Advanced Decision-Making Methods and Applications in System Safety and Reliability Problems: Approaches, Case Studies, Multi-criteria Decision-Making, ... in Systems, Decision and Control, 211) [1st ed. 2022] 3031074297, 9783031074295

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Studies in Systems, Decision and Control 211

He Li Mohammad Yazdi

Advanced Decision-Making Methods and Applications in System Safety and Reliability Problems Approaches, Case Studies, Multi-criteria Decision-Making, Multi-objective Decision-Making, Fuzzy Risk-Based Models

Studies in Systems, Decision and Control Volume 211

Series Editor Janusz Kacprzyk, Systems Research Institute, Polish Academy of Sciences, Warsaw, Poland

The series “Studies in Systems, Decision and Control” (SSDC) covers both new developments and advances, as well as the state of the art, in the various areas of broadly perceived systems, decision making and control–quickly, up to date and with a high quality. The intent is to cover the theory, applications, and perspectives on the state of the art and future developments relevant to systems, decision making, control, complex processes and related areas, as embedded in the fields of engineering, computer science, physics, economics, social and life sciences, as well as the paradigms and methodologies behind them. The series contains monographs, textbooks, lecture notes and edited volumes in systems, decision making and control spanning the areas of Cyber-Physical Systems, Autonomous Systems, Sensor Networks, Control Systems, Energy Systems, Automotive Systems, Biological Systems, Vehicular Networking and Connected Vehicles, Aerospace Systems, Automation, Manufacturing, Smart Grids, Nonlinear Systems, Power Systems, Robotics, Social Systems, Economic Systems and other. Of particular value to both the contributors and the readership are the short publication timeframe and the worldwide distribution and exposure which enable both a wide and rapid dissemination of research output. Indexed by SCOPUS, DBLP, WTI Frankfurt eG, zbMATH, SCImago. All books published in the series are submitted for consideration in Web of Science.

He Li · Mohammad Yazdi

Advanced Decision-Making Methods and Applications in System Safety and Reliability Problems Approaches, Case Studies, Multi-criteria Decision-Making, Multi-objective Decision-Making, Fuzzy Risk-Based Models

He Li School of Intelligent Systems Engineering Sun Yat-Sen University Shenzhen, China

Mohammad Yazdi Faculty of Engineering and Applied Science Memorial University of Newfoundland St. John’s, NL, Canada

ISSN 2198-4182 ISSN 2198-4190 (electronic) Studies in Systems, Decision and Control ISBN 978-3-031-07429-5 ISBN 978-3-031-07430-1 (eBook) https://doi.org/10.1007/978-3-031-07430-1 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 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, expressed 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 Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

System safety and reliability analysis subjects become more complex over time as several parameters are now required for decision-making [1, 2]. The existing decision-making methods are not effective or efficient for improving the system’s safety performance. Here, the decision-makers utilize decision-making tools that play a critical role in system safety and reliability analysis. However, the sources of uncertainty, such as interdependency between the contributing factors during the decision-making process, dynamic features of the factors, and other reasons, can make decision-making a challenging task. Proposing and utilizing advanced decision-making tools are essential to address the complexity and multi-dimensional variability of decision-making problems in system safety and reliability analysis. This book provides valuable insight into practical and advanced decision-making methods under different mathematical models in system safety and reliability analysis. The explained case studies highlight the applicability of each proposed approach, which can be helpful for complex systems as well as risk assessment, evaluation, and management processes. This book is organized to include ten chapters. Chapter 1 introduces a novel dynamic decision-making trial and evaluation laboratory (DEMATEL) for improving the safety management system. The main idea of this chapter and its challenges is how to include the parameter time in the original formulation of the DEMATEL technique. In this case, the dynamic features contributing to system safety and reliability analysis decision-making problem are adequately addressed. Chapter 2 transforms a fault tree analysis model to construct a Bayesian network approach with higher superiority than conventional analysis correlations. The system reliability of the main drive systems, critical failure modes and elements, and sensitivity analysis of critical failures of the main drive systems are highlighted. Moreover, modeling patterns with continuous life and discrete fault data are demonstrated. Overall, this chapter shows a detailed reliability modeling and analysis process with correlated failures of complex systems. Chapter 3 proposes an advanced decision-making tool to minimize the risk of well drilling to identify different practical barriers (essential factors) and evaluate their causality and interdependencies. The necessary weights of essential factors are v

vi

Preface

integrated within the subjectivity of a decision-maker’s opinions and data variability distributions, where the outcomes are more practical. As an application of study, selective good drilling blowouts barriers are analyzed and evaluated to show the efficiency and rationality of the progressive DEMATEL–game theory approach Chapter 4 develops an advanced failure mode and effect analysis (FMEA) technique to analyze offshore wind turbines (OWTs) by predicting the weakest links of the system and improving the system’s safety and reliability performance with additional corrective actions when required. We put forward to integrate FMEA with probabilistic linguistic preference relations (PLPRs), the multiplicative consistencybased weight tool, and the best-worst method (BWM). The proposed method is then applied to real case OWT to show its potential and applicability. Chapter 5 investigates the best sustainable hydrogen storage technologies considering different contributing factors in large metropolitan areas. In this chapter, integration of the Bayesian network approach and interval type-2 fuzzy sets are utilized to deal with the causality and uncertainty during the investigation process, respectively. A practical application is provided for site selection of the best sustainable hydrogen storage technologies in large metropolitan areas of Iran. Chapter 6 presents the fuzzy cognitive map (FCM) method helps to improve human system health and wellbeing in front of toxic people over time as a system safety decision-making problem. The FCM tool is systematically constructed to map the relationships between toxic people (targeted persons), toxic behaviors, toxic terrors, and toxic handling behaviors. Chapter 7 proposes an advanced Technique for Order Preference by Similarity to an Ideal Solution (TOPSIS), which takes from the underlying the idea of a spherical fuzzy set (SFS) to improve human system reliability despite subjective uncertainty over time. The advanced TOPSIS-PFS method can adequately identify the reliable index of a complex system (human–environment–machine) and help decisionmakers prioritize the critical index layers. Performing the proper intervention actions can improve and establish a robust human reliability system. Chapter 8 aims to model the system safety and reliability decision-making problems into a strategic game and solve them by utilizing the non-cooperative game-theoretic concept. The stochastic game theory suggests the non-dominated outcomes and solutions to decision-making problems. Moreover, different types of non-cooperative stability definitions solve the game theory-based system safety and reliability decision-making problems. Chapter 9 presents a hybrid model to extend the BWM as an acceptable decisionmaking method. In this case, the neutrosophic fuzzy set and evidence theory is integrated with BWM by dealing with subjective and objective uncertainties from decision-makers’ opinions and natural variabilities. By evaluating the risks of wildfire, the shortages of the model are obtained, and preventive and mitigative actions are proposed to enhance the system’s performance. Chapter 10 argues that the decision-makers neglect the probability in the risk assessment method, as such arguments would provide decision-makers with viable insight into the system safety and reliability decision-making process. As a result, the outcomes will be much more trustable and reliable.

Preface

vii

This book will be one of the most important guidance books for professionals and researchers working in system safety and reliability. It also aims to become a valuable reference book for postgraduate and undergraduate students. Finally, as the authors of the present book, we are grateful to our family and friends for their constant love, patience, and support. You have all contributed through your words of encouragement for the period of present work, with the best wishes that the book will be helpful to all concerned. Shenzhen, China St. John’s, Canada April 2022

He Li Mohammad Yazdi

References 1.

2.

Yazdi, M.: A brief review of using linguistic terms in system safety and reliability analysis BT—linguistic methods under fuzzy information in system safety and reliability analysis. In: Yazdi, M. (ed.), pp. 1–4. https://doi.org/10.1007/978-3-030-93352-4_1. Springer, Cham, (2022) Rausand, M., Haugen, S.: Risk Assessment: Theory, Methods, and Applications. Wiley, (2020)

Contents

1

2

3

Dynamic Decision-Making Trial and Evaluation Laboratory (DEMATEL): Improving Safety Management System . . . . . . . . . . . . 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Proposed Methodology: Dynamic-DEMATEL . . . . . . . . . . . . . . . . 1.3 Application of Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 5 7 11 11

Reliability Analysis of Correlated Failure Modes by Transforming Fault Tree Model to Bayesian Network: A Case Study of the MDS of a CNC Machine Tool . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Methodologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Case Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 The MDS and Failure Criterion . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Establish the Fault Tree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Failure Data of Components . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.4 Reliability Analysis with Discrete Failure Data . . . . . . . . . 2.3.5 Reliability Analysis with Continuous Life Data . . . . . . . . 2.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15 15 16 19 19 19 21 22 24 26 26

What Are the Critical Well-Drilling Blowouts Barriers? A Progressive DEMATEL-Game Theory . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Preliminary: DEMATEL “Decision-Making Trial and Evaluation Laboratory” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Preliminary: Game Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 The Progressive DEMATEL-Game Theory Approach . . . . . . . . . 3.5 Case Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

29 29 30 31 32 35 37 42 ix

x

4

5

Contents

Developing Failure Modes and Effect Analysis on Offshore Wind Turbines Using Two-Stage Optimization Probabilistic Linguistic Preference Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Preliminary: Probabilistic Linguistic Preference Relations (PLPRs) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 The Proposed Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Application of Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Integration of the Bayesian Network Approach and Interval Type-2 Fuzzy Sets for Developing Sustainable Hydrogen Storage Technology in Large Metropolitan Areas . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Preliminary: Bayesian Network Approach . . . . . . . . . . . . . . . . . . . 5.3 Preliminary: Interval Type-2 Fuzzy Sets (IT2FSs) . . . . . . . . . . . . . 5.4 Case Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Results and Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

47 47 50 52 59 62 62

69 69 70 72 75 79 81 82

6

How to Deal with Toxic People Using a Fuzzy Cognitive Map: Improving the Health and Wellbeing of the Human System . . . . . . . 87 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 6.2 Preliminary: Fuzzy Cognitive Map . . . . . . . . . . . . . . . . . . . . . . . . . . 89 6.3 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 6.4 Case Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 6.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

7

An Advanced TOPSIS-PFS Method to Improve Human System Reliability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Preliminary: Spherical Fuzzy Set (PFS) . . . . . . . . . . . . . . . . . . . . . . 7.3 Methodology: An Advanced TOPSIS-PFS Method . . . . . . . . . . . . 7.4 Case Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

Stochastic Game Theory Approach to Solve System Safety and Reliability Decision-Making Problem Under Uncertainty . . . . . 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Preliminary: MCDM as a Game Theory . . . . . . . . . . . . . . . . . . . . . 8.3 The Bouali Sina Fire Accident Decision-Making Problem . . . . . . 8.4 The Stochastic Game Theory of Bouali Sina Fire Accident Decision-Making Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

109 109 110 112 114 122 124 127 127 129 134 140

Contents

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8.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 9

Advanced Decision-Making Neutrosophic Fuzzy Evidence-Based Best–Worst Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Preliminary: Neutrosophic Fuzzy Set . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Preliminary. Evidence Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 The Proposed Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5 Application of Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5.1 Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5.2 Comparison Analysis Using Fuzzy AHP, BWM, TOPSIS, and the Simple Average Method . . . . . . . . . . . . . 9.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10 A Holistic Question: Is It Correct that Decision-Makers Neglect the Probability in the Risk Assessment Method? . . . . . . . . . . 10.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Open Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

153 153 155 157 159 165 174 174 178 178 185 185 186 188

Chapter 1

Dynamic Decision-Making Trial and Evaluation Laboratory (DEMATEL): Improving Safety Management System

1.1 Introduction In the literature, there are three types of dependency methods, including (i) structural dependency (e.g., TOPSIS (Technique for Order of Preference by Similarity to Ideal Solution) [1–3], BWM (best worse method [4–6], etc.), (ii) casual dependency (e.g., Bayesian networks [7–10], Fuzzy cognitive map [11], DEMATEL (decision-making trial and evaluation laboratory) [12–15], and (iii) preferential dependency such as conditional preference networks [16]. DEMATEL, like all structural dependency methods, has enough potential to determine the cause and effects of dependency between a group of contributing factors. Moreover, DEMATEL can develop an influence diagram to illustrate the interdependencies between the contributing factors. DEMATEL reflects the direct and indirect relationships of the contributing factors in a decision-making problem. DEMATEL also has enough potential to classify the relationships of contributing factors in two cause and effect classes. Besides, DEMATEL recognizes the critical factors in the much more complicated system. The DEMATEL method has been widely applied in different applications and domains, such as computer science artificial intelligence [17–19], operations research management science [17, 18, 20, 21], green sustainable science technology [22–25], and industrial engineering [26–29], among others. Figure 1.1 illustrates the number of publications on DEMATEL utilization based on all application domains since 2000. This trend showed the highest increase in 2018; it is predictable that DEMATEL utilizations in different case studies, especially system safety and reliability analysis, will continue to evolve in the next years. According to Table 1.1, the first ten highly cited documents consider the “Average Citations per Year” of each publication in the area of DEMATEL. The “Average Citations per Year” indicates the “WoS citation index” for a paper by the end of December 2021.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 H. Li and M. Yazdi, Advanced Decision-Making Methods and Applications in System Safety and Reliability Problems, Studies in Systems, Decision and Control 211, https://doi.org/10.1007/978-3-031-07430-1_1

1

2

1 Dynamic Decision-Making Trial and Evaluation Laboratory (DEMATEL) …

Fig. 1.1 Distribution of “published works per year till the end of December 2021” in DEMATEL utilizations based on all application domains since 2000 (according to the web of science (WoS) database)

The performance procedure of the DEMATEL method is provided as the following: • Generating the group direct-influence matrix Let us consider that there would be n number of contributing factors “F = {F1 , F2 , F3 , . . . , Fn }”. To begin the assessment process with the DEMATEL, several decision-makers l as a group “DM = {DM1 , DM2 , DM3 , . . . , DMl }” are employed highlighting that the influence of factor “Fi ” on factor “Fj ” within five integer-scales “very high (VH) influence (4), high (H) influence (3), moderate (M) influence  (2),  low (L) influence (1), and no (N) influence (0)”. Hence, consider that ” is the single influential matrix, and is constructed by k th decision“Z k = z ikj n×n

maker, where “z ikj ” signifies that the decision elicited from “D Ml ” on “Fi ” influences “Fj ”. By uniting the l input from “D Ml ”, the group direct-influence matrix is derived  as “Z i j = 1l lk=1 z ikj ”, where i, j = 1, 2, …, n. • Implementing the normalized direct-influence matrix   The direct-influence matrix, “X = xi j n×n ” is normalized as “X = max 1 n aij × Z” j=1 in which 1≤ i ≤ n . It should be added that any parts of matrix X are in [0,1].  Besides, “ nj=1 xi j ≥ 0” or “ nj=1 xi j ≤ 1”. Subsequently, the total direct-influence   matrix “T = ti j n×n ” is determined as: “T = X + X 2 + X 3 + . . . + X h ”. The indirect influence i over j is represented by “ti j ”. Thus, the entire relationship among the contributing factors would be represented with “total direct-influence matrix”. Since “h → ∞”, the “total direct-influence matrix” can be formulated as “T = X (I − X )−1 ”. In which, “I” denotes the unit matrix according to the Markov chain theory, the “X h ” indicates matrix X power.

1.1 Introduction

3

Table 1.1 The “highly cited papers based on citation measures” in the field of DEMATEL till the end of December 2021 (according to the WoS database) Row

Descriptions

Reference

Total citations

Average per year

1

Evaluating intertwined effects in e-learning programs: A novel hybrid MCDM model based on factor analysis and DEMATEL

Tzeng et al. [30]

646

43.07

2

Developing global managers’ competencies using the fuzzy DEMATEL method

Wu and Lee [31]

585

39

3

A novel hybrid MCDM approach based on fuzzy DEMATEL, fuzzy ANP and fuzzy TOPSIS to evaluate green suppliers

Gulcin and Gizem [32]

512

51.2

4

Choosing knowledge management strategies by using a combined ANP and DEMATEL approach

Wu [33]

324

23.14

5

Using fuzzy DEMATEL Lin [34] to evaluate the green supply chain management practices

316

35.11

6

Fuzzy DEMATEL method Chang et al. [17] for developing supplier selection criteria

313

28.45

7

Using DEMATEL to Hsu et al. [35] develop a carbon management model of supplier selection in green supply chain management

305

33.89

8

A DEMATEL method in identifying key success factors of hospital service quality

Shieh et al. [36]

300

25

9

A causal analytical method for group decision-making under fuzzy environment

Lin and Wu [37]

291

20.79

10

A causal and effect Tseng [38] decision-making model of service quality expectation using grey-fuzzy DEMATEL approach

243

18.69

4

1 Dynamic Decision-Making Trial and Evaluation Laboratory (DEMATEL) …

• Producing the “influential relation map”

 n   Next is producing the “influential relation map”, “C = i=1 ti j 1×n = t. j 1×n ” n  and “R = i=1 ti j n×1 = [ti. ]n×1 ” are the columns and rows summation of “total direct-influence matrix”. In which, “t. j ” indicates the jth column summation  “T = X × (I − X )−1 ”, exemplifying factor influences “F j ” summation receiving influence from other factors. Likewise, “t. j ” is the row summation in the “total direct-influence matrix” and presents the influences summation dispatching from factor  Fi  to the other factors, directly or indirectly. Consider that “ j = i, i, j ∈ {1, 2, . . . , n}”, “R + C” is named as “Prominence,” meaning that the horizontal axis vector is the relative importance that every single contributing factor receives influence from itself or others. The “R + C” is used for the specific control degree for the complex system, such as system safety and reliability analysis. Likewise, “R − C” named “Relation” is subsequently  for  obtained the vertical axis highlighting the net effect. To satisfy the condition “ t. j − ti. > 0”, (net) on the contributing factor becomes the cause factor since the “F j ” influences   the other contributing factors. In the case of satisfying condition “ t. j − ti. < 0”, the contributing factor “F j ” is influenced by other contributing factors. It is then considered for the effect group. The “influential relation map” is drawn within the score of “R + C” and “R − C”, as depicted in Fig. 1.2. The drawn influence diagram would specify the insights for decision-makers in system safety and reliability analysis decision-making problem. Four classes (1, 2, 3, and 4) can be highlighted in the drawn influence diagram for the contributing factors according to their location in the diagram. The contributing factors belonging to zone 1 are the critical factors (i.e., givers). The contributing factors belonging to zone 2 contain the driving factors (i.e., autonomous givers). The contributing factors belonging to zone 3 are the autonomous receivers (i.e., independent factors). The contributing factors belonging to zone 4 are impact factors (i.e., receivers). The critical point here is that the contributing factors belonging to zone 4 cannot be improved directly due to the fact that other factors influence them.

Fig. 1.2 The four classes of “influential relation map”

1.2 Proposed Methodology: Dynamic-DEMATEL

5

• Computing the importance weight of all factors The importance weights of the contributing factors are derived. The factors’ weight is according to the “Prominence” score is formulated by normalization process as t +t “wi = n i.n .(tj i. +t. j ) ”, in which i, j = 1, 2, . . . , n. i=1 j=1 There is still room for further development of the DEMATEL technique according to the current literature. Thus, the main contribution of the present work is developing Dynamic DEMATEL in the system safety, and reliability analysis decision-making problems since a practical system safety and reliability analysis decision-making problems are time-dependent. The organization of the present chapter is provided as the following. First, in Sect. 1.2, the DEMATEL is developed in a dynamic manner. In Sect. 1.3, an example illustration of a safety management system is presented. A conclusion is explained in the Sect. 1.4.

1.2 Proposed Methodology: Dynamic-DEMATEL To propose dynamic-DEMATEL, the main idea and challenges include the parameter time τ in the original formulation of the DEMATEL technique, as explained in the introduction section. The proposed dynamic-DEMATEL framework is depicted in Fig. 1.3. The methodology has four key steps: (i) determining the decision-making contributing factors, (ii) constructing the time-dependent group direct influence matrix, (iii) creating the “influential relation map”, and (iv) deriving the importance weights of all contributing factors. An explanation of all steps is specified as the following.

Fig. 1.3 The proposed dynamic-DEMATEL framework

6

(i)

1 Dynamic Decision-Making Trial and Evaluation Laboratory (DEMATEL) …

Determining the decision-making contributing factors

All potential decision-making contributing factors connecting to the system safety and reliability analysis decision-making problem must be recognized. Then, the contributing factors can be obtained according to the decision-makers’ understanding to the problem and using the existing literature such as reports and published papers. (ii)

Constructing the time-dependent group direct influence matrix

The main contribution of dynamic-DEMATEL is performed in this step. Conventionally, a survey is required to be completed by questioning a group-based of decision-makers to share their individual judgements regarding the influence of the contributing factors on each other with the scale of 0–4. However, as discussed earlier, and from the published state of arts such as but not limited to [39–42], DEMATEL, like all Multiple-criteria decision-making tools, is independent of time and cannot provide reliable and valuable insights for system safety and reliability analysis decision-making problems. It also increases the inconsistency of decision-making problems. To justify the 0–4 scale-based score in the original form of DEMATEL and to seek the simplicity, in the present study, we assumed that the time dependency is linear (δτ + C), in which δ = 0, 1, 2, 3, and 4, τ is time, and C is the constant value according to the decision-makers’ opinions. As an example, DM1 shares her opinions regarding   the influence of factor A on B as 4τ + 56. Hence, consider ˜ ” is a single influential matrix in dynamic-DEMATEL, and is that “ Z k = z˜ ikj n×n

provided by kth decision-maker, where  z˜ ikj  signifies that the decision elicited from “D Ml ” on “Fi ” influences “Fj ”. By integrating the l input from “D Ml ”, the group  direct-influence matrix of dynamic-DEMATEL is derived as “ Z˜ i j = 1l lk=1 z˜ ikj ”, where i, j = 1, 2, …, n. It should be added that the time τ is defined based on decision-makers and can be considered as seconds, minutes, hours, days, weeks, months, or years. The only point is that some decision-making tools such as the Markov Chain [43–45] and the Bayesian network [46–48] can consider parameter time during the assessment process. However, the main advantage of dynamic-DEMATEL over such methods is the potential capability of DEMATEL in the classification of contributing factors. (iii)

Creating the “influential relation map”

Considering the unit value for time τ in step (ii), the “influential relation map” can ˜ and “Relation, R˜ − C”. ˜ be drawn by computing the “Prominence, R˜ + C” direct-influence matrix” is defined as “T˜ = The “total dynamic −1 X˜ × I − X˜ ”. Similar to the original formulation of DEMATEL, in the dynamic-DEMATEL,  and rows n  summation of columns   are computed as  n ˜ ˜ ˜ ˜ R = = t ” and “ = t˜i. n×1 ”, respectively. t t “C˜ = . j 1×n i=1 i j 1×n i=1 i j n×1 ˜ In which, “t. j ” indicates the jth column summation “total dynamic direct-influence matrix”, demonstrating factor influences “F j ” summation receiving influence from

1.3 Application of Study

other factors. Likewise, “t˜. j ” is the row summation in the “T˜ =

7



−1 X˜ × I − X˜ ”

and gives influences summation dispatching from factor “Fi ” to the other factors, directly or indirectly. Once, the values of R˜ + C˜ and R˜ − C˜ are derived, the dynamic “influential relation map” can be drawn. Subsequently, the contributing factors are categorized into four (1, 2, 3, and 4) classes at time slice τt . According to obtained results, different strategies as intervention actions can help the system’s safety performance. For example, applying strategy A* reduces the score of factor A to be influenced by other factors and makes it an independent factor. Therefore, it is required to reevaluate the dynamic-DEMATEL at different time slices to see whether the conducted strategy is effective. (iv)

Deriving the importance weights of all contributing factors

The importance weights of the contributing factors are according to the “Prominence” t˜ +t˜ score as formulated by the normalization process as “w˜ i = n i.n .(j t˜i. +t˜. j ) ”, in which i=1 j=1 i, j = 1, 2, . . . , n. This step is optional and can provide further information for system safety decision-makers and planners. As studied in the next section, the proposed dynamicDEMATEL is applied in a safety management system for improvement purposes.

1.3 Application of Study The simple application of a safety management system is taken into account to examine the causality and interdependency among a group of contributing factors that affect the safety performance of a small industrial sector. This application’s selection is to establish the introduced dynamic-DEMATEL in practice. The outcomes of the dynamic-DEMATEL can be used for long-term decision-making strategies. The following safety management system example shows how the dynamic-DEMATEL can be applied: (i)

Determining the decision-making contributing factors

All potential contributing factors relevant to the safety management system must be recognized. To make a clear understanding of the decision-making problem, the following considerations have been made: • There are only five contributing factors in the understudy of the safety management system, as “Training (F1), Reputation of System under study (F2), Cost (F3), Number of accidents (F4), and Flexibility of system (F5)”, • The causality and interdependency among the five mentioned contributing factors in the understudy of safety management are exclusive in this application and might be different in other examples,

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1 Dynamic Decision-Making Trial and Evaluation Laboratory (DEMATEL) …

• The employed group of decision-makers in the present work are not real decision-makers, merely named as such to demonstrate the introduced dynamicDEMATEL applicability, and • We considered that the importance weights of all five mentioned contributing factors and decision-makers in the understudy of safety management are equivalent. In the next step, the time-dependent group direct influence matrix is constructed. (ii)

Constructing the time-dependent group direct influence matrix

It is considered that there are three decision-makers (DM1, DM2, and DM3) to investigate the dynamic causality and interdependency among the negative and positive contributing factors. The assessments of decision-makers at time τ1 are presented in Tables 1.2, 1.3 and 1.4. The time-dependent group direct influence matrix is combined to the aggregated matrix as expressed in Table 1.5. (iii)

Creating the “influential relation map”

Considering the unit value for time τ = 1, the “influential relation map” is drawn by ˜ and “Relation, R˜ − C”, ˜ resulted in Tables 1.6 computing the “Prominence, R˜ + C” and 1.7 and Fig. 1.4. The summation of the columns and rows’ in the “total relation risk matrix” presents the vectors’ scores. Similarly, considering the unit value for time τ = 25, the “influential relation ˜ and “Relation, R˜ − C”, ˜ map” can be drawn by computing the “Prominence, R˜ + C” resulted in Tables 1.8 and 1.9 and Fig. 1.5. Table 1.2 The assessment of DM1 at the time τ1 DM1

F1

F2

F3

F4

F5

F1

7τ1 + 34

τ1 + 9

11τ1 + 44

100τ1 + 5

0

F2

0

0

0

0

16τ1 + 3

F3

0

0

0

0

0

F4

16τ1 + 22

23τ1

5τ1 + 12

0

0

F5

0

15τ1 + 20

50τ1 + 8

2τ1 + 4

0

Table 1.3 The assessment of DM2 at time τ1 DM2

F1

F2

F3

F4

F5

F1

5τ1 + 10

4τ1 + 5

25τ1 + 50

6τ1 + 13

0

F2

0

0

0

0

33τ1 + 5

F3

0

0

0

0

0

F4

2τ1 + 30

10τ1 + 10

3τ1 + 8

0

0

F5

0

8τ1 + 5

12τ1 + 9

7τ1 + 11

0

1.3 Application of Study

9

Table 1.4 The assessment of DM3 at the time τ1 DM3

F1

F2

F3

F4

F5

F1

24τ1 + 30

5τ1 + 19

7τ1 + 8

9τ1 + 4

0

F2

0

0

0

0

20τ1 − 20

F3

0

0

0

0

0

F4

5τ1 − 5

14τ1 − 10

8τ1 − 6

0

0

F5

0

10τ1 + 35

32τ1 + 15

8τ1 + 8

0

Table 1.5 The aggregated matrix three DM1 to DM3 opinions at the time τ1 Aggregated matrix

F1

F2

F3

F4

F5

F1

12τ1 + 24.7

3.4τ1 + 11

38.4τ1 + 44

38.4τ1 + 7.4

0

F2

0

0

0

0

23τ1 − 12

F3

0

0

0

0

0

F4

7.7τ1 + 15.7

15.7τ1

5.4τ1 + 4.7

0

0

F5

0

11τ1 + 16.7

31.4τ1 + 10.7

5.7τ1 + 7.7

0

Table 1.6 The total direct-influence matrix for time τ = 1 F1

F2

F3

F4

F5

F1

0.210

0.082

0.446

0.262

0.000

F2

0.000

0.000

0.000

0.000

0.063

F3

0.000

0.000

0.000

0.000

0.000

F4

0.134

0.090

0.029

0.000

0.000

F5

0.000

0.158

0.241

0.077

0.000

Table 1.7 The total relation matrix for time τ = 1 T˜ R˜ Sum C˜ Sum R˜ + C˜

R˜ − C˜

Causality

F1

1.425

0.516

1.942

0.909

Cause

F2

0.096

0.430

0.526

-0.334

Effect

F3

0.000

0.982

0.982

-0.982

Effect

F4

0.452

0.481

0.933

-0.029

Effect

F5

0.526

0.090

0.615

0.436

Cause

Figures 1.4 and 1.5 show that considering the same linear causality and interdependency among the contributing factors, the four contributing factors were categorized in different classes over time from τ = 1 to τ = 25. Thus, we can see how performing a single strategy can change the category of contributing factors in a period.

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1 Dynamic Decision-Making Trial and Evaluation Laboratory (DEMATEL) …

Fig. 1.4 The influential diagram of examining the causality and interdependency among the group of contributing factors for time τ = 1 Table 1.8 Total direct-influence matrix for time τ = 25 F1

F1

F2

F3

F4

F5

0.136

0.040

0.420

0.404

0.000

F2

0.000

0.000

0.000

0.000

0.235

F3

0.000

0.000

0.000

0.000

0.000

F4

0.013

0.164

0.058

0.000

0.000

F5

0.000

0.122

0.335

0.063

0.000

Table 1.9 The total relation matrix for time τ = 25 T˜ R˜ Sum C˜ Sum R˜ + C˜

R˜ − C˜

Causality

F1

1.321

0.181

1.502

1.141

Cause

F2

0.373

0.468

0.841

-0.095

Effect

F3

0.000

1.037

1.037

-1.037

Effect

F4

0.314

0.562

0.876

-0.248

Effect

F5

0.585

0.345

0.931

0.240

Cause

(iv)

Deriving the importance weights of all contributing factors

The importance weights of the contributing factors for time τ = 25 are based on the “Prominence” score formulated by the normalization process for F1, F2, F3, F4, and F5 are 0.2897, 0.1620, 0.2000, 0.1688, and 0.1688 in exact accordance.

References

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Fig. 1.5 The influential diagram of examining the causality and interdependency among the group of contributing factors for time τ = 25

1.4 Conclusion Many contributing factors play a significant role in system safety and reliability. Generally, the contributing factors have interdependency, and direct and indirect relationships that vary over time. Therefore, decision-makers need to use a reliable decision-making tool to improve system safety performance continually. In the present study, we developed the DEMATEL technique dynamically to be much more capable of dealing with system safety and reliability analysis decision-making problems in the period. The simple structure of dynamic-DEMATEL compared to the existing probabilistic methods such as Markov Chain and Bayesian network causes the proposed dynamic-DEMATEL to be attractive. However, it is an initial step in this regard, and therefore it needs further developments as a direction for future studies, such as considering the non-linear time dependency and a mathematical representation to obtain the constant value.

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Chapter 2

Reliability Analysis of Correlated Failure Modes by Transforming Fault Tree Model to Bayesian Network: A Case Study of the MDS of a CNC Machine Tool

2.1 Introduction CNC machine tools are basic and crucial equipment of the modern manufacturing industry [1]. Control, precision, completeness of functions, as well as reliability and quality are core attributes of CNC machine tools [2]. The investigation of the reliability of a CNC machine tool can track back to the 1970s, when the former Soviet Union machine expert Plonekph [3] published the accuracy and reliability of the numerical control machine [4]. Vershinin [5] considered the early failures of CNC machine tools, investigated the maintenance measures of such devices under specific conditions. Methods for reliability allocation, prediction, and early failure elimination have become hot topics in recent years [6]. For instance, Jones [7] analyzed the operation data of 35 CNC machine tools and concluded that the failure time of the CNC machine tools follows Weibull distributions. Keller [8] used fuzzy theory for failure analysis and reliability evaluation, which can deal with the issues with fuzzy and uncertainty information. Gupta et al. [9] applied time series analysis models to improve the reliability analysis and maintenance decision-making for CNC machine tools. Arts et al. [10] studied the competition risk of the mixed distribution model for reliability and risk of CNC machine tools, where a new parameter estimation method was given. Yoshi [11] collected the testing data of 45 machine centers and evaluated the reliability index of these machine tools by analyzing the failure data. Koshy et al. [12] evaluated the lifetimes of milling cutters to determine the operational performance and quality of the equipment. More recently, Li et al. [2] pointed out that early failures restrict the quality and reliability of CNC machine tools and proposed a Bayesian network model with a failure rates transformation mechanism. The feasibility of such a model is validated using failure data collected from several operating CNC machine tools. Advances in technology and rapidly accumulated operational experiences led to an increase in the quality and reliability of CNC machine tools. For instance, the mean time between failure (MTBF) of Chinese CNC machine tools have been improve from © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 H. Li and M. Yazdi, Advanced Decision-Making Methods and Applications in System Safety and Reliability Problems, Studies in Systems, Decision and Control 211, https://doi.org/10.1007/978-3-031-07430-1_2

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2 Reliability Analysis of Correlated Failure Modes …

200 to 300 h (1991) to 900 h (2006) [2, 13, 14]. Timing curtailed reliability test, TypeII censored test, Group sequential test, and Bayesian theory-based experiment were implemented to handle the problems on data analysis and reliability assessment of CNC machine tools [15]. Statistic-based, model-based, and degradation-based methods are applied in the reliability analysis of CNC machine tools. Specifically, Keller et al. [16] collected failure data from 35 operating CNC machine tools, and bases on which the reliability of such devices was examined from a data analysis point of view. You et al. [17] based on the lifetime of 20 CNC machine tools, identified several failure-prone systems of CNC machine tools: drive and servo units, motherboards, electrical systems, control panels, and position detecting units. Liu et al. [18] constructed a Fault Tree (FT) model to represent failures of the hydraulic system of a CNC machine tool. Mi et al. [19] developed a Fuzzy Fault Tree (FFT) model, which is more applicable to consider multiple uncertainties associated with reliability analysis, to access the reliability of a hydraulic system of a CNC machine tool. Peng et al. [20] extended the degradation analysis of heavy CNC machine tools by considering multiple degradation indicators, accordingly, the remaining useful life of a group of heavy-duty machine tools operating in different environmental and operational conditions were accessed. However, CNC machine tools are complicated pieces of equipment that are composed of multiple components and elements, both independent of, and dependent on, each other. The complicated configuration and function designs, and the integrated power and medium (electricity, lubricating oil) determine that the failures of such devices are dependent on each other. To this end, the reliability analysis of CNC machine tools should be deepened to correlation modeling and analysis. This chapter introduces a method of analysing correlated failure modes of CNC machine tools to further finalize the reliability analysis of such equipment by transforming the Fault Tree model into Bayesian network. This chapter is arranged as follows: Sect. 2.2 discusses the methodologies; Sect. 2.3 lists the case studies; and Sect. 2.4 provides the conclusions of the study.

2.2 Methodologies FTA is widely applied in complex system reliability and safety analysis [21]. It is a common modeling tool for the reliability analysis of CNC machine tools because it is easy to model the relations among the CNC machine tool as a whole, its components, elements, and failure affecting factors, such as human and environmental factors [22]. Li et al. [23] proposed a fuzzy fault tree analysis model of the Auto Drive Axle System. Methods such as triangular fuzzy number and normal fuzzy numbers are used to describe the probability of basic events. A further work was conducted in Ref. [24], where a new system reliability analysis method was created to analyze the reliability and failures of a 900 T pressing system. Bobbio [25] applied Bayesian networks (BNs) in system reliability analysis through transforming fault tree into BN. It concluded that any fault tree can be transformed to the corresponding Bayesian network, and it demonstrated that all

2.2 Methodologies

17

fault tree analysis parameters can be obtained through Bayesian network inference. Aiming at overcoming limitations of fault tree analysis, Yang et al. [26] combined BN and fault tree analysis, reporting a common procedure for transforming fault tree into BN. Pearl [27] introduced Bayesian networks to represent the uncertain knowledges [28]. Owing to their flexible predictive and diagnostic mechanisms, BNs have been widely applied to data mining [29], computer vision [30]. It was also used in the reliability analysis, risk identification and maintainability [31]. A Bayesian network is expressed using a directed acyclic graph and conditional probability Table [32]. The directed acyclic graph includes nodes and directed edges (or arcs). Nodes represent variables [32, 33] which can be either continuous random variable or discrete random variable [34, 35]. A directed edge describes correlations between variables, directly from a parent node to the corresponding child node and provides the logic dependency between two nodes. Conditional Probability Table (CPT) quantify the correlation strength between two nodes through values of conditional probability [36]. The BNs’ capability of integrating multiple information makes they are wildly used in various engineering fields [37, 38]. It allows one to combine discrete data and continuous data to characterize an actual system. BNs can effectively simplify the complexity of the reference compared with other methods such as Markov Chains and Monte Carlo method, owing probi=nto conditional 2| pi(xi )| parameters ability assumption [38]. For example, for n-variable system, i=1 are needed to describe the whole system, but 2n − 1 are needed to describe the whole system under  the circumstance of non-conditional probability assumption. It is easy i=n | pi(xi )| 2 ≤ 2n − 1. Hence, BN can avoid the state space explosion to prove that, i=1 problem that is generally encountered by other methods such as a Markov Chain and Monte Carlo method. This chapter, presents a way of transforming a Bayesian network from a fault tree. And accordingly, the reliability prediction, reliability diagnosis, and sensitivity analysis of critical failures for the MDS can be finalized in the situations of considering both continuous life data and discrete fault data. The way of transforming procedure for discrete failure data includes the following steps, see Table 2.1: Table 2.1 Transformation algorithm for discrete failure data Logic gates

C

A

B

AND

C

A

B

OR

BN structure

Conditional probability tables

Conditional probability formula

A

0

B

0

1

0

1

C

0

0

0

1

P(C P(C P(C P(C

= 1|A = 0, B = 0) = 0 = 1|A = 0, B = 1) = 0 = 1|A = 1, B = 0) = 0 = 1|A = 1, B = 1) = 1

P(C P(C P(C P(C

= 1|A = 0, B = 0) = 0 = 1|A = 0, B = 1) = 1 = 1|A = 1, B = 0) = 1 = 1|A = 1, B = 1) = 1

1

A

0

B

0

1

0

1 1

C

0

1

1

1

18

2 Reliability Analysis of Correlated Failure Modes …

Step 1: Map events in FT into nodes in BN. Create nodes in BN for all events in FT. Labels and represented meanings of nodes are in line with that of events in FT. Step 2: Map causalities in FT into edges of the BN. Link nodes in BN according to the causalities among events of FT. Step 3: Map logic of gates of FT into CPTs in BN. Transform gates in FT into CPTs of child nodes (output events of the gate) in BN. Specifically, for an AND gate, the failure probability of a child node equal to 1 only if all parent nodes fail. While, for an OR gate, the failure probability of a child node equal to 1 if one of the parent nodes fails. The way of transforming procedure for continuous lifetime data includes the following steps, see as in Table 2.2: Steps 1 and 2 are the same as that of the transforming algorithm of discrete failure data. Step 3 Abstract a system as series system under the circumstance that nodes connected by “or gate”, hence, the system’s life time is the minimum of each component’s life time, as:   L S = min L X 1 , L X 2 , · · · , L X n

(2.1)

where, L S is the system’s life time, and L X 1 , L X 2 , . . . , L X n are components life time. Step 4 Abstract a system as parallel system under the circumstance that nodes connected by “AND gate”, hence, the system’s life time is the maximum of each component’s life time, as:   L S = max L X 1 , L X 2 , · · · , L X n

(2.2)

where, L S is the system’s life time, and L X 1 , L X 2 , . . . , L X n are components life time. Table 2.2 Transformation algorithm for continuous life data Logic gates C

A

Conditional probability formula   L S = max L X 1 , L X 2 , . . . , L X n

B

AND C

A

BN structure

B

OR

  L S = min L X 1 , L X 2 , . . . , L X n

Note L S : system life time L X i : component life time

2.3 Case Studies

19

Fig. 2.1 Simplified structural relationship of the main drive system

2.3 Case Studies 2.3.1 The MDS and Failure Criterion The failure criterions for the MDS of the CNC machine tool includes: Components of the MDS cannot function properly to provide and transmit precise and required drive power for the main spindle; Temperature of the MDS exceed 20 °C above normal level; Noise of the MDS exceed 4 dB above normal level; Vibration range of the MDS exceed the threshold value. The main drive system is declared fail when at least one of the above requirements is violated. The simplified structure of the MDS is displayed in Fig. 2.1. Gears’ parameters of the MDS are listed in Table 2.3. Bearings’ parameters are showed in Table 2.4.

2.3.2 Establish the Fault Tree The MDS failure is defined as the top event of the fault tree, which can be caused by motor and power failure (A1 ) or transmission system failure ( A2 ). A2 can be characterized as spindle drive system no power (B1 ), lacking of power (B2 ), no speed change (B3 ), imbalance of the output (B4 ), too much noise or abnormal vibration (B5 ). The lubrication system insufficient is represented by C1 , which can be caused by oil seal failure (X 8 ). According to the logic above, a fault tree for the main drive system is established and presented in Fig. 2.2.

Gear number

25

58

53

25

27

52

Code

01

02

03

04

05

06

5

5

5

5

4

4

Gear modulus

Table 2.3 Gears of the MDS

20

20

20

20

20

20

Pressure angle (°)

III

II

III

II

II

I

Fitting axis

11

10

09

08

07

Code

28

25

39

29

23

Gear number

6

6

5

5

5

Gear modulus

20

20

20

20

20

Pressure angle (°)

VI

V

V

IV

III

Fitting axis

20 2 Reliability Analysis of Correlated Failure Modes …

2.3 Case Studies

21

Table 2.4 Bearings of the MDS Code

Size

Quantity

Fitting axis

Bearing types

D310

50 × 110 × 27

3

III, IV

Deep groove ball bearing

D410

50 × 130 × 31

1

III

D311

55 × 120 × 29

2

I, V

D312

60 × 130 × 31

1

III

D313

65 × 140 × 33

1

II

D215

75 × 130 × 25

1

I

D316

80 × 170 × 39

2

II

D36311

55 × 120 × 29

2

VI

D36314

70 × 150 × 35

1

V

D36316

80 × 170 × 39

1

V

D2007928E

140 × 190 × 32

2

V

Deep groove ball bearing

Angular contact ball bearing Tapered roller bearing

System Failure A1

X1

X3

A2

X2

X4

B1

X5

B2

X4

X6

X7

B5

B4

B3

X4

X5

X4

X5

X6

C1 X8

Fig. 2.2 The fault tree model of the MDS of the CNC machine tool

2.3.3 Failure Data of Components Failure rates of elements are assumed to follow Exponential distributions, see Table 2.5. Furthermore, the mean time to failure (MTTF) of the elements can be attained. Failure probability of basic events are listed in Table 2.6 with the service time of 1000 h. An exponential distribution with the parameter of 5 × 10−6 was assumed for the failure of overload which is caused by the unprofessional operations.

22

2 Reliability Analysis of Correlated Failure Modes …

Table 2.5 Basic event lifetime distribution [39] Basic event

Exponential distribution Parameter λ(h−1 )

Bearing Axis

2×

10−5

0.2 ×

10−6

10−5

Gear

1×

Oil seal

0.2 × 10−6

Basic event

5×

104

5×

106

1×

105

Exponential distribution Parameter λ(h−1 )

MTTF (h) Motor Fork-lever Switch

MTTF (h)

1×

10−5

1 × 105

1×

10−5

1 × 105

0.2 ×

10−6

5 × 106

5 × 106

Table 2.6 Failure probability of basic events Event code

Event name

Failure probability (1000 h)

Event code

Event name

Failure probability (1000 h)

X1

Motor failure

0.0100

X5

Axis failure

0.0004

X2

No electricity

0.0004

X6

Bearing failure

0.0198

X3

Overload

0.0050

X7

Fork-lever failure

0.0100

X4

Gear failure

0.0100

X8

Oil seal failure

0.0004

2.3.4 Reliability Analysis with Discrete Failure Data A Bayesian networks was transformed from the fault tree presented in Fig. 2.3. Results of the Bayesian network with discrete failure data are listed in Table 2.7. The Agenarisk software [35, 36, 38] is used to reasoning the BN model. The probability of the MD’S failure is 0.0548 in 1000 h. The probability of failure of X 1 , X 4 , X 6 , X 7 , are higher than 0.1. And probability of failure of X 6 is larger than

Fig. 2.3 Bayesian network transformed from fault tree

1

0.0100

0.0004

0.0050

0.0100

X1

X2

X3

X4

1

1

1

P(T = 1|X i = 1)

Failure rate for events

Event Code

Table 2.7 Bayesian network reference results

0.1825

0.0912

0.0073

0.1825

P(X i = 1|T = 1)

X8

X7

X6

X5

Event Code

0.0004

0.0100

0.0198

0.0004

Failure rate for events

1

1

1

1

P(T = 1|X i = 1)

0.0073

0.1825

0.3613

0.0073

P(X i = 1|T = 1)

2.3 Case Studies 23

24

2 Reliability Analysis of Correlated Failure Modes …

0.3. Hence, the weak links of the MDS are Motor failure (X 1 ), Gear failure (X 4 ), Bearing failure (X 6 ), Fork-lever failure (X 7 ), following the failure criterion that the failure rate should be no more than 0.1. The weak link is Bearing failure (X 6 ) if the criterion that the failure rate should be less than 0.3. It indicates that it is hard failures other than soft failures that effect the reliability of the MDS of the CNC machine tool. In addition, the bearing failure is the most critical failure. Elements of the MDS connects directly through the mechanical structure (e.g., shaft and bearing) and are sharing the same working conditions (e.g., lubricating oil and temperature), which introduce correlated failures (modes) to the MDS. For example, in CNC machine tools the failure mode insufficient power is often accompanied with output noise, and imbalance of the output is often together with abnormal vibration. Hence, it is of significance for repair and maintenance of the MDS by diagnosing correlation failures. The correlations among failure modes of the MDS is quantified by the correlation matrix: ⎛ ⎞ 1 0.6586 0.0100 0.6775 0.6840 B1 ⎟ B2 ⎜ ⎜ 0.3413 1 0.0100 0.3380 1 ⎟ ⎜ ⎟ (2.3) i = B3 ⎜ 0.0153 0.0296 1 0.0104 0.0304 ⎟ ⎜ ⎟ B4 ⎝ 1 0.9627 0.0100 1 1 ⎠ B5 0.3455 0.9745 0.0100 0.3422 1 B1

B2

B3

B4

B5 = j

where ai j represent P(X j = 1|X i = 1).

2.3.5 Reliability Analysis with Continuous Life Data Fault tree can model a system, a component and even a single failure mode. According to quantitative matrix above, the failure mode B5 is the most complicated one. Accordingly, B5 was regarded as the key failure mode of the MDS. The fault tree of failure mode B5 is established and transformed into Bayesian networks in Fig. 2.4. The failure properties are as listed in Table 2.5. With the model, the MTTF of failure mode B5 is 14,250 h with the medium value of 9867 h. The possibility that B5 would not happen at the service time of 1000 h is 0.93195. The possibility that B5 would not happen of the MDS at different system service time is listed in Table 2.8. The elements’ reliability at different service time is shown in Table 2.9. Overall, X 4 , X 6 are weak links of the MDS which restrict the reliability improvement and should be handled according to the results above. Hence, sensitivity analysis for X 4 is conducted to investigate that how X 4 influence the reliability of the MDS. The sensitivity analysis result is shown in Fig. 2.5. It can be concluded that X 4 is the main cause of the MDS failures during the early period within 50,000 h, and which is the primary factor of early failures of the MDS.

2.3 Case Studies

25

Fig. 2.4 Bayesian network of the failure mode B5 with continuous lifetime data Table 2.8 The possibility that B5 would not happen in different service time (P B5 ) Service time (h)

P B5

Service time (h)

P B5

Service time (h)

P B5

200

0.98628

800

0.94604

5000

0.70478

300

0.97896

900

0.93934

6000

0.65556

400

0.97226

1000

0.93195

7000

0.61104

500

0.96555

2000

0.87164

8000

0.57027

600

0.95945

3000

0.91193

700

0.95275

4000

0.75759

Table 2.9 Failure probability diagnosis of the failure mode B5 Event code

X4

X5

X6

X8

Reliability (500 h)

0.64552

0.99869

0.85871

0.99869

Reliability (1000 h)

0.64601

0.99856

0.85731

0.99856

Reliability (2000 h)

0.64347

0.99953

0.85775

0.99952

26

2 Reliability Analysis of Correlated Failure Modes …

Fig. 2.5 Sensitivity analysis of X 4 /the curve represents P(B5 |X 4 ); the abscissa is the life time of X 4 ; the ordinate is the life time of B5

2.4 Conclusions A general method for transforming fault tree into Bayesian networks is used to finalize the reliability analysis of the main drive system of a CNC machine tool with both discrete fault data and the continuous lifetime data. The purpose is to provide a framework of reliability analysis of complex systems with correlated failures (modes). The reliability and failure probability of the MDS, its elements, and its failure modes (only failure probability) are obtained. Bearings are ascertained to be the most crucial element of the MDS and are the weak links of the MDS. The correlations among failure modes are identified and then quantified based on the discrete failure data. too much noise or abnormal vibration is the commonly happened failure modes. Critical failure causes that are Gear and bearing failures that is confirmed by the BN model with continuous lifetime data and the sensitivity analysis is completed.

References 1. Li, Y., Zhang, X., Ran, Y., Zhang, G.: Reliability modeling and analysis for CNC machine tool based on meta-action. Qual. Reliab. Eng. Int. 37(4), 1451–1467 (2021) 2. Li, H., Deng, Z.M., Golilarz, N.A., Soares, C.G.: Reliability analysis of the main drive system of a CNC machine tool including early failures. Reliab. Eng. Syst. Saf. 215, 107846 (2021) 3. Hao, Q.B.: Fuzzy Comprehensive Allocation and Prediction for NC Machine Tools Reliability and Maintainability. Ph.D. Dissertation, Jilin University, Jilin (2012). (In Chinese) 4. Wu, J., Shao, X.Y., Deng, C., Mao, K.M.: Reliability prediction system development based on performance degradation. Comput. Integr. Manuf. Syst. 17(1), 191–197 (2011) 5. Vershinin, B.V., Sharin, Y.S.: Ways of increasing the reliability of NC equipment. Sov. Eng. Res. 7(5), 71–71 (1987) 6. He, X.: Recent development in reliability analysis of NC machine tools. Int. J. Adv. Manuf. Technol. 85(1), 115–131 (2016) 7. Jones, J.A., Hayes, J.A.: Use of a field failure database for improvement of product reliability. Reliab. Eng. Syst. Saf. 55(2), 131–134 (1997) 8. Keller, A.Z., Kara-Zaitri, C.: Further applications of fuzzy logic to reliability assessment and safety analysis. Microelectron. Reliab. 29(3), 399–404 (1989)

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Chapter 3

What Are the Critical Well-Drilling Blowouts Barriers? A Progressive DEMATEL-Game Theory

3.1 Introduction In order to reduce the number of accidents in well drilling blowouts, it is vital to maintain the well-drilling integrity by performing reliable operational measures (e.g., organizational, technical, and managerial well-drilling blowouts barriers (WDBBs)). The adequately implemented WEDDs could mitigate the risk of well-drilling loss, such as kill operation, kick detection and prevention, and blowouts prevention [1–3]. Evaluating the performance of essential factors in WDBBs identifies any potential abnormal and non-conformities in the well-drilling operations leading to blowout accidents [4]. Hence, there is a requirement to address the direct and indirect interdependency of essential factors and improve the system safety performance of well-drilling blowouts preventions by recommending different intervention actions [5–9]. Using the essential factors based on WDBBs provides critical information for decision-makers in risk reduction into an acceptable level or ALARP [10–12]. The decision-making tools, particularly multi-criteria decision-making (MCDM) techniques and their extensions, have been used for risk mitigations in different application domains, such as TOPSIS (“Technique for Order of Preference by Similarity to Ideal Solution”) [13–17], AHP (“analytical hierarchy process”) [18–22], BWM (“best–worst method”) [23–25], and more. Among all MCDM tools, DEMATEL (“Decision-making trial and evaluation laboratory”) [26, 27] enables decisionmakers to study “causes and effects” of relationships among the essential factors in the system. DEMATEL supports the interdependency of factors by a map-illustrations, reflecting their relationships in the complicated system decision-making problem. However, there would always be conflicts of interest between the decision-makers (e.g., players, parties, participants, etc.). Thus, the existing decision-making method can be integrated within the game-theoretic context to deal with this challenge. The game-theoretic context has been widely used in different domains, such as: engineering electrical electronics [28–31], operations research management science [29, © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 H. Li and M. Yazdi, Advanced Decision-Making Methods and Applications in System Safety and Reliability Problems, Studies in Systems, Decision and Control 211, https://doi.org/10.1007/978-3-031-07430-1_3

29

30

3 What Are the Critical Well-Drilling Blowouts Barriers? A Progressive …

Fig. 3.1 The capabilities and shortages of both DEMATEL and game theory

32, 33], safety science and safety systems [34–39], computer science artificial intelligence [40, 41], green sustainable science technology [42, 43], transportation science technology [44, 45], and business finance [46–48]. Figure 3.1 presents the capabilities and shortages of DEMATEL and game theory (justified after Golestani [49]). The main contribution of the present chapter is utilizing the integration of DEMATEL and Game theory to determine the comprehensive criticality weight of essential factors in well-drilling blowouts barriers. The organization of the present chapter is constructed as the following. In Sect. 3.2, the Preliminary of DEMATEL is explained. In Sect. 3.3, the Preliminary game theory is explained. In Sect. 3.4, the progressive DEMATEL-Game theory approach is introduced. An application of study (well-drilling blowouts barriers) is illustrated in Sect. 3.5. Finally, the conclusion is provided in Sect. 3.6.

3.2 Preliminary: DEMATEL “Decision-Making Trial and Evaluation Laboratory” The DEMATEL technique is proposed by the “Battelle Memorial Institute of Geneva Research Center” [50]. The DEMATEL would potentially help decision-makers in system safety problems to have insights and measure the interdependency and causality of the system safety essential factors. In addition, DEMATEL plays as an

3.3 Preliminary: Game Theory

31

influence diagram illustrating direct and indirect relationships among the group of factors [51–56]. The original form of DEMATEL can be applied to a decision-making problem as the following: 1.

Determining the direct-relation matrix using measurement scales

The measurement scale is determined in this step to evaluate the causality among the essential factors in the decision-making problem within five suggested scores as {0, 1, 2, 3, 4, and 5}. The scores mean that there amount of impact from factor A to factor B, as “no impact, little impact, low impact, high impact, and very high impact”, respectively. Considering the scores, the decision-makers can determine the direct matrix based on their individual opinions, and it is demonstrated   relation as D = dij n×m , where i, j = 1, 2, 3, . . . , n. Here, the di j is indicated on the direct relation of factor A (Fi ) on factor B (F j ). In addition, the di j = 0 if and only of i = j. 2.

Normalizing the determined direct-relation matrix in step 1

For the set of n essential factors {F 1 , F 2 , …, F n } in the decision-making problem, the di j is mapped into [0,1], the normalized direct-relation matrix (N ) is computed as the following equation: N = D/ max 3.

n j=1

di j

n i=1

 di j

(3.1)

Computing the total-relation matrix

The total-relation matrix (T ), including direct/indirect interdependency among the essential factors, is derived as the following equation:  T = N (I − N )

(3.2)

where I is the “unit matrix”. 4.

Computing the prominence score of all essential factors

According to the summation of each row and column in the T matrix, denoted as Ri (i = 1, 2, . . . , n) and C j ( j = 1, 2, . . . , m), respectively, the prominence (P) is computed as P = Ri + Cj . In a simple word, the score of P shows the influence of ith essential factor and its degree of being influenced by other essential factors, including itself.

3.3 Preliminary: Game Theory The concept of game theory is constructed based on “Nash equilibrium” [57, 58]. The point here is that any possible game outcomes are feasible and optimum and

32

3 What Are the Critical Well-Drilling Blowouts Barriers? A Progressive …

would equal 1 when none of the players have reasons and motivations to differ from his/her selected strategy following another player strategy selection [59–62]. In fact, a player cannot receive any incremental advantage from diverting his/her strategy selection, considering that the other players would remain constant on their strategy selections. Assuming that there are n players in the particular game (, ∇), in which i is the set of strategies for player “i”,  = 1 × 2 × . . . × n is the strategy profile set and ∇(x) = (∇1 (x), ∇2 (x), . . . ∇n (x)) is the payoff vector. Besides, let us assume that θi is the profile strategy of player “i” and θ−i indicates that the rest of players’ profile strategies (excluding player “i”). Then, every single player selects a strategy, θi = (θ1 , θ2 , . . . θn ). Subsequently, the player “i” determines the payoff vector as ∇i (θi ). According to this point, a strategy profile θ ∗ ∈  falls into a “Nash equilibrium” [57, 58]; therefore, if there is no one-sided deviation in the strategy selection by any players, the following equation should be satisfied:   ∗ ∗ ≥ i θi , θ−i ∀i = {1, 2, 3, . . . , n} and θi ∈ ∇i i θi∗ , θ−i

(3.3)

3.4 The Progressive DEMATEL-Game Theory Approach This chapter assumed that there would be subjective and objective weights from decision-makers opinions [63–66], and variability distributions [5, 67–69], respectively. In fact, using a single-weight method (i.e., objective-based or subjectivebased) suffers from a couple of limitations. For example, the objective-based weight disregards the decision-makers’ knowledge and actual uncertain circumstances. In contrast, subjective weight depends on human cognitive judgments and is influenced by individual experiences. Thus, the progressive DEMATEL-Game theory approach explained in detail in the literature [70] integrates the subjective and objective weights. As mentioned earlier, the DEMATEL method can study interactions and feedback among a group of essential factors. In addition, the outcomes of game theory are feasible and optimal. Therefore, the results are much more reasonable and reliable. It should be added that there are many methods to obtain the subjective/objective weights in the state of arts, such as entropy [71–73], AHP (analytical hierarchy process) [74–76], BWM (best–worst method) [77, 78], simple averaging method, etc. Before introducing the progressive DEMATEL-Game theory approach, let us rereview the game theory as a strategic interaction that obtains the optimal equilibrium among a set of opposing conflicts of interest. In the game theory, the optimal outcome can maximize the “payoff utility” from the expectations of players [79, 80]. Thus, in the progressive DEMATEL-Game theory, both objective and subjective weight methods are considered the players in the game. Subsequently, according to the “Nash equilibrium” (Eq. 3.3), the integrated, feasible, and optimal weights of essential factors are derived.

3.4 The Progressive DEMATEL-Game Theory Approach

33

Moreover, the essential factors in the system safety decision-making problem have direct and indirect interdependency. Using the DEMATEL method, the prominence score of essential factors would be obtained, and a higher value shows the importance of the factor in the system safety. Hence, the progressive DEMATEL-Game theory better reflects the importance weights of factors. As shown in Fig. 3.2, the progressive DEMATEL-Game theory approach is constructed in six key steps, and the detail of the framework is provided as the following. Step 1: Determining the direct-relation matrix using measurement scales This step is performed based on what is provided in the first step original form DEMATEL technique   in the preliminary section. Thus, the direct-relation matrix is denoted by D = dij n×m , where i, j = 1, 2, 3, . . . , n. Here, the di j is indicated on the direct relation of factor A (Fi ) on factor B (F j ). In addition, the di j = 0 if and only of i = j. Fig. 3.2 The progressive DEMATEL-Game theory framework

34

3 What Are the Critical Well-Drilling Blowouts Barriers? A Progressive …

Step 2: Normalizing the determined direct-relation matrix and computing the total-relation matrix. The normalized direct-relation matrix (N ) is derived for the set of n essential factors {F 1 , F 2 , …, F n } using Eq. 3.1. Besides, the total-relation matrix (T ), including direct and indirect interdependency among the essential factors, is obtained using Eq. 3.2. Step 3: Computing the prominence score of all essential factors In this step, the prominence (P = Ri + Cj ) score is computed as a degree of being influenced by other essential factors, including itself. The Ri “(i = 1, 2, . . . , n) and C j ( j = 1, 2, . . . , m)” indicate the summation of each row and column, respectively. Step 4: Justifying the objective and subjective-based weighting methods As mentioned earlier, we considered two weighting methods (σ ) in this chapter. The vector can be established as a set of σ = (σ1 , σ2 , . . . , σm ). For both weighting methods, a pre-treatment is conducted utilizing the “operator .×”. This pre-treatment shows that the two vectors are combined to derive a new single-vector with the same-dimension. νi =

σi . × P sum(σi . × P)

(3.4)

For example, we assumed that the normalization and justification in the schematic σ ve +σi,subjective ).×P . format, which is different from the study of [70] as νi = m( i,objecti i=1 (σi,objecti ve +σi,subjecti ve ).×P Therefore, a probable weight vector set is constructed by σ vectors, as the following “arbitrary linear” equation: W =

m 

λ j ν Tj



λj > 0



(3.5)

j=1

In which the W is the potential weight vector and λ j is the “weight coefficient” and requires to be determined. Step 5: Computing the weight coefficient in the game-theoretic context Decision-makers can obtain the optimal equilibrium weight vector W ∗ based on the game-theoretic context once a consensus value is determined within m weights. This consensus value is considered a linear combination of optimal “weight coefficient” λ j . Thus, the mathematical programming model aims to minimize the variation between W and σi by the following engaging equation: min

m  k=1

λk × vkT − σiT

2

i = 1, 2, 3, . . . , m

(3.6)

3.5 Case Study

35

According to the features of matrix differentiation, the optimum first-order derivative is derived as the following equation: m 

λk × νkT × σi = σi × σiT i = 1, 2, 3, . . . , m

(3.7)

k=1

where, the optimum first-order derivative includes m linear equations related to m variables [λ1 , λ2 , . . . , λn ], and can be solved with a simple optimization model and programming software, such as Lindo (www.lindo.com). Once the “weight coefficient” λ j = [λ1 , λ2 , . . . , λn ] is obtained, the normalized “weight coefficient” can be determined as the following equation: λj λ∗j = m j=1

λj

(3.8)

Step 6: Computing the optimal equilibrium weight Finally, the optimal equilibrium weight can be computed as the following equation: W∗ =

m 

 λ∗j .σiT λ j > 0

(3.9)

j=1

In the following section, an application is illustrated to respond to what are the critical well drilling blowouts barriers.

3.5 Case Study In the existing state of arts, many researchers have investigated essential factors in process-based operational units, such as [81–85]. In the present chapter, the welldrilling blowouts barriers are taken into account for further investigation to improve the system safety performance of the process-based sector. To seek the simplicity of computations, only four and ten factors and sub-factors are identified, respectively. The considered essential factors in the present work are presented in Fig. 3.3. The descriptions of each factor and sub-factor are provided in the study of Mirderikvand et al. [3]. The introduced progressive DEMATEL-Game theory in the methodology section is conducted to obtain the comprehensive essential factors’ weights of WDBBs, as the following steps. Step 1: Determining the direct-relation matrix using measurement scales The causality and interdependency among the essential factors (particularly subfactors), either negatively or positively, are examined, as presented in Table 3.1. It is

36

3 What Are the Critical Well-Drilling Blowouts Barriers? A Progressive …

Fig. 3.3 The schematics illustration of WDBBs (essential factors, simplified after Khan et al. [86]) Table 3.1 The direct-relation matrix using measurement scales Tag

F1

F2

F3

F4

SF1

SF2

SF3

SF4

SF5

SF6

SF7

SF8

SF9

SF10

F1

1

2

4

2

2

5

5

1

3

4

3

2

0

1

F2

1

5

4

2

4

2

2

3

0

2

2

1

4

5

F3

5

2

3

0

4

2

3

1

3

1

3

2

5

5

F4

2

4

5

0

3

5

0

3

2

1

4

1

1

3

SF1

0

4

4

5

5

3

3

1

1

0

2

3

4

1

SF2

5

1

5

4

1

3

1

1

5

1

1

3

2

3

SF3

0

3

1

2

4

1

1

2

4

0

5

3

4

0

SF4

1

3

0

5

5

5

4

4

3

2

2

2

4

1

SF5

5

0

5

5

1

1

1

3

5

5

1

2

4

5

SF6

0

1

4

2

4

4

0

2

3

5

5

1

1

1

SF7

3

1

1

5

5

2

1

4

5

5

0

2

4

3

SF8

2

4

5

0

1

5

3

4

1

2

5

3

1

2

SF9

2

4

0

4

2

4

3

2

0

0

5

0

4

3

SF10

0

4

0

5

2

5

4

3

5

4

3

4

2

1

3.6 Conclusion

37

assumed that a single decision-maker shares his opinions for evaluating the essential factors. Step 2: Normalizing the determined direct-relation matrix and computing the total-relation matrix The normalized direct-relation matrix (N ) is derived for the essential factors using Eq. 3.1, provided in Table 3.2. Besides, the total-relation matrix (T ), including direct and indirect interdependency among the essential factors, is obtained using Eq. 3.2 and presented in Table 3.3. Step 3: Computing the prominence score of all essential factors According to the summation of each row and column in the T matrix, denoted as Ri “(i = 1, 2, . . . , n) and C j ( j = 1, 2, . . . , m)”, respectively, the prominence (P) is computed as P = Ri + Cj , presented in Table 3.4. Step 4: Justifying the objective and subjective-based weighting methods The prominence results of essential factors indicated that F3, SF5, and SF2 have a high prominence in WDBBs. Thus, the essential factors in the following steps play a significant role. In this chapter, we assumed that there are two types of arbitrary subjective and objective weights. Using Eq. 3.4, the objective and subjective-based weighting methods are justified (normalized) with prominence, presented in Table 3.5. Step 5: Computing the weight coefficient in the game-theoretic context In this step, using Eqs. 3.7 and 3.8, the “weight coefficient” of subjective and objective weights are determined as λ∗ = [0.5707, 0.4293], respectively. Step 6: Computing the optimal equilibrium weight

Finally, the optimal equilibrium weight can be computed by W ∗ = mj=1 λ∗j .σiT . The results are presented in Table 3.6. The differences in the subjective, objective, and equilibrium weight are depicted in Fig. 3.4.

3.6 Conclusion In the present study, a progressive DEMATEL-Game theory is utilized to investigate the well-drilling blowouts barriers as it can be a most severe disaster in the well-drilling process, which may lead to devastating damage. Thus, the WDBBs are adequately studied, and adequate intervention actions are proposed to improve system safety over time. For this purpose, the four categories of essential factors as “policy-making, operational, personal, and mechanical factors” and their sub-factors are described and taken into account for further investigation. The weights of WDBBs

0.0698

0.0698

0.0233

0.0930

0.0233

0.1163

0.0465

0.0000

0.1163

0.0000

0.0233

0.1163

0.0000

0.0698

0.0465

0.0465

0.0000

F2

F3

F4

SF1

SF2

SF3

SF4

SF5

SF6

SF7

SF8

SF9

SF10

F2

0.0930

0.0930

0.0233

0.0000

0.0233

0.0930

0.0930

0.0465

0.1163

0.0465

F1

0.0233

Tag

F1

0.0000

0.0000

0.1163

0.0233

0.0930

0.1163

0.0000

0.0233

0.1163

0.0930

0.1163

0.0698

0.0930

0.0930

F3

0.1163

0.0930

0.0000

0.1163

0.0465

0.1163

0.1163

0.0465

0.0930

0.1163

0.0000

0.0000

0.0465

0.0465

F4

0.0465

0.0465

0.0233

0.1163

0.0930

0.0233

0.1163

0.0930

0.0233

0.1163

0.0698

0.0930

0.0930

0.0465

SF1

Table 3.2 The normalized direct-relation matrix (N) SF2

0.1163

0.0930

0.1163

0.0465

0.0930

0.0233

0.1163

0.0233

0.0698

0.0698

0.1163

0.0465

0.0465

0.1163

SF3

0.0930

0.0698

0.0698

0.0233

0.0000

0.0233

0.0930

0.0233

0.0233

0.0698

0.0000

0.0698

0.0465

0.1163

SF4

0.0698

0.0465

0.0930

0.0930

0.0465

0.0698

0.0930

0.0465

0.0233

0.0233

0.0698

0.0233

0.0698

0.0233

SF5

0.1163

0.0000

0.0233

0.1163

0.0698

0.1163

0.0698

0.0930

0.1163

0.0233

0.0465

0.0698

0.0000

0.0698

SF6

0.0930

0.0000

0.0465

0.1163

0.1163

0.1163

0.0465

0.0000

0.0233

0.0000

0.0233

0.0233

0.0465

0.0930

SF7

0.0698

0.1163

0.1163

0.0000

0.1163

0.0233

0.0465

0.1163

0.0233

0.0465

0.0930

0.0698

0.0465

0.0698

SF8

0.0930

0.0000

0.0698

0.0465

0.0233

0.0465

0.0465

0.0698

0.0698

0.0698

0.0233

0.0465

0.0233

0.0465

SF9

0.0465

0.0930

0.0233

0.0930

0.0233

0.0930

0.0930

0.0930

0.0465

0.0930

0.0233

0.1163

0.0930

0.0000

SF10

0.0233

0.0698

0.0465

0.0698

0.0233

0.1163

0.0233

0.0000

0.0698

0.0233

0.0698

0.1163

0.1163

0.0233

38 3 What Are the Critical Well-Drilling Blowouts Barriers? A Progressive …

0.3963

0.5025

0.4558

0.4971

0.3143

0.4295

0.3394

0.2892

0.4263

0.2507

0.3490

0.4713

0.2813

0.3972

0.3654

0.3075

0.3444

F2

F3

F4

SF1

SF2

SF3

SF4

SF5

SF6

SF7

SF8

SF9

SF10

F2

0.5342

0.4459

0.3671

0.4518

0.4036

0.4819

0.4608

0.4619

0.5212

0.4047

F1

0.3129

Tag

F1

0.4985

0.3711

0.5606

0.5054

0.4902

0.6219

0.4724

0.3763

0.5523

0.5075

0.5237

0.5157

0.5118

0.5054

F3

Table 3.3 The total-relation matrix (T )

F4

0.6099

0.4850

0.4539

0.6024

0.4471

0.6230

0.5974

0.4140

0.5156

0.5272

0.4188

0.4679

0.4859

0.4515

SF1

0.5489

0.4429

0.4913

0.6071

0.5030

0.5393

0.6042

0.4636

0.4519

0.5471

0.4935

0.5598

0.5461

0.4661

SF2

0.6563

0.5158

0.6101

0.5836

0.5237

0.5930

0.6386

0.4135

0.5452

0.5253

0.5658

0.5510

0.5262

0.5565

SF3

0.4346

0.3435

0.3980

0.3679

0.2760

0.3928

0.4289

0.2800

0.3374

0.3690

0.3017

0.4100

0.3651

0.4011

SF4

0.4689

0.3547

0.4553

0.4753

0.3588

0.4748

0.4716

0.3382

0.3626

0.3535

0.3953

0.3855

0.4157

0.3455

SF5

0.5902

0.3658

0.4623

0.5734

0.4543

0.6116

0.5183

0.4282

0.5355

0.4078

0.4452

0.5097

0.4065

0.4710

SF6

0.4582

0.2763

0.3813

0.4722

0.4145

0.5046

0.3829

0.2613

0.3468

0.2863

0.3282

0.3620

0.3569

0.3958

SF7

0.5329

0.4720

0.5358

0.4568

0.4795

0.5103

0.4922

0.4477

0.4282

0.4408

0.4679

0.5028

0.4567

0.4533

SF8

0.4295

0.2611

0.3833

0.3731

0.2930

0.3935

0.3702

0.3119

0.3664

0.3540

0.3087

0.3639

0.3209

0.3298

SF9

0.5291

0.4719

0.4736

0.5600

0.4137

0.5857

0.5619

0.4548

0.4627

0.5109

0.4338

0.5693

0.5281

0.4033

SF10

0.4471

0.3965

0.4352

0.4757

0.3641

0.5509

0.4258

0.3127

0.4436

0.3853

0.4281

0.5041

0.4872

0.3733

3.6 Conclusion 39

40

3 What Are the Critical Well-Drilling Blowouts Barriers? A Progressive …

Table 3.4 The prominence (P = Ri + Cj ) of essential factors

Table 3.5 The justified the objective and subjective-based weighting methods

Prominence

Tag F1

10.749

F2

12.627

F3

13.606

F4

13.011

SF1

13.251

SF2

13.983

SF3

10.255

SF4

12.472

SF5

14.104

SF6

10.894

SF7

13.583

SF8

11.362

SF9

12.469

SF10

13.112

Tag

Objective

Subjective

Justified

F1

0.72

0.51

0.08242

F2

0.19

0.11

0.00576

F3

0.7

0.14

0.04866

F4

0.2

0.92

0.08272

SF1

0.65

0.5

0.08882

SF2

0.43

0.12

0.02144

SF3

0.89

0.85

0.15737

SF4

0.97

0.12

0.07510

SF5

0.94

0.77

0.20904

SF6

0.52

0.79

0.09475

SF7

0.82

0.98

0.22306

SF8

0.1

0.14

0.00332

SF9

0.88

0.55

0.12924

SF10

0.99

0.27

0.10551

are comprehensively computed with the introduced progressive DEMATEL-Game theory approach dealing with both objective and subjective weights.

3.6 Conclusion

41

Table 3.6 The equilibrium weight (W *) Tag

Objective

Subjective

0.4293

0.5707

Equilibrium weight (W *)

F1

0.72

0.51

0.6002

F2

0.19

0.11

0.1443

F3

0.7

0.14

0.3804

F4

0.2

0.92

0.6109

SF1

0.65

0.5

0.5644

SF2

0.43

0.12

0.2531

SF3

0.89

0.85

0.8672

SF4

0.97

0.12

0.4849

SF5

0.94

0.77

0.8430

SF6

0.52

0.79

0.6741

SF7

0.82

0.98

0.9113

SF8

0.1

0.14

0.1228

SF9

0.88

0.55

0.6917

SF10

0.99

0.27

0.5791

Fig. 3.4 The difference illustration of subjective, objective, and equilibrium weight

42

3 What Are the Critical Well-Drilling Blowouts Barriers? A Progressive …

Thus, the advantages of the present study can be highlighted as the following: • The model is applied in the case study with four factors and ten sub-factors, considered a medium case. The model, therefore, has the capabilities to solve much-complicated decision-making problems with more factors and sub-factors, • The model could integrate objective/subjective importance weights, reflecting more meaningful and reasonable results. The decision-makers can freely choose how they are willing to obtain the objective and subjective importance weights of the factors, • According to Liu et al. [70], it is stated that the DEMATEL is used to optimize game theory, “the criterion with high prominence will play an important role in Nash equilibrium, in which way we can obtain more reasonable comprehensive weights”, and • Finally, by increasing the input information to the model, the uncertainty of system safety performance would be reduced, and this helps decision-makers to make a much more reliable decision.

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Chapter 4

Developing Failure Modes and Effect Analysis on Offshore Wind Turbines Using Two-Stage Optimization Probabilistic Linguistic Preference Relations

4.1 Introduction Offshore wind turbines (OWTs) have become acceptable worldwide to produce sustainable energy over the last decade [1, 2]. By increasing the demand for implementing OWTs in the offshore market, the decision-makers in the energy sectors have concerns about the cost, vulnerabilities, and reliability of OWTs since they need periodic maintenance after challenging procedures and incur substantial costs. The OWTs are typically located 10–40 km far away from shore [3]. Such distance causes the operation and maintenance cost and time to be increased [3–7]. As an example, the maintenance cost of OWTs is almost five times greater than onshore ones [6]. Therefore, it is vital for decision-makers to guarantee effective and efficient OWTs maintenance and shorten the maintenance costs by utilizing system safety and reliability analysis methods [5, 8–12]. In the system safety and reliability analysis techniques, the critical components with a high level of risks are identified, and by utilizing intervention actions, including preventive, mitigative, control, and corrective actions, the risk level is declined to the acceptable level or ALARP (“As low as reasonably practicable”) [13–19]. Among all system safety and reliability analysis methods, the failure mode and effect analysis (FMEA) as a proactive risk management technique has been extensively utilized to identify the failure mods of the system before failure occurrence [20–22, 22–27]. The proper FMEA implementation could shorten the product time, decrease maintenance cost, as well as increase the viability of the system by satisfying customers. Considering the advantages of FEMA as mentioned above, this becomes a core risk management method in system safety reliability performance improvement in the different application domains, including [28], chemical [29], healthcare and medical system [30], manufacturing [31], defense service [32], electronic [33–35], nuclear [36, 37], automotive [38, 39], agriculture [40], and environment [41, 42]. In conventional FMEA techniques, three risk factors for each failure mode (known as Severity (S), Occurrence (O), and Detection (D)) are scored on a scale of 1 to 10. The © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 H. Li and M. Yazdi, Advanced Decision-Making Methods and Applications in System Safety and Reliability Problems, Studies in Systems, Decision and Control 211, https://doi.org/10.1007/978-3-031-07430-1_4

47

48

4 Developing Failure Modes and Effect Analysis on Offshore Wind …

score of 1 and 10 correspond to the lowest and highest importance in terms of risk factors, respectively. By multiplying the risk factor score, the risk priority number (RPN) for each failure mode is determined, and the failure modes are then prioritized in descending order to receive the intervention actions [43–49]. However, the conventional FMEA method suffers from several limitations. These shortcomings, as well as proposed improvements to this method, are outlined by scholars as to the following: (i) The importance of Severity, Occurrence, and Detection are assumed to be equal [20, 50]; (ii) risk priority numbers only take into account Severity, Occurrence, and Detection [51]; (iii) different combinations of Severity, Occurrence, and Detection produce the same risk priority number [22, 26, 52, 53]; (iv) there is not enough information to judge risk priority number computation [23, 51, 54, 55]; (v) in some systems, failure modes can receive some intervention actions on which they are entirely dependent; and (iv) conventional FMEA (like many risk management methods) generally assesses the risk of a system statically and does not consider the effect of time in the system profile [56–59]. A complete overview of the comprehensive limitations of conventional FMEA is reported in [20]. Figure 4.1 illustrates the number of publications on FMEA utilization based on all application domains and any turbines since 2000. This trend showed the highest increase in 2019; it can be intensely predictable that FMEA utilizations in different case studies, especially turbines, will continue to evolve in the coming years. In Table 4.1, the first ten highly cited papers are provided by analyzing the “Average Citations per Year” of each publication in the field of FMEA and “Wind turbines”. The “Average Citations per Year” indicates the number of “WoS citation index” for a paper by the end of December 2021. Considering risk factors in FMEA methods, using multi-criteria decision-making (MCDM) tools can properly handle importance weights of risk factors as well

Fig. 4.1 Distribution of “published works per year till the end of December 2021” in FMEA utilizations based on all application domains and any turbines since 2000 (according to the web of science (WoS) database)

4.1 Introduction

49

Table 4.1 The “highly cited papers based on citation measures” in the field of failure analysis and “Wind turbines” (according to the WoS database) Row Descriptions

Reference

Total Average citations per Year

1

Failure Modes and Effects Analysis (FMEA) for wind turbines

Arabian-Hoseynabadi 221 et al. [60]

18.42

2

Reliability-Centered Maintenance for Wind Turbines Based on Statistical Analysis and Practical Experience

Fischer, Katharina et al. [61]

131

13.1

3

An FMEA-Based Risk Assessment Approach Shafiee and for Wind Turbine Systems: A Comparative Dinmohammadi [9] Study of Onshore and Offshore

76

9.5

4

Risk assessment of floating offshore wind turbine based on correlation-FMEA

Kang et al. [62]

51

10.2

5

Using a Hybrid Cost-FMEA Analysis for Wind Turbine Reliability Analysis

Tazi et al. [63]

36

7.2

6

Techno-economic, environmental, and safety Mukherjee et al. [64] assessment of hydrogen-powered community microgrids; a case study in Canada

31

6.2

7

Reliability prediction of an offshore wind turbine gearbox

27

9

8

An integrated model using SWOT analysis Adem et al. [66] and Hesitant fuzzy linguistic term set for evaluation occupational safety risks in the life cycle of wind turbine

20

5

9

Failure Modes, Effects and Criticality Ozturk et al. [67] Analysis for Wind Turbines Considering Climatic Regions and Comparing Geared and Direct Drive Wind Turbines

19

4.75

10

Life cycle reliability and maintenance analyses of wind turbines

19

3.8

Bhardwaj et al. [65]

Chan and Mo [68]

as prioritizing the failure modes RPNs. As a result, much multi-criteria decisionmaking (MCDM) tools (e.g., TOPSIS “technique for Order of Preference by Similarity to Ideal Solution” [69–71], DEMATEL “decision-making and evaluation laboratory” [72–74], PROMETHEE “Preference Ranking Organization METHod for Enrichment of Evaluations” [75–80], BWM “best–worst method” [81, 82]) have been applied to deal with the shortages of classical FMEA, including [47, 83–92]. However, most of them cannot consider the failure modes relationships over risk factors. In addition, it would be a much more practical way to evaluate the risk of failure mode by comparing one to one, utilizing preference relations. Therefore, in this study, probabilistic linguistic preference relations (PLPRs) as a new and interesting topic can represent the qualitative judgment of decision-makers with pairwise comparison among criteria and alternatives in MCDM tools underlying the concepts probabilistic linguistic term sets (PLTSs) [93, 94]. The PLTSs can

50

4 Developing Failure Modes and Effect Analysis on Offshore Wind …

adequately reflect the decision-makers’ complicated and uncertain linguistic expressions, which have been applied in different applications [95–97]. Besides, it would enhance the elicited preferences formalization process [98]. In the present work, we aim to develop the conventional FMEA technique by integrating it with the advanced best–worst method (BWM) underlying the concept of PLTSs to assess the risk of OWTs. The main contributions of the study can be highlighted as the following. First, the preference relations of risk assessment information in PLTSs format are collected from a group of decision-makers. Second, a multiplicative consistency-based weight tool as the two-step mathematical programming technique is utilized to obtain the importance weights of risk factors. Third, an advanced BWM method is proposed to deal with the inherent shortages that have not been covered yet, as well as ranking the failure modes pair wisely. Finally, the developed model is applied to OWTs to show its applicability and effectiveness. The organization of the present work is provided as the following. In Sect. 4.2, the probabilistic linguistic preference relations and best–worst method are preliminary. In Sect. 4.3, a new framework to develop FMEA is proposed. In Sect. 4.4, an application of a case study is explained. Finally, the conclusion and directions for future study are provided in Sect. 4.5.

4.2 Preliminary: Probabilistic Linguistic Preference Relations (PLPRs) In the following, the brief fundamental concepts of PLPRs are explained, providing insights into the proposed methodology section. Definition 4.1 [100] Let us assume that the δi ∈  is a fixed value and S = {st |t = −τ, . . . , −1, 0, 1, . . . , τ } is a linguistic linguistic   term set. A probabilistic term set (PLTS) S would be HS ( p) = δi , h iS ( p)|δi ∈  and the following equation:   h iS ( p) = sλi(k) pi(k) |sλi(k) ∈ S ≥ 0 λ = −τ, . . . , −1, 0, 1, . . . , τ, h iS ( p)

k=

1, 2, . . . ., #h iS ( p),



pi (k) ≤ 1

(4.1)

k=1

  where sλi(k) pi(k) is the k th linguistic term of sλi(k) and connected with the pi (k) ) probability, and #h iS ( p) indicates all linguistic terms in the h iS ( p). To seek simplicity, the sλi(k) , k = 1, 2, . . . ., #h iS ( p) are sorted in ascending ranking. In addition, the h iS ( p) is named the probabilistic linguistic element (PLE). Definition 4.2 [99] Let us assume that that h 1S ( p) and h 2S ( p) do two PLEs, which are #h 1S ( p) and #h 2S ( p). Therefore, the corresponding operation laws are as the

4.2 Preliminary: Probabilistic Linguistic Preference …

51

following equation:  1(k)   2(k)  p ⊕ sλ2(k) p h 1S ( p) ⊕ h 2S ( p) = sλ1(k) 1 2  1(k) 2(k)  +p p = s (k) |k = 1, 2, . . . ., #h iS ( p) λ1 +λ2 2 2

p1(k)

p2(k) 2(k) × s |k = 1, 2, . . . ., #h iS ( p) h 1S ( p) ⊗ h 2S ( p) = sλ1(k) 1 λ2  1(k)  k = 1, 2, . . . ., #h i ( p) and A 0 p Ah 1S ( p) = s 1(k) S Aλ1

(4.2) (4.3) (4.4)

Definition 4.3 [99] Let us assume that the h S ( p) is a PLE. Thus, the expected value, as well as the variance, can be computed from Eqs. 4.5 and 4.6, respectively. #h S ( p)

E(h S ( p)) =



p (k)



k=1

σ (h S ( p)) =

#h ( p) S  k=1

p

(k)



λ(k) + τ 2τ

λ(k) + τ 2τ 



2 0.5 − E(h S ( p))

(4.5)

(4.6)

Definition 4.4 [99] Let us assume that that h 1S ( p) and h 2S ( p) do two PLEs. Thus, the fundamental law comparisons can be described as the following:     • If E h 1S ( p) > E h 2S ( p), then h 1S ( p)  h 2S ( p); • If E h 1S ( p)  < E h 2S ( p) ,then h 1S ( p) ≺ h 2S ( p);   • If E h 1S ( p) = E h 2S ( p) , then (i)if σ h 1S ( p) > σ h 2S ( p) , then h 1S ( p) ≺  h 2S ( p); and (ii) h 1S ( p) = σ h 2S ( p) , then h 1S ( p) ∼ h 2S ( p). Definition 4.5 [99] Let us assume that h iS ( p) = sλ1(k) |k = 1, 2, . . . ., #h iS ( p), (i = 1, 2, . . . ., n) is the n PLEs. Thus, the linguistic probability average (PLWA) operator is determined as the following equation:   n nwi h iS ( p) P L W A h 1S ( p), h 2S ( p), . . . , h nS ( p) = ⊕i=1

     

= nw1 sλ1(k) p 1(k) ⊕ nw2 sλ2(k) p 2(k) sλ1(k) p1(k) ∈h 1S

⊕ ... ⊕



sλ2(k) p2(k) ∈h 2S

 

nwn sλn(k) p n(k)



(4.7)

sλn(k) pn(k) ∈h nS

where = (w1 , w2 , . . . , wn )T is the related importance weight vector with wi ∈ 0, 1 w n and i=1 wi = 1.

52

4 Developing Failure Modes and Effect Analysis on Offshore Wind …

4.3 The Proposed Methodology This section proposes a new framework to make viable decision-making in improving system safety and reliability performance (see Fig. 4.2). The proposed decisionmaking framework has four main steps, including (Pre-step) identifying and defining the list of failure modes and risk factors, respectively, (Step 1) computing the group-based preference evaluation of decision-makers, (Step 2) obtaining the importance weights of risk factors, and (Step 3) prioritizing the risk of failure modes in descending order. In this present work, a hybrid and collaborative-based decisionmaking method is introduced, considering the advantages and merits rather than a single-based decision-making tool, some of them can be highlighted as: • Choosing a proper and reliable decision-making tool is a difficult task since each of them would provide a different ranking with the same input information. Consequently, using a hybrid method with the integration of these tools resulted in the final decision-making step [100, 101],

Fig. 4.2 The proposed framework of decision-making in improving FMEA performance

4.3 The Proposed Methodology

53

• Using a hybrid methodology provides an opportunity to integrate both Epistemic and Aleatory input information [102, 103], • The PLPRs can also assist decision-makers to deal with uncertainties arising from humans cognitively, opinionated, and biased preferences [99], and • A hybrid model provides closer results for the actual case application representation [20, 104, 105]. Therefore, needs to be handled with the main drawback of the typical decisionmaking tools, such as the MCDM one. According to this point, the detail of each step to improve the FMEA method in the present work is provided as the following. Let us assume that the system safety and reliability decision-making problems with m number of failure modes (FMs) and n number of risk factors. These can be reformed and presented as F M = {#F M1 , #F M2 , . . . , #F Mm } and R F = {# R F1 , # R F2 , . . . , # R Fn } in the exact accordance. Then, the identified failure modes would be examined as pairs, underlying all risk factors with the l “crossfunctional” decision-makers as D M = {# D M1 , # D M2 , . . . , # D Ml }. In addition, η = {η1 , η2 , . . . , ηl } is the importance weights of the decision-makers, in which

l  uj ηu = 1. Let us assume that Ruj = rig is the matrix risk ηu ∈ [0, 1] and u=1

m×m

evaluation of preferences elicited from a group of decision-makers D Mu , in which

  rigu j is a PLE over F Mi −F Mg underlying R F j . Assume that the P u = piuh n×n m×m

is the importance weight of matrix risk evaluation of preferences elicited from D Mu over the set of risk factors. Accordingly, the introduced extension of the FMEA method is performed within the following steps: Step 1: Computing the group-based preference evaluation of decision-makers In the pre-step, all failure modes and risk factors are identified and defined, respectively. Step 1.1: Examining the “multiplicative consistency of preference evaluation matrices” To examine the “multiplicative consistency of preference evaluation matrices” Ruj = ( j = 1, 2, ..n, andu = 1, 2, . . . l), the following equation is utilized to compute the consistency index:

C I Ruj =

1 6 m(m − 1)(m − 2) #ru j ig  





uj   u j(k) u j(k) u j(k) # r ig  ln N S r m   + ln N S + ln 1 − N S r r  ih hg ig   ×





   − ln 1 − N S ru j(k) + ln N S ru j(k) + ln 1 − N S ru j (k)  i >h>h k=1   ih ig hg

(4.8)

54

4 Developing Failure Modes and Effect Analysis on Offshore Wind …



Let us consider that C is a “predefined consistency threshold”. IF C I Ruj ≤ C, therefore, the Ruj is an acceptable level of matrix risk evaluation of preferences. In contrast, the matrix Ruj must be justified interactively for satisfying the multiplicative consistency. The importance matrix risk evaluation of preferences P u (u = 1, 2, . . . , l) could be examined similarly. It should be added that, in this study, it is assumed that the N S is a numerical scale function as the sigmoid function, therefore, for a linguistic term st , N S(st ) = 1+e1 −t [106, 107]. In fact, as a human behavior role “the law of diminishing marginal utility,” the sigmoid function “right-skewed S-shaped utility function” can adequately express the diminish of utility property function [106]. Step 1.2: Aggregating the preference evaluation metrics of FMEA decisionmakers. The PLWA operator is used to construct the matrix risk evaluation of group-based

j , in which: preferences of the j th risk factor as R j = rig m×m



rigj = P L W A rig1 j , rig2 j , . . . , rligj



= rlu=1lηu rig





1 j (k) 2 j(k) m j (k) = ∪ lη1 rig ⊕ ∪ lη2 rig ⊕ . . . ⊕ ∪ lηl rig rig1 j (k) ∈rig1 j rig2 j (k) ∈rig2 j rligj (k) ∈rligj uj

(4.9)

  In addition, the group-based matrix risk evaluation of preferences P u = piuh n×n is calculated with the following information:   Pi j = P L W A Pi1j , Pi2j , . . . , Pil j = ⊕lu=1lηu Piuj



= ∪ lηu Pi1(k) ⊕ ∪ lηu Pi2(k) ⊕ ... ⊕ j j 1 Pi1(k) j ∈Pi j

2 Pi2(k) j ∈Pi j

∪

l Pil(k) j ∈Pi j



(4.10) lηu Pil(k) j

This matrix is only structured for risk factors and can be constructed for failure modes of limited number. Due to the fact that the pairwise comparison would be a much more complicated task and time-consuming when the number of failures modes is high. Step 2: Obtaining the importance weights of risk factors In this step, the two-step mathematical programming technique [106, 108] is utilized to obtain the importance weights of risk factors. This is a multiplicative consistencybased weight tool,  and it is extended in this paper underlying the idea of PLPR. For the P = pi j n×n as the group importance evaluation of matrix preference, it is     assumed P = E = ði j n×n |ði j ∈ pi j representing a set of PLPR according to the P. Step 2.1: Determining the best PLPR matrix for risk factors

4.3 The Proposed Methodology

55

In the introduced FMEA method, the multiplicative consistency index must have the least value possible as much as possible. Thus, the following optimization model is constructed to derive the PLPR risk factors matrix:     n    6  ln N S ði j + ln N S(ðit ) + ln(1 − N S(ðit ))  min         − ln 1 − N S ði j − ln N S(ðit ) − ln 1 − N S ð jt  n(n − 1)(n − 2) i < j 0}, and μ A˜ (x, u) is the upper membership function  with μ A˜ (x, u) = sup{u, u ∈ [0, 1], μ A˜ (x, u) > 0}.

To seek simplicity, the pair of A˜ = A L , AU = ((a L , b L , c L , d L ; h), (a U , bU , cU , d U )) is named an interval value of type-2 trapezoidal fuzzy numbers, in which a L , b L , c L , d L , a U , bU , and cU are non-negative real values, and h(0 ≤ h ≤ 1) is the height of A L . It is known that fuzzy triangular numbers are the particular case of trapezoidal numbers. If one assumes that b L = c L , and bU = cU , then the interval type-2 trapezoidal fuzzy numbers of changes to the interval type-2 fuzzy triangular numbers (IT2TFNs). In the present book chapter, the IT2TFNs are taken into account for expert elicitation process due to the following reasons: (i) the fuzzy triangular numbers are commonly utilized form of fuzzy numbers and has many straightforward operation rules, and (ii) the combination and division of two trapezoidal fuzzy numbers is not the trapezoidal fuzzy number.



Definition 5.3 [31] Let us assume that A˜ 1 = a1L , b1L , d1L ; h 1 , a1U , b1U , d1U and



A˜ 2 = a2L , b2L , d2L ; h 2 , a2U , b2U , d2U are two IT2TFNs. Thus, the corresponding fundamental operations can be expressed as the following:

A˜ 1 ⊕ A˜ 2 = a1L + a2L , b1L + b2L , d1L + d2L ; min(h 1 , h 2 ) , • ,,

a1U + a2U , b1U + b2U , d1U + d2U L L L L L L

U U U U U U

• A˜ 1 ⊗ A˜ 2 = a1 .a2 , b1 .b2 , d1 .d2 ; min(h 1 , h 2 ) , a1 .a2 , b1 .b2 , d1 .d2 ,



• λ A˜ 1 = λa1L , λb1L , λd1L ; h 1 , λa1U , λb1U , λd1U , λ ≥ 0.



Definition 5.4 [32] Let us assume that A˜ 1 = a1L , b1L , d1L ; h 1 , a1U , b1U , d1U



and A˜ 2 = a2L , b2L , d2L ; h 2 , a2U , b2U , d2U are two IT2TFNs. Thus, the distance between A˜ 1 and A˜ 2 can be computed as the following equation:     

 L   a2 − a1L + 2b2L − 2b1L + d2L − d1L + 4 a1u − a2u + 4 b1U − b2U 1   D A˜ 1 , A˜ 2 =         8  + 4 d U − d U + 3 2bU − a U − d U h − 3 2bU − a U − d U h   1 2 1 2 1 1 1 2 2 2 



(5.10) Particularly:   1

 D A˜ 1 , 0 = a1L + 2b1L + d1L + 4a1u + 4b1U + 4d1U + 3 2b1U − a1U − d1U h 1  8 (5.11) Definition 5.5 Let us assume that A˜ 1 and A˜ 2 are two IT2TFNs. Therefore, the comparison between the A˜ 1 and A˜ 2 are defined in the following circumstances:

74

5 Integration of the Bayesian Network Approach and Interval …

    • A˜ 1 is superior to A˜ 2 if D A˜ 1 , 0 > D A˜ 2 , 0 , denoted as A˜ 1 > A˜ 2 ;     • A˜ 1 is indifferent to A˜ 2 if D A˜ 1 , 0 = D A˜ 2 , 0 , denoted as A˜ 1 = A˜ 2 ;     • A˜ 1 is inferior to A˜ 2 if D A˜ 1 , 0 < D A˜ 2 , 0 , denoted as A˜ 1 < A˜ 2 .



Definition 5.6 [33] Let us assume that A˜ = a L , b L , d L ; h , a U , bU , d U is an IT2FN. The entropy of A˜ is defined as the following equation: ⎛

⎞   ⎝ μ A˜ (x), 1 − μ A˜ (x) ⎠ x min    ⎛ ⎞ E A˜ =     ⎝ μ A˜ (x), 1 − μ A˜ (x) ⎠ x max  



(5.12)

In which:  μ (x) =  A˜  μ A˜ (x) =



 L L a , b and a L = b L ;  L L b , d and b L = d L .

(5.13)

 U U a , b and a U = bU ;  U U b , d and bU = d U .

(5.14)

L h x − b Lha−a L , x ∈ b L −a L 1 hd L − d L −b L x − d L −b L , x ∈

U 1 x − bUa−aU , x ∈ bU −a U dU 1 − d U −b U x − d U −bU , x ∈

Definition Let us assume that A˜ i =

5.7 [33]

Considering the aiL , biL , diL ; h i , aiU , biU , diU (i = 1, 2, .., n) is n IT2FN. n wi = 1. Thus, weight vector w = (w1 , w2 , . . . , wn ), where wi ∈ [0, 1] and i=1 the interval type-2 fuzzy weighted average (IT2FWA) operator is derived as:   I T 2F W A = A˜ 1 , A˜ 2 , . . . , A˜ n ⎛⎛ ⎞ ⎛ ⎞⎞ n n n n n      L L L U U U ˜ ⎝ ⎝ ⎠ ⎝ wi Ai wi ai , wi bi , wi di ; min h i , wi ai , wi bi , wi di ⎠⎠ = i=1

i=1

1≤i≤n

i=1

i=1

(5.15)

i=1

and the interval type-2 fuzzy ordered weighted average (IT2FOWA) operator is obtained from:   I T 2F O W A = A˜ 1 , A˜ 2 , . . . , A˜ n

5.4 Case Study

75

Table 5.1 The fuzzy ratings obtained from decision-makers can be characterized by IT2TFNs

Linguistic terms

IT2TFNs

Very low

((0, 0, 0.05; 0.95), (0, 0, 0.1))

Low

((0.05, 0.1, 0.2; 0.95), (0, 0.1, 0.3))

Medium low

((0.2, 0.3, 0.4; 0.95), (0.15, 0.35, 0.5))

Medium

((0.4, 0.5, 0.6; 0.95), (0.3, 0.5, 0.7))

Medium high

((0.6, 0.7, 0.8; 0.95), (0.5, 0.7, 0.9))

High

((0.8, 0.85, 0.95; 0.95), (0.7, 0.9, 1))

Very high

((0.95, 1, 1; 0.95), (0.9, 1, 1))

⎛⎛

⎞⎞

n 

⎜⎝ ω j aσL( j ) , ω j bσL( j) , ω j dσL( j) ; min h σ ( j ) ⎠,⎟ ⎟ ⎜ 1≤ j≤n ⎟ ⎜ j=1 ⎟ ⎜ ˜ ⎞ = ω j Aσ ( j ) ⎜ ⎛ ⎟ n n n ⎟ ⎜    j=1 U U U ⎠ ⎠ ⎝⎝ ω j aσ ( j) , ω j bσ ( j) , ω j dσ ( j ) n 

j=1

j=1

j=1

(5.16) In which ω j ( j = 1, 2, . . . , n) are connected with ordered weights and ˜ ˜ (σ (1), σ (2), . . . , σ (n)) is the permutation of 1, 2, . . . , n, satisfying

Aσ ( j−1) ≥ Aσ ( j) for all j = 2, . . . , n. It should be added that if w = n1 , n1 , . . . . n1 , then the IT2FWA operator will change into the interval type-2 fuzzy average (IT2FA) operator. The fuzzy ratings obtained from decision-makers can be characterized by IT2TFNs as presented in Table 5.1 [34]. In the next section, a sustainability assessment of hydrogen energy storage systems in large metropolitan areas of Iran using an integration of IT2TFNs-BN is studied.

5.4 Case Study This section gives an actual application for site selection of the best sustainable hydrogen storage technologies in large metropolitan areas of Iran. To select the best site location for sustainable hydrogen storage technologies in large metropolitan areas (see Fig. 5.2), it is necessary to identify all potential contributing factors. Such contributing factors accomplish the sustainability and technical perspective necessities of making sustainable hydrogen storage technologies. The technical contributing factors significantly influence the best sustainable hydrogen storage technologies. The present study predominantly focused on the technical contributing factors like “reliability shortage expectation”, commonly ignored by published works.

76

5 Integration of the Bayesian Network Approach and Interval …

Fig. 5.2 The potential metropolitan areas in Iran for implementing the sustainable hydrogen storage technologies

Consequently, the sub-contributing factors measured in this example are accomplished, in which the associated literature with sustainable hydrogen storage technologies are gathered and evaluated, such as but not limited to [35, 36]. Thus, the initial sub-contributing factors are measured. Afterward, a group of decision-makers with a background in sustainable hydrogen storage technologies and working experience in hydrogen system safety and control areas are asked to provide an evaluation of initial sub-contributing factors. The decision-makers eliminate the less critical sub-contributing factors. Figure 5.3 illustrates all recognized contributing factors, sub-contributing factors, and causality among the factors considered to determine site selection of the best sustainable hydrogen storage technologies in large metropolitan areas of Iran. In this regard, this section has been constructed according to the following research questions: (i) which metropolitan areas in Iran have priority to be invested for implementing the sustainable hydrogen storage technologies, and (ii) which sustainable hydrogen storage technologies have priority to be implemented? A short description of contributing factors, sub-contributing factors, and alternatives are presented in the following:

5.4 Case Study

77

Fig. 5.3 All recognized contributing factors and sub-contributing factors

• CF1-Social aspects: The social aspects affect an individual’s lifestyle. – SCF11-Perceived risk level: This refers to the “spirit cost associated with customers’ purchasing behavior” [30]. – SCF12-Job creation opportunities: This helps society by enhancing gross domestic product. Once a person is employed, he or she would be paid by an employer resulting from spending money in his or her society, such as “food, clothing, entertainment, and a variety of other areas”. – SCF13-Social acceptance: This can be defined as “tolerating and welcoming the differences and diversity in others because most people attempt to look and act like others do in order to fit in” [37]. – SCF14-Acceptable health level: It is defined as “the probability of a potential hazard-related incident or exposure occurring and the severity of harm or damage that may result are as low as reasonably practicable (ALARP)” [38], or to a low-risk level [39–41]. • CF2-Economic aspects: This is referred to as the “importance of social responsibility, broad consideration of society and businesses, contribution to the public interest, and corporate voluntary participation” [42]. – SCF21-Costs: It includes but is not limited to the cost of land, operation, maintenance, investments, employees, and taxes. – SCF22-Benefits: This refers to the benefits that can be quantified in terms of money generated, such as net income and revenues. – SCF23-Lifetime: It refers to the age of the system that will work over a while. A higher lifetime is preferred as much as possible. • CF3-Technical circumstances: This provides the “physically immersive qualities of an experience. They generally are the parameters that lead us to label an experience belonging to a particular medium and the degree of sensory details within that medium” [43].

78

5 Integration of the Bayesian Network Approach and Interval …

– SCF31-Response time: This is the ability of hydrogen equipment to respond sufficiently quickly. – SCF32-Durability: It refers to when technical circumstance lasts a long time. – SCF33-Reliability: This is the ability of an item to perform a required function under given environmental and operational conditions and over a period [44]. – SCF34-Storage capacity: Hydrogen can be stored physically as a gas or a liquid. Storage of hydrogen as a gas typically requires high-pressure tanks and further requirement which needs to be taken into account carefully. – SCF35-Energy sources: Hydrogen can be produced from diverse domestic resources, including fossil fuels and natural gas. In addition, biomass, geothermal, solar, or wind are also used to produce hydrogen [45]. – SCF36-Efficiency: Hydrogen is also considered high efficiency, low polluting fuel that can be used for transportation, heating, and power generation in places where it is difficult to use electricity [46]. However, the efficiency may vary to the type of energy source. – SCF37-Volume: It is also essential to consider the storage volume of hydrogen in terms of maintenance, safety, and lifetime. • CF4-Environmental issues: These issues refer to the emissions of hydrogen that can lead to “increased burdens of methane and ozone and hence to an increase in global warming. Therefore, hydrogen can be considered an indirect greenhouse gas with the potential to increase global warming” [47]. – SCF41-Greenhouse gas emissions: Hydrogen is an indirect greenhouse gas with a global warming potential GWP of 5.8 over a 100-year time horizon [47]. – SCF42-Water consumption: Looking at hydrogen production, the “minimum water electrolysis can consume about 9 kg of water per kg of hydrogen. However, considering the process of water de-mineralization, the ratio can range between 18 and 24 kg of water per kg of hydrogen or even up to 25.7–30.2” [48]. – SCF43-General water impacts: Hydrogen production will also require secure, long-term access to water. “Large amounts of water are also required for hydrogen production using fossil fuels, with the current dominant technology of ’steam reforming’ using water for the reaction stage, process water and cooling” [49]. In this study, the six potential sustainable hydrogen storage technologies are considered [36, 50–52]: Liquid H2 (A1), Metal hydrides (A2), Chemical hydrogen (A3), Compressed gas (A4), Cryo compression (A5), and Carbon nanotubes (A6). The potential metropolitan areas of Iran to implement sustainable hydrogen storage technologies are Isfahan (S1), Tehran (S2), Tabriz (S3), Ahvaz (S4), and Bandar Abbas (S5).  In the next section, the results and discussions are presented.

5.5 Results and Discussions

79

Fig. 5.4 The initial aggregated results for “S1-Isfahan”

5.5 Results and Discussions In order to perform an integration of the Bayesian Network approach and Interval type-2 fuzzy sets, four decision-makers within relevant background and experience in the hydrogen storage system and hydrogen safety are employed. Using an extension of the best–worst method [53–55], the importance weights of four decision-makers are determined.1 Then, the decision-makers shared their opinions regarding the importance weights of contributing factors and aggregated them as C F1 = 0.25, C F2 = 0.21, C F3 = 0.32, andC F4 = 0.22. In the next step, all decision-makers share their opinions regarding fuzzy ratings characterized by IT2TFNs (Table 5.1) to evaluate sub-contributing factors. The interval type-2 fuzzy weighted average operator presented in definition 5.7 is utilized to obtain the “True” (False = 1 − True) value in the Bayesian Network approach. Similarly, the same process was conducted for determining the “True” value of sustainable hydrogen storage technologies (A1–A6). As an example, the initial aggregated results for “S1-Isfahan” are presented in Fig. 5.4. As much as the “True” value of the node sustainable hydrogen storage technology is high, the metropolitan area would have greater priority to be invested in for implementing the sustainable hydrogen storage technologies. The priority of metropolitan areas to be invested in for implementing sustainable hydrogen storage technologies are presented in Table 5.2. Using backward propagation analysis in the Bayesian network approach, the best sustainable hydrogen storage technologies can be derived for every single metropolitan area. The backward propagation analysis for “S1-Isfahan” is presented in Fig. 5.5. The results for the other metropolitan areas are provided in Table 5.3. As it can be seen from the provided results, the ranking of metropolitan areas is S5  S4  S3  S1  S2. In addition, for the first rank metropolitan area (Bandar Abbas-S5), the ranking of sustainable hydrogen storage technologies is: A2  A1  A6  A5  A3  A4, in which best sustainable hydrogen storage technology for “Bandar Abbas” is “Metal hydrides”. 1 All input data used to support the findings of the present chapter might be available from the corresponding author upon request.

80

5 Integration of the Bayesian Network Approach and Interval …

Table 5.2 The priority of metropolitan areas to be invested for implementing the sustainable hydrogen storage technologies Metropolitan areas

True value %

Ranking

Isfahan (S1)

45

4

Tehran (S2)

26

5

Tabriz (S3)

68

3

Ahvaz (S4)

75

2

Bandar Abbas (S5)

89

1

Fig. 5.5 The backward propagation analysis for “S1-Isfahan”

Table 5.3 The results of backward propagation analysis to obtain the ranking sustainable hydrogen storage technologies Metropolitan areas

Sustainable hydrogen storage technologies

True value %

Ranking

S1

A1

23

1

A2

38

3

A3

20

2

A4

40

5

A5

29

4

A6

79

6

A1

34

3

A2

48

1

A3

41

2

A4

12

6

A5

37

3

A6

33

S2

5 (continued)

5.6 Conclusion

81

Table 5.3 (continued) Metropolitan areas

Sustainable hydrogen storage technologies

True value %

Ranking

S3

A1

78

1

A2

65

3

A3

71

2

A4

32

5

A5

60

4

A6

26

6

A1

39

6

A2

54

5

A3

72

2

A4

59

4

A5

60

3

A6

79

1

A1

67

2

A2

77

1

A3

41

5

A4

34

6

A5

56

4

A6

64

3

S4

S5

5.6 Conclusion In recent years, hydrogen energy has received much attention in academia and the industrial sectors, since hydrogen energy is clean, decreases carbon emission to the atmosphere, and handling energy demand fluctuations is vital for stack holder planers and governments. Thus, performing the best hydrogen storage technologiesbased site location should be investigated considering environmental, operational, technological, and social contributing factors. In the present chapter, an integration of the Bayesian Network approach and Interval type-2 fuzzy sets are utilized to deal with the causality and uncertainty during the investigation process, respectively. The introduced approach is then utilized to investigate the hydrogen energy storage technologies in Iran’s large metropolitan areas. The results indicated that the Metal hydrides for Bandar Abbas city are the most suitable hydrogen energy storage alternative for Iran. As a direction for future study, the introduced approach can be justified considering the more complicated extension of fuzzy set theory, including intuitionistic fuzzy sets [56–58], and Pythagorean fuzzy sets [59–61], as well as probabilistic representations [62, 63] to reflect a decision-makers opinions.

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Chapter 6

How to Deal with Toxic People Using a Fuzzy Cognitive Map: Improving the Health and Wellbeing of the Human System

6.1 Introduction Human health and wellbeing are crucial concerns for every individual. To be specific, as a human, we may have faced many harmful words throughout our life, and we are unbelievably emotionally tender and fragile. The people surrounding us can play the role of bullies by throwing out hurtful words. It is enough to experience one of the following situations once talking with a person: (i) “Do you feel emotionally numb after you talk to this person?”, (ii) “Do you lack energy after being around this person?”, (iii) “Do you feel that this person constantly puts you down or belittles you, especially in front of others?”, (iv) “Does the person try to sabotage you by doing things behind your back?”, (v) “Do you act uncharacteristically submissive or aggressive in the person’s presence?” and seventy-three questions more presented in [1]. Dr. Lillian Glass called when people say “yes” to the seventy-three questions are the "toxic people” and can be anyone, including a childhood classmate, brother/sister, parent, lover, husband/wife, boss or coworker [1, 2]. With the descriptions mentioned above, the human is a mental and physical system. Thus, the human system’s health and wellbeing, particularly mentally, can be considerably improved if we use reliable techniques to improve our system performance over time. As an example, addicted smokers use tobacco frequently due to the fact that smokers have abused the chemical substance nicotine [3]. They are willingly intoxicated by different types of chemical substances, even when the negative impacts such toxins are obvious. However, the health and wellbeing system performance of an addicted smoker can be improved to some extent using different improvement plans such as quitting smoking stepwisely. We believe that all programs Toxic people, “A toxic person is anyone who has poisoned your life, who is not supportive, who is not happy to see you grow to see you succeed, who does not wish you well. In essence, he or she sabotages your efforts to lead a happy and productive life” defined by Dr. Lillian Glass [1].

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 H. Li and M. Yazdi, Advanced Decision-Making Methods and Applications in System Safety and Reliability Problems, Studies in Systems, Decision and Control 211, https://doi.org/10.1007/978-3-031-07430-1_6

87

88

6 How to Deal with Toxic People Using a Fuzzy Cognitive Map …

Table 6.1 The summarized decision-making tools and application domains regarding factors’ causality, influence, interdependency, and dynamic variation Row

Developed tools and application domains

Causal dependency

“Structural equation modeling” [6] “DEMATEL (decision-making trial and evaluation laboratory)” [7–13] “Interpretative structural modeling” [14] “System dynamics” [15] “Fuzzy cognitive map” [16–18] “Causal maps” [19] “Bayesian networks” [20–25]

Preferential dependency

“Conditional preference networks” [26] “Multi-attribute utility theory/multi-attribute value theory” [27]

Structural dependency

“AHP (analytic hierarchy process)” [28–33] “ANP (analytic network process)” [34–36] “VIKOR (VlseKriterijumska Optimizacija I Kompromisno Resenje)” [37–42] “TOPSIS (technique for order of preference by similarity to ideal solution)” [43–49] “BWM (Best–Worst method)” [50–53] “ELECTRE (elimination and choice expressing reality)” [54–56]

to improve human health and wellbeing system performance fall under the complicated decision-making process. In such problems, many factors must be taken into account regarding their causality, influence, interdependency, and dynamic variation. A couple of works reviewed some of the common decision-making tools in [4, 5]. Table 6.1 summarizes the decision-making tools incorporating the factors that go into a complex decision-making problem and provides examples of application domains. Among all mentioned methods in Table 6.1, the fuzzy cognitive map (FCM) has enough potential to examine risk-based contributing factors in a complex system considering their causes and effects on relationships. Hence, the FCM is considered a robust and efficient decision-making technique, which considerably saves assessment time in the complex system within several factors and limited decision-makers interactions [57, 58]. The FCM method assists decision-makers in recognizing the human cognitive mechanism and provides a set of intervention actions to improve system performance [59–61]. The FCM technique can also model the complex system when there is restricted subjective and objective input data, or the data collection is expensive for decision-makers [62]. In addition, the FCM can minimize the dependency of the assessment process on decision-makers’ opinions [63]. Applying the FCM tool to a complex decision-making problem enables assessors to predict the current and future states of the complex system within learning algorithms and conduct different scenarios.

6.2 Preliminary: Fuzzy Cognitive Map

89

Fig. 6.1 Distribution of “published works per year till the end of December 2021” in “fuzzy cognitive map” utilizations based on all application domains since 2000 (according to the web of science (WoS) database)

Figure 6.1 illustrates the number of publications on a “fuzzy cognitive map” utilization based on all application domains and any turbines since 2000. This trend showed the highest increase in 2019; it can be intensely predictable that “fuzzy cognitive map” utilizations in different case studies will continue evolving in the following years. According to Table 6.2, the first ten highly cited documents are presented considering the “Average Citations per Year” of each publication in the area of “fuzzy cognitive map.” The “Average Citations per Year” indicates the “WoS citation index” for a paper by the end of December 2021. The critical contribution of the present work is to use an FCM as a reliable artificial intelligence-based method to model human system health and wellbeing by dealing with toxic people. The organization of the present chapter is provided as the following. In Sect. 6.2, the preliminary fuzzy cognitive map is introduced. In Sect. 6.3, the methodology to investigate how to deal with toxic people and improve human system health and wellbeing is proposed. In Sect. 6.4, the application of the study, corresponding objectives, results, and discussion are presented. Finally, the conclusion and study remarks are highlighted in Sect. 6.5.

6.2 Preliminary: Fuzzy Cognitive Map The cognitive map [74] includes a set of nodes (i.e., concepts), directed arcs, and adjective weights. It is initially introduced as an essential role to improve the decisionmaking process in political, social, and economic aspects. The nodes indicate the “events, status, objects, or trends”, and the directed arcs represent the “causality relationships” in the decision-making system. The cognitive map utilizes a set of discrete values “[−1, 0, 1]” to show the different “causality relationships” in the

90

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Table 6.2 The “highly cited papers based on citation measures” in the field of “Fuzzy cognitive map” till the end of December 2021 (according to the WoS database) Row Descriptions

Reference

“Total citations” “Average per year”

1

“Active Hebbian learning algorithm to train fuzzy cognitive maps”

Papageorgiou et al. [64]

207

10.89

2

“Comparing the inference capabilities of binary, trivalent and sigmoid fuzzy cognitive maps”

Tsadiras [65]

159

10.6

3

“Fuzzy Cognitive Map learning based on nonlinear Hebbian rule”

Papageorgiou et al. [66]

158

7.9

4

“Fuzzy Cognitive Maps Stylios and Groumpos 147 in modeling [67] supervisory control systems”

6.39

5

“Unsupervised learning Papageorgiou et al. techniques for [68] fine-tuning fuzzy cognitive map causal links”

128

7.53

6

“Dynamical cognitive Miao et al. [69] network—an extension of the fuzzy cognitive map”

127

5.77

7

“A fuzzy cognitive map Georgopoulos et al. approach to the [70] differential diagnosis of specific language impairment”

119

5.95

8

“The first study of Fuzzy Cognitive Maps learning using particle swarm optimization”

Parsopoulos et al. [71] 114

5.7

9

“Fuzzy cognitive map architectures for medical decision support systems”

Stylios et al. [72]

7.47

112

(continued)

6.2 Preliminary: Fuzzy Cognitive Map

91

Table 6.2 (continued) Row Descriptions 10

Reference

“Fuzzy Cognitive Maps Jetter and Kok [73] for futures studies A-methodological assessment of concepts and methods”

“Total citations” “Average per year” 110

12.22

system as negatively, no interaction, and positive influence, respectively. It is further developed in the interval “[−1, 1]” using fuzzy set theory and corresponding interactions [75]. The adjective weights changing in the interval “[−1, 1]” reflects the nodes’ influential strengths together. Thus, the cognitive map is extended into the fuzzy cognitive map (FCM). Besides, the cyclic loops of weighted diagraphs present the non-linear relationships as well as feedback amongst nodes. The 2-tuple representations as (C, E) are used in a common FCM, in which C = {C1, C2, . . . , Cn } as a set of FCM nodes, and the Ci value is specified as Ai ; E = wi j n×n , adjective matrix, and wi j denotes the adjective weights,Ci into C j . Once the nodes and interactions are determined in the decision-making system, the system is modeled as two different types of figures (i) adjective matrix, and (ii) directed graph. Hence, it would be clearer to have insight into the influences of nodes on each other and how much their strength is. The node value Ai indicates the activation level Ci , and it can be different fuzzy numbers varying in the interval [0, 1]. The zero value shows inactivity, and one value illustrates the activity, and any other value between zero and one denotes activity and inactivity partially. The input information (objective or subjective predictions of nodes/concepts data) into the model should then be transferred into the interval [0,1] within FCM initial value simulation. Moreover, the adjective weight between two nodes illustrates three kinds of performance, including “wi j ∈ [−1, 0) meaning that the increase (decrease) of Ci will result in a decrease (increase) of C j , in other words, the influence of Ci on C j is negative”, “wi j ∈ (0, 1] meaning that the increase (decrease) of Ci will result in an increase (decrease) of C j , in other words, the influence of Ci on C j is positive”, and the Ci and C j ” [16]. “wi j = 0 meaning that there is no  interaction between  An activating vector A0 = A01 , A02 , . . . , A0n is the input data of a decisionmaking system, the simulation process is settled, and concepts values begin to iterate dynamically. The transformation function values can be derived as the following equation: ⎡

⎤

⎥ ⎢ ⎥ ⎢ n ⎥ ⎢ k−1 k−1 k ⎢ Ai · w ji ⎥ Ai = f ⎢ Ai + ⎥ ⎥ ⎢ ⎦ ⎣ j =1 j = i

(6.1)

92

6 How to Deal with Toxic People Using a Fuzzy Cognitive Map …

where Aik indicates the Ci value after the k number of iterations, likewise, Aik−1 indicates the Ci value after the k − 1 number of iterations, w ji is the adjective weights Ci into C j , f is named “threshold function,” and “Sigmoid” is the most common function as f (x) = (1 + exp(−λx))−1 . The parameter λ is the shape of the “Sigmoid” function and is commonly considered a positive value in the interval [1, 5]. The f (x) is also between 1 and 5. In the existing literature, it is noted that if Ai0 = 0 as the initial value of an active node in the model, the value would be changed to 0.5 in the next iteration. This would impact other nodes and might raise an opposing conflict in case of where initial value becomes 0.5 [76]. To handle this situation and acquire much more realistic results with minor deviation, Eq. 6.1 is upgraded with Eq. 6.2 as the following [77]: ⎡

⎤

⎥ ⎢ ⎥ ⎢ n ⎥ ⎢     k−1 k−1 ⎥ 2 A − 1 + − 1 · w Aik = f ⎢ 2 A ji ⎥ i i ⎢ ⎥ ⎢ ⎦ ⎣ j =1 j = i

(6.2)

The success of each iteration in FCMs was demonstrated in the literature [78]. In every iteration, the concepts’ values Ak−1 are combined with an adjective matrix to update the outcomes, indicating the causality impacts. It should be added that the memory of the previous iteration is stored in the above mentioned equation by the accumulation of the concepts’ value in the last iteration. The simulation would be stopped since it gains an equilibrium state in which no new information is produced and considered (Ak = Ak−1 ). In other words, the limit state circumstance is yielded prior as Ak − Ak−1 < θ , and θ is a residual value. Subsequently, the concepts’ value Ak denotes the outcomes state of the decision-making system.

6.3 Methodology As shown in Fig. 6.2, the one pre-step and six key steps should be conducted to investigate how to deal with toxic people and improve human system health and wellbeing using the advanced FCM technique. The description of each step is provided as the following: • Pre-step: Before starting, the decision-making problem should be adequately defined. The study’s objective must be clarified, and challenges, assumptions, and further clarification have to be highlighted. • Step 1: The nodes (concepts) are identified in the mathematical formulation. Historical information about the system (human health behavior) and subjective opinions elicited from the assessor (e.g., consular) are utilized together to investigate and define all potential concepts in the complex human health system.

6.3 Methodology

93

Fig. 6.2 The FCM framework to improve human system health and wellbeing to deal with the toxic people

• Step 2: The causality analysis is carried out in this step. The connections between the nodes are investigated by interdependency analysis, and the FCM structure is designed. • Step 3: The scores of the nodes are assigned in this step. To reflect the human cognitive behavior, the boundaries’ interval (e.g., upper and lower) can be used within a different fuzzy set based on the situation that the assessor is studying about, such as Pythagorean fuzzy sets (PFS) [79], intuitionistic fuzzy sets (IFS) [80], etc. • Step 4: The assessor provides his/her opinions regarding the strength of the causality connections between the nodes. The adjective weights are computed by aggregating all opinions (if more than one). If there is more than one assessor, many aggregation methods can be utilized for this purpose including evidence theory [81–83], Similarity Aggregation Method [84–86], simple averaging method, etc.

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6 How to Deal with Toxic People Using a Fuzzy Cognitive Map …

• Step 5: The outcomes from the previous step are normalized. The normalized scores with several scenarios are then fed into the FCM simulator to activate the concepts. Afterward, the non-linear Hebbian learning algorithm (NHL) is utilized to process the inference. Finally, the simulation results are gained once the transition circumstances are achieved. The robustness and efficiency of the NHL can be prioritized by comparing the equilibrium scores of human health and wellbeing system conditions. • Step 6: The sensitivity analysis is conducted to highlight the validation, robustness, and inevitability of the FCM in improving the human system’s health and wellbeing according to the natural human thinking mode.

6.4 Case Study In this section, a decision-making problem is defined for how to deal with toxic people and tries to improve the system’s health and wellbeing. The story of this problem came up to my mind when I was on holiday. I decided to read different success-based books which could help my health and wellbeing over time including “Atomic Habits: An Easy & Proven Way to Build Good Habits & Break by James Clear”, “The Steve Jobs Way: Leadership for a New Generation by Jay Elliot”, “Leading: Learning from Life and My Years at Manchester United by Alex Ferguson and Michael Moritz”, and “Toxic People: 10 Ways Of Dealing With People Who Make Your Life Miserable by Lillian Glass”, etc. I can indeed attest that the latter book was much more beneficial and provided a better understanding of the individuals with whom we all communicate, work, and spend joyful time. Agreeing that there is some cause in our lives to be unhappy, worthless, inadequate, physically/mentally distressed called “toxic people.” Thirty types of “toxic terrors” are identified in the book. However, these “toxic terrors” are not toxic to everyone; however, they might be toxic to only you, specifically. Lillian provided some straightforward strategies to recognize who can be toxic to you. Following that, ten critical demonstrated practices were introduced for successfully dealing with “toxic people.” The truth is that human cognitive behavior can be assessed and improved qualitatively. However, having knowledge and experience in decision-making science, we can at least do our best and go through such cases in more detail using mathematical formulations (i.e., quantitative methods). Therefore, in this case, the decision-making problem is defined as follows. • Pre-step: Determination of problem scope It should be added that this is a fictional case, and the people included are not real. A client visits a psychologist and explains his challenges and feelings about his miserable lifestyle since last year. He shared his failures in career, relationships, and personal interests. At the end of the day, the psychologist found that the client had been receiving toxicity from “toxic terrors,” which were thrown by “toxic people.”

6.4 Case Study

95

Fig. 6.3 The hierarchical structure of relationships among DM, PS, and TPs

Thus, a decision-making problem is defined, in which the client (DM) should recognize the potential “toxic people” (TPs) considering different “toxic terrors” (TTs) that continually communicate with him over the last year. The psychologist (PS) provides cause and effect influence map to examine the interdependency between the DM, TPs, TRs, and intervention strategies (ISs) to deal with TPs. The DM could identify only 8 TPs underlying PS supervision and pieces of advice. The relationships among DM, PS, and TPs are depicted in Figure 6.3. • Step 1: Identification of the concepts To seek simplicity in the present work, 15 concepts (i.e., TTs) out of 30 are identified and taken into account based on Dr. Lillian Glass’ recommendation (see Fig. 6.4). Likewise, six out of ten potential ISs are presented in Fig. 6.5 for handling the TPs. In the following steps, the DM with the assistance of the PS would understand how to deal with each single TP, individually. A further description is provided in Step 5. • Step 2: Constructing the map The cause-effect analysis is performed. The connections between the nodes are examined by interdependency analysis. It is based on TPs expressions underlying PS supervision and pieces of advice, and the structure of the FCM is designed as depicted in Fig. 6.6.

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Fig. 6.4 The 15 TTs considered (to obtain more details, please refer to [1])

• Step 3: Assigning the score of concepts The score of concepts is assigned in this step. To reflect human cognitive behavior, the boundaries’ interval is used as fuzzy D-numbers. The detailed computations are provided in [87, 88]. The results are presented in Table 6.3. • Step 4: Computing the adjective weights Similar to step 3, the adjective weights are computed to show the strength of causality connections between the concepts. The DM and PS interactively provide opinions regarding the strength of causality connections between the nodes. The adjective weights are obtained in a probabilistic manner. As presented in Fig. 6.6, the “Sigmoid” function is considered with slope 1.5 and an offset 0.1. The result is illustrated in Fig. 6.7.

6.4 Case Study

97

Fig. 6.5 The six ISs considered for handling the TPs (to obtain more details, please refer to [1])

Fig. 6.6 The FCM structure with an illustration of the “Sigmoid” function

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Table 6.3 The assigned concept scores Concepts (TTs)

Score

TT.1

0.231

TT.2

0.354

TT.3

0.214

TT.4

0.541

TT.5

0.651

TT.6

0.689

TT.7

0.256

TT.8

0.114

TT.9

0.498

T.10

0.769

T.11

0.431

T.12

0.890

T.13

0.145

T.14

0.156

T.15

0.452

Fig. 6.7 The toxic level of TPs (an example for TP.1)

A similar procedure is carried out for all TPs, and toxic levels (TL) with corresponding rankings presented in Table 6.4. It should be added that graphical presentation and computation are conducted using FCM EXPERT software developed by Nápoles et al. [89].

6.4 Case Study

99

Table 6.4 The toxic level of TPs with corresponding rankings

TPs

TLs

TP.1

0.459

6

TP.2

0.341

7

Ranking

TP.3

0.145

9

TP.4

0.521

4

TP.5

0.891

1

TP.6

0.781

2

TP.7

0.489

5

TP.8

0.259

8

TP.9

0.548

3

TP.10

0.119

10

Assuming that the acceptability threshold is 20% (according to the interactive communication between the PS and DM); therefore, only TP.3 and TP.10 do not require receiving any further ISs for handling the TTs of TPs. Moreover, the TPs toxic level is high, and it is necessary to perform more ISs. For example, TP.5 with (T Ls = 0.891) needs all six ISs such as “the mirror technique”. • Step 5: Simulating FCM with NHL The idea behind the FCM training is to deal with a lack and uncertainty in acquiring and sharing tacit knowledge for initial adjective weights. Then, the FCM simulation causes the results fall into adequate intervals. The “unsupervised learning” facilitates the adjective weights’ viability and creates precise reasoning [90]. In addition, “Hebbian learning” is the key algorithm used in “unsupervised systems”. The “Hebbian learning” refines the adjective weights based on the interconnection between the input and output data. In the literature [91], The recommended algorithm to improve non-linear function is as follows:

 k−1 A w ji = η Aik−1 Aik−1 − wk−1 ji i

(6.3)

In which w ji indicates the variation of w ji from time slice “k − 1” to “k”, the indicates the score of the parameter η shows the learning coefficient, Aik−1 and, Ak−1 j k−1 nodes Ci in time slice “k − 1”, w ji is the adjective weights from C j into Ci in the previous time slice. The adjective weights’ algorithm is later justified [92] considering the “decay parameter” and “sing function” with the following equation:

  k−1 k−1 − sgn wk−1 Ak−1 wk−1 w kji = γ wk−1 ji + η Ai j ji ji Ai

(6.4)

100

6 How to Deal with Toxic People Using a Fuzzy Cognitive Map …

Fig. 6.8 The simulation of the FCM with NHL using FCM EXPERT software



In which γ presents the weighted “decay parameter”, the sgn w k−1 indicates ji the physical meaning of the weights once it is updating, satisfying w k−1 > 0 and ji  

k−1 k−1 k−1 = 1 and sgn w ji = −1, in the same w ji < 0 which, leads to the sgn w ji order. According to the earlier description, equation (6.4) is utilized (see Fig. 6.8). The simulation result of the FCM with NHL using FCM EXPERT software in a 6-time slice (irritation) is depicted in Fig. 6.9. • Step 6: Conducting a sensitivity analysis According to the natural human thinking mode, the sensitivity analysis is performed to show the validation, robustness, and inevitability of the FCM in such decision-making problems. In this regard, the toxic level of the TPs with corresponding rankings are investigated by eliminating each TT individually. The results are provided in Table 6.5. It can be seen that eliminating a TT in each case cannot significantly change the original ranking. This simply means that the TPs have other TT and interdependencies between them. Therefore, one TT cannot meaningfully change the toxic level.

6.5 Conclusion This paper aims to develop a framework utilizing an advanced FCM technique, which provides insight and understanding about the effectiveness of intervention strategies for handling toxic people’s toxic terrors. A graphical representation of the model is formulated to model human cognitive behavior. The fuzzy D-numbers are then utilized to deal with the interaction and uncertainty of the input data. Additionally, the process of human system health and wellbeing is determined by FCM reasoning

6.5 Conclusion

101

Fig. 6.9 The simulation result of the FCM with NHL using FCM EXPERT software in a 6-time slice

and the non-linear representation of toxic terrors into the toxic levels of toxic people. The outcome of a toxic level for a group of toxic people is obtained, indicating the effectiveness of FCM reasoning. It can be understood from the FCM simulation mechanism, and the utilized methodology has straightforward merits and considerable learning capability. The study’s application demonstrates how human system health and well-being can adequately be improved when with toxic people. However, the critical challenge is determining the initial adjective weights based on interactive communication between the PS and DM. Since the structured model is deeply based on the pieces of knowledge and experiences of the PS and DM, objective input data can rarely be engaged in such cases. As a direction for future study, different “supervised learning and digging algorithms” can be utilized to improve the FCM performance in the case of objective data availability. It should be added that the main elements of FCM must be close to the actual case problem.

6

3

9

6

7

9

4

1

2

5

8

3

10

TP.1

TP.2

TP.3

TP.4

TP.5

TP.6

TP.7

TP.8

TP.9

TP.10

8

5

2

1

4

10

7

TT.2

Ranking

TT.1

TPs

10

3

8

5

2

1

4

9

6

7

TT.3

Table 6.5 The sensitivity analysis

10

3

8

5

2

1

4

9

7

6

TT.4

10

3

8

5

2

1

4

9

7

6

TT.5

9

3

8

5

2

1

4

10

7

6

TT.6

10

3

8

5

2

1

4

9

7

6

TT.6

10

3

8

5

2

1

4

9

7

6

TT.7

10

3

8

5

2

1

4

9

7

6

TT.8

10

3

8

5

2

1

4

7

9

6

TT.9

10

4

8

5

2

1

3

9

7

6

TT.10

10

3

8

5

2

1

4

9

7

6

TT.11

10

3

9

5

2

1

4

8

7

6

TT.12

10

1

8

5

2

3

4

9

7

6

TT.13

10

3

8

5

1

2

4

9

7

6

TT.14

10

3

9

5

1

2

4

8

7

6

TT.15

102 6 How to Deal with Toxic People Using a Fuzzy Cognitive Map …

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60. Bevilacqua, M., Ciarapica, F.E., Mazzuto, G.: Fuzzy cognitive maps for adverse drug event risk management. Saf. Sci. 102, 194–210 (2018). https://doi.org/10.1016/j.ssci.2017.10.022 61. Adesina, K.A., Nedjati, A., Yazdi, M.: A Short communication Improving marine safety management system by addressing common safety program. Res. Mar. Sci. 5, 671–680 (2020) 62. Özesmi, U., Özesmi, S.: A participatory approach to ecosystem conservation: fuzzy cognitive maps and stakeholder group analysis in Uluabat Lake, Turkey. Environ. Manage. 31, 518–531 (2003). https://doi.org/10.1007/s00267-002-2841-1 63. Bakhtavar, E., Valipour, M., Yousefi, S., Sadiq, R., Hewage, K.: Fuzzy cognitive maps in systems risk analysis: a comprehensive review. Complex Intell. Syst. 7, 621–637 (2021). https://doi. org/10.1007/s40747-020-00228-2 64. Papageorgiou, E.I., Stylios, C.D., Groumpos, P.P.: Active hebbian learning algorithm to train fuzzy cognitive maps. Int. J. Approx. Reason. 37, 219–249 (2004). https://doi.org/10.1016/j. ijar.2004.01.001 65. Tsadiras, A.K.: Comparing the inference capabilities of binary, trivalent and sigmoid fuzzy cognitive maps. Inf. Sci. (Ny). 178, 3880–3894 (2008). https://doi.org/10.1016/j.ins.2008. 05.015 66. Papageorgiou, E., Stylios, C., Groumpos, P.: Fuzzy cognitive map learning based on nonlinear hebbian rule. In: (Tom) D. Gedeon, T., Fung L.C.C. (eds.) BT—AI 2003: Advances in Artificial Intelligence. Springer Berlin Heidelberg, 2003, pp. 256–268 67. Stylios, C.D., Groumpos, P.P.: Fuzzy cognitive maps in modeling supervisory control systems. J. Intell. Fuzzy Syst. 8, 83–98 (2000) 68. Papageorgiou, E.I., Stylios, C., Groumpos, P.P.: Unsupervised learning techniques for finetuning fuzzy cognitive map causal links. Int. J. Hum. Comput. Stud. 64, 727–743 (2006). https://doi.org/10.1016/j.ijhcs.2006.02.009 69. Miao, Y., Liu, Z.-Q., Li, S., Siew, C.K.: Dynamical cognitive network-an extension of fuzzy cognitive map. In: Proceedings 11th International Conference on Tools with Artificial Intelligence, 1999, pp. 43–46. https://doi.org/10.1109/TAI.1999.809764 70. Georgopoulos, V.C., Malandraki, G.A., Stylios, C.D.: A fuzzy cognitive map approach to differential diagnosis of specific language impairment. Artif. Intell. Med. 29, 261–278 (2003). https://doi.org/10.1016/S0933-3657(02)00076-3 71. Parsopoulos, K.E., Papageorgiou, E.I., Groumpos, P.P., Vrahatis, M.N.: A first study of fuzzy cognitive maps learning using particle swarm optimization. Congr. Evol. Comput. 2, 1440–1447 (2003). https://doi.org/10.1109/CEC.2003.1299840 72. Stylios, C.D., Georgopoulos, V.C., Malandraki, G.A., Chouliara, S.: Fuzzy cognitive map architectures for medical decision support systems, Appl. Soft Comput. 8, 1243–1251 (2008). https://doi.org/10.1016/j.asoc.2007.02.022. 73. Jetter, A.J., Kok, K.: Fuzzy cognitive maps for futures studies—a methodological assessment of concepts and methods. Futures. 61, 45–57 (2014). https://doi.org/10.1016/j.futures.2014. 05.002 74. Axelrod, R. (ed.): Structure of Decision: The Cognitive Maps of Political Elites. Princeton University Press (1976). http://www.jstor.org/stable/j.ctt13x0vw3 75. Kosko, B.: Fuzzy cognitive maps. Int. J. Man. Mach. Stud. 24, 65–75 (1986). https://doi.org/ 10.1016/S0020-7373(86)80040-2 76. Kannappan, A., Tamilarasi, A., Papageorgiou, E.I.: Analyzing the performance of fuzzy cognitive maps with non-linear hebbian learning algorithm in predicting autistic disorder, Expert Syst. Appl. 38 (2011) 1282–1292. https://doi.org/10.1016/j.eswa.2010.06.069. 77. Papageorgiou, E.I.: A new methodology for decisions in medical informatics using fuzzy cognitive maps based on fuzzy rule-extraction techniques. Appl. Soft Comput. 11, 500–513 (2011). https://doi.org/10.1016/j.asoc.2009.12.010 78. Stylios, C.D., Groumpos, P.P.: Modeling complex systems using fuzzy cognitive maps. IEEE Trans. Syst. Man, Cybern.—Part A Syst. Humans. 34, 155–162 (2004). https://doi.org/10.1109/ TSMCA.2003.818878 79. Yager, R.R.: Pythagorean membership grades in multicriteria decision making. IEEE Trans. Fuzzy Syst. 22, 958–965 (2014). https://doi.org/10.1109/TFUZZ.2013.2278989

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Chapter 7

An Advanced TOPSIS-PFS Method to Improve Human System Reliability

7.1 Introduction In complex and high-tech process-based system operations, human errors can lead to system failure and result in severe accidents [1, 2]. Thus, decision-makers always need to develop a human reliability model by reflecting human cognitive behavior, investigating the operational mechanisms, and evaluating and preventing potential failures. Due to subjective uncertainty of human reliability inference with human– environment-machine, a mathematical model integrating with decision-making tools is suggested [3–5]. Therefore, decision-making methods are growing with introducing many MCDM (multi-criteria decision-making) tools and extensions to the system’s safety performance over time. Among MCDM tools, TOPSIS (“Technique for Order of Preference by Similarity to Ideal Solution”) proposed by Ching-Lai Hwang and Yoon in 1981 have been widely utilized by many scholars in different applications domains, such as but not limited to [6–15]. The idea of the TOPSIS method is to solve an MCDM problem by obtaining the shortest and farthest distances from the positive and negative ideal solutions, respectively. A couple of extensions of the TOPSIS method is integrating with the rough cloud approach [16], fuzzy set theory [13, 17], intuitionistic fuzzy hybrid [15, 18], goal programming-based [19], interval-valued-fuzzy [20], hesitant-fuzzy [21], hesitant-fuzzy-linguistic [22, 23], Pythagorean-fuzzy [24] Interval-valued-Pythagorean-fuzzy [25], and Neutrosophic [26, 27]. The TOPSIS technique has been extended with a single and interval-valued spherical fuzzy set (PFS) due to its ability to deal with the subjectivity of human cognitive behaviors [28, 29]. Thus, in this chapter, an advanced integration of the TOPSIS method and PFS is justified to identify the reliability index of a complex system. The results help decision-makers to prioritize the critical index layers (human–environmentmachine). Subsequently, performing the appropriate intervention actions (i.e., corrective, mitigative, preventive, and control) can establish and improve a robust human reliability system over time. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 H. Li and M. Yazdi, Advanced Decision-Making Methods and Applications in System Safety and Reliability Problems, Studies in Systems, Decision and Control 211, https://doi.org/10.1007/978-3-031-07430-1_7

109

110

7 An Advanced TOPSIS-PFS Method to Improve Human System Reliability

The organization of this chapter is as follows. In Sect. 7.2, the preliminary PFS is presented. In Sect. 7.3, a framework of the advanced TOPSIS-PFS method is introduced. In Sect. 7.4, an example is illustrated. Finally, a conclusion and research remarks are highlighted in Sect. 7.5.

7.2 Preliminary: Spherical Fuzzy Set (PFS) The main concept of the Spherical fuzzy set (PFS) is provided following [30]. Definition 1 A single value of SFS denoted as A˜ s is in the universe of discourse U and presented as the following equation: A˜ s =

    u, μ A˜ s (u), ν A˜ s (u), π A˜ s (u) u ∈ U

(7.1)

where μ A˜ s (u) : U → [0, 1], ν A˜ s (u) : U → [0, 1], π A˜ s (u) : U → [0, 1], and 0 ≤ μ2A˜ (u) + ν 2A˜ (u) + π A2˜ (u) ≤ 1, ∀u ∈ U . s

s

s

In this regard, the values μ A˜ s (u), ν A˜ s (u), and π A˜ s (u) are called the “membership, in-membership, and hesitancy” degree of u into the A˜ s in the same order. In addition,  0.5 the “refusal degree” refers to X A˜ s (u) = 1 − μ2A˜ (u) − ν 2A˜ (u) − π A2˜ (u) . s

s

s

Definition 2 The fundamental operators of a “Single valued spherical fuzzy set (SV-SFS)” can be defined for two SFS ( A˜ s , D˜ s ) as the following:

  0.5 , ν 2A˜ ν D2˜ , 1 − μ2D˜ π A2˜ A˜ s ⊕ D˜ s = μ2A˜ + μ2D˜ − μ2A˜ μ2D˜ s s s s s s s s •  0.5  + 1 − μ2A˜ π D2˜ − π A2˜ π D2˜ , s s s s

 0.5   A˜ s ⊗ D˜ s = μ2A˜ μ2D˜ , ν 2A˜ + ν D2˜ − ν 2A˜ ν D2˜ , 1 − ν D2 s π A2˜ s s s s s s s •  + 1 − ν 2A˜ π D2˜ − π A2˜ π D2˜ , s s s s

    λ 0.5 λ  λ 0.5 2 λ 2 2 2 ˜ • λ. As = 1 − 1 − μ A˜ , ν A˜ , 1 − μ A˜ − 1 − μ A˜ − π A˜ s

∀λ ≥ 0, • A˜ λS

=

∀λ ≥ 0.

 μλA˜ , s



1− 1−

s

ν 2A˜ s

s

s

s

 λ 0.5  λ  λ 0.5 2 2 2 , 1 − ν A˜ − 1 − ν A˜ − π A˜ s

s

s

7.2 Preliminary: Spherical Fuzzy Set (PFS)

111

Definition 3 For two SFS ( A˜ s , D˜ s ) the following conditions are satisfied: • A˜ s ⊕ D˜ s = D˜ s ⊕ A˜ s , • A˜s ⊗ D˜ s = D˜ s ⊗ A˜ s , • λ A˜ s ⊕ D˜ s = λ A˜ s ⊕ λ D˜ s , • λ1 A˜ s ⊕ λ2 A˜ s = (λ1 + λ2 ) A˜ s ,  λ • A˜ s ⊕ D˜ s = A˜ λs ⊗ D˜ sλ , • A˜ λs 1 ⊗ A˜ λs 2 = A˜ λs 1 +λ2 . Definition 4 The “Single-valued Spherical Weighted Arithmetic Mean (SWAM)” is defined as the following:  SWAMw A˜ s1 , A˜ s2 , . . . A˜ sn = w1 A˜ s1 , w2 A˜ s2 , . . . wn A˜ sn ⎧  n  n  n ⎨ wi 0.5   wi   wi  2 ν A˜ si , = 1− , 1 − μ A˜ 1 − μ2A˜ si si ⎩ i=1 i=1 i=1 ⎫  n  wi 0.5 ⎬  2 2 − 1 − μ A˜ − π A˜ si si ⎭

(7.2)

i=1

In which, w = (w1 , w2 , . . . , wn ), wi ∈ [0, 1], and satisfying

n i=1

wi = 1.

Definition 5 The “Single-valued Spherical Weighted Geometric Mean (SWGM)” is defined as the following:  SW G Mw A˜ s1 , A˜ s2 , . . . A˜ sn i 2 ˜ wn + A˜ w = A˜ w s2 + · · · + Asn ⎧s1    n n  n  ⎨ wi wi 0.5   wi  2 2 = , 1− , μ A˜ 1 − ν A˜ 1 − ν 2A˜ si si si ⎩ i=1 i=1 i=1 ⎫  n  wi 0.5 ⎬  2 2 − 1 − ν A˜ − π A˜ si si ⎭

(7.3)

i=1

Definition 6 The “score function (SF)” and “accuracy function (AF)” of the set of SFS can be defined as the following equations:  π A˜ 2  π A˜ 2 S F( A˜ s ) = 2μ A˜ s − s − 2ν A˜ s − s 2 2

(7.4)

112

7 An Advanced TOPSIS-PFS Method to Improve Human System Reliability

AF( A˜ s ) = μ2A˜ + ν 2A˜ + π A2˜ s

s

s

(7.5)

It should be added that the A˜ s < D˜ s if (i) S F( A˜ s ) < S F( D˜ s ), or (ii) S F( A˜ s ) = S F( D˜ s ) and AF( A˜ s ) = AF( D˜ s ).

7.3 Methodology: An Advanced TOPSIS-PFS Method Let us assume that there are three hierarchical layers to assess the human system reliability, and they are (i) target layer, (i) criterion layer, and (iii) index layer. In this regard, X = {X 1 , X 2 , . . . X m }, (m ≥ 2) is defined as a set of index layers, C = {C1 , C2 , . . . Cn } is defined as a set of criterion layers, and w = {w1 , w2 , . . . wn } is vector weight of criterion layers satisfying the condition 0 ≤ wn ≤ 1 and the n w j = 1. The following eight critical steps proposed in [31] are justified in the j=1 present chapter to assess the human system reliability decision-making problem. • Step 1. In this step, the human system reliability, including three hierarchical layers are identified. Then a group of decision-makers is employed for the evaluation process based on the qualitative linguistic terms and corresponding spherical fuzzy numbers (PFNs) depicted in Fig. 7.1. • Step 2. Once decision-makers share their opinions regarding the importance of the index layer over each criterion layer, the aggregated opinions can be determined  SWAM with consideration of the importance weight of decision-makers using w j . The exact process should be performed for the criterion layer over the target layer. • Step 3. The aggregated “spherical fuzzy decision matrix” is constructed in this step. Let us assume that the X i (i = 1, 2, . . . , m) is the values of the   index layer criterion layer, by C 1, 2, . . . , n) = μi j , υi j , πi j . underlying the C j (i = (X ) j i   In addition, the D = C j (X i ) m×n is called a “spherical fuzzy decision-matrix”.   Thus, the D = C j (X i ) m×n can be constructed as the following equation: ⎤ (μ11 , υ11 , π11 ) · · · (μ1n , υ1n , π1n ) ⎥ ⎢ .. .. .. =⎣ ⎦ . . . (μm1 , υm1 , πm1 ) · · · (μmn , υmn , πmn ) ⎡

  D = C j (X i ) m×n

(7.6)

• Step 4. The aggregated weighted “spherical fuzzy decision matrix” is computed in this step as the following equation: ⎡   ⎢ D = C j (X iw ) m×n = ⎣

⎤ (μ11w , υ11w , π11w ) · · · (μ1nw , υ1nw , π1nw ) ⎥ .. .. .. ⎦ (7.7) . . . (μm1w , υm1w , πm1w ) · · · (μmnw , υmnw , πmnw )

7.3 Methodology: An Advanced TOPSIS-PFS Method

113

Fig. 7.1 The engaged linguistic terms and the relevant PFS

• Step 5. The SF is utilized to defuzzify the aggregated weighted “spherical fuzzy decision matrix” as the following equation:    πi jw 2 πi jw 2  − 2νi jw − S F C j (X iw ) = 2μi jw − 2 2

(7.8)

• Step 6. The “spherical fuzzy negative ideal solution (SF-NIS)” and “spherical fuzzy positive ideal solution (SF-PIS)” are determined considering the SF values derived in the previous step.     S F − N I S : X − = C j , min S F C j (X i w ) >  j = 1, 2, . . . , n    − − −  − − −   − − = C1 , μ− 1 , ν1 , π1 , C 2 , μ2 , ν2 , π2 , . . . , C n , μn , νn , πn

(7.9)

    S F − P I S : X ∗ = C j , max S F C j (X i w ) >  j = 1, 2, . . . , n        = C1 , μ∗1 , ν1∗ , π1∗ , C2 , μ∗2 , ν2∗ , π2∗ , . . . , Cn , μ∗n , νn∗ , πn∗

(7.10)

• Step 7. The normalized Euclidean distance between the X i and SF-NIS/SF-PIS are derived using the following equations [32]:

114

7 An Advanced TOPSIS-PFS Method to Improve Human System Reliability n " " " " "  1 !"" 2 " " " " " D X˜ i j , X −j = "μ X˜ i j − μ2X − " + "ν X2˜ i j − ν X2 − " + "π X2˜ i j − π X2 − " (7.11) j j j 2n j=1 n " " " " "  1 !"" 2 " " " " " D X˜ i j , X ∗j = (7.12) "μ X˜ i j − μ2X ∗j " + "ν X2˜ i j − ν X2 ∗j " + "π X2˜ i j − π X2 ∗j " 2n j=1

• Step 8. The closeness ratio (CR) of each single index layer is obtained in this step as the following equation:  D X˜ i j , X −j  CR =  D X˜ i j , X ∗j + D X˜ i j , X −j

(7.13)

It should be added that the layer index is prioritized in descending order.

7.4 Case Study A CNC machine with a specific the operating system is considered to show the effectiveness of the introduced advanced TOPSIS-PFS method. Based on the existing literature [33] and to the best of the author’s knowledge, the three hierarchical human system reliability layers are provided and depicted in Fig. 7.2.

Fig. 7.2 The three hierarchical human system reliability layers

7.4 Case Study

115

Table 7.1 The aggregated importance weight of the criterion layer Criterion layer

Decision-maker 1

Decision-maker 2

Decision-maker 3

Aggregated

C1

HI

SMI

LI

(0.558, 0.211, 0.342)

C2

VHI

HI

VHI

(0.775, 0.051, 0.229)

C3

EI

HI

ALI

(0.708, 0.088, 0.318)

C4

EI

LI

HI

(0.565, 0.203, 0.375)

C5

EI

LI

HI

(0.565, 0.203, 0.375)

C6

HI

AMI

SMI

(0.627, 0.150, 0.306)

C7

VHI

HI

LI

(0.649, 0.139, 0.271)

According to step 1, the human system reliability, including three hierarchical layers, is identified. Then three decision-makers are employed for the evaluation process using the qualitative linguistic presented in Fig. 7.1. In step 2, the decision-makers share their opinions for the index layer under each criterion and criterion layer. The importance weights of decision-makers are assumed to be 0.3, 0.3, and 0.4. The aggregated importance weight of the criterion layer is presented in Table 7.1. In addition, the “spherical fuzzy decision matrix” in qualitative terms and “aggregated weighted spherical fuzzy decision matrix” are presented in Tables 7.2 and 7.3, respectively. The SF values are computed in the next step, and the result is presented in Table 7.4. Afterward, the SF-NIS and SF-PIS are provided in Table 7.5. In the next step, the normalized Euclidean distance between the X i and SF-NIS/SF-PIS are derived using Eqs. 7.11 and 7.12. The results are provided in Table 7.6. Finally, the CR is determined from normalized Euclidean distance and presented in Table 7.7. Thus, the critical index layers are identified as C25  C15  C14  C11  C66. The best index layers are identified as C43  C74  C75  C73  C55. That is while in literature [33], the index layers C26, C53, C62, and C74 were identified as important layers, and needed to be improved with proper intervention actions.

116

7 An Advanced TOPSIS-PFS Method to Improve Human System Reliability

Table 7.2 The “aggregated weighted spherical fuzzy decision matrix” C4

C5

C11 EI, AMI, AMI

ALI, LI, LI VHI, EI, HI

VLI, VLI, VLI

ALI, LI, EI VLI, VLI, EI

ALI, LI, EI

C12 HI, EI, HI

VLI, EI, HI

VLI, EI, HI

EI, EI, HI

VLI, VLI, EI

VHI, HI, HI

VLI, EI, HI

C13 VHI, HI, HI

VLI, VLI, EI

EI, AMI, AMI

VLI, VLI, EI

VLI, EI, HI

VLI, HI, EI

VHI, HI, HI

C14 HI, EI, HI

VLI, VLI, VLI

VLI, VLI, EI

VLI, VLI, VLI

VLI, HI, EI

ALI, LI, EI VLI, EI, HI

C15 VLI, VLI, EI

VLI, VLI, EI

VLI, VLI, VLI

EI, EI, HI

SLI, LI, SLI

VLI, EI, HI

VLI, VLI, VLI

C21 SLI, HI, EI EI, EI, HI

VLI, HI, EI

LI, LI, LI

VLI, VLI, EI

HI, EI, HI

EI, EI, HI

C22 VLI, VLI, EI

VLI, VLI, EI

VLI, VLI, VLI

VLI, HI, EI

EI, AMI, AMI

VLI, HI, EI

VLI, HI, EI

C23 EI, EI, HI

VLI, VLI, EI

HI, HI, HI

VLI, VLI, VLI

VHI, EI, HI

VLI, VLI, EI

EI, EI, HI

C24 VLI, VLI, EI

HI, HI, HI

VLI, VLI, VLI

VHI, EI, HI

VLI, VLI, VLI

VLI, VLI, EI

VLI, HI, EI

C25 VLI, VLI, VLI

VLI, VLI, VLI

VLI, VLI, EI

LI, LI, LI

HI, EI, HI

VLI, HI, EI

VLI, VLI, VLI

C26 VLI, VLI, EI

VLI, VLI, EI

ALI, LI, EI VLI, VLI, EI

VLI, HI, EI

LI, LI, LI

EI, EI, HI

C31 HI, HI, HI

ALI, LI, EI VLI, VLI, EI

VLI, VLI, EI

VLI, VLI, EI

VLI, EI, HI

C32 VLI, VLI, EI

VLI, VLI, VLI

Tag

C1

C2

C33 ALI, LI, LI VLI, VLI, EI

C3

VHI, EI, HI

C6

C7

VHI, VHI, VHI, VHI, ALI, LI, EI LI, LI, LI VHI VHI

VLI, VLI, EI

VLI, VLI, EI

VLI, HI, EI

EI, EI, HI

VLI, HI, EI

C34 ALI, LI, LI ALI, LI, EI ALI, LI, EI VLI, EI, HI

VLI, VLI, EI

VLI, HI, EI

VLI, VLI, EI

C35 VLI, VLI, EI

LI, LI, LI

VLI, EI, HI

EI, EI, HI

SLI, LI, SLI

C41 VLI, HI, EI

ALI, LI, EI VLI, VLI, EI

VHI, VHI, EI, EI, HI VHI

VHI, EI, HI

LI, LI, LI

C42 VLI, EI, HI

VLI, HI, EI

SLI, LI, SLI

VLI, HI, EI

VLI, HI, EI

VLI, EI, HI

VLI, VLI, EI

C43 VHI, VHI, VHI, VHI, VLI, HI, VHI VHI EI

VLI, HI, EI

VHI, VHI, VHI, VHI, VHI, VHI, VHI VHI VHI

C44 VLI, HI, EI

VLI, VLI, EI

VLI, HI, EI

VHI, VHI, VLI, HI, VHI EI

VLI, VLI, EI

C45 HI, HI, HI

VLI, HI, EI

VLI, VLI, EI

ALI, LI, EI VHI, EI, HI

VHI, VHI, VLI, VLI, VHI EI

VLI, VLI, EI

VHI, HI, HI

VLI, VLI, EI

VHI, VHI, VHI

(continued)

7.4 Case Study

117

Table 7.2 (continued) C2

C3

C4

C46 VLI, VLI, EI

VLI, HI, EI

VHI, HI, HI

VHI, VHI, SLI, LI, VHI SLI

C51 SLI, LI, SLI

ALI, LI, EI SLI, HI, EI VHI, EI, HI

SLI, HI, EI VHI, EI, HI

C52 LI, LI, LI

VHI, EI, HI

Tag

C1

C5

C6

C7

VLI, VLI, EI

VHI, VHI, VHI VLI, VLI, EI

VLI, HI, EI

VLI, VLI, EI

VHI, VHI, VHI, VHI, HI, HI, HI VHI VHI

C53 VHI, VHI, LI, LI, LI VHI

SLI, LI, SLI

LI, LI, LI

VLI, VLI, EI

C54 VHI, EI, HI

LI, LI, LI

VHI, HI, HI

VHI, EI, HI

ALI, LI, EI VHI, HI, HI

C55 SLI, LI, SLI

VHI, EI, HI

VHI, VHI, VLI, HI, VHI EI

C56 SLI, LI, SLI

HI, HI, HI

SLI, LI, SLI

VHI, VHI, SLI, LI, VHI SLI

HI, EI, HI

VHI, VHI, VHI

C57 VLI, VLI, EI

SLI, LI, SLI

SLI, LI, SLI

VHI, HI, HI

VLI, VLI, EI

HI, HI, HI

SLI, LI, SLI

C61 LI, LI, LI

VLI, VLI, EI

VHI, HI, HI

VLI, VLI, EI

VLI, HI, EI

HI, EI, HI

SLI, LI, SLI

C62 SLI, LI, SLI

LI, LI, LI

LI, LI, LI

ALI, LI, EI VHI, VHI, VLI, VLI, VHI EI

HI, EI, HI

C63 ALI, LI, EI VLI, VLI, EI

HI, HI, HI

SLI, LI, SLI

VLI, HI, EI

VLI, VLI, EI

VHI, EI, HI

C64 SLI, LI, SLI

EI, EI, HI

VLI, VLI, EI

VLI, HI, EI

SLI, LI, SLI

HI, HI, HI

HI, HI, HI

C65 SLI, LI, SLI

VLI, EI, HI

EI, EI, HI

VLI, VLI, VLI

VLI, EI, HI

VLI, VLI, EI

HI, EI, HI

C66 VLI, EI, HI

SLI, LI, SLI

SLI, LI, SLI

SLI, SLI, SLI

SLI, LI, SLI

SLI, LI, SLI

VLI, VLI, EI

C71 HI, HI, HI

EI, AMI, AMI

VLI, EI, HI

SLI, SLI, SLI

VLI, VLI, VLI

HI, EI, HI

HI, HI, HI

C72 SLI, LI, SLI

VLI, VLI, EI

VLI, EI, HI

EI, AMI, AMI

VLI, VLI, EI

VHI, EI, HI

VLI, VLI, EI

C73 SLI, LI, SLI

SLI, LI, SLI

VLI, HI, EI

VLI, EI, HI

SLI, LI, SLI

VLI, EI, HI

EI, AMI, AMI

C74 EI, EI, HI

VLI, EI, HI

HI, EI, HI

VHI, EI, HI

HI, EI, HI

VLI, VLI, EI

HI, HI, HI

C75 VLI, HI, EI

VLI, HI, EI

HI, EI, HI

HI, HI, HI

VLI, EI, HI

VLI, VLI, EI

VHI, EI, HI

VHI, HI, HI

VHI, EI, HI

HI, HI, HI SLI, HI, EI

ALI, LI, EI VHI, VHI, VHI

μ

ν

π

μ

C3 ν

π

C4 μ

ν

π

C5 μ

ν

π

C6 μ

ν

π

C7 μ

ν

π

(continued)

C44 0.299 0.314 0.505 0.204 0.475 0.485 0.299 0.314 0.505 0.452 0.282 0.410 0.299 0.314 0.505 0.204 0.475 0.485 0.452 0.282 0.410

C43 0.452 0.282 0.410 0.452 0.282 0.410 0.299 0.314 0.505 0.299 0.314 0.505 0.452 0.282 0.410 0.452 0.282 0.410 0.452 0.282 0.410

C42 0.314 0.296 0.492 0.299 0.314 0.505 0.189 0.387 0.466 0.299 0.314 0.505 0.299 0.314 0.505 0.314 0.296 0.492 0.204 0.475 0.485

C41 0.299 0.314 0.505 0.208 0.472 0.490 0.204 0.475 0.485 0.452 0.282 0.410 0.338 0.260 0.530 0.393 0.224 0.477 0.170 0.715 0.381

C35 0.256 0.446 0.463 0.212 0.703 0.363 0.256 0.446 0.463 0.256 0.446 0.463 0.393 0.236 0.463 0.423 0.187 0.506 0.236 0.454 0.426

C34 0.183 0.575 0.359 0.261 0.442 0.469 0.261 0.442 0.469 0.393 0.236 0.463 0.256 0.446 0.463 0.375 0.258 0.478 0.256 0.446 0.463

C33 0.183 0.575 0.359 0.256 0.446 0.463 0.256 0.446 0.463 0.520 0.113 0.406 0.375 0.258 0.478 0.423 0.187 0.506 0.375 0.258 0.478

C32 0.256 0.446 0.463 0.142 0.802 0.268 0.566 0.218 0.364 0.566 0.218 0.364 0.261 0.442 0.469 0.212 0.703 0.363 0.256 0.446 0.463

C31 0.496 0.312 0.415 0.261 0.442 0.469 0.256 0.446 0.463 0.493 0.130 0.443 0.256 0.394 0.468 0.256 0.446 0.463 0.393 0.236 0.463

C26 0.280 0.441 0.428 0.280 0.441 0.428 0.286 0.437 0.434 0.280 0.441 0.428 0.411 0.249 0.437 0.233 0.701 0.334 0.463 0.173 0.467

C25 0.155 0.801 0.238 0.155 0.801 0.238 0.280 0.441 0.428 0.233 0.701 0.334 0.507 0.132 0.418 0.411 0.249 0.437 0.155 0.801 0.238

C24 0.280 0.441 0.428 0.543 0.304 0.364 0.155 0.801 0.238 0.539 0.109 0.392 0.155 0.801 0.238 0.280 0.441 0.428 0.411 0.249 0.437

C23 0.463 0.173 0.467 0.280 0.441 0.428 0.543 0.304 0.364 0.155 0.801 0.238 0.539 0.109 0.392 0.280 0.441 0.428 0.463 0.173 0.467

C22 0.280 0.441 0.428 0.280 0.441 0.428 0.155 0.801 0.238 0.411 0.249 0.437 0.654 0.057 0.304 0.411 0.249 0.437 0.411 0.249 0.437

C21 0.431 0.211 0.460 0.463 0.173 0.467 0.411 0.249 0.437 0.233 0.701 0.334 0.280 0.441 0.428 0.280 0.441 0.428 0.463 0.173 0.467

C15 0.201 0.536 0.457 0.201 0.536 0.457 0.112 0.826 0.270 0.334 0.376 0.499 0.186 0.542 0.424 0.310 0.400 0.461 0.112 0.826 0.270

C14 0.365 0.361 0.480 0.112 0.826 0.278 0.201 0.536 0.475 0.112 0.826 0.278 0.296 0.412 0.492 0.206 0.533 0.480 0.310 0.400 0.477

C13 0.410 0.348 0.424 0.201 0.536 0.475 0.471 0.343 0.393 0.201 0.536 0.475 0.310 0.400 0.477 0.296 0.412 0.492 0.410 0.348 0.424

C12 0.365 0.361 0.480 0.310 0.400 0.477 0.310 0.400 0.477 0.093 0.376 0.519 0.201 0.536 0.475 0.410 0.348 0.424 0.310 0.400 0.477

C11 0.471 0.343 0.387 0.144 0.636 0.362 0.388 0.354 0.445 0.112 0.826 0.270 0.206 0.533 0.462 0.201 0.536 0.457 0.206 0.533 0.462

C2

π

μ

ν

Tag C1

Table 7.3 The “spherical fuzzy decision matrix” in qualitative terms is expressed by three decision-makers, using A˜ s ⊗ D˜ s from definition

118 7 An Advanced TOPSIS-PFS Method to Improve Human System Reliability

μ

ν

π

μ

C3 ν

π

C4 μ

ν

π

C5 μ

ν

π

C6 μ

ν

π

C7 μ

ν

π

C75 0.344 0.279 0.453 0.344 0.279 0.453 0.424 0.384 0.427 0.454 0.328 0.385 0.360 0.258 0.436 0.234 0.456 0.441 0.452 0.168 0.413

C74 0.388 0.215 0.481 0.360 0.258 0.436 0.424 0.184 0.437 0.452 0.168 0.413 0.424 0.184 0.437 0.234 0.456 0.441 0.454 0.328 0.385

C73 0.217 0.464 0.401 0.217 0.464 0.401 0.344 0.279 0.453 0.360 0.258 0.436 0.217 0.464 0.401 0.360 0.258 0.436 0.548 0.141 0.335

C72 0.217 0.464 0.401 0.234 0.456 0.441 0.360 0.258 0.436 0.548 0.141 0.335 0.234 0.456 0.441 0.452 0.168 0.413 0.234 0.456 0.441

C71 0.454 0.328 0.385 0.548 0.141 0.335 0.360 0.258 0.436 0.260 0.610 0.438 0.130 0.804 0.250 0.424 0.184 0.437 0.454 0.328 0.385

C66 0.348 0.264 0.454 0.209 0.467 0.418 0.209 0.467 0.418 0.251 0.612 0.449 0.209 0.467 0.418 0.209 0.467 0.418 0.226 0.467 0.455

C65 0.209 0.467 0.418 0.228 0.390 0.445 0.375 0.222 0.497 0.125 0.805 0.263 0.348 0.264 0.454 0.226 0.459 0.455 0.410 0.192 0.455

C64 0.209 0.467 0.418 0.375 0.222 0.497 0.226 0.459 0.455 0.332 0.284 0.470 0.280 0.467 0.418 0.439 0.332 0.406 0.439 0.332 0.406

C63 0.231 0.455 0.461 0.226 0.459 0.455 0.188 0.708 0.357 0.209 0.467 0.418 0.332 0.284 0.470 0.226 0.459 0.455 0.436 0.178 0.433

C62 0.209 0.467 0.418 0.188 0.708 0.357 0.188 0.708 0.357 0.231 0.455 0.461 0.502 0.248 0.354 0.226 0.459 0.455 0.410 0.192 0.455

C61 0.188 0.708 0.357 0.226 0.459 0.455 0.461 0.166 0.396 0.226 0.459 0.455 0.332 0.284 0.470 0.410 0.192 0.455 0.209 0.467 0.418

C57 0.204 0.475 0.485 0.189 0.482 0.453 0.189 0.482 0.453 0.415 0.215 0.446 0.204 0.475 0.485 0.396 0.357 0.449 0.189 0.482 0.453

C56 0.189 0.482 0.453 0.396 0.357 0.449 0.189 0.482 0.453 0.452 0.282 0.410 0.189 0.482 0.453 0.370 0.236 0.495 0.452 0.282 0.410

C55 0.189 0.482 0.453 0.393 0.224 0.477 0.452 0.282 0.410 0.299 0.314 0.505 0.415 0.215 0.446 0.208 0.472 0.490 0.452 0.282 0.410

C54 0.393 0.224 0.477 0.170 0.715 0.381 0.415 0.215 0.446 0.393 0.224 0.477 0.208 0.472 0.490 0.415 0.215 0.446 0.314 0.285 0.524

C53 0.452 0.282 0.410 0.170 0.715 0.381 0.189 0.482 0.453 0.170 0.715 0.381 0.204 0.475 0.485 0.393 0.224 0.477 0.396 0.357 0.449

C52 0.170 0.715 0.381 0.393 0.224 0.477 0.299 0.314 0.505 0.204 0.475 0.485 0.452 0.282 0.410 0.452 0.282 0.410 0.396 0.357 0.449

C51 0.189 0.482 0.453 0.208 0.472 0.490 0.314 0.285 0.524 0.393 0.224 0.477 0.314 0.285 0.524 0.393 0.224 0.477 0.452 0.282 0.410

C46 0.204 0.475 0.485 0.299 0.314 0.505 0.415 0.215 0.446 0.452 0.282 0.410 0.189 0.482 0.453 0.204 0.475 0.485 0.452 0.282 0.410

C45 0.396 0.357 0.449 0.299 0.314 0.505 0.204 0.475 0.485 0.208 0.472 0.490 0.393 0.224 0.477 0.452 0.282 0.410 0.204 0.475 0.485

C2

π

μ

ν

Tag C1

Table 7.3 (continued)

7.4 Case Study 119

120

7 An Advanced TOPSIS-PFS Method to Improve Human System Reliability

Table 7.4 The result of SF values Tag C11

C1 0.538

C2 −0.306

C3 0.290

C4

C5

C6

C7

−0.470

−0.059

−0.064

−0.059

C12

0.226

0.085

0.119

−0.008

−0.062

0.351

0.119

C13

0.351

−0.162

0.534

−0.062

0.119

0.092

0.351

C14

0.226

−0.586

−0.062

−0.465

0.092

−0.056

0.119

−0.159

−0.470

C15

−0.064

C21

0.399

0.478

0.363

C22

0.068

0.029

−0.428

C23

0.478

0.029

0.800

C24

0.068

0.814

−0.428

C25

−0.428

−0.486

0.068

0.159

−0.083

0.123

−0.470

−0.196

0.068

0.068

0.478

0.363

1.328

0.363

0.363

−0.428

0.772

0.068

0.478

0.772

−0.428

0.068

0.363

−0.196

0.642

0.363

−0.428

C26

0.068

0.029

0.077

0.068

0.363

−0.196

0.478

C31

0.603

−0.014

0.032

0.576

0.051

0.032

0.307

C32

0.032

−0.512

0.902

0.902

0.040

−0.213

0.032

C33

−0.122

−0.023

0.032

0.693

0.261

0.349

0.261

C34

−0.122

−0.014

0.040

0.307

0.032

0.261

0.032

C35

0.032

−0.297

0.032

0.032

0.307

0.349

0.010

C41

0.116

−0.106

−0.027

0.483

0.169

0.300

−0.253

C42

0.143

0.093

−0.003

0.116

0.116

0.143

−0.027

C43

0.483

0.486

0.116

0.116

0.483

0.483

0.483

C44

0.116

−0.112

C45

0.303

0.093

−0.027

C46

−0.027

0.093

0.369

0.116

0.483

0.116

−0.027

0.483

−0.022

0.300

0.483

−0.027

0.483

−0.043

−0.027

0.483

C51

−0.043

−0.106

0.134

0.300

0.134

0.300

0.483

C52

−0.253

0.300

0.116

−0.027

0.483

0.483

0.303

C53

0.483

−0.043

−0.253

−0.027

0.300

0.303

−0.375

C54

0.300

−0.375

0.369

0.300

−0.022

0.369

0.134

C55

−0.043

0.300

0.483

0.116

0.369

−0.022

0.483

C56

−0.043

0.296

−0.043

0.483

−0.043

0.241

0.483

C57

−0.027

−0.128

−0.043

C61

−0.242

−0.069

0.522

C62

−0.022

−0.338

−0.242

0.369

−0.027

0.303

−0.043

−0.003

0.182

0.350

−0.022

0.003

0.678

−0.003

0.350

C63

0.003

−0.069

−0.242

−0.022

0.182

−0.003

0.429

C64

−0.022

0.250

−0.003

0.182

0.057

0.439

0.439

C65

−0.022

−0.021

0.251

−0.440

0.219

−0.003

0.350

−0.022

−0.073

−0.022

−0.022

−0.007

0.251

−0.063

−0.443

0.397

0.494

C66

0.219

−0.087

C71

0.494

0.844

(continued)

7.4 Case Study

121

Table 7.4 (continued) Tag

C1

C2

C72

−0.015

−0.054

C3 0.251

C4

C5

C6

C7

0.861

0.006

0.484

0.006

C73

−0.015

−0.072

0.210

0.251

−0.015

0.251

0.861

C74

0.286

0.246

0.397

0.484

0.397

0.006

0.494

C75

0.210

0.202

0.375

0.494

0.251

0.006

0.484

SF-NIS

0.603

SF-PIS

−0.428

0.844

0.902

−0.586

−0.470

0.902

1.328

0.484

0.861

−0.470

−0.443

−0.213

−0.470

Table 7.5 The SF-NIS and SF-PIS Tag

C1

C2

C3

C4

C5

C6

C7

SF-NIS

(0.155, 0.801, 0.238)

(0.112, 0.826, 0.278)

(0.112, 0.826, 0.270)

(0.112, 0.826, 0.270)

(0.130, 0.804, 0.250)

(0.212, 0.703, 0.363)

(0.112, 0.826, 0.270)

SF-PIS

(0.496, 0.312, 0.415)

(0.543, 0.304, 0.364)

(0.566, 0.218, 0.364)

(0.566, 0.218, 0.364)

(0.654, 0.057, 0.304)

(0.452, 0.168, 0.413)

(0.548, 0.141, 0.335)

Table 7.6 The normalized Euclidean distance between the X i and SF-NIS/SF-PIS

Tag

SF-NIS

SF-PIS

C11

0.2408

0.2743

C12

0.3503

0.2102

C13

0.3445

0.1935

C14

0.2267

0.3096

C15

0.2035

0.3157

C21

0.3636

0.1767

C22

0.3460

0.1556

C23

0.3576

0.1617

C24

0.2822

0.2117

C25

0.1910

0.3041

C26

0.3230

0.2010

C31

0.3762

0.1683

C32

0.2884

0.2193

C33

0.3750

0.1742

C34

0.3302

0.2147

C35

0.3213

0.2155

C41

0.3430

0.2117

C42

0.3734

0.2040 (continued)

122

7 An Advanced TOPSIS-PFS Method to Improve Human System Reliability

Table 7.6 (continued)

Tag

SF-NIS

SF-PIS

C43

0.4131

0.1179

C44

0.3758

0.1820

C45

0.3638

0.1928

C46

0.3644

0.1785

C51

0.3883

0.1755

C52

0.3526

0.1907

C53

0.2906

0.2352

C54

0.3685

0.1875

C55

0.3849

0.1631

C56

0.3567

0.1758

C57

0.3221

0.2234

C61

0.3226

0.2056

C62

0.2794

0.2239

C63

0.3029

0.2218

C64

0.3567

0.1765

C65

0.3110

0.2187

C66

0.2872

0.2374

C71

0.3304

0.1735

C72

0.3525

0.1627

C73

0.3536

0.1496

C74

0.4089

0.1276

C75

0.3873

0.1356

7.5 Conclusion Human reliability is one of the critical parts of system safety and reliability analysis. This is considered the main challenge for system safety due to human cognitive behavior and subjective uncertainty. In the present chapter, the spherical fuzzy set, with its excellent capability to deal with subjective uncertainty, improves human system reliability. The existing literature supports the spherical fuzzy set as a reliable trend to extend MCDM methods in a fuzzy environment. Thus, the number of publications has increased over time. The TOPSIS method as an effective and robust MCDM tool is utilized and extended underlying the spherical fuzzy set to investigate the human system reliability. It helps to identify the critical layers and justifying them by suggesting adequate intervention actions. This advanced TOPSIS method extended with a spherical fuzzy set could provide reliable results to improve human system reliability significantly.

7.5 Conclusion Table 7.7 The CR is determined from normalized Euclidean distance

123 Tag

CR

C11

0.4675

C12

0.6250

C13

0.6403

C14

0.4227

C15

0.3919

C21

0.6730

C22

0.6898

C23

0.6887

C24

0.5714

C25

0.3858

C26

0.6164

C31

0.6909

C32

0.5680

C33

0.6829

C34

0.6060

C35

0.5985

C41

0.6184

C42

0.6466

C43

0.7780

C44

0.6736

C45

0.6536

C46

0.6712

C51

0.6887

C52

0.6490

C53

0.5527

C54

0.6627

C55

0.7023

C56

0.6698

C57

0.5905

C61

0.6108

C62

0.5552

C63

0.5773

C64

0.6689

C65

0.5871

C66

0.5475

C71

0.6557

C72

0.6842 (continued)

124

7 An Advanced TOPSIS-PFS Method to Improve Human System Reliability

Table 7.7 (continued)

Tag

CR

C73

0.7027

C74

0.7621

C75

0.7406

References 1. Zhu, S.-P., Huang, H.-Z., Peng, W., Wang, H.-K., Mahadevan, S.: Probabilistic physics of failure-based framework for fatigue life prediction of aircraft gas turbine discs under uncertainty. Reliab. Eng. Syst. Saf. 146, 1–12 (2016). https://doi.org/10.1016/j.ress.2015. 10.002 2. Zhu, S.-P., Liu, Q., Lei, Q., Wang, Q.: Probabilistic fatigue life prediction and reliability assessment of a high pressure turbine disc considering load variations. Int. J. Damage Mech. 27, 1569–1588 (2017). https://doi.org/10.1177/1056789517737132 3. Rausand, M., Haugen, S.: Risk Assessment: Theory, Methods, and Applications. Wiley (2020) 4. Yazdi, M., Golilarz, N.A., Nedjati, A., Adesina, K.A.: An improved lasso regression model for evaluating the efficiency of intervention actions in a system reliability analysis. Neural Comput. Appl. (2021). https://doi.org/10.1007/s00521-020-05537-8 5. Zarei, E., Yazdi, M., Abbassi, R., Khan, F.: A hybrid model for human factor analysis in process accidents: FBN-HFACS. J. Loss Prev. Process Ind. 57, 142–155 (2019). https://doi. org/10.1016/j.jlp.2018.11.015 6. Bakioglu, G., Atahan, A.O.: AHP integrated TOPSIS and VIKOR methods with Pythagorean fuzzy sets to prioritize risks in self-driving vehicles. Appl. Soft Comput. 99 (2021). https://doi. org/10.1016/j.asoc.2020.106948 7. Jozaghi, A., Alizadeh, B., Hatami, M., Flood, I., Khorrami, M., Khodaei, N., Tousi, E.G.: A comparative study of the AHP and TOPSIS techniques for dam site selection using GIS: a case study of Sistan and Baluchestan Province, Iran. Geosciences 8, 1–23 (2018). https://doi.org/ 10.3390/geosciences8120494 8. Liu, X., Zhou, X., Zhu, B., He, K., Wang, P.: Measuring the maturity of carbon market in China: an entropy-based TOPSIS approach. J. Clean. Prod. 229, 94–103 (2019). https://doi. org/10.1016/j.jclepro.2019.04.380 9. Büyüközkan, G., Çifçi, G.: A novel hybrid MCDM approach based on fuzzy DEMATEL, fuzzy ANP and fuzzy TOPSIS to evaluate green suppliers. Expert Syst. Appl. 39, 3000–3011 (2012). https://doi.org/10.1016/j.eswa.2011.08.162 10. Song, W., Ming, X., Wu, Z., Zhu, B.: Failure modes and effects analysis using integrated weight-based fuzzy TOPSIS. Int. J. Comput. Integr. Manuf. 26, 1172–1186 (2013). https://doi. org/10.1080/0951192X.2013.785027 11. Vahdani, B., Mousavi, S.M., Tavakkoli-Moghaddam, R.: Group decision making based on novel fuzzy modified TOPSIS method. Appl. Math. Model. 35, 4257–4269 (2011). https://doi. org/10.1016/j.apm.2011.02.040 12. Ramya, S., Devadas, V.: Integration of GIS, AHP and TOPSIS in evaluating suitable locations for industrial development: a case of Tehri Garhwal district, Uttarakhand, India. J. Clean. Prod. 238, 117872 (2019). https://doi.org/10.1016/j.jclepro.2019.117872 13. Jiang, G.-J., Chen, H.-X., Sun, H.-H., Yazdi, M., Nedjati, A., Adesina, K.A.: An improved multi-criteria emergency decision-making method in environmental disasters. Soft Comput. (2021). https://doi.org/10.1007/s00500-021-05826-x 14. Yazdi, M., Korhan, O., Daneshvar, S.: Application of fuzzy fault tree analysis based on modified fuzzy AHP and fuzzy TOPSIS for fire and explosion in the process industry. Int. J. Occup. Saf. Ergon. 26, 319–335 (2020) 15. Yazdi, M.: Risk assessment based on novel intuitionistic fuzzy-hybrid-modified TOPSIS approach. Saf. Sci. 110, 438–448 (2018). https://doi.org/10.1016/j.ssci.2018.03.005

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16. Li, J., Fang, H., Song, W.: Sustainable supplier selection based on SSCM practices: a rough cloud TOPSIS approach. J. Clean. Prod. 222, 606–621 (2019). https://doi.org/10.1016/j.jcl epro.2019.03.070 17. Selim, H., Yunusoglu, M.G., Yilmaz Balaman, S.: ¸ A dynamic maintenance planning framework based on fuzzy TOPSIS and FMEA: application in an international food company. Qual. Reliab. Eng. Int. 32, 795–804 (2016). https://doi.org/10.1002/qre.1791 18. Liu, H.C., You, J.X., Shan, M.M., Shao, L.N.: Failure mode and effects analysis using intuitionistic fuzzy hybrid TOPSIS approach. Soft Comput. 19, 1085–1098 (2015). https://doi.org/ 10.1007/s00500-014-1321-x 19. Ramezani, M., Bashiri, M., Atkinson, A.C.: A goal programming-TOPSIS approach to multiple response optimization using the concepts of non-dominated solutions and prediction intervals. Expert Syst. Appl. 38, 9557–9563 (2011). https://doi.org/10.1016/j.eswa.2011.01.139 20. Ye, F.: An extended TOPSIS method with interval-valued intuitionistic fuzzy numbers for virtual enterprise partner selection. Expert Syst. Appl. 37, 7050–7055 (2010). https://doi.org/ 10.1016/j.eswa.2010.03.013 21. Xu, Z., Zhang, X.: Hesitant fuzzy multi-attribute decision making based on TOPSIS with incomplete weight information. Knowl Based Syst. 52, 53–64 (2013). https://doi.org/10.1016/ j.knosys.2013.05.011 22. Wu, Z., Xu, J., Jiang, X., Zhong, L.: Two MAGDM models based on hesitant fuzzy linguistic term sets with possibility distributions: VIKOR and TOPSIS. Inf. Sci. (Ny) 473, 101–120 (2019). https://doi.org/10.1016/j.ins.2018.09.038 23. Beg, I., Rashid, T.: TOPSIS for hesitant fuzzy linguistic term sets. Int. J. Intell. Syst. 28, 1162–1171 (2013). https://doi.org/10.1002/int.21623 24. Onar, S.C., Öztaysi, B., Kahraman, C.: Multicriteria evaluation of cloud service providers using Pythagorean fuzzy TOPSIS. J. Mult. Valued Log. Soft Comput. 30, 263–283 (2018) 25. Sajjad Ali Khan, M., Abdullah, S., Yousaf Ali, M., Hussain, I., Farooq, M.: Extension of TOPSIS method base on Choquet integral under interval-valued Pythagorean fuzzy environment. J. Intell. Fuzzy Syst. 34, 267–282 (2018). https://doi.org/10.3233/JIFS-171164 26. Elhassouny, A., Smarandache, F.: Neutrosophic-simplified-TOPSIS multi-criteria decisionmaking using combined simplified-TOPSIS method and neutrosophics. In: 2016 IEEE International Conference on Fuzzy Systems, 2016: pp. 2468–2474. https://doi.org/10.1109/FUZZIEEE.2016.7738003 27. Biswas, P., Pramanik, S., Giri, B.C.: TOPSIS method for multi-attribute group decision-making under single-valued neutrosophic environment. Neural Comput. Appl. 27, 727–737 (2016). https://doi.org/10.1007/s00521-015-1891-2 28. Gündo˘gdu, F.K.: Principals of spherical fuzzy sets. In: Kahraman, C., Cebi, S., Cevik Onar, S., Oztaysi, B., Tolga, A.C., Sari, I.U. (eds.) BT—intelligent and fuzzy techniques in Big Data analytics and decision making, pp. 15–23. Springer International Publishing, Cham (2020) 29. Kutlu Gündo˘gdu, F., Kahraman, C.: A novel fuzzy TOPSIS method using emerging intervalvalued spherical fuzzy sets. Eng. Appl. Artif. Intell. 85, 307–323 (2019). https://doi.org/10. 1016/j.engappai.2019.06.003 30. Kutlu Gündo˘gdu, F., Kahraman, C.: Optimal site selection of electric vehicle charging station by using spherical fuzzy TOPSIS method. In: Kahraman, C., Kutlu Gündo˘gdu F. (eds.) BT— decision making with spherical fuzzy sets: theory and applications, pp. 201–216. Springer International Publishing, Cham (2021). https://doi.org/10.1007/978-3-030-45461-6_8 31. Kutlu Gündo˘gdu, F., Kahraman, C.: Spherical fuzzy sets and spherical fuzzy TOPSIS method. J. Intell. Fuzzy Syst. 36, 337–352 (2019). https://doi.org/10.3233/JIFS-181401 32. Ejegwa, P.A.: Modified Zhang and Xu’s distance measure for Pythagorean fuzzy sets and its application to pattern recognition problems. Neural Comput. Appl. 32, 10199–10208 (2020). https://doi.org/10.1007/s00521-019-04554-6 33. Liu, X., Liu, Z., Chen, P.Q., Xie, Z.Y., Lai, B.J., Zhan, B., Lao, J.R.: Human reliability evaluation based on objective and subjective comprehensive method used for ergonomic interface design. Math. Probl. Eng. 2021 (2021). https://doi.org/10.1155/2021/5560519

Chapter 8

Stochastic Game Theory Approach to Solve System Safety and Reliability Decision-Making Problem Under Uncertainty

8.1 Introduction Increasing industrial accidents, particularly in the hydrocarbon and petrochemicalbased sectors, highlights that the existing system safety and reliability analysis methods have not succeeded in predicting system failure by improving system safety performance. Consequently, the system safety has faced several catastrophic accidents with a considerable amount of loss. As an example, the fire and explosion accident at Bouali Sina petrochemical plant (2016) resulted in approximately $108 million to resolve the accident losses, including assets, environmental, revenues, and reputation loss of the system [1]. One can say that the effectiveness of system safety and reliability analysis methods in predicting the failure and preventing and mitigating it from an accident accurately are questionable and controversial. This demands further attempts to understand how we can improve the efficiency and effectiveness of these system safety and reliability analysis methods. In this regard, decision-making methods are growing by introducing many MCDM (multi-criteria decision-making) tools and extensions to the system’s safety performance over time, such as TOPSIS (Technique for Order of Preference by Similarity to Ideal Solution) proposed Ching-Lai Hwang and Yoon in 1981. A couple of extensions to the TOPSIS method are integrating with the rough cloud approach [2], fuzzy set theory [3, 4], Pythagorean Fuzzy Sets [5], intuitionistic fuzzy hybrid [6, 7], and goal programming-based TOPSIS [8]. Furthermore, AHP is a structured technique for organizing and analyzing complex decisions based on mathematics and psychology. Saaty developed AHP, and a couple of extensions of the AHP method are integrating it with D numbers [9], Neutrosophic AHP-Delphi Group decision-making [10], fuzzy set theory [11, 12], interval-valued intuitionistic fuzzy sets [13], and Combining Fuzzy AHP and Fuzzy TOPSIS [14, 15]. DEMATEL (decision-making and evaluation laboratory) was first developed by the Geneva Research Centre of the Battelle Memorial Institute [16] to visualize the structure of complicated causal relationships through matrixes or digraphs [17]. Extensions of the DEMATEL method include © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 H. Li and M. Yazdi, Advanced Decision-Making Methods and Applications in System Safety and Reliability Problems, Studies in Systems, Decision and Control 211, https://doi.org/10.1007/978-3-031-07430-1_8

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integrating the Bayesian network [18, 19], Intuitionistic fuzzy set, [20], Pythagorean Fuzzy Sets [21], D numbers [22], and a gray-based appeached [23]. The Preference Ranking Organization METHod for Enrichment of Evaluations (PROMETHEE) was developed at the beginning of the 1980s by Jean-Pierre Brans and has been extensively studied and refined since then. Extensions of the PROMETHEE method are the hesitant fuzzy linguistic information [24], Pythagorean Fuzzy Information [25], probabilistic Linguistic Environment [26], and cloud model theory [27]. Serafim Opricovic originally developed the VIKOR method to solve conflicting decision problems and into commensurable [28]. A couple of extensions of the VIKOR method are with 2-dimension uncertain linguistic information [29], Pythagorean fuzzy set [30], entropy measure [31], spherical fuzzy sets [32], interval-valued intuitionistic fuzzy information [33], and perspective of regret theory [34]. The best worst method (BWM) is proposed as a proper alternative to AHP to minimize the comparisons and improve the consistency ratio [35, 36]. A couple of extension of the BWM is integrating the Bayesian network [37], intuitionistic fuzzy set [38, 39], Z-number [40], hesitant fuzzy linguistic terms [41], intuitionistic fuzzy preference relations [42], a group multi-criteria decision-making based on the best–worst method [43], fuzzy weighted averaging operator [44], interval-valued fuzzy-rough numbers best worst method [45], and possibility degree with probabilistic linguistic information [46]. However, most MCDM tools are normative, and decision makers often prefer using rational statements to find solutions. This would result in different MCDM tools by revealing the subjectivity of justice and fairness in the specific type of decision-making problem. The final results depend on using MCDM tools, as no method can guarantee reliable results and decision-maker’s preferences [47–49]. Thus, such normative MCDM tools assume that there would be perfect cooperation between the decision-makers, and they would determine the non-dominated as Pareto optimal solutions for the decision-making problems. According to this point, these methods cannot be suitable when the decision-makers have conflicts, and there is no perfect cooperation between the decision-makers. Hence, the solution results do not necessarily non-dominated. In order to better understand and gain insight into the system safety and reliability analysis decision-making problem, the “Game theory” concept has enough potential to study the “strategic actions” from the decision-makers to obtain much more reliable, efficient, and effective results. Like MCDM tools, researchers have been attempted to integrate “Game theory” in the system safety and reliability analysis by highlighting that the “Game theory” model has considerable advantages rather than conventional system safety and reliability techniques, including but not limited to [50–61]. The three examples are, the dependent failures at the early life of a system are analyzed by cooperative “Game theory” [62]. Gao et al. assessed the impact of decision-making errors on strategies of software detection [53]. In another study, urban pipeline accidents are analyzed by integration of “Game theory” approaches and safety engineering systems [60]. The potential merits of game theory compared to the MCDM tools are: (i) its capabilities for reflecting and addressing several socioeconomic, engineered, and policy features of system safety and reliability analysis

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decision-making problem, similar to environmental resources [63–65], and (ii) it does not require a piece of descriptive information as quantitative or objective data. The contribution of the present work is suggesting and utilizing an approach to solve the system safety and reliability analysis decision-making problem to deal with the uncertainty. A conventional MCDM problem is mapped into the game-theoretic context. Then, using Monte-Carlo simulation, stochastic game theory is used to better understand the decision-making problem than common MCDM tools. Finally, the utilized approach can help decision-makers predict the system failure precisely and improve the system safety performance by developing the effectiveness and efficiency of intervened actions, including mitigative, preventive, and control actions. It should be added that the idea behind the present study comes from the study of Dr. Kaveh Madani [63], as “A Monte-Carlo game theoretic approach for Multi-Criteria Decision Making under uncertainty” It then provides insight and motivates us to develop the game-theoretic context in the SSRA. The organization of the paper is constructed as the following. In Sect. 8.2, the Preliminary of MCDM as a Game theory is explained. In Sect. 8.3, real system safety and reliability-based examples such as the Bouali Sina fire accident decisionmaking problem are studied. In Sect. 8.4, the stochastic game theory of the Bouali Sina fire accident decision-making problem is developed and examined. In the last Section, a conclusion of the study is provided by highlighting the challenge of current work and the direction for future research tasks.

8.2 Preliminary: MCDM as a Game Theory A single MCDM problem with m and n number of alternatives and criteria respectively can be represented in cardinal form as the following: MC D MCar dinal = [Pmn ]m×n

(8.1)

where Pi j indicates the performance of ith alternative underlying the jth criterion for i = 1, 2, 3, . . . , m and j = 1, 2, 3, . . . n. A similar matrix would be used where there is a group of decision-makers (k DMs), in which a single DM has a criterion, meaning that n equal to k(n = k). In this case, the Pi j is substituted with Ui j , the utility of jth DM among a set of alternatives can be represented in cardinal form as the following: MC D MCar dinal = [Umn ]m×k

(8.2)

for i = 1, 2, 3, . . . , m and j = 1, 2, 3, . . . n. Both Eqs. 8.1 and 8.2 are the same in the concepts and computations process; therefore, they can mathematically be treated in the same way [66]. For the general format of MCDM problems with the m number of alternatives and k DMs, if a single DM has n q criteria, the MCDM matrix can be developed into Eq. 8.3.

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MC D MCar dinal = [Pmnk ]m×l

(8.3)

where Pi jq is the performance of ith alternative underlying the jth criterion of DM (player) q, for i = 1, 2, 3, . . . , m and j = 1, 2, 3, . . . n q , q = 1, 2, . . . k, and l = k q=1 nq. According to Eq. 8.3, it is suggested that a decision-making problem with a group of decision-makers can mainly be solved as a decision-making problem with a single decision-maker, in which there is l number of DMs where every single DM has only a criterion. In order to solve such decision-making problems using the available decision-making tools, the procedure would be straightforward since the criteria are evaluated within the equal importance weight. However, it is not practical circumstances in realistic decision-making as all decision-makers have a different quality profile and therefore may have different influential powers. In addition, the criteria also may not have an equal importance weight. Thus, the highlighted differences would make the solution to the decision-making problem less reliable using the existing decision-making tools because the uncertainty increases by variation of criteria and decision-makers’ importance weights. In this regard, reliable solutions can be obtained by modeling and studying Eq. 8.3 as a strategic game in the concept of Game theory. Game theory can handle the MCDM problems [64]. The strategic game (i.e., conflict) is defined as a decision condition that contains more than an independent DM. This means that the individual choices can determine the possible outcomes, having the individual preferences over the possible conflicts’ outcome [67]. Thus, the non-cooperative game is suggested to study such decision-making problems by utilizing the ordinal ranking instead of the classical cardinal one (i.e., performance ranking) [64, 68]. Therefore, in case the DMs are uncertain about sharing the performance of the alternatives, the decision-making problem can be changed into the ordinal form. It can be derived by substituting Eqs. 8.1 and 8.2 with the following Equation. MC D M Or dinal = [Rmn ]m×n

(8.4)

Besides, the general decision-making matrix, Eq. 8.3 can be replaced by Eq. 8.5 as the following. MC D M Or dinal = [Rmnk ]m×l

(8.5)

where Ri j is the ranking of ith alternative regarding the jth criterion, and Ri jq is the ranking of ith alternative regarding the jth criterion of decision-maker (player) q in his point of view. The fact is that soliciting the ordinal alternatives’ ranking from DMs would be a less challenging task rather than cardinal information. If there is a piece of ordinal information, the outputs are much less sensitive to uncertainties based on the provided

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information by DMs. Therefore, the outcomes are not sensitive to performance variation since the same ranking. Moreover, another advantage of considering MCDM as a strategic game is the lack of necessity for the importance weights of DMs and criteria, reducing the uncertainties in the outcomes. The fundamental elements of any MCDM problems are (i) criteria, (ii) alternatives, and (iii) the performance of alternatives for every criterion. The elements of MCDM problems correspond to the fundamental elements of the strategic game as (i) players, (ii) strategies, and (iii) players’ payoffs from possible outcomes. The outcomes for the possible combinations of players’ strategies are illustrated in Fig. 8.1. The common MCDM tools only focus on group decisions, and an “alternative” in the context of MCDM would be the possible corporative result. It is considered that there is excellent cooperation between all parties, allowing an agreement on an alternative and possible cooperative outcome and disregarding the non-cooperative situation when the parties may not agree. The latter causes the non-cooperative be the overall outcome. For example, there are two different industrial workshops close to each other within a shared power transformer. In each workshop, a couple of evacuation systems are available and based on the workload, and they can be turned on, such as split coolers, fans, and chillers. In the provided FTA (fault tree analysis), the top event (TE) is the failure of the power transformer. The system failure occurs workshops’ workload is high, and then all of their evacuation systems are working (AND logical gate). Both workshops have been working with shared power transformers for a long-time (let us say over 20 years). The profit payoff for each workshop is their revenues from selling their production minus employees’ salaries, raw materials, and electricity. The MCDM tools can only consider the two possible outcomes, each occurring once both parties agree on the same alternatives. Thus, the MCDM problem is conventionally defined as the 2 * 2 matrix, which provides a return of two workshops from two outcomes/alternatives. The non-cooperative game theory considers more comprehensive views for the decision-making problems by developing a set of feasible results, including both non-cooperative and cooperative outcomes. Under the non-cooperative game, there is no pressure and enforcement Fig. 8.1 Mapping MCDM problems into game theory problems

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on the DMs to select cooperative outcomes. The game theory considers that all parties are somehow “self-optimizers” and trying to maximize their benefits considering different constraints imposed from other DMs’ decisions and actions [69, 70]. Therefore, the structure of decision-making problems may lead to non-cooperative, and some DMs prefer cooperation. In order to convert an MCDM problem into its classical form as a 2 * 2 matrix and to a game theory form, a transition matrix is required, which contains both cooperative and non-cooperative outcomes. This matrix includes all possible combinations of every DMs’ strategy and provides the information derived from all DMs. For the problems of the workshop with two parties (player 1, player 2), each player has two alternatives: {Low Electricity Load (LEL), and High Electricity Load (HEL)), and four possible outcomes: {LEL-LEL, LEL, HEL, HEL, LEL, HEL, HEL}. Accordingly, the transition matrix for this decision-making problem is an ordinal form, as illustrated in Fig. 8.2 (left-hand side). It is a four-two-based matrix indicating the payoffs of two players from four possible outcomes. Each row and column denote an outcome and player, respectively. This, the numbers in column 1, shows the player one payoff from the four possible outcomes, and numbers in column 2 denote the player 2 payoffs from the four possible outcomes. This matrix is based on the 2 * 2 industrial workshops game and presented on the right-hand side of Fig. 8.2. To simulate the behaviors of DMs in the game, it is necessary to estimate how the game is playing and find out the possible outcomes (equilibria) of the games. The game’s models can be applied by stability definitions, reflecting different kinds of people at different levels of risks, attitudes, and opponents’ preferences’ knowledge. Games with discrete strategies like industrial workshops can be analyzed using definitions of non-cooperative, including (i) “Nash stability” [71], (ii) “General Metarationality” [69], (iii) “Symmetric Meta-rationality”, (iv)“Sequential Stability” [70], (v) “Non-Myopic Stability”, (vii) and limited move stability [72, 73]. Different applications highlighted that these stability definitions could predict the outcomes of MCDM problems in a game theory framework. The characteristics features of these stability definitions are already discussed in the existing literature, such as [74], which reviewed the characteristics of the non-cooperative stability definitions. In addition, it is highlighted the “utility finding resolution”. The six mentioned stability definitions can be characterized according to the following features [68]: (i) Foresight meaning the “number of moves or degree of reflection, this is the total number of moves and countermoves considered by the DM before unilaterally changing his decision. The first move is always assumed to be the DM’s departure from a given

Fig. 8.2 An example of industrial workshops, mapping MCDM (left-hand-side) into the game theory (right-hand-side) problems

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Fig. 8.3 The definitions of stability and behavior od decision-makers

state (outcome). The second move is the reaction by an opponent”, (ii) Willingness to dis-improve meaning that the “dis-improvement is a unilateral move to a state which is less preferred than the current state; based on different stability definitions, different DMs might be willing to make dis-improvements during the game”, and (iii) Knowledge of preferences meaning that “a decision-maker in the game might only be aware of his preferences or may be aware of the preferences of all DMs in the game”. A qualitative comparison of the mentioned six non-cooperative stability definitions is presented in Fig. 8.3. It illustrates how the six stability definitions could present different types of DMs’ behavior within diverse characters, reflecting a wide range of DMs’ behaviors in conflict circumstances, from cautious and conservative to strategic and proactive. The six mentioned stability definitions comprehensively explained in the literature; one can refer to the [68, 75, 76]. As the characteristics of DMs are taken into account in the decision-making resolution process besides their preferences matrices in the non-cooperative stability definitions game, the game theory” can adequately reflect the behaviors of all players in the game, which typically ignored in the common MCDM tools [64]. Therefore, the results obtained from game theory would be much closer to practical situations. In the MCDM tools, the effect of the behavior of DMs on the outcomes is neglected. Game theory outcomes may not be “Pareto optimal solution”, as it also assumes that the DMs’ possible non-cooperative behavior is leading to “Pareto inferior” outcomes in the context of group decision-making [63].

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To find all possible game outcomes, a stability analysis is conducted. Once all DMs obtain an outcome be stable under stability definition, the game’s outcome would be equilibrium underlying that stability definition. An outcome (state) is stable for a given player underlying provided stability definition if the player finds out changing a state is an incorrect act. All stability definitions reflect the specific DM behavior. Therefore, to make a better decision-making simulation with a group of DMs and increase the reliability of outcomes, mainly when there is a lack of data, and enough information concerning DMs’ behaviors characteristics, a range of stability definitions are suggested. Finally, an outcome with is the equilibrium under more solutions has a greater chance of being the final game outcome.

8.3 The Bouali Sina Fire Accident Decision-Making Problem A fire accident occurred in 2016 at Bouali Sina petrochemical plant (“Bouali Sina is also known Ibn Sina, Abu Ali Sina, Avicenna, Pour Sina, and was a Persian polymath”). The fire was started by leaking the hydrocarbon into the environment because the “blind flange gasket” was a rupture in the pump line (P-8001A) [77]. The history of the accident is the highlight as the following: “The fire started around 17:15 and escalated quickly than ever imagined. The workers in Unit 900 discovered the fire and evacuated the scene immediately. Pumps P-8001A/B/C are responsible for transporting the mixture of xylene liquids into the furnace H-8001. The mechanical seal of the pump P-8001A was reported to have broken down a few weeks after the accident and was sent to the workshop for repair and overhauling. However, two blinded plates were put on the line to improvise the seal to keep the plant operating. The investigation equally captured that at about or around 7 am, the workers at the para-xylene unit reported to the control room when there was a hydrocarbon leakage from the pump’s mechanical seal (P-8001C). The pump was isolated, and a manual gearbox was adapted and managed instead of the pump (P-8001C). With this, there was pressure built up beyond the capacity of the remaining pump (P-8001B). During the manual transferring gearbox to the location at 5:15 PM, the blind flange gasket installed on the pump line (P-8001A) has been ruptured and was leaking the hydrocarbon to the environment at a pressure and temperature of 10.5 bar and 300 °C, respectively. The vaporized hydrocarbon reached the ignition source of the furnace (H-8001), located 10 m north of the leakage point leading to the ignition and burning around that area. Para-xylene is an aromatic hydrocarbon, a highly flammable and toxic compound with a boiling point of more than 138 °C, a non-volatile compound having its vapor heavier than air and highly explosive. The combustion led to the production of a large cloud of combustible gases, which diffused through the area and re-ignited into further burning with a large pool of fire. Figures 8.4 and 8.5 depict the happenings within the furnace (H-8001), as explained above, with how it got through the fluid line into the Tower (8001). The continuous flow of the fluid line aggravated

8.3 The Bouali Sina Fire Accident Decision-Making Problem

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Fig. 8.4 The performance range of intervention actions under two risk factors

Fig. 8.5 The transition matrix for the Bouali Sina petrochemical plant decision-making problem

the burning beyond the capacity of the firefighters. The hot insulation cover of the tower to the open/floating roof of the naphtha tanks (TK-2001A/C) was blown off by the wind, and the released gases resulted in the further explosion at the naphtha tank [77]. The extent of the damage reported by an Iranian insurance company was estimated to revolve around 94 million euros and was said to be the most serious fire accident in the history of the Iranian petrochemical industry [78]”. The schematic history of the Bouali Sina petrochemical plant accident is provided in published works. In the literature, two studies were conducted toward learning the different aspects of this fire accident [1, 79]. In [1], the authors explained how to recognize and deal with different uncertainties of this process safety system, especially this fire accident. In another study [79], authors utilized a complicated mathematical LASSO (most minor absolute shrinkage and selection operator) to prevent this fire accident’s failure probability correctly. Therefore, the proposed methodology can be used for

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similar targets. However, on-site decision-makers prefer to follow specific strategies or rules (alternative) to improve system performance in a more practical situation. In this study, we are assumed that the Bouali Sina petrochemical plant as a system has only four alternatives (intervention actions) to deal with the potential risk of installed component (such as P-8001A, B, and C), underlying two risk factors, Likelihood (the probability of occurrence of an undesired event, in this case, fire), and cost (cost of intervention actions plus the cost of loss after the accident) [80, 81]. The four intervention actions are (i) Prevents actions (“those actions or measures are taken to prevent injuries before they happen”), (ii) Mitigative actions (“a mitigation action is a specific action, project, activity, or process taken to reduce or eliminate long-term risk to people and property from hazards and their impacts”, (iii) Control actions (“reference input (desired value), determines the deviation, and produces a control signal, that will reduce the deviation to zero or a small value”), and No actions till failure (after failure (accident) repair or replace actions might be conducted). In addition, it is assumed that two main decision-makers (parties) are playing their roles in this MCDM problem, (i) the safety department and (ii) the operating department. It is also assumed that both departments have several conflicts in the work tasks. In this study, players, parties, and decision-makers are identical. Figure 8.4 summarizes the performance of each intervention action under two risk factors for selecting the future intervention actions to deal with the potential risk of an installed component. In the study conducted by Yazdi et al. [79], it is highlighted that the Bouali Sina petrochemical plant fire accident has other shortages in the applied intervention actions. Therefore, we assumed that the efficacy of intervention actions’ performance is ideal in the present work due to seeking simplicity. This paper only focuses on the intervention actions, which are applicable and listed in the previously published works. Therefore, the aim is to determine the best intervention actions among four possible alternatives suggested by Yazdi et al. Resolving the complicated and multi-aspects of Bouali Sina petrochemical plant decision-making problem conflicts is not the objective of this paper. It is discussed that some parties in the system may not be interested in developing a cooperative to resolve the Bouali Sina petrochemical plant decision-making problem. The performance of each single intervention action under two risk factors involves considerable uncertainty, reflected by large performance ranges. The performance ranges of intervention actions have been approximated through an analysis containing decision trees and surveys, as discussed in [82]. The current uncertainty in performance ranges causes a more challenging task to select feasible and optimum intervention actions. Therefore, it is necessary to have a reliable method that suggests a decision to estimate the outcomes of decision-making problems with uncertain performance. To the best of the authors’ understanding of decision-making problems and supports in literature [83–86], it is believed that the “Monte-Carlo Game Theory” provides a better reflection of and valuable insights into the decision-making problems. In the Bouali Sina petrochemical plant decision-making problem, the best intervention actions would have lower risk levels, meaning that the lowest likelihood and cost. However, there is always a tradeoff between each pair of intervention actions, with no alternative being the best under both risk factors. In order to define

8.3 The Bouali Sina Fire Accident Decision-Making Problem

137

the Bouali Sina petrochemical plant decision-making problem in a matrix form, the performance values are needed. Table 8.1 provides performance ranges of intervention actions within uncertainty for each intervention action under corresponding risk factors. One way to simplify the computation process is averaging the interval values, in which the stochastic decision-making problem is translated into a deterministic decision-making problem. In this regard, the cardinal and ordinal form of the Bouali Sina petrochemical plant decision-making problem is derived as the following. ⎡

MC D MCar dinal

1.05 ⎢ 0.6 =⎢ ⎣ 0.7 2.25

⎤ 25 32.5 ⎥ ⎥ , 37.5 ⎦ 65 4×2

and the ordinal form as: ⎡

MC D M Or dinal

3 ⎢1 =⎢ ⎣2 4

⎤ 1 2⎥ ⎥ . 3⎦ 4 4×2

The numbers in both matrices’ left and right columns indicate the average of risk factors (likelihood and cost), respectively, for all intervention actions. In other words, it represents the average performance of four intervention actions. The values denote the rank of intervention actions based on decision-makers opinions regarding the relevant criterion column in the ordinal matrix. The lower rank would be preferred for decision-makers corresponding to that criterion. The transition matrix can be further developed to study the Bouali Sina petrochemical plant decision-making problem as a non-cooperative game, which contains all possible combinations of the alternatives. As there are four intervention actions, the matrix would have 16 rows (two power four). The ranking of intervention actions in the transition matrix is conducted for similar reasons. It should be added that participating two DMs with Table 8.1 The performance range of intervention actions under two risk factors Tags

Intervention actions (alternatives)

Risk factors Likelihood (%)

Unit cost (million $/year)

A

Preventive actions

10–40

0.5–1.6

B

Mitigative actions

15–50

0.3–0.9

C

Control actions

20–55

0.2–1.4

D

No actions till failure (then repair replace)

45–85

1.5–3

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8 Stochastic Game Theory Approach to Solve System Safety …

different intervention actions selection would not change the content of the decisionmaking problem. Thus, such non-cooperative outcomes and the corresponding game can be developed, as illustrated in Fig. 8.5. As it can be seen, there are four highlighted cells on the right-hand side representing the possible cooperative outcomes when both parties elect the same strategy far from what actually is. Under other outcomes, the payoffs are the same as the actual case, as the on-site circumstances for intervention actions cannot be changed unless both DMs select the same strategies. However, both DMs have the freedom to select an independent strategy, which might not be identical. In this regard, the Bouali Sina petrochemical plant decision-making problem in the context of the game theory problem can be further explained in the deterministic form. Meanwhile, the definitions of non-cooperative stability can be utilized to discover all possible outcomes and understand decision-making purposes. To analyze the Bouali Sina petrochemical plant decision-making problem by utilizing the six mentioned non-cooperative stability definitions (Fig. 8.3), the decision-making supports packages “GMCR II, Graph Model for Conflict Resolution, Decision Support System II” [87, 88] according to the Graph model for conflict resolutions is engaged [63, 68, 89]. The stability analysis outcomes in terms of stability definitions are provided in Table 8.2. The results are based on the deterministic Bouali Sina petrochemical plant game, in which it is considered the performance average of DMs for each intervention action (average performance of each intervention action under each criterion). The second column of the table indicates how many stability definitions would be satisfied for that outcome (i.e., the outcome would be stable for both DMs). For instant, the intervention action D as an outcome was not an equilibrium under any of the stability definitions. Thus, these intervention actions are unlikely and are not recommended due to the fact that it has a high cost and seems to be unstable over the lifetime of the system. The robustness of the stabilities is provided in the third column. The fact is that an equilibrium satisfying several stability definitions is taken into account as a robust and decisive outcome. Therefore, they are more likely to be suggested [63, 64]. It should be added that there is no room for deviation in a robust equilibrium. Selecting the intervention actions C or B are likely and can provide a robust equilibrium. However, both are not that robust and powerful compared to intervention actions A. The decision-makers can suggest intervention actions B and C; however, the Table 8.2 The results of stability analysis for the deterministic Bouali Sina petrochemical plant game within average performance Tags

Intervention actions (alternatives)

Satisfying stability definitions?

Stability robustness?

Stability cooperation?

A

Preventive actions

6

Robust and powerful

YES

B

Mitigative actions

4

Weak

NO

C

Control actions

3

Weak

NO

D

No actions till failure (then repair replace)

0

Unstable

NO

8.3 The Bouali Sina Fire Accident Decision-Making Problem

139

interest conflicts can only be ended by suggesting intervention action A. In addition, the recognized equilibria might not be optimum since they are assumed as the noncooperative potential for parties. The game theory can recommend the non-dominated outcomes by cooperation similar to most MCDM methods. The DMs can simultaneously increase the payoffs by agreeing or modifying their shared opinions (strategies) during the assessment process. The cooperation can be obtained merely between two parties, not necessarily all parties’ agreement. There are coalitions between two or more parties, which are created once the partners would like to modify their strategies concerning the threats/actions from other parties. In this regard, the “coalition analysis in group decision” [67] can be derived to obtain all possible cooperative outcomes utilizing GMCR II. This analysis result assists decision-makers in recognizing all potential motivations and opportunities among parties to create different coalitions. The Bouali Sina petrochemical plant game has only two players. Thus, cooperative outcomes can only be reached when both players cooperate. In Table 8.2, the last column illustrates which Bouali Sina petrochemical plant game outcomes would be stable with the parties’ cooperation. As it can be seen, the only interaction action A would be the stable outcome for this game. Let us assume that there is only one player in the Bouali Sina petrochemical plant game. Therefore, a cooperative outcome has to be taken into account, and the outcome is always non-dominated, like MCDM problems with an individual or a group of DMs. In addition, if there is another external player (e.g., the Iran National Petrochemical Company) who is the only and only player in this game, intervention action A would be the most possible outcome for this decision-making problem. That is while when a group has made a decision like Bouali Sina petrochemical plant game, the tendencies of DMs in non-cooperative game, the shortages of objective data regarding the future, lack of vision-related to the dynamic behavior of DMs, and the problem, and deficiency of trust between the players causes that the non-cooperative results would have lower probability in a short period of system lifetime. In the Bouali Sina petrochemical plant game, the players might prefer to use intervention actions A, based on self-interest (individually or coalition). However, the history of the Bouali Sina petrochemical plant accident [80, 81] somehow supports the finding of the deterministic game theory approach, as the company spent time and invested in preventive actions rather than control and mitigative. The analysis also indicated that the intervention actions B and C are equilibria according to the average performance and might be considered in the game course process. However, the “blind flange gasket” accident as a rupture in the pump line (P-8001A) highlights that the outcomes of even deterministic game theory cannot be reliable since the suggested outcomes are not sustainable (robust) over time. In a more extended period of system lifetime, the intervention action A would still be acceptable in case of continued non-cooperative behavior of players. That is while the reliable results can be derived by resolving the players’ conflicts in advance by developing a much more powerful, robust, and strong equilibrium considering all reliable subjective and objective information with the agreements on most of the potential outcomes. In the next section, the stochastic

140

8 Stochastic Game Theory Approach to Solve System Safety …

game theory as an extension of the common MCDM problem is further developed to provide much more reliable outcomes by satisfying the history of the Bouali Sina petrochemical plant accident.

8.4 The Stochastic Game Theory of Bouali Sina Fire Accident Decision-Making Problem As mentioned earlier, the valuable information would be discarded using the average performance of intervention actions, resulting in fewer or unreliable outcomes. This would have occurred especially there is a non-linear relationship between performance and parameters of the decision-making problem. In addition, the uncertainty of input variables into the decision-making problem can impact the ranking of intervention actions. The stochastic game-theoretic approach can adequately handle the uncertainties of input variables into outcomes and the subjective preferences obtained from decision-makers about the likelihood of all possible ranking of the outcomes. In Fig. 8.6, a straightforward framework is provided by utilizing simple Monte-Carlo simulation, which helps decision-makers solve the decision-making game under uncertainty. In the beginning, a random number is set to the preference performance of each decision-maker according to the selected range of performance (e.g., what is already mentioned as interval values). It is assumed that the random selection is equal to λ. Each simulation of λ generates a single independent cardinal value of deterministic decision-making problem (i.e., MCDM), and the corresponding probability is Pr = λ1 . In order to solve λ in which randomly generated in the decision-making problem, the relevant ordinal transition matrix(cs) needs to be derived. The point is that the different cardinal-based rankings might be related to a single ordinal-based ranking (i.e., the λ cardinal-based is relevant into the δ ordinal-based transition matrix, in which δ ≤ λ). It can decline the mathematical computations considerably since the δ matrix just analyzed. According to this point, the ordinal transition matrices’ probabilities can be presented as PrT = RλT , where T = 1, 2, 3, . . . δ, and RT is the total number of the relevant cardinal matrix(es). In addition, δ strategic games are further developed according to the relevant matrices. All games are solved then in a deterministic manner, and the corresponding equilibrium is obtained by engaging the six aforementioned non-cooperative stability definitions. In this regard, the probabilities of equilibriums can be presented as PrTE where E is the intervention actions in the Bouali Sina fire accident decision-making problem. Finally, the probability  of each outcome in the overall stochastic game can be computed as PrE = δT=1 PrTE . Stochastic game theory using Monte-Carlo simulation converts the stochastic decision-making problem into the deterministic one, in which all the decision-making problem is mapped into the δ deterministic games. Once all δ games are solved, the probabilities of each game are computed, and the stochastic results are determined.

8.4 The Stochastic Game Theory of Bouali Sina Fire Accident …

141

Fig. 8.6 The stochastic game theory approach to solve the Bouali Sina fire accident decision-making problem under uncertainty

In fact, the probabilistic-based outcomes better reflect input uncertainties and corresponding impacts on the outcome preferences’ performance [90, 91]. The MonteCarlo simulation’s stochastic game theory can also provide sensitivity analysis (SA) without changing the variables one by one, and SA for the input variables can be conducted at the same time, and the changes of all possible outcomes can be explored. This Monte-Carlo simulation would be time-consuming for the game having several players within cardinal input. However, the Monte-Carlo simulation declines the computations and time process using ordinal-based information. In this case, the number of generated λ cardinal matrices are clustered into the less δ ordinal games. That is while, it depends on the ranges of uncertainty, in which δ might be considerably smaller than λ. The Bouali Sina fire accident decision-making problem is solved

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8 Stochastic Game Theory Approach to Solve System Safety …

utilizing the Monte-Carlo simulation to determine the preference performances that impact the outcomes of the decision-making problem.  To solve the stochastic game theory of the Bouali Sina fire accident decisionmaking problem, a set of 100,000 random numbers (to reach convergence) as points are taken into account. It is considered that all numbers have an equal chance to be selected (uniform distributions). However, the probability distributions can be fitted and used in the Monte-Carlo simulation process once the new information becomes available. Therefore, there would be 8 numbers in every run, indicating the preference performance of all intervention actions under the risk facts (4 * 2). Afterward, the selected numbers from two players and the four intervention actions were clustered and ranked into the proper transition matrices. In this case, the 100,000 (n) deterministic cardinal matrices are clustered into the 128 (m) transition matrices and cause the decision-making problem becomes almost 782 times smaller. The Bouali Sina accident game structures were developed according to the transition matrices. All games are then manually fed into the GMCR II package and resolved considering six identified non-cooperative stability definitions. The simulation of 128 games in the GMCR II package is time-consuming. Therefore, the games within probability occurrence over 40% are considered as the candidates for examinations, which contains 48 games having the probability of occurrence of 93.5%. Thus, the 80 structured games with cumulation probability of occurrence 6.5% are left behind and unstudied. Table 8.3 provides the results of the Monte-Carlo simulation for the Bouali Sina fire accident decision-making problem. In the same manner with deterministic game outcomes based on average preference performance, in a probabilistic way, the game has three possible stale outcomes as (i) the probability of being the result, (ii) the probability of being the highest chance of results, and (iii) the probability of being stable results. As a feature of the Monte-Carlo simulation, there is no dependency between the probability of any two pair of outcomes. The intervention action D was never an equilibrium outcome. The intervention actions A, B, and C have had almost the same outcomes. This means that none of them has priority and superiority over other ones. The system should take care of win–win strategies, with no losers here. Integrating A and (B and Table 8.3 The results of the Monte-Carlo simulation for the Bouali Sina fire accident decisionmaking problem Tags Probability of being Probability of being Probability of being Probability of being the resulta the highest chance of stable resultsa unstable results resultsa A

74.91

22.78

22.10

66.42

B

72.80

51.54

62.18

9.15

C

73.54

31.35

55.93

29.70

D

0

0

Not applicable

Not applicable

a It

is an equilibrium outcome

8.4 The Stochastic Game Theory of Bouali Sina Fire Accident …

143

C) interventions would be the best possible outcome for the Bouali Sina fire accident decision-making problem. The third column presents the cumulative probability occurrence of the most possible outcome by satisfying all six stability definitions. The mitigative action is the most stable outcome as it is stable by satisfying all six stability definitions over 55% (51.54% form 93.5%). The control action is the next most stable outcome. That is while the cumulative probability occurrence control action does not have a considerable difference compared with preventive action, almost 10%. Thus, the result may change if the remaining probability occurrence (1–93.5 = 6.5%) is modeled into the GMCR II package. In this regard, it can be said that the probability occurrence of both control and preventive actions would be increased and become closer to the probability occurrence of mitigative action. The fourth column presents the coalition analysis results. As can be seen, intervention action B was the best outcome when both players cooperated most of the time. We can see that the probability occurrence of intervention action C is less than B; however, it still has a big chance (55% from 93.5%) to be the game’s outcome in case of the coalition between the players. The intervention action A cannot be that much sable or unstable most of the time (the status of Bouali Sina fire accident decision-making problem), highlighting that the continuation of preventive action is a temporary solution and would cause an accident. Therefore, the preventive action should be combined with the other two intervention actions, especially the mitigative one. It is discussed that the same equally selected intervention actions on the deterministic game might be difficult be obtained when the numbers are randomly generated. Therefore, the variability of intervention actions within identical preferences performance might change the game’s outcome. To deal with this challenge and validate the results, a sensitivity analysis (SA) is conducted by developing an identical preference performance of intervention actions in the random selection phase, in which the randomly selected preference performances are rounded. In this regard, the generated numbers are rounded into two decimals. This change caused the number of possible games to increase from 128 to 452. The only games with a probability occurrence of more than 40% are then taken into account for modeling purposes, which contained 53 games within a cumulative probability of 89.45%. Table 8.4 presents the results of Table 8.4 The results of sensitivity analysis for the Bouali Sina fire accident decision-making problem using the Monte-Carlo simulation Tags Probability of being Probability of being Probability of being Probability of being the resulta the highest chance of stable resultsa unstable results resultsa A

65.18

18.53

21.19

60.65

B

62.78

29.37

58.86

8.12

C

59.93

23.90

47.21

28.11

D

0

0

Not applicable

Not applicable

a It

is an equilibrium outcome

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8 Stochastic Game Theory Approach to Solve System Safety …

Table 8.5 The results comparison of game analysis with sensitivity analysis Tags Probability of being Probability of being Probability of being Probability of being the resulta the highest chance of stable resultsa unstable results resultsa A

2.40

1.05

0.22

1.42

B

2.47

5.47

0.82

0.25

C

3.36

1.84

2.15

0.39

D

0.00

0.00

Not applicable

Not applicable

a It

is an equilibrium outcome

SA for the stochastic game theory of the Bouali Sina fire accident decision-making problem using Monte-Carlo simulation. As it can be seen, again, intervention actions A, B, and C have had almost the same chance of being outcomes, and intervention action A never would be a stable solution for both players. Therefore, the results indicated that a combination of intervention actions could solve the Bouali Sina fire accident decision-making problem. Intervention B is the most stable outcome with cooperation and non-cooperation, and intervention action C is the second most stable solution with and without cooperation. The results of cumulative probability occurrence in Tables 8.3 and 8.4 are different. Thus, Table 8.5 presents that the differences in the results are negligible. The considerable change is the robustness outcome of intervention actions B since it identically preferences performance outcomes were developed. Therefore, the equilibrium satisfied all six stability definitions; however, it has less chance than a regular game analysis. All in all, the corresponding possible outcome is approximately five times greater than intervention action C. Thus, it still would have an outstanding possibility of becoming stable with cooperation. As can be understood from the deterministic and stochastic game-theoretic analysis, intervention actions A is still suggested. However, the insights provided from stochastic-based game theory and support from the state of arts [63, 64, 68] highlighted that the obtained information could not be reachable through conventional one as the history of the Bouali Sina fire accident deterministic game-theoretic analysis. For example, intervention actions B and C in a deterministic game-theoretic analysis have weak stability robustness. That is while the results of the stochastic gametheoretic analysis indicated that mitigative actions had priority to control actions, even under the coalition. In addition, using the stochastic game-theoretic analysis escalates the trust and decreases the possible conflicts between the players.

8.5 Conclusion

145

8.5 Conclusion The present work developed a game-theoretic approach to solve system safety and reliability analysis decision-making problems, where there is uncertainty in input variables, either be subjective or objective. Considering the stochastic nature of a particular decision-making problem, a Monte-Carlo simulation is performed to convert the MCDM problem into the deterministic game theory context. The noncooperative concepts of game theory are used to solve the problem. The outcomes are further converted into the stochastic game theory and engaged the probabilities to reflect the uncertainty of decision-making problems properly. To the best understandings of authors from the developed a game-theoretic approach and having supports from the existing state of arts [63, 64, 68, 92], the following advantages of game-theoretic approach rather than common decision-making tools can be highlighted: • The game-theoretic approach, the importance weights of criteria, as well as decision-makers are not required in the assessment process, • The game-theoretic approach, Monte-Carlo simulation can adequately deal with the uncertainties of decision-making problems, even when there is no precise quantitative data, • In the game-theoretic approach, there is no requirement of the complex objective functions of decision-making problems, • The game-theoretic approach can use ordinal ranking instead of complicated and time-consuming cardinal ranking, • The game-theoretic approach can be applied in all decision-making problems since they are looking for feasible and optimum outcomes, • The game-theoretic approach does not necessarily consider that the decisionmakers have perfect cooperation, as it considers that the “self-optimizing” tendencies of decision-makers, • In the game-theoretic approach, the Monte-Carlo simulation could explain all possible outcomes, if decision-makers may or may not cooperate, • In the game-theoretic approach, the Monte-Carlo simulation could provide a set of recommendations to table non-dominated solutions, • Analysis of Bouali Sina petrochemical plant decision-making problem utilizing the developed game-theoretic approach suggests that the engaged intervention actions to prevent any potential undesired events were not effective and stable and should be updated and replaced with the more stable intervention actions, and • The history of the Bouali Sina petrochemical plant accident confirms developing the game-theoretic approach. However, during the present study, a critical challenge has been faced, which requires to be studied as a direction for future works. In most system safety and reliability analysis decision-making problems, more players are precipitated with different conflicts of interest and may create much more coalitions. Therefore, to

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8 Stochastic Game Theory Approach to Solve System Safety …

deal with such a complex decision-making problem, it is further required to develop Monte-Carlo simulations into the Markov Chain Monte-Carlo (MCMC) simulations. MCMC has much more advantages than Monte-Carlo simulations and is well-known as a robust tool with a considerable capacity to calculate the problematical posterior distribution with two or more dimensions [91, 93, 94].

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Chapter 9

Advanced Decision-Making Neutrosophic Fuzzy Evidence-Based Best–Worst Method

9.1 Introduction Iran is mainly a dry land, and the region’s subtropic weather leads the pastures and forests to be much susceptible and vulnerable to fire. Environmental and human parameters, including solar radiation, arson, and improvidence, can set a considerable part of Iran’s forests and pastures are on fire hazard. Hundreds of fire accidents every year burn bushes, pastures, plants, and forests in Iran [1]. Consequently, such fires destroy valuable environmental resources, produce air pollutions, kill different kinds of animals, harm plant species, and seriously endanger people living near forests and pastures [2, 3]. The latest fires in Iran’s Zagros Forests have imposed considerable losses to the environment, forests, villages, and forest inhabitants, resulting in an extensive financial problem. Thus, increasing the risk of fires and subsequent losses have become vital for decision-makers designing and developing efficient fire prevention and mitigation intervention actions. Researchers have conducted a few studies on Iran forests’ fire vulnerability, that the most proposed decision-making approaches are at the initial phases. For example, the fire risk of Boroujerd rangeland was examined using AHP (analytical hierarchy process) [4] and a graphical risk-based illustration was obtained according to the different vegetation types, land usage, elevation, slope, standard topographic maps, and the average of objective data. The fires of Mazandaran Province forests in a period were studied utilizing the “spatiotemporal shot-noise Cox process” [5], and fire risk in district 3 of the Neka-Zalemroud forests are also studied in order to predict future fires in the regions [6]. Finally, the three main parameters as environment, climatic, and anthropogenic impacting on the fires’ severity were studied in Golestan forests [7] by utilizing the statistical function and logistical regressions models. However, decision-making problems become more complex over time; therefore, it is necessary to develop a methodology with high capability features. This should be dynamic and has proper adaptability over time. According to the type of decisionmaking problem, a tool or combination of decision-making tools as a hybrid model © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 H. Li and M. Yazdi, Advanced Decision-Making Methods and Applications in System Safety and Reliability Problems, Studies in Systems, Decision and Control 211, https://doi.org/10.1007/978-3-031-07430-1_9

153

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could be integrated to solve the problem. Moreover, the reliable decision-making methods have to be much more realistic by considering both objective and subjective uncertainties. In general, fuzzy set theory and its extensions have been widely used to deal with subjective uncertainty, which is based on decision makers’ opinions [8–13]. Probabilistic methods, evidential theories, and simulation-based approaches have been extensively engaged [14–18]. In this regard, decision-making methods are growing by introducing many MCDM (multi-criteria decision-making) tools and their extensions, such as TOPSIS (Technique for Order of Preference by Similarity to Ideal Solution) proposed by Ching-Lai Hwang and Yoon in 1981. A couple of extension of the TOPSIS method is integrating with the rough cloud approach [19], fuzzy set theory [20], Pythagorean Fuzzy Sets [21], intuitionistic fuzzy hybrid [22, 23], and goal programming-based TOPSIS [24]. Furthermore, AHP is a structured technique for organizing and analyzing complex decisions based on mathematics and psychology. Saaty developed AHP in the 1970s and a couple of extensions of the AHP method is integrating with D numbers [25], Neutrosophic AHP-Delphi Group decision-making [26], fuzzy set theory [27], interval-valued intuitionistic fuzzy sets [28], and Combining Fuzzy AHP and Fuzzy TOPSIS [29]. Besides, DEMATEL (decision-making and evaluation laboratory) was first developed by the Geneva Research Centre of the Battelle Memorial Institute [30] to visualize the structure of complicated causal relationships through matrixes or digraphs [31]. A couple of extension of the DEMATEL method is integrating the Bayesian network [32, 33], Intuitionistic fuzzy set, [34], Pythagorean Fuzzy Sets [35], D numbers [36], and a gray-based appeach [37]. The Preference Ranking Organization METHod for Enrichment of Evaluations (PROMETHEE) was developed at the beginning of the 1980s by Jean-Pierre Brans and has been extensively studied and refined since then. Extensions of PROMETHEE method are the hesitant fuzzy linguistic information [38], Pythagorean Fuzzy Information [39], probabilistic Linguistic Environment [40], and cloud model theory [41]. Serafim Opricovic originally developed the VIKOR method to solve conflicting decision problems and into commensurable [42]. A couple of extensions of the VIKOR method is with 2-dimension uncertain linguistic information [43], Pythagorean fuzzy set [44], entropy measure [45], spherical fuzzy sets [46], interval-valued intuitionistic fuzzy information [47], and perspective of regret theory [48]. Among all MCDM tools, BWM (best–worst method) proposed by Rezaei [49] as a novel MCDM technique has considerable advantages compared to other methods such as (i) the number of pairwise comparisons is shortened, (ii) the obtained values are much more reliable as they would be obtained from min–max optimization problem; therefore, the results are optimum, (iii) in BWM, decision-makers would only use the numbers with 1–9, (iv) BWM has better dealing with inconsistency of decision making problem using consistency ratio. BWM has been widely applied in many different domains [13, 50, 51]. A couple of extension of the BWM method in chronological order are with Z-number [52], hesitant fuzzy linguistic terms [53], interval-valued multi-granular 2-tuple linguistic-based approach [54], Entropy-TOPSIS method under an intuitionistic fuzzy environment [55] evaluation based on the distance from average solution [56], hybrid uncertain sustainability

9.2 Preliminary: Neutrosophic Fuzzy Set

155

indicators based on rough-fuzzy set [57], fuzzy set theory [58], Bayesian best–worst method [59, 60], intuitionistic fuzzy preference relations [61], a group multi-criteria decision-making based on the best–worst method [62], fuzzy weighted averaging operator [63], interval-valued fuzzy-rough numbers best worst method [64], possibility degree with probabilistic linguistic information [65], and two-dimensional uncertain linguistic variables and alternative queuing method [66]. Considering all proposed extensions of BWM by scholars, however, there is still room for improvement, such as (i) it is a complicated task for decision-makers to obtain the best and worst option when the amount of the options is high, (ii) the dynamic features have not considered in the proposed methods, and (iii) there is still a question to deal with subjective uncertainty of decision-making problem. Considering the shortages mentioned above and the high flexibility of the original form of BWM, the practical method is required to solve the decision-making problem, which is also the primary motivation of the present work. Therefore, in this study, BWM is extended using evidence theory and Neutrosophic fuzzy set, called advanced Neutrosophic fuzzy evidence best–worst method. The evidence theory [67, 68] is commonly used to model uncertainties with lack of information. In addition, a Neutrosophic fuzzy set is utilized to deal with subjective uncertainties from decision makers’ opinions [69–71]. Therefore, the contribution of this study is threefold: First of all, the integration of evidence theory and Neutrosophic fuzzy set is proposed to deal with subjective and objective uncertainties. Secondly, the original form of BWM is developed to deal with uncertain environments, and finally, a framework is proposed with high capability to solve a decision-making problem effectively, which is preventive and mitigative wildfire risks. The outcomes provide significant insights into other decision-making problems in different application domains. The paper is organized as follows. In Sects. 9.2 and 9.3, the preliminary of fundamental definitions, operations, and some basic features of Neutrosophic fuzzy set and evidence theory are described, respectively. In Sect. 9.4, the methodology is proposed to extend BWM, which is advanced Neutrosophic fuzzy evidence BWM. In Sect. 9.5, a case study is considered as an application of the research to show the effectiveness of the proposed method. Finally, Sect. 9.6 is the conclusion that summarizes the work by highlighting the challenges of current work and direction for future studies.

9.2 Preliminary: Neutrosophic Fuzzy Set Since 1965, the first proposal of fuzzy set theory was presented, it has been widely applied in different application domains [72]. That is why the first introduction of fuzzy theory has resulted in many arguments and criticisms from several outstanding researchers. As an example, Professor William Kahan said, “Fuzzy theory is wrong, wrong, and pernicious …” [73]. However, the theory has been vastly spread into science in ways that are widely accepted nowadays. The classical fuzzy set, which is the original form of this theory, is parameterized by a membership function such as

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the trapezoidal, triangular, combination of them, etc. On following fuzzy set improvement, a single value of the Neutrosophic set [74, 75] and highlighted with the truth membership function Tn˜ (x), indeterminacy membership function In˜ (x), and a falsity membership function Fn˜ (x). In such Neutrosophic set, all value of x, Tn˜ (x),In˜ (x), and Fn˜ (x) are located in the interval of zero and 1. In addition, the summation of membership functions of a single Neutrosophic set satisfies the 0 ≤ Tn˜ (x)+In˜ (x)+Fn˜ (x) ≤ 3 for all the values of x. Let us assume that the n˜ = Tn˜ (x), In˜ (x), Fn˜ (x) and s˜ = Ts˜ (x), Is˜ (x), Fs˜ (x) are the two-single value Neutrosophic set, the summation, multiplication, union, and intersection of the the two single value Neutrosophic set can be obtained as the following equations [26, 76]: n˜ ⊕ s˜ = Tn˜ (x) + Ts˜ (x) − Tn˜ (x) × Ts˜ (x), In˜ (x) × Is˜ (x), Fn˜ (x) × Fs˜ (x) (9.1) for all value of x n˜ ⊗ s˜ = Tn˜ (x) × Ts˜ (x), In˜ (x) + Is˜ (x) − In˜ (x) × Is˜ (x), Fn˜ (x) + Fs˜ (x) − Fn˜ (x) × Fs˜ (x)

(9.2)

n˜ ∪ s˜ = max(Tn˜ (x), Ts˜ (x)), min(In˜ (x), Is˜ (x)), min(Fn˜ (x), Fs˜ (x)) for all value of x

(9.3)

n˜ ∩ s˜ = min(Tn˜ (x), Ts˜ (x)), max(In˜ (x), Is˜ (x)), max(Fn˜ (x), Fs˜ (x)) (9.4)

for all value of x

Let us assume that n˜ = {x1 |Tn˜ (x1 ), In˜ (x1 ), Fn˜ (x1 ), . . . ., xm |Tn˜ (xm ), In˜ (xm ), Fn˜ (xm )} and n˜ = {x1 |Ts˜ (x1 ), Is˜ (x1 ), Fs˜ (x1 ), . . . ., xm |Ts˜ (xm ), Is˜ (xm ), Fs˜ (xm )} are two single value of Neutrosophic sets for xi ∈ X (i = 1, 2, 3, . . . , m), the normalized Euclidean distance within the single value of Neutrosophic sets and the crisp value of Neutrosophic fuzzy number can be obtained as the following equations [77]: d E,nor m    m   1   = × (Tn˜ (xi ) − Ts˜ (xi ))2 + (In˜ (xi ) − Is˜ (xi ))2 + (Fn˜ (xi ) − Fs˜ (xi ))2 3 i=1 (9.5)



  1 − Tn˜ (x)2 + In˜ (x)2 + Fn˜ (x)2 /3 1−   C = p  2 2 2 /3 k=1 1 − Tn˜ (x i ) + In˜ (x i ) + Fn˜ (x i )

(9.6)

9.3 Preliminary. Evidence Theory

157

Let us assume that in a decision-making problem like BWM, the decision matrix is built using the single value of Neutrosophic sets according to decision makers’ opinions, and the aggregated decision-making matrix can therefore be constructed using the following equation using single value Neutrosophic weight average operator (AO):  AO di1j , di2j , . . . , dirj = w1 di1j ⊕ w2 di2j ⊕ · · · ⊕ wr dirj ⎛ ⎞ r r r       w w w =1−⎝ 1 − Tirj k , 1 − Iirj k , 1 − Firj k ,⎠ p=1

p=1

(9.7)

p=1

In which, there are k decision-makers, i criteria, and j alternatives, with consideration of Ti j , Ii j , Fi j representing a single value of Neutrosophic set.

9.3 Preliminary. Evidence Theory In 1967 the evidence theory was introduced, and later in the 1980s, it was developed by Shafer [78, 79]. The development was according to the milestones on the upper and lower bounds of belief assignment. The theory is named Dempster-Shafer theory (D-S) that is dealt with both uncertain and imprecision information in reallife applications such as but not limited to [80, 81]. Three fundamental parameters of and plausibility measure (Pl), basic belief (probability) assignment (BBA), and belief measure (Bel) are used by D-S to characterize uncertainty in a belief structure to deal with subjective uncertainty of decision makers’ opinions. Assume that  be a set of N elements that is a finite nonempty exhaustive set of mutually exclusive options in evidence theory called frame of discernment. (PS) as the power set of  could have all possible subsets. The subsets are 2 which includes 2N members in 2 . If  = {X, Y, Z }, and N = 3, then the power set is ψ = 2 = {{φ}, {X }, {Y }, {Z }, {X, Y }, {X, Z }, {Y, Z }, {X, Y, Z }}, where φ represents a null set and the evidence theory could begin by introducing the frame of discernment. BBA as a belief mass is denoted by m( psi ) and subsequently the power set: m( A) : 2 → [0, 1], and satisfying the Eq. (9.9): m(∅) = 0 

m(A) = 1

Ai ∈ψ

where 0 ≤ m(A) ≤ 1, and A ∈ 2 . A describes the given opinions coming from decision makers’ views.

(9.8) (9.9)

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9 Advanced Decision-Making Neutrosophic Fuzzy Evidence-Based …

The prior probability assignment function defines the belief measure (Bel), a lower bound for set A. The relationship between them could be described as follows: Bel(A) =



m(A)

(9.10)

Ak ⊆Ai

where Bel(φ) = 0 and Bel() = 1. Shafer represents that for natural number n, A ⊆ : Bel( A1 ∪ A2 ∪ . . . ∪ An ) ≥ −





Bel( Ai )

i

 Bel Ar ∩ A j + · · · + (−1)n Bel(A1 ∩ A2 ∩ . . . ∩ An )

(9.11)

r> j

where i, r, j = 1, 2, . . . , n. The upper bound for a set psi which is named plausibility measure (PL), could be presented as follows: 

 Pl( Ai ) = 1 − Bel Ai =

m(Ak )

(9.12)

Ak ∩ Ai =φ

where Ai denotes the negation of psi . Interested readers could refer to Ross [82, 83] to obtain detailed descriptions. Multiple sources of evidence could be aggregated by D-S evidence theory. These sources are elicited from different related experts. D-S combination rule is introduced Dempster & Shafer (D-S) as the most useful one. By combining the DM’ results, the following statistically independent summation is considered as follows: m 1−n = m 1 ⊕ m 2 ⊕ m 3 ⊕ · · · ⊕ m n

(9.13)

where ⊕ denotes summation. By m(A I ) and m( A I I ) as basic belief assignments and sets of evidence for a known event, obtained from two independent experts, then the D-S combination rule can be presented as follows: m1 =

q 

wke m k1

(9.14)

k=1

where there is q number of decision-makers expressing their opinions, wke is the corresponding importance weights of decision-makers, and m k1 is an opinion collected from decision-maker q.

9.4 The Proposed Methodology

159

9.4 The Proposed Methodology A framework based on BWM for making a proper and reliable decision in emergency decision-making of wildfire is proposed (see Fig. 9.1). The Best worst method (BWM) is an appropriate alternative of the hierarchical analytical process (AHP) to compute the importance weight of criteria [49, 84]. BWM needs fewer comparison data compared to the AHP and accordingly much more viable and consistent results based on its unique pairwise comparison procedure. The BWM method has been broadly applied in different areas based on various decision-making problems [85–92]. In the present study, an advanced Neutrosophic fuzzy evidence BWM is proposed and developed to evaluate the importance of identified criteria in the emergency decision-making procedure. The proposed decision-making procedure has seven key stages: Fig. 9.1 The proposed methodology to deal, prevent, and mitigate wildfire

160

1. 2. 3. 4. 5. 6. 7.

9 Advanced Decision-Making Neutrosophic Fuzzy Evidence-Based …

Identifying the decision criteria Obtaining the best criterion using Neutrosophic fuzzy evidence representation Obtaining the worst criterion using Neutrosophic fuzzy evidence representation Defining the vector of best to others Determining the vector others to worst Obtaining the optimal importance weight of criteria Decision making by providing preventive and mitigative actions.

Let us assume that the specified decision-making problem has several criteria. The Neutrosophic fuzzy evidence pairwise comparison corresponding to the number of criteria can be performed using qualitative terms as depicted in Fig. 9.2. These qualitative terms represent the degree of belief, which indicates the DMs’ confidence level. Therefore, both confidence level and qualitative terms are taken into account to make a Neutrosophic fuzzy evidence matrix having a row vector of Er ow = cn (en1 , en2 , . . . ., enm ) and column vector of E column = cm (e1m , e2m , . . . ., enm ), in Fig. 9.2 The qualitative terms for pairwise comparison purposes

9.4 The Proposed Methodology

161

which enm is denoting the relative Neutrosophic fuzzy evidence preference of criterion n over m. As an example, for a BBA, enm ([E I ]) is equal to corresponding Neutrosophic fuzzy numbers, where n = m. Considering the main concepts and features of BWM [49, 84], only 2n-1 comparisons are required to be performed instead of n^2 in the methods like AHP to construct the make a Neutrosophic fuzzy evidence matrix. It should be added that the reasons of extending Neutrosophic fuzzy numbers with BWM can be highlighted as the following: • The Neutrosophic fuzzy numbers are one of the most recent extensions of fuzzy numbers and could adequately deal with the shortages of classical fuzzy numbers, • BWM is the only technique among all MCDM tools, which can provide the optimum results. Thus, the outcomes from BWM and its corresponding extension are reliable and do not need further validations. The description of each step is summarized as follows: (1)

Identifying the decision criteria

Let us assume that there is n number of criteria as [C1 , C2 , ..., Cn ], and there are critical to making a proper decision considering different alternatives in the decision-making problem. As an example, in case of preventing wildfire, the decision criteria can be topographic, climate conditions, and so on. (2)

Obtaining the best criterion using Neutrosophic fuzzy evidence representation

In this step, the best criterion for Neutrosophic fuzzy evidence representation would be identified among a set of alternatives. Due to subjective uncertainty from decision makers’ opinions, two types of BBAs should be utilized to evaluate the decision makers’ opinions on the best and worst criteria. According to this point, the set of all criteria based on FOP is signified as  = [C1 , C2 , ..., Cn ]. In this case, the best criterion is denoted by C B . Therefore, considering the plausibility function definition Pl(A) could be used as the upper probability, which is the proposition of A. In literature [93], it is used Pl(A) to support the decision-making in classification. Because the Pl(.) considers all possible beliefs connected to the different classes. Thus, to obtain the C B , it should be identified as an element of , having the highest value of plausibility. The C B can therefore be defined consideration set of criteria [C1 , C2 , ..., Cn ] and the Neutrosophic fuzzy evidence representation of C B as the best criterion using the following equation: C B = arg max (Pl B (Ci ))

(9.15)

where i ∈ {1, 2, 3, . . . , n}, Pl B (.) is the plausibility transform of C B and defined as the following equation:

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9 Advanced Decision-Making Neutrosophic Fuzzy Evidence-Based …

Pl B (Ci ) =



C B (Ci ) ∀ Ci ⊆ 

(9.16)

Ci ∩C j =,i= j

(3)

Obtaining the worst criterion using Neutrosophic fuzzy evidence representation

Similar to step (2), the worst criterion CW should be identified as an element of , having the lowest value of plausibility. The CW can therefore be defined consideration set of criteria [C1 , C2 , . . . , Cn ] and the Neutrosophic fuzzy evidence representation of CW as the worst criterion using the following equation: CW = arg max (Pl W (Ci ))

(9.17)

where i ∈ {1, 2, 3, . . . , n}, Pl W (.) is the plausibility transform of CW and defined as the following equation: Plw (Ci ) =



CW (Ci ) ∀ Ci ⊆ 

(9.18)

Ci ∩C j =,i= j

According to the definitions, the obtained best and worst criteria based on the evaluation technique can overcome the subjective uncertainty of decision makers’ opinions. Subsequently, a practical technique should be proposed the final best and worst criteria underlying the ideal of evidence theory. It should be highlighted that for C B and CW within the same plausibility value; one can be selected as the best and worst criterion, respectively. (4)

Determining the vector best for others

In this step, the Neutrosophic fuzzy evidence representation is determined to obtain the preference of the best criterion over all other criteria using BBAs. According to this point, the output of the vector best to others can be obtained as the following equation: CB = [C B1 , C B2 , . . . , C Bn ]

(9.19)

In which CB signifies that the Neutrosophic fuzzy evidence representation vector best to others, C B j describe the Neutrosophic fuzzy evidence preference of the best criterion C B over other criteria j, j = [1, 2, . . . , n], and it is clear that C B B ([E I ]) in terms of Neutrosophic fuzzy numbers is equal to 1. (5)

Determining the vector others to worst

Similar to step (4), the Neutrosophic fuzzy evidence representation is determined to obtain the precedence of all criteria compared to the worst one using BBAs. According to this point, the output of the vector others to worst can be obtained as the following equation:

9.4 The Proposed Methodology

163

CW = [C1W , C2W , . . . , CnW ]T

(9.20)

In which, CW signifies that the Neutrosophic fuzzy evidence representation vector others to worst, C j W describe the Neutrosophic fuzzy evidence preference of criteria j to the worstest criterion CW over, j = [1, 2, . . . ., n], and it is clear that CW W ([E I ]) in terms of Neutrosophic fuzzy numbers is equal to 1. (6)

Optimal importance weight

According to steps (4) and (5), the Neutrosophic fuzzy evidence representation preference comparisons for the best and worst criteria have been performed based on BBAs. To find out the optimal importance weights of criteria, the mentioned Neutrosophic fuzzy evidence representation preference comparisons should be converted into the numerical, in which by given best to others vector, CB = [C B1 , C B2 , . . . , C Bn ], for C B j , the corresponding probability distributions should  be computed within Bet PC , and can be obtained as PB j = PB1 j , PB2 j , . . . .PBθ j , where θ = 9 in the present study based on Fig. 9.2, the number of qualitative terms. Accordingly, the numerical form of C B j can be determined as the following equation: ϑB j =

θ 

i PBθ j

(9.21)

i=1

In which, i shows the relevant value of qualitative terms based on Fig. 9.2. Therefore, the numerical best to others vector can be defined as C˜B = [ϑ B1 , ϑ B2 , . . . , ϑ Bn ]. Similarly, for the given others-to-worst vector, CW = [C1W , C2W , . . . , CnW ], for C j W , the corresponding probability  distributions should  be computed within Bet PC , and can be obtained as P j W = P j1W , P j2W , . . . .P jθW , where θ = 9 in the present study based on Fig. 9.2, the number of qualitative terms. Accordingly, the numerical form of C j W can be determined as the following equation: ϑ jW =

θ 

i P jθW

(9.22)

i=1

In which, i shows the relevant value of qualitative terms based on Fig. 9.2. Therefore, the numerical best to others vector can be defined as C˜W = [ϑ1W , ϑ2W , . . . , ϑnW ]. Underlying the main principles of BWM proposed by Rezaei [49], a mathematical of criteria. formulation is required to optimize the importance weights  This solution     w  can be utilized by maximum absolute differences as  wWBj − ϑ B j , and  W j Wj − ϑ j W  for all j are minimized. The mentioned maximum absolute differences are the constraints of an optimization model. Therefore, model 2 is defined to find the optimal importance weights (w1∗ , w2∗ , . . . ., wn∗ ) as the following: Model 2:

164

9 Advanced Decision-Making Neutrosophic Fuzzy Evidence-Based …

     wB   wj  Min max  − ϑ B j  ,  − ϑ j W  . W W j

jW

Subject to. n 

w j = 1,

j=1

w j ≥ 0, j = {1, 2, . . . , n}. According to Rezaei [49], the re-establishment of Model 2 can be defined into Model 3: min ξ Subject to.     WB   − ϑ B j  ≤ ξ, W j     ϑ j W − W j  ≤ ξ,  WW  n  w j = 1, j=1

w j ≥ 0, j = 1, 2, . . . n.  The optimal importance weights are denoted as w ∗ = w1∗ , w2∗ , . . . , wn∗ . The consistency of the results must be computed. The consistency ratio is shown in equation (9.23): CR =

ξ∗ CI

(9.23)

where CR is known as a consistency index based on the maximum value of ξ . Better consistency is shown by smaller CR values. In this paper, the acceptance condition is C R ≤ 0.2. (7)

Making the decisions by providing preventive and mitigative actions.

Up to this point, the optimum importance weight of criteria is obtained. A similar process should be performed in the present study for the alternatives. Next is ranking the alternatives in descending order to see which one needs to receive preventive or mitigative actions. To evaluate the alternative, we first defined a factor called the Quality Index (QI) of the wildfire. QI here has two advantages, including (i)

9.5 Application of Study

165

comparison between different wildfires based on geographical location and (ii) backward propagation to obtain the critical alternative to receive preventive to mitigative actions. Thus, the QI can be defined as the following equation: Q Ix =

β α  

wi∗j(C) w(∗A)i j

(9.24)

j=1 i=1

In which α and β are the numbers of criteria and alternatives, respectively. The number of x denoted the location of wildfire in case of comparison purposes. In order to rank the alternatives, the total weight of the alternatives is required using Eq. (9.25) as the following: w(TA)i =

α 

w(∗A)i wi∗j (C)

(9.25)

j=1

According to what are obtained so far, decision-makers can find out the strength and weak points of the understudy system so that by proposing and making intervention preventive and mitigative, the system’s performance before and after wildfire would be improved.

9.5 Application of Study Increasing the risk of wildfire and its consequences as air pollution, negative effects on rivers and lakes, and green forest loss become a much more critical concern, which needs to develop a framework to prevent and mitigate wildfire risk. As an example, in recent years. The Zagros forests in Iran have inflicted severe, widespread damage to the environment, forests, villages, and forest inhabitants. This would result in an extensive cost for the country to control the fire [1]. Wildfire is a destructive phenomenon of Zagros’ vegetation cradle, which considerably harms the thickness and coverage of the pastures and forests region. Wildfire may cause burn whole of the mentioned region ultimately. Considering the statistics, over 374 wildfires have been spread in Iran’s Zagros forests since March 2020, which caused the burning of more than 50,000 h, including old-age trees and a wide variety of wildlife nature such as Persian squirrels, Persian leopard, Persian fallow deer, and wolfs. It is also important to highlight that the Zagros oak trees are growing gradually; therefore, they require more than 100 years to recover [94]. As shown in Fig. 9.3, Zagros forests contain the conservable coverage in Iran, and also this feeds over 50% of livestock and some most essential rivers as a source of freshwater. Thus, it is critical to take the requirements into account by preventing the occurrence of wildfire and mitigating the consequence of wildfire in Zagros forests.

166

9 Advanced Decision-Making Neutrosophic Fuzzy Evidence-Based …

Fig. 9.3 Distributions of Zagros Forest in Iran (these figures are provided from (after) studies of [95–98])

The qualitative terms are presented in Table 9.1 to evaluate the preferences. In addition, the criteria and sub-criteria to assess the risks of wildfire are provided in Table 9.2. Four independent decision-makers with experience and expertise in risk assessment and environmental assessment are employed to perform the study. The descriptions of evaluator decision-makers are given in Table 9.3. The importance weights of the four criteria are evaluated first. As provided in Table 9.3, four decision-makers are employed for the evaluation of wildfire risks. An approach provided is utilized to obtain the importance weights [100, 101], and the results for the decision-makers #DM1, #DM2, #DM3, and #DM4 are 0.181, 0.247, 0.304, and 0.231, respectively. To specify the best and worst criteria, the evaluation obtained from decision-makers expressed in BBA forms in terms of Neutrosophic Table 9.1 The qualitative terms and their corresponding Neutrosophic fuzzy numbers in order to evaluate the preferences [77]

Qualitative terms

Neutrosophic fuzzy numbers (Tn˜ (x),In˜ (x), and Fn˜ (x))

Extremely high (EH)

(1.00, 0.00, 0.00)

Very high (VH)

(0.90, 0.10, 0.05)

High (H)

(0.80, 0.20, 0.15)

Medium high (MH)

(0.65, 0.35, 0.30)

Medium (M)

(0.50, 0.50, 0.45)

Medium Low (ML)

(0.35, 0.65, 0.60)

Low (L)

(0.20, 0.75, 0.80)

Very low (VL)

(0.10, 0.85, 0.90)

Extremely low (EL)

(0.05, 0.90, 0.95)

9.5 Application of Study

167

Table 9.2 The description of criteria and sub-criteria used in wildfire risks determination (adapted from [99]) #C1

Climate

#C11

Average annual temperature

#C12

Average annual precipitation

#C 2

Socio-economic

#C21

Distance to human settlement

#C22

Distance to power lines

#C23

Distance to roads

#C24

Population density

#C25

Land over types

#C 3

Stand structure

#C31

Tree species mixture

#C32

Canopy closure

#C33

Development stage

#C 4

Topographic

#C41

Solar radiation

#C42

Elevation

#C43

Slope

#C44

Aspect

#C45

Topographic wetness

Table 9.3 Information for the evaluator group of decision-makers Decision-makers tag

Title, degree, and experience

#DM1

Reliability assessor, PhD, and 15-year experience in reliability analysis in different industrial sectors

#DM2

Environment assessor, PhD, and 12-year experience in environmental assessment of high-tech industrial sectors

#DM3

Environment assessor, PhD, and 18-year experience fire risks of forests

#DM4

Instructor, PhD, and 25-year experience teaching environmental assessment courses

{#C2 } =(0.90, 0.10, 0.05), m #DM1 {#C2 } = (0.65, 0.35, 0.30), fuzzy numbers m #DM1 B B #DM2 {#C } = = m B {#C1 } (1.00, 0.00, 0.00), m #DM2 (0.90, 0.10, 0.05), 3 B #DM3 {#C } {#C } = 0.00, 0.00), = m m #DM3 (1.00, (0.90, 0.10, 0.05), 2 1 B B #DM4 {#C } {#C } = 0.35, 0.30), m = 0.10, 0.05), and m #DM4 (0.65, (0.90, 1 2 B B #DM1 {#C } {#C } = 0.85, 0.90), = 0.75, 0.80), m m #DM1 (0.10, (0.20, 3 3 w W #DM2 {#C } {#C } = 0.85, 0.90), = 0.75, 0.80), m m #DM2 (0.10, (0.20, 4 3 w W #DM3 {#C } {#C } m = 0.85, 0.90), = 0.75, 0.80), m #DM3 (0.10, (0.20, 2 4 W W

168

9 Advanced Decision-Making Neutrosophic Fuzzy Evidence-Based …

{#C3 } = (0.20, 0.75, 0.80), m #DM4 {#C4 } = (0.10, 0.85, 0.90). The aggrem #DM4 W W gated BBA values can be obtained using Eq. (9.14) in terms of Neutrosophic fuzzy numbers, m B {#C1 } = 0.279, m B {#C2 } = 0.309, m B {#C3 } = 0.210, m B {#C4 } = 0.202, m W {#C1 } = 0.349, m W {#C2 } = 0.323, m W {#C3 } = 0.164, and m W {#C4 } = 0.164. Therefore, the best and worst criteria are #C2 #C1 , respectively. The vectors of best to others and others to worst based on four different decisionmakers are provided in Tables 9.4 and 9.5, respectively. The outcomes of aggregated evaluation based on all collected opinions from decision-makers using aggregator equation in Sects. 1 and 2, are provided in Tables 9.6 and 9.7. In order to obtain optimal Neutrosophic fuzzy evidence weights for all criteria, the non-linear optimization model can be constructed underlying the idea of BWM as the following model 3: Model 3 min ξ Subject to.      0.8623      ≤ ξ,  0.8623 − 0.8623 − 0.5846  W    W 1 2       0.8623   0.8623 − 0.8773 ≤ ξ,  − 0.6758 ≤ ξ, ≤ ξ,  W3 W4         0.8706 − W1  ≤ ξ, 0.7035 − W2    0.8706  0.8706        W3  W4    ≤ ξ, 0.6673 − ≤ ξ, 0.7585 − ≤ ξ, 0.8706  0.8706  4 

w j = 1,

j=1

w j ≥ 0, j = 1, 2, . . . , 4. The optimal Neutrosophic fuzzy evidence weights for all criteria are determined by solving the model. The initial results within continuous interactive contact with decision-makers, are the optimal weights w1∗ = 0.0665, w2∗ = 0.6401, w3∗ = 0.1354, and w4∗ = 0.1580. To evaluate the consistency ratio, Ksi* (https://bestworstmethod. com/software) value is obtained as 0.3076, in which the closer value of ksi* to zero would provide much more reliable results. This means that this model and the corresponding results have sufficient consistency. Also, the Neutrosophic fuzzy evidence preferences for sub-criteria #C11 to #C45 can be constructed, and the corresponding results concerning every single criterion

#C1

#C2

#C3

#C4

Best criterion m{M} =(0.50, 0.50, 0.45)

#C2

Best criterion m{V L} =(0.10, 0.85, 0.90) m{E I } = (1.00, 0.00, 0.00) m{E I } =(0.65, 0.35, 0.30) m{H } =(0.80, 0.20, 0.15)

#C2

#DM3

0.304

#DM4

0.231

m{E I } =(0.65, 0.35, 0.30)

#C2

0.247

m{E I } = (1.00, 0.00, 0.00) m{H } =(0.80, 0.20, 0.15)

#C2

m{V L} =(0.10, 0.85, 0.90)

Best criterion m{V L} =(0.10, 0.85, 0.90) m{E I } = (1.00, 0.00, 0.00) m{E I } =(0.65, 0.35, 0.30) m{E I } =(0.65, 0.35, 0.30)

m{E I } = (1.00, 0.00, 0.00) m{H } =(0.80, 0.20, 0.15)

0.181

Best criterion m{H } =(0.80, 0.20, 0.15)

#DM2

#DM1

Decision-makers tag

Table 9.4 The Neutrosophic fuzzy evidence priority of the best criterion over other criteria for different DMs

9.5 Application of Study 169

170

9 Advanced Decision-Making Neutrosophic Fuzzy Evidence-Based …

Table 9.5 The Neutrosophic fuzzy evidence preference of other criteria over the worst criterion based on different decision-makers Decision makers tag

Over other criteria

Worst criterion #C1

#DM1

#C1

m{E I } = (1.00, 0.00, 0.00)

0.181

#C2

m{H } = ((0.80, 0.20, 0.15),

#C3

m{M} =(0.50, 0.50, 0.45)

#C4

m{M} =(0.50, 0.50, 0.45)

#DM2

#C1

m{E I } = (1.00, 0.00, 0.00)

0.247

#C2

m{M L} = (0.35, 0.65, 0.60)

#C3

m{M H }=(0.65, 0.35, 0.30)

#C4

m{M H } = (0.65, 0.35, 0.30)

#DM3

#C1

m{E I } = (1.00, 0.00, 0.00)

0.304

#C2

m{V L} =(0.10, 0.85, 0.90)

#C3

m{V L} =(0.10, 0.85, 0.90)

#C4

m{M L} = (0.35, 0.65, 0.60)

#DM4

#C1

m{E I } = (1.00, 0.00, 0.00)

0.231

#C2

m{H } =(0.80, 0.20, 0.15)

#C3

m{M L} = (0.35, 0.65, 0.60)

#C4

m{H } = ((0.80, 0.20, 0.15)

Table 9.6 The aggregated Neutrosophic fuzzy evidence preference of the best criterion over other criteria based on different decision-makers #C1

#C2

#C3

#C4

Best criterion #C2

(0.6347, 0.0313, 0.0122)

(1.0000, 1.0000, 1.0000)

(0.0286, 0.7005, 0.7807)

(0.2031, 0.0839, 0.0446)

ϑB j

0.5846

0.8623

0.8773

0.6758

Table 9.7 The aggregated Neutrosophic fuzzy evidence preference of other criteria over the worst criterion based on different decision-makers

Criteria

Worst criterion #C1

ϑ jW

#C1

(1.0000, 1.0000, 1.0000)

0.8706

#C2

(0.4430, 0.3439, 0.2920)

0.7035

#C3

(0.5794, 0.3137, 0.2723)

0.6673

#C4

(0.3985, 0.4817, 0.4729)

0.7585

after the aggregation process of Neutrosophic fuzzy numbers from four decisionmakers are presented in Table 9.8. The ranking of all sub-criteria is illustrated in Fig. 9.4. In order to highlight the merits of the proposed method in two aspects as (i) expression and (ii) dealing with uncertain information, the consistency in the computation process of importance weights of sub-factors is evaluated and presented in Table 9.9. As can be understood from Table 9.9, the advanced Neutrosophic fuzzy

9.5 Application of Study

171

Table 9.8 The importance weights of all sub-criteria with respect to each single criterion Criteria

#C1 0.0665

#C2 0.6401

#C3 0.1354

#C4 0.1580

Weight

Sub-criteria

–

–

–

#C11

0.2534

–

–

0.0496

#C12

0.7466

–

–

0.1499

#C21

–

0.2342

–

0.0831

#C22

–

0.1298

–

0.2077

#C23

–

0.3245

–

0.0632

#C24

–

0.0987

–

0.1362

#C25

–

0.2128

–

0.0590

#C31

–

–

0.4356

0.0175

#C32

–

–

0.1289

0.0590

#C33

–

–

0.4355

0.0146

#C41

–

–

–

0.0923

0.0107

#C42

–

–

–

0.0678

0.0720

#C43

–

–

–

0.4559

0.0366

#C44

–

–

–

0.2314

0.0241

#C45

–

–

–

0.1526

0.0169

Fig. 9.4 The ranking of sub-criteria

172

9 Advanced Decision-Making Neutrosophic Fuzzy Evidence-Based …

Table 9.9 The evaluating of consistency ratio among criteria and sub-criteria Criteria

Proposed advanced Neutrosophic fuzzy evidence BWM

Criteria {#C1 , #C2 , #C3 , #C4 }

0.3076

Sub-criteria {#C11 , #C12 }

0.2371

Sub-criteria {#C21 , #C22 , #C23 , #C24 , #C25 } 0.3563 Sub-criteria {#C31 , #C32 , #C33 }

0.1693

Sub-criteria {#C41 , #C42 , #C43 , #C44 , #C45 } 0.2643

evidence BWM can provide a low consistency ratio, highlighting that the proposed approach has a high capability to deal with uncertainties. As it can be seen, the worse sub-criteria are identified #C42 (Elevation),#C33 (Development stage), and #C11 (Average annual temperature) are the most subcritical. According to the sub-criteria, they are rather objective-based and stochastic. Therefore, it required that many parameters are integrated to control the risk of wildfire in Zagros Forest. Another analysis is the evaluating of risk levels among a couple of suspected wildfire points in Zagros Forest in the order received preventive and mitigative actions. As illustrated in Fig. 9.5, the five suspected locations for wildfire in Zagros Forest are considered to evaluate wildfire risks. Table 9.10 provides the risk level priority

Fig. 9.5 The five suspected locations for wildfire in Zagros Forest (this figure is provided after studies of [95–98])

0.6436

2

Risk level weights

Ranking

0.0721

0.0169

#C45

0.0239

0.3913

0.0366

0.0241

0.9966

#C43

0.0720

#C42

0.7130

0.6841

#C44

0.0146

0.0107

#C33

#C41

0.6687

0.0590

#C32

0.2247

0.2007

0.0590

0.0175

0.4292

#C25

0.1362

#C24

0.8198

0.8914

0.5397

0.9369

0.1146

0.0012

0.0094

0.0009

0.0718

0.0073

0.0104

0.0395

0.0035

0.0133

0.0585

0.0518

0.1851

0.0448

0.1404

0.0057

1

0.7038

0.9559

0.9275

0.8885

0.4748

0.6399

0.0500

0.7359

0.8823

0.1310

0.7406

0.8533

0.9996

0.0080

0.8945

0.5490

L2 w2∗

Optimum

LA

w1∗

Various locations in Zagros Forest

#C31

0.2077

0.0632

#C22

0.0831

#C21

#C23

0.0496

0.1499

#C11

#C12

weights

Sub-criteria

0.0162

0.0224

0.0325

0.0342

0.0068

0.0007

0.0434

0.0154

0.0077

0.1009

0.0539

0.2076

0.0007

0.1341

0.0272

Optimum

5

0.4084

0.3564

0.2099

0.5401

0.0847

0.0149

0.7147

0.1156

0.5713

0.2721

0.6874

0.6170

0.2772

0.6983

0.4329

0.3000

w3∗

L3

0.0060

0.0051

0.0198

0.0061

0.0002

0.0104

0.0068

0.0100

0.0161

0.0936

0.0390

0.0576

0.0580

0.0649

0.0149

Optimum

4

0.4476

0.5191

0.0435

0.2348

0.3811

0.8753

0.7227

0.4044

0.5566

0.7374

0.2761

0.4552

0.2766

0.5391

0.8884

0.0597

w4∗

L4

Table 9.10 The risks level of priority for a different location in Zagros Forest with respect to all sub-criteria

0.0088

0.0010

0.0086

0.0274

0.0094

0.0106

0.0239

0.0097

0.0435

0.0376

0.0288

0.0574

0.0448

0.1332

0.0030

Optimum

L5

3

0.5592

0.3703

0.3756

0.7932

0.0369

0.6792

0.0272

0.3180

0.9217

0.5141

0.6029

0.5315

0.7728

0.7775

0.4421

0.6497

w5∗

0.0063

0.0091

0.0290

0.0027

0.0073

0.0004

0.0188

0.0161

0.0303

0.0821

0.0336

0.1605

0.0646

0.0663

0.0322

Optimum

9.5 Application of Study 173

174

9 Advanced Decision-Making Neutrosophic Fuzzy Evidence-Based …

of all five locations in Zagros Forest. In this case, there are five potential locations (i.e., L1, L2, L3, L4, and L5), which need to be evaluated. With consideration of the importance weights of every single sub-criteria, the relative importance weights of all potential five locations are calculated by Eq. (9.25), and the outcomes are provided in Table 9.10 as L2 > L1 > L5 > L4 > L3. It highlights that the risk level of location 2 is higher than other locations. Therefore, location should be taken into consideration and further evolutions to prevent and mitigate wildfire risk.

9.5.1 Sensitivity Analysis According to Saltelli (2002) [102], sensitivity analysis (SA) can be defined as a study of how the uncertainty in the output can be assigned to several sources of uncertainty in corresponding inputs. SA can show how much the prediction of results is accurate. It helps decision-makers assess the riskiness of a decision by evaluating how much the output depends on a specific input parameter. In this study, SA has been performed by considering the original form of decision-makers opinions, in which each single SA performance of the sub-criteria is eliminated. In this case, it can be understood how the variation of sub-criteria can change the outcomes as the best and worst sub-criteria. Table 9.11 presents the importance weights of sub-criteria in each performance. Figure 9.6 illustrates the ranking of sub-criteria according to the importance weights in each performance. As it can be seen from SA, the ranking of sub-criteria does not have meaningful variations. This highlights that the ranking of sub-criteria is not sensitive to the decision-makers opinions about sub-criteria comparisons. Therefore, as mentioned in the literature [52, 103–106] and our understanding from this performed SA, consideration of opinions and their corresponding confidence level would be necessary for any decision-making elicitation procedure.

9.5.2 Comparison Analysis Using Fuzzy AHP, BWM, TOPSIS, and the Simple Average Method This subsection aims to determine the feasibility and practicality of the introduced methodology via comparison analysis with Fuzzy AHP, BWM, TOPSIS, and the Simple Average Method. All mentioned methods have been widely accepted and applied tools in the MCDM field. All four methods determine the ranking of subcriteria, and the results are presented in Table 9.12. As shown in Fig. 9.7, the ranking of sub-criteria is entirely consistent with the first highest best, and worst sub-criteria. This means that the best and worst sub-criteria in all approaches are similar. Thus, decision-makers based on some realistic restrictions such as time, complexity, etc., can rely on other methods. However, most optimal results are different in the selected four methods. By computing the Spearman rank correlation coefficient between

0.12944 0.11578 0.12488 0.10791 0.12759 –

#C24

0.01388 0.01241 0.01339 0.01157 0.01368 0.01261 0.01374 0.01434 0.01374 –

#C33

0.02290 0.02049 0.02210 0.01909 0.02258 0.02082 0.02268 0.02368 0.02268 0.02375 0.02384 0.02236 0.02322 –

0.01606 0.01437 0.01550 0.01339 0.01583 0.01460 0.01590 0.01660 0.01590 0.01665 0.01672 0.01568 0.01628 0.01649 –

#C44

#C45

0.02369

0.03572 0.03598

0.06936 0.07026 0.07078

0.03478 0.03111 0.03356 0.02900 0.03429 0.03162 0.03444 0.03596 0.03444 0.03607 0.03621 0.03396 –

#C43

0.00993 0.01031 0.01044 0.01052

0.01017 0.00910 0.00981 0.00848 0.01002 0.00924 0.01007 0.01051 0.01007 0.01054 –

0.06843 0.06121 0.06602 0.05705 0.06745 0.06219 0.06775 0.07074 0.06775 0.07095 0.07123 –

#C41

#C42

0.01444 0.01355 0.01407 0.01425 0.01435

0.05814 0.05837 0.05475 0.05684 0.05758 0.05800

0.01647 0.01724 0.01731 0.01624 0.01686 0.01708 0.01720

0.05607 0.05016 0.05410 0.04675 0.05527 0.05096 0.05552 0.05797 –

#C32

0.05797 0.05552 0.05814 0.05837 0.05475 0.05684 0.05758 0.05800

0.05607 0.05016 0.05410 0.04675 0.05527 0.05096 –

0.01663 0.01488 0.01605 0.01387 0.01639 0.01512 0.01647 –

#C25

#C31

0.12816 0.13382 0.12816 0.13421 0.13474 0.12639 0.13122 0.13292 0.13390

0.05459 0.05947 0.06209 0.05947 0.06228 0.06252 0.05865 0.06089 0.06168 0.06213

0.19457 0.17941 0.19545 0.20407 0.19545 0.20467 0.20548 0.19275 0.20010 0.20269 0.20419

0.06007 0.05373 0.05795 0.05007 –

#C23

0.06584 0.07785 0.07178 0.07820 0.08165 0.07820 0.08189 0.08221 0.07712 0.08006 0.08110 0.08170

0.07898 0.07064 –

0.19740 0.17657 0.19044 –

#C21

#C22

0.13744 0.11877 0.14043 0.12948 0.14106 0.14728 0.14106 0.14771 0.14830 0.13911 0.14441 0.14629 0.14737

0.04216 0.04548 0.03930 0.04647 0.04284 0.04667 0.04873 0.04667 0.04888 0.04907 0.04603 0.04778 0.04840 0.04876

–

0.14246 –

#C11

#C12

Sub-criteria Importance weights

Table 9.11 The importance weights of all sub-criteria with respect to each single SA performance

9.5 Application of Study 175

176

9 Advanced Decision-Making Neutrosophic Fuzzy Evidence-Based …

Fig. 9.6 The ranking of sub-criteria according to the importance weights in each performance

Table 9.12 The ranking results based on different approaches Sub-criteria

Proposed advanced Neutrosophic fuzzy evidence BWM

BWM

Fuzzy AHP

TOPSIS

Simple average method

#C11

1

1

2

2

5

#C12

14

13

15

13

11

#C21

12

12

14

10

15

#C22

15

14

11

8

6

#C23

10

10

12

7

14

#C24

13

9

10

12

7

#C25

8

7

9

11

8

#C31

5

8

8

6

9

#C32

9

11

10

15

4

#C33

3

3

4

5

5

#C41

2

4

3

1

12

#C42

11

5

7

9

13

#C43

7

6

5

4

3

#C44

6

15

14

14

2

#C45

4

3

6

3

1

each pair of methods, the conformity priority of methods is precisely reflected and presented in Table 9.13. The Spearman rank correlation coefficient between “proposed methodology” with other methods is considerably better. Thus, considering the “Proposed methodology” in such decision-making problems among the existing decision methods, it could be concluded that this study has enough merits to be used

9.5 Application of Study

177

Fig. 9.7 The ranking results based on different approaches

Table 9.13 Spearman rank correlation coefficient between “proposed methodology” and other approaches Proposed advanced Neutrosophic fuzzy evidence BWM

Proposed advanced Neutrosophic fuzzy evidence BWM

0.7179

Proposed advanced Neutrosophic fuzzy evidence BWM

BWM

0.7533

Proposed advanced Neutrosophic fuzzy evidence BWM

TOPSIS

0.6536

Proposed advanced Neutrosophic fuzzy evidence BWM

Simple Average Method

0.3692

BWM

Fuzzy AHP

0.9237

BWM

TOPSIS

0.7653

BWM

Simple Average Method

0.1389

Fuzzy AHP

TOPSIS

0.8035

Fuzzy AHP

Simple Average Method

0.2778

TOPSIS

Simple Average Method

0.0426

in such similar cases. BWM, TOPSIS, and Simple Average Method are standard methods, and therefore they are considered for comparison purposes. Compared with the existing approaches in terms of simplicity in computation, one can say that the proposed approach in this study is complex and time-consuming.

178

9 Advanced Decision-Making Neutrosophic Fuzzy Evidence-Based …

9.6 Conclusion In nowadays, evaluating the risks of wildfire is a vital and sophisticated task for decision-makers. It is a quiet significant multi-criteria decision-making problem. The present work offers an extension to conventional BWM by integrating evidence theory and a Neutrosophic fuzzy set. Accordingly, the pairwise comparison in BWM could be defined by qualitative terms and taken into account both types of subjective and objective uncreatively. According to the proposed method (called advanced Neutrosophic fuzzy evidence best–worst method), a flexible framework with a different type of advantages is constructed including (i) similar to the traditional form of BWM, the advanced Neutrosophic fuzzy evidence best–worst method has fewer pairwise comparisons compared to AHP, (ii) using evidence theory and Neutrosophic fuzzy set, the uncertainties are further tolerated, (iii) considering the applicability of the proposed method in wildfire risks assessment, it could be extended to various application domains. The results indicated that the proposed method could adequately identify the critical factors to mitigate and prevent the consequences and likelihood of wildfire occurrence. However, a couple of challenges had arisen during this study, which required further investigations as a direction for future study. First of all, it is still a challenging task using crisp values to describe a factor, parameter, criteria, or alternative with stochastic feature. Therefore, it is required to consider the advantages of probability methods such as Bayesian Network, Petri nets, Markov process, etc., and integrate them with MCDM tools. Secondly, confidence factor could be considered for future studies. For this purpose, a Neutrosophic fuzzy set can be combined with a Z number [104, 107]. Finally, sensitivity analysis could be conducted by different types of aggregation methods.

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Chapter 10

A Holistic Question: Is It Correct that Decision-Makers Neglect the Probability in the Risk Assessment Method?

10.1 Problem Statement A psychologist’s experiment held in the early 1970s [1] adequately showed how decision-makers neglect the term “probability” in the risk assessment procedures. Two groups of individuals were asked to participate in this experiment. The first group was told they faced negligible electric shock. The second group was told that such occurrence had a probability of 0.5. Shortly after that, both groups faced evaluated anxiety, such as nervousness, a high heart rate, and other symptoms. The results mesmerized the psychologists as the anxiety of both groups was identical. In the second experiment, psychologists declared that there would be two sets of electric shock reductions from 0.5 to 0.2 and 0.1 to 0.05. The results interestingly showed that there was no difference between the two groups. In the third step of the experiment, psychologists announced that the electric shock would increase. As a result, the anxiety of both groups increased to the same degree. We can conclude from this experience that humans, as decision-makers in different decision-making problems, respond to the expected magnitude of an event occurrence, not to its probability. In a simple work, the decision-makers are challenged with a “lack of an intuitive grasp of probability”. One more example is presented in Table 10.1. Keeping such drawbacks in mind, how one can perform any risk assessment methods with meaningful and sensible results? How can one trust such risk assessment outcomes? We know that risk is defined as a combination of severity (consequence, magnitude) and likelihood (probability) [2]. When there is no response from decision-makers to the probability of an event occurrence, we should not consider the risk more equivalent to the severity rather than probability. This occurs because, in recent years, the number of scholars that has attempted to develop probabilistic risk assessment methods in different application domains has significantly increased, such as but not limited to Refs. [3–12]. Figure 10.1 illustrates the number of publications on probabilistic risk assessment utilization based on all application domains since 2000. This trend shows the © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 H. Li and M. Yazdi, Advanced Decision-Making Methods and Applications in System Safety and Reliability Problems, Studies in Systems, Decision and Control 211, https://doi.org/10.1007/978-3-031-07430-1_10

185

186

10 A Holistic Question: Is It Correct that Decision-Makers …

Table 10.1 Example of decision-makers challenged with a “lack of an intuitive grasp of probability” Descriptions

Outcomes

Two games of chance: in the first game, one can win 10 million USD, and in the second game, 10 K USD!

Which one would decision-makers select? The probability of winning in the first and second game are 10−9 , and 10−4 , respectively

Fig. 10.1 Distribution of “published works per year till the end of February 2022” in probabilistic risk assessment utilizations based on all application domains since 2000 (according to the web of science (WoS) database)

highest increase in 2020. It is expected that probabilistic risk assessment utilization in different case studies, especially system safety and reliability analysis, will continue to evolve in the following years. According to Table 10.2, the first ten highly cited documents consider the “Average Citations per Year” of each publication in the area of probabilistic risk assessment. The “Average Citations per Year” indicates the “WoS citation index” for a paper by the end of February 2022.

10.2 Open Discussion Our papers highlight several times that one of the most important goals of system safety and reliability analysis methods is improving the system safety performance over time [2, 22, 23]. This can be achieved by reducing a hazardous event’s occurrence probability, consequence loss, or both. The challenging part in the risk assessment is the probability, particularly with the existing uncertainty. Another example regarding decision-makers’ cognitive behaviors is the urn that contains 90 balls: 30 balls are red, and the other 60 are either black or yellow in

10.2 Open Discussion

187

Table 10.2 The “highly cited papers based on citation measures” in the field of probabilistic risk assessment till the end of February 2022 (according to the WoS database) Row Descriptions

Reference

Total citations Average per year

1

Latin hypercube sampling Helton and Davis [13] and the propagation of uncertainty in analyses of complex systems

1230

61.5

2

Survey of sampling-based Helton et al. [14] methods for uncertainty and sensitivity analysis

706

41.53

3

A new uncertainty importance measure

533

33.31

4

Health risk assessment on Chen and Liao [16] human exposed to environmental polycyclic aromatic hydrocarbons pollution sources

401

23.59

5

The respiratory health Horwell and Baxter [17] 322 hazards of volcanic ash: a review for volcanic risk mitigation

18.94

6

Dose–response modeling of continuous endpoints

Slob [18]

282

13.43

7

A discrete-time Bayesian network reliability modeling and analysis framework

Boudali and Dugan [19]

248

13.78

8

Distribution and toxicity of sediment-associated pesticides in agriculture-dominated water bodies of California’s Central Valley

Weston et al. [19]

244

12.84

9

Rice is a major exposure Mondal and Polya [20] route for arsenic in Chakdaha block, Nadia district, West Bengal, India: A probabilistic risk assessment

231

15.4

10

Mitigating risk from abnormal loads and progressive collapse

217

12.76

Borgonovo [15]

Ellingwood [21]

188

10 A Holistic Question: Is It Correct that Decision-Makers …

unknown proportions. The balls are well mixed so that each ball is as likely to be drawn as any other. The decision-makers must choose between gamble A: decisionmakers will receive $200 USD if they draw a red ball, and gamble B: decision-makers will receive $200 USD if they draw a black ball. Faced with these choices, most of them would select Gamble A. In order to play this game again with the following difference as if decision-makers draw a black ball. Which one would decision-makers select? Most of them would select Gamble A again. However, it is not logical, as, in the first round, they considered that urn B had fewer red balls and more black ones. Therefore, they would rationally have drawn B for the second game. This situation is called “Ellsberg paradox,” defined as a “paradox of choice in which people’s decisions produce inconsistencies with subjective expected utility theory, and generally taken to be evidence for ambiguity aversion, in which a person tends to prefer choices with quantifiable risks over those with unknown risks” [24]. According to the Ellsberg conclusions, the selection of risk levels is favored in occurrences in which the risk probability is straightforward compared to the ones in which risk probability is unknown. The decision-makers strongly favor a selection with transparent risk probability, and even the occurrences might facilitate better utility. Considering a set of alternatives that each selection can carry out the known risk levels, the decision-makers would still prefer to select the computable risk even with lower utility results. As an open discussion here, We conclude that by determining that the “risk” based on decision-makers contributions should be more equivalent to the severity than probability in the risk assessment methods.

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