Contents

#### **Preface XI**

Preface

of these disorders.

tional, and medical aspects of ergonomics.

of this book interesting and very valuable.

effective feedback process that we had with our authors.

wife, Bakiye Yalınç, for her support and encouragement.

I have been working on musculoskeletal disorders (MSDs) for almost two decades. It was a long-term aim to work on such a book. InTech's invitation to edit this book has motivated me to reflect and share all the knowledge and experience I collected throughout the years. This book covers topics on the anatomy, posture, prevalence, pain, treatment, and interven‐ tions for musculoskeletal disorders. Thus, it can be utilized as a guide to identify and ana‐ lyze the risk factors, reveal the impact of prevention and intervention, and discuss treatment

In these chapters, you will find most of the significant aspects of MSDs. To this extent, a wide array of extensive and comprehensive discussions is provided on occupational, educa‐

The first step to cover was to select and disseminate the keywords, which helped us receive numerous abstracts. These abstracts were then carefully evaluated, and those that were be‐ lieved to contribute significantly to the literature were invited to take their place in this book. It took several months to edit each and every word of the sent chapters. It was a very

In this book, you will read chapters from authors in Egypt, Finland, Germany, Italy, and the USA. They discussed the MSD-related topics from their perspectives and reflected and pro‐ vided solution to the problems experienced from their own eyes. Thus, it makes the contents

I, hereby, would like to thank Ms. Martina Usljebrka who helped me with positive attitude at every single step of the publication process. Moreover, I would like to thank my dear

**Orhan Korhan, PhD**

Eastern Mediterranean University, Famagusta

Turkish Republic of Northern Cyprus (TRNC), Turkey


**Author details**

Auwalu Faisal Koko1,2\*, Muhammed Bello3 and Muhammad Abubakar Sadiq4

1 College of Civil Engineering and Architecture, Zhejiang University, China

2 International Center for Architecture and Urban Development Studies, Zhejiang University, Hangzhou, China

3 Department of Architecture, Kaduna Polytechnic, Kaduna State, Nigeria

4 Nigerian Building and Road Research Institute (NBRRI), Abuja, Nigeria

\*Address all correspondence to<EMAIL_ADDRESS>
© 2023 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


**3.3 Nanofluid as the working fluid**

Thermo-physical properties of the Nanofluids are superior among various working fluids, so which are suitable for high heat transfer applications.

**295**

**Figure 20.**

*kg/s [75].*

**Figure 19.**

*Recent Advancements in Thermal Performance Enhancement in Microchannel Heatsinks…*

Significant research has been done on MCHS with Nanofluids using experimental and single and multi-phase CFD models. Multi-phase methods are noticed to be more precise compared to the single-phase numerical methods [68]. The inaccuracy of the 2-phase mixture method and the 1-phase methods for 1% Nanofluid compared to the experimental results were found to be 11.39% and

Ayoub Abdollahi et al. [70] proved that the water-based SiO2 Nanofluid has the best performance among the four water-based SiO2, CuO, Al2O3, and ZnO Nanofluids. In a similar study conducted on the hexagonal channeled MCHS using various water-based Nanofluids, it was observed that Al2O3-water Nanofluid has the highest heat transfer coefficient [71]. The analysis on CuO-Water Nanofluid flow in trapezoidal channeled MCHS proved that the thermal performance of heat sink

*The variation of average nu with Reynolds for various nanofluid volume fractions [72].*

*Distribution of nanofluid thermal conductivity at the outlet with various mass Flux (i) 0.0001 kg/s, (ii) 0.0003*

*DOI: http://dx.doi.org/10.5772/intechopen.97087*

32.6% respectively [69].

**Figure 18.** *Schematic diagram of single nozzle with convex dimple arrangement [67].*

*Recent Advancements in Thermal Performance Enhancement in Microchannel Heatsinks… DOI: http://dx.doi.org/10.5772/intechopen.97087*

Significant research has been done on MCHS with Nanofluids using experimental and single and multi-phase CFD models. Multi-phase methods are noticed to be more precise compared to the single-phase numerical methods [68]. The inaccuracy of the 2-phase mixture method and the 1-phase methods for 1% Nanofluid compared to the experimental results were found to be 11.39% and 32.6% respectively [69].

Ayoub Abdollahi et al. [70] proved that the water-based SiO2 Nanofluid has the best performance among the four water-based SiO2, CuO, Al2O3, and ZnO Nanofluids. In a similar study conducted on the hexagonal channeled MCHS using various water-based Nanofluids, it was observed that Al2O3-water Nanofluid has the highest heat transfer coefficient [71]. The analysis on CuO-Water Nanofluid flow in trapezoidal channeled MCHS proved that the thermal performance of heat sink

**Figure 19.** *The variation of average nu with Reynolds for various nanofluid volume fractions [72].*

**Figure 20.**

*Distribution of nanofluid thermal conductivity at the outlet with various mass Flux (i) 0.0001 kg/s, (ii) 0.0003 kg/s [75].*

*Heat Transfer - Design, Experimentation and Applications*

dimple arrangement is presented in **Figure 18**.

*Schematic model of* **(a)** *free-surface jet-arrays and* **(b)** *submerged jet-arrays [66].*

*Schematic diagram of single nozzle with convex dimple arrangement [67].*

**3.3 Nanofluid as the working fluid**

has superior overall performance among the three arrangements studied and it has a high heat transfer rate and lowest pressure loss. The single nozzle with a convex

Thermo-physical properties of the Nanofluids are superior among various working fluids, so which are suitable for high heat transfer applications.

**294**

**Figure 18.**

**Figure 17.**

was enhanced by increasing the Nanofluid volume fraction with a penalty of high pumping power [72]. The variation of Average Nu with the Reynolds number (Re) is shown in **Figure 19**.

The numerical investigation on the Nanofluid-based triangular and trapezoidal grooved MCHS revealed that the water-based Al2O3 Nanofluid was the best coolant for the trapezoidal grooved MCHS [73, 74]. In a numerical study conducted on Al2O3-water Nanofluid-based MCHS, the rise of Nanoparticles' concentration at the walls and non-uniform distribution of thermal conductivity was observed with increasing Discharge [75]. **Figure 20** shows the distribution of the Nanofluid thermal conductivity at the outlet.

