Mathematics and Computer Science [Volume 2] 9781119896326

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Mathematics and Computer Science Volume 2

Scrivener Publishing 100 Cummings Center, Suite 541J Beverly, MA 01915-6106

Advances in Data Engineering and Machine Learning Series Editors: Niranjanamurthy M, PhD, Juanying XIE, PhD, and Ramiz Aliguliyev, PhD Scope: Data engineering is the aspect of data science that focuses on practical applications of data collection and analysis. For all the work that data scientists do to answer questions using large sets of information, there have to be mechanisms for collecting and validating that information. Data engineers are responsible for finding trends in data sets and developing algorithms to help make raw data more useful to the enterprise. It is important to have business goals in line when working with data, especially for companies that handle large and complex datasets and databases. Data Engineering Contains DevOps, Data Science, and Machine Learning Engineering. DevOps (development and operations) is an enterprise software development phrase used to mean a type of agile relationship between development and IT operations. The goal of DevOps is to change and improve the relationship by advocating better communication and collaboration between these two business units. Data science is the study of data. It involves developing methods of recording, storing, and analyzing data to effectively extract useful information. The goal of data science is to gain insights and knowledge from any type of data — both structured and unstructured. Machine learning engineers are sophisticated programmers who develop machines and systems that can learn and apply knowledge without specific direction. Machine learning engineering is the process of using software engineering principles, and analytical and data science knowledge, and combining both of those in order to take an ML model that’s created and making it available for use by the product or the consumers. “Advances in Data Engineering and Machine Learning Engineering” will reach a wide audience including data scientists, engineers, industry, researchers and students working in the field of Data Engineering and Machine Learning Engineering.

Publishers at Scrivener Martin Scrivener ([email protected]) Phillip Carmical ([email protected])

Mathematics and Computer Science Volume 2

Edited by

Sharmistha Ghosh M. Niranjanamurthy Krishanu Deyasi Biswadip Basu Mallik and

Santanu Das

This edition first published 2023 by John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, USA and Scrivener Publishing LLC, 100 Cummings Center, Suite 541J, Beverly, MA 01915, USA © 2023 Scrivener Publishing LLC For more information about Scrivener publications please visit www.scrivenerpublishing.com. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, except as permitted by law. Advice on how to obtain permission to reuse material from this title is available at http://www.wiley.com/go/permissions. Wiley Global Headquarters 111 River Street, Hoboken, NJ 07030, USA For details of our global editorial offices, customer services, and more information about Wiley products visit us at www.wiley.com. Limit of Liability/Disclaimer of Warranty While the publisher and authors have used their best efforts in preparing this work, they make no rep­ resentations or warranties with respect to the accuracy or completeness of the contents of this work and specifically disclaim all warranties, including without limitation any implied warranties of merchant-­ ability or fitness for a particular purpose. No warranty may be created or extended by sales representa­ tives, written sales materials, or promotional statements for this work. The fact that an organization, website, or product is referred to in this work as a citation and/or potential source of further informa­ tion does not mean that the publisher and authors endorse the information or services the organiza­ tion, website, or product may provide or recommendations it may make. This work is sold with the understanding that the publisher is not engaged in rendering professional services. The advice and strategies contained herein may not be suitable for your situation. You should consult with a specialist where appropriate. Neither the publisher nor authors shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages. Further, readers should be aware that websites listed in this work may have changed or disappeared between when this work was written and when it is read. Library of Congress Cataloging-in-Publication Data ISBN 978-1-119-89632-6 Front cover images supplied by Wikimedia Commons Cover design by Russell Richardson Set in size of 11pt and Minion Pro by Manila Typesetting Company, Makati, Philippines Printed in the USA 10 9 8 7 6 5 4 3 2 1

Contents Preface xvii 1 A Comprehensive Review on Text Classification and Text Mining Techniques Using Spam Dataset Detection 1 Tamannas Siddiqui and Abdullah Yahya Abdullah Amer 1.1 Introduction 2 1.2 Text Mining Techniques 3 1.2.1 Data Mining 3 1.2.2 Information Retrieval 4 1.2.3 Natural Language Processing (NLP) 5 1.2.4 Information Extraction 5 1.2.5 Text Summarization 6 1.2.6 Text Categorization 7 1.2.7 Clustering 7 1.2.8 Information Visualization 7 1.2.9 Question Answer 8 1.3 Dataset and Preprocessing Steps 9 1.3.1 Text Preprocess 9 1.4 Feature Extraction 9 1.4.1 Term Frequency – Inverse Document Frequency 10 1.4.2 Bag of Words (BoW) 10 1.5 Supervised Machine Learning Classification 11 1.6 Evaluation 11 1.7 Experimentation and Discussion Results for Spam Detection Data 11 1.8 Text Mining Applications 13 1.9 Text Classification Support 13 1.9.1 Health 13 1.9.2 Business and Marketing 14 1.9.3 Law 14

v

vi  Contents 1.10 Conclusions References

14 15

2 Study of Lidar Signals of the Atmospheric Boundary Layer Using Statistical Technique 19 Kamana Mishra and Bhavani Kumar Yellapragada 2.1 Introduction 19 2.2 Methodology 21 2.2.1 A Statistical Approach to Determine the CBLH 21 2.2.2 A Statistical Approach to Determine the Best Fit Distribution to the Backscatter Signals of the Lidar Dataset 22 2.3 Mathematical Background of Method 23 2.4 Example and Result 24 2.5 Conclusion and Future Scope 27 Acknowledgement 28 References 28 Annexure 30 3 Optimal Personalized Therapies in Colon Cancer Induced Immune Response using a Fokker-Planck Framework Souvik Roy and Suvra Pal 3.1 Introduction 3.2 The Control Framework Based on Fokker-Planck Equations 3.3 Theoretical Results 3.4 Numerical Schemes 3.5 Results 3.6 Conclusion Acknowledgments References 4 Detection and Classification of Leaf Blast Disease using Decision Tree Algorithm in Rice Crop Sarvesh Vishwakarma and Bhavna Chilwal 4.1 Introduction 4.2 Proposed Methodology 4.3 Result Analysis 4.4 Conclusion 4.5 Future Work References

33 33 35 39 41 43 44 45 45 49 49 51 52 55 55 56

Contents  vii 5 Novel Hybrid Optimal Deep Network and Optimization Approach for Human Face Emotion Recognition J. Seetha, M. Ayyadurai and M. Mary Victoria Florence 5.1 Introduction 5.2 Related Work 5.3 System Model and Problem Statement 5.4 Proposed Model 5.4.1 Preprocessing Stage 5.4.2 Knowledge Based Face Detection 5.4.3 Image Resizing 5.4.4 Feature Extraction 5.5 Proposed HDC-GEN Classification 5.6 Result and Discussion 5.6.1 Performance Metrics Evaluation 5.6.2 Comparative Analysis 5.7 Conclusion References 6 An Application of Information Technology in Adaptive Leadership of Ministry of Ayush During Pandemic of Covid 19: A Case Study Vikram Singh, Shikha Kapoor and Sandeep Kumar Gupta 6.1 Introduction 6.2 Ministry of AYUSH 6.3 Leadership Principles and Practices by Ministry of AYUSH During Covid-19 6.4 Effective Communication 6.5 Sharing of Resources 6.6 Shared Decision Making 6.7 Training of Manpower 6.8 Use of IT Platform 6.9 Finding Opportunities for R&D During the Crisis 6.10 Collaborating with Stakeholders for International Day of Yoga (IDY) 6.11 Providing Hope When Nothing Seemed to be Working 6.12 Leveraging Old Knowledge 6.13 Conclusion References

59 60 61 62 63 64 64 64 64 65 68 69 70 74 74

77 77 78 79 79 80 81 81 81 83 84 87 87 88 88

viii  Contents 7 Encoder-Decoder Models for Protein Secondary Structure Prediction 91 Ashish Kumar Sharma and Rajeev Srivastava 7.1 Introduction 91 7.2 Literature Review 93 7.3 Experimental Work 93 7.3.1 Data Set 93 7.3.2 Proposed Methodology 94 7.3.3 Data Preprocessing 94 7.3.4 Long Short Term Memory 94 7.4 Results and Discussion 96 7.5 Conclusion 97 References 98 8 Hesitancy, Awareness, and Vaccination: A Computational Analysis on Complex Networks Dibyajyoti Mallick, Aniruddha Ray, Ankita Das and Sayantari Ghosh 8.1 Introduction 8.2 Model Formulation 8.3 Model Analysis on Complex Network 8.3.1 Effect of Vaccination Rate 8.3.2 Effect of Negative Rumors 8.3.3 Effect of Positive Peer Influence 8.4 Conclusions and Perspectives References 9 Propagation of Seismic Waves in Porous Thermoelastic Semi-Infinite Space with Impedance Boundary Conditions Annu Rani and Dinesh Kumar Madan 9.1 Introduction 9.2 Basic Equations 9.3 Problem Formulation 9.4 Reflection at the Free Surface 9.4.1 Boundary Conditions 9.4.2 Energy Ratios 9.5 Numerical Results and Discussion 9.6 Conclusion References

101 101 103 106 107 108 108 111 112 115 115 116 118 121 124 126 127 133 134

Contents  ix 10 IoT Based Ensemble Predictive Techniques to Determine the Student Observing Analysis through E-Learning Rufia Thaseen I., S. Shahar Banu and Sudha Rajesh 10.1 Introduction 10.2 Review of Literature 10.2.1 Objectives of the Study 10.3 Methodology 10.4 Analysis and Interpretation 10.5 Findings and Conclusion References

137 138 140 142 142 143 147 148

11 Modelling and Analysis of a Congestion Dependent Queue with Bernoulli Scheduled Vacation Interruption and Client Impatience 151 K. Jyothsna and P. Vijaya Kumar 11.1 Introduction 152 11.2 Model Overview 154 11.3 Model Analysis 155 11.3.1 Pre-Arrival Epoch Probabilities 157 11.3.2 Arbitrary Epoch Probabilities 162 11.4 Special Cases 163 11.5 Performance Metrics 163 11.6 Numerical Outcomes 164 11.7 Conclusion 169 References 169 12 Resource Allocation Determines Alternate Cell Fate in Bistable Genetic Switch 173 Priya Chakraborty and Sayantari Ghosh 12.1 Introduction 173 12.2 Model Formulation 176 12.3 Result Section 178 12.3.1 Pitchfork Bifurcation in Genetic Toggle 178 12.3.1.1 Resource Affinity Regulates the Symmetry of Pitchfork Bifurcation 178 12.3.1.2 Availability of Total mRNA Pool Regulates the Symmetry of Pitchfork Bifurcation 180 12.3.1.3 Total Resource Availability Regulates the Point of Bifurcation in the System 180

x  Contents 12.3.2 Saddle Node Bifurcation in Genetic Toggle 12.3.2.1 Resource Distribution Regulates the Point of Bifurcation in Toggle Switch 12.3.2.2 Region of Interest in Toggle Switch is Significantly Regulated by Resource Allocation 12.3.2.3 Total Resource Availability T Regulates Saddle Node Bifurcation Curve 12.4 Conclusion Acknowledgement References 13 A Hybrid Approach to Ontology Evaluation Aastha Mishra and Preetvanti Singh 13.1 Introduction 13.2 Background 13.3 The Developed OntoEva Method 13.4 Ontology Selection for Epilepsy Disorder 13.4.1 Accuracy 13.4.2 Adaptability 13.4.3 Clarity 13.4.4 Completeness 13.4.5 Conciseness 13.4.6 Consistency 13.4.7 Organizational Fitness 13.5 Results 13.6 Comparison of Ontologies 13.7 Conclusion References

180 182 182 183 183 184 184 187 187 188 189 190 191 191 191 192 192 192 192 201 201 202 203

14 Smart Health Care Waste Segregation and Safe Disposal 205 R.M. Bommi, Sami Venkata Sai Rajeev, Sarvepalli Navya, Veluru Sai Teja and Uppala Supriya 14.1 Introduction 206 14.2 Related Works 207 14.3 System Architecture 210 14.3.1 Wrapping 212 14.3.2 Incinerator 213 14.3.3 Conveyor Cleaning System 213 14.3.4 Circuit Diagram 213 14.3.5 Optimal Path Planning Algorithm for Waste Collection 214

Contents  xi 14.4 Methodology 14.5 Mobile App 14.6 Conclusions and Future Works Declarations Availability of Data and Materials Competing Interests Author’s Contribution References

214 217 218 219 219 219 219 219

15 Investigation of Viscoelastic Magnetohydrodynamics (MHD) Flow Over an Expanded Lamina Surrounded in a Permeable Media 223 Hiranmoy Mondal, Arindam Sarkar and Raj Nandkeolyar 15.1 Introduction 223 15.1.1 Literature Review 223 15.1.2 Nomenclature 224 15.2 Formulation of the Problem 226 15.2.1 Analytical Solution 227 15.2.2 Numerical Methods (Spectral Quasi-Linearization Methods) 228 15.3 Result and Argument 230 15.4 Conclusion 235 References 236 16 Quickest Multi-Commodity Contraflow with Non-Symmetric Traversal Times Shiva Prakash Gupta, Urmila Pyakurel and Tanka Nath Dhamala 16.1 Introduction 16.2 Preliminaries with Flow Models 16.2.1 Mathematical Model with Contraflow 16.3 QMCCF with Non-Symmetric Transit Times 16.3.1 Approximation Approach for the QMCCF 16.4 Conclusions Acknowledgments References

239 239 242 242 244 245 248 248 248

17 A Mathematical Representation for Deteriorating Goods with a Trapezoidal-Type Demand, Shortages and Time Dependent Holding Cost 251 Ruma Roy Chowdhury 17.1 Introduction 251

xii  Contents 17.2 Assumptions and Notations 17.2.1 Assumptions 17.2.2 Notation 17.2.3 Decision Variables 17.3 Formulation and Solution 17.3.1 Case 1 i) 0 < ν < t1 < t2 < t3 < T 17.3.2 17.3.3

Case 2 ii) 0 < ta < ν < tb < tc < T Case 3 iii) 0 < ta < tb < ν < tc < T

17.4 Numerical Example 17.5 Discussion 17.6 Inference References

253 253 254 255 255 255 257 259 261 261 262 262

18 An Amended Moth Flame Optimization Algorithm Based on Fibonacci Search Approach for Solving Engineering Design Problems 265 Saroj Kumar Sahoo and Apu Kumar Saha 18.1 Introduction 266 18.2 Classical MFO Algorithm 268 18.3 Proposed Method 270 18.4 Results and Discussions on IEEE CEC 2019 Benchmark Problems 273 18.5 Real-Life Applications 277 18.5.1 Optimal Gas Production Capacity Problem 277 18.5.2 Three-Bar Truss Design (TSD) Problem 277 18.6 Conclusion with Future Studies 278 References 279 19 Image Segmentation of Neuronal Cell with Ensemble Unet Architecture 283 Kirtan Kanani, Aditya K. Gupta, Ankit Kumar Nikum, Prashant Gupta and Dharmik Raval 19.1 Introduction 284 19.2 Methods 285 19.3 Dataset 285 19.4 Implementation Details 286 19.5 Evaluation Metrics 286 19.6 Result 286 19.7 Conclusion 289 References 289

Contents  xiii 20 Automorphisms of Some Non-Abelian p−Groups of Order p4 291 Muniya Sansanwal, Harsha Arora and Mahender Singh 20.1 Introduction 291 20.2 Categorization of p-Groups with Order p4 292 20.3 Number of Automorphisms of Some Non-Abelian Groups of Order p4 293 20.3.1 Computation of Automorphisms of Group G9 293 20.3.2 Computation of Automorphisms of Group G10 297 20.3.3 Computation of Automorphisms of G11 300 References 304 21 Viscoelastic Equation of p-Laplacian Hyperbolic Type with Logarithmic Source Term Nazlı Irkıl and Erhan Pişkin 21.1 Introduction 21.2 Preliminaries 21.3 Global Existence Result 21.4 Blow Up Results of the Solution for Equation (21.1) References

305 305 307 314 317 324

22 Flow Dynamics in Continuous-Time with Average Arc Capacities 327 Badri Prasad Pangeni and Tanka Nath Dhamala 22.1 Introduction 327 22.2 Literature Review 329 22.3 Failure in Extension of AP to AAP 329 22.4 Formulation 331 22.5 Conclusion 334 Acknowledgment 334 References 334 23 Analysis of a Multiserver System of Queue-Dependent Channel Using Genetic Algorithm Anupama and Chandan Kumar 23.1 Introduction 23.2 Description of the Model 23.3 Notations 23.4 Steady State Equations 23.4.1 Performance Characteristics of the System 23.4.2 Queue Length Evaluations at Different Epochs 23.4.3 Leisure Period and Working Period Length

337 337 338 338 340 342 343 343

xiv  Contents 23.4.4 Cost of the System 23.5 Conclusions References 24 An Approach to Ranking of Single Valued Neutrosophic Fuzzy Numbers Based on (α, β, γ) Cut Sets Sunayana Saikia 24.1 Introduction 24.2 Definition and Representations 24.3 Proposed Method 24.4 Theorems 24.5 Numerical Examples 24.6 Conclusion References

343 347 347 349 349 351 355 356 357 358 359

25 Performance Analysis of Database Models Based on Fuzzy and Vague Sets for Uncertain Query Processing 363 Sharmistha Ghosh and Surath Roy 25.1 Introduction 363 25.2 Basic Definitions 364 25.2.1 Fuzzy Set 365 25.2.2 Vague Set 365 25.2.3 Similarity Measure 365 25.3 Algorithm to Generate Membership Values 366 25.4 Real Life Applications 368 25.5 Conclusion 382 References 382 26 Estimating Error of Signals by Product Means ( N , pn , qn ) (C, 2) of the Fourier Series in a r W(L , ξ (t))(r ≥ 1) Class 385 Pankaj Tiwari and Aradhana Dutt Jauhari 26.1 Introduction 385 26.1.1 Definition 1 386 26.1.2 Definition 2 386 26.1.3 Definition 3 386 26.1.4 Definition 4 387 26.1.5 Remark 1 387 26.2 Known Result 388 26.2.1 Theorem 1 388 26.3 Main Theorem 389

Contents  xv 26.3.1 Theorem 2 389 26.4 Some Auxiliary Results 390 26.4.1 Lemma-1 390 26.4.2 Lemma-2 390 26.5 Theorem’s Proof 391 26.6 Applications 394 26.6.1 Cor. 1 394 26.6.2 Cor. 2 394 26.7 Conclusion 394 Acknowledgement 395 References 395 About the Editors

397

Index 401

Preface The mathematical sciences are part of nearly all aspects of everyday life. The discipline has underpinned such beneficial modern capabilities as internet searching, medical imaging, computer animation, weather prediction, and all types of digital communications. Mathematics is an essential component of computer science. Without it, you would find it challenging to make sense of abstract language, algorithms, data structures, or differential equations, all of which are necessary to fully appreciate how computers work. In a sense, computer science is just another field of mathematics. It does incorporate various other fields of mathematics, but then focuses those other fields on their use in computer science. Mathematics matters for computer science because it teaches readers how to use abstract language, work with algorithms, self-analyze their computational thinking, and accurately model real-world solutions. Algebra is used in computer programming to develop algorithms and software for working with math functions. It is also involved in design programs for numerical programs. Statistics is a field of math that deploys quantified models, representations, and synopses to conclude from data sets. This book focuses on mathematics, computer science, and where the two intersect, including heir concepts and applications. It also represents how to apply mathematical models in various areas with case studies. The contents include 29 peer-reviewed papers, selected by the editorial team.

xvii

1 A Comprehensive Review on Text Classification and Text Mining Techniques Using Spam Dataset Detection Tamannas Siddiqui and Abdullah Yahya Abdullah Amer* Department of Computer Science, Aligarh Muslim University, Aligarh, UP, India

Abstract

Text data mining techniques are an essential tool for dealing with raw text data (future fortune). The Text data mining process of securing exceptional knowledge and information from the unstructured text is a fundamental principle of Text data mining to facilitate relevant insights by analyzing a huge volume of raw data in association with Artificial Intelligence natural language processing NLP Machine Learning algorithms. The salient features of text data mining are attracted by the contemporary business applications to have their extraordinary benefits in global area operations. In this, a brief review of text mining techniques, such as clustering, information extraction, text preprocessing, information retrieval, text classification, and text mining applications, that demonstrate the significance of text mining, the predominant text mining techniques, and the predominant contemporary applications that are using text mining. This review includes various existing algorithms, text feature extractions, compression methods, and evaluation techniques. Finally, we used a spam dataset for classification detection data and a three classifier algorithm with TF-IDF feature extraction and through that model achieved higher accuracy with Naïve Bayes. Illustrations of text classification as an application in areas such as medicine, law, education, etc., are also presented. Keywords:  Text mining, text classification, spam detection, text preprocessing, text analysis

*Corresponding author: [email protected] Sharmistha Ghosh, M. Niranjanamurthy, Krishanu Deyasi, Biswadip Basu Mallik, and Santanu Das (eds.) Mathematics and Computer Science Volume 2, (1–18) © 2023 Scrivener Publishing LLC

1

2  Mathematics and Computer Science Volume 2

1.1 Introduction Text data mining techniques are predominantly used for extracting relevant and associated patterns based on specific words or sets of phrases. Text data mining is associated with text clustering, text classification, and the product of granular taxonomy, sentiment analysis, entity relation modeling, and document summarization [1]. Prominent techniques in text mining techniques include extraction, summarization, categorization, retrieval, and clustering. These techniques are used to infer distinguished, quality knowledge from text from previously unknown information and different written resources obtained from books, emails, reviews, emails, and articles with the help of information retrieval, linguistic analysis, pattern recognition, information extraction, or information extraction tagging and annotation [2]. Text preprocessing is the predominant functionality in text data mining. Text preprocessing is essential to bring the text into a form that can be predictable and analyzable for text data mining. Text preprocessing can be done in different phases to formulate the text into predictable and analyzable forms. These are namely lowercasing, lemmatization, stemming, stop word removal, and tokenization. These important text preprocess steps are predominantly performed by machine learning algorithms for natural language processing tasks. These preprocessing steps implement data cleaning and transformation to eliminate outliers and make it standardized to create a suitable model to incorporate the text data mining process [3]. Text data mining techniques are predominantly used for records management, distinct document searches, e-discovery, organizing a large set of a text data, analysis and monitoring of understandable online text in internet communication and blogs, identification of large textual datasets associated with patients during a clinical area, and clarification of knowledge for the readers with more extraordinary search experience [4]. Text data mining techniques are predominantly used in scientific literature mining, business, biomedical, and security applications, computational sociology, and digital humanities as shown in Figure 1.1 below. Dimensionality reduction

Features extraction

classification learn model

Predict test data Evaluate model

Figure 1.1  Overview of text classification.

Review on Text Classification and TM Techniques   3 Table 1.1  Text classification compared model classifiers. Model classifiers

Authors

Architecture

Features extraction

SVM and KNN

C. W. Lee et al. [7]

Gravity Inverse Moment

Similarity TF-IDF vectorizer

Wikipedia

Logistic Regression

L. Kumar et al. [13]

Bayesian Logistic Regression

TF-IDF

RCV1-v2

Naïve Bayes NB

A. Swapna et al. [9]

Weight Enhancing Method

Weights words

Reuters-31678

SVM

T. Singh et al. [11]

String Subsequence Kernel

TF-IDF vectorizer

20

Corpus

Newsgroups

The paper reviews text data mining techniques, various steps involved in text preprocessing, and multiple applications that implement text data mining methods discussed in Table 1.1.

1.2 Text Mining Techniques Text Mining (TM) indicates informational content involved in several sources like newspapers, books, social media posts, email, and URLs. Text data summary and classification are typical applications of text mining, particularly among different fields. It is appropriate to discuss some of the techniques applied to achieve them through the step set shown in Figure 1.2 below.

1.2.1 Data Mining Text mining is empowered in big data analytics to analyze unstructured textual data to extract new knowledge and distinguish significant patterns and correlations hidden in the huge amount of data sets. Big data analytics are predominantly used for extracting the information and patterns that are hidden implicitly in the data sets in the form of automatic or semi-­ automatic unstructured formats or natural language texts. To perform this test, mining operations, unsupervised learning algorithms, and supervised learning algorithms or methods are predominantly used. These methods’

4  Mathematics and Computer Science Volume 2 Text collection

Preprocessing Text data Feature Selection

Classification

TM Techniques

Information Extraction

Summarization

Machine Learning Algorithms

Clustering

Information Retrieve

Knowledge

Figure 1.2  Text data mining techniques.

functionality is used for classification and prediction by using a set of predictors to reveal hidden structures in the information database [5]. In this process, text mining is performed using pattern matching on regular documents and unstructured manuscripts [6].

1.2.2 Information Retrieval Information Retrieval [IR] is a prominent method in text data mining techniques. The fundamental principle of IR is identifying documents stored in the database in unstructured formats, which meets the requirements of the information needed from the large collection of documents stored in the datasets. IR is available in three models: Boolean Model, Vector Model, and Probabilistic Model. In text data mining techniques, IR plays a vital role with the indexing system and collection of documents [7]. This method is predominantly used for locating a specific item in natural language documents. IR is used for learned knowledge extraction to convert text within structured data for interesting mining relationships [8]. It has been identified as a big issue to discover the appropriate designs and analyze text records from huge amounts of data. Text data mining technique IR has resolved the issue and successfully selected attractive patterns

Review on Text Classification and TM Techniques   5 from the greatest knowledge data sets. IR techniques are predominantly used for choosing the appropriate text documents from the huge volume of databases with enhanced speed within a short period. The text data mining technique IR extracts the exact required text documents from the greatest databases and presents the accuracy and relevance of results [9].

1.2.3 Natural Language Processing (NLP) NLP linguistics is subfield of computer science and AI. The fundamental principle of NLP is to deal with the connection between computer machines and humans with an assistant of NLP to read, interpret, learn, and make sense of languages spoken by humans in a valuable way. It is powered by AI, which can facilitate the machines to read, understand, interpret, manipulate, and derive meaning from human languages [10]. It is a prominent AI technology used in text data mining to transform the unstructured text depicted in documents and databases into normalized, structured data suitable for performing analysis or implementing machine learning algorithms [11]. Long Short-Term Memory [LSTM] is one of the predominant AI Machine Learning algorithms to remember values with a recurrent neural network’s help. Seq2seq model is another predominant model used in the NLP technique, which works with encoder-decoder structure. In this model, it initially built the vocabulary list to identify the correct grammar syntax. It works with some tags to identify the structured and unstructured language identified in the documents. The named entity recognition model is another predominant model to identify relevant names and classify names by their entity. It is used to find the names of people, names of places, and any other important entity in the given dataset in text or documents. The NLP process features a Preferences’ Graph [12]. It is utilized to build a set of user preferences. While the document is written, the repetitively chosen tense, adjectives, conjunctions, and prepositions are identified and NLP creates a User Preference Graph. Based on the graph, it predicts the next word of the sentence to calculate the probability theory. Word embedding is another inbuilt mechanism obtained from the feature training and language model of NLP. The terms and idioms are planned into a vector of the actual number of graphs during this process. Th ​ e model is designed from big unstructured text to generate the most probable output from the input text searched from the documents’ database [13].

1.2.4 Information Extraction Information Extraction [IE] is the beginning point for a computer to estimate unstructured language documents. IE is predominantly used to

6  Mathematics and Computer Science Volume 2 Unstructured Web Text The second sign of the Zodiac is Taurus. Strokes are the third most common cause of death in America today. No study would be complete without mentioning the largest rodent in the world, the Capybara.

Structured Sequences Sign of the Zodiac: 1. Aries 2. Taurus 3. Gemini... Most Common Cause of Death in America: 1. Heart Disease 2. Cancer 3. Stroke... Largest rodent in the world: 1. Capybara 2. Beaver 3. Patagonian Cavies

Figure 1.3  Information extraction.

distinguish the important relationships involved in the document as shown in Figure 1.3. It is working to distinguish the predefined arrangements in a text with the pattern matching method. Information that cannot be utilized for mining is processed and evaluated by Information Extraction when the documents consist of information. IE is rich with a fundamental mechanism to distinguish the unstructured text available in the articles, blogs, emails, reviews, and other documents with predefined arrangements for post-processing. Post-processing is essential for performing web mining and searching tools. IE is associated with the quality of extracting knowledge from text. IE performs the extracting knowledge from text operations from the unstructured text instead of abstract knowledge. IE performs text mining tasks and further methods to explore the information from the text data in hand [14].

1.2.5 Text Summarization Text summarization is the part of text data mining techniques used to distill the predominant data from a source to generate an abridged version for a specific user and related task. The process of summarization is a stepby-step process. Step one is converting the paragraph into sentences. Step two is text processing. Step three is tokenization. Step four evaluates the weighted occurrence frequency of the words and finally, step five substitutes the words with the respective weighted frequencies. The process of summarization occurs in two ways. These are extractive summarization and abstractive summarization. The fundamental utility of summarization is to reduce the text’s content while preserving the meaning of the text. Abstractive summarization is a technique the summary produces by generating new sentences by rephrasing or using novel words. The process finally extracts the important sentences from the document. The extractive

Review on Text Classification and TM Techniques   7 summarization extracts the important information or sentence from the given text document or original document. A statistical method is needed to be implemented on the original document [19].

1.2.6 Text Categorization The fundamental objective of text categorization in text mining is to sort the documents into groups. This can be done automatically by employing the technique to perform the classification using natural language processing and machine learning. This process categorizes consumer reviews, customer support tickets, blogs, complaints, and other content-rich text documents [20]. Text categorization has several features made from attributes of the documents. Predominantly, the features are developed from words and broader content of the corpus for higher dimensional features. Text categorization plays a vital role in performing news classification, sentiment analysis, and web page classification. Text categorization can be implemented along with Linear Support Vector Machine algorithms to achieve the best results [21].

1.2.7 Clustering In clustering, the main objective is to perform cluster analysis of content-­ based text records. It uses ML and NLP to recognize and classify unstructured text data using extracted descriptors from a targeted document available in the database. Word clustering is part and parceling of text clustering for part sets of information in subsets of semantically related words. It is aimed to perform duties ranging from word sense or basic disambiguation to knowledge retrieval [22]. Unsupervised text clustering is widely used in NLP to group similar data. It obtains the distance between the points. The predominant clustering methods are namely soft clustering and hard clustering. Hard clustering performs the grouping task for every object that belongs to exactly one cluster. Hard clustering allows an object to be grouped with one or more clusters based on its meaning and nature [23].

1.2.8 Information Visualization Information Visualization is meant for information analysis and enables knowledge discovery via interactive graphical representations of textual data. It enables exploration and understanding. It can be formulated in the form of a tag or a word with varied size, position based on frequency,

8  Mathematics and Computer Science Volume 2 categorization, or importance of the tag. Information visualization or data visualization can be performed by employing seeing factors as maps, graphs, and charts to understand the pattern extracted from text data. These main objectives of text visualization are used to summarize large amounts of text, formulate text data that is easy to understand, identify the insights in qualitative data, and discover hidden trends and patterns of the text. Text visualization can be done with the help of machine learning. This can be performed to make sense of qualitative data quickly, easily, and at scale. Word clouds are a great initiation for visualizing qualitative data. World clouds facilitate insights useful for exploratory analysis to learn the insights of the dataset and define labeling criteria for more advanced text analysis. AI Machine Learning algorithms are used for sorting the data into categories. Text visualization enables the extraction of the data’s actionable insights and reveals the trends of the data [24].

1.2.9 Question Answer Question answering is a subfield of information retrieval and NLP linguistic computer science is concerned with creating systems in order to automatically answer questions modeled through human language. Implementing machine learning algorithms for the process of Question/Answer in a full data-driven architecture can produce a practical solution for answering the question with the right short answer rather than giving a list of possible answers. This can be implemented in the NLP to generate an accurate answer with the knowledge database’s text similarity. Implementing machine learning algorithms can produce the administration of the documents, analyze profiles and phrases, and perform marketing activities to generate alerts to post the company branding information. A QA system can be built with the help of python code and facilitate the user to post the question, developing the internal mechanism to extract the question and search for the matching documents with approximate string-matching function from the knowledge base and extract the exact answer for the question as shown in Figure 1.4 below [25]. key words extraction

User Interface

Answering handel

Figure 1.4  Question answering system.

Review on Text Classification and TM Techniques   9

1.3 Dataset and Preprocessing Steps In this review, we are utilizing a ‘spam_ham_dataset’ from Kaggle. This is data set contains a total of 5172 documents with four columns. In the dataset, emails were considered spam (1) or not (0), i.e., unwanted business e-mail. The dataset is split into two parts: 70% is the training set and 30% is the testing set. The experiment was conducted on the required dataset of both the training set and testing set [15].

1.3.1 Text Preprocess Text preprocessing is predominant in text data mining for dimensionality reduction. Once the text is available in the knowledge database, it should be preprocessed to implement the Machine Learning model. It is essential to perform preprocessing on the data with essential steps, namely Tokenization, Lower Casing, Filtering, Stemming, and Lemmatization [16]. The documents available in the knowledge database are described as a vector in a multi-dimensional area. Every single word has a unique dimension discussed below in Table 1.2 [33].

1.4 Feature Extraction Text feature extraction involves carrying out a dictionary of terms from the textual data then converting them into a feature set available to the Table 1.2  Preprocessing steps. Preprocessing

Description

1. Tokenization

Is the method of dividing the sentence within terms [26]

2. Filtering

Token filtering is the process of filtering out any tokens that are not useful for application. The token filtering process eliminates the digits, punctuation marks, stop-word tokens, and other unnecessary tokens in the text [27].

3. Lemmatization

Lemmatization is the process of resolving the term to Lemma, meaning the part of speech of the term. Lemmatization transforms the word into a proper root form with a part of the speech tagger [28].

4. Stemming

Is the method used for transforming a term to its root; this process is associated with the normalization task associated with bluntly removing word affixes [29]

10  Mathematics and Computer Science Volume 2 classifier. Next, we will show some techniques that can be applied to extract features from text data.

1.4.1 Term Frequency – Inverse Document Frequency A difficulty with the BoW method is that the terms with high-frequency display control in the data cannot provide much information about the model. Also, due to that difficulty, control-specified words that have a lower score may be ignored. To resolve that difficulty, the frequency of words is rescaled by analyzing where the words frequently happen compared to the total text document. Here, we applied TF-IDF. Term Frequency measures frequency of the word in a popular text and the Inverse Document Frequency IDF number represents words within the whole text document [30].

1.4.2 Bag of Words (BoW) BoW is common among different feature extraction techniques and it forms a word as the port feature set of total word instance. It is then recognized as a “bag” of words for a process that does not mind the orders of the words or how many times a word occurs, but just represents words already within a record of words discussed below in Table 1.3 [31]. Table 1.3  Advantages and disadvantages of feature extraction. Model

Advantages

Disadvantages

TF-IDF

Simple to calculate Simple to divide the similarity among recognized TFIDF Easy to extract a descriptive word within the text document General items do not impact an outcome because of IDF (for example, “are”, “is”, etc.)

BoW does not catch situations in the text (syntactic) BoW does not catch purpose meaning in the text (semantics)

BoW

Simple to compute Simple to count a similarity among documents using the BoW Easy extraction and metric descriptive words by a document Task within unknown terms

It does not capture the place in the text (syntactic) • BoW does not capture purpose in the text (semantics) • Common terms affect outcomes (for example, “am”, “is”, etc.)

Review on Text Classification and TM Techniques   11

1.5 Supervised Machine Learning Classification In this part, we summarize existing text dataset classification algorithms. We define the KNN [17] algorithm which is utilized for data classification. Then, we describe other methods like logistic regression and Naïve Bayes [18], which are commonly utilized in the scientific community as classification techniques. Classification algorithms associated with the linear support vector machine have achieved high accuracy in the classification of data and performance evaluation text data mining operations.

1.6 Evaluation The experimental evaluation of text classifiers measures efficiency (i.e., capacity to execute the correct classification). Accuracy, recall, and precision are generally employed to measure the effectiveness of a text classifier. Accuracy (FP+FN/TP+TN+FP+FN = 1-accuracy), on the other hand, is not w idely used for text classification applications because it is insensitive to variations in the number of correct decisions due to the large value of the denominator (TP + TN), as discussed below in Table 1.4.

1.7 Experimentation and Discussion Results for Spam Detection Data In the experimental part of the review study, we used spam data to classify files as either spam or non-spam by using machine learning algorithms. Our experimental was done on the Windows system in a Python virtual environment with Anaconda Jupyter to implement the code to build a machine learning model. These steps are executed in five phases. The first phase is to import the library, the second is to load and preprocess the Table 1.4  Confusion matrix in machine learning. Accuracy

TP+TN/TP+TN+FP+FN

Recall

TP/TP+FN

Precision

TP/TP+FP

F1-Measure

2(P*R)/(P+R)

12  Mathematics and Computer Science Volume 2 dataset, the third phase uses TF-IDF feature extraction techniques, the fourth phase is to train and test our model (after dividing the dataset into two parts: 70% training set and 30% testing set to fit the model), and the last phase is to predict and evaluate the model by using amusement precision, recall, accuracy, and f score confusion matrix, as shown in Table 1.5 and Figure 1.5 below. Table 1.5 shows the outcome of three classifier models. Through accuracy, recall, precision, and F1-Measure, Naïve Bayes achieved the higher accuracy at around 97.06% among classifier models for detection of spam files from a data set. As shown in the Figure 1.5, Naïve Bayes achieved higher results and higher improvement accuracy with a data trained spam model using K-Nearest Neighbor and Logistic Regression algorithms.

Table 1.5  Spam dataset classification. Algorithm used

Accuracy (%)

Recall (%)

Precision (%)

F1-measure

Naïve Bayes

97.06

97.76

95.65

97.78

K-Nearest Neighbor

93.12

92.55

92.01

93.09

Logistic Regression

94.89

91.32

93.21

94.76

Spam Dataset classification

98 97 96 95 94 93 92 91 90 89 88

Naïve Bayes Knearest Neighbor Logistic Regression

Accuracy (%)

Recall (%)

Precision (%)

Figure 1.5  Spam dataset classification.

F1‐Measure

Review on Text Classification and TM Techniques   13

1.8 Text Mining Applications Text data mining is predominantly used in different applications. Text mining’s fundamental objective is information extraction, information retrieval, categorization, clustering, and summarization. These activities are helping business applications with extensive operational capacity. The main applications have already implemented text data mining techniques. These are namely risk management applications, knowledge management applications, content enrichment applications, business intelligence applications, contextual advertising applications, customer care services applications, cybercrime prevention applications, spam filtering, data analysis from social media, and fraud detection through claims investigation applications, as shown in Figure 1.6 below [32, 34].

1.9 Text Classification Support 1.9.1 Health Most text data in the health and medical domain displays unstructured patient information by vague terms with typographical errors. Here, the role of text classification process data and dealing with it in order to predict illness uses machine learning algorithms. The medical field consists of selecting medical diagnoses to special class values taken from many categories and this is a field of healthcare where text classification techniques can be deeply valuable [24].

Risk Management

Fraud Detection

Applications of Text Mining

Business Intelligence

Customer Care Services

Social Media Analysis

Figure 1.6  Applications of text data mining [32].

14  Mathematics and Computer Science Volume 2

1.9.2 Business and Marketing Profitable organizations and companies use social networks such as Facebook and Twitter for marketing purposes such as shopping, buying, selling, etc. Through mining and sentiments analysis, businesses know what a customer desires about the producer in order to improve and increase product from the producer. Additionally, text classification in organizations and companies is an important tool for businesses to obtain more customers efficiently.

1.9.3 Law Government institutions have generated large volumes of legal record text documents. Analysis and retrieving that data manually is so difficult. Here, a system is required to deal with information and process it automatically in order to help lawyers and their clients [22]. Organization of those record text documents is the foremost difficulty to the law community. Building a system to classify documents is helpful for the law community.

1.10 Conclusions In this review paper, we present a review of text data mining techniques with relevant descriptions. Text data mining techniques are applied to extract readable knowledge from raw text data sets regularly into the unstructured data. So, in this paper we presented the gist of various previous research works presented on the predominant text mining techniques, namely Data Mining, Text Classification, Information Extraction, Question Answering, Topic Tracking, Natural Language Processing, Information Retrieval, Text Summarization, Text Categorization, Clustering, and Information Visualization. The paper has presented the important mechanism of preprocessing text data mining with four steps: Tokenization, Filtering, Lemmatization, and Stemming. We have discussed the process of text data mining by presenting the fields that use text mining applications. Lastly, we compared the most popular text classification algorithms. Finally, we used a spam dataset for classification detection data and a three classifier algorithm with TF-IDF feature extraction, achieving higher accuracy with Naïve Bayes around 97.06%. Illustrations of the usage of text classification as support for applications in medicine, law, education, etc. are related in a separate part.

Review on Text Classification and TM Techniques   15

References 1. S. S. Kermode and V. B. Bhagat, “A Review: Detection and Blocking Social Media Malicious Posts,” Int. J. Mod. Trends Eng. Res., vol. 3, no. 11, pp. 130– 136, 2016, doi: 10.21884/IJMTER.2016.3133.Q4M8O. 2. M. Allahyari et al., “A Brief Survey of Text Mining: Classification, Clustering and Extraction Techniques,” 2017, [Online]. Available: http://arxiv.org/ abs/1707.02919. 3. C. C. Aggarwal and C. X. Zhai, “Mining text data,” Min. Text Data, vol. 9781461432, pp. 1–522, 2013, doi: 10.1007/978-1-4614-3223-4. 4. M. A. Zende, M. B. Tuplondhe, S. B. Walunj, and S. V. Parulekar, “Text mining using python,” no. 3, pp. 2393–2395, 2016. 5. N. F. F. Da Silva, E. R. Hruschka, and E. R. Hruschka, “Tweet sentiment analysis with classifier ensembles,” Decis. Support Syst., vol. 66, pp. 170–179, 2014, doi: 10.1016/j.dss.2014.07.003. 6. A. Nielsen, “Python programming — text and web mining,” Aust. J. Plant Physiol., vol. 21, pp. 507–516, 2011, [Online]. Available: http://www2.imm. dtu.dk/pubdb/views/edoc_download.php/5781/pdf/imm5781.pdf. 7. C. W. Lee, S. A. Licorish, B. T. R. Savarimuthu, and S. G. Macdonell, “Augmenting text mining approaches with social network analysis to understand the complex relationships among users’ requests: A case study of the android operating system,” Proc. Annu. Hawaii Int. Conf. Syst. Sci., vol. 2016-March, pp. 1144–1153, 2016, doi: 10.1109/HICSS.2016.145. 8. N. U. Pannala, C. P. Nawarathna, J. T. K. Jayakody, L. Rupasinghe, and K. Krishnadeva, “Supervised Learning-Based Approach to Aspect Based Sentiment Analysis,” 2016 IEEE Int. Conf. Comput. Inf. Technol., pp. 662– 666, 2016, doi: 10.1109/CIT.2016.107. 9. A. Swapna, K. G. Guptha, and K. Geetha, “Efficient Approach for Web Search Personalization in User Behavior Supported Web Server Log Files Using Web Usage Mining,” vol. 2, no. 4, pp. 189–196, 10. G. Barbier, “Social Network Data Analytics,” 2011, doi: 10.1007/978-14419-8462-3. 11. T. Singh, M. Kumari, T. L. Pal, and A. Chauhan, “Current Trends in Text Mining for Social Media,” Int. J. Grid Distrib. Comput., vol. 10, no. 6, pp. 11–28, 2017, doi: 10.14257/ijgdc.2017.10.6.02. 12. A. Shahrestani, M. Feily, R. Ahmad, and S. Ramadass, “Architecture for applying data mining and visualization on network flow for botnet traffic detection,” ICCTD 2009 - 2009 Int. Conf. Comput. Technol. Dev., vol. 1, pp. 33–37, 2009, doi: 10.1109/ICCTD.2009.82. 13. L. Kumar and P. K. Bhatia, “Available Online at www.jgrcs.info TEXT MINING: CONCEPTS, PROCESS, AND APPLICATIONS,” vol. 4, no. 3, pp. 36–39, 2013.

16  Mathematics and Computer Science Volume 2 14. K. N. S. S. V Prasad, S. K. Saritha, and D. Saxena, “A Survey Paper on Concept Mining in Text Documents,” Int. J. Comput. Appl., vol. 166, no. 11, pp. 7–10, 2017. 15. p. Anitha and C. Guru Rao, “ Email Spam Filtering Using Machine Learning Based Xgboost Classifier Method,” Int. J. Adv. Comput. Sci. Appl., Vol.12 No.11 (2021), pp. 2182-2190, doi: 10.14569/IJACSA.2015.060121. 16. N. T. and A. M., “Investigating Crimes using Text Mining and Network Analysis,” Int. J. Comput. Appl., vol. 126, no. 8, pp. 19–25, 2015, doi: 10.5120/ ijca2015906134. 17. M. Zareapoor and S. K. R, “Feature Extraction or Feature Selection for Text Classification: A Case Study on Phishing Email Detection,” Int. J. Inf. Eng. Electron. Bus., vol. 7, no. 2, pp. 60–65, 2015, doi: 10.5815/ijieeb.2015.02.08. 18. D. Saxena, S. K. Saritha, and K. N. S. S. V. Prasad, “Survey on Feature Extraction methods in Object,” Int. J. Comput. Appl., vol. 166, no. 11, pp. 11–17, 2017. 19. S. A. Salloum, M. Al-Emran, A. A. Monem, and K. Shaalan, “A Survey of Text Mining in Social Media: Facebook and Twitter Perspectives,” Adv. Sci. Technol. Eng. Syst. J., vol. 2, no. 1, pp. 127–133, 2017, doi: 10.25046/aj020115. 20. M. Neeraja and J. Prakash, “Detecting Malicious Posts in Social Networks Using Text Analysis,” vol. 5, no. 6, pp. 2015–2017, 2016. 21. G. S. L. Vishal Gupta, “A Survey of Text Mining Techniques and Applications,” J. Emerg. Technol. web Intell., vol. 1, no. 1, p. 17, 2009, doi: 10.4304/jetwi.1.1.60-76. 22. M. Sukanya and S. Biruntha, “Techniques on text mining,” Proc. 2012 IEEE Int. Conf. Adv. Commun. Control Comput. Technol. ICACCCT 2012, no. 978, pp. 269–271, 2012, doi: 10.1109/ICACCCT.2012.6320784. 23. T. Siddiqui, A. Y. A. Amer, and N. A. Khan, “Criminal Activity Detection in Social Network by Text Mining: Comprehensive Analysis,” 2019 4th Int. Conf. Inf. Syst. Comput. Networks, ISCON 2019, pp. 224–229, 2019, doi: 10.1109/ISCON47742.2019.9036157. 24. A. Yahya, A. Amer, and T. Siddiqui, “Detection of Covid-19 Fake News text data using Random Forest and Decision tree Classifiers Abstract:,” vol. 18, no. 12, pp. 88–100, 2020. 25. M. R. Begam, “Survey: Tools and Techniques implemented in Crime Data Sets,” vol. 2, no. 6, pp. 707–710, 2015. 26. M. Ramageri, “DATA MINING TECHNIQUES AND APPLICATIONS,” vol. 1, no. 4, pp. 301. 27. G. Nandi and a. Das, “A Survey on Using Data Mining Techniques for Online Social Network Analysis.,” Int. J. Comput. Sci. Issues, vol. 10, no. 6, pp. 162–167, 2013, [Online]. Available: http://search.ebscohost.com/login.aspx?direct= true&profile=ehost&scope=site&authtype=crawler&jrnl=16940784&AN= 93404019&h=MWYzsNVeP2Q8klAFiWFHW3PUpgJLZxRIpB1jfSK4qJfBbaMUEp4nY/oJdYPRHc4xHL0dBYfuGxhZsmiP7ToLBg==&crl=c. 28. T. Siddiqui, N. A. Khan, and M. A. Khan, “PMKBEA: A Process Model Using Knowledge Base Software Engineering Approach,” pp. 5–7, 2011.

Review on Text Classification and TM Techniques   17 29. S. Alami and O. Elbeqqali, “Cybercrime profiling: Text mining techniques to detect and predict criminal activities in microblog posts,” 2015 10th Int. Conf. Intell. Syst. Theor. Appl. SITA 2015, 2015, doi: 10.1109/SITA.2015.7358435. 30. “Build Your First Text Classifier in Python with Logistic Regression | Kavita Ganesan.” https://kavita-ganesan.com/news-classifier-with-logistic-regression-​ in-python/#.X4xyddD7Q7e (accessed Oct. 18, 2020). 31. V. Kumar and L. Velide, “A DATA MINING APPROACH FOR PREDICTION AND TREATMENT Supervised machine learning algorithm:” vol. 3, no. 1. pp. 73–79, 2014. 32. P. C. Thirumal and N. Nagarajan, “Utilization of data mining techniques for the diagnosis of diabetes mellitus - A case study,” ARPN Journal of Engineering and Applied Sciences, vol. 10, no. 1. pp. 8–13, 2015. 33. E. M. G. Younis, “Sentiment Analysis and Text Mining for Social Media Microblogs using Open Source Tools: An Empirical Study,” Int. J. Comput. Appl., vol. 112, no. 5, pp. 44–48, 2015, doi: 10.1093/ojls/gqi017. 34. Amer A.Y.A., Siddiqui T. (2021) Detecting Text-Bullying on Twitter Using Machine Learning Algorithms. In: Bhattacharya M., Kharb L., Chahal D. (eds) Information, Communication and Computing Technology. ICICCT 2021. Communications in Computer and Information Science, vol 1417. Springer, Cham. https://doi.org/10.1007/978-3-030-88378-2_17

2 Study of Lidar Signals of the Atmospheric Boundary Layer Using Statistical Technique Kamana Mishra1* and Bhavani Kumar Yellapragada2 School of Mathematical and Statistical Sciences, Indian Institute of Technology Mandi, Himachal Pradesh, India 2 Department of Space, National Atmospheric Research Laboratory Tirupati, India 1

Abstract

The rotational turbulence caused by mixing the layers of air, wind shear components, mountain waves, aerosol particles, and other pollutants affects the lowest and densest layer of the earth’s surface troposphere. Due to the turbulence, the height of the convective boundary layer (CBLH) changes over the day dramatically. We observe the variation in peak positions of lidar backscatter signals by performing a statistical technique for analyzing the behavior of the convective boundary layer (CBL). After that, to examine the behavior of the whole boundary layer, a distribution method and histogram plots will be used. We provide the statistical method for getting the best fit distribution to show how the result leads to the physical observation of data. Keywords:  Statistical technique, lidar data, distribution plots, convective boundary layer

2.1 Introduction Surface forcing of the planetary boundary layer influences the closest and deepest part of the earth’s atmosphere, i.e., the troposphere. The turbulence in the troposphere causes the high variability of the temperature, moisture, wind shear, and pollutant particles. Due to surface friction, *Corresponding author: [email protected] Sharmistha Ghosh, M. Niranjanamurthy, Krishanu Deyasi, Biswadip Basu Mallik, and Santanu Das (eds.) Mathematics and Computer Science Volume 2, (19–32) © 2023 Scrivener Publishing LLC

19

20  Mathematics and Computer Science Volume 2 winds in ABL are weaker than above and tend to blow towards the area where pressure is low. It is noticeable over the Indian subcontinent that the height of the convective boundary layer (CBLH) is low during the winter and monsoon and high in the sunny days as the warmer lower-layer air mixes with the cool air and convection arises in the summertime only. This behavior indicates that solar heating affects the CBLH, and hence, CBL is also named the daytime planetary boundary layer. The convection occurs by mixing of airs forming bubbles of warmer air or eddies. The small eddies generated by the local wind shear cause turbulence in the surface layer and have a completely random behavior, however, the turbulence in the mixed layer is caused by large bubbles of warmer air, and hence, it is not completely random. Since the turbulent processes are found to be nondeterministic, it is beneficial to use a robust technique like statistical methods to find out the variability in the boundary layer and its key parameter as CBLH and has been studied by Pal, S., Behrendt, A., and Wulfmeyer V. in Elastic-backscatter-lidar-based characterization of the convective boundary layer and investigation of related statistics [3]. The signal which will be used in the study of CBLH is the total backscatter from the atmosphere that represents combined aerosol and molecular backscatter as mentioned earlier by Dang R, Yang Y, Hu X-M, Wang Z, and Zhang S [2]. For using the statistical method, we need the information in terms of data and we choose an instrument with high resolution power named Lidar (Light Detection and Ranging), which is a remote sensing method used in measuring the ranges to the earth and the concept of using Lidar to detect ABL height relies on the assumption that there is a strong gradient in the concentration of aerosols in the CBL versus the free atmosphere. The importance of defining a new technique is that this method works for single scan data and uses spatial average, however, the variance method needs multiple numbers of scans to identify the variance and use time average. After finding the CBLH through statistical technique, we will try to examine the behavior of the whole boundary layer rather than just the CBL part. The histogram plots and distribution method can serve this purpose very well, meaning a list of best fit distribution along with each signal can be used to examine the behavior of the boundary layer, used before by the authors in ‘The Determination of Aerosol Distribution by a No-Blind-Zone Scanning Lidar” [6]. After getting the list of all possible distributions, it will be checked whether there is any range of the signals which follow a specified distribution or if the signals are following the distributions randomly. Now, to find the distribution of the backscatter signals manually for any structure of the boundary layer, a new technique needs to be introduced

Study of Lidar Signals of the ABL Using Statistical Technique  21 which holds in practical life. In this new technique, we will try to use the statistical approach. Lastly, we check the consistency for applying both the statistical techniques by using two different datasets and relate the mathematical result with the physical observations to examine the behavior of ABL.

2.2 Methodology 2.2.1 A Statistical Approach to Determine the CBLH Due to the high-resolution power of Lidar, signals of Lidar can be used in finding the CBLH as Georgoulias, A.K., Papanastasiou, D.K., and Melas, D. et al. used in “Statistical analysis of boundary layer heights in a suburban environment” [4]. There are different types of Lidar techniques that serve a specific purpose, such as temperature profile using vibrational Raman (VR) and rotational Raman (RR). Backscattering Lidar and Elastic-backscatter Lidar (EBL) were used in ‘New Technique to Retrieve Tropospheric Temperature Using Vibrational and Rotational Raman Backscattering’ [5]. In our statistical technique, we do simple differentiation of the signal concerning the height, i.e., tracking the local maxima points, then we do some restrictions on the points of local maxima to indicate the real position of peaks of backscatter signal. The peak position changes with change in values of the restriction set. The peak position changes with change in values of the restriction set given below in Table 2.1. Table 2.1  Restriction set used in algorithm for detecting peak position of lidar signals. Elements of set of restriction

Meaning

Default value

mph

Minimum peak height

None

mpd

Minimum distance between two peaks

1 (positive integer)

threshold

Noise level

0

kpsh

Keep peaks with the same height even if they are closer than `mpd`

False

valley

Local minima

False

22  Mathematics and Computer Science Volume 2

2.2.2 A Statistical Approach to Determine the Best Fit Distribution to the Backscatter Signals of the Lidar Dataset In this technique, we start with the histogram plots which help in deciding which type of distributions can take place and after that, a list of distributions will be prepared, which can be discrete or continuous. Now, to get the best fit distribution one needs to use tests under the null hypothesis to prove that this particular distribution holds well. There are many tests like the chi-square test for independence, ANOVA (analysis of variance), Homogeneity of Variance (HOV), Mood’s Median, Distance Covariance Test (d cov), Kolmogorov-Smirnov test (KS test), Welch’s T-test, Kruskal-Wallis H Test, etc. that are available to use but choosing a test is a crucial part of the procedure. So, after roughly looking at the histogram plots of the data and examining the previous research defined in “Analysis of ocean clutter for wide-band radar based on real data” [7], “A Speckle Filtering Method Based on Hypothesis Testing for Time-Series SAR Images” [8], “Structural Dynamics of Tropical Moist Forest Gaps” [9], and “Non-cooperative signal detection in alpha stable noise via Kolmogorov-Smirnov test” [10], it can be observed that the data follows continuous distribution and since the Kolmogorov-Smirnov test (KS test) can be applied only for the data which follows continuous distribution and has a limitation that the distribution must be fully specified, the k-s test is the best option according to our dataset. Now, after defining the list of possible distributions and choosing the test, the percentile bins will be defined so that we can make use of a particular range of the data rather than the exact dataset. An increment in the number of percentile bins results in a more accurate distribution type. That is why we check whether the result holds for a large number of bins or not. After that, observed frequency and expected frequency is calculated to formulate the chi-square statistics. The fitness of distributions will be sorted based on p-value, which is calculated from the hypothesis and the chisquare statistics. Since all the parameters are defined here, to use this test for a large number of columns a function can be created and by giving the number of columns, one can get the name of the best fit distribution. Now, we can plot the histogram with the best-fit distribution curve to check whether our formulation of function holds or not. One can also get distribution with its value of scale and shape parameters using the format function which will be helpful to make a graph of the distribution corresponding to its parametric values.

Study of Lidar Signals of the ABL Using Statistical Technique  23

2.3 Mathematical Background of Method This new technique of finding CBLH is defined on the behalf of mathematics and it will pave the way for using the statistical methods more often in the future for observing the behavior of atmospheric data. To use the statistical approach, a dataset of backscatter signals is required and Lidar helps us get this dataset. Detecting the peaks of the signals implies getting a set of those points from the Lidar dataset, at which there is a steep gradient and differentiation serves this purpose very well. Since the values of this dataset are completely random and defining a function is not possible at all, we find the points of local maxima through the graph. The slope is zero at the points where there is local maxima or minima. Let us define this collection of local maxima and local minima points by A and set of local maxima points by B. Now, let a ∈ A. If the slope is positive ∀ y ∈ (a-δ, a) and negative ∀ y ∈ (a, a+ δ), then a ∈ B and if the slope is negative ∀ y ∈ (a-δ, a) and positive ∀ y ∈ (a, a+ δ), then a ∈ A\B. After getting elements of B, we will move to the next step which means checking whether the point of local maxima is peak or not because the first derivative has a downward zero-crossing at peak maximum. But, in real-life experiments there is some random noise that produces the false peak points and the result will be affected if we include these false positions of peaks also. To avoid this problem, one has to put some restrictions which are defined earlier in the methodology. After that, a statistical technique will be used to find the distribution which fits best to the given data. The histogram is a helpful tool to figure out what kind of distribution data we have. Hence, by roughly analyzing the histogram plot, one can define a set of possible distributions (continuous or discrete) which can fit the data and choose the test accordingly. We will make use of the KS test here, which decides whether the sample comes from a population with specific distribution and is based on the empirical distribution function (ECDF). If N data points are given, then firstly order them from smallest to largest value, which can be denoted as X1, X2, …, XN. Then, ECDF can be defined as:



EN = n(i)/N

where n(i) is the number of points that are less than Xi. K-S Test: (Null hypothesis) H0: data follows given distribution (Alternative hypothesis) H1: data does not follow the given distribution function

(2.1)

24  Mathematics and Computer Science Volume 2 Test Statistic: This test is defined as:



i −1 i   = D max  F ( Xi ) − , − F ( Xi )  1≤i ≤N  N N 

(2.2)

where F is the cumulative distribution function of the distribution that will be tested with the condition that it must be continuous distribution, as well as fully specified, which means parameters (such as local, shape, and scale) cannot be calculated from the dataset. Significance level: α Critical Value: We reject the null hypothesis if the value of D is greater than the critical value obtained from the table and this value can be written as a p-value. So, a p-value based on the approximation of the distribution of the test statistics under the null hypothesis is calculated. Based on the p-values and chi-square statistics, we will sort the distribution starting from the best fit distribution. Since all the parameters have been termed, we can define a function using all these, send the value of the data column-wise, and then get the desired best fit distribution. One remark needs to be stated here: more than one distribution can be fit to the given pattern of data with a very small difference in their p-values. In that case, we can take the top three distributions and get a distribution that fits for some particular range of columns. This analysis emphasizes the databases with various forms of utterances in Tamil language for the applied classification technique and the obtained accuracy with the technique used to extract the feature. The number of samples which have been taken for this are tabulated in escalating order.

2.4 Example and Result We will now consider an example of the Lidar dataset of backscatter signals of the boundary layer and scans are taken for every second for the time interval of ten minutes. In Figure 2.1, the temporal variation of Lidar signals shows the rise of boundary layer thermals. We have taken here the scans for a dataset in the interval 09:24-09:34. In ‘Analyzing the atmospheric boundary layer using high-order moments obtained from multiwavelength lidar data: impact of wavelength choice’ the authors have made

Study of Lidar Signals of the ABL Using Statistical Technique  25 3

Range (km)

2.5 2 1.5 1 0.5 09:24 09:25 09:26 09:27 09:28 09:29 09:30 09:31 09:32 09:33 Time (IST)

×106 9 8 7 6 5 4 3 2 1 0

Figure 2.1  Temporal variation of lidar signal for dataset-1.

the same figures for the larger time period from 13:00 to 19:00 UTC and from 17:00 to 21:00 UTC [1]. The reason for taking this time interval is that the turbulence in the CBL increases with the rise in sunrays, i.e., in the morning and due to the temperature variation boundary layer thermal increases. However, if we take the time-interval in the night, then turbulence ceases with the decrease in the amount of solar heating. After using the statistical technique for all the backscatter signals, it is observed that after a certain height of signal, data behaves similarly for all the backscatter signals, which means the point at which there is a steep gradient approximately has the same location for each signal. This means that after a certain height of the layer, less presence of aerosols and other pollutant particle ceases the turbulence. Hence, a part of the backscatter signal in which the turbulence is very high is used rather than using the whole range. After getting the peak points for each scan, we make a graph of peak points along with scan number for a time-interval. Now, if we compare these two figures, then it can be seen that the graph of peak points detected using the statistical technique matches the figure containing the rise of boundary layer thermals. When one observes graphs of the peak points for each signal in the included excel sheet, it can be easily seen that in starting there is a peak point near the first peak and after some signals, there is no point of the peak near the first peak point. Again, there is a peak point found near the first one which shows the dissimilarity in the location of peaks of the backscatter signals. From Figure 2.2, one can notice that peak points are randomly distributed over the interval. Also, the changes in peak height are not following any particular pattern. This result shows that the changes in the part of the surface layer of CBL do not follow any identified pattern and hence cannot be organized into any predefined structure.

26  Mathematics and Computer Science Volume 2

Altitude (m)

2000

1500

0924 0925 0926 0927 0928 0929 0930 0931 0932 0933 Local Time (LT)

Figure 2.2  Peak points with scan numbers for dataset-1.

However, after excluding a certain range, peak points attain the approximately same position, which physically shows that the turbulence above this shallow layer in the mixed layer is not completely random and can be organized into some predefined structure like plums and thermals. This method seems to be consistent and to confirm this we will take another dataset with the time interval 09:34-09:44, having the same interval length. The same process will be used here and after getting the peak points for each scan, we will make a graph of peak points for dataset-2 along with scan number. This method seems to be consistent and to confirm this we will take another dataset with the time interval 09:34-09:44, having the same interval length as shown in Figure 2.3. The same process will be used here and after getting the peak points for each scan, we will make a graph of peak points for dataset-2 along with the scan number as shown in Figure 2.4. 3

Range (km)

2.5 2 1.5 1 0.5 09:34 09:35 09:36 09:37 09:38 09:39 09:40 09:41 09:42 09:43 09:44 Time (IST)

Figure 2.3  Temporal variation of lidar signal for dataset-2.

×106 9 8 7 6 5 4 3 2 1 0

Study of Lidar Signals of the ABL Using Statistical Technique  27

Altitude (m)

2000

1500

0934 0935 0936 0937 0938 0939 0940 0941 0942 0943 LOCAL TIME (LT)

Figure 2.4  Peak points with scan numbers for dataset-2.

In this, if we compare both the figures then the peak points clearly show the temporal variation of Lidar signals. Hence, the result shows the consistency of the statistical method used here and this method can be utilized in detecting peak positions of backscatter signals. After using the statistical technique (k-s test statistic), the distributions which fit the dataset of the backscatter signals have been found and a list of them along with the histogram plot, as well as the graph of best-fit distribution in an excel file are included in the link. Now, if we analyze the graphs up to the two-thirds part of the dataset, gamma or beta distribution fits in its maximum part, which has an almost same graph, and in the remaining one-third of the dataset, t-distribution fits perfectly. However, a few datasets perform the log norm, chi-square, inverse Gaussian, and Pearson 3 distributions. If we use the same technique for dataset-2 and analyze the graphs, the whole dataset follows t-distribution. This is probably happening because we have taken this dataset for the time-interval which is after the time for the first dataset.

2.5 Conclusion and Future Scope The observations we have obtained using a statistical technique for the scans are the same as the temporal variation of Lidar signals. We have checked it using another dataset. Hence, the method gives a consistent result and can be utilized in detecting peak positions of backscatter signals. After that, we move to analyze the behavior of the whole boundary layer rather than only the CBL part using the distribution method for scans. Then, it is noticeable

28  Mathematics and Computer Science Volume 2 that the changes in the distribution with some particular ranges of signals are happening due to the physical process involved in that. Sometimes we get a broader spectrum and sometimes a narrower spectrum. The reason for this change is that the broader spectrum indicates turbulence process and wind variability, whereas the narrower spectrum indicates temperature inversion and is associated with stable layer phenomenon. The fitting of distribution is affected by thermal heating, water vapor condensation, and other physical processes involved during the day. So, we have examined the natural environment and now this study leads us to examine the changes in the manmade environment. The consistency of this method is paving the way to use it in different fields such as hydrodynamic modeling, storm surge modeling, etc.

Acknowledgement I would like to express my gratitude to the team of Lidar project of NARL, Tirupati, India for providing the data for the research work.

References 1. de Arruda Moreira, G., da Silva Lopes, F. J., Guerrero-Rascado, J. L., da Silva, J. J., Arleques Gomes, A., Landulfo, E., and Alados-Arboledas, L., 07 Aug 2019. Analyzing the atmospheric boundary layer using high-order moments obtained from multi- wavelength lidar data: impact of wavelength choice, Atmos. Meas. Tech., 12, 4261– 4276, https://doi.org/10.5194/ amt-12-4261-2019. 2. Dang R, Yang Y, Hu X-M, Wang Z, Zhang S, 2019. A Review of Techniques for Diagnosing the Atmospheric Boundary Layer Height (ABLH) Using Aerosol Lidar Data. Remote Sensing. 11(13):1590. https://doi.org/10.3390/ rs11131590. 3. Pal, S., Behrendt, A., and Wulfmeyer, V., 2010. Elastic-backscatter-lidarbased characterization of the convective boundary layer and investigation of related statistics, Ann. Geophys., 28, 825–847, https://doi.org/10.5194/ angeo-28-825-2010. 4. Georgoulias, A.K., Papanastasiou, D.K., Melas, D. et al., 2009. Statistical analysis of boundary layer heights in a suburban environment. Meteorol Atmos Phys 104, 103– 111. https://doi.org/10.1007/s00703-009-0021-z. 5. Su, J., McCormick, M. P., & Lei, L. (2020). New Technique to Retrieve Tropo­ spheric Temperature Using Vibrational and Rotational Raman Backscattering. Earth and Space Science, 7. https://doi. org/10.1029/2019EA000817.

Study of Lidar Signals of the ABL Using Statistical Technique  29 6. Wang, J.; Liu, W.; Liu, C.; Zhang, T.; Liu, J.; Chen, Z.; Xiang, Y.; Meng, X, 2020. The Determination of Aerosol Distribution by a No-Blind-Zone Scanning Lidar. Remote Sens. 12, 626. https://doi.org/10.3390/rs12040626 7. QingWeiPing, 2011. Analysis of ocean clutter for wide-band radar based on real data https://doi.org/10.1145/2071639.2071669 8. Yuan, J.; Lv, X.; Li, R, 2018. A Speckle Filtering Method Based on Hypothesis Testing for Time-Series SAR Images. Remote Sensing, 10, 1383. https://doi. org/10.3390/rs10091383. 9. Hunter MO, Keller M, Morton D, Cook B, Lefsky M, Ducey M, et al. (2015) Structural Dynamics of Tropical Moist Forest Gaps. PLoS ONE 10(7): e0132144. https://doi.org/10.1371/journal.pone.0132144 10. J. Luo, S. Wang, E. Zhang and J. Luo, 2015. “Non-cooperative signal detection in alpha stable noise via Kolmogorov-Smirnov test” 2015 8th International Congress on Image and Signal Processing (CISP), 2015, pp. 1464-1468, doi: 10.1109/CISP.2015.7408114.

30  Mathematics and Computer Science Volume 2

Annexure Flowchart and Algorithm for peak detection Flowchart: START

Input mph, mpd

No for j=1, j≤=n, Yes Read data of jth column

x=[data], size = size(x), dx= x[1:]-x[:1]

No for i=1, jmph Yes ZK = value Yes ZK+1 - ZK ≥mpd

No Yes Exit

Peak_value = value

Study of Lidar Signals of the ABL Using Statistical Technique  31 Algorithm: Step-1. Start Step-2. Input the minimum peak height (mph) and minimum distance between two peaks (mpd). Step-3. Start a for loop to read columns of multidimensional data. Step-4. Read the data of the column if ‘for’ loop is true else, go to step 11. Step-5. Introduce a variable x to store the data in an array form. Then, use the size function to find the size of x and after that, use the differentiation function to differentiate the variable x. Step-6. Again, start a ‘for’ loop for data stored in x. Step-7. Track local maxima of data if for loop is true else, go to step 4 and restart the process for the next column. Step-8. If it is a point of maxima then, check whether its value is greater than mph or not. If yes, go to step otherwise go to step 7. Step-9. Introduce a new variable Zk and store the value in it. Step-10. If difference between two local maxima i.e. Zk+1 - Zk is greater than mpd then, assign this value as peak value else go to step 7. Step-11. Exit. Algorithm for finding the distribution: Step-1. Start. Step-2. Make a function in which column number will be passed. Step-3. Define a variable which takes the data of the column and size of the data. Step-4. Input the list of possible distributions to find the best one among them. Step-5. Input the number of percentile bins. Step-6. Calculate the observed and expected frequency. Step-7. Calculate the p-value by using the predefined Kolmogorov-Smirnov (KS) Test Step-8. Calculate the value of chi-square statistics by using the formula as: χ2 =

∑

(Oi − ei )2 where Oi and are observed and expected frequencies Oi

respectively. Step-9. Pass the number of columns in the function. Step-10. Plot the histogram with the graph of best fit distribution for cross-checking the result. Step-11. Stop.

3 Optimal Personalized Therapies in Colon Cancer Induced Immune Response using a Fokker-Planck Framework Souvik Roy* and Suvra Pal Department of Mathematics, The University of Texas at Arlington, Arlington, TX, USA

Abstract

In this paper, a new stochastic framework to determine optimal combination therapies in colon cancer-induced immune response is presented. The dynamics of colon cancer are described through an It¨o stochastic process, whose probability density function evolution is governed by the Fokker-Planck equation. An open-loop control optimization problem is pro- posed to determine the optimal combination therapies. Numerical results with combination therapies comprising of the chemotherapy drug Doxorubicin and immunotherapy drug IL-2 validate the proposed framework. Keywords:  Fokker-Planck optimization, non-linear conjugate gradient, immunotherapy, chemotherapy

3.1 Introduction Colon cancer is a leading cause of global cancer related deaths [11]. The lack of early symptoms forces the detection of colon cancer to take place at the metastatic phase of the cancer [29]. Thus, it becomes important to devise fast and accurate treatment strategies for a cure. In this context, combination therapies have been clinically shown to be an effective strategy for combating cancer in comparison to monotherapy (see [18]). Conventional monotherapeutic techniques are indiscriminate in choosing actively growing cells that lead to the death of not only cancerous cells but also healthy cells. E.g., chemotherapy drugs can be toxic and lead to multiple side effects and risks by weakening the *Corresponding author: [email protected]; ORCID: 0000-0002-6134-1381 Sharmistha Ghosh, M. Niranjanamurthy, Krishanu Deyasi, Biswadip Basu Mallik, and Santanu Das (eds.) Mathematics and Computer Science Volume 2, (33–48) © 2023 Scrivener Publishing LLC

33

34  Mathematics and Computer Science Volume 2 patient’s immune system, targeting the bone-marrow cells [22]. This leads to increased susceptibility to secondary infections. However, with combination therapies, even though it might still be toxic if there is a presence of a chemotherapy drug, the toxicity effect is significantly diminished due to different targets being affected. Moreover, since the effect of combination therapies work in synergy, lower dosages of the individual drugs in the combination therapy are required for treatment, which further reduces the toxic effects [19]. To develop optimal combination therapies in colon cancer, it is important to understand relevant biomarkers that govern the progression of the cancer [8]. One such biomarker is the immune response to the cancerous cells. It has been observed that onsite immune reactions at the tumor locations and prognosis are highly correlated and independent of the size of the tumor [12]. Furthermore, a high expression of various immune pathways like Th1 and Th17 are associated with a poor prognosis or prolonged disease-free survival for patients with colon cancer [32]. Thus, it becomes important to develop optimal personalized treatments for colon cancer patients, taking into account the various types of immune cells, their numbers, and interaction with the cancer cells and each other. Since drugs for treatment of colon cancer like chemotherapeutic drugs and immunotherapies are quite costly, it is expensive to perform in-vitro and vivo experimental studies for testing effects of such drugs. An alternative cost effective option is to develop computational frameworks for testing optimal combination drug dosages [7]. This can be done using pharmacokinetic models that are given by a set of ordinary or partial differential equations (ODE or PDE). There are several dynamic models that have been developed to describe the immune response and interactions in colon cancer and associated therapies. In [4], a combined compart- mental model with that of irinotecan is used to describe the physiology of colon cancer. A set of ODEs were used to represent a 3D structure of colon cancer in [14]. In [15], the authors use describe the phenomenon of tumorigenesis in colon cancer using mathematical modeling. The authors in [17], describe the initiation of colon cancer and its link with colitis using a mathematical framework. In [23], the authors use a pharmacokinetic cellular automata model to incorporate the cytotoxic effects of chemotherapy drugs. In [13], dynamical systems are used to describe multiple pathways in colon cancer. For a detailed review of various mathematical models for colon cancer-induced immune response, we refer the reader to [3]. We contribute to the field of pharmacokinetic cancer research by presenting an effective approach to develop personalized therapies for colon cancerinduced immune response. The starting point of this estimation process is the dynamic model for colon cancer induced immune response, given in [10]. The model describes the evolution of four variables: the tumor cell count, concentration of the Natural Killer cells, concentration of the CD8+ cells, and

Therapies in Colon Cancer using FP Framework  35 concentration of the remaining lymphocytes. We extend the dynamic model in [10] to an Itô stochastic ODE system that takes into account the randomness of the colon cancer-induced immune response dynamics. We also model a combination therapy comprising of a chemotherapy drug and an immunotherapy drug as a control in the stochastic ODE system. The goal is to obtain an optimal combination therapy strategy that drives the cancer-induced immune response state. Since solving an optimal control problem using stochastic state equations is difficult, we use a more convenient framework for obtaining optimal combination therapies through the Fokker-Planck (FP) equations that govern the evolution of the joint probability density function (PDF) associated with the random variables in the stochastic process. Such FP control frameworks have been used for problems arising in control of collective and crowd motion [24, 25], investigating pedestrian motion from a game theoretic perspective [26], reconstructing cell-membrane potentials and mean field control problems [5], and controlling production of Subtilin [31]. Very recently, in [27, 28], FP frameworks were used for parameter estimation in colon cancer induced dynamic systems. Until now, this is the first work that considers the FP framework to devise optimal combination treatment strategies for colon cancer induced immune response. In the next section, we describe an FP control framework for devising optimal treatments in colon cancer induced immune response systems. Section 3.3 is concerned with the theoretical properties of the FP optimization problems. In Section 3.4, we describe the numerical discretization of the optimality system. In Section 3.5, we obtain the optimal combination therapies comprising of Doxorubicin and IL-2 for two types of simulated colon cancer patients and show the correspondence of the results with experimental findings. We end with a section of conclusions.

3.2 The Control Framework Based on Fokker-Planck Equations The interaction between immune cells and colon cancer growth in a patient can be modeled using a coupled system of ODEs. The starting point is the set of equations given in the paper by dePillis [10]. The following are the variables associated with different types of cell populations that appear as model variables. 1. T  (τ)- tumor cell population number (cells). 2. N(τ)- natural killer (NK) cells concentration per liter of blood (cells/L).

36  Mathematics and Computer Science Volume 2 3. L  (τ)- cytotoxic T lymphocytes (CD8+) concentration per liter of blood (cells/L). 4. C(τ)- other lymphocytes concentration per liter of blood (cells/L). The governing system of ODEs representing the dynamics of the above defined cell populations is given as follows.



dT T0 , = aT − abT 2 − DT − cNT − α1u1 (t )T , T (0) = dτ dN = eC − fN − pNT − α 2u1(t )N + β2u2 (t )N , N (0) = N0 , dτ T dL L − qLT + (r1N + r2C )T − α 3u1(t )L = mL + j k +T dτ + β3u2 (t )L, L(0) = L0 , dC = α − β C − α 4u1(t )C, C(0) = C0 , dτ

(3.1)

1 ,T0 , N 0 , C0 represent the initial conditions for T, N, s(T /L )l + 1 L, and C, respectively, and u1, u2 represent dosages of Doxorubicin and IL-2, respectively. The parameters of the system (3.1) are defined in [10]. To stabilize the solutions of the numerical methods, we non-dimensionalize the ODE system (3.1) using the following non-dimensionalized state variables and parameters: where D = d

= T k= k2= N , L k= k4C , t = k5τ , 1T , N 3 L, C d c b f a ek 1010 p = ,b = ,c = ,d , e = 2= ,f = ,p , k5 k2k5 k1k5 k1 k5 k5 k4 k5 j m r2k3 r1k3 108 q k q m = , r2 , = = ,j = , k k= , , r1 = = 1 k5k1 k5 k5 k1k4 k5 k1k2k5 α k4 β αi βi s 250 s, α ,β l l, α= , i 1= ,. . ., 4, βi i = 1, 2. = = = = i = k5 k5 k5 k5 = a

(3.2)

Therapies in Colon Cancer using FP Framework  37 Then, the transformed non-dimensional ODE system is given as follows:

dT = aT − abT 2 − DT − cNT − α1u1(t )T , T (0) = T0 dt dN = eC − fN − 10−10 pNT − α 2u1(t )N + β2 u2 (t )N , = N (0) = N0 , dt T dL L − 10−8 qLT + ( r1 N + r2 C)T − α 3 u1(t) L = mL + j k +T dt + β 3 u 2(t) L , L (0) = L0 , dC = α − β C − α 4 u1 (t )C , C (0) = C0 , dt

(3.3)



(L / T )ll where D = d . 4 s ⋅10 ⋅ (k1 / k3 ) + (L / T )

The compact form of the aforementioned system of ODEs, given in (3.3), is as follows:



dX = F ( X , U ), dt X (0) = X 0 ,

(3.4)

T where X (t ) = (T (t ), N (t ), L (t ), C(t )) . We extend the ODE system (3.3) to include stochasticity present in the dynamics. For this purpose, we consider the Itô stochastic differential equation corresponding to (3.3).

