49] , [0.61, 0.67]i h[0.70, 0.80] , [0.30, 0.35] , [0.42, 0.50]i h[0.10, 0.20] , [0.20, 0.25] , [0.30, 0.35]i

d3 h[0.45, 0.54] , [0.30, 0.45] , [0.41, 0.51]i h[0.70, 0.80] , [0.31, 0.35] , [0.45, 0.50]i h[0.47, 0.55] , [0.44, 0.54] , [0.51, 0.65]i h[0.84, 0.87] , [0.35, 0.45] , [0.41, 0.48]i h[0.91, 0.94] , [0.45, 0.55] , [0.31, 0.35]i

d5 h[0.40, 0.50] , [0.41, 0.50] , [0.65, 0.70]i h[0.35, 0.47] , [0.25, 0.34] , [0.36, 0.41]i h[0.50, 0.60] , [0.33, 0.40] , [0.65, 0.70]i h[0.31, 0.42] , [0.25, 0.34] , [0.41, 0.47]i h[0.20, 0.30] , [0.25, 0.30] , [0.35, 0.45]i

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V. Chinnadurai and A. Bobin Table 26. (I˜3 )3 shows the IV N SS data for the third question

w1 w2 w3 w4 w5

w1 w2 w3 w4 w5

d1 h[0.45, 0.50] , [0.70, 0.77] , [0.42, 0.45]i h[0.35, 0.40] , [0.60, 0.66] , [0.62, 0.65]i h[0.60, 0.65] , [0.56, 0.66] , [0.21, 0.25]i h[0.45, 0.50] , [0.65, 0.70] , [0.35, 0.55]i h[0.40, 0.45] , [0.71, 0.81] , [0.46, 0.52]i

d2 h[0.43, 0.55] , [0.20, 0.25] , [0.40, 0.50]i h[0.45, 0.50] , [0.30, 0.35] , [0.35, 0.40]i h[0.20, 0.30] , [0.20, 0.25] , [0.35, 0.45]i h[0.44, 0.54] , [0.30, 0.40] , [0.45, 0.50]i h[0.54, 0.64] , [0.25, 0.30] , [0.65, 0.70]i

d4 h[0.65, 0.85] , [0.35, 0.48] , [0.45, 0.56]i h[0.35, 0.55] , [0.45, 0.58] , [0.35, 0.46]i h[0.35, 0.37] , [0.45, 0.58] , [0.65, 0.75]i h[0.75, 0.85] , [0.35, 0.45] , [0.45, 0.58]i h[0.38, 0.40] , [0.25, 0.35] , [0.35, 0.48]i

d3 h[0.40, 0.45] , [0.30, 0.35] , [0.40, 0.45]i h[0.65, 0.70] , [0.45, 0.50] , [0.45, 0.55]i h[0.45, 0.50] , [0.40, 0.54] , [0.50, 0.55]i h[0.85, 0.90] , [0.32, 0.45] , [0.40, 0.45]i h[0.90, 0.95] , [0.42, 0.52] , [0.30, 0.35]i

d5 h[0.44, 0.50] , [0.44, 0.50] , [0.65, 0.75]i h[0.35, 0.40] , [0.25, 0.35] , [0.30, 0.35]i h[0.53, 0.58] , [0.35, 0.40] , [0.65, 0.75]i h[0.45, 0.55] , [0.65, 0.70] , [0.55, 0.60]i h[0.28, 0.30] , [0.21, 0.25] , [0.38, 0.45]i

Table 27. (I˜4 )3 shows the IV N SS data for the fourth question

w1 w2 w3 w4 w5

w1 w2 w3 w4 w5

d1 h[0.15, 0.25] , [0.95, 1.00] , [0.40, 0.45]i h[0.25, 0.35] , [0.85, 0.90] , [0.45, 0.50]i h[0.35, 0.45] , [0.75, 0.80] , [0.50, 0.55]i h[0.24, 0.45] , [0.42, 0.45] , [0.35, 0.45]i h[0.55, 0.65] , [0.55, 0.60] , [0.60, 0.65]i

d2 h[0.55, 0.57] , [0.25, 0.38] , [0.40, 0.60]i h[0.45, 0.47] , [0.35, 0.48] , [0.45, 0.55]i h[0.15, 0.30] , [0.25, 0.38] , [0.38, 0.48]i h[0.65, 0.67] , [0.35, 0.48] , [0.45, 0.65]i h[0.75, 0.77] , [0.35, 0.55] , [0.65, 0.75]i

d4 h[0.65, 0.70] , [0.30, 0.35] , [0.45, 0.50]i h[0.15, 0.20] , [0.40, 0.45] , [0.35, 0.40]i h[0.54, 0.64] , [0.40, 0.50] , [0.55, 0.60]i h[0.70, 0.80] , [0.30, 0.35] , [0.42, 0.50]i h[0.10, 0.20] , [0.20, 0.25] , [0.30, 0.35]i

d3 h[0.55, 0.60] , [0.35, 0.45] , [0.40, 0.45]i h[0.73, 0.74] , [0.34, 0.45] , [0.45, 0.55]i h[0.45, 0.55] , [0.25, 0.30] , [0.50, 0.65]i h[0.40, 0.45] , [0.35, 0.45] , [0.40, 0.55]i h[0.25, 0.35] , [0.45, 0.55] , [0.30, 0.45]i

d5 h[0.45, 0.50] , [0.35, 0.45] , [0.65, 0.75]i h[0.25, 0.30] , [0.20, 0.32] , [0.30, 0.45]i h[0.55, 0.60] , [0.55, 0.60] , [0.65, 0.75]i h[0.30, 0.35] , [0.45, 0.54] , [0.41, 0.45]i h[0.25, 0.30] , [0.75, 0.80] , [0.35, 0.50]i

Table 28. (I˜5 )3 shows the IV N SS data for the fifth question

w1 w2 w3 w4 w5

d1 h[0.41, 0.62] , [0.35, 0.45] , [0.45, 0.50]i h[0.51, 0.63] , [0.33, 0.55] , [0.55, 0.70]i h[0.43, 0.53] , [0.38, 0.48] , [0.20, 0.30]i h[0.20, 0.23] , [0.45, 0.52] , [0.32, 0.40]i h[0.31, 0.42] , [0.35, 0.45] , [0.40, 0.45]i

d2 h[0.55, 0.70] , [0.25, 0.36] , [0.45, 0.50]i h[0.45, 0.57] , [0.35, 0.46] , [0.34, 0.50]i h[0.67, 0.77] , [0.20, 0.30] , [0.38, 0.46]i h[0.60, 0.65] , [0.35, 0.66] , [0.43, 0.50]i h[0.50, 0.60] , [0.33, 0.46] , [0.54, 0.70]i

d3 h[0.72, 0.80] , [0.38, 0.40] , [0.45, 0.48]i h[0.75, 0.82] , [0.35, 0.55] , [0.42, 0.50]i h[0.32, 0.40] , [0.20, 0.30] , [0.55, 0.68]i h[0.70, 0.85] , [0.32, 0.40] , [0.45, 0.58]i h[0.24, 0.55] , [0.44, 0.50] , [0.35, 0.44]i

Applications to Assess the Resilience of Workers w1 w2 w3 w4 w5

d4 h[0.55, 0.65] , [0.30, 0.35] , [0.45, 0.55]i h[0.45, 0.55] , [0.40, 0.45] , [0.35, 0.45]i h[0.55, 0.67] , [0.45, 0.56] , [0.54, 0.70]i h[0.78, 0.88] , [0.30, 0.35] , [0.45, 0.56]i h[0.90, 0.92] , [0.20, 0.25] , [0.35, 0.44]i

243

d5 h[0.45, 0.66] , [0.45, 0.55] , [0.52, 0.70]i h[0.22, 0.30] , [0.28, 0.30] , [0.35, 0.48]i h[0.55, 0.62] , [0.33, 0.45] , [0.62, 0.70]i h[0.61, 0.72] , [0.25, 0.35] , [0.35, 0.50]i h[0.20, 0.33] , [0.25, 0.32] , [0.42, 0.52]i

Table 29. (I˜6 )3 shows the IV N SS data for the sixth question

w1 w2 w3 w4 w5

w1 w2 w3 w4 w5

d1 h[0.85, 0.90] , [0.35, 0.50] , [0.45, 0.50]i h[0.55, 0.60] , [0.33, 0.40] , [0.65, 0.70]i h[0.75, 0.85] , [0.38, 0.40] , [0.25, 0.30]i h[0.50, 0.55] , [0.45, 0.50] , [0.35, 0.40]i h[0.80, 0.85] , [0.35, 0.40] , [0.45, 0.50]i

d2 h[0.65, 0.70] , [0.25, 0.30] , [0.40, 0.50]i h[0.80, 0.84] , [0.12, 0.15] , [0.10, 0.15]i h[0.75, 0.85] , [0.25, 0.40] , [0.38, 0.40]i h[0.45, 0.65] , [0.35, 0.50] , [0.45, 0.50]i h[0.38, 0.48] , [0.33, 0.40] , [0.25, 0.30]i

d4 h[0.25, 0.35] , [0.35, 0.40] , [0.45, 0.50]i h[0.45, 0.55] , [0.35, 0.45] , [0.25, 0.35]i h[0.25, 0.35] , [0.55, 0.65] , [0.65, 0.70]i h[0.55, 0.65] , [0.35, 0.40] , [0.45, 0.50]i h[0.25, 0.30] , [0.25, 0.30] , [0.35, 0.50]i

d3 h[0.75, 0.80] , [0.35, 0.40] , [0.45, 0.50]i h[0.75, 0.85] , [0.35, 0.40] , [0.45, 0.50]i h[0.81, 0.85] , [0.15, 0.30] , [0.10, 0.12]i h[0.75, 0.85] , [0.35, 0.40] , [0.45, 0.50]i h[0.20, 0.55] , [0.45, 0.50] , [0.35, 0.40]i

d5 h[0.65, 0.75] , [0.45, 0.50] , [0.65, 0.70]i h[0.25, 0.30] , [0.25, 0.30] , [0.35, 0.50]i h[0.79, 0.82] , [0.15, 0.30] , [0.10, 0.12]i h[0.65, 0.70] , [0.25, 0.30] , [0.40, 0.50]i h[0.55, 0.60] , [0.25, 0.40] , [0.38, 0.40]i

Step 4: Apply SF Definition 2.9, and form the matrices S(I1∗ )1 , S(I2∗ )1 , S(I3∗)1 , S(I4∗)1 , S(I5∗)1 and S(I6∗ )1 for the first session.

S(I1∗)1 =

w1 w2 w3 w4 w5

d1 20.600 60.755 60.590 6 40.550 0.400

d2 0.500 0.810 0.275 0.680 0.525

d3 0.675 0.125 0.650 0.700 0.475

d4 0.725 0.475 0.185 0.705 0.125

d5 0.5003 0.2757 0.2657 7 0.8255 0.185

S(I2∗)1 =

w1 w2 w3 w4 w5

d1 20.675 60.190 60.575 6 40.715 0.175

d2 1.020 0.725 0.025 0.735 0.995

d3 0.360 0.555 0.370 0.760 1.045

d4 0.475 0.200 0.235 0.800 0.025

d5 0.6353 0.1707 0.1907 7 0.4205 0.075

S(I3∗)1 =

w1 w2 w3 w4 w4

d1 20.725 60.320 60.945 6 40.660 0.625

d2 0.945 0.060 0.000 0.710 1.070

d3 0.275 0.600 0.300 0.785 1.020

d4 0.475 0.175 0.415 0.685 0.040

d5 0.7253 0.3757 0.4007 7 0.5655 0.575

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V. Chinnadurai and A. Bobin

S(I4∗)1 =

w1 w2 w3 w4 w5

d1 20.685 60.650 60.600 6 40.595 0.500

d2 0.250 0.300 0.230 0.615 0.325

d3 0.525 0.595 0.475 0.300 0.300

d4 0.610 0.525 0.155 0.675 0.250

d5 0.7003 0.4007 0.2157 7 0.6405 0.050

S(I5∗)1 =

w1 w2 w3 w4 w5

d1 20.365 60.335 60.610 6 40.520 0.290

d2 0.580 0.110 0.480 0.620 0.575

d3 0.610 0.735 0.600 0.610 0.525

d4 0.400 0.500 0.465 0.625 0.665

d5 0.3903 0.4457 0.2857 7 0.6105 0.030

S(I6∗)1 =

w1 w2 w3 w4 w5

d1 20.475 60.275 60.845 6 40.600 0.625

d2 0.600 0.100 0.650 0.625 0.025

d3 0.650 0.625 0.675 0.650 0.475

d4 0.175 0.525 0.250 0.475 0.000

d5 0.8003 0.7957 0.8757 7 0.4755 0.575

Likewise, apply SF Definition 2.9, and form the S(I1∗ )2 , S(I2∗)2 , S(I3∗ )2 , S(I4∗)2 , S(I5∗)2 and S(I6∗ )2 for the second session.

S(I1∗)2 =

w1 w2 w3 w4 w5

d1 20.580

60.210 60.475 6 40.540 0.200

d2 0.565 0.305 0.240 0.535 0.215

d3 0.635 0.630 −0.005 0.710 0.520

d4 0.760 0.300 0.320 0.700 0.120

d5 0.0503 0.2157 0.1907 7 0.1205 0.100

S(I2∗)2 =

w1 w2 w3 w4 w5

d1 20.630 60.200 60.625 6 40.525 0.235

d2 0.375 0.205 0.025 0.385 0.405

d3 0.340 0.505 0.345 0.715 0.950

d4 0.400 0.225 0.285 0.615 0.075

d5 0.0303 0.2207 0.2407 7 0.1555 0.025

S(I3∗)2 =

w1 w2 w3 w4 w5

d1 20.740 60.340 60.990 6 40.550 0.630

d2 0.265 0.475 0.025 0.315 0.170

d3 0.275 0.585 0.465 0.785 0.925

d4 0.500 0.350 0.420 0.615 0.090

d5 0.2003 0.1107 0.4007 7 0.3205 0.575

S(I4∗)2 =

w1 w2 w3 w4 w5

d1 20.725 60.650 60.625 6 40.485 0.450

d2 0.375 0.350 0.305 0.655 0.495

d3 0.515 0.715 0.295 0.345 0.295

d4 0.555 0.490 0.130 0.605 0.355

d5 0.3253 0.2757 0.2007 7 0.7255 0.050

Applications to Assess the Resilience of Workers

S(I5∗)2 =

w1 w2 w3 w4 w5

S(I6∗)2 =

w1 w2 w3 w4 w5

d1 20.330

d2 0.350 0.505 0.425 0.510 0.335

d3 0.540 0.705 −0.030 0.565 0.545

d1 20.700

d2 0.550 0.825 0.625 0.475 0.350

d3 0.700 0.525 0.975 0.575 0.350

60.370 60.580 6 40.290 0.295

60.350 60.770 6 40.550 0.625

d4 0.350 0.450 0.425 0.600 0.675

d5 0.4103 0.2007 0.3257 7 0.6155 0.065

d4 0.200 0.450 0.275 0.425 −0.050

d5 0.4503 0.1507 0.8257 7 0.5755 0.650

245

Similarly, apply SF Definition 2.9, and form the matrices S(I1∗ )3 , S(I2∗)3 , S(I3∗)3 , S(I4∗)3 , S(I5∗)3 and S(I6∗)3 for the third session.

S(I1∗)3 =

w1 w2 w3 w4 w5

d1 20.630 60.235 60.525 6 40.540 0.225

d2 0.530 0.255 0.185 0.485 0.165

d3 0.685 0.695 0.645 0.730 0.545

d4 0.710 0.325 0.295 0.695 0.145

d5 0.2003 0.2057 0.2407 7 0.1405 0.130

S(I2∗)3 =

w1 w2 w3 w4 w5

d1 20.685 60.240 60.625 6 40.475 0.225

d2 0.390 0.220 0.075 0.500 0.330

d3 0.410 0.605 0.420 0.810 1.095

d4 0.525 0.225 0.280 0.615 0.050

d5 0.2303 0.3207 0.2407 7 0.2205 0.125

S(I3∗)3 =

w1 w2 w3 w4 w5

d1 20.775 60.370 61.005 6 40.700 0.695

d2 0.265 0.425 0.075 0.365 0.190

d3 0.325 0.650 0.420 0.835 1.070

d4 0.525 0.225 0.465 0.615 0.050

d5 0.1753 0.1607 0.4507 7 0.3905 0.625

S(I4∗)3 =

w1 w2 w3 w4 w5

d1 20.750 60.700 60.650 6 40.380 0.550

d2 0.375 0.375 0.110 0.525 0.510

d3 0.550 0.630 0.200 0.350 0.425

d4 0.660 0.560 0.175 0.685 0.275

d5 0.2403 0.3507 0.2307 7 0.6005 0.105

S(I5∗)3 =

w1 w2 w3 w4 w5

d1 20.440

60.385 60.660 6 40.340 0.340

d2 0.455 0.495 0.550 0.665 0.325

d3 0.685 0.775 −0.005 0.620 0.470

d4 0.425 0.525 0.495 0.650 0.740

d5 0.4453 0.1357 0.3157 7 0.5405 0.080

246

V. Chinnadurai and A. Bobin

S(I6∗)3 =

w1 w2 w3 w4 w5

d1 20.825 60.265 60.915 6 40.625 0.725

d2 0.500 0.830 0.735 0.500 0.520

d3 0.675 0.700 0.945 0.700 0.475

d4 0.200 0.600 0.225 0.500 0.125

d5 0.5003 0.1257 0.9207 7 0.5005 0.510

Step 5: Compute the QHDIIV N N SM s from the beginning of the first session by applying Definition 6.3. * " #+ 2 X
1 Ir∗ (5)1 = gpij , (wij )1 + 0.5 0.9s .(wij )1+s , (5) 3 s=1

where r = 1, 2, ..., 6, i and j = 1, 2, ..., 5.

I1∗ (5)1 =

w1 w2 w3 w4 w5

I2∗ (5)1 =

w1 w2 w3 w4 w5

I3∗ (5)1 =

w1 w2 w3 w4 w5

I4∗ (5)1 =

w1 w2 w3 w4 w5

I5∗ (5)1 =

w1 w2 w3 w4 w5

I6∗ (5)1 =

w1 w2 w3 w4 w5

d1 2h3, 0.37i

d2 h3, 0.32i h3, 0.35i h4, 0.15i h3, 0.37i h4, 0.23i

d3 h3, 0.41i h4, 0.23i h3, 0.30i h3, 0.44i h3, 0.31i

d4 h3, 0.45i h4, 0.25i h4, 0.15i h3, 0.43i h4, 0.08i

d5 h4, 0.20i 3 h4, 0.15i 7 h4, 0.15i 7 7 h3, 0.31i 5 h4, 0.09i

d1 2h3, 0.41i

d2 h3, 0.45i h3, 0.30i h4, 0.02i h3, 0.37i h3, 0.44i

d3 h4, 0.23i h3, 0.34i h4, 0.23i h3, 0.47i h2, 0.64i

d4 h4, 0.29i h4, 0.13i h4, 0.16i h3, 0.44i h4, 0.03i

d5 h4, 0.25i 3 h4, 0.13i 7 h4, 0.13i 7 7 h4, 0.19i 5 h4, 0.05i

d1 2h3, 0.46i

d2 h3, 0.39i h4, 0.15i h4, 0.01i h3, 0.33i h3, 0.41i

d3 h4, 0.18i h3, 0.38i h4, 0.23i h3, 0.49i h2, 0.62i

d4 h3, 0.30i h4, 0.14i h4, 0.26i h3, 0.40i h4, 0.03i

d5 h4, 0.30i 3 h4, 0.16i 7 h4, 0.25i 7 7 h4, 0.29i 5 h3, 0.36i

d1 2h3, 0.44i

d2 h4, 0.19i h4, 0.20i h4, 0.14i h3, 0.37i h4, 0.25i

d3 h3, 0.33i h3, 0.39i h4, 0.23i h4, 0.20i h4, 0.20i

d4 h3, 0.38i h3, 0.32i h4, 0.09i h3, 0.41i h4, 0.17i

d5 h3, 0.31i 3 h4, 0.22i 7 h4, 0.13i 7 7 h3, 0.40i 5 h4, 0.04i

d1 2h4, 0.23i

d2 h3, 0.31i h4, 0.18i h4, 0.30i h3, 0.37i h4, 0.29i

d3 h3, 0.38i h3, 0.46i h4, 0.19i h3, 0.37i h3, 0.32i

d4 h4, 0.24i h3, 0.31i h4, 0.29i h3, 0.39i h3, 0.42i

d5 h4, 0.25i 3 h4, 0.20i 7 h4, 0.19i 7 7 h3, 0.37i 5 h4, 0.03i

d1 2h3, 0.37i

d2 h3, 0.35i h4, 0.27i h3, 0.41i h3, 0.35i h4, 0.13i

d3 h3, 0.41i h3, 0.38i h3, 0.50i h3, 0.40i h4, 0.27i

d4 h4, 0.12i h3, 0.32i h4, 0.15i h4, 0.29i h4, 0.01i

d5 h3, 0.40i 3 h3, 0.30i 7 h3, 0.54i 7 7 h3, 0.31i 5 h3, 0.36i

6h3, 0.31i 6h3, 0.34i 6 4h3, 0.34i h4, 0.19i

6h4, 0.13i 6h3, 0.37i 6 4h3, 0.38i h4, 0.12i

6h4, 0.21i 6h3, 0.60i 6 4h3, 0.40i h3, 0.40i

6h3, 0.41i 6h3, 0.38i 6 4h3, 0.32i h3, 0.31i

6h4, 0.22i 6h3, 0.38i 6 4h4, 0.26i h4, 0.19i

6h4, 0.18i 6h3, 0.52i 6 4h3, 0.37i h3, 0.40i

Applications to Assess the Resilience of Workers

247

Next, compute the QHDIIV N N SM from the beginning of the second session by applying Definition 6.3. * + 
 1 Ir∗ (5)2 = gpij , (wij )2 + 0.5 0.91 .(wij )3 , (6) 2 where r = 1, 2, ..., 6, i and j = 1, 2, ..., 5.