Few studies used the advantages of both the geometries and the Nanofluids in the MCHS. Water-based Diamond Nanofluid has the best performance among the various Nanofluids used in the analysis of the wavy channeled MCHS [76]. M.M. Sarafraz et al. [77] witnessed a 76% improvement in the heat transfer performance of MCHS with a 20% increase in the pumping power at the Re higher than 1376. The variation of the Nu obtained from the experiments and existing correlations is shown in **Figure 21**.

The efficiency of an energy system can understand by finding its entropy generation. Some studies investigated the entropy generation of the MCHS to analyze its performance using various working fluids. In the first and second law thermodynamic analysis of offset strip-fin MCHS with CuO Nanofluid as coolant, it was found that the thermal characteristics of the MCHS improved with Re but the frictional entropy generation was also increased [78]**.** The rate of entropy generation of MCHS in the flow direction concerning the number of fins is presented in **Figure 22**.

### **3.4 Magneto-hydrodynamics (MHD) influence**

MHD is an interdisciplinary subject that has been used in various engineering problems like the design of MHD pumps and flows meters and cooling of nuclear reactors etc. In the field of microfluidics, MHD pimping is very favorable because of its uncomplicated design and less power consumption [79–82]. In the starting, conventional heat

**297**

**Figure 23.**

**Figure 22.**

*Recent Advancements in Thermal Performance Enhancement in Microchannel Heatsinks…*

exchangers with magnetic fluid are investigated to know its performance by applying it. The proportionality of friction factor and Nusselts number with the applied magnetic field was noticed in the study of the MHD effect on circular tube flow of Fe3O4-Water Nanofluid [83]. Variation of Nusselt number with Magnetic field effects is presented in **Figure 23**. The working fluid velocity was observed to be reduced with Lorentz force generated because of magnetic flux applied. The increase of the local heat transfer coefficient was witnessed experimentally when an external magnetic field was applied to the W-40 (magnetic Nanofluid) flow in a rectangular duct [84]. Few research studies about the MHD effect on micro-channels noticed the improvement of their hydraulic and thermal behavior. The decrease of the gradient of velocity at the wall with a rise in the index of Power-law flow was observed in the flow of non-Newtonian fluid flow in a microchannel under a transfer magnetic field [85]. The increment of the gradient of velocity near the wall and reduction of maximum velocity was found with the rising Hartmann number. The increase in the Joule heating and the viscous dissipation was observed to decrease the Nusselt number. Chunhong Yang et al. [86] studied the thermal behavior of an incompressible MHD flow in a rectangular microchannel by taking the combined influence of the viscous dissipation

*The entropy generation of MCHS in the flow direction with the number of fins [78].*

*The nu variation with the magnetic field (a) in the range Re, (b) along the channel [83].*

*DOI: http://dx.doi.org/10.5772/intechopen.97087*

**Figure 21.** *The variation of the nu obtained from the experiments and existing correlations [77].*

*Recent Advancements in Thermal Performance Enhancement in Microchannel Heatsinks… DOI: http://dx.doi.org/10.5772/intechopen.97087*

#### **Figure 22.**

*Heat Transfer - Design, Experimentation and Applications*

**3.4 Magneto-hydrodynamics (MHD) influence**

*The variation of the nu obtained from the experiments and existing correlations [77].*

is shown in **Figure 19**.

shown in **Figure 21**.

thermal conductivity at the outlet.

was enhanced by increasing the Nanofluid volume fraction with a penalty of high pumping power [72]. The variation of Average Nu with the Reynolds number (Re)

The numerical investigation on the Nanofluid-based triangular and trapezoidal grooved MCHS revealed that the water-based Al2O3 Nanofluid was the best coolant for the trapezoidal grooved MCHS [73, 74]. In a numerical study conducted on Al2O3-water Nanofluid-based MCHS, the rise of Nanoparticles' concentration at the walls and non-uniform distribution of thermal conductivity was observed with increasing Discharge [75]. **Figure 20** shows the distribution of the Nanofluid

Few studies used the advantages of both the geometries and the Nanofluids in the MCHS. Water-based Diamond Nanofluid has the best performance among the various Nanofluids used in the analysis of the wavy channeled MCHS [76]. M.M. Sarafraz et al. [77] witnessed a 76% improvement in the heat transfer performance of MCHS with a 20% increase in the pumping power at the Re higher than 1376. The variation of the Nu obtained from the experiments and existing correlations is

The efficiency of an energy system can understand by finding its entropy generation. Some studies investigated the entropy generation of the MCHS to analyze its performance using various working fluids. In the first and second law thermodynamic analysis of offset strip-fin MCHS with CuO Nanofluid as coolant, it was found that the thermal characteristics of the MCHS improved with Re but the frictional entropy generation was also increased [78]**.** The rate of entropy generation of MCHS in the flow direction concerning the number of fins is presented in **Figure 22**.

MHD is an interdisciplinary subject that has been used in various engineering problems like the design of MHD pumps and flows meters and cooling of nuclear reactors etc. In the field of microfluidics, MHD pimping is very favorable because of its uncomplicated design and less power consumption [79–82]. In the starting, conventional heat

**296**

**Figure 21.**

exchangers with magnetic fluid are investigated to know its performance by applying it. The proportionality of friction factor and Nusselts number with the applied magnetic field was noticed in the study of the MHD effect on circular tube flow of Fe3O4-Water Nanofluid [83]. Variation of Nusselt number with Magnetic field effects is presented in **Figure 23**. The working fluid velocity was observed to be reduced with Lorentz force generated because of magnetic flux applied. The increase of the local heat transfer coefficient was witnessed experimentally when an external magnetic field was applied to the W-40 (magnetic Nanofluid) flow in a rectangular duct [84].