= dT (aT (1 − bT ) − cNT − DT − α1u1(t )T )dt + σ 1(T )dW1(t ), T (0) = T0 , dN = (eC − fN − 10−10 pNT − α 2u1(t )N + β2u2 (t )N )dt + σ 2 (N )dW2 (t ), N (0) = N 0 , T L − 10−8 qLT + (r1N + r2C )T − α 3u1(t )L dL = (mL + jT k +T + β3u2 (t )L)dt + σ 3 (L )dW3 (t ), L (0) = L0 , dC = (α − β C − α 4u1(t )C )dt + σ 4 (C )dW4 (t ), C (0) = C0 , 



(3.5)

38  Mathematics and Computer Science Volume 2 where dWi, i = 1, 2, 3, 4 are one-dimensional Wiener processes and σi, i = 1, 2, 3, 4 are positive constants. Equation (3.5) can be written using a compact notation as follows:



dX = F (X, U) dt + σ(X) dW (t),



X(0) = X0,

(3.6)

where



dW (t) = (dW1(t)  dW2(t)  dW3(t)  dW4(t))T

is a four-dimensional Wiener process vector with stochastically independent components and

σ = diag (σ1  σ2  σ3  σ4)



is the dispersion matrix. We now describe the PDF of the stochastic process (3.6), confined in a Lipschitz domain Ω by virtue of a reflecting barrier on ∂Ω. This is motivated by the maximum cell carrying capacity. Thus, X(t) ∈ Ω ⊂ R4+= {x ∈ R4 : xi ≥ 0, i = 1, 2, 3, 4}, Let x = (x1, x2, x3, x4)T. Define f (x, t) as the PDF for the stochastic process described by (3.6), i.e., f (x, t) is the probability of X(t) assuming the value x at time t. Then, the PDF of X(t) evolves through the following Fokker-Planck (FP) equations.

∂ t f (x , t ) + ∇ ⋅ (F (x , U ) f (x , t ))=

f (x , 0) = f 0 (x ),

1 ∇ ⋅ (σ 2 (x )∇f (x , t )), 2

(3.7)

where f0(x) is non-negative with mass equals one and U (t ) = (u1 (t ), u2 (t )) in the admissible set



U ad = {U ∈L2 ([0,T ]): 0 ≤ ui (t ) ≤ Di , Di > 0, ∀t ∈[0,T ], i = 1,2}

Here, the FP domain is Q = Ω × (0, Tf ), where Tf is the final time and f0(x) represents the distribution of the initial state X0 of the process. The FP equation (3.7) is associated with the no-flux boundary conditions. To describe this, we write (3.7) in flux form as

Therapies in Colon Cancer using FP Framework  39

∂tf (x, t) − ∇ · H = 0, f (x, 0) = f0(x),

(3.8)

where the components of the flux H is given as follows:



H j (x , t ; f= )

σ 2j 2

∂ x j f − Fj (x , U) f , j = 1, 2, 3, 4.

(3.9)

Then, the no-flux boundary conditions are



H ⋅n = 0 on ∂Ω × (0, T f ), ^

(3.10)

with nˆ as the unit outward normal on ∂Ω. To obtain the optimal combination therapy function vector U, we solve the following optimization problem: = U * arg = minU∈Uad J ( f , U) :



ν T ν T α ( f (x , t ) − f * (x , t ))2 dx + 1 ∫ u1 (t ) 2 dt + 2 ∫ u2 (t ) 2 dt , ∫ 2Q 20 2 0



(3.11) subject to the FP system (3.7),(3.10), where the desired PDF is f *(x, t).

3.3 Theoretical Results In this section we describe some theoretical results related to the minimization problem (3.11). One can also find similar results in [1, 24, 25]. For this purpose, we denote the FP system (3.7),(3.10) as E(f0, U) = 0. The existence and uniqueness of solutions of (3.7) is given in the following proposition. Proposition 3.1. Assume f0 ∈ H1(Ω) with f0 non-negative and U ∈ Uad. Then, there exists a unique non-negative solution of E(f0, U ) = 0, given by f ∈ L2([0, Tf ]; H1(Ω)) ∩ C([0, Tf ]; L2(Ω)). Under the assumptions of higher regularity of ∂Ω and the boundary of Ω, one can also obtain H2(Ω) regularity of the solution of (3.7) (see [30]). Next, we state the conservativeness property of (3.7), which can be proved using straightforward applications of weak formulation, integration by parts, and divergence theorem for the flux H.

40  Mathematics and Computer Science Volume 2 Proposition 3.2. The FP system given in (3.7),(3.10) is conservative. The next proposition states and proves the L2 stability property of (3.7). Proposition 3.3. The FP system (3.7),(3.10) solution, given by f, satisfies the following L2 stability property:



|| f (t )||L2 ( Ω ) ≤ || f0 ||L2 ( Ω ) exp(|| σ −1 ||22 N 2t ),



(3.12)

where N = supΩ×U |F (x, θ)|. Proof. Multiplying (3.7) with the test function ψ = f (·, t) and an integration by parts gives:

∂ = − ||σ∇f (t )||2 2 + 2Ω (F (t )) ⋅ σ −1σ∇f (t )dx. (3.13) || f (t )||2 2 L ( Ω) L (Ω) ∂t The last term in (3.13) can be estimated using the Young’s inequality, 2bd ≤ kb2 + d2/k, with k = || σ−1 ||2, which is the L2 matrix norm of σ−1. We then obtain the following:



∂ || f (t )||22 ≤ ||σ −1 ||2 N 2 || f (t )||22 2 L (Ω) L (Ω) ∂t

Applying the Gronwall’s inequality gives the desired result. The above results imply that the map Λ : Uad → C([0, Tf ]; H1(Ω)), given by f = Λ(U), is continuous and Fréchet differentiable. The main theoretical result in this work, which shows that there is an optimal open-loop control U*, is given below: Theorem 3.1. Let f0 ∈ H1(Ω) and let J be given as in (3.11). Then, there exists (f ∗, U ∗) ∈ C([0, Tf ]; H1(Ω)) × Uad with f ∗ being a solution to E(f0, U ∗) = 0 and U ∗ minimizing J in Uad. Proof. Since J is bounded, a minimizing sequence (Um) exists in Uad. Moreover, J is being coercive and sequentially weakens lower semi-continuous in Uad, implying the boundedness of this sequence. Due to the fact that Uad is a closed and convex subset of a Hilbert space, the sequence (Um) contains a convergent subsequence (U ml) in Uad, such that U ml → U ∗. Correspondingly, the sequences (f ml) and (∂t  f ml), where fml = Λ(θml) are bounded in L2([0, Tf ]; H1(Ω)), L2([0, Tf ]; H−1(Ω)), respectively. This implies the weak convergence of the sequences to f ∗ and ∂t f ∗, respectively.

Therapies in Colon Cancer using FP Framework  41 We next use the compactness result of Aubin-Lions [16] to obtain strongly convergent subsequence (f mk) in L2([0, Tf], L2(Ω)). Thus, the sequence (F (U mk)fmk) in L2([0, Tf], L2(Ω)) is weakly convergent. This implies that f ∗ = Λ(U ∗) and (f ∗, U ∗) are minimizers of J. For the minimization problem (3.11), the optimality system can now be written as ∂ t f ( x , t ) + ∇ ⋅ (F ( x , U ) f (x , t ))= f (x , 0) = f 0 ( x ), in Ω, ^ 0, H ⋅n =

1 2

∇ ⋅ (σ 2∇f (x , t )), in Ω × (0, Tf ), (FOR )

on ∂Ω × (0, Tf ).



1 −∂t p(x , t ) − f (x , t )(F (x ,θ ) ⋅∇p(x , t )) − ∇ ⋅ (σ 2∇p(x , t )) 2 * = −α ( f (x , t ) − fi (x , t )), in Ω × (0,T f ), p(x ,T f ) = 0, in Ω,

∂p = 0, on ∂Ω × (0,T f ). ∂n

∫

βU − ∇U F ⋅∇p,ψ − U

L2 ([0,T ])

≥ 0, ∀ψ ∈U ad .



(ADJ)

(OPT)

Ω

The optimality system comprises of three sets of equations: the forward or state equation that governs the FP dynamics (FOR), the adjoint equation (ADJ), and the optimality condition (OPT). In the next section, we describe numerical schemes to implement the optimality system.

3.4 Numerical Schemes We consider the mesh {Ωh}h>0 given by

h = {(x1, x2, x3, x4) ∈ R4 : (x1i, x2j, x3k, x4l) = (x10 + ih, x20 + jh, x30 + kh, Ω x40 + lh)}, where (i, j, k, l) ∈ {0, , N x1 } × {0,  , N x2 } × {0, , N x3 } × {0,  , N x4 } ∩ Ω, and Nxi is the number of discretization points along the ith coordinate

42  Mathematics and Computer Science Volume 2 direction. Define δt = Tf /Nt to be the temporal discretization step, where Nt denotes the maximum number of temporal steps. This gives us the discretized domain for Ω as follows

Qh,δt = {(x1i, x2j, x3k, x4l, tm) : (x1i, x2j, x3k, x4l) ∈ Ωh, tm = mδt, 0 ≤ m ≤ Nt}. The value of f (x, t) on Qh,δt is denoted as fim, j . To solve the forward Fokker-Planck equation (3.7), we use the scheme described in [27], that is comprised of the spatial discretization using the Chang-Cooper (CC) method [6]. The temporal derivative discretization is done using the fourstep, alternate direction implicit Douglas-Gunn (DG4) method. The combined scheme is referred to as the DG4-CC scheme. It has been shown in [27] that this scheme is positive, conservative, stable, and second order convergent in the L1 norm. Furthermore, we use the temporal D-G scheme for the time discretization in the first term, one sided finite difference discretization for the second term, and central difference for the third term on the left hand side of the adjoint equation (ADJ). To solve the optimization problem (3.11), we use a projected non-­ linear conjugate gradient scheme (PNCG), as described in [28, 27]. Such a scheme has also been used for solving optimization problems related to controlling stochastic crowd motion [24, 25], studying avoidance behavior of pedestrians using game theory [26], and cure rate models [20, 21]. The PNCG scheme is described below. Algorithm 3.1 (PNCG Scheme). 1. Input: Initial guess u0. Compute d0 = −∇Jˆ(u0 )H1 . Set k = 0 and maximum number of iterations kmax, tolerance TOL. 2. If (k < kmax), do 3. Compute uk+1 = PU [uk + αk dk] with αk obtained by the Armijo line-search method. 4. Evaluate g k+1 = ∇Jˆ(uk+1 )H1 . T

|| y k ||2  1  5. Evaluate β 2 = y − d  k  g k +1. k dkT y k  dkT y k  HG k

6. 7. 8. 9.

Compute dk+1 = − g k+1 β kHG dk . Check for ||uk+1 − uk||2 < TOL. If yes, terminate. Update k = k + 1. End if.

Therapies in Colon Cancer using FP Framework  43

3.5 Results This section describes the numerical results for obtaining optimal treatment strategies with the aforementioned FP framework. We choose our domain Ω = (0, 6)4 and discretize it using N xi = 51 points for i = 1, 2, 3, 4. The final time Tf is chosen to be 10 and the maximum number of time steps Nt is chosen to be 200. The values of the constants used in converting the ODE system (3.1) to its non-dimensional form given in (3.3) are given as k1 = 10−10, k2 = 10−5, k3 = 10−7, k4 = 10−8, and k5 = 1. To obtain the target PDF f ∗(x, t), we simulate the ODE system (3.3) with the u1 , u2 set to 0 and with the value of the non-dimensional parameters (d , l , s ) = (2.1,1.1,1.25) that represents a patient with strong immune response [10]. The values of the other parameters are taken from [10]. After we obtain the trajectories of T¯, N¯, L¯, and C, we then choose 20 time points ti, and at each point (T¯(ti), N¯ (ti), L¯(ti), C¯(ti)) assign a Gaussian PDF given by a normal distribution with variance 0.05. We finally perform a 5D interpolation to obtain the desired PDF f ∗(x, t). Based on a statistical analysis of the dataset given in [10], we choose



σ i ( x ) = 0.5( xi1.2 + 0.001), i = 1, 2, 3, 4.

The regularization parameters are chosen to be α = 1 and ν1 = ν2 = 0.01. The maximum tolerable dosage for Doxorubicin is taken to be 7 mg/day and for IL-2 is taken to be 7.2×105 IU IL-2/l/day, which implies D1 = 7 and D2 = 0.072. In Test Case 1, we simulate a tumor patient with a value of (T (0), N (0), L(0), C(0)) = (2 × 109, 105, 107, 108). This implies (T¯(0), N¯ (0), L¯(0), C¯(0)) = (0.2, 1, 1, 1)). We also choose the values of the non-dimensional parameters (d , l , s ) = (1.3, 2 , 10) that represent patients with weak immune systems. The goal is to determine optimal drug dosages ui , i = 1,2 such that the tumor profile is given by the target PDF f ∗. Figure 3.1 presents the simulation results obtained with our framework. We observe that without the combination therapy, the mean tumor cell count will keep on increasing until it reaches the cell carrying capacity, ultimately leading to patient death. With the combination therapy, the mean tumor cell count is brought back to diminishing levels. We also note the optimal dosage patterns for both Doxorubicin and IL-2 over time. Traditionally, Doxorubicin is administered every 21 days with a dosage of 142.5 mg [9], that translates to a total dosage of 70 mg for 10 days. However, from Figure 3.1, we note that the total dosage of Doxorubicin over 10 days

44  Mathematics and Computer Science Volume 2 4

×105 2

Plot of u1

1.6

3 2.5 2 1.5

Plot of u2

1.8

3.5 u2 (IUIL-2/l)

u1 (mg)

×109 Plot of T (with treatment) 2 1.8 3.5 1.6 3 1.4 2.5 1.2 1 2 0.8 1.5 0.6 1 0.4 0.5 0.2 0 0 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 t (days) t (days) (b) T (with treatment) (a) T (without treatment) T(cells)

T(cells)

×1010 Plot of T (without treatment) 4

1.4 1.2 1 0.8

1

0.6

0.5

0.4

0.2 0 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 t (days) t (days) (c) Doxorubicin profile (d) IL-2 profile

Figure 3.1  Test case 1: plots of mean tumor profiles and drug dosages over 10 day period. ×104 Plot of u2 Plot of u1 ×109 Plot of T (with treatment) 10 2 3 9 1.8 3 2.5 8 1.6 7 2.5 1.4 2 6 1.2 2 5 1.5 1 1.5 4 0.8 1 3 0.6 1 2 0.4 0.5 0.5 1 0.2 0 0 0 0 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 t (days) t (days) t (days) t (days) (d) IL-2 (c) Doxorubicin (b) T (with treatment) (a) T (without treatment) u2 (IUIL-2/l)

u1 (mg)

T(cells)

T(cells)

×1010 Plot of T (without treatment) 3.5

Figure 3.2  Test case 2: plots of mean tumor profiles and drug dosages over 10 day period.

is far less than 70 mg. Moreover, the observed daily dosage of IL-2 from Figure 3.1 is less than the standard dosage of IL-2 over a day, which is 2.1 × 106 IU IL-2/l. This suggests that optimal combination therapies administered daily leads to lower total dosages and, thus, lower toxicity effects. In Test Case 2, we choose the same initial values of T , N , L , and C, along with the values of the non-dimensional parameters (d , l , s ) = (1.6 , 1.4 , 2) that represent patients with moderately strong immune systems. Figure 3.2 shows the plots of the tumor profiles without and with treatment and the optimal daily dosages of Doxorubicin and IL-2. We observe a similar behavior as Test Case 1. In addition, we note that the dosages are smaller compared to the patient in Test Case 1 since the immune system is moderately strong. This shows the effectiveness of our framework in obtaining optimal dosages in colon cancer.

3.6 Conclusion In this paper, we presented a new stochastic frameworks to determine optimal combination therapies in colon cancer-induced immune response. We

Therapies in Colon Cancer using FP Framework  45 considered the tumor and immune system response dynamics as proposed in [10] and extend it to a stochastic process to account for incorporating randomness in the dynamics. We characterized the state of the stochastic process using the PDF, whose evolution is governed by the FP equation. We then solved an optimal control problem with open loop controls to obtain the optimal combination dosages involving chemotherapy and immunotherapy. Numerical results demonstrate the feasibility of our proposed framework to obtain small dosages of the combination drugs, leading to lower toxicity whilst preserving the effectiveness to eliminate the tumor.

Acknowledgments The authors were supported by the National Institutes of Health (Grant Number: R21CA242933). S. Roy was also partially supported by the Interdisciplinary Research Program, University of Texas at Arlington, Grant number: 2021-772. The content is solely the responsibility of the authors and does not necessarily represent the official views of the National Institutes of Health.

References 1. M. Annunziato and A. Borzì. A fokker–planck control framework for multidimensional stochastic processes. Journal of Computational and Applied Mathematics, 237(1):487–507, 2013. 2. M. Annunziato and A. Borzì. A fokker–planck approach to the reconstruction of a cell membrane potential. SIAM Journal on Scientific Computing, 43(3):B623–B649, 2021. 3. A. Ballesta and J. Clairambault. Physiologically based mathematical models to optimize therapies against metastatic colorectal cancer: a mini-review. Current pharmaceutical design, 20(1):37–48, 2014. 4. A. Ballesta, J. Clairambault, S. Dulong, and F. Levi. A systems biomedicine approach for chronotherapeutics optimization: focus on the anticancer drug irinotecan. In New Challenges for Cancer Systems Biomedicine, pages 301– 327. Springer, 2012. 5. A. Borzì and L. Grüne. Towards a solution of mean-field control problems using model predictive control. IFAC-PapersOnLine, 53(2):4973–4978, 2020. 6. J. Chang and G. Cooper. A practical difference scheme for fokker-planck equations. Journal of Computational Physics, 6(1):1–16, 1970. 7. Y. Chang, M. Funk, S. Roy, E. Stephenson, S. Choi, H. V. Kojouharov, B. Chen, and Z. Pan. Developing a mathematical model of intracellular calcium

46  Mathematics and Computer Science Volume 2 dynamics for evaluating combined anticancer effects of afatinib and rp4010 in esophageal cancer. International Journal of Molecular Sciences, 23(3):1763, 2022. 8. Y. Chang, S. Roy, and Z. Pan. Store-operated calcium channels as drug target in gastroesophageal cancers. Frontiers in pharmacology, 12:944, 2021. 9. L. de Pillis, K. Renee Fister, W. Gu, C. Collins, M. Daub, D. Gross, J. Moore, and B. Preskill. Mathematical model creation for cancer chemo-immunotherapy. Computational and Mathematical Methods in Medicine, 10(3):165– 184, 2009. 10. L. DePillis, H. Savage, and A. Radunskaya. Mathematical model of colorectal cancer with monoclonal antibody treatments. arXiv preprint arXiv:1312.3023, 2013. 11. C. Fitzmaurice, D. Dicker, A. Pain, H. Hamavid, M. Moradi-Lakeh, M. F. MacIntyre, C. Allen, G. Hansen, R. Woodbrook, C. Wolfe, et al. The global burden of cancer 2013. JAMA oncology, 1(4):505–527, 2015. 12. J. Galon, A. Costes, F. Sanchez-Cabo, A. Kirilovsky, B. Mlecnik, C. LagorcePagès, M. Tosolini, M. Camus, A. Berger, P. Wind, et al. Type, density, and location of immune cells within human colorectal tumors predict clinical outcome. Science, 313(5795):1960–1964, 2006. 13. S. Haupt, A. Zeilmann, A. Ahadova, H. Bläker, M. von Knebel Doeberitz, M. Kloor, and V. Heuveline. Mathematical modeling of multiple pathways in colorectal carcinogenesis using dynamical systems with kronecker structure. PLoS computational biology, 17(5):e1008970, 2021. 14. M. D. Johnston, C. M. Edwards, W. F. Bodmer, P. K. Maini, and S. J. Chapman. Mathematical modeling of cell population dynamics in the colonic crypt and in colorectal cancer. Proceedings of the National Academy of Sciences, 104(10):4008–4013, 2007. 15. N. L. Komarova, C. Lengauer, B. Vogelstein, and M. A. Nowak. Dynamics of genetic instability in sporadic and familial colorectal cancer. Cancer biology & therapy, 1(6):685– 692, 2002. 16. J.-L. Lions. Quelques méthodes de résolution de problemes aux limites non linéaires. Paris, Dunod-Gauth. Vill., 1969. 17. W.-C. Lo, E. W. Martin Jr, C. L. Hitchcock, and A. Friedman. Mathematical model of colitis-associated colon cancer. Journal of theoretical biology, 317:20–29, 2013. 18. R. B. Mokhtari, T. S. Homayouni, N. Baluch, E. Morgatskaya, S. Kumar, B. Das, and H. Yeger. Combination therapy in combating cancer. Oncotarget, 8(23):38022, 2017. 19. R. B. Mokhtari, S. Kumar, S. S. Islam, M. Yazdanpanah, K. Adeli, E. Cutz, and H. Yeger. Combination of carbonic anhydrase inhibitor, acetazolamide, and sulforaphane, reduces the viability and growth of bronchial carcinoid cell lines. BMC cancer, 13(1):1–18, 2013.

Therapies in Colon Cancer using FP Framework  47 20. S. Pal and S. Roy. A new non-linear conjugate gradient algorithm for destructive cure rate model and a simulation study: illustration with negative binomial competing risks. Communications in Statistics-Simulation and Computation, pages 1–15, 2020. 21. S. Pal and S. Roy. On the estimation of destructive cure rate model: A new study with exponentially weighted poisson competing risks. Statistica Neerlandica, 75(3):324–342, 2021. 22. A. H. Partridge, H. J. Burstein, and E. P. Winer. Side effects of chemotherapy and combined chemohormonal therapy in women with early-stage breast cancer. JNCI Monographs, 2001(30):135–142, 2001. 23. G. G. Powathil, D. J. Adamson, and M. A. Chaplain. Towards predicting the response of a solid tumour to chemotherapy and radiotherapy treatments: clinical insights from a computational model. PLoS computational biology, 9(7):e1003120, 2013. 24. S. Roy, M. Annunziato, and A. Borzì. A fokker–planck feedback controlconstrained approach for modelling crowd motion. Journal of Computational and Theoretical Transport, 45(6):442–458, 2016. 25. S. Roy, M. Annunziato, A. Borzì, and C. Klingenberg. A fokker–planck approach to control collective motion. Computational Optimization and Applications, 69(2):423–459, 2018. 26. S. Roy, A. Borzì, and A. Habbal. Pedestrian motion constrained by fp-­ constrained nash games. Royal Society Open Science, 4:170648, 2017. 27. S. Roy, S. Pal, A. Manoj, S. Kakarla, J. Villegas, and M. Alajmi. A fokker-planck framework for parameter estimation and sensitivity analysis in colon cancer. AIP Conference Proceedings (accepted, in press), 2522, 070005, 2022. 28. S. Roy, Z. Pan, and S. Pal. A fokker-planck feedback control framework for optimal personalized therapies in colon cancer-induced angiogenesis. Journal of Mathematical Biology, 84:23, 2022. 29. H. Schmoll, E. Van Cutsem, A. Stein, V. Valentini, B. Glimelius, K. Haustermans, B. Nordlinger, C. Van de Velde, J. Balmana, J. Regula, et al. Esmo consensus guidelines for management of patients with colon and rectal cancer. a personalized approach to clinical decision making. Annals of oncology, 23(10):2479–2516, 2012. 30. T. Tao. Nonlinear dispersive equations: local and global analysis. Number 106. American Mathematical Soc., 2006. 31. V. Thalhofer, M. Annunziato, and A. Borzì. Stochastic modelling and control of antibiotic subtilin production. Journal of mathematical biology, 73(3):727– 749, 2016. 32. M. Tosolini, A. Kirilovsky, B. Mlecnik, T. Fredriksen, S. Mauger, G. Bindea, A. Berger, P. Bruneval, W.-H. Fridman, F. Pagès, et al. Clinical impact of different classes of infiltrating t cytotoxic and helper cells (th1, th2, treg, th17) in patients with colorectal cancer. Cancer research, 71(4):1263–1271, 2011.

4 Detection and Classification of Leaf Blast Disease using Decision Tree Algorithm in Rice Crop Sarvesh Vishwakarma1* and Bhavna Chilwal2 Department of CSE, Graphic Era (Deemed to be University), Dehradun, India 2 Department of CSE, DIT University Dehradun, Uttarakhand, India

1

Abstract

The agricultural field is the most important field for any nation but some issues prevail and affect agricultural products every year. Agricultural diseases are the main concern for yield loss. This chapter uses the Decision Tree technique to form a tree structure for leaf blast disease level detection in rice crops. A Decision Tree is used as a classification technique and here disease levels are classified based on symptoms that occur during infection. The Iterative Dichotomiser 3 (ID3) algorithm is one of the important methods to form a Decision Tree based on entropy and information gain. The nodes in the tree are the symptoms that have different labels for disease occurrence. This decision tree will help detect the occurrence of disease as per the symptoms and help farmers get information about the severity level of a disease so that they can take required measures on time to save the crop from loss. Keywords:  Decision tree, ID3 algorithm, entropy, information gain, leaf blast disease

4.1 Introduction There are so many fungal diseases that prevail in rice, but this chapter focuses on leaf blast disease in rice crops. Blast disease is also known as rice fever. Agricultural scientists find out about the disease by checking their symptoms, similarly, this paper uses three particular symptoms for leaf blast [5]. These symptoms will be used to form a decision tree and *Corresponding author: [email protected] Sharmistha Ghosh, M. Niranjanamurthy, Krishanu Deyasi, Biswadip Basu Mallik, and Santanu Das (eds.) Mathematics and Computer Science Volume 2, (49–58) © 2023 Scrivener Publishing LLC

49

50  Mathematics and Computer Science Volume 2

(a) Healthy Leaf

(b) Disease Leaf

Figure 4.1  Comparison between healthy and diseased leaf.

detect the labels of disease occurrence (as shown in Figure 4.1(a) and Figure 4.1(b)). The decision tree uses attributes to classify the leaves according to symptoms. Every decision tree comprises of two types of nodes: a Decision Node and the Decision Leaf Node. Different classification algorithms are used nowadays to implement decision trees. Here, an ID3 classification algorithm is used for implementation. This algorithm stands for Iterative Dichotomiser 3. This algorithm forms the smallest decision tree possible. The ID3 algorithm uses the Entropy and Information Gain [6–8, 13]. The role of Entropy is to control the splitting of the data for the decision tree. With the help of entropy, the boundaries of the decision tree are decided. The formula for class entropy is given below:



T=

−P N  P   N  log 2  log 2  −  P+N P+N  P+N P+N 

(4.1)

For a sample of negative N- and positive P+ Entropy for each attribute:





IG (Pi , N i ) =

−Pi  Pi log 2  Pi + N i  Pi + N i

E(S) =

Ni   Ni   − P + N log 2  P + N  (4.2) i i   i  i

∑ (P + N ) × I (P , N ) i

P+N

i

G

i

i

(4.3)

The information gain calculates the decrease in the value of entropy after data gets split on different attribute values. The decision tree finds the

Detection and Classification of Leaf Blast Disease  51 parameter value which has the highest information gain. The three symptoms taken for disease detection are the growth stage of the rice crop, disease index, and lesion type [1]. Linguistic variables are taken for different ranges of these variables. The ID3 algorithm will detect the leaf blast disease in rice crops by using these symptoms as the attributes and decide from which class the rice leaf belongs. The implementation of a decision tree in different research is very vast. It framed different rules to form a decision tree that is influenced by climatic parameters for the productivity of soybeans [2]. The tree takes symptoms and forms forty rules for fuzzy logic for detecting leaf blast disease by using regression methodology [3]. A decision tree classifier is used by measuring the temperature and soil moisture as parameters to form a system for predicting cotton crop disease [4]. Medical data mining for the prediction of heart disease is done by using Naïve Bayes and decision tree algorithms [8, 11, 12] and this research work shows a prediction accuracy of 99%. A prototype was used to evaluate the severity of disease on rice crops by using computational intelligence and machine learning. K-Mean segmentation has been used with fuzzy logic to calculate the degree of the disease that occurred in rice plants [9, 10]. The prototype has 86% accuracy.

4.2 Proposed Methodology The symptoms and rule table are considered based on the work described in [14, 15], which presents fuzzy rule techniques for different combinations of symptoms [16–18]. First of all, we take the attributes of a dataset and form a table that comprises linguistic values regarding each attribute and response column with positive or negative responses [19, 20]. Then, we calculate the class entropy for the whole table, i.e., Entropy (T). But among all the attributes we have to find the root attribute for the tree, we calculate the information gain for each attribute which is then used to find the entropy value of that specific attribute, i.e., Entropy (X). The computation of gain has to be performed after getting the attributed entropy values and comparing them against the class entropy (as shown in Equation 4.4).



GainAttribute = T − E (S)

(4.4)

The attribute which has maximum gain value becomes the root node and again the iteration of the whole process starts for expanding the tree [21–23]. So, to form the table we have provided linguistic terms for different values of each attribute or symptom. The rice plant has three important growth stages and during these stages, the plant has some height between

52  Mathematics and Computer Science Volume 2 1 cm to 100 cm on average. Therefore, the linguistic variables [24–26] for the Growth Stage are mentioned below: • Germinating Stage – Small (S) • Vegetative Stage – Moderate (M) • Reproduction Stage – Tall (T) For the disease index, the range of input scores is varying from 0 to 100%. Scores are predefined in the standard evaluation system by IRRI 2015. The linguistic variables for different scales are provided below: • Score 1 and 3 – Low • Score 5 and 7 – Mid • Score 9 – High Lesion type (standard evaluation system of rice (SES), IRRI 2015) values are taken from scale 5 to scale 9 because as per the experts, the lesions are noticeable properly. • Scale 5 and 6 – Low • Scale 7 and 8 – Mid • Scale 9 – High The result column has two specific classes for which two values are used to provide disease labels: 1. Resistive R (+) for healthy leaves or leaves with a very small effect 2. Susceptive S (-) for diseased leaves which need proper care and are harmful for crop production

4.3 Result Analysis After deciding the linguistic values for different symptoms, we formed a rule base table (as shown in Table 4.1), which has fourteen combinations of these symptoms with their disease occurrence labels. The computation of class entropy for the disease occurrence has been done as below: Resistive (+) = 4 Susceptive (-) = 10

Detection and Classification of Leaf Blast Disease  53 Table 4.1  Rule-based disease occurrence. Growth stage

Disease index

Lesion type

Occurrence label

Small

No

No

R (+)

Moderate

Low

No

R (+)

Moderate

Low

Low

R (+)

Tall

Mid

Mid

S (-)

Moderate

Low

Low

S (-)

Tall

Mid

Mid

S (-)

Moderate

Mid

Mid

S (-)

Tall

Mid

Mid

S (-)

Tall

High

High

S (-)

Tall

Mid

Mid

S (-)

Moderate

Low

Mid

S (-)

Moderate

Low

Low

R (+)

Moderate

High

Low

S (-)

E (Disease Occurrence) = E (4, 10) E (4, 10) = − (0.714 log2 0.714) − (0.286 log2 0.286) = 0.863 This value is used to find the Gain for each attribute separately, as shown in Table 4.2. The choice of parameter is done to reflect the highest gain value for the decision tree. Therefore, the ID3 algorithm is applied for creating a decision tree and we detect the symptoms which have maximum impact and Table 4.2  Gain value for each attribute. Growth stage Growth stage

R(+)

S(-)

IG

Entropy

Gain

Low

0

0

0

0.496

0.367

Moderate

4

3

0.951

Tall

0

7

0

54  Mathematics and Computer Science Volume 2 that will become the root of the tree, i.e., lesions type as it has the highest information gain value. The attribute which has the greatest gain as the decision node becomes the root of the decision tree so the tree would look like a three-child leaf (Low, Mid, and High) with root node Lesions Type. From Table 4.3, we get all mid and high-range lesion types having disease occurrence response as the susceptive (+) class label response, meaning the farmers have to take measures to stop the severity of the disease. But the low range of lesions type has two responses, both resistive and susceptive labels, for some rows so we take the next attribute branch which has the second greatest information gain, i.e., disease index which is 0.46, as depicted in Table 4.4. Table 4.3  Lesion types for each attribute. Lesion type Lesion type

R(+)

S(-)

IG

Entropy

Gain

Low

4

1

0.721

0.281

0.582

Mid

0

7

0

High

0

2

0

Table 4.4  Disease index for each attribute. Disease type Disease index

R(+)

S(-)

IG

Entropy

Gain

Low

4

2

0.9198

0.397

0.466

Mid

0

5

0

High

0

3

0

Lesion Type Low Disease Index Low

Mid

Mid Susceptive High

Figure 4.2  Detection of root node for decision tree.

High Susceptive

Detection and Classification of Leaf Blast Disease  55 Lesion Type Low Disease Index Low

Mid

Growth Stage Small

Moderate

No Infection Resistive

Susceptive

Mid Susceptive

High Susceptive

High Susceptive

Tall Susceptive

Susceptive

Figure 4.3  Final decision tree for leaf blast disease.

Figure 4.2 has been derived by using two major symptoms of leaf blast disease in which lesion type is the root node. Now, in the next iteration we take the third greatest information gain attribute branch, i.e., the growth stage with a gain value of 0.367. The tree would take the shape, as shown in Figure 4.3.

4.4 Conclusion Disease in crops and plants is a major problem faced by farmers. The traditional methods to detect the occurrence of diseases are tiresome, costly, and are not time efficient in comparison with the new soft computing techniques which provide a digital era in the agricultural sector. The decision tree provides an easier view of the whole process and how to detect the occurrence level of disease. We can say that the crop has specific symptoms during a particular type of disease. This means by making use of symptoms, the risk level could be detected which is very helpful for farmers and agricultural scientists to save the yield loss.

4.5 Future Work The future scope for this work is to implement the tree by using different algorithms, except for ID3. Also, we can form the decision tree for the classification of other diseases. We can also extend this work by using a Random Forest algorithm by making more than one decision tree.

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References 1. Pari Skamnioti, Sarah J. Gurr, Against the grain: safeguarding rice from rice blast disease, Trends in Biotechnology, Volume 27, Issue 3, 2009, Pages 141150, ISSN 0167-7799, https://doi.org/10.1016/j.tibtech.2008.12.002. 2. P. Udupa, S. Vishwakarma, A survey of MRI segmentation Techniques for Brain Tumor Studies, Bonfring International Journal of Advances in Image Processing, Vol. 6, No. 3, August 2016, pp. 22-27. Doi: http://www.journal. bonfring.org/papers/aip/volume6/BIJ-10467.pdf 3. F. M. Shamim, S. Vishwakarma, Exploiting the Motion Learning Paradigm for Recognizing Human Actions, Bonfring Internation Journal of Advances in Image Processing, Vol. 6, No. 3, August 2016, pp. 11-16. http://www.journal.​ bonfring.org/papers/aip/volume6/BIJ-10465.pdf 4. Chopda, J., Raveshiya, H., Nakum, S., Nakrani, V., 2018. Cotton crop disease detection using decision tree classifier. International Conference on Smart City and Emerging Technology (ICSCET), 1–6. 5. Donatelli, M., Magarey, R.D., Bregaglio, S., L., W., Whish, J.P.M., Savary, S., 2017. Modelling the impacts of pests and diseases on agricultural systems. Agricultural Systems 155, 213–224. 6. Edwards-Murphy, F., Magno, M., Whelan, P.M., O’Halloran, J., Popovici, E.M., 2016. b+wsn: Smart beehive with preliminary decision tree analysis for agriculture and honey bee health monitoring. Computers and Electronics in Agriculture 124, 211–219. 7. Ilic, M., Ilic, S., Jovic, S., Panic, S., 2018. Early cherry fruit pathogen disease detection based on data mining prediction. Computers and Electronics in Agriculture (Elsevier) 150, 418–425. 8. Karthiga, A.S., Mary, M.S., Yogasini, M., 2011. Early prediction of heart disease using decision tree algorithm. International Journal of Advanced Research in Basic Engineering Sciences and Technology (IJARBEST) 3(03). 9. Nithya, A., Sundaram, V., 2011. Wheat disease identification using classification rules. International Journal of Scientific & Engineering Research 2(09), 1–5. 10. Sethy, P.K., Negi, B.S., 2018. Measurement of disease severity of rice crop using machine learning and computational intelligence. Cognitive Science and Artificial Intelligence (Springer) 2, 1–11. 11. Shouman, M., Turner, T., Stocker, R., 2011. Using decision tree for diagnosing heart disease patients. 9th Australasian Data Mining Conference (AusDM 2011) 121, 23–29. 12. Singh, P., Jagyasi, B., Rai, N., Gharge, S., 1–6. Decision tree-based mobile crowdsourcing for agriculture advisory system. 2014 Annual IEEE India Conference (INDICON), 2014. 13. Zhang, Y.D., Wang, S., Dong, Z., 2014. Classification of alzheimer’s disease based on structural magnetic resonance imaging by kernel support vector machine decision tree. Progress In Electromagnetics Research 144, 171–184.

Detection and Classification of Leaf Blast Disease  57 14. Vishwakarma, S., Agrawal, A. A survey on activity recognition and behavior understanding in video surveillance. Vis Comput  29, 983–1009 (2013). https://doi.org/10.1007/s00371-012-0752-6 15. S. Vishwakarma and A. Agrawal, A novel approach for feature quantization using one-dimensional histogram, 2011 Annual IEEE India Conference, 2011, pp. 1-4, doi: 10.1109/INDCON.2011.6139391. 16. S. Vishwakarma, A. Sapre and A. Agrawal, Action recognition using cuboids of interest points, 2011 IEEE International Conference on Signal Processing, Communications and Computing (ICSPCC), 2011, pp. 1-6, doi: 10.1109/ ICSPCC.2011.6061680. 17. V. Shetty, S. Vishwakarma, Harisha and A. Agrawal, Design and implementation of video synopsis using online video inpainting, 2017 2nd IEEE International Conference on Recent Trends in Electronics, Information & Communication Technology (RTEICT), 2017, pp. 1208-1212, doi: 10.1109/ RTEICT.2017.8256790. 18. S. Vishwakarma and A. Agrawal, Framework for human action recognition using spatial temporal based cuboids, 2011 International Conference on Image Information Processing, 2011, pp. 1-6, doi: 10.1109/ICIIP.2011.6108881. 19. A. Genovese, V. Piuri, F. Scotti and S. Vishwakarma, Touchless Palmprint and Finger Texture Recognition: A Deep Learning Fusion Approach, 2019 IEEE International Conference on Computational Intelligence and Virtual Environments for Measurement Systems and Applications (CIVEMSA), 2019, pp. 1-6, doi: 10.1109/CIVEMSA45640.2019.9071620. 20. R. Singh, S. Vishwakarma, A. Agrawal, and M.D. Tiwari, Unusual activity detection for video surveillance, IITM ‘10: Proceedings of the First International Conference on Intelligent Interactive Technologies and Multimedia, December 2010 Pages 297–305 https://doi.org/10.1145/1963564.1963616 21. Donida Labati R., Genovese A., Piuri V., Scotti F., Vishwakarma S. (2020) Computational Intelligence in Cloud Computing. In: Kovács L., Haidegger T., Szakál A. (eds) Recent Advances in Intelligent Engineering. Topics in Intelligent Engineering and Informatics, vol 14. Springer, Cham. https:// doi.org/10.1007/978-3-030-14350-3_6 22. Vishwakarma, S. and Agrawal, A., 2010. A Robust Algorithm for Shadow Removal of Moving Object Detection in Video Surveillance System.  International Journal of Recent Trends in Engineering and Technology, ACEEE, AMAE, and ACEE, USA, 4(1), pp.73-75. 23. Donida Labati, Angelo Genovese, Vincenzo Piuri, Fabio Scotti, Sarvesh Vishwakarma, I-SOCIAL-DB: A labeled database of images collected from websites and social media for Iris recognition, Image and Vision Computing, Volume 105, 2021, 104058, ISSN 0262-8856, https://doi. org/10.1016/j.imavis.2020.104058 24. Vishwakarma S., Agrawal A. (2012) Representing Feature Quantization Approach Using Spatial-Temporal Relation for Action Recognition. In: Kundu M.K., Mitra S., Mazumdar D., Pal S.K. (eds) Perception

58  Mathematics and Computer Science Volume 2 and Machine Intelligence. PerMIn 2012. Lecture Notes in Computer Science, vol 7143. Springer, Berlin, Heidelberg. https://doi.org/10.1007/​ 978-3-642-27387-2_13 25. Acharya P., Vishwakarma S., Abnormal Crowd Behavior Detection using Structural Context Descriptor, Bonfring Internation Journal of Advances in Image Processing, Vol. 6, No. 3, August 2016, pp. 17-21 DOI: 10.9756/ BIJAIP.10466 26. Udupa P., Vishwakarma S., Segmentation of Brain MRI to differentiate Healthy and Tumor Affected Tissues along with Subsequent Tumor Grading, International Journal of Image Processing and Applications, Vol. 1, Issue 01, October 2016, pp. 1-6.

5 Novel Hybrid Optimal Deep Network and Optimization Approach for Human Face Emotion Recognition J. Seetha1, M. Ayyadurai2* and M. Mary Victoria Florence3 Department of Computer Science and Business Systems, Panimalar Engineering College, Chennai, India 2 Department of CSE, SRM Institute of Science and Technology, Ramapuram, Chennai, India 3 Department of Mathematics, Panimalar Engineering College, Chennai, India 1

Abstract

Emotion is significant in deciding a human’s ideas, behavior, and feelings. Using the advantages of deep learning, an emotion detection system can be constructed and various applications such as face unlocking, feedback analysis, and so on are executed with high accuracy. Artificial intelligence’s fast development has made a significant contribution to the technological world. However, it has several difficulties in achieving optimal recognition. Interpersonal differences, the intricacy of facial emotions, posture, and lighting, among other factors, provide significant obstacles. To resolve these issues, a novel Hybrid Deep Convolutional based Golden Eagle Network (HDC-GEN) model algorithm is proposed for the effective recognition of human emotions. The main goal of this research is to create hybrid optimal strategies that classify five diverse human facial reactions. The feature extraction of this research is carried out using Heap Coupled Bat Optimization (HBO) method. The execution of this research is performed by MATLAB software. The simulation outcomes are compared with the conventional methods in terms of accuracy, recall, precision, and F-measure and the comparison shows the effective performance of proposed approaches in facial emotion recognition. Keywords:  Face emotion recognition, deep learning, optimization, hybrid model, golden eagle optimization, feature extraction, and bat optimization *Corresponding author: [email protected] Sharmistha Ghosh, M. Niranjanamurthy, Krishanu Deyasi, Biswadip Basu Mallik, and Santanu Das (eds.) Mathematics and Computer Science Volume 2, (59–76) © 2023 Scrivener Publishing LLC

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5.1 Introduction Emotional analytics combine psychology and technology in an intriguing way. The examination of facial gestures is one of the most used methods for recognition of emotions. Basic emotions are thought to be physiologically fixed, intrinsic, and universal to all individuals and many animals [1]. Severe reactions are either a collection of fundamental emotions or a collection of unusual feelings. The major issue is figuring out which emotions are fundamental and which are complicated. While analyzing the information acquired by the ears and sight, people may detect these signals even when they are softly expressed. Based on psychological research that illustrates that visual data affects speech intelligibility, it is reasonable to infer that human emotion interpretation shares a predictable pattern [2]. The practice of recognizing human emotions from facial gestures is known as recognition of facial expression. Facial expression is a universal signal that all humans use to communicate their mood. In this era, facial expression detection systems are very important because they can record people’s behavior, sentiments, and intentions [3]. Computer networks, software, and networking are rapidly evolving and becoming more widely used. These systems play a vital part in our daily lives and make life much easier for us. As a result, face expression detection as a method of image analysis is quickly expanding. Human–computer interface, psychological assessments, driving enabled businesses, automation, drunk driver identification, healthcare, and, most importantly, lie detection are all conceivable uses [4]. The human brain perceives emotions instinctively and technology that can recognize emotions has recently been developed. Furthermore, Artificial Intelligence (AI) can identify emotions by understanding the meaning of each face expression and learn to adapt to fresh input. Artificial Neural Networks (ANN) are being employed in AI systems currently [5]. Long Short Term Memory (LSTM) based RNN [6], Nave Bayes, K-Nearest Neighbors (KNN), and Convolutional Neural Networks [7] are used to tackle a variety of challenges such as excessive makeup, position, and expression [8]. The properties of the swarm intelligence optimization method, which can execute parallel computation and sharing of information, have caught our interest in order to make good use of excess computational resources and enhance optimization effectiveness [9]. The Particle Swarm Optimization (PSO) and Ant Colony Optimization (ACO) algorithms, as well as their upgraded variants, constitute the swarm intelligence algorithm [10]. The traditional approaches are slow and inaccurate, but a face expression detection system based on deep learning has proven

Human Face Emotion Recognition  61 to be superior. Furthermore, typical techniques largely focus on face analysis while keeping the backdrop intact, resulting in a large number of irrelevant and inaccurate characteristics that complicate CNN training. In this framework, this paper proposes a novel for facial emotion recognition using a Hybrid Deep Convolutional based Golden Eagle Network (HDC-GEN) model algorithm. Moreover, the finest features are extracted using HBO method. The current article comprises of four basic facial expression categories that have been reported: displeasure/anger, sadness/unhappiness, happiness/smile, fear, and surprise/astonishment. Investigations have been carried out and the findings reveal that the suggested network outperforms the benchmark and state-of-the-art solutions for emotion recognition based on facial expression images. It offers several advantages over manual correction in terms of training efficiency and performance, especially when computational resources are sufficient. When compared to other standard emotion detection approaches, this proposed method produced better results in analyzing and classifying emotional data gathered from investigations. The following is an outline of the article’s layout: The traditional methods used for emotion classification are explained in Section 5.2. In Section 5.3, the system model along with problem definition is provided. The proposed method is described in Section 5.4. Finally, the findings are provided in Section 5.5 and the implications are explored in Section 5.6.

5.2 Related Work Some of the recent works correlated to this research are articulated as follows: Humans have traditionally had an easy time detecting emotions from facial expression but performing the same performance with a computer program is rather difficult. In this study, Mehendale and Ninad [11] presented a unique face emotion identification approach based on convolutional neural networks. Using an EV of length 24 data, it was feasible to appropriately emphasize the emotion with 96 percent accuracy. Furthermore, before to the formation of EV, a unique background removal approach is used to prevent dealing with many issues that may arise. Researchers were able to attain cutting-edge performance in emotion identification by employing deeper architectures. For this reason, Mungra, Dhara, et al. [12] developed the PRATIT model which employs certain image preprocessing processes and a CNN design for face emotion identification. To manage differences in the photos, preprocessing methods including gray scaling, resizing, cropping, and normalization are utilized in PRATIT, which obtains an accuracy rate of 78.52%.

62  Mathematics and Computer Science Volume 2 A multimodal attention based Bidirectional Long Short-Term Memory (BLSTM)  network architecture for effective emotion identification is described by Li, Chao, et al. [13]. The optimal temporal properties are automatically learned using BLSTM with Recurrent Neural Networks (LSTMNNs). The learnt deep characteristics are then given into a DNN, which predicts the probability of emotional response for every stream. Furthermore, the overall feeling is predicted using a decision level hybrid method. Estrada, et al. [14] compared numerous sentiment classification models for the categorization of learning attitudes in an Artificial Learning Environment termed ILE-Java, employing three distinct methodologies including deep learning, machine learning, and an evolutionary technique named EvoMSA. It was also found that the EvoMSA produced the best findings, with an accuracy of 93% for the sample sentiTEXT and 84% for the sample eduSERE, depending on the outcomes of the mentioned methods. In reality, optimizing hyper-parameters continues to be a problem when developing machine learning models like CNNs. Therefore, Gao, Zhongke, et al. [15] offered an autonomously optimized model that selects the structure utilizing a binary coding scheme and GPSO with gradient consequences. The GPSO-based algorithm effectively explores the optimum solution, enabling CNNs to outperform well in classification throughout the dataset. The findings demonstrate that this strategy, which is centered on a GPSO-optimized Classification algorithm, reached a high level of recognition rate.

5.3 System Model and Problem Statement Humans accomplish the task of identifying expressions on a daily basis with ease, yet automated expression identification remains tough due to the difficulties in separating the emotions’ feature set. Face detection and preprocessing, face image extraction, and classification techniques are the three phases involved in human recognition utilizing facial images. There are major differences in how people articulate themselves. Due to numerous elements such as backdrop, lighting, and position, photographs of the same subject with the same expression change. Emotion recognition is difficult owing to a variety of input modalities that play a vital part in interpreting information. The task of recognizing emotions is complicated by the lack of a big collection of training images and the difficulty of distinguishing emotion depending on whether the input image is fixed or an evolving clip into face features. The challenge faced is mostly for real-time identification of facial emotions that vary drastically.

Human Face Emotion Recognition  63

5.4 Proposed Model Emotion artificial intelligence is a system that can detect, imitate, understand, and deal with human facial gestures and feelings. Artificial intelligence systems have grown in popularity in the contemporary world as a result of their extensive capabilities and ease of use. However, the traditional methods have certain drawbacks for the accurate recognition of emotions. Thus, a novel HDC-GEN classification method is proposed in this work for significant human facial emotion identification. Primarily, the input images are preprocessed using a median filter because the raw dataset has more noise and unwanted elements are placed. Furthermore, the knowledge based face detection function and image cropping are provided using a MATLAB function. The feature extraction is employed by HBO algorithm to improve the classification. Then, the HDC-GEN algorithm improves the facial expression recognition accuracy. The proposed model of emotion recognition diagrammatic illustration is provided in Figure 5.1.

Human emotion dataset

Input image

Feature extraction by HBO method

Resize image

HDC-GEN method

Pre-processing stage by median filter

Face detection // knowledge based algorithm

Emotion Parameter tuning

Figure 5.1  Proposed model of emotion recognition illustration.

Performance measures

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5.4.1 Preprocessing Stage Preprocessing is the initial step in the process of extracting features from images. Moreover, preprocessing is the phase at which noise is removed via filtering. The median filter is employed to remove noise in this case. Preprocessing is necessary because it removes unnecessary noise and improves the sharpness of the images.

5.4.2 Knowledge Based Face Detection The aim of face detection in this research is to give the position and size of the face in the entire image. Learning-based face detection techniques are the most successful methods in terms of detection accuracy and speed. There could be a variety of items in the background of the input image, such as a building, a people, or trees. This face detection framework can process images very quickly while attaining good detection rates. In this study, the goal of face detection is to determine the size and position of the face in the full image. In terms of detection speed and accuracy, knowledge-based image detection algorithms are the most successful. Rectangular characteristics can be generated quickly using this cascade object detector in MATLAB. Any position’s image representation is the summation of the pixels beyond and to the side of it. For example, the sum of the pixels beyond and to the side of m and n makes up the integral image at point m and n.

5.4.3 Image Resizing The cropping function is used to crop the face area from the input human image once the bounding box has detected the face region using MATLAB. The facial region inside each sample image is clipped at this point.

5.4.4 Feature Extraction A right and left eye, nose, and mouth are among the traits examined in the suggested technique. The images are then enlarged to maintain the same distance between the midpoints of the left and right eyes and rotated such that the line connecting the two is horizontal. Next, we initialize all feature parameters to the HBO method. The feature extraction is done by the maximum and minimum point estimation using Equation 5.1:

di = dmin (dmax − dmin) × δ

(5.1)

Human Face Emotion Recognition  65 where di is the feature point, minimum distance rate is denoted as dmin, maximum distance rate is denoted as dmax, and random path of selection is considered as δ. Furthermore, the priority of feature extraction estimated by the fitness in Equations 5.2 and 5.3:



= f1n  x1n −1 − X∗  × fi + f1n −1NT

(5.2)



= x1n x1n −1 + f1n

(5.3)

where n is denoted as time step, f1n is the optimal fitness speed, and the best point is considered as X*. The extraction condition for each feature is evaluated by the fitness of heap using Equation 5.4:

n

fi ( t 1)

^

f f in (t )

^

f fin (t )

Mni

n

Mni

fin (t) , f Mni

f in

n

Mni

fin (t ) , f Mni



(5.4)

where f is represented as objective function, t is the current iteration, n is the number of feature vectors, and βn is the nth feature vector. The intensity value and row and column position scores are computed for each pixel in the fixed region (nose, right eye, left eye, and mouth). The neutral image as well as the emotion image is used to calculate the feature quality. For the purpose of training, the proposed hybrid network is provided the feature values for all of the images.

5.5 Proposed HDC-GEN Classification Classification is the process of predicting specific outcomes based on a set of inputs. The method uses a training set, which consists of a collection of qualities and their related outcomes, sometimes referred to as the target or prediction attribute, to predict the outcome. The program tries to figure out if there are any correlations between the qualities that might help predict the outcome. The HDC-GEN layer consists of three layers named the input layer, output layer, and hidden layer. The HDC-GEN algorithm is the combination of improved deep CNN and Golden Eagle Optimization. A learning function in the hidden units is operated by the golden eagle fitness function.

66  Mathematics and Computer Science Volume 2 Initialization: Initialize the featured data as (aj, bj, cj). The kernel function is executed to the input function using Equation 5.5: +∞

∑ m(τ )n(t − δ )

z= (t ) = m(t ) ∗ n(t )

(5.5)

δ = −∞

where m is the input image feature and n is the convolution kernel performance, τ is denoted as time delay, and t is symbolized as time, respectively. The tangent hyperplane is evaluated for the kernel exploration by Equation 5.6: a

h1c1 + h2c2 + ..haca= n ⇒

∑h c = n i i

(5.6)

i =1

where h1, h2,……ha are the normal vector and c1, c2,……ca are the ith node decision vector. Max-Pooling Function: Once the kernel function is applied, then the max-pooling operation is executed. The invariant features are extracted in the kernel layer using the max-pooling function, turning the extracted features into various images. The max-pooling function is applied in all portions of the featured images using Equation 5.7:



x zj ,a = max = p

1

(x

z −1, a ( j −1)×o + p

)

(5.7)

where o is denoted as the section range and p is considered as the width window. Optimized Fully Connected Training Layer: The fully connected layer of HDC-GEN is used to train the non-linear combination of data. In addition, the images are flattened into column vectors. For each training iteration, it is then fed into a feed-forward based deep net as well as a backward propagation model. The training set is teaching the right class size in this stage. Thus, the feed-forward propagation is evaluated in Equation 5.8:

x zj =

∑h

α ( x zj1−1,a ) + baz −1

z −1 k, j

a

(5.8)

Human Face Emotion Recognition  67 The training may be characterized using a vector that starts at the current position of the data and ends at the ideal weight points in the data memory. The training vector for classification is computed using Equation 5.9: Start

Initialize the different human emotion data

Apply median filter for preprocessing

Compute face detection function using knowledge based approach

Apply face cropping function for image resizing

Extract the features from the image using HBO method

Initialized the features to HDC-GEN for classification

Contribute the backward propagation in the optimized fully connected training phase

Apply max-pooling function

Compute the softmax operation for classification

No If criteria met?

New emotion exploitation is apply

Yes

Optimal human emotion is recognized

Stop

Figure 5.2  Flowchart for proposed human emotion recognition.

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t az ≡ b zfa−1 − baz −1

(5.9)

where t az is the training exploitation vector, b z−1 is the best location of fa data visited for the emotion, and baz−1 is the present position of the feature point. Furthermore, the trained data is regularized by the Softmax operation for the further function of classification. Classification: Based on the probabilities from the optimized training stage, the emotions are classified further using Equation 5.10: t

= f (x ) arg = max m y(h | x ) = arg max m

e x za

N

∑e

xt z p



(5.10)

p =1

where the weight vector is denoted as z, y is represented as the label variable, the features of sample are considered as x, and the label of class is denoted as m, respectively. As a result, the actions are classified using the trained network. Once the best feeling has been identified, the criterion ceases to work until the best result has been achieved. Finally, the suggested approach’s performance is assessed using several metrics. Figure 5.2 shows the flowchart for the suggested approach of human emotion recognition.

5.6 Result and Discussion All of the applications were tested on a Windows platform with an Intel i5 powered by an octa processor and 8 GB of RAM. Then, MATLAB 2018a was used to conduct the experiment. Initially, the expanded Cohn– Kanade expression sample was used to assess the algorithm’s efficiency. As the  amount of images in the collection improves, so does the accuracy. The dataset was firstly divided into two parts: training set with 80% of the data as well as a testing set with 20% of the data. Several of the training networks were uploaded and supplied the whole testing set with a single image at a moment during the validation process. This was a brand-new image that the model had not seen before. In the beginning, the image input to the model was preprocessed. As a result, the model did not know what the proper output was and had to forecast it accurately based on its own training. It preceded to categorize the emotion depicted in the image

Human Face Emotion Recognition  69 only based about what it had previously learned and the image’s features. As a result, for each image, it generated a list of identified emotion probabilities. The number of correct forecasts was calculated by comparing the highest probability reaction for each image with the real feelings associated with the visuals.

5.6.1 Performance Metrics Evaluation The performance of the proposed method is validated with the subsequent metrics such as accuracy, precision, F-measure, recall, error, and processing time. Precision: The precision metrics are one of the significant evaluation measures for the performance analysis of proposed emotion classification. This is evaluated by the ratio of particular predicted positive emotions to the overall predicted positive emotion, which is expressed using Equation 5.11: ∧

Tp

Precision =

∧

∧

T p+ Fp





(5.11)

∧

where the true positive is denoted as T p and false positive is signified as F p. Accuracy: The accuracy is another important parameter for the evaluation measures in emotion classification. It characterizes the accurate percentage of classified emotion and when the accuracy reaches 100%, then it is referred to as the classification of emotion at its finest and is estimated using Equation 5.12: ∧

Accuracy =

∧

∧

∧

∧

T p+Tn ∧

∧

T p +Tn + F p + Fn



(5.12)

Recall: It defines the particular amount of accurate positives and is classified particularly as optimistic, expressed using Equation 5.13: ∧



Sensitivity =

Tp ∧

∧

T p + Fn



(5.13)

70  Mathematics and Computer Science Volume 2 F-Measure: The metrics of F-measure are estimated by the ratio of mean weight of precision and recall using Equation 5.14: ∧

F-Measure =

2T p ∧

∧

∧

2T p + F p + F n



(5.14)

Error Rate: The proposed emotion classification error rate is evaluated using Equation 5.15 as:

Error = 1 − Accuracy

(5.15)

5.6.2 Comparative Analysis The performance measures attained from the proposed approach are compared with the traditional models GPSO-CNN [15], BLSTM-RNN [13], and SSA-DCNN in terms of precision, accuracy, recall, error rate, F-measure, and processing time. The precision value of the proposed approach for anger, sadness, fear, surprise, and happiness is compared with the conventional methods, GPSO-CNN [15], BLSTM-RNN [13] and SSA-DCNN, and is illustrated in Figure 5.3. From this, the happy emotion achieved the highest precision value by the proposed approach (98.9%) when compared with the existing methods GPSO-CNN (98%), BLSTMRNN (74%), and SSA-DCNN (96.45%).

100

Precision (%)

80 GPSO-CNN BLSTM-RNN SSA-DCNN Proposed

60 40 20 0 Angry

Sadness

Fear

Surprise

Figure 5.3  Comparison of precision over existing methods.

Happy

Human Face Emotion Recognition  71 The accuracy value of the proposed approach for anger, sadness, fear, surprise, and happiness is compared with the existing methods GPSOCNN [15], BLSTM-RNN [13], and SSA-DCNN and is illustrated in Figure 5.4. From this, the happy emotion achieved the highest accuracy value by the proposed approach (99.2%) when compared with the existing methods GPSO-CNN (69%), BLSTM-RNN (86%), and SSA-DCNN (95.33%). Moreover, the recall value of the proposed approach for anger, sadness, fear, surprise, and happiness is compared with existing methods such as GPSO-CNN [15], BLSTM-RNN [13], and SSA-DCNN and is illustrated in Figure 5.5. From this, the happy emotion achieved the highest recall value by the proposed method (97.6%) when compared with the existing methods GPSO-CNN (80%), BLSTM-RNN (88%), and SSA-DCNN (96%). The F-measure value for the five different emotions compared with the existing methods such as GPSO-CNN [15], BLSTM-RNN [13], and SSADCNN is illustrated in Figure 5.6. From this, the happy emotion achieved the highest F-measure value by the proposed method (97.98%) when compared with the existing methods GPSO-CNN (85%), BLSTM-RNN (88%), and SSA-DCNN (95%). Also, the estimation of processing time is the foremost measure in this study because less processing time is more significant than high time. The processing time for the five different emotions compared with the existing

100

Accuracy (%)

80

60 GPSO-CNN BLSTM-RNN SSA-DCNN Proposed

40

20

0 Angry

Sadness

Fear

Surprise

Figure 5.4  Comparison of accuracy over existing models.

Happy

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Recall (%)

80

60 GPSO-CNN BLSTM-RNN SSA-DCNN Proposed

40

20

0 Angry

Sadness

Fear

Surprise

Happy

Figure 5.5  Comparison of recall with existing methods.

Proposed SSA-DCNN BLSTM-RNN GPSO-CNN

Happy

Surprise

Fear Sadness

Angry 0

20

40 60 F-measure (%)

80

100

Figure 5.6  Comparison of F-measure over conventional models.

methods such as GPSO-CNN [15], BLSTM-RNN [13], and SSA-DCNN is illustrated in Figure 5.7. From this, the happy emotion achieved less processing time by the proposed method (14ms) compared with the existing methods GPSO-CNN (60ms), BLSTM-RNN (35ms), and SSA-DCNN (50ms).

Human Face Emotion Recognition  73

Processing time (ms)

60

40 GPSO-CNN BLSTM-RNN SSA-DCNN Proposed

20

0 Angry

Sadness

Fear

Surprise

Happy

Figure 5.7  Comparison of processing time over existing models.

The error value obtained from the proposed classification model for the five different emotions is compared with the existing methods such as GPSO-CNN [15], BLSTM-RNN, [13] and SSA-DCNN and illustrated in Figure 5.8. From this, the happy emotion achieved less error than the proposed method (0.01%) compared with the existing methods GPSO-CNN (1%), BLSTM-RNN (0.2%), and SSA-DCNN (0.18%). GPSO-CNN BLSTM-RNN SSA-DCNN Proposed

1.8 1.6 1.4 Error rate (%)

1.2 1.0 0.8 0.6 0.4 0.2 0.0 Angry

Sadness

Fear

Surprise

Happy

Figure 5.8  Comparison of error rate over conventional approaches.

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5.7 Conclusion Emotion recognition is a difficult problem that is rising in relevance due to its numerous applications. Facial expressions can be useful in assessing a human’s emotion or mental state. The proposed design solutions were executed effectively and a computational simulation verified their validity. The development of a hybrid model and a program for identifying emotions by facial expression is the major result of this study. Each category of emotions has its own set of recognition accuracy metrics. Images with the emotion “happy” (99.2%) have the best recognition accuracy, while images with the emotion “sadness” have the worst (97%). Also, the comparative analysis proves that the proposed method has achieved superior performance in emotion recognition over the existing models in terms of high accuracy, precision, F-measure, recall, and less processing time as well as error rate. In the future, a new intelligent algorithm can develop with proper validation of various human emotions.

References 1. Ghanem, Bilal, Paolo Rosso, and Francisco Rangel. “An emotional analysis of false information in social media and news articles.” ACM Transactions on Internet Technology (TOIT) 20.2 (2020): 1-18. 2. Dasgupta, Poorna Banerjee. “Detection and analysis of human emotions through voice and speech pattern processing.”  arXiv preprint arXiv:1710.10198 (2017). 3. Chen, Caihua. “An analysis of Mandarin emotional tendency recognition based on expression spatiotemporal feature recognition.”  International Journal of Biometrics 13.2-3 (2021): 211-228. 4. Sánchez-Gordón, Mary, and Ricardo Colomo-Palacios. “Taking the emotional pulse of software engineering—A systematic literature review of empirical studies.” Information and Software Technology 115 (2019): 23-43. 5. Hemanth, D. Jude, and J. Anitha. “Brain signal based human emotion analysis by circular back propagation and Deep Kohonen Neural Networks.” Computers & Electrical Engineering 68 (2018): 170-180. 6. Du, Guanglong, et al. “A convolution bidirectional long short-term memory neural network for driver emotion recognition.” IEEE Transactions on Intelligent Transportation Systems (2020). 7. Ashwin, T. S., and Guddeti Ram Mohana Reddy. “Automatic detection of students’ affective states in classroom environment using hybrid convolutional neural networks.”  Education and Information Technologies  25.2 (2020): 1387-1415.

Human Face Emotion Recognition  75 8. Nonis, Francesca, et al. “Understanding Abstraction in Deep CNN: An Application on Facial Emotion Recognition.”  Progresses in Artificial Intelligence and Neural Systems. Springer, Singapore, 2021. 281-290. 9. Sarmah, Dipti Kapoor. “A survey on the latest development of machine learning in genetic algorithm and particle swarm optimization.” Optimization in Machine Learning and Applications. Springer, Singapore, 2020. 91-112. 10. Liu, Junjie. “Automatic Film Label Acquisition Method Based on Improved Neural Networks Optimized by Mutation Ant Colony Algorithm.” Computational Intelligence and Neuroscience 2021 (2021). 11. Mehendale, Ninad. “Facial emotion recognition using convolutional neural networks (FERC).” SN Applied Sciences 2.3 (2020): 1-8. 12. Mungra, Dhara, et al. “PRATIT: a CNN-based emotion recognition system using histogram equalization and data augmentation.” Multimedia Tools and Applications 79.3 (2020): 2285-2307. 13. Li, Chao, et al. “Exploring temporal representations by leveraging attention-based bidirectional LSTM-RNNs for multi-modal emotion recognition.” Information Processing & Management 57.3 (2020): 102185. 14. Estrada, María Lucía Barrón, et al. “Opinion mining and emotion recognition applied to learning environments.”  Expert Systems with Applications  150 (2020): 113265. 15. Gao, Zhongke, et al. “A GPSO-optimized convolutional neural networks for EEG-based emotion recognition.” Neurocomputing 380 (2020): 225-235.

6 An Application of Information Technology in Adaptive Leadership of Ministry of Ayush During Pandemic of Covid 19: A Case Study Vikram Singh1, Shikha Kapoor1 and Sandeep Kumar Gupta2* AIBS, Amity University, Noida, India 2 AMET University, Chennai, India

1

Abstract

The pandemic of COVID-19, since its beginning, has been impacting the lives of people the world over. However, in India, the impact would have been catastrophic had Ayush’s intervention not been there. This paper examines the initiatives and interventions by the leadership of the Ministry of Ayush (MoAyush) to manage the pandemic situation through adaptive leadership. Effective communication, use of IT platform, HR training, engagement with stakeholders, and adaptation of R&D activities are the factors that helped the MoAyush to establish itself as the leader in the fight against the pandemic of COVID-19. Keywords:  COVID-19, leadership, intervention, stakeholders, communication, training, R&D

6.1 Introduction The very first case of COVID-19 was detected in China on 12th December 2019. A comprehensive package of technical guidance was issued by the World Health Organization (WHO) with advice to all countries about detection, testing, and managing potential cases on 10th January 2020. The Govt. of India immediately directed the states and UTs to take all necessary action for health sector preparedness following WHO’s advisory. *Corresponding author: [email protected] Sharmistha Ghosh, M. Niranjanamurthy, Krishanu Deyasi, Biswadip Basu Mallik, and Santanu Das (eds.) Mathematics and Computer Science Volume 2, (77–90) © 2023 Scrivener Publishing LLC

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78  Mathematics and Computer Science Volume 2 The screening of incoming passengers at the airports of Delhi, Mumbai, and Kolkatta commenced on the 17th January 2020 and further extended to airports at Chennai, Cochin, Bengaluru, and Hyderabad on 21st Jan 2020. The first COVID-19 positive case in the country was detected on 30th Jan 2020 in Kerala. During times of crisis, there is a generally widespread sense of volatility, uncertainty, complexity, and ambiguity which need very prompt and high impact decisions in a limited information setting. Leaders have to manage these elements and adapt to new situations along the way. The crisis management is the set of activities targeted at reducing the negative consequence of crises. Effective crisis management saves lives, safeguards infrastructure, and restores public faith (Boin et al., 2013). Several characteristics and practices that effective leaders embody during the time of crises are communication, decision making, humanism, innovation, realism, and core values (Kaul et al., 2020). This paper examines the main characteristics and practices the MoAyush adopted during the crises. The Ministry of Ayush, while taking stock of the situation, issued an advisory on ways to protect the general public from COVID-19 and stay healthy (PIB release 29th Jan 2020). The advisory comprised of simple home remedies, including the use of common Ayurvedic, Homeopathic, and Unani medicines. The Ayush Ministry kept a close watch on the pandemic spread and responded with enhanced vigour and foresightedness. It took various initiatives including an awareness generation campaign, Research and Development (R&D), and pieces of training of Ayush manpower in clinical practices to handle the future eventualities, etc. Various important initiatives have helped mitigate the impact that COVID-19 could have brought to the nation.

6.2 Ministry of AYUSH On the 9th of November 2014, the Department of Ayush was converted into a full-fledged Ministry by the Government. It was done with a vision for reviving the profound knowledge of ancient systems of medicine and ensuring the optimal development and propagation of the Ayush systems of healthcare. The budget of the Department of Ayush was around Rs 10 crore in 1992 and Rs. 1272 crores in 2014-2015, which became to Rs. 2970.30 crores for the year 2021-2022. There has been a significant increase in the work undertaken by the Ministry over the years. It is worth mentioning that the budget of the Ayush Ministry is a small fraction of

Application of IT in Adaptive Leadership of MoAyush  79 the health budget but the impact among stakeholders has been large (PIB release 7th Feb 2021). The present government is very supportive of Ayush systems. Various new and innovative programmes have been launched since 2014. International Day of Yoga (IDY), the establishment of IT infrastructure in the form of AyushGrid, the launch of National Ayush Mission, and certification of yoga professionals and accreditation of Yoga Institutions were some of the significant contributions of the Ministry apart from regular Central Sector Schemes. The Ayush services are available in all major government hospitals and Ayush treatment is part of the insurance policy now. Out of 125,000 Health and Wellness Centres (HWCs) the government is planning to establish, 10% would be dedicated to Ayush systems.

6.3 Leadership Principles and Practices by Ministry of AYUSH During Covid-19 In the changing environment, the study of leadership during a crisis has become more relevant because crises are more unpredictable, longer-­ lasting, and costly than in the past (Pinsdorf, 2004). Leaders all across the world are grappling with the difficulties of anticipating, responding to, and trying to recover from the pandemic of COVID-19. The pandemic of COVID-19 impacted every sector, but it put the health sector to the test. The foundations of India’s healthcare have been shaken. The public sector performed miserably compared to the private sector. However, the Ayush sector came to the rescue during this crisis through timely intervention and adaptive leadership. The following principles and practices are worth noticing.

6.4 Effective Communication Effective communication is very crucial during any crisis. Communication done with urgency, transparency, and empathy is effective. It helps the public to adjust to the tremendously changing conditions during any crisis like the pandemic of COVID-19 has shown. The Ministry of Ayush, without waiting for the widespread of Corona Virus, provided effective communication in the form of press releases and advisories to Chief Secretaries of the States/Union Territories and various stakeholders in January 2020.

80  Mathematics and Computer Science Volume 2 The Ministry launched a dashboard on its website which provided COVID-19 related information in English as well as in Hindi. Immunity boosting measures for subject care were placed at a very conspicuous place and visible immediately on the landing page. The Ministry also opened an outlet for an immunity booster kit at its Headquarters and also made it available on the Amazon online platform. One of the aspects of effective communication during a crisis is the efficacy of communication. The Ministry through its website, social media, and autonomous bodies regularly communicated with the stakeholders and public at large. Various campaigns like “Ayush for Immunity” and “Yoga for Wellness” were launched on a mass scale. The Ministry also invited suggestions and proposals from various organizations under its Extra Mural Research Scheme to fight COVID-19. Hundreds of proposals were received and considered. Further, MoA also issued directives to all State/UT licensing authorities and drug controllers of Ayush to increase the production of immunity booster medicines. On one side, efforts were on for promoting immunity efforts but some unscrupulous people were advertising and promoting misleading information about Ayush intervention and were adverting claims for treatment of Covid-19 and cheating people. The ministry took this misleading information very seriously and directed all Regulatory Authorities in the States/Union Territories, through its circular dated 1st April 2020, to stop and prevent publicity and advertisement of Ayushrelated claims for COVID-19 treatment in print, TV, and electronic media and to take necessary action against the persons/agencies involved in contravening the relevant legal provisions and the aforesaid National Disaster Management guidelines (NMMA).

6.5 Sharing of Resources COVID-19 drastically changed the way people work, shop, eat, travel, and meet. The pandemic set up a paradigm in employment landscaping that required re-scaling and re-training. The Ayush practitioners, though equally competent, were only a fraction of those engaged in clinical management. With conventional health infrastructure coming under huge stress, Ayush infrastructure provided the balance in supply and demand for clinical management. The Ministry enabled the availability of infrastructure in all Ayurveda, Siddha, Unani, and Homoeopathy institutes to act like hospitals, labs, and ICUs along with medical and Para-medical staff to compete with the COVID-19 pandemic. The States and the UTs were suggested to use Ayush infrastructure, which comprises of 50000

Application of IT in Adaptive Leadership of MoAyush  81 beds attached to 750 Ayush Colleges and 86 clinical facilities and Research Councils under the Ministry. All the hospitals, colleges, centres, and facilities were shared.

6.6 Shared Decision Making Shared decision making involves the bidirectional exchange of information and values, better implementation of the decisions, and enhancement of management strategies (Abrams et al., 2020). Before all the advisories and directions, MoAyush not only consulted the stakeholders but also made shared decisions. The India Institute of Ayurveda (AIIA) launched the Ayuraksha programme for Delhi Police and their families (PIB, 30th April 2020). All the research councils and national institutes were consulted by the Ministry on regular basis. Ayush-64, an Ayurveda medicine found effective in mild and moderate cases, was freely distributed for COVID-19 positive cases in the second wave. Further, as part of AzadikaAmrutMahotsav, the immunity booster medicines would be distributed to 75 lakh people across the country, with special attention on the geriatric population and front line workers (PIB release 2nd Sept 2021).

6.7 Training of Manpower The Ministry of Ayush and the Ministry of Health and Family Welfare jointly organized training of around 33000 Ayush Master Trainers for further training of the Ayush workforce, considering the future requirement to manage the pandemic in April 2020. Over 66000 Ayush personnel were trained at the Integrated Government Online Training (iGOT) portal. The effectiveness of training on iGOT can be seen in that over 37000 Ayush staff were successfully deployed for COVID-19 related health services. In order to attain a high degree of uniformity in sickness care, the ministry has set rules for registered practitioners of distinct Ayush systems (Guidelines for Ayush Practitioners for COVID 19).

6.8 Use of IT Platform The complete lockdown was announced in the country on 24th March 2020, commencing at midnight. All services including healthcare services in OPD were at a standstill. There was an urgent need for guidelines on

82  Mathematics and Computer Science Volume 2 Telemedicine for Ayush systems to the community of service. The ministry issued the guidelines on 7th April 2020 on its website. On May 7, 2020, the Ministry launched the “Ayush Sanjeevani” mobile app to collect data on experience, usage of Ayush advocacies within the population, and its impact on COVID-19 prevention. This app would gather information on the measures applied by the people for enhancing immunity and keeping themselves healthy during COVID-19 situations (PIB release dated 7th May 2020). Data from over 723 thousand respondents for three months, May-July 2020, revealed that 85.2% used Ayush methods for maintenance of body immunity during the COVID-19 pandemic and around 89.8% of them benefitted from AYUSH measures. (Srikanth et al., 2021). Further, the Ayush Clinical Case Repository (ACCR) portal (https:// accr.ayush.gov.in) was established and developed by the Ministry as a platform to support both Ayush practitioners and the public to portray the strength of the Ayush System. In this portal, Ayush practitioners from anywhere in the world can enrol themselves and share relevant information about their successful treatment for the benefit of all. It was launched on 27th May 2021 (PIB release dated 26th May 2021). As of now, it has 28 cases including 12 on COVID-19. The Ministry of Ayush already initiated the Ayush Grid Project in the year 2018 for creating a comprehensive IT backbone for the entire Ayush Sector (Yadava & Ps, n.d.). Through the Ayush Grid, the entire Ayush Sector would be digitalized and it would lead to sector transformation in the field of healthcare delivery at all levels and would help research, education, drug development, and regulation of various health programmes of the Government. The Ayush Grid would be beneficial for all the stakeholders of Ayush at the National and Global levels. Under the Ayush Grid, the project of the Ayush Health Management Information System (AHMIS) had been developed into a quality based information system and over 100 clinical establishments of the Ministry are using it successfully. The Ministry also offered the AHMIS to all the Ayush units outside the Government umbrella so that the entire Ayush Sector benefits. The components of Ayush Grid initiatives would gather and connect all the folks and stakeholders of the Ayush Sector. The Ayush Grid is expected to emerge as a comprehensive IT backbone serving 0.85 million Ayush physicians and over 500 million citizens within 3 years. The Ayush Grid has been integrated with the National Digital Health Mission (NDHM) and will be useful to the public for varied options of health needs. The integration of the Ayush Grid into NDHM will speed up the mainstreaming of Ayush systems of healthcare (PIB release dated 2nd Oct 2020).

Application of IT in Adaptive Leadership of MoAyush  83 The Ministry also launched a dedicated yoga portal (https://yoga.ayush. gov.in/) to meet the demand of Yoga in the country as well as abroad. The portal contains information on Common Yoga Protocol (CYP), the audiovisual source material for training and practice on CYP, and information related to IDY. Further, a Yoga Break (Y-Break) mobile app was also launched for practising 5 minutes of yoga at the workplace. The protocol has a set of stretches, pranayama, and meditation developed by experts. Y-break, when practised twice a day, would reduce stress and refresh and refocus the workforce. A pilot project of Telemedicine was launched in the Siddha System of medicine in 100 villages in Tamil Nadu in Nov 2019 by the Ministry with the help of Common Service Centres. Detailed guidelines of telemetry of Ayurveda, Unani, and Siddha practitioners were issued by the Central Council for Indian Medicines on 7th April 2020 and these guidelines are also made available on the website of the Ministry in line with India’s digital health policy. The Ministry of Ayush also developed a dashboard (https:// dashboard.ayush.gov.in/) for various information in the Ayush Sector like Ayush health infrastructure, health, drug industry, institute, research, budget, etc. Another project launched by the Ministry of Ayush is “Ayush Next” to support Ayush education. It is a digital platform intended to enhance the realm of information exchange by providing career recommendations, an interactive forum, quizzes, and more. The group of professionals and domain experts is well-versed in resolving concerns about careers and employment in the Ayush sector. Information on various degrees and certifications is also available. It is a place for young people of the Ayush sector to share their thoughts and knowledge, seek guidance and counselling, discuss, and continue to learn.

6.9 Finding Opportunities for R&D During the Crisis Ayush Systems have been criticized for not being practically scientific and lacking evidence. The pandemic brought a different type of opportunity before the Ayush Sector to collect fresh evidence and re-establish itself with modern medical science. An Interdisciplinary Ayush R&D Task Force was constituted with Prof. Bhushan Patwardhan as chairperson and other members from the Council of Scientific and Industrial Research (CSIR), the Indian Council of Medical Research (ICMR), Department of Biotechnology (DBT), All India Institute of Medical Sciences (AIIMS), and prominent Ayush Institutions in the first week of April 2020. It designed and formulated clinical research protocols for prophylactic

84  Mathematics and Computer Science Volume 2 studies and add-on interventions in COVID-19 positive cases. Further, The Ministry of Ayush solicited research ideas from diverse stakeholders in order to develop scientific proof. Over 125 clinical research projects and basic experimental investigations were initiated in over 150 centres across the country based on the recommendations of the Interdisciplinary task group. To name a few, these studies were conducted by King George Medical University Lucknow, All India Medical Institute New Delhi, Institute of Medical Sciences BHU, Govt. Medical College Nagpur, KEM Hospital Pune, National Institute of Pharmaceutical Education and Research Kolkatta, DMIMS Wardha Indian Institute of Integrative Medicine Jammu, and Institute of Genomics and Integrative Biology (IGIB) New Delhi. In addition, the Central Council for Study in Yoga and Naturopathy (CCRYN) has initiated a research initiative to validate the efficacy of Yoga in supporting early recovery of COVID– 9 patients in collaboration with Rajiv Gandhi Super Speciality Hospital, Delhi, AIIMS Delhi, AIIMS Rishikesh, and RML Hospital, Delhi. The All India Institute of Ayurveda (AIIA) discovered that Ayurvedic therapies such as Ayush kwatha and Fifatrol pills might be helpful in mild to severe COVID-19 infection in a “quite short period” with “total regression of symptoms.” As per the case report published in the AIIA journal ‘Ayurveda Case Report’ in October 2021, the use of four Ayurvedic interventions, Sanshamanivati, Ayush kwatha, Laxmivilasa rasa, and Fifatrol tablets, not only improved the condition of the COVID-19 infected patient but also gave a negative report on the fast antigen test within six days of medication.

6.10 Collaborating with Stakeholders for International Day of Yoga (IDY) The United Nations General Assembly (UNGA) unanimously recognised 21st June as the International Day of Yoga (IDY) on the appeal of PM Narendra Modi. The day has been celebrated around the world in everincreasing numbers. The Ministry of Ayush is the nodal ministry responsible for managing and promoting the practice of Yoga and coordination activities for the observation of IDY every year. The IDY observance for the year 2021 could not be more timely, as the world and India at large reel from a deadly wave of the pandemic as new variants emerged. Hence, in that spirit, the theme of this year was kept as “Yoga for Wellness” to remind

Application of IT in Adaptive Leadership of MoAyush  85 the world about Yoga and the holistic approach to health and wellbeing it takes. Despite IDY observances, though overcast by restrictions due to the pandemic for a second time, the Ministry learned from previous years’ experience. The Ministry followed a three-pronged strategy which included taking a digital-first approach, activating Government of India stakeholders and their networks which have a wide-ranging reach, and collaborating with the private sector in a greater way to pave the road for the adoption of Yoga at a larger scale and development of the sector in the years to come. In the digital-first approach, the Ministry leveraged its existing IT platforms and social media including Facebook, Twitter, Instagram, YouTube, the Ministry’s website, and the Yoga Portal; it also leveraged new platforms such as Koo, Josh, and Bolo India to reach wider viewers which consume diverse kinds of content and prefer local and regional languages. The Ministry released additional digital assets apart from improvements in the existing ones, namely the Namaste Yoga application. Further, the Government’s MyGov platform was used for the first time to launch citizen-centred awareness building and engaging activities. People were encouraged to practice yoga based on Common Yoga Protocol (CYP) virtually across the country. Various IT tools and online resources were created during the pandemic. Yoga has been very useful for the well-being of people during the pandemic. The Prime Minister of India himself leads the nation for IDY celebration. He launched the “mYoga” application, developed in house by the Morarji Desai National Institute of Yoga in collaboration with the World Health Organisation (WHO) on 7th IDY on 21st June 2021. The app can be used by all to promote one world, one health. (PIB release 21st June 2021). The mYoga application focuses on WHO’s theme of ‘Be He@lthy, Be Mobile (BHBM)’, which is in line with the United Nations’ Sustainable Development Goal of achieving Universal Health Coverage by 2030. The app addresses diseases and risk factors with an opportunity to navigate upon the m-Health programme for Yoga. It offers a collection of learning modules and practice sessions in both audio and video formats, which enable users to practice Yoga anytime from the comfort of their homes. The mYoga app is available in the Google Play Store and Apple App Store. As per the AppBrain ranking report, (https://www.appbrain. com/), the WHO mYoga app is placed at a Google Play store rating of 4.75. It has over 50+ thousand downloads.

86  Mathematics and Computer Science Volume 2 The yoga portal of the ministry had a viewership of over 1.05 million. The Yoga Certification Board (YCB) for the Yoga Appreciation Programme and Yoga Volunteer Training (YVT) Course, reported an outreach of about 1.214 million. The Prime Minister addressed the nation on IDY through various DD channels, emphasising the role of maintaining health and well-being in the lives of people, especially at a time when the world was reeling from the COVID pandemic. All the Central Ministries, State Governments, public sector undertakings, autonomous bodies, various yoga organizations, Universities, National Cadet Corps (NCC), National Social Service (NSS), Nehru Yuva Kendra (NYKs), schools, hospitals, railways, postal services, and GramPradhans were brought on board. To reach the nooks and corners of the country, MoAyush also used services of various Common Service Centres (CSC) and Community Radio Stations to promote IDY activities and motivate people to practice yoga. The Ministry also had a partnership with Nickelodeon and HealthifyMe to reach different segments of people. Indian Missions Abroad were also equally active in the promotion and observation of IDY. The coordinated efforts of MoAyush have made Yoga a mass movement and around 156.86 million people participated during IDY 2021 (Table 6.1). Table 6.1  Social media outreach during International Day of Yoga 2021. Social media platform

Outreach (in thousands)

YouTube

2,100

Facebook

11,551

Twitter

8,910

Instagram

4,100

KOO

6,000

JOSH

3,57,300

Bolo India

700

Total Social Media

3,90,661

Summary of Platform-wise Outreach of IDY 2021 (Source MoAyush).

Application of IT in Adaptive Leadership of MoAyush  87

6.11 Providing Hope When Nothing Seemed to be Working Ayush, through its communications, continued to inform the citizens to use Ayush products, particularly home remedies for improving immunity in the fight against COVID-19. This hope has helped millions, which is evident from the demand for Ayush immunity booster products, including Chavanprash. The social media of the Ayush Ministry kept engaging people on the use of Ayush products. Prime Minister spoke on Ayush Kadha in mannkibaat. The Secretary of the Ministry, Rajesh Kotecha, who himself is a prominent Ayurveda expert and a Padma Shri awardee, on almost all forums emphasized that Ayush systems were effective tools in the management of COVID-19. Yoga sessions were arranged in Covid Care Centres and were well received.

6.12 Leveraging Old Knowledge Yoga and Naturopathy have been known for their long term positive impact on the body, mind, and spirit. In the lockdown period, commencing in March 2020, more than physical fitness, the mental and emotional health of people was put to the test. Yoga came as a saviour. The Ministry of Ayush through its autonomous bodies, the Morarji Desai National Institute of Yoga, and the National Institute of Naturopathy (NIN), conducted regular yoga sessions for health and wellbeing. Further considering the impact on mental well-being, the Ministry of Ayush in collaboration with SVYASA and NIMHANS also released a protocol for psychosocial rehabilitation of COVID-19 infected patients. In October 2020, the Ayush Ministry issued suggestions to incorporate “Ayurveda and Yoga therapies” into India’s national clinical management plan for COVID-19 infection. It suggested treating COVID-19 with warm water gargles, medicinal ghee in nostrils, steam inhalation, drinking “golden milk” (turmeric mixed with hot milk), and kashayam/kadha/ kwath (hot infusion with Ayurvedic herbs) in addition to a healthy diet, sleep, and exercise. The clinical protocol advised patients to take Ayurvedic formulations such as Guduchi and Guduchi Ghana Vati, with Asvagandha and Pipalli, even if they were suffering from hypoxia (a lack of oxygen in the body) and dyspnea caused by COVID-19 infection.

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6.13 Conclusion The crisis of COVID-19 has impacted the whole country, but for the Ayush sector, it brought tremendous opportunities. The ministry of Ayush provided timely intervention and has helped reposition Ayush therapy not only at the national level but worldwide. It was made possible with IT tools and the use of the digital platform. Effective and regular communication during the continuity of the crises, sharing of resources with other stakeholders, use of IT platform regressively, training and preparedness of existing manpower, and leveraging old wisdom were some of the practices which helped MoAyush to establish itself as a true leader at the time of crisis.

References 1. Abrams, E. M., Shaker, M., Oppenheimer, J., Davis, R. S., Bukstein, D. A., &Greenhawt, M. (2020). The Challenges and Opportunities for Shared Decision Making Highlighted by COVID-19. In Journal of Allergy and Clinical Immunology: In Practice. American Academy of Allergy, Asthma and Immunology. https://doi.org/10.1016/j.jaip.2020.07.003. 2. Boin, A., Kuipers, S., & Overdijk, W. (2013). Leadership in times of crisis: A framework for assessment. International Review of Public Administration, 18(1), 79–91. https://doi.org/10.1080/12294659.2013.10805241 3. Kaul, V., Shah, V. H., & El-Serag, H. (2020). Leadership During Crisis: Lessons and Applications from the COVID-19 Pandemic. Gastroenterology. https://doi.org/10.1053/j.gastro.2020.04.076 4. Kaul V, Shah VH, EI-Serag H (2020). Leadership During Crisis: Lessons and Applications from the COVID-19 Pandemic, Gastroenterology. 5. Lee A. Wuhan novel coronavirus (COVID-19): why global control is challenging? Vol. 179, Public Health. Elsevier B.V.; 2020. p. A1-2 6. Ministry of AYUSH, Govt. of India 2020. D.O. No. S.16030/18/2019-NAM dated 6th March 2020, Ministry of AYUSH, Govt. of India 2020. Ayurveda’s immunity-boosting measures for self-care during COVID 19 crisis. https:// www.ayush.gov.in/docs/123.pdf 7. Ministry of AYUSH, Govt. of India 2020.Guidelines for AYUSH Practitioners for COVID 19. https://www.ayush.gov.in/ayush-guidelines.html 8. Pinsdorf,  M.  K.  (2004).  All Crises are Global: Managing to Escape Chaos. United States: Fordham University Press. 9. Srikanth N, Rana R, Singhal R, Jameela S, Singh R, Khanduri S, Tripathi A, Goel S, Chhatre L, Chandra A, Rao B, Dhiman K (2021). Mobile App– Reported Use of Traditional Medicine for Maintenance of Health in India

Application of IT in Adaptive Leadership of MoAyush  89 During the COVID-19 Pandemic: Cross-sectional Questionnaire Study, JMIRx Med 2021;2(2):e25703, URL: https://xmed.jmir.org/2021/2/e25703, DOI: 10.2196/25703 10. Yadava, R., & Ps, A. (n.d.). Development and contributions of AYUSH sector A Review. 11. https://pib.gov.in/PressReleasePage.aspx?PRID=1621787 dated 7th May 2020 12. https://pib.gov.in/Pressreleaseshare.aspx?PRID=1695897 dated 7th Feb 2021 13. https://pib.gov.in/PressReleasePage.aspx?PRID=1619678 dated 30th April 2020 14. https://pib.gov.in/PressReleasePage.aspx?PRID=1721962 dated 26th May 2021 15. https://pib.gov.in/PressReleasePage.aspx?PRID=1729128dated 21st June 2021 16. https://pib.gov.in/PressReleaseIframePage.aspx?PRID=1751452 dated 2nd Sept 2021 17. https://www.appbrain.com/app/who-myoga-app/org.who.APPMYOGA 18. https://pib.gov.in/PressReleasePage.aspx?PRID=1600895, dated 29.1.2020

7 Encoder-Decoder Models for Protein Secondary Structure Prediction Ashish Kumar Sharma* and Rajeev Srivastava Department of Computer Science and Engineering, Indian Institute of Technology (BHU), Varanasi, Uttar Pradesh, India

Abstract

Proteins are arranged in a linear sequence due to peptide bonds. In proteins, a peptide bond combines the amino group of one protein with the carboxyl group of another protein. Protein secondary structure formation results from their biophysical and biochemical properties, like natural languages which depend on their grammatical rule. So, the proposed model predicts a secondary structure from protein primary sequences using the encoder-decoder based machine translation method. The proposed model uses an encoder-decoder model based on long shortterm memory network. The proposed work uses training and testing performed on available public datasets, namely CullPDB and data1199. The proposed model has better Q3 accuracy of 84.87% and 87.39% for CullPDB and data1199, respectively. Further, the proposed work was evaluated by comparing their performance with other methods which predict secondary structure only from a single sequence. The Encoder-Decoder Model for predicting secondary structure from a single primary sequence is performing better than other single sequence-based methods. Keywords:  Protein structure prediction, amino acids sequence, proteomics, one hot encoding, encoder-decoder, long short-term memory

7.1 Introduction Protein is important biomolecules which help in developing the cells of all living organisms. Proteins mainly perform the role of transporters, catalysts, receptors, and hormones within all living organisms [1]. The *Corresponding author: [email protected] Sharmistha Ghosh, M. Niranjanamurthy, Krishanu Deyasi, Biswadip Basu Mallik, and Santanu Das (eds.) Mathematics and Computer Science Volume 2, (91–100) © 2023 Scrivener Publishing LLC

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92  Mathematics and Computer Science Volume 2 primary sequences of proteins are a linear sequence of twenty naturally occurring amino acids. The amino acids are represented by one or three letter symbols called codons. Three levels of structure mainly describe the Protein structure [2]. The primary structure is linearly connected amino acids with peptide bonds. The protein secondary structure is defined by the local segments which are categorized into eight classes. These eight categories of secondary structure are the alpha-helix (H), 3-10 helix (G), pi helix (I), beta-sheet (E), beta bridge (B), turn (T), bend (S), and loop (L) [3]. Further, the eight classes of secondary structure are simplified into three classes: helix, strand, and loop/coil. The tertiary structure is the three-dimensional structure. To predict the secondary structure of a protein, the primary sequence is a sequence-labeling task. The protein secondary structure helps in estimating its functions, which is essential for drug designing and protein engineering. In natural language processing, there are several character-based methods for classification and prediction. These character-based methods are mainly classified on the basis of their character-level information into three categories. These three character-based models are bag-of-n-gram, tokenization-based, and end-to-end [4]. The neural machine translation has become popular with better performance of character-level models. The encoder-decoder models, which use character-based deep neural networks, are applied to several problems such as machine translation, question answering, and speech recognition [5]. The protein sequences are formed with amino acids which are defined by biophysical and biochemical principles. The biophysical and biochemical principles of proteins are similar to the grammar of natural languages. This similarity motivates protein sequences that are the output of a specific biological language and develops a method based on natural language processing such as the encoder-decoder method to discover functions encoded within protein sequences. The contributions made by this work are: (1) in the proposed model, the primary sequence map to the secondary structure sequence as language translation; (2) one LSTM layer encodes the primary sequences and returns its internal state to input in the decoder and another LSTM layer works as a decoder for secondary structure prediction; and (3) the proposed encoder–decoder model for secondary structure prediction was evaluated on two datasets, cullpdb and data1199. The experiments performed show that the proposed encoder-decoder model better captures the features from amino acid sequences to predict their secondary structure.

Encoder-Decoder for Protein Secondary Structure Prediction  93

7.2 Literature Review In the last few decades, various methods have been proposed which are mapping a protein’s primary sequence to their secondary structure sequence. Protein secondary structure prediction methods combine sequence homology searches [6–8] with basic features such as physicochemical properties of the primary sequence [9] and backbone torsion angles [10]. Further, these combined features are fed into neural networks [11, 12] or deep neural networks [13, 14] that produce secondary structures of protein. Recently, methods used for protein secondary structure prediction are PSIPRED [15] and JPRED [16], which have an average Q3 accuracy of approximately 80–85% for benchmark datasets. PSIPRED was the first secondary structure prediction tool that uses the PSI-BLAST search to improve prediction accuracy. PSIPRED utilize the protein database UNIREF90 to obtain sequence profiles and pass to the neural network with two-layers. JPRED integrates the PSI-BLAST sequence profile and HMMer [17] sequence profiles to the neural network. The currently available tools which have Q3 accuracy around 85–90% include SSpro [18], DeepCNF [19], PORTER [11], and PSRSM [20]. SSpro uses template data by utilizing sequence homology sequences. If homology sequences are not available, then the neural network predicts their secondary structure class. Two methods, DeepCNF and PORTER, combine the protein sequence profiles with deep conditional neural fields and convolutional neural networks. The protein secondary structure prediction methods only from a single sequence are PSIPRED-Single and SPIDER3-Single [21–24]. These methods utilize the information extracted from a single sequence to find the protein secondary structure.

7.3 Experimental Work 7.3.1 Data Set To train the model, protein data was obtained by downloading from cullpdb [25], which has a resolution value below 2.5 Å, R-factor below 1.0, and 30% non-redundant sequences in February 2017. The sequences with incomplete information, having a length value below 30 residues, and with find similarity more than 30% are removed by BlastClust [26]. We used publicly available dataset data1199 as a testing set. The test set is 1199 sequences with no redundancy [27].

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7.3.2 Proposed Methodology The proposed protein secondary structure prediction model is a long shortterm memory network-based encoder-decoder architecture. The input is the primary sequence and the output is the protein’s secondary structure. Naturally, both the input protein primary sequences and output secondary structure sequences are of equal lengths since each amino acid character has a corresponding secondary structure character. In the proposed model, the conditional probability of a secondary structure sequence (y1, . . . , ym) estimated is given as an input amino acid sequence (x1, . . . , xn).



p ( y1 , y2 ........ yn | x1 , x2 ,..........xn )

(7.1)

The proposed encoder-decoder model first encodes the input protein primary sequence and one amino acid character in the primary sequence is represented using a latent vector. Then, it decodes to a protein secondary structure sequence.

7.3.3 Data Preprocessing The protein sequences are split into twenty amino acid characters. Each amino acid is numbered with an integer value in the range of 1-20. The amino acid characters are represented as one-hot vector. The protein sequences are of varying length, but the deep learning model accepts the fixed length. If any sequence exceeds in size, then the remaining character is discarded. We padded with zeros in the shorter sequence.

7.3.4 Long Short Term Memory Long short term memory (LSTM) [28] uses the capability of input gate (it), forget gate (ft), and output gate (ot) to control the flow of information to operate selective read, selective forget, and selective write. To utilize the information efficiently and discard the unnecessary information, the three gates in LSTM use the current input, previous state, and output selectively. The activation function depends on the gates used in the LSTM. The logistic and sigmoid function are used as an activation function. The flow of information depends on memory blocks used in the hidden layer. The governing equations for LSTM are as follows:



= at σ ( w xa .xt + uha .Outt −1 + ba )

(7.2)

Encoder-Decoder for Protein Secondary Structure Prediction  95



= it σ ( w xi .xt + uhi .Outt −1 + bi )

(7.3)

= ft σ ( whf .xt + uhf .Outt −1 + b f )

(7.4)

= ot σ ( who .xt + uho .Outt −1 + bo )

(7.5)

state = atΘit + ftΘstatet−1

(7.6)

= out statet ΘOt



(7.7)

One hot Encoding

0 0

0 0

0

h3

hfinal

y2

y3

y4

Secondary structure Probability

Secondary structure Probability

Secondary structure Probability

Secondary structure Probability

Predicted Secondary Structure

Softmax Function

Decorder

h2

y1

hinitial

0

0 0

0 0

1 0

0 0

1

1 0

0

0

0

0

0

0

0

1

1

0

0

1 0 0

1

0

0

0

0

0

1

0

x1

x2

x3

x4

y1

y2

y3

y4

Primary Sequences

Supervised Secondary Structure

0

One hot Encoding

h1

STATE VECTOR

Encorder

where Wxa, Wxi, Whf, and Who are weight matrices for input vectors xt, Uha, Uhi, Uhf, and Uho, which are weight matrices for previous state Outt-1, ba, bi, bf, and bo, which are the bias terms for each memory gate at, it, ft, ot. ʘ signifies the Hadamard product (element-wise product) operation and σ is the sigmoid function. The encoder-decoder model for the primary protein sequence to the secondary structure is depicted in the Figure 7.1. In the model, the encoder and decoder both use LSTM having 256 units. The state vector representation obtained from the encoder layer works as the input to the decoder

Figure 7.1  Encoder–decoder model for protein primary sequence to secondary structure.

96  Mathematics and Computer Science Volume 2 LSTM. The encoder LSTM encoded the primary sequences, while the second decoder LSTM predicts the secondary structure sequences. The encoder LSTM layer processes the individual amino acid characters in a sequence and finally, produce a state vector. The decoder receives the state vectors from the encoder. After processing the complete input primary sequence, the decoder layer uses the secondary structure character to decode the state vectors into a secondary structure sequence. The decoder layer maximizes the value of log-likelihood of predicted secondary structure sequence on the basis of encoder layer state vectors and past protein secondary structure characters. The encoder layer processes the protein primary sequences by utilizing individual amino acid characters and finally, produce the state vectors, while the decoder layer uses the secondary structure sequence and the encoder layer state vectors.

7.4 Results and Discussion The performance metrics from Q3 accuracy are used for evaluating secondary structure prediction. Python version 3.6.7 is used for implementing the proposed sequence-to-sequence model. To implement the model at the front end, keras is used while Tensorflow is used as the back end. Keras is an open-source application programming interface for deep learning. Tensorflow has an excellent computational ability. The statistical method used to sample the dataset was 70 to 30 for the training set and testing set to develop a generalized model. The encoder-­ decoder model is based on a long short-term memory network, which has several hidden layers and each layer has a number of processing cells, so a large amount of data is used for processing. The RmsProp optimization has a small minibatch of 64 to more quickly process the model. The weights and biases are updated using categorical cross-entropy, which is the negative log likelihood loss. The proposed encoder-decoder model performance compared with other methods such as SPIDER3-Single [22] and PSIpred-Single [21], which only use a single sequence for the cullpdb dataset, are listed in Table 7.1. We find that the proposed model performance with single LSTM is higher than other methods, i.e., SPIDER3-Single [22] and PSIpred-Single [21]. SPIDER3-Single combines the one-hot feature vector with Bidirectional LSTM for protein secondary structure prediction and PSIpred-Single considers statics of significant amino acid by calculating their correlation at each segment.

Encoder-Decoder for Protein Secondary Structure Prediction  97 Table 7.1  Comparison of performance of various single-sequence based prediction on Cullpdb. Methods

Q3 (%)

Seq2Seq (LSTM)

84.87

SPIDER3-Single

73.24

PSIpred-Single

70.21

In order to show the effectiveness of the sequence-to-sequence model, the proposed model performance is compared with single sequence based methods SPIDER3 [22], JPred4 [29], and RaptorX [30] to predict protein secondary structure for dataset data1199, are listed in Table 7.2. Table 7.2  Comparison of various single-sequence based prediction on data1199. Methods

Q3 (%)

Seq2Seq Model

87.39

SPIDER3

83.3

JPred4

79.3

RaptorX

81.5

Spider3 methods used a bidirectional long short-term memory network. JPred4 used the JNet procedure and RaptorX used deep convolutional neural fields for predicting the protein secondary structure. To estimate the performance of our model, the protein sequences in the testing sets and training sets have low similarity. The Q3 accuracy of our sequence-tosequence model is 87.39%.

7.5 Conclusion In the proposed work, primary protein sequences are translated to their secondary structure using the sequence-to-sequence model. Protein primary sequences are represented as one-hot encoding and directly feed to the LSTM based sequence-to-sequence model. The proposed model has comparatively better performance on two publicly available datasets, CullPDB and data1199. Despite its simplicity, the LSTM based sequence-to-sequence

98  Mathematics and Computer Science Volume 2 model easily captures the complex relationship between amino acids and their secondary structure.

References 1. Z. Wang, F. Zhao, J. Peng, J. Xu, Protein 8-class secondary structure prediction using conditional neural fields., Proteomics. 11 (2011) 3786–92. https:// doi.org/10.1002/pmic.201100196. 2. Q. Jiang, X. Jin, S.-J. Lee, S. Yao, Protein secondary structure prediction: A survey of the state of the art., J. Mol. Graph. Model. 76 (2017) 379–402. https://doi.org/10.1016/j.jmgm.2017.07.015. 3. Y. Wang, H. Mao, Z. Yi, Protein secondary structure prediction by using deep learning method, Knowledge-Based Syst. (2017). https://doi.org/10.1016/j. knosys.2016.11.015. 4. H. Schütze, S. Schütze, H. Adel, E. Asgari, arXiv:1610.00479v3 [cs.CL] 1 May 2017, 2017. 5. E. Vylomova, T. Cohn, X. He, G. Haffari, Word Representation Models for Morphologically Rich Languages in Neural Machine Translation, in: Proc. First Work. Subword Character Lev. Model. NLP, Association for Computational Linguistics (ACL), 2018: pp. 103–108. https://doi. org/10.18653/v1/w17-4115. 6. A. Drozdetskiy, C. Cole, J. Procter, G.J. Barton, JPred4: A protein secondary structure prediction server, Nucleic Acids Res. 43 (2015) W389–W394. https://doi.org/10.1093/nar/gkv332. 7. D. Ganguly, D. Roy, M. Mitra, G.J.F. Jones, Word Embedding based Generalized Language Model for Information Retrieval, in: 2015. https://doi. org/10.1145/2766462.2767780. 8. M. Ashburner, C.A. Ball, J.A. Blake, D. Botstein, H. Butler, J.M. Cherry, A.P. Davis, K. Dolinski, S.S. Dwight, J.T. Eppig, M.A. Harris, D.P. Hill, L. IsselTarver, A. Kasarskis, S. Lewis, J.C. Matese, J.E. Richardson, M. Ringwald, G.M. Rubin, G. Sherlock, Gene ontology: Tool for the unification of biology, Nat. Genet. (2000). https://doi.org/10.1038/75556. 9. J. Chen, G. Liu, V. Pantalone, Q. Zhong, Physicochemical properties of proteins extracted from four new Tennessee soybean lines, J. Agric. Food Res. 2 (2020) 100022. https://doi.org/10.1016/j.jafr.2020.100022. 10. M.N. Faraggi, A. Arnau, V.M. Silkin, Role of band structure and local-field effects in the low-energy collective electronic excitation spectra of 2H-NbSe 2, Phys. Rev. B - Condens. Matter Mater. Phys. 86 (2012) 035115. https://doi. org/10.1103/PhysRevB.86.035115. 11. C. Mirabello, G. Pollastri, Porter, PaleAle 4.0: High-accuracy prediction of protein secondary structure and relative solvent accessibility, Bioinformatics. 29 (2013) 2056–2058. https://doi.org/10.1093/bioinformatics/btt344.

Encoder-Decoder for Protein Secondary Structure Prediction  99 12. A. Yaseen, Y. Li, Context-based features enhance protein secondary structure prediction accuracy, J. Chem. Inf. Model. 54 (2014) 992–1002. https://doi. org/10.1021/ci400647u. 13. R. Heffernan, K. Paliwal, J. Lyons, A. Dehzangi, A. Sharma, J. Wang, A. Sattar, Y. Yang, Y. Zhou, Improving prediction of secondary structure, local backbone angles, and solvent accessible surface area of proteins by iterative deep learning, Sci. Rep. 5 (2015) 1–11. https://doi.org/10.1038/srep11476. 14. S. Wang, J. Peng, J. Ma, J. Xu, Protein Secondary Structure Prediction Using Deep Convolutional Neural Fields, Sci. Rep. 6 (2016). https://doi.org/10.1038/ srep18962. 15. L.J. McGuffin, K. Bryson, D.T. Jones, The PSIPRED protein structure prediction server, Bioinformatics. 16 (2000) 404–405. https://doi.org/10.1093/ bioinformatics/16.4.404. 16. J.A. Cuff, M.E. Clamp, A.S. Siddiqui, M. Finlay, G.J. Barton, JPred: A consensus secondary structure prediction server, Bioinformatics. 14 (1998) 892– 893. https://doi.org/10.1093/bioinformatics/14.10.892. 17. R.D. Finn, J. Clements, S.R. Eddy, HMMER web server: Interactive sequence similarity searching, Nucleic Acids Res. 39 (2011) W29. https://doi. org/10.1093/nar/gkr367. 18. C.N. Magnan, P. Baldi, SSpro/ACCpro 5: Almost perfect prediction of protein secondary structure and relative solvent accessibility using profiles, machine learning and structural similarity, Bioinformatics. 30 (2014) 2592– 2597. https://doi.org/10.1093/bioinformatics/btu352. 19. S. Wang, J. Peng, J. Ma, J. Xu, Protein Secondary Structure Prediction Using Deep Convolutional Neural Fields, Sci. Rep. 6 (2016). https://doi.org/10.1038/ srep18962. 20. Y. Ma, Y. Liu, J. Cheng, Protein secondary structure prediction based on data partition and semi-random subspace method, Sci. Rep. 8 (2018). https://doi. org/10.1038/s41598-018-28084-8. 21. Z. Aydin, Y. Altunbasak, M. Borodovsky, Protein secondary structure prediction for a single-sequence using hidden semi-Markov models, BMC Bioinformatics. 7 (2006). https://doi.org/10.1186/1471-2105-7-178. 22. R. Heffernan, K. Paliwal, J. Lyons, J. Singh, Y. Yang, Y. Zhou, Single-sequencebased prediction of protein secondary structures and solvent accessibility by deep whole-sequence learning, J. Comput. Chem. 39 (2018) 2210–2216. https://doi.org/10.1002/jcc.25534. 23. A.K. Sharma, R. Srivastava, Protein Secondary Structure Prediction using Character bi-gram Embedding and Bi-LSTM, (2019) 1–7. 24. A.K. Sharma, R. Srivastava, Variable Length Character N-Gram Embedding of Protein Sequences for Secondary Structure Prediction, Protein Pept. Lett. 28 (2020) 501–507. https://doi.org/10.2174/0929866527666201103145635. 25. G. Wang, R.L. Dunbrack, PISCES: a protein sequence culling server, Bioinforma. Appl. NOTE. 19 (2003) 1589–1591. https://doi.org/10.1093/ bioinformatics/btg224.

100  Mathematics and Computer Science Volume 2 26. D. Wei, Q. Jiang, Y. Wei, S. Wang, A novel hierarchical clustering algorithm for gene sequences, BMC Bioinformatics. 13 (2012) 174. https://doi. org/10.1186/1471-2105-13-174. 27. R. Heffernan, Y. Yang, K. Paliwal, Y. Zhou, Capturing non-local interactions by long short-term memory bidirectional recurrent neural networks for improving prediction of protein secondary structure, backbone angles, contact numbers and solvent accessibility, Bioinformatics. 33 (2017) 2842–2849. https://doi.org/10.1093/bioinformatics/btx218. 28. S. Hochreiter, J. Schmidhuber, Long Short-Term Memory, Neural Comput. (1997). https://doi.org/10.1162/neco.1997.9.8.1735. 29. A. Drozdetskiy, C. Cole, J. Procter, G.J. Barton, JPred4: A protein secondary structure prediction server, Nucleic Acids Res. (2015). https://doi. org/10.1093/nar/gkv332. 30. S. Wang, W. Li, S. Liu, J. Xu, RaptorX-Property: a web server for protein structure property prediction, Nucleic Acids Res. 44 (2016) W430–W435. https://doi.org/10.1093/nar/gkw306.

8 Hesitancy, Awareness, and Vaccination: A Computational Analysis on Complex Networks Dibyajyoti Mallick, Aniruddha Ray, Ankita Das and Sayantari Ghosh* Department of Physics, National Institute of Technology Durgapur, Durgapur, West Bengal, India

Abstract

The vaccines for preventing COVID-19 are being considered as the most effective way to reduce the high pandemic burden on the global health infrastructure. However, public hesitancy towards vaccination is a crucial and pressing problem. Our study has been designed to determine the parameters affecting the vaccination decisions of common individuals. Using the platforms of the compartmental model and network simulation, we categorize people and observe their motivation towards vaccinations in a mathematical social contagion process. We consider peer influence as an important factor in this dynamic and study how individuals influence their peers regarding vaccination decisions. The efficiency of the vaccination process is estimated by the period of time required to vaccinate a substantial fraction of the total population. We discovered the major barriers and drivers of this dynamics and concluded that it is required to formulate specific strategies targeting specifically the undecided and vaccine hesitant people. Keywords:  Complex network, mathematical model, social epidemic, vaccination strategies

8.1 Introduction The novel coronavirus disease 2019 (COVID-19), caused by severe acute respiratory syndrome Coronavirus 2 (SARS-CoV-2), was first reported in *Corresponding author: [email protected] Sharmistha Ghosh, M. Niranjanamurthy, Krishanu Deyasi, Biswadip Basu Mallik, and Santanu Das (eds.) Mathematics and Computer Science Volume 2, (101–114) © 2023 Scrivener Publishing LLC

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102  Mathematics and Computer Science Volume 2 Wuhan, Hubei Province of China. This disease is now a global pandemic which is spreading rapidly from person to person, causing major public health concerns and economic crisis [1, 14]. A variety of active intervention policies have been introduced to suppress the spreading of this disease such as hand sanitizing, social distancing, travel restrictions, partial or complete lockdown, wearing masks, quarantining, etc. After the declaration of the pandemic by WHO (World Health Organization) in March 2020, pharmaceutical companies and scientists have encountered a race against time to develop vaccines [6]. The recent availability of multiple vaccines against coronavirus has brought hope for prevention of the spreading and a rapid recovery of our badly affected economy with a promise of a sooner resumption of normal life. However, the widespread hesitancy about vaccines is becoming a major obstacle for global health. Many people have strong hesitation towards vaccination, which is defined as the confusion about safety and effectiveness of the vaccine. The origin of this usually lies in some rumors regarding vaccines concerning side effects. This has become a huge challenge for governments and public health authorities for reaching the expected and required vaccination coverage [3, 9, 19]. The key priority now is ensuring vaccine acceptance because the lag in vaccination may provide a window for spreading the new variants and can also be a major obstacle in developing society-wide herd immunity. Until now, many researchers and scientists have tried to conduct different surveys to understand the behavior towards vaccine acceptance and hesitancy [2, 8, 10]. Some studies [7] have applied different theoretical models to explain vaccine acceptance, hesitancy, and willingness of individuals towards vaccines as well as refusal to vaccinate, which may vary depending upon personal decisions and epidemiological conditions [18]. Many advanced countries are trying to conduct country specific surveys [11], while several reports are being prepared by different countries such as the United Kingdom (UK), United States of America (USA), China, India, and Saudi Arabia to understand this vaccine acceptance behaviour [5, 12, 16]. However, most of these studies are based on heuristic arguments rather than mathematical analysis. A computational framework having a mathematical foundation helps to draw quantitative conclusions and is much more effective for predictive modeling purposes. This paper represents a new mathematical model regarding the vaccination process which incorporates behavioral changes of every individual in a society, driven by global and local factors. The main assumption here is that the vaccination dynamics can be considered as a social contagion process. This study aimed to identify the acceptability of the Covid-19 vaccine and information in support or opposing vaccination that is flowing in a society like a viral infection. We take into account refusal and hesitancy factors as

Hesitancy, Awareness, and Vaccination  103 well as the positive attitude of people towards vaccination [17]. In our study, we focused on identifying the effects of the factors that could increase the vaccination coverage through numerical simulations on an artificial society. Moreover, we could justify several results found by survey based studies where this kind of question has already been explored using survey results [4, 15]. Our results address the vaccination acceptance problem using a comprehensive mathematical and computational framework that would help us figure out correct strategies to encourage community for vaccine uptake and to stop further spreading of this pandemic.

8.2 Model Formulation To understand this kind of problem, we have considered a set of differential equations to depict the possible transitions. To describe the vaccine dynamics, we have a compartmental model, as shown in Figure 8.1. At any time t, the total population N(t) is subdivided into four states: Ignorant (I), Hesitant (H), Unwilling (U), and Vaccinated (V). • Ignorant (I) − As the name suggests, in this group people are ignorant about the vaccine. They have no idea about vaccination and have no clear opinion. In general, these people will not involve themselves in the vaccination process. But, they may be influenced by hesitant and vaccinated groups as well. Each person in this group is represented by I(t). • Hesitant (H) − In this group people know about vaccines, nevertheless they feel hesitation while making decisions -μU

ΥUV µ

α(H+V)I

βH

I

- µl

U

H

-μH

λH V -μV

Figure 8.1  Compartmental diagram of proposed model (I-H-U-V). All the transitions considered in the proposed model are shown through blue arrows. Subpopulations are denoted by yellow boxes.

104  Mathematics and Computer Science Volume 2 regarding vaccination, being affected by the rumors. This group may be influenced by the global advertising campaigns in favor of vaccination and take vaccines eventually. On the other hand, they could also be influenced further by rumors and transit to an unwilling state. Each person in this group is represented by H(t). • Unwilling (U) − In this group people believe in negative rumors regarding vaccines and they push themselves from vaccination. Again, some Unwilling people may return to the Hesitant class if they cannot decide whether to get vaccine or not, by influence through the vaccinated sub-population. Each person in this group is represented by U(t). • Vaccinated (V) – In this group, people are completely vaccinated and they are trying to influence other groups to join them. Each person in this group is represented by V(t). In this model, the key variable we have is the vaccinated population who are influencing other populations to get vaccinated. This is also the target population which has to be maximized over a certain period of time. Here, this group of people is playing an effective role to control the epidemic and spread the infodemic. Considering the total population as 1 (normalized form),



I+H+U+V=1

Now, we will discuss the following possibilities of transition of people from one sub-population to another over a given time period ‘t’. Let us discuss all the possible transitions considered in the model categorically. • Possible transitions related to I sub-population: In this population, µ is the rate at which people are entering into the ignorant population. This is added to consider the demographic variations. Now, the members of I may be influenced by the H and V population. H and V people might spread words on their inclination and decision and encourage others for vaccination. Let us consider α to be the effective contact rate. If each member of I group is going to be influenced by a hesitant and vaccinated group with that rate,

Hesitancy, Awareness, and Vaccination  105 then α(H + V)I amount will be subtracted from the I group and must be added to the hesitant group. In the normal course, there must be some people who can die from that population, hence the term µ should also be deducted from group I in the rate equation. • Possible transitions related to H sub-population: As mentioned in the previous case, due to the effect of influence from group H and V, the term α(H + V)I should be added in this rate equation. As usual, people from Group 3 can also die, hence the term µH can also be deducted from H group in its rate equation. Due to some reasons (influence from other groups or due to development of self-awareness among people), they are going for vaccination at a rate λ and some are entering into the Unwilling group at a rate β. Hence, the term βH and λH should be deducted from the rate equation. Again, from the Unwilling group some people are coming back into the Hesitant group by believing in some rumors, hence the term γUV should be added to the rate equation of H. • Possible transitions related to U sub-population: From the previous case discussed above it should be noted that the terms γUV and µU should be deducted from the rate equation due to their hesitant behavior and natural death from this group, respectively. • Possible transitions related to V sub-population: From the discussion of Case 2, it is evident that the term λH must be added to its rate equation due to direct wishing for vaccination from group H and µV must be deducted due to natural death of people from this group. Hence, all the rate equations of this model are compiled as follows:



dI = µ − α ( H + V )I − µ I dt

(8.1)



dH = α ( H + V )I − β H − λ H + γ UV − µ H dt

(8.2)

106  Mathematics and Computer Science Volume 2



dU = β H − γ UV − µU dt

(8.3)



dV = λ H − µV dt

(8.4)

8.3 Model Analysis on Complex Network In ODE-based models, one of the major issues is homogeneous mixing, which indicates every individual in a population has the same probability of having contact with each other [20]. Our society is highly heterogeneous and to accommodate that fact into our findings, we study the model on the heterogeneous setting of a complex network. The simulations are performed on a random network having 10000 nodes with an average degree of 5. Here, we choose the EoN module in Networkx from python [13] and run these following simulations in Google Colaboratory. Varying the transition rates of each parameter we see the effects of that particular parameter on the time evolution of the dynamics through our model. The general dynamics have been depicted in Figure 8.2. It shows that eventually most of the people in the population get vaccinated, however the time of coverage might be different depending on the parameter values. Thus, to quantify this growth curve, we define vaccination coverage time as ζV:

10000

VACCINATED HESITANT UNWILLING

8000 6000 4000 2000 0 0.0

2.5

5.0

7.5

10.0 12.5 15.0 17.5 20.0

Figure 8.2  Population of different classes vs. time. Eventually most of the population gets vaccinated, but time of coverage depends on parameters.

Hesitancy, Awareness, and Vaccination  107

ζV = {t|V (t) = 0.9 Vmax} This means that we consider a timescale, ζV, within which 90% of maximum vaccination, Vmax has been achieved. This will give us an estimate of the vaccination coverage speed.

8.3.1 Effect of Vaccination Rate Vaccination rate is a major parameter of this dynamic. If you carefully observe different sub-populations, the effect could be prominently observed, as reported in Figure 8.3.

8000 6000 4000 2000 0

4000 3500 3000 2500 2000 1500 1000 500 0

λ=0.2 λ=0.5 λ=0.8

0

5 10 15 20 25 30 35 40 time (a) λ=0.2 λ=0.5 λ=0.8

0

5

10 15 20 25 30 35 40 time (c)

Total unwilling population

10000

Total hesistant population

Total vaccinated population

• Effect on vaccinated people: As we can see from the results in Figure 8.3(a), increasing the rate does not change the maximum number of vaccinated people, but it shifts the saturation point leftwards which means the system is reaching the saturation point faster. • Effect on hesitant people: We can see from the plots shown in Figure 8.3(b) that increasing the rate shifts the peak point λ=0.2 λ=0.5 λ=0.8

3000 2500 2000 1500 1000 500 0 0

5

10

24 22 20 18 16 14 12 10 8 0.2

0.3

0.4

15 20 time (b)

0.5 (d)

25

0.6

30

0.7

35

40

0.8

Figure 8.3  (a) Vaccination coverage with time for different values of vaccination rate λ. (b) Time evolution of unwilling people in population with different values of vaccination rate parameter. (c) Time evolution of hesitant people in population with different values of vaccination rate parameter. (d) Vaccination coverage time (ζV) for different values of vaccination rate.

108  Mathematics and Computer Science Volume 2 slightly leftwards, which means the system is reaching the peak point a bit faster and we also can see that the maximum number of people reaching in a hesitant class decreases a little bit. • Effect on unwilling people: As we can see from the plots shown in Figure 8.3(c), increasing the parameter flattens the curve which means that the maximum number of people going to the unwilling class decreases, which can be explained by the success rate of the vaccination process. As shown in Figure 8.3(d), we can see that the saturation time decreases with increasing vaccination rate, that is λ, which states that this particular parameter will make the system reach the saturation point earlier.

8.3.2 Effect of Negative Rumors Negative rumors often sway people towards some risky and dangerous activity. In this vaccination process, some negative rumors about the vaccine exist that concern matters like several side effects, cost effectiveness, fear, and misinformation regarding the vaccine. Hesitant people are often influenced by them. The parameter that takes into account this phenomena is β. The results related to this are shown in Figure 8.4. • Effect on vaccinated class: As we can see in Figure 8.4(a), increasing the parameter, β, shifts the saturation point of vaccinated people rightwards. This means as more people feel negative about vaccination, more time will be taken by the system to reach the saturation point. • Effect on unwilling class: In Figure 8.4(b), we can see that many people are motivated by their neighbors, friends, or relatives to believe some negative rumors about vaccination and refuse to get vaccines and along the way the maximum number of unwilling people increases.

8.3.3 Effect of Positive Peer Influence Peer influence occurs when people are motivated or influenced towards something by seeing their neighbor or friends. When someone’s peers influence them to do something positive, it is considered as positive peer influence. In the vaccination context, positive peer influence occurs when

Total vaccinated population

Hesitancy, Awareness, and Vaccination  109 Total unwilling population

10000 8000 6000 4000

Total hesitant population

2000

β = 0.2 β = 0.5 β = 0.8

0 0

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30

35

40

β = 0.2 β = 0.5 β = 0.8

2000 1500 1000 500 0 0

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β = 0.2 β = 0.5 β = 0.8

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2500

20

time (b)

25

30

35

40

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3000

12.0 11.5

2000

11.0 10.5 10.0

1000

9.5

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time (c)

25

30

35

40

9.0

0.2

0.3

0.4

0.5

0.6

0.7

0.8

(d)

Figure 8.4  (a) Vaccination coverage with time for different values of negative rumor influence rate β. (b) Time evolution of unwilling people in population with different values of negative rumor influence rate parameter. (c) Time evolution of hesitant people in population with different values of negative rumor influence rate parameter. (d) Vaccination coverage time (ζV) for different values of negative rumor influence rate.

someone who is vaccinated is motivating others who are not sure about vaccination. Here, the vaccinated people influence the hesitant people to get vaccinated. Here we are considering two parameters α and γ that take into account this phenomenon. The results related to this are shown in Figure 8.5 and Figure 8.6. • Effect on Vaccinated people: As we can see in Figure 8.5(a) and 8.6(a), increasing the parameter shifts the saturation point of vaccinated people leftwards. This means the more people that feel positive about vaccination, then the sooner the system will reach the saturation point. • Effect on Unwilling people: We can also see from Figures 8.5(b) and 8.6(b) that along the way, the maximum number of unwilling people decreases as they are getting motivated by some positive rumors towards vaccination. So, we can draw the conclusion that if more people are well informed about the success rate of vaccination, then that would speed up the

10000 8000 6000 4000 2000

Y=0.2 Y=0.5 Y=0.8

0 0

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15

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Total unwilling population

Total vaccination population

110  Mathematics and Computer Science Volume 2 Y=0.2 Y=0.5 Y=0.8

2500 2000 1500 1000 500 0 0

5

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15

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(a)

25

30

35

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(b)

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Total hesitant population

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time Y=0.2 Y=0.5 Y=0.8

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20

25

30

35

0.2

40

0.3

0.4

0.5

(d)

time

(c)

0.6

0.7

0.8

Figure 8.5  (a) Vaccination coverage with time for different values of positive peer influence rate γ. (b) Time evolution of unwilling people in population with different values of positive peer influence rate parameter. (c) Time evolution of hesitant people in population with different values of positive peer influence rate parameter. (d) Vaccination coverage time (ζV) for different values of positive peer influence rate. Total unwilling population

Total vaccination population

10000 8000 6000 4000 2000

α= 0.2 α= 0.5 α= 0.8

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5

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α= 0.2 α= 0.5 α= 0.8

2000 1500 1000 500 0

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Total hesitant population

(a)

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time

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(b)

16

α= 0.2 α= 0.5 α= 0.8

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15

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14 13

2000

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11 10

0 0

5

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20

time

(c)

25

30

35

40

0.2

0.3

0.4

0.5

(d)

0.6

0.7

0.8

Figure 8.6  (a) Vaccination coverage with time for different values of positive peer influence rate α. (b) Time evolution of unwilling people in population with different values of positive peer influence rate α. (c) Time evolution of hesitant people in population with different values of positive peer influence rate α. (d) Vaccination coverage time (ζV) for different values of positive peer influence rate α.

Hesitancy, Awareness, and Vaccination  111 vaccination process and also decrease the amount of unwilling people. As shown in Figures 8.5(d) and 8.6(d), for both parameters of positive peer influence, that is, α and γ, the saturation time decreases with increasing parameters, which means positive peer influence will make the system reach the saturation point earlier.

8.4 Conclusions and Perspectives Along with the infectious spread of SARS-COV2, the information and misinformation regarding available vaccines is also spreading person-to-person, causing a large-scale effect on vaccination coverage. Through computational analysis and network simulations, we have explored the contagion dynamics and proposed a framework to analyze this decision process. From the model it has been observed that the presence of the efficient vaccination systems can run the entire dynamic with positive feedback. This is a significant result, as we can see a lot of people not only in India but around the world are hesitant about getting vaccinated. In our model, we have shown that if the peer influence is overall positive, then the saturation point shifts leftwards, so we can conclude that the spreading of awareness about the positive effects of vaccination should be carried out on the ground level. We have also shown in our model that if vaccination rate increases, then the saturation point of V class shift leftwards. In real life we can explain it like this: if the vaccination rate in a particular area can be increased by awareness and smooth execution of the vaccination process by a competent authority, then the system would reach its saturation point sooner. A strong peer influence also might signify a strongly connected society. The shift of the saturation point leftwards for high positive peer influence means that physically and virtually everyone is well connected. This may also indicate high vaccination rates in urban areas. In the future we will focus on implementing further realistic terms in our model. For example, if someone from a hesitant class goes to the unwilling class, it can be considered that the person was under the influence of someone who has been vaccinated, thus we can bring the non-linearity. Mathematical explorations are used to find out the fixed points and bifurcations if there are any. Considering coupled dynamics or delayed dynamics of disease along with the infodemic might be another interesting future study.

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References 1. W. Bank. The economy in the time of covid-19, 2020. 2. S. Bhattacharyya, A. Vutha, and C. T. Bauch. The impact of rare but severe vaccine adverse events on behaviour-disease dynamics: a network model. Scientific reports, 9(1):1–13, 2019. 3. A. A. Dror, N. Eisenbach, S. Taiber, N. G. Morozov, M. Mizrachi, A. Zigron, S. Srouji, and E. Sela. Vaccine hesitancy: the next challenge in the fight against covid-19. European journal of epidemiology, 35(8):775–779, 2020. 4. E. Dubé, M. Vivion, and N. E. MacDonald. Vaccine hesitancy, vaccine refusal and the anti-vaccine movement: influence, impact and implications. Expert review of vaccines, 14(1):99–117, 2015. 5. J. Eskola, P. Duclos, M. Schuster, N. E. MacDonald, et al. How to deal with vaccine hesitancy? Vaccine, 33(34):4215–4217, 2015. 6. K. B. Habersaat and C. Jackson. Understanding vaccine acceptance and demand—and ways to increase them. BundesgesundheitsblattGesundheitsforschung-Gesundheitsschutz, 63(1):32–39, 2020. 7. C. Jarrett, R. Wilson, M. O’Leary, E. Eckersberger, H. J. Larson, et al. Strategies for addressing vaccine hesitancy–a systematic review. Vaccine, 33(34):4180– 4190, 2015. 8. K. Kalimeri, M. G. Beiró, A. Urbinati, A. Bonanomi, A. Rosina, and C. Cattuto. Human values and attitudes towards vaccination in social media. In Companion Proceedings of The 2019 World Wide Web Conference, pages 248–254, 2019. 9. D. Kumar, R. Chandra, M. Mathur, S. Samdariya, and N. Kapoor. Vaccine hesitancy: understanding better to address better. Israel journal of health policy research, 5(1):1–8, 2016. 10. H. J. Larson, C. Jarrett, W. S. Schulz, M. Chaudhuri, Y. Zhou, E. Dube, M. Schuster, N. E. MacDonald, R. Wilson, et al. Measuring vaccine hesitancy: the development of a survey tool. Vaccine, 33(34):4165–4175, 2015. 11. J. V. Lazarus, S. C. Ratzan, A. Palayew, L. O. Gostin, H. J. Larson, K. Rabin, S. Kimball, and A. El-Mohandes. A global survey of potential acceptance of a covid-19 vaccine. Nature medicine, 27(2):225–228, 2021. 12. J. Luyten, L. Bruyneel, and A. J. van Hoek. Assessing vaccine hesitancy in the UK population using a generalized vaccine hesitancy survey instrument. Vaccine, 37(18):2494–2501, 2019. 13. J. C. Miller and T. Ting. Eon (epidemics on networks): a fast, flexible python package for simulation, analytic approximation, and analysis of epidemics on networks. arXiv preprint arXiv:2001.02436, 2020. 14. A. Nishi, G. Dewey, A. Endo, S. Neman, S. K. Iwamoto, M. Y. Ni, Y. Tsugawa, G. Iosifidis, J. D. Smith, and S. D. Young. Network interventions for managing the covid-19 pandemic and sustaining economy. Proceedings of the National Academy of Sciences, 117(48):30285–30294, 2020.

Hesitancy, Awareness, and Vaccination  113 15. P. Peretti-Watel, H. J. Larson, J. K. Ward, W. S. Schulz, and P. Verger. Vaccine hesitancy: clarifying a theoretical framework for an ambiguous notion. PLoS currents, 7, 2015. 16. P. Pronyk, A. Sugihantono, V. Sitohang, T. Moran, S. Kadandale, S. Muller, C. Whetham, and R. Kezaala. Vaccine hesitancy in indonesia. The Lancet Planetary Health, 3(3):e114–e115, 2019. 17. M. Sallam. Covid-19 vaccine hesitancy worldwide: a concise systematic review of vaccine acceptance rates. Vaccines, 9(2):160, 2021. 18. M. Siddiqui, D. A. Salmon, and S. B. Omer. Epidemiology of vaccine hesitancy in the United states. Human vaccines & immunotherapeutics, 9(12):2643–2648, 2013. 19. G. Troiano and A. Nardi. Vaccine hesitancy in the era of covid-19. Public Health, 2021. 20. N. Wang, Y. Fu, H. Zhang, and H. Shi. An evaluation of mathematical models for the outbreak of covid-19. Precision Clinical Medicine, 3(2):85–93, 2020.

9 Propagation of Seismic Waves in Porous Thermoelastic Semi-Infinite Space with Impedance Boundary Conditions Annu Rani and Dinesh Kumar Madan* Department of Mathematics, Chaudhary Bansi Lal University, Bhiwani, Haryana, India

Abstract

This work investigates the waves propagation in an isotropic saturated porousthermoelastic medium. The incidence of a seismic wave is considered at the free surface of a thermoelastic semi-infinite space. The surface of semi-infinite space is taken as isothermal. Reflection ratios of waves for the incidence of longitudinal waves are derived for impedance boundary conditions. The graph depicts the effect of porosity on reflection ratios versus incident angle and impedance parameter. Graph plotting is done with the help of the MATLAB programming language. Results are presented for a particular model in this paper. Keywords:  Porous, thermoelastic, reflection, impedance, isothermal

9.1 Introduction Thermoelasticity is the study of the change in shape and dimension of a solid object when the temperature of the object varies. With the theory of elasticity, Madan et al. [1] and Kumari and Madan [2] explore the propagation of waves. By extending the classical coupled theory of thermoelasticity by introducing relaxation time, Lord and Shulman [3] and Green and Lindsay [4] presented the generalized theory of thermoelasticity. A number of researchers [5–11] have discussed the phenomena of waves *Corresponding author: [email protected] Sharmistha Ghosh, M. Niranjanamurthy, Krishanu Deyasi, Biswadip Basu Mallik, and Santanu Das (eds.) Mathematics and Computer Science Volume 2, (115–136) © 2023 Scrivener Publishing LLC

115

116  Mathematics and Computer Science Volume 2 reflecting on a free surface. Rani and Madan [12] investigated the effect of initial stress and imperfect interface on waves. In porous media, the thermo-poroelasticity theory describes the relationship between temperature and stress components. A thermoelastic saturated porous material with solid-fluid pores is a thermally conducting porous solid and these materials are generally found in reservoir rocks of the earth’s crust. The phenomenon of co-existence of thermoelasticity and porosity plays a crucial role in the non-destructive evaluation of composite materials and structures. The propagation of thermoelastic waves in fluidsaturated soil was studied by Haibing [13]. Kumar et al. [14] investigated the Rayleigh waves propagation in thermoelastic semi-infinite space with double porosity under isothermal and insulated boundary conditions in thermoelastic semi-infinite space with double porosity. Wilson et al. [15] discuss the propagation of acoustic waves. In a thermo-poroelastic medium, Wang et al. [16] explored the reflection phenomenon of inhomogeneous elastic waves. The reflection phenomenon of seismic waves at insulated and free surfaces of thermo-poroelastic material was explored by Zorammuana and Singh [17]. Impedance boundary-conditions are defined as a linear combination of unknown parameters and corresponding derivatives that are predictable on the surface. Electromagnetism, acoustics, seismology, science, and technology all benefit from these conditions. The propagation of waves for impedance boundary-conditions was studied by Singh [18], Vinh and Hue [19], Vinh and Xuan [20], and others. To conclude the wave propagation effects, a quantitative analysis of reflection at the surface of a thermo-poroelastic medium is required. This work examines the waves propagation at the surface of a saturated porous-thermoelastic isothermal medium with an impedance boundary condition. The influence of solid porosity and the Biot parameter for the coupling of fluid and solid phases on the reflection ratios versus impedance parameter and incident angle of a longitudinal wave is studied.

9.2 Basic Equations The stress-tensors in a non-viscous thermal-porous saturated solid are given by [21]:



σij = τij + β (−Pf) δij,

where τij: components of stress, Pf: pressure of fluid,

(9.1)

Propagation of Seismic Waves  117 β: Biot parameter, δij: Kronecker delta The stress-strain relation is given by [22, 23]



τij = λekk δij + 2μeij − γs (ϴ − ϴ0) δij,

Pf = −βNekk − Nvk, k + γs (ϴ − ϴ0) δij, (i, j, k = 1, 2, 3)

(9.2) (9.3)

where λ, μ: lame’s constants, eij: components of strain, ϴ: temperature, γs: coefficient of thermal stress corresponding to solid, γf: coefficient of thermal stress corresponding to fluid, N: elastic parameter for bulk-coupling of fluid, vi = f (Ui − ui), f: solid porosity, Ui: component of displacement corresponding to fluid, ui: component of displacement corresponding to solid, 1  ∂u ∂u  Also, the strain components are emn =  m + n  , m, n = 1, 2, 3. 2  ∂xn ∂xm  The equation of motion in an isotropic thermal-porous solid saturated with a non-viscous fluid are given by [24]:



σ= ρ ui + ρ f vi , ij , j

(9.4)



(P= ρ f ui + qvi , f ), i

(9.5)



∂  Kθ,ii=  1 + t 0  ( ρ ceθ + γθ0 (u k , k + v k , k ) ) , ∂t  

(9.6)

where ρ: porous-aggregate density, ρf: pore-fluid density, q: coupling parameter between porous-aggregate solid and pore-fluid, γ = γs + βγf

118  Mathematics and Computer Science Volume 2

9.3 Problem Formulation Using Equations (9.1)-(9.3) in Equation (9.4), we have:

σ ij = λ ekkδ ij + 2 µeij − γ s (θ − θ 0 )δ ij + β ( β Nekk + Nv k ,k

− γ f (θ − θ 0 )δ ij ,(λ + µ + β 2 N )uk ,ki + µuk ,ii − γθ ,i − ρvk + β Nv k ,ki − ρ f vk = 0

(9.7)

Using Equations (9.1)-(9.3) in Equation (9.5), we have:



β Nuk , ki + Nvk , ki − γ f θ ,i −qvk − ρ f uk = 0

(9.8)

Using Equations (9.1)-(9.3) in Equation (9.6), we have:



∂  Kθ,ii=  1 + t 0  ( ρ ceθ + γθ0 (u k , k + v k , k ) ) ∂t  

(9.9)

Displacement components in terms of potentials are



u1 =

∂p f ∂q f ∂ps ∂qs − , u3 = + ∂x ∂z ∂z ∂x

(9.10)



v1 =

∂p f ∂q f ∂p f ∂q f − , v3 = + ∂x ∂z ∂z ∂x

(9.11)

Using Equations (9.10) and (9.11) in Equations (9.7)-(9.9), we have:

(λ + µ + β 2 N )∇ 2 ps + µ∇ 2 ps − γθ − ρ ps + β N ∇ 2 p f − ρ f p f = 0 (9.12)

µ∇ 2qs − ρqs − ρ f q f = 0

(9.13)



β N ∇2 ps − γ f θ − qp f + N ∇2 p f − ρ f ps = 0

(9.14)

Propagation of Seismic Waves  119



qq f + ρ f qs = 0

(9.15)

ρ ce (θ + t 0θ) + γθ0 (∇2 p s + t 0∇2 ps + ∇2 p f + t 0∇2 p f ) − K ∇2θ = 0 (9.16) ∂2 ∂2 where ∇2 = 2 + 2 is the Laplace operator. ∂x ∂z The coupling structure of potentials pf , ps with thermal wave can be easily observed from Equations (9.12), (9.14), and (9.16), while the coupling structure of potentials qf and qs can be easily observed from Equations (9.13) and (9.15). The velocities of waves are [17] 1

3  H1 H 2 − H 3   6 2   1 3  H   2 2 H12  2  V1 =   H 2 − H1  + − −   9    3 9   H13   3 − +  3 2 27    H H H H   1 1 2 3   − +      27 6 2  1



3  H1 H 2 − H 3  6  2   1 3    H 2 2   H 2 − H1  +  − 1 , (9.17)  3 3   3 9   H1  − +  3  2 27   H 1 − H1 H 2 + H 3        27  6 2 

120  Mathematics and Computer Science Volume 2

( ) 2

H2

V2 =

3

  H H 2 6 1

 H H  6 1

−

2

−

    H H  6



9

( )   

H1

−

1

2

H3 2

1

+

H1

−

−

{( ) (

H3

H2

+

2

H1

−

3

+

−

H1 H2

−

27

H3

+

6

2

)}

1

3

2

2

3

H1

H1

9

27

3

H1

+

−

)} 2

3

9

{( ) ( H2

3

2

H1 H2

+

H3

6

2

2

1

H1

−

27

1

  −  27

3

3

3

2

 

3

3

H1

2

1

3

2

{ } H2

−

3

9

2

−

H3 2

{( ) ( H2

+

−

3

H1

+

9

)}

1

3

2

3

H1

−

H1 H2

27

 H H +  6 1

1

2

  2 H  −  27   i 1

2

H1

H3

+

6

2

2

  −  27

2

−

H3 2

{( ) ( H2

+

2

−

3

1

3

H1

+

9

3

H1

)} 2

−

H1 H2

27

+

6

H3 2

3

3

H1

3

3

1



2

(9.18)

H2 3

V3 =

  2  H H   6 1

 H H  6 − 1



2

−

H3 2

2

−

H3 2

 H  3

+ 

 H +   3

2

2

2

−

H1 9

   H    −    27  1

2

−

H1 9

  

3

H  27

+

3

1

−

H1H 2 6

+

H3 2

2

2

−

H1 9

  

H +  27

3

1

2

−

H1H 2 6

+

H3 2

2

H1 3

1

   H   −    27  2

3

−

3

2

1

3

1

1 3

3

1

−



Propagation of Seismic Waves  121

      H H   6     H H   6 1

1

1

32

2

2

−

−

H3 2 H3 2

 H +   3

2

 H +   3

2



2

−

H1 9

{   

H2

3

3

2

−

H1 9

H +  27

}

3

1

−

H1H 2 6

+

H3 2

   2

1

   27 

2

−

9

  

3

H +  27

3

1

−

H1H 2 6

+

H3

−

H1

   H   −    27 

2

2

2

3

3

2

1

H1

1

3

1

1 3

  +   i     

2



Vs =

= H1 where,

µ , ρ 2f ρ− q

(9.19)

(9.20)

η1 η2 η3 i = , H2 ,= H3 = , t t0 + , η0 ω η0 η0

η1 = ρCet (Dq + N ρ − 2 β N ρ f ) + K ( ρq − ρ 2f ) + θ 0tγγ f ( ρ − ρ f ) +γ 2θ 0t (q − ρ f ),

η2 = − K (Dq + ρ N − 2β N ρ f ) − ρtCe (DN − β 2 N 2 ) − θ 0tγγ f (D − β N ) +θ 0tγ 2 ( β N − N ),

2 η3 = (λ + 2 µ )KN ,η0 = ρCet ( ρ f − qρ )



(9.21)

9.4 Reflection at the Free Surface Consider an isotropic porous-thermoelastic semi-infinite space, z > 0 (Figure 9.1). If a longitudinal wave is incident with an angle e at the surface, then it will give four reflected waves. Let e1, e2, e3 be angles which reflect coupled longitudinal waves, making them normal, and es be the angle which reflected transverse waves makes normal in medium M.

122  Mathematics and Computer Science Volume 2 x axis Isotropic porous thermoelastic semiinfinite space (M)

e

e1 e2 e3

D1

es

C3 C2

C0

C1 z axis

Figure 9.1  Geometry of problem for reflection of waves when longitudinal wave is incident.

= ps C0e i (k0 ( xsin(e ) + zcos (e )) −ωt ) + C1e i (k1 ( xsin(e1 ) − zcos(e1 )) −ωt )

+C2e i (k2 ( xsin(e2 ) − zcos (e2 )) −ωt ) + C3e i (k3 ( xsin(e3 ) − zcos(e3 )) −ωt ) , (9.22)

= p f a0C0e i (k0 ( xsin(e ) + zcos (e )) −ωt ) + a1C1e i (k1 ( xsin(e1 ) − zcos(e1 )) −ωt ) i ( k xsin ( e2 ) − zcos ( e2 ) ) −ωt ) +a2C2e 2 ( + a3C3e i (k3 ( xsin(e3 ) − zcos (e3 )) −ωt ) (9.23)

qs = D1e i (ks ( xsin(es ) − zcos (es )) −ωt ) ,

(9.24)



q f = d1D1e i (ks ( xsin(es ) − zcos (es )) −ωt ) ,

(9.25)

θ − θ0 = b0C0e i (k0 ( xsin(e ) + zcos (e )) −ωt ) + b1C1e i (k1 ( xsin(e1 ) + zcos (e1 )) −ωt )

+b2C2e i (k2 ( xsin(e2 ) − zcos (e2 )) −ωt ) + b3C3e i (k3 )( xsin(e3 ) − zcos (e3 )) −ωt ) , (9.26)

Propagation of Seismic Waves  123 where

a0 =

( D − ρV02 ) γγ f − ( β N − ρ f V02 )

( N − qV02 ) − ( β N − ρ f V02 )

γf γ

,

(( N − qV )( D − ρV ) − ( β N − ρ V ) ) k b = 2 0

0



2 0

f

2 2 0

2 0

 2 γ f 2   ( β N − ρ f V0 ) γ − ( N − qV0 ) γ

ai =

( D − ρVi2 ) γγ f − ( β N − ρ f Vi2 )

( N − qVi ) − ( β N − ρ f Vi ) γγ f 2

2

(9.27)

,

(( N − qV )( D − ρV ) − ( β N − ρ V ) ) k b = i

i

2

i

2

f

i

2 2

 2 γ f β ρ − N V ( ) γ − ( N − qVi2 ) γ f i 

d1 =

,

2 i

, i = 1,2,3,

µ − ρVs2 2 2 , D = λ + 2µ + β N ρ f Vs

Using Snell’s Law, angles and corresponding wave vectors are associated by:



k1 sin (e) = k1 sin (e1) = k2 sin (e2) = k3 sin (e3) = ks sin (es) (9.28)

and in terms of velocities, are given by:



sin(e) sin(e1 ) sin(e2 ) sin(e3 ) sin(es ) = = = = V0 V1 V2 V3 Vs

(9.29)

124  Mathematics and Computer Science Volume 2

9.4.1 Boundary Conditions At surface z = 0, impedance boundary conditions are given by [18]: 1. σ33 = 0, which gives:

γ b1  k12  2 2 ( ) 2 cos N e a N λ + β + µ + β + 1 1  k12  k02 γ b  k2  X1 +  (λ + β 2 N ) + 2 µ cos 2 e2 + a2 β N + 22  22  k2  k0 k2 γ b  k2  X 2 +  (λ + β 2 N ) + 2 µ cos 2 e3 + a3 β N + 23  32 X 3 − 2 µ sin es cos es s2  k3  k0 k0

γb   X 4 =  (λ + β 2 N ) + 2 µ cos 2 e + a0 β N + 20   k0 



(9.30) 2. σ31 + ωZu1 = 0 gives:

 2k12   2k22  k1 k sin cos − sin sin e2 cos e2 − iω Z 22 sin e2  e X e e i Z ω + 1 1  1 1  2 2 2 k0 k0  k0   k0   2k 2  k X2 +  23 sin e3 cos e3 − iω Z 32 sin e3  k0   k0 2   k k X3 +  (cos2 es − sin2 es ) s2 − iω Z 2s cos es  k0 k0   iω Z X 4 = 2 sin e0 cos e0 + sin e0 k0 (9.31) 3. v 3 = 0 gives:



k k k k1 a1 cos e1 X1 + 2 a2 cos e2 X2 + 3 a3 cos e3 X3 − s d sin es k0 k0 k0 k0 (9.32) X 4 = a0 cos e,



Propagation of Seismic Waves  125 4.

∂θ 0 (h → 0 for thermally-insulated surface, h → ∞ + hθ = ∂y for isothermal surface) gives

−ik1 a1 X1 cos (e1) − ik2 a2 X2 cos (e2) − ik3 a3 X3 cos (e3) + h (b1 X1 + b2 X2 + b3 X3) = −b0h + ik0 cos e.

(9.33)

where

= X1



C1 C2 C3 D1 = , X2 = , X3 = , X4 , C0 C0 C0 C0

(9.34)

The matrix equation from these equations can be written as

BX = Y,



(9.35)

where

 z1    X1    X    , X =  2  ,Y =  z 2   z3   X3         z 4   X 4   γ b1  k12  2 2 b= ( λ + β N ) + 2 µ cos e + a β N + , 11 1 1  k12  k02  b11 b 21 B= b31  b41

b12 b22 b32 b42

b13 b23 b33 b43

b14 b24 b34 b44

γ b2  k22  2 2 b= + N + 2 ( ) µ cos β , e + a N + λ β 12 2 2  k22  k02  γ b3  k32  2 2 b= ( N ) 2 cos e a N + + + + , λ β µ β 13 3 3  k32  k02   2k 2  k2 k b14 = −2 µ sin es cos es s2 , b21 =  21 sin n e1 cos e1 − iω Z 12 sin e1  , k0 k0  k0   2k22  k = b22  2 sin e2 cos e2 − iω Z 22 sin e2  , k0  k0 



126  Mathematics and Computer Science Volume 2

 2k 2  k b23 =  23 sin e3 cos e3 − iω Z 32 sin e3  , k0  k0    k k2 b24 =  (cos 2 es − sin 2 es ) s2 − iω Z s2 cos es  ,   k0 k0 k k k1 a1 cos e1 , b32 = 2 a2 cos e2 , b33 = 3 a3 cos e3 , k0 k0 k0 k b34 = − s d sin es , b41 = ( −ik1a1 cos(e1 ) + hb1 ), k0 b42 = ( −ik2a2 cos(e2 ) + hb2 ), b43 = ( −ik3a3 cos(e3 ) + hb3 ), b31 =

γb b44 = 0, z1 =  (λ + β 2 N ) + 2µ cos 2 e + a0 β N + 20  ,  k0  iω Z sin e0 , z3 = a0 cos e , k0 z 4 = −b0h + ik0 cos e.

z 2 = 2sin e0 cos e0 +



(9.36) Here, Xi, i = 1, 2, 3, 4 represents reflection ratios in isotropic thermoelastic porous medium.

9.4.2 Energy Ratios The average energy transmission per unit surface area per unit time is given by [25]:

= P σ 33u 3 + σ 31u1 − Pf β v 3



(9.37)

Using Equations (9.22)-(9.28) and (9.37),



E1 =

PrefL1 R1 2 PrefL 2 R2 2 = X1 , E 2 = = X2 , Pinc L1 R0 Pinc L 2 R0

E3 =

PrefL 3 R3 2 PrefT R4 2 X4 , = X 3 , E4 = = Pinc L 3 R0 Pinc T R0

(9.38)

Propagation of Seismic Waves  127 where PrefL1, PrefL2, PrefL3, and PrefT are the average energy of reflected coupled longitudinal waves at e1, e2, e3, and transverse wave at e4, respectively, and

γ b γ f b0a0 β  3  k0 cos e , R0 = −  λ + 2 µ + β 2a0 N + a02 β N + s 2 0 +  k02  k0 γ b γ f b1a1 β  3  R1 =  λ + 2 µ + β 2a1 N + a12 β N + s 2 1 + k1 cos e1 ,  k1 k12 

γ b γ f b2a2 β  3  R2 =  λ + 2 µ + β 2a2 N + a22 β N + s 2 2 + k2 cos e2 ,  k2 k22  γ b γ f b3a3 β  3  R3 =  λ + 2 µ + β 2a3 N + a32 β N + s 2 3 + k3 cos e3 ,  k3 k32  R4 = µ ks3 cos es .

(9.39)

9.5 Numerical Results and Discussion For the numerical computation of amplitude ratios, we employed the following parameters [24]:

λ = 3.7 GPa, µ = 7.9 GPa, N = 6GPa, Ce = 1040

J W , K = 170 , kg K m

ρf kg kg , 3 , ρ f = 950 3 , q = 1.05 m m f GPa β f = 2.37 × 10 −3 , β s = 2 β f ,ω = 2s −1 , K ° −10 T0 = 300 K , t0 = 10 s ρ = 2216

The problem of isothermal surfaces has been studied here. To plot the graphs and execute numerical calculations, the MATLAB programming language is employed. Figure 9.2 explored the impact of porous-parameter f on the reflection ratio of a coupled longitudinal wave at an angle e1 to the incident angle e of the longitudinal wave. It is concluded that as the value of f increases, the reflection ratio of a longitudinal wave decreases for e 70°, attaining the same value for f = 0.03 (circle-line), f = 0.04 (star-line), and f = 0.05 (triangle-line). Variation of the reflection ratios of coupled longitudinal waves at an angle e2, e3 against the incident angle for β = 0.4, Z = 10, f = 0.03 (circle-line), f = 0.04 (star-line), f = 0.05 (triangle-line) is shown in Figures 9.3 and 9.4, respectively. It is concluded that the reflection ratio of a coupled-longitudinal wave against the incident angle attains minimum value at e = 71°, then e > 71° attains the same value for distinct values of f. Qualitatively, the

9

×10–8

8 7 6 X2

5 4 3 2 1 0

0

10

20

30

40

50 e (in degrees)

60

70

80

90

Figure 9.3  Reflection ratio X2 of longitudinal wave (at angle e2) against incident angle e for β = 0.04, Z = 10, f = 0.03 (circle-line), f = 0.04 (star-line), f = 0.05 (triangle-line).

Propagation of Seismic Waves  129 0.045 0.04 0.035 0.03

X3

0.025 0.02 0.015 0.01 0.005 0

0

10

20

30

40

50 e (in degrees)

60

70

80

90

Figure 9.4  Reflection ratio X3 of longitudinal wave (at angle e3) against incident angle e for β = 0.04, Z = 10, f = 0.03 (circle-line), f = 0.04 (star-line), f = 0.05 (triangle-line).

same behavior of the reflection ratios of a longitudinal wave at an angle e2, e3 against the incident angle is observed. Figure 9.5 explores the variation in amplitude ratio of transverse wave against incident angle e of the longitudinal wave. The reflection ratio of a coupled longitudinal wave attains a maximum value at e = 8° and then decreases gradually. The reflection ratio of a transverse wave behaves qualitatively the same as a coupled longitudinal wave at an angle e2, e3. Figures 9.6 and 9.7 exhibit variation in the amplitude ratio of a coupled longitudinal wave at angle e1, e2 against impedance parameter Z for f = 0.01, β = 0.1 (represented by circle-line), β = 0.2 (represented by triangle-line), and β = 0.3 (represented by star-line). The magnitude of 4 3.5 3 X4

2.5 2 1.5 1 0.5 0

0

10

20

30

40 50 e (in degrees)

60

70

80

90

Figure 9.5  Reflection ratio X4 of transverse wave against incident angle e (in degrees) for β = 0.04, Z = 10, f = 0.03 (circle-line), f = 0.04 (star-line), f = 0.05 (triangle-line).

130  Mathematics and Computer Science Volume 2 1.08 1.075 1.07 1.065 X1

1.06 1.055 1.05 1.045 1.04 0

5

10

15

20

25 Z

30

35

40

45

50

Figure 9.6  Reflection ratio X1 of longitudinal wave at e1 against Z (impedance parameter) for f = 0.01, e = 30°, β = 0.1 (circle-line), β = 0.2 (triangle-line), β = 0.3 (star-line).

6

×10-8

5 X2

4 3 2 1 0

0

5

10

15

20

Z

25

30

35

40

45

50

Figure 9.7  Reflection ratio X2 of longitudinal wave at e2 against Z (impedance parameter) for f = 0.01, e = 30°, β = 0.1 (circle-line), β = 0.2 (triangle-line), β = 0.3 (star-line).

amplitude ratios of coupled longitudinal wave at angle e1, e2 increases as the value of β increases and attains a maximum value at Z = 50. The amplitude ratios of a coupled longitudinal wave at an angle e3 and transverse wave increases against the impedance parameter Z and attains a maximum value at Z = 50 in Figures 9.8 and 9.9, respectively. It is noticed that the quantitative value of the reflection ratio decreases as the value of β increases.

Propagation of Seismic Waves  131 0.014 0.013 0.012 X3

0.011 0.01 0.009 0.008 0.007

0

5

10

15

20

25 Z

30

35

40

45

50

Figure 9.8  Reflection ratio X3 of longitudinal wave at e3 against Z (impedance parameters) for f = 0.01, e = 30°, β = 0.1 (circle-line), β = 0.2 (triangle-line), β = 0.3 (star-line).

0.5 0.45 0.4 X4

0.35 0.3 0.25 0.2 0.15 0.1 0

5

10

15

20

25 Z

30

35

40

45

50

Figure 9.9  Reflection ratio X4 of transverse wave against Z (impedance parameter) for f = 0.01, e = 30°, β = 0.1 (circle-line), β = 0.2 (triangle-line), β = 0.3 (star-line).

Effect of Biot parameter on energy ratios of transverse and longitudinal waves against incident angle of longitudinal wave is shown in Figures 9.10-9.13. It is noticed that the energy ratios of longitudinal and transverse waves achieve a fixed value for incident angle e > 70°. As the value of the Biot parameter grows, the energy ratio of longitudinal waves reflected at an angle e1 decreases, while the energy ratio of longitudinal and transverse waves reflected at angles e1, e2 and e3, respectively, increases against the incident angle of longitudinal waves.

132  Mathematics and Computer Science Volume 2 1.1 1 0.9 0.8 E1

0.7 0.6 0.5 0.4 0.3

0

10

20

30

40

50 60 e (in degrees)

70

80

90

Figure 9.10  Energy ratio E1 of longitudinal wave at e1 against incident angle e for f = 0.05, Z = 10, β = 0.3 (circle-line), β = 0.4 (star-line), β = 0.5 (triangle-line). 4.5

×10-8

4 3.5 3 E2

2.5 2 1.5 1 0.5 0

0

10

20

30

40

50 60 e (in degrees)

70

80

90

Figure 9.11  Energy ratio E2 of longitudinal wave at e2 against incident angle e for f = 0.05, Z = 10, β = 0.3 (circle-line), β = 0.4 (star-line), β = 0.5 (triangle-line). 0.5 0.45 0.4 0.35 0.3 E3

0.25 0.2 0.15 0.1 0.05 0 0

10

20

30

40 50 e (in degrees)

60

70

80

90

Figure 9.12  Energy ratio E3 of longitudinal wave at e3 against incident angle e for f = 0.05, Z = 10, β = 0.3 (circle-line), β = 0.4 (star-line), β = 0.5 (triangle-line).

Propagation of Seismic Waves  133 3.5

×10-3

3 2.5 E4

2

1.5 1 0.5 0

0

10

20

30

40

50 60 e (in degrees)

70

80

90

Figure 9.13  Energy ratio E4 of transverse wave against incident angle e for f = 0.05, Z = 10, β = 0.3 (circle-line), β = 0.4 (star-line), β = 0.5 (triangle-line).

9.6 Conclusion The effects of porosity and impedance on waves propagating in isotropicporous thermoelastic semi-infinite space is investigated in this work. The following key points are drawn from graphs in this paper: • As solid porosity increases, the amplitude ratio of the longitudinal wave reflected at an angle e1, drops against the incident angle of longitudinal waves, reaching a minimum value at e = 65° and subsequently, a fixed value for e > 70° • The reflection ratios of longitudinal waves (at angle e2, e3) and transverse wave falls against incident angle of longitudinal wave, as solid porosity increases, reaching a minimum for e > 70° • Amplitude ratios of reflected waves in isotropic porousthermoelastic medium increases with the impedance parameter • The reflection ratio of the coupled longitudinal waves at angle e1, e2, increases against the impedance parameter with increment of Biot parameter • The reflection ratio of the longitudinal wave at e3 and transverse wave decreases against the impedance parameter with increment of Biot parameter • The Biot parameter has less impact on the reflection ratio of the transverse waves • The energy ratio of the reflected longitudinal waves at an angle e1 drops against the incident angle of longitudinal

134  Mathematics and Computer Science Volume 2 waves, achieving a minimum value at e = 65° and subsequently a fixed value for e > 70° • The sum of all energy ratios approaches unity

References 1. D.K. Madan, A. Rani and M. Punia, A note on the effect of rigidity and initial stress on the propagation of Rayleigh waves in pre-stressed orthotropic elastic layered medium, Proc. Indian Natl. Sci. Acad., Vol. 87, pp. 487–498, 2021. 2. A. Kumari and D.K. Madan, Deformation field due to seismic sources with imperfect Interface, J. Earth Syst. Sci., Vol. 130, pp. 1-19, 2021. 3. H. Lord and Y. Shulman, A generalized dynamical theory of thermoelasticity, J. Mech. Phys. Solids, Vol. 15, pp. 299-309, 1967. 4. A.E. Green and A. Lindsay, Thermoelasticity, J. Elast., Vol. 2, pp. 1-7, 1972. 5. B. Singh, Reflection of P and SV waves from free surface of an elastic solid with generalized thermodiffusion, J. Earth Syst. Sci., Vol. 114, pp. 159-168, 2005. 6. A. N. Abd-Alla and A.A.S. Al-Dawy, The reflection phenomenon of SV waves in a generalized thermoelastic medium, Int. J. Math. Sci., Vol. 23, pp. 529546, 2000. 7. J. N. Sharma, V. Kumar and D. Chand, Reflection of generalized thermoelastic waves from the boundary of a half-space, J. Thermal Stresses, Vol. 26, pp. 925-942, 2003. 8. S. Abo-dahab, R.A. Mohamed and B. Singh, Rotation and magnetic field effects on P wave reflection from a stress–free surface of elastic half-space with voids under one thermal relaxation time, Journal of Vibration and Control, Vol. 17, pp. 1827-1839, 2010. 9. R. Bijarnia and B. Singh, Propagation of plane waves in an anisotropic generalized thermoelastic solid with diffusion, Journal of Engineering Physics and Thermophysics, Vol. 85, pp. 442-448, 2012. 10. S. B. Sinha and K. A. Elsibai, Reflection of thermoelastic waves at a solid halfspace with two thermal relaxation times, Journal of Thermal Stresses, Vol. 20, pp. 129-146, 1996. 11. A.N. Sinha and S.B. Sinha, Reflection of thermoelastic waves at a solid half-space with thermal relaxation, Journal of Physics of the Earth, Vol. 22, pp. 237-244, 1974. 12. A. Rani and D.K. Madan, Effect of initial stress and imperfect interface on love waves propagation in prestressed orthotropic layer coated over a prestressed orthotropic semi-infinite space, Journal of Rajasthan Academy of Physical Sciences, Vol. 20, pp. 219-228, 2021.

Propagation of Seismic Waves  135 13. T. Haibing, L. Ganbin, X. Kanghe, Z. Rongyue and D. Yuebao, Characteristics of wave propagation in the saturated thermoelastic porous medium, Transp Porous Med., Vol. 103, pp. 47-68, 2014. 14. R. Kumar, R. Vohra and S. M. Abo-Dahab, Rayleigh waves in thermoelastic medium with double porosity, MOJ Civil engineering. Vol. 4, pp. 143-148, 2018. 15. R.K. Wilson and E. S. Aifantis, A double porosity model for acoustic wave propagation in fractured porous rock, Int. J. Engg. Sci., Vol. 22, pp. 12091227, 1984. 16. E. Wang, J.M. Carcoine, Y. Yuan and J. Ba, Reflection of inhomogeneous plane waves at the surface of a thermo-poroelastic medium, Geophysical Journal International, Vol. 224, pp. 1621-1639, 2021. 17. C. Zorammuana and S. S. Singh, Elastic waves in thermoelastic saturated porous Medium, Meccanica, Vol. 51, pp. 593-609, 2016. 18. B. Singh, Reflection of plane waves from surface of a generalized thermoviscoelastic porous solid half-space with impedance boundary conditions, Mechanics and Mechanical Engineering, Vol. 22, pp. 1483-1496, 2018. 19. P.C. Vinh and T.T. Hue, Rayleigh waves with impedance boundary conditions in anisotropic solids, Wave Motion, Vol. 51, pp. 1082-1092, 2014. 20. P.C. Vinh and N.Q. Xuan, Rayleigh waves with impedance boundary condition: Formula for the velocity, existence and uniqueness, Eur. J. Mech. A Solids, Vol. 61, pp. 180-185, 2017. 21. M. A. Biot, Mechanics of deformation and acoustic propagation in porousmedia, J Appl Phys, Vol. 33, pp. 1482-1498, 1962. 22. J. Bear, S. Sorek, G. Ben-Dor and G. Mazor, Displacement waves in saturated thermoelastic porous media, I. Basic Equations. Fluid Dyn Res, Vol. 9, pp. 155-164, 1992. 23. A. Levy, S. Sorek, G. Ben-Dor and J. Bear, Evolution of the balance equations in saturated thermoelastic porous media following abrupt simultaneous changes in pressure and temperature, Trans Porous Media, Vol. 21, pp. 241268, 1995. 24. M.D. Sharma, Wave propagation in thermoelastic saturated porous medium, J Earth Syst Sci., Vol. 117, pp. 951-958, 2008. 25. J.D. Achenbach, Wave propagation in elastic solids, North-Holland, Amsterdam, 1973.

10 IoT Based Ensemble Predictive Techniques to Determine the Student Observing Analysis through E-Learning Rufia Thaseen I.1, S. Shahar Banu1* and Sudha Rajesh2 Department of Computer Applications, B. S. Abdur Rahman Crescent Institute of Science & Technology, Vandalur, Tamil Nādu, India 2 Department of Computational Intelligence, SRMIST, Kattankulathur, Tamil Nādu, India

1

Abstract

The objective was to make a comparative study of the IoT based Ensemble Predictive system with real-life teacher predictions on student observation during E-Learning. Data is collected from 46 faculties for 188 periods through an opinion-based survey using a questionnaire. Similarly, for the 188 periods the data was collected from the created IoT based Ensemble Predictive System. The system is designed in such a way that it uses five variables, namely: Level of Interaction, No. of Questions Raised, No. of Students in the Class, No. of Concepts Taught in a Period, and Responsiveness of the Students to Questions during Class Hours, to perform the student observation analysis. From observation, it is interpreted that there is no major difference in the solution provided by the faculties and output generated by the IoT Ensemble Predictive system. Further, it was found that there is a remarkable relationship between the opinion provided by the faculties and output generated by the IoT Ensemble Predictive system. But for the item (No. of Students in the Class), there is no significant relationship between the opinion provided by the faculties and output generated by the IoT Ensemble Predictive system. Therefore, it can be interpreted that when the number of students increased or decreased beyond ideal conditions, there is the possibility of deviation in the opinion provided by the faculties and output generated by the system. Also, the calculated R-value is 0.788, meaning there is a 78.8% strong positive relationship between the opinion provided by the faculties and output generated by the system. *Corresponding author: [email protected] Sharmistha Ghosh, M. Niranjanamurthy, Krishanu Deyasi, Biswadip Basu Mallik, and Santanu Das (eds.) Mathematics and Computer Science Volume 2, (137–150) © 2023 Scrivener Publishing LLC

137

138  Mathematics and Computer Science Volume 2 The estimated R-Square Value (0.621) indicates that it is possible to forecast an IoT based Ensemble Predictive system in the real world with 62.1% efficiency. Furthermore, the evaluated. Coefficients’ significance value is not more than 0.05 and this confirms that an IoT based Ensemble Predictive system can be forecasted with real-world information and vice-versa. Keywords:  IoT, education system, E-learning

10.1 Introduction The World Wide Web was very efficient in terms of posting and retrieving data in the beginning stages. The World Wide Web was only a collection of static web pages with links to other sites and users could access the information they needed by surfing and moving from one page to the next [1]. After some time, the situation changed and web 2.0 was born. A user of a website could do significantly more with it than they could with a web 1.0 site, such as reading material and navigating through one page to another page through hyperlinks. The normal person’s engagement with online resources is easily available. Social networking sites like Facebook, YouTube, and Flickr, among others, are examples of web 2.0 applications. We are currently living in the age of semantic web. Computers in web 3.0 are capable of intelligently evaluating information available on web sites, as well as developing and propagating the information they contain [2]. Until recently, the internet was designed to bring people together, enabling them to communicate, exchange information, send messages, and hold virtual conferences. We now have machine to machine interactions in addition to human to human and human to machine interactions thanks to the introduction of web 3.0. This is a huge step forward. System-to-system communication can be shown through a communication linking a temperature sensor and AC [3]. Consider the following scenario: a temperature sensor monitors the room temperature and when it rises beyond a certain threshold, it sends an indication to the AC prompting it to immediately turn on the switch. If the temperature goes down a specific point [4], the temperature sensor will send a second signal to the air conditioner, this time instructing it to turn off the machine. M2M is a technology that is used in the IOT. With the growth of technology, it is now feasible for people to engage with and contribute information on a website in a conventional manner. The information that can be utilized in the reception and transmission of analyzed data may be used to its full capacity by the specific kind of connection of different things that can be achieved via the internet [5].

IoT Based Ensemble Predictive Techniques  139 As examples, a lot of web 2.0 apps are available. YouTube, Flickr, Facebook, and a slew of other sites are among them. Consumers are increasingly familiar with web 3.0, which is a semantic web that enables them to explore with more sophistication. Because of the existence of web 3.0, persons who use computers can understand the availability of information in a more intellectual manner [6]. Web 3.0 can both send and develop information depending on the data it receives from its data sources. The internet may be considered one of the available resources for individuals to interact and communicate with one another, as well as for the spread and acquisition of relevant information. It may also be used for information transfer and in the sophisticated version, video conferencing can be utilized to make information accessible. Apart from the advancement of man-to-man communication, the promotion of web 3.0 may also be beneficial to the advancement of man-to-machine communication. Web 3.0 also allows for machine-to-machine communications [7]. Every day, more smart items that are tailored to certain situations are produced and they are becoming increasingly common in many fields of education, including higher education. Bright education is the most significant component in establishing IoT bright cities that is made possible by virtual learning and digital transformation when it comes to smart services that are backed by developing technologies such as the IoT. Applying IoT in online learning, when utilized in bright cities, allows e-citizens to be connected and imaginative with a higher degree of involvement and teamwork in the learning activity as well as other decisions made by their peers and the community [8]. The implementation of IoT in smart cities has a great impact on online learning processes by offering a comprehensive electronic approach to the online educational community’s resources through a centrally integrated system in a smart fashion. Academic work shows the ability and effect of the IoT in smart city e-education and e-learning processes are expected, measurable, and cannot be overlooked. Hyper-connection linking items, higher degree of access, scalability, and integration of communication networks (RFID and WSNs) are Internet of Things features that might improve the effectiveness of e-learning techniques in “smart settings,” including brighter cities. It can be considered as the most essential tool for supporting the Internet of Things-based learning system, which is paving the way for more efficient ways in the online-education of smart cities in the not-too-distant future, despite its existing challenges. Future research has to stay focused on present online learning methods and their similarity with smart cities, as well as the creation of new approaches based on this fact.

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10.2 Review of Literature Professionals who want to further their careers and students who want to acquire new skills while maintaining their existing work status, as well as individuals who want to obtain training or prepare for certification exams, are increasingly turning to online education. Institutions are finding it increasingly challenging to keep careful monitoring and evaluation of their courses as the number of persons enrolling in online courses grows [9]. However, owing to a shortage of supervisors and tutors, educational institutions are investing extensively in online teaching and learning to allow their students to utilize the network as a platform. High-bandwidth applications like animations, video conferencing, and simulations, all of which may be provided over the Internet, are available to a worldwide audience of networked learners through the Internet [10]. The website offers a collaborative environment for sharing materials with others, as well as online course courses. Sensors are utilized in the Internet of Things (IoT) to collect data from learners when they interact with virtual platforms like (MOOCs) and the data is then transferred to a central database system to be analyzed. The goal of this investigation is to see whether the IoT can be used to improve online learning and teaching. Several data-mining technologies will be deployed inside the central database systems to filter, organize, integrate, and analyze data in order to provide reports for various management levels [11]. The word “e-learning” refers to the application of technology in the field of education and learning. Learners may utilize virtual classrooms, which are given via e-learning, to expand their knowledge without needing to employ traditional learning methods. As a consequence of the fast proliferation of information and communication technology in today’s environment, the letter “e” has become the symbol of the information technology age we are currently living in [12]. The letter “e” is a frequent abbreviation for electrical components. As seen by the emergence of phrases like “e-learning,” “e-health,” “e-business,” “e-government,” and many others, the prefix “e” is becoming increasingly popular in every industry. As a consequence of globalization, networking, and information technology reaching their pinnacle, e-learning is becoming increasingly significant in the field of education [13]. Today’s world is controlled by globalization, networking, and IT. In eLearning, social media plays an essential role in enhancing the learner experience. In recent years, the use of social media to promote course material, training programs, learning tools, and registration in new courses has grown in popularity. The most extensively used social media

IoT Based Ensemble Predictive Techniques  141 sites for marketing include YouTube, LinkedIn, Twitter, Facebook, and Google Plus. The below table gives you a detailed number of online social platforms in India. IoT is a widely utilized technology that allows communication and collaboration through the Internet. It grows in both size and dimension as it develops, affecting many aspects of our life, including educational chances [14]. According to the Internet of Things’ basic premise, any devices having a unique IP address are allowed to connect with one another in the future, both physically and digitally. The data acquired by sensors, tags, or actuators and transferred to a cloud system through a gateway is the most basic structure of the IoT. In the IoT, machine-to-machine with object-to-machine interactions are prevalent. The IoT has facilitated the expansion of a wide range of applications, from simple domestic appliances to complex medical equipment [15]. The Internet of Things covers a broad variety of human activities, including smart cities, smart businesses, and smart energy use to name a few. Education is one of the most important activities of humans that the Internet of Things has influenced, with the shape of education in the nottoo-distant future likely to be converted into a more imaginative framework. IoT is a relatively new phenomenon that promotes innovation across a broad variety of businesses. The area of education is one of these sectors (sometimes known as e-education). IoT has the potential to deliver a broad variety of e-educational technologies that will have a major influence on the future of educational institutions since it can be combined with other information technology (IT) technologies. Intelligent technologies will be installed in the future educational facility [16]. Students and teachers must verify their position as users by showing their finger prints and RFID ID cards in front of the reader, as well as mobile checking, whether visiting the rooms in person or gaining permission to the school’s automated system administration. In the future, sensors will be deployed in IoT classrooms to ensure that only authorized teachers and students have access to the space. The RFID or WSN devices that will be mounted on the smart whiteboards and desks will be able to detect and identify the persons who use them physically. Students and teachers in the smart classroom may connect with one another in a mutually advantageous manner. The Internet of Things (IoT) has the potential to provide secure interaction, linking all physical and technological objects. As a consequence of this technology, students may connect online to labs, libraries, didactic materials, tests, and instructional messages while also doing administrative tasks more efficiently in a large virtual classroom. Another benefit of this kind of virtual learning is that in this format, all learning assignments and activities will be described as objects. E-learning also refers to the incorporation of electronic tools,

142  Mathematics and Computer Science Volume 2 software and hardware applications, and web-based activities into a learning system or system of learning, and it dates back to the 1980s. In fact, due to the fast development of technology and communication technologies, it is now feasible to study online using a large learning environment, which is becoming more popular (Internet) [17]. The Internet of Things (IoT) allows learning environments to be expanded by integrating actual and virtual things without interfering with the learning process. Traditional e-learning has the ability to provide learners with a wide virtual access environment as a digital strategy, but it also has important limitations, as noted below [18]. The most major constraints in e-learning settings are geographical location, face-to-face communication between objects, and effective cooperation between virtual and real agents. The usage of smart objects in the learning environment is one potential answer to the issues listed above. The Internet of Things (IoT) is commonly recognized as the most important source of smart agents for e-learning. The Internet of Things (IoT) has the ability to include two key aspects into traditional e-learning: intelligence and object interaction [19]. IoT has the potential to establish a vast platform for students and instructors to access a variety of distant learning devices and products. A high degree of interaction between virtual and physical goods may lead to the creation of a wide range of collaborative situations.

10.2.1 Objectives of the Study The objective was to make a comparative study of an IoT based Ensemble Predictive system with real-life teacher predictions on student observation during E-Learning and further, to identify the forecasting efficiency of the IoT based Ensemble Predictive system with the real world.

10.3 Methodology The descriptive and experimental research design is adopted for further study. The data is collected from 46 faculties for 188 periods through an opinion-based survey using a questionnaire. Similarly, the 188 periods the data was collected from created an IoT based Ensemble Predictive System. The system is designed in such a way that it uses five variables, namely: Level of Interaction, No. of Questions Raised, No. of Students in the Class, No. of Concepts Taught in a Period, and Responsiveness of the Students to Questions during Class Hours, to perform the student observing analysis.

IoT Based Ensemble Predictive Techniques  143

10.4 Analysis and Interpretation Analysis is done to find out the average score provided by the system and opinions provided by the faculties during 188 hours. Table 10.1  Group statistics - teacher & system feedback. Group statistics Teacher/system feedback

N

Mean

Std. deviation

Std. error mean

Level of interaction

Teacher feedback

188

4.3564

.69806

.05091

System feedback

188

4.4255

.70883

.05170

Teacher feedback

188

4.4255

.70125

.05114

System feedback

188

4.3457

.70342

.05130

Teacher feedback

188

4.3723

.66194

.04828

System feedback

188

4.4362

.60406

.04406

Teacher feedback

188

4.3883

.73366

.05351

System feedback

188

4.4043

.65111

.04749

Teacher feedback

188

4.3989

.67452

.04919

System feedback

188

4.3404

.67107

.04894

No. of questions raised

No. of students in the class

No. of concepts taught in a period Responsiveness of the students to questions

Source: (Primary data)

From the Table 10.1 Group statistics - teacher & system feedback, mean score calculated using the opinion provided by the faculties and output of the analysis made by the system, it can be interpreted that students have a high level of observation during the class hours, as the mean score lies between (4.3404 - 4.4255). Analysis is done to check the difference in opinion between the faculties and system.

144  Mathematics and Computer Science Volume 2 From Table 10.2 Independent sample test - teacher & system feedback, the approximated value is more than 0.05, meaning the null hypothesis is accepted. Therefore, there is no major difference in the answers provided by the faculties and output generated by the IoT Ensemble Predictive system. Therefore, analysis is taken forward to identify whether there is any significant relationship between the opinion provided by the faculties and output generated by the IoT Ensemble Predictive system. From Table 10.3 Correlation analysis - teacher & system feedback, the approximate value is less than 0.05 for most of the items, meaning the null hypothesis is rejected. Therefore, there is a closed relationship between the opinion provided by the faculties and output generated by the IoT Ensemble Predictive system. But for the item (No. of Students in the Class), the evaluated value is greater than 0.05 for most of the items, meaning the null hypothesis is accepted. But, there is no serious relationship between the answers provided by the faculties and output generated by the IoT Ensemble Predictive system when the No. of students increased or decreased beyond normal ideal conditions. Table 10.2  Independent sample test - teacher & system feedback. Independent samples test

Level of interaction

Equal variances assumed

Levene’s test for equality of variances

t-test for equality of means

F

Sig.

t

df

Sig. (2-tailed)

.371

.543

-.953

374

.341

-.953

373.912

.341

1.101

374

.271

1.101

373.996

.271

Equal variances not assumed No. of questions raised

Equal variances assumed Equal variances not assumed

.171

.680

(Continued)

IoT Based Ensemble Predictive Techniques  145 Table 10.2  Independent sample test - teacher & system feedback. (Continued) Independent samples test

No. of students in the class

Equal variances assumed

Levene’s test for equality of variances

t-test for equality of means

F

Sig.

t

df

Sig. (2-tailed)

.164

.686

-.977

374

.329

-.977

370.912

.329

-.223

374

.824

-.223

368.794

.824

.843

374

.400

.843

373.990

.400

Equal variances not assumed No. of concepts taught in a period

Equal variances assumed

.974

.324

Equal variances not assumed Responsiveness of the students to questions

Equal variances assumed Equal variances not assumed

.052

.820

Source: (Primary data)

Herein, an analysis was carried out to identify the forecasting efficiency of the IoT based Ensemble Predictive system with the real world. From Table 10.4 Regression analysis - teacher & system feedback, the Calculated R-value is 0.788, meaning there is a 78.8% strong positive relationship between the opinion provided by the faculties and output generated by the system. The estimated R-Square Value (0.621) indicates that it is possible to forecast an IoT based Ensemble Predictive system with in the real world with 62.1% efficiency. Also, the ANOVA significance value is less than 0.05, illustrating the model is fit. Furthermore, the estimated coefficients’ significance value is less than 0.05 and this confirms that an IoT based Ensemble Predictive system can be forecasted with real-world information and vice-versa.

146  Mathematics and Computer Science Volume 2 Table 10.3  Correlation analysis - teacher & system feedback. Correlation System feedback - level of interaction Teacher feedback level of interaction

Pearson correlation

.384

Sig. (2-tailed)

.000

N

188 System feedback - no. of questions Raised

Teacher feedback no. of questions raised

Pearson correlation

.307

Sig. (2-tailed)

.000

N

188 System feedback - no. of students in the class

Teacher feedback no. of students in the class

Pearson correlation

.140

Sig. (2-tailed)

.055

N

188 System feedback - no. of concepts taught in a period

Teacher feedback no. of concepts taught in a period

Pearson correlation

.162

Sig. (2-tailed)

.026

N

188 System feedback - responsiveness of the students to questions (Continued)

IoT Based Ensemble Predictive Techniques  147 Table 10.3  Correlation analysis - teacher & system feedback. (Continued) Correlation Teacher feedback responsiveness of the students to questions

Pearson correlation

.171

Sig. (2-tailed)

.019

N

188 Overall system feedback

Overall teacher feedback

Pearson correlation

.788

Sig. (2-tailed)

.000

N

188

Source: (Primary data)

10.5 Findings and Conclusion From the evaluation, it can be interpreted that there is no big difference in the opinion provided by the faculties and output generated by the IoT Ensemble Predictive system. Further, it was identified that there is a significant relationship between the opinion provided by the faculties and output generated by the IoT Ensemble Predictive system. But for the item (No. of Students in the Class), there is no significant relationship between the opinion provided by the faculties and output generated by the IoT Ensemble Predictive system. Therefore, it can be interpreted that when the No. of students increased or decreases beyond normal ideal conditions, there is the possibility of deviation in the opinion provided by the faculties and output generated by the system. Also, The Calculated R-value is 0.788, meaning there is a 78.8% strong positive relationship between the opinion provided by the faculties and output generated by the system. The estimated R-Square Value (0.621) indicates that it is possible to forecast an IoT based Ensemble Predictive system in the real world with 62.1% efficiency. Furthermore, the evaluated coefficients significance value is less than 0.05 and this confirms that IoT based Ensemble Predictive system can be forecasted with real-world information and vice-versa.

148  Mathematics and Computer Science Volume 2 Table 10.4  Regression analysis - teacher & system feedback. Model summary Model

R

R square

Adjusted R square

Std. error of the estimate

1

.788a

.621

.378

.32915

a. Predictors: (constant), overall teacher feedback ANOVA Sum of squares

df

Mean square

F

Sig.

Regression

12.432

1

12.432

114.747

.000b

Residual

20.151

186

.108

Total

32.583

187

Model 1

a. Dependent variable: overall system feedback b. Predictors: (constant), overall teacher feedback Coefficientsa Unstandardized coefficients

Model B 1

Standardized coefficients

Std. error

Beta

t

Sig.

(Constant)

1.961

.228

8.602

.000

Overall teacher feedback

.554

.052

10.712

.000

.618

a. Dependent variable: overall system feedback Source: (Primary data)

References 1. Berecz, “Overview of E-learning Strategies from the Point of View of Higher Education,” J. Appl. Multimed., vol. 13, no. 4, pp. 117–127, 2019, doi: 10.26648/jam.2018.4.003. 2. Hammi, R. Khatoun, S. Zeadally, A. Fayad, and L. Khoukhi, “IoT technologies for smart cities,” IET Networks, vol. 7, no. 1, pp. 1–13, 2018, doi: 10.1049/ iet-net.2017.0163.

IoT Based Ensemble Predictive Techniques  149 3. M. Njeru, M. S. Omar, S. Yi, S. Paracha, and M. Wannous, “Using IoT technology to improve online education through data mining,” Proc. 2017 IEEE Int. Conf. Appl. Syst. Innov. Appl. Syst. Innov. Mod. Technol. ICASI 2017, no. May, pp. 515–518, 2017, doi: 10.1109/ICASI.2017.7988469. 4. S. Kusuma and D. Kasi Viswanath, “IOT and Big data analytics in e-Learning: A technological perspective and review,” Int. J. Eng. Technol., vol. 7, no. 1, pp. 164– 167, 2018, doi: 10.14419/ijet.v7i1.8.11540. 5. M. Bayani, K. Leiton, and M. Loaiza, “Internet of things (IOT) Advantages on e-learning in the smart cities IoT-based library automation & monitoring system view project Internet of things (IoT) advantages on e-learning in the smart cities,” Int. J. Dev. Res., vol. 07, no. January, pp. 17747–17753, 2017, [Online]. Available: http://www.journalijdr.com. 6. S. Madakam, R. Ramaswamy, and S. Tripathi, “Internet of Things (IoT): A Literature Review,” J. Comput. Commun., vol. 03, no. 05, pp. 164–173, 2015, doi: 10.4236/jcc.2015.35021. 7. Y. S. Sai and K. K. Kumar, “Internet of things and its applications,” Int. J. Eng. Technol., vol. 7, pp. 422–427, 2018, doi: 10.14419/ijet.v7i2.7.10758. 8. V. D. Soni, “IOT connected with e-learning,” Int. J. Integr. Educ., vol. 2, no. 5, pp. 273–277, 2019. 9. K. Balakrishna, F. Mohammed, C. R. Ullas, C. M. Hema, and S. K. Sonakshi, “Application of IOT and machine learning in crop protection against animal intrusion,” Glob. Transitions Proc., vol. 2, no. 2, pp. 169–174, 2021, doi: 10.1016/j.gltp.2021.08.061. 10. Abdellatif et al., “Communication-Efficient Hierarchical Federated Learning for IoT Heterogeneous Systems with Imbalanced Data,” Futur. Gener. Comput. Syst., vol. 128, pp. 406–419, 2021, doi: 10.1016/j.future.2021.10.016. 11. K. F. Muteba, K. Djouani, and T. O. Olwal, “Deep reinforcement learning based resource allocation for narrowband cognitive radio-Iot systems,” Procedia Comput. Sci., vol. 175, no. 2019, pp. 315–324, 2020, doi: 10.1016/j. procs.2020.07.046. 12. J. Su, S. He, and Y. Wu, “Features Selection and Prediction for IoT Attacks,” High- Confidence Comput., p. 100047, 2021, doi: 10.1016/j.hcc.2021.100047. 13. E. Anthi, L. Williams, A. Javed, and P. Burnap, “Hardening machine learning denial of service (DoS) defences against adversarial attacks in IoT smart home networks,” Comput. Secur., vol. 108, p. 102352, 2021, doi: 10.1016/j. cose.2021.102352. 14. M. W. Rahman, R. Islam, A. Hasan, N. I. Bithi, M. M. Hasan, and M. M. Rahman, “Intelligent waste management system using deep learning with IoT,” J. King Saud Univ. - Comput. Inf. Sci., no. xxxx, 2020, doi: 10.1016/j. jksuci.2020.08.016. 15. B. Song et al., “ScienceDirect IoT and Machine learning for in-situ process control using Laser Based IoT and Machine learning for in-situ process using Laser Based Additive study Additive Manufacturing (LBAM) case study new methodology and physical architecture of a , * analyze the functional

150  Mathematics and Computer Science Volume 2 existing products for an assembly oriented Tiwari a product family identification,” Procedia CIRP, vol. 104, pp. 1813– 1818, 2017, doi: 10.1016/j. procir.2021.11.306. 16. D. Deng, X. Li, V. Menon, M. J. Piran, H. Chen, and M. A. Jan, “Learningbased joint UAV trajectory and power allocation optimization for secure IoT networks,” Digit. Commun. Networks, no. August 2020, 2021, doi: 10.1016/j. dcan.2021.07.007. 17. R. Akhter and S. A. Sofi, “Precision agriculture using IoT data analytics and machine learning,” J. King Saud Univ. - Comput. Inf. Sci., no. xxxx, 2021, doi: 10.1016/j.jksuci.2021.05.013. 18. V. R. Kebande, R. A. Ikuesan, N. M. Karie, S. Alawadi, K. K. R. Choo, and A. Al-Dhaqm, “Quantifying the need for supervised machine learning in conducting live forensic analysis of emergent configurations (ECO) in IoT environments,” Forensic Sci. Int. Reports, vol. 2, no. July, p. 100122, 2020, doi: 10.1016/j.fsir.2020.100122. 19. M. Esmail Karar, A.-H. Abdel-Aty, F. Algarni, M. Fadzil Hassan, M. A. Abdou, and O. Reyad, “Smart IoT-based system for detecting RPW larvae in date palms using mixed depthwise convolutional networks,” Alexandria Eng. J., 2021, doi: 10.1016/j.aej.2021.10.050.

11 Modelling and Analysis of a Congestion Dependent Queue with Bernoulli Scheduled Vacation Interruption and Client Impatience K. Jyothsna1* and P. Vijaya Kumar2 Department of Mathematics, Anil Neerukonda Institute of Technology and Sciences, Visakhapatnam, Andhra Pradesh, India 2 Department of Mathematics, GITAM (Deemed to be University), Visakhapatnam, Andhra Pradesh, India 1

Abstract

This paper’s purpose is to model and analyze an impatient client renewal input Bernoulli scheduled vacation interruption queue. Two types of congestion dependent client impatience, namely balking and reneging, have been incorporated. The server shifts to working vacation mode whenever the system is completely depleted. The server follows Bernoulli’s vacation interruption schedule after executing a service during working vacation, i.e., the server either continues working during the vacation with certain probability or interrupts the working vacation with the complementary probability. Regular service durations, working vacation service durations, and vacation durations are expected to be congestion dependent and exponentially distributed. The stationary probability distributions at different epochs were estimated utilizing supplementary variable and iterative approaches. A few performance metrics and the model parameters’ influence on the performance metrics have been provided through numerical outcomes. Keywords:  Balking, reneging, congestion dependent, multiple working vacations, vacation interruption, Bernoulli schedule

*Corresponding author: [email protected] Sharmistha Ghosh, M. Niranjanamurthy, Krishanu Deyasi, Biswadip Basu Mallik, and Santanu Das (eds.) Mathematics and Computer Science Volume 2, (151–172) © 2023 Scrivener Publishing LLC

151

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11.1 Introduction Queues or waiting lines are prominent in normal life and even in a range of corporate and manufacturing contexts. Waiting for service is inescapable, resulting in dissatisfaction, annoyance, and anxiety among clients. Clients are strained for time in today’s fast-paced environment. They would prefer not to be involved in the act of waiting for service. Clients’ behavior is affected by such time limitations. Balking and reneging are two terms used in queuing parlance to describe this type of impatience. If a client on arrival discovers the queueing system is not empty, he or she may elect not to join it. Balking is the term used in queueing theory to describe this behavior. Clients who queue up but leave before the service is completed are termed to be reneged. Clients’ balking behavior in an exponential queueing situation has been initially proposed by Haight [1]. For the single server Markovian problem, Haight [2] introduced the reneging behavior of clients. The research of a balking queue under renewal input process has been conducted by Finch [3]. Wang et al. [4] carried out an optimum and sensitivity analysis as well as a number of numerical experiments on a machine maintenance issue including multiple servers and balking. Ward [5] examined the outcomes as well as the reneging behavior of the queueing problem under traditional high traffic. Queuing models with server vacations are especially significant for laying the groundwork for effective design and research in a number of real life scenarios like commercial and industrial manufacturing units, software and information networking, and other sectors. Clients cannot access the service facility for a predefined duration of time in vacation queueing approaches. The server may be clearing surplus tasks left by departing clients, undergoing restoration, or taking a break during vacation. Several authors have explored queueing scenarios where the service facility takes a break, see [6−16]. Working vacation queues have gained popularity in the modern generations as a result of possible uses in data transmission, internet technology, industrial systems, and many other associated systems. Servi and Finn [17] presented the idea of working vacations (WV), wherein rather than shutting down altogether, the service facility delivers service to the arrivals at a decreased rate. This process is continued until a waiting client is discovered at a vacation termination instant. A series of such WV is stated as multiple working vacations (MWV). Working vacation queues are more convenient to utilize since they are more accessible than the traditional vacation queues. Due to this, working vacation queueing systems have sparked a lot of interest. Banik et al. [18] published the analysis of an MWV queue, wherein the arrivals follow a renewal process, utilizing a supplementary

Modelling and Analysis of a Congestion Dependent Queue  153 variable method. A finite buffer balking queue under renewal input process has been reported by Vijaya Laxmi and Jyothsna [19]. Majid [20] worked on the study of client impatience in a working vacation waiting line. When some system indices reach a specific threshold in WV queues, the server may restore regular service, interrupting the working vacation. This sort of action is referred to as vacation interruption (VI) in working vacation queues. Li and Tian [21] initially came up with the idea of VI in a Markovian queue. Li et al. [22] extended Li and Tian [21]’s research to a VI queue wherein the arrivals follow a renewal process. Majid and Manoharan [23] published the research of a Markovian queue with VI wherein the server is expected to interrupt the WV if the system detects at least N waiting clients at a service finishing moment during the WV. At the moment the service is completed in WV, the server determines whether to extend the WV with a probability of q or to disrupt the WV with the complimentary probability of 1 − q, referred to as Bernoulli scheduled vacation interruption. Initially, Zhang and Shi [24] explored VI under the Bernoulli rule in an exponential queue. The investigations of an impatient client VI queue under Bernoulli rule and renewal input process has been published by Vijaya Laxmi et al. [25]. More information regarding working vacation queues with client impatience may be found in Bouchentouf and Yahiaoui [26], Majid et al. [27], as well as references therein. The majority of queueing system research is predicated on the notion that the system’s state has no effect on the input and service parameters. In practice, however, many of the queueing system’s service and input parameters are based on the system’s state and reducing the clients wait time. Consider the case of telecommunications network routing. Congestion independent routing probabilities cannot prevent tasks from being sent to already overburdened disc stations in such networks, whereas jobs are routed to the least loaded discs first when congestion dependent routing probabilities are used. State dependent routing outperforms state independent routing in terms of throughput. For literature on state dependent queues, see [28−30] and the references therein. To our knowledge, no research has ever examined queues that include congestion dependence, working vacations, Bernoulli schedule VI, and impatient clients; this gap in the literature drives the current research. Also, the uncorrelated arrival process often yields a better approximation than the exponential distribution since the memoryless quality of the aforementioned distribution might not necessarily fit the aims of applications and it can incorporate Erlang, deterministic distributions, etc. Hence, investigating the impact of client impatience in a congestion dependent Bernoulli scheduled VI queue under renewal input process is worthwhile.

154  Mathematics and Computer Science Volume 2 Service, as well as the vacation durations of the service facility, are expected to be exponentially distributed and congestion dependent. The concept is researched employing supplementary variable and iterative techniques. The stationary distribution of the number of clients has been computed at pre-arrival and arbitrary epochs. Performance metrics and numerical data indicating the change in the performance metrics with the model parameters are provided in table and graph forms. The following is the layout of the article’s main body. Section 11.2 presents the model’s overview. The computation of the stationary distributions at diverse time epochs is in Section 11.3 and special cases of the model in Section 11.4. In Section 11.5, several model performance metrics are provided and numerical calculations to show the parametric influence on system performance metrics are disposed in Section 11.6. Finally, Section 11.7 brings the article to a conclusion.

11.2 Model Overview A GI/M/1/N/MWV queue that is congestion dependent and having balking, reneging, vacation interruption under a Bernoulli schedule is studied. The time between two subsequent arrivals are assumed to be distributed identically and are independent random variables with probability density function a(z), z ≥ 0, Laplace-Stieltjes transformation (LST) Aˆ (θ ), and mean inter-arrival duration 1/λ = −Aˆ (1)(0). When a client views j clients in the system, it can queue up with probability βj or balk with the complimentary probability 1 − βj. We also presume that β0 = 1, 0 ≤ βj+1 < βj ≤ 1, 1 ≤ j ≤ N – 1, and βN = 0. If an arrival does not join the queue, he or she will be permanently lost. If an arrival has to wait longer than expected for service after entering the queue, he may renege. With parameter α, the client’s waiting period preceding reneging is exponentially distributed. (j − 1)α is taken as the average reneging rate, as an impatient client’s arrival and departure are unrelated. A single service facility offers services on a first-come, first-served (FCFS) basis. The service facility departs for a WV as and when the system is depleted. Unlike the queueing paradigms with conventional vacation policy, the service is provided to the waiting clients at a slower rate during WV. At the moment, the service is completed during WV, the server determines whether to extend the vacation with the probability of q or to disrupt the WV with a complimentary probability of 1 − q, designated as Bernoulli scheduled vacation interruption. When there are j clients in the system, the regular service durations, WV service durations, and vacation durations are all considered to be

Modelling and Analysis of a Congestion Dependent Queue  155 exponentially distributed with congestion dependent rates µj, ηj and ϕj, 1 ≤ j ≤ N, respectively. Further, define the average regular service rate (µm), average vacation service rate (ηm), and average vacation rate (ϕm) as

∑

∑

N

N

ηj µj =j 1 =j 1 = µm = ; ηm ; φm = N N

∑

N j =1

N

φj

At time t, let • Ns(t) = The total count of clients, counting the one who is currently undergoing service • Z(t) = Upcoming arrival’s remaining inter-arrival time. At steady state, define:

= π j (z )dz lim = Pr {N s (t ) j, z ≤ Z (t ) < z + dz}, z ≥ 0, 0 ≤ j ≤ N , t →∞



ω j (z )dz = lim Pr {N s (= t ) j, z ≤ Z (t ) < z + dz }, z ≥ 0, 1 ≤ j ≤ N , t →∞



The Laplace-Stieltjes transformations that follow are introduced:

= π j (θ ) ^

ω j (θ ) = ^



∫ ∫

∞

0 ∞ 0

e −θ zπ j (z )dz , 0 ≤ j ≤ N , e −θ zω j (z )dz 1 ≤ j ≤ N .



At an arbitrary instant, while the server is in WV and regular service, the probability of j clients in the system can be denoted by π j ≡ πˆ j (0),(0 ≤ j ≤ N ) and ω j ≡ ωˆ j (0),(1 ≤ j ≤ N ), respectively.

11.3 Model Analysis To determine the system length probabilities at arbitrary instants, differential difference equations are formulated that associate the distribution of number of clients in the system at the completion of the vacation and regular service period. We accomplish this by employing the supplementary variable technique, which connects the system’s state at two time instants t and t+dt. The differential difference equations under steady-state

156  Mathematics and Computer Science Volume 2 conditions can be constructed utilizing probabilistic reasonings and taking limit t → ∞ as

−π 0(1) (z ) = µ1ω1(z ) + η1π 1(z ), −π (j1) (z ) = −r ( j)π j (z ) + a(z )((1 − β j )π j (0) + β j−1π j−1(0)) + (qη j+1 + jα )π j+1(z ), 1 ≤ j ≤ N − 1, −π N(1) (z ) = −r (N )π N (z ) + a(z )(π N (0) + β N −1π N −1(0)), −ω1(1) (z ) = −s(1)ω1(z ) + a(z )(1 − β1 )ω1(0) + φ1π 1(z ) + s(2)ω2 (z )qη2π 2 (z ), −ω (j1) (z ) = −s( j)ω j (z ) + a(z )((1 − β j )ω j (0) + β j−1ω j−1(0)) + φ jπ j (z ) + qηπ j+1(z ) + s( j + 1)ω j+1(z ), 2 ≤ j ≤ N − 1, −ω (1) (z ) = −s(N )ωN (z ) + φN π N (z ) + a(z )(β N −1ωN −1(0) + ωN (0)), N where πj(0) and ωj(0) are respective rates of arrivals and for 1 ≤ j ≤ N, r(j) = ϕj + ηj +(j − 1)α, s(j) = µj + (j − 1)α. The preceding set of equations is multiplied with e−θz and integrated over z for 0 to ∞ to get

−θ π 0 (θ ) = −π 0 (0) + µ1 ω1(θ ) + η1 π 1(θ ), ^

^

^

(11.1)

^

(r ( j) − θ )π j (θ ) = −π j (0) + A(θ )((1 − β j )π j (0) + β j −1π j −1(0)) ^

+ (qη j +1 + jα )π j + 1(θ ), ^



1 ≤ j ≤ N – 1,



(11.2)

(r ( N ) − θ )πˆ N (θ ) = −π N (0) + Aˆ (θ )(π N (0) + β N −1π N −1 (0)), (11.3) ^

(s(1) − θ ))ω1(θ ) = −ω1(0) + A(θ )(1 − β1 )ω1(0) + φ1 π 1(θ ) ^

^

+ s(2)ω 2 (θ ) + qη2π 2 (θ ), ^

(11.4)

^

(s( j) − θ )ω j (θ ) = −ω j (0) + A(θ )((1 − β j )ω j (0) + β j−1ω j−1(0)) ^



+ φ j π j (θ ) + qηπ j+1(θ ) ^

+ s( j + 1)ωˆ j +1 (θ ),2 ≤ j ≤ N − 1,

(11.5)

Modelling and Analysis of a Congestion Dependent Queue  157

ˆ ˆ ˆ (s( N ) − θ )ω N (θ ) = −ω N (0) + φ N π N (θ ) + A(θ )( β N −1ω N −1 (0) + ω N (0)). (11.6) Adding the above equations results as:



∑

N j =0

πˆ j (θ ) +

∑

1 − Aˆ (θ ) ωˆ j (θ ) = j =1 θ

N

∑

N j =0

π j (0) +

∑

Assigning θ → 0 and utilizing the normality condition

∑

N j =1

N j =1

ω j (0 )

∑

N j =0



πj +

ω j = 1 produces

∑

N

π j (0) +

∑

N

=j 0=j 1

ωj = λ

(11.7)

11.3.1 Pre-Arrival Epoch Probabilities Let π j− , 0 ≤ j ≤ N , and ω j− ≤ j ≤ N symbolize the probability of j clients in the system during WV and during regular service period, respectively, at pre-arrival instant. Applying Bayes’ theorem, we have



1 1 = π j (0); ω j− ω j (0). λ λ

= π j−

(11.8)

To obtain the pre-arrival instant probabilities (πˆj, ωˆj), we need to evaluate the rate probabilities (πj(0), ωj(0)) as shown below. Taking θ = r(N) in Equation (11.3) gives

πN−1(0) = ϱN−1πN(0),



(11.9)

^

where ϱN = 0, ϱN−1 =



1 − A(r (N )) ^

A(r (N ))

. Substituting πN−1 (0) in (11.3), we get

π N (θ ) = Ψ N ,θ π N (0), ^

158  Mathematics and Computer Science Volume 2 where ^

Ψ N ,θ

{A(θ )(β N −1N −1+N )−N = , if θ ≠ r (N ), r (N ) − θ ^ (1)

r (N ) − A (θ )(β N −1N −1 + N ), if θ =





with

Ψ N ,θ =

^ (l )

{A (θ )(β N −1N −1+N ) + lψ l −1 , if θ ≠ r (N ), r (N ) − θ ^ (1)

r (N ) − A (θ )(β N −1N −1 + N ), if θ =





Taking θ = r(j) in (11.2) and proceeding as above, we obtain

πj(0) = ϱjπN (0), j = N − 2, N − 3, . . . , 0,



(11.10)

where



j =

 j +1 (1 − Aˆ (r ( j + 1))(1 − β j +1 )) − (qη j +2 + ( j + 1)α )Ψ j +2,r ( j + 1) A(r ( j + 1))β j



and

πˆ j (θ ) = Ψ j ,θ π N (0), j = N − 1, N − 2,…,1,

where ^

Ψ j ,θ

A(θ )(β j−1 j−1 + (1 − β j ) j ) + (qη j+1 + jα )Ψ j+1,θ −j , if θ ≠ r ( j) = r (N ) − θ ^



(1)

− ( A (θ )(β j−1 j−1 + (1 − β j ) j ) + (qη j+1 + jα )Ψ (j1+)1,θ ), if θ = r ( j)



Modelling and Analysis of a Congestion Dependent Queue  159 with ∧

Ψ j ,θ =

A(θ )(β j −1 j −1 + (1 − β j ) j ) + (qη j +1 + jα )Ψ (jl+)1,θ + lΨ (jl,θ−1) r ( j) − θ ∧

−

as

A

( l +1)

, if θ ≠ r ( j)

(θ )(β j −1 j −1 + (1 − β j ) j ) + (qη j +1 + jα )Ψ ((lj++11),θ ) l +1

, if θ = r ( j)

Taking θ = s(N) in (11.6), we have ωN−1(0) in terms of πN (0) and ωN (0)

ωN−1(0) = τN−1(0)ωN(0) + νN−1πN(0),

(11.11)

where ^

= τ N 1,= ν N 0,τ N −1 =



φ Ψ 1 − A(s(N )) , ν N −1 = − ^ N N ,s ( N ) ^ A(s(N ))β N −1 A(s(N ))β N −1

Substituting (11.11) in (11.6) gives

= ω N (θ ) ∆ N, θ ωN (0) + Ω N ,θ π N (0), ^





where ∧

∆ N ,θ

{A(θ )(β N −1τ N −1 + τ N ) − τ N = if θ ≠ s(N ) s( N ) − θ ∧

(1)

s(N ), − A (θ )(β N −1τ N −1 + τ N ), if θ = ∧

Ω N ,θ

A(θ )(β N −1ν N −1 + φN Ψ N ,θ ) = if θ ≠ s(N ), s( N ) − θ ∧



(1)

− ( A (θ )β N −1ν N −1 + φN Ψ (N1),θ ), if θ = s(N ),



160  Mathematics and Computer Science Volume 2 with ∧

∆

(l ) N ,θ

(l )

A (θ )(β N −1τ N −1 + τ N ) − l∆(Nl −,θ1) = if θ ≠ s(N ), s( N ) − θ ∧

− ∧

Ω

(l ) N ,θ

A

(θ )(β N −1τ N −1 + τ N ) , if θ = s(N ), l +1

(l )

A (θ )(β N −1ν N −1 + φN Ψ (Nl ),θ + lΩ(Nl −,θ1) ) = if θ ≠ s(N ), s( N ) − θ ∧

−



(l +1)

A

(l +1)

(θ )(β N −1ν N −1 + φN Ψ (Nl +,θ1) ) , if θ = s(N ) l +1



From Equation (11.5), we get ωj(0), ωˆ j (θ ) in terms of πN (0) and ωN (0) as

ωj−1(0) = τj−1ωN(0) + νj−1πN(0), j = N − 1, . . . , 2,



(11.12)

where for 2 ≤ j ≤ N − 1,



τ j −1 =

(1 − Aˆ (s( j ))(1 − β j ))τ j − s( j + 1) Aˆ (s( j ))β j −1

ν j −1 =

ˆ ( s( j ))(1 − β j ))ν j − s( j + 1)Ω j +1,s ( j ) − φ j Ψ j ,s ( j ) − q − η j +1Ψ j +1,s ( j ) (1 − A Aˆ (s( j ))β j −1

j +1, s ( j )



and



= ωˆ j (θ ) ∆ j, θ ω N (0) + Ω j ,θ π N (0), = j N − 1,…, 2,



where for 2 ≤ j ≤ N − 1, ∧

∆ j ,θ

∧

s( j + 1)∆ j +1,θ + A(θ )(β j −1τ j −1 − (1 − A(θ )(1 − β j ))τ j = , if θ ≠ s( j), s( j ) − θ ∧

(1)

∧

(1)

∆ = −s( j + 1)∆(j1+)1,θ + A (θ )(β j −1τ j −1 + A (θ )(1− β j ))τ j , if θ = s( j), j ,θ

Modelling and Analysis of a Congestion Dependent Queue  161 ∧

∧

Ω j ,θ =

s ( j + 1)Ω j + 1 ,θ + A(θ ) β j −1ν j −1 − (1 − A(θ )(1 − β j ))ν j + φ j Ψ j ,θ + q η j + 1 Ψ j + 1 ,θ s( j) − θ ∧

(1)

(1)

∧

(1)

(1)

, if θ ≠ s ( j ),

(1)

− s ( j + 1)Ω j + 1 ,θ + A (θ )( β j −1ν j −1 + A (θ )(1 − β j ))ν j + φ j Ψ j ,θ + q η j + 1 Ψ j + 1 ,θ , if θ = s ( j ),





with ∧

∆ j ,θ

(l )

s( j + 1)∆(jl+)1,θ + A (θ )(β j −1τ j −1 + (1 − β j )τ j ) + l ∆(jl,θ−1) = , if θ ≠ s( j), s(l) − θ ∧

−



s( j + 1)∆(jl++11,θ) + A

(l )

(θ )(β j −1τ j −1 + (1 − β j )τ j ) , if θ = s( j), l +1

(l )

(l )

(l )

s( j + 1)Ω j +1,θ + A (θ )( β j −1ν j −1 + (1 − β j )ν j ) + φ j Ψ j ,θ l Ω l j ,θ + q η j +1 Ψ j +1,θ

(l )

Ω j ,θ =

s( j ) − θ ( l +1 )



∧

(l +1)

−

∧

s( j + 1)Ω j +1,θ + A

( l +1 )

( l +1 )

, if θ ≠ s( j ),

( l +1 )

(θ )( β j −1ν j −1 + (1 − β j )ν j ) + φΨ j ,θ + q η j +1 Ψ j +1,θ l +1

, if θ = s( j ),

Setting θ = s(1) in (11.4) yields ωN (0) in terms of πN (0) as

ωN(0) = kπN(0),

where ∧



ν ( A(s(1))(1 − β1 ) − 1) + s(2)Ω2, s(1) + φ1Ψ1, s(1) + qη2 Ψ 2, s(1) k= 1 τ 1(1 − A(s(1)(1 − β1 )) − s(2)∆ 2, s(1) Lastly, (11.7) yields the remaining unknown πN(0) as



∑

= π N (0) λ (

N j =0

j +

∑

N j =1

(kτ j +ν j ))−1



162  Mathematics and Computer Science Volume 2 Theorem 11.1. The probabilities π −j (0 ≤ j ≤ N ) and ω −j (1 ≤ j ≤ N ) at pre-​ arrival epoch are determined as follows:

∑

= π j−  j (

− (kτ j ω = j

∑ (kτ +ν )) +ν )(∑  + ∑ (kτ N

j =0

N

j +

j =1

N

j

j =0

j

j

−1

N

j

j =1

j

,0 ≤ j ≤ N, +ν j ))−1 ,1 ≤ j ≤ N .



Proof. Using (11.8) and (11.13) in (11.9) to (11.12), we obtain the result of the theorem.

11.3.2 Arbitrary Epoch Probabilities In this subsection, the pre-arrival probabilities π −j and ω −j are used to derive the arbitrary epoch probabilities πj (0 ≤ j ≤ N ) and ωj (1 ≤ j ≤ N ). This is treated as a theorem given below. Theorem 11.2. The probabilities at the arbitrary epoch are obtained as

(λβ N −1 ) − π N −1 , r (N ) 1 = ((qη j+1 + jα )π j+1 + λ (β j−1π j−−1 − β jπ j− )), =j N − 1,. . .,1, πj r ( j) 1 (φN π N + λβ N −1ωN− −1 ), ωN = s( N ) 1 (s( j + 1)ω j+1 + φ jπ j + qη jπ j+1 + λ (β j−1π j−−1 − β jπ j− )), ωj = s( j ) =j N − 1,. . ., 2 1 (s(2)ω2 + φ1π 1 + qη2π 2 − λβ1π 1− ) ω1 = s(1)

πN =

π0 = 1−

N

∑(π j =1

j

+ ω j ).

Modelling and Analysis of a Congestion Dependent Queue  163 Proof. The intended outcome of the theorem is obtained by placing θ = 0 in Equations (11.2) to (11.6), utilizing (11.8) and the normalization condition.

11.4 Special Cases Case 1: Taking q = 1, α = 0, and βj = 1 for 1 ≤ j ≤ N − 1, the present model reduces to GI/M (n)/1/N queue with MWV and the findings are consistent with Goswami et al. [28]. Case 2: Setting q = 1, µj = µ, ηj = η, ϕj = ϕ ∀ j = 1, 2, . . . , N and setting α = 0, the current model is reduced to a GI/M/1/N/MWV queue with balking. Our findings are identical to Vijaya Laxmi and Jyothsna [19] in this situation. Case 3: Taking q = 1, the current model gets reduced to GI/M (n)/1/N/ MWV queue with balking and reneging and the outcomes are consistent with Vijaya Laxmi and Jyothsna [29]. Case 4: Taking µj = µ, ηj = η, ϕj = ϕ ∀ j = 1, 2, . . . , N brings down our model to a GI/M/1/N/MWV queue with balking, reneging, and Bernoulli scheduled VI. The results are consistent with Vijaya Laxmi et al. [25]. Case 5: Taking q = 1, µj = µ, ηj = η, ϕj = ϕ, βj = 1 ∀ j = 1, 2, . . . , N and α = 0 lowers the current model to GI/M/1/N/MWV queue. The results match with Banik et al. [18]. Case 6: Setting q = 1, µj = µ, ηj = 0, ϕj = ϕ ∀ j = 1, 2, . . . , N and taking α = 0, βj = 1 for 1 ≤ j ≤ N, our numerical findings correspond to GI/M/1/N queue with multiple vacations (Karaes-man and Gupta) [7]. Case 7: Taking q = 1, µj = µ, ηj = 0, ϕj → ∞ ∀ j = 1, 2, . . . , N and setting α = 0 the current model becomes a renewal input balking queue. The working vacation probabilities πj(1 ≤ j ≤ N) do not occur in this situation and our outcomes agree with Finch [3] (GI/M/1/N queue with balking).

11.5 Performance Metrics The performance metrics of the model can be assessed upon obtaining the state probabilities at pre- arrival and arbitrary instants. The expected system length (Ls), the blocking probability (Pl), and the expected waiting duration in the system (Ws) of a client employing Little’s Rule are, respectively, given by:

164  Mathematics and Computer Science Volume 2 N

Ls =

∑ j(π + ω ); P = π j

j

l

− j

+ ω −j ; Ws = Ls /λe ,

j =1

where λe = 1 − Pl represents the effective arrival rate. The mean rate of balking (br), the mean rate of reneging (rr), and the mean rate of client loss (lr) are measured by N

br =

N

∑ λ(1 − β )(π + ω ); rr = ∑ ( j − 1)α (π + ω ); lr = br + rr. j

j

j

j =1

j

j

j =1



11.6 Numerical Outcomes This section shows how the performance metrics respond to the changes in model parameters. The model parameters are selected at random as N = 10, λ = 1.5, α = 0.7, and q = 0.4 and the balking function is considered as βj = 1−j/N 2 with the assumption that β0 = 1 and βN = 0. The congestion dependent rates are assumed to be µj = 0.4j, ηj = 0.2j, and ϕj = 0.1j with averages µm = 2.2, ηm = 1.1, and ϕm = 0.55, respectively. For HE2 inter-­arrival time distribution, we have chosen σ1 = 0.149883, σ2 = 0.850117, λ1 = 0.484237, and λ2 = 2.380330. In Table 11.1, we have presented the values of Ls, Ws, Pl, br, rr, and lr for the following balking functions: βj = 1 − j/N2, βj = 1/(j + 1), and βj = e−j. The mean balking rate (br) is lowest for the balking function βj = 1 − j/N2, as Table 11.1  Numerical values of performance metrics for different balking functions. βj = 1 − j/N2

βj = 1/(j + 1)

βj = e−j

Ls

2.008288

1.343753

1.191490

Ws

1.338865

0.895836

0.794326

Pl

0.000004

0.000000

0.000000

br

0.030131

0.772889

0.940714

rr

0.754335

0.314381

0.214521

lr

0.784466

1.087270

1.155234

Modelling and Analysis of a Congestion Dependent Queue  165 noticed in the table, which in turn results in least mean rate of client loss (lr). This supports our decision to use βj = 1 − j/N 2 as the balking function. Figure 11.1 and Figure 11.2 show the influence of λ on the expected system length (Ls) and the mean rate of client loss (lr) for exponential inter-­ arrival time distribution, respectively, for different q. The images show that as λ is increased, both Ls and lr grow for any q. Furthermore, for a fixed λ, 2.6 2.4

q=0.0 q=0.4

2.2

q=0.8 q=1.0

2

Ls

1.8 1.6 1.4 1.2 1 0.8 0.5

1.5

1

2

λ

Figure 11.1  Influence of λ on Ls. 1.8 q=0.0 q=0.4 q=0.8 q=1.0

1.6 1.4

lr

1.2 1 0.8 0.6 0.4

1

2

1.5 λ

Figure 11.2  Effect of λ on lr.

2.5

166  Mathematics and Computer Science Volume 2 Ls and lr rise as q increases, meaning that Ls and lr are the least in VI queues when compared to queues with no vacation interruption. The impact of mean regular service rate (µm) on the expected system length (Ls) and on the expected waiting duration (Ws) are displayed in Figure 11.3 and Figure 11.4, respectively, for different mean vacation rates (ϕm), wherein the inter-arrival durations are considered to be Erlang-5 distributed. As expected, Ls and Ws deplete with the increase of µm for any ϕm. A similar effect is seen on Ls and Ws with the increase of ϕm for fixed µm. Figures 11.5 and 11.6 show the changes in lr and Ws, respectively, while the inter-arrival times are distributed according to deterministic distribution for different ηm and ϕm. It is worth noting that as ηm and ϕm grow, lr and Ws deplete. For different inter-arrival time distributions, Figure 11.7 displays the system capacity (N) versus the blocking probability (Pl). The blocking probability (Pl) is lowest when the inter-arrival durations are distributed according to deterministic distribution and highest when they are distributed according to HE2 distribution. When inter-arrival durations are exponentially distributed, the impact of the reneging rate (α) on the expected system length (Ls) and on the mean rate of client loss (lr) is depicted in Figure 11.8. As visualized in the graph, the expected system length falls as α grows, but the mean rate of client loss increases with α, as would be predicted. 2.25 φm = 0.55 φm = 0.82 φm = 1.10

2.2 2.15

Ls

2.1 2.05 2 1.95 1.9 1.85

1.7

1.8

Figure 11.3  Impact of µm on Ls.

1.9

2

2.1

2.2 µm

2.3

2.4

2.5

2.6

2.7

Modelling and Analysis of a Congestion Dependent Queue  167 1.46 φm=0.55 φm=0.82 φm=1.10

1.44 1.42 1.4

Ws

1.38 1.36 1.34 1.32 1.3 1.28 1.26 1.7

1.8

1.9

2

2.1

2.2

µm

2.3

2.4

2.5

2.6

2.7

Figure 11.4  Impact of µm on Ws.

0.81 φm = 0.55 φm = 0.82 φm = 1.10

0.8 0.79

lr

0.78 0.77 0.76 0.75 0.74 0.73

0.6

0.7

0.8

Figure 11.5  Change in lr with ηm.

0.9

1

1.1 ηm

1.2

1.3

1.4

1.5

1.6

168  Mathematics and Computer Science Volume 2 1.4 φm = 0.55

1.39

φm = 0.82 φm = 1.10

1.38

Ws

1.37 1.36 1.35 1.34 1.33 1.32

0.7

0.8

0.9

1

1.1 ηm

1.2

1.3

1.4

1.5

1.6

Figure 11.6  Change in Ws with ηm.

0.12 HE2 Exponential Erlang 3 Deterministic

0.1

Pl

0.08

0.06

0.04

0.02 0 4

5

6

7

8

9

10 N

Figure 11.7  N versus Pl.

11

12

13

14

15

Modelling and Analysis of a Congestion Dependent Queue  169 3.5 Ls Ir

3 2.5 2 1.5 1 0.5 0

0.5

1

α

1.5

2

2.5

Figure 11.8  α versus Ls and lr.

11.7 Conclusion The current research work examines a finite capacity renewal input congestion dependent queue with Bernoulli scheduled vacation interruption and client impatience. The regular service rates, working vacation service rates, and vacation rates are considered to be exponentially distributed and are congestion dependent. To obtain the stationary probabilities at various epochs, we have adopted supplementary variable and iterative techniques. We also presented a few numerical simulations to show the consequence of the model parameters on the key metrics, which are provided in table and graph forms. The current findings would be beneficial and significant for modelling the transportation business, manufacturing systems, medical administration, etc. The future direction of this work would be to expand the model under examination by incorporating state dependent inter-­ arrival times.

References 1. F. A. Haight, Queueing with balking (1957), Biometrika, vol.44, issue.3-4, pp.360-369, https://doi.org/10.1093/biomet/44.3-4.360

170  Mathematics and Computer Science Volume 2 2. F. A. Haight, Queueing with reneging (1959), Metrika, vol.2, isuue.1, pp.186197, https://doi.org/10.1007/BF02613734 3. Finch, P.D., Balking in the queueing system GI/M/1 (1959), Acta Mathematica Hungarica, vol.10, issue.1, pp.241-247, https://doi.org/10.1007/BF02063302 4. Kuo-Hsiung Wang, Yuh-Ching Liou and Dong-Yuh Yang, Cost optimization and sensitivity analysis of the machine repair problem with variable servers and balking (2011), Procedia–Social and Behavioral Sciences, vol.25, issue.14, pp.178-188, DOI: 10.1016/j.sbspro.2011.10.539 5. Amy R. Ward, Asymptotic analysis of queueing systems with reneging: A survey of results for FIFO, single class models, (2012) Surveys in Operations Research and Management Science, vol. 17, issue.1, pp.1-14, https://doi. org/10.1016/j.sorms.2011.08.002 6. Takagi, H. (1991). Queueing Analysis: A Foundation of Performance Evaluation, Vacation and Priority Systems, Vol. 1. Elsevier, Amsterdam. 7. Fikri Karaesmen, Surendra M. Gupta . The finite capacity GI/M/1 queue with server vacations (1996), Journal of the Operational Research Society, vol.47, issue.6, pp.817-828, https://doi.org/10.1057/jors.1996.101 8. Naishuo Tian, Zhe George Zhang (2006). Vacation Queueing Models: Theory and Applications. International Series in Operations Research and Management Science, Springer, New York. 9. Eitan Altman, Uri Yechiali, Analysis of customers’ impatience in queues with server vacations (2006), Queueing Systems, vol.52, issue.4, pp.261-279, https://doi.org/10.1007/s11134-006-6134- x 10. Dequan Yue, Wuyi Yue, Yanping Sun, ‘Performance analysis of an M/M/c/N queueing system with balking, reneging and synchronous vacations of partial servers’ in 2006 The Sixth Inter- national Symposium on Operations Research and Its Applications, Xinjiang, China, ORSC and APORC, 128-143. 11. Uri Yechiali, Queues with system disasters and impatient customers when system is down (2007), Queueing Systems, vol.56, pp.195-202, http://dx.doi. org/10.1007/s11134-007-9031-z 12. Eitan Altman, Uri Yechiali, Infinite-server queues with system’s additional task and impatient customers (2008), Probability in the Engineering and Information Sciences, vol.22, issue.4, pp.477-493, http://dx.doi.org/10.1017/ S0269964808000296 13. Nir Perel, Uri Yechiali, Queues with slow servers and impatient customers (2010), European Journal of Operational Research, vol.201, issue.1, pp.247258, doi:10.1016/j.ejor.2009.02.024 14. Dequan Yue, Wuyi Yue, Zsolt Saffer and Xiaohong Chen, Analysis of an M/M/1 queueing system with impatient customers and a variant of multiple vacation policy (2014), Journal of Industrial and Management Optimization, vol.10, issue.1, pp.89-112, http://dx.doi.org/10.3934/jimo.2014.10.89 15. Amina Angelika Bouchentouf, Abdelhak Guendouzi, Cost optimization analysis for an MX/M/c vacation queueing system with waiting servers and

Modelling and Analysis of a Congestion Dependent Queue  171 impatient customers (2019), SeMA Journal, vol.76, issue.2, pp.309-341, http://dx.doi.org/10.1007 16. M. I. G. Suranga Sampath, Jicheng Liu, Impact of customers’ impatience on an M/M/1 queueing system subject to differentiated vacations with a waiting server (2020), Quality Technology and Quantitative Management, vol.17, issue.2, pp.125-148, https://doi.org/10.1080/16843703.2018.1555877 17. L. D. Servi, S. G. Finn, M/M/1 queue with working vacations (M/M/1/ WV) (2002), Performance Evaluation, vol.50, issue.1, pp.41–52, https://doi. org/10.1016/S0166-5316(02)00057-3 18. A. D. Banik, U. C. Gupta, S. S. Pathak, On the GI/M/1/N queue with multiple working vacations - analytic analysis and computation (2007), Applied Mathematical Modelling, vol.31, issue.9, pp.1701-1710, https://doi. org/10.1016/j.apm.2006.05.010 19. P. Vijaya Laxmi, K. Jyothsna, Analysis of finite buffer renewal input queue with balking and multiple working vacations (2013), OPSEARCH, vol.50, issue.(4), pp. 548-565. http://dx.doi.org/10.1007/s12597-013-0123-8 20. Shakir Majid, Performance analysis of a Markovian queue with impatient customers and working vacation (2021), Journal of the Operations Research Society of China, https://doi.org/10.1007/s40305-021-00361-w. 21. Jihong Li, Naishuo Tian, The M/M/1 queue with working vacations and vacation interruption (2007), Journal of Systems Science and Systems Engineering, vol.16, pp.121-127, https://doi.org/10.1007/s11518-006-5030-6 22. Ji-HongLi, Nai-Shuo Tian, Zhan-You Ma, Performance analysis of GI/M/1 queue with working vacations and vacation interruption (2008), Applied Mathematical Modelling, vol.32, issue.12, pp.2715-2730, https://doi. org/10.1016/j.apm.2007.09.017 23. Shakir Majid and P. Manoharan, Analysis of an M/M/1 queue with working vacation and vacation interruption (2019), Applications and Applied Mathematics: An International Journal, vol.14, issue.1, pp.19-33, 24. Hongbo Zhang, Dinghua Shi, The M/M/1 queue with Bernoulli-schedulecontrolled vacation and vacation interruption (2009), International Journal of Information and Management Sciences, vol.20, issue.(4), pp.579-587. 25. Pikkala Vijaya Laxmi, Singuluri Indira and Kanithi Jyothsna, Ant colony optimization for optimum service times in a Bernoulli schedule vacation interruption queue with balking and reneging (2016), Journal of Industrial and Management Optimization, vol.12, issue.4, pp.1199–1214, http://dx.doi. org/10.3934/jimo.2016.12.1199 26. Bochentouf, A. A., Yahiaoui, L. (2017). On feedback queueing system with reneging and retention of reneged customers, multiple working vacations and Bernoulli schedule vacation interruption, Arabian Journal of Mathematics, 6(1), 1-11. 27. Shakir Majid, P. Manoharan and A. Ashok, Analysis of an M/M/1 queueing system with working vacation and impatient customers, American

172  Mathematics and Computer Science Volume 2 International Journal of Research in Science, Technology, Engineering & Mathematics (2019), Special Issue, pp.314-322, 28. V. Goswami, P. Vijaya Laxmi, K. Jyothsna, Analysis of GI/M (n)/1/N queue with state-dependent multiple working vacations (2013), OPSEARCH, vol.50, issue.1, pp.106-124, http://dx.doi.org/10.1007/s12597-012-0096-z 29. Pikkala Vijaya Laxmi and Kanithi Jyothsna, On renewal input state dependent working vacations queue with impatient customers (2015), International Journal of Mathematics in Operational Research, vol.7, issue.6, pp.661–680. 30. Doo Il Choi, Dae-Eun Lim, Analysis of the state-dependent queueing model and its application to battery swapping and charging stations (2020), Sustainability, vol.12(6), issue.2343, pp.1-15, https://doi.org/10.3390/su12062343

12 Resource Allocation Determines Alternate Cell Fate in Bistable Genetic Switch Priya Chakraborty and Sayantari Ghosh* Department of Physics, National Institute of Technology Durgapur, Durgapur, West Bengal, India

Abstract

Living cells need a constant availability of certain resources to have a sustained gene expression process. Limited availability of cellular resources for gene expression, like ribosomes and RNA Polymerase, significantly modifies the system dynamics. Factors like the variation in rate of binding or variation in efficiency of the recruited resource have the potential to affect crucial dynamic phenomena like cell fate determination. In this paper we have taken a very important motif, a bistable genetic toggle switch, and explored the effect of resource imbalance in this circuit in terms of the bifurcations taking place. We show that initial asymmetric biasing to resources via resource affinity or gene copy number significantly modifies the cell fate transition, both in pitchfork and saddle node type bifurcation. Our study establishes that in a limited resource environment, controlled resource allocation can be an important factor for robust functioning of the synthetic or cellular genetic switches. Keywords:  Gene regulation, resource allocation, cell fate decision, pitchfork bifurcation, saddle node bifurcation, genetic toggle switch

12.1 Introduction Proteins that govern all functionalities in a living cell are produced by two major steps: transcription and translation, which indeed are a combination of several intermediate processes. In order to understand intriguing cellular processes like cellular decision making or to operate a synthetic circuit inside the cell, a detailed mathematical study of cellular or synthetic gene *Corresponding author: [email protected] Sharmistha Ghosh, M. Niranjanamurthy, Krishanu Deyasi, Biswadip Basu Mallik, and Santanu Das (eds.) Mathematics and Computer Science Volume 2, (173–186) © 2023 Scrivener Publishing LLC

173

174  Mathematics and Computer Science Volume 2 regulatory dynamics is of prime concern. Extreme non-linearity inside cellular systems and the nature of coupling of one ongoing process with another makes this understanding of the dynamics extremely difficult. However, some re-occurring genetic pattern, called a motif, can be found in living organisms performing some typical tasks for the cell. Scientists are focusing largely to explore repeatedly occurring motifs to understand the cell dynamics as well as cellular decision making from the last few decades. Genetic toggle is one of the most extensively studied biological motifs [7, 16, 18], where two genes (lacI and tetR) mutually repress each other’s expression. Dynamics of this genetic motif are usually studied from the perspective of saddle-node bifurcation. Here, in the case of saddle node bifurcation of toggle, the system undergoes a transition to a bistable state from a monostable one and after a specific range of parameters, again, the system becomes monostable. Thus, a specific region of bistability separates two otherwise monostable regions in the phase plane. The system can also undergo a pitchfork bifurcation for some specific symmetry, as studied in some recent works [2]. In case of pitchfork type bifurcation, a monostable region converts to a bistable one at a transition point and remains bistable for rest. Bistabilty introduces some irreversibility in biological systems, that once the system attains its steady state, it retains its same steady state even on application of some output perturbations on a small scale. Thus, in the case of cell fate differentiation and cellular decision making, the genetic toggle motif is taken as a canonical circuit for understanding [1]. From its early invention as a synthetic circuit by Gardner et al. [7], different approaches for robust controlling of genetic toggle are proposed. Some of them involve real time feedback control [12], auto regulation, noise [19], the addition of some diffusible molecules like isopropyl-β-D-thiogalactopyranoside (IPTG), and anhydrotetracycline (aTc) to control the promoter activity as well. The search for a novel control parameter is still going on and in some recent studies it was established that the limited availability of cellular ingredients serving in the process of protein production can act as a robust parameter in cell dynamics. In the intermediate steps of protein production, majorly transcription and translation, the genes collect resources for successful completion of its expression from the cell. RNAP, transcription factor (TF), ribosome, degradation machinery, etc. are various resources that the cell supplies to the synthetic genetic circuit implemented in it or uses for its endogenous gene functionalities. It is experimentally verified that the cell does not contain these resources abundantly and depending upon the mode of operation, the availability of resources in different cells varies significantly. In the translation process, the ribosome is considered to be the most important

Alternate Cell Fate in Bistable Genetic Switch  175 resource the gene circuit collects from the cell. Though the presence of this essential cellular resource in all the prokaryotes and eukaryotes is a fact, the amount of free ribosome in different living beings is different. Even for different cells having different functionalities, they differ in availability of free ribosomes. Like the pancreatic cells, in eukaryotes, they are dedicated for most of the protein production over other cells and contain an unusually high number of ribosomes. On the contrary, the smooth endoplasmic reticulum (SER) does not have ribosomes on its surface and thus does not participate in protein production. This limited availability of the essential translational resource inside the cell significantly affects the ongoing dynamics. In a low protein activity or for a demand of low resource for the implemented synthetic construct, this limited availability may not affect things, but in higher protein activity or for a larger resource demand for the synthetic construct, unprecedented resource competition comes into the picture. Different experimental and theoretical studies establish that ribosome limitation, significantly modifying the circuit dynamics and at a larger scale the system chooses the favorable state. In a recent study, it is established that the protein production curve can be largely modified in terms of sensitivity and amplification by controlling ribosome availability and its distribution in the system [5]. Due to the nature of coupling of different ongoing processes, the competition in outer motifs, which seems unimportant in the study of motif interest, significantly modifies the dynamic behavior and can ruin the entire system, resulting in some emergent responses in output [4]. Not only limited to ribosomes, this competition can arise for RNAP, gene copy numbers, degradation machinery, and many more. Yuriy Mileiko and team in a recent study have shown that gene copy number variation brings a significant change in the dynamics of some well-known motifs [13]. The effect of decoy binding is also established by some recent papers [9, 11]. This trend of exploring the consequences of resource availability and distribution in cell dynamics is new and scientists are getting exciting novel mechanisms of system modelling from it. Cellular decision-making, environmental sensing, and cell to cell communication are three key processes underlying pattern formation and development in microscopic and complex organisms. From a theoretical point of view, though the cellular decision making seems to be reversible, in practice most of the time this is irreversible due to some secondary effects arising from the process. The biologically programmed cell death (apoptis) or cell death in response to injury (lysis) is most the promising reason here. Several approaches in determining cell fate like feedback-controlled regulation, cell size, and growth rate dependency [10, 17] are established in the recent past. The effect of noise, especially when a group of cells

176  Mathematics and Computer Science Volume 2 is participating, statistically predicts the most probable result, as shown in [1]. Though the presence of these various efforts of determining cell fate transition, the effect of limited availability of resource or asymmetry in resource affinity inside the cell and in the cell fate dynamics is not explored much. This asymmetry serving as initial biasness has the capacity to pre-pattern the cell fate. In this paper, we have taken a simple genetic toggle motif and considered, along with the mutual repression of each other, that both the genes are collecting resources (say ribosome here) from the same pool with different affinities. We also explore the condition for variability in gene copy numbers for the pitchfork type of bifurcation here. Some of the major findings of this paper are: • Variability in resource affinity as well as gene copy number introduces asymmetry in pitchfork bifurcation and prepatterns the cell fate. The greater the asymmetry, the more the system pre-patterns itself in determining an alternate cell fate. • Asymmetry in resource affinity regulates the range of bistability and thus robustness of the switch in saddle node bifurcation. • Total availability of resource regulates the point of bifurcation and region of interest in the system.

12.2 Model Formulation Let us consider X and Y are the two proteins repressing each others promoter activity, forming a toggle switch, as shown in Figure 12.1. The hammerhead symbols represent the repression here. We also consider both these proteins are collecting resources from the same pool for their expression. We particularly focus on the pool of ribosome here, the essential resource for the translation process, for the transcriptional complex to be translated as a protein product. Moving a step closer to cell dynamics, we consider that the ribosomes are distributed over small several cytoplasmic compartments inside the cell. Let T represent this local pool of resources available in the immediate vicinity of the toggle switch. Thus, our consideration of the two participants of the concerned toggle switch collects resource ribosomes from the pool T, which stands from its biological relevancy without any doubt. The available mRNA pool for translation is represented by gx and gy for protein X and Y, respectively.

Alternate Cell Fate in Bistable Genetic Switch  177

Y

X

resx

resy Resource T

(b)

Bistable region

Monostable region

Monostable region

Bifurcation parameter (c)

Steady states of protein

Steady states of protein

(a)

Bistable region

Monostable region

Bifurcation parameter (d)

Figure 12.1  Model Motif. (a) Genetic toggle, two proteins X and Y mutually repress each others promoter activity. Hammerhead symbol represents the repression here. Also, both the participants of the toggle collect resources from the same pool T with the affinities resx and resy respectively. (b) Schematic diagram of inter cellular competition determining cell fate. Strength of repression drives the cell to a particular fate. (c) Saddle node bifurcation in genetic toggle. A bistable region separates two monostable regions in phase space. (d) Pitchfork bifurcation in genetic toggle. Monostable region switches to a bistable region at bifurcation point.

The available mRNA pool gx and gy, collecting ribosomes from the pool T, with affinities resx and resy, makes a ribosome bound complex cx and cy which will be translated to protein X and Y at a rate of ∈x and ∈y, respectively. This asymmetry in resource allocation is very insightful here. The polycistronic mRNA pool in most of the bacterial organisms contains multiple ribosome binding sites (RBS) [3] and the rate of translation depends on the rate of recruitment of ribosomes to this RBS, as well as on the rate of translation initiation. The rate of ribosome recruitment also depends

178  Mathematics and Computer Science Volume 2 upon many factors. Including all these rates, we generalize the resource allocation or the protein production, which can have different rates as well. As mentioned, from the total pool T, the ribosome bound complexes are presented by cx and cy, thus further free pool of resource ribosome for translation is estimated by (T − cx − cy). The mutual repression is captured by a Michaelis–Menten type term in our model and the hill function co-operativity n is taken as 2. The ODE representing the scenario is given by Equation 12.1 below.



dc x res x (T − c x − c y ) g x − c x = dt dX c x ∈x −X = dt 1 + Y n dc y = res y (T − c x − c y ) g y − c y dt dY c y ∈y −Y = dt 1 + X n

(12.1)

In a steady state, all the rates of change are equal to 0 and we investigated the system.

12.3 Result Section 12.3.1 Pitchfork Bifurcation in Genetic Toggle Pitchfork bifurcation occurs at specific equilibrium with perfect symmetry conditions of the toggle system. For one variable dynamic system, several studies are present, while for a two variable dynamic system, a few studies in the recent past [2] investigated this phenomena in brief. For a conventional pitchfork model, we take the production rates of the two proteins to be equal, thus here we take ∈x, the production rate of X from its complex cx, and, ∈y, the rate of production of Y from its complex cy is equal.

12.3.1.1 Resource Affinity Regulates the Symmetry of Pitchfork Bifurcation Following the conventional way of pitchfork bifurcation in a toggle system, the protein production rates are equal and a detailed literature review

Alternate Cell Fate in Bistable Genetic Switch  179

7 6

10

Concentration of protein X

Concentration of protein X

strongly supports our consideration of taking different resource affinity values without any loss of generality. We find that the resource affinity value regulates the symmetry of pitchfork bifurcation significantly. We investigated the model for fixed values of n = 2, gx = gy = 5, T = 5, resy = 2, and changing ∈x so that for every point ∈x = ∈y for four different values of resx. When resx = resy, we find a beautiful symmetric pitchfork bifurcation in the output, while more interestingly asymmetry in resource affinity values destroys the symmetry in the output pitchfork with a significant impression. Starting with a lower resource affinity for X, resx < resy, there is a smooth transition of the pitchfork to a low state of the system, as shown in Figure 12.2a, while a higher state is only accessible for a very large perturbation in the system. While starting from a higher resource affinity, resx > resy for Figure 12.2d, the continuous accessible state is the higher production state, while the lower production states are only accessible for a large perturbation in the system. The same resource affinity of X and Y

rex = 1

5 4 3 2 1 1 3 4 2 Rate of X production (Єx)

8

rex = 1.8

6 4 2 1 3 4 2 Rate of X production (Єx)

5

(a)

(b) 14

rex = 2

Concentration of protein X

Concentration of protein X

12 10

5

12

rex = 3

10

8 6 4 2 1

3

2

4

5

8 6 4 2 1

3

2

4

Rate of X production (Єx)

Rate of X production (Єx)

(c)

(d)

5

Figure 12.2  Resource Affinity Regulates Symmetry of Pitchfork Bifurcation. Concentration of protein X wrt the rate of X production ∈x plot. n = 2, resy = 2, gx = gy = 5, T = 5, for all the plots. Resource affinity for X production, i.e., resx =1 for (a), resx=1.8 for (b), resx = 2 for (c), and resx = 3 for (d).

180  Mathematics and Computer Science Volume 2 (resx = resy = 2 for Figure 12.2c) results in a symmetric pitchfork. Also, it is interesting to note that the larger the asymmetry, the higher the stability of the chosen transitioned state and the lower the chances for its transition to another steady state even when perturbation is present from the outside (comparing Figure 12.2a and Figure 12.2d with Figure 12.2c). So, the results indicate to the conclusion that an initial asymmetry in resource allocation pre-patterns the cells to a higher production regime or in a lower production regime and determines the cell fate.

12.3.1.2 Availability of Total mRNA Pool Regulates the Symmetry of Pitchfork Bifurcation We find similar results with respect to the gene copy number available for translation for a particular protein. The number of copies of a particular gene present in the genotype is usually called the gene copy number. A symmetry in the presence of gene copy number with symmetry in other system parameters shows a perfect pitchfork in the output, while asymmetry in the initial condition of gene copy number pre-patterns the system to a higher or lower production state, as shown in Figure 12.3. It is interesting to note that the greater the asymmetry in the initial gene copy number, the greater the perturbation the system demands to transit from its continuous steady state to the other.

12.3.1.3 Total Resource Availability Regulates the Point of Bifurcation in the System We find that total resource availability significantly regulates the bifurcation point of the system, as shown in Figure 12.4. We investigate the model for two fixed values of T, increasing the total resource available for translation and the position of bifurcation comes to a lower value of input signal. Also, the range of steady states drastically changes.

12.3.2 Saddle Node Bifurcation in Genetic Toggle The genetic toggle, most conventionally known as the genetic toggle switch, is biologically most important for its on/off switch like behavior which plays a significant role in determining cell fate. From its early invention, researchers are deliberately searching for the ways of robust controlling of the toggle switch. We find beautiful control on the toggle switch by controlling resource distribution.

6

Concentration of protein X

Concentration of protein X

Alternate Cell Fate in Bistable Genetic Switch  181

gx = 3

5 4 3 2 1 1 2 3 Rate of X production (Єx)

8

gx = 4.5

6 4 2 1 2 3 Rate of X production (Єx)

4

(b)

10

Concentration of protein X

Concentration of protein X

(a)

4

8 gx = 5 6 4 2 1 2 3 Rate of X production (Єx)

8

gx = 6

6 4 2 1 2 3 Rate of X production (Єx)

4

(c)

4

(d)

7 6 5

Concentration of protein X

Concentration of protein X

Figure 12.3  Availability of Total mRNA Pool Regulates Symmetry of Pitchfork Bifurcation and Cell Fate Transition. Concentration of protein X vs. rate of X production ∈x plot. resx = resy = 2, n = 2, T = 5, gy = 5 is fixed for all the plots and ∈x = ∈y. Gene copy numbers available for production of X, i.e., gx =3 for (a), gx = 4.5 for (b), gx = 5 for (c), and gx = 6 for (d).

T=5

4 3 2 1 0.5 1.0 1.5 2.0 2.5 Rate of X production (Єx) (a)

3.0

14 12

T=10

10 8 6 4 2 0.5

1.0 1.5 2.0 2.5 Rate of X production (Єx)

3.0

(b)

Figure 12.4  Total Resource T Availability Regulates Point of Pitchfork Bifurcation in Genetic Toggle. Concentration of protein X vs. rate of X production ∈x plot. n = 2, gx = gy = 5, ∈x = ∈y, resx = resy = 2 for both the plots. T = 5 for (a) plot and T = 10 for (b) plot.

182  Mathematics and Computer Science Volume 2

12.3.2.1 Resource Distribution Regulates the Point of Bifurcation in Toggle Switch We find the resource distribution significantly regulates the point of bifurcation in the saddle node of genetic toggle as well, as shown in Figure 12.5. We investigate the system for 3 different values of resx, keeping all other parameters fixed at resy = 2, n = 2, gx = gy = 5, T = 5, and ∈y = 2 and plot the concentration of protein X with respect to the activator of X production ∈x. Considering the blue line (continuous and dashed) primarily, which shows the scenario when resx = resy = 2 and that resource allocation for X to Y is the same, we find a change in resx shifts the curve left to the green curve (or right to the red curve) for a higher affinity for resources to X than Y, resx = 3, > resy = 2 (for a lower affinity for resource to X than Y, resx = 1, < resy = 2).

12.3.2.2 Region of Interest in Toggle Switch is Significantly Regulated by Resource Allocation For a saddle node bifurcation, the most interesting region is the range of input signal for which the output protein concentration attains two drastically different concentrations depending upon the mode of forward or backward operation. When investigated in a bifurcation diagram, a set of stable equilibrium points is separated by unstable equilibrium points, the system cannot achieve physically. From Figure 12.5, we also get that the range of interest increases (or decreases) for X with a lower resource allocation resx = 1 < resy = 2 (for X getting higher resource than Y, resx = 1.6 1.5

Range of bistable region (Єx)

Concentration of protein X

10

1.4 1.3

8

1.2

resx = 1 resx = 2 resx = 3

6 4

1.1 1 0.9

2 0 0

2 4 6 Rate of X production (Єx) (a)

8

0.8 0.7

5 5.5 6 6.5 7 7.5 8 8.5 9 5.5 10 Available Total resource T (b)

Figure 12.5  (a). Rate of X production ∈x vs. concentration of protein X plot in case of saddle node bifurcation in genetic toggle. resy = 2, gx = gy = 5, T = 5, n = 2, ∈y = 2. (b). Phase plot for range of bistable region ∈x, the rate of X production from its complex vs. total resource availability T. resx = resy = 1, n = 2, gx = gy = 5, ∈y = 1.

Alternate Cell Fate in Bistable Genetic Switch  183 3 > resy =2). It is interesting to note that lower resource availability to X than Y stabilises the bifurcation curve for larger fluctuation. This is not only giving us an opportunity for robust controlling of the system, but also signifies that initial biasing of the system towards the resource significantly modifies cell fate in terms of stability.

12.3.2.3 Total Resource Availability T Regulates Saddle Node Bifurcation Curve Along with the variation in resource affinity, the availability of the total nutrients here significantly regulates the saddle node bifurcation curve, as shown in Figure 12.5b. The regulation is quite positive here though. Greater resources stabilize the system for a larger range of bistability and greater switch robustness, while with less availability of resources the switch response is not stable and the steady states can alter even for low fluctuation in the system.

12.4 Conclusion Cellular decision making is a fundamental biological phenomenon by which a cell opts the different states prior to environmental conditions, leading to asymmetric cell differentiation. The underlying reason behind this is still not entirely explored. We take a simple genetic motif, genetic toggle here, and show that resource affinity asymmetry of the toggle participants, both for the saddle node and in pitchfork bifurcation, significantly biases the cell fate. This mutually repressing motif is very common in nature [6, 14], where in output the system shows patterning by choosing one cell fate over another. The availability of total resource pool significantly regulates the bifurcation point in the motif. In the case of a synthetic circuit, this resource limitation is very true because the gene circuit implemented in the host entirely depends upon the host’s resource for its expression and in the case of a cell, the limitation in cellular resource ribosome is a major factor, indicating our findings to be true also. We also investigated the effect of gene copy number in the case of pitchfork bifurcation, indicating an initial asymmetry that biases the cell fate to lower or higher production states accordingly. Here, it is important to note that our entire consideration is valid for a low growth state of the system. The effect of growth rate on cell dynamics is well established [17]. Overexpression of endogenous genes, or adding

184  Mathematics and Computer Science Volume 2 some synthetic construct in the cell, destabilizes the resource distribution and makes the growth rate smaller, while growth causing dilution enhances protein degradation. So, some researchers point out that growth is a significantly regulatory parameter in every cellular phenomenon, but some experimental results also pointed out that these effects only depend upon experimental conditions, causing some momentary changes in dynamics. Our study mostly follows the experimental situation [8, 15] when the growth rate is low and competition effect is significant. A perfect noise free environment in the cell is impracticable. For a single cell, though the consideration does not violate the reality, working with a group of cells, the predictability can vary significantly. The addition of noise in existing dynamics will give a result close to reality. Also, the limitation of other cellular nutrients in the way of gene expression can regulate the alternate protein production, stabilizing one state over other, regulating cell fate. In the future, we would like to extend our work for a complete scenario including transcriptional, translational, and degradation machinery competition in a noisy cell environment.

Acknowledgement PC and SG acknowledge the support by DST-INSPIRE, India, vide sanction Letter No. DST/INSPIRE/04/2017/002765 dated- 13.03.2019.

References 1. G. Balázsi, A. van Oudenaarden, and J. J. Collins. Cellular decision making and biological noise: from microbes to mammals. Cell, 144(6):910–925, 2011. 2. I. Bose and S. Ghosh. Bifurcation and criticality. Journal of Statistical Mechanics: Theory and Experiment, 2019(4):043403, 2019. 3. D. H. Burkhardt, S. Rouskin, Y. Zhang, G.-W. Li, J. S. Weissman, and C. A. Gross. Operon mrnas are organized into orf-centric structures that predict translation efficiency. Elife, 6:e22037, 2017. 4. P. Chakraborty and S. Ghosh. Emergent correlations in gene expression dynamics as foot-prints of resource competition. The European Physical Journal E, 44(10):1–12, 2021. 5. P. Chakraborty and S. Ghosh. Emergent regulatory response and shift of half induction point under resource competition in genetic circuits. arXiv preprint arXiv:2112.04985, 2021.

Alternate Cell Fate in Bistable Genetic Switch  185 6. E. Clark. Dynamic patterning by the drosophila pair-rule network reconciles long-germ and short-germ segmentation. PLoS biology, 15(9):e2002439, 2017. 7. T. S. Gardner, C. R. Cantor, and J. J. Collins. Construction of a genetic toggle switch in escherichia coli. Nature, 403(6767):339–342, 2000. 8. A. Gyorgy, J. I. Jiménez, J. Yazbek, H.-H. Huang, H. Chung, R. Weiss, and D. Del Vecchio. Isocost lines describe the cellular economy of genetic circuits. Biophysical journal, 109(3):639–646, 2015. 9. S. Jayanthi, K. S. Nilgiriwala, and D. Del Vecchio. Retroactivity controls the temporal dynamics of gene transcription. ACS synthetic biology, 2(8):431– 441, 2013. 10. S. Klumpp, Z. Zhang, and T. Hwa. Growth rate-dependent global effects on gene expression in bacteria. Cell, 139(7):1366–1375, 2009. 11. T.-H. Lee and N. Maheshri. A regulatory role for repeated decoy transcription factor binding sites in target gene expression. Molecular systems biology, 8(1):576, 2012. 12. J.-B. Lugagne, S. S. Carrillo, M. Kirch, A. Köhler, G. Batt, and P. Hersen. Balancing a genetic toggle switch by real-time feedback control and periodic forcing. Nature commu- nications, 8(1):1–8, 2017. 13. Y. Mileyko, R. I. Joh, and J. S. Weitz. Small-scale copy number variation and large- scale changes in gene expression. Proceedings of the National Academy of Sciences, 105(43):16659–16664, 2008. 14. Y. Saka and J. C. Smith. A mechanism for the sharp transition of morphogen gradient interpretation in xenopus. BMC developmental biology, 7(1):1–9, 2007. 15. I. Shachrai, A. Zaslaver, U. Alon, and E. Dekel. Cost of unneeded proteins in e. coli is reduced after several generations in exponential growth. Molecular cell, 38(5):758–767, 2010. 16. M. Strasser, F. J. Theis, and C. Marr. Stability and multiattractor dynamics of a toggle switch based on a two-stage model of stochastic gene expression. Biophysical journal, 102(1):19–29, 2012. 17. C. Tan, P. Marguet, and L. You. Emergent bistability by a growth-modulating positive feedback circuit. Nature chemical biology, 5(11):842–848, 2009. 18. T. Tian and K. Burrage. Bistability and switching in the lysis/lysogeny genetic regulatory network of bacteriophage λ. Journal of Theoretical Biology, 227(2):229–237, 2004. 19. J. Wang, J. Zhang, Z. Yuan, and T. Zhou. Noise-induced switches in network systems of the genetic toggle switch. BMC systems biology, 1(1):1–14, 2007.

13 A Hybrid Approach to Ontology Evaluation Aastha Mishra* and Preetvanti Singh Department of Physics & Computer Science, Dayalbagh Education Institute, Agra, Uttar Pradesh, India

Abstract

In last few decades, researchers have been motivated to facilitate ontology in several fields. In medical science, ontology is used to describe the theory of medical vocabularies and the correlation shared among them, thus permitting the sharing of medical knowledge. Definition based on ontology of disease allows each class of disease to be classified in a formalized structure singularly and along with the discussion of ontological realism for the treatment and diagnosis of disease. This paper uses a hybrid approach to evaluate ontology for epilepsy disease. A multi-criteria decision making (MCDM) method is used to decide the best ontology provided by Bio-portal to select the best suitable characteristics for epilepsy disease. Keywords:  Ontology, ontology evaluation, AHP, epilepsy, MCDM

13.1 Introduction Nowadays, the single resource most critical for top management in an organization to sustain competitive advantage is knowledge. An organization can have competitive edge over its competitors by building an excellent process to manage knowledge. Ontology is a static knowledge representation method that includes definitions of basic concepts in a domain and relations among them which are machine interpretable. The information retrieval quality is improved from a keyword-based retrieval to knowledge-based search. Users should have a way for assessing ontologies and deciding which one fits their requirements to best face a multitude *Corresponding author: [email protected] Sharmistha Ghosh, M. Niranjanamurthy, Krishanu Deyasi, Biswadip Basu Mallik, and Santanu Das (eds.) Mathematics and Computer Science Volume 2, (187–204) © 2023 Scrivener Publishing LLC

187

188  Mathematics and Computer Science Volume 2 of ontologies. The main benefit of developing an ontology is that it enables sharing common understanding of the information structure and efficient reuse of domain knowledge. Ontologies are an elementary data structure for knowledge conceptualization, but many ontologies are built for conceptualizing the same body of knowledge. Ontology selection is thus an important issue that must be addressed to access these and decide which one fits the requirements of the user best. Evaluating characteristics of ontologies has also become necessary because of the increasing number of candidates for reuse in a domain and complexities of ontologies. Various approaches for ontology selection have been considered in literature [6, 7, 9, 11, 20, 23]. A multi-criteria decision-making approach to ontology selection deals with the problem of selecting the ontology best suited to the needs of the decision maker. Multi-criteria decision making  (MCDM) assigns the  selection of the choice of a best alternative from several available options in a  decision, subject to several  vague or concrete criteria  or  attributes. It helps people make decisions according to their choices in cases where more than one conflicting criterion exists. Analytical Hierarchy Process (AHP), an MCDM process, is a structured decision-making tool for quantifying the weights of decision criteria. Experience of experts is utilized to estimate the relative magnitude of parameters through pairwise comparisons. This paper presents a multi-criteria decision making approach based method to evaluate ontologies and select the best ontology in the medical domain.

13.2 Background Methods have been developed to evaluate and select the best ontology. [16] explored the gain utilization of Semantic Web technologies in the domain of recruitment and developed an ontology based recruitment process. [17] used ontology evaluation techniques for agent cooperation to measure the quality of ontology. A criteria selection framework was proposed by [10] for guiding the selection of suitable criteria for various levels of ontology evaluation. [5] presented a method for ontology evaluation based on the goal, question, and metric approach for empirical evaluation. [15] proposed a method to integrate ontologies to select and recommend adapted internship seekers. [22] presented metrics, approaches, and other similar aspects of ontology evaluation in a concise manner. [12] presented a scalable data-driven framework for ontology evaluation, targeting Big Data scenarios and use cases. [21] presented the outstanding

A Hybrid Approach to Ontology Evaluation  189 contribution to ontology evaluation by considering social and community related themes. [2] organized four-categorical schemes to evaluate ontology in the existing literature. 200 ontology samples are considered, which are taken from the National Center for Biomedical Ontology (NCBO) Bio Portal. [4] evaluated the U Ontology by ontology evaluation methodology to evaluate the quality of the developed ontology. [11] discussed already present evaluation metrics that support the ontology evaluation process for offering guidance useful to knowledge handling, representation, and conceptualization. [13] explored different metrics used for the evaluation of quality of ontology from different dimensions for ontology evaluation. An approach to evaluate quality of reused parts was proposed by [24]. The model represented evaluation information with semantic properties. [1] presented OnToology, a web-based application to manage ontology engineering support activities. [3] presented the 5 ontologist’s results, revealing high system usability of OntoKeeper and use-cases. [8] evaluated a set of consistent and objective ontology structural metrics. Ontology repositories used for evaluation have been used as corpora. [14] developed a decision support system for manufacturing process selection based on ontology-enabled case-based reasoning. From the review, it was observed that MCDM techniques are used to provide an efficient way to evaluate and select ontologies. This paper develops a method for evaluating and selecting ontologies in the medical domain based on the characteristics of a disease.

13.3 The Developed OntoEva Method Ontology selection is the method of identifying the ontologies or ontology models that best suit the requirements of a decision maker. The precondition of ontology selection lies in evaluating all considered ontologies on the basis of certain criteria. The same body of knowledge is conceptualized by different ontologies, so the method developed in this paper will enable a user to select the one that best suits the requirements of the decision maker. The OntoEva method was developed to evaluate and select the ontologies using AHP. The use of AHP will enable a decision maker in evaluating the ontologies based one some prioritized criteria. The proposed method determines the most suitable ontology by considering a set of characteristics and evaluation criteria. Weights are generated for each evaluation criteria according to pairwise comparison of the criteria and the criterion with the highest weight is considered as the most important criteria. Computation of the weights is guided by the experience of decision makers. OntoEva

190  Mathematics and Computer Science Volume 2 first identifies the characteristics of ontologies for a domain and computes weights for each criterion. The class mapping is then observed for prioritized factors, which helps in determining the best ontology for the domain. The methodology steps are as follows: Step 1: Identify the information need. Step 1.1: Identify the domain and the information need. Step 1.2: Identify ontology structures that satisfy the information need. Step 2: Specify the selection criteria. These criteria are similar to ontology popularity, topic coverage, or ontology structure. Step 3: Identify n characteristics of ontology for a domain. Step 4: Categorize these characteristics in m clusters. Step 5: Construct a pairwise comparison matrix for each of the m clusters using the 9-point scale [18] and calculate (Inw) importance weights for the characteristics (criteria). Table 13.1 shows the 9-point scale is change to AHP pairwise com­parison table. Table 13.1  9-point scale [18]. Value of ajk

Interpretation

1

Same importance

3

Average importance of one over another

5

Important or strong importance

7

Extreme importance

9

Very extreme importance

2,4,6,8

Middle values between the two adjacent judgments

Step 6: For the prioritized characteristics in each cluster, perform class mappings. Step 7: Based on class mapping, compute importance weights for the considered ontologies.

13.4 Ontology Selection for Epilepsy Disorder Epilepsy is a neurological disorder that affects people of all ages. According to [19], of the total population approximately 2% suffers from epilepsy, which ranks it as the second most common neurologic disorder. A person

A Hybrid Approach to Ontology Evaluation  191 with epilepsy experiences symptoms like sensations, irritability, headache, depression, ‘funny feeling’, abnormal behavior, confusion, and sometimes loss of consciousness. The main reason of the suffering is lack of awareness about the disease. Thus, it becomes necessary that a commoner is provided a method to efficiently diagnose the disease at early stages. The three ontology structures considered in this study for this goal are: • Extended Syndromic Surveillance Ontology (ESSO): To facilitate the mining of free-text clinical documents, an opensource terminological ontology is designed in English. It consists of epilepsy syndromes, seizure types, and data elements associated with them. • Epilepsy and Seizure Ontology (EPSO): To support epilepsy focused informatics, tools this ontology are developed for patient care and clinical research. • Epilepsy Ontology (EPILONT): It is ontology about the epilepsy domain and epileptic seizure based on the diagnosis proposed by the ILAE (International League Against Epilepsy). The fundamental properties considered for the selection process are as follows.

13.4.1 Accuracy To obtain higher accuracy, correct definitions and descriptions of classes, properties, and individuals which clearly define the domain are required, i.e., the considered ontologies should specify epilepsy disease.

13.4.2 Adaptability Adaptability defines how long the ontology predicts its uses. Ontologies for epilepsy are originally designed to describe the criteria on epilepsy disease and its vocabulary also allows formalizing symptoms of all kinds and differentiating between diseases.

13.4.3 Clarity Clarity calculates how productively the ontology can communicate with the deliberate meaning of the given terms. The name of the ontology should clearly explain the content and its function.

192  Mathematics and Computer Science Volume 2

13.4.4 Completeness For completeness, the domain of interest and the thickness and richness of the ontology should be properly covered. In order to identify the disorder, the list of all relevant characteristics is provided by the three ontologies.

13.4.5 Conciseness Conciseness is the evaluation criteria that states if the ontology includes irrelevant elements regarding to the domain to be covered. For example, ontology about epilepsy disease may take an important view on what the disease actually is. It is not important to state if a person suffering from other disease and any related information about that.

13.4.6 Consistency Consistency explains that the ontology should not involve or allow for any discrepancies, for example: Confused being Confused Memory is the one of the symptoms of epilepsy, but having a logical axiom, for example: calling confused a mental state will contradict the statement.

13.4.7 Organizational Fitness Organizational fitness involves various factors deciding the ease of how ontology can be deployed within an organization. A hospital may decide that all used ontologies align to the Bio-portal. This will help the organization in reducing costs when integrating data from different sources to align the ontologies. The three ontologies, EPSO, EPILONT, and ESSO, fulfill all the above mentioned properties and are suitable for selecting the best ontology for epilepsy disease. After selecting the ontologies, the topic coverage of these ontologies was analyzed from the Bio-portal, as given in Tables 13.2, 13.3, and 13.4. The characteristics analyzed were the ontology metric classes, properties, individuals, maximum depth, and maximum number of children. In order to evaluate ontologies for epilepsy disease, the characteristics of epilepsy are selected after having discussions with doctors. Considering their opinions, the selected criteria are clustered under m=4 groups, as shown in Table 13.5, and were decomposed into hierarchy as illustrated in Figure 13.1.

A Hybrid Approach to Ontology Evaluation  193 Table 13.2  EPSO ontology. Classes

1357

Individuals

2

Properties

29

Maximum depth

17

Maximum number of children

146

Table 13.3  EPILONT ontology. Classes

138

Individuals

0

Properties

10

Maximum depth

4

Maximum number of children

28

Table 13.4  ESSO ontology. Classes

2705

Individuals

0

Properties

166

Maximum depth

12

Maximum number of children

214

For each of the clusters, a pairwise comparison matrix is developed (Table 13.6). These matrices were consistent, as the CI and CR values were < 0.1. The computed weight for each characteristic is given in Table 13.7. From the computed weights it can be seen that the focal type (A3) is the highest rated characteristic, AED (B2) is the highest weighted treatment, Head Trauma (C1) is the highest rated cause of epilepsy, and Confused Memory (D2) is the most noticeable symptom due to the highest weight value.

194  Mathematics and Computer Science Volume 2 Table 13.5  Selected criteria. Cluster

Description

Criteria

[A] Types

Epilepsy is diagnosed in people when they have two or more seizures.

A1 – Absence A2 – Atomic A3 – Focal A4 – Generalized A5 – Tonic

[B] Treatment

The goal of treatment in patients suffering from epileptic seizures is to achieve a seizure-free status without any side effects.

B1 – Ketonic Diet B2 – AED B3 – Physical Exercise B4 – Surgery B5 – Nerve Stimulation

[C] Causes

The causes can be complex and sometimes hard to identify.

C1 – Head Trauma C2 – Brain Condition C3 – Prenatal Injury C4 – Alcohol Consumption C5 – Genetics

[D] Symptoms

Symptoms differ from person to person and according to the type of seizure.

D1 – High Fever D2 – Confused Memory D3 – Fainting D4 – Narcolepsy D5 – Cataplexy D6 – Panic Attack D7 – Breathing Difficulty

EPILEPSY

Treatment

Types

A1

A2

A3

A4

A5

B1

EPSO

B2

B3

B4

Causes

B5

C1

ESSO

Figure 13.1  Hierarchy for epilepsy disorder.

C2

C3

Symptoms

C4

C5

D1

EPILONT

D2

D3

D4

D5

D6

D7

A Hybrid Approach to Ontology Evaluation  195 Table 13.6  Pairwise comparison matrix for clusters. Cluster: Type A1

A2

A3

A4

A5

A1

1.000

0.333

0.200

0.333

0.333

A2

3.000

1.000

0.250

0.333

0.500

A3

5.000

4.000

1.000

4.000

4.000

A4

3.000

3.000

0.250

1.000

0.500

A5

3.000

2.000

0.250

2.000

1.000

Cluster: Treatment B1

B2

B3

B4

B5

B1

1.000

0.250

3.000

0.500

0.500

B2

4.000

1.000

3.000

2.000

2.000

B3

0.333

0.333

1.000

0.500

0.333

B4

2.000

0.500

2.000

1.000

2.000

B5

2.000

0.500

3.000

0.500

1.000

C1

C2

C3

C4

C5

C1

1.000

2.000

2.000

3.000

5.000

C2

0.500

1.000

2.000

2.000

3.000

C3

0.500

0.500

2.000

2.000

3.000

C4

0.333

0.500

1.000

1.000

2.000

C5

0.200

0.333

0.500

0.500

1.000

Cluster: Causes

Cluster: Symptoms D1

D2

D3

D4

D5

D6

D7

D1

1.000

0.333

0.500

2.000

3.000

4.000

2.000

D2

3.000

1.000

2.000

2.000

3.000

4.000

5.000

D3

2.000

0.500

1.000

3.000

2.000

4.000

3.000 (Continued)

196  Mathematics and Computer Science Volume 2 Table 13.6  Pairwise comparison matrix for clusters. (Continued) D4

0.500

0.500

0.333

1.000

0.500

3.000

2.000

D5

0.333

0.333

0.500

2.000

1.000

3.000

2.000

D6

0.250

0.250

0.250

0.333

0.333

1.000

3.000

D7

0.500

0.200

0.333

0.500

0.500

0.333

1.000

After identifying the prioritized characteristics for each cluster, each of these characteristics was searched in each of the three ontologies. For Cluster I, Type, Focal was present in the class mapping. These diagrams illustrate the presence of characteristic class in each of the ontologies. EPSO Ontology (7-mapping) Continuant Dependent Continuant Diease Epilepsy Focal Epilepsy

ESSO Ontology (16-mapping) Classification_System Blume_2001_ILAE_Glossary Blume_2001_I_General_Terms Blume_2001_I_5.0_Focal Focal

A Hybrid Approach to Ontology Evaluation  197 Table 13.7  Computed weights. Cluster

Criteria

Inw

A

A1

0.058

A2

0.103

A3

0.488

A4

0.162

A5

0.191

B1

0.128

B2

0.373

B3

0.081

B4

0.230

B5

0.188

C1

0.382

C2

0.246

C3

0.185

C4

0.118

C5

0.068

D1

0.166

D2

0.298

D3

0.213

D4

0.098

D5

0.113

D6

0.060

D7

0.052

B

C

D

198  Mathematics and Computer Science Volume 2 EPILONT Ontology (0-mapping) Seizure Types Crises Continuas Focal Status Epilepticus

According to above data, the focal type of epilepsy exists in all the three ontologies. The focal type of epilepsy is the most prominent type among all the epilepsy types. Most patients suffering from epilepsy suffer from this type of the disease. In EPSO type presents with 7-class mapping, ESSO presents a focal class with 16-class mapping, and EPILONT presents with 0-mappping. Considering the above data, pairwise comparison matrixes are generated and Ontology Weights (Ow) are computed using the Eigenvector approach.

EPSO EPILONT

EPSO

EPILONT

ESSO

1.000

5.000

0.333

1.000

0.143

ESSO

1.000

The results are consistent, as the values of CI and CR were less than 0.1. The same procedure is performed for all other selected criteria. For Cluster II, Treatment, the class mapping of each of the considered Epilepsy Ontology was observed. The Treatment AED is present in class mapping. EPILONT (0 – mapping) Sindromes Epilepticos Crises que nao obrigam neccessariamente ao diagnostic de Epilepsia Drug or other chemically - induced seizures

A Hybrid Approach to Ontology Evaluation  199 EPSO (7- mapping) Continuant Independent Continuant Clinical Drug Component DrugBrandName

ESSO (1- mapping) Treatment Medication Medication_Name

In EPSO, the AED class is present as a clinical drug component and its subclass drug brand name with 7-class mapping, ESSO consist of Medication_ Name class with 1-class mapping, and EPILONT consists of 0-mappping.

EPSO

EPSO

EPILONT

ESSO

1.000

5.000

3.000

1.000

0.500

EPILONT ESSO

1.000

For Cluster III, Causes, Head Trauma is present in class mapping as follows: EPILONT – not found EPSO (18 – mapping) Continuant Dependent Continuant Etiology Structural Cause Selerosis Traumatic Brain Injury

200  Mathematics and Computer Science Volume 2 ESSO (20 – mapping) Hidden Terms Buchhalter Meeting Head Injury Traumatic Brain Injury

In EPSO Ontology, Traumatic Brain Injury class is present with 20-class mapping, ESSO consists of the same class name with 18-class mapping, and in EPILONT it is not present.

EPSO

EPSO

ESSO

1.000

1.000

ESSO

1.000

For Cluster IV Symptoms, Confused Memory is present in class mapping: EPILONT (71- mapping) General Concepts Syncope

Oncology mapping

Symptom Ontology Neurological and Physiological Symptom Confusion

EPSO (0-mapping) Occurrent Processual Entity Bodily Process Physical Process Physical Pathological Process Seizure Feature Seizure Consciousness State

A Hybrid Approach to Ontology Evaluation  201 ESSO – Not Found In EPSO Ontology, class seizure consciousness state is present with 0-class mapping, ESSO does not have that class, and in EPILONT it consists of a Confusion class with 71-class mapping.

EPSO

EPSO

EPILONT

1.000

0.250

EPILONT

1.000

13.5 Results On the basis of above analysis, the computed weights are shown in Table 13.8. Finally, the average weights of each of the 3 ontologies are computed:



EPSO = 0.279 + 0.648 + 0.500+0.200 = 1.627/ 4 = 0.406



EPILONT = 0.072+0.122+ 0.500+0.800 =1.494/4 = 0.373



ESSO = 0.649 + 0.230+0+0 = 0.879/4 = 0.219

It can be seen that EPSO is the suitable ontology for diagnosing epilepsy disease.

13.6 Comparison of Ontologies When these ontologies are compared from the Bio-portal platform on the basis of ontology metrics like classes, properties, individuals, maximum depth, and maximum number of children, the most suited ontology is ESSO. However, from the computed weights, as shown in Table 13.8, EPSO was considered as the best ontology for epilepsy disease. The result was accepted by neurologists, as the EPSO provides all the required information a patient needs and it contains all the classes that best defined the epilepsy disease.

202  Mathematics and Computer Science Volume 2 Table 13.8  Computed ontology importance weights. Types – Focal Ontology

OW

EPSO

0.279

EPILONT

0.072

ESSO

0.649

Treatment – AED Ontology

OW

EPSO

0.648

EPILONT

0.122

ESSO

0.230

Causes – Head Trauma Ontology

OW

EPSO

0.500

EPILONT

0.500

ESSO

0.000

Symptoms – Confused Memory Ontology

OW

EPSO

0.200

EPILONT

0.800

ESSO

0.000

13.7 Conclusion This paper presents a hybrid method for ontology selection. The method uses AHP technique to compute weights for determining the best suited ontology for diagnosing a disease. The study demonstrates its use for diagnosing epilepsy. Three ontologies, EPSO, ESSO, and EPILONT are considered for the study. Focal, AED, Head Trauma, and Confused Memory were evaluated as the important for determining the best ontology among these three.

A Hybrid Approach to Ontology Evaluation  203 For future reference, a more improved ontology for epilepsy disease can be developed. Ontology Evaluation and Selection can be done using other decision making techniques.

References 1. Alobaid, A., Garijo, D., Poveda-Villalón, M., Santana-Perez, I., FernándezIzquierdo, A., & Corcho, O. (2019). Automating ontology engineering support activities with OnToology. Journal of Web Semantics, 57, 100472. 2. Amith, M., He, Z., Bian, J., Lossio-Ventura, J. A., & Tao, C. (2018). Assessing the practice of biomedical ontology evaluation: Gaps and opportunities. Journal of biomedical informatics, 80, 1-13. 3. Amith, M., Manion, F., Liang, C., Harris, M., Wang, D., He, Y., & Tao, C. (2019). Architecture and usability of OntoKeeper, an ontology evaluation tool. BMC medical informatics and decision making, 19(4), 152. 4. Ashraf, J., Hussain, O. K., Hussain, F. K., & Chang, E. J. (2018). Representation Phase: Ontology Usage Ontology (U Ontology). In Measuring and Analysing the Use of Ontologies (pp. 171-203). Springer, Cham. 5. Bandeira, J., Bittencourt, I. I., Espinheira, P., & Isotani, S. (2016). FOCA: A methodology for ontology evaluation. ArXiv preprint arXiv:1612.03353. 6. Brank, J., Grobelnik, M., & Mladenic, D. (2005, October). A survey of ontology evaluation techniques. In Proceedings of the conference on data mining and data warehouses (SiKDD 2005)  (pp. 166-170). Citeseer Ljubljana, Slovenia. 7. Fonou-Dombeu, J. V. (2019, June). A Comparative Application of Multicriteria Decision Making in Ontology Ranking. In International Conference on Business Information Systems (pp. 55-69). Springer, Cham. 8. Franco, M., Vivo, J. M., Quesada-Martínez, M., Duque-Ramos, A., & Fernández-Breis, J. T. (2019). Evaluation of ontology structural metrics based on public repository data. Briefings in bioinformatics. 9. Haghighi, P. D., Burstein, F., Zaslavsky, A., & Arbon, P. (2013). Development and evaluation of ontology for intelligent decision support in medical emergency management for mass gatherings. Decision Support Systems, 54(2), 1192-1204. 10. Hooi, Y. K., Hassan, M. F., & Shariff, A. M. (2015, May). Ontology evaluation—A criteria selection framework. In 2015 International Symposium on Mathematical Sciences and Computing Research (iSMSC)  (pp. 298-303). IEEE. 11. Jarosław, W. (2018). An Attempt to Knowledge Conceptualization of Methods and Tools Supporting Ontology Evaluation Process. Procedia computer science, 126, 2238-2247. 12. Knoell, D., Atzmueller, M., Rieder, C., & Scherer, K. P. (2017). A Scalable Framework for Data-Driven Ontology Evaluation. In WM (pp. 97-106).

204  Mathematics and Computer Science Volume 2 13. Lourdusamy, R., & John, A. (2018, January). A review on metrics for ontology evaluation. In 2018 2nd International Conference on Inventive Systems and Control (ICISC) (pp. 1415-1421). IEEE. 14. Mabkhot, M. M., Al-Samhan, A. M., & Hidri, L. (2019). An OntologyEnabled Case-Based Reasoning Decision Support System for Manufacturing Process Selection. Advances in Materials Science and Engineering, 2019. 15. M’Baya, A., Laval, J., Moalla, N., Ouzrout, Y., & Bouras, A. (2016, December). Ontology based multi criteria recomanded system to guide internship assignment process. 3rd IEEE International Conference on Computational Science and Computational Intelligence (CSCI 2016), Dec 2016, Las Vegas, United States. ffhal-01788507f 16. Mochol, M., Oldakowski, R., & Heese, R. (2004). Ontology based recruitment process. Informatik 2004, Informatik verbindet, Band 2, Beiträge der 34. Jahrestagung der Gesellschaft für Informatik eV (GI). 17. Plinere, D., & Borisov, A. (2014). Evaluation of the Ontological Knowledge Model/Ontoloģiskā zināšanu modeļa novērtēšana/Оценка качества онтологической модели знаний. Information Technology and Management Science, 17(1), 81-85. 18. Saaty, T. L. (1988). What is the analytic hierarchy process? In Mathematical models for decision support (pp. 109-121). Springer, Berlin, Heidelberg. 19. Stafstrom, C. E. (1998). The pathophysiology of epileptic seizures: a primer for pediatricians. Pediatrics in review, 19(10), 342. 20. Sun, L., Ma, J., Zhang, Y., Dong, H., & Hussain, F. K. (2016). Cloud-FuSeR: Fuzzy ontology and MCDM based cloud service selection. Future Generation Computer Systems, 57, 42-55. 21. Talebpour, M., Sykora, M., & Jackson, T. (2017, November). Social and Community Related Themes in Ontology Evaluation: Findings from an Interview Study. In International Joint Conference on Knowledge Discovery, Knowledge Engineering, and Knowledge Management  (pp. 320-336). Springer, Cham. 22. Verma, A. (2016, August). An abstract framework for ontology evaluation. In  2016 International Conference on Data Science and Engineering (ICDSE) (pp. 1-6). IEEE. 23. Vrandečić, D. (2009). Ontology evaluation. In  Handbook on ontologies (pp. 293-313). Springer, Berlin, Heidelberg. 24. Xu, F. X., Liu, X. H., Chen, W., Zhou, C., & Cao, B. W. (2018). An Ontology and AHP Based Quality Evaluation Approach for Reuse Parts of End-of-Life Construction Machinery. Mathematical Problems in Engineering, 2018.

14 Smart Health Care Waste Segregation and Safe Disposal R.M. Bommi1*, Sami Venkata Sai Rajeev2, Sarvepalli Navya2, Veluru Sai Teja2 and Uppala Supriya2 Saveetha School of Engineering, SIMATS, Chennai, India Department of CSE, Chennai Institute of Technology, Chennai, India 1

2

Abstract

Medical waste disposal has been a big issue due to an exponentially growing population and the COVID-19 pandemic. Increased waste generation per person has resulted from urbanization, industrialization, and economic development. Substandard medical waste separation at the site of origin might have a cascading effect on the environment, putting humans, wildlife, soil, and water bodies at risk. If hazardous airborne pollutants are not effectively controlled, separated, and burned by on-site or off-site incineration, environmental concerns linked with inadequate clinical waste may pollute the air we breathe. This paper proposes an IoT based smart healthcare waste segregator which segregates the waste into five kinds. The sensors detect the type of waste and the waste gets disposed into the smart bins accordingly. Using artificial intelligence, the status of filling of the bin is indicated through LEDs. When the bin reaches the maximum-level, an alert message is sent to the municipal authorities. The filled waste gets wrapped automatically. The wastes which need to be incinerated is burnt in the incinerator chamber available in the system. Therefore, this system will capably make the environment smart, clean, and safe. Keywords:  Sensors, IoT, artificial intelligence, incinerator, garbage, smart bins

*Corresponding author: [email protected] Sharmistha Ghosh, M. Niranjanamurthy, Krishanu Deyasi, Biswadip Basu Mallik, and Santanu Das (eds.) Mathematics and Computer Science Volume 2, (205–222) © 2023 Scrivener Publishing LLC

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14.1 Introduction The Internet plays an important role in today’s world by linking computers to the worldwide web (www), which permits users to access data from anywhere in the world [1]. The Internet of Things (IoT) refers to things that are connected to the internet and can often be managed from there [2]. Garbage is described as solid substances generated as a result of human activities that are removed from the system [3] because they are no longer useful in the respective economic, biomedical, or technical method. In a wider context, solid waste refers to all products that are used in the home, industry, or agriculture. Municipal solid waste (MSM) is described as waste that accrues in areas maintained by municipalities that are responsible for its disposal and recycling. People can throw garbage in waste bins, which is why they are valuable in life [3]. If this did not happen, the future would be a mess. Because a business or household has a garbage disposal device, it becomes a valuable piece of equipment. The dustbin’s position as a conciliator of changing waste practices has barely been regarded, despite its importance in our daily lives. Bins, it is believed, are providing a telling indicator of new garbage relationships in society as they are repurposed as environmental technologies for modern recycling schemes. Garbage and litter are all over the television these days, with disturbing statistics of debris filling the world. Despite the grim news, a number of people and policymakers are trying to change the trend by creative waste management practises. The Government has encouraged city-based schemes and public-private collaboration projects to improve waste management systems, but these have proven to be troublesome. The shortage of financial capital, sufficient expertise, and technical competencies in the public sector are the main obstacles to developing solid waste management systems around the world. Governments have begun to look at PPPs as a possible solution. The amount of change and development made was minimal. Medical waste management is tough and complicated, especially during pandemics like COVID. Due to the apparent forte of the global outbreak, contemporary waste centers are adapting to reveal the abnormal scientific waste and its affect on viral propagation with particular statistics on the amount of scientific waste generated, waste warm spots, and remedy centers. Technological knowhow [24] on inspection, segregation, transportation, storage, and reliable waste control structures are required to optimize contemporary sources and centers to satisfy the crisis, as healthcare waste portions are anticipated to increase rapidly. For patients, healthcare workers, and waste

Smart Health Care Waste Segregation and Safe Disposal  207 collectors, improper medical waste disposal can lead to accidents, diseases, harmful consequences, and air pollution. Bio-dangerous cloth and its opposite numbers encompass non-dangerous waste, infectious waste, radioactive waste, bacterial waste, chemical waste, cytotoxic waste, sharps waste, and pharmaceutical trash. A pandemic has identified an unusual amount of medical waste. As a result of this research, some serious problems were discovered and some important suggestions were made. To prevent the spread of any dangerous diseases, a proper waste management system is needed. This paper proposes a waste management strategy that is both effective and successful. This garbage sorter is made with sensors and an Arduino microcontroller. The suggested scheme satisfies the need for continuous garbage material tests in the bins. It aids in the disposal of garbage until the bins become overflowing. As a result, the device is helpful in waste management when it is monitored and informed on a daily basis. This translates to a cleaner city and a higher quality of life.

14.2 Related Works Garbage and litter are all over the television these days, with disturbing statistics of debris filling the world. Despite the grim news, a number of people and policymakers are trying to change the trend by creative waste management practices. These five forward-thinking countries are taking a novel approach to waste management in order to make the environment a safer, healthier place. Germany is first, followed by Austria, South Korea, Wales, and Indonesia. Clean Harbors, Stericycle Inc., Covanta Holding, and others are among the best waste management firms in the world. The Government of India has encouraged city-based schemes and publicprivate collaboration projects to improve waste management systems, but these have proven to be troublesome. The lack of financial resources, appropriate skills, and technological competencies with the public sector are the main obstacles to improving solid waste management services in India. Governments have begun to look at PPPs as a possible solution. The amount of change and development made was minimal. Some serious problems have been discovered as a result of this research and some significant proposals have been made. This article [4] presents a system (hardware, software, and communications) to improve trash handling while also involving citizens. The model employs an IoT method in which the discharged trash from the trash container is continuously monitored by sensors that provide real-time

208  Mathematics and Computer Science Volume 2 information upon those filling levels of each chamber. An IoT-based smart trash system [5] (SGS) is presented in this study to decrease food waste. Wireless mesh networks enable battery-powered smart trash bins (SGBs) to interact with one another in an SGS, while a router and server collect and analyse data for application services [6], suggesting a unique approach for achieving waste management that is both vigorous and efficient by forecasting the likelihood of waste levels in trash bins in this article. Combining machine learning and graph theory, the system can optimise trash collection via the shortest path. This article describes an investigative case that was carried out on the Ton Duc Thang University (Vietnam) campus to assess the system’s performance and viability. A cross-domain robust distributed trust management [7] (Robust Trust) system is suggested in this work that also makes a system suitable for independently evaluating faith towards various devices. The credibility in this approach is split into three security technologies that enable IoT nodes be resilient against hacked and malicious devices/nodes. This study [8] proposes a strategy for achieving this ambitious goal. In this article, a microcontroller is used to interface an ultrasonic sensor and GSM modem to construct an intelligent container. The highest point of the trashcan is fitted with an ultrasonic sensor, which measures the dustbin’s height. [9] A novel waste disposal based on an Android mobile app and a Bluetooth-enabled smart dustbin is proposed in this article. Through the lines painted on the floor, this android application controlled the bot. The lines are usually specified along the path. A white line on a black surface or vice versa might be one of the predetermined routes. In this study [10], a novel architecture is suggested with the goal of improving waste disposal, on-site handling, and transfer optimization. The network is based on a sensor network and Data Transfer Nodes (DTN) are used to communicate data from garbage bin filling to a remote server. [4] This article presents a system (hardware, software, and communications) to improve trash management while also involving citizens. The system uses an IoT method in which the discarded trash from the smart bin is continually monitored by sensors that provide real-time information on the filling level of each compartment [11]. This article proposes a revolutionary sensor node design based on the use of low-cost, high-efficiency components such as water level, soil moisture, temperature, humidity, and rain sensors. The transmitting module, in particular, is based on the LoRa LPWAN method, ensuring overall system performance. The principal circuit board of the system is optimised by combining two layers and doing code optimization [12]. The concept of a garbage surveillance system for smart campuses, colleges, clinics, and bus stops is proposed in this article.

Smart Health Care Waste Segregation and Safe Disposal  209 The Garbage Monitoring System is a clever dustbin that makes use of sensors to raise it above the trendy dustbin [13]. In the proposed system, public trash cans would be equipped with an integrated device that allows for real-time tracking of waste levels in the bins. The information on waste levels would be used to create an optimal path for trash collection vans, which will save money on fuel. The load sensors will enhance the accuracy of trash level information, while humidity sensors would give data on waste segregation in a dust bin. [14] The suggested model, which employs a convolutional neural network (CNN), a prominent machine learning approach, to separate biodegradable and insoluble waste, is presented in this study. The plan also includes an architectural concept for a smart garbage can that incorporates a microprocessor and several sensors [15]. They presented an Internet of Things-based management of solid waste in this article, which allows garbage bin monitoring, dynamic scheduling, and tracking of garbage collector vehicles in a city of the future. Garbage bins fitted with moderate embedded devices are placed around the city under the proposed model. [16] To combat COVID-19, this study suggests a leader-follower strategy for hazardous infectious waste collection and government aid distribution. We suggest a smart waste material classification based on the 50-layer residual net pre-train (ResNet-50) Convolutional Neural Network model, which would be a machine learning tool that represents the extractor and a Support Vector Machine (SVM) that is used to categorize waste into useful groups/types including glass, metal, paper, and plastic among others. As proposed by [17, 18], the “Smart Dustbin” in this article would be a cellular bin that autonomously monitors rubbish levels and transmits messages to the appropriate city officials to update the bin’s status [19]. This article discusses an automated system that allows depositors to dispose of their e-waste and be compensated for it. In terms of e-waste collectors, an online bidding session is held with the winner receiving ownership of the electronic trash that he or she won in the online bidding process. For this, an interactive digital bin with multiple sensors and modules is required. After user verification, the bin must be able to lock and unlock itself (Urlagunta). The smart bin in this article is based on the Microcontroller system, which is connected to a GSM modem and also an ultrasonic sensor. An ultrasonic sensor is mounted on the top of the trashcan to measure the dustbin’s height. The minimum height is set at 10cm. The Microcontroller will be designed such that when the trash fills, the remaining height above the threshold height will be shown. [20] This paper describes the design and implementation of an Internet of Things (IoT) based Arduino microcontroller that uses ultrasonic

210  Mathematics and Computer Science Volume 2 sensors to detect the amount of trash in garbage bins, reveal the information, update it as “empty,” “half-filled,” or “filled” on an LCD screen at periodic intervals, and also transmit the data level information [21]. An Internet of Things (IoT) architecture for real-time garbage monitoring and collection is presented in this study, with the goal of improving and optimizing solid waste collection in cities. The Netlogo multi-agent platform was used to mimic real-time monitoring and intelligent waste management decisions. [22] An automatic trash segregator is proposed in this study. When medical waste is detected, the conveyor belt is moved by an external motor. The trash will be sent to the sensing and classification units. The input picture is captured, pre-processed, median filtered, contrast enhanced, and segmented in five phases [23]. The article creates the ‘Waste HierarchyTechnology Readiness Levels’ framework and evaluates each waste management plan against it, showing the technological maturity and the strategy’s place in the Waste Framework directive, as well as its relative adherence to circular economy concepts [3]. This article presents a concept for a smart garbage bin surveillance system  in Ethiopian cities that uses the Global System for Mobile Communication (GSM) to organize trash collection networks in residential and commercial sectors. The level of waste material in the trash bin has been monitored using an ultrasonic sensor in this suggested system and it will continually connect with the authorized clean-up person’s cell phone via GSM modem. This research will be crucial not only for individuals whose professions primarily entail the disposal of garbage cans, but also for the general public who will be following the smart city rules for a higher quality of life. The implementation of this suggested technique would go a long way forward into ensuring effective and efficient real-time garbage disposal and resource utilization, as well as improving life for smart city residents. The actual implementation of the functional conceptual model will serve as an advanced as well as instructive technique of sustaining and improving the wellbeing of smart urban residents in long run.

14.3 System Architecture Figure 14.1 illustrates the overall system architecture module. Inlet, Split Rollers, Pipeline System, Dihydrogen Monoxide Outlet System, Exhaust System, Chained Conveyor Belt, Incinerator, Solar Panel, Brushes, Arduino Uno Microcontroller, IR Sensor, Ultrasonic Sensor, Moisture Sensor, Glass Detector sensor, Inductive Proximity Sensor, Node-MCU

Smart Health Care Waste Segregation and Safe Disposal  211

STEP DOWN

BRIDGE RECTIFIER

TRANSFORMER (230 TO 12V AC)

FILTER CIRCUIT

VOLTAGE REGULATOR (IC 7805 & 7812)

POWER SUPPLY UNIT

DC MOTOR (CONVEYOR BELT)

GLASS SENSOR ULTRASONIC SENSOR MOISTURE SENSOR

PROXIMITY SENSOR

LEDS

ARDUINO

ESP8266

MOBILE APP

RELAY

MOTOR

Figure 14.1  System architecture.

ESP8266, Wrapper, 3 chambers for dihydrogen monoxide, 2 for contaminant dihydrogen monoxide, and 1 for pristine dihydrogen monoxide, Thermo-engenderer and Electrodes, ECU Board, Breadboards, Jumper Wires, OLED Exhibit, LED Lights, and a DC Motor are a few of the main components utilized for making the module. If medical waste is disposed of in the inlet, it is transferred to the chain conveyor belt, which triggers the DC motor, which causes the conveyor belt to move. Metal, dry, wet, glass, and incinerated wastes are dissevered into five categories. The sensors detect the waste predicated on the type of waste and the waste is then sorted into concrete bins. The state of filling the bin is immediately signaled by LEDs, utilizing artificial perspicacity. The red LED on the bin commences flashing as it approaches full capacity and a caveat note is sent to the local ascendant entities. The waste that has been filed is immediately bundled. The wastes that must be incinerated are burned in the contrivance’s incinerator chamber. The thermo-­engenderer converts the heat engendered in the incinerator chamber into electrical energy. The conveyor belt’s DC motor consumes the engendered electrical energy. The conveyor belt is automatically washed until the whole operation is consummated utilizing a chain conveyor belt cleaning machine.

212  Mathematics and Computer Science Volume 2 IoT, AI, and web/app development are the three innovations utilized in this scheme. To control the flow of waste into the conveyor, the inlet segment has an open and close function. To detect metallic waste, an inductive proximity sensor is utilized. Arduino Uno is in charge of the conveyor belt’s pacing and rotation. An incinerator is additionally part of this contrivance. When a human pushes the trigger, the controls are turned off and the waste on the conveyor belt is sent to the incinerator container. The walls of the incinerator are composed of clay and covered with aluminum foil. Inside the incinerator are the ECU board electrodes and thermoengenderer, which are habituated to engender electricity from thermal energy. The DC motor is driven by the engendered electricity. The sensors are deactivated until the procedure is done. This contrivance withal has the capability of sending an admonishment SMS if a bin is loaded. Action is detected by an infrared sensor. Moisture sensors for dry and wet waste, Inductive Proximity sensors for metal waste, Ultrasonic sensors for bin filling tracking, warning messaging via Node MCU ESP8266-12E, visual exhibit via OLED exhibit, and waste packaging via hotwire sealer are all utilized. The Biomedical Waste Segregator is a piece of automated machinery that sorts waste into four categories: metal, glass, dry, and wet. The suggested framework would be capable of monitoring and managing the solid waste amassment process as well as the total amassment process. To detect metallic waste, an inductive proximity sensor is utilized. Dry and wet waste is disunited utilizing a blower system. A microcontroller controls the timing and rotation of the conveyor belt. This contrivance additionally has a feature that sends an admonition SMS if a bin is loaded.

14.3.1 Wrapping The Ultrasonic sensor causes the door of the respective container to close as the waste is filled in it. The container’s door is made up of two semi-circular panels. One of the circu lar plates is connected to the other by a thin rod, while the other is connected by an insulating rod (namely wood). A very low volt current is passing through each rod as the doors close. The rods (the rod with electric flow heats up) come next to each other until the door is fully locked. The heat and compression are just enough to melt and seal the trash container. Once the doors of the container are closed, the current in the rod flows till the rod reaches a high enough temperature.

Smart Health Care Waste Segregation and Safe Disposal  213

14.3.2 Incinerator To burn the garbage, an incinerator is required. Medical waste must be incinerated in some cases. When a human pushes the trigger, the controls are turned off and the waste on the conveyor belt is sent to the incinerator container. The walls of the incinerator are made of clay and covered with aluminum foil. Inside the incinerator are the ECU board electrodes and the thermos-generator, which are used to generate electricity from thermal energy. The DC motor is driven by the produced electricity.

14.3.3 Conveyor Cleaning System The conveyor cleaner makes contact with the returned conveyor and uses a rotating, pre-moistened, heavy-duty belt to scrub it continuously. For ease of product handling, the clean conveying surface is allowed to dry. Conveyor cleaners are made of heavy-duty stainless steel and have long-lasting scrubbing belts that need no maintenance. A 1/4 inch NTT-thread is used to connect to the water inlet and a manual shutoff is included. A three-quarter inch ID hose is used to empty the drain pan. The belt friction corrections, as well as the drain tub, are also conveniently accessible for normal maintenance.The CC series of conveyor cleaners are enticing solutions for reduced tension line downtime because of these characteristics, as well as ease of operation and easy insulation.

14.3.4 Circuit Diagram In Figure 14.2, the left most side ultrasonic sensor is connected trigg pin = 38 (output) and echo = 40 (input). On the left, the 2nd ultrasonic sensor is connected trigg pin = 42 (output) and echo = 44 (input). The 3rd ultrasonic sensor is connected trigg pin = 46 (output) and echo = 48 (input). On the right most side, the ultrasonic sensor is connected trigg pin = 52 (output) and echo = 50 (input). M2, M3, M4, and M5 are servo motors. At M2, pwm=12. At M3, pwm=11. At M4, pwm=10. At M5, pwm=9. LED 2=51 LED 1 =49 LED 3 = 47 LED 4 =45. We use esp 8266. So, the tx is connected to rx 0 and the rx is connected to tx 0, then the other pin of the 16 *2 LED display.

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Figure 14.2  Circuit diagram.

14.3.5 Optimal Path Planning Algorithm for Waste Collection Figure 14.3 illustrates the flow chart of proposed methodology and the following algorithm supports the same. Step 1: Set up the microcontroller and all of the sensors. Step 2: Switch on the ESP8266 and initialise the SIM. Step 3: When Wi-Fi is open, the mobile device connects to the network using an IP address. Step 4: When the height and weight of the bins exceed the margin, an SMS message is sent. Step 5: Using the IP Address on the HTML tab, you can check the status of the bins.

14.4 Methodology The conveyor belt motor engages as the waste arrives and the conveyor belt commences to move. Many of the motors and controls, as well as the

Smart Health Care Waste Segregation and Safe Disposal  215 Start

Initializing all modules

Start taking data of sensors Continuously monitor the level of the dustbin

No If know status

No

Yes

If dry dust bin/wet dustbin is full

Yes Enter IP Address (192.168.4.1)

No

Height is full

Yes HTML page displayed with details

No If load is full Yes

Sending information to authorized persons & server (IoT)

Sending information to authorized persons & server (IoT)

Figure 14.3  Optimal path planning algorithm.

microcontroller, have been switched on. With a significantly growing population and in this COVID pandemic, it is extensively more basic to be benevolent concerning how well we, as individuals, manage our prosperity and environment. Considering the insights, it is seen that authentic clinical trash evacuation is especially expected for a spotless environment. The modernized waste segregator is a capable and monetary waste combination structure with a base proportion of human mediation with no risk to human life. Using a vehicle line makes the system significantly more accurate, economic, and clearer to put in and use at a local level. Segregating these misfortunes at a local level will be timesaving. The proposed structure fulfils the requirement for reliable monitoring of garbage content in the containers. It helps with disposing of the waste material before it floods from the canisters. So, standard observation and recommendation make the structure significant in waste across the board. This prompts an immaculate city for better living.

216  Mathematics and Computer Science Volume 2 If the trash is not metallic, the conveyer belt sensor decides if it is a wet or dry waste by estimating the dampness content of the waste. If there is moisture in the waste, it is classified as wet waste and the conveyor belt is switched off, the wet waste motor is turned on, and the garbage is deposited in the wet waste container. Counter 2 is also increased. If the conveyor belt is not carrying any wet trash, the waste is deposited into the dry waste bin at the top of the conveyor belt. When a waste containing glass particles is identified as glass waste, the ergo conveyor belt is turned off, the glass waste motor is turned on, and the garbage is forced into the glass waste bin. In consequence, Counter 3 is raised. The Figure 14.4 brings about the flow diagram of overall process. Conclusively, the wastes are deposited in the congruous containers, consummating the segregation process. When the dustbin level reaches 50%, the yellow LED which is annexed to the bins gets turned on. At 75%, the green LED gets turned on. At 90%, the red LED commences blinking and at 100%, the red LED gets turned on and then the SMS alert will be activated. When the bin gets filled, by utilizing the wrapper, the waste gets wrapped automatically. This system has a self-cleaning system with a conveyor belt. The conveyor chain enters the conveyor cleaner and is Medical waste. Split roller splits the Medical waste. Waste is passed onto the conveyer belt. The sensors detect the waste and send into the particular bin. Once the bin is filled, waste get warped. Self cleaning of the conveyer belt will be done automatically. The thermal energy which comes from the increnator will produce the electricity. When the bin is filled GMC get intimated automatically. It produces high accuracy in segregating the waste. Eco-friendly & hygenic.

Figure 14.4  Flow diagram of overall process.

Smart Health Care Waste Segregation and Safe Disposal  217 engaged by the hold-down bracket. The hold-down bracket guarantees that the chain makes full contact with the scrubbing belt, which rotates in the antithesis direction. The conveyor chain is squeegeed by a spring-loaded neoprene blade as it exits the conveyor cleaner before being returned to the conveyor line. The conveyor belt is swept in this manner.

14.5 Mobile App

Figure 14.5  Web page inerface to display status of smart bins.

218  Mathematics and Computer Science Volume 2 This Mobile App as seen in Figure 14.5 describes about detecting the amount of waste in each bin. First, we need to select the state, district, and area. Using this information, it displays the number of bins present in that area and the amont of waste in them. As mentioned, LEDs are used for easy identification of the amount of waste. If it is of red color, it indicates that the bin is completely filled and needs to be cleaned. If it is yellow the bin is partially filled and if it is green it is empty. This information is sent to the nearest municipal corporations, so that the waste disposal becomes easy.

14.6 Conclusions and Future Works With a dramatically expanding populace and in this Coronavirus pandemic, it is considerably more critical to be kind with regards to how well we, the people, deal with our wellbeing and climate. In view of the perceptions, it is perceived that legitimate clinical garbage removal is particularly required for a clean climate. The computerized squander segregator is a proficient and financial waste assortment framework with a base measure of human intercession and furthermore makes no peril for human existence. Utilizing a transport line makes the framework substantially more exact, financially savvy, and furthermore more straightforward to place in and use at a homegrown level. Isolating these losses at a homegrown level, likewise, will be timesaving. The proposed framework satisfies the need for consistently keeping an eye on trash content in the receptacles. It assists with discarding the waste material before it floods from the canisters. So, standard observation suggests that the framework is valuable across the board. This prompts a spotless city for better living. The automatic waste segregator is a cost-efficacious and reliable waste accumulation contrivance that requires no human involution and poses little risk to human safety. The utilization of a conveyor belt ameliorates the precision, cost-efficacy, and facilitation of installation and utilization of the system on a domestic substructure. Dissevering these wastes at the household level would preserve time. The suggested scheme is slated for perpetual monitoring of the medical waste material in the bins. It avails in the disposal of garbage until the bins become overflowing. This translates to a cleaner city and a higher standard of living. Restorative treatment is basic to our survival and supportability. Incomprehensible restorative squander isolation at the point of inchoation may have a domino impact on the environment, posturing dangers to people, natural life, and soil, as well as dihydrogen monoxide bodies. Natural dangers related with destitute healthcare squander administration

Smart Health Care Waste Segregation and Safe Disposal  219 may sully the air we breathe through poisonous airborne contaminants in the event that they are not satisfactorily contained, isolated, and burned by on-site or off-site burning. As a result, such squander needs uncommon care and support before being disposed of. It is vital that healthcare labourers get it the esteem of therapeutic squander control.

Declarations Availability of Data and Materials Information accessibility is not appropriate for this composition, as no unused information was made or analyzed in this ponder.

Competing Interests The Author(s) declare(s) that there is no conflict of interest.

Author’s Contribution All the authors contributed indistinguishably to the work. All authors examined and endorsed the ultimate composition.

References 1. B. Bharadwaj, M. Kumudha, N. Gowri Chandra, G. Chaithra, Automation of Smart waste management using IoT to support “Swachh Bharat Abhiyan” A practical approach, Proc. 2017 2nd Int. Conf. Comput. Commun. Technol. ICCCT 2017. (2017) 318–320. https://doi.org/10.1109/ICCCT2.2017.7972300. 2. A. Sharma, R. Kumar, V. Mansotra, Proposed Stemming Algorithm for Hindi Information Retrieval, Int. J. Innov. Res. Comput. Commun. Eng. (An ISO Certif. Organ. 3297 (2016) 11449–11455. https://doi.org/10.15680/ IJIRCCE.2016. 3. M. Karthikeyan, D. Eligo, S. Gebrehiyot, E. Bekele, D. Daniel, G. Dansa, GSM Based Smart Waste Bin Monitoring System In Ethiopia, VI (2019) 796–804. 4. K. Pardini, J.J.P.C. Rodrigues, O. Diallo, A.K. Das, V.H.C. de Albuquerque, S.A. Kozlov, A smart waste management solution geared towards citizens, Sensors (Switzerland). 20 (2020) 1–15. https://doi.org/10.3390/s20082380. 5. I. Hong, S. Park, B. Lee, J. Lee, D. Jeong, S. Park, IoT-Based Smart Garbage System for Efficient Food Waste Management, Sci. World J. 2014 (2014). https://doi.org/10.1155/2014/646953.

220  Mathematics and Computer Science Volume 2 6. T. Anh Khoa, C.H. Phuc, P.D. Lam, L.M.B. Nhu, N.M. Trong, N.T.H. Phuong, N. Van Dung, N. Tan-Y, H.N. Nguyen, D.N.M. Duc, Waste Management System Using IoT-Based Machine Learning in University, Wirel. Commun. Mob. Comput. 2020 (2020). https://doi.org/10.1155/2020/6138637. 7. K.A. Awan, I. Ud Din, A. Almogren, M. Guizani, A. Altameem, S.U. Jadoon, RobustTrust - A Pro-Privacy Robust Distributed Trust Management Mechanism for Internet of Things, IEEE Access. 7 (2019) 62095–62106. https://doi.org/10.1109/ACCESS.2019.2916340. 8. V.P. Vijaynaidu, T. Dhikhi, Smart garbage management system, Int. J. Pharm. Technol. 8 (2016) 21204–21211. https://doi.org/10.17577/ijertv4is031175. 9. C.S. Anilkumar, G. Suhas, S. Sushma, A smart dustbin using mobile application, Int. J. Innov. Technol. Explor. Eng. 8 (2019) 3964–3967. https://doi. org/10.35940/ijitee.K2129.0981119. 10. S. Longhi, D. Marzioni, E. Alidori, G. Di Buò, M. Prist, M. Grisostomi, M. Pirro, Solid waste management architecture using wireless sensor network technology, 2012 5th Int. Conf. New Technol. Mobil. Secur. - Proc. NTMS 2012 Conf. Work. (2012). https://doi.org/10.1109/NTMS.2012.6208764. 11. T.A. Khoa, M.M. Man, T.Y. Nguyen, V.D. Nguyen, N.H. Nam, Smart agriculture using IoT multi-sensors: A novel watering management system, J. Sens. Actuator Networks. 8 (2019). https://doi.org/10.3390/jsan8030045. 12. R. Asha, N. Mahajan, A. Yadav, K.S. Balamurugan, Garbage Managing Smart System using-IOT, Int. J. Recent Technol. Eng. 8 (2019) 3646–3649. https:// doi.org/10.35940/ijrte.d7849.118419. 13. P.S.A. Mahajan, A. Kokane, A. Shewale, M. Shinde, S. Ingale, Smart Waste Management System using IoT, Int. J. Adv. Eng. Res. Sci. 4 (2017) 93–95. https://doi.org/10.22161/ijaers.4.4.12. 14. M.W. Rahman, R. Islam, A. Hasan, N.I. Bithi, M.M. Hasan, M.M. Rahman, Intelligent waste management system using deep learning with IoT, J. King Saud Univ. - Comput. Inf. Sci. (2020). https://doi.org/10.1016/j. jksuci.2020.08.016. 15. S.S. Chaudhari, V.Y. Bhole, Solid Waste Collection as a Service using IoTSolution for Smart Cities, 2018 Int. Conf. Smart City Emerg. Technol. ICSCET 2018. (2018) 1–5. https://doi.org/10.1109/ICSCET.2018.8537326. 16. J. Valizadeh, A. Hafezalkotob, S.M. Seyed Alizadeh, P. Mozafari, Hazardous infectious waste collection and government aid distribution during COVID19: A robust mathematical leader-follower model approach, Sustain. Cities Soc. 69 (2021) 102814. https://doi.org/10.1016/j.scs.2021.102814. 17. O. Adedeji, Z. Wang, Intelligent waste classification system using deep learning convolutional neural network, Procedia Manuf. 35 (2019) 607–612. https://doi.org/10.1016/j.promfg.2019.05.086. 18. Aniqa Bano, Ikram Ud Din, Asma A. Al-Huqail, “AIoT-Based Smart Bin for Real-Time Monitoring and Management of Solid Waste”, Scientific Programming, vol. 2020, Article ID 6613263, 13 pages, 2020. https://doi. org/10.1155/2020/6613263.

Smart Health Care Waste Segregation and Safe Disposal  221 19. K.B. Kumar R, S.B. Patil, S. Manjula, H.N. Kumar, An IoT Based Approach for Efficient Collection and Disposal of E-waste, Int. J. Eng. Tech. 4 (2018) 379–383. 20. S. Abba, C. Light, IoT-Based Framework For Smart Waste Monitoring And Control System, A Case Study of Smart Cities., (2020) 8224. https://doi. org/10.3390/ecsa-7-08224. 21. E.D. Likotiko, D. Nyambo, J. Mwangoka, M Ulti -a Gent B Ased I Ot S Mart W Aste M Onitoring a Nd C Ollection, International Journal of Computer Science, Engineering and Information Technology (IJCSEIT), Vol.7, No.5, October 2017. 22. K.G. Devi, K. Yasoda, M. Dhivya, B. Kishore, Automatic Health Care Waste Segregation and Disposal System, J. Xidian Univ. 14 (2020) 5281–5290. https://doi.org/10.37896/jxu14.5/573. 23. C.A. Fletcher, R. St. Clair, M. Sharmina, A framework for assessing the circularity and technological maturity of plastic waste management strategies in hospitals, J. Clean. Prod. 306 (2021) 127169. https://doi.org/10.1016/j. jclepro.2021.127169. 24. Sarkodie, S.A., Owusu, P.A. Global assessment of environment, health and economic impact of the novel coronavirus (COVID-19). Environ Dev Sustain 23, 5005–5015 (2021). https://doi.org/10.1007/s10668-020-00801-2.

15 Investigation of Viscoelastic Magnetohydrodynamics (MHD) Flow Over an Expanded Lamina Surrounded in a Permeable Media Hiranmoy Mondal1*, Arindam Sarkar2 and Raj Nandkeolyar2 Department of Applied Science, Maulana Abul Kalam Azad University of Technology, West Bengal, Kolkata, India 2 Department of Mathematics, National Institute of Technology Jamshedpur, Jamshedpur, India 1

Abstract

The results of the boundary value problem of MHD nanofluid micropolar fluid flow have been carried out throughout this study. Using the spectral quasilinearization method (SQLM), the mass as well as heat transfer effects of fluid have been investigated. A stream consists of viscous dissipation, bouncy force, thermal radiation, and Joule heating. It has been scrutinized that the existence of magnetic parameter Mn enhances concentration and temperature gradient, whereas the appearance of permeable media turns down the velocity profile. Keywords:  Stretching surface, MHD flow, darcy dissipation, viscoelastic liquid, nanofluids, chemical reaction, spectral quasi linearization method

15.1 Introduction 15.1.1 Literature Review Viscoelastic Magneto-hydrodynamic nanofluids through a stretching sheet have a broad range of practical applications in industries such as paper production, metal spinning, relegation of plastic lamina, continuous *Corresponding author: [email protected] Sharmistha Ghosh, M. Niranjanamurthy, Krishanu Deyasi, Biswadip Basu Mallik, and Santanu Das (eds.) Mathematics and Computer Science Volume 2, (223–238) © 2023 Scrivener Publishing LLC

223

224  Mathematics and Computer Science Volume 2 welding of metals, production of synthetic sheets, removal of heat from metallic sheets, etc. The study of magnetohydrodynamics (MHD) takes place as an essential part in thermodynamics, especially during the determination of boundary layer flow for cooling of nuclear reactors, high-temperature plasmas, etc. [1]. Pal and Mondal [2] reported that the stretching parameter and velocity profile are inversely related. Baag et al. [3] investigated the entropy generation through an electrically-conducting viscoelastic MHD fluid stream upon an exponentially expanded porous lamina. The boundary layer equations have been solved via Kummer’s function. The viscoelastic and magnetic parameters were shown to be inversely related to velocity, whereas the elastic parameter was linearly proportional to velocity. Furthermore, porous matrix and the inverse of the Prandtl number were found to be directly proportional to temperature [3]. Also, Pal and Mondal [4] studied laminar, unwavering, and convective boundary layer Newtonian fluid streams. They examined the heat absorbing/producing effect of the fluid which is passing through a non-­ isothermal along with the thermal radiation immersed in a permeable surface.

15.1.2 Nomenclature B0 Cp f K Mn R T T∞ ρ Kp k0 K ′p s r D DB DT µ ν σ Rc Pr

Strength of uniform magnetic field Specific heat of the fluid Stream function Thermal conductivity of the fluid Magnetic parameter Radiation parameter Non-dimensional temperature Ambient temperature/temperature far from sheet Density of the fluid Porosity parameter Dimensionless elastic parameter Porosity parameter Plate concentration parameter Plate temperature parameter Molecular diffusivity Brownian motion coefficient Thermophoretic diffusion coefficient Dynamic viscosity Kinematic coefficient of viscosity Electrical conductivity Elastic parameter Prandtl number

Viscoelastic MHD Flow Over an Expanded Lamina  225 Nt Nb Le

Thermophoresis parameter Brownian motion parameter Lewis number

At present, nanofluids are very significant in engineering applications as well as biomedical science, as it increases heat transfer and energy efficiency in various kinds of thermal systems. Nanofluid is a special kind of heat transfer medium containing oxides or metals, e.g., zinc oxide, copper oxide, alumina or iron oxide having a radius less than 100 nm, which are equally and stably spread out in a base fluid like water, oil, acetone, etc. Over the last few years, nanofluids have drawn a substantial amount of attention for their unique properties which are very beneficial in the chemical and metallurgical area, cooling, ventilation, thermal therapy for cancer treatment, power generation, micro-manufacturing, and air conditioning. The literature survey shows that during the past few decades, curiosity around fluid flowing through a stretching surface has grown. Crane [5] investigated constant stretching surface temperature problems. Later on, a small number of researchers investigated the outcomes of chemical repercussions, heat and mass alternation, thermal slip conditions of various non-Newtonian fluids, or probable combined effects of the above [6–11]. Due to countless industrial applications like power generation system and modern metallurgical processes, many researches have been attracted to the MHD electrically conducting fluid flow. Hayat et al. [12] considered an unperturbed incompressible MHD Walter-B nanofluid flow that is passing through a nonlinear stretching surface. Using homotopy analysis they deciphered their mathematical model, whereas Daniel et al. [13] did his research in the heat alternation effect of MHD nanofluid stream take across a porous stretching surface. Hayat et al. [14] studied the heat transmission of a nanofluid stream with various dissipation and Joule heating along a permeable stretched cylinder and the stream was subjected to nonlinear thermal diffusion with homogeneous-heterogeneous reactions. They used an Explicit Euler formula to resolve their fluid model. Hayat et al. [15] calculated the effect of melting heat distribution of a MHD nanofluid genre with nonlinear thermal diffusion in regards to the stagnation point along an expanded lamina. Using homotopy analysis they deciphered their flow equations. Kishan and Maripala [16] presented a mathematical model of the Darcy Forchheimer MHD boundary layer stream by a permeable expanded lamina. They also calculated the thermophoresis and viscous dissipation. The perusal of a hydromagnetic stream of an electrically conducting fluid has drawn a significant observation due to its applications in recent metallurgical and metal-working processes. The effect of a magnetic field is an important part to manage the momentum as well as heat conduction in the

226  Mathematics and Computer Science Volume 2 boundary layer stream of various kind of fluids. Several researchers (Ibrahim et al. [17], Turkyilmazoglu [18], Farooq et al. [19], Sheikholeslami et al. [20], Baag et al. [21], Sheikholeslami et al. [22], Abdul Hakeem et al. [23]) studied about the impact of magnetic field on MHD fluid. It is immensely important to scrutinize the behavior of heat absorption/production on account of several kinds of physical models in modern days. Eldahab and Aziz [24] have investigated the results of uneven heat exploitation/whiff effect of viscous fluid with thermal radiation. Abel et al. [25] examined the non-Newtonian MHD viscoelastic fluid upon a flat lamina. Ganga et al. [26] investigated the entropy generation on nanofluid flow through an expanded lamina by considering velocity slip condition and nonlinear thermal radiation. The study of heat distribution analysis for multiphase magnetic fluid passing through a flat lamina with heat exploitation/production and thermal diffusion was scrutinized by Zeeshan et al. [27]. The spectral quasi-linearization method (SQLM) is an important numerical method to solve highly nonlinear differential equation in fluid dynamics. Mondal et al. [28] and Motsa [29] used this method to solved the mathematical problem of the fluid flow. The originality of the recent perusal is to consider the chemical reaction and thermophoresis effect on the viscoelastic MHD nanofluid flow. The values of (Cf), (Nu), and (Sh) for different parameters has been discussed numerically. The outcome of this study will provide useful information for applications in several engineering processes along with a complement to the prior perusal.

15.2 Formulation of the Problem Along the way of the main stream of the lamina, the x − axis has been considered and the direction of the y – axis is perpendicular to the lamina with velocity components u, v along the x – axis and y – axis, respectively as demonstrated in Figure 15.1. The ruling equations of two-dimensional stream considering the boundary layer approximation are presented in this fashion:

∂u ∂v + =0 ∂x ∂y

u



(15.1)

∂u ∂u ∂ 2u B 2u v +v = v 2 −σ 0 − u ∂x ∂r K P′ ρ ∂y −

k0  ∂  ∂ 2u  ∂ 3u ∂u ∂ 2u  u v − +     ρ  ∂x  ∂y 2  ∂y 3 ∂y ∂x∂y 

(15.2)

Viscoelastic MHD Flow Over an Expanded Lamina  227 Vetical Magnetic field

B0

Momentum Boundary Layer

C∞

T∞

Thermal Boundary Layer y Porous Medium

Concentration Boundary layer

Slit

up = λx

Stretching sheet Tw

Cw

x

Figure 15.1  Schematic diagram.





 ∂T ∂C DT  ∂T 2  ∂ 2T ∂T   ∂T + = k 2 + τ  DB +v ρC p  u  (15.3) ∂y ∂y T∞  ∂y   ∂y  ∂y   ∂x u

∂C ∂C ∂ 2C D ∂ 2T +v = D 2 + T 2 − R(C − C∞ ) T∞ ∂y ∂x ∂y ∂y

(15.4)

The relevant boundary conditions are r

x = u u= λ x, = v 0= , T Tp (x ) = A   + T∞ , p l s



x = C C= B   + C∞ , at y = 0 p (x ) l ∂u u = 0, = 0, T → T∞ , C → C∞ as y → ∞. ∂x



(15.5)

15.2.1 Analytical Solution The stream function Ψ(x, y) is worthwhile for the continuity equation

228  Mathematics and Computer Science Volume 2



u=

∂Ψ ∂Ψ ,v = − . ∂x ∂y

(15.6)

Considering the similarity conversion

λ = η y= , Ψ (x , y ) x vλ f (η ) ν Substituting all the above values, Equations (15.2), (15.3), and (15.4) are transformed as follows:



1  f ′′′ + ff ′′ − f ′2 − Rc{2 f ′ f ′′′ − ff ′′2 − ff iv } −  Mn + KP 

  f ′ = 0 (15.7) 



θ″ + {Nbθ′∅′ + Ntθ′2 – (rf′θ − fθ′)}Pr = 0

(15.8)



1 1 Nt sf ′∅ − f ∅′ = ∅′′ + θ ′′ − R1∅ Le Le Lb

(15.9)

f(0) = 0,  f′(0) = 1,   f′(∞) = 0,  f″(∞) = 0,

θ(0) = 1, θ(∞) = 0



∅′(0) = −1, ∅ → 0 at η = ∞

(15.10)

Wherein f′(η) stand for the differentiation of f with respect to η, τ D ∆T τ DB ∆C Nb = represents Brownian motion parameter, N t = T repre­ vT∞ v v sents the thermophoresis parameter, Pr = is the Prandtl number, k v R Le = is the Lewis number, and R1 = . D λ

15.2.2 Numerical Methods (Spectral Quasi-Linearization Methods) Amalgamated with boundary condition (15.10), the nonlinear-coupled ordinary differential Equations (15.7), (15.8), and (15.9) are solved numerically, applying the spectral quasi-linearization method (SQLM).

Viscoelastic MHD Flow Over an Expanded Lamina  229

1  F ≡ f ′′′ + ff ′′ − f ′2 − Rc{2 f ′f ′′′ − f ′′2 − ff iv } −  Mn + KP  ∂F a0, r = iv = Rcf r ∂f ∂F a1, r = = 1 − 2Rcf r′ ∂f ′′′ ∂F a2, r = = f r + 2Rcf r′′ ∂f ′′ a3, r =



  f′=0 

1  ∂F  = −2 f r′ − 2Rcf r′′′−  Mn + ∂f ′ K P   ∂F = f r′′+ Rcf riv a4 , r = ∂f

The linear term is as follows:



a0, r f riv+1 + a1, r f r′′′+1 + a2, r f r′′+1 + a3, r f r′+1 + a4, r f r = RF The nonlinear term of the energy equation is:

θ ≡ θ ′′ + {N bθ ′∅′ + N tθ ′2 − (rf ′θ − f θ ′)}Pr = 0 b0 , r = b1, r =

∂θ = (N b∅′r + 2N tθr′ + f r )Pr ∂θ ′ ∂θ = −Pr rf ′ b2 , r = ∂θ ∂θ = −Pr rθr b 3, r = ∂f ′ ∂θ = Prθ ′ ∂f ∂θ = = N b Prθ ′ ∂∅′

b4, r =

∂θ =1 ∂θ ′′

b5, r

230  Mathematics and Computer Science Volume 2 The converted linear term is as follows:



b0, rθr′′+1 + b1, rθr′+1 + b2, rθr +1 + b3, r f r′+1 + b 4, r f r +1 + b5, r ∅′r +1 = Rθ The nonlinear term of the concentration equation is:

Nt θ ′′ − LeR1∅ − Le(sf ′∅ − f ∅′) Nb ∂∅ =1 c0, r = ∂∅′′ ∂∅ = Lef r c1, r = ∂∅′ ∂∅ = −LeR1 − sLe f r′ c2, r = ∂∅ ∂∅ = −sLeθr c 3, r = ∂f ′ ∂∅ = Le∅′r c 4, r = ∂f ∂∅ N b = c 5, r = ∂θ ′′ N t

∅ ≡ ∅′′ +



The converted linear term is as follows:



c0, r ∅′′r +1 + c1, r ∅′r +1 + c2, r ∅r +1 + c3, r f r′+1 + c 4, r f r +1 + c5, rθr′′+1 = R∅

15.3 Result and Argument The effect of ruling parameters on the skin-friction coefficient, Nusselt number, and Sherwood number has been presented in the following, Table 15.1 from tabular data we observe that (Cf) enhances with a rise in both magnetic parameter and thermophoresis parameter, whereas (Cf) reduces with enhancement in Soret number, chemical repercussion, and Brownian motion parameter. Nu reduces monotonically with enhancement in Soret number and rises with enhancement in magnetic, Brownian, and

Viscoelastic MHD Flow Over an Expanded Lamina  231 Table 15.1  Values (Cf), (Nu), and (Sh) for different parameters. Mn

Sr

R1

Nb

Nt

−f″(0)

−θ′(0)

−Ø′(0)

0.0

0.7

0.2

0.5

0.2

6.47747836

-1.24792674

-0.16130247

0.2

6.68965205

-1.23881769

-0.15872752

0.6

7.09600494

-1.22067600

-0.15482334

6.68965205

-1.23881769

-0.15872752

1.6

6.68959350

-2.06833985

0.15987681

5.0

6.69009405

-4.33945643

1.04495979

6.68959327

-1.57062405

0.21775166

0.4

6.69037312

-1.09209497

-0.36582335

0.6

6.69002619

-1.00077201

-0.51685468

6.68988106

-1.67046286

0.80664545

0.6

6.68993581

-1.12522670

-0.25586930

0.8

6.69012545

-0.93800745

-0.36733042

0.1

6.68959286

-1.32703668

-0.32973517

0.4

6.68990575

-1.09355804

0.11126215

0.7

6.69039059

-0.93732032

0.40484395

0.2

0.2

0.2

0.2

0.7

0.7

0.7

0.7

0.2

0.0

0.2

0.2

0.5

0.5

0.2

0.5

0.2

0.2

0.2

thermophoresis parameters and chemical reaction. Further, it has been audited that the sequel of magnetic parameter, thermophoresis parameter, and Soret number is to enhance the Sherwood number and reverse the effects of chemical reactions. The Brownian motion parameter has been observed on the Sherwood number. Figures 15.2 and 15.3 show a graph of Mn on f′(η) and f(η), respectively. Through Figure 15.4 it can be audited that the result of Mn enhances the concentration at all points. Figures 15.5 and 15.6 portray the temperature gradient θ(η) for several values of the magnetic parameter Mn and Prandtl number Pr. It has been audited that in Figure 15.5, θ(η) increases with η and it does not depend on the value of the Prandtl number, thermophoresis parameter, Lewis parameter, etc., whereas Figure 15.7 portrays an enhancement in the elastic parameter Rc, which decreases the transverse velocity f(η) for Mn = 2.

232  Mathematics and Computer Science Volume 2 1 0.8 Rc = 0.1; Kp = 0.5; Sr = 0.2; S = 0.5; Pr = 8; Nb = 0.5; Nt = 0.2; Le = 0.6; R1 = 0.5;

0.6 f’(η)

Mn = 0.1 Mn = 2.5 Mn = 3.5 Mn = 4.7

0.4 0.2 0

0

0.5

1

1.5

2

2.5 η

3

3.5

4.5

4

5

Figure 15.2  Result of Mn on f′(η) for Rc = 0.1, Pr = 8. 0.6 0.5 0.4 f(η)

0.3 Rc = 0.1; Kp = 0.5; Sr = 0.2; S = 0.5 Pr = 8; Nb = 0.5; Nt = 0.2; Le = 0.6; Che = 0.5

0.2

Mn = 0.1 Mn = 2.5 Mn = 3.5 Mn = 4.7

0.1 0 -0.1 0

1

2

3

4

5 η

6

7

8

9

10

9

10

Figure 15.3  Results of Mn on f(η) for Rc = 0.1, Pr = 8. 1 0.8

Rc = 0.1; Kp = 0.5; Sr = 0.2; S = 0.5; Pr = 8; Nb = 0.5; Nt = 0.2; Le = 0.6; Che = 0.5

Φ(η)

0.6

Mn = 0.1 Mn = 2.5 Mn = 3.5 Mn = 4.7

0.4 0.2 0

-0.2

0

1

2

3

4

5 η

6

Figure 15.4  Results of Mn on Ø(η) for Rc = 0.1, Pr = 8.

7

8

Viscoelastic MHD Flow Over an Expanded Lamina  233 1 0.8

Rc = 0.1; Kp = 0.5; Sr = 0.2; S = 0.5; Pr = 8; Nb = 0.5; Nt = 0.2; Le = 0.6; Che = 0.5

θ(η)

0.6

Mn = 0.1 Mn = 2.5 Mn = 3.5 Mn = 4.7

0.4 0.2 0

0

0.5

1

1.5

2.5 η

2

3.5

3

4

4.5

5

8

9

10

Figure 15.5  Results of Mn on θ(η) for Rc = 0.1, Pr = 8.

1 0.8 Mn = 2; Kp = 0.5; Sr = 0.2; S = 0.5 Rc = 0.1; Nb = 0.5; Nt = 0.2; Le = 0.6; Che = 0.5

θ(η)

0.6

Pr = 6.8 Pr = 8.9 Pr = 10 Pr = 12

0.4 0.2 0

0

1

2

3

4

5 η

6

7

Figure 15.6  Results of Pr on θ(η) for Mn = 2.

0.5 0.4 Mn = 2; Kp = 0.5; Sr = 0.2; S = 0.5; Pr = 8; Nb = 0.5; Nt = 0.2; Le = 0.6; Che = 0.5;

f(η)

0.3

Rc = 0 Rc = 0.01 Rc = 0.1 Rc = 0.2

0.2 0.1 0 -0.1 0

1

2

3

4

Figure 15.7  Results of Rc on f(η) for Mn = 2.

5 η

6

7

8

9

10

234  Mathematics and Computer Science Volume 2 The concentration profiles Ø(η) for various values of chemical repercussion parameter Che are portrayed in Figure 15.8. The temperature and chemical reaction parameters are inversely related. The influence of the plate concentration parameter S, Lewis parameter Le, and Porosity parameter Kp on the concentration profile is shown in Figure 15.9, 15.10, and 15.11. From Figures 15.8 and 15.9, it is audited that concentration gradient parameter Ø(η) diminishes with η for Mn keeping constant, Mn = 2, i.e., the plate concentration parameter and Lewis parameter are inversely proportional with concentration gradient. It is also audited that with the enhancement on η, after a certain time there will be no effect of plate concentration parameters on concentration gradient. Figure 15.11 illustrates the result of porosity parameter on velocity gradient and it is seen that appearance of porous matrix increases the velocity 1

Φ(η)

0.8 Mn = 2; Kp = 0.5; Sr = 0.2; S = 0.5; Pr = 8; Nb = 0.5; Nt = 0.2; Le = 0.6; Rc = 0.1

0.6

Che = 0.5 Che = 2 Che = 5 Che = 8

0.4 0.2 0 0

1

2

3

4

5 η

6

7

8

9

10

Figure 15.8  Results of chemical reaction on Ø(η) for Mn = 2, Rc = 0.1, S = 0.5, Pr = 8. 1 0.8 Mn = 2; Kp = 0.5; Sr = 0.2 Pr = 8; Nb = 0.5; Nt = 0.2; Le = 0.6; Rc = 0.1; Che 0.5

Φ(η)

0.6

S = 0.5 S=2 S=8 S = 10

0.4 0.2 0 -0.2

0

1

2

3

4

5 η

6

7

8

9

10

Figure 15.9  Results of plate concentration on Ø(η) for Mn = 2, Rc = 0.1, Che = 0.5.

Viscoelastic MHD Flow Over an Expanded Lamina  235 1 0.8 Mn = 2; Kp = 0.5; Sr = 0.2; Pr = 8; Nb = 0.5; Nt = 0.2; Le = 0.6; Rc = 0.1; Che = 0.5, S = 0.5

Φ(η)

0.6

Le = 0.6 Le = 1 Le = 2 Le = 5

0.4 0.2 0 -0.2 0

1

2

3

4

5 η

6

7

8

9

10

Figure 15.10  Results of Lewis parameter Le on Ø(η) for Mn = 2, Rc = 0.1, Che = 0.5.

0.6 0.5

f(η)

0.4 0.3

Mn = 2; Sr = 0.2; Pr = 8; Nb = 0.5; Nt = 0.2; Le = 0.6; Rc = 0.1; Che = 0.5, S = 0.5

0.2

Kp = 0.5 Kp = 1 Kp = 1.6 Kp = 2.5

0.1 0 -0.1 0

1

2

3

4

5 η

6

7

8

9

10

Figure 15.11  Results of Kp on f(η) for Mn = 2, Rc = 0.1, Che = 0.5.

at each point and after a certain time, transverse velocity becomes parallel to the axis of η.

15.4 Conclusion The solutions for a steady boundary layer stream and heat distribution for a nanofluid throughout an exponentially expanded lamina in the presence of a chemical reaction is analysed. Results of the magnetic parameter and the viscoelastic parameter on various velocities are discussed. The impression of the magnetic parameter, Prandtl number, and the heat sink/source parameter on the temperature profiles are presented. The main findings of this research may be shortened as follows:

236  Mathematics and Computer Science Volume 2 a) Concentration and temperature are directly proportional to the magnetic parameter b) Elastic parameter decreases the temperature at every point c) Plate concentration parameter and Lewis parameter are inversely proportional with concentration profile d) Presence of magnetic parameter and porous matrix decreases the velocity at all points

References 1. D. Pal and H. Mondal, Influence of thermophoresis and Soret-Dufour on magnetohydrodynamic heat and mass transfer over a non-isothermal wedge with thermal radiation and Ohmic dissipation, J. Magn. Magn. Mater., 331, 250–255, 2013. 2. D. Pal and H. Mondal, Influence of Soret and Dufour on MHD buoyancy-­ driven heat and mass transfer over a stretching sheet in porous media with temperature-dependent viscosity, Nucl. Eng. Des., 256, 350–357, 1970. 3. S. Baag, S. R. Mishra, G. C. Dash, and M. R. Acharya, Entropy generation analysis for viscoelastic MHD flow over a stretching sheet embedded in a porous medium, Ain Shams Eng. J., 8, no. 4, 623–632, 2017. 4. D. Pal and H. Mondal, Influence of temperature-dependent viscosity and thermal radiation on MHD forced convection over a non-isothermal wedge, Appl. Math. Comput., 212, no. 1, 194–208, 2009. 5. Crane LJ. Flow past a stretching plane. Jape Math Phys (ZAMP), 21, 645–7, 1970. 6. Ishak A, Nazar R, Pop I. Heat transfer over a stretching surface with variable heat flux in micropolar fluids. Phys Lett A, 372, 559–61, 2008. 7. Ishak A, Nazar R, Arifin NM, Pop I. Mixed convection of the stagnation point flow towards a stretching vertical permeable sheet. Malaysian J Math Sci 2, 217–226, 2007. 8. Ishak A, Nazar R, Pop I. Mixed convection boundary layers in the stagnation point flow towards a stretching vertical permeable sheet. Mechanica 41, 509–518, 2006. 9. Yao S, Fang T, Zhong Y. Heat transfer of a generalized stretching/shrinking wall problem with convection boundary conditions. Commun Nonlinear Sci Numer Simul 16, 752–760, 2011. 10. Quasim M, Hayat T, Obaidat S. Radiation effect on the mixed convection flow of a viscoelastic fluid along an inclined stretching sheet. Z Naturforsch 67, 195–202, 2012. 11. Hayat T, Quasim M, Mesloub S. MHD flow and heat transfer over a stretching sheet with Newtonian heating with slip condition. Int J Numer Methods Fluid 66, 963–975, 2011.

Viscoelastic MHD Flow Over an Expanded Lamina  237 12. Hayat, T., Qayyum, S., Alsaedi, A. and Ahmad, B., Magnetohydrodynamic (MHD) nonlinear convective flow of Walters-B nanofluid over a nonlinear stretching sheet with variable thickness, International Journal of Heat and Mass Transfer, 110, 506-514, 2017. 13. Daniel, Y.S., Aziz, Z.A., Ismail, Z. and Salah, F., Double stratification effects on unsteady electrical MHD mixed convection flow of nanofluid with viscous dissipation and Joule heating, Journal of Applied Research and Technology, 15(5), 464-476, 2017. 14. Hayat, T., Qayyum, S., Khan, M.I. and Alsaedi, A., Entropy generation in magnetohydrodynamic radiative flow due to rotating disk in presence of viscous dissipation and Joule heating, Physics of Fluids, 30(1), 017101, 2018. 15. Hayat, T., Qayyum, S., Alsaedi, A. and Shafiq, A., Inclined magnetic field and heat source/sink aspects in flow of nanofluid with nonlinear thermal radiation, International Journal of Heat and Mass Transfer, 103, 99-107, 2016. 16. Kishan, N. and Maripala, S., Thermophoresis and viscous dissipation effects on Darcy-Forchheimer MHD mixed convection in a fluid saturated porous media, Advances in Applied Science Research, 3(1), 60-74, 2012. 17. Ibrahim, W., Shankar, B. and Nandeppanavar, M.M., MHD stagnation point flow and heat transfer due to nanofluid towards a stretching sheet, Int. J. Heat Mass Transfer., 56, 1-9, 2013. 18. Turkyilmazoglu, M., Exact analytical solutions for heat and mass transfer of MHD slip flow in nanofluids, Chem. Eng. Sci., 84, 182-187, 2012. 19. Farooq, M., Ijaz Khan, M., Waqas, M., Hayat, T., Alsaedi, A. and Imran Khan, M., MHD stagnation point flow of viscoelastic nanofluid with non-linear radiation effects, Journal of Molecular Liquids., 221, 1097-1103, 2016. 20. Sheikholeslami, M., Hari Kataria, R. and Akhil Mittal, S., Effect of thermal diffusion and heat-generation on MHD nanofluid flow past an oscillating vertical plate through porous medium, Journal of Molecular Liquids, 257, 12-25, 2018. 21. Baag, S., Mishra, S. R., Dash, G. C. and Acharya, M. R., Entropy generation analysis for viscoelastic MHD flow over a stretching sheet embedded in a porous medium, Ain Shams Engineering Journal, 8(4), 623-632, 2017. 22. Sheikholeslami, M. and Rokni, H. B., Influence of melting surface on MHD nanofluid flow by means of two-phase model, Chinese J. Physics, 55(4), 1352-1360, 2017. 23. Abdul Hakeem, A.K., Govindaraju, M. and Ganga, B., Influence of inclined Lorentz forces on entropy generation analysis for viscoelastic fluid over a stretching sheet with nonlinear thermal radiation and heat source/sink, J. Heat Mass Transfer Research., 6(1), 1-10, 2019. 24. Emad M. Abo-Eladahab, Mohamed A. El Aziz, Blowing/suction effect on hydromagnetic heat transfer by mixed convection from an inclined continuously stretching surface with internal heat generation/absorption, Int. J. Therm. Sci., 43, 709-719, 2004.

238  Mathematics and Computer Science Volume 2 25. Subhas Abel, M. and Mahesha, N., Heat transfer in MHD viscoelastic fluid flow over a stretching sheet with variable thermal conductivity, non-uniform heat source and radiation, Appl. Math. Model., 32, 1965-1983, 2008. 26. Ganga, B., Govindaraju, M. and Abdul Hakeem, A.K., Effects of inclined magnetic field on entropy generation in nanofluid over a stretching sheet with partial slip and nonlinear thermal radiation, Iranian Journal of Science and Technology: Transactions of Mechanical Engineering, 1139, 2018. 27. Zeeshan A, Majeed A. Effect of magnetic dipole on radiative non-­Darcian mixed convective flow over a stretching sheet in porous medium. J Nanofluids 4, 617–626, 2016. 28. H Mondal, S Ghosh, PK Roy, S Chatterjee, Effects of Ion-Slip and Hall Currents on Magnetohydrodynamic Nanofluid Flow with Thermal Diffusion Using Spectral Quasi-Linearization Method, Journal of Nanofluids 10 (4), 608-615. 29. S. S. Motsa. A new spectral local linearization method for nonlinear boundary layer flow problems. Journal of Applied Mathematics, 2013.

16 Quickest Multi-Commodity Contraflow with Non-Symmetric Traversal Times Shiva Prakash Gupta1*, Urmila Pyakurel2 and Tanka Nath Dhamala2 Tri-Chandra Multiple Campus, Tribhuvan University, Kathmandu, Nepal Central Department of Mathematics, Tribhuvan University, Kathmandu, Nepal 1

2

Abstract

Routing multiple distinct commodities from particular supply points to appropriate destination points via the arcs of a network topology while preserving capacity limits is the most challenging issue in operations research. The quickest multi-commodity flow issue reduces the delivery time to complete the process. Computationally, this problem is NP-hard. We consider asymmetric transit times on anti-parallel lanes due to uneven road conditions and flow dependency. Outbound lane capacity can be boosted by reorienting lanes towards demand locations. We incorporate the contraflow strategy to the quickest multicommodity flow problem and provide an approximation scheme utilizing lengthbounded flow with non-symmetric traversal times on oppositely oriented lanes. Keywords:  Network flow, quickest multi-commodity, contraflow, non-symmetric traversal times, length-bound

16.1 Introduction The topology of a transport system is described as a network that correlates to the transshipment of diverse commodities, with distribution centers, demand centers, and intersections of road segments serving as nodes and arcs serving as linkages between nodes. The starting and ending destinations of commodities are known as supply points and demand points, respectively. Flow refers to the collection of goods that have been *Corresponding author: [email protected] Sharmistha Ghosh, M. Niranjanamurthy, Krishanu Deyasi, Biswadip Basu Mallik, and Santanu Das (eds.) Mathematics and Computer Science Volume 2, (239–250) © 2023 Scrivener Publishing LLC

239

240  Mathematics and Computer Science Volume 2 transported via the network. In networks with temporal dimensions capacity and travel times are allocated to the arcs. For more details, we refer to [1−5]. The multi-commodity network flow problem entails transporting several commodities from specified delivery sites to appropriate destination points while staying within arc capacity restrictions. Static flow and dynamic flow are two types of flow that can occur in this scenario. Maximum, maximum concurrent, and least cost flow issues are the three categories of static flow problems. Maximum flow issues emerge when the aggregate of all commodity flows is maximized. The maximum flow problem wherein the proportion of total demand is maximized is a maximum concurrent flow problem. The minimal cost flow problem is described as determining the flow value that meets all commodity demands at the lowest cost while adhering to capacity constraints on all arcs. Furthermore, dynamic multi-­commodity flow problems may be classified as maximal dynamic, quickest, minimum cost, and earliest arrival flow problems (see [4−8]). Ford and Fulkerson [9] are credited with inventing network flow over time in the late 1950s. In the theory of supply and demand, the aim is to compute the quickest way to fulfill the needs for something, i.e., to get it from origin to destination is the quickest flow problem. The first polynomial-time bound was computed by Burkard et al. [10] for this problem using a binary search. They reduced the complexity and gave strong polynomial-time bounds with the help of a parametric search. The quickest transshipment problem is one in which a flow of commodities fulfills all supplies and demands in the least amount of time. Authors in [11] generalize the naive method to solve a dynamic flow problem to the situation of multi-commodities. Furthermore, the authors used length-bounded static flows instead of methodology, which used a static minimal cost flow computation. Even though static flows do not contain a time component, Fleischer and Skutella [11] looked at flows that offer feasible routes using travel time as a measure of length. They divide static flows into pathways, with each path starting at an origin node and ending at a corresponding destination node. Multi-commodity flow problems are more difficult than single-commodity flow problems, even in graphs with series-parallel composition or with only two commodities. The author in [12] found that the flow of several commodities over time is NP-hard. NP-hardness exists in multiple products with and without intermediate node storage for the quickest flow problem with a simple flow path. Because this issue is NP-hard, two techniques to obtain an approximate solution have been presented in [11, 13].

Multi-commodity Contraflow with Non-symmetric Traversal Times  241 Lozovanu et al. [14] and Kappmeir [15] used a time expansion approach for solving the maximum dynamic multi-commodity flow problem. This approach is extended to multiple sources and a single sink multicommodity earliest arrival transshipment in [15]. Contraflow refers to the switching arc orientations to boost traffic flow value and minimize traversal time by increasing the capacity of lanes in a network. For a single-source and a single-sink, the authors in [16] proposed models and strongly polynomial-time methods to solve maximum and quickest flow problems. At time zero, these contraflows are made and then rectified. Nath et al. [17] investigated the problem with asymmetric traversal times on oppositely directed lanes and modified the algorithm of [16] so that traversal time of a non-reversed arc is taken after lane reversals and solved the problem in polynomial-time for a single-commodities. In [18], the authors reported a polynomial-time solution to the earliest arrival transshipment problem and the maximum flow problem with priority on terminals. Gupta et al. [19] investigated the non-polynomial solution of the generalized maximum flow issue on a lossy network with nonsymmetric traversal times on anti-parallel lanes. Recently, the authors in [20] extended this to the case of multi-commodity with symmetric transit times and the same authors in [21] provided FPTAS to a maximum dynamic multi-commodity contraflow problem with non-symmetric traversal times on oppositely oriented lanes. The fundamental objective of partial contraflow is to use the capacity of underutilized lanes and was introduced by Pyakurel et al. [22]. In times of emergency, the stored capacity of unoccupied arcs might be utilized for logistical assistance and facility placement. We present the quickest multi-commodity contraflow (QMCCF) problem with non-symmetric traversal times on anti-parallel lanes in this paper. Fleischer and Skutella [11, 13] developed an approximation algorithm to this problem without contraflow using a T-length bounded function. The authors in [23, 24] introduced a partial lane reversal approach with symmetric traversal times on oppositely directed lanes. We provide the polynomial-time approximate solution of the problem by utilizing a T-length bound approximation. The travel time for transshipping commodities from origin nodes to destination nodes is reduced by using the lane reversal strategy in the routing problem. The study’s main relevance is the decrease in delivery time. The rest of the article is arranged as follows. Section 16.2 describes fundamental concepts with flow models. The QMCCF problem with non-­ symmetric traversal times on oppositely oriented lanes is introduced in

242  Mathematics and Computer Science Volume 2 Section 16.3. We provide a method for solving the issue in polynomial-time in this section. The final portion concludes the paper.

16.2 Preliminaries with Flow Models The multi-commodity flow problem entails distributing several commodities from their origin-destination pairs across a specified network to meet the entire demand for each commodity. We define the basic notations and provide mathematical formulations for this problem, in which arc reversals are allowed to optimize the objective value by reverting their directions as needed.

16.2.1 Mathematical Model with Contraflow Suppose a network architecture N = (V, A, K, u, τ, di, S+, S−, T) is composed of a collection of nodes V and arcs A. The collection of commodities is denoted by K = {1, 2, ..., k}, with |V | = n and |A| = m. The demand di is set and it is delivered via a particular origin-destination path si-ti, where si ∈ S+ and ti ∈ S−, ∀i ∈ K. The capacity function u : A → R+ regulates the flow and traversal time function. τ : A → R+ computes the time it takes to carry the flow on each arc, e. In discrete and continuous-time contexts, the time period T is indicated as T = {0, 1, ..., T} and T = [0, T], respectively. The sets δ+(v) = {e : e = (v, .) | . ∈ V} and δ−(v) = {e : e = (., v) | . ∈ V } represent a set of arcs moving out from v and entering into v, respectively, so that δ+(S−) = δ−(S+) = ∅ besides contraflow network. The auxiliary network for network N is represented by Na = (V, Aa, K, a a u , τ , di, S+, S−, T), with the edges having no direction. The arc er represents the backward arc of e. The capacity is re-defined as the total of the capacities of the anti-parallel lanes, in Na with ue = 0, if e ∉A. The traversal time τa along with arc a in Na is the same as the traversal time of non-reversed arcs. For a single lane, we assume τ a = τ e = τ er .. Other network parameters are unaltered. A static network is one that lacks a time dimension and is defined by N = (V, A, K, u, di, S+, S−). The static flow function is defined by x : A → R+. To compute the solution of real world dynamic flow problems, many useful properties of static network topology are essential tools. A multi-commodity flow over time Φ in a network N with contraflow is a collection of commodities described by Φi : Aa × T → R+ meeting the requirements (16.2−16.4).



min  T

(16.1)

Multi-commodity Contraflow with Non-symmetric Traversal Times  243 subject to,

 di if v = si  Φ (θ ) − Φ (θ − τ e ) =  − di if= v = ti ∀i ∈ K + −  0 otherwise θ =0 e∈δ (v ) θ =τ e e∈δ (v )  T

∑∑

i e



∑∑

i e

(16.2)

σ



T

∑∑

θ =0 e∈δ + (v )

σ

Φie (θ ) −

∑ ∑ Φ (θ −τ ) ≤ 0, ∀v ∉ {s ,t },i ∈ K ,σ ∈T i e

e

i

i

θ =τ e e∈δ − (v )



(16.3)

∑

0 ≤ Φ e (σ ) =Φie (σ ) ≤ ue + uer , ∀e ∈ Aa , i ∈ K ,σ ∈ T (16.4) e∈δ + (v )

In this case, the third equation in constraint (16.2) denotes flow conservation restrictions at time T, but the constraints in (16.3) denote non-­ conservation needs at intermediate time points T. Likewise, capacity with lane reversals limits the bundle restrictions in (16.4). The purpose is to deliver a predetermined quantity of flow to fulfill the demand for each commodity from supply points to destination locations in the least amount of time, as specified by the first two conditions of (16.2). The cost is defined by

c(Φ) =

T

∑ ∑ ∑ Φ (θ ) cei

i e

θ= 0

e∈A i∈K

(16.5)



The cost bound for each commodity i is determined as:



T

∑ ∑ Φ (θ ) ≤ C cei

e∈A

i e

θ= 0

(16.6)

i



Example 1. Suppose a two-commodity network, as illustrated in Figure 16.1(a), has capacity and travel times on arcs. The pathways s1 − t1 and s2 − t2

244  Mathematics and Computer Science Volume 2 s1

3,1

3,1

4,1

2,1

4,2

x 2,2

s2

t2

v

6,2(1) 2,2 6,1

4,1 y

3,2

w

3,1

t2

v

5,1

3,1

4,1 x

3,2

4,2

y 10,2(1)

5,2 t1

(a)

s1

s2

6,1

w

7,1 8,2

3,2

t1 (b)

Figure 16.1  (a) Network with multi-commodity having capacity and traversal time on arcs. (b) Arc reversals with symmetric and non-symmetric traversal times.

are used to transport the first and second commodities, respectively. A contraflow network is depicted in Figure 16.1(b). Authors in [25] modeled the quickest contraflow problem for a single commodity as an integer programming problem. Furthermore, they proposed greedy and bottle-neck relaxation heuristics to provide a numerical solution. Rebennack et al. [16] presented its polynomial-time solution. However, the problem with numerous sources and/or sinks is NP-complete via a reduction from 3-SAT [16]. In [26, 27], the continuous-time variant of a single-source single-sink maximum and the quickest contraflow problems are polynomially solvable. The authors in [22] introduced and solved the problem with partial lane reversals.

16.3 QMCCF with Non-Symmetric Transit Times The lane reversal approach for the QMCCF problem with non-symmetric traversal times on oppositely oriented lanes is discussed in this section. In a network architecture where arc reversals are allowed, a solution to this problem meets given demands at defined nodes in the shortest time feasible. We present a polynomial-time approximation scheme (PTAS) to compute the solution. Furthermore, our technique may be used to save lanes that do not need to be reversed to shorten the time. Instead of symmetric traversal times on oppositely oriented lanes in [23, 24], we consider asymmetric traversal times on oppositely oriented lanes and present Problem 1. Problem 1. Assume a network N = (V, A, K, u, τ, S+, S−, di, T) has non-­ symmetric traversal times on oppositely oriented lanes. The QMCCF problem aims to find the minimum feasible time that meets the requirements (16.2), (16.3), and (16.4) to transport a specified number of demands di, ∀i ∈ K from

Multi-commodity Contraflow with Non-symmetric Traversal Times  245 supply points to corresponding destination points at a bounded cost by reverting the orientation of only essential arcs from the beginning. Even with a series-parallel network composition or network having two commodities only, the multi-commodity flow over time problem is NP-hard [12]. The proof is based on reductions from the NP-hard PARTITION and 3-PARTITION issues. In addition, the problem of maximal multicommodity flow with no limits on intermediate node storage is NP-hard. A time expansion is a tool that provides the pseudo-polynomial time solution. The same hardness exists to the quickest multi-commodity flow problem having a simple path and without intermediary node storage according to [12]. Authors in [25] used 3-SAT to show that lane reversal is NP-complete. As a result, we have Theorem 1. Theorem 1. The QMCCF problem with bounded cost having asymmetric traversal times on anti-parallel arcs is NP-hard. As the quickest multi-commodity flow problem is NP-hard, two techniques are proposed to provide an approximate solution in [13]. One is the discretization of bigger time steps as opposed to single time steps, while the other is the length-bounded flow. The authors in [23, 24] incorporated partial lane reversals with equal traversal times on oppositely oriented lanes and provided the solution using the same approach. We consider non-symmetric traversal times on oppositely oriented lanes and provide a PTAS to Problem 1 with the help of length-bounded flow.

16.3.1 Approximation Approach for the QMCCF A single or multi-commodity path flow is considered as a length-bounded flow, where each used path is subjected to a length restriction. Let Pi represent the sum of all si − ti routes ∀i ∈ K in the network Na. A static flow x is known as a T-length bounded if every constituent flow xi for each x Pi i ∈ K can be broken down into the amount of flows x Pi , i.e., x i = with x Pi > 0 and the length τ P =

∑ e ∈P

∑ P ∈Pi

τ e of any path P ∈ Pi is at most T.

A T-length, restricted static flow x that matches multi-commodity constraints is NP-hard to discover [13]. This issue, on the other hand, is polynomial-time approximated. We compute an approximate solution to the QMCCF problem with non-symmetric traversal times on anti-parallel lanes by applying lengthbounded flows as described [23, 24] and presented in Algorithm 1. In Step 2 of our approach, we first review the solution technique of [11, 13]. All

246  Mathematics and Computer Science Volume 2 other steps of the algorithm are feasible according to [23]. The removal of the cycle flow in Step 3 suggests that flow only travels in one direction, not both. In Step 5, the capacity of the unutilized arcs is preserved. We argue that a possible approximate solution to the QMCCF with non-symmetric traversal time on oppositely oriented lanes on network N would likewise be a feasible approximate solution to the quickest multi-commodity flow issue on modified network Na. The authors in [17] have shown that the complexity of the lane reversal problems having non-symmetric travel times on oppositely oriented lanes is the same as symmetric travel time on oppositely oriented lanes. As a result, an approximate QMCCF with non-symmetric traversal time on oppositely oriented lanes may be optimally calculated on network N. Hence, we have Theorem 2. Algorithm 1 T-length Bound Algorithm for QMCCF with Asymmetric Transit Times Input: A network N = (V, A, K, u, τ, di, S+, S−, T ) with non-symmetric travel times Output: The QMCCF 1. Construct the modified network by adding capacities of anti-parallel arcs on Na = (V, Aa, K, ua, τa, di, S+, S−, T) as



u= ue + uer ,, a

r reverted towards e e re is τ e if arc if arc is reverted towards e τ a : = τa: =  er er . if arc e eisreverted τ er if arc reverteetowards d towards

2. Calculate the flow on the modified network Na by applying an algorithm of [13] to the problem having bounded cost. 3. Remove cycles by decomposing the flow along the origin-destination pathways and cycles ∀i ∈ K. 4. Revert er ∈ A to the capacity xe − ue iff xe > ue, ue substituted by 0 whenever e ∉A, ∀i, where x e =

k

∑ i =1

x ei and ue =

k

∑u

i e

i =1

5. For each e ∈ A, if er is reverted, the saved capacity of arc er is ua − xe and the saved capacity of arc e is zero. If none of the arcs are e and er is reversed, sc(e) = ue − xe > 0.

Multi-commodity Contraflow with Non-symmetric Traversal Times  247 Theorem 2. An approximate solution to the QMCCF problem with nonsymmetric traversal times on oppositely oriented lanes can be determined in polynomial-time by using T-length bound. Example 2. Consider a network as illustrated in Figure (16.1)(a) with two commodities having demands d1 = 12 and d2 = 8. The quickest time without contraflow with symmetric and asymmetric traversal times on lanes can be computed by using Figure 16.1(a), and the quickest time with contraflow having symmetric and asymmetric traversal times on lanes can be computed by using Figure 16.1(b), respectively. There is only one path for commodity-1 from s1 to t1 and only one path for commodity-2, i.e., s2 to t2 before contraflow with symmetric or asymmetric traversal times (cf. Figure 16.1(a)). A 5-length bound is required for computing the solution of Problem 1 to the quickest multi-commodity without contraflow having symmetric traversal times (cf. Figure 16.1(a)), and it is repeated four times. As a result, T1 = 8 units of time are required to meet the indicated demand d1 = 12 of commodity-1. Similarly, a 5-length bound is required and is repeated four times for commodity-2. As a result, it takes T2 = 8 units of time to meet demand d2 = 8 of commodity-2. Hence, the shortest possible time needed to fulfill both needs is T = max{T1, T2} = {8, 8} = 8. If we take asymmetric traversal times on oppositely oriented lanes before contraflow (cf. Figure 16.1(a)), a 4-length bound is essential to meet the demand of commodity-1 and it is repeated 4-times. So, T1 = 7 is the shortest time needed to fulfill the demand of commodity-1. Similarly, a 4-length bound is essential to meet the need of commodity-2 and it is repeated 4-times. So, to fulfill the need of commodity-2, T2 = 7 is the quickest time. Hence, the shortest possible time needed to fulfill both needs is T = max{T1, T2} = {7, 7} = 7. However, if contraflow is used (see Figure 16.1(b)), and traversal time is symmetric, the time taken to meet both needs is T = 6 units. In the same way, if lane reversal is used (see Figure 16.1(b)), and traversal time is non-symmetric on oppositely oriented lanes, it requires T = 5 units of time to meet both demands. Thus, contraflow with symmetric traversal times saves approximately 25 percent of the time, whereas contraflow with non-symmetric traversal times on oppositely oriented lanes saves approximately 28.7 percent of the time. Table 16.1 and Table 16.2 summarize the data obtained from Example 2. Table 16.1  Quickest time without contraflow. LB without contraflow ST 8

LB without contraflow AT 7

248  Mathematics and Computer Science Volume 2 Table 16.2  quickest time with contraflow. LB with contraflow ST 6

LB with contraflow AT 5

LB=length bound, ST=symmetric transit times, AT=asymmetric transit times.

16.4 Conclusions One of the primary issues in operations research is routing many commodities from their delivery points to target points over a shared network. A crucial concern is the reduction of time (cost). A well-known quickest flow problem was addressed to deliver the requisite demand in the shortest feasible period. For a single-commodity, this issue was solved in strongly polynomial-time, but for several commodities it is NP-hard. However, a polynomial-time approximation technique based on a length-bounded function was found. The lane reversals technique is an essential tool for improving the quickest time in the two-way network. We incorporated this approach in length-bounded approximation. In this work, we investigated the QMCCF problem with non-symmetric traversal times on oppositely oriented lanes. We presented its mathematical model and provided a polynomial-time approximation algorithm. This study minimizes the routing time significantly. By analyzing the results of the previously studied problem with constant transit time, it would be interesting to extend these strategies to flow-dependent transit times. The findings in this research have both theoretical and practical implications.

Acknowledgments The first author wishes to thank the University Grant Commission of Nepal for the Ph.D. research grant and the second author wishes to thank the Alexander Von Humboldt Foundation for the Digital Cooperation Fellowship (August 1, 2021 to January 31, 2022).

References 1. R. K. Ahuja, T. L. Mangati, and J. B. Orlin, Network flows: theory, algorithm, and applications, Prentice Hall, Englewood Cliffs, 1993.

Multi-commodity Contraflow with Non-symmetric Traversal Times  249 2. A. Assad, Multi-commodity network flows a survey, Networks, Vol. 8(1), pp. 37-91, 1978. 3. J. Kennington, A survey of linear cost multi-commodity network flows, Operations Research, Vol. 26(2), pp. 209-236, 1978. 4. K. Salimifard and S. Bigharaz, The multi-commodity network flow problem: state of the art classification, applications, and solution methods, Springer, pp. 1-47, 2020. 5. I.-L. Wang, Multi-commodity network flows: A survey, part I: Applications and formulations, International Journal of Operations Research, Vol. 15(4), pp. 145-153, 2018. 6. A. Ali, R. Helgason, J. Kennington, and H. Lall, Computational comparison among three multi-commodity network flow algorithms, Operations Research, Vol. 28(4), pp. 995-1000, 1980. 7. J. A. Tomlin, Minimum cost multi-commodity network flows, Operational Research, Vol. 14, pp. 45-51, 1966. 8. U. Pyakurel, S. P. Gupta, D. P. Khanal, and T. N. Dhamala, Efficient algorithms on multi-commodity flow over time problems with partial lane reversals, International Journal of Mathematics and Mathematical Sciences, Hindawi, pp. 1-13, 2020. Article ID 2676378, https://doi.org/10.1155/2020/2676378. 9. L. R. Ford and D. R. Fulkerson, Flows in networks, Princeton University Press, Princeton, New jersey, 1962. 10. R. E. Burkard, K. Dlaska, and B. Klinz, The quickest flow problem. ZORMethods and Models of Operations Research, Vol. 37, pp. 31-58, 1993. 11. L. Fleischer and M. Skutella, The quickest multicommodity flow problem, In International Conference on Integer Programming and Combinatorial Optimization, Springer, pp. 36-53, 2002. 12. A. Hall, S. Hippler, and M. Skutella, Multi-commodity flows over time: efficient algorithms and complexity. Science Direct, Vol. 379, pp. 387-404, 2007. 13. L. Fleischer and M. Skutella, Quickest flows over time, SIAM Journal on Computing, Vol. 36(6), pp. 1600-1630, 2007. 14. D. Lozovanu and M. Fonoberova, Optimal dynamic multicommodity flows in networks, Electron Notes Discrete Math, Vol. 25, pp. 93-100, 2006. 15. P. W. Kappmeier, Generalizations of flows over time with application in evacuation optimization, PhD Thesis, Technical University, Berlin, Germany, 2015. 16. S. Rebennack, A. Arulselvan, L. Elefteriadou, and P. M. Pardalos, Complexity analysis for maximum flow problems with arc reversals, Journal of Combinatorial Optimization, Vol. 19, pp. 200-216, 2010. 17. H. N. Nath, U. Pyakurel, and T. N. Dhamala, Network reconfiguration with orientation dependent transit times, International Journal of Mathematics and Mathematical Sciences, Hindawi, pp. 1-11, 2021. Article ID 6613622, https://doi.org/10.1155/2021/6613622. 18. S. P. Gupta, U. Pyakurel, and T. N. Dhamala, Network flows with arc reversals and non-symmetric transit times, American Journal of Mathematics and Statistics, Vol. 11(2), pp. 27-33, 2021.

250  Mathematics and Computer Science Volume 2 19. S. P. Gupta, U. Pyakurel, and T. N. Dhamala, Generalized dynamic contraflow with non-symmetric transit times, American Journal of Computational and Applied Mathematics, Vol. 11(1), pp. 12-17, 2021. 20. S. P. Gupta, U. Pyakurel, and T. N. Dhamala, Multi-commodity flow problem on lossy network with partial lane reversals. Annals of Operations Research (ANOR), 323(1-2), 45-63, 2023, https://doi.org/10.1007/s10479-023-05210-y 21. S. P. Gupta, U. Pyakurel, and T. N. Dhamala, Dynamic multi-­commodity contraflow problem with asymmetric transit times, Journal of Applied Mathematics, Hindawi, pp. 1-8, 2022. Article ID 3697141, https://doi. org/10.1155/2022/3697141. 22. U. Pyakurel, H. N. Nath, S. Dempe, and T. N. Dhamala, Efficient dynamic flow algorithms for evacuation planning problems with partial lane reversal, Mathematics, Vol. 7, pp. 1-29, 2019. 23. T. N. Dhamala, S. P. Gupta, D. P. Khanal, and U. Pyakurel, Quickest multi-­ commodity flow over time with partial lane reversals, Journal of Mathematics and Statistics, Vol. 16(1), pp. 198-211, 2020. 24. S. P. Gupta, D. P. Khanal, U. Pyakurel, and T. N. Dhamala, Approximate algorithms for continuous-time quickest multi-commodity contraflow problem, The Nepali Mathematical Sciences Report, Vol. 37(1-2), pp. 30-46, 2020. 25. S. Kim, S. Shekhar, and M. Min, Contraflow transportation network reconfiguration for evacuation route planning, IEEE Trans. Knowl. Data Eng., Vol. 20, pp. 1115-1129, 2008. 26. U. Pyakurel and T. N. Dhamala, Evacuation planning by earliest arrival contraflow, Journal of Industrial and Management Optimization, Vol. 13, pp. 487501, 2017. 27. U. Pyakurel, T. N. Dhamala, and S. Dempe, Efficient continuous contraflow algorithms for evacuation planning problems. Annals of Operations Research (ANOR), Vol. 254, pp. 335-364, 2017.

17 A Mathematical Representation for Deteriorating Goods with a Trapezoidal-Type Demand, Shortages and Time Dependent Holding Cost Ruma Roy Chowdhury

*

Department of Basic Science and Humanities, University of Engineering & Management, Kolkata, India

Abstract

A production inventory model for a perishable good decaying at a constant rate has been devised in this research article. A ramp-type demand function has been considered, that is, the demand follows an accelerated growth initially for some time and finally becomes constant. The model allows shortage and is completely backlogged. The cost of holding the inventory is considered to vary linearly with time. Four cases have been developed depending on the position of time point at which the demand becomes constant in a trapezoidal-type demand. The production cycle restarts after a certain time. Optimal production stopping time and resuming time are calculated to optimize the expense of the production set-up per unit. The model is illustrated with a numerical. Keywords:  Trapezoidal demand, holding cost varying with time, completely backlogged short-ages, constant deterioration

17.1 Introduction Demand is always fluctuating according to the requirements and preferences of customers and is quite unpredictable. In case of new launches of items like cosmetics, electronic gadgets with new technologies, trendy Email: ruma [email protected] Sharmistha Ghosh, M. Niranjanamurthy, Krishanu Deyasi, Biswadip Basu Mallik, and Santanu Das (eds.) Mathematics and Computer Science Volume 2, (251–264) © 2023 Scrivener Publishing LLC

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252  Mathematics and Computer Science Volume 2 garments, etc. in the market, the items are in high demand initially. But, with the passage of time as more improvised items come into the market, the demand for the old ones become constant and stabilized. In order to deal with this kind of demand trend, trapezoidal demand has been dealt with in various studies since Ritchie (1980) [1]. Shaikh et al. (2020) [2] devised a stock optimization model for perishing goods with a preservation facility and a trapezoidal demand. Yang (2019) [3] developed an optimization model for a trapezoidal order trend having bi-level business credit type financing while placing orders. Chauhan et al. (2021) [4] developed an EOQ model for a trapezoidal type ordering system. During the stock out or shortage duration, backlogging is considered, also allowing certain discount strategies. Sana (2010) [5] considered deterioration and partial backlogging. Kawakatsu (2010) [6] devised an optimal refilling strategy for a retailer. A seasonal good with a trapezoidal order trend is considered with focus on Special Display Goods. Avinadav et al. (2013) [7] have developed a model where the demand varies with price of the item. Wang and Huang (2014) [8] devised a model for periodic items having price and trapezoidal-type order trends and deterioration. Ahmed et al. (2013) [9] proposed a fresh method for investigating the EOQ (Economic Order Quantity) assuming a trapezoidal demand, partial backlogging, and a general perish rate. Perishing of items is an unavoidable phenomenon that should not be neglected while formulating an inventory system. But, it definitely depends on the kind of item taken into consideration. For example, for highly deteriorating items like fruits, vegetables, etc., deteriorating items should be taken as a variable function of time, quantity, etc. But, items such as metals (steel utensils, iron rods, etc.), medicines (expiry), etc. deteriorate extremely slow and hence the rate of deterioration might be considered constant. In this study, the items being studied are of the non-depreciating type like electronic goods, medicines, COVID protective gears, etc. In recent time, Halima et al. (2021) [10] devised an overtime production model for goods with a constant rate of perish and the demand depending on non-linear price and stock dependent order trend. Mandal and Pal (1998) [11] have developed inventory models for perishing goods that follows a trapezoidal demand. Shortages of items are very common in the market system. When shortages occur, it is completely a customer’s will whether to wait for the backorder or to go to some other seller for instant availability. Backlogging rate varies with the holdback time for the next refilling. Goyal (1988) [12] established a model for replenishment of trendy/fashionable inventories including shortages. Giri et al. (2000) [13] devised a model on lot sizing

Mathematical Representation for Deteriorating Goods   253 study for perishing goods with stockout situations and demand depending on time. Wang (2002) [14] devised a replenishment strategy for perishing items with inventory running with shortages and thus, partial backlogging. Ghosh and Chaudhuri (2005) [15] developed an economic order quantity model for a perishable good with trended demand and stockout situations in each cycle. Pentico and Drake (2011) [16] did a study on deterministic models for the economic order quantity models and economic production quantity models considering backordering. Sharma and Singh (2013) [17] devised a replica for perishable goods, accounting for consumption rate and backlogging. Roy Chowdhury et al. (2014) [18] designed an order-level inventory model for a decaying good having a quadratic type ordering pattern and backlogging during a stock out situation in every cycle. Tripathi (2015) [19] developed a model for perishing goods with an escalating order trend and stock out situation under inflationary spiral. In the present work, a production-inventory model for a constantly perishable good with a trapezoidal-type order trend has been considered. The production starts at an initial time and is proportional to demand. The cost of holding inventory varies linearly with time. Shortage is allowed and is completely backlogged. Three cases have been dealt with depending on the position of the time at which the trapezoidal-type demand gains stability. We have illustrated the model numerically.

17.2 Assumptions and Notations 17.2.1 Assumptions 1. A single item is considered 2. Order for the item is supposed to be a ramp-type 3. The holding expense varies linearly with time, that is Ch = h + st(h, s > 0) 4. The production rate P is proportional to demand 5. Shortages occur and are completely backlogged 6. The stock is subjected to constant deterioration, θ 7. The time horizon, T, for the model is finite 8. The perished items are neither repaired nor replaced 9. Lead time is negligible

254  Mathematics and Computer Science Volume 2

17.2.2 Notation A: The unit ordering cost per cycle ($/unit) P (t): Production rate varies directly with demand D(t), that is, P (t) = βD(t); this means that the higher the demand, the greater the production; this is required for ramp-type demand as initially demand is more, hence production should also be more to satisfy the customer demand, but the demand stabilizes after sometime and the production has to be reduced accordingly in order to avoid wastage of items and loss ν: The parameter of the trapezoidal demand function (time point) where demand becomes constant D(t): In case of certain items like COVID gear, with the outbreak of COVID-19 in India in the year 2020, the demand of masks and PPEs saw an increasing demand in the beginning but after some time the demand became stabilized. This kind of demand is a ramp type demand which can be mathematically expressed as: Demand at the time t ≥ 0



D(t) = a + bt − (t − ν)H(t − ν)

(17.1)

where H(t − ν) is a Heviside function defined as



H (t − v ) = 1 = 0,

(t ≥ ν ) (t < ν )

(17.2)

CO : Overall ordering expense CH : Overall holding expense CP : Overall production expense CD : Overall deterioration expense CS : Overall shortage expense ch : Time dependent Holding Cost (h+st) in the time interval [0, t2] cp : Unit production expenses per item per unit time ($/unit/unittime) cd : Expenses of deterioration per item per unit time ($/unit) cs : Shortage expenses per unit per unit time θ : Constant deterioration rate of the item, where 0 < θ 0 at tm, tm, tm, where tm, tm, and tm are the solution of



 ∂ 2 ATC  *2  ∂t1  ∂ 2 ATC H= * *  ∂t 2∂t1  ∂ 2 ATC  ∂t *∂t *  3 1

∂ 2 ATC ∂t1*∂t 2* ∂ 2 ATC ∂t 2*2 ∂ 2 ATC ∂t 3*∂t 2*

∂ 2 ATC   ∂t1*∂t 3*  ∂ 2 ATC  ∂t 2*∂t 3*  ∂ 2 ATC  ∂t 3*2 

(17.21)

Mathematical Representation for Deteriorating Goods   261

17.4 Numerical Example Example 1: Case 1 (0 < µ < ta < tb < tc < T) We now illustrate the above developed production inventory model with a numerical example. Random data taken are as follows: A = 100, Cd = 10, a = 70, b = 20, β = 0.001, θ = 0.001, Cs = 15, T = 8, ν = 5.6, Cp = 10, e = 2.718, h = 5, s = 0.05 (with appropriate units). We use MATHEMATICA 7.0 to calculate the optimized values m m m tam , tbm , tcm . The optimized values of ta, tb, tc are t a = 7.2, t b 7.38, t c 7.6 units and the optimized average expense is ATC = 15, 613 units. Example 2: Case 2 (0 < ta < µ < tb < tc < T) We now illustrate the above developed production inventory model with a numerical example. Random data taken are as follows: A = 100, Cd = 10, a = 70, b = 20, β = 0.001, θ = 0.001, Cs = 15, T = 8, ν = 5.7, Cp = 10, e = 2.718, h = 4, s = 0.04 (with appropriate units). We use MATHEMATICA 11.3 to calculate the optimized values m m m tam , tbm , tcm . The optimal values of ta, tb, tc are ta = 5.618, tb 8.05, tc = 8.08 units and the optimized average expense is ATC = 16, 041 units. Example 3: Case 3 (0 < ta < tb < µ < tc < T ) We now illustrate the above developed production inventory model with a numerical example. Random data taken are as follows: A = 100, Cd = 10, a = 80, b = 10, β = 0.001, θ = 0.001, Cs = 20, Ch = 4, T = 8, ν = 7.42, Cp = 10, e = 2.718, h = 5, s = 0.04 (with appropriate units). We use MATHEMATICA 11.3 to calculate the optimized values m m m tam , tbm , tcm . The optimal values of ta, tb, tc are ta = 7.2954, tb 7.3939, tc = 7.496 units and the optimized average expense is ATC = 1, 47, 896 units.

17.5 Discussion Case 1: (0 < µ < ta < tb < tc < T) Here, ν < ta and this implies that the demand curve becomes constant before the production stops (ta = 7.2). Hence, the manufactured items will take some time to sell and the inventory level will fall to zero after some delay. This aspect is clear from the optimum values received in Example 1. Case 2: (0 < ta < µ < tb < tc < T) Here, ta < ν < tb and this implies that the demand curve becomes constant after the production stops (ta = 5.618). After the stabilization of demand,

262  Mathematics and Computer Science Volume 2 the level of the inventory reduces at a very slow rate. Hence, the level reduces to zero after a long duration at tb = 8.05. This is clear from the optimum values of tb, tc received in Example 2. Case 3: (0 < ta < tb < µ < tc < T) Here, tb < ν and this implies that the demand curve is stabilized after the inventory level dips below the zero mark and the shortage begins. Because the inventory is running in shortage, therefore the optimal time to start the production is almost immediately. This aspect is clear from the optimum values in Example 3.

17.6 Inference This study deals with a production-reserve (stock) model with a trapezoidal-type demand for an item that perishes constantly and inventory that is stored at a flexible cost. Shortages have been considered. The items considered are the ones like COVID gear (masks, PPE kits, etc.). Three cases have been discussed related to the demand change time and the inventory levels and production start and stop times. The decision variables are the production halt/break off time (ta), the time at which the stock gets exhausted (tb), and production restart time (tc). In the future, this study can be taken further by considering a flexible deterioration rate.

References 1. E. Ritchie Practical inventory replenishment policies for a linear trend in demand followed by a period of steady demand J. Oper. Res. Soc., 31 (7) (1980), pp. 605-613. 2. An inventory model for deteriorating items with preservation facility of ramp type demand and trade credit Ali Akbar Shaikh, Gobinda Chandra Panda, Md. Al-Amin Khan, Abu Hashan Md. Mashud, Amiya Biswas https://doi. org/10.1504/IJMOR.2020.110895 International Journal of Mathematics in Operational Research Volume 17, Issue 4, 2020. 3. Open Journal of Business and Management, Vol.7 No.2, April 2019 An Inventory Model for Ramp-Type Demand with Two-Level Trade Credit Financing Linked to Order Quantity Hui-Ling Yang DOI: 10.4236/ ojbm.2019.72029 4. Chauhan et al. [2021] An order quantity scheme for ramp type demand and back-logging during stock out with discount strategy, Anand Chauhan,

Mathematical Representation for Deteriorating Goods   263 Shilpy Tayal, https://doi.org/10.1504/IJSOI.2021.114113 International Journal of Services Operations and Informatics Volume 11, Issue 1 5. Sana, S., (2010), ‘Optimal selling price and lotsize with time varying deterioration and partial backlogging’, Appl. Math. Comput., Vol. 217, pp. 185-194. 6. H. Kawakatsu, Optimal retailers replenishment policy for seasonal products with ramp-type demand rate, IAENG Int. J. Appl. Math. 40 (4) (2010) 17. 7. Avinadav, T., Herbon, A., Spiegel, U. 2013. Optimal inventory policy for a perishable item with demand function sensitive to price and time. Int. J. Production Economics 144, 497506. 8. Pricing for seasonal deteriorating products with price-and ramp-type time-dependent demand C Wang, R Huang - Computers & Industrial Engineering, 2014 - Elsevier 9. M.A. Ahmed, T.A. Al-Khamis, L. Benkherouf Inventory models with ramp type demand rate, partial backlogging and general deterioration rate Appl. Math. Comput., 219 (2013), pp. 4288-4307. 10. Alexandria Engineering Journal Volume 60, Issue 3, June 2021, Pages 27792786 An over- time production inventory model for deteriorating items with nonlinear price and stock dependent demand Mohammad Abdul Halima A.Paul MonaMahmoud B.Alshahranic Atheelah Y.M.Alazzawi Gamal M.Ismaile 11. Mandal, B., Pal, A.K., 1998. Order level inventory system with ramp type demand rate for deteriorating items. Journal of Interdisciplinary Mathematics 1 (1), 4966. 12. Goyal, S.K., (1988), ‘A heuristic for replenishment of trended inventories considering shortages’, Journal of the Operational Research, Vol.39, pp.885-887. 13. Giri, B.C., Chakraborty, T., Chaudhuri, K.S., (2000), ‘A note on a lot sizing heuristic for deteriorating items with time-varying demands and shortages’, Computers and Operations Research, Vol.27, pp.495-505. 14. Wang, S.P., (2002), ‘An inventory replenishment policy for deteriorating items with shortages and partial backlogging’, Computers & Operations Research, Vol.29(14), pp.2043-2051. 15. Ghosh, S.K., Chaudhuri, K.S., (2005), ‘An EOQ model for a deteriorating item with trended demand and variable backlogging with shortages in all cycles’, Int. J. Adv. Model. Optim. (Buchar. Rom.), Vol.7(1), pp.57-68. 16. Pentico, D.W., Drake, M.J., (2011), ‘A survey of deterministic models for the EOQ and EPQ with partial backordering’, European Journal of Operations Research, Vol. 214(2), pp. 179-198. 17. Sharma, S., Singh, S.R., (2013), ‘An inventory model for decaying items, considering multi variate consumption rate with partial backlogging’, Indian Journal of Science and Technology, Vol. 6(7), pp.4870-4880.

264  Mathematics and Computer Science Volume 2 18. Roy Chowdhury, R., Ghosh, S.K., Chaudhuri, K. S., (2014), ‘An Order-Level Inventory Model for a Deteriorating Item with Time-Quadratic Demand and Time-Dependent Partial Back-logging with Shortages in All Cycles’, American Journal of Mathematical and Management Sciences, vol.33(2), pp.75-97. 19. Tripathi, R.P., (2015), ‘Economic order quantity for deteriorating item with nondecreasing demand and shortages under inflation and time discounting’, International Journal of Engineering, Vol.28(9), pp.1295-1302.

18 An Amended Moth Flame Optimization Algorithm Based on Fibonacci Search Approach for Solving Engineering Design Problems Saroj Kumar Sahoo* and Apu Kumar Saha Department of Mathematics, National Institute of Technology Agartala, Tripura, India

Abstract

The moth flame optimization (MFO) algorithm is a swarm intelligence (SI) based algorithm which gained popularity among researchers due to a special kind of movement mechanism, namely, a transverse orientation mechanism of the moth in nature. Like other SI based algorithms, it also suffers from good quality solution and slow convergence speed. To avoid the drawbacks, a new variant of MFO algorithm, namely, a Fibonacci technique based MFO algorithm (in short Ft-MFO) is presented in this paper. We merged the concept of Fibonacci search method in the classical MFO algorithm to improve the search quality and accelerate the convergence speed of the MFO algorithm. To validate the performance of the proposed algorithm, Ft-MFO is compared with six popular stochastic optimization algorithms on an IEEE CEC2019 test suite and two constraint engineering design problems. Experimental results demonstrate that the proposed Ft-MFO algorithm is superior to the other stochastic algorithms in terms of solution quality and convergence rate. Keywords:  Moth flame optimization algorithm, Fibonacci search method, benchmark functions

*Corresponding author: [email protected] Sharmistha Ghosh, M. Niranjanamurthy, Krishanu Deyasi, Biswadip Basu Mallik, and Santanu Das (eds.) Mathematics and Computer Science Volume 2, (265–282) © 2023 Scrivener Publishing LLC

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18.1 Introduction The challenge of finding the better solution in optimization problems is an interesting topic of research due to its importance in both academia and industry. If the number of optimization parameters increases, the optimization problem complexity also increases. In recent decades, researchers are very much interested in machine learning and artificial intelligence (AI) techniques as real-life difficulties, such as constrained or unconstrained, linear or nonlinear, continuous or discontinuous, that can be easily tackle by AI and machine learning techniques [1, 2]. Due to the aforementioned characteristics, there are various levels of difficulty to handle such types of difficulties using conventional techniques through numerical or mathematical programming, namely quasi-Newton approach, quadratic programming, conjugate gradient, fast steepest method [3], etc. In various existing research [4], it has been experimentally proved that the above-mentioned approaches are not efficient enough to handle non-differentiable and real-life multimodal issues. In contrast, the nature-based algorithms have played a substantial role in tackling these issues, as these algorithms are simple and can be easily applied. A few of them are the Genetic Algorithm (GA) [5], Particle Swarm Optimizer (PSO) [6], Differential Evolution (DE) [7], Spotted Hyena Optimization Algorithm (SHO) [8], Butterfly Optimization Algorithm (BOA) [9], Whale Optimization Algorithm (WOA) [10], Sine Cosine Algorithm (SCA) [11], Moth Flame Optimizer (MFO) [12], Salp Swarm Algorithm (SSA) [13], etc. Typically, these algorithms initiate by a set of randomly chosen initial solutions and continue until they get the optimal solution of a problem. When the algorithm exceeds the pre-determined number of iterations it will automatically be terminated. There is an increasing interest for the efficient, low-cost, and effective implementation of such metaheuristic algorithms these days. The MFO algorithm is the subject of this article. In 2015, Mirjalili discovered MFO, a swarm intelligence-based algorithm. Transverse orientation is used by moths to navigate in the wild and served as an inspiration for MFO. Spiral flight search and simple flame generation (SFG) are two of the most important MFO tactics. In the SFG technique, the best moths and flames collected thus far can be used to directly manufacture flames. By following moths’ transverse direction, the SFS technique allows moths to spiral toward the flames to update the positions in an iterative way. It is possible that MFO will find the best option in the available search space. The transverse orientation of moths in particular is critical to the effectiveness of MFO. The key advantage of MFO on other algorithms is its ability to tackle several tough issues involving confined and unknown search spaces such as optical network unit placement [14], automatic generation

An Amended Moth Flame Optimization Algorithm  267 control problem [15], image segmentation [16], feature selection [17], medical diagnoses [18], and smart grid system [19]. As a new population-based optimization method, MFO’s performance still needs to be improved and studied in some dimensions, including convergence speed and global search capabilities. Various academics have come up with a variety of ways to fix the MFO algorithm’s flaws and a few of these are presented here. In order to tackle the shortcomings of the MFO method, Hongwei et al. [20] presented a new form of the algorithm called chaos-enhanced MFO, which incorporates a chaotic map. Yueting et al. [21] proposed a series of new MFO algorithm variants by integrating MFO with Gaussian, Cauchy, and Levy mutations to reduce the disadvantages of the MFO and expand the search capability of MFO. To achieve a more stable balance between diversity and intensification in the MFO algorithm by embedding Gaussian mutation and chaotic local search, Xu et al. [22] developed CLSGMFO. Three additional adjustments were introduced to the MFO and suggested E-MFO by Kaur et al. [23] to keep a favorable balance between diversification and intensification and boost exploration and exploitation, respectively. For the prediction of software errors, Tumar et al. [24] implemented a modified MFO method and presented an extended binary moth-flame optimization algorithm (EBMFO). To strike a compromise between global and local search capabilities, Wei Gu and Gan Xiang [25] suggested a new modified MFO algorithm termed as a “multi operator MFO algorithm” (MOMFO). An updated version of the MFO algorithm was created by Ma and colleagues [26] to alleviate the MFO’s shortcomings, including delayed and local minimum convergence. A new version of the MFO, namely EMFO, based on the mutualism phase of symbiotic organism search, has been proposed by Sahoo et al. [27]. Few recent upgraded versions of MFO algorithms are discussed in [28–34]. Researchers have also developed more efficient algorithms in addition to the ones listed above. For example, Chakraborty et al. [35] introduced WOAmM, where the authors embedded the modified mutualism phase in WOA to alleviate inherent drawbacks of WOA. Nama et al. [36] proposed the hybrid SOS (HSOS) by integrating Simple Quadratic Interpolation (SQI) with SOS to enhance the robustness of the process. An effective hybrid method called m-MBOA has been developed by Sharma and Saha [37]. Because of this, BOA’s overall performance was enhanced by using mutualism in the exploring part. Chakraborty et al. [38] introduced an efficient hybrid method called HSWOA by hybridizing the HGS algorithm into the WOA algorithm and applied it to solve different engineering design problems. Sharma et al. [39] introduced a different type of modification in BOA named mLBOA in which Lagrange interpolation and SQI are used in exploration and exploitation phases, respectively, to improve the original BOA algorithm. In the present study, a modified

268  Mathematics and Computer Science Volume 2 MFO is formulated, namely Ft-MFO with the help of the Fibonacci technique. The major steps involved in this work are as follows: • Firstly, a non-linear function is embedded into the classical MFO algorithm to maintain exploration and exploitation capability of the suggested Ft-MFO. • Secondly, we merged the concept of Fibonacci search method in MFO to boost the solution quality of the proposed Ft-MFO. • The efficiency of the new Ft-MFO is examined with six popular stochastic optimization algorithms on an IEEE CEC 2019 test suite and two constrained engineering problems. • A Friedman rank test was carried out in order to investigate the performance of the newly suggested Ft-MFO algorithm. The rest of this article is structured as follows. Section 18.2 provides an overview of the MFO algorithm. Section 18.3 shows the suggested Ft-MFO algorithm. Section 18.4 presents the simulation outcomes and performance metrics. Statistical tests and a convergence analysis are discussed in in Section 18.5, as well as real life problems. Finally, conclusions with future enhancements are discussed in Section 18.6.

18.2 Classical MFO Algorithm This section presents the origin of the MFO algorithm and its working process with the mathematical formulation presented below. Moths are members of the Arthropoda family. Moth navigation techniques are one-of-a-kind, which draws researchers’ attention. In the subsections that follow, the MFO algorithm is represented mathematically. The positions of all moths are represented using a vector of choice variables. Take a peek at the moth matrices below.



 x1,1  X1      x2,1 X2 X= =         x N −1,1  X N    x N ,1

x1,2    x N ,2

    

where Xi = [ xi ,1 , xi ,2 ,…, xi ,n ] , i ∈{1,2,…, N }.

x1,n −1    x N ,n −1

x1,n   x 2,n    (18.1)  x N −1,n  x N ,n 

An Amended Moth Flame Optimization Algorithm  269 N is the number of moths in the original population, while n denotes variable numbers. The flame matrix (FMx) is the second crucial factor of the MFO algorithm. Mathematically, it can be represented as follows:

 Fm1,1  FMx1     Fm2,1  FMx 2 =  FMx =         FmN −1,1   FMx N    FmN ,1

Fm1,2    FmN ,2

    

Fm1,n −1    FmN −1

Fm1,n   Fm2,n     FmN −1,n  FmN ,n 

(18.2)

Moths move spirally when they are nearer to the flame, therefore the author used a logarithmic spiral function which is as follows:

x

K +1 i

 δ ⋅ e bt ⋅ cos(2π t ) + Fm (k ), i i = bt  δ i ⋅ e ⋅ cos(2π t ) + FmN . FM (k ),

i ≤ N .FM i ≥ N .FM

(18.3)

δ i xiK − Fmi represents the distance of the moth at ith place and where = its specific flame (Fmi) and t can be any random number between −1 and 1. Here, b is a fixed constant used to recognize the spiral flight shape. A moth moves like a helix towards the flame with a one-dimensional approach and a discrete value of t and represented as follows:



 −1 a1 = −1 + current iter   maxiter

t = (a1 – 1) × r + 1

  

(18.4) (18.5)

where maxiter and a1 indicate the maximum iterations and the constant of convergence, respectively, which decreases linearly from (−1) to (−2). In every iteration, flame positions for the current and last iterations are collected and arranged as per the fitness value for the global and local search. The number of flames (N.FM) that have been lowered over the iteration can be calculated using the formula below.

(N .FM Lst it − 1)   = N .FM round  N .FM Lst it − crnt .it  (18.6) max it  

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18.3 Proposed Method The main characteristics of generalized optimization algorithms include exploration and exploitation. Exploration involves searching the entire search region. However, exploitation is characterized as examining limited areas within a large search field. These effects contribute to the algorithm’s ability to prevent local minima stagnation (as a result of exploitation) and to promote convergence and solution variety (from exploration). The equilibrium between these two occurrences is also vital. Any algorithm that achieves these three criteria is deemed as a state-of-the-art algorithm. In MFO, the spiral motion of moths around the flame provides exploration and exploitation. It is easier to understand exploration and exploitation when the exponent factor ‘t’ is used to explain it. Iteratively, in traditional Ft-MFO, the parameter t is taken from the linearly decreasing range of (−1) to 2, but in our new approach we have introduced a non-linear decreasing range of (−1) to 2, which helps maintain an equilibrium of global and local search by first exploring the search space and then gradually shrinking and exploiting the region found. The following is a mathematical formula for the parameter ‘r.’ 3



r = −2 + e

 −iteration     k ×maxiter 



(18.7)

where k is a constant and its value is 0.55, which was suitably chosen so that it helps in both global and local searches and is represented in Figure 18.1. Fibonacci Search Method (FSM) The FSM is a mathematical process that shifts and narrows down the search range by using Fibonacci numbers to obtain the extreme value of unimodal functions. The optimal point is always contained within the range being narrowed. Shifting can take place in both directions. The values of that function at two experiment points determine the changing direction. The Fibonacci numbers, which are defined as follows, are the foundation of the FSM. Fib = [F1, F2, F3, … .., Fn], where, Fi ∀ i = 1,2,..n are Fibonacci numbers and generated by the following equation:

F0 = 1 = F1, Fno = Fno−1 + Fno−2, no = 2,3,4, … . n

(18.8)

An Amended Moth Flame Optimization Algorithm  271 Parameter (r) vs Iterations

-1 -1.1

MFO

Ft-MFO

-1.2

Parameter (r)

-1.3 -1.4 -1.5 -1.6 -1.7 -1.8 -1.9 -2

0

100

200

300

400 500 600 700 Number of iterations

800

900 1000

Figure 18.1  Non-linear adaption curve.

Let t1 and t2 be two experimental numbers of any finite length of interval with upper and lower limits as UL and LL, respectively. Calculate two initial points:



 F  F UL −  no − 2  (UL − LL) (18.9) = t1 LL +  no − 2  (UL − LL), t 2 =  Fno   Fno 

The range is moved to the right because the function’s value at t2 is greater than that at t1 and to the left if t1 is greater than that at t2. The new value of t3 and t4 are generated using the Fibonacci search formula as t3 = t1 and t4 = t2. If two functional values are unequal, then only one (t3 or t4) will be considered as a new experimental point, whereas the other will be the same as either of t1 and t2 depending on the contracting direction. In the case of two equal function values, then both t3 and t3 form new experimental points. Due to the high computational efficiency of FSM, the author of [40] used a modified Fibonacci search method for the partially shaded solar PV array. Recently, Yazici et al. [41] applied a modified Fibonacci search method for conversion systems of wind energy. We embedded the concept of the Fibonacci search method after the position update phase of MFO. The pseudocode of the projected Ft-MFO method is presented below as Algorithm 18.1.

272  Mathematics and Computer Science Volume 2 .

Algorithm 18.1: Pseudocode of Ft-MFO Algorithm Objective function f(X), X = (X1 X2……....Xdimension); for i = 1: No. of search agents for j = 1: dimension Generate n organism solutions Xi(i = 1, 2, … ., n) using equation X (i, j) = L .B(i) + (U .B (i) – LB(i)) *rand end end While Current iteration < Maximum iteration if Iteration == 1 Enter N .FM = N in initial population else Employ Eq. (18.6). end FM = Fitness Function f(X); if Iteration == 1 Sort the moths according to FM Update the Flames Iteration = 0; else Sort the moths based on FM from last iteration Update the Flames end Reduce the converge constant for j = 1: No. of search agents for k = 1: dimension Find r and t using Eq. (18.5) & (18.7) Update moths position as to their particular flame using Eq. (18.3) end end Apply Fibonacci search process using Eq. (18.8) and Eq. (18.9) Current iteration = Current iteration + 1 End while Output: The best solution with the minimum fitness

An Amended Moth Flame Optimization Algorithm  273

18.4 Results and Discussions on IEEE CEC 2019 Benchmark Problems It should be noted that the code for the proposed Ft-MFO algorithm has also been written in the MATLAB environment and run on a 1.80 GHz i5 8th generation computer with 8.00GB of RAM and MATLAB R2015a. In this paper, the proposed Ft-MFO has been applied on IEEE CEC 2019 test problems and the performance results are compared with a wide variety of state-of-the-art methods. The selected parameters of all population-based optimization algorithms involve the maximum iteration and number of populations as 1000 and 30, respectively. To determine the best set of parameters for Ft-MFO and all other algorithms, 30 trials are performed for each possible set of parameters. The studied cases are presented below. The CEC 2019 benchmark function has stored more complex type functions than other CEC test suits. The author [42] developed some complex single objective optimization problems, namely, “The 100-digit Challenge”. There is a total of ten number functions that are multimodal and inseparable in nature. The first three functions (F1-F3) of CEC 2019 have different search ranges and different dimensions but the other seven functions (F4F10) have the same search range with the same dimensions. The optimal value of all ten complex functions (F1-F10) is set to one. Table 18.1 shows the performance results of 10 (ten) IEEE CEEC’2019 test suites using Ft-MFO and six basic current state-of-the-art optimization techniques, including MFO, DE, SOS, JAYA, BOA, and WOA. Table 18.2 shows how many times the efficiency of Ft-MFO is higher, similar to, and worse than that of other methods. In 8, 7, 8, 9, 7, and 8 benchmark functions, respectively, Ft-MFO outperforms MFO, DE, SOS, JAYA, BOA, and WOA; comparable results are seen in 2, 0, 0, 0, 2, and 0 benchmark functions, while poorer results are shown in 0, 3, 2, 1, 1, and 2 benchmark functions. The Friedman Test, introduced by Milton Friedman [43], is a non-parametric statistical test. It is used to spot treatment variations across several test runs. The Friedman rank test has been employed in this study to equate the mean results of the algorithms for each benchmark issue. Table 18.3 shows that Ft-MFO has the lowest rank which is highlighted in bold, implying that its performance is superior to other algorithms. The convergence performance of Ft-MFO with other algorithms is shown in Figure 18.2, which is very competitive with respect to other methods.

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Table 18.1  Simulation results of Ft-MFO and other algorithms on CEC’2019 suite. Algorithm

F1

F2

F3

F4

F5

AVG

STD

AVG

STD

AVG

STD

AVG

STD

AVG

STD

Ft-MFO

1

0

5

0

8.40

1.59

3.34e+01

3.39e+01

1.64

1.68

MFO

1

0

5

0

9.62

9.60e-01

1.05e+02

1.56e+01

1.00e+02

2.59e+01

DE

1.82E+06

8.25E+05

1.93e+03

3.40e+02

4.58

6.11e-01

8.54

1.99

1.77

2.64e-02

SOS

4.75E+03

6.88E+03

2.65e+02

1.96e+02

4.61

9.78e-01

2.08e+72

3.42e+72

1.19e+73

4.40e+73

JAYA

6.53E+06

2.83e+06

4.68e+03

7.78e+02

8.09

7.57e-01

4.10e+01

4.90

2.60

1.67e-01

BOA

1

0

5

0

6.79

1.15

1.26e+02

2.15e+01

1.09e+02

2.33e+01

WOA

1.45E+08

1.17E+08

1.10E+04

3.36E+03

8.16

1.60

6.62E+01

2.40E+01

3.83

1.43

Algorithm

F6

Ft-MFO

F7

F8

F9

F10

AVG

STD

AVG

STD

AVG

STD

AVG

STD

AVG

3.07

1.21

8.22e+02

2.23e+02

4.25

2.41e-01

1.26

9.43e-02

2.15e+01

STD 1.75e-01 (Continued)

An Amended Moth Flame Optimization Algorithm  275

Table 18.1  Simulation results of Ft-MFO and other algorithms on CEC’2019 suite. (Continued) Algorithm

F6

F7

F8

F9

F10

MFO

1.21e+01

1.09

2.29e+03

2.51e+02

5.13

1.79e-01

3.86

6.23e-01

2.16e+01

1.43e-01

DE

1.11

1.48e-01

1.07e+03

1.24e+02

4.48

2.71e-01

1.29

4.20e-02

2.25e+01

2.52

SOS

2.15

2.22e-01

4.55e+71

4.09e+71

5.50

0

1.59e+71

2.34e+71

2.18e+01

8.74e-04

JAYA

5.90

9.23e-01

1.51e+03

2.09e+02

4.28

2.08e-01

1.56

1.11e-01

2.16e+01

1.27e-01

BOA

1.36e+01

1.15

2.07e+03

2.89e+02

5.12

2.83e-01

4.14

5.51e-01

2.16e+01

1.85e-01

WOA

9.94

1.84

1.49E+03

2.85E+02

4.73

3.09E-01

1.48

1.69E-01

21.46

1.12E-01

276  Mathematics and Computer Science Volume 2 Table 18.2  Experimental results of Ft-MFO with other algorithms on IEEE CEC’2019 test suite. MFO

DE

SOS

JAYA

BOA

WOA

Superior to

8

7

8

9

7

8

Similar to

2

0

0

0

2

0

Inferior to

0

3

2

1

1

2

Table 18.3  Friedman rank test. Algorithm

Mean rank

Rank

Ft-MFO

3.40

1

SOS

3.76

2

BOA

4.08

3

MFO

4.66

4

DE

4.69

5

WOA

4.67

6

JAYA

4.79

7

Converges graph of function number 5

log(f(x)-f(x*))

5 4 3 2

DE Ft-MFO JAYA BOA WOA MFO SOS

2.5 2 1.5 1 0.5

1

8 7.5 7

0 200 400 600 800 1000 1200 1400 1600 1800 2000 Function Evaluation Converges graph of function number 7 DE Ft-MFO JAYA BOA WOA MFO SOS

6.5

200 400 600 800 1000 1200 1400 1600 1800 2000 Function Evaluation Converges graph of function number 8 DE Ft-MFO JAYA BOA WOA MFO SOS

1.6 1.5 1.4 1.3 1.2 1.1 1

6

0.9 5.5 0

0

1.8 1.7 log(f(x)-f(x*))

0 0

log(f(x)-f(x*))

Converges graph of function number 6

3 DE Ft-MFO JAYA BOA WOA MFO SOS

log(f(x)-f(x*))

6

200 400 600 800 1000 1200 1400 1600 1800 2000 Function Evaluation

0

200 400 600 800 1000 1200 1400 1600 1800 2000 Function Evaluation

Figure 18.2  Convergence graph of Ft-MFO with other competative algorithms.

An Amended Moth Flame Optimization Algorithm  277

18.5 Real-Life Applications Optimal gas production capacity and three-bar truss issues are utilized as real-world problems to show how well the proposed Ft-MFO algorithm works on real-life problems.

18.5.1 Optimal Gas Production Capacity Problem The above problem is an unconstrained problem and details with mathematical representation are elaborated in [44]. The simulation results of the above problem are presented in Table 18.4. In this table, the results of DE, GSA, and DE-GSA are taken from [45]. From Table 18.4, we can conclude that the performance of the proposed Ft-MFO method is shown to be superior to that of other methods.

18.5.2 Three-Bar Truss Design (TSD) Problem The challenge of designing a three-bar truss is one of structural optimization in civil engineering. Complex limited search space makes this challenge useful [46]. Two design criteria, buckling stress and deflection, have been modified in order to produce the lowest possible weight. The multiple components of the three-bar truss design issue are depicted in Figure 18.3. Our new Ft-MFO algorithm is used to solve the TSD problem, which is then compared to the CS, DEDS, PSO-DE, Tsa, and MBA algorithms taken from the literature [12]. The comparative findings, as well as optimal weights and optimal variables, are provided in Table 18.5. Our suggested Ft-MFO technique outperforms the other three algorithms, as shown in Table 18.5.

Table 18.4  Simulation results for optimal gas production problem. Item

DE

MFO

GSA

DE-GSA

Ft-MFO

x1

17.5

17.5

17.5

17.5

17.5

x2

600

600

600

600

600

f(x)

169.844

71.4495

169.844

169.844

71.4459

278  Mathematics and Computer Science Volume 2

1

D

3

2 A

D A 4

A

P

Figure 18.3  Three-bar truss design problem.

Table 18.5  Simulation results for three-bar truss problem. Algorithm

Optimal variables

Optimal weight

x1

x2

Ft-MFO

0.408966

0.288146

174.2762166

MFO

0.788244770931922

0.788244770931922

263.895979682

CS

0.78867

0.40902

263.9716

DEDS

0.78867513

0.40824828

263.8958434

PSO-DE

0.7886751

0.4082482

263.8958433

Tsa

0.788

0.408

263.68

MBA

0.7885650

0.4085597

263.8958522

18.6 Conclusion with Future Studies This paper presents an upgraded variety of the classic MFO algorithm, namely an amended MFO (Ft-MFO) which uses a Fibonacci search concept and a non-linear adaption formula to improve the MFO algorithm and make a good tradeoff between diversification and intensification. To evaluate the performance of Ft-MFO, IEEE CEC 2019 benchmark functions have been considered for experimentations and compared with the basic MFO, DE, JAYA, BOA, SOS, and WOA. The Friedman Test is used to measure the effectiveness of the suggested Ft-MFO algorithm. It has also been used to solve two engineering issues, providing a better result than previous algorithms to validate the proposed Ft-MFO. According to simulation

An Amended Moth Flame Optimization Algorithm  279 results, using the global best solution in the optimization process quickly brought the proposed method into focus. To avoid local optima traps and premature convergence, the proposed technique is helpful. In the future we can extend it to multi-objective optimization, apply it to solve higher constraint optimization problems like car-side crash problems, robot gripper problems, welded beam design problems, etc. We can generate an efficient metaheuristic algorithm by hybridizing our suggested approach, Ft-MFO, with any other meta-heuristic algorithm.

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19 Image Segmentation of Neuronal Cell with Ensemble Unet Architecture Kirtan Kanani1, Aditya K. Gupta1, Ankit Kumar Nikum1*, Prashant Gupta2 and Dharmik Raval1 Department of Mechanical, Sardar Vallabhbhai National Institute of Technology, Surat, Gujarat, India 2 Licious, Banglore, India

1

Abstract

Medical image segmentation consists of heterogeneous pixel intensities, noisy/ ill-defined borders, and high variability, which are significant technical obstacles for segmentation. Also, generally the requirement of annotated samples by the networks is significantly large to achieve high accuracy. Gathering this dataset for the particular application and annotating new images is both time-consuming and costly. Unet solves this problem by not requiring vast datasets for picture segmentation. The present work describes the use of a network that depends on augmentation of the existing annotated dataset to make better use of these examples and a comparison of encoder accuracy on Unet is presented. The encoder principal function is to reduce image dimensionality while keeping as much information as possible. EfficientNets tackles both of these issues and utilizing it as an encoder of Unet can further enhance its accuracy. The test dataset highest F1-Score and IoU were 0.7655 and 0.6201 on neuronal data values, respectively. It outperforms Inception and ResNet encoder networks with considerably more parameters and a higher inference time. Keywords:  Image segmentation, computer vision, deep learning, neural cell

*Corresponding author: [email protected] Sharmistha Ghosh, M. Niranjanamurthy, Krishanu Deyasi, Biswadip Basu Mallik, and Santanu Das (eds.) Mathematics and Computer Science Volume 2, (283–290) © 2023 Scrivener Publishing LLC

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19.1 Introduction Neurodegenerative illnesses cover a diverse spectrum of ailments resulting from progressive damage to cells and nervous system connections. These illnesses eventually lead to diseases such as Alzheimer’s and Parkinson’s disease [1], which are among the six highest causes of death. According to the WHO report [2] on neurological disorders, neurological illnesses account for 12% of all fatalities worldwide. Figure 19.1 shows the architecture of the modified U-Net model used for training. The goal of finding treatments and solutions for neurodegenerative illnesses is becoming more urgent with the increase in population. In 2015, neurological illnesses caused 250.7 million disability-adjusted life years (DALYs) and 9.4 million deaths, accounting for 10.2% of global DALYs and 16.8% of global deaths [3]. Our work is focused on assisting doctors in determining the effects of medical treatments in neuron cells using machine learning and computer vision. For biomedical image analysis, the segmentation of 2D pictures is a critical problem [4]. We have proposed an Unet based neural network that upgrades the FCN model for biomedical application, i.e., RIC Unet, for nuclei segmentation. Following Unet architecture, RIC-Unet uses a channel attention mechanism, residual blocks, and multi-scale ENCODER

I N P U T

DECODER

(8x8x1824) (16x16x768) (16x16x1056) (32x32x384) (64x64x240)

(128x128x144) (128x128x48)

(16x16x2048) (16x16x3104)

(32x32x640) (32x32x256) (64x64x128) (64x64x368) (128x128x64) (128x128x208) (256x256x32)

O U T P U T

Concatenate

Figure 19.1  Architecture of Unet with EfficentNet as encoder. The decoder is similar to the original Unet model but with cropping and contamination from various encoder elements.

Image Segmentation of Neuronal Cell with Unet Architecture  285 techniques for segmentation of nuclei accurately. This network is superior to other techniques in terms of analysing indicators and cost-­ effectiveness in assisting doctors in diagnosing the details of these histological images. In many medical picture segmentation issues U-Net provides a state-of-the-art performance. U-Net with residual blocks or blocks with dense connections, Attention U-Net, and recurrent residual convolutional U-Net are a few examples of U-Net adaptations. All of these adaptations have some changes in common such as the ways in which the path of information flow is altered and an encoder-decoder structure with skip connections. In this work, the author was inspired by the success of ResNet and R2-UNet [5] employing modified residual blocks. Here, the two convolutional layers in one block are sharing the same weights [5]. For the segmentation of cell nuclei images [6], the intersection over union (IoU) is used as a quality matrix for increasing the performance of Unet, ResUNet, and DeepLab. The proposed model has a mean IoU value of 0.9005 for validation dataset.

19.2 Methods The Unet framework mainly inspired our architecture. We utilized EfficientNet-B0 as an encoding part. Like Unet, encoder branch and decoder branch features are concatenated. We chose EfficientNet as the encoder because it proposes an effective compound scaling strategy that effectively scales up a baseline ConvNet to any target resource limitations in a more principled manner while retaining model efficiency. The ImageNet dataset containing color images was used for training the original EfficientNet. The images had a resolution of 224×224. In the present study, a pre-trained EfficientNet model was implemented as an encoder part of Unet to reduce the training time. The EfficientNet-B0 with low parameter quantity was applied owing to limited complexity and an easily identifiable nature of the neuronal images.

19.3 Dataset The dataset consists of 606 grayscale images with an image size of 520 × 704 pixels. We chose 364 photos at random for training, 61 for validation, and 60 for testing. Image count of ‘shsy5y’, ‘Astro’, ‘Cort’ was 155, 131, and

286  Mathematics and Computer Science Volume 2 320, respectively. The neuroblastoma cell line Cort consistently has the lowest precision scores of the three cancer cell lines. This could be because neuronal cells have a distinct, uneven, and concave look, making them difficult to segment using standard mask heads.

19.4 Implementation Details Data augmentation is being performed which includes random cropping, random horizontal/vertical flipping, 35° rotation, and further enhanced accuracy normalization is used on the training images with various schemes. In this study, the call back function was implemented, which monetarized the validation loss for 150 training epochs. If there is not a significant change in this value, then training is terminated. The training and testing images have a resolution of 256 × 256 pixels. We used binary cross-entropy (BCE) with sigmoid as a loss function and Adam as an optimizer. The initial learning rate was set to 1e-4 to optimize the model weights and PyTorch on Tesla P100-PCIE-16GB was used to implement the model.

19.5 Evaluation Metrics Average Precision (AP) [7] is used as a metric to evaluate both instances’ segmentation performance. For instance, in segmentation, we employ APmask in Equation 19.1 to represent the AP at the mask and the IoU between the ground truth mask and each predicted segmentation mask.



AIOUαmask=

1 N ∑i =1 IOUαmask N

(19.1)

19.6 Result For this work of semantic segmentation, we used a variety of designs and the results are given in Table 19.1. Our investigation began with the DeepLabV3+ framework. For extraction of features, we initially utilized EfficientNet-B0 as an encoder. We experimented with several encoder backbones such as EfficientNet-B0 [8], ResNet-50 [9], InceptionResNetV2 [10],

Image Segmentation of Neuronal Cell with Unet Architecture  287 Table 19.1  Result comparison in terms of mean intersection over union (IoU), F1-score, and accuracy on the neuronal dataset. Model

F1-score

IoU

Accuracy

Unet with EfficientNet-B0 as encoder (binary cross entropy logits)

0.7286

0.573

0.9466

Unet with ResNet-50 as encoder (binary cross entropy logits)

0.7053

0.5447

0.9254

Unet with Inception-ResNet-v2 as encoder (binary cross entropy logits)

0.7264

0.5704

0.9550

Unet with EfficientNet-B0 as encoder (dice loss)

0.7655

0.6201

0.9493

PSPNet with EfficientNet-B2 as encoder (binary cross entropy logits)

0.6075

0.4362

0.9321

PSPNet with InceptionNet as encoder (binary cross entropy logits)

0.6579

0.4902

0.9311

DeeplabV3+ with EfficientNet-B5 as encoder (binary cross entropy logits)

0.7089

0.5490

0.9351

Unet with Inception-ResNet-v2 as encoder (dice loss)

0.7283

0.5727

0.9166

DeeplabV3+ with EfficientNet-B5 as encoder (dice loss)

0.7265

0.5705

0.9351

and EfficientNet-B5 with the  renowned Unet decoder. From Table 19.1, it can be shown that the Unet decoder outperforms the DeepLabV3+ decoder. That is because Unet now has additional low-level encoder characteristics that are essential in analyzing complicated scenes with multiple dense objects. We further analyze the class-wise segmentation performance of the best-performing model on the three classes. We obtain class-wise IoU on the validation dataset, shown in Table 19.2. It can be observed that the Cort Table 19.2  Class wise IoU on neuronal dataset with efficient Unet as model. Neuronal cells

shsy5y

Astro

Cort

Mean IoU

0.4580

0.5474

0.2948

288  Mathematics and Computer Science Volume 2 cell has the lowest IoU. The Cort cell image contains different geometry types compared to Astro and sphy5y. Besides the circular geometry of Cort cells, thread-like structures are also present in the image, which is similar to Astro and shyishy5 geometry. This makes model prediction clouded. The model’s accuracy is further hampered by a small dataset. The ground truth segmentation maps of a few images, as well as their anticipated segmentation maps, are shown in Figure 19.2. The first column displays the input photos for a single cell. The ground truth and anticipated segmentation maps are shown in the second and third columns, respectively. The cell type is in the order Astro, Cort, Shsy5y from top to bottom.

Figure 19.2  Result of semantic segmentation on neuronal validation dataset with efficient Unet as architecture.

Image Segmentation of Neuronal Cell with Unet Architecture  289

19.7 Conclusion This study employs multiple kinds of encoders for image segmentation and is based on Unet architecture. It provides a comprehensive procedure that includes selecting a data set, acquiring a training set, training a deep convolutional neural network, and segmenting cell images using the convolution neural network. Finally, following cell image segmentation, the Unet with various encoders was utilized to generate the resultant image, and a basic analysis of the acquired values was performed. This technique achieves high-precision semantic segmentation of shsy5y cells and Astro pictures compared to traditional image segmentation.

References 1. S. Przedborski, M. Vila, and V. Jackson-Lewis, ‘Series Introduction: Neurodegeneration: What is it and where are we?’, J. Clin. Invest., vol. 111, no. 1, p. 3, Jan. 2003, doi: 10.1172/JCI17522. 2. W. H. Organization, ‘Neurological disorders: public health challenges’, World Health Organization, 2006. 3. V. L. Feigin et al., ‘Global, regional, and national burden of neurological disorders, 1990–2016: a systematic analysis for the Global Burden of Disease Study 2016’, Lancet Neurol., vol. 18, no. 5, pp. 459–480, May 2019, doi: 10.1016/S1474-4422(18)30499-X. 4. Z. Zeng, W. Xie, Y. Zhang, and Y. Lu, ‘RIC-Unet: An Improved Neural Network Based on Unet for Nuclei Segmentation in Histology Images’, IEEE Access, vol. 7, pp. 21420–21428, 2019, doi: 10.1109/ACCESS.2019. 2896920. 5. J. Zhuang, ‘LadderNet: Multi-path networks based on U-Net for medical image segmentation’, Oct. 2018, Accessed: Dec. 13, 2021. [Online]. Available: https://arxiv.org/abs/1810.07810v4. 6. C. A. R. Goyzueta, J. E. C. De la Cruz, and W. A. M. Machaca, ‘Integration of U-Net, ResU-Net and DeepLab Architectures with Intersection Over Union metric for Cells Nuclei Image Segmentation’, pp. 1–4, Nov. 2021, doi: 10.1109/EIRCON52903.2021.9613150. 7. M. Everingham, L. Van Gool, C. K. I. Williams, J. Winn, and A. Zisserman, ‘The pascal visual object classes (VOC) challenge’, Int. J. Comput. Vis., vol. 88, no. 2, pp. 303–338, Sep. 2010, doi: 10.1007/s11263-009-0275-4. 8. M. Tan and Q. V. Le, ‘EfficientNet: Rethinking Model Scaling for Convolutional Neural Networks’, 36th Int. Conf. Mach. Learn. ICML 2019, vol. 2019-June, pp. 10691–10700, May 2019, Accessed: Dec. 13, 2021. [Online]. Available: https://arxiv.org/abs/1905.11946v5.

290  Mathematics and Computer Science Volume 2 9. K. He, X. Zhang, S. Ren, and J. Sun, ‘Deep Residual Learning for Image Recognition’, Proc. IEEE Comput. Soc. Conf. Comput. Vis. Pattern Recognit., vol. 2016-December, pp. 770–778, Dec. 2015, doi: 10.1109/CVPR.2016.90. 10. C. Szegedy, S. Ioffe, V. Vanhoucke, and A. A. Alemi, ‘Inception-v4, InceptionResNet and the Impact of Residual Connections on Learning’, 31st AAAI Conf. Artif. Intell. AAAI 2017, pp. 4278–4284, Feb. 2016, Accessed: Dec. 15, 2021. [Online]. Available: https://arxiv.org/abs/1602.07261v2.

20 Automorphisms of Some Non-Abelian p−Groups of Order p4 Muniya Sansanwal1*, Harsha Arora2 and Mahender Singh3 Department of Mathematics, JJTU Jhunjhunu, Rajasthan, India Department of Mathematics, Govt. College for Women Hisar, Haryana, India 3 Department of Mathematics, Om Sterling Global University, Hisar, Haryana, India 1

2

Abstract

This paper contains the number of automorphisms of several non-Abelian groups of order p4, p-odd prime are computed and GAP (Group Algorithm Programming) software has been used for the verification of a number of automorphisms. Keywords:  -p-groups, automorphism, semi-direct product MSC subject classification 2000:  20D45, 20D60

20.1 Introduction Let G be a p-group of order p4, p-odd prime and Aut(G) represent the group of all automorphisms of a given group G. There are many research papers in the literature related to the automorphisms of p-groups, for instance [1, 3, 4], etc. In [1], the automorphisms of groups of order p3 are computed along with the automorphisms of abelian groups of order p4. The present paper is an extension of the research work in [1]. In this paper, we will compute the automorphisms of groups of order p4. William Burnside [2] classified all groups having order p4. By using these classifications we derive the automorphisms of some groups having order p4. We divide the derived results into two parts. In these two parts, one part is dedicated to the categorization of all p−groups of order p4 and the other part is dedicated to investigating the number of automorphisms of several non-­Abelian groups with order p4. *Corresponding author: [email protected] Sharmistha Ghosh, M. Niranjanamurthy, Krishanu Deyasi, Biswadip Basu Mallik, and Santanu Das (eds.) Mathematics and Computer Science Volume 2, (291–304) © 2023 Scrivener Publishing LLC

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20.2 Categorization of p-Groups with Order p4 According to the categorization derived by Burnside [2], if p is an odd prime, then there will be 15 groups of order p4. Five of these are abelian and the rest are non-abelian, as given below. Abelian Groups: 4

G1 = < x1 : x1p = 1 > ≅ Z p4 . 3

p G2 = < x1 , x2 : x1p= x= 1, x= x2 x1 > ≅ Z p3 × Z p . 1x 2 2 2

2

p p 1, x= G3 = < x1 , x2 : x= x= x2 x1 > ≅ Z p2 × Z p2 . 1x 2 1 1 2

p p x= 1, x1x2 = x2 x1 , x2 x3 G4 = < x1 , x2 , x3 : x1p= x= 2 3 = x= x3 x1 > ≅ Z p2 × Z p × Z p . 3 x 2 , x1x 3 p p p p G5 = < x1 , x2 , x3 , x 4 : x= x= x= x= x3 x1 , 1, x1x2 x= 2 x1 , x1x 3 1 2 3 4= x1x 4 = x= x= x 4 x2 , x3 x 4 = x 4 x3 4 x1 , x 2 x 3 3 x2 , x2 x 4

> ≅ Zp × Zp × Zp × Zp.

Non-Abelian Groups: 3

2

p p G6 =< x1 , x2 : x1= x= 1= , x2 x1 x11+ p x2 > ≅ Z p3 φ Z p , 2

φ ( y ) ↔ (1 + p2 ) y . 2

2

p p G7 =< x1 , x2 : x= x= 1= , x2 x1 x11+ p x2 > ≅ Z p2 φ Z p2 , 1 2

φ ( y ) ↔ (1 + p) y . 2

p p G8 =< x1 , x2 , x3 : x= x= x3p 1= , x1x2 x2 x1 , x2 x3 = x3 x2 , 1 2 = x3 x1 = x1x2 x3 > ≅ (Z p × Z p ) Z p2 . p p p p G9 =< x1 , x2 , x3 , x 4 : x= x= x= x= 1, x 4 x3 = x3 x 4 , x2 x 4 = x 4 x2 , 1 2 3 4 = x1x 4 x= x3 x2 , x2 x1x3 = x3 x1 , 4 x1 , x 2 x 3 x= x2 x1 > ≅ ((Z Z p × Z p ) Z p ) × Z p . 1x 2 p p p p > 33,,G G1010 = = = = x= = 1,, xx44xx33 xx= = 1 2 3 4 = 2 x3 x 4 , x 4 x2 1 2 4 1 2 3 4

x3 x= x1x= xx= xx= xx2 xx1 >>≅≅ ((ZZp ×× ZZp ×× ZZp )Z ZZp ,, 2 x3 x 4 , x 4 x2 2 x 4 , x1x 4 4x 1 , x2 x3 3x 2 , x1x 3 3x 1 , x1x 2 = x1x 4 xx= = = = 4 x1 , x 2 x 3 3 x 2 ,2 x1x 3 3 x1 , x1x 2 2 1 p p p )Z p p p −p > ≅ (Z p × Z p × Z p )Z Z p ,If p  = 3 then < x1 , x2 x3 : x1p= x= x = = 1 , x x x = x , x x x x x , 1 2 2 1 1 3 3 1 2 x 2 x 3 = x 3 x1 x 2 > . 2 3

x= x3 x1x2 , x2 x3 = x3 x1− p x2 > . 2 x1 , x1x 3

Automorphisms of Some Non-Abelian p−Groups of Order p4  293 2

p p 1+ p G11 = < x1 , x2 , x3 : x1p= x= x= = 1, x2 x1 x= x3 x1 , 2 3 1 x 2 , x1x 3 x 2 x 3 = x 3 x 2 > ≅ ( Z p2  Z p ) × Z p . 2

p p G12 = < x1p= x= x= 1, x2 x1 x= x3 x1 , 1x 2 , x1x 3 2 3=

= x3 x2 x1p x2 x3 > ≅ (Z p2 × Z p ) Z p . 2

p p 1+ p = 1, x2 x1 x= x1x2 x3 , G13 = < x1 , x2 , x3 : x1p= x= x= 2 3 1 x 2 , x 3 x1

x= x3 x2 > ≅ (Z p2 × Z p )φ1 Z p , φ1(z ) ↔ (1,1, 0)z . 2 x3 2



p p p 1+ p For p > 3,G14 = < x1 , x2 , x3 : x1= x= x= = 1, x2 x1 x= x11+ p 2 3 1 x 2 , x 3 x1

x2 x3= , x3 x2 x1p x2 x3 > ≅ (Z p2  Z p )φ2 Z p , φ2 (z ) ↔ (1,1,1)z , 2

p If p = 3 then < x1 , x2 , x3 : x1p = x2p = = 1, x3p x= x2 x11+ p , 1 , x1x 2

x2 x3 > . x1x3 = x3 x1x2−1 , x= 3 x2 2

1+ p p p 2 < >3,3G,G = 1, x2 x1 x= 15 = 1 2 3 2 3 3 2 1 2 3 1 x 2 , x 3 x1 1 1+ p 1+dp dp z x1 x2 , x3 x1 x1 x2 x3= , x3 x2 x1 dp where d ≢ x x > ≅ Z Z Z , z ↔ , , d (  )  φ ( ) ( 1 1 ) z 2 3 p )φ p 3 p 2 Z = x1 2x22x2 > ≅ (Z φ3 Z p , φ3 (z ) ↔ (1,1,d ) p p z 0,1(mod p), 1,d ) 2 p −p 1, x3p x= x2 x11+ p , x1x3 = x3 x1x2−1 ,x= x2 x3 > . If p= 3,< x1 , x2 , x3 ; x1p= x= 3 x2 2= 1 , x1 x 2 1+ p −1 x2 x3 > . 3 x2 1 , x1x 3 = x 3 x1x 2 ,x=

20.3 Number of Automorphisms of Some Non-Abelian Groups of Order p4 The automorphisms of five abelian groups, namely G1, G2, G3, G4, and G5, of order p4 are already computed in the paper [1]. In this paper, we compute the automorphisms of several non-abelian p−groups with order p4, namely G9, G10, and G11 denoted above.

20.3.1 Computation of Automorphisms of Group G9 p p p G9 = < x1 , x2 , x3 , x 4 ; x1p = x= x= x= 1, x 4 x3 x= x 4 x2 , x1x 4 3 x 4 , x2 x 4 2 3 4= == x= x3 x= x= 4 x1 , x 2 x 3 2 , x 2 x1x 3 3 x1 , x1x 2 x 2 x1 > ≅ ( ( Z p × Z p ) Z p ) × Z p .

294  Mathematics and Computer Science Volume 2 We can deduce relations by using generator relations from the structure description of G9 and elementary calculations:

x3i x1j = x1j x2(ij ) x3i , (

n(n−1) ij ) (nj ) 2 3

(x1i x3j )n = x1(ni ) x2



x

.



Now, we shall investigate the automorphism group ( (Z p × Z p ) Z p ) × Z p. Let ψ ∈ Aut ( (Z p × Z p ) Z p ) × Z p be defined by:

(



)

 x1 → x1i x2j x3k x 4l ;  m n q r  x2 → x1 x2 x3 x 4 ; ψ : s t u v  x3 → x1 x2 x3 x 4 ; x → x w x x x y x z ; 1 2 3 4  4

i, j, k, l ∈ Z p , m, n, q , r ∈ Z p , s, t , u, v ∈ Z p , w, ,x , y , z ∈ Zp.



As ψ is automorphism, ψ should preserve the relation x2x1x3 = x3x1, that is

ψ (x2 )ψ (x1 )ψ (x3 ) = ψ (x3 )ψ (x1 ) m n q r i j k l s t u v 1 2 3 4 1 2 3 4 1 2 3 4

⇒ x x x x x x x x x x x x = x1s x2t x3u x 4v x1i x2j x3k x 4l ⇒ x1(m+i +s ) x2(n+qi + j+qs +ks +t ) x3(q+k +u ) x 4(r +l +v ) = x1( s +i ) x2(t +ui + j ) x3(u+k ) x 4(v +l )



⇒ x1m x2(n+qi +qs +ks ) x3q x 4r = x2(ui ) ⇒ m ≡ 0(mod p), n + qi + qs + ks ≡ ui(mod p), q ≡ 0(mod p) and r ≡ 0(mod p) ⇒ n + ks ≡ ui(mod p) or n ≡ iu − sk(mod p)



Automorphisms of Some Non-Abelian p−Groups of Order p4  295 As n ≢ 0(mod p), because if n ≡ 0(mod p) then order of (ψ (x2 ) ) ≠ p.

⇒ n = iu - sk ≢ 0(mod p).



Also, x4x3 = x3x4 then ψ(x4)ψ(x3) = ψ(x3)ψ(x4)

⇒ x1w x2x x3y x 4z x1s x2t x3u x 4v = x1s x2t x3u x 4v x1w x2x x3y x 4z ⇒ x1(w +s ) x2( x + ys +t ) x3( y +u ) x 4( z +v ) = x1( s +w ) x2(t +uw + x ) x3(u+ y ) x 4(v +z ) ⇒ x2( ys ) = x2(uw ) ⇒ ys ≡ uw(mod p) ⇒ ys − uw ≡ 0(mod p).





Further, x1x4 = x4x1 implies ψ(x1)ψ(x4) = ψ(x4)ψ(x1)

⇒ x1i x2j x3k x 4l x1w x2x x3y x 4z = x1w x2x x3y x 4z x1i x2j x3k x 4l ⇒ x1(i +w ) x2( j+kw + x ) x3(k + y ) x 4(l +z ) = x1(w +i ) x2( x + yi + j ) x3( y +k ) x 4( z +l ) ⇒ x2(kw ) = x2( yi ) ⇒ kw ≡ yi(mod p) ⇒ yi − kw ≡ 0(mod p).





Now, ys − uw ≡ 0(mod p) and yi − kw ≡ 0(mod p) are two homogeneous equations with determinant iu − sk ≠ 0. Hence, these equations have only trivial solutions. ⇒ y = w ≡ 0(mod p). Next, we check the kernel of ψ. Let x1h1 x2h2 x3h3 x 4h4 be any element of ((Z p × Z p ) Z p ) × Z p so that it is mapped to identity.

ψ (x1h1 x1h2 x1h3 x1h4 ) = 1 ⇒ ψ (x1 )h1ψ (x2 )h2ψ (x3 )h3ψ (x 4 )h4 = 1 ⇒ (x1i x2j x3k x 4l )h1 (x1m x2n x3q x 4r )h2 (x1s x2t x3u x 4v )h3 (x1w x2x x3y x 4z )h4 = 1 h (h −1) h (h −1) ( jh1 + 1 1 ik +nh2 +th3 + 3 3 su + xh4 +kh1sh3 ) ( kh1 +uh3 ) (lh1 +vh3 + zh4 ) 2 2 3 4

⇒ x1(ih1 +sh3 ) x2

x ⇒ ih1 + sh3 ≡ 0(mod p),

x

=1

(20.1)



296  Mathematics and Computer Science Volume 2



jh1 +

h1(h1 − 1) h (h − 1) ik + nh2 + th3 + 3 3 su + xh4 + kh1sh3 ≡ 0(mod p), 2 2

(20.2)

kh1 + uh3 ≡ 0(mod p),

(20.3)

lh1 + vh3 + zh4 ≡ 0(mod p).



(20.4)

Equations (20.1) and (20.3) are the system of two homogeneous equations with iu − sk ≠ 0 determinant, which gives only a trivial solution. Hence, h1 = h3 = 0. From (20.4), we get

zh4 ≡ 0(mod p) As z ≢ 0(mod p),

we get

h4 = 0.



Further, from (20.2) we get nh2 ≡ 0(mod p), but n ≢ 0(mod p) ⇒ h2 = 0. So, if h ∈ ( (Z p × Z p ) Z p ) × Z p and ψ(h) = 1, it must be that h = 1. Hence, the kernel is trivial and ψ is an automorphism with the constraints on the parameters already deduced. Now it is very easy to calculate the order of the automorphism group. We have p choices for all j, l, t, v, x, and p − 1 choices for z and i, u, s, and k will in some sense be equivalent to a matrix in GL(Fp), which gives (p2 −p)(p2 − 1) choices for the elements. Therefore, we have that ψ ∈ Aut ( (Z p × Z p ) Z p ) × Z p is really defined as:

(

)

(



 x1 → x1i x2j x3k x 4l ;  iu −sk  x2 → x2 , ψ : s t u v  x3 → x1 x2 x3 x 4 ; x → x x x z ; 2 4  4

)

j, l ∈ Z p , t , v ∈ Zp , x , z ∈ Z p , p  z.

Aut ( ( (Z p × Z p ) Z p ) × Z= p6 ( p − 1)2 ( p2 − 1). p)



Automorphisms of Some Non-Abelian p−Groups of Order p4  297

20.3.2 Computation of Automorphisms of Group G10



p p p p G10 = < x1 , x2 , x3 , x 4 ; x= x= x= x= 1, x 4 x3 = x2 x3 x 4 , x 4 x2 1 2 3 4 = x1x= x= x= x= x2 x1 2 x 4 , x1x 4 4 x1 , x 2 x 3 3 x 2 , x1x 3 3 x1 , x1x 2 > ≅ (Z p × Z p × Z p ) Z p

By using our relations from the structure of G10 and elementary calculations, we can deduce some useful relations:

x 4i x2j = x1(ij ) x2j x 4i , ij (i −1) ) (ij ) j i 2 1 2 3 4

i j 4 3

x x =x

i j k l n 1 2 3 4

(

x xx ,

n(n−1) n(n−1)l (l −1) n(n−1)l (n−2) 2   jl + k+ kl   ni + 2 4 6   1

(x x x x ) = x

x

n(n−1)   kl   jn+ 2   ( kn) ( jn) 3 2 4

x

x

.



We begin our study of automorphism group (Zp × Zp × Zp) ⋊ Zp by using the above relations. Let ψ ∈ Aut ( (Z p × Z p × Z p ) Z p ) be defined by:

 x1 → x1i x2j x3k x 4l ;  m n q r  x2 → x1 x2 x3 x 4 ; ψ : s t u v  x3 → x1 x2 x3 x 4 ; x → x w x x x y x z ; 1 2 3 4  4



i, j, k, l ∈ Z p , m, n, q , r ∈ Z p , s, t , u, v ∈ Z p , w, x, y, z ∈ Zp.



Since ψ is an automorphism, ψ should preserve the relation x4x3 = x2x3x4,

hence

ψ(x4)ψ(x3) = ψ(x2)ψ(x3)ψ(x4) ψ (x 4 )ψ (x3 ) = ψ (x2 )ψ (x3 )ψ (x 4 ) ⇒x x x x x x x x = x1m x2n x3q x 4r x1s x2t x3u x 4v x1w x2x x3y x 4z w x y z s t u v 1 2 3 4 1 2 3 4

( zt +

⇒ x1

zu ( z −1) ) ( zu ) 2 2

x

ru (r −1) vy (v −1) ry (r −1) ) + +rx +rvy + (m+rt +vx + (n+ru +vy +ry ) q r 2 2 2 2 3 4 1

=x x x x zu(z − 1) ru(r − 1) vy(v − 1) ry(r − 1) ⇒ zt + ≡ m + rt + vx + + + rx + rvy + (mod p), 2 2 2 2

(20.5)

298  Mathematics and Computer Science Volume 2



zu ≡ n + ru + vy + ry (mod p),

(20.6)



q ≡ 0(mod p) and

(20.7)



r ≡ 0 (mod p)

(20.8)

using (20.7) and (20.8) in (20.5) and (20.6), we get

⇒ zt +



zu(z − 1) vy(v − 1) (mod p), ≡ m + vx + 2 2

(20.9)

zu ≡ n + vy(mod p)

(20.10)

Also,

x 4 x2 = x1x2 x 4 ⇒ ψ (x 4 )ψ (x2 ) = ψ (x1 )ψ (x2 )ψ (x 4 ) ⇒ x1w x2x x3y x 4z x1m x2n x3q x 4r = x1i x2j x3k x 4l x1m x2n x3q x 4r x1w x2x x3y x 4z (w +m+ zn) ( x +n) y z 1 2 3 4

⇒x

x

ly (l −1)    i +m+w +ln+lx +  2  ( j +n+ x +ly ) ( k + y ) (l + z )  4 1 2 3

x x =x

x

x

x

ly (l −1)    i +ln+lx +  2  ( j +ly ) k l  3 4 1 2

x ⇒ x1( zn) =

⇒ zn ≡ i + ln + lx +

x

x x

ly(l − 1) (mod p), 2

(20.11)



j + ly ≡ 0(mod p),

(20.12)

k ≡ 0(mod p) and

(20.13)

l ≡ 0(mod p)

(20.14)

Using Equations (20.13) and (20.14) in (20.11) and (20.12), we get



⇒ zn ≡ i(mod p) or i ≡ zn(mod p) and

(20.15)



j ≡ 0(mod p).

(20.16)

Since j, k, l ≡ 0(mod p), therefore i ≢ 0(mod p) because if i ≡ 0(mod p), then Ord(x1) ≠ p. Further, from (20.15), i = zn ≢ 0(mod p), we have z ≢ 0(mod p) and n ≢ 0(mod p).

Automorphisms of Some Non-Abelian p−Groups of Order p4  299 Next, we have x2x3 = x3x2

⇒ ψ (x2 )ψ (x3 ) = ψ (x3 )ψ (x2 ) ⇒ x1m x2n x3q x 4r x1s x2t x3u x 4v = x1s x2t x3u x 4v x1m x2n x3q x 4r ⇒ x1(m+s ) x2(n+t ) x3u x 4v = x1( s +m+vn) x2(t +n) x3u x 4v ⇒ vn ≡ 0(mod p).





Now, n ≢ 0(mod p) gives v ≡ 0(mod p) From (20.10), we can see zu ≡ n + vy(mod p), therefore zu ≡ n(mod p) or n ≡ zu(mod p). Also, from (20.15) i ≡ zn(mod p), we get i ≡ zzu(mod p). Thus, we have

I ≡ z2u(mod p). Further, as n ≡ zu(mod p) and n ≢ 0(mod p), therefore u ≢ 0 (mod p). zu(z − 1) ≡ m(modp) ≡ Substituting the value of v in (20.9), we get zt + 2 zu(z − 1) m(mod p) or m ≡ zt + . Now, we shall find the constraints on the 2 parameters for which ψ has a trivial kernel. Let x1h1 x2h2 x3h3 x 4h4 ∈ ker (ψ ).

⇒ ψ (x1h1 x2h2 x3h3 x 4h4 ) = 1 ⇒ ψ (x1 )h1ψ (x2 )h2ψ (x3 )h3ψ (x 4 )h4 = 1 ⇒ (x1i x2j x3k x 4l )h1 (x1m x2n x3q x 4r )h2 (x1s x2t x3u x 4v )h3 (x1w x2x x3y x 4z )h4 = 1 h (h −1) h (h −1) z ( z −1) h4 (h4 −1)(h4 −2) 2 yz ) (ih1 +mh2 + sh3 +wh4 + 4 4 xz + 4 4 y+ 6 2 4

⇒ x1

x

h (h −1) yz ) (nh2 +th3 + xh4 + 4 4 (uh3 + yh4 ) (zzh4 ) 2 4 2 3

x

x

=1

h4 (h4 − 1) h (h − 1)z (z − 1) xz + 4 4 2 4 h (h − 1)(h4 − 2) 2 yz ≡ 0(mod p), y+ 4 4 (20.17) 6

⇒ ih1 + mh2 + sh3 + wh4 +



nh2 + th3 + xh4 +

h4 (h4 − 1) yz ≡ 0(mod p), 2

(20.18)

300  Mathematics and Computer Science Volume 2

uh3 + yh4 ≡ 0(mod p),

(20.19)

zh4 ≡ 0(mod p).

(20.20)

From (20.20), we have h4 ≡ 0(mod p) for z ≢ 0(mod p) and from (20.19), we get uh3 ≡ 0(mod p). Since u ≢ 0(mod p), therefore h3 = 0. Further, from (20.18), we have nh2 ≡ 0(mod p). As n ≢ 0(mod p), we have h2 = 0. From (20.17), ih1 ≡ 0(mod p), but i ≢ 0(mod p), thus h1 = 0. So, if h ∈ ( (Z p × Z p × Z p ) Z p ) and ψ(h) = 1, it must be that h = 1, hence the kernel is trivial and ψ is an automorphism with the constraints we have already deduced. Thus, for ψ to be an automorphism there are p choices for all s, t, w, x, y, and p − 1 choices for both u and z. Therefore, we have that ψ ∈ Aut ((Z p × Z p × Z p ) Z p ) is really defined as:



 x1 → x1nz ;  zu ( z −1) zt +  z x2zu , ψ :  x2 → x1  x3 → x1s x2t x3u ;  w x y z  x 4 → x1 x2 x3 x 4 ;

nz  0(mod p),

s, t , u ∈ Z p , p  u, w , x , y , z ∈ Z p , p  z.



Aut ( (Z p × Z p × Z p ) Z= p5 ( p − 1)2 . p) 20.3.3 Computation of Automorphisms of G11 2



p p 1+ p G11 = < x1 , x2 , x3 ; x1p= x= x= = 1, x2 x1 x= x3 x1 , x2 x3 2 3 1 x 2 , x1x 3 = x 3 x 2 > ≅ ( Z p2  Z p ) × Z p

We can deduce some useful relations by using the generator relations of G11 and elementary calculations:

x2j x1i = x1(i + jip) x2j , (ni +



(x1i x2j )n = x1

n(n−1) ijp ) (nj ) 2 2

x

.



Automorphisms of Some Non-Abelian p−Groups of Order p4  301 With these relations, we begin our study of the automorphisms of ( Z p2  Z p ) × Z p . Let ψ ∈ Aut (Z p2  Z p ) × Z p , be defined by:

(



)

 x1 → x1i x2j x3k ;  ψ :  x2 → x1l x2m x3n ;  q r s  x3 → x1 x2 x3 ;

i ∈ Z p2 , j , k ∈ Z p , l ∈ Z p2 , m , n ∈ Z p , q ∈ Z p2 , r , s ∈ Z p .



Now, ord(x1) = ord(ψ(x1)) = p2. Therefore, (x1i x2j x3k ) p ≠ 1. Hence



x1pi ≠ 1 ⇒ p  i.

Also, the order of x2 ∈ (Z p2  Z p ) × Z p is p, so the order of ψ(x2) is also p. ⇒ (x1l x2m x3n ) p = 1



1 ⇒ x1lp = ⇒ p/l ⇒ l ⇒ pt ; for some t ∈ Z p .



Order of x3 is also p, so Order of ψ(x3) is p

⇒ (x1q x2r x3s ) p = 1



1 ⇒ x1qp = ⇒ p/q pu; for some u ∈ Z p . ⇒q=



As ψ is an automorphism, ψ should preserve the relation x2 x1 = x11+ p x2 , thus we get

302  Mathematics and Computer Science Volume 2

ψ (x2 )ψ (x1 ) = ψ (x1 )1+ pψ (x2 ) ⇒ x1l x2m x3n x1i x2j x3k = (x1i x2j x3k )1+ p x1l x2m x3n x1( pi +i +l ) x2( j+m) x3(k +n) ⇒ x1(l +i +mip) x2(m+ j ) x3(n+k ) = ⇒ x1(mip) = x1( pi ) ⇒ imp ≡ pi(mod p2 ) ⇒ im ≡ i(mod p) Since p  i ⇒ i ≡ 0(mod p) ⇒ m ≡ 1(mod p). And x3 x1 = x1x3 , then ψ (x3 )ψ (x1 ) = ψ (x1 )ψ (x3 ) ⇒ x1q x2r x3s x1i x2j x3k = x1i x2j x3k x1q x2r x3s ⇒ x1(q+i +rip) x2(r + j ) x3( s +k ) = x1(i +q+ jqp) x2( j+r ) x3(k +s ) ⇒ x1(rip) = x1( jqp) ⇒ x1(rip) = x1( jpup) ⇒ x1(rip) = 1 ⇒ p/ri but p  i ⇒ p/r ⇒ r ≡ 0(mod p).



Next, we find the constraints for parameters using the kernel. Let x1x x2y x3z be any element of (Z p2  Z p ) × Z p that it is mapped to identity.

⇒ ψ (x1x x2y x3z ) = 1 ⇒ ψ (x1 )xψ (x2 ) yψ (x3 )z = 1 ⇒ (x1i x2j x3k )x (x1( pt ) x2m x3n ) y (x1( pu ) x3s )z = 1 ( xi +

⇒ x1

⇒ xi +

x ( x −1) ijp+ pty + puz ) ( xj +my ) ( xk +ny + sz ) 2 2 3

x

x

=1

x(x − 1) ijp + pty + puz = 0(mod p2 ), 2

(20.21)

Automorphisms of Some Non-Abelian p−Groups of Order p4  303

xj + my = 0(mod p),

(20.22)

xk + ny + sz = 0(mod p).

(20.23)

From (20.21), we get

x(x − 1) ijp + pty + puz 2 x(x − 1) ⇒ p/p2 /xi + ijp + pty + puz 2 x(x − 1) ijp + pty + puz ⇒ p/xi + 2 ⇒ p/xi But p  i ⇒ p/x ⇒x= 0(mod p). p2 /xi +



Next, from (20.22) we have



my ≡ 0(mod p)

But p f m

⇒ y ≡ 0(mod p).

Also, from (20.23) we obtain sz ≡ 0(mod p), but p f s, therefore z ≡ 0(mod p). So, if h ∈ (Z p2  Z p ) × Z p and ψ(h) = 1, it must be that h = 1, hence the kernel is trivial and ψ is an automorphism with the restrictions we have already deduced on the parameters. Now, we have here p2 − p choices for i and p choices for all l, q, j, k, n, and p − 1 choices for s. Therefore, we have that ψ ∈ Aut (Z p2  Z p ) × Z p is really defined as:

304  Mathematics and Computer Science Volume 2



 x1 → x1i x2j x3k ;  ψ :  x2 → x1pt x2 x3n ;  pu s  x3 → x1 x3 ;

p  i , j, k ∈ Z p , t, n ∈ Zp , u ∈ Z p , p  s.



|Aut (Z p2  Z p ) × Z p | = p6 ( p − 1)2 .

References 1. H. Arora and R. Karan: What is the probability an automorphism fixes a group element?, Communications in Algebra, 45(3), 1141–1150 (2017). 2. William Burnside: Theory of groups of finite order, Cambridge University Press, first edition, 1897. Reprinted 2010 through Nabu Press. 3. Geir T. Helleloid Automorphism Groups of Finite p-Groups: Structure and Applications, arxiv: 0711.2816 (2007). 4. Hans Liebeck: The Automorphism Group of finite p-groups, Journal of Algebra, 4, 426–432 (1966).

21 Viscoelastic Equation of p-Laplacian Hyperbolic Type with Logarithmic Source Term Nazlı Irkıl* and Erhan Pişkin† Department of Mathematics, Dicle University, Diyarbakır, Turkey

Abstract

This paper aims to address the problem with viscolelastic p-Laplacian type equations with logarithmic nonlinearity, t

u tt +

∫

g(t − s)∆ u(s) ds − div((1 + ∇ p − 2 )∇u) − ∆u t = |u|p − 2 uln|u|

0

in which p ≥ 2, under a convenient hypotheses on g (t). Under suitable conditions, we discuss global existence and blow up the results. Keywords:  Existence, blow up, viscoelastic equation, p–Laplacian type equation, logarithmic nonlinearity

21.1 Introduction Let Ω be a bounded domain in Rn (for n ≥ 1) with smooth boundary Σ = ∂Ω. We deal with the following problem in the initial boundary problem for (x, t) ∈ Ω × R+:

*Corresponding author: [email protected] † Corresponding author: [email protected] Sharmistha Ghosh, M. Niranjanamurthy, Krishanu Deyasi, Biswadip Basu Mallik, and Santanu Das (eds.) Mathematics and Computer Science Volume 2, (305–326) © 2023 Scrivener Publishing LLC

305

306  Mathematics and Computer Science Volume 2 t   u + g(t − s) ∆ u(s) ds − div((1 + ∇( p − 2) )∇u) − ∆u = |u|( p − 2)uln|u|, t  tt 0   u(x,0) = u 0 (x), u t (x , 0) = u1 (x),  x ∈∑ × R + ,  u(x , t) = 0,

∫

The memory kernel g(t) is a real function which has typical properties. In (21.1), if we take ln |u| = 1 and replace the second order as the fourth order, we have t

∫

utt + ∆ u + g(t − s) ∆ u(s) ds − div(∇ p − 2∇u) − ∆u t = f (u). 2



0



Equation (21.2) was named as a fourth order weak viscoelastic plate equation with a lower order perturbation of p-Laplacian type and it can be taken as a general form of the one-dimensional nonlinear equation of elastoplastic microstructure flows. This type of equation was first considered by Andrade et al. [3]. They deal with interplay of the p-Laplacian operator with viscoelastic terms. The authors obtained existence and decay results of solutions with gt(t) ≤ cg (t) . Later results about qualitative theory results of solutions to (21.2) type equations were obtained by different authors (see [4, 13, 14, 18, 19, 26]). The logarithmic wave equations which were introduced by BialynickiBirula and Mycielski in [5] are related with many areas of physics. Logarithmic source term occurs naturally in inflation cosmology and quantum mechanics, supersymmetric field theories, and nuclear physics [7, 17]. Later, by the motivation of these works, analysis of logarithmic source terms attracted many authors (see [8−10, 16, 23, 25, 32, 35]). There is some work about mathematical behavior in viscoelasticity with logarithmic source terms. The references were given in our work for interested readers (see [2, 6, 11, 12, 20, 31]). The model (21.1) for g = 0 and without ∆u was studied, but there is not a substantial number of work related with p-Laplacian hyperbolic type equations. For contact problems on p-Laplacian type equations with logarithmic source terms see ([15, 24, 30, 33, 34]). The goal of our work is to prove the global existence of a weak solution to the problem (21.1). Blow up results were established for suitable conditions at E (0) < d. As far as know, there is no work which is related on pLaplacian hyperbolic type equations with viscoelastic terms and logarithmic source terms.

Viscoelastic Equation of p-Laplacian Hyperbolic Type  307 The structure of the work is as follows: to understand the expression in the work more easily, firstly we give some definitions, notations, energy functions, and some lemmas which will be used in our proof in Section 21.1. In part 2 and part 3, global existence results and blow up results were stated for the solution of problem (21.1), respectively.

21.2 Preliminaries In this part, we consider the notation and some well-known lemmas. We define the inner product as



∫ v(x )u(x )dx.

(v , u) =

in Standard Lebesque Space L2 (Ω) and we denote:

(∇v, ∇u) = ∫ ∇v(x)∇u(x)dx

in H 01 (Ω ). The norm in L2 (Ω) and Lp (Ω) will be defined by ǁ.ǁ and ǁ.ǁ p. And C will be various positive constants. Lemma 21.1. [1, 22] Let q be a positive number, which implies that

2 ≤ q < ∞, for n = 1,2

2≤q≤

2n , for n ≥ 3. n−2

So, for ∀ u ∈ H 01 (Ω ), we have

ǁuǁq ≤ Cq ǁuǁ

(21.3)

where Cq is a positive constant. Now, we will consider some assumptions based on the kernel function g(t): (A1) In accordance with g: [0, ∞) → R+ (g(t) is a nonnegative function), we suppose that the function g ∈ C1 [0, ∞) and ∞

g (0) ≥ 0 ,

∫ g (s)ds = 1 − l > 0,

(21.4)

0

(A2) Let ϑ be a positive number which satisfies for t ≥ 0



g′ (t) ≤ ϑg(t).

(21.5)

308  Mathematics and Computer Science Volume 2 E(t) will be denoted total energy functional for the model investigated in (21.1). It follows that

1 1 1 E(t ) = ||ut ||2 + ||∇ u||pp +  1 − 2  2 p 1 1 + ( g °∇ u)(t ) − 2 p

t

∫

0

 1 g (s )ds  ||∇ u||2 + 2 ||u||pp p 

∫ u ln|u| dx p

(21.6)



Ω

We introduce the J(u) as a potential energy functional so that

1 1 p J (u) = ||∇ u||p +  1 − 2  p 1 1 + ( g °∇ u)(t ) − 2 p



t

∫

0

 1 g (s )ds  ||∇ u||2 + 2 ||u||pp p 

∫ u ln|u| dx. p

(21.7)



Ω

We obtain for u ∈W01, p (Ω ),

 I (u) = ||∇ u|| +  1 −  p p

t

∫

0

 g (s )ds  ||∇ u||2 + ( g °∇ u)(t ) − 

∫ u ln|u| dx , p

Ω



(21.8) where



t

( g °∇ u)(t ) = ∫ 0 g (t − s )||∇ u(s ) −∇ u(t )||2 ds

(21.9)

Then, it is clearly concluded that for u ∈W01,p (Ω ),

1 1  J (u) =  −   1 −  2 p  



t

∫

0

 I (u) 1 g (s ) ||∇ u||2 + + 2 ||u||pp p p 

1 1 +  −  ( g °∇ u)(t ),  2 p



(21.10)

Viscoelastic Equation of p-Laplacian Hyperbolic Type  309

1 E(t ) = ||ut ||2 + J (u). 2



(21.11)

According to potential well theory, the set stability for the problem (21.1) was considered as:

U = {u ∈W01,p (Ω )| J (u) < d,I (u) > 0} ∪ {0},



(21.12)

So, we denote outer space of the potential well by

V = {u ∈W01,p (Ω )| J (u) < d,I (u) < 0}.



(21.13)

The potential well depth can be considered as:

d=



inf

1, p

u∈W0 ( Ω )\{0} λ >0

sup(λu).

(21.14)

Similar to the result in [29], one has:

0 < d = inf J (u),



u∈N

Where according to the potential functional, well-known Nehari Manifold N is given by:

N = {u ∈W01,p (Ω )\ {0}: I (u) = 0}.



Now, we state some definitions and lemmas. Lemma 21.5 and Lemma 21.6 are related with potential well theory results. Lemma 21.2. [21]. Suppose that u is a function which satisfies u ∈W01,p (Ω ) and a is defined as positive number. Then,



∫

Ω

u p ln |u|dx
0 and u be defined as the solution of problem (21.1). If u holds

u ∈C([0 ,T ];W01,p (Ω )),ut ∈C([0 ,T ]; H 01 (Ω ),utt ∈([0 ,T ];W0−1,p ¢ (Ω )),

310  Mathematics and Computer Science Volume 2 and

∫

utt vdx +

Ω

∫

t

(1 +∇ p−2 )∇ u∇ vdx −

Ω

∫ ∫ ∫ ∇ u∇ vdsdx + ∫ ∇ u∇ vdx = ∫ u t

Ω

0

Ω

Ω

p−2

uln |u|vdx,

Ω



for each test functions v ∈W01,p (Ω ), then u will be taken as a weak solution for the (21.1). Lemma 21.4. Assume that (A1) and (A2) conditions, which are related with g(t) functional, hold. Then, the total energy of problem (21.1) decreases over respect to t because

g (t ) 1 1 ||∇ u(t )||2 E ′(t ) = ( g ′ °∇ u)(t ) − ||∇ ut ||2 − 2 2 2 1 ≤ − ||∇ ut ||2 ≤ 0. 2



(21.15)

where t

( g ′ °∇ u)(t ) =

∫ g ′(t − s)||∇ u(s) −∇ u(t )|| dt 2

(21.16)

0

Proof. First, we multiply both sides of the equation by ut in the (21.1) equation. Later, by integrating over t it yields that

1 E(t ) = E(0) − ||∇ ut ||2 − 2

t

∫

0

1 g (t )||∇ u(t )|| + 2 2

t

∫ ( g ′ °∇ u)(t ) 0

1 ≤ ||∇ ut ||2 2



Here, by direct calculations we obtained (21.16). 1 ,p

Lemma 21.5. For any u ∈W0 (Ω )\{0}, and ||u||pp ≠ 0, it gives

i) lim J (λu) = 0 , lim J (λu) = −∞ , λ →0

λ →∞

(21.17)

Viscoelastic Equation of p-Laplacian Hyperbolic Type  311 ii) For 0 < λ < ∞ we find a unique λ* such that

dJ (λu) =0 d λ λ = λ*



iii) J (λu) functional is strictly decreases on λ* < λ < ∞, increases on 0 < λ < λ* and attains the maximum at λ = λ* and I (λu) > 0 for 0 < λ < λ*, I (λu) < 0 for λ* < λ < ∞ and the Nehari functional

dJ (λu) = I (λu). dλ Proof. i) Thanks to the definition of J(u), we conclude that I(λ*u) = 0, where λ

1 J ( λu ) = p −



1 p

∫

Ω

1 (λ∇ u) dx +  1 − 2  p

t

∫

0

 g (s )ds  

1

∫ (λ∇ u) dx 2

Ω

1

∫ (λu) ln |λu|dx + p ∫ (λu) dx + 2 λ ( g °∇ (λu))(t ) p

p

2

Ω

2

Ω

t

=

λ2  1 − 2 

+

λ λp p ||∇ u||p − p p

∫

0

 1 λ2 p g (s )ds  || u||2 + ( g °∇ u)(t ) + 2 λ p ||u||p 2 p 

p

∫

(u) p ln|u| dx −

Ω

1 p λ lnλ p

∫ (u) dx. p

(21.18)

Ω



p

It is easy to get from ||u||p ≠ 0 that is lim J (λu) = 0 , lim J (λu) = −∞. λ →0 λ →∞ ii)  Now, differentiating J (λu) with respect to λ, it yields that

 dJ (λu) = λ  1 − dλ   −λ p−2 



t

 g (s )ds  || u||2 +( g °∇ u)(t ) + λ p − 2 ||∇ u||pp 

∫

0

∫ (u) ln|u|dx − λ p

Ω

Let λ −1

dJ (λu) = φ (λu), so we obtain dλ

p−2

lnλ||∇ u||pp  .  

(21.19)



312  Mathematics and Computer Science Volume 2

dφ (λu) = − λ p −3[(2 − p)||∇ u||pp +( p − 2) dλ p p

∫ (u) ln|u|dx p

Ω

p p

+ ( p − 2)lnλ||u|| + ||u|| ].

Let



dφ (λu) = 0 , which implies dλ

 (2 − p)||∇ u||pp +( p − 2) ∫ Ω (u) p ln|u| dx + ||u||pp  λ1 = 0 and λ2 = exp   (2 − p)||u||pp  An elementary calculation shows that dφ (λu) dφ (λu) > 0 on (0 , λ2 ); < 0 on (λ2 , ∞). dλ dλ



We observe from 2 ≤ p that lim φ (λu) = −∞. so that there is exactly λ →∞

dJ (λu) = 0. ϕ(λ*u) = 0, which means d λ λ = λ* iii)  A simple corollary of the (ii) directly follows from



I ( λu ) = λ

dJ (λu) . dλ

(21.20)

Lemma 21.6. Assume that u ∈W01,p (Ω )\{0}, and ||u||pp ≠ 0. Then, we obtain sup λu = d where λ >0



n 2

d = [2π (l + 1)] e

2(n + p( l +1))− p 2 ( l +1) 2

.

Proof. If I(λ*u) = 0 and ||u||pp ≠ 0, from Lemma 21.4 and Equation (21.10), we obtain

Viscoelastic Equation of p-Laplacian Hyperbolic Type  313

I (λ *u) ( p − 2)  sup λu = J (λ *u) = + 1 − p 2 p  λ >0



Since 0
2 [2π (l + 1)]2 e p

2(n + p ( l +1))− p 2 ( l +1)) 2

=

h . p2



21.3 Global Existence Result This part is related to global existence results. We established that the solutions will exist globally for problem (21.1) for the case 0 < E(0) < d by considering the following lemma to state proof of Theorem 21.8. Lemma 21.7. Assume that u0 ∈ U, u1 ∈ H 01 (Ω ). Let initial energy holds 0 < E(0) < d. Moreover, u ∈ U where u is taken as a weak solution of Equation (21.1). Proof. For 0 < E(0) < d, we claim that u was taken as a weak solution for Equation (21.1). Now, we will use the contradiction method. We suppose that u ∉ U and t* are the smallest time of u (t*) ∉ U. By continuity of u(t), we know clearly that u(t*) ∈ ∂U and u ∈ U for t ∈ [0, t*). By virtue of the definition of U and the continuity of I(u) and J(u) about time, we have either J(u(t*)) = d or I(u(t*)) = 0. Thanks to Lemma 21.3, it follows that

Viscoelastic Equation of p-Laplacian Hyperbolic Type  315

d > E(0) ≥ E (u(t *)) ≥ J(u(t *)) .



(21.26)

Therefore, it is clear that the former case is not possible. Suppose that I(u(t*)) = 0, then u (t*) ∈ N. By the definition of d, J(u (t*)) ≥ d was obtained, which is in contradiction with J(u (t*)) < E(0) < d. So, we showed that the other case is not possible as well. Theorem 21.8. Let u0 ∈ U, u1 ∈ H 01 (Ω ). Then, Equation (21.1) with initial boundary conditions has a global weak solution u(t ) ∈ L∞ (0 , ∞;W01,p (Ω )) with utt ∈ ∞ u(t ) ∈ L (0 , ∞;W01,p (Ω )) with utt ∈ L∞ (0 , ∞; H 01 ((Ω )) under the condition 0 < E (0) < d. Proof. Our purpose is to prove that ||∇ u||pp +l ||∇ u||2 + ||∇ u||pp is bounded for not depending on t. By using Lemma 21.4, (21.11), definition of E(t), and the condition 0 < E(0) < d, we obtain

1 E(t ) = ||ut ||2 + J(u(t )) ≤ E(t ) < d . 2



(21.27)

According to conditions in Theorem 21.8, by using Lemma 21.7, for t ≥ 0 we obtain



u(t) ∈ U.

(21.28)

We conclude from (21.10) and (21.28) that

I (u)  1 1   + − 1 − d > J (u) = p  2 p  



t

∫

0

 g (s )ds  ||∇ u||2 

+

1 1 p 1 −  ( g °∇ u)(t ) 2 ||u||p +  p  2 p

>

( p − 2)l 1 p ||∇ u||2 + ||u||p 2p p2



(21.29)

which implies that



2 pd p > ||∇ u||2 and p 2d > ||u||p . ( p − 2)l



(21.30)

316  Mathematics and Computer Science Volume 2 It follows from (A1), the definition of J(u), (21.11), and (21.27) that

1 1 p p ||ut ||2 + 2||∇ u||pp + pl ||∇ u||2 + ||u||p ≤ 2 pd + p p

∫ u ln|u| dx. p

Ω

(21.31)

By using Lemma 21.1, (21.31) shows that

1 p p ||ut ||2 + 2||∇ u||pp + pl ||∇ u||2 + ||u||p p ( p − 2)a 2 a2 p p ||u||p + ||∇ u||p ≤ 2 pd + 4π 2π n p p − (1 + ln a)||u||p + ||u||p +ln ||u||p . p



(21.32)

Let a = 2π (1 + l ), then by using Lemma 21.6 and (21.30), (21.32) becomes

p ||ut ||2 + (1 − l )||∇ u||pp + pl ||∇ u||2 1

( p − 2)(1 + l ) ||u||pp + ||u||pp ln( p 2d ) p } ≤ 2 pd + 2 1 n − (1 + ln 2π (1 + l ))||u||pp − ||u||pp p p p −1 1 (n +1) n ( p − 2)(1+ l )  2 pdp  p 2p p p 2 2 (2π (1 + l ))  ||u||pp + ln( p d ) − ln e ≤ e + e ( p − 1)  



=

4 pd ln p ( p − 1)



(21.33)



which satisfies that

 1 1 1  1   4 pd p ||ut ||2 + (1 − l )||∇ u||pp + pl ||∇ u||2 ≤ max  , ,   ln p.  p 1 − l pl  pl   ( p − 1) (21.34)

Viscoelastic Equation of p-Laplacian Hyperbolic Type  317 Therefore, we have solutions (u) for the Equation (21.1) that exist globally, by using (21.34) and by the same argument, the continuation theorem [27, 28].

21.4 Blow Up Results of the Solution for Equation (21.1) This section is related to blow up results for Equation (21.1) at infinity. We state some lemma which will be useful for our proof. Proof Lemma 21.9 can be obtained by same argument as Lemma 21.7. Lemma 21.9. Let u be taken as a weak solution of Equation (21.1). Suppose that u0 ∈ V, u1 ∈ H 01 (Ω ) and 0 < E(0) < d. Therefore, we have u ∈ V and E(t) < d. Theorem 21.10. Let u ∈W01,p (Ω ) and u1 ∈ H 01 (Ω ) and ∫ Ω u0u1dx > 0. If u0 ∈ V and E (0) < d, then the solution u of Equation (21.1) goes to the infinity when t → ∞, i.e.,

lim u(t ) = ∞.



t →∞

Proof. Under the conditions of Theorem 21.10, we conclude that I(u) is decreasing with respect to t, thank to results of the Lemma 21.9. Thus, we can find for all t > 0

 I (u) = ||∇ u|| +  1 −  p p

t

∫

0

 g (s )ds  ||∇ u||2 + ( g °∇ u)(t ) − 

∫ u ln|u| dx < 0. p

Ω



(21.35) By using the Logarithmic Sobolev inequality, (21.35) can be written as

  a2  p  1 −  ||∇ u||p +  1 − 2π  − 

t

∫

0

 g (s )ds  ||∇ u||2 + ( g °∇ u)(t ) 

2

( p − 2)a n ||u||pp +  (1 + ln a) − ln ||u||p  ||u||pp 4π p 

< 0.

(21.36)

318  Mathematics and Computer Science Volume 2 Since 0
2, a = (2π (1 + l ), (21.37), (21.38), and (21.15), (21.49) can be found as t

2

2

Λ(t ) ≥ −2 p E(t ) − ( p + 2)

∫ |∇ u (s)| ds + 2 p d t

2

2

0

 ≥ −2 p  E(0) −  2

t

∫

0

 |∇ ut (s )| ds  + 2 p 2d − ( p 2 + 2)  2

t

∫ |∇ u (s)| ds t

2

0

t

= −2 p 2 E(0) + 2 p 2d + ( p 2 − 2)

∫ |∇ u (s)| ds. t

2

(21.50)

0





Therefore, we obtain from the condition of Theorem 21.10 (E (0) < d)



Λ(t) ≥ 0.

(21.51)

Therefore, by Inequality (21.51) and condition of the functional G(t), (21.48) yields that



G′′G −

p2 + 2 (G′)2 ≥ 0. 4

(21.52)

Viscoelastic Equation of p-Laplacian Hyperbolic Type  323 By directly calculation of (21.52), it follows that

ln|G ′(t )| − ln|G(t )|



p2 + 2 4

≥ 0.

(21.53)

By integrating both sides of Inequality (21.53) over [0, t] for p > 2, it yields that

G(t ) ≥

G(0)  p2 − 2 G(0) 1 −  4

p2 − 2 4

 t 

4 p −2

.

(21.54)

2

p2 − 2

 G ′(0)  4 < 1 where From definition of G(t), it is clear to conclude that   G(0)  G(0) =|| u ||2 + T || ∇u ||2 . Consequently, we can rewrite (21.54) so that

G(t ) ≥

G(0) 2

 ( p − 2)  G ′(0)   1 − 4  G(0)   

4 p2 − 2

.

Therefore, lim G(t ) = ∞ for some T* satisfies t →T *



0 uij in AAP, go to Step 4 (capacity violation) Formulate AAP Use NP-complete PARTITION problem to reduction Reduce AAP to AP by averaging the capacity of arcs where violation occurs 7. Feasibility of AP implies the feasibility of AAP [10] 8. Construct hybrid network, i.e, mixed capacity network - Output: Feasible flow

334  Mathematics and Computer Science Volume 2

22.5 Conclusion When AP is transformed into the time-expanded network, it violates capacity constraints in AAP. This infeasibility of flow leads to the linear formulation of AAP. Using the well-known NP-complete PARTITION problem, AAP is reduced to AP by some modification on arc capacities, i.e, averaging the capacities by the corresponding transit times. The feasibility of reduced AP can be obtained by the algorithm introduced by Hoppe and Tardos [10], implying the feasibility of the original AAP. Practically, the hybrid network would be functioning simultaneously. After observing the feasibility of the flow, the optimality can be obtained by the suitable optimality seeking method (such as network simplex method, etc.).

Acknowledgment The first author acknowledges University Grants Commission, Nepal for PhD fellowship.

References 1. E. J. Anderson, P. Nash and A. B. Philpott (1982). A class of continuous network flow problems. Mathematics of Operation Research, 7: 501-514. 2. J. E. Aronson (1989). A survey of dynamic network flows. Annals of Operations Research, 20: 1-66. 3. N. Baumann and M. Skutella (2006). Solving evacuation problems efficiently–earliest arrival flows with multiple sources. In: 47th annual IEEE symposium on foundations of computer science (FOCS’06), 399–410. 4. R. Burkard, K. Dlaska and B. Klinz (1993). The quickest flow problem. ZOR Meth Models Oper Res, 37: 31–58. 5. L. R. Ford and D. R. Fulkerson (1958). Constructing maximal dynamic flows from static flows. Operations Research Letters, 6: 419-433. 6. L. R. Ford and D. R. Fulkerson (1962). Flows in networks. Princeton University Press, New Jersey. 7. D. Gale(1959). Transient flows in networks. Michigan Math J, 6(1): 59–63. 8. H. Hamacher and S. Tjandra (2001). Mathematical modeling of evacuation problems: a state of art. Berichte des Frauenhofer ITWM, Nr. 24. http:// www.itwm.fraunhofer.de/fileadmin/ITWMMedia/Zentral/Pdf/Berichte ITWM/2001/bericht24.pdf. Accessed 18 Oct 2006. 9. B. Hoppe and E. Tardos (1994). Polynomial time algorithms for some evacuation problems. In: Proceedings of the fifth annual ACM-SIAM symposium

Flow Dynamics in Continuous-Time with Average Arc Capacities  335 on discrete algorithms, Society for Industrial and Applied Mathematics Philadelphia, PA, 433–441. 10. B. Hoppe and É. Tardos (2000). The quickest transshipment problem. Mathematics of Operations Research 25: 36–62. 11. E. Kohler, R. H. Mohring, and M. Skutella (2009). Traffic networks and flows over time. In: Lerner J, Wagner D, Zweig K (eds) Algorithmics of large and complex networks. Lecture notes in computer science, vol 5515. Springer, Berlin, 166–196. 12. B. Kotnyek (2003). An annotated overview of dynamic network flows. INRIA, Sophia Antipolis, France. 13. V. Melkonian (2007). Flows in dynamic networks with aggregate arc capacities. Information Processing Letters 101: 30–35. 14. E. Nasrabadi and S. M. Hashemi (2010). Minimum cost time-varying network flow problems. Optim Meth Software, 25(3): 429–447. 15. B. P. Pangeni and T. N. Dhamala (2021). A brief survey on dynamic network flows in continuous-time model. Journal of Mathematical Sciences and Computational Mathematics 2(4): 467-477. 16. A. B. Philpott (1990). Continuous-time flows in networks. Mathematics of Operation Research, 15: 640-661. 17. A. B. Philpott (1982). Algorithms for continuous network flow problems. Ph.D. thesis, University of Cambridge UK. 18. M. Skutella (2009). An introduction to network flows over time. In W. Cook, L. Lovasz and J. Vygen (eds.): Research Trends in Combinatorial Optimization, Springer, Berlin, 451-482.

23 Analysis of a Multiserver System of Queue-Dependent Channel Using Genetic Algorithm Anupama1* and Chandan Kumar2 Department of Mathematics, Darbhanga College of Engineering, Darbhanga, Bihar, India 2 Department of Mechanical Engineering, BIT Sindri, Jharkhand, India

1

Abstract

In the present work, a M/M/3 Queuing system with a Queue-dependent multi-server has been considered. Here, a number of failed machines form a queue and repairmen consider a service provider or service channel which starts its service when the queue length is N. We found a generating function for breakdown machines. Afterwards, some performance measures including idle time and busy time for the system have been evaluated. At last, cost is optimized using a genetic algorithm. Keywords:  M/M/3, repairmen, queue dependent server, genetic algorithm

23.1 Introduction The concept of a multiserver queueing system with queue-dependent servers is not new. Hahn and Sivazlian, in 1990, developed the M/M/2 model with service stations [5]. Yadin and Naor, in 1963, invented the N policy concept for the single server system [9]. The (0, K, N, M) rule in a two server system used by Rhee and Sivazlian in 1990 helps to derive the working period distribution [12]. Natarajan studied a system for warm stand-bys in 1968 [13]. With one online unit and two standbys with failure and repair time exponentially distributed, this model assumes only one condition, namely that the system will fail if spares are not present for the breakdown machine. There is *Corresponding author: [email protected] Sharmistha Ghosh, M. Niranjanamurthy, Krishanu Deyasi, Biswadip Basu Mallik, and Santanu Das (eds.) Mathematics and Computer Science Volume 2, (337–348) © 2023 Scrivener Publishing LLC

337

338  Mathematics and Computer Science Volume 2 an analytical closed-form solution for the M/Ek/1 system [15]. The server system for standbys and 2 types of failure time distribution is also examined. Wang and Ke (2000) introduce a supplementary and recursive technique for developing a solutions to a single server queueing system working with the N policy [16]. The current system is that the line length determines the server value in the system. In the event that no machines fail in the system, all the fixers are temporarily inactive until the length of the line reaches a predetermined level, i.e., W, which is also a decision change. For this single server period, it will be activated instantly. Sometime later, if the line length in the system goes up to the aforementioned level, it says S, which is less than W and then the other available server will also run immediately. Similarly, if the failure machines in the system go up to a next level, say T, which is less than S, then all the servers will be activated immediately. The objectives of this paper are: i. To achieve a sustainable distribution of waiting line opportunities ii. Finding average count of failed machines iii. To design the total cost of the model iv. The genetic algorithm method is used to optimize costs

23.2 Description of the Model The states for the model depicted by [i, n], i = 0, 1, 2, 3 represent that no repairman is working, one of the repairmen is working, two of the repairmen working, or three of the repairmen working, respectively. n ϵ [0, L] is the number of failed machines in the model.

23.3 Notations ϸ[0, n] = Occurrence of n customers in the model where there are no operators, when n ϵ [0, W-1] ϸ[1, n] = Occurrence of n customers in the model when one modifier works, when n ϵ [0, S-1] ϸ[2, n] = Occurrence of n customers in the model, where two moderators are working, when n ϵ [q + 1, T-1]

Analysis of a Multiserver System of Queue-Dependent Channel   339 ϸ[3, n] = Probability of n customers in the system, where three moderators are working, when n ϵ [ r + 1, L]. α = Average arrival rate β = Average service level ϸ0 = Zero server probability in the system ϸ1 = One server probability in the system ϸ2 = Two server probability in the system ϸ3 = Three server probability in the system L0 = Line length of the system if all fixers are on vacation L1 = Length of the line in the system where a single server is working in the system L2 = Line length in the system where two modifiers are working in the system L3 = Line length in the system where all three fixers work in the system Ls = Line length in system EI =Idle time expected value E B1 = Busy time expected length during which any repairman is working on the system E B2 = Busy time expected length during which any repairman is working on the system E B3 = Expected length of busy time during which every repairman works on the system E C = Busy time expected length Ch = Cost of each failed machine during the unit available in the system Cv1 = Cost savings per failed machine during each unit in the system when 1 mechanic is on vacation Cv2 = Cost of handling per failed machine during each unit present in the system when 2 technicians are on vacation Cv3 = Cost savings per failed machine during each unit present in the system when 3 technicians are on vacation C01 = Holding cost of single technician functioning in the system during each unit time C02 = Holding cost of 2 technicians functioning in the system during unit time C03 = Holding cost of 3 technicians functioning in the system during each unit time

340  Mathematics and Computer Science Volume 2

23.4 Steady State Equations A diagram of the rate of change is discussed in Anupama and Solanki (Nov 2014). A diagram of the arrival and departure of the failed machines is shown. Also, server installation is based entirely on line length in the specified system. The steady state probability equations for the system are given below:



α ∗ ϸ[0,0] = β ∗ ϸ[1,1]

(23.1)



α * ϸ[0, n] = α * ϸ[0, n–1]   n є [1, W – 1]

(23.2)



[α + β] ∗ ϸ[1,1] = β ∗ ϸ[1,2]

(23.3)



[α + β] * ϸ[1, n] = α * ϸ[1, n–1] + β * ϸ[1, n+1]   n є [2, q – 1] (23.4)



[α + β] ∗ ϸ[1,q] = α ∗ ϸ[1,q−1] + β ∗ ϸ[1,q+1] + 2 ∗ β ∗ ϸ[2,q+1] (23.5)



[α + β] * ϸ[1, n] = α * ϸ[1, n–1] + β * ϸ[1, n+1]   n є [q + 1, W – 1] (23.6)



[α + β] ∗ ϸ[1,W] = α ∗ ϸ[1,W−1] + β ∗ ϸ[1,W+1] + α ∗ ϸ[0,W−1] (23.7)



[α + β] * ϸ[1, n] = α * ϸ[1, n–1] + β * ϸ[1, n+1]   n є [W + 1, S – 2] (23.8)



[α + β] ∗ ϸ[1,S−1] = α ∗ ϸ[1,S−2]

(23.9)



[α + 2 β] ∗ ϸ[2,q+1] = 2 ∗ β ∗ ϸ[2,q+2]

(23.10)

[α + 2β] * ϸ[2, n] = α * ϸ[2, n–1] + 2 * β * ϸ[2, n+1]  n є [q + 2, r – 1] (23.11)

[α + 2 β] ∗ ϸ[2,r] = α ∗ ϸ[2,r−1] + 2 ∗ β ∗ ϸ[2,r+1] + 3 ∗ β ∗ ϸ[3,r+1] (23.12)

[α + 2β] * ϸ[2, n] = α * ϸ[2, n–1] + 2 * β * ϸ[2, n+1]  n є [r + 1, S – 1] (23.13)

[α + 2 β] ∗ ϸ[2,S] = α ∗ ϸ[2,S−1] + 2 ∗ β ∗ ϸ[2,S+1] + α ∗ ϸ[1,S−1] (23.14)

Analysis of a Multiserver System of Queue-Dependent Channel   341



[α + 2β] * ϸ[2, n] = α * ϸ[2, n–1] + 2 * β * ϸ[2, n+1]  n є [S + 1, T – 2] (23.15)



[α + 2 β] ∗ ϸ[2,T−1] = α ∗ ϸ[2,T−2]

(23.16)



[α + 3 β] ∗ ϸ[3,r+1] = 3 ∗ β ∗ ϸ[3,r+2]

(23.17)

[α + 3β] * ϸ[3, n] = α * ϸ[3, n–1] + 3 * β * ϸ[3, n+1]   n є [r + 2, T – 1] (23.18)



[α + 3 β] ∗ þ[3,T] = α ∗ þ[3,T−1] + 3 ∗ β ∗ þ[3,T+1] + α ∗ þ[2,T−1] (23.19)

[α + 3β] * þ[3, n] = α * þ[3, n–1] + 3 * β * þ[2, n+1]   n є [T + 1, L – 1]

(23.20)



α ∗ þ[3,L−1] = 3 ∗ β ∗ þ[3,L]

(23.21)

The probability ϸ[i,n] (i = 0-3) from equation 23.1-23.21 are found by using recursive method

þ[0, n] = þ[0,0],    n є [1, W – 1]



 [1,n] =



[1 − n ]  [0,0] , (1 − )

n  [1, q]

(23.22)

(23.23)

 (1 − n ) (S − S−W )(1 − n−Q )  +  [1,n] =    [0,0] n  [q + 1, W] (1 − )(1 − S−Q )   (1 − ) 

(23.24)

 (1 − n )(n−W − S−W )   [1,n] =   [0,0] , S−Q ( 1 − )( 1 − )    



 [ 2 ,n ] =



   n−Q  (1 −  ) 1 −      2   , [0,0] (2 − )(1 − S−Q )

S−W −1

n  [W + 1, S −1] (23.25)

W

n  [q +1, S] (23.26)

342  Mathematics and Computer Science Volume 2







 [ 2 ,n ] =



 [3 ,n] =

 [ 3 ,n ] =



S−W −1



n−S

S−Q

  

[0 ,0]

, n  [S, S + 1] (23.27)

   n−r  (1 −  ) 1 −      3    , n  [r +1,T ] [0 ,0] (3 − )(1 − ρ T−r )

T−S−1

T−S−1

Where  =

   (1 −  )   1 −   2  2 (2 − )(1 − S−Q ) W

S

n −T T−r     (1 −  )   1 −     [0 ,0] 3  3  , n  [ T + 1, L] (23.29) (3 − )(1 − T−r ) S

α β

23.4.1 Performance Characteristics of the System 0 = zero server probability = ∑ nW=−01 [0,n] , S −1

1 = One server probability =

∑

[1,n]

,

n =1

T −1

2 = Two server probability =

∑

[2,n]

,

n =Q +1 L

3 = Three server probability =

(23.28)

∑

n = r +1

[3,n]

,

Analysis of a Multiserver System of Queue-Dependent Channel   343

23.4.2 Queue Length Evaluations at Different Epochs W −1

L 0 = Length of queue when all server are on vacation =

∑ n

[0,n]

,

n =1

S −1

L1 = Length of queue when one server is working in the system =

∑ n

[1,n]

,

n =1

T −1

L 2 = Length of queue when two server is working in the system =

∑ n

[2,n]

n = Q +1 L

L 3 = Length of queue when one server is working in the system =

∑ n

[3,n]

n = r +1

23.4.3 Leisure Period and Working Period Length

Expected Leisure period , EI = W / α

(23.30)

The queue length of a busy one-time server one, busy two-time server, and busy three-time server are calculated. Since a working period is the addition of the leisure time, the busy time of 1 fixer, the busy time of 2 fixers, and the busy time of 3 fixers, EC = EB1 + EB2 + EB3 + EI

23.4.4 Cost of the System Total expected cost during each unit of the M \ M \ 3 line system with a limited power L is measured. Here, {0, (q, r), W, S, T} are variable resolutions. In line with our goal of determining the maximum number of control variables, {(q, r), W, S, T} states (q*, r*, W*, S*, T*) to decrease the cost of the model. The cost of the model is below.

Expected cost ,E{(q , r ),W , S,T } = Ch Ls + (CV 1 + CV 2 + CV 3 ) 

+ C03

EB3 + α Cl P (3, l) EC

EB EI EB + C01 1 + C02 2 EC EC EC (23.31)



,

344  Mathematics and Computer Science Volume 2 Since the total amount of expected cost of work is unstructured, it is very tough to the correct values by analyzing reduction in the total cost of the expected cost over time for each unit. A specific search method can be used to find important results. In a search method, it is mandatory to have high limits for dynamic decisions and the search method can be discontinued after a complete global search in the internal area. The upper bound is L for the optimal values (q*, r*, W*, S*, T*). Total cost function given in (23.31) is minimized by formulating the optimization problem

E(q* ,r * ,W * ,S* ,T *) =

min

Q λ1 ,λ2 ,λ3 (B), then A ≻ B; b)  λ1 ,λ2 ,λ3 ( A) <  λ1 ,λ2 ,λ3 (B), then A ≺ B; c)  λ1 ,λ2 ,λ3 ( A) =  λ1 ,λ2 ,λ3 (B), then A ~ B, where (0 ≤ λ1, λ2, λ3 ≤ 1).

24.4 Theorems Some crucial and fundamental theorems are discussed here. Theorem 24.4.1. Let  λ1 ,λ2 ,λ3 be linear. Proof. Let, A = ⟨(a1, b1, c1), (i1, j1, k1), (p1, q1, r1)⟩ and B = ⟨(a2, b2, c2), (i2, j2, k2), (p2, q2, r2)⟩ be two SVTNNs and κ be any real number. Then, it has to  λ1 ,λ2 ,λ3 ( A) + κ  λ1 ,λ2 ,λ3 (B). Then, prove that  λ1 ,λ2 ,λ3 ( A + κ B) = 1

1

 λ1 ,λ2 ,λ3 ( A + κ B) is equal to λ1 ∫ f (α )Lα ( A + κ B)dα + (1 − λ1 ) ∫ f (α ) 0

0

1

1

Rα ( A + κ B)dα + λ2 ∫ g (β )Lβ ( A + κ B)d β + (1 − λ2 ) ∫ g (β )R β ( A + κ B)d β + λ3 0

0

1

1

0

0

1

∫ h(γ )Lγ ( A + κ B)dγ + (1 − λ3 ) ∫ h(γ )Rγ ( A + κ B)dγ ,which leads to λ1 ∫

0 1

1

f (α )(Lα ( A) + κ Lα (B))dα + (1 − λ1 ) ∫ f (α )(Rα ( A) + κ Rα (B))dα + λ2 ∫ g (β ) 0

0

1

1

0

0

(Lβ ( A) + κ Lβ (B))d β + (1 − λ2 ) ∫ g (β )(R β ( A) + κ R β (B))d β + λ3 ∫ h(γ )(Lγ ( A) + 1

1

κ Lγ (B))dγ + (1 − λ3 ) ∫ h(γ )(Rγ ( A) + κ Rγ (B))dγ , then it is true that(λ1 ∫ f (α ) 0

1

α

0

1

α

1

β

L ( A)dα + (1 − λ1 ) ∫ f (α )R ( A)dα + λ2 ∫ g (β )L ( A)d β + (1 − λ2 ) ∫ g (β )R 0

0

β

0

1

1

1

0

0

0

( A)d β + λ3 ∫ h(γ )Lγ ( A)dγ + (1 − λ3 ) ∫ h(γ )Rγ ( A)dγ ) + κ (λ1 ∫ f (α )Lα (B)dα +



3

0

1

0

0 1

1

f (α )(Lα ( A) + κ Lα (B))dα + (1 − λ1 ) ∫ f (α )(Rα ( A) + κ Rα (B))dα + λ2 ∫ g (β ) 0

0

1

1

0

0

(Lβ ( A) + κ Lβ (B))d β + (1 − λ2 ) ∫ g (β )(R β ( A) + κ R β (B))d β + λ3 ∫ h(γ )(Lγ ( A) + 1 Ranking of Single Valued Neutrosophic Fuzzy Numbers  1 357 κ Lγ (B))dγ + (1 − λ3 ) ∫ h(γ )(Rγ ( A) + κ Rγ (B))dγ , then it is true that(λ1 ∫ f (α ) 0

0

1

1

1

Lα ( A)dα + (1 − λ1 ) ∫ f (α )Rα ( A)dα + λ2 ∫ g (β )Lβ ( A)d β + (1 − λ2 ) ∫ g (β )R β 0

0

1

0

1

1

( A)d β + λ3 ∫ h(γ )Lγ ( A)dγ + (1 − λ3 ) ∫ h(γ )Rγ ( A)dγ ) + κ (λ1 ∫ f (α )Lα (B)dα + 0

0

0

1

1

1

0

0 1

0

1 0

0

(1 − λ1 ) ∫ f (α )Rα (B)dα + λ2 ∫ g (β )Lβ (B)d β + (1 − λ2 ) ∫ g (β )R β (B)d β +λ3 ∫ h(γ )Lγ (B)dγ + (1 − λ3 ) ∫ h(γ )Rγ (B)dγ ), then the result follows that  λ1 ,λ2 ,λ3 ( A) + k  λ1 ,λ2 ,λ3 (B). Theorem 24.4.2. If A, B, and C are three arbitrary SVTNNs and A ≲ B and B ≲ C are true, then A ≲ C is true. Proof. Given that A ≲ B, then  λ1 ,λ2 ,λ3 ( A) ≤  λ1 ,λ2 ,λ3 (B), for 0 ≤ λ1, λ2, λ3 ≤ 1. Also, given that B ≲ C, then  λ1 ,λ2 ,λ3 (B) ≤  λ1 ,λ2 ,λ3 (C ), for 0 ≤ λ1, λ2, λ3 ≤ 1. This implies that  λ1 ,λ2 ,λ3 ( A) ≤  λ1 ,λ2 ,λ3 (C ), for 0 ≤ λ1, λ2, λ3 ≤ 1. Hence, the result follows that A ≲ C.

24.5 Numerical Examples In this section, some numerical examples are discussed to assess how well the proposed method performs. Example 24.5.1. Consider two SVTNNs A = ⟨(0.50, 0.65, 0.80), (0.10, 0.15, 0.30), (0.10, 0.20, 0.30)⟩, B = ⟨(0.10, 0.20, 0.30), (0.20, 0.30, 0.40), (0.40, 0.50, 0.70)⟩ and three weighted functions be P = (λ1 = 0.3, λ2 = 0.4, λ3 = 0.5), Q = (λ1 = 0.5, λ2 = 0.5, λ3 = 0.5) and S = (λ1 = 1.0, λ2 = 1.0, λ3 = 0.9). Then, the comparison is between two SVTNNs, A and B, as shown in Table 24.1. Example 24.5.2. Consider two SVTNNs A = ⟨(0.30, 0.45, 0.50), (0.10, 0.20, 0.40), (0.10, 0.20, 0.30)⟩, B = ⟨(0.20, 0.30, 0.35), (0.10, 0.10, 0.10), (0.60, 0.70, 0.80)⟩ and three weighted functions P = (λ1 = 1.0, λ2 = 1.0, λ3 = 1.0), Q = (λ1 = 0.5, λ2 = 0.5, λ3 = 0.5) and S = (λ1 = 0.1, λ2 = 0.1, λ3 = 0.1). Then, the comparison is between the two SVTNNs, A and B, as shown in Table 24.2. Example 24.5.3. Consider two SVTNNs A = ⟨(0.6, 0.7, 0.9), (0.1, 0.3, 0.4), (0.1, 0.4, 0.6)⟩, B = ⟨(0.1, 0.2, 0.3), (0.4, 0.5, 0.6), (0.7, 0.8, 0.9)⟩ and three weighted functions P = (λ1 = 0.3, λ2 = 0.4, λ3 = 0.5), Q = (λ1 = 0.3, λ2 = 0.6, λ3 = 0.5) and S = (λ1 = 0.6, λ2 = 0.4, λ3 = 0.7). Then, the comparison is between the two SVTNNs, A and B, as shown in Table 24.3.



358  Mathematics and Computer Science Volume 2 Table 24.1  Ranking of SVNNs in Example 24.5.1. λ1,λ2,λ3

 λ1 , λ2 , λ3 (A)

 λ1 , λ2 , λ3 (B)

Results

0.3,0.4,0.5

0.5217

0.5384

A≺B

0.5,0.5,0.5

0.5083

0.5083

A ~B

1.0,1.0,0.9

0.4534

0.45502

A≺ B

Table 24.2  Ranking of SVNNs in Example 24.5.2. λ1,λ2,λ3

 λ1 , λ2 , λ3 (A)

 λ1 , λ2 , λ3 (B)

Results

1.0,1.0,1.0

0.0669

0.5168

A≺ B

0.5,0.5,0.5

0.4251

0.5460

A≺ B

0.1,0.1,0.1

0.4719

0.5693

A≺ B

Table 24.3  Ranking of SVNNs in Example 24.5.3. λ1,λ2,λ3

 λ1 , λ2 , λ3 (A)

 λ1 , λ2 , λ3 (B)

Results

0.3,0.4,0.5

0.6584

0.8101

A≺ B

0.3,0.6,0.5

0.7550

0.7500

A≻B

0.6,0.4,0.7

0.7000

0.7401

A≺ B

24.6 Conclusion The majority of real-life decision-making is based on fuzzy numbers due to their language connotations. Ordering fuzzy numbers is much more important in order to solve such difficulties. There are several documented ranking systems for single-valued fuzzy numbers. This paper developed a novel ranking approach for single-valued fuzzy numbers based on convex combinations of the fuzzy numbers’ α, β, and γ cut sets. A number of theorems have also been proved using the proposed method. Furthermore, numerical examples have been provided to demonstrate the suggested strategy’s superior performance. This ranking method may be used to look at the rankings of intuitionistic fuzzy numbers, type-2 fuzzy numbers, hesitant fuzzy numbers, and other fuzzy numbers. Further, this proposed

Ranking of Single Valued Neutrosophic Fuzzy Numbers  359 method can be applied to real-life problems like medicine, business, economics, computer sciences, etc.

References 1. S. I. A. Aal, M. M. A. A. Ellatif, and M. M. Hassan. Two Ranking Methods of Single Valued Triangular Neutrosophic Numbers to Rank and Evaluate Information Systems Quality, 2018. 2. A. Bhaumik, S. Roy, and D.-F. Li. (α, β, γ)-cut set based ranking approach to solving bi-matrix games in neutrosophic environment. Soft Computing, 25:1–11, 02 2021. 3. P. Biswas, S. Pramanik, and B. Giri. Value and ambiguity index based ranking method of single-valued trapezoidal neutrosophic numbers and its application to multi-attribute decision making. Neutrosophic Sets and Systems, 12:127–138, 07 2016. 4. G. Bortolan and R. Degani. A review of some methods for ranking fuzzy subsets. Fuzzy Sets and Systems, 15(1):1 – 19, 1985. 5. M. Brunelli and J. Mezei. How different are ranking methods for fuzzy numbers? a numerical study. International Journal of Approximate Reasoning, 54(5):627 – 639, 2013. 6. S. M. Chen. New methods for subjective mental workload assessment and fuzzy risk analysis. Cybernetics and Systems, 27(5):449–472, 1996. 7. S.-M. Chen and K. Sanguansat. Analyzing fuzzy risk based on a new fuzzy ranking method between generalized fuzzy numbers. Expert Systems with Applications, 38(3):2163 – 2171, 2011. 8. Z. Chen, G. Huang, and A. Chakma. Hybrid fuzzy-stochastic modeling approach for assessing environmental risks at contaminated groundwater systems. Journal of Environmental Engineering, 129(1):79–88, 2003. 9. F. Choobineh and H. Li. An index for ordering fuzzy numbers. Fuzzy Sets and Systems, 54(3):287 – 294, 1993. 10. R. Chutia. Ranking of fuzzy numbers by using value and angle in the epsilon-deviation degree method. Applied Soft Computing, 60:706 – 721, 2017. 11. R. Chutia and B. Chutia. A new method of ranking parametric form of fuzzy numbers using value and ambiguity. Applied Soft Computing, 52:1154 – 1168, 2017. 12. R. Chutia, R. Gogoi, and D. Datta. Ranking p-norm generalised fuzzy numbers with different left height and right height using integral values. Mathematical Sciences, 9(1):1–9, 2015. 13. I. Deli. Operators on single valued trapezoidal neutrosophic numbers and svtn-group decision making. Neutrosophic Sets and Systems, 22:131–150.

360  Mathematics and Computer Science Volume 2 14. I. Deli and Y. uba. A ranking method of single valued neutrosophic numbers and its applications to multi-attribute decision making problems. International Journal of Machine Learning and Cybernetics, 8, 08 2017. 15. D. Dubois and H. Prade. Fuzzy Sets and Systems: Theory and Applications. Academic Press, Inc., Orlando, FL, USA, 1980. 16. T. Garai, H. Garg, and T. Roy. A ranking method based on possibility mean for multi-attribute decision making with single valued neutrosophic numbers. Journal of Ambient Intelligence and Humanized Computing, 11, 11 2020. 17. P. Grzegorzewski. Algorithms for trapezoidal approximations of fuzzy numbers preserving the expected interval. Fuzzy Sets and Systems, 159:1354– 1364, 06 2008. 18. R. Jain. Decisionmaking in the presence of fuzzy variables. IEEE Transactions on Systems, Man and Cybernetics, SMC-6(10):698–703, Oct 1976. 19. R. Jain. A procedure for multiple-aspect decision making using fuzzy sets. International Journal of Systems Science, 8(1):1–7, 1977. 20. C. Jana, G. Muhiuddin, and M. Pal. Multiple-attribute decision-making problems based on svtnh methods. Journal of Ambient Intelligence and Humanized Computing, 11, 09 2020. 21. C. Jana and M. Pal. A robust single-valued neutrosophic soft aggregation operators in multi-criteria decision making. Symmetry, 11(1), 2019. 22. C. Jana, M. Pal, F. Karaaslan, and J. Wang. Trapezoidal neutrosophic aggregation operators and their application to the multi-attribute decision-making process. Scientia Iranica, 27(3):1655–1673, 2020. 23. F. KARAASLAN. Gaussian single-valued neutrosophic numbers and its application in multi- attribute decision making. Infinite Study, 2018. 24. J. Peng, J. Wang, H.-Y. Zhang, and X.-h. Chen. An outranking approach for multi-criteria decision-making problems with simplified neutrosophic sets. Applied Soft Computing, 25:336–346, 12 2014. 25. R. A. Shureshjani and M. Darehmiraki. A new parametric method for ranking fuzzy numbers. Indagationes Mathematicae, 24(3):518 – 529, 2013. 26. F. Smarandache. Neutrosophy: Neutrosophic Probability, Set, and Logic: Analytic Synthesis & Synthetic Analysis. Analytic Synthesis and Synthetic Analysis Series. American Research Press, 1998. 27. F. Smarandache. A unifying field in logics: Neutrosophic logic., 1999. 28. F. Smarandache. Neutrosophic set a generalization of the intuitionistic fuzzy set. International Journal of Pure and Applied Mathematics, 24:38–42, 01 2004. 29. V. Ulucay, A. Kilic, I. Yildiz, and M. ahin. A new approach for multi-attribute decision-making problems in bipolar neutrosophic sets. Infinite Study, 2018. 30. X. Wang and E. E. Kerre. Reasonable properties for the ordering of fuzzy quantities (i). Fuzzy Sets and Systems, 118(3):375 – 385, 2001. 31. X. Wang and E. E. Kerre. Reasonable properties for the ordering of fuzzy quantities (ii). Fuzzy Sets and Systems, 118(3):387 – 405, 2001.

Ranking of Single Valued Neutrosophic Fuzzy Numbers  361 32. V. F. Yu and L. Q. Dat. An improved ranking method for fuzzy numbers with integral values. Applied Soft Computing, 14, Part C:603 – 608, 2014. 33. L. A. Zadeh. Fuzzy sets. Information and Control, 8(3):338–353, 1965. 34. H.-Y. Zhang, J. Wang, and X. Chen. An outranking approach for multi-­ criteria decision-making problems with interval-valued neutrosophic sets. Neural Computing and Applications, 27, 04 2015.

25 Performance Analysis of Database Models Based on Fuzzy and Vague Sets for Uncertain Query Processing Sharmistha Ghosh1* and Surath Roy2 Department of Basic Science & Humanities, Institute of Engineering & Management, Kolkata, West Bengal, India 2 Department of Mathematics, Brainware University, Kolkata, West Bengal, India 1

Abstract

One of the primary aspects of utilization of any database model lies in its potential in processing information and queries accurately. In the present work, the authors intend to make a comparative analysis on the capability of fuzzy and vague relational database models in treating uncertain queries. A new algorithm is proposed and query testing related to a real life example is performed. The investigation demonstrates that a relational data model based on vague set theory produces more refined decisions than a fuzzy data model. It may thus be asserted that a relational database management system (RDBMS) using vague theoretic concept might lead to better software fabrication than the presently accessible ones. Keywords:  Fuzzy set, vague set, database model, similarity measure, SQL, fuzzy SQL, vague SQL

25.1 Introduction In real life, information is very often imprecise or incomplete. The conventional relational database system fails to treat such uncertain data. The theory of fuzzy sets, as formalized by Zadeh [20] in 1965, is extensively applied to handle such inexact or imprecise data. Several authors [1, 6, 10, 11, 14, 15, 17–19] have also worked on formulation of a query language for a database representation based on fuzzy sets. Bosc et al. [1] and Nakajima et al. [15] *Corresponding author: [email protected] Sharmistha Ghosh, M. Niranjanamurthy, Krishanu Deyasi, Biswadip Basu Mallik, and Santanu Das (eds.) Mathematics and Computer Science Volume 2, (363–384) © 2023 Scrivener Publishing LLC

363

364  Mathematics and Computer Science Volume 2 have outstretched the familiar SQL language in the framework of fuzzy set theory, namely, SQLF. More recently, Moreau et al. [14] have discussed a procedure wherein one may fetch data with no prior information about the database constitution or precise query language. The potential benefits and efficacy of the use of fuzzy query in a traditional data model have been thoroughly explained in [17]. An intelligent approach to extend SQL language to treat flexible conditions in queries was proposed by Mama et al. [11] in 2021. Gau and Buehrer [4] initiated the theory of vague sets in 1993 as an abstraction of a fuzzy set theoretic approach. It is believed that a vague set that employs interval-based membership values may process uncertain information in a more efficient manner compared to a fuzzy set. The vague theoretic concept was further embodied in relations by Lu and Ng [8, 9] and a new query language called VSQL evolved. A vague database model was designed by Zhao and Ma [21] and vague querying strategies with SQL were also investigated. In [3], Dutta et al. have used vague sets to rejuvenate “vague search”, a new method of intelligent search that is capable of answering any uncertain query put forth by the user. In [12], the detailed design aspects of a vague database model have been thoroughly discussed. An architecture for processing of hesitant queries has been designed by Mishra et al. [13] along with a comparison of fuzzy and vague sets in handling imprecise queries. In the present work, the authors also aspire to analyze the ability of fuzzy and vague database models in relation to uncertain query processing. It has been noticed that the algorithms used in literature (see refs. [10, 13, 16]) to obtain membership values possess certain drawbacks. In the current investigation, the authors present a novel algorithm which is devoid of such deficiencies. The proposed algorithm generates membership functions for fuzzy or vague sets for the calculation of membership values that are independent of the attribute. However, it is worthwhile to mention that the membership functions employed earlier in literature [10, 16] depend on the attribute type. The framework of the chapter is as follows. The definitions of fuzzy and vague sets as well as related fundamental concepts are presented in Sections 25.2.1 and 25.2.2. The similarity measure formulae used in this study appear in Section 25.2.3. The algorithm designed in the present analysis to generate membership values is proposed in Section 25.3. Real life examples are demonstrated in Section 25.4 to observe that a vague theoretic approach is better suited in processing queries that are not precise. The concluding remarks are reported in Section 25.5.

25.2 Basic Definitions Let X denote the universe of discourse and x represent an element of X.

Performance Analysis of Database Models  365

25.2.1 Fuzzy Set Definition 25.2.1.1: A fuzzy set S, defined in the universe of discourse X, is a set of ordered pairs S = {( x , µ S ( x )) : x ∈ X } , where µS : X → [0,1] denotes the grade of membership of x in S. It may be easily observed that an ordinary subset A of X may be treated as a fuzzy set with membership function µA that takes binary values, i.e.,

 1 µA =  0 



if

x ∈A

if

x ∉A

25.2.2 Vague Set Definition 25.2.2.1: A vague set S, in the universe of discourse X, is characterized by two membership functions, namely: (i) a truth function tS : X → [0,1] and (ii) a false function fS : X → [0,1]. Here, tS(x) represents a lower bound on the grade of membership of x as obtained from the ‘evidence in favor of x’, whereas fS(x) denotes a lower bound on the negation of x as deduced from the ‘evidence against x’ and tS(x) + fS(x) ≤ 1. The grade of membership µS(x) of x in the vague set S is then bounded by a subinterval [t S (x ), 1 − f s (x )] of [0, 1], i.e., tS(x) ≤ µS(x) ≤ 1 – fS(x). = x ,[t S (x ), 1 − f S (x )] : Now, the vague set S may be represented as S : x ∈ X } . The interval [t S (x ), 1 − f s (x )] is termed as the vague value of the element x. It may be noted that if tS(x) is equal to (1 − f S (x ) ) , the information about object x is precise and the theory of vague set degenerates to that of a fuzzy set. In case both tS(x) and (1 − f S (x ) ) are 1, which confirms that x belongs to S, the information about x is exact, and the theory reverts to that of an ordinary set. Similarly, if tS(x) and (1 − f S (x ) ) both take the value 0, the knowledge about x is again exact which relates to the situation that x in not in S. Hence, a crisp set as well as fuzzy set can be contemplated to be a specific case of vague set.

{

25.2.3 Similarity Measure The concept of similarity of vague sets has been studied by several researchers [2, 5, 7, 8]. The similarity measure proposed by Lu et al. in [8] was

366  Mathematics and Computer Science Volume 2 shown to be more effective in general cases. This measure has been used in this work, which is presented below: Definition 25.2.3.1: Similarity Measure for Vague Values u [t1 , 1 − f1 ] and= v [t 2 , 1 − f 2 ] , Let u and v be vague values such that= where 0 ≤ t1 ≤ 1 – f1 ≤ 1 and 0 ≤ t2 ≤ 1 – f2 ≤ 1. If SM(u, v) denotes the similarity measure between u and v, then



SM (u,= v)

 (t1 − t2 ) − ( f1 − f2 )  1 −  1 − ( t1 − t 2 ) + ( f1 − f 2 ) . 2  

(

)

Definition 25.2.3.2: Similarity Measure for Vague Sets Let X = {x1 , x2 , x3, ..., xn } be the universe. Let S1 and S2 be two vague sets of X, such that



= S1

{ x ,[t i

S1

(xi ),1 − f S1 (xi )] : xi ∈ X} ,

where t S1 ( xi ) ≤ µ S1 ( xi ) ≤ 1 − f S1 ( xi ) and 1 ≤ i ≤ n and



S2 = { xi ,[t S2 ( xi ),1 − f S2 ( xi )] : xi ∈ X } ,

where t S2 ( xi ) ≤ µ S2 ( xi ) ≤ 1 − f S2 ( xi ) and 1 ≤ i ≤ n. Now, the similarity measure between S1 and S2, denoted by SM (S1 , S2 ) , is defined as:



1 SM ( S1 , S2 ) = n

n

∑ SM ([t

S1

( xi ),1 − f S1 ( xi )],[t S2 ( xi ),1 − f S2 ( xi )])

i =1

25.3 Algorithm to Generate Membership Values In the literature, various membership functions have been employed by different authors for finding the membership values for different fuzzy attributes. For an EMPLOYEE database, the membership function deployed by Raju et al. [16] corresponding to a fuzzy set ‘close to u’ and fuzzy attribute Salary is the following:

Performance Analysis of Database Models  367

µclose to u ( x ) =

1 . |x − u| 1+ 20000

However, for the Experience attribute, Raju et al. defined the membership function as

µclose to u ( x ) =

1 . |x − u| 1+ 4

On the contrary, Ma et al. [10] used the following membership functions for the same fuzzy attributes, Salary and Experience:

µclose to u ( x ) =

µclose to u ( x ) =

1 2 ;  |x − u|  1+  20000  1 2.  |x − u| 1+  4 

It is obvious that the above membership functions depend on the specific attribute under investigation. In this study, the authors have made an effort to devise an algorithm for calculation of membership values for different fuzzy/vague attributes. It may be pointed out that the membership function generated by said algorithm is attributed as independent for a fuzzy/vague set. The membership values generated by the proposed algorithm also compare well with that of others obtained in the literature. A similar algorithm was designed by Mishra et al. [13] for the purpose of estimation of membership values. It is observed that the same algorithm has been utilized by Yadav in [19] for SQL query processing. But, the formula put forth in [13] is found to possess the following deficiencies: (i) Negative membership value is generated in certain instances. (ii) Addition of new records in the given data leads to change of membership values for the existing data that is not sensible.

368  Mathematics and Computer Science Volume 2 The algorithm devised in this study is presented below. It may be noted that it is devoid of such anomalies. Algorithm 25.3.1 Input: Fuzzy or vague data and attributes in an imprecise query Output: Membership value lying in [0,1] Method: Fetch attributes from the query for each fuzzy or vague attribute do begin f_data = data value of the attribute from the query range = maximum domain value – minimum domain value for every tuple in the relation do begin t_value = tuple value from the domain of fuzzy or vague attribute if (|f_data – t_value | ≤ range) |f _ data − t _ value| membership _ value = 1 − range else membership_value = 0 end for loop for tuple end for loop for fuzzy or vague attribute

25.4 Real Life Applications We now use the following EMPLOYEE relation (Table 25.1) as a real life example and process certain imprecise queries with one or more attributes. A comparative analysis is presented in this section for each of the queries on the framework of fuzzy and vague models. Query 1: “To fetch the employees with age close to 53 years”. Query 2: “To retrieve the data of the employees whose age is more or less than 53 years with work experience close to 22 years.” The fuzzy and vague database models were used to execute the above queries with one or more attributes. In each case, the vague database model produced better results than the fuzzy data model, as may be observed from the analysis presented herein. Query 1: “To fetch the employees with age close to 53 years”.

Performance Analysis of Database Models  369 Table 25.1  EMPLOYEE relation. Name

Age (years)

Experience (years)

Remuneration (₹)

Mr. Roy

30

4

32000

Mr. Roychowdhury

31

6

32500

Mr. Pramanik

34

8

34000

Mr. Barik

51

18

58000

Mr. Bose

57

22

77900

Mr. Ghosh

49

17

57400

Mr. Samanta

50

16

56800

Mr. Bhowmick

41

13

46500

Mr. Banerjee

39

12

42000

Mr. Chattopadhyay

53

20

80900

Mr. Ganguly

52

19

80200

Mr. Kher

56

21

81100

Mr. Mukhopadhyay

54

23

82200

Mr. Patnaik

55

27

82000

Mr. Sahoo

62

35

122000

Solution with Fuzzy Model: The fuzzy characteristic here is Age and the fuzzy data is close to 53 years. We employ the algorithm discussed in Section 25.3 to find the membership value for each domain value of the Age attribute. Here, Domain of Age attribute = {30, 31, 34, 51, 57, 49, 50, 41, 39, 53, 52, 56, 54, 55, 62} f_data = 53 Range = 62 – 30 = 32 We now deploy the formula provided in Algorithm 25.3.1 to determine the membership values for every domain value of Age attribute as follows: 1st tuple: Membership Value = 1 - (|53–30| / 32) = 0.28125 2nd tuple: Membership Value = 1 - (|53–31| / 32) = 0.3125

370  Mathematics and Computer Science Volume 2 The complete record of membership values for each tuple is shown in Table 25.2. Table 25.3 displays the complete representation of this relation corresponding to Query 1. For the fuzzy attribute Age, the fuzzy representation appears in column 3 and its equivalent vague formulation occurs in column 4. This vague formulation is now utilized to compute the similarity measures (SM) with the fuzzy data ‘close to 53’, which may be represented as in vague notation. The formula for similarity measure, as introduced in Definition 25.2.3.1, is then applied. If one considers the vague data u = and v = , then

t1 = 1, f1 = 0, t2 = 0.28125, f2 = 0.71875

Table 25.2  Membership values. Name

Age (years)

Membership value

Mr. Roy

30

0.28125

Mr. Roychowdhury

31

0.3125

Mr. Pramanik

34

0.40625

Mr. Barik

51

0.9375

Mr. Bose

57

0.875

Mr. Ghosh

49

0.875

Mr. Samanta

50

0.90625

Mr. Bhowmick

41

0.625

Mr. Banerjee

39

0.5625

Mr. Chattopadhyay

53

1.0

Mr. Ganguly

52

0.96875

Mr. Kher

56

0.90625

Mr. Mukhopadhyay

54

0.96875

Mr. Patnaik

55

0.9375

Mr. Sahoo

62

0.71875

Performance Analysis of Database Models  371

Table 25.3  Similarity measures for Query 1 with Fuzzy Model.

Name

Age (years)

Fuzzy representation of age

Vague representation of age

Mr. Roy

30

Mr. Roychowdhury

31

Mr. Pramanik

SM of age with

Experience (years)

Remuneration (₹)

SM (tuple)

0.53033

4

32000

0.53033



0.55902

6

32500

0.55902

34



0.63738

8

34000

0.63738

Mr. Barik

51



0.96825

18

58000

0.96825

Mr. Bose

57



0.93541

22

77900

0.93541

Mr. Ghosh

49



0.93541

17

57400

0.93541

Mr. Samanta

50



0.95197

16

56800

0.95197

Mr. Bhowmick

41



0.79057

13

46500

0.79057

Mr. Banerjee

39



0.75000

12

42000

0.75000

Mr. Chattopadhyay

53



1

20

80900

1 (Continued)

372  Mathematics and Computer Science Volume 2

Table 25.3  Similarity measures for Query 1 with Fuzzy Model. (Continued)

Name

Age (years)

Fuzzy representation of age

Vague representation of age

Mr. Ganguly

52

Mr. Kher

56

Mr. Mukhopadhyay

SM of age with

Experience (years)

Remuneration (₹)

SM (tuple)

0.98425

19

80200

0.98425



0.95197

21

81100

0.95197

54



0.9842

23

82200

0.9842

Mr. Patnaik

55



0.96825

27

82000

0.96825

Mr. Sahoo

62



0.84779

35

122000

0.84779

Performance Analysis of Database Models  373 and thus, SM (u, v ) =



=

 (1 − 0.28125) − (0 − 0.71875)  1 −  (1 − (1 − 0.28125) + (0 − 0..71875) ) 2   1 −= 0.71875

= 0.28125 0.53033

Again, for u = and v = ,

t1 = 1, f1 = 0, t2 = 0.3125, f2 = 0.6875. Then, SM (u, v )



= =

 (1 − 0.3125) − (0 − 0.6875)   (1 − (1 − 0.3125) + (0 − 0.68775) ) 1 −   2 1 −= 0.6875

0.3125 0.55902 =

and so on. The complete results are shown in Table 25.3. Now, the following SQL statement is generated to execute the given query at an α-cut or threshold value 0.95, provided by the decision maker: SELECT * FROM EMPLOYEE WHERE SM(tuple) ≥ 0.95. Table 25.4 now presents the resultant tuples retrieved from the EMPLOYEE database. Table 25.4  Resulting tuples for Query 1 with Fuzzy Model at α= 0.95. Name

Age (years)

Experience (years)

Remuneration (₹)

Mr. Barik

51

18

58000

Mr. Samanta

50

16

56800

Mr. Chattopadhyay

53

20

80900

Mr. Ganguly

52

19

80200

Mr. Kher

56

21

81100

Mr. Mukhopadhyay

54

23

82200

Mr. Patnaik

55

27

82000

374  Mathematics and Computer Science Volume 2

Table 25.5  Similarity measures for Query 1 with Vague Model. Name

Age (years)

Vague representation of age

SM of age with

Experience (years)

Remuneration (₹)

SM (tuple)

Mr. Roy

30

0 (26.7)  r

This is also applicable to any function that increases in a positive way ξ(t), g(x) ∈ Lip(ξ(t), r), if



(∫

2π 0

)

1/r

|( g (x + t ) − g (x= )) |r dx O(ξ (t )), r ≥ 1,t > 0

(26.8)

388  Mathematics and Computer Science Volume 2 This is also applicable to any function that increases in a positive way r ξ(t), an integer r ≥ 1, f ∈ W(L , ξ(t))



(∫

2π 0

)

1/r

β |( g (x + t ) − g (x ))}sin= x |r dx O(ξ (t )), β ≥ 0 ,t > 0. (26.9)

We will use the notations below as a guide:

ψ= (t ) g (x + t ) + g (x − t ) − 2 g (x )

P = n (t )

1 p q (n − k + 1) sin2 (n − k + 2)t /2 ∑nk =0 k k sk (26.10) 2π Rn (k + 1)(k + 2) sin2t /2 r

Then, the weighted W(L , ξ(t)) class is a subset of the Lipα, Lip(α, r), and Lip(ξ(t), r) classes. As a result, the following additions have been made:

Lipα ⊂ Lip(α, r) ⊂ Lip(ξ(t), r) ⊂ W(Lr, ξ(t)) for all 0< α ≤ 1 and r≥1.

26.2 Known Result In 2011, Nigam [1] proved a theorem on (C,1)(E,q) means of the Fourier series. Proceeding the work in 2014, Mishra et al. [4], the product (E,s) (N,pn, qn) - summability mean of the Fourier series showed a theorem on the degree of approximation. Pradhan [5] proposed the following theorem in 2016.

26.2.1 Theorem 1 If f ∈ W(Lr, ξ(t)) class is a 2π-periodic function that is integrable in the Lebesgue sense in [0,2π], then the degree of approximation is given by



NE n

T

1  β+  1  − gr = O  (n + 1) r ξ    (n + 1)   

where TnNE is the (N , pn , qn ) (E,s) transform of {sn}, provided ξ(t) has the following requirements:

{ }

ξ(t ) is a decreasing sequence, t

Estimating Error of Signals  389

     



∫

1

(n +1)

0

1/r

r   t |φ (t )|   1  βr sin t dt  = O  ξ (t )  n + 1    1/r

r  t −δ |φ (t )|  δ  ξ (t )  dt  = O(n + 1) 1  (n +1)

∫

π

26.3 Main Theorem Various mathematicians including Nigam [2], Deger [6], and Mishra et al. [7,  8] studied the degree of approximation using various summability approaches. By using the Riesz-Cesaro product summability approach, we r obtain a novel result on the degree of approximation of function g ∈ W(L , ξ(t)) class. The following theorem is established.

26.3.1 Theorem 2

1 If g ∈ W(Lr, ξ(t)) with 0 ≤ β ≤ 1 − , the degree of approximation by Rieszr Cesaro product mean of the Fourier series satisfies for n = 0,1,2… 1   1  β+  r + 2 TnNC − g (x )r = O ξ  n ( n )     (n + 2)  



(26.11)

where TnNC is the (N , pn , qn ) (C,2) transform of sn, supposing that function ξ(t) satisfies the required criteria, ξ (t ) is a function that does not increase and t

{ }





     

∫

1 0

(n + 2 )

1/r

r  t |ψ (t )sin β t|   n  dt  = O     n + 2 ξ (t )    1/r

r  t −δ |ψ (t )| sin β t   dt  = O(n + 2)δ   1 ξ (t )   (n + 2 )

∫

π

(26.12)

(26.13)

where δ is an arbitrarily defined s.t. qs(β − δ) – 1 > 0 r−1 + s−1 = 1, r ≥ 1 and (26.12) and (26.13) uniformly hold in x.

390  Mathematics and Computer Science Volume 2

26.4 Some Auxiliary Results Uniformly holding the following lemmas is necessary to prove the above theorem.

26.4.1 Lemma-1 For



0≤t ≤

1 1 π π , ≤ for 0 < t < , (n + 2) sint 2t 2

we get Pn(t) = O(n + 2)

1 Proof : |Pn (t )| = 2π Rn

n

∑ k =0

pk qk (n − k + 1) sin2 (n − k + 2)t /2 sk (k + 1)(k + 2) sin2t /2

1 n sk (n − k + 1) (n − k + 2)t 2 /π 2 ∑ k =0 (k + 1)(k + 2) 2π t 2 /π 2 = O (n+ 2)

≤

26.4.2 Lemma-2 For (n + 2)−1 < t < π, n ) we get Pn (t ) = O( (n + 2)t 2



pk qk (n − k + 1) sin 2 (n − k + 2)t /2 1 n Proof : |Pn (t )| = ∑ k =0 sk 2π Rn (k + 1)(k + 2) sin 2t /2 t t Using Jordan’s Lemma sin ≥ and sinkt ≤, we have 2 π

≤



1  π 2  n pk qk (n − k + 1) sk   ∑ k =0 2π Rn  t 2  (k + 1)(k + 2)

n  = O   (n + 2)t 2 

Estimating Error of Signals  391

26.5 Theorem’s Proof Using Riemann-Lebesgue theorem, we have

1 sn ( g ) − g ( x ) = 2π



∫

π 0

sin 2 (n − k + 2)t /2 ψ (t ) dt sin 2t /2

The Riesz Cesaro transform of the sequence is given by

1 p q (n − k + 1) π sin2 (n − k + 2)t /2 {TnNC − g (x )} =∑nk =0 k k sk ∫ 0 ψ (t ) dt 2π Rn (k + 1)(k + 2) sin2t /2 =

π

∫ ψ (t )P (t )dt n

0

1 (n+2 )

= ∫0

ψ (t )Pn (t )dt + ∫

π 1 (n+2 )

ψ (t )Pn (t )dt (26.14)

= I1 + I 2  1 ( n+ 2)

where |I1| ≤ ∫ 0 |ψ (t )||Pn (t )| dt Applying Minikowski’s inequality, since

|ψ(x, t) – ψ(x)| ≤ |g(u + x + t) – g(u + x)| + |g(u – x – t) – g(u – x)|  

∫

2π 0

1/r

 |ψ ( x + t ) − ψ ( x )sin x| dx   β

 ≤ 

∫

2π 0

r

1/r

 |g (u + x + t ) − g (u + x )sin x| dx   β

r

∫



1/r

 2π  |g (u − x − t ) − g (u − x )sin β x|r dx  +  0  = O{ξ (t )}

and the fact that g ∈W ( Lr , ξ (t ) ) ⇒ ψ (t ) ∈W ( Lr , ξ (t ) ) and Lemma 1, we have



392  Mathematics and Computer Science Volume 2

 I1 ≤  

∫

1 (n + 2) 0

  t |ψ (t )| sin β t r     dt   ξ (t )     

n    = O   (n + 2)   n    =   (n + 2)  

∫

∫

1 (n + 2) 0

1 (n + 2) 0

{

I /r

} }

  

∫

1 (n + 2) 0

{

}

1/ s

 ξ (t )|Pn (t )| s   dt    β  tsin t  

 ξ (t )O(n + 2)  tsin β t 

{

1/s

 ξ (t )| Pn (t ) s   dt    β  tsin t   {using (26.12)}

1/ s

  dt    

s

{using Lemma 1}

Using the second mean value theorem for integrals, ξ(t) is a decreasing function and we get

1     ≤ O  nξ      (n + 2)      1    = O  nξ     (n + 2)   

∫ ∫

1 (n + 2) 0 1 (n + 2) 0

1/s

  dt    t ( β +1)s  

0 ≤ε
0 and are both rising functions, the second mean (1 / x ) value theorem is used and

1     = O n(n + 2)δ −1 ξ    (n + 2)     

∫

(n + 2)

1/π

1/ s

 δ s − β s −2s+2   x  dx

1/ s

1     x β s + 2 s −δ s − 2+1    δ −1  = O n(n + 2) ξ     (n + 2)     β s + 2s − δ s − 1   

(n + 2)



1     x ( β −δ + 2)−1 s    = O n(n + 2)δ −1 ξ   (n + 2)     ( β − δ + 2) − 1       s   1





π



1    = O ξ  n(n + 2)β +1/r      (n + 2) 

(26.16)

394  Mathematics and Computer Science Volume 2 from (26.14) and (26.15), (26.16) becomes



  1   TnNC − g (x ) r = O ξ  n(n + 2)β +1/ r     (n + 2)  

26.6 Applications From our main theorem, we can derive the following corollaries.

26.6.1 Cor. 1

If ξ(t) = tα, then weighted class W ( Lr , ξ (t ) ) , 1 ≤ r < ∞ reduces to the group of class Lip(α, r) and then a function f is a degree of approximation belonging to the class Lip(α, r), r−1 ≤ α ≤ 1, given by

TnNC − g (x )r =+ O{n(n 2)1/ r }



Proof: Setting produces, the result is β = 0 in (26.5).

26.6.2 Cor. 2 For r → ∞ in Cor. 1 for the class function, g ∈ Lip(α, r) reduces to the class Lip α and a function g belonging to the class Lipα, 0 < α < 1 whose degree of the approximation is given by



TnNC − g (x )r =+ O{n(n 2)α }

26.7 Conclusion The previous finding on function degree of approximation demonstrated that the result is frequently true in nature and can be simplified to a few special cases. As a result, the current research can be used for a series of issues in the fields of analysis, technology, and engineering. Figure 26.1 shows some unique behaviour in the Fourier series approximations [11].

Estimating Error of Signals  395 1

n=1

0 −1 1

t

n=5

0

t

−1 1

n=11 t

0 −1 1 0

n=49 t

−1

Figure 26.1  Approximations of such Fourier Series to sq (t). Each plot indicates the number of terms in the Fourier sum and the square wave is displayed as a dashed line over two periods.

Acknowledgement The sincere gratitude goes to the anonymous referees for their careful reading, remarks, and valuable comments, as well as many other useful suggestions for improved presentation. The author wishes to express his appreciation to the members of the editorial board.

References 1. H. K. Nigam: “Degree of approximation of a function belonging to weighted (Lip (ξ(t), r) class by (C,1) (E,q) means”, Tamkang Journal of Mathematics, 42, 31-37 (2011). 2. H. K. Nigam: “On Approximation of functions by product operators”, Survey in Mathematics and its Applications ISSN 1842-6298(Electronic), 18437265(print) Volume 8 (2013), 125-136. 3. K. Sharma. and S.S. Malik: “Degree of Approximation by Product means of the Fourier Series in a (W ( Lr , ξ(t ) ) (r > 1) − class by (E,1)(C,2) product summability Transform”, International Journal of Mathematical Trends and Technology (IJMTT), Volume-8 Number-2,ISSN 2231-5373,2014, 10.14445/22315373/IJMTT-V8P518.

396  Mathematics and Computer Science Volume 2 4. M. Mishra et al.: “Approximation of Fourier series of a function of Lipschitz class by product means” Journal of advances in Mathematics, 9 2475-2484 (2014). 5. T. Pradhan et al.: “Approximation of signals belonging to generalised Lipschitz class using ( N , pn , qn )( E , s ) -summability mean of Fourier series”, Cogent Mathematics (2016),3: 1250343.http://dx.doi.org/10.1080/23311835.2016.1 250343.  “On Approximation to functions in the W ( Lp , ξ(t ) ) class by a new 6. U. Deger: matrix mean”, Novi Sad J. Math, Vol.46 No.1, 2016, 1-14. 7. V. N. Mishra et al.: “Approximation of Signals by Product Summability Transform”, Asian Journal of Mathematics and Statistics 6(1): 12-22, 2013, ISSN 1994-5418/ DOI: 10.3923/AJMS.2013,12.22. 8. V. N. Mishra, K. Khatri, L. N. Mishra: “Product Summability Transform of Conjugate series of Fourier series”, Hindawi Publishing Corporation, International Journal of Mathematics and Mathematical Sciences, Volume 2012, Article 1D 298923, 13 pg doi: 10.1155/2012/298923. 9. A. Zygmund: “Trigonometric Series”, second ed., Cambridge University Press, Cambridge, 1959. 10. G. H. Hardy: Divergent Series Oxford University Press 1st Ed. 1949. 11. https://eng.libretexts.org/Bookshelves/Electrical_Engineering.

About the Editors Biswadip Basu Mallik  is presently a Senior Assistant Professor of Mathematics in the Department of Basic Sciences & Humanities at the Institute of Engineering & Management, Kolkata, India. He has been involved in teaching and research for more than 21 years and has published several research papers in various scientific journals and book chapters with reputed publishers. He has authored five books at undergraduate levels in the areas of Engineering Mathematics, Quantitative Methods, and Computational Intelligence. He has also published five Indian patents along with nine edited books. His fields of research work are Computational Fluid Dynamics & Mathematical Modelling. Prof. Basu Mallik is a Managing editor of the Journal of Mathematical Sciences & Computational Mathematics (JMSCM), USA. He is also an Editorial board member and reviewer of several international journals. He is a senior life member of the Operational Research Society of India (ORSI) and a life member of the Calcutta Mathematical Society (CMS), Indian Statistical Institute (ISI), Indian Science Congress Association (ISCA), and International Association of Engineers (IAENG). Niranjanamurthy M., Ph.D., is working as an Assistant Professor at the Department of Artificial Intelligence and Machine Learning, BMS Institute of Technology and Management, Bangalore, INDIA. He completed Ph.D. in Computer Science, M.Tech. - Master of Computer Technology (Software Engineering), M.Phil.-Computer Science, MCA at VTU, Belgaum, Karnataka,  BCA from Kuvempu University with University 5th Rank. Dr. Niranjanamurthy is having 13 years of teaching experience and 2 years of industry experience as a Software Engineer. He has published 20 books and 85 articles in various National/International Conferences/ International Journals. He has filed 30 Patents out of which 6 has been granted. Four research scholars are pursuing Ph.D. work under his supervision in the area of Data Science, ML, and Networking. He is the reviewer/ editorial board member of 22 International Journals and is also a National/ International  Ph.D. thesis examiner. Dr. Niranjanamurthy received the 397

398  About the Editors Best Research journal reviewer award twice. He had also received earlier Young Researcher award in Computer Science and Engineering. He obtained Patent Award - 2021 from KSCST, Karnataka, India.  He conducted various National and International Conferences, National Level workshops and delivered lectures. Dr. Niranjanamurthy is associated with various professional bodies as follows: IEEE Member, Life Member of “The International Association of Engineers” (IAENG),  and Member of “Computer Science Teachers Association” (CSTA). His areas of interest are Data Science, ML, E-Commerce, and M-Commerce related to Industry Internal tool enhancement, Software Testing, Software Engineering, Web Services, Web-Technologies, Cloud Computing, Big Data Analytics, and Networking. Sharmistha Ghosh, Ph.D., is currently working as a Professor at the Institute of Engineering & Management, Kolkata, India. She obtained B.Sc.(H) and M.Sc. degrees in Mathematics from Jadavpur University, Kolkata, India in the years 1992 and 1994 respectively as University Gold Medalist. Dr. Ghosh received Ph.D. degree in Mathematics from Indian Institute of Technology, Kharagpur, India in 1999. She also obtained M.Sc. nat. degree in Industrial Mathematics from University of Kaiserslautern, Germany in 2000 and is a DAAD fellow. Her major field of study includes Fuzzy & Vague Databases as well as Computational Fluid Dynamics. She has published several research articles in International Journals/Conferences, book chapters in these domains and also one edited book. She is the editor of renowned international journals and also works as reviewer of journals as well as doctoral theses in India and abroad. She has delivered invited talks in several national as well as international forum and organized many national and international conferences and workshops. Krishanu Deyasi, Ph.D., is currently an Associate Professor in the Department of Basic Science & Humanities, Institute of Engineering & Management Kolkata, India. He did Ph.D. from Indian Institute of Science Education and Research (IISER) Kolkata. Dr. Deyasi obtained his Master Degree in Mathematics from Indian Institute of Technology (IIT), Kharagpur. He has Post-Doctoral experience from The Institute of Mathematical Sciences (IMSc) Chennai. His major area of research is Spectral Graph Theory and Complex Networks. He has written 3 books in the area of Engineering Mathematics and has published one edited book. He has published papers in National and International Journals. He is also the editor of some Scientific Journals.

About the Editors  399 Santanu  Das is currently working as an Assistant Professor at the Department of Basic Science and Humanities at Institute of Engineering & Management, Kolkata, India. He completed his under-graduation and post-graduation in Mathematics from Jadavpur University and is currently pursuing his PhD from the same University. His field of research work is Cosmology and Dynamical system. He is the Faculty coordinator of Society for Data science IEM student chapter. He takes classes of various Mathematics Courses for B.Tech,, M.Tech. and MCA students. He is also a Development Content Editor of the Journal of Mathematical Sciences & Computational Mathematics (JMSCM), USA.

Index Accreditation, 79 Accrues, 206 Activation, 94 Adaptive, 77, 79 Adjoint equation, 41, 42 Adoption, 85 AED, 193, 194, 198, 199, 202 Agricultural, 49, 55 Ambient temperature, 224 Amino Acid, 91, 92, 94, 96, 98 Amino Acids Sequence, 91 Amplitude ratio, 127, 129, 130, 133 Analytical hierarchy process, 188 Animations, 140 Ant Colony Optimization (ACO), 60 Anti-parallel, 239, 241, 242, 245, 246 Approximation, 24, 153, 226, 239, 241, 244, 245, 248, 385, 388, 389, 394, 395 Arc problem, 327, 328, 330 Arrival time, 155, 164, 165, 166, 169, 332 Artificial Intelligence (AI), 1, 59, 60, 63, 205, 266 Artificial Neural Networks (ANN), 60 Asvagandha, 87 Asymmetric, 173, 183, 239, 241, 244, 245, 246, 247, 248, 250 Atmospheric boundary layer (ABL), 19, 24 Attribute, 7, 50, 51, 53, 54, 55, 65, 188, 350, 364, 366, 367, 368, 369, 370, 375, 378, 382 Aubin-Lions, 41

Automorphism, 291, 293, 294, 296, 297, 300, 301, 303 Average arc problem, 327, 328 Awareness, 78, 85, 101, 105, 111, 191 Ayurveda, 80, 81, 83, 84, 87 Ayushgrid, 79 Azadikaamrutmahotsav, 81 Backward Propagation Model, 66 Balking, 151, 152, 153, 154, 163, 164, 165 Bernoulli Schedule, 151, 153, 154, 163, 169 Best fit distribution, 19, 20, 22, 24, 31 Bidirectional Long Short-Term Memory (BLSTM), 62 Big data, 3, 188 Binary Coding Scheme, 62 Biomarkers, 34 Bio-Portal, 186, 192, 201 Biot parameter, 116, 117, 131, 133 Blow up, 305, 306, 307, 317 Boundary condition, 38, 39, 115, 116, 124, 227, 228, 256, 257, 259, 315 Bounding box, 64 Breakdown Machines, 337, 347 Bright Cities, 139 Brownian motion coefficient, 224 Brownian motion parameter, 225, 228, 230, 231 Capacity violation, 333 Catastrophic, 77 Cell count, 34, 43

401

402  Index Cellular automata, 34 Central difference, 42 Centre, 79 Certification, 83, 86, 140 Chang-Cooper, 42 Characteristics, 61, 62, 64, 78, 187, 188, 189, 190, 192, 196, 213, 266, 270, 287, 342, 345, 375 Chemical reaction, 226, 231, 234, 235 Chemotherapy, 33, 34, 35, 45 Chi-square statistics, 22, 24, 31 Citizens, 82, 87, 139, 207, 208 Class Mapping, 190, 196, 198, 199, 200, 201 Classification, 1, 2, 3, 4, 7, 11, 12, 13, 14, 24, 49, 50, 55, 61, 62, 63, 65, 67, 68, 69, 70, 73, 92, 196, 209, 210, 291 Client, 14, 151, 152, 153, 154, 155, 157, 163, 164, 165, 166, 169 Cluster, 7, 194, 196, 197, 198, 199, 200 Coercive, 40 Cohn-Kanade expression, 68 Coil, 92 Collaboration, 84, 85, 87, 141, 206, 207 Colon cancer, 33, 34, 35, 44 Combination therapies, 33, 34, 35, 44 Comparative analysis, 70, 74, 363, 368 Complex Network, 101, 106 Complexity, 78, 240, 246, 266, 285, 329 Composite, 116, 387 Computational analysis, 101, 111 Computer networks, 60 Computer vision, 284 Computers, 138, 139, 206 Concentration boundary layer, 227 Concentration gradient, 234 Confused Memory, 192, 193, 194, 199, 202 Congestion, 151, 153, 154, 155, 164, 169, 329 Congestion dependent, 151, 153, 154, 155, 164, 169 Conservation constraint, 332

Conservativeness, 39 Conspicuous, 80 Constant deterioration rate, 254 Continuous-time, 242, 327, 328, 329, 331 Contraflow, 239, 241, 242, 244, 247, 248 Convective boundary layer height (CBLH), 19, 20, 21, 23 Convergent, 40, 41, 42 Convex combination, 349, 350, 351, 358 Conveyor, 210, 211, 212, 213, 214, 216, 217, 218 Convolutional Neural Network (CNN), 60, 61, 62, 65, 93, 209, 289 Coronavirus, 101, 102, 218 Correlation, 3, 65, 96, 144, 146, 147, 187 Course Material, 140 Covanta, 207 COVID-19, 77, 78, 79, 80, 81, 82, 84, 87, 88, 101, 102, 205, 209, 254 Criteria, 8, 67, 187, 188, 189, 190, 191, 192, 194, 197, 198, 270, 277, 350 Cropping, 61, 63, 64, 67, 284, 286 Crowd motion, 35, 42 Crust, 116 Cullpdb, 91, 92, 93, 96, 97 Cytotoxic effects, 34 Dashboard, 80, 83 Data Analysis, 13 Data Analytics, 3 Data Visualization, 8 Data1199, 91, 92, 93, 97 Database Model, 363, 364, 368, 382 Data-driven, 8, 188 DBT, 83 Decision level hybrid method, 62 Decision Tree, 49, 50, 51, 53, 54, 55 Deep learning, 59, 60, 62, 94, 96 Degree of Approximation, 388, 389, 394

Index  403 Density of the fluid, 224 Despite, 85, 97, 139, 206, 207 Detection, 1, 11, 12, 13, 14, 20, 22, 30, 33, 49, 51, 54, 59, 60, 61, 62, 63, 64, 67, 77 Differential equations, 103, 228, 256, 257, 259 Digital, 2, 55, 82, 83, 85, 88, 139, 141, 142, 209, 248 Digital Transformation, 139 Dimensionless elastic parameter, 224 Dimensions, 240, 267, 273 Directives, 80 Disaster, 80 Disease, 6, 34, 49, 50, 51, 52, 53, 54, 55, 85, 101, 102, 111, 187, 189, 191, 192, 198, 201, 202, 203, 207, 284 Distributed, 25, 81, 151, 154, 155, 166, 169, 176, 208, 337 Diverse, 59, 84, 85, 154, 239, 284 Doctors, 192, 284, 285 Domain knowledge, 188 Dosages, 34, 36, 43, 44, 45 Douglas-Gunn, 42 Doxorubicin, 33, 35, 36, 43, 44 Dynamic, 22, 28, 33, 34, 35, 36, 37, 41, 45, 101, 102, 103, 106, 107, 111, 173, 174, 175, 176, 178, 183, 209, 226, 240, 241, 242, 327, 328, 329, 330, 331, 333, 344 Dynamic Model, 28, 34, 35 Dynamic network, 327, 329, 330 Dynamic viscosity, 224 Dynamical systems, 34 E-Business, 140 E-Citizens, 139 Education, 1, 14, 82, 83, 84, 139, 140, 141 EduSERE, 62 Efficacy, 80, 84, 218, 364 E-Government, 140 E-Health, 140

Elastic parameter, 117, 224, 231, 235, 236 E-Learning, 137, 138, 139, 140, 141, 142 Electrical conductivity, 224 Electronic, 80, 139, 141, 209, 251, 252 Encoder-Decoder, 5, 91, 92, 94, 95, 96 Energy ratios, 126, 131, 134 Entropy, 49, 50, 51, 52, 53, 54, 96, 224, 226, 286, 287 Environment, 11, 21, 28, 62, 79, 140, 142, 152, 173, 175, 183, 184, 205, 206, 207, 215, 218, 273, 349 Epilepsy disease, 187, 191, 192, 201, 203 EPILONT, 191, 192, 193, 194, 198, 199, 200, 201, 202 EPSO, 191, 192, 193, 196, 198, 199, 200, 201, 202 Equipment, 141, 206 ESSO, 193, 194, 196, 198, 199, 200, 201, 202 Evaluation, 1, 11, 52, 69, 116, 140, 147, 187, 188, 189, 192, 203, 276, 286, 343 EvoMSA, 62 Existence, 39, 116, 139, 218, 223, 305, 306, 307, 314, 333, 349 Exploitation vector, 68 Facebook, 14, 85, 86, 138, 139 Face-To-Face Communication, 142 Failed Machines, 337, 338, 340, 347 Feasibility of flow, 331, 332, 333, 334 Fibonacci Search Method, 265, 268, 270, 271 Finite difference, 42 Flickr, 138, 139 Flow capacity, 327 Fluid, 116, 117, 223, 224, 225, 226 F-measure, 59, 69, 70, 71, 72, 74 Focal, 193, 194, 196, 198, 202 Fokker-Planck, 33, 35, 38, 42

404  Index Fourier series, 385, 388, 389, 394, 395 Free surface, 115, 116, 121 Friedman Rank Test, 268, 273, 276 Fuzzy Logic, 51 Fuzzy set, 350, 351, 363, 364, 365, 366 Garbage, 206, 207, 208, 209, 210, 213, 215, 216, 218 Genetic algorithm, 266, 337, 338, 344, 347 Genetic toggle switch, 173, 180 Genomics 84 Geographical Location, 142 Geriatric, 81 Gestures, 60, 63 Ghee, 87 Gray scaling, 61 Growth Stage, 51, 52, 53, 55 Guduchi, 87 He@lthy, 85 Head Trauma, 193, 194, 199, 202 Health, 13, 33, 50, 77, 79, 80, 81, 82, 83, 85, 86, 87, 101, 102, 205 Health Care, 205 Healthcare, 13, 60, 78, 79, 81, 82, 205, 206, 218, 219 Heap coupled bat Optimization (HBO), 59 Helix, 92, 269 Hierarchy, 188, 192, 194, 210 High-Bandwidth, 140 Homotopy analysis, 225 Hybrid 59, 61, 62, 65, 74, 187, 202, 267, 333, 334 Hybrid Deep Convolutional based Golden Eagle Network (HDCGEN), 59 Hybrid network, 65, 333, 334 Hyper-Connection Linking, 139 Hypothesis testing, 22 IEEE CEC’2019, 276 IL-2, 33, 35, 36, 43, 44

ILE-JAVA, 62 Image pre-processing 63 Image segmentation 267, 283, 289 Immune pathways 34 Immunity 80, 81, 82, 87, 102 Immunotherapy 33, 35, 45 Impatience 151, 152, 153, 169 Impedance 115, 116, 124, 129, 130, 131, 133 Imprecise Query 368, 381, 382 Incident 115, 116, 121, 122, 127, 128, 129, 131, 132, 133 Incinerator 205, 210, 211, 212, 213 Information Gain 49, 50, 51, 54, 55 Infusion 87 Inhalation 87 Initial condition 36, 180 Initiatives 77, 78, 82 Inlet 210, 211, 212, 213 Innovation 78, 141, 212 Insulated 116, 125 Integration Of Communication Networks 139 Intelligently Evaluating 138 Intelligibility 60 Interdisciplinary 83, 84 International 79, 84, 86, 191 Internet 2, 138, 139, 140, 141, 142, 152, 206, 209, 210 Internet Of Things 139, 140, 141, 142, 206, 209, 210 Intervention 77, 79, 80, 84, 88, 102 IoT 137, 138, 139, 140, 141, 142, 144, 145, 147, 205, 207, 208 Isothermal 115, 116, 125, 127, 224 Isotropic 115, 117, 121, 122, 126, 133 Iterative Dichotomiser 3 49, 50 Jupyter 11 Keras 96 Kernel function 66, 307 Kinematic coefficient of viscosity 224 K-Nearest neighbors (KNN) 60

Index  405 Knowledge management 13 Kolmogorov-Smirnov (KS) test, 31 Kwath, 87 Lane reversal, 241, 243, 244, 245, 246, 247, 248 Lane, 239, 241, 243, 244, 245, 246, 247, 248 Leadership, 77, 79 Leaf Blast, 49, 51, 55 Learning Tools, 140 Length-bounded, 140, 245, 248 Lesions Type, 54 Lewis number, 225, 228 Lidar Backscatter signals, 19 Linear programming, 327, 331 Linguistic Variable, 51, 52 LinkedIn, 141 Lipschitz domain, 38 Logarithmic Sobolev inequality, 313, 317, 321, Long Short-Term Memory Network, 96, 97 Longitudinal, 116, 121, 122, 127, 128, 129, 130, 131, 132, 133 Lymphocytes, 35, 36 Machine Interactions, 138, 141 Machine learning, 4, 5, 7, 8, 9, 11, 13, 51, 62, 208, 209, 266, 284 Magnetic parameter, 224, 230, 231, 235, 236 Mannkibaat, 87 Materials, 116, 140, 141, 219 Mathematical Model, 34, 102, 225, 242, 248, MATLAB, 63, 64, 68, 127, 273, 345 Maximum dynamic, 241 Maximum flow, 240, 241, 329 MCU, 210, 212 Medical Science, 83, 84, 225 Meditation, 83 Metrics, 68, 69, 70, 74, 96, 154, 163, 164, 188, 189, 201, 268, 286

Midnight, 81 Minimization problem, 39, 41 Minimum time, 331 Mobile Communication, 210 Model, 2, 3, 4, 5, 8, 9, 10, 11, 12, 34, 35, 42, 61, 62, 63, 66, 68, 70-74, 92-97, 102-110, 148, 152, 154, 155, 163, 164, 178, 179, 189, 209, 242, 252-261, 285-288, 308, 329, 331, 338, 343, 364, 368, 369-381, Moderators, 338, 339 Molecular diffusivity, 224 Momentum boundary layer, 227 MSM, 206 Multi-commodity, 241, 242, 244, 245, 246, 247 Multi-commodity flow, 241, 245, 246 Multi-criteria decision making (MCDM), 188 Multiserver system, 337 NAMASTE, 85 Nanofluids, 223, 225 Natural Killer cells, 34 Natural Language Processing, 2, 5, 7, 14, 92 Naturopathy, 84, 87 Nave Bayes, 60 Nehari functional, 311 Network, 5, 14, 60, 61, 62, 65, 68, 85, 92, 93, 94, 96, 97, 106, 111, 139, 140, 152, 208, 209, 210, 214, 240248, 266, 284, 285, 289, 328, 329, 330, 333, 334, Neural cell, 283 Neural Network, 60, 61, 92, 93, 209, 284, 289 Neurological disorder, 190, 284 Neurologists, 201 Nickelodeon, 86 Non-abelian groups, 291, 292, 293 Non-conservation constraint, 332 Non-dimensional temperature, 224

406  Index Non-symmetric, 241, 244, 245, 246, 247, 248 NP-complete, 244, 245, 333, 334 Nusselt number, 230 One Hot Encoding, 95 Ontology Engineering, 189 Ontology Selection, 188, 189, 190, 202 Open-loop, 40 Optimal control, 35, 45 Optimality condition, 41 Optimality system, 41 Optimistic, 69 Optimization, 35, 39, 42, 60, 65, 96, 208, 252, 266, 267, 268, 270, 273, 277, 279, 344 Groups, 291, 292, 293 Pandemic, 78-86, 102, 103, 206, 207, 215, 218 Paradigm, 80, 154 Parallel computation, 60 Parameter estimation, 35 Particle Swarm Optimization (PSO), 60 PARTITION, 245, 332, 333, 334 Peptide, 92 Performance, 11, 61, 63, 66, 68, 69, 70, 74,  92, 96, 97, 154, 163, 164, 208, 267, 268, 273, 277, 285, 286, 287, 342, 358, 366 Personalized therapies, 34 Pharmaceutical, 84, 102, 207 Pharmacokinetic models, 34 Pipeline, 102 Pitchfork bifurcation, 174-183 p-Laplacian type, 306 Plate concentration parameter, 224, 234, 236 Plate temperature parameter, 224 Polynomial, 240-245, 247, 248, 328, 329 Polynomial-time, 240, 241, 242, 245, 247, 248 Populations, 35, 36, 103, 104, 107, 273,

Pores, 116 Porosity parameter, 224, 234 Porosity, 116, 117, 133, 234 Porous, 116, 117, 121, 122, 126, 127, 133, 224, 227, 234, 236 Porous medium, 126, 227 Potential well, 309 Potential, 35, 77, 118, 119, 141, 142, 308, 309, 364 Pranayama, 83 Prandtl number, 224, 228, 231, 235 Preprocessing, 2, 3, 4, 9, 14, 61, 62, 64, 67, 94 Primary Structure, 92 Probabilities, 68, 69, 153, 155, 157, 162, 163, 169 Probability density function, 35, 154 Prophylactic, 83 Proteins, 91, 92, 176, 177, 178 Protocol, 83, 85, 87 Psychological, 60 Psychosocial, 87 Python, 96, 106 Q3 Accuracy, 93, 96, 97 Query processing, 364, 367 Questionnaire, 142 Queue, 152, 153, 154, 163, 166, 169, 337, 338, 343, 347 Quickest, 240, 241, 244, 245, 246, 247, 248, 329 Radiation parameter, 224 Random Forest Algorithm, 55 Randomness, 35, 45 Reflecting barrier, 38 Reflection, 116, 121, 122, 127, 128, 130, 131, 133 Related Works, 207 Relaxation time, 115 Reneging, 152, 154, 163, 164, 166 Repairmen, 338 Resource allocation, 177, 178, 180, 182 Restorative, 218

Index  407 Rice Crop, 49, 51, Rocks, 116 Saddle node bifurcation, 174, 176, 177, 180, 182, 183 Scalability, 139 Screen, 210 Segmentation, 51, 284-289 Seismology, 116 Seizure, 191, 194, 198, 200, 201 Semantic Web, 138, 139, 188 Sensors, 140, 141, 207-214, 216, SentiTEXT, 62 Sequence-Labeling, 92 Sherwood number, 230, 231 Similarity Measure, 364, 365, 366, 370, 371, 373, 374, 375, 376, 377, 378, 379, 380 Single valued neutrosophic fuzzy numbers, 351 Sink, 235, 241, 244, 328, 329, 332, 333 Skin-friction coefficient, 230 Smart Bins, 217 Softmax, 67, 68, 95 Soil, 51, 116, 208, 218 Solid, 115-117, 133, 206, 207, 209, 210, 212 Soret number, 230 Source, 3, 6, 83, 86, 96, 142, 143, 145, 147, 191, 192, 206, 225, 241, 244, 306, 328, 329, 330, 332 Specific heat, 224 Split Rollers, 210 SQL, 226, 228, 364, 367, 373, 375, 381 SSA-DCNN, 70-73 State equation, 35, 41, 340 Static, 96, 138, 187, 240, 242, 245, 329 Stationary Distribution, 154 Statistical Techniques, 21 Stericycle, 207 Stochastic, 35, 37, 38, 42, 44, 45, 268 Storage capacities, 329 Strand, 92 Stream function, 224, 227

Strength of uniform magnetic field, 224 Stress, 80, 83, 116, 117, 277 Stretching surface, 225 Strongly polynomial, 241, 248, 329 Susceptive, 52, 54, 55 Symmetric, 179, 180, 241, 244, 246-248 Symptoms, 33, 49, 50, 51, 52, 55, 84, 191, 192, 194, 195, 200, 202 System Architecture, 210, 211 Technicians, 339 Temperature gradient, 231 Temperature sensor, 138 Temporal variation of lidar signals, 24, 27 Tensor flow, 96 Thermal boundary layer, 227 Thermal conductivity of the fluid, 224 Thermodynamics, 224 Thermoelastic, 115, 116, 121, 122, 126, 133, Thermophoresis parameter, 225, 228, 230, 231 Thermophoretic diffusion coefficient, 224 Threshold, 21, 138, 153, 209, 347, 373, 378, 381 Time dependent holding cost, 254 Time expansion, 241, 245 Time horizon, 253, 255, 329, 333 Time-expanded network, 328, 330, 333, 334 Toxicity, 34, 44, 45 Transit time, 241, 244, 246, 248, 328, 329, 330, 333, 334 Transverse, 121, 127, 129, 130, 131, 133, 231, 235, 266 Trapezoidal demand, 252, 254 Trash, 207-210, 212, 215, 216, 218 TSD problem, 277 Tumorigenesis, 34 Turbulence, 19, 20, 25, 26, 28 Twitter, 14, 85, 86, 141

408  Index Unani, 78, 80, 83 User Interface, 8 Vacation Interruption, 153, 154, 166, 169 Vacations, 152, 153, 163 Vaccination, 102-111 Vague set, 364-367, 378, 382 Vague SQL, 381 Velocity, 224, 226, 231, 234, 235, 236 Video Conferencing, 139, 140 Virtual Learning, 141 Viscoelastic, 223, 224, 226, 235, 306 Viscous, 116, 117, 225, 226 Wales, 207 Wave propagation, 116 Weak convergence, 40

Web 2.0, 138, 139 Web 3.0, 138, 139 Weight vector, 68 Weights, 3, 96, 188, 189, 190, 193, 197, 198, 201, 202, 285, 286 Width window, 66 Wiener process, 38 Working Vacations, 152, 153, World wide web, 138 Wrapping, 212 Yoga, 79-87 YouTube, 85, 86, 138, 139, 141 α- cut set, 355 β- cut set, 355 γ- cut set, 355

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