Generalized Intuitionistic Multiplicative Fuzzy Calculus Theory and Applications [1st ed.] 9789811556111, 9789811556128

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Uncertainty and Operations Research

Shan Yu Zeshui Xu

Generalized Intuitionistic Multiplicative Fuzzy Calculus Theory and Applications

Uncertainty and Operations Research Editor-in-Chief Xiang Li, Beijing University of Chemical Technology, Beijing, China Series Editor Xiaofeng Xu, Economics and Management School, China University of Petroleum, Qingdao, Shandong, China

Decision analysis based on uncertain data is natural in many real-world applications, and sometimes such an analysis is inevitable. In the past years, researchers have proposed many efficient operations research models and methods, which have been widely applied to real-life problems, such as finance, management, manufacturing, supply chain, transportation, among others. This book series aims to provide a global forum for advancing the analysis, understanding, development, and practice of uncertainty theory and operations research for solving economic, engineering, management, and social problems.

More information about this series at http://www.springer.com/series/11709

Shan Yu Zeshui Xu •

Generalized Intuitionistic Multiplicative Fuzzy Calculus Theory and Applications

123

Shan Yu Department of General Education Army Engineering University of PLA Nanjing, Jiangsu, China

Zeshui Xu Business School Sichuan University Chengdu, Sichuan, China

ISSN 2195-996X ISSN 2195-9978 (electronic) Uncertainty and Operations Research ISBN 978-981-15-5611-1 ISBN 978-981-15-5612-8 (eBook) https://doi.org/10.1007/978-981-15-5612-8 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2020 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Preface

Decision-making is not only a frequent activity in management, but also a common behavior in politics, economy, technology and life. The subject of decision-making is human, and we live in a world of complex practical problems, lack of human cognition and full of mental uncertainty. In this case, the fuzzy set theory, proposed by Zadeh in 1965, as an effective tool for describing uncertainty, has reached the stage of history. Over the last decades, a variety of generalizations of classical fuzzy set have been derived from various angles and the intuitionistic fuzzy set, given by Atanassov in 1983, has been its typical promotion form. With the deepening of the research, in order to describe more comprehensively and objectively the decision information of asymmetrical distribution in phenomena, such as diminishing marginal effect, non-uniform attenuation of signal with rainfall intensity in satellite communication and so on, Xia et al. (2013) proposed the intuitionistic multiplicative fuzzy set on the basis of Saaty’s 1–9 scale. Furthermore, we consider the closure of limit operation on this set and extend it to the generalized intuitionistic multiplicative fuzzy set. The establishment of calculus theory has been a great achievement of human civilization, and it has created an important decision-making method for humans to solve practical problems with mathematical tools. However, traditional calculus theory is based on accurate real numbers and exact distributions in real number field. With the rapid development and vast application of the fuzzy set, the integration of fuzzy mathematics into the already complete classical calculus theory has opened up a new path for the development of modern decision science. In 2017, Lei et al. constructed the framework of intuitionistic fuzzy calculus theory. Motivated by this, we sincerely wish that the generalized intuitionistic multiplicative fuzzy calculus models introduced in this book will enrich and develop the classical fuzzy calculus theory as well as the intuitionistic fuzzy calculus theory. The objective of this book is to introduce the latest development of generalized intuitionistic multiplicative fuzzy calculus and its application. To achieve this, the construction of the calculus models with concrete mathematical expressions for generalized intuitionistic multiplicative fuzzy information, new information fusion methods based on the definite integral models and the military case study will be v

vi

Preface

introduced in this book. These new advances will help the researchers who have interest in the generalized intuitionistic multiplicative fuzzy set and stimulate more research interests of this research field. The book is mostly self-contained, and the main prerequisites are provided in Chap. 1. You can read the five chapters sequentially to have a clear logic of their organizations, which essentially focus on two issues (manifested more clearly in the following content logic flowchart), one of which is to build the definite integral models under generalized intuitionistic multiplicative fuzzy environment (introduced in Chaps. 2–4, which constitute an orderly and unified whole), and the other is about how to aggregate not only discrete or continuous but also correlative (generalized) intuitionistic (multiplicative) fuzzy information (methods and applications are introduced in Chap. 5).