Few research studies about the MHD effect on micro-channels noticed the improvement of their hydraulic and thermal behavior. The decrease of the gradient of velocity at the wall with a rise in the index of Power-law flow was observed in the flow of non-Newtonian fluid flow in a microchannel under a transfer magnetic field [85]. The increment of the gradient of velocity near the wall and reduction of maximum velocity was found with the rising Hartmann number. The increase in the Joule heating and the viscous dissipation was observed to decrease the Nusselt number. Chunhong Yang et al. [86] studied the thermal behavior of an incompressible MHD flow in a rectangular microchannel by taking the combined influence of the viscous dissipation

**Figure 23.** *The nu variation with the magnetic field (a) in the range Re, (b) along the channel [83].*

*The entropy generation of MCHS in the flow direction with the number of fins [78].*

and the Joule heating. This analysis shows the decrease of normal velocity with increment in the Hartmann number (Ha) without any applied lateral electric field. The increasing-decreasing trend of Velocity and temperature profiles was noticed with the rising the Hartmann number under the applied lateral electric field. The generation of entropy in the MCHS was diminished as the influence of the Lorentz force generated by the injected electric current and magnetic field applied transversely [87].

### **3.5 Flow boiling in MCHS**

Flow boiling in the MCHS can vanish the heat fluxes in the range of 30 to 100 W/cm2 with the acceptable channel surface temperatures [88]. The flow boiling implemented MCHSs were used for a variety of applications, like cooling of PEM fuel cells, thermal management of the IGBTs (insulated gate bipolar transistors), refrigeration systems, etc. [89]. Most of the researchers' attention is on estimating the impact of the mass flow rate, heat flux, vapor quality, and surface characteristics on the boiling heat transfer [90, 91]. Stable flow boiling is also one of the best way to enhance the heat transfer in the MCHS. The experimental analysis done by John Mathew et al. [92] on the copper hybrid MCHS with flow boiling revealed that the local heat transfer coefficient is consistent increases with the heat flux and becomes sensitive to the heat flux. The pressure loss was also found to increase in the 2-phase flow with heat flux under all mass flow rate conditions. **Figure 24** shows the variation of upstream heat transfer coefficient in the microchannel concerning the effective heat flux. A numerical study of Yang Luo et al. [93] on manifold MCHS with subcooled two-phase flow boiling proved that heat flux and the manifold ratio significantly influence the pressure drop and thermal resistance (Rth) of the microchannel. The authors suggested that the manifold ratios should be between 1 and 2 for low-pressure drop and the better thermal performance of the manifold heatsink.

The impact of the surface characteristics on the flow boiling of regasified and deionized water micro-channel was experimentally studied and found that the characteristics of the Cu microchannel surface are transient [88]. The heat transfer coefficients of the aged Cu microchannel surface were unchanged and even enhanced

#### **Figure 24.**

*Variation of upstream HT coefficient in the microchannel with respect to the heat flux with the images of flow visualization [92].*

**299**

**4. Conclusions**

**Figure 25.**

*Recent Advancements in Thermal Performance Enhancement in Microchannel Heatsinks…*

after proper cleaning. Hongzhao Wang et al. [94] examined the wettability patterned microchannel and homogeneous hydrophilic microchannel and found a 22% higher heat transfer coefficient for wettability patterned microchannel. The heat transfer coefficient was noticed to be improved for the wettability patterned channel with the mass flux. The transient variation of Heaters' temperature variation is shown in **Figure 25**. The experimental and simulation analysis on microchannel with 2-phase continuous boiling revealed that heat transfer was improved at the fixed heat flux with increment in mass flux but it may tend to unstable boiling [95]. For the unstable boiling in a microchannel, the oscillation amplitude was observed to be influenced by the structural parameters of the microchannel and the thermal conductivity. In the experimental investigation on 2-phase flow regimes in a microchannel, the formation of the wave on the liquid film was observed in film flow regimes [96] and the wavelength of the waves on the liquid film is depending on flow rate of the gas and liquid. Along with the MCHS performance improvement methods discussed above, a few other inventive methods were noticed in the literature. C.J. Ho et al. [97] examined the microencapsulated PCM (MEPCM) based MCHS under the sudden pulsed heat flux and disclosed that the layer of MEPCM layer not effective in controlling the temperature rise in the MCHS. At the high amplitude of heat flux pulse, the MEPCM layer has the improved cooling performance. Soumya Bandyopadhyay, Suman Chakraborty [98] investigated the thermophoretic force effect and the interfacial tension by studying Newtonian fluid dynamics in a microchannel with the consideration of temperature dependency of viscosity. Linda Arsenjuk et al. [99] investigated the slug flow in parallelized microchannel and obtained static fluid distribution with high-pressure loss. Zan Wu et al. [100] analyzed the slug flow in a square microchannel and correlated the velocity of the slug in terms of the

*Transient variation of heaters' temperature (lines represents the average temperature) [94].*

Capillary number using bulk velocity and continuous phase viscosity.

The advancements in the thermal performance enhancement methods for microchannel heat sinks are discussed so far. Each method is selected based on

*DOI: http://dx.doi.org/10.5772/intechopen.97087*

*Recent Advancements in Thermal Performance Enhancement in Microchannel Heatsinks… DOI: http://dx.doi.org/10.5772/intechopen.97087*

#### **Figure 25.**

*Heat Transfer - Design, Experimentation and Applications*

**3.5 Flow boiling in MCHS**

100 W/cm2

and the Joule heating. This analysis shows the decrease of normal velocity with increment in the Hartmann number (Ha) without any applied lateral electric field. The increasing-decreasing trend of Velocity and temperature profiles was noticed with the rising the Hartmann number under the applied lateral electric field. The generation of entropy in the MCHS was diminished as the influence of the Lorentz force generated

by the injected electric current and magnetic field applied transversely [87].