I1∗ (5)2 =

w1 w2 w3 w4 w5

I2∗ (5)2 =

w1 w2 w3 w4 w5

I3∗ (5)2 =

w1 w2 w3 w4 w5

I4∗ (5)2 =

w1 w2 w3 w4 w5

I5∗ (5)2 =

I6∗ (5)2 =

w1 w2 w3 w4 w5

w1 w2 w3 w4 w5

d1 2h3, 0.43i

d2 h3, 0.40i h4, 0.21i h4, 0.16i h3, 0.38i h4, 0.14i

d3 h3, 0.47i h3, 0.47i h4, 0.14i h3, 0.52i h3, 0.38i

d4 h3, 0.54i h4, 0.22i h4, 0.23i h3, 0.51i h4, 0.09i

d5 h4, 0.07i 3 h4, 0.15i 7 h4, 0.15i 7 7 h4, 0.09i 5 h4, 0.08i

d1 2h3, 0.47i

d2 h4, 0.28i h4, 0.15i h4, 0.03i h3, 0.31i h4, 0.28i

d3 h4, 0.26i h3, 0.39i h4, 0.27i h3, 0.54i h2, 0.72i

d4 h3, 0.32i h4, 0.16i h4, 0.21i h3, 0.45i h4, 0.05i

d5 h4, 0.07i 3 h4, 0.18i 7 h4, 0.17i 7 7 h4, 0.13i 5 h4, 0.04i

d1 2h3, 0.54i

d2 h4, 0.19i h3, 0.33i h4, 0.03i h4, 0.24i h4, 0.13i

d3 h4, 0.21i h3, 0.44i h3, 0.33i h3, 0.58i h2, 0.70i

d4 h3, 0.37i h4, 0.23i h3, 0.31i h3, 0.45i h4, 0.06i

d5 h4, 0.14i 3 h4, 0.09i 7 h3, 0.30i 7 7 h4, 0.25i 5 h3, 0.43i

d1 2h3, 0.53i

d2 h4, 0.27i h4, 0.26i h4, 0.18i h3, 0.45i h3, 0.36i

d3 h3, 0.38i h3, 0.50i h4, 0.19i h4, 0.25i h4, 0.24i

d4 h3, 0.43i h3, 0.37i h4, 0.10i h3, 0.46i h4, 0.24i

d5 h4, 0.22i 3 h4, 0.22i 7 h4, 0.15i 7 7 h3, 0.50i 5 h4, 0.05i

6h4, 0.16i 6h3, 0.36i 6 4h3, 0.39i h4, 0.15i

6h4, 0.15i 6h3, 0.45i 6 4h3, 0.37i h4, 0.17i

6h4, 0.25i 6h2, 0.72i 6 4h3, 0.43i h3, 0.47i

6h3, 0.48i 6h3, 0.46i 6 4h3, 0.33i h3, 0.35i d1 2h4, 0.26i

6h4, 0.27i 6h3, 0.44i 6 4h4, 0.22i h4, 0.22i d1 2h3, 0.54i

6h4, 0.23i 6h3, 0.59i 6 4h3, 0.42i h3, 0.48i

d2 h4, 0.28i h3, 0.36i h3, 0.34i h3, 0.40i h4, 0.24i

d2 h3, 0.39i h3, 0.60i h3, 0.48i h3, 0.35i h4, 0.29i

d3 h3, 0.42i h3, 0.53i h5, −0.02i h3, 0.42i h3, 0.38i

d3 h3, 0.50i h3, 0.42i h2, 0.70i h3, 0.45i h4, 0.28i

d4 h4, 0.27i h3, 0.34i h3, 0.32i h3, 0.45i h3, 0.50i

d4 h4, 0.15i h3, 0.36i h4, 0.19i h3, 0.33i h4, 0.00i

d5 h3, 0.31i 3 h4, 0.13i 7 h4, 0.23i 7 7 h3, 0.43i 5 h4, 0.05i d5 h3, 0.34i 3 h4, 0.10i 7 h2, 0.62i 7 7 h3, 0.40i 5 h3, 0.44i

Similarly, compute the QHDIIV N N SM at the beginning of the third session by applying Definition 6.3. D E Ir∗ (5)3 = gpij , (wij )3 , (7)

248

V. Chinnadurai and A. Bobin

where r = 1, 2, ..., 6, i and j = 1, 2, ..., 5.

I1∗ (5)3 =

w1 w2 w3 w4 w5

I2∗ (5)3 =

w1 w2 w3 w4 w5

I3∗ (5)3 =

w1 w2 w3 w4 w5

I4∗ (5)3 =

w1 w2 w3 w4 w5

I5∗ (5)3 =

I6∗ (5)3 =

w1 w2 w3 w4 w5

w1 w2 w3 w4 w5

d1 2h2, 0.63i

d2 h3, 0.53i h4, 0.26i h4, 0.19i h3, 0.49i h4, 0.17i

d3 h2, 0.69i h2, 0.70i h2, 0.65i h2, 0.73i h3, 0.55i

d4 h2, 0.71i h3, 0.33i h4, 0.30i h2, 0.70i h4, 0.15i

d5 h4, 0.20i 3 h4, 0.21i 7 h4, 0.24i 7 7 h4, 0.14i 5 h4, 0.13i

d1 2h2, 0.69i

d2 h3, 0.39i h4, 0.22i h4, 0.08i h3, 0.50i h3, 0.33i

d3 h3, 0.41i h2, 0.61i h3, 0.42i h1, 0.81i h1, 1.10i

d4 h3, 0.53i h4, 0.23i h4, 0.28i h2, 0.62i h4, 0.05i

d5 h4, 0.23i 3 h3, 0.32i 7 h4, 0.24i 7 7 h4, 0.22i 5 h4, 0.13i

d1 2h2, 0.78i

d2 h4, 0.27i h3, 0.43i h4, 0.08i h3, 0.37i h4, 0.19i

d3 h3, 0.33i h2, 0.65i h3, 0.42i h1, 0.84i h1, 1.07i

d4 h3, 0.53i h4, 0.23i h3, 0.47i h2, 0.62i h4, 0.05i

d5 h4, 0.18i 3 h4, 0.16i 7 h3, 0.45i 7 7 h3, 0.39i 5 h2, 0.63i

d1 2h2, 0.75i

d2 h3, 0.38i h3, 0.38i h4, 0.11i h3, 0.53i h3, 0.51i

d3 h3, 0.55i h2, 0.63i h4, 0.20i h3, 0.35i h3, 0.43i

d4 h2, 0.66i h3, 0.56i h4, 0.18i h2, 0.69i h4, 0.28i

d5 h4, 0.24i 3 h3, 0.35i 7 h4, 0.23i 7 7 h2, 0.60i 5 h4, 0.11i

6h4, 0.24i 6h3, 0.53i 6 4h3, 0.54i h4, 0.23i

6h4, 0.24i 6h2, 0.63i 6 4h3, 0.48i h4, 0.23i

6h3, 0.37i 6h1, 1.01i 6 4h2, 0.70i h2, 0.70i

6h2, 0.70i 6h2, 0.65i 6 4h3, 0.38i h3, 0.55i d1 2h3, 0.44i

d2 h3, 0.46i h3, 0.50i h3, 0.55i h2, 0.67i h3, 0.33i

6h3, 0.39i 6h2, 0.66i 6 4h3, 0.34i h3, 0.34i d1 2h1, 0.83i

d2 h3, 0.50i h1, 0.83i h2, 0.74i h3, 0.50i h3, 0.52i

6h4, 0.27i 6h1, 0.92i 6 4h2, 0.63i h2, 0.73i

∗ Step 6: Determine I+ (5)1 I1∗ (5)1 , I2∗ (5)1, ..., I6∗(5)1 matrices.

∗ I+ (5)1 =

by

w1 w2 w3 w4 w5

∗ Likewise, form I+ (5)2 by ∗ ∗ ∗ I1 (5)2 , I2 (5)2, ..., I6 (5)2 matrices.

d3 h2, 0.69i h2, 0.78i h5, −0.01i h2, 0.62i h3, 0.47i

d3 h2, 0.68i h2, 0.70i h1, 0.95i h2, 0.70i h3, 0.48i

summing d1 219 622 618 6 419 21

d2 19 22 23 18 22

summing

d3 20 19 22 19 18

d4 h3, 0.43i h3, 0.53i h3, 0.50i h2, 0.65i h2, 0.74i

d4 h4, 0.20i h2, 0.60i h4, 0.23i h3, 0.50i h4, 0.13i

the d4 21 21 24 19 23

the

d5 h3, 0.45i 3 h4, 0.14i 7 h3, 0.32i 7 7 h3, 0.54i 5 h4, 0.08i d5 h3, 0.50i 3 h4, 0.13i 7 h1, 0.92i 7 7 h3, 0.50i 5 h3, 0.51i

corresponding

entries

of

entries

of

d5 223 237 237 7 205 22

corresponding

249

Applications to Assess the Resilience of Workers

∗ I+ (5)2 =

w1 w2 w3 w4 w5

∗ Similarly, form I+ (5)3 by ∗ ∗ ∗ I1 (5)3 , I2 (5)3, ..., I6 (5)3 matrices.

∗ I+ (5)3 =

d1 219 623 617 6 419 21

d2 22 21 22 19 23

d3 20 18 22 19 18

summing

w1 w2 w3 w4 w5

d1 212 620 611 6 416 18

d2 19 18 21 17 20

d4 20 21 22 18 23

the d3 15 12 18 11 14

d5 223 247 217 7 215 22

corresponding

d4 17 19 22 13 22

entries

of

d5 223 227 197 7 195 21

Step 7: Tabulate the details as in Table 30 and assess the resilience of the workers during the lockdown. The first column shows the data of from the beginning of the first session(1, 2, and 3 sessions), the second column shows the data of from the beginning of the second session(2 and 3 sessions), and the third column shows the data of the third session(3session). Table 30. Shows the consolidated information for all the questions

w1 w2 w3 w4 w5

1, 2, and 3 sessions d1 d2 d3 d4 19 19 20 21 22 22 19 21 18 23 22 24 19 18 19 19 21 22 18 23

d5 22 23 23 20 22

d1 19 23 17 19 21

2 and 3 sessions d2 d3 d4 22 20 20 21 18 21 22 22 22 19 19 18 23 18 23

d5 22 24 21 21 22

d1 12 20 11 16 18

d2 19 18 21 17 20

3-session d3 d4 15 17 12 19 18 22 11 13 14 22

d5 22 22 19 19 21

Analysis: Consider the first column data (Table 30) if the psychiatrist wishes to examine the resilience of the workers from the beginning of session 1. We understand from the first column that the workers w1 , w2 , and w4 have reached the acceptance stage. Similarly, the workers w3 , and w5 are in the depression stage and may require psychological treatments. Likewise, consider the second column data if the psychiatrist examines the resilience of the workers from the beginning of session 2. We infer from the second column that the workers w2 , and w4 have reached the acceptance stage. The worker w1 shows a dominance towards the anger and acceptance stage. The worker w3 shares the anger, bargaining, and depression stage. Consider the third column data if the psychiatrist examines the resilience of the workers from the beginning of session 3. We conclude from the third column that the workers w1 , w2 , and w4 have reached the acceptance stage. The workers w3 , and w5 are in the depression stage and may require psychological treatments. Table 30 provides holistic details of the workers based on the data collected during the longitudinal study. Also, we understand that workers w3 , and w5 are yet to reach the acceptance level and may require the help of the therapist.

250

9.

V. Chinnadurai and A. Bobin

C OMPARISON STUDY

This section shows the significance of QHDIIV N N SM . We present the superiority of QHDIIV N N SM by comparing it with a simple average method. Consider the IV N SSs mentioned in Tables 18, 19, 20, 21, 22, and 23. Now to understand the risk level associated with each worker, compute the stages as in Table 31 by using Algorithm 4.2. Similarly, consider the data in the third column(Table 30). Now, compute the average by considering the corresponding entries of these two tables and tabulate the details in Table 32. Table 31. Shows the consolidated information for all the questions

w1 w2 w3 w4 w5

d1 14 19 13 19 19

d2 19 17 20 17 20

d3 17 15 21 15 17

d4 18 19 21 13 22

d5 21 24 19 18 21

Table 32. Shows the average of sessions 2 and 3

w1 w2 w3 w4 w5

d1 13.00 19.50 12.00 17.50 18.50

d2 19.00 17.50 20.50 17.00 20.00

d3 16.00 13.50 19.50 13.00 15.50

d4 17.50 19.00 21.50 13.00 22.00

d5 21.50 23.00 19.00 18.50 21.00

Analysis: From Tables 32 and 30 (second column - 2 and 3 sessions), we infer that the stage levels are different for the same workers. From Table 32, we understand that w1 , w2 and w4 are in acceptance stage, w3 and w5 are in depression stage. Now let us analyze the data available in the second column in Table 30 for the same workers. In this case, w1 shows a dominance towards anger and acceptance stages. w3 shows a shared risk-level of anger, bargaining, and depression. w5 shows the dominance towards anger and depression levels. We know that the simple average method computes the average of the resultant values, whereas, in the QHDIIV N N SM method, it enhances the effect of SF value at the second session; and decreases the effect of SF value at the third session. Hence, the stage levels are different in each of the discussed methods. In general, the tabulated result helps the psychiatrist to take up a decision in a longitudinal study; hence QHDIIV N N SM method proves to be a useful tool for the psychiatrist.

Applications to Assess the Resilience of Workers

10.

251

L IMITATIONS , C ONCLUSION AND F UTURE WORKS

The restrictions of this proposed study are i) may need a certified psychological counselor or a therapist to execute the case studies. ii) may cause different remedy methods if over one expert is examining the workers. iii) negative preferences for psychological applications. In this study, we suggest an acceptable workaround for two vital problems. These problems act as a barrier for the psychiatrist from applying neutrosophic theory in the psychological field. i) Most of the psychological studies manage questionnaires, and psychiatrists would like to follow the standard method of handling rating scale construction and classical theory to access the conditions. But it is not advisable to capture the data when the research area is subject to uncertainty, especially in psychological perspectives. ii) Also, psychiatrists would like to record the data in a longitudinal study and analyze the behavior modification of humans during every session. We present solutions to these arguments by applying a combination of IV N SS, N SS, and QHDF . We can relate the concepts of IV N N SS and QHDIIV N N SS to the psychological principles. These notions support the psychiatrists to capture the data using neutrosophic theory and also follow the traditional rating method. IV N N SS helps the psychiatrists to use their ancient evaluation method (positive and negative evaluation norms). QHDIIV N N SS helps to analyze the data from a longitudinal perspective. We may extend this concept to other notable operators like Einstein geometry and Einstein weighted-geometry [50], single-valued neutrosophic power averaging operators [51], and cubic number generalized weighted Heronian mean operator [52] that are used widely in the linguistic neutrosophic environment. Also, we may extend these notions to other fuzzy hybrid sets and determine the significance of the same with a real-life case study.

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[19] Nancy, and Harish Garg. An Improved Score Function For Ranking Neutrosophic Sets And Its Application To Decision-Making Process. International Journal for Uncertainty Quantification, vol. 6, no. 5, 2016, pp. 377-85. [20] Garg, Harish and Nancy. Multi-Criteria Decision-Making Method Based on Prioritized Muirhead Mean Aggregation Operator under Neutrosophic Set Environment. Symmetry, vol. 10, no. 7, July 2018, p. 280. [21] Garg, H.; Nancy. Multiple criteria decision making based on Frank choquet heronian mean operator for single-valued neutrosophic sets. Applied and computational mathematics vol. 18, 2019, pp. 163-188. [22] Garg, Harish. Novel Neutrality Aggregation Operator-Based Multiattribute Group Decision-Making Method for Single-Valued Neutrosophic Numbers. Soft Computing, vol. 24, no. 14, July 2020, pp. 10327-49. [23] Garg Harish. Decision making analysis based on sine trigonometric operational laws for single-valued neutrosophic sets and their applications. Applied and Computational Mathematics An international journal, vol. 19, no.2, Aug 2020, pp. 255-276. [24] Garg, Harish and Nancy. Multiple Attribute Decision Making Based on Immediate Probabilities Aggregation Operators for Single-Valued and Interval Neutrosophic Sets. Journal of Applied Mathematics and Computing, vol. 63, no. 1-2, June 2020, pp. 619-653. [25] Garg, Harish and Nancy. Algorithms for single-valued neutrosophic decision making based on topsis and clustering methods with new distance measure. AIMS Mathematics, vol. 5, no. 3, March 2020, pp. 2671-2693. [26] Basumatary, Bhimraj and Said, Broumi. Interval-Valued Triangular Neutrosophic Linear Programming Problem. International Journal of Neutrosophic Science (IJNS), vol. 10, no. 2, 2020, pp. 105-115. [27] Ajay, D., Said Broumi, and Aldring. J. An MCDM Method under Neutrosophic Cubic Fuzzy Sets with Geometric Bonferroni Mean Operator. Neutrosophic Sets and Systems, vol. 32, no. 1, 2020, pp. 187-202. [28] Broumi, Said et al. Intelligent Algorithm for Trapezoidal Interval Valued Neutrosophic Network Analysis. CAAI Transactions on Intelligence Technology, vol. 5, no. 2, June 2020, pp. 88-93. [29] Smithson M., Oden G.C. Fuzzy Set Theory and Applications in Psychology. In: Zimmermann HJ. (eds) Practical Applications of Fuzzy Technologies. The Handbooks of Fuzzy Sets Series, Springer, Boston, MA. vol 6, 1999, pp. 557-585. [30] Rosch, Eleanor. Cognitive Representations of Semantic Categories. Journal of Experimental Psychology: General, vol. 104, no. 3, Sept. 1975, pp. 192-233.

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[45] Phelps, E. S., and R. A. Pollak. On Second-Best National Saving and GameEquilibrium Growth. The Review of Economic Studies, vol. 35, no. 2, Apr. 1968, p. 185. [46] Laibson, David. Golden Eggs and Hyperbolic Discounting. The Quarterly Journal of Economics, vol. 112, no. 2, May 1997, pp. 443-78. [47] Diamond, Peter, and Botond Koszegi. Quasi-Hyperbolic Discounting and Retirement. Journal of Public Economics, vol. 87, no. 9-10, Sept. 2003, pp. 1839-72. [48] Liu, Yaya, et al. Intertemporal Hesitant Fuzzy Soft Sets: Application to Group Decision Making. International Journal of Fuzzy Systems, vol. 22, no. 2, Mar. 2020, pp. 619-35. [49] Alcantud, Jose, and Maria Munoz Torrecillas. Intertemporal Choice of Fuzzy Soft Sets. Symmetry, vol. 10, no. 9, Sept. 2018, p. 371. [50] Fan,Changxing et al. Linguistic Neutrosophic Numbers Einstein Operator and Its Application in Decision Making. Mathematics, vol. 7, no. 5, Apr. 2019, p. 389. [51] Garg, Harish and Nancy. Linguistic Single-Valued Neutrosophic Power Aggregation Operators and Their Applications to Group Decision-Making Problems. IEEE/CAA Journal of Automatica Sinica, vol. 7, no. 2, Mar. 2020, pp. 546-558. [52] Fan, Changxing, and Jun Ye. Heronian Mean Operator of Linguistic Neutrosophic Cubic Numbers and Their Multiple Attribute Decision-Making Methods. Mathematical Problems in Engineering, vol. 2018, July 2018, pp. 1-13.

SECTION III. EXTENSION OF THE NEUTROSOPHIC SET

In: Decision-Making with Neutrosophic Set Editor: Harish Garg

ISBN: 978-1-53619-419-7 c 2021 Nova Science Publishers, Inc.

Chapter 11

2-A DDITIVE C HOQUET C OSINE S IMILARITY M EASURES FOR S IMPLIFIED N EUTROSOPHIC S ETS AND A PPLICATIONS TO M EDICAL D IAGNOSIS 2 ¨ Ezgi T¨urkarslan1 , Murat Olgun2,∗, Mehmet Unver and S¸eyhmus Yardımcı2 1 TED University, Faculty of Arts And Science, Department of Mathematics, Ankara, Turkey 2 Ankara University, Faculty of Science, Department of Mathematics, Ankara, Turkey

Abstract As an extension of the concepts of fuzzy set, intuitionistic fuzzy set, interval valued fuzzy set and interval valued intuitionistic fuzzy set, the concept of neutrosophic set has been developed to represent uncertain, imprecise, incomplete and inconsistent information that exists in the real life problems. The class of simplified neutrosophic sets is a subclass of the class of neutrosophic sets and includes the class of single valued neutrosophic sets and the class of interval valued neutrosophic sets. A cosine similarity measure plays an important role to determine the degree of similarity between two object such as fuzzy sets, intuitionistic fuzzy sets, neutrosophic sets, simplified neutrosophic sets etc. and it is based on an average or a weighted model. However, such models have some drawback when modeling real-life problems due to the lack of the interaction between the criteria. In this correspondence, to overcome this problem, six new cosine similarity measures for simplified neutrosophic sets are proposed by considering 2-additive Choquet integral model. Thus, more sensitive models than the cosine similarity measures in the literature are proposed via consideration of the interaction between the criteria. Moreover, the effort of calculating the sensitivity is decreased with the help of 2-additivity. To demonstrate the efficiency of the proposed cosine similarity measures we apply them to the medical diagnosis problems.

Keywords and phrases: neutrosophic sets, Choquet integral, similarity measure, medical diagnosis 2010 Mathematics Subject Classification: 28E10, 68T10, 94D05 ∗

Corresponding Author’s Email: [email protected] (Corresponding author).

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INTRODUCTION

Fuzzy set theory was introduced by Zadeh [1] in 1965 and it has profoundly influenced to various disciplines such as medicine, engineering, basic sciences, management sciences, social sciences etc., as it can successfully model some real-life problems [2, 3, 4, 5, 6, 7, 8, 9]. Fuzzy set theory is very successful in handling uncertainties arising from vagueness or partial belonging of an element in a set. However, it cannot represent and handle all sorts of uncertainties prevailing in different real-life problems such as problems involving incomplete and inconsistent information. Moreover, it is unable to represent the indeterminacy independently. Thus, in order to overcome all these shortcomings, we need some more general concepts than the concept of fuzzy set. Neutrosophy is a branch of philosophy that studies the origin, nature, and scope of neutralities, as well as their interactions with different ideational spectra [10]. Smarandache [10] proposed the concept of neutrosophic set from a philosophical point of view in 1998. The concept of neutrosophic set is a generalization of the concept of fuzzy set based on neutrosophy. A neutrosophic set is characterized by a truth membership function, an indeterminacy membership function and a falsity membership function and each membership degree, the value of the function at a point, is a real standard or non-standard subset of the non-standard unit interval ]− 0, 1+[. Besides, there is no restriction both on the membership functions and on the sum of the membership degrees. In a neutrosophic set, indeterminacy is quantified explicitly and truth membership, indeterminacy membership and falsity membership are independent. However, the set theoretical operators such as intersection, union and inclusion cannot be defined on the non-standard unit interval, so it is very difficult to perform the applications of neutrosofic sets to technical and engineering fields. Therefore, single and interval-valued neutrosophic sets have been proposed to overcome this deficiency. If the membership degrees are singletons then the neutrosophic set is called a single-valued neutrosophic set (SVNS) [11] and if the membership degrees are intervals then the neutrosophic set is called an interval neutrosophic set (INS) [12]. Ye [13] proposed the concept of simplified neutrosophic set (SNS) that contains the INSs and SVNSs when the truth, indeterminacy and falsity membership degrees are constrained in the real standard unit interval [0, 1] as a simplified form of the neutrosophic set and he defined an aggregation function for simplified neutrosophic set. The concept of neutrosophic set is a powerful tool to deal with incomplete, indeterminate and inconsistent information that exists in the real life. This situation has increased researchers’ interest in these sets and many studies have been conducted on these sets and their applications on real life problems. Researchers have made efforts to develop the best ranking methods in neutrosophic environment. For example, most researchers have used the ranking methods to solve real life applications such as the distance measure [14, 15, 16, 17, 18], similarity measure [19, 20, 21, 13, 22], mean operator [23, 24], aggregation operators [25, 26, 27, 28, 29] and applied them to different fields. One of these applications is to solve medical diagnosis problems using the similarity measure for SVNSs and INSs. Many researchers proposed similarity measures using various methods to increase the sensitivity of the solution of the application and used these measures to solve medical diagnosis problems. For example, Ye and Fu [30] proposed similarity measures based on tangent functions for multi-period medical diagnosis method. Chou et al. [31] defined new

2-Additive Choquet Cosine Similarity Measures ...

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similarity measures that are similar to Ye and Fu’s method. Liu et al. [32] presented several similarity measures for both SVNSs and INSs. Sindhu et al. [33] proposed similarity measures for SVNSs and INSs by using an exponential function. Sahin and Karabacak [34] presented a similarity measure based on matrix norm. A cosine similarity measure is a useful tool that is used in information retrieval and that determines the degree of similarity between two objects. This notion is applied to various real life problems such as medical diagnosis, pattern recognition, decision making [35, 36, 37, 38, 39, 40]. Most of the cosine similarity measures are based on an average model or a weighted average model that do not consider the interactions among elements of a given universe. Therefore, they are not always reasonable in some cases. It means an average or a weighted average model does not work well in many real life problems. In this chapter, to overcome this problem, we use the concept of Choquet integral that is a non-linear continuous aggregation function. The concept of Choquet integral and the concept of capacity were introduced by Choquet [41] in 1953. Later, the concept of capacity was expanded to the concept of fuzzy measure by Sugeno in 1974 [42]. The Choquet integral is defined with respect to a fuzzy measure (or non-additive monotonic measure) and it can be thought as a generalization of the weighted mean, since a fuzzy measure is able to model the interaction between criteria in many situations. However, a fuzzy measure is defined on the power set and this situation exponentially increases the number of subsets that need to be measured and this complicates the fuzzy measure identification process in a set with excessive elements. To overcome this difficult process, many authors studied various fuzzy measure identification methods (see, e.g., [43, 44, 45, 46]). Grabisch [43] proposed the concept of k-additive fuzzy measure that seriously increases the effort in a fuzzy measure identification process. Particularly, any interaction among two elements can be represented and interpreted by a Choquet integral with respect to a 2-additive fuzzy measure [47]. The remainder of this chapter is organized as follows: In Section 2, some basic properties of SNSs, some existing cosine similarity measures and weighted cosine similarity measures for SNSs are recalled. In Section 3, the concepts of fuzzy measure, M¨obius representation of a fuzzy measure and the concepts of 2-additive fuzzy measure, Choquet integral with respect to 2-additive fuzzy measure and theorems about these concepts are given. In Section 4, six new cosine similarity measures between SNSs based on the 2-additive Choquet fuzzy integral are proposed. In Section 5, to demonstrate the effectiveness of the proposed similarity measure, applications on medical diagnosis problems are given. In section 6, we recall some more existing similarity measures for SVNSs and INSs and we give a comparative analysis between proposed cosine similarity measures and these existing similarity measures.