Content logic flowchart of this book

This book can be used as a reference for researchers and practitioners working in the fields of fuzzy mathematics, operations research, information science, etc. It can also be used as a textbook for postgraduate and senior-year undergraduate students. In fact, the calculus models in generalized intuitionistic multiplicative fuzzy environment have been studied, and a new way to aggregate not only discrete or continuous but also correlative information as well as a military case have been

Preface

vii

offered to the readers accordingly. With this vision in mind, we hope the readers would be inspiring when they happen to study this book some day in the future. We appreciate the financial support of this book: The National Natural Science Foundation of China (No. 71771155). Nanjing, China Chengdu, China March 2020

Shan Yu Zeshui Xu

Introduction

In this book, based on the generalized intuitionistic multiplicative fuzzy set, we give a thorough and systematic introduction to the latest research results on definite integration models and their applications in decision-making for generalized intuitionistic multiplicative fuzzy information. The detailed research work is summarized as follows: (1) The generalized intuitionistic multiplicative fuzzy set is studied. Firstly, the concepts of generalized intuitionistic multiplicative fuzzy set and generalized intuitionistic multiplicative fuzzy number are introduced based on consideration of the closure of calculus operations in intuitionistic multiplicative set. Secondly, in order to describe derivatives and differentials conveniently, two basic operations (subtraction and division) of generalized intuitionistic multiplicative fuzzy numbers are supplemented and a region partition method on generalized intuitionistic multiplicative fuzzy sets is provided. Finally, the concept of generalized intuitionistic multiplicative fuzzy sequence is introduced to characterize the “proximity” of one generalized intuitionistic multiplicative fuzzy number to another by the limit tool. Considering the generalized intuitionistic multiplicative fuzzy number as a variable, the change values are classified, analogies are drawn to the concept of sequence convergence in the real number field and four kinds of convergence forms of generalized intuitionistic multiplicative fuzzy sequence are described to prepare the ground for the subsequent chapters. (2) The theoretical framework of generalized intuitionistic multiplicative fuzzy calculus is preliminarily constructed (demonstrated more clearly in the following organizational structure map: The definite integral model for generalized intuitionistic multiplicative fuzzy information). Firstly, we make a very thorough analysis on the commonly used intuitionistic multiplicative fuzzy information fusion operators to provide some relevant assumptions for the construction of definite integral models in generalized intuitionistic multiplicative fuzzy environment. Besides, we also give the concept of generalized intuitionistic multiplicative fuzzy function by considering the membership

ix

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Introduction

function and the non-membership function as a whole and discuss its continuity, derivability and differentiability by taking four basic operation rules together with the definition of sequence convergence into account. After that, we point out linkages between the derivative and the elastic coefficient in economics. Moreover, we study the connections between the increment of dependent variable and the differential of generalized intuitionistic multiplicative fuzzy function to offer convenience to future practical application. Secondly, based on the two kinds of derivatives for generalized intuitionistic multiplicative fuzzy functions, we consider their corresponding anti-derivatives by solving two ordinary differential equations, introduce the concepts of their primitive functions and indefinite integrals and study their related properties. Thirdly, we seek the concrete mathematical expressions of the definite integral models for generalized intuitionistic multiplicative fuzzy information from two different angles. On the one hand, following the idea of microelement and the limiting procedure as Riemann Integral’s definition (that is, three steps of “partitioning, summing and taking the limit”) in real number field, we make a detailed description of the intervals and the partitions in generalized intuitionistic multiplicative fuzzy environment by graphics, give the definitions of generalized intuitionistic multiplicative fuzzy subtraction and division definite integrals and guess their mathematical expressions. On the other hand, by constructing the generalized intuitionistic multiplicative fuzzy subtraction (division) primitive function with the variable upper limit, based on the definition of the subtraction (division) derivative, we build bridges between the primitive functions and the definite integral models, introduce the fundamental theorem of calculus in generalized intuitionistic multiplicative fuzzy environment, deduce the concrete mathematical expressions of the definite integral models for generalized intuitionistic multiplicative fuzzy information and study their related properties subsequently. Finally, from four aspects, we analyze, verify and extend the two kinds of the definite integral models for generalized intuitionistic multiplicative fuzzy information. Firstly, in addition to the subtraction definite integral model which is based on the subtraction and division laws, we follow the addition and multiplication laws to model the division definite integral. Secondly, through defining the integrands in subtraction (division) definite integral model, we show that the operation of scalar multiplication (exponentiation) can be replaced by the operations of the addition and the multiplication among an infinite number of generalized intuitionistic multiplicative fuzzy numbers. Thirdly, we point out that the definite integral models for generalized intuitionistic multiplicative fuzzy information are additive to integrands and intervals, the two kinds of generalized intuitionistic multiplicative fuzzy primitive functions with the variable upper limit are monotonous and the assumptions which have already been provided were reasonable. Fourthly, by constructing suitable integrands and selecting proper integral limits, we illustrate that the definite integral models for generalized intuitionistic multiplicative fuzzy information are the expanding forms of the existing commonly used discrete fusion operators.