Flow boiling in the MCHS can vanish the heat fluxes in the range of 30 to

implemented MCHSs were used for a variety of applications, like cooling of PEM fuel cells, thermal management of the IGBTs (insulated gate bipolar transistors), refrigeration systems, etc. [89]. Most of the researchers' attention is on estimating the impact of the mass flow rate, heat flux, vapor quality, and surface characteristics on the boiling heat transfer [90, 91]. Stable flow boiling is also one of the best way to enhance the heat transfer in the MCHS. The experimental analysis done by John Mathew et al. [92] on the copper hybrid MCHS with flow boiling revealed that the local heat transfer coefficient is consistent increases with the heat flux and becomes sensitive to the heat flux. The pressure loss was also found to increase in the 2-phase flow with heat flux under all mass flow rate conditions. **Figure 24** shows the variation of upstream heat transfer coefficient in the microchannel concerning the effective heat flux. A numerical study of Yang Luo et al. [93] on manifold MCHS with subcooled two-phase flow boiling proved that heat flux and the manifold ratio significantly influence the pressure drop and thermal resistance (Rth) of the microchannel. The authors suggested that the manifold ratios should be between 1 and 2 for low-pressure drop and the better thermal performance of the manifold heatsink. The impact of the surface characteristics on the flow boiling of regasified and deionized water micro-channel was experimentally studied and found that the characteristics of the Cu microchannel surface are transient [88]. The heat transfer coefficients of the aged Cu microchannel surface were unchanged and even enhanced

*Variation of upstream HT coefficient in the microchannel with respect to the heat flux with the images of flow*

with the acceptable channel surface temperatures [88]. The flow boiling

**298**

**Figure 24.**

*visualization [92].*

*Transient variation of heaters' temperature (lines represents the average temperature) [94].*

after proper cleaning. Hongzhao Wang et al. [94] examined the wettability patterned microchannel and homogeneous hydrophilic microchannel and found a 22% higher heat transfer coefficient for wettability patterned microchannel. The heat transfer coefficient was noticed to be improved for the wettability patterned channel with the mass flux. The transient variation of Heaters' temperature variation is shown in **Figure 25**.

The experimental and simulation analysis on microchannel with 2-phase continuous boiling revealed that heat transfer was improved at the fixed heat flux with increment in mass flux but it may tend to unstable boiling [95]. For the unstable boiling in a microchannel, the oscillation amplitude was observed to be influenced by the structural parameters of the microchannel and the thermal conductivity. In the experimental investigation on 2-phase flow regimes in a microchannel, the formation of the wave on the liquid film was observed in film flow regimes [96] and the wavelength of the waves on the liquid film is depending on flow rate of the gas and liquid.

Along with the MCHS performance improvement methods discussed above, a few other inventive methods were noticed in the literature. C.J. Ho et al. [97] examined the microencapsulated PCM (MEPCM) based MCHS under the sudden pulsed heat flux and disclosed that the layer of MEPCM layer not effective in controlling the temperature rise in the MCHS. At the high amplitude of heat flux pulse, the MEPCM layer has the improved cooling performance. Soumya Bandyopadhyay, Suman Chakraborty [98] investigated the thermophoretic force effect and the interfacial tension by studying Newtonian fluid dynamics in a microchannel with the consideration of temperature dependency of viscosity. Linda Arsenjuk et al. [99] investigated the slug flow in parallelized microchannel and obtained static fluid distribution with high-pressure loss. Zan Wu et al. [100] analyzed the slug flow in a square microchannel and correlated the velocity of the slug in terms of the Capillary number using bulk velocity and continuous phase viscosity.

#### **4. Conclusions**

The advancements in the thermal performance enhancement methods for microchannel heat sinks are discussed so far. Each method is selected based on heatsink application, the heat flux needs to be dissipated, space availability, etc. The primary conclusions drawn from this chapter are,


IntechOpen Book Series Education and Human Development

## Volume 7

## Aims and Scope of the Series

Education and Human Development is an interdisciplinary research area that aims to shed light on topics related to both learning and development. This Series is intended for researchers, practitioners, and students who are interested in understanding more about these fields and their applications.

## Meet the Series Editor

Katherine Stavropoulos received her BA in Psychology from Trinity College, in Connecticut, USA and her Ph.D. in Experimental Psychology from the University of California, San Diego. She completed her postdoctoral work at the Yale Child Study Center with Dr. James McPartland. Dr. Stavropoulos' doctoral dissertation explored neural correlates of reward anticipation to social versus nonsocial stimuli in children with and without autism spectrum

disorders (ASD). She has been a faculty member at the University of California, Riverside in the School of Education since 2016. Her research focuses on translational studies to explore the reward system in ASD, as well as how anxiety contributes to social challenges in ASD. She also investigates how behavioral interventions affect neural activity, behavior, and school performance in children with ASD. She is also involved in the diagnosis of children with ASD and is a licensed clinical psychologist in California. She is the Assistant Director of the SEARCH Center at UCR and is a faculty member in the Graduate Program in Neuroscience.

## Meet the Volume Editor

Dr. Teresa Silva is an Associate Professor in Criminology at Mid Sweden University. With a background in health sciences and psychology, the focus of her research has been on child development and the risk factors for children's and adolescents' mental health and problematic behaviour, including delinquency and violence. Initially in Spain, and after in Canada and Sweden, Dr. Silva researched different parenting areas such as child maltreatment,

dysfunctional families, harsh and warm parenting, parental monitoring, parental alienation, and family migration processes, among other topics. In her academic career, she has always advocated for children's rights and the necessity of improving the support and assistance that public administration and private organisations provide for them and their families.