2.

BASIC C ONCEPTS OF SNS S AND SOME E XISTING C OSINE SIMILARITY MEASURES F OR SNS S

Smarandache [10] proposed the concept of neutrosophic set from a philosophical point of view that is a part of neutrosophy and that extends the concepts of fuzzy set [1], interval valued fuzzy set [48, 49], intuitionistic fuzzy set [50], and interval-valued intuitionistic

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fuzzy set [51]. To apply a neutrosophic set to the science and engineering areas, Ye [52, 53] introduced the concept of SNS as a simplified form/subclass of the neutrosophic sets. The class of SNSs is a subclass of the class of neutrosophic sets and it includes the classes of INSs and SVNSs. Definition 1. [52, 53] Let X 6= ∅ be a set. A simplified neutrosophic set A of X is given with A = {hx, TA(x), IA(x), FA (x)i : x ∈ X} , (2.1) where TA , IA and FA are functions from X to closed interval [0, 1] or the class of all sub-intervals of the closed interval [0, 1]. The values TA (x), IA(x) and FA (x) indicate the truth membership degree, the indeterminacy membership degree and the falsity membership degree of the element x to the set A, respectively. If the values of the functions are singletons, i.e., the functions are to [0, 1], then the neutrosophic set reduces to SVNS (see, e.g., [39]). If the values of the functions are sub-intervals, i.e., the functions are to the class of sub-intervals of [0, 1], then the neutrosophic set reduces to INS (see, e.g., [39]). For a SVNS A, TA (x), IA(x), FA(x) ∈ [0, 1] with 0 ≤ TA (x) + IA (x) + FA (x) ≤ 3 for all x ∈ X [52, 53, 40]. For an INS A, TA (x) := [TAL(x), TAU (x)], IA (x) := [IAL (x), IAU (x)], FA (x) := [FAL (x), FAU (x)] ⊆ [0, 1] and 0 ≤ TAU (x) + IAU (x) + FAU (x) ≤ 3, for any x ∈ X [53, 40]. Let X = {x1 , x2 , ..., xn} be a finite set and let A = {hx, TA(x), IA(x), FA (x)i : x ∈ X} and B = {hx, TB (x), IB (x), FB (x)i : x ∈ X} be two SVNSs. Ye [53] proposed a cosine similarity measure between A and B by using the arithmetic mean: n

C1 (A, B) :=

1X TA (xi )TB (xi ) + IA (xi )IB (xi ) + FA(xi )FB (xi ) q q . n i=1 T 2 (x ) + I 2 (x ) + F 2 (x ) T 2 (x ) + I 2 (x ) + F 2 (x ) i i A i A i A i B i B B

(2.2)

Ye [40] also proposed two cosine similarity measures via cosine function between A and B by using the arithmetic mean: SC1 (A, B) :=

n hπ i 1X cos (|TA (xi ) − TB (xi )| ∨ |IA (xi ) − IB (xi )| ∨ |FA (xi ) − FB (xi )|) , n i=1 2

(2.3)

SC2 (A, B) :=

n hπ i 1X cos (|TA (xi ) − TB (xi)| + |IA (xi ) − IB (xi )| + |FA (xi ) − FB (xi )|) , n i=1 6

(2.4)

and

2-Additive Choquet Cosine Similarity Measures ...

263

where “∨” denotes the maximum operator. Weighted versions of these similarity measures n X are also defined by taking weights 0 ≤ ω1 , ω2 , ..., ωn ≤ 1 with ωi = 1 instead of 1/n i=1

(see, [53, 40]). Let X = {x1 , x2 , ..., xn} be a finite set and let

A = {hx, [TAL(x), TAU (x)], [IAL(x), IAU (x)], [FAL(x), FAU (x)]i : x ∈ X} and B = {hx, [TBL(x), TBU (x)], [IBL(x), IBU (x)], [FBL(x), FBU (x)]i : x ∈ X} be two INSs. Ye [53] proposed a cosine similarity measure between A and B by using the arithmetic mean: „ « TAL (xi)TBL (xi ) + IAL (xi )IBL (xi ) + FAL (xi )FBL (xi) n X +TAU (xi )TBU (xi ) + IAU (xi )IBU (xi) + FAU (xi )FBU (xi ) 1 C2 (A, B) := 0 q n i=1 2 2 2 2 2 2 T (x ) + IAL (xi ) + FAL (xi ) + TAU (xi ) + IAU (xi) + FAU (xi ) @ q AL i 2 2 2 2 2 2 TBL (xi ) + IBL (xi) + FBL (xi ) + TBU (xi ) + IBU (xi ) + FBU (xi)

1.

(2.5)

A

Ye [40] also proposed two cosine similarity measures via cosine function between A and B by using the arithmetic mean:

and

2 0 13 n (|TAL (xi ) − TBL (xi )| ∨ |IAL (xi ) − IBL (xi )| ∨ 1X π SC3 (A, B) := cos 4 @ |FAL (xi ) − FBL (xi )|) + (|TAU (xi ) − TBU (xi )| ∨ A5 , n i=1 4 |IAU (xi ) − IBU (xi )| ∨ |FAU (xi ) − FBU (xi )|) 2 0 13 n |TAL (xi ) − TBL (xi )| + |IAL (xi ) − IBL (xi )| + 1X π @ |FAL (xi ) − FBL (xi )| + |TAU (xi ) − TBU (xi )| + A5 . SC4 (A, B) := cos 4 n i=1 12 |IAU (xi ) − IBU (xi )| + |FAU (xi ) − FBU (xi )|

(2.6)

(2.7)

Similarly, weighted versions of these similarity measures were also defined by taking n X weights 0 ≤ ω1 , ω2 , ..., ωn ≤ 1 with ωi = 1 instead of 1/n (see, [53, 40]). i=1

3.

C HOQUET INTEGRAL A ND 2-A DDITIVE F UZZY MEASURES

The Choquet integral is a basic tool for modeling of decisions under risk and uncertainty in real life. In this chapter, we consider a particular case of the Choquet integral where the fuzzy measure is 2-additive. First of all, let us recall some basic notions of fuzzy measure theory that are used in this correspondence. Definition 2. [41] Let X 6= ∅ be a finite set and let P (X) be the family of all subsets of X. If i.) σ(∅) = 0 ii.) σ(X) = 1 iii.) A ⊆ B implies σ(A) ≤ σ(B) (monotonicity), then the set function σ : P (X) → [0, 1] is called a fuzzy measure on X .

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Definition 3. [54] The Mbius representation of a set function σ on X is a function m : P (X) → R that is defined by m(A) :=

X

(−1)|A\B|σ(B)

(3.1)

B⊂A

for all A ∈ P (X). A fuzzy measure σ is expressed as follows: σ(A) =

X

m(B)

(3.2)

B⊂A

for all A ∈ P (X) whenever its Mbius representation m is given. As a result, the Mbius representation over singletons is equal to the fuzzy measure itself. Definition 4. [43] Let X 6= ∅ be a finite set and let σ be a fuzzy measure on X. σ is said to be 2-additive if its Mbius transform m satisfies m(A) = 0 for all A ⊂ X such that |A| > 2 and there exist at least one subset A ⊂ X with |A| = 2 such that m(A) 6= 0 . The following theorem gives an important property of the Mbius representation. Theorem 1. [54] Let X 6= ∅ be a finite set and let σ : P (X) → R be a function. σ is a fuzzy measure on X if and only if its Mbius representation m satisfies (i) m(∅) = 0, (ii)

X

m(B) = 1,

B⊂X

(iii)

X

m(B) ≥ 0, for all A ⊂ X and for all x ∈ A.

x∈B⊂A

A fuzzy measure is defined on the power set. Therefore, we need to determine the fuzzy measure values of 2n number of sets. Thus, the process of determining a fuzzy measure over a set with the excess number of elements is quite difficult. Grabisch introduced a crucial kind of fuzzy measure that is named k-additive fuzzy measure to facilitate the process of determining a fuzzy measure in a set with large elements [43]. For instance, if k = 2, it is enough to determine the fuzzy measure values of n(n − 1)/2 subsets in order to determine the fuzzy measure. Here, the number n(n − 1)/2 is the number of subsets of 2 elements of a set with n elements. Another important concept to represent a fuzzy measure is interaction index (see, e.g., [43]). Definition 5. [43] Let X 6= ∅ be a finite set and let σ be a fuzzy measure over X. The interaction index I(T ) of a subset T of X is defined by n−|T |

I(T ) :=

X k=0

|T |

ϑk

X

X

K⊂X\T L⊂T |K|=k

(−1)|T \L|σ(L ∪ K)

(3.3)

2-Additive Choquet Cosine Similarity Measures ... where p

ϑk :=

(n − k − p)!k! . (n − p + 1)!

265

(3.4)

The following theorem gives the connection between the interaction index and the Mbius representation of a fuzzy measure. Theorem 2. [43] Let X 6= ∅ be a finite set and let m be the Mbius representation of a fuzzy measure σ on X. The interaction index I(T ) corresponding to fuzzy measure σ satisfies the following equality: |X\T | X 1 X I(T ) = m(T ∪ K) (3.5) k+1 k=0

K⊂X\T |K|=k

for any T ⊂ X . Consequently; when σ is 2-additive, we have  m(T ), |T | = 2 I(T ) = 0, |T | > 2

(3.6)

[43]. Interaction between only two criteria can exist whenever the fuzzy measure is 2-additive. That is, there is no interaction between more than two criteria. Let X = {x1 , x2 , ..., xn} be a finite set and let Iij := I(xi , xj ). 1. If Iij > 0, then there is a positive interaction between the criteria xi and xj , and when they come together, their severity increases. 2. If Iij < 0, then there is negative interaction between the criteria i and j, and one of the criteria is more redundant. When these two criteria come together, their severity decreases. 3. If Iij = 0, then there is no interaction between the criteria i and j and they are independent from each other. The concept of the Choquet integral can be considered as a generalization of the weighted average by assigning a weight to each subset of the universal set by fuzzy measure. Definition 6. [41] Let X = {x1 , x2 , ..., xn} be a finite set and let σ be a fuzzy measure on X. The Choquet integral of a function f : X → [0, 1] with respect to σ is defined by (C)

Z X

f dσ :=

n X k=1


f (x(k)) − f (x(k−1)) σ(E(k)),

(3.7)

where the sequence {xk }nk=0 is a new permutation of the sequence {xk }nk=0 such that 0 :=  f (x(0)) ≤ f (x(1)) ≤ f (x(2)) ≤ ... ≤ f (x(n) ) and E(k) := x(k), x(k+1), ..., x(n) .

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In the case of Choquet integral with respect to a 2-additive fuzzy measure σ, the Definition 6 is equivalent to following expressions: (C2−add. )

Z

f dσ :=

X

mi f (xi) +

xi ∈X

X

X

mij min(f (xi ), f (xj ))

(3.8)

{xi ,xj }⊆X

where m is the Mbius representation of σ on X and mi := m(xi), mij := m(xi , xj ) [55].

4.

MAIN R ESULTS

In this section, we propose six new cosine similarity measures for SNSs by using the 2additive Choquet integral and we give some propositions about these similarity measures. Definition 7. Let X = {x1 , x2 , ..., xn} be a finite set, let A and B be two SVNSs and let σ be a 2-additive fuzzy measure on X. A 2-additive Choquet cosine similarity measure between A and B is given with Z (C2−add. ,σ) WC1 (A, B) := (C2−add. ) fA,B dσ (4.1) X

where TA (xi)TB (xi ) + IA (xi )IB (xi ) + FA (xi )FB (xi ) q fA,B (xi ) := q , TA2 (xi ) + IA2 (xi) + FA2 (xi ) TB2 (xi ) + IB2 (xi ) + FB2 (xi )

for i = 1, 2, ..., n.

Proposition 1. Let X be a finite set, let A and B be two SVNSs on X. A 2-additive (C ,σ) Choquet cosine similarity measure WC1 2−add. (A, B) between A and B satisfies the following properties: (C

(P1 ) 0 ≤ WC1 2−add. (C

(P2 ) WC1 2−add.

,σ)

,σ)

(A, B) ≤ 1; (C

(A, B) = WC1 2−add. (C

(P3 ) If A = B then WC1 2−add.

,σ)

,σ)

(B, A);

(A, B) = 1.

Proof. (P1 ) Since fA,B (xi ) ∈ [0, 1] for any i = 1, 2, ..., n and the Choquet (C ,σ) integral is monotone, we have 0 ≤ WC1 2−add. (A, B) ≤ 1 . (P2 ) It is trivial since fA,B (xi ) = fB,A (xi ) for any i = 1, 2, ..., n. (P3 ) If A = B then TA (xi ) = TB (xi ), IA (xi ) = IB (xi ) and FA (xi ) = FB (xi) for i = 1, 2, ..., n. Then, we have fA,B (xi ) = 1. Now, we obtain from Theorem 1 that X (C ,σ) WC1 2−add. (A, B) = m(D) = 1. D⊂X

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Hence, the proof is completed. Definition 8. Let X = {x1 , x2 , ..., xn} be a finite set, let A and B be two SVNSs on X and let σ be a 2-additive fuzzy measure on X. Two new 2-additive Choquet cosine similarity measure based on cosine function between A and B is given with Z (C2−add. ,σ) (1) WSC1 (A, B) := (C2−add. ) fA,B dσ (4.2) X (C ,σ) WSC2−add. (A, B) 2

Z

:= (C2−add. )

(2)

fA,B dσ

(4.3)

X

where (1)

fA,B (xi ) := cos

hπ

(|TA (xi ) − TB (xi )| ∨ |IA (xi ) − IB (xi )| ∨ |FA (xi ) − FB (xi )|)

hπ

(|TA (xi ) − TB (xi )| + |IA (xi ) − IB (xi )| + |FA(xi ) − FB (xi )|)

2

i

and (2)

fA,B (xi ) := cos

6

i

for i = 1, 2, ..., n. Proposition 2. A 2-additive Choquet cosine similarity measure based on cosine function (C ,σ) WSC2−add. (A, B) for (k = 1, 2) between SVNS A and SVNS B satisfies P1 , P2 and the k following properties: (C ,σ) (P*3 ) A = B if and only if WCk 2−add. (A, B) = 1; (P4 ) If C is a SVNS on X and A ⊆ B ⊆ C, then (C

,σ)

(C

,σ)

WSC2−add. k and

WSC2−add. k

(C

,σ)

(C

,σ)

(A, C) ≤ WSC2−add. k

(A, C) ≤ WSC2−add. k

(A, B)

(B, C).

Proof. (P1 ) and (P2 ) can be proved similar to Proposition 1. (P*3 ) For any SVNS A and SVNS B, if A = B then TA (xi) = TB (xi ), IA (xi) = IB (xi) and FA (xi) = FB (xi ) for all i = 1, 2, ..., n. Thus, we obtain that |TA (xi ) − TB (xi )| = 0, (k) |IA (xi ) − IB (xi )| = 0 and |FA (xi) − FB (xi )| = 0. Therefore, we have fA,B (xi ) = 1 and (C

so WSC2−add. k

,σ)

(A, B) = 1, for k = 1, 2. (C

,σ)

Conversely, assume that WSC2−add. (A, B) = 1, for k = 1, 2. Then, from Theorem 1 k X (k) we have m(D) = 1. This implies fA,B (xi) = 1, for k = 1, 2.. Thus, we obtain D⊂X

|TA (xi) − TB (xi)| = 0, |IA (xi ) − IB (xi)| = 0 and |FA (xi ) − FB (xi )| = 0, since cos 0 = 1. Therefore, we have TA (xi ) = TB (xi), IA (xi ) = IB (xi ) and FA (xi ) = FB (xi ), for all i = 1, 2, ..., n. Hence, A = B.

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268

(P4 ) If A ⊆ B ⊆ C then TA (xi ) ≤ TB (xi ) ≤ TC (xi ), IA (xi ) ≥ IB (xi ) ≥ IC (xi ) and FA (xi) ≥ FB (xi ) ≥ FC (xi ), for all i = 1, 2, ..., n. Thus, we have |TA (xi ) − TB (xi )| ≤ |TA (xi ) − TC (xi )| , |TB (xi ) − TC (xi )| ≤ |TA (xi ) − TC (xi )| , |IA (xi ) − IB (xi )| ≤ |IA (xi ) − IC (xi )| , |IB (xi ) − IC (xi )| ≤ |IA (xi ) − IC (xi )| , |FA (xi ) − FB (xi )| ≤ |FA (xi ) − FC (xi )| , |FB (xi ) − FC (xi )| ≤ |FA(xi ) − FC (xi )| .

Since the cosine function is decreasing within the interval [0, π/2], we obtain (k) (k) (k) (k) fA,C (xi ) ≤ fA,B (xi ) and fA,C (xi ) ≤ fB,C (xi), for k = 1, 2. Now, using the mono(C

tonicity of the Choquet integral we have WSC2−add. k

(C ,σ) WSC2−add. (A, C) k

≤

(C ,σ) WSC2−add. (B, C). k

,σ)

(C

(A, C) ≤ WSC2−add. k

,σ)

(A, B) and

Hence, the proof is completed.

Definition 9. Let X = {x1 , x2 , ..., xn} be a finite set, let A and B be two INSs and let σ be a 2-additive fuzzy measure on X. A 2-additive Choquet cosine similarity measure between A and B is given with Z (C2−add. ,σ) (4.4) WSC (A, B) := (C2−add. ) gA,B dσ X

where „

TAL(xi )TBL (xi) + IAL (xi )IBL (xi ) + FAL (xi )FBL (xi ) +TAU (xi )TBU (xi ) + IAU (xi)IBU (xi ) + FAU (xi )FBU (xi)

«

gA,B (xi ) := 0 q 2 2 2 2 2 T 2 (x ) + IAL (xi ) + FAL (xi ) + TAU (xi) + IAU (xi ) + FAU (xi ) @ q AL i 2 2 2 2 2 2 TBL (xi ) + IBL (xi ) + FBL (xi) + TBU (xi) + IBU (xi) + FBU (xi )

1, A

for i = 1, 2, ..., n.

(C

Remark 1. Similar to the proof of Proposition 1 it can proved that WSC2−add. the conditions P1 − P3 .

,σ)

satisfies

Definition 10. Let X = {x1 , x2 , ..., xn} be a finite set, let A and B be two INSs and let σ be a 2-additive fuzzy measure on X. Two 2-additive Choquet cosine similarity measure based on cosine function between A and B are given with Z (C ,σ) (3) WSC2−add. (A, B) := (C ) fA,B dσ (4.5) 2−add. 3 X (C

WSC2−add. 4

,σ)

(A, B) := (C2−add. )

Z

(4)

fA,B dσ

X

where (3) fA,B (xi )

2

0 13 (|TAL (xi ) − TBL (xi)| ∨ |IAL (xi ) − IBL (xi)| ∨ π := cos 4 @ |FAL (xi ) − FBL (xi )|) + (|TAU (xi ) − TBU (xi )| ∨ A5 , 4 |IAU (xi) − IBU (xi )| ∨ |FAU (xi) − FBU (xi )|)

(4.6)

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269

and (4) fA,B (xi )

2

0 |TAL (xi) − TBL (xi )| + |IAL (xi ) − IBL (xi )| + π @ |FAL (xi) − FBL (xi )| + |TAU (xi ) − TBU (xi )| + := cos 4 12 |IAU (xi ) − IBU (xi )| + |FAU (xi ) − FBU (xi )|

for i = 1, 2, ..., n, respectively.

(C

Proposition 3. A cosine similarity measure WSC2−add. k tions P1 , P2 , P3∗ , P4 .

,σ)

13

A5 ,

for k = 3, 4 satisfies the condi-

Proof. P1 , P2 and (P3* ) can be proved similar to Proposition 1. (P4 ) If A ⊆ B ⊆ C, then TAL (xi ) ≤ TBL(xi ) ≤ TCL (xi ), TAU (xi ) ≤ TBU (xi) ≤ TCU (xi), IAL (xi ) ≥ IBL (xi) ≥ ICL (xi ), IAU (xi ) ≥ IBU (xi ) ≥ ICU (xi ) and FAL (xi) ≥ FBL(xi ) ≥ FCL (xi ), FAU (xi ) ≥ FBU (xi ) ≥ FCU (xi), for all i = 1, 2, ..., n. Thus, we have |TAL (xi ) − TBL (xi)| ≤ |TAL (xi) − TCL (xi)| , |TBL (xi ) − TCL(xi )| ≤ |TAL (xi ) − TCL(xi )| , |TAU (xi) − TBU (xi)| ≤ |TAU (xi ) − TCU (xi)| , |TBU (xi) − TCU (xi )| ≤ |TAU (xi ) − TCU (xi )| , |IAL (xi ) − IBL (xi )| ≤ |IAL (xi ) − ICL (xi)| , |IBL (xi ) − ICL (xi )| ≤ |IAL (xi ) − ICL (xi )| , |IAU (xi ) − IBU (xi )| ≤ |IAU (xi ) − ICU (xi)| , |IBU (xi ) − ICU (xi)| ≤ |IAU (xi ) − ICU (xi )| , |FAL (xi ) − FBL (xi )| ≤ |FAL (xi ) − FCL (xi )| , |FBL (xi ) − FCL (xi )| ≤ |FAL (xi) − FCL (xi )| . |FAU (xi ) − FBU (xi )| ≤ |FAU (xi ) − FCU (xi)| , |FBU (xi) − FCU (xi )| ≤ |FAU (xi ) − FCU (xi )| .

(k)

Since the cosine function is decreasing within the interval [0, π/2], we obtain fA,C (xi ) ≤ (k)

(k)

(k)

fA,B (xi ) and fA,C (xi ) ≤ fB,C (xi ), for k = 3, 4. On the other hand, using the mono(C

tonicity of the Choquet integral, we have WSC2−add. k

(C ,σ) WSC2−add. (A, C) k

5.

≤

(C ,σ) WSC2−add. (B, C). k

,σ)

(C

(A, C) ≤ WSC2−add. k

,σ)

(A, B) and

Hence, the proof is completed.

Some Applications

A medical diagnosis problem contains incomplete, uncertain, and inconsistent information. Since neutrosophic sets can model uncertain and inconsistent information in the real life, we choose the medical diagnosis problems to apply the proposed cosine similarity measures for SNSs. Moreover, we consider the interaction between symptoms and use 2-additive fuzzy measure to reduce the calculation effort when the number of symptoms is high. To show the effectiveness of the proposed similarity measures and to compare our results with the results in the literature we consider the medical diagnosis problem adapted from [40]. The target of this problems is to diagnose the disease of patients P1 and P2 among Q1 , Q2 , Q3 , Q4 and Q5 . Example 1. Let us consider the set of diagnoses and symptoms as follows, respectively:   Q1 (Viral fever), Q2(Malaria), Q3(Typhoid), Q= Q4 (Gastritis), Q5(Stenocardia)

270

¨ Ezgi T¨urkarslan, Murat Olgun, Mehmet Unver et al.   s1 (Temperature), s2 (Headache), s3 (Stomach pain), S= s4 (Cough), s5 (Chest pain)

Assume that a patient P1 that has all the symptoms is represented by the following SVNS:    hs1 , 0.8, 0.2, 0.1i, hs2 , 0.6, 0.3, 0.1i,  hs3 , 0.2, 0.1, 0.8i, hs4 , 0.6, 0.5, 0.1i, P1 (Patient) =   hs5 , 0.1, 0.4, 0.6i Moreover, assume that each diagnosis Qi , i = 1, 2, 3, 4, 5, is given as SVNSs:    hs1 , 0.4, 0.6, 0.0i, hs2 , 0.3, 0.2, 0.5i,  Q1 (Viral fever) = hs3 , 0.1, 0.3, 0.7i, hs4 , 0.4, 0.3, 0.3i, ,   hs5 , 0.1, 0.2, 0.7i    hs1 , 0.7, 0.3, 0.0i, hs2 , 0.2, 0.2, 0.6i,  hs3 , 0.0, 0.1, 0.9i, hs4 , 0.7, 0.3, 0.0i, Q2 (Malaria) = ,   hs5 , 0.1, 0.1, 0.8i    hs1 , 0.3, 0.4, 0.3i, hs2 , 0.6, 0.3, 0.1i,  Q3 (Typhoid) = hs , 0.2, 0.1, 0.7i, hs4 , 0.2, 0.2, 0.6i, ,  3  hs5 , 0.1, 0.0, 0.9i    hs1 , 0.1, 0.2, 0.7i, hs2 , 0.2, 0.4, 0.4i  Q4 (Gastritis) = , hs3 , 0.8, 0.2, 0.0i, hs4 , 0.2, 0.1, 0.7i, ,   hs5 , 0.2, 0.1, 0.7i    hs1 , 0.1, 0.1, 0.8i, hs2 , 0.0, 0.2, 0.8i,  Q5 (Stenocardia) = hs , 0.2, 0.0, 0.8i, hs4 , 0.2, 0.0, 0.8i, .  3  hs5 , 0.8, 0.1, 0.1i

Let us construct a 2-additive fuzzy measure. For this purpose, the weights of all criteria are taken equally. To determine the measure of a subset of two elements we use the interaction index that is equal to Mbius representation (see (3.6)). As the sum of the Mbius representations (fuzzy measures) of singletons is equal to 1 we have from (ii) of Theorem 1 that the sum of Mbius representations of subsets of 2 elements should be equal to zero. Now considering interaction of the symptoms we have Table 1 of Mbius representation that is enough to calculate the Choquet integrals (see, 3.8). However, for the sake of completeness we give the 2-additive fuzzy measure σ in Table 2 that corresponds the Mbius representation. For example, since there is a redundancy between the symptoms s1 , s2 we assign a negative value for I1,2 = m1,2 . (C

,σ)

(C

,σ)

(C

,σ)

The proposed cosine similarity measures WC1 2−add. , WSC2−add. and WSC2−add. 1 2 are computed for P1 and each Qk , (k = 1, 2, 3, 4, 5) and the comparison table with the results obtained in [40] are given in Table 3.

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271

Table 1. Mbius representation of σ

m({s1 , s2 }) = −0.06 m({s1 , s5 }) = 0 m({s2 , s5 }) = 0.08 m({s4 , s5 }) = 0.01

m({s1 , s3 }) = 0 m({s2 , s3 }) = 0 m({s3 , s4 }) = 0

m({s1 , s4 }) = −0.12 m({s2 , s4 }) = 0 m({s3 , s5 }) = 0.09

Table 2. 2-Additive fuzzy measure

σ(∅) = 0 σ({s3 }) = 0.20 σ({s1 , s2 }) = 0.34 σ({s1 , s5 }) = 0.40 σ({s2 , s5 }) = 0.48 σ({s4 , s5 }) = 0.41

σ({s1 }) = 0.20 σ({s4 }) = 0.20 σ({s1 , s3 }) = 0.40 σ({s2 , s3 }) = 0.40 σ({s3 , s4 }) = 0.40

σ({s2 }) = 0.20 σ({s5 }) = 0.20 σ({s1 , s4 }) = 0.28 σ({s2 , s4 }) = 0.40 σ({s3 , s5 }) = 0.49

Table 3. Similarity scores for SVNSs

Similarity Measures C1 SC1 SC2 (C2−add. ,σ) WC1

(Q1 , P1 )

(Q2 , P1)

(Q3 , P1 )

(Q4 , P1 )

(Q5 , P1 )

0.8505 0.8942 0.9443 0.8710

0.8661 0.8976 0.9571 0.8522

0.8185 0.8422 0.9264 0.8579

0.5148 0.6102 0.8214 0.5415

0.4244 0.5607 0.7650 0.4278

WSC2−add. 1

0.8913

0.8866

0.8594

0.6268

0.5606

0.9544

0.9537

0.9352

0.8296

0.7691

(C

,σ)

(C

,σ)

WSC2−add. 2

According to the recognition principle of maximum degree of similarity between SVNSs, the process of assigning the pattern P1 to Qi , (k = 1, 2, 3, 4, 5) is described by

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272

n o (C ,σ) k = arg max WSV 2−add. (Q , P ) . i 1 NS

(5.1)

1≤i≤5

The numerical result presented above and (5.1) show that k = Q1 for each 2-additive Cho(C ,σ) (C ,σ) (C ,σ) quet similarity measure WC1 2−add. , WSC2−add. and WSC2−add. . Namely, we conclude 1 2 that the patient P1 suffers from the disease Q1 (Viral fever).

Example 2. Let us consider Example 1 again. A patient P2 with the all symptoms can be represented by the following INS: 9 8 < hs1 , [0.3, 0.5], [0.2, 0.3], [0.4, 0.5]i , hs2 , [0.7, 0.9], [0.1, 0.2], [0.1, 0.2]i , = P2 (Patient) = hs3 , [0.4, 0.6], [0.2, 0.3], [0.3, 0.4]i , hs4 , [0.3, 0.6], [0.1, 0.3], [0.4, 0.7]i , : ; hs5 , [0.5, 0.8], [0.1, 0.4], [0.1, 0.3]i (C

,σ)

(C

,σ)

(C

,σ)

and WSC2−add. The proposed cosine similarity measures WC2 2−add. , WSC2−add. 4 3 are computed for P2 and each Qk , (k = 1, 2, 3, 4, 5). The comparison table of the results with the results obtained in [40] are given in Table 4. Table 4. Similarity scores for INSs

(Q1 , P2 )

(Q2 , P2 )

(Q3, P2 )

(Q4 , P2 )

(Q5 , P2 )

0.6775 0.7283 0.8941 0.6490

0.5613 0.6079 0.8459 0.5273

0.7741 0.7915 0.9086 0.6755

0.7198 0.7380 0.9056 0.7106

0.6872 0.7157 0.8797 0.6669

,σ)

0.6925

0.5763

0.6980

0.7290

0.7013

(C ,σ) WSC2−add. 4

0.8802

0.8284

0.8612

0.8944

0.8714

Similarity Measures C2 SC3 SC4 (C2−add. ,σ) WC2 (C

WSC2−add. 3

According to the recognition principle of maximum degree of similarity between SVNSs, the process of assigning the pattern P1 to Qi , (k = 1, 2, 3, 4, 5), is described by n o (C ,σ) k = arg max WSV 2−add. (Q , P ) . (5.2) i 2 NS 1≤i≤5

The numerical result presented above and (5.2) show that k = Q4 for each 2-additive (C ,σ) (C ,σ) (C ,σ) Choquet similarity measure WC2 2−add. , WSC2−add. and WSC2−add. . Namely, we con3 4 clude that the patient P2 suffers from the disease Q4 (gastritis).

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273

Consequently, both results obtained in Example 1 and Example 2 are different from those obtained in [40]. In this chapter, while solving the medical diagnosis problems, the interactions between the symptoms are taken into account by using fuzzy measure and its Mbius representation. Thus, the sensitivity of the results obtained is increased thanks to the proposed similarity measure based on 2-additive Choquet integral method.

6.

T HE C OMPARATIVE A NALYSIS OF P ROPOSED C HOQUET INTEGRAL METHOD WITH SOME E XISTING SIMILARITY MEASURES

In this section, we recall some similarity measures that is applied to the medical diagnosis problem studied in Section 5. Then, we compare them with the proposed Choquet cosine similarity measures. Liu et al. [32] improved a similarity measure for SVNSs by combining the similarity and the distance measure proposed in [21]. Let X = {x1 , x2 , ..., xn} be a finite set and let A = {hx, TA(x), IA(x), FA (x)i : x ∈ X} and B = {hx, TB (x), IB (x), FB (x)i : x ∈ X}

be two SVNSs. A similarity measure between A and B is given with ∗ S1SV N S (A, B) :=

1 (S1SV N S (A, B) + 1 − DSV N S (A, B)) 2

(6.1)

where n X

S1SV N S (A, B) :=

i=1 n X

((TA (xi ) ∧ TB (xi )) + (IA (xi ) ∧ IB (xi )) + (FA (xi) ∧ FB (xi ))) (6.2) ((TA (xi ) ∨ TB (xi )) + (IA (xi ) ∨ IB (xi )) + (FA (xi) ∨ FB (xi )))

i=1

and

DSV N S (A, B) :=

!1 n i 2 1 Xh 2 2 2 |TA(xi ) − TB (xi )| + |IA (xi ) − IB (xi )| + |FA (xi ) − FB (xi )| 3n i=1

(6.3)

are the similarity and the distance measures given in [21], respectively. Moreover, Liu et al. [32] proposed a similarity measure for SVNSs by combining the distance measure DSV N S and the cosine similarity measure C1 recalled in Section 2:

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274

∗ S2SV N S (A, B) :=

1 (C1 (A, B) + 1 − DSV N S (A, B)) . 2

(6.4)

Ye and Fu [30] proposed two similarity measures based on the tangent function for SVNSs: T1 (A, B) := 1 −

T2 (A, B) := 1 −

n hπ i 1 X tan (|TA(xi ) − TB (xi )| ∨ |IA (xi) − IB (xi )| ∨ |FA (xi) − FB (xi )|) , n i=1 4

(6.5)

n hπ i 1 X tan (|TA(xi ) − TB (xi )| + |IA (xi) − IB (xi )| + |FA (xi ) − FB (xi)|) . n i=1 12

(6.6)

Weighted versions of these similarity measures are also defined by taking weights 0 ≤ n X ω1 , ω2 , ..., ωn ≤ 1 with ωi = 1 instead of 1/n (see, [30]). In [30], a patient was evali=1

uated according to a single time period by using similarity measures based on the tangent function for SVNSs. However, as time changes, a patient’s symptoms and the effects of these symptoms may change. Therefore, Ye and Fu [30] proposed a multi-period similarity measure based on the tangent function that evaluates to patients at different times. The similarity measure between a patient and the considered disease in each period tk (k = 1, 2, ..., q) is given by the following: Wi (tk ) := 1 −

m X j=1

ωj tan

hπ i (|Tj (tk ) − Tij | + |Ij (tk ) − Iij | + |Fj (tk ) − Fij |) 12

where the weight vector of symptoms is 0 ≤ ω1 , ω2 , ..., ωm ≤ 1 with

m X

(6.7)

ωj = 1 and

j=1

the characteristic values between a patient and symptoms are denoted by the form of a single valued neutrosophic value (SVNV) hTj (tk ), Ij (tk ), Fj (tk )i and the characteristic values between symptoms and the considered diseases are denoted by the form of a SVNV hTij , Iij , Fij i for convenience. Then, weighted measure Mi is obtained for i = 1, 2, ..., n with the following formula: Mi =

q X

Wi (tk )w(tk )

(6.8)

j=1

where weight vector of the periods is 0 ≤ w(t1 ), w(t2 ), ..., w(tq) ≤ 1 with q X w(tk ) = 1. As a weighted measure value gives a proper diagnosis, in [30] a pak=1

tient is evaluated at 3 different moments according to the symptoms that the patient had (see, Table 5 in [30]).

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275

Chou et al. [31] defined two similarity measures for SVNSs based on maximum norm and arithmetic mean instead of tangent function and applied them to a multiperiod health diagnostics problem by using same algorithm in [30]. These similarity measures are given with Mω1 (A, B) := 1 −

m X

ωj (|TA (xi ) − TB (xi )| ∨ |IA (xi) − IB (xi )| ∨ |FA (xi ) − FB (xi )|) ,

m X

ωj

(6.9)

j=1

and Mω2 (A, B) := 1 −

j=1

„

|TA(xi ) − TB (xi )| + |IA (xi ) − IB (xi)| + |FA (xi ) − FB (xi)| 3

where weight vector of symptoms is 0 ≤ ω1 , ω2 , ..., ωm ≤ 1 with

m X

«

,

(6.10)

ωj = 1. Then, they

j=1

used the weighted measure WiM ωl for l = 1, 2 and for i = 1, 2, ..., n that is calculated with the same algorithm in (6.8.)

Table 5 shows the comparison of the results of the proposed similarity measures given with (4.1), (4.2) and (4.3) and the existing similarity measures are discussed in this section. The results are compatible with the results of the methods that evaluated the patients according to time, whereas they are incompatible with the results of other similarity measures. The reason why the results are not compatible is that these measures do not take into account the relationship between symptoms. The reason why the results are compatible with the measures that evaluate the patient according to time is that we take the same weights with Chou [31] and Ye and Fu [30] for the symptoms. However, it is an important deficiency that these studies do not take into account the interaction between symptoms. Now, we recall some existing similarity measures for INSs. Let X = {x1 , x2 , ..., xn} be a finite set and let A = {hx, [TAL(x), TAU (x)], [IAL(x), IAU (x)], [FAL(x), FAU (x)]i : x ∈ X} and B = {hx, [TBL(x), TBU (x)], [IBL(x), IBU (x)], [FBL(x), FBU (x)]i : x ∈ X} be two INSs. Liu et al.[32] defined a similarity measure for INSs by combining the similarity measure and the distance measure proposed in [19] by ∗ S1IV N S (A, B) :=

where

1 (S1IV N S (A, B) + 1 − DIV N S (A, B)) 2

(6.11)

¨ Ezgi T¨urkarslan, Murat Olgun, Mehmet Unver et al.

276

Table 5. Similarity scores for SVNSs

Similarity Measures ∗ S1SV NS ∗ S2SV NS Mi WiM ω1 WiM ω2 (C ,σ) WC1 2−add.

(Q1 , P1 )

(Q2 , P1 )

(Q3 , P1 )

(Q4 , P1 )

(Q5 , P1 )

0.6663 0.8223 0.8183 0.6730 0.7770 0.8710

0.7188 0.8378 0.7852 0.5990 0.7323 0.8522

0.5387 0.6377 0.7966 0.6560 0.7483 0.8579

0.4594 0.5500 0.7427 0.5260 0.6810 0.5415

0.4336 0.4881 0.7167 0.5270 0.6510 0.4278

WSC2−add. 1

0.8913

0.8866

0.8594

0.6268

0.5606

0.9544

0.9537

0.9352

0.8296

0.7691

(C

,σ)

(C ,σ) WSC2−add. 2

Table 6. Similarity scores for INSs

Similarity Measures ∗ S1IV NS

(Q1 , P1 )

(Q2 , P1 )

(Q3 , P1 )

(Q4 , P1 )

(Q5 , P1 )

0.5783

0.4610

0.6273

0.5772

0.5401

∗ S2IV NS

0.6804

0.5729

0.7503

0.7061

0.6734

(C

,σ)

0.6490

0.5273

0.6755

0.7106

0.6669

(C

,σ)

0.6925

0.5763

0.6980

0.7290

0.7013

(C

,σ)

0.8802

0.8284

0.8612

0.8944

0.8714

WC2 2−add. WSC2−add. 3 WSC2−add. 4

1 (TAL(xi ) ∧ TBL (xi)) + (TAU (xi ) ∧ TBU (xi ))+ @ (IAL (xi ) ∧ IBL (xi )) + (IAU (xi ) ∧ IBU (xi ))+ A (FAL (xi ) ∧ FBL (xi )) + (FAU (xi ) ∧ FBU (xi )) i=1 0 1 S1IV N S (A, B) := n (TAL(xi ) ∨ TBL (xi)) + (TAU (xi ) ∨ TBU (xi ))+ X @ (IAL (xi ) ∨ IBL (xi )) + (IAU (xi ) ∨ IBU (xi ))+ A (FAL (xi ) ∨ FBL (xi )) + (FAU (xi ) ∨ FBU (xi )) i=1 n X

0

(6.12)

and

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1 2 31 n |TAL (xi) − TBL (xi )|2 + |TAU (xi ) − TBU (xi )|2 + 2 X 1 4 |IAL (xi ) − IBL (xi)|2 + |IAU (xi ) − IBU (xi )|2 + 5A DIV N S (A, B) := @ 6n i=1 |FAL (xi ) − FBL (xi)|2 + |FAL (xi) − FBL (xi )|2

(6.13)

0

are the similarity and the distance measures given in [19], respectively. Moreover, they proposed another similarity measure for INSs by combining the distance measure DIV N S and the cosine similarity measure C2 given in Section 2: ∗ S2IV N S (A, B) :=

1 (C2 (A, B) + 1 − DIV N S (A, B)) . 2

(6.14)

Table 6 shows the comparison of the results of the proposed similarity measures given with (4.4), (4.5) and (4.6) and the existing methods discussed in this section. We get incompatible results with the previous ones. However, this is not unexpected because we consider the interaction among symptoms that makes our analysis more sensitive.

C ONCLUSION In this chapter, we propose six new cosine similarity measures based on 2-additive Choquet integral model for SNSs and we give some theoretical approach. Since we know the direction of the relationship between symptoms concretely and clearly, by considering the fuzzy measure theory we increase the sensitivity during the medical diagnosis with the help of the interaction between the symptoms and we reduce the computational effort by using 2-additivity. As a result, we reduce the computational effort and we allow more symptoms to be evaluated. In the future, the results can be extended to a linguistic environment [5, 27] or the proposed Choquet integral model can be applied to the soft set theory [56].

ACKNOWLEDGMENTS The authors are grateful to the Reviewers for carefully reading the chapter and for offering substantial comments and suggestions that enabled them to improve the presentation.

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In: Decision-Making with Neutrosophic Set Editor: Harish Garg

ISBN: 978-1-53619-419-7 © 2021 Nova Science Publishers, Inc.

Chapter 12

MULTI-ATTRIBUTE GROUP DECISION-MAKING BASED ON UNCERTAIN LINGUISTIC NEUTROSOPHIC SETS AND POWER HAMY MEAN OPERATOR Yuan Xu1, Xiaopu Shang1 and Jun Wang2, 1

School of Economics and Management, Beijing Jiaotong University, Beijing, China 2 School of Economics and Management, Beijing University of Chemical Technology, Beijing, China

ABSTRACT The topic of this chapter is to investigate multi-attribute group decision-making (MAGDM) method, which provides a new manner for decision makers to help them to select the best alternatives. Considering the fact that decision makers would like to provide their evaluation information in the form of natural language, this paper proposes a new tool, called the uncertain linguistic neutrosophic sets (ULNSs). In ULNS, truthmembership, falsity-membership, and indeterminacy-membership are denoted as uncertain linguistic variables, and hence the proposed ULNSs are powerful to depict decision makers’ preferred information over alternatives. In this paper, firstly the definition, basic operational rules, comparison method and distance measure of ULNSs are presented and discussed. Second, a series of aggregation operators for ULNSs based on power average operator and Hamy mean operator are proposed. The desirable characteristics of the introduced operators make them useful to deal with MAGDM problems based on ULNSs. Third, a novel MAGDM method based on ULNSs is proposed. Numerical examples have demonstrated the validity of our proposed decisionmaking method. We further illustrate the advantages of our MAGDM method through comparative analysis.

* Corresponding Author’s Email: [email protected]

284

Yuan Xu, Xiaopu Shang and Jun Wang

Keywords: uncertain linguistic neutrosophic sets, power average, Hamy mean, uncertain linguistic neutrosophic power Hamy mean, multi-attribute group decision-making

1. INTRODUCTION Aggregation operators (AOs) are acknowledged as a useful tool when determining the optimal alternatives in the procedure of multi-attribute group decision-making (MAGDM) [1-8]. AOs refer to a series of special functions, which are employed to fuse attribute values to calculate the overall evaluation values. By ranking comprehensive evaluations, the corresponding ranking results of alternatives are derived. The classical weighted average (WA) and weighted geometric (WG) are two commonly used AOs, which effectively produce overall evaluation results. More and more scientists have noticed the importance and necessity of considering the interrelationship among fused values, however, WA and WG operators do not have such capability. The Bonferroni mean (BM) [9] AO has gained more and more attention, as it has the ability of capturing the interrelationship among arguments. The characteristic has successfully received great interests and investigators started to study BM-based MAGDM method. In [10-15], for example, scholars studied MAGDM methods under intuitionistic fuzzy sets (IFSs), hesitant fuzzy sets, dual hesitant fuzzy sets, Pythagorean fuzzy sets, q-rung orthopair fuzzy sets, hesitant fuzzy linguistic terms set, etc. Similar to BM, the Heronian mean [16], Maclaurin symmetric mean [17], generalized Maclaurin symmetric mean [18], Hamy mean [19] and Muirhead mean [20] have also been extensively applied in MAGDM problems, due to their ability of dealing with the complex interrelationship among attributes. Take Muirhead mean (MM) operator as an example, it is powerful in information fusion process owing to its ability of reflecting the interrelationship among any numbers of input variables. Liu and Li [21] applied MM under IFSs and introduced a new MAGDM method. Tang et al. [22] generalized MM into Pythagorean fuzzy sets to evaluate emerging technology commercialization performance. Garg and Nancy [23] proposed a MAGDM method, which considers not only the priority relationship, but also interrelationship among attributes. Wang et al. [24] studied MM under q-rung orthopair fuzzy sets and proposed a powerful MAGDM method. Liu and Teng [25] put forward a series of probabilistic linguistic MM operators and applied them to some practical MAGDM instances. For more recent developments of MM-based decision-making methods, readers are suggested to refer [26-30]. As realistic MAGDM problems are too complicated, sometimes it is insufficient to merely consider the interrelationship among attributes. In general, decision makers (DMs) usually come from different fields and they probably have different background and expertise. Hence, there exist situations wherein DMs provide unduly high or low evaluation, due to different background and time shortage or information loss. Obviously,

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unreasonable or extreme evaluation values have bad influence on the final decision results and to gain a reasonable ranking order of alternatives, such kind of negative effect should be eliminated. The power average (PA) [31] operator, proposed by professor Yager, has been popular and famous, because of its ability of effectively treating unreasonable evaluation values. Due to this characteristic, PA has been extensively applied in MAGDM with intuitionistic fuzzy information [32], hesitant fuzzy information [33], dual hesitant fuzzy information [34], Pythagorean fuzzy information [35], q-rung orthopair fuzzy information [36], and so forth. In [37], He et al. proposed a compound AO, viz. power Bonferroni mean (PBM), by combining PA with BM. The information fusion performance of PBM is conspicuous as it takes advantages of both PA and BM operators. After the introduction of PBM, Liu and Liu [38] used PBM to aggregate attribute values which are given in the form of linguistic intuitionistic fuzzy numbers. Liu and Li [39] proposed an interval-value intuitionistic fuzzy PBM-based MAGDM method and used to select the more suitable supplier. Wang and Li [40] proposed a hybrid Pythagorean fuzzy MAGDM method based on PBM and interactive operational rules. Liu and Gao [41] introduced novel intuitionistic fuzzy PBM operators and corresponding decision-making method under the Dempster-Shafer framework. Liu and Liu [42] introduced PBM-based MAGDM method under linguistic q-rung orthopair fuzzy environment. Recently, Liu et al. [43] combined PA with HM and proposed power Hamy mean (PHM) operator. It is remarkable that PHM is more powerful and flexible than PBM as PHM effectively considers the interrelationship among multiple arguments. In [43], authors studied PHM in interval neutrosophic sets, proposed a new MAGDM method and investigated its applications in reality. Afterwards, Liu and Li [44], and Liu et al. [45] proposed trapezoidal fuzzy two-dimensional linguistic PHM and normal wiggly hesitant fuzzy linguistic PHM operators, respectively, and applied them in real MAGDM problems. Up to present, MAGDM methods based on PHM are still fewer, which needs further investigation. IFSs, originated by Prof. Atanassov [46], are an improved version of the classical fuzzy sets. The main advantage of IFSs is reflected by their ability of describing fuzzy information or data from both membership degree and non-membership degree. Recent studies on IFSs-based MAGDM methods have illustrated their high efficiency in solving real decision-making problems [47-50]. In [51], Zhang proposed an extension of IFSs,

i.e., linguistic intuitionistic fuzzy sets (LIFSs), which use two linguistic terms s  l p , lq  , where l p and lq stand for the linguistic variables of the truth/membership and falsity/nonmembership degrees, respectively, to denote the membership and non-membership degrees. It is obvious that LIFSs take the advantages of natural languages and are more suitable for DMs to express their evaluation values. Evidences have indicated the good performance of LIFSs in handling actual-life MAGDM problems [52, 53]. However, the disadvantage of LIFSs is also obvious. On the one hand, when facing with the MAGDM

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problems with high degree of complexity, for instance, the DMs feel fuzzy and uncertainty when providing their linguistic evaluation information owing to the limitation of human thought and the uncertainty of decision environment, the LIFSs will no longer be applicable because they can only provide one value for each element. On the other hand, the LIFSs can only provide the linguistic membership degrees and linguistic nonmembership degrees in the evaluation process and cannot describe indeterminate and inconsistent linguistic information. In order to solve the shortcomings of LIFSs, Fang ang Ye [54] extended neutrosophic set (NS) [55], which is a famous in collecting inconsistent information and has been proved by many studies [56-64], and they provided the definition of linguistic neutrosophic sets (LNSs), where truth-membership, falsitymembership, and indeterminacy-membership are composed of three linguistic terms. Compared with LIFSs, LNSs can not only address imprecise information, but also the inconsistent information in the decision-making process. As LNS is designed to handle the incomplete, imprecise, and inconsistent information, it has attracted a lot of attention once defined. Fan et al. [65] combined LNSs with BM operators and proposed the linguistic neutrosophic weighted BM (LNWBM) and linguistic neutrosophic weighted geometric BM (LNWGBM) operators and applied them to MAGDM problems. Wang and Liu [66] introduced the linguistic neutrosophic generalized weighted partitioned BM (LNGWPBM) operator. Liu and You [67] developed some Hamy averaging operators with LNSs to deal with the MAGDM problems. In addition, LNN-weighted arithmetic averaging (LNNWAA) operator [54], the weighted linguistic neutrosophic Maclaurin symmetric mean (WLNMSM) operator [68], weighted linguistic neutrosophic power Muirhead mean (WLNPMM) operators [69], Linguistic neutrosophic power weighted Heronian aggregation (LNPWHA) operator [70] and so on were proposed one after another. Other than the above, to provide the experts with the maximum freedom in the decision-making process, uncertain linguistic sets (ULSs) was introduced in [71], which replace the individual sematic value in linguistic terms with interval values. For more information about ULSs, readers are suggested to refer [72, 73]. To sum up, there are several aspects that have not been taken into consideration in the existing studies: (1) When the problem is with high degree of complexity and needs to consider DMs’ uncertain, imprecise, and inconsistent information at the same time, the existing studies may be inadequate. (2) Although PHM operator has been proposed by [43], its application scope should be promoted to make it more valuable. Based on which, we summarize the motivations and objectives of this chapter as follows: (1) In real-world decision-making problems, it is necessary to depict DMs’ evaluation information from both quantitative and qualitative aspects, also the uncertainty, indeterminate and inconsistent of DMs’ information should also be considered. In light of these, we propose the uncertain linguistic neutrosophic set, which can not only consider the qualitative analysis of DMs, but also solve the problem of inconsistency between evaluation information. (2) The connections among things are universal and objective. In MAGDM

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problems, it is a common phenomenon that there are interrelationships among attributes. Among the existing AOs, PHM is known as an efficient tool to consider the interrelationship among multiple attributes with a parameter variable k and also can alleviate the unduly high or low values of the evaluation information. Thus, PHM operator may be a good choice to aggregate the assessments of high-complexity MAGDM problems. In summary, this chapter tends to introduce a new method to describe the uncertainty, imprecise and inconsistent information as well as assemble the input arguments with multiple interrelationships. The main objectives of this paper are mainly manifested in three aspects: (1) Propose the definition of uncertain linguistic neutrosophic sets (ULNs) and some properties of ULNs. (2) Apply the PHM operator to uncertain linguistic neutrosophic information environment and provide a family of uncertain linguistic neutrosophic PHM operators. (3) Illustrate the application and advantages of the proposed MAGDM method by giving some calculation cases. Consequently, an uncertain linguistic neutrosophic set decision-making method based on PHM operator is proposed to select the best alternative in this chapter. Compared with other methods, the proposed work mainly has the following advantages and contributions: (1) New fuzzy set, the uncertain linguistic neutrosophic set (ULNS) is introduced. Based on which the operational rules, comparison method and distance measure are defined. ULNS is efficient at dealing with the uncertainty, imprecise and inconsistent evaluation information and more general and flexible than many existing fuzzy sets. (2) A new decision-making method is proposed by extending the PHM operator to ULNS. Subsequently, the uncertain linguistic neutrosophic power average (ULNPA) operator, the uncertain linguistic neutrosophic power weighted average (ULNPWA) operator, the uncertain linguistic neutrosophic power Hamy mean (ULNPHM) operator, and the uncertain linguistic neutrosophic power weighted Hamy mean (ULNPWHM) operator are introduced. In the proposed method, not only the interrelationship among attributes, but also the unreasonable and impractical assessments are considered. The rest part of this chapter is organized as follows. Section 2 recalls some basic concepts and notions. Section 3 proposed the definition of ULNSs and uncertain linguistic neutrosophic numbers (ULNNs), the operational rules, comparison method and distance measure are also introduced. In section 4, a family of ULN PHM operators are defined to aggregate the evaluation values of MAGDM problems. To further illustrate the detailed application steps of the propose method, we give a novel MAGDM method under ULNNs in section 5. In section 6, some numerical examples are given to demonstrate the advantages and priorities of the proposed method. Finally, conclusion remarks are provided.

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2. PRELIMINARIES 2.1. Neutrosophic Sets Neutrosophic set (NS) is a part of neutrosophy, which studies the origin, nature, and scope of neutralities, as well as their interactions with different ideational spectra [55]. NS is a power general formal framework which can be regarded as a generalization of fuzzy, intuitionistic fuzzy, and interval-valued fuzzy sets, because it allows an uncertainty and/or hesitation independent of the membership and non-membership information [74]. Smarandache [55] first gave the definition of NS, and we recall it as follows: Definition 1 [55]. Let X be a space of points (objects), with a genetic element in X denoted by x. A neutrosophic set A in X is a characterized by a truth-membership function TA  x  , a indeterminacy-membership function I A  x  and a falsity-membership function

FA  x  . TA  x  , I A  x  and FA  x  are real standard or nonstandard subsets of 0 ,1  . That is

TA  x  : X  0 ,1  , I A  x  : X  0 ,1  , and FA  x  : X  0 ,1  . There is no restriction

on the sum of TA  x  , I A  x  and FA  x  , so 0  sup TA  x   sup I A  x  +sup FA  x   3 . Definition 2 [55]. The complement of a neutrosophic set A is denoted by Ac and is

defined as TAc  x   1  TA  x  , I Ac  x   1  I A  x  and FAc  x   1  FA  x  for every x in X. Definition 3 [55]. A neutrosophic set A is contained in the other neutrosophic set B,

A  B if and only if inf TA  x   inf TB  x  , sup TA  x   sup TB  x  , inf I A  x   inf I B  x  , sup I A  x   sup I B  x  , inf FA  x   inf FB  x  , and sup FA  x   sup FB  x  for every x in X.

2.2. Linguistic Neutrosophic Sets Definition 4 [54]. Let X be a given ordinary fixed set and S  s0 , s1 ,..., st  be a given linguistic term set with odd cardinality t 1 , then a linguistic neutrosophic sets A defined on X is expressed as A

  x, s

  x

, s  x  , s  x 





x X ,

(1)

where s  x  , s  x  , s  x   s0,t  , s  x  , s  x  and s  x  represent the truth-membership degree, the indeterminacy-membership degree and the falsity-membership degree. For convenience,

Multi-Attribute Group Decision-Making …



289



the ordered pair s  x , s  x , s  x is called a linguistic neutrosophic number (LNN), which can be denoted as    s , s , s  for simplicity. Some basic operations of LNNs are presented as follows.

Definition 5 [54]. Let 1   s1 , s1 , s1  ,  2   s2 , s2 , s 2  and    s , s , s  be any three LNNs defined on S  s0 , s1 ,..., st  , and  be a positive real number, then   1   2 =  s    1

1 2 2 t

 , s 12 , s1 2  ; t t 

  1   2 =  s  , s  t  + 1 2

1

2

1  2 t

,s

1  2 



   s  

  t  t 1   t 



,s



,s



,s

  t  t 

  t  t



 

t  t 

  t  t 1   t 

  ; 

 ;  



    s    , s

 1 2 t

(2)

  t  t 1   t



(3)

(4)

 .  

(5)

The comparison method for comparing two LNNs is presented as follows. Definition 6 [54]. Let    s , s , s  be an LNN, the score function of  is defined as S   

2t       , 3t

(6)

And the accuracy function of  is defined as H   

  . t

For any two LNNs 1   s1 , s1 , s1  and  2   s2 , s2 , s 2  , If S 1   S  2  , then 1   2 ; If S 1   S  2  , then If H  1   H  2  , then 1   2 ;

(7)

290

Yuan Xu, Xiaopu Shang and Jun Wang If H  1   H  2  , then 1   2 .

2.3. The Power Average Operator, Hamy Mean Operator and Power Hamy Mean Operator Definition 7 [31]. Let ai  i  1, 2,..., n  be a set of crisp numbers, then the power average (PA) operator is expressed as n

PA  a1 , a2 ,..., an  

 1  T  a  a i

i 1 n

 1  T  a 

n



i 1, i  j

,

(8)

i

i 1

where T  ai  

i





Sup  ai , a j  and Sup ai , a j denotes the support degree for ai from a j ,

satisfying the following properties: 1) 0  Sup  ai , a j   1 ;

(9)

2) Sup  ai , a j   Sup  a j , ai  ;

(10)

3) Sup  a, b   Sup  c, d  , if a  b  c  d .

(11)

Definition 8 [19]. Let ai  i  1, 2,..., n  be a collection of crisp numbers and k  1, 2,..., n . If

HM

k 

1  a1 , a2 ,..., an   k Cn

1k

 k    ai j   1i1 ...ik  n  j 1 

,

(12)

then HM  k  is the Hamy mean (HM) operator, where  i1 , i2 ,..., ik  traverses all the k-tuple combination of 1, 2,..., n  , and Cnk is the binomial coefficient. Definition 9 [43]. Let ai  i  1, 2,..., n  be a collection of crisp numbers, then the power Hamy mean (PHM) operator is defined as

Multi-Attribute Group Decision-Making … 1k

  k n 1  T  ai   ai 1 j PHM  k   a1 , a2 ,..., an   k    n Cn 1i1 ...ik  n  j 1 1  T  ai     i 1 

where T  ai  



291

     

,

(13)



Sup  ai , a j  and Sup ai , a j denotes the support degree for ai from a j ,

n



i 1, i  j

satisfying the properties presented in Definition 7.

3. UNCERTAIN LINGUISTIC NEUTROSOPHIC SETS 3.1. Definition of ULNSs and ULNNs Definition 10. Let X be a given ordinary fixed set and S  s0 , s1 ,..., st  be a given linguistic term set, then a linguistic neutrosophic sets A defined on X is expressed as A

  x, s

  x

, s  x  ,  s  x , s  x   ,  s  x , s  x  





x X ,

(14)

where  s  x , s  x  ,  s  x , s  x   ,  s  x  , s  x   S are uncertain linguistic variables, denoted the truth-membership degree, the indeterminacy-membership degree and the falsitymembership degree, respectively, such that    ,    and    . For convenience, the

 s

ordered pair

  x

, s  x   ,  s  x  , s  x   ,  s  x  , s  x  



is called an uncertain linguistic





neutrosophic number (ULNN), which can be denoted as    s , s  ,  s , s  ,  s , s  for simplicity. Especially, the ULNN reduced to the LNN when    ,    and    .

3.2. Operational Rules of ULNNs







Definition 11. Let 1   s1 , s1  ,  s1 , s1  ,  s1 , s1  ,  2   s2 , s2  ,  s2 , s2  ,  s 2 , s2 







and    s , s  ,  s , s  ,  s , s  be any three ULNNs and  be a positive real number, then   

1   2 =   s

1  2 

1 2 t

,s

1 2 

12 t

      ,  s 12 , s12  ,  s1 2 , s12   ; t   t t    t

(15)

292

Yuan Xu, Xiaopu Shang and Jun Wang 

 

1   2 =   s  , s   ,  s   + 

1 2

1 2

t

t

 

1



    s  

  t  t 1   t 



,s



  t  t 1   t



12

,s

t

1 +2 

t  t 

t  t

   

12 t

    ,  s   1 2 , s  12   ; 1 2 t    1 2 t

      , s  , s   , s  , s    ;   t  t  t  t    t  t  t  t        

 

     s    , s     ,  s  

2



 t  t 1   t 



,s



 t  t 1   t



    , s  .  ,s      t t 1 t  t  t 1 t      

(16)

(17)

(18)

Example 1. Let 1   s6 , s7  ,  s3 , s4 ,  s4 , s5  and  2   s4 , s5  ,  s3 , s4  ,  s4 , s6  be two ULNNs defined on a LTS S  si i  0,1,...,8 . According to Definition 11, we can obtain the following results (1) 1   2   s7 , s7.625  ,  s1.125 , s2  ,  s2 , s3.75  ;

(19)

(2) 1  2   s3 , s4.375  ,  s4.875 , s6  ,  s6 , s7.25  ;

(20)

(3) 21   s4.5 , s6.125  ,  s4.875 , s6  , s6 , s6.875  ;

(21)

(4) 12   s7.5 , s7.875  ,  s1.125 , s2  , s2 , s3.125  .

(22)

Theorem 1 Let 1 ,  2 and  be any three ULNNs, then (1) 1   2   2  1 ;

(23)

(2) 1   2   2  1 ;

(24)

(3)  1   2    2  1 ,   0 ;

(25)

(4) 1  2   1  2   , 1 , 2  0 ;

(26)

(5) 1   2   2  1  ,   0 ;

(27)



Multi-Attribute Group Decision-Making … (6)         1

1  2

2

293

, 1 , 2  0 .

(28)

3.3. Comparison Method of ULNNs





Definition 12. Let    s , s  ,  s , s  ,  s , s  be a ULNN, then the score function of

 is defined as 4t            , 6t

S   

(29)

and the accuracy function is defined as

H   

For



2t        . 4t

any

(30)

two

ULNNs

 2   s , s  ,  s , s  ,  s , s  2

2

2

2

2

2



1   s , s  ,  s , s  ,  s , s  1

1

1

1

1

1



and



If S 1   S  2  , then 1   2 ; If S 1   S  2  , then If H 1   H  2  , then 1   2 ; If H 1   H  2  , then 1   2 . Example 2. Suppose that there are two ULNNs defined on an LTS S  si i  0,1,...,8 ,

1   s6 , s7  ,  s3 , s4 ,  s4 , s5  ,  2   s4 , s5  ,  s3 , s4  ,  s4 , s6  , 3   s5 , s8  ,  s2 , s5  ,  s4 , s5  . According to Definition 12, we can obtain that 48  6  7  3  4  4  5 28  6  7  4  5 S 1    0.6250 .  0.6042 , H 1   48 68 Similarly, we can get S  2  =0.5000 , S  3  =0.6042 , H  2  =0.4688 , H  3  =0.6250 . Obviously, we can find that S 1   S  2  , then 1   2 ; that S 1   S  3  and H 1   H  3  , then 1   3 .

294

Yuan Xu, Xiaopu Shang and Jun Wang

3.4. Distance Measure between two ULNNs





Definition 13. Let 1   s , s  ,  s , s  ,  s , s  and  2    s , s  ,  s , s  ,  s , s   be any two ULNNs, then the distance between 1 and  2 is defined as d 1 ,  2  

1

1

1

1

1

1

2

1   2  1  2  1  2  1  2   1   2  1  2 6t

2

2

2

,

2

2

(31)

Example 3. Suppose that there are two ULNNs defined on an LTS S  si i  0,1,...,8 ,

1   s6 , s7  ,  s3 , s4 ,  s4 , s5  ,  2   s4 , s5  ,  s3 , s4  ,  s4 , s6  . According to Definition 13, we can obtain that d 1 ,  2  

6  4  7 5  33  4 4  4 4  5 6 68

 0.1042 .

4. AGGREGATION OPERATORS FOR ULNNS In the section, we propose some series of aggregation operators for ULNNs and study their properties.

4.1. The Uncertain Linguistic Neutrosophic Power Average Operator Definition 14. Let  i  i  1, 2,..., n  be a collection of ULNNs, then the uncertain linguistic neutrosophic power average (ULNPA) operator is defined as  1  T  i    i n

ULNPA 1 ,  2 ,...,  n  

i 1 n

 1  T  i  

,

(32)

i 1

where T  i  

n



i 1, i  j





Sup  i ,  j  and Sup i ,  j denotes the support degree for  i from  j ,

satisfying the following properties: 1) 0  Sup i ,  j   1 ;

(33)

2) Sup i ,  j   Sup  j , i  ;

(34)

Multi-Attribute Group Decision-Making … 3) Sup i ,  j   Sup  s , t  , if d i ,  j   d  s , t  .

295 (35)

To simplify Eq. (32), let i 

1  T  i   1  T  i   n

,

(36)

i 1

then, Eq. (32) can be written as n

ULNPA 1 ,  2 ,...,  n     i i ,

(37)

i 1

n

where   i  1 and 0  i  1 . i 1





Theorem 2. Let i   si , si  ,  si , si  ,  s i , si  i  1, 2,..., n  be a collection of ULNNs, the aggregated value by the ULNPA operator is also an ULNN and       , ULNPA 1 ,  2 ,...,  n     s n  i , s n  i  ,  s n  i , s n  i  ,  s n  i , s n  i     t t  1 i  t t  1 i    t   i  t   i    t   i  t   i    t    i 1  t  t    i 1  t  t   i 1  i 1  i 1    i1  t  

Proof. According to Definition 11, we can obtain that 

 i i    s  

i

   t  t 1 i  t  

,s

i

   t  t 1 i   t 

      ,  s i , s i  ,  s i , s i   ,   t  i  t  i    t   i  t  i     t     t   t     t

then,       n   i  i    s n  i , s n  i  ,  s n  i , s n  i  ,  s n  i , s n  i     t t  1 i  t t  1 i    t   i  t   i    t   i  t   i    i 1 t    i 1  t  t    i 1  t  t   i 1  i 1  i 1    i1  t  

Therefore, the proof of Theorem 2 is completed.

(38)

296

Yuan Xu, Xiaopu Shang and Jun Wang Theorem 3 (Idempotency).





Let i   si , si  ,  si , si  ,  s i , si  i  1, 2,..., n  be



a

series

of

ULNNs,

if



i     s , s  ,  s , s  ,  s , s  holds for all i, then ULNPA 1 ,  2 ,...,  n    ,

(39)

Theorem 4 (Boundedness).





Let i   si , si  ,  si , si  ,  s i , si  i  1, 2,..., n  be a series of ULNNs, then ULNPA  ,   ,...,     ULNPA1 ,  2 ,...,  n   ULNPA  ,   ,...,    ,

(40)

where   

    s

n

max i

,s

n

max i

i 1

i 1

     , s n  , s n , s n   , n  ,  smin i min i min  i min i  i 1 i 1   i1    i1

and 

    s  

n

min i i 1

     , s n , s n  , s n , s n   .  min i i max i i maxi  i 1   max i 1 i 1 i 1 i 1   max 

,s

n

4.2. The Uncertain Linguistic Neutrosophic Power Weighted Average Operator Definition 15. Let  i  i  1, 2,..., n  be a collection of ULNNs and w   w1 , w2 ,..., wn  be T

the weight vector, such that 0  wi  1 and



n i 1

wi  1 . The uncertain linguistic

neutrosophic power weighted average (ULNPWA) operator is defined as n

ULNPWA 1 ,  2 ,...,  n  

 w 1  T    i 1 n

i

i

 w 1  T   i 1

i

i

i

,

(41)

Multi-Attribute Group Decision-Making … where T  i  

n



i 1, i  j



297



Sup  i ,  j  and Sup i ,  j denotes the support degree for  i from  j ,

satisfying the properties presented in Definition 14. To simplify Eq. (41), let

i 

wi 1  T  i   n

 wi 1  T  i  

,

(42)

i 1

then, Eq. (17) can be written as n

ULNPWA 1 ,  2 ,...,  n    i i ,

(43)

i 1

n

where  i  1 and 0  i  1 . i 1





Theorem 5. Let i   si , si  ,  si , si  ,  s i , si  i  1, 2,..., n  be a series of ULNNs, then the aggregated value by the ULNPWA operator is also an ULNN and  ULNPWA 1 ,  2 ,...,  n     s n  i , s n  i   t t  1 i  t t  1 i  t  i 1    i1  t 

    ,  , s n  , s n   ,s n   ,s n   i  i    i  i  i  i     t   i  i t  t  t     t t t t         i 1 i 1   i1   i1 

(44)

4.3. The Uncertain Linguistic Neutrosophic Power Hamy Mean (ULNPHM) Operator Definition 16. Let  i  i  1, 2,..., n  be a collection of ULNNs and k  1, 2,..., n , the uncertain linguistic neutrosophic power Hamy mean (ULNPHM) operator is expressed as   k n 1  T  i    i 1 j ULNPHM  k  1 ,  2 ,...,  n   k   n Cn 1i1 ...ik  n  j 1 1  T  i     i 1 

where T  i  

n



i 1, i  j



1k

   ,   



(45)

Sup  i ,  j  and Sup i ,  j denotes the support degree for  i from  j ,

satisfying the properties presented in Definition 14. If we assume

298

Yuan Xu, Xiaopu Shang and Jun Wang i 

1  T  i  n

 1  T    i 1

,

(46)

i

then Eq. (45) can be written as ULNPHM  k  1 ,  2 ,...,  n  

1 Cnk

1k

 k    n i j  i j  1 i1 ... ik  n  j 1  

,

(47)

where we call   1 , 2 ,..., n  be the power weight vector, such that 0   i  1 and T



i  1 .

n

i 1





Theorem 6. Let i   si , si  ,  si , si  ,  s i , si  i  1, 2,..., n  be a collection of ULNNs, the aggregated value by the ULNPHM operator is also an ULNN and    k  ULNPHM 1 ,  2 ,...,  n  =   s 1 ,s 1 1 k  k 1 k  k      Cn Cn n n       k    i j  i j       k   i j  i j             t t   t t   1  11 t       1  1 1 t       1i ...i n   j 1        1i1...ik n   j1      1 k               

   s 1 ,s 1 1 k  k 1 k  k      Cn Cn n i   n    j k           k   i j  i j     ij        t   1   1  1   1   t  1i           1i ...i n   j 1   t     t      1 k        1 ...ik n   j1          

   ,   

   s 1 ,s 1 1 k  k 1 k  k      Cn Cn n i   n    j k     ij       k   i j  i j            t   1   1  1   1   t  1i           1i ...i n   j 1   t     t      1 k        1 ...ik n   j1          

   .    

   ,   

(48)

Proof. According to Definition 11, we can obtain that   n i j  i j    s n , s n  i j  i j  i j  i j t  t 1    t t 1 t   t     

and,

     ,  s  i j ni j , s  i j ni j  t   t  t   t   

     ,  s   i j ni j , s  i j ni j  t   t  t   t   

   ,   

Multi-Attribute Group Decision-Making …      n i j  i j    s k  n i  , s n i   ,  j j k j 1   t  11 i j   t  11 i j      j1   t   j1   t         k

         s k   i n i j  , s k   i n i j   ,  s k    i n i j  , s k   i n i j    .  t  t  1 t j   t  t  1 t j     t  t  1 t j   t  t  1 t j         j 1  j 1  j 1      j1              

Therefore,

1k

  n i j  i j   j  1   k

    ,   s 1k ,s 1k  k    n i j      k   i j ni j   ij      t   1 1  1 1 t       t   t     j 1   j 1              

        s    , s 1k ,s 1k 1k ,s 1k  t t  k 1 i j n i j   t t  k 1 i j n i j     t t  k 1  i j n i j   t t  k 1 i j n i j       t    t    t    t                  j 1    j 1    j 1   j 1                      

and,



1 i1 ... ik  n

1k

  n i j  i j    j 1  k

    ,    s 1k ,s 1k    n n    k   i j  i j      k   i j  i j           t t  1  11 t     t t  1  11 t        j 1    1i1...ik n   j1   1i1...ik  n                 

    s , ,s  1 k  1 k      n n       k   i j  i j       k   i j  i j      t   1  1 t      t   1  1 t         1i1...ik n   j1        1i1...ik n   j1             s  . ,s  1 k  1 k      n i   n i        k k   i  j  i  j      t   1  1 t j      t   1  1 t j          1i1...ik n   j1        1i1...ik n   j1         

Finally, we can get

299

300

Yuan Xu, Xiaopu Shang and Jun Wang

1 Cnk

1k



1 i1 ... ik  n

 k  n i j  i j    j 1 

     s 1 ,s 1 1 k  k 1 k  k      Cn Cn n n    k    i  i j        k   i j  i j      t  t   1   1 1     t t   1  11 t j        t   1i ...i n   j 1          1i1...ik n   j 1     1 k             

   s 1 ,s 1 1 k  k 1 k  k      Cn Cn n n      k   i j  i j       k   i j  i j         t   1   1  1   1   t  1i           1i ...i n   j 1   t     t      1 k        1 ...ik n   j1          

   ,   

   s 1 ,s 1 1 k  k 1 k  k      Cn Cn n n      k    i j  i j       k   i j  i j         t   1   1  1   1   t  1i           1i ...i n   j 1   t     t      1 k        1 ...ik n   j1          

   .    

   ,   

Moreover, the ULNPHM operator have the following properties. Theorem 7 (Idempotency).





Let i   si , si  ,  si , si  ,  s i , si  i  1, 2,..., n  be a collection of ULNNs, if





i     s , s  ,  s , s  ,  s , s  for all i, then ULNPHM  k  1 ,  2 ,...,  n  = .



Proof. Since i     s , s  ,  s , s  ,  s , s 

(49)



holds for all i, we can get

Sup i ,  j   1 for i, j  1, 2,..., n ; i  j .Thus, we have  i  1 n  i  1, 2, , n  . According to

Theorem 6, we can obtain

1 Cnk



1 i1 ... ik  n

1k

  n i j  i j    j 1  k

     s 1 ,s 1 1 k  k 1 k  k      Cn Cn n n       k    i j  i j       k   i j  i j      t  t   1   1 1    t t   1  11 t          t   1i ...i n   j 1          1i1...ik n   j 1     1 k             

   s 1 ,s 1 1 k  k 1 k  k      Cn Cn n i   n    j k   i j       k   i j  i j            t   1   1  1   1   t  1i           1i ...i n   j 1   t     t      1 k        1 ...ik n   j1          

   ,   

   ,   

Multi-Attribute Group Decision-Making …    s 1 ,s 1 1 k  k 1 k  k      Cn Cn n n      k    i j  i j       k   i j  i j         t   1   1  1   1   t  1i           1i ...i n   j 1   t     t      1 k        1 ...ik n   j1          

     s 1 ,s 1   1 k  k     k    1 k   Cnk Cn k    ij ij         t t   1      t  t   1         1i1...ik n   j 1  t      1i1...ik n   j 1  t          

  s 1 ,s 1      k    1 k   Cnk   k    1 k   Cnk i ij       t   1  1 j         t   1   1  t      1i1...ik  n   j 1     1i1...ik n   j1  t      

  ,   

  s 1 ,s 1      k    1 k   Cnk   k    1 k   Cnk i ij     t   1  1 j         t   1   1  t      1i1...ik n   j 1     1i1...ik n   j1  t      

      

    s 1 ,s 1      i j   Cnk  i j   Cnk        t t 1  t  t 1    1i       t    1i1...ik n      1 ...ik n  t     s 1 ,s 1   i j   Cnk  i j   Cnk  t     t     t    1i1...ik n  t     1i1...ik n 

301

       

  ,   

  ,  

     , s 1 ,s 1    i j   Cnk  i j   Cnk   t     t     t      1i1...ik n  t     1i1...ik n 

     

           s  i  , s  i   ,  s  i  , s  i   ,  s   i  , s  i     t t 1 j  t t 1 j    t  j  t  j    t  j  t  j     t    t    t   t  t     t              





  si , si  ,  si , si  ,  s i , si    i j   . j   j j   j j   j

Theorem 8 (Boundedness).





Let i   si , si  ,  si , si  ,  s i , si  i  1, 2,..., n  be a collection of ULNNs, then ULNPHM   ,   ,...,     ULNPHM 1 ,  2 ,...,  n   ULNPHM   ,   ,...,    ,

where

(50)

302

Yuan Xu, Xiaopu Shang and Jun Wang   

    s

n

max i

,s

n

max i

i 1

i 1

     , s n  , s n , s n   , n  ,  smin i min i min  i min i  i 1 i 1   i1    i1

and   

    s

     , s n , s n  , s n , s n   .  min i i max i i maxi  i 1   max i 1 i 1 i 1 i 1   max 

,s

n

min i i 1

n

Proof. According to Theorem 6 and Theorem 7, it is easy to obtain that s

      t t  1  1i ...i n    1 k    



k





 11 j 1





i j

1 1 k  k Cn n  ij  

 t 

   

n

s

         t t  1  1i1...ik n         

   



  n   min  i 1 1 i 1 j   t j 1     k



 min  i j .

1 1 k  k Cn n  ij  

      

    

i 1

     

Similarly, we have n

n

s

      t t  1  1i ...i n    1 k    



1 1 k  k Cn n  ij  

   1 1 i j    t  j 1    k



   

 min i j , s i 1

      t 1  1i ...i n    1 k    



   

1 1 k  k Cn n  ij  

  1 i j    t  j 1     k



   

      t 1  1i ...i n    1 k    



k

1 1 k  k Cn n  ij  

k



1 1 k  k Cn n  ij  

j 1



  1  i j    t  j 1    



   

n

 max i j , s i 1

      t 1  1i ...i n    1 k    



   

i 1

   

n

s

 max i j ,

1 1 k  k Cn n  ij  

  1  i j    t  j 1     k



   

 max  i j , i 1

   

n

s

      t 1  1i ...i n    1 k    



 i j

 1

   t 

   

 max i j . i 1

   

According to Definition 12, it is easy to obtain that ULNPHM   ,   ,...,     ULNPHM

1 ,  2 ,...,  n  . Similarly, we have

ULNPHM 1 ,  2 ,...,  n   ULNPHM   ,   ,...,    .

Therefore, ULNPHM   ,   ,...,     ULNPHM 1 ,  2 ,...,  n   ULNPHM   ,   ,...,    . In the following, we discuss some special cases of the ULNPHM operator with respect to the parameter k. Special Case 1: When k = 1, the ULNPHM operator is reduced to the uncertain linguistic neutrosophic power average (ULNPA) operator.

Multi-Attribute Group Decision-Making … ULNPHM k k1 1 ,  2 ,...,  n  

1 Cnk

1k

 k    n i j  i j  1 i1 ... ik  n  j 1  

303

n

   i i  i 1

        , s    .   s  n i  , s i  i  , s  n i  , s  n i  , s  n i  n n      t   i   t   i     t    i   t   i      i    i  1   t  t   1         t t            t   i 1      i1  t    i1  t      i1  t    i1  t        i1  t  

(51)

In this case, if Sup i ,  j   t  0 for all i  j , then the ULNPHM operator reduces to the uncertain linguistic neutrosophic average (ULNA) operator, i.e., ULNPHM k k1 1 ,  2 ,...,  n  

1 Cnk

1k

 k    n i j  i j  1 i1 ... ik  n  j 1  



1 n  i n i 1

            .   s 1 ,s 1  , s 1 ,s 1  , s 1 ,s 1  n n n n n n   t t   1 i   n t t   1 i   n   t   i  n t  i  n   t    i  n t  i  n              i 1  t      i 1   i 1     i 1   i 1       i1  t  

(52)

Special Case 2: When k = n, the ULNPHM operator is reduced to the uncertain linguistic neutrosophic power geometric (ULNPG) operator. ULNPHM k kn 1 ,  2 ,...,  n  

1 Cnk

1k

 k    n i j  i j  1 i1 ... ik  n  j 1  

1n

 n     n j  j   j 1 

         , s ,   s 1 ,s 1 1 ,s 1    n    j n j   n  n    j n j   n    n    j n j   n  n    n j   n  j                   t   11 t    t   11 t      t t   1 t    t t   1 t             j 1   j 1   j 1     j 1      s  . 1 ,s 1  n    n j   n     n    j n j   n j  1   t  t   1    t t             t     j 1      j 1   t   

(53)

In this case, if Sup i ,  j   t  0 for all i  j , then the ULNPHM operator reduces to the uncertain linguistic neutrosophic geometric (ULNG) operator. ULNPHM k kn 1 ,  2 ,...,  n  

1 Cnk

1k

 k    n i j  i j  1 i1 ... ik  n  j 1  

1

   j  n n

j 1

            .   s 1 ,s 1  , s 1 ,s 1  , s 1 ,s 1  n n n n n n   j n   j n   j n   j n   j n   j n   t    t      t t  1  t t  1    t t  1  t t  1    t  t  t  j 1  j 1  j 1    j1  t    j1  t     j1  t 

(54)

304

Yuan Xu, Xiaopu Shang and Jun Wang

4.4. The Uncertain Linguistic Neutrosophic Power Weighted Hamy Mean (ULNPWHM) Operator Definition 17. Let  i  i  1, 2,..., n  be a collection of ULNNs and k  1, 2,..., n . Let w   w1 , w2 ,..., wn  be the weight vector, such that 0  wi  1 and i 1 wi  1 . The uncertain T

n

linguistic neutrosophic power weighted Hamy mean (ULNPWHM) operator is expressed as   k nwi 1  T  i    i 1 j ULNPWHM  k  1 ,  2 ,...,  n   k   n j Cn 1i1 ...ik  n  j 1 wi 1  T  i     i 1 

where T  i  

n



i 1, i  j



1k

   ,   

(55)



Sup  i ,  j  and Sup i ,  j denotes the support degree for  i from  j ,

satisfying the properties presented in Definition 14. If we assume that

i 

wi 1  T  i   n

 wi 1  T  i  

,

(56)

i 1

then Eq. (55) can be written as ULNPWHM  k  1 ,  2 ,...,  n  

1 Cnk

1k

 k    nwi j  i j  , 1 i1 ... ik  n  j 1  

(57)

where we call w   w1 , w2 ,..., wn  be the power weight vector, such that 0  wi  1 and T



n i 1

wi  1 .





Theorem 9. Let i   si , si  ,  si , si  ,  s i , si  i  1, 2,..., n  be a collection of ULNNs, the aggregated value by the ULNPWHM operator is also an ULNN and     k ULNPWHM   1 ,  2 ,...,  n     s 1 ,s 1   1  k 1  k    Cn Cn    k    nwi j   k     k    nwi j   k      ij ij                  t  t   1   1 1    t t  1i      1   1 1 t   t        1i1...ik  n   j 1          1 ...ik n   j 1          

   ,    

Multi-Attribute Group Decision-Making …    s 1 ,s 1   1  k 1  k    C C nwi   k   n nwi   k   n         j j n n     i j      i j      1   t   1   1         1  t   t  1i1      j 1    j 1   t  ...ik  n  1i1...ik  n                          s 1 ,s 1   1  k 1  k    Cn Cn    n    nwi j   k     n    nwi j   k     ij     ij      1   t   1   1         1  t   t  1i1  t       j 1   ...ik  n     1i1...ik n   j 1                   

305

   ,    

    .     

(58)

Proof. According to Definition 11, we can obtain that   nwi j  i j    s nw , s nw  i j  i j  i j  i j t  t 1    t t 1 t    t     

     ,  s  i j nwi j , s  i j nwi j  t   t  t   t   

     ,  s   i j nwi j , s  i j nwi j  t   t  t   t   

Therefore,      nwi j  i j    s k  ,s k  , nw nw i j  i j  i j  i j    j 1   t  11   t  11      j1   t   j1   t         k

         s n   i nwi j  , s n   i nwi j   ,  s n    i nwi j  , s n   i nwi j    .  t t  1 t j   t t  1 t j     t t  1 t j   t t  1 t j         j 1  j 1  j 1      j1              

and,

1k

 k  nwi  i    j 1 j j 

    ,    s 1 ,s 1   k    nwi j   k  k    nwi j   k    t   11 i j    t   11 i j        j1   t     j1   t          

   .   

306

Yuan Xu, Xiaopu Shang and Jun Wang         s    , ,s , s 1 1 ,s 1    n    nwi j   1k   nw nw nw k k k i i i  n     j   n     j   n    j    t t   1 ti j    t t   1 ti j      t t   1 ti j    t t   1 ti j             j 1    j 1    j 1        j1                  

Furthermore,

1k

 k    nwi j  i j  1 i1 ... ik  n  j 1  

        s 1 ,s 1  ,     k    nwi j   k    k    nwi j   k    ij ij        1    t  t  1   1 1   t t 1i          1 1 t  t         j 1   ...  i  n j  1 1  i  ...  i  n  1 k  1 k                    

             s 1 ,s 1   , s 1 ,s 1       nw nw nw nw   n     i j k    n     i j k      n     i j k    n    i j k     ij    1  i j    1  i j    1  i j    1  1  1   t  1   1  t t t               1i1...ik n   j1   t     1i1...ik n   j1   t       1i1...ik n   j1   t     1i1...ik n   j 1   t                                       

and,

1 Cnk

1k



1 i1 ... ik  n

 k  nwi  i    j 1 j j 

      s 1 ,s 1   1  k 1  k    Cn Cn    k    nwi j   k     k    nwi j   k      i i j j         1   t  t   1   1 1    t t  1i         1 1 t   t        1i1...ik n   j 1          1 ...ik n   j 1          

   s 1 ,s 1   1  k 1  k    C C nwi   k   n nwi   k   n         j j n n     i j      i j      1   t   1   1         1  t   t  1i1      j 1    j 1   t  ...ik  n  1i1...ik  n                          s 1 ,s 1   1  k 1  k    Cn Cn    n    nwi j   k     n    nwi j   k     ij     ij      1   t   1   1         1  t   t  1i1  t       j 1   ...ik  n     1i1...ik n   j 1                   

   ,    

   ,    

    .     

Thus, the proof of Theorem 9 is completed. In addition, it is easy to prove that the ULNPWHM operator has the property of boundedness.

Multi-Attribute Group Decision-Making …

307

5. A NOVEL MAGDM METHOD UNDER ULNNS In the above sections, we have demonstrated the powerfulness and strongness of ULNSs and their AOs. In this section, we investigate the application of ULNSs and their AOs in MAGDM. Let’s consider a MAGDM problem in which DMs depict their evaluations in the form of uncertain linguistic neutrosophic information. Suppose that there are m alternatives that to be evaluated, which can be denoted as A   A1 , A2 ,..., Am  . Let C = C1 , C2 ,..., Cn  be a set of attributes, whose weight vector is w   w1 , w2 ,..., wn  , T

n satisfying the condition that i 1 wi  1 and 0  wi  1 . Let Dh  h  1, 2, , l  be a set of DMs

with the weight vector  = 1 ,2 , ,l  such that 0  h  1 and h1h  1 . The DM Dh T



l



employs an ULNN  ijh   sh , sh  ,  sh , sh  ,  sijh , sijh  to express his/her evaluation value ij

ij

ij

ij

with regard to alternative Ai  i  1, 2,..., m  under attribute C j  j  1, 2, .n  based on the LTS, such as S = {s0 = extremely poor, s1 = very poor, s2 = poor, s3 = slightly poor, s4 = medium, s5 = slightly better, s6 = good, s7 = very good, s8 = prefect}. Hence, a set of ULN decision matrices Rh  ijh mn are obtained. In the following, we provide the main steps of choosing the optimal alternative based on the proposed AOs. Step 1: Standardize the original ULN decision-making information. Generally, there are two types of attributes, benefit type and cost type. Hence, before determining the best alternative, the original decision matrices should be normalized according to the following formula





  sh , sh  ,  sh , sh  ,  sh , sh  for the benefit attribute C j  ij ij   ij ij   ij ij     ,   sthij , sthij  ,  sthij , sth ij  ,  sthij , sth ij  for the cost attribute C j        h ij





(59)

where h  1, 2, , l ; i  1,2, , m ; j  1, 2, , n . Step 2: Calculate the support Sup ijk , ijd  according to the following formula Sup ijk , ijd   1  d ijk , ijd  ,

(60)

where k , d  1, 2, , l; k  d ; i  1, 2 , m; j  1, 2, , n and d ijk , ijd  is the Hamming distance between  ijk and  ijd .

308

Yuan Xu, Xiaopu Shang and Jun Wang Step 3: Calculate T ijk  by T  ijk  

n



k 1, k  d

Sup  ijk ,  ijd  ,

(61)

where k , d  1, 2, , l; k  d ; i  1, 2 , m; j  1, 2, , n . Step 4: Compute the power weight ijk associated with the ULNN  ijk by

ijk 



k 1  T  ijk 





 k 1  T  ijk  t

k 1



,

(62)

l where k , d  1, 2, , l; k  d ; i  1, 2 , m; j  1, 2, , n , ijk  0 and  k 1ijk  1 .

Step 5: Utilize the ULNNPWHM operator to aggregate individual decision matrix, i.e.,

ij  ULNNPWHM ij1 , ij2 , , ijt  ,

(63)

Step 6: Calculate the supports Sup ipk , ifd  by Sup ipk , ifd   1  d ipk , ifd  ,

(64)

where i  1,2, , m ; p, f  1, 2, , n ; p  f and d ip , if  is the Hamming distance between  ip and  if .

Step 7: Compute T  ij  by T  ijk  

n



f , p 1, f  p

Sup  ip ,  if  ,

(65)

where i  1, 2, , m; p, f  1, 2, , n; p  f . Step 8: Calculate the power weight  ij associated with ULNN  ij according to the following formula

Multi-Attribute Group Decision-Making …

 ij 



w j 1  T  ij 



 w 1  T    n

j 1

j

309

,

(66)

ij

where i  1, 2, , m; j  1, 2, , n . Step 9: For each alternative, utilize the proposed ULNNPWHM operator to compute the overall evaluation values, i.e.,  ij  ULNPWHM  k   i1 ,  i 2 ,...,  in  ,

(67)

and a series of overall evaluation values  i  i  1, 2, , m  are obtained. Step

10:

Compute

the

scores

of

 i  i  1, 2, , m 

according

to

Definition 12. Step 11: Rank the corresponding alternatives and select the optimal one.

6. NUMERICAL EXAMPLES In this subsection, we utilize some examples to show the application process and the advantages of the proposed method. Example 4. There are four companies as a set of alternatives A   A1 , A2 , A3 , A4  , where A1 is a car company; A2 is a food company; A3 is a toy company and A4 is a arms company. There are five attributes to be considered: (1) C1 is the geological risk analysis; (2) C2 is the production risk analysis; (3) C3 is the market risk analysis; (4) C4 is the management risk analysis; (5) C5 is the social environment analysis, whose weighted vector is w   0.3,0.2,0.1,0.3,0.1 . A group of three DMs Dh  h  1, 2,3 are invited to T

evaluate the attribute values by the LTSs S = {s0 = extremely poor, s1 = very poor, s2 = poor, s3 = slightly poor, s4 = medium, s5 = slightly better, s6 = good, s7 = very good, s8 = prefect}and assume the weight vector of the three DMs is  =  0.35,0.35,0.3 . By the T

above information, the company needs to consider the interrelationship among attributes through parameter k. Suppose that parameter k = 2 and DMs Dh  h  1, 2,3 give the evaluate values of alternatives Ai  i  1, 2,3, 4  with respect to the attributes C j  j  1, 2,3, 4,5  by ULNN, then the decision matrices Rh  ijh mn are constructed and listed in Table 1-3 (See Appendixes).

310

Yuan Xu, Xiaopu Shang and Jun Wang

6.1. Procedure of Decision Making Based on the ULNNPWHM Operator Step 1: As all the attributes are cost type, then we normalize the evaluation values by Eq. (59). The normalized decision-making matrices are shown in Tables 4-6 (See Appendixes). Step 2: Calculate the Sup ijk , ijd  according to Eq. (60). For convenience, we utilize the symbol

S dk

to

represent

the

support

between

 ijk

and

 ijd

 i  1, 2,3, 4; j  1, 2,3, 4,5; k , d  1, 2,3; k  d  . Hence, we obtain the following results 0.7917  0.8958 S21  S12    0.8542  0.7917

0.8125 0.6458 0.8125 0.8125  0.7500 0.8333 0.7500 0.8750  , 0.8125 0.8958 0.8542 0.6458   0.8750 0.8542 0.9167 0.8333

 0.8958 0.7917 S31  S13    0.8333  0.9167

0.8125 0.6458 0.8958 0.9583 0.9167 0.7292 0.8542 0.8333 , 0.8542 0.8333 0.7708 0.8333  0.8333 0.8958 0.9375 0.8750 

 0.8542  0.8542 S32  S23    0.8542  0.7917

0.8750 0.7500 0.8333 0.8125 0.7917 0.7708 0.8125 0.7917  . 0.9583 0.7708 0.8750 0.8125  0.8333 0.8750 0.8958 0.8333

Step 3: Calculate T ijk  according to Eq. (61). For convenience, we use the symbol T k to represent the values T ijk   i, j  1, 2,3, 4; k  1, 2,3 1.6875 1.6875 T1   1.6875  1.7083

1.6250 1.2917 1.7083 1.7708 1.6667 1.5625 1.6042 1.7083 , 1.6667 1.7292 1.6250 1.4792   1.7083 1.7500 1.8542 1.7083

1.6458 1.7500 T2   1.7083  1.5833

1.6875 1.3958 1.6458 1.6250 1.5417 1.6042 1.5625 1.6667  , 1.7708 1.6667 1.7292 1.4583  1.7083 1.7292 1.8125 1.6667 

1.7500 1.6458 T3   1.6875  1.7083

1.6875 1.3958 1.7292 1.7708 1.7083 1.5000 1.6667 1.6250  . 1.8125 1.6042 1.6458 1.6458  1.6667 1.7708 1.8333 1.7083

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Step 4: For DM Dk , calculate his/her power weight associated with the ULNN  ijk on the basis of his/her weight k according to Eq. (62). For convenience, we use the symbol  k to represent the values  ijk  i, j  1, 2,3, 4; k  1, 2,3 . Therefore, we can obtain the following results  0.3495  0.3488 1=   0.3491  0.3557

0.3447 0.3400 0.3520 0.3566 0.3542 0.3506 0.3494 0.3552 , 0.3398 0.3578 0.3444 0.3441  0.3516 0.3501 0.3526 0.3519 

0.3440 0.3569 2=  0.3518   0.3393

0.3529 0.3554 0.3439 0.3378  0.3375 0.3563 0.3438 0.3497  , 0.3531 0.3496 0.3581 0.3412   0.3516 0.3475 0.3474 0.3465 

 0.3065  0.2943 3=  0.2992  0.3049

0.3041 0.3056  0.3083 0.2932 0.3067 0.2951 . 0.3072 0.2926 0.2975 0.3147   0.2968 0.3024 0.3000 0.3016  0.3025 0.3046

Step 5: Utilize the ULNPWA operator to aggregate individual decision matrices into a collective one, as shown in Table 7 (See Appendixes). The calculation process of the ULNPWA operator can be found as Eq. (63). Step 6: Calculate the support between  ip and  if , that is, Sup ip , if  , according to Eq. (64). For convenience, we utilize the symbol S pf to represent the value Sup ip , if 

 i  1, 2,3, 4; p, f

 1, 2,3, 4,5; p  f  . Hence, we can obtain the following results

S 12  S 21   0.8796 0.9152 0.9554 0.9606  , S 13  S 31   0.7302 0.8149 0.9485 0.9788  , S 14  S 41   0.8940 0.9415 0.8930 0.9493 ,

S 15  S 51   0.8375 0.8618 0.8229 0.9834  , S 23  S 32   0.8094 0.8815 0.9311 0.9403 , S 24  S 42   0.9271 0.9130 0.9154 0.9121 , S 25  S 52   0.9143 0.9316 0.7809 0.9458  ,

S 34  S 43   0.7365 0.7983 0.9441 0.9438  ,

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Yuan Xu, Xiaopu Shang and Jun Wang S 35  S 53   0.7378 0.9094 0.8253 0.9650  , S 45  S 54   0.9290 0.8452 0.8099 0.9482  .

Step 7: Calculate the support T  ij  according to Eq. (65). Similarly, we utilize the symbol Tij to denote the value T  ij  for simplicity, and we can obtain the following matrix 3.3412 3.5334 T  3.6198  3.8722

3.5304 3.0139 3.4866 3.4186  3.6414 3.4040 3.4980 3.5480  . 3.5827 3.6491 3.5625 3.2389   3.7587 3.8278 3.7535 3.8424 

Step 8: Calculate the power weight  ij associated with the ULNN  ij according to Eq. (66), and we have 0.2962 0.3000  0.3039   0.3041

0.2060 0.0913 0.3061 0.1005 0.2048 0.0972 0.2977 0.1003 . 0.2010 0.1020 0.3002 0.0930   0.1980 0.1004 0.2967 0.1007 

Step 9: For alternative Ai  i  1, 2,3, 4  , utilize the ULNPWHM operator to calculate the overall evaluation  i  i  1, 2,3, 4  . Without the loss of generality, let k  2 , and the overall evaluation values are shown as follows:

1   s4.6189 , s5.6866 ,  s3.3085 , s4.2580 ,  s2.6801 , s3.5121  ,

2   s4.7708 , s5.9383 ,  s2.8127 , s4.0242 ,  s2.1058 , s2.7995  , 3   s4.7404 , s6.5504 ,  s3.2580 , s4.3729 ,  s2.1217 , s3.0372  ,

4   s5.3587 , s6.4666  ,  s2.0668 , s3.0811 ,  s2.2541 , s3.2930  . Step 10: Calculate the score values S  i  i  1, 2,3, 4  of the overall evaluation values, and we can obtain S 1   0.5947 , S  2   0.6451 , S  3   0.6354 , S  4   0.6902 .

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Step 11: According to the score values S  i  i  1, 2,3, 4  , the ranking orders of the alternatives can be determined, that is, A4

A2

A3

A1 . Therefore, A4 is the best

alternative.

6.2. Sensitivity Analysis As aforementioned, the parameter variable k of the proposed method plays an important role in the final results. In this subsection, we attempt to explain the sensitivity of the proposed method by assigning different values to k based on ULNPWHM operator and present the score values and the corresponding ranking orders in Table 8 (See Appendixes). From Table 8 (See Appendixes), we can find that the ranking orders of different parameters are the same, i.e., A4 A2 A3 A1 . However, different score values can be derived with different parameters of k with the ULNPWHM operator, and the larger the value of k, the smaller the score values of  j  j  1, 2,3, 4  . In practical applications, the value of parameter k represents the number of interacted decision attributes, expressed in the operator, that is, the number of interdependent ULNNs. When k  1 , the ULNPWHM operator cannot take any interrelationships among any attributes into consideration. When k  2 , the proposed method considers the interrelationship between any two input arguments. When k  3 , the interrelationship among any three attributes is taken and when k  4 , the ULNPWHM operator reflects the interrelationship among any four attributes. With this property, DMs can assign a proper value to parameter k according to the needs of actual application situations, which reflects the flexibility and universality of our proposed method. In real-world decision-making problems, DMs can choose the appropriate value of k according to their preferences. If the expert prefers risk, he/she can take the value of k as small as possible; whereas, he/she can take the parameter k as large as possible. Objectively, we usually take k   n 2 to solve problems, where symbol [] is a round function and n is the number of elements that need to be aggregated.

6.3. Validity Analysis To prove the validity and the effectiveness of the proposed method, we utilize our proposed method based on ULNPWHM operator, that method proposed by Fang and Ye [54] based on linguistic neutrosophic number weighted arithmetic average (LNNWAA) operator to solve Example 5 described in the following.

314

Yuan Xu, Xiaopu Shang and Jun Wang Example 5. There is a supplier selection problem in supply chain management, and

four prospective suppliers Ai  i  1, 2,3, 4  are required to be evaluated with three attributes C j  j  1, 2,3 : (1) relationship closeness C1 ; (2) product quality C 2 ; (3) price

competitiveness C3 , whose weight vector is w   0.35,0.25,0.4 . Three DMs Dh h  1, 2,3 T

T

1 1 1 , whose weighted vector is    , ,  , are invited to evaluate each alternative with 3 3 3

respect to each attribute using LNNs from the linguistic term set S = {s0 = extremely poor, s1 = very poor, s2 = poor, s3 = slightly poor, s4 = medium, s5 = slightly better, s6 = good, s7 = very good, s8 = prefect}, and the decision matrix is shown in Table 9-11 (See Appendixes). When applying the data showed in Table 9-11 (See Appendixes) to the proposed method, we find that it does not match the ULNPWHM operator very well. Therefore, we do some changes to the data to accommodate the proposed method, i.e., the assessment information R111 is changed from s6 , s1 , s2 to  s6 , s6  ,  s1 , s1 ,  s2 , s2  , and R121 is changed from s7 , s2 , s1 to  s7 , s7  ,  s2 , s2 ,  s1 , s1  . It is obvious to find that the transformed data is

theoretically equal to the original data and fits well with the proposed method. Then, the score values are calculated and the ranking orders of these three methods are shown in Table 12 (See Appendixes). From Table 12 (See Appendixes), we can find that although the score values of different methods are slightly different, the ranking orders have always been consistent, i.e., A4 A2 A3 A1 , which illustrate the validity and effectiveness of the proposed method.

6.4. Advantages of Our Proposed Method In order to illustrate the advantages and superiorities of the proposed method, we compare it with Fang and Ye’s [54] method with LNNWAA operator, and Liu and You’s [67] method with WLNHM operator. We utilize these methods to deal with the following examples and compare their ranking results to explain the advantages of the proposed method.

6.4.1. The Flexibility of Aggregating DMs’ Hesitant Evaluate Information As aforementioned, our proposed method is based on the uncertain linguistic neutrosophic sets which can expand the description scope of the evaluation information provided by DMs. In this subsection, we utilize the methods proposed by Fang and Ye [54] and Liu and You [67] which are based on linguistic neutrosophic sets for comparative analysis to illustrate the flexibility of the proposed method when aggregating DMs’ hesitant information in decision-making process.

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Affected by life experience and knowledge structure, DMs may feel indecisive and hesitant when faced with multiple decision-making problems, in which conditions they may prefer to provide more ambiguous assessment information. Therefore, a method that can accommodate more fuzzy information can be more friendly to DMs and help them provide information more freely. According to Definition 4 and Definition 10 we can easily know that, compared with LNNs, ULNNs emphasize the description of interval values information and provide DMs more freedom to depict their hesitation and uncertainty, which is more suitable for the application scenarios with high degree of complexity and fuzziness. Thus, our proposed method can be more humanization and generalization than Fang and Ye’s [54] and Liu and You’s [67] methods. To describe this advantage more clearly, we give the following example. Example 6. As we have mentioned in Definition 5, the ULNN will be reduced to LNN when the upper limit and the lower limit are all equal in truth-membership, falsitymembership, and indeterminacy-membership, and a review of Example 5 can help us understand this point more clearly. Therefore, any decision information under linguistic neutrosophic environment can be solved with our proposed method. However, if the DMs prefer or are required to utilize the ULNN to express their preference, methods based on linguistic neutrosophic will no longer be applicable. For a more thoroughly understand, we utilize the proposed method, Fang and Ye’s [54] and Liu and You’s [67] methods to solve Example 4, respectively, and the calculation results are shown in Table 13 (See Appendixes). From Table 13 (See Appendixes), we can clearly see that the methods based on linguistic neutrosophic sets [54, 67] cannot be utilized to solve the problem in Example 6 because they can only solve the information under linguistic neutrosophic environment. On the contrary, the proposed method done well in dealing with this kind of problem, which illustrates that our proposed method is more flexibility and superiority than the other two methods [54, 67].

6.4.2. The Ability of Reducing the Negative Influence of Unreasonable Information Due to the differences in personal preferences and the reserve level of prior knowledge among DMs, in practical decision-making problems, experts may be likely to provide some unreasonable or unrealistic evaluation values. These unduly assessments have negative impacts on the final results, leading to unreasonable final decision, and may bring huge profit loss to the enterprise. Therefore, it is essential to reduce the impact of unreasonable factors in the assessment process. As mentioned above, our proposed method is based on the PA and the PHM operators, which has the power to allow argument values to support each other, reducing the decision bias caused by unduly high or low evaluation values. To better illustrate this advantage, we utilize our proposed

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Yuan Xu, Xiaopu Shang and Jun Wang

method and Fang and Ye’s [54] method to deal with the following example and make a comparative analysis of the final ranking orders of the two methods. Example 7. In this example, we directly refer to Example 5 for convenience and make some changes to the original data. Firstly, we change the LNNs to ULNNs in the way introduced in Example 5 to fit the proposed method. Furthermore, assume that DM D3 prefers alternative A2 , and for the attribute C 2 of A2 , he/she provides an extremely high value s8 , s1 , s1 for his/her evaluation information. The other attribute values remain unchanged. We utilize our method and that presented in [54] to solve this example and list the results in Table 14 (See Appendixes). Compared Table 12 (See Appendixes) and Table 14 (See Appendixes), we can find that the ranking orders calculated by Fang and Ye’s [54] method changed from A4 A2 A3 A1 to A2 A4 A3 A1 under the influence of the new data. This is because the method proposed by Fang and Ye [54] fail to deal with DMs’ biased evaluation values. Furthermore, it is worth noting that our proposed method adopts objective weight measure to determine the weight vector of input arguments, which makes it possible to neutralize the decision bias caused by unreasonable information. Thus, although one DM provide an unduly high evaluation value of  22 , our proposed method maintains a certain degree of stability, i.e., the ranking order keeps the same as that in Example 5 and the best alternative is always A4 . Therefore, we can conclude that the proposed method has the ability of reducing the negative influences of unreasonable evaluation values and is more robust in practical application.

6.4.3. The Ability of Considering the Interrelationship Among Multiple Attributes In practical MAGDM problems, there are often interdependent relationships between the attributes. For instance, attribute C3 (market risk) and C5 (social environment risk) of Example 4 are theoretically related. Changes in the social environment are usually reflected in the market, leading to the increase of market risks. Therefore, considering the interrelationship between attributes in the decision-making process can help us to improve the superiority of the proposed method. By reviewing the previous literature, we find that LNS-based MAGDM methods are mostly established on the operators, such as BM operator [9], Heronian mean operator [16], which usually assume that the input arguments are independent of each other, or there is only a interrelationship between any two attributes. When encountering decision-making problems with complex relationships among attributes, these methods will be no longer applicable. By contrast, the proposed method is based on PHM operator and PWHM operator, which have a parameter variable k to determine the number of interdependent input arguments included in the calculation process. In addition, the proposed method also contains the advantages of the above

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317

methods. If the parameter k is equal to 1, then our method takes no interrelationship among any attribute, which is the same as [54]; If k is equal to 2, the interrelationship between any two attributes are considered, which is the same as [9, 16]. Besides, the advantage of the proposed method is that when k takes values of 3 or 4, it can consider the relationship among any three or four attributes, respectively, making it more general and extensive than the methods in [9, 16, 54].

CONCLUSION Decision-making problems in the real world regularly with high dimensions, multiple factors, and strong relevance, resulting it difficult to make a proper choice. This chapter introduced a new MAGDM method by combining ULNSs and PHM operator. We first recalled the basic concepts of NSs, LNSs, PA operator, HM operator, and PHM operator, and then gave the definition of ULNSs as well as its operational rules, the comparison rules, and the distance measure between any two ULNSs. Subsequently, we extended the PHM operator to ULN environment and proposed a family of ULN aggregation operators, the ULNPA operator, the ULNPWA operator, the ULNPHM operator, the ULNPWHM operator. These operators can not only determine the number of interdependent input arguments included in the calculation process by a parameter k, but also eliminate the negative influences of DMs’ preference or bias on the alternatives, and effectively solve the uncertainty, imprecise and inconsistent evaluation information given by DMs. Then, to illustrate the application process of the proposed method, we gave the main steps of solving the MAGDM problem with ULN information. To further illustrate the effectiveness of the proposed method, we applied the proposed method in some real decision-making problems and compare it with some existing methods. Case study results show that our proposed method is more efficient and superior to the others, and can provide DMs more freedom to give the evaluation information according to their personal preferences. In summary, the advantages of our proposed method are mainly reflected in three aspects: (1) The flexibility of aggregating DMs’ hesitant and inconsistent evaluate information; (2) The ability of reducing the negative influence of unreasonable information; (3) The ability of considering the interrelationship among multiple attributes. To better illustrate these advantages, we conducted several comparative analyses. Given that the MAGDM problems in real world is more and more complex and the good performance of the proposed method, in future works, we shall apply it to some other practical MAGDM problems, such as investment selection, low carbon selection, medical diagnosis, etc. In addition, considering that ULNS is an effective and powerful technology, we will unitize it to combine with other aggregation operators, such as the power Bonferroni mean operator, the power Heronian mean operator, the power Muirhead mean operator and so forth.

A4

A3

A2

A1

A4

A3

A2

A1

1

4

3

7

4

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7

5

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6

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6

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0

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3

3

3

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4

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3

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4

4

6

6

7

5

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7

7

2

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1

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3

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3

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6

5

5

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7

6

5

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7

6

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2

3

2

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3

2

7

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7

3

5

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6

4

6

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7

2

0

3

3

3

1

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3

3

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0

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2

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7

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5

 s , s  ,  s , s  ,  s , s   s , s  ,  s , s  ,  s , s   s , s  ,  s , s  ,  s , s   s , s  ,  s , s  ,  s , s 

1

 s , s  ,  s , s  ,  s , s   s , s  ,  s , s  ,  s , s   s , s  ,  s , s  ,  s , s   s , s  ,  s , s  ,  s , s 

Table 2. Uncertain linguistic neutrosophic decision matrix R 2 given by D2

1

3

2

4

 s , s  ,  s , s  ,  s , s   s , s  ,  s , s  ,  s , s   s , s  ,  s , s  ,  s , s   s , s  ,  s , s  ,  s , s 

7

6

7

6

 s , s  ,  s , s  ,  s , s   s , s  ,  s , s  ,  s , s   s , s  ,  s , s  ,  s , s   s , s  ,  s , s  ,  s , s 

5

5

7

5

C4

6

4

6

2

C3

4

3

5

1

C2

3

3

4

6

C1

1

2

2

5

 s , s  ,  s , s  ,  s , s   s , s  ,  s , s  ,  s , s   s , s  ,  s , s  ,  s , s   s , s  ,  s , s  ,  s , s 

3

 s , s  ,  s , s  ,  s , s   s , s  ,  s , s  ,  s , s   s , s  ,  s , s  ,  s , s   s , s  ,  s , s  ,  s , s 

2

 s , s  ,  s , s  ,  s , s   s , s  ,  s , s  ,  s , s   s , s  ,  s , s  ,  s , s   s , s  ,  s , s  ,  s , s 

2

 s , s  ,  s , s  ,  s , s   s , s  ,  s , s  ,  s , s   s , s  ,  s , s  ,  s , s   s , s  ,  s , s  ,  s , s 

1

C4

C3

C2

C1

Table 1. Uncertain linguistic neutrosophic decision matrix R1 given by D1

APPENDIXES

2

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 s , s  ,  s , s  ,  s , s   s , s  ,  s , s  ,  s , s   s , s  ,  s , s  ,  s , s   s , s  ,  s , s  ,  s , s 

C5

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5

2

 s , s  ,  s , s  ,  s , s   s , s  ,  s , s  ,  s , s   s , s  ,  s , s  ,  s , s   s , s  ,  s , s  ,  s , s 

C5

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5

1

 s , s  ,  s , s  ,  s , s   s , s  ,  s , s  ,  s , s   s , s  ,  s , s  ,  s , s   s , s  ,  s , s  ,  s , s  4

1

2

0

 s , s  ,  s , s  ,  s , s   s , s  ,  s , s  ,  s , s   s , s  ,  s , s  ,  s , s   s , s  ,  s , s  ,  s , s  3

6

6

7

7

C5 7

5

4

5

7

C4 6

5

4

6

6

 s , s  ,  s , s  ,  s , s   s , s  ,  s , s  ,  s , s   s , s  ,  s , s  ,  s , s   s , s  ,  s , s  ,  s , s 

2

 s , s  ,  s , s  ,  s , s   s , s  ,  s , s  ,  s , s   s , s  ,  s , s  ,  s , s   s , s  ,  s , s  ,  s , s  2

C5

C4

Table 4. The normalized decision matrix R1 given by D1

4

5

4

5

 s , s  ,  s , s  ,  s , s   s , s  ,  s , s  ,  s , s   s , s  ,  s , s  ,  s , s   s , s  ,  s , s  ,  s , s 

7

7

7

4

 s , s  ,  s , s  ,  s , s   s , s  ,  s , s  ,  s , s   s , s  ,  s , s  ,  s , s   s , s  ,  s , s  ,  s , s 

7

6

6

4

 s , s  ,  s , s  ,  s , s   s , s  ,  s , s  ,  s , s   s , s  ,  s , s  ,  s , s   s , s  ,  s , s  ,  s , s 

6

6

3

3

C3

5

5

2

7

C2

2

3

2

7

C1

1

0

1

4

 s , s  ,  s , s  ,  s , s   s , s  ,  s , s  ,  s , s   s , s  ,  s , s  ,  s , s   s , s  ,  s , s  ,  s , s 

3

 s , s  ,  s , s  ,  s , s   s , s  ,  s , s  ,  s , s   s , s  ,  s , s  ,  s , s   s , s  ,  s , s  ,  s , s 

2

 s , s  ,  s , s  ,  s , s   s , s  ,  s , s  ,  s , s   s , s  ,  s , s  ,  s , s   s , s  ,  s , s  ,  s , s 

1

C3

C2

C1

Table 3. Uncertain linguistic neutrosophic decision matrix R3 given by D3

A4

A3

A2

A1

A4

A3

A2

A1

3

3

5

2

1

1

2

2

1

6

4

5

5

8

7

7

6

1

2

1

2

2

3

2

3

1

1

3

6

4

1

3

7

5

6

3

6

8

4

5

1

3

5

2

2

5

6

3

3

4

6

5

6

6

7

8

7

7

2

2

5

4

3

3

6

5

1

1

1

1

1

2

2

1

7

6

6

4

8

7

6

5

3

2

1

3

4

3

4

4

1

1

2

1

2

2

3

2

6

6

6

3

7

7

7

5

2

4

3

7

4

5

5

7

2

3

5

6

3

4

6

7

4

7

1

3

4

5

5

7

4

6

6

8

5

6

3

4

2

2

4

5

3

3

1

1

1

2

1

3

1

3

2

2

1

3

4

3

5

6

5

8

6

2

2

2

1

3

3

3

2

1

1

3

3

2

2

4

4

6

5

7

7

6

8

1

4

2

1

6

4

2

3

2

3

5

2

 s , s  ,  s , s  ,  s , s   s , s  ,  s , s  ,  s , s   s , s  ,  s , s  ,  s , s   s , s  ,  s , s  ,  s , s 

C5

5

6

5

 s , s  ,  s , s  ,  s , s   s , s  ,  s , s  ,  s , s   s , s  ,  s , s  ,  s , s   s , s  ,  s , s  ,  s , s 

4

 s , s  ,  s , s  ,  s , s   s , s  ,  s , s  ,  s , s   s , s  ,  s , s  ,  s , s   s , s  ,  s , s  ,  s , s 

Table 6. The normalized decision matrix R3 given by D3

1

2

4

4

 s , s  ,  s , s  ,  s , s   s , s  ,  s , s  ,  s , s   s , s  ,  s , s  ,  s , s   s , s  ,  s , s  ,  s , s 

6

8

5

3

 s , s  ,  s , s  ,  s , s   s , s  ,  s , s  ,  s , s   s , s  ,  s , s  ,  s , s   s , s  ,  s , s  ,  s , s 

5

7

4

5

 s , s  ,  s , s  ,  s , s   s , s  ,  s , s  ,  s , s   s , s  ,  s , s  ,  s , s   s , s  ,  s , s  ,  s , s 

5

3

2

4

 s , s  ,  s , s  ,  s , s   s , s  ,  s , s  ,  s , s   s , s  ,  s , s  ,  s , s   s , s  ,  s , s  ,  s , s 

4

2

1

5

C4

1

2

4

4

C3

1

1

3

2

C2

7

6

5

1

C5

C4

C1

6

4

5

2

 s , s  ,  s , s  ,  s , s   s , s  ,  s , s  ,  s , s   s , s  ,  s , s  ,  s , s   s , s  ,  s , s  ,  s , s 

1

 s , s  ,  s , s  ,  s , s   s , s  ,  s , s  ,  s , s   s , s  ,  s , s  ,  s , s   s , s  ,  s , s  ,  s , s 

7

 s , s  ,  s , s  ,  s , s   s , s  ,  s , s  ,  s , s   s , s  ,  s , s  ,  s , s   s , s  ,  s , s  ,  s , s 

6

C3

C2

C1

Table 5. The normalized decision matrix R 2 given by D2

5.6597

, s7.1424  ,  s1.6507 , s2.7334  ,  s1.6536 , s2.3437 

, s5.6291  ,  s3.7633 , s5.2007  ,  s3.3445 , s4.6876 

, s6.7861  ,  s1.6876 , s2.6915  ,  s2.0314 , s2.3776 

, s6.3255  ,  s1.6745 , s2.6879  ,  s2.3207 , s3.0559 

4.9786

, s6.7125  ,  s2.3408 , s3.3482  ,  s1.3266 , s2.0065 

A4

A4 A4

S 1   0.5947 , S  2   0.6451 , S  3   0.6354 , S  4   0.6902 S 1  =0.5491 , S  2  =0.6172 , S  3  =0.5921 , S  4  =0.6661 S 1  =0.5200 , S  2  =0.6091 , S  3  =0.5684 , S  4  =0.6545

k 2

k 3

k 4

A2

A2

A2

A3

A3

A3

A3

A4

S 1  =0.6968 , S  2  =0.7468 , S  3  =0.7343 , S  4  =0.7749

k 1

A2

Ranking orders

A1

A1

A1

A1

, s7.0096  ,  s1.6822 , s3.0318  ,  s2.0196 , s3.6817 

, s7.2993  ,  s3.0363 , s4.0483  ,  s1.6540 , s2.3492 

, s7.3143  ,  s2.0281, s3.4004  ,  s3.0921, s3.7937 

Table 8. The score values and ranking results with different parameter values k

, s6.7096  ,  s1.6536 , s2.3272  ,  s2.0191, s3.0370 

4.1726

, s8.0000  ,  s3.3530 , s4.6927  ,  s1.3446 , s2.3583  5.6581

5.6755

, s5.2596  ,  s3.3915 , s4.4555  ,  s1.3245 , s2.3766 

, s5.9902  ,  s2.0171, s3.0325  ,  s2.0208 , s3.0089 

4.6505

, s5.9766  ,  s2.3390 , s3.3481  ,  s1.9833 , s2.3206 

5.9722

, s5.6590  ,  s1.9523 , s3.5155  ,  s1.6731, s2.3622 

, s5.3124  ,  s4.6747 , s5.2905  ,  s4.7852 , s5.8624 

Score functions S  i  i  1,2,3,4 

4.9911

5.6803

3.9478

5.3130

 s  s  s  s

6.3608

, s6.9925  ,  s1.6757 , s2.7296  ,  s1.9544 , s3.0564 

 s  s  s  s

5.6338

, s6.7752  ,  s2.3708 , s3.4196  ,  s1.6535 , s2.6721 

C5

4.9818

, s6.0186  ,  s3.3962 , s4.4437  ,  s1.0130 , s1.6874 

C4

5.6620

4.6493

4.9792

3.9522

, s5.6277  ,  s3.0358 , s4.0537  ,  s1.9824 , s3.3672 

 s  s  s  s

4.6121

, s6.9939  ,  s3.5033 , s4.5757  ,  s1.3219 , s1.9834 

 s  s  s  s

5.9936

 s  s  s  s

Parameters

A4

A3

A2

A1

A4

A3

A2

A1

C3

C2

C1

Table 7. The comprehensive decision matrix R

s5 , s1 , s2

s6 , s1 , s1

A3

A4

s7 , s3 , s4

s6 , s3 , s4

s7 , s2 , s4

s7 , s2 , s3

A1

A2

A3

A4

C1

s7 , s2 , s3

A2

s7 , s1 , s2

A4

s6 , s1 , s2

s6 , s2 , s2

A3

A1

s7 , s1 , s1

A2

C1

s6 , s1 , s2

A1

C1

s5 , s2 , s3

s5 , s4 , s2

s4 , s2 , s3

s4 , s2 , s3

C3

s5 , s2 , s1

s6 , s1 , s2

s5 , s1 , s2

s7 , s3 , s3

C2

s6 , s1 , s1

s7 , s2 , s4

s6 , s2 , s3

s5 , s2 , s5

C3

Table 11. Linguistic neutrosophic decision matrix R3 given by D3

s5 , s1 , s1

s5 , s1 , s2

s6 , s1 , s1

s6 , s1 , s1

C2

Table 10. Linguistic neutrosophic decision matrix R 2 given by D2

s7 , s2 , s3

s7 , s1 , s1

s7 , s3 , s2

s7 , s2 , s1

C2

Table 9. Linguistic neutrosophic decision matrix R1 given by D1

s7 , s2 , s1

s6 , s2 , s2

s7 , s2 , s1

s6 , s2 , s2

C3

S 1   0.5947 , S  2   0.6451 , S  3   0.6354 , S  4   0.6902 .

Cannot be calculated Cannot be calculated

Score functions S  i  i  1,2,3,4 

A3

A4

S 1  =0.7255 , S  2  =0.7644 , S  3  =0.7396 , S  4  =0.7747 .

The proposed method with ULNPWHM method  k  2 

A2

A3

A3

A2

S 1  =0.7489 , S  2  =0.8467 , S  3  =0.7599 , S  4  =0.8049 .

Fang and Ye’s [54] method with LNNWAA operator

A4

Ranking orders

A2

Score functions S  i  i  1,2,3,4 

A4

—— ——

A1

A1

A3

Ranking orders

A2

Methods

Table 14. The score values and ranking orders of Example 7 by different methods

The proposed method with ULNPWHM method  k  2 

Liu and You’s [67] method with WLNHM operator  k  2 

Fang and Ye’s [54] method with LNNWAA operator

Methods

Table 13. The score values and ranking orders of Example 6 by different methods

A4

S 1   0.7255 , S  2   0.7425 , S  3   0.7396 , S  4   0.7747 .

The proposed method with ULNPWHM method  k  2 

A3

A4

S 1   0.7489 , S  2   0.7728 , S  3   0.7599 , S  4   0.8049 .

Fang and Ye’s [54] method with LNNWAA operator

A2

Ranking orders

Score functions S  i  i  1,2,3,4 

Methods

Table 12. The score values and ranking orders of Example 5 by different methods

A1

A1

A1

324

Yuan Xu, Xiaopu Shang and Jun Wang

ACKNOWLEDGMENTS This work was supported by National Natural Science Foundation of China (61702023), Humanities and Social Science Foundation of Ministry of Education of China (17YJC870015), Beijing Natural Science Foundation (7192107), and the Beijing Social Science Foundation (19JDGLB022).

CONFLICT OF INTERESTS We declare that there is no conflict of interest with regard to the publication of this chapter.

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In: Decision-Making with Neutrosophic Set Editor: Harish Garg

ISBN: 978-1-53619-419-7 © 2021 Nova Science Publishers, Inc.

Chapter 13

AN N-DIMENSIONAL NEUTROSOPHIC LINGUISTIC APPROACH TO POVERTY ANALYSIS WITH AN EMPIRICAL STUDY D. Ajay1,*, J. Aldring2 and S. Nivetha2 1

Department of Mathematics, Sacred Heart College (Autonomous), Tirupattur, Tamilnadu (State), India 2 Department of Mathematics, Sacred Heart College (Autonomous), Tirupattur, Tamilnadu (State), India

ABSTRACT Poverty measure is an important social determinant on which the policy makers of a country rely very much. It takes a variety of field study to come under the frame of measuring the poverty levels of the target people and it is a difficult task due to its nature. Fuzzy approach has admitted measuring poverty under its purview since analysing poverty requires a multidimensional approach involving uncertainty and ambiguity. So, this paper makes a new attempt to analyzing poverty among the target group of people using n-number of linguistic variables approach in neutrosophic environment. It potentially deliberates the measure of poverty levels of the households using poverty indicators like income, education, employment and assets. Moreover neutrosophic membership functions are defined according to the nature of the poverty indicators. The proposed method is examined with a case study of ten households in a town in India and we categorize the levels of poverty of these households.

Keywords: fuzzy sets, neutrosophic sets, poverty analysis, fuzzy poverty analysis, linguistic variables, uncertainty

*

Corresponding Author’s Email: [email protected].

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ABBREVIATIONS NSs FSs

neutrosophic sets fuzzy sets

1. INTRODUCTION Fuzzy sets have a proven record of handling uncertainty and amibiguity. Empirical studies in various fields suggest that applying fuzzy sets surpass the application of classical sets in those fields where machines need to work like human brain. To fulfill the needs of the time and technology, many extensions to fuzzy sets have been suggested. One such extension that concerns the positive, negative and indeterminate dimesions of the factors involved in human like processors, is the notion of neutrosophic sets (Smarandache, 1999). Since its inception, neutrosophic sets have been employed in many research fields and especially in decision making where human brain like system or direct human intervention is involved (Ajay et al., 2019; Ajay & Aldring, 2019; Deli & Şubaş, 2017; Ajay et al., 2020b; Deli & Öztürk, 2020; Ajay & Chellamani, 2020). In this work we extend the application of neutrosophic sets to the analysis of poverty. The important social measurement is the poverty measure of the people. It can be used by the policy makers to make identification of the “poor” in order to fulfill social security. In the recent literature many researchers developed and proposed a variety of poverty measures and methods to determine the poverty line (Abdulla, 2011; Belhadj & Matoussi, 2010; Chatterjee et al., 2014). Many discussions have come to the fore on poverty measure due to its nature. Measuring poverty is not an easy task and it involves different indicators. Particularly income, education, employment, and assets are the important indicators. Categorizing people into economically poor and non-poor involves many objective as well as subjective factors. Therefore a relative approach scales poverty based on whether a person’s income is below or above the national average. Yet another approach, called subjective approach, states that an individual can evaluate his or her own situation (Van Praag, 1971; Vero & Werquin, 1997). Most of the poverty measures predict the poverty levels from the inadequacy of income (Wagle, 2007). Therefore in this research we additionally encounter with educational status, employment, and assets of the people to measure the poverty levels of the target people and show this approach gives more efficient results.

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2. LITERATURE REVIEW The debate over poverty analysis took a decisive turn with the introduction of fuzzy logic in its analysis which resulted in a multidimensional approch to poverty. Since then, several researchers have employed fuzzy logic and have shown that the results are more meaningful. Giordani & Giorgi (2010) used fuzzy logic to extend poverty measures based on the two inequality indices namely Gini and Bonferroni indices. The inclusion of fuzzy logic in their analysis enabled them to categorize observations based on degrees of being poor and non-poor. Mukherjee et al. (2011) adopted a fuzzy approach to poor household identification and in their empirical example showed that this method enriched the process of identifying poor households. Caramuta & Contiggiani (2006) provided a new approach to measurement of poverty by fuzzifying poverty line approach in contrast to the existing approaches. Using a household survey in Argentina they demonstrated their two indices of poverty based on fuzzy membership functions. Betti et al. (2004) proposed a statistical model and employed it to poverty analysis in Great Britain with the help of two statistical tools, namely Dynamic Indices and Average Transition Matrices. In contrast to the traditional unidimensional approch that is largely based on income and expenditure, many researchers proposed multidimensional approaches based on fuzzy logic. Such an approach was adopted for the data from Camaroon in which the researcher combined monetary and non-monetary indicators to estimate fuzzy poverty index (Siani, 2015). A similar approach has also been adopted by many researchers (Betti & Verma, 2004; Berenger & Celestini, 2006; Oyekale & Okunmadewa, 2008; Lemmi & Betti, 2006; Kim, 2012; Betti & Verma, 1999; Nasri & Belhadj, 2018; Betti et al., 2006; Betti et al., 2008; Maggio, 2004; Miceli, 2006; Mussard & Noel, 2005; Kouassi & Seka, 2017; Tarditi, 2007; Chatterjee et al., 2014). Deutsch & Silber (2005) presented a useful comparison of various approaches available for multidimensional poverty measurement using data from Israel and showed that there was a fair agreement among these approaches. Delalić et al. (2017) went on to establish the differences between unidimensional and multidimensional approaches to measurement of poverty and declared the insufficiency of unidimensional approaches. On the contrary, Belhadj & Matoussi (2010) developed a fuzzy approach based only on monetary variable. A closer look at the available literature reveals that most of the researchers who wanted to avoid entering into decades-old dichotomy of poor versus non-poor and wanted to provide social reality that has several stages between being poor and non-poor, adopted either a multi-attribute or a multi-criteria or a multivariate approach to poverty analysis. Annoni et al. (2008) proposed a fuzzy multi-criteria analysis and used this analysis to provide the structural representation of poverty in Sicily, Italy. Such representation of poverty brought to light the inexplicable relation between different poverty indicators. Dagum (2002) showed the advantages of a multivariate fuzzy analysis and empirically

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proves using the data from Italy that multivariate analysis would be appropriate to understand the dimensions of poverty and take measures to address it. In another effort to analyse poverty Kumar & Pathinathan (2014) advocated a method using fuzzy membership ranking model in which they included the status of children as an indicator with the other economic indicators. Using fuzzy triangular numbers in fuzzy AHP, they also proposed a new multi-criteria method and depicted it through a case study in Nalanda District in India (Kumar & Pathinathan, 2015a). They advanced this approach in yet another seminal work to introduce pentagonal fuzzy numbers and utilized stratified fuzzy AHP method for representation of poverty level of households (Kumar & Pathinathan, 2015b). In the background of the major economic slowdown, Ciani et al. (2018) ventured to adopt a fuzzy approach to measure poverty for a period of eight years in the Mediterranean area with a special focus on the financial dimension of poverty. The steps and the method involved in deriving a multidimensional poverty index using fuzzy sets is depicted with an example from Indian data by Neff (2013) who also presents how advantageous such a method could be when it comes to poverty analysis. Pabuccu (2017) assigned fuzzy weights to poverty determinants based on which the households were classified according to their poverty index. Basing themselves on the logic of fuzzy set theory, Jing et al., (2019) came out with the fuzzy proximity method and showed the efficacy of their method through an empirical study in China.

3. METHODOLOGY In this research, neutrosophic approach (Smarandache, 2005) has been implemented. This notion is very new to measuring poverty using different poverty indicators/factors. A variety of neutrosophic membership functions are framed with the help of linguistic variables. Some of the neutrosophic operations (Smarandache, 2016) and an aggregation operator are used to formulate an algorithm. This research methodology was motivated by the following literature: Gulistan et al. (2018) in which some linguistic neutrosophic cubic mean operators and entropy measure are used in the application of choosing an area supervisor. Neutrosophic cubic sets have been implemented in multi-criteria decisionmaking with an application (Jianming Zhan et al. 2017). Different types of neutrosophic cubic heronian mean operators have been used in MCDM with the help of cosine similarity functions (Gulistan et al. 2019). New logarithmic operational laws have been used to multi attribute decision making for single-valued neutrosophic numbers (Garg & Nancy, 2018). Recently, a ranking method based on possibility mean for multi-attribute decision making with single valued neutrosophic numbers has been introduced (Garai et al., 2020). Neutrosophic sets have been extended thorough complex fuzzy sets (Gulistan & Khan, 2020). Novel neutrality aggregation operator-based multi attribute group decision-making method for single-valued neutrosophic numbers has also been

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introduced based on new neutrality operational laws (Garg, 2020). Multiple attribute decision making based on immediate probabilities aggregation operators for singlevalued and interval neutrosophic sets was developed (Garg & Nancy, 2020). Further, in this chapter a new poverty measure approach has been proposed based on neutrosophic sets and is executed with a case study. The rest of the chapter is organized as follows. In section 4, we recall some of the basic fuzzy definitions and operations. Then a neutrosophic approach to poverty measurement has been discussed in detail in section 5. In section 6, a case study has been conducted and demonstrated with calculation. Finally, the conclusion part is given in section 7.

4. BASIC CONCEPTS Definition 1. Fuzzy Sets (Zadeh, 1965) If 𝑥 is a particular element of universe of discourse X, then a fuzzy set A is defined by a fuzzy membership function (𝜇𝐴 ) which is associated to each 𝑥 a membership value in the closed unit interval of zero and one. i.e., 𝜇𝐴 (𝑥) ∶ 𝑋 → [0,1]

Definition 2. Neutrosophic Sets (Smarandache, 2005) Let B be a neutrosophic set in the universal discourse or non-empty set X and any object x in B has the form 𝐵 = {〈𝑥, 𝜇𝐵 (𝑥), 𝜎𝐵 (𝑥), 𝜗𝐵 (𝑥) 〉; 𝑥𝜖𝑋} where 𝜇𝐵 (𝑥), 𝜎𝐵 (𝑥) and 𝜗𝐵 (𝑥) represent the degree of truth, the degree of indeterminacy and the degree of falsity membership functions respectively which take their values in the unit closed interval of zero and one, and satisfy the relation 0 ≤ 𝜇𝐵 (𝑥) + 𝜎𝐵 (𝑥) + 𝜗𝐵 (𝑥) ≤ 3.

Definition 3. Single Valued Neutrosophic Sets (Wang, et al. 2012) Let X be a space of points (objects), with a generic element in X denoted by 𝑥. A single valued neutrosophic set (SVNS) B in X is characterized by truth-membership function 𝜇𝐵 (𝑥) , indeterminacy-membership function 𝜎𝐵 (𝑥) and falsity-membership function 𝜗𝐵 (𝑥). For each point 𝑥 in X, 𝜇𝐵 (𝑥), 𝜎𝐵 (𝑥), 𝜗𝐵 (𝑥) ∈ [0,1].

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D. Ajay, J. Aldring and S. Nivetha When X is continuous, a SVNS B can be written as 𝐵 = ∫〈𝑥, 𝜇𝐵 (𝑥), 𝜎𝐵 (𝑥), 𝜗𝐵 (𝑥) 〉 ; 𝑥𝜖𝑋 When X is discrete, a SVNS B can be written as 𝑛

𝐵 = ∑〈𝑥, 𝜇𝐵 (𝑥), 𝜎𝐵 (𝑥), 𝜗𝐵 (𝑥) 〉 ; 𝑥𝜖𝑋 𝑖=1

Definition 3. Subtraction of Two Neutrosophic Numbers (Smarandache, 2016) Let 𝑁1 = 〈𝑇1 , 𝐼1 , 𝐹1 〉 and 𝑁2 = 〈𝑇2 , 𝐼2 , 𝐹2 〉 be the two neutrosophic numbers, then the subtraction of two neutrosophic numbers is defined by

=(

𝑁1 ⊖ 𝑁2 = 𝑇1 −𝑇2 𝐼1 𝐹1

〈𝑇1 , 𝐼1 , 𝐹1 〉 ⊖ 〈𝑇2 , 𝐼2 , 𝐹2 〉

, , ) ∀ 𝑇1 , 𝐼1 , 𝐹1 , 𝑇2 , 𝐼2 , 𝐹2 ∈ [0,1]

1−𝑇2 𝐼2 𝐹2

with the restrictions that: 𝑇2 ≠ 1, 𝐼2 ≠ 0 and 𝐹2 ≠ 0.

Definition 4. Division of Two Neutrosophic Numbers (Smarandache, 2016) Let 𝑁1 = 〈𝑇1 , 𝐼1 , 𝐹1 〉 and 𝑁2 = 〈𝑇2 , 𝐼2 , 𝐹2 〉 be the two neutrosophic numbers, then the division of 𝑁1 by 𝑁2 is defined by 𝑁1 ⊘ 𝑁2 =

〈𝑇1 , 𝐼1 , 𝐹1 〉 𝑇1 𝐼1 − 𝐼2 𝐹1 − 𝐹2 ) ∀ 𝑇1 , 𝐼1 , 𝐹1 , 𝑇2 , 𝐼2 , 𝐹2 ∈ [0,1] = ( , , 〈𝑇2 , 𝐼2 , 𝐹2 〉 𝑇2 1 − 𝐼2 1 − 𝐹2

with the restrictions that: 𝑇2 ≠ 0, 𝐼2 ≠ 1 and 𝐹2 ≠ 1.

Definition 5. Direct Sum The direct sum of two neutrosophic numbers is defined by 𝑁1 ⊕ 𝑁2 = (𝑇1 + 𝑇2 − 𝑇1 ∗ 𝑇2 , 𝐼1 + 𝐼2 − 𝐼1 ∗ 𝐼2 , 𝐹1 ∗ 𝐹2 )

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Definition 6. Score Function of Neutrosophic Numbers Let A = 〈𝑇, I, F〉 be a single valued neutrosophic number. A score function of neutrosophic number is defined based on truth membership, indeterminacy membership, and falsity membership function as follows: 𝑆(𝐴) =

1 + 𝑇 − 2𝐼 − 𝐹 2

where 𝑆(𝐴) ∈ [−1,1].

5. NEUTROSOPHIC APPROACH TO POVERTY MEASUREMENT Neutrosophic membership functions of poverty have been defined for four poverty indicators namely income, education, employment and assets in order to measure the poverty level of target group of people. These neutrosophic membership functions are defuzzied by neutrosophic numbers. And this section proposes an algorithm for poverty measure with neutrosophic aggregation operator.

5.1. Linguistic Neutrosophic Membership Function of Poverty (LNMFP) 𝑁𝑝 = { i ; 〈𝑇𝑃 (i), 𝐼𝑝 (i), 𝐹𝑝 (i)〉 /i ∈ 𝑁} with i = 1,2,3,...,n and 𝑁𝑃 is neutrosophic membership function of each individual i to the neutrosophic set of the poor (𝑁). The neutrosophic membership function assigns one of the following values: 〈0,0,1〉, if the individual i is poor 𝑁𝑝 (i) = {〈1,1,0〉, if the individual i is not poor 〈0,0,1〉 ≤ 𝑁𝑝 (i) ≤ 〈1,1,0〉

Further, LNMFP is extended to n dimension that means the word “poor” or “not poor” itself is a fuzzy linguistic word or uncertain. So, we split the word into n dimensional linguistic word to calculate more accurate values. The following definition explains this in more detail with the help of neutrosophic membership function.

5.2. n-Number of Linguistic Neutrosophic Membership Function of Poverty 〈1,0,0〉 if 0 ≤ 𝑙𝑘 ≤ 𝑙1 , 𝑙 −𝑙

2 𝐾 𝑁𝑝 (i) = {𝑙𝑘 ⊕ 𝑙 −𝑙 if 𝑙1 ≤ 𝑙𝑘 ≤ 𝑙2, 2

1

〈0,1,1〉 if 𝑙2 ≤ 𝑙𝑘 .

(1)

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D. Ajay, J. Aldring and S. Nivetha

where 𝑁𝑝 (i) defines the degree of neutrosophic membership to the set of the poor according to the linguistic neutrosophic variable of 𝑙𝑘 over poverty indicators like income, education, expenditure, land, etc. 𝑙1 and 𝑙2 are lower and upper limits of poverty linguistic neutrosophic variables, and also 𝑙𝑘 is a linguistic neutrosophic variable with k = 1,2,3,...,n and it should be in the ascending order.

Algorithm for Measuring the Poverty Levels of Householders or Target People Step 1. Form neutrosophic membership functions for poverty indicators. Step 2. Find combined neutrosophic decision matrix Step 3. Aggregate the neutrosophic information Step 4. Calculate score values to aggregate neutrosophic numbers Step 5. Rank the alternatives according to their score values

5.3. Neutrosophic Membership Function for Poverty Indicators 5.3.1. Income Here we categorize the income levels with their corresponding neutrosophic numbers in order to measure levels of poverty of households in the target group as shown in Table 1. Table 1. Neutrosophic numbers for income levels Linguistic Variables (𝒍𝒌 ) (in INR) ≤ 5000 ≤ 7000 ≤ 9000 ≤ 10500 ≤ 12000 ≤ 15000 ≤ 18000 ≤ 25000 ≤ 35000 ≤ 40000 ≥ 40000

Neutrosophic Numbers (1,0,0) (0.9, 0.1, 0.1) (0.8,0.15,0.20) (0.70,0.25,0.30) (0.60,0.35,0.40) (0.50,0.50,0.50) (0.40,0.65,0.60) (0.30,0.75,0.70) (0.20,0.85,0.80) (0.10,0.90,0.90) (0,1,1)

We take a lower limit of poverty level 𝑙1 to be 12000 and upper limit of poverty level 𝑙2 as 25000. Here we define the membership function for the linguistic variables whose values lie between the lower and upper limits. The remaining variables are assigned neutrosophic numbers directly by experts. By using the equation 1, we define the following:

An n-Dimensional Neutrosophic Linguistic Approach to Poverty …

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〈1,0,0〉 𝑖𝑓 0 ≤ 𝑙𝑘 ≤ 5000 〈0.9,0.1,0.1〉 𝑖𝑓 5000 ≤ 𝑙𝑘 ≤ 5000, 〈0.8,0.15,0.20〉 𝑖𝑓 7000 ≤ 𝑙𝑘 ≤ 9000, 〈0.7,0.25,0.30〉 𝑖𝑓 9000 ≤ 𝑙𝑘 ≤ 10500, 〈0.3,0.75,0.7〉 − 𝑙𝑘 𝑙𝑘 ⊕ 〈0.3,0.75,0.7〉 − 〈0.6,0.35,0.4〉

𝐼𝑛𝑐𝑜𝑚𝑒: 𝑁𝑝 (i) =

if 12000 ≤ 𝑙𝑘 ≤ 25000, 〈0.2,0.85,0.80〉 𝑖𝑓 25000 ≤ 𝑙𝑘 ≤ 35000, 〈0.1,0.90,0.90〉 𝑖𝑓 35000 ≤ 𝑙𝑘 ≤ 40000, 〈0,1,1〉 𝑖𝑓 𝑔𝑟𝑒𝑎𝑡𝑒𝑟 𝑡ℎ𝑎𝑛 40000. {

5.3.2. Education The education level of people is another very influential indicator in measuring the poverty of the target group. So, we formulate neutrosophic membership function as shown in Table 2. Table 2. Neutrosophic numbers for Education level Linguistic Variables (𝒍𝒌 ) Illiterate (𝒍𝟏 ) Middle school 8th grade (𝒍𝟐 ) Matric Pass/10th grade (𝒍𝟑 ) Intermediate Pass/10+2 grade (𝒍𝟒 ) Graduate (𝒍𝟓 ) Post Graduate (𝒍𝟔 ) Doctorate (𝒍𝟕 ) Post-Doctorate (𝒍𝟖 )

Neutrosophic Numbers (1,0,0) (0.9, 0.1, 0.1) (0.8,0.15,0.20) (0.70,0.25,0.30) (0.50,0.50,0.50) (0.20,0.85,0.80) (0.10,0.90,0.90) (0,1,1)

The linguistic variables are in the ascending order i.e., 〖 l_1