Introduction

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(3) Taking the intuitionistic fuzzy definite integral model for example, we discuss its application in decision-making and big data research. Beyond that, we do further research for the generalized intuitionistic multiplicative fuzzy environment to arrive at the parallel results. Firstly, based on the intuitionistic fuzzy definite integral model, we provide a fusion method for discrete and interrelated intuitionistic fuzzy information; Secondly, based on the intuitionistic fuzzy definite integral model, we provide a novel fusion method for continuous intuitionistic fuzzy information which is distributed in bounded regions of different shapes. Thirdly, we provide an approach to multi-criteria decision-making with not only discrete but also continuous intuitionistic fuzzy information, and the results are performed in parallel with the generalized intuitionistic multiplicative fuzzy environment. Finally, on the basis of the above results, we study the two important processes of the interaction by the external agents of the network public opinion in violence and terrorist incidents, i.e., internet public opinion and public opinion of other external agents, construct three key risk scenarios related to these two processes, simulate and quantify the formation of the representative collective will or emotion of general netizens’ in the internet public opinion, dynamically demonstrate the strange images of other external agents influencing the trend of public opinion and comprehensively quantify the interaction effect by the external agents of the network public opinion.

The definite integral model for generalized intuitionistic multiplicative fuzzy information

xii Introduction

Contents

1 Basic Operations Between Generalized Intuitionistic Multiplicative Fuzzy Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Generalized Intuitionistic Multiplicative Fuzzy Set . . . . . . . . . . 1.2 Some Operational Laws Related to Any GIMFNs . . . . . . . . . . 1.3 The Regional Divisions Related to a GIMFN . . . . . . . . . . . . . . 1.4 A New Order of GIMFNs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 The Convergences of Sequences of GIMFNs . . . . . . . . . . . . . . 1.6 Chapter Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1 2 4 6 7 12 14 15

2 Derivatives and Differentials for Generalized Intuitionistic Multiplicative Fuzzy Information . . . . . . . . . . . . . . . . . . . . 2.1 Generalized Intuitionistic Multiplicative Fuzzy Function . 2.2 The Derivability of GIMFFs . . . . . . . . . . . . . . . . . . . . . 2.3 The Differentials of GIMFFs . . . . . . . . . . . . . . . . . . . . . 2.4 Chapter Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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17 18 21 30 36 37

3 Indefinite Integrals of Generalized Intuitionistic Multiplicative Fuzzy Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 The Subtraction Indefinite Integral of GIMFF . . . . . . . . . . . 3.2 The Properties of the Subtraction Indefinite Integral . . . . . . . 3.3 The Division Indefinite Integral and Its Properties . . . . . . . . 3.4 Chapter Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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39 39 44 46 50 50

4 Definite Integrals of Generalized Intuitionistic Multiplicative Fuzzy Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Subtraction Definite Integral of GIMFFs . . . . . . . . . . . . . . . . . . . 4.2 The Properties of Subtraction Definite Integrals . . . . . . . . . . . . . .

51 51 60

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4.3 Division Definite Integrals and Their Properties . . . . . . . . . . . . . . 4.4 Chapter Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Several Applications Based on the Definite Integral Models for (Generalized) Intuitionistic (Multiplicative) Fuzzy Information . . 5.1 Preparing Knowledge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Discussion on the Relationships Among the Existing Orders for IFNs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Definite Integrals for Discrete Correlative Intuitionistic Fuzzy Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Definite Integrals for Continuous Intuitionistic Fuzzy Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Application in Intuitionistic Fuzzy Multi-criteria Decision Making . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Application in Generalized Intuitionistic Multiplicative Fuzzy Environment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7 Study on the Interaction Effect by the External Agents of the Network Public Opinion in Violence and Terrorist Incidents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7.1 The Meaning of Research . . . . . . . . . . . . . . . . . . . . . 5.7.2 Analysis of the Applicability to the Problem . . . . . . . . 5.7.3 Simulation Experiment . . . . . . . . . . . . . . . . . . . . . . . . 5.8 Chapter Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Appendix A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 Appendix B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

Chapter 1

Basic Operations Between Generalized Intuitionistic Multiplicative Fuzzy Information

Since Zadeh (1965) introduced the fuzzy set in 1965, it has been widely applied in various fields of modern society (Bustince et al. 2008; Yager and Kacprzyk 1997). Later on, after analyzing the shortcomings and deficiencies of the fuzzy set, Atanassov (1986) extended it by introducing the intuitionistic fuzzy set (IFS), in which the symmetrical scale is used. But in real-life, there exist the problems that need to assess their variables with the grades that are not uniformly and symmetrically distributed (Degani and Bortolan 1988; Herrera and Herrera-Viedma 2008; Herrera et al. 2008; Liu et al. 2004; Torra 2000, 2001). Here we take the law of diminishing marginal utility in economics for example, if we invest the same resources to a company with bad performance and to a company with good performance respectively, the former yields more utility than the latter. To deal with such a situation, one of the most basic concepts in the famous Analytic hierarchy process (AHP) (Saaty 1977), Saaty’s 1–9 scale (Saaty 1980), was first proposed by professor Saaty, an American operations researcher. The intuitionistic multiplicative set (IMS) (Xia et al. 2013), is the typical concept put forward in 2013 on the basis of this scale, drawing on the advantages of intuitionistic fuzzy set in the representation of information as well. Recently, the intuitionistic multiplicative set theory has received great attention and has been applied to many practical fields, including preference relations (Jiang et al. 2013; Xia et al. 2013; Yu et al. 2013), compatibility measures and consensus models (Jiang et al. 2013), decision making (Liao et al. 2014; Xia and Xu 2013; Xu 2013; Yu and Liao 2016; Yu and Xu 2014), the priority weights deriving (Xu 2013), and aggregation techniques (Xia and Xu 2013; Xia et al. 2013; Yu and Shi 2015), etc. To study calculus, we must first study its domain of definition, limits and other related concepts. The existing  IMS is defined based on Saaty’s 1–9 scale, and enclosed  by ρ = 1 9, ρ = 9, σ = 1 9 and σ = 9. When looking at the derivatives over it, we find that its result may no longer belong to the IMS, even the abnormal limits (0, +∞) and (+∞, 0). So, it is necessary to consider the closure of the existing IMS to the derivative operation by extending its boundary to ρ = 0 and σ = 0, and the new set is denoted as the generalized intuitionistic multiplicative fuzzy set (GIMFS) © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2020 S. Yu and Z. Xu, Generalized Intuitionistic Multiplicative Fuzzy Calculus Theory and Applications, Uncertainty and Operations Research, https://doi.org/10.1007/978-981-15-5612-8_1

1

2

1 Basic Operations Between Generalized Intuitionistic …

(Yu et al. 2017). As such, the operational laws, comparison and sorting methods and so on, which are already defined on it will be re-validated, and the related concepts of calculus theory in real number field, such as intervals, sequences, convergence forms, etc. never existed in the IMS theory, are also necessary to be discussed to make full preparation for constructing the definite integral model. This chapter is based on Yu et al. (2017) as well as Yu and Xu (2016a) and mainly introduces some basic concepts and algorithms related to the generalized intuitionistic multiplicative fuzzy information, including the generalized intuitionistic multiplicative fuzzy number (GIMFN) (Yu et al. 2017), the basic operational laws, the methods of comparison and sorting, the regional divisions related to a GIMFN, the sequences of GIMFNs and their convergence forms, etc., which are the preliminary knowledge used in the follow-up discussion.

1.1 Generalized Intuitionistic Multiplicative Fuzzy Set Based on Saaty’s 1–9 scale (See Table 1.1 (Saaty 1980) for more details), Xia et al. (2013) extended the intuitionistic fuzzy set and introduced the IMS as follows: Definition 1.1 (Xia et al. 2013). Let X be a fixed set, a intuitionistic multiplicative set (IMS) is defined as: A = { (x, ρ A (x), σ A (x))|x ∈ X } which assigns to each element x ∈ X a membership information ρ A (x) and a nonmembership information σ A (x), with the conditions:  1 9 ≤ ρ A (x), σ A (x) ≤ 9, ρ A (x)σ A (x) ≤ 1, ∀x ∈ X, Table 1.1 The 1/9–9 scale

1/9–9 scale

Meaning

1/9

Extremely not accepted

1/7

Very strongly not accepted

1/5

Strongly not accepted

1/3

Moderately not accepted

1

Equally accepted

3

Moderately accepted

5

Strongly accepted

7

Very strongly accepted

9

Extremely accepted

Other values between 1/9 and 9

Intermediate values used to present compromise

1.1 Generalized Intuitionistic Multiplicative Fuzzy Set

3

Fig. 1.1 Plane representation of intuitionistic multiplicative set

For simplicity, Xia et al. (2013) called the pair α = (ρα , σα ) a intuitionistic multiplicative number (IMN) (In the following discussions, we denote it as α=(ρ, σ ), unless it is expressly stated) and M the set of all the IMNs, which is demonstrated in Fig. 1.1.  From Fig. 1.1, for each IMN, it is a pair α = (ρα , σα ) with the conditions: 1 9 ≤ ρα , σα ≤ 9, ρα σα ≤ 1, that  is to say, every  IMN is located in a close area bounded by two straight lines (ρ = 1 9 and σ = 1 9) and a curve (ρσ = 1). Considering the closure of limit operation in future discussion, we give M an extended range, which is bounded by two infinity points (O = (0, +∞) and ∞ = (+∞, 0)), coordinates (ρ and σ ), and a curve (ρσ = 1) and denote it as G(GIMFS, called generalized intuitionistic multiplicative fuzzy set). The specific definition is as follows: Definition 1.2 (Yu et al. 2017). Let X be a fixed set, a generalized intuitionistic multiplicative fuzzy set (GIMFS) is defined as: D = { (x, ρ D (x), σ D (x))|x ∈ X } ∪ {O = (0, +∞), ∞ = (+∞, 0)} which assigns to each element x ∈ X a membership information ρ D (x) and a nonmembership information σ D (x), with the conditions: 0 < ρ D (x), σ D (x) < +∞, ρ D (x)σ D (x) ≤ 1, ∀x ∈ X

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1 Basic Operations Between Generalized Intuitionistic …

For convenience, Yu et al. (2017) called the ordered pair α = (ρ, σ ) a generalized intuitionistic multiplicative fuzzy number (GIMFN) and G the set of all the GIMFNs. Obviously, any α should satisfy: 0 < ρ, σ < +∞, ρσ ≤ 1. In the following discussions, for any α ∈ G, we take α = O and α = ∞ as default, unless it is expressly stated.

1.2 Some Operational Laws Related to Any GIMFNs Below we introduce some operational laws related to any GIMFNs and discuss their corresponding properties. Definition 1.3 (Yu et al. 2017). Let α = (ρ, σ ), α1 = (ρ1 , σ1 ), and α2 = (ρ2 , σ2 ) be three GIMFNs, and λ > 0, then   2σ1 σ2 2 )−1 (1) α1 ⊕ α2 = (1+2ρ1 )(1+2ρ ; , 2 (2+σ1 )(2+σ2 )−σ1 σ2   2ρ1 ρ2 2 )−1 (2) α1 ⊗ α2 = (2+ρ1 )(2+ρ ; , (1+2σ1 )(1+2σ 2 2 )−ρ1 ρ2   λ λ −1 (3) λα = (1+2ρ) , (2+σ2σ)λ −σ λ ; 2   2ρ λ (1+2σ )λ −1 . (4) α λ = (2+ρ) λ λ 2 −ρ Motivated by the addition and multiplication operations for GIMFNs, we introduce two new basic operational laws on GIMFS, which are the subtraction law and the division law. Definition 1.4 (Yu et al. 2017). For any two GIMFSs: A = {x, ρ A (x), σ A (x)|x ∈ X } and B = {x, ρ B (x), σ B (x)|x ∈ X }, the subtraction and division laws have the following forms: (1) A B = {x, ρ A B (x), σ A B (x)|x ∈ X }, where ⎧ ρ A (x)−ρ B (x) , i f 0 < ρ B (x) < ρ A (x) < +∞ ⎪ 1+2ρ B (x) ⎪ ⎪ ⎪ and 0 < σ A (x)< σ B (x) < +∞ ⎨ ρ A B (x) = and (ρ A (x) − ρ B(x)) (1 + 2ρ B (x)) ≤ ⎪ ⎪ ⎪ (σ B (x) − σ A (x)) (σ A (x)(2 + σ B (x))) ⎪ ⎩ 0, other wise ⎧ σ (x)(2+σ (x)) A B ⎪ , i f 0 < ρ B (x) < ρ A (x) < +∞ ⎪ σ B (x)−σ A (x) ⎪ ⎪ ⎪ and 0 < σ A (x) < σ B (x) ⎨  < +∞ σ A B (x) = and (ρ A (x) − ρ B(x)) (1 + 2ρ B (x)) ≤ ⎪ ⎪ ⎪ (σ B (x) − σ A (x)) (σ A (x)(2 + σ B (x))) ⎪ ⎪ ⎩ +∞, other wise

1.2 Some Operational Laws Related to Any GIMFNs

5

(2) A B = {x, ρ A B (x), σ A B (x)|x ∈ X }, where ⎧ ρ (x)(2+ρ (x)) A B ⎪ , i f 0 < ρ A (x) < ρ B (x) < +∞ ⎪ ρ B (x)−ρ A (x) ⎪ ⎪ ⎪ and 0 < σ B (x) < σ A (x) < +∞ ⎨ ρ A B (x) = and (σ A (x) − σ B(x))/(1 + 2σ B (x)) ≤ ⎪ ⎪ ⎪ (ρ B (x) − ρ A (x)) (ρ A (x)(2 + ρ B (x))) ⎪ ⎪ ⎩ +∞, other wise ⎧ σ (x)−σ (x) A B ⎪ , i f 0 < ρ A (x) < ρ B (x) < +∞ ⎪ 1+2σ B (x) ⎪ ⎪ ⎪ and 0 < σ B (x) < σ A (x) ⎨  < +∞ σ A B (x) = and (σ A (x) − σ B(x)) (1 + 2σ B (x)) ≤ ⎪ ⎪ ⎪ (ρ B (x) − ρ A (x)) (ρ A (x)(2 + ρ B (x))) ⎪ ⎪ ⎩ 0, other wise By Definition 1.4, the operation “ ” is the inverse operation of “⊕”, which means α ⊕ β β = α, ∀α, β ∈ G. The operation “ ” is the

operation of “⊗”. Take inverse “ ” for example, if we let α = (ρα ,  , which σα ) and β = , σ ρ β β



 satisfy

1 + 2ρβ ≤ σβ − σα σα 2 + σβ ρβ < ρα , σα < σβ and ρα − ρβ then 

ρα − ρβ σα 2 + σβ , α β = 1 + 2ρβ σβ − σα which is still a GIMFN. Inversely, if two GIMFNs α and β don’t meet such conditions, then we get that α β isn’t a GIMFN. In addition, we take account of the closure of the operation by defining α β = (0, +∞). However, (0, +∞) is almost meaningless. Similarly, the same analysis works in the operation “ ”. According to Definitions 1.3 and 1.4, the above algorithm satisfies the following properties: Theorem 1.1 (Yu et al. 2017). Let α, α1 , and α2 be three GIMFNs, and λ, λ1 , λ2 > 0, then α1 ⊕ α2 , α1 ⊗ α2 , λα, and α λ are GIMFNs, and (1) (2) (3) (4) (5) (6)

α1 ⊕ α2 = α2 ⊕ α1 ; α1 ⊗ α2 = α2 ⊗ α1 ; λ(α1 ⊕ α2 ) = λα1 ⊕ λα2 , λ > 0; (α1 ⊗ α2 )λ = α1λ ⊗ α2λ , λ > 0; λ1 α ⊕ λ2 α = (λ1 + λ2 )α, λ1 , λ2 > 0; α λ1 ⊗ α λ2 = α λ1 +λ2 , λ1 , λ2 > 0.

The following is a brief comparison of the algorithms on the generalized intuitionistic multiplicative fuzzy set and the real number field:

6

1 Basic Operations Between Generalized Intuitionistic …

In real number field, we know that any two real numbers can be connected by using the four basic operations (“+, −, ×, /”). For example, 2.5 = 5/2, 3 = 2 + 1, etc. Hence, it is natural to ask whether or not a given GIMFN α can be changed to any GIMFN β by using the four basic operations (“⊕, , ⊗, ”). From now on, we consider ∀α0 ∈ G as a variable and discuss the change region related to α0 according to the four basic operational laws (“⊕, , ⊗, ”) as below.

1.3 The Regional Divisions Related to a GIMFN Definition 1.5 (Yu et al. 2017). Let α, α0 , and β be three GIMFNs, if α = α0 ∗ β, where ∗ ∈ {⊕, , ⊗, }, then we call α the change value related to α0 . In the two-dimensional Cartesian coordinate system of Fig. 1.2 (Yu et al. 2017), we take M ⊂ G for example and divide the whole domain of the seemingly triangular area:  M = {(ρ , σ )|1 9 ≤ ρ, σ ≤ 9, ρσ ≤ 1} into eight parts (S1 − S8 , as shown in Fig. 1.2 (Yu et al. 2017)), which are related to α0 = (ρα0 , σα0 ) according to the four basic operational laws. From Fig. 1.2 (Yu et al. 2017), if α = (ρ, σ ) ∈ S1 ∪ S2 , then ρ ≥ ρ0 , σ ≤ σ0 . By the definition of the subtraction law, we can get

α α0 =

ρ − ρ0 σ (2 + σ0 ) , 1 + 2ρ0 σ0 − σ

 ∈M

That is, for any β ∈ M, we can get α0 ⊕ β ∈ S1 ∪ S2 . Similarly, the corresponding eight parts can be depicted as: Fig. 1.2 The regional divisions related to α0

1.3 The Regional Divisions Related to a GIMFN

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S1 = {α | ther e exists β ∈ M, such that α = α0 ⊕ β} S2 = {α | ther e exists β ∈ M, such that α = α0 ⊕ β or α = α0 β} S3 = {α | ther e exists β ∈ M, such that α = α0 β} S4 = {α | ther e does not exist β ∈ M, such that α = α0 ∗ β, ∗ ∈ {⊕, ⊗, , }} S5 = {α | ther e exists β ∈ M, such that α = α0 β} S6 = {α | ther e exists β ∈ M, such that α = α0 β or α = α0 ⊗ β} S7 = {α | ther e exists β ∈ M, such that α = α0 ⊗ β} S8 = {α | ther e does not exist β ∈ M, such that α = α0 ∗ β, ∗ ∈ {⊕, ⊗, , }} For convenience, we continue to use the same symbols as Lei and Xu (2015) to complete the following discussion, that is, Si which is depicted above and A∗α0 which is discussed below. Here we give the following definitions: ⊗ Definition 1.6 (Yu et al. 2017). Let A⊕ α0 = S1 ∪ S2 , Aα0 = S5 ∪ S6 , Aα0 = S6 ∪ S7 , and

⊕ Aα0 = S2 ∪ S3 , then we call Aα0 the addition region related to α0 , Aα0 the subtraction

region related to α0 , A⊗ α0 the multiplication region related to α0 , and Aα0 the division ⊕

region related to α0 , respectively. Additionally, Aα0 = Aα0 ∪ Aα0 ∪ A⊗ α0 ∪ Aα0 is ¯ called the change region related to α0 , and Aα0 = S4 ∪ S8 is called the non-change region related to α0 .

Compared with the two change directions of a real number, there are four change directions related to ∀α0 ∈ M (as shown in Fig. 1.3). For example, if α1 ∈ A⊕ α0 , then there exists β1 ∈ M, such that α1 = α0 ⊕ β1 . Similarly, if α2 ∈ A , α ∈ A 3 α0 α0 , and ⊗ α4 ∈ Aα0 , then there exist β2 , β3 and β4 , such that α2 = α0 β2 , α3 = α0 β3 , and α4 = α0 ⊗ β4 . For convenience, we take ∀α0 ∈ G for example, there is a brief illustration of A⊕ α0 and A α0 as shown in Fig. 1.4 (Yu and Xu 2016a).

1.4 A New Order of GIMFNs Limit is the basis of the calculus theory and the ultimate description of the “degree of approach”. In this process, mutual comparison is a necessary link, which is often achieved by sorting. The existing orders can be divided into partial order and full order. Here, it is necessary to introduce the basic order theory briefly.

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1 Basic Operations Between Generalized Intuitionistic …

A

Fig. 1.3 Four change directions related to α0

A

A

A Fig. 1.4 The addition region and subtraction region of α0

Definition 1.7 (Lei and Xu 2017). Let P be a set with a binary relation R. The relation R consists of some ordered pairs, these basic elements of which are both in P. For example, for any two elements p1 and p2 ( p1 ∈ P and p2 ∈ P), if the order pair ( p1 , p2 ) ∈ R, then it is denoted by p1 Rp2 . In addition, if the binary relation R satisfies the following three conditions: (1) (Reflexivity) For any elements p ∈ P, there is p Rp. (2) (Antisymmetry) If p1 Rp2 and p2 Rp1 , then p1 = p2 . (3) (Transitivity) If p1 Rp2 and p2 Rp3 , there is p1 Rp3 . then we call the binary relation R as a partial order, and call the set P as a poset. In addition, if there must be p1 Rp2 or p2 Rp1 for any two given p1 and p2 ( p1 ∈ P and

1.4 A New Order of GIMFNs

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p2 ∈ P), which means that any two elements in P are always comparable, then we call the partial order R as a total order or a linear order. We notice that there are various methods for ordering intuitionistic multiplicative numbers in existing researches. Among them, the following partial order proposed by Xia et al. (2013) is the earliest and most commonly used one in intuitionistic multiplicative set: Definition 1.8 (Xia et al. 2013). The partial order α1 ≥ P α2 means that for αi = (ρi , σi ) ∈ IMNs, i = 1, 2, which satisfy the conditions ρ1 ≥ ρ2 and σ1 ≤ σ2 ; the partial order α1 > P α2 means that the two IMNs meet one of the following conditions: (1) ρ1 > ρ2 , σ1 ≤ σ2 ; (2) ρ1 > ρ2 , σ1 < σ2 ; and (3) ρ1 ≥ ρ2 , σ1 < σ2 respectively. Similarly, we can get the partial orders α1 ≤ P α2 and α1 < P α2 . Furthermore, Xia et al. (2013) proposed a total order comparison method based on score function and accuracy function in intuitionistic multiplicative set M. In the following discussion, for ∀α = (ρ,  σ ) ∈ M, the score function and the accuracy function are denoted as s(α) = ρ σ and h(α) = ρσ respectively. Definition 1.9 (Xia et al. 2013). Let α1 = (ρ1 , σ1 ) and α2 = (ρ2 , σ2 ) be two intuitionistic multiplicative numbers. For the comparison of two IMNs α1 and α2 , the following laws can be given: (1) If s(α1 ) > s(α2 ), then α1 > α2 ; (2) If s(α1 ) = s(α2 ), then a. If h(α1 ) > h(α2 ), then α1 > α2 ; b. If h(α1 ) = h(α2 ), then α1 = α2 . According to Definition 1.8 and 1.9, if α1 ≥ P α2 , then we have α1 ≥ α2 . The only pity is that these two orders have nothing to do with the basic operations defined on the intuitionistic multiplicative set, which is quite different from the comparison of numbers in real number domain and the definition of derivatives and other common concepts in traditional calculus theory. For one thing, from the equations 9 = 6+3 and  9 6 = 1.5, we know that 9 is bigger than 6. For another thing, the classical derivative is actually the limit of the ratio of the increment of the function to the increment of the independent variable, in which both the increment and the ratio are related to the four basic operations (“+, −, ×, /”). Furthermore, as we know, Riemann integral in real number field contains three basic parts: the integral sign, the integral limits and the integrand. For integral limits, if we can define the integral on a total order according to the four basic operational laws (“⊕, , ⊗, ”) in GIMFS, then it would be perfect. Therefore, before making a discussion of the definite integrals in GIMFS, a new order of GIMFNs is introduced to denote the integral limits sufficiently as follows: Definition 1.10 (Yu and Xu 2016a). If there exists γ ∈ G, satisfying α ⊕ γ = β, then we denote α ≤⊕ β. Furthermore, if α = β, then we denote α