## Contents

### **Chapter 7 83**

Mitigating the Unmet Psychosocial Support Needs of Left-behind Learners in Child-Headed Households: Exploring 'Educare' Support at a Zimbabwean School *by Ricanos Jaure and Alfred Henry Makura*

### **Chapter 8 97**

The Effectiveness of Foster Care Arrangements for Unaccompanied Migrant Children in Zimbabwe: A Case of Children under Foster Care in Dulibadzimu, Beitbrigde *by Mulwayini Mundau and Maxwell Chikowe*


Entropy Diversity in Multi-Objective Particle Swarm Optimization

Eduardo J. Solteiro Pires 1, \*, José A. Tenreiro Machado 2 and Paulo B. de Moura Oliveira 1

*Received: 30 August 2013; in revised form: 30 November 2013 / Accepted: 3 December 2013 / Published: 10 December 2013*

Abstract: Multi-objective particle swarm optimization (MOPSO) is a search algorithm based on social behavior. Most of the existing multi-objective particle swarm optimization schemes are based on Pareto optimality and aim to obtain a representative non-dominated Pareto front for a given problem. Several approaches have been proposed to study the convergence and performance of the algorithm, particularly by accessing the final results. In the present paper, a different approach is proposed, by using Shannon entropy to analyze the MOPSO dynamics along the algorithm execution. The results indicate that Shannon entropy can be used as an indicator of diversity and convergence for MOPSO problems.

Keywords: multi-objective particle swarm optimization; Shannon entropy; diversity

#### 1. Introduction

Particle swarm optimization (PSO) is a metaheuristic algorithm based on social species behavior. PSO is a popular method that has been used successfully to solve a myriad of search and optimization problems [1]. The PSO is inspired in the behavior of bird blocking or fish schooling [2]. Each bird or fish is represented by a particle with two components, namely by its position and velocity. A set of particles forms the swarm that evolves during several iterations giving rise to a powerful optimization method.

The simplicity and success of the PSO led the algorithm to be employed in problems where more than one optimization criterion is considered. Many techniques, such as those inspired in genetic algorithms (GA) [3,4], have been developed to find a set of non-dominated solutions belonging to the Pareto optimal front. Since the multi-objective particle swarm optimization (MOPSO) proposal [5], the algorithm has been used in a wide range of applications [1,6]. Moreover, a considerable number of variants of refined MOPSO were developed in order to improve the algorithm performance, e.g., [7].

In single objective problem the performance of the algorithms can be easily evaluated by comparing the values obtained by each one. Moreover, when the performance over time is required the evolution of the best fitness value of the population is normally used. Advanced studies can be accomplished by means of the dynamic analysis [8,9] of the evolution. Many indexes were introduced to measure the performance of multi-objective algorithms according to the solution set produced by them [10–12]. In those cases, when is difficult to identify the best algorithm, nonparametic statistical tests are crucial [13,14].

Shannon entropy has been applied in several fields, such as communications, economics, sociology, and biology among others, but in evolutionary computation it has not been fully explored. For an example of research work in this area we can refer to Galaviz-Casas [15], which studies the entropy reduction during the GA selection at the chromosome level. Masisi *et al.* [16] used the (Renyi and Shannon) entropy to measure the structural diversity of classifiers based in neural networks. The measuring index is obtained by evaluating the parameter differences and the GA optimizes the accuracy of 21 classifiers ensemble. Myers and Hancock [17] predict the behavior of GA formulating appropriate parameter values. They suggested the population Shannon entropy for run-time performance measurement, and applied the technique to labeling problems. Shannon entropy provides useful information about the algorithm state. The entropy is measured in the parameter space. It was shown that populations with entropy smaller than a given threshold become saturated and the population diversity disappears. Shapiro and Bennett [18,19] adopted the maximum entropy method to find out equations describing the GA dynamics. Kita *et al.* [20] proposed a multi-objective genetic algorithm (MOGA) based on a thermodynamical GA. They used entropy and temperature concepts in the selection operator.

Farhang-Mehr and Azarm [21] formulated an entropy based MOGA inspired by the statistical theory of gases, which can be advantageous in improving the solution coverage and uniformity along the front. Indeed, in a enclosed environment, when an ideal gas undergoes an expansion, the molecules move randomly, archiving a homogeneous and uniform equilibrium stated with maximum entropy. This phenomenon occurs regardless of the geometry of the closed environment.

Qin *et al.* [22] presented an entropy based strategy for maintaining diversity. The method maintains the non-dominated number of solutions by deleting those with the worst distribution, one by one, using the entropy based strategy. Wang *et al.* [23] developed an entropy-based performance metric. They pointed out several advantages, namely that (i) the computational effort increases linearly with the solution number, (ii) the metric qualifies the combination of uniformity and coverage of Pareto set and (iii) it determines when the evolution has reached maturity.

LinLin and Yunfang [24] proposed a diversity metric based on entropy to measure the performance of multi-objective problems. They not only show when the algorithm can be stopped, but also compare the performance of some multi-objective algorithms. The entropy is evaluated from the solution density of a grid space. These researchers compare a set of MOGA algorithms performance with different optimization functions.

In spite of having MOPSO used in a wide range of applications, there are a limited number of studies about its dynamics and how particles self-organize across the Pareto front. In this paper the dynamic and self-organization of particles along MOPSO algorithm iterations is analyzed. The study considers several optimization functions and different population sizes using the Shannon entropy for evaluating MOPSO performance.

Bearing these ideas in mind, the remaining of the paper is organized as follows. Section 2 describes the MOPSO adopted in the experiments. Section 3 presents several concepts related with entropy. Section 4 addresses five functions that are used to study the dynamic evolution of MOPSO using entropy. Finally, Section 5 outlines the main conclusions and discusses future work.

#### 2. Multiobjective Particle Swarm Optimization

The PSO algorithm is based on a series of biological mechanisms, particularly in the social behavior of animal groups [2]. PSO consists of particles movement guided by the most promising particle and the best location visited by each particle. The fact that particles work with stochastic operators and several potential solutions, provides PSO the ability to escape from local optima and to maintain a population with diversity. Moreover, the ability to work with a population of solutions, introduces a global horizon and a wider search variety, making possible a more comprehensive assessment of the search space in each iteration. These characteristics ensure a high ability to find the global optimum in problems that have multiple local optima.

Most real world applications have more than a single objective to be optimized, and therefore, several techniques were proposed to solve those problems. Due to these reasons, in the last years many of the approaches and principles that were explored in different types of evolutionary algorithms have been adapted to the MOPSO [5].

Multi-objective optimization problem solving aims to find an acceptable set of solutions, in contrast with uni-objective problems where there is only one solution (except in cases where uni-objective functions have more than one global optimum). Solutions in multi-objective optimization problems intend to achieve a compromise between different criteria, enabling the existence of several optimal solutions. It is common to use the concept of dominance to compare the various solutions of the population. The final set of solutions may be represented graphically by one or more fronts.

Algorithm 1 illustrates a standard MOPSO algorithm. After the swarm initialization, several loops are performed in order to increase the quality of both the population and the archive. In iteration loop t, each particle in the population selects a particle guide from the archive A(t). Based on the guide and personal best, each particle moves using simple PSO formulas. At the end of each loop (Line 12) the archive A(t + 1) is updated by selecting the non-dominant solutions among the population, P(t), and the archive A(t). When the non-dominant solution number is greater than the size of the archive, the solutions with best diversity and extension are selected. The process comes to an end, usually after a certain number of iterations.

#### 3. Entropy

Many entropy interpretations have been suggested over the years. The best known are *disorder*, *mixing*, *chaos*, *spreading*, *freedom* and *information* [25]. The first description of entropy was proposed by Boltzmann to describe systems that evolve from ordered to disordered states. *Spreading* was used by Guggenheim to indicate the diffusion of a energy system from a smaller to a larger volume. Lewis stated that, in a spontaneous expansion gas in an isolated system, *information* regarding particles locations decreases while, the missing information or, *uncertainty* increases.

Shannon [26] developed the information theory to quantify the information loss in the transmission of a given message. The study was carried out in a communication channel and Shannon focused in physical and statistical constraints that limit the message transmission. Moreover, the measure does not addresses, in this way, the meaning of the message. Shannon defined H as a measure of information, choice and uncertainty:

$$H(X) = -K \sum\_{x \in X} p\_i(x) \log p\_i(x) \tag{1}$$

The parameter K is a positive constant, often set to value 1, and is used to express H in an unit of measure. Equation (1) considers a discrete random variable x ∈ X characterized by the probability distribution p(x).

Shannon entropy can be easily extended to multivariate random variables. For two random variables (x, y) ∈ (X, Y ) entropy is defined as:

$$H(X,Y) = -K \sum\_{x \in X} \sum\_{y \in Y} p\_i(x,y) \log p\_i(x,y) \tag{2}$$

#### 4. Simulations Results

This section presents five functions to be optimized with 2 and 3 objectives, involving the use of entropy during the optimization process. The optimization functions F1 to F4, defined by Equations (3) to (6), are known as Z2, Z3, DTLZ4 and DTLZ2 [27,28], respectively, and F5 is known as UF8, from CEC 2009 special session competition [29].

$$F\_1 = \begin{cases} f\_1(X) &=& x\_1 \\ g(X) &=& 1 + 9 \sum\_{i=2}^m \frac{x\_i}{m-1} \\ h(f\_1, g) &=& 1 - \left(\frac{f\_1}{g}\right)^2 \\ f\_2(X) &=& g(X)h(f\_1, g) \end{cases} \tag{3}$$

$$F\_2 = \begin{cases} f\_1(X) &=& x\_1 \\ g(X) &=& 1 + \frac{g}{m} \sum\_{i=1}^m x\_i \\ f\_2(X) &=& g(X) - \sqrt{g(X)x\_1} - 10x\_1 \sin \pi x\_1 \end{cases} \tag{4}$$

$$F\_3 = \begin{cases} f\_1(X) &=& [1 + g(X)] \cos(x\_1^\alpha \pi/2) \cos(x\_2^\alpha \pi/2) \\ f\_2(X) &=& [1 + g(X)] \cos(x\_1^\alpha \pi/2) \sin(x\_2^\alpha \pi/2) \\ f\_3(X) &=& [1 + g(X)] \sin(x\_1^\alpha \pi/2) \\ g(X) &=& 1 + 9 \sum\_{i=3}^m (x\_i^\alpha - 0.5)^2 \end{cases} \tag{5}$$

$$F\_4 = \begin{cases} f\_1(X) &=& [1 + g(X)] \cos(x\_1 \pi/2) \cos(x\_2 \pi/2) \\ f\_2(X) &=& [1 + g(X)] \cos(x\_1 \pi/2) \sin(x\_2 \pi/2) \\ f\_3(X) &=& [1 + g(X)] \sin(x\_1 \pi/2) \\ g(X) &=& [1 + g(X)] \sin(x\_1 \pi/2) \end{cases} \tag{6}$$

$$F\_{5} = \begin{cases} f\_{1}(X) &= \cos(0.5x\_{1}\pi)\cos(0.5x\_{2}\pi) + \frac{2}{|J\_{1}|}\sum\_{j\in J\_{1}}\left(x\_{j} - 2x\_{2}\sin(2\pi x\_{1} + \frac{j\pi}{m})\right)^{2} \\\ f\_{2}(X) &= \cos(0.5x\_{1}\pi)\sin(0.5x\_{2}\pi) + \frac{2}{|J\_{2}|}\sum\_{j\in J\_{2}}\left(x\_{j} - 2x\_{2}\sin(2\pi x\_{1} + \frac{j\pi}{m})\right)^{2} \\\ f\_{3}(X) &= \sin(0.5x\_{1}\pi) + \frac{2}{|J\_{3}|}\sum\_{j\in J\_{3}}\left(x\_{j} - 2x\_{2}\sin(2\pi x\_{1} + \frac{j\pi}{m})\right)^{2} \\\ J\_{1} &= \quad \{j|3 \le j \le m,\text{ and } j-1 \text{ is a multiplication of } 3\} \\\ J\_{2} &= \quad \{j|3 \le j \le m,\text{ and } j-2 \text{ is a multiplication of } 3\} \end{cases} (7)$$

These functions are to be optimized using a MOPSO with a constant inertia coefficient w = 0.7 and acceleration coefficients φ1 = 0.8 and φ2 = 0.8. The experiments adopt t = 1000 iterations and the archive has a size of 50 particles. Furthermore, the number of particles is maintained constant during each experiment and its value is predefined at the begging of each execution.

To evaluate the Shannon entropy the objective space is divided into cells forming a grid. In the case of 2 objectives, the grid is divided into 1024 cells, nf1 ×nf2 = 32×32, where nfi is the number of cells in objective i. On the other hand, when 3 objectives are considered the grid is divided in 1000 cells, so that nf1 × nf2 × nf3 = 10 × 10 × 10. The size in each dimension is divided according to the maximum and minimum values obtained during the experiments. Therefore, the size si of dimension i is given by:

$$s\_i = \frac{f\_i^{\max} - f\_i^{\min}}{n\_{f\_i}} \tag{8}$$

The Shannon entropy is evaluated by means of the expressions:

$$H\_2(O) = \sum\_{i}^{n\_{f\_1}} \sum\_{j}^{n\_{f\_2}} \frac{n\_{ij}}{N} \log \frac{n\_{ij}}{N} \tag{9}$$

$$H\_3(O) = \sum\_{i}^{n\_{f\_1}} \sum\_{j}^{n\_{f\_2}} \sum\_{k}^{n\_{f\_3}} \frac{n\_{ijk}}{N} \log \frac{n\_{ijk}}{N} \tag{10}$$

where nijk is the number of solutions in the range of cell with indexes ijk.

The dynamical analysis considers only the elements of the archive A(t) and, therefore, the Shannon entropy is evaluated using that set of particles.

### *4.1. Results of* F1 *Optimization*

The first optimization function to be considered is F1, with 2 objectives, represented in Equation (3). For measuring the entropy, Equation (9) is adopted (*i.e.*, H2). The results depicted in Figure 1 illustrate several experiments with different population sizes Np = {50, 100, 150, 200, 250}. The number of parameters is maintained constant, namely with value m = 30.

Figure 1. Entropy H(f1, f2) during the MPSO evolution for F1 function.

In Figure 1 it is verified that, in general, entropy has a value that hardly varies over the MOPSO execution. At the beginning, outside the transient, the entropy measure is H2 ≈ 3.7. This transient tends to dissipate as the PSO converges and the particles became organized. Additionally, from Figure 1 it can be seen that the archive size does not influence the PSO convergence rate. Indeed, MOPSO is an algorithm very popular to find optimal Pareto fronts in multi-objective problems, particularly with two objectives.

Figure 2. Non-dominated solutions at iteration t = 1 for F1 function and Np = 150.

Figure 3. Non-dominated solutions at iteration t = 90 for F1 function and Np = 150.

Figure 4. Non-dominated solutions at iteration t = 1000 for F1 function and Np = 150.

Figures 2–4 show that during all the evolution process the MOPSO presents a good diversity. Therefore, is expected that entropy has a small variation throughout iterations. Moreover, after generation 90, entropy presents minor variations revealing the convergence of the algorithm.

### *4.2. Results of* F2 *Optimization*

Figure 5 illustrates the entropy evolution during the optimization of F2. This function includes 2 objectives and leads to a discontinuous Pareto front represented in Figure 6. The experiments were executed with the same population sizes as for F1. It was verified that experiments with a low number of population solutions have a poor (low) initial entropy, revealing a nonuniform front solution at early iterations.

Figure 5. Entropy H(f1, f2) during the MPSO evolution for F2 function.

Figure 6. Non-dominated solutions at iteration t = 1000 for F2.

### *4.3. Results of* F3 *Optimization*

In the case of the optimization of function F3 in Equation (5), three objectives are considered. The entropy evolution is plotted in Figure 7 for Np = {100, 150, 200, 250, 300, 350, 400}. Moreover, is considered m = 12 and α = 100.

For experiments with a small population size, the convergence of the algorithm reveals some problems. Indeed, for populations with Np = 50 particles the algorithm does not converge to the Pareto optimal front. With Np = 100 particles the algorithm takes some time to start converging. This behavior is shown in Figure 7 where pools with many particles (*i.e.*, 350 and 400 particles) reach faster the maximum entropy. In other words, a maximum entropy corresponds to a maximum diversity.

Figure 7. Entropy H(f1, f2, f3) during the MPSO evolution for F3 function.

Figure 8. Non-dominated solutions at iteration t = 1 for F3.

Figure 9. Non-dominated solutions at iteration t = 30 for F3.

Solution

In Figure 7 three search phases are denoted by SP1, SP2 and SP3. The SP1 phase corresponds to a initial transient where the particles are spread for all over the search space with a low entropy. For the experiment with Np = 300, phase SP1 corresponds to the first 30 iterations (see Figures 8 and 9). The second phase, SP2, occurs between iterations 40 and 200, where the particles search the f1 × f3 plane, finding mainly a 2-dimensional front (Figures 10 and 11). Finally, in the SP3 phase (e.g., steady state) the algorithm approaches the maximum entropy. In this phase, particles move in the entire front and are organized in order to give a representative front with good diversity (see Figures 12 and 13).

Figure 10. Non-dominated solutions at iteration t = 40 for F3.

Figure 11. Non-dominated solutions at iteration t = 200 for F3.

Solution

Figure 12. Non-dominated solutions at iteration t = 300 for F3.

Figure 13. Non-dominated solutions at iteration t = 1000 for F3.

Solution

For experiments considering populations with more particles, these phases are not so clearly defined. This effect is due to the large number of particles that allows the algorithm to perform a more comprehensive search. In other words, the MOPSO stores more representative space points helping, in this way, the searching procedure.

### *4.4. Results of* F4 *Optimization*

The results for the optimization function F4 are depicted in Figure 14. The function has 3 objectives and the Pareto front is similar to the one for function F3. It can be observed that optimization with a larger number of solutions presents a regular convergence, as was verified for F3.

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 2 4 6 8 10 H(f1, f2, f3) 100 1000 t Np = 150 Np = 200 Np = 250 Np = 300 Np = 350 Np = 400

Figure 14. Entropy H(f1, f2, f3) during the MPSO evolution for F4 function.

### *4.5. Results of the* F3*,* F4 *and* F5 *Medians*

MOPSO is a stochastic algorithm and each time it is executed is obtained a different convergence path for the best particle. In this line of thought, for each test group, 22 distinct simulations were performed and the median taken as representing the entropy evolution. This section presents the entropy evolution for 12 cases test set, 6 for F3 and 6 for F4, with population sizes of Np = {150, 200,..., 350, 400} particles.

Figures 15 and 16 show evolution of the median of the 6 cases test sets for F3 and F4, respectively. It can be seen that the larger the population size, the faster the convergence of the algorithm in finding an uniform spreading covering the Pareto front. The only exception is the case of Np = 200 particles and F4 function, that leads to a faster convergence than the case with Np = 250 particles.

Figure 17 presents the evolution of the median of 6 cases test sets for F5. It can also be observed that population size affects the diversity, and consequently the space exploration, of the algorithm at early iterations.

At initial iterations, it is natural to observe a entropy peak because the particles are scattered throughout the objective space, and it is difficult to find near particles among the others. In these stages spread was not maximum (entropy) because the distribution is not uniform.

Figure 15. Median entropy H(f1, f2, f3) during the MPSO evolution for F3 function.

Figure 16. Median entropy H(f1, f2, f3) during the MPSO evolution for F4 function.

Figure 17. Median entropy H(f1, f2, f3) during the MPSO evolution for F5 function.

*4.6. Results of the* F3*,* F4 *and* F5 *Medians for MSPSO Algorithm*

In this section, the functions F3, F4 and F5 are optimized using the Spreed-constrained Multi-Objective PSO (SMPSO) [30]. The algorithm was downloaded from jMetal website [31]. The algorithm was slightly modified in order to save the archive solutions during the algorithm evolution. This SMPSO has the particularity of producing new effective particles positions in cases where the velocity becomes too high and uses a polynomial mutation as a turbulence factor.

Figures 18 and 19 present the evolution of the median of the 12 cases test sets, 6 for F3 and 6 for F4, with population sizes of Np = {150, 200,..., 350, 400} particles. From these functions it can be seen that the algorithm maintains a good spreed during its entire evolution, even in initial iterations. This is due to the polynomial mutation and velocity effect.

Figure 18. Median entropy H(f1, f2, f3) during the MSPSO evolution for F3 function.

Figure 19. Median entropy H(f1, f2, f3) during the MSPSO evolution for F4 function.

When optimizing F5 (Figure 20) it can be observed a low entropy at early iterations that increases over the iterations, meaning that the spread also improves throughout the execution of the algorithm.

Figure 20. Median entropy H(f1, f2, f3) during the MSPSO evolution for F5 function.

The MSPSO maintains a good diversity during the search process. This phenomena, does not occur in the standard MOPSO used previously, where diversity decreases in initial stages of the algorithm.

#### 5. Conclusions and Future Work

This paper addressed the application of the entropy concept for representing the evolution behavior of a MOPSO. One interpretation of entropy is to express the spreading of a system energy from a small to a large state. This work adopted this idea and transposed it to measure the diversity during the evolution of a multi-objective problem. Moreover, this measure is able to capture the convergence rate of the algorithm. The optimization of four functions was carried-out. According to the entropy index, the F1 and F2 functions, with two objectives, are easily and early reached, independently of the number of population particles. When three objectives are considered, for functions F3, F4 and F5, the number of population particles has an important role during the algorithm search. It was verified that entropy can be used to measure the convergence and the MOPSO diversity during the algorithm evolution. On the other hand, when using the MSPSO algorithm an high entropy/diversity was observed during the entire evolution. Therefore, the deployed entropy based index can be used to compare the solution diversity during evolution among different algorithms.

In conclusion, the entropy diversity can be used as a evolution index for multi-objective algorithms in the same way the best swarm element is used in single objective optimization problems.

Future work will be devoted to incorporating entropy based indexes to evaluate the swarm diversity in the algorithm run time. Indeed, as the analysis results presented in this paper confirm, swarm diversity can be evaluated along evolution time, and if the entropy index is lower than a problem function specific threshold, then measures can be adopted to reverse the diversity population decrease. The use of online entropy based indexes can also be applied to the swarm population as well as to the non-dominated archive. This will allow evaluate both populations diversity in order to prevent the MOPSO premature convergence. The former research lines are currently under research and their results will be submitted for another publication soon.

#### Conflicts of Interest

The authors declare no conflict of interest.

#### References

Reprinted from *Entropy*. Cite as: Couceiro, M.S.; Clemente, F.M.; Martins, F.M.L.; Machado, J.A.T. Dynamical Stability and Predictability of Football Players: The Study of One Match. *Entropy* **2014**, *16*, 645-674.


*Review* **The Equilibrium Concept, or . . . (Mis)concept in Beaches**

**Enzo Pranzini 1,\* and Allan T. Williams 2**

**Abstract:** Beaches, as deposits of unconsolidated material at the land/water interface, are open systems where input and output items constitute the sediment budget. Beach evolution depends on the difference between the input/output to the system; if positive the beach advances, if negative the beach retreats. Is it possible that this difference is zero and the beach is stable? The various processes responsible for sediment input and output in any beach system are here considered by taking examples from the literature. Results show that this can involve movement of a volume of sediments ranging from few, to over a million cubic meters per year, with figures continuously changing so that the statistical possibility for the budget being equal can be considered zero. This can be attributed to the fact that very few processes are feedback-regulated, which is the only possibility for a natural system to be in equilibrium. Usage of the term "beach equilibrium" must be reconsidered and used with great caution.

**Keywords:** beach sediment budget; coastal dynamics; beach evolution; coastal morphology; feedback processes


**2. Recycling and disposal options of modern permanent magnets**

In recent years, researchers have been exploring new ways to make permanent magnets that are more environmentally friendly. These approaches include using alternative materials, such as iron and cobalt, and developing new manufacturing processes that reduce waste and energy consumption [5, 6]. By adopting these new methods, we can create sustainable permanent magnets that support our technological advances without damage. New ways to reduce environmental damage from permanent magnets